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\footline={\hfill}
\rightline{FTUAM 02--28}
\rightline{December, 27,  2002}
\rightline{}
\bigskip
\hrule height .3mm
\vskip.6cm
\centerline{{\bigfib Low Energy Pion Physics}}
\medskip
\centerrule{.7cm}
\vskip1cm

\setbox9=\vbox{\hsize65mm {\noindent\fib 
F. J. Yndur\'ain} 
\vskip .1cm
\noindent{\addressfont Departamento de F\'{\i}sica Te\'orica, C-XI,\hb
 Universidad Aut\'onoma de Madrid,\hb
 Canto Blanco,\hb
E-28049, Madrid, Spain.}\hb}
\smallskip
\centerline{\box9}
\vskip1truecm
\setbox0=\vbox{\abstracttype{Abstract} In these notes we present an introductory   
review on various topics about low energy pion physics (some kaon  physics is discussed as well). 
Among these, we include the uses of analyticity and unitarity to describe partial
 wave amplitudes (for which we give accurate and economical parametrizations)
 and form factors; 
(forward) dispersion relations; and
the use of the Froissart--Gribov representation to evaluate 
accurately the low energy parameters (scattering lengths and effective ranges)
 for higher ($l\geq1$) waves. 
Finally, we describe some pion physics in QCD and then pass on to
study  the nonlinear sigma model, and the chiral 
perturbation theory approach to low energy pion interactions.\hb
\indent
Most of the results presented are known, but we also give a set of new,  precise 
determinations of some scattering lengths as well as an independent calculation of 
three of the parameters  $\bar{l}_2$, $\bar{l}_6$ (to one loop), and one $\bar{f}_2$,
  to  two loops,  that appear in chiral perturbation
theory: 
$$\bar{l}_2=6.37\pm0.22,\;\bar{l}_6=16.35\pm0.14,\quad \bar{f}_2=5.520\pm0.056.
$$
The  S waves are  discussed   and compared 
with chiral perturbation theory expectations, in particular in connection with higher 
scattering lengths.
Also new is the evaluation of some electromagnetic corrections.
}
\centerline{\box0} 
\vskip3truecm
\hrule
\smallskip
{\petit\noindent
Work supported in part by CICYT, Spain.
\hb
This review is based on a graduate course  at the UAM for 
the academic year 2002-2003.}
\vfill\eject
{Typeset with \physmatex}
\vfill\eject


\noindent {\medfib Contents}
\bigskip
{\petit 
\item{1. }{\fib Introduction}
\itemitem{1.1. }{Foreword\leaderfill 1}
\itemitem{1.2. }{Normalization; kinematics; isospin\leaderfill 2}
\itemitem{}{1.2.1. Conventions\leaderfill 2}
\itemitem{}{1.2.2. Isospin\leaderfill 3}
\itemitem{1.3. }{Field-theoretic models\leaderfill 4}
\smallskip
\item{2. }{\fib Analyticity properties of scattering amplitudes, p.w. 
amplitudes, form factors\hb and correlators. Bounds}
\itemitem{2.1. }{Scattering amplitudes and partial waves\leaderfill 5}
\itemitem{2.2. }{Form factors\leaderfill 7}
\itemitem{2.3. }{Correlators\leaderfill 8}
\smallskip
\item{3. }{\fib The effective range formalism for p.w. amplitudes. 
Resonances}
\itemitem{3.1. }{Effective range formalism\leaderfill 9}
\itemitem{3.2. }{Resonances in (nonrelativistic) potential scattering\leaderfill 10}
\smallskip
\item{4. }{\fib The P p.w. amplitude for  $\pi\pi$ scattering in the elementary rho model}
\itemitem{4.1. }{The $\rho$ propagator and the $\pi^+\pi^0$ scattering amplitude\leaderfill 15}
\itemitem{4.2. }{The weak coupling approximation\leaderfill 17}
\itemitem{4.3. }{Low energy scattering\leaderfill 18}
\itemitem{4.4. }{The chiral rho model\leaderfill 19}
\smallskip
\item{5. }{\fib The effective range formalism for p.w. amplitudes; 
resonances (multichannel formalism).\hb Unitarity and form factors; correlators}
\itemitem{5.1. }{General formalism. Eigenphases\leaderfill 21}
\itemitem{5.2. }{The $K$-matrix and the effective range matrix. 
Resonances\leaderfill 22}
\itemitem{5.3. }{Resonance parametrizations in the two-channel case\leaderfill 24}
\itemitem{5.4. }{Reduction to a single channel. Weakly coupled channels\leaderfill 24}
\itemitem{5.5. }{Unitarity for the form factors\leaderfill 26}
\itemitem{5.6. }{Unitarity for correlators\leaderfill 27}
\smallskip
\item{6. }{\fib Extraction and parametrizations of p.w. amplitudes for $\pi\pi$ scattering.\hb
Form factors}
\itemitem{6.1. }{$\pi\pi$ scattering\leaderfill 31}
\itemitem{6.2. }{Form factors\leaderfill 32}
\itemitem{ }{6.2.1. The pion form factor\leaderfill 32}
\itemitem{ }{6.2.2. Form factor of the pion in $\tau$ decay\leaderfill 33}
\itemitem{ }{6.2.3.  $K_{l4}$ decay\leaderfill 34}
\itemitem{6.3. }{The P  wave\leaderfill 34}
\itemitem{ }{6.3.1. The P wave in the elastic approximation\leaderfill35}
\itemitem{ }{6.3.2. The $\rho$ and weakly coupled channels: 
$\omega-\rho$ interference \leaderfill37}
\itemitem{ }{6.3.3. The P wave for $1\,\gev\leq s^{1/2}\leq 1.3\,\gev$\leaderfill 37}
\itemitem{6.4. }{The D and F waves\leaderfill 38}
\itemitem{ }{6.4.1. Parametrization of the  $I=2$ D wave \leaderfill 38}
\itemitem{ }{6.4.2. Parametrization of the  $I=0$ D wave\leaderfill 39}
\itemitem{ }{6.4.3. The F wave \leaderfill 40}
\itemitem{6.5. }{The S wave\leaderfill 40}
\itemitem{ }{6.5.1. Parametrization of the S wave for  $I=2$\leaderfill 40}
\itemitem{ }{6.5.2. Parametrization of the S wave for  $I=0$\leaderfill 42}
\itemitem{ }{6.5.3. The $I=0$ S wave between $960\,\mev$ and $1300\,\mev$\leaderfill 45}
\smallskip
\item{7. }{\fib Analyticity; dispersion relations and the 
Froissart--Gribov representation.\hb
Form factors: the Omn\`es--Muskhelishvili method}
\itemitem{7.1. }{The Omn\`es--Muskhelishvili method\leaderfill 47}
\itemitem{ }{7.1.1. The full Omn\`es--Muskhelishvili problem \leaderfill 47}
\itemitem{ }{7.1.2. The incomplete Omn\`es--Muskhelishvili  problem
\leaderfill 48}
\itemitem{7.2. }{Application to the pion form factor of the Omn\`es--Muskhelishvili\hb method\leaderfill 50}
\itemitem{7.3. }{Dispersion relations and Roy equations\leaderfill 53}
\itemitem{ }{7.3.1. Fixed $t$ dispersion relations\leaderfill 53}
\itemitem{ }{7.3.2. Forward dispersion relations\leaderfill 54}
\itemitem{ }{7.3.3. The Roy equations\leaderfill 55}
\itemitem{ }{7.3.4. Reggeology\leaderfill 56}
\itemitem{7.4. }{Evaluation of  forward dispersion relations for $\pi\pi$ 
 scattering 
\leaderfill 58}
\itemitem{ }{7.4.1.   $\pi^0\pi^+$ \leaderfill 58}
\itemitem{ }{7.4.2.   $\pi^0\pi^0$ \leaderfill 60}
\itemitem{ }{7.4.3. The Olsson sum rule\leaderfill 60}
\itemitem{7.5.}{The Froissart--Gribov representation and low energy P and D wave parameters\leaderfill 61}
\itemitem{ }{7.5.1. Generalities\leaderfill 61}
\itemitem{ }{7.5.2. D waves\leaderfill 63}
\itemitem{ }{7.5.3. Scattering lengths of P and F waves\leaderfill64}
\itemitem{7.6.}{Summary and conclusions\leaderfill 65}
\smallskip
\goodbreak
\item{8. }{\fib QCD, PCAC and chiral symmetry for pions and kaons}
\itemitem{8.1. }{The QCD Lagrangian. Global symmetries; conserved currents\leaderfill 67}
\itemitem{8.2. }{Mass terms and invariances; chiral invariance\leaderfill 67} 
\itemitem{8.3. }{Wigner--Weyl and Nambu--Goldston 
realization of symmetries\leaderfill 74}
\itemitem{8.4. }{PCAC, $\pi^+$ decay, the pion propagator and light quark mass ratios\leaderfill 76}
\itemitem{8.5. }{Bounds and estimates of light quark masses in terms of the pion and kaon masses
\leaderfill 80}
\itemitem{8.6. }{The triangle anomaly; $\pi^0$ decay. 
The gluon anomaly. The $U(1)$ problem\leaderfill 81}
\itemitem{ }{8.6.1. The triangle anomaly and $\pi^0$ decay\leaderfill 81}
\itemitem{ }{8.6.2. The $U(1)$ problem and the gluon anomaly\leaderfill 86}
\smallskip
\item{9. }{\fib Chiral perturbation theory}
\itemitem{9.1. }{Chiral Lagrangians\leaderfill 89}
\itemitem{ }{9.1.1. The $\sigma$ model\leaderfill 89}
\itemitem{ }{9.1.2. Exponential formulation\leaderfill 91}
\itemitem{9.2. }{Connection with PCAC, and a first application\leaderfill 92}
\itemitem{9.3. }{Chiral perturbation theory: general formulation\leaderfill 94}
\itemitem{ }{9.3.1. Gauge extension of chiral invariance\leaderfill 94}
\itemitem{ }{9.3.2. Effective Lagrangians in the chiral limit\leaderfill 95}
\itemitem{ }{9.3.3. Finite pion mass corrections\leaderfill 96}
\itemitem{ }{9.3.4. Renormalized effective theory\leaderfill 97}
\itemitem{ }{9.3.5. The parameters of chiral perturbation theory \leaderfill 98}
\itemitem{9.4. }{Comparison of chiral perturbation theory to one loop 
with experiment\leaderfill 99}
\itemitem{9.5. }{Weak and electromagnetic interactions.\hb The 
accuracy of chiral perturbation theory calculations\leaderfill 102}
\smallskip
\noindent
{{\fib Appendix: the conformal mapping method}\leaderfill 105}
\medskip

\noindent{{\fib Acknowledgements}\leaderfill 107}

\noindent{{\fib References}  \leaderfill 107} 

\noindent{\phantom{x}} 
}
\vfill\eject 

%\brochureendcover{}

\pageno=1
\brochureb{\smallsc chapter 1}{\smallsc introduction}{1}

\bookchapter{1. Introduction}
\vskip-0.5truecm
\booksection{1.1. Foreword}
The matter of parametrizations and uses of pion-pion partial waves (p.w.), 
form factors and correlators, in particular in connection with
 resonances, received a great deal of attention in
the late fifties, sixties and  early seventies of last century --until 
QCD emerged as the theory of strong interactions and such studies were 
relegated to a secondary plane. 
In recent times a renewed interest has arisen in this subject and this due to, at least, 
the following  reasons. 
One is the popularity of chiral perturbation theory calculations
 (to which the last part of these notes is
devoted), in  particular of low energy $\pi\pi$ 
parameters: 
scattering lengths and ranges, pion charge radius, etc. 
A second reason is the use of low energy calculations of the pion form factor to get precise
estimates of  the muon magnetic moment or the value of the QED charge on the 
$Z$ particle. And last, but certainly not least, we have the appearance of new 
experimental data on hadronic $\tau$ decay, the pion electromagnetic form factor and on $K_{e4}$ decay. 
The existence of these data allow a much improved determination of low energy pionic observables.

Unfortunately, some of the old lore appears to be lost and indeed  
modern calculators seem to be unaware of parts of it. 
In the present review we do not present much new knowledge, but mostly intend 
to give an introductory, easily accessible reference to the 
studies of scattering amplitudes, form factors and correlators 
involving pions in the low energy region: our 
aim is, in this respect, mainly pedagogical. 
For this reason we put special emphasis on some of the topics {\sl not} discussed on 
the more recent works on the subject.
 
Nevertheless, some novel results are reported. 
These include parametrizations of the lowest waves (S, P, D, F) in $\pi\pi$ scattering 
which are compatible with analyticity and unitarity and, of course, experimental data, 
depending only on a few (two to four) parameters per wave. 
This is used to evaluate forward dispersion relations and 
the Froissart--Gribov representation of the P, D and higher waves. 
From this there follow very precise determinations of the corresponding 
scattering lengths and effective range parameters. 
Using this, as well as the results on the P wave following from the electromagnetic pion form 
factor and the decay $\tau\to \nu \pi^0\pi^+$, 
we obtain, in particular, 
 a precise
determination (to one loop) of some of  the $\bar{l}$ parameters in chiral perturbation theory, as 
well as an evaluation  
 of some electromagnetic corrections.

The plan of this review is as follows. 
In Chapters 1 and 2 we describe briefly the analyticity properties of various quantities 
(correlators, form factors, partial waves and scattering amplitudes). 
The elements of the effective range formalism and the characterization of resonances 
are given in Chapter~3. These topics are illustrated in a simple model in Chapter~4,
 while in Chapter~5 we extend the previous analyses (including the requirements of unitarity) 
to the multichannel case.

The core of the review is contained in the last four chapters. In  chapters~6 and 7  
we apply the tools described before to the study of 
partial wave amplitudes and scattering amplitudes 
for $\pi\pi$ scattering and to fit the pion form factor. 
Here we implement simple parametrizations of partial wave amplitudes consistent with 
analyticity and unitarity, and fitting experimental data; 
this should be useful to people needing manageable 
representations of  $\pi\pi$ phases, as happens e.g. for 
$J/\psi\to\gamma\pi\pi$ studies. 
Then we discuss (Chapter~7) how the various theoretical requirements (fixed $t$ 
dispersion relations and the Froissart--Gribov representation) may be used 
to check compatibility of the results found with crossing symmetry and 
analyticity for the scattering amplitudes. 
The  Froissart--Gribov representation is also used to get precise determinations of 
low energy parameters for the waves with $l=1$ and higher.
With respect to form factors, the Omn\`es--Muskhelishvili method is 
employed to perform an accurate fit to the 
pion form factor, obtaining in particular precise values of the 
corresponding low energy parameters.
Something which is missing in this review is the Roy equations analysis; 
there are in the literature two recent papers (Ananthanarayan 
et al., 2001 and Colangelo, Gasser and Leutwyler, 2001) that 
fill this gap.

The results we obtain are summarized in \sect~7.6 where, in particular, we present a set 
of simple parametrizations for the low energy 
S, P, D, F waves, compatible with 
experimental data and theoretical constraints.

In Chapter~8 we remember that pions are made of quarks, and that we have a theory for 
the interactions of these, QCD. We discuss invariance properties of the QCD 
Lagrangian, in particular chiral invariance that plays a key role for 
the dynamics of pions. 
We use this and PCAC to derive relations between the masses of
 the quarks and the pion and kaon masses, and to
study pion decay. Finally, in Chapter~9 we develop the consistent description of 
pion dynamics based on chiral invariance,  known as chiral perturbation theory. 
In the last sections of this chapter we use the results obtained 
in Chapters~6 and 7 to test the predictions of chiral perturbation theory, 
and show how  to obtain values for the parameters on which it depends.



Before entering into the main body of these notes, it is convenient to clarify what is to be 
understood as ``low energy." 
Above energies $s^{1/2}$ of, say, $1.3\;\sim\;2\,\gev$, perturbative 
QCD (or Regge theory, as the case may be)  is applicable; we will   
be very little concerned with these energies. 
At very low energies, $s^{1/2}\ll\Lambdav_0$, where $\Lambdav_0$ 
is a scale parameter that (depending on the process) may vary 
from $\Lambdav\simeq600\,\mev$ to $4\pi f_\pi\sim 1.1\,\gev$, 
chiral perturbation theory is applicable; this we treat 
in detail in Chapters~8 and 9. Between the two energy scales, analyticity and 
unitarity allow at least an {\sl understanding} of pionic observables. 
This understanding certainly holds until inelastic production begins to become important. 
This means that we are able, at most, to cover the energy range of   $s^{1/2}$ 
below 1.3 \gev; in some cases (like the isospin 0 S-wave in $\pi\pi$ scattering), 
this means only up to $s^{1/2}\simeq0.9\,\gev$; in others 
 (like the isospin 0 D-wave in the same process) we are able to go up to 
 $s^{1/2}\simeq1.4\,\gev$. 

These notes are primarily about pions. However, in some cases 
kaons and (to a lesser extent) etas are treated as well.


\booksection{1.2. Normalization; kinematics; isospin}
\vskip-0.5truecm
\booksubsection{1.2.1. Conventions}

\noindent
Before entering into specific discussions we will say a few words on 
our normalization conventions.\fnote{We assume here a basic knowledge of $S$ matrix theory, 
in particular of crossing symmetry or partial wave expansions, and of isospin invariance, 
at the level of the first chapters of the texts of 
Martin, Morgan and Shaw~(1976) or Pilkuhn~(1967).} 
If $S$ is the relativistically invariant scattering matrix we 
define the scattering amplitude $F$ for particles $A$, $B$ to give particles $C_i$ 
by
$$\langle C_1,\dots,C_n|S|A,B\rangle=
\ii \delta(P_f-P_i)F(A+B\to C_1+\cdots+C_n).
\equn{(1.2.1)}$$
We take the states to be normalized in a relativistically invariant manner:  
if $p$ is the four-momentum, and $\lambda$ the helicity of a particle, then
$$\langle p,\lambda|p',\lambda'\rangle=2\delta_{\lambda\lambda'}p_0\delta({\bf p}-{\bf p}').
\equn{(1.2.2)}$$
We will seldom consider particles with spin in these notes. 
Particles with spin pose problems of their own; the generalization of 
our discussions to spinning particles is not trivial.


It is the function $F$, defined as in (1.2.1), with the states 
normalized as in (1.2.2), the one which is free of 
kinematical singularities and zeros. 
That is to say, any discontinuity or pole of $F$ is associated with 
dynamical effects. 
If we had used a nonrelativistic normalization 
and defined a corresponding scattering amplitude $T_{\rm NR}$, 
we could write
$$T_{\rm NR}=
\dfrac{1}{\sqrt{2p^0_{A}}}\dfrac{1}{\sqrt{2p^0_{B}}}
\dfrac{1}{\sqrt{2p^0_{{C_1}}}}\dots\dfrac{1}{\sqrt{2p^0_{{C_n}}}}\,F.
\equn{(1.2.3)}$$
Then, no matter which field-theoretic interaction we assumed, 
 $T_{\rm NR}$ would show the branch cuts associated with the factors $1/\sqrt{p^0}$ in (1.2.3).

In what regards form factors care has to be exercised to get 
form factors without kinematic cuts. 
For the simple case of the e.m. (electromagnetic) 
form factor of the pion 
(or any other spinless particle) such form factor is that defined by
$$\langle p_1|J^{\rm e.m.}_\mu(0)|p_2\rangle=
(2\pi)^{-3}(p_1+p_2)_\mu F_\pi(t),\quad t=(p_1-p_2)^2.
\equn{(1.2.4a)}$$
Note that, with this definition, $F_\pi(0)=1$. 
\equn{(1.2.4a)} is valid for spacelike $t\leq0$. 
For timelike $t\geq 4\mu^2$ we write
$$\langle p_1,p_2|J^{\rm e.m.}_\mu(0)|0\rangle
=(2\pi)^{-3}(p_1-p_2)_\mu F_\pi(t),\quad t=(p_1+p_2)^2.
\equn{(1.2.4b)}$$
Both values of $F_\pi$ are particular cases of a single function, $F_\pi(t)$, 
that can be defined for arbitrary, real or complex values 
of the variable $t$ (see below). 


We finish this subsection with a few more definitions. 
Let us consider scattering of two pions:
$$\pi_1(p_1)+\pi_2(p_2)\to \pi'_1(p'_1)+\pi'_2(p'_2).$$
We define the Mandelstam variables
$$s=(p_1+p_2)^2,\quad u=(p_1-p'_2)^2,\quad t=(p_1-p'_1)^2.$$
They satisfy the equality, for pions on their mass shells,
$$s+u+t=4\mu^2.$$
(Here, and throughout these notes, $\mu\equiv138\,\mev$ is the {\sl average} mass of the pions. 
When referring specifically to neutral or charged pion masses 
we will write $m_{\pi^0}$ or $m_{\pi^{\pm}}$).

In terms of these variables the modulus of the 
three-momentum, $k$, and the cosine of the 
scattering angle (both in the center of mass) 
are given by
$$k=\dfrac{\sqrt{s-4\mu^2}}{2},\quad \cos\theta=1+\dfrac{2t}{s-4\mu^2}.$$


With our definitions, the two body differential cross 
section in the center of mass is given in terms of $F$ as
$$\left.\dfrac{\dd \sigma}{\dd\omegav}\right|_{\rm c.m.}=\dfrac{\pi^2}{4s}
\dfrac{k'}{k}|F(i\to f)|^2,
\equn{(1.2.5)}$$
with $k$, $k'$ the moduli of the three-momenta of 
initial, final particles. For particles with arbitrary masses $m_i$  
(1.2.5) is still valid but now
$$k=\dfrac{1}{2s^{1/2}}\sqrt{[s-(m_1-m_2)^2][s-(m_1+m_2)^2]}$$ 

The total cross section, also with the same conventions, is
$$\sigma_{\rm tot}(s)=\dfrac{4\pi^2}{\lambda^{1/2}(s,m_1,m_2)}\imag F(s,0).
\equn{(1.2.6)}$$
Here we define K\"all\'en's quadratic form
$$\lambda(a,b,c)=a^2+b^2+c^2-2ab-2ac-2bc=
\big[a-(\sqrt{b}-\sqrt{c})^2\big]\,\big[a-(\sqrt{b}+\sqrt{c})^2\big].$$

\booksubsection{1.2.2. Isospin}

\noindent
As we know, there are three kinds of pions: two charged ones, $\pi^\pm$ with a mass
$m_{\pi^\pm}=139.57\,\mev$, and a neutral pion, with mass $m_{\pi^0}=134.98\,\mev$. 
If we neglected electromagnetic interactions, and the 
mass difference between $u$, $d$ quarks, then the interactions of the three pions would be 
identical, and they would have a common mass, that we denote by $\mu$ and take equal to the average: 
$\mu=138\,\mev$.
The invariance under rotations of the three pions, called {\sl isospin invariance}, 
is best described by 
introducing a different basis to describe the pions, $|\pi_i\rangle$, $i=1,\,2,\,3$, 
related to the physical pions by
$$|\pi^0\rangle=|\pi_3\rangle,\quad
 |\pi^\pm\rangle=\dfrac{\mp1}{\sqrt{2}}\Big\{|\pi_1\rangle\pm\ii|\pi_2\rangle\Big\}.
$$
Isospin transformations are then just rotations, $|\pi_j\rangle\to\sum_kR_{jk}|\pi_k\rangle$ 
with $R$ a rotation matrix. 
We can then, in the limit of exact isospin invariance, diagonalize the total isospin and its third
component and consider e.g.  scattering amplitudes corresponding to fixed isospin.

\booksection{1.3. Field-theoretic models}

\noindent 
It is always convenient to illustrate abstract arguments with model calculations 
in which one can see how the general properties are realized in explicit examples. 
We will take as a very convenient model one in which pions and rho are 
realized as elementary fields, $\phi_a$, $\rho_a$ ($a$ is an 
isospin index). 
The corresponding Lagrangian will be
$${\cal L}=g_{\mu\nu}(\delta_{ab}\partial_\mu-\ii g_\rho\epsilon_{abc}\rho^c_\mu)\phi_b
(\delta_{ad}\partial_\nu-\ii g_\rho\epsilon_{ade}\rho^e_\nu)\phi_d
-\mu^2 \phi_a\phi_a+{\cal L}_\rho.
\equn{(1.3.1)}
$$
Here $\mu$ is the pion mass and ${\cal L}_\rho$ is the pure rho Lagrangian, that need not be specified. 
The mass of the rho particle, $M_\rho$, can be assumed to be introduced by a Higgs-type 
mechanism, with the mass of the associated Higgs particle so large that it will have no 
influence on calculations for energies of the order of $M_\rho$ or lower. 

 The interactions in (1.3.1) induce pion-rho vertices: 
a $\pi\pi\rho$ vertex, which is  
associated with the factor $\ii g_\rho (p_1-p_2)_\mu \epsilon_{abc}$, 
and a seagull one proportional to $\ii g^2_\rho g_{\mu\nu}$.
This model is {\sl not} chiral invariant (see later for a chiral invariant version) 
but it is very simple and thus will be used to illustrate general properties, 
such as analyticity or unitarity, 
independent of the underlying dynamics.

\bookendchapter
\brochureb{\smallsc chapter 2}{\smallsc analyticity properties of 
scattering
 amplitudes, etc. bounds}{5}
\bookchapter{2. Analyticity properties of 
scattering\hb
 amplitudes, p.w. amplitudes,\hb form factors and correlators.\hb Bounds}
\vskip-0.5truecm
\booksection{2.1. Scattering amplitudes and partial waves}

\noindent
Analyticity of partial waves follows from unitarity and causality. 
In local field theories (such as QCD) 
both properties are, of course, satisfied, but 
locality at least would be violated in a theory of strings; although this 
would occur at energies much higher than the ones in which we are interested here.
In the case of the $\pi\pi$ scattering amplitude, 
$F(s,t)$, one can prove that it is, for  $t$ in the 
Martin--Lehmann 
ellipse,\fnote{With foci at $t=0$ and $t=4\mu^2-s$ (for pions) 
and right extremity at $t=4\mu^2$. This includes the 
physical region, $4\mu^2-s\leq t\leq 0$. For the proof, 
see Martin~(1969) and references there.} 
analytic in the complex $s$ plane with the exception of two cuts: 
a r.h. (right hand) cut, from 
$s=4\mu^2$ to $+\infty$, and a l.h. (left hand) 
cut from $-\infty$ to $-t$.
In addition, if there existed bound states, there would appear poles at the values 
of $s$ or $u$ given by the square of the mass of the bound state. 
For scattering of particles with masses $m_i$, there is a relation among the 
Mandelstam variables: 
 $$s+u+t=\sum_im_i^2.$$

The p.w. (partial wave) amplitudes, $f_l(s)$, are related to 
$F$ though the  expansion,
$$F(s,t)=\sum_{l=0}^\infty (2l+1)P_l(cos\theta)f_l(s)
\equn{(2.1.1a)}$$
with inverse
$$f_l(s)=\tfrac{1}{2}\int_{-1}^{+1}\dd \cos\theta P_l(cos\theta)F(s,t).
\equn{(2.1.1b)}$$
Here $\theta$ is the scattering angle in the center of mass, and 
$P_l$ are the 
Legendre polynomials. 
We note that the restriction of, say, $s$ to physical values 
produces the {\sl physical} $F(s,t)$ and $f_l(s)$ 
{\sl provided} we take the limit of $s$ real from the upper half plane. 
That is to say, if $s$ is real and physical (and so is $t$), 
the physical values of scattering amplitude and partial waves 
are obtained for
$$F(s,t)=\lim_{\epsilon\to+0}F(s+\ii\epsilon,t);
\quad f_l(s)=\lim_{\epsilon\to+0}f_l(s+\ii\epsilon).$$

For elastic scattering at {\sl physical} $s$ (which, for $\pi\pi$ scattering means $s$ 
real and larger than or equal to $4\mu^2$) and below the opening of the 
first inelastic threshold, that we will denote by $s_0$, one can express the $f_l$ 
in terms of phase shifts:
$$f_l(s)=\dfrac{2s^{1/2}}{\pi k}\sin\delta_l(s)\ee^{\ii\delta_l(s)}
=\dfrac{2s^{1/2}}{\pi k}\dfrac{1}{\cot\delta_l(s)-\ii};\quad
4\mu^2\leq s\leq s_0.
\equn{(2.1.2)}$$

The previous equations are valid assuming that the scattering 
particles are {\sl distinguishable}. 
For $\pi\pi$ scattering, however, the situation is a bit 
complicated. 
One may still write (2.1.1) and (2.1.2) for the processes 
$\pi^0\pi^+\to\pi^0\pi^+$, but not for $\pi^0\pi^0\to\pi^0\pi^0$ or 
$\pi^+\pi^+\to\pi^+\pi^+$. 
The general recipe is the following: if $F^{(I_s)}$ 
is an amplitude with isospin $I_s$ in channel $s$, 
one has to replace (2.1.1) by
$$\eqalign{
F^{(I_s)}(s,t)=&\,2\times\sum_{l={\rm even}}(2l+1)P_l(\cos\theta)f_l^{(I_s)}(s),\quad I_s=\hbox{even},\cr
F^{(I_s)}(s,t)=&\,2\times\sum_{l={\rm odd}}(2l+1)P_l(\cos\theta)f_l^{(I_s)}(s),\quad I_s=\hbox{odd,}\cr
f_l^{(I)}(s)=&\,\dfrac{2s^{1/2}}{\pi k}\sin\delta_l^{(I)}(s)\ee^{\ii\delta_l^{(I)}(s)}.\cr 
\cr}
\equn{(2.1.3)}$$
Due to Bose statistics, even waves only exist with isospin $I=0,\,2$ and odd waves must 
necessarily have isospin $I=1$. 
For this reason, we will often omit the 
isospin index for odd waves, 
writing e.g. $f_1$, $f_3$ instead of $f_1^{(1)}$, 
$f_3^{(1)}$.


When inelastic channels  open (2.1.2) is no more valid, 
but one can still write
$$f_l(s)=\dfrac{2s^{1/2}}{\pi k}
\left[\dfrac{\eta\,\ee^{2\ii\delta_l}-1}{2\ii}\right].
\equn{(2.1.4)}$$
Here $\eta$, called the {\sl inelasticity parameter},
 is real positive and smaller than or equal to unity; 
$1-\eta$ is proportional to the amount of inelasticity present.
For $\pi\pi$ scattering $s_0=16\mu^2$, but the approximation of 
neglecting inelasticity is valid at the 2\% level or better 
below $s\simeq 1\,\gev$, the precise value depending on 
the particular wave.

The cut structure is more complicated for other processes. 
For example, for $\pi K$ scattering the r.h. cut starts at 
$s=(\mu+m_K)^2$ and the l.h cut also begins at $u=(\mu+m_K)^2$. 
But, since now $s+u+t=2\mu^2+m^2_K$, the l.h. cut in the variable $s$ 
runs from $-\infty$ to $-t+(m_K-\mu)^2$ for the scattering amplitude, and
 from $-\infty$ to $(m_K-\mu)^2$ for the p.w. amplitudes.

For $\bar{K}K\to\pi\pi$ or $\pi\pi\to\bar{K}K$ scattering, 
the  $u$-channel is $\pi K$ scattering. Therefore, the 
r.h. cut starts at the (unphysical) value $s=4\mu^2$, and the l.h. cut at 
$u=(\mu+m_K)^2$, hence the l.h. cut 
runs from $-\infty$ to $-t+(m_K-\mu)^2$ for the scattering amplitude, and
 from $-\infty$ to $(m_K-\mu)^2$ for the p.w. amplitudes.

 
\topinsert{
\setbox0=\vbox{\hsize9.truecm{\epsfxsize 7truecm\epsfbox{cut_plane2.eps}}} 
\setbox6=\vbox{\hsize 6.5truecm\captiontype\figurasc{Figure 2.1.1. }{The  domain of 
analyticity for $\pi\pi$ partial waves (shadowed region)
 with the cuts of $f_l(s)$ in the complex $s$ plane.\hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\line{
%\tightboxit
{\box0}\hfil\box6}
\medskip
}\endinsert
  
Finally, for $\bar{K}K$ scattering, the l.h. cut runs, as for 
$\pi\pi$, from $-\infty$ to 0 for p.w. amplitudes; but there is 
a r.h. cut, both for the amplitude and for p.w.'s 
associated with the unphysical channel $\bar{K}K\to\pi\pi$, and  
starting at $4\mu^2$.  

Note that from (2.1.1b) it follows that it is $f_l(s)$ 
that satisfies analyticity properties without kinematical zeros or singularities.
These analyticity properties are rather complicated in general; 
in the simple case of $\pi\pi$ scattering 
we have that $f_l(s)$ is analytic in the complex $s$ plane except for two 
cuts, one from $4\mu^2$ to $+\infty$ and another from 
$-\infty$ to 0 (\fig~2.1.1), inherited respectively from the 
r.h. and l.h. cuts of $F(s,t)$.
If there existed bound states (which is not the case 
for $\pi\pi$), there would appear poles at the values 
of $s$ given by the square of the mass of the bound state.


Another property that follows from (2.1.1b) plus the assumption 
(realized in the real world for $\pi\pi$ scattering) that there is no bound state with zero 
energy is the behaviour, as $k\to0$,
$$f_l(s)\simeqsub_{k\to0}\dfrac{4\mu}{\pi}k^{2l}a_l
\equn{(2.1.5)}$$
where $a_l$ is the so-called $l$-th wave {\sl scattering length}. 

Analyticity is seldom of any use without bounds. 
Again, on very general grounds one knows that, for
$t$ physical, 
$|F(s,t)|$ is bounded by 
$C|s|\log^2|s|$ (the Froissart bound). 
Specific behaviours, particularly those that hold in Regge theory, will be discussed in \subsect~6.2.4.
For the proof of the Froissart and related bounds we 
require unitarity, causality and the assumption that Green's functions grow, at 
most, like polynomials of the momenta. This last 
assumption holds in renormalizable field theories, 
to all orders in perturbation theory; but may fail 
for nonrenormalizable ones. General discussions of 
bounds and expected high energy behaviour of 
scattering amplitudes may be found in Martin~(1969), Barger and Cline~(1969), Sommer~(1970), 
Yndur\'ain~(1972), etc. In connection with Roy equations, see 
Pennington~(1975) and Roy~(1990).

\booksection{2.2. Form factors}
Analyticity of form factors, such as the pion or kaon form factors,
 can be proved quite generally using only
causality and  unitarity. 
In particular the pion form factor $F_\pi(t)$ turns out to 
be analytic in the complex $t$ plane cut from $t=4\mu^2$ to 
$+\infty$ (\fig~2.2.1). 
This analyticity, in particular, provides the link between both 
definitions of $F_\pi$, \equs~(1.2.4). 
For timelike, physical $t$, the physical value 
should in fact be defined as
$$F_\pi(t)=\lim_{\epsilon\to+0}F_\pi(t+\ii\epsilon);\quad t\geq 4\mu^2;$$
if we had taken $\lim_{\epsilon\to+0}F_\pi(t-\ii\epsilon)$
 we would have obtained $F^*_\pi(t)$. 



Unlike scattering amplitudes, for which bounds hold in {\sl any} 
local field theory, one cannot prove bounds for 
form factors in general. 
However, bounds can be obtained in QCD, where we can even find the high 
momentum behaviour with the Brodsky--Farrar counting rules. 
In particular, for $F_\pi$ one has the Farrar--Jackson~(1979) behaviour
$$F_\pi(t)\simeqsub_{t\to\infty}\;\dfrac{12C_F\pi f^2_\pi\alpha_s}{-t},
\equn{(2.2.1)}$$
where $f_\pi$ is the pion decay constant, $f_\pi\simeq 93\,\mev$, 
$\alpha_s$ the QCD coupling, and $C_F=4/3$ is a colour factor. 


%\topinsert{
\setbox0=\vbox{\hsize9.truecm{\epsfxsize 7truecm\epsfbox{cut_plane1.eps}}} 
\setbox6=\vbox{\hsize 6.5truecm\captiontype\figurasc{Figure 2.2.1. }{The  domain of 
analyticity (shadowed region)
 for $F_\pi(t)$, or $\piv(t)$, in the complex $t$ plane.\hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\line{
%\tightboxit
{\box0}\hfil\box6}
\bigskip
%}\endinsert
The corrections to (2.2.1), however, cannot be calculated. 
This asymptotic behaviour holds, in principle, only on the real axis, but the 
Phragm\'en--Lindel\"of theorem ensures its 
validity in all directions of the complex plane.

\booksection{2.3. Correlators}
Consider for example the vector current operator $V_\mu(x)=\bar{q}\gamma_\mu q'$
with $q,\,q'$ quark fields. 
We associate to it the correlator
$$\piv_{\mu\nu}(p)=\ii\int\dd^4x\;\ee^{\ii p\cdot x}
\langle0|{\rm T}V^{\dag}_\mu(x) V_\nu(0)|0\rangle=
\left(-g_{\mu\nu}t+p_\mu p_\nu\right)\piv_{\rm tr}(t)+
p_\mu p_\nu\piv_S(t),\quad t=p^2.
\equn{(2.3.1)}$$

The $\piv(t)$ can be shown, again using only unitarity and causality, to 
be analytic in the complex $t$ plane with a cut from $t_0$ to $+\infty$ 
where $t_0$ is the squared invariant mass of the lightest state with the 
quantum numbers of the current $V_\mu$.  
If $V_\mu$ is the e.m. (electromagnetic) current, that we denote by 
$J_\mu$, and we neglect weak and e.m. interactions, 
then $\piv_S=0$ and $\piv_{\rm tr}$ is analytic except for a cut 
from $t=4\mu^2$ to $+\infty$.


There is no bound with validity for arbitrary field theories for the 
correlators; but, in QCD, we can actually calculate the 
behaviour for large momentum; it is given by 
the parton model result for the $\piv$.

\bookendchapter

\brochureb{\smallsc chapter 3}{\smallsc the effective range formalism for p.w.
 amplitudes. 
resonances}{9}
\bookchapter{3. The effective range formalism for p.w.\hb
 amplitudes. 
Resonances}
\vskip-0.5truecm
\booksection{3.1. Effective range formalism}

\noindent
We will consider here only the pion-pion case. 
The discontinuity of $f_l(s)$ across the {\sl elastic} 
cut is very easily evaluated. Because all functions (scattering amplitudes, 
form factors and correlators) are real analytic\fnote{A complex function
 $f(z)$ is 
{\sl real analytic} if it satisfies $f^*(z^*)=f(z)$. 
The theorem of Painlev\'e ensures that, if a function 
analytic for $\imag z\neq0$  is real analytic, and is real on a segment
$[a,b]$ of the real axis, it is also 
analytic on the segment. For more information on 
matters of complex analysis, we recommend the texts of Ahlfors~(1953) and Titchmarsh~(1939).} 
we can calculate their discontinuity as
$${\rm disc} f(s)=2\ii \imag f(s)=
\lim_{\epsilon\to+0} \left\{f(s+\ii \epsilon)-f(s-\ii \epsilon)\right\}=
\lim_{\epsilon\to+0} \left\{f(s+\ii \epsilon)-f^*(s+\ii \epsilon)\right\}.
\equn{(3.1.1)}$$
For p.w. amplitudes, and for  physical $s$ below the inelastic threshold 
$s_0$,  we have
$$\imag f_l(s)=\dfrac{\pi k}{2s^{1/2}}|f_l(s)|^2,\quad 4\mu^2\leq s\leq s_0.
\equn{(3.1.2)}$$

This suggests how we can form from $f_l$
 a function in which this elastic cut is absent.
This is the function $\phiv_l(s)$ defined for arbitrary complex $s$ by
$$\phiv_l(s)=
\dfrac{\ii k^{2l+1}}{2\sqrt{s}}+\dfrac{k^{2l}}{\pi f_l(s)}.
\equn{(3.1.3a)}$$
We assume
that 
$f_l(s)$ does not vanish for $0\leq s< 4\mu^2$, or for 
$4\mu^2<s\leq s_0$. If $f_l$ vanished below 
threshold, or on the elastic cut, the function $\phiv_l$ would have poles at such zeros; 
the analysis can be generalized quite easily to cope with this, and, for 
$\pi\pi$ scattering, we will 
show explicitly how in the cases of the S waves and the $I=2$ D wave. 

We can rewrite (3.1.3a) as
$$\phiv_l(s)=
-2^{-2-2l}(s-4\mu^2)^l\left(\dfrac{4\mu^2}{s}-1\right)^{1/2}+2^{-2l}
\dfrac{(s-4\mu^2)^l}{\pi f_l(s)}.
\equn{(3.1.3b)}$$
In this second form it is obvious that the first term in the r.h. side 
is analytic for all $s$, except for a (kinematic) cut running from 
$-\infty$ to $s=0$ and a cut for $s\geq4\mu^2$. The second term is also analytic 
over the segment $0\leq s< 4\mu^2$, 
and it presents a dynamical cut from $-\infty$ to 0 due to the l.h. cut of $f_l$ 
(\fig~3.1.1).
 
\midinsert{
\setbox0=\vbox{\hsize16.truecm{\epsfxsize 14.truecm\epsfbox{cut2.eps}}} 
\setbox6=\vbox{\hsize 14.5truecm\captiontype\figurasc{Figure 3.1.1. }{The
 cuts in the complex $s$ plane for $\phiv_l(s)$. 
The dotted line shows the absent elastic cut. 
We have taken $s_0=1\,\gev^2$, and the drawing is to scale. \hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\centerline{\tightboxit{\box0}}
\bigskip
\centerline{\box6}
\medskip
}\endinsert

 We next check that $\phiv_l(s)$ is
analytic over the  elastic cut. 
We have, for $4\mu^2<s\leq s_0$,
$$\imag\phiv_l(s)=\dfrac{ k^{2l+1}}{2\sqrt{s}}+k^{2l}\dfrac{-\imag f_l(s)}{\pi f_l^*(s)f_l(s)}.$$
Using then (3.1.2), the r.h. side is seen to vanish. 
The only point which appears dangerous is the threshold, $s=4\mu^2$, because 
here $f_l$ vanishes for $l\geq 1$; 
but  this zero is exactly compensated by the zero of the factor $k^{2l}$; 
cf.~(2.1.5). 
Therefore it follows that the function $\phiv_l(s)$ is analytic along the elastic cut. 
Its only singularities  are thus (apart from poles due to zeros of $f_l$),
 a r.h. cut from $s=s_0$ to 
$+\infty$; and a l.h. cut, formed by two superimposed cuts, namely, 
the kinematic cut of 
$$2^{-2-2l}(s-4\mu^2)^l\sqrt{\dfrac{4\mu^2}{s}-1}, $$
and the dynamical cut of
$$\dfrac{k^{2l}}{\pi f_l(s)}$$
due to the l.h. cut of $f_l(s)$. 

Eq.~(3.1.3b) defines $\phiv_l(s)$ for all complex $s$; in the particular case where $s$ is 
on the elastic cut, we can use \equn{(2.1.2)} to get
$$\phiv_l(s)=\dfrac{k^{2l+1}}{2\sqrt{s}}\cot \delta_l(s),
\quad 4\mu^2\leq s\leq s_0.
\equn{(3.1.4)}$$
In general, i.e., for any value (complex or real) of $s$, we can solve (3.1.2) and write
$$f_l(s)=\dfrac{2s^{1/2}}{\pi k}\dfrac{1}{2s^{1/2}k^{-2l-1}\phiv_l(s)-\ii}
=\dfrac{k^{2l}}{\pi}\dfrac{1}{\phiv_l(s)-\ii k^{2l+1}/2s^{1/2}} .
\equn{(3.1.5)}$$
 $\phiv_l(s)$ is real on the segment $0\leq s\leq s_0$, but it
 will be {\sl complex} above the 
inelastic threshold, $s_0$, and also for $s\leq0$.

The fact that $\phiv_l(s)$ is analytic across the elastic region is valid not 
only for $\pi\pi$, but also for other p.w. amplitudes; 
for example, for pion-nucleon,  nucleon-nucleon or even nucleon-nucleus. 
This implies that, at low energies ($k\to 0$), one can expand 
$$\phiv_l(s)=\dfrac{1}{4\mu a_l}+R_0k^2+R_1k^4+\cdots.$$
This is the so-called effective range formalism, widely used in low 
energy nucleon and nuclear physics. 
The quantity $a_l$ is  the scattering length (cf.~ \equn{(2.1.4)}) and the $R_i$ 
are related to the range of the potential 
(if the scattering is caused by a short-range potential). 
For $\pi\pi$ scattering, the expansion is convergent in the 
disk $|s-4\mu^2|<1$, shown  shaded in \fig~3.1.2.

 
\topinsert{
\setbox0=\vbox{\hsize16.truecm{\epsfxsize 14.truecm\epsfbox{cut_disc2.eps}}} 
\setbox6=\vbox{\hsize 14.5truecm\captiontype\figurasc{Figure 3.1.2. }{The
circle of convergence for the effective range expansion for $\phiv_l(s)$;
 $s_0=1\,\gev^2$. \hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\centerline{\tightboxit{\box0}}
\bigskip
\centerline{\box6}
\medskip
}\endinsert

\booksection{3.2. Resonances in (nonrelativistic) potential scattering}

\noindent
Consider scattering by a  spherical potential, $V(r)$, 
that we assume to be of short range.  
We will simplify the discussion by working  
in the nonrelativistic approximation. 
The nonrelativistic energy $E$ is $E=s^{1/2}-m_1-m_2$ 
with $m_i$ the masses of the particles, and we shall let $m$ be the 
reduced mass. 
To lighten notation, we take mass units so that $2m=1$.

 The $l$-wave Schr\"odinger equation is
$$\dfrac{\dd^2\psi_l(r)}{\dd r^2}+\left[k^2-V(r)-\dfrac{l(l+1)}{r^2}\right]\psi_l(r)=0.
\equn{(3.2.1)}$$
One may express its solutions as
$$\psi_l(r)\simeqsub_{r\to\infty}\dfrac{1}{2\ii}
\left\{\ee^{\ii kr-\ii l\pi/2+\ii \delta_l(E)}-
\ee^{-\ii kr+\ii l\pi/2-\ii \delta_l(E)}\right\}.
\equn{(3.2.2a)}$$
In principle, (3.2.2a) is valid only for 
physical $k\geq0$. However, because (3.2.1) depends explicitly on $k$, we can take the 
solution to be valid for arbitrary, even complex $k$. 

From (3.2.2a) we can find the p.w. amplitudes. First, we rewrite it as
$$\psi_l(r)\simeqsub_{r\to\infty}j^-(k,l)\ee^{\ii kr}+j^+(k,l)\ee^{-\ii kr};
\equn{(3.2.2b)}$$
the $j^\pm$, known as the Jost functions, are identified, 
at large $r$, comparing with (3.2.2a). 
In terms of these we can write the $S$-matrix element,
$$s_l(E)\equiv\ee^{2\ii \delta_l},$$
as
$$s_l(E)=(-1)^{l+1} \dfrac{j^-(k,l)}{j^+(k,l)}.
\equn{(3.2.3)}$$

Now, the exchange of $k\to-k$ does not alter the Schr\"odinger equation, but it 
exchanges the exponentials in (3.2.2a). Therefore, one must have
$$j^-(-k,l)=j^+(k,l).
\equn{(3.2.4)}$$

If we start from the $k$ plane, then the energy plane will be a two-sheeted 
Riemann surface (\fig~3.2.1). We designate {\sl physical} sheet (sheet I) to 
that coming from $\imag k>0$, and {\sl unphysical} sheet (sheet II) 
to that obtained from $\imag k<0$. 
When considering $s_l(E)$ as a function of $E$ it then follows that 
we have two determinations. 
Since obviously $k^{\rm II}=-k^{\rm I}$, we find 
$$s_l^{\rm II}(E)=\left[s_l^{\rm I}(E)\right]^{-1}.
\equn{(3.2.5)}$$
The physical value is  
$$s_l(E)=\lim_{\epsilon\to+0}s_l^{\rm I}(E+\ii\epsilon)=
\lim_{\epsilon\to+0}s_l^{\rm II}(E-\ii\epsilon).$$ 

We shall now look for singularities of $s_l(E)$. For 
physical $E>0$ we cannot have poles because $|s_l(E)|=1$. 
For $E=-E_B<0$, a pole of $s_l(E)$ means a zero of $j^+$. 
If the pole occurs in the first sheet, this 
means that the corresponding value of the momentum will be
$k_B=\ii|k|=\ii\sqrt{E_B}$ and hence (3.2.2b) becomes
$$\psi_l(r)\simeqsub_{r\to\infty} j^-(k_B,l)\ee^{-|k|r},$$
i.e., the wave function of a bound state with binding energy $E_B$. 
We thus conclude that poles of the 
$S$-matrix in the physical sheet 
for energies below threshold correspond to bound states.\fnote{We will not be interested in poles in the 
unphysical sheet with
 negative energies, known as {\sl antibound} states. 
More details on the subject of this section may 
be found in the treatises of Omn\`es and Froissart~(1963) and Goldberger and Watson~(1964).}


 
\topinsert{
\setbox0=\vbox{\hsize13.truecm{\epsfxsize 11.5truecm\epsfbox{riemann.eps}}} 
\setbox6=\vbox{\hsize 9truecm\captiontype\figurasc{Figure 3.2.1. }{The
Riemann sheet for p.w. amplitudes. \hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\centerline{{\box0}}
\vskip-1truecm
\centerline{\box6}
\medskip
}\endinsert

We next investigate the meaning of poles in the lower half-plane in the 
unphysical sheet, that is to say, poles located at $E^{\rm II}=E_R=E_0-\ii \gammav/2$ 
with $E_0,\,\gammav>0$. If there is a pole of $s_l^{\rm II}(E)$ for 
$E=E_R-\ii \gammav/2$, then (3.2.5) implies that 
the physical $S$-matrix element has a zero in the same location:
$$s_l^{\rm I}(E_R-\ii\gammav/2)=0$$
(\fig~3.2.1).
The corresponding wave function is not as easily obtained as for the bound state case; 
a detailed discussion may be found in Godberger and Watson (1966) or Galindo and 
Pascual (1978), but 
an essentially correct result may be obtained by replacing, in the standard 
time dependent wave function for stationary states 
$$\psiv=\ee^{-\ii Et}\psi({\bf r}),$$
$E$ by the complex value $E_R-\ii\gammav/2$. So  we get
$$\psiv=\ee^{-\gammav t/2}\ee^{-\ii E_Rt}\psi({\bf r}):$$
 the probability to find the state decreases with time as 
$|\psiv|^2=\ee^{-\gammav t}$, which can be interpreted as 
a metastable state that decays with a lifetime $\tau=1/\gammav$; that is to say, 
a {\sl
resonance}. 
$\gammav$ is called the {\sl width} of the resonance, and is 
equal to the indetermination in energy of the metastable state.

Let us consider now the corresponding physical phase shift. 
The pole and zero of $s^{\rm II}_l,\,s^{\rm I}_l$ 
imply corresponding zeros of the Jost functions. 
We will assume that $\gammav$ is very small; then, 
 in the neighbourhood of $E_R$ we can write
$$s^{\rm I}_l(E)\simeqsub_{E\sim E_R}
\dfrac{E-E_R-\ii\gammav/2}{E-E_R+\ii\gammav/2}.
\equn{(3.2.6a)}$$
For the phase shift this implies
$$\cot\delta_l(E)\simeqsub_{E\sim E_R}\dfrac{E_R-E}{\gammav/2}.
\equn{(3.2.6b)}$$
This means that at $E_R$ the phase shift goes, {\sl growing}, through $\pi/2$ 
and that it varies rapidly.

We can write the corresponding formulas for the p.w. amplitudes, 
now for the relativistic case. 
We profit from the analyticity of the effective range 
function over the elastic cut to conclude from (3.2.6b) 
and the proportionality between $\cot\delta_l$ and 
$\phiv_l$ that,  for 
$s=M_R^2$ (where $M_R$ is the invariant mass corresponding to 
the energy $E_R$), $\phiv_l(s)$ must have a zero: 
$$\phiv_l(s)\simeqsub_{s\simeq M_R^2}\dfrac{M_R^2-s}{\gamma}.
\equn{(3.2.7a)}$$ 
So we may write the p.w. amplitude in its neighbourhood as
$$f_l(s)\simeqsub_{s\simeq M_R^2}\dfrac{1}{\pi}
\,\dfrac{k^{2l}\gamma}{M_R^2-s-\ii k^{2l+1}\gamma/2s^{1/2}}.
\equn{(3.2.7b)}$$
The residue of $\phiv_l$, $\gamma$, can be related to the 
width of the resonance:
$$\gammav=\gamma k_R^{2l+1}/2M^2_R.
\equn{(3.2.7c)}$$
\equn{(3.2.7)} is the (relativistic) {\sl Breit--Wigner} formula for the 
p.w. scattering amplitude near a resonance. 
Note however that Eqs.~(3.2.7) are only valid in the vicinity of the resonance; 
away from it, the ratio $\phiv_l(s)/(M_R^2-s)$ will not 
be a constant, so in general we will have to admit  
a dependence of $\gamma$ (and $\gammav$) on $s$.

Let us consider another characterization of a resonance. 
Returning to  nonrelativistic scattering, one can prove 
that the time delay that the interaction causes in 
the scattering of two particles in angular momentum 
$l$ and with energy $E$ is 
$$\Delta t=2\dfrac{\dd\delta_l(E)}{\dd E}.$$
We can say that the particles resonate when this time delay is maximum. 
In the vicinity of a zero of the effective range, we can use 
(3.2.7) to show that $\Delta t(s)$ is maximum for 
$s=M_R^2$ and then the time delay equals $1/\gammav$. 

We have therefore three definitions of an {\sl elastic} resonance: a pole of 
the scattering amplitude in the unphysical Riemann sheet; a zero of the 
effective range function; or a maximum of the quantity
$$\dd\delta_l(s)/\dd s.$$
These three definitions agree, to order $\gamma^2$, when $\gamma$ is small, and neglecting 
variations of $\gamma$; but
a precise description of broad resonances requires 
discussion of these variations. 
In these notes, however, we will only give the value of $s^{1/2}$ at which the phase crosses $\pi/2$. 
Since we will also give explicit parametrizations, to find e.g. the 
location of the poles should not be a difficult matter for the interested reader.

Unstable 
 elementary particles may also be considered a special case of 
resonances; thus, for example, one may treat the
$Z$ particle as a  fermion-antifermion resonance. 
We  discuss this  for a simple model in \sect~4.2.

\bookendchapter
\brochureb{\smallsc chapter 4}{\smallsc the p p.w. amplitude for  $\pi\pi$
 scattering in the elementary rho model}{15}
\bookchapter{4. The P p.w. amplitude for  $\pi\pi$ scattering in the elementary rho model}
\vskip-0.5truecm
\booksection{4.1. The $\rho$ propagator and the $\pi^+\pi^0$ scattering amplitude}

\noindent
Before continuing with general properties of pion interactions, 
it is convenient to illustrate what we have already seen with 
a simple, explicit model.
In the present chapter we do precisely this; 
specifically, we consider the elementary rho model 
and take $\pi^0$, $\pi^+$ interactions to be given by 
the Lagrangian given in (1.3.1).
We start by calculating the $\rho^+$ propagator in dimensional regularization, to lowest order and neglecting
the  rho self-interactions. We therefore consider only the diagrams  in \fig~(4.1.1). 
The corresponding vacuum polarization function is then
$$\eqalign{\piv^{(\rho)}_{\mu\nu}(q)=&\,
\ii^2g^2_\rho\int\dd^D\hat{p}\,(2p+q)_\mu(2p+q)_\nu\dfrac{\ii}{p^2-\mu^2}\dfrac{\ii}{(p+q)^2-\mu^2}\cr
+&\,2\ii g^2_\rho g_{\mu\nu}\int\dd^D\hat{p}\,\dfrac{\ii}{p^2-\mu^2}; \cr}
$$ 
 we have defined
$$\dd^D\hat{p}\equiv\dfrac{\dd^D p}{(2\pi)^D}\nu^{4-D}_0,
$$
and $\nu_0$ is an arbitrary mass parameter. After standard manipulations, and with $D=4-\epsilon$, we find
$$\piv^{(\rho)}_{\mu\nu}(q)=(-q^2g_{\mu\nu}+q_\mu q_\nu)\dfrac{\ii g^2_\rho}{16\pi^2}
\left\{\tfrac{1}{3}\left(\dfrac{2}{\epsilon}-\gammae+\log4\pi-\log \nu^2_0\right)
-\int_0^1\dd x(1-2x)^2\log(\mu^2-x(1-x)q^2)\right\}.
\equn{(4.1.1a)}$$
Here $\gammae\simeq0.5772$ is Euler's constant.

 
\midinsert{
\medskip
\setbox0=\vbox{\hsize16.5truecm{\epsfxsize 15.7truecm\epsfbox{rho_prop.eps}}} 
\setbox6=\vbox{\hsize 11truecm\captiontype\figurasc{Figure 4.1.1. }{The
sum of one loop corrections to the rho propagator. \hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\centerline{{\box0}}
\vskip-1.5truecm
\centerline{\box6}
\medskip
}\endinsert

We then calculate the dressed rho propagator. For this, we first rewrite 
(4.1.1a) as
$$\piv^{(\rho)}_{\mu\nu}(q)=(-q^2g_{\mu\nu}+q_\mu q_\nu)\ii\piv_D(s),\quad s=q^2.
\equn{(4.1.1b)}$$
The dressed propagator is then,
$$D_{\mu\nu}^{(\rho;0)}=\dfrac{-\ii g_{\mu\nu}}{s-M_0^2}+
\dfrac{-\ii g_{\mu\nu}}{s-M_0^2}(-\ii s\piv_D)\dfrac{-\ii}{s-M_0^2}+\cdots+
\hbox{gauge terms.}.$$
The gauge terms are terms proportional to $q_\mu q_\nu$. Summing this we find
$$D_{\mu\nu}^{(\rho;0)}=\dfrac{-\ii g_{\mu\nu}}{s-M_0^2+s\piv_D}+\hbox{gauge terms}.$$



This is still unrenormalized, and $M_0$ is the unrenormalized rho mass. 
We renormalize in the \msbar\ scheme, with scale parameter the (renormalized) rho mass, 
$\nu^2_0=M^2$. 
Thus,
$$\piv_{\rm ren.}(s)=
-\dfrac{g^2_\rho}{16\pi^2}\int_0^1\dd x(1-2x)^2\log\dfrac{\mu^2-x(1-x)s}{\bar{M}^2}
\equn{(4.1.2a)}$$
and the renormalized, dressed rho propagator is
$$D_{\mu\nu}^{(\rho)}=\dfrac{-\ii g_{\mu\nu}}{s-\bar{M}^2+s\piv_{\rm ren.}(s)}+\hbox{gauge terms};
\quad \bar{M}=\bar{M}(M^2).
\equn{(4.1.2b)}$$
For $s$ real and larger than $4\mu^2$ we can split $\piv_{\rm ren.}$ into a real and 
an imaginary part as follows:
$$\piv_{\rm ren.}(s)=
-\dfrac{g^2_\rho}{16\pi^2}\int_0^1\dd x(1-2x)^2\log\left|\dfrac{\mu^2-x(1-x)s}{\bar{M}^2}\right|
+\ii \dfrac{g^2_\rho}{16\pi^2}\dfrac{8k^3}{3s^{3/2}},\quad s\geq 4\mu^2.
\equn{(4.1.3)}$$

We next evaluate the scattering amplitude, with the fully dressed propagator. 
We have to calculate the amplitudes $
F^{(s)}$ and  $F^{(u)}$   associated with diagrams 
(s), (u) in \fig~4.1.2, so that the scattering amplitude is
 $F=
F^{(s)}+F^{(u)}$. 
For the first we find,
$$
F^{(s)}=-16\dfrac{g^2_\rho}{16\pi^2}\,\dfrac{k^2}{s-\bar{M}^2+s\piv_{\rm ren.}}\cos\theta,
\equn{(4.1.4)}$$
where $\theta$ is the scattering angle in the c.m.


Projecting $F^{(s)}$ onto the P wave we get
$$
f^{(s)}_1(s)=\tfrac{16}{3}\dfrac{g^2_\rho}{16\pi^2}\,\dfrac{k^2}{\bar{M}^2-s-s\piv_{\rm ren.}(s)}.
\equn{(4.1.5)}$$

We have to add the contribution of diagram (u); note that, in this model, there is no 
contribution from the $t$ channel, at leading order, because you cannot make a $\rho$ 
with two $\pi^0$s.  
We have,
$$F^{(u)}=4\dfrac{g^2_\rho}{16\pi^2}\left(\dfrac{3s-4\mu^2}{2}-2k^2\cos\theta\right)
\dfrac{1}{u-\bar{M}^2+u\piv_{\rm ren.}(u)}
\equn{(4.1.6)}$$
and we recall that $u=4\mu^2-s-t=-2k^2(3+\cos\theta)$. 
Projecting into the P wave we find
$$f^{(u)}_1(s)=\tfrac{1}{2}\dfrac{4g^2_\rho}{16\pi^2}\int_{-1}^{+1}\dd\cos\theta\,\cos\theta
\left(\dfrac{3s-4\mu^2}{2}-2k^2\cos\theta\right)
\dfrac{1}{u-\bar{M}^2+u\piv_{\rm ren.}(u)}.
\equn{(4.1.7a)}$$


 
\topinsert{
\medskip
\setbox1=\vbox{\epsfxsize 10.5truecm\epsfbox{pipiscatter.eps}}
\centerline{\kern16em\box1} 
\setbox6=\vbox{\hsize 10truecm\captiontype\figurasc{Figure 4.1.2. }{Diagrams 
for $\pi^0\pi^+$ scattering mediated by the $\rho$.}} 
\bigskip
\centerline{\box6}
\bigskip
}\endinsert
\bigskip


The complete partial wave amplitude is
$$f_1(s)=
f^{(s)}_1(s)+f_1^{(u)}(s).
\equn{(4.1.7b)}$$

This p.w. amplitude does not satisfy 
unitarity. 
This is a general fact; perturbation theory only verifies {\sl perturbative}
 unitarity, that is to say, 
unitarity up to corrections of higher orders. 
We will see in this example how this works; this will allow us to 
see explicitly how in this model the rho behaves as a resonance.

First of all we check that $f_1$ verifies the expected analyticity properties. 
$
f^{(s)}_1(s)$ has a right hand cut, due to that of $\piv_{\rm ren.}(s)$ 
which is the only piece in (4.1.5) which is nonanalytic. 
From \equn{(4.1.3)} we see that it extends from 
$4\mu^2$ to $+\infty$. 
The l.h. cut of $f_1$ comes from the l.h. cut of $F^{(u)}_1(s,t)$. 
From (4.1.7), the only discontinuity occurs when $\piv_{\rm ren.}(u)$  
is discontinuous, which happens when $u\geq 4\mu^2$. 
In terms of $s,\,\cos\theta$ this condition becomes
$$u-4\mu^2=-\tfrac{1}{2}\left[s+4\mu^2
+(s-4\mu^2)\cos\theta\right]\geq0.$$
Therefore, $F^{(u)}(s,t)$ has a discontinuity for $s$ in the range from $-\infty$ to 
$$s_\theta=\dfrac{4\mu^2(\cos\theta-1)}{1+\cos\theta}.$$
Because in (4.1.7) we integrate for $\cos\theta$ between $-1$ and $+1$, it follows 
that the cut of $f^{(u)}_1(s)$, and hence of $f_1(s)$, runs from 
$-\infty$ to 0, as was to be expected on general grounds.


\booksection{4.2. The weak coupling approximation}
Let us now make further approximations. 
If we calculate the rho decay width in our model 
to lowest order  
we obtain, after a simple calculation,
$$\gammav(\rho^+\to\pi^+\pi^0)=\dfrac{g^2_\rho k^3_\rho}{6\pi M^2},
\quad k_\rho=\dfrac{\sqrt{M^2-4\mu^2}}{2}.$$
Putting numbers for the rho mass and width it follows that
$$\dfrac{g^2_\rho}{16\pi^2}\simeq 0.23$$
so the approximation of considering this quantity to be small 
is not too bad.

For $s$ physical, and in particular for $s\sim M^2$, the 
piece $s\piv_{\rm ren.}(s)$ in the expression 
for $
f^{(s)}_1(s)$, although of nominal order 
$g^2_\rho$ (see (4.1.2)) cannot be neglected; else, $
f^{(s)}_1(s)$ 
would be infinite around $s=M^2$. 
However, $u\piv_{\rm ren.}(u)$ can be neglected to a first approximation in $g^2_\rho$ in
 the expression (4.1.7). 
If we do this, $f_1^{(u)}$ can be easily integrated explicitly  and becomes
$$f_1^{(u)}(s)\simeq\dfrac{4g^2_\rho}{16\pi^2}\dfrac{2s-4\mu^2+\bar{M}^2}{k^2}
\left\{1-\dfrac{2k^2+\bar{M}^2}{4k^2}\log\left(1+\dfrac{4k^2}{\bar{M}^2}\right)\right\}.
\equn{(4.2.1)}$$
In this approximation the l.h. cut only runs up to $s=4\mu^2-M^2$: 
the discontinuity across the piece $[4\mu^2-M^2,0]$ is of 
order $(g^2/16\pi^2)^2$, and can be neglected (within the model) in a first approximation. 
With the value found for $g_\rho$, we expect this to be valid to some 6\%.


If we consider the region near $s=\bar{M}^2$, then $|f_1^{(s)}(s)|\sim 1$ 
while $f_1^{(u)}(s)$ is of order $g^2/16\pi^2$. 
We can further approximate $f_1$ by neglecting the whole of $f_1^{(u)}(s)$, 
 and thus write
$$f_1(s)\simeqsub_{s\sim M^2}
f^{(s)}_1(s)=
\tfrac{16}{3}\dfrac{g^2_\rho}{16\pi^2}\dfrac{k^2}{\bar{M}^2-s-s\piv_{\rm ren.}(s)}.
\equn{(4.2.2)}$$
It is important to notice that 
this approximation 
is only valid when $f_1^{(s)}(s)$ is of order unity; otherwise, both $s$ and $u$ channel 
pieces are of comparable order of magnitude.
Another interesting point is that 
this approximation is unitary and indeed it is very similar to the
Breit--Wigner approximation.  To see this more clearly, we define the 
(resonance) mass of the rho as the solution of the 
 equation
$$\bar{M}^2=M^2_\rho\left\{1+\dfrac{g^2_\rho}{16\pi^2}\int_0^1\dd
x(1-2x)^2\log\left|\dfrac{\mu^2-x(1-x)\bar{M}^2}{\bar{M}^2}\right|\right\}.
\equn{(4.2.3)}$$
From (4.1.3) and (4.1.5) it follows that, in the present approximation,
we can identify, for physical $s$,
$$\cot\delta_1(s)=\ii+\dfrac{6\pi}{k^3g^2_\rho}
\left[\bar{M}^2-s-s\piv_{\rm ren}(s)\right]=
\dfrac{6\pi s^{1/2}}{k^3g^2_\rho}
\left(\bar{M}^2-s-s\real\piv_{\rm ren}(s)\right),$$
which, because of (4.2.3), vanishes at $s=M^2_\rho$.

In this model, the effective range function is
$$\phiv_1(s)=\dfrac{s-4\mu^2}{4}\sqrt{\dfrac{4\mu^2}{s}-1}
+\dfrac{3\pi}{g^2_\rho}\left[\bar{M}^2-s-s\piv_{\rm ren}(s)\right].$$

\booksection{4.3. Low energy scattering}
We now calculate the low energy scattering in the elementary rho model; 
to be precise, we will evaluate the scattering length, $a_1$. 
For this calculation a number of approximations can be made. 
Although, in the real world, $\gammav_\rho$ and $\mu$ 
are very similar, it is believed that they 
have a different origin. 
$\mu^2$ is supposed to 
be proportional to the sum of $u$ and $d$ quark masses, whereas $\gammav_\rho$ 
is related to the QCD parameter $\lambdav$.
We will thus  make a calculation neglecting the $u$ channel contribution 
and evaluating $\real\piv$ in leading order in $\log M^2_\rho/\mu^2$. 
In this approximation we have (cf. Eqs.~(2.1.4), (4.1.5)) 
$$a_1^{(s)}=\dfrac{g^2_\rho}{12\pi\mu M^2_\rho}\,
\dfrac{1}{1-(4\mu^2/M^2_\rho)\left[1+\dfrac{g^2_\rho}{48\pi^2}\log\dfrac{M^2_\rho}{\mu^2}\right]}
\simeq36\times10^{-3}\,\mu^{-3}.
\equn{(4.3.1)}$$

The experimental value is 
$$a_1=(39.1\pm2.4)\times10^{-3}\,\mu^{-3}.$$
We see that, for such a crude model, the 
agreement with experiment is quite good; in fact, as we will see in Chapter~8, 
comparable to what one 
gets with sophisticated calculations. 
On the other hand, of course, the model is only valid for the P 
wave; 
for example, it gives zero (to order $g^2_\rho/16\pi$) for $\pi^0\pi^0$ 
scattering, although the interaction here is very strong.

\booksection{4.4. The chiral rho model}
The model we have developed for $\rho$ mediated pion interactions is not compatible 
with chiral symmetry. 
A model compatible with this has been developed by Gasser and Leutwyler;\fnote{In fact, 
the chiral rho model is much older; see e.g. Coleman, Wess and Zumino~(1969) or Weinberg~(1968b).} 
in it the $\rho$ is   coupled through the field strengths, 
$F^{(a)}_{\mu\nu}$, with $a$ an isospin index, to the 
pions. The model is rather complicated and  can be found in the paper of these authors 
(Gasser and Leutwyler,~1984; see also Ecker et al.~1989 where it is further developed). 
This coupling produces a nonrenormalizable interaction (as opposed to the 
previous rho model, which was renormalizable) so 
only tree level calculations are, in principle, allowed with it. 

In fact, it is possible to make loop calculations with this model, but 
to get finite results we will have to add extra interactions (and extra 
coupling constants) every time we go to a higher order in the number of 
loops taken into account; the model soon loses 
its predictive power and, in this respect, it is inferior to the nonchiral model we have studied 
before. 
Moreover, it cannot satisfy rigorous unitarity (that requires 
an infinite number of loops), although Dyson resumed versions 
of it are available in the literature (Guerrero and Pich,~1997).  
Its main interest lies in providing an {\sl explicit} realization 
for chiral perturbation theory calculations, and a way to extrapolate these to the resonance region. 

We will not give the details of such calculations here, that 
the interested reader may find  in  
the literature quoted.

\bookendchapter
\brochureb{\smallsc chapter 5}{\smallsc the effective range formalism. resonances (multichannel), etc.}{21}
\bookchapter{5. The effective range formalism for\hb p.w. amplitudes;
resonances \hb
(multichannel formalism).\hb
Unitarity and form factors; correlators}
\vskip-0.5truecm
\booksection{5.1. General formalism. Eigenphases}

\noindent
The extension of the developments of the previous section to the case where we have 
several channels open is very simple, provided 
these channels are all two-particle channels. 
To a good approximation this is the case for pion-pion scattering up to energies 
of about $s^{1/2}\simeq 1.3\,\gev$.

In the general case, we label the various two-body channels 
by  letters $a,b,\dots$, each with values $1,2,\dots,n$ (for $n$ channels). 
So, we have the p.w. amplitudes\fnote{We put in this Chapter
 the angular momentum variable $l$ as 
an index or superindex, according to convenience. 
So we write $f_l$ or $f^{(l)}$, $\delta^{(l)}$ or $\delta_l$.}
$f^{(l)}_{ab}(s)$ that describe scattering of particles\fnote{Note the reversed order; 
this is because the S matrix elements are usually defined by 
$$\langle P_1(a),P_2(a)|S|P_1(b),P_2(b)\rangle.$$}  $P_1(b)+P_2(b)\to
P_1(a)+P_2(a)$.  

As an example, we may have the channels
$$\matrix{\pi^+\pi^-,\quad &a=1\cr
\pi^0\pi^0,\quad &a=2\cr
K^+K^-,\quad &a=3\cr
K^0\bar{K}^0,\quad &a=4.\cr}$$
This would be simplified to two uncoupled two-channel problems 
(for isospin 0 and 1)  
if assuming isospin invariance.


We  define the (modulus of the) three-momentum, in channel $a$, as 
$k_a$. 
Then, the unitarity condition may be written  as
$$\imag f^{(l)}_{ab}(s)=\dfrac{\pi}{2s^{1/2}}\sum_ck_cf^{(l)}_{ac}(s)f^{(l)}_{bc}(s)^*,
\equn{(5.1.1)}$$
and we have used time-reversal invariance which implies that
$$f^{(l)}_{ab}=f^{(l)}_{ba}.$$ 
If we had only one channel, or if there were only diagonal interactions 
($f^{(l)}_{ab}=f^{(l)}_a\delta_{ab}$), (5.1.1) would tell us that
one can write
$$f^{(l)}_a=\dfrac{2s^{1/2}}{\pi k_a}\sin\delta^{(l)}_a\ee^{\ii \delta^{(l)}_a},
$$ 
i.e., \equn{(2.1.2)}.

To treat the general case it is convenient to use a matrix formalism. 
Denoting the matrices by boldface letters, we define 
$$
{\bf f}_l=\left(f^{(l)}_{ab}\right),\quad {\bf k}=(k_a\delta_{ab}).$$
We will also define the multichannel $S$-matrix elements,
$$s^{(l)}_{ab}(s)=\dfrac{2s^{1/2} }{\pi k_a}\delta_{ab}+2\ii f^{(l)}_{ab}(s)
\equn{(5.1.2a)}$$
or, in matrix notation,
$${\bf s}_l=(2s^{1/2}/\pi){\bf k}^{-1}+
{\bf f}_l.
\equn{(5.1.2b)}$$
If we had uncoupled channels, (2.1.2) would tell us immediately that
$$s^{(l)}_a=\dfrac{2s^{1/2}}{\pi k}\ee^{2\ii \delta_a^{(l)}}.$$
To see what the unitarity relations imply in the multichannel case,
 it is convenient to form the matrix $\bf u$
with
$$u_{ab}\equiv k_a^{1/2}s^{(l)}_{ab}k_b^{1/2}.$$
After a simple calculation, using (5.1.1) and 
time reversal invariance, we find that
$$\sum_c u^*_{ac}u_{bc}=\dfrac{4s}{\pi^2}\delta_{ab}.$$
Therefore, $(\pi/2s^{1/2}){\bf u}={\bf D}_l$ is a unitary matrix. 
We let ${\bf C}_l$ be the unitary matrix 
that diagonalizes it, and denote by $\widetilde{\bf D}_l$
to the diagonalized matrix, with elements
$(\exp{2\ii\widetilde{\delta}^{(l)}_a})\delta_{ab}$. 
The $\widetilde{\delta}^{(l)}_a(s)$ are called the {\sl eigenphases}, and are the
generalization to the multichannel case of the 
ordinary phase shifts.  We find that we can write:
$${\bf s}_l=
\dfrac{2s^{1/2}}{\pi}\,{\bf k}^{-1/2}{\bf C}_l\widetilde{\bf D}_l
{\bf C}_l^{-1}{\bf k}^{-1/2}.
\equn{(5.1.3)}$$
Note that, because of time reversal invariance, the matrix $\bf C$ may in fact be chosen to be 
{\sl real}. 

Inverting these relations we obtain the general form for the p.w. amplitudes,
$$\eqalign{
{\bf f}_l=&\dfrac{2s^{1/2}}{\pi}
{\bf k}^{-1/2}{\bf C}_l
\widetilde{\bf f}_l{\bf C}_l^{-1}{\bf k}^{-1/2},\cr
\widetilde{\bf f}_l=&\pmatrix{
\sin\widetilde{\delta}^{(l)}_1\ee^{\ii\widetilde{\delta}^{(l)}_1}&0&\dots&0\cr
0&\sin\widetilde{\delta}^{(l)}_2\ee^{\ii\widetilde{\delta}^{(l)}_2}&\dots&0\cr
\vdots&\vdots&\ddots&\vdots\cr
0&0&\dots&\sin\widetilde{\delta}^{(l)}_n\ee^{\ii\widetilde{\delta}^{(l)}_n}\cr}.
\cr}
\equn{(5.1.4)}$$
This is the generalization of (2.1.2) to the quasi-elastic multichannel case.

\booksection{5.2. The $K$-matrix and the effective range matrix. 
Resonances}

\noindent
We define the $K$-matrix, ${\bf K}_l$, 
such that
$$
{\bf f}_l=\left\{{\bf K}_l^{-1}-\dfrac{\ii\pi}{2s^{1/2}}{\bf k}\right\}^{-1}.
\equn{(5.2.1)}$$
In terms of it we can write the matrix ${\bf D}_l$ as
$${\bf D}_l=\dfrac{
1+\ii(\pi/2s^{1/2})\,{\bf k}^{1/2}{\bf K}_l{\bf k}^{1/2}}
{1-\ii(\pi/2s^{1/2})\,{\bf k}^{1/2}{\bf K}_l{\bf k}^{1/2}}.
\equn{(5.2.1)}$$
The unitarity and symmetry of ${\bf D}_l$ in the quasi elastic region
 means that ${\bf K}_l$ 
will be hermitean and symmetric there, hence it will be {\sl real} 
across the two particle cuts: 
$${\bf K}_l={\bf K}_l^{\dag}={\bf K}_l^*.$$

The definition of ${\bf K}_l$ does not take into account the 
behaviour at the thresholds. To do so we define the 
{\sl effective range} matrix, ${\bf \Phi}_l$ by
$${\bf \Phi}_l=\dfrac{1}{\pi}{\bf k}^l{\bf K}_l^{-1}{\bf k}^l.$$
In terms of it we find
$$
{\bf f}_l=\dfrac{1}{\pi}
{\bf k}^l\left({\bf \Phi}_l-\dfrac{\ii}{2s^{1/2}}{\bf k}^{2l+1}\right)^{-1}{\bf k}^l,
\equn{(5.2.2)}$$
an obvious generalization of (3.1.5). 
${\bf \Phi}_l$ is  real and symmetric. It is therefore analytic 
except for the l.h. cut of the $f^{(l)}_{ab}$, and for 
the r.h. cut that occurs when $s$ is above a true inelastic (multiparticle) threshold, 
$s>s_{\rm mult.}$.

Let us now discuss resonances in the multichannel case. 
It is clear that the  eigenstates of the time evolution operator 
will correspond to the eigenphases, as they are eigenstates of the 
$S$-matrix. 
We will therefore identify resonances with a resonant-like behaviour of the 
eigenphases: we will say that we have a resonance at $s=M^2$ 
provided one of the eigenphases crosses $\pi/2$ and varies rapidly there. 
We will assume that resonances are simple, i.e., only one eigenphase resonates 
at a given $s=M^2$, and moreover we suppose that $M$ does not 
coincide with the thresholds. 
The resonance condition, in eigenchannel $r$, is then
$$\widetilde{\delta}^{(l)}_r(s=M^2)=\pi/2;\quad
\left.\dfrac{\dd\widetilde{\delta}^{(l)}_r(s)}{\dd s}\right|_{s=M^2}=\hbox{maximum},
\equn{(5.2.3a)}$$
but
$$\widetilde{\delta}^{(l)}_{i\neq r}(s=M^2)\neq\pi/2.
\equn{(5.2.3b)}$$

Let us see what this implies in terms of ${\bf\Phi}_l$. From (5.1.4), (5.2.2) 
we can write
$$\widetilde{\bf f}_l=2s^{1/2}
\left({\bf C}_l^{-1}{\bf k}^{-l-1/2}{\bf \Phi}_l{\bf k}^{-l-1/2}{\bf C}_l-\ii\right)^{-1}.$$
Because $\widetilde{\bf f}_l$ and $\ii$ are diagonal, so must be
${\bf g}_l\equiv{\bf C}_l^{-1}{\bf k}^{-l-1/2}{\bf \Phi}_l{\bf k}^{-l-1/2}{\bf C}_l$. 
Recalling again (5.1.4), it follows that its elements are such that
$$(2s^{1/2}g^{(l)}_a-\ii)^{-1}=\sin\widetilde{\delta}^{(l)}_a\ee^{\ii\widetilde{\delta}^{(l)}_a},$$
i.e., one can write 
$$2s^{1/2}g^{(l)}_a=\cot\widetilde{\delta}^{(l)}_a.$$
The resonance condition then is equivalent (forgetting for the 
moment the requisite of rapid variation of the derivative of the phase) to 
the condition
$$g^{(l)}_r(s=M^2)=0;\quad g^{(l)}_{a\neq r}(s=M^2)\neq0.$$
Therefore, the quantity $\det({\bf g}_l(s))$ has a simple zero at $s=M^2$. 
Since, for this value of $s$, the determinants of ${\bf k},\,{\bf C}_l$ are finite, 
we have obtained that the condition of resonant behaviour 
(above all thresholds) is that the 
determinant of the  effective 
range matrix,
$$\det({\bf\Phi}_l(s))$$
has a simple zero at $s=M^2$.

We will next incorporate the condition of rapid variation, and calculate the 
{\sl partial widths}, that generalize the quantity $\gammav$ 
of the one-channel case. Near $s=M^2$ we write
$$\cot\widetilde{\delta}^{(l)}_r(s)\simeq \dfrac{M^2-s}{M\gammav}.\equn{(5.2.4)}$$
The condition of rapid variation is that $\gammav$ be small. 
Next, and using (5.1.4), we have
$$f^{(l)}_{ab}\simeqsub_{s\sim M^2}\dfrac{2s^{1/2}}{\pi}
\dfrac{1}{\sqrt{k_ak_b}}\left\{C^{(l)}_{ar}C^{(l)}_{br}\dfrac{M\gammav}{M^2-s-\ii M\gammav}+
\sum_{i\neq r}C^{(l)}_{ai}C^{(l)}_{bi}
\sin\widetilde{\delta}^{(l)}_i\ee^{\ii \widetilde{\delta}^{(l)}_i}\right\}.
\equn{(5.2.5a)}$$
We have profited from the unitarity and reality of ${\bf C}_l$ to write 
${\bf C}_l^{-1}={\bf C}_l^{\rm T}$.


We then define the {\sl partial widths}, $\gammav_a$, and {\sl inelasticity parameters} 
$x_a$ as
$$\gammav_a^{1/2}\equiv C^{(l)}_{ar}\gammav^{(1/2)};\quad
x_a=\gammav_a/\gammav.$$
Since the matrix ${\bf C}_l$ is orthogonal, 
one has $\sum_a\gammav_a=\gammav$. 
In terms of the $\gammav_a$ we can rewrite (5.2.5a) as
$$f^{(l)}_{ab}\simeqsub_{s\sim M^2}\dfrac{2s^{1/2}}{\pi}
\dfrac{1}{\sqrt{k_ak_b}}\left\{\dfrac{M\gammav_a^{1/2}\gammav_b^{1/2}}{M^2-s-\ii M\gammav}+
\sum_{i\neq r}C^{(l)}_{ai}C^{(l)}_{bi}\sin\widetilde{\delta}^{(l)}_i
\ee^{\ii \widetilde{\delta}^{(l)}_i}\right\}.
\equn{(5.2.5b)}$$
Thus we see that in the presence of a resonance all channels show a Breit--Wigner 
behaviour, plus a background due to the reflection of all the nonresonant eigenphases.

If, for a given channel, $x_a\simeq 1$, then we say that, in this channel, the resonance is
{\sl elastic}; if $x_a<1/2$, we say that it is {\sl inelastic}. 
For elastic resonances, and if
 the phase is near $\pi/2$ at the resonance, the parameter $\eta$ of (2.1.4) 
is related to $x$ by
$$\eta=2x-1.
\equn{(5.2.6)}$$ 

\booksection{5.3. Resonance parametrizations in the two-channel case}

\noindent
We will now present explicit formulas for parametrizations of resonances in 
the important case where only two channels are open. 
We start by changing a little bit the notation, writing, for obvious reasons, 
$g_l^{(\pm)}$ for the two 
eigenvalues of ${\bf g}_l$. 

We want to present parametrizations that profit from the analyticity of 
${\bf \Phi}_l$ so that they are not only valid on the resonance; thus,  
we will write our formulas in terms of ${\bf \Phi}_l$. 
Actually, we will use as parameters the 
diagonal elements of ${\bf \Phi}_l$,  $\phiv^{(l)}_{11}(s)$, $\phiv^{(l)}_{22}(s)$, and 
its determinant, that, because we have a resonance at $s=M^2$, we may write as  
$\det{\bf \Phi}_l(s)=\gamma(s)(s-M^2)$, with 
$\gamma(s)$ a smooth function (that can in most cases be approximated by a constant).

Next, we express the $g_l^{(\pm)}$ in terms of these parameters. 
We let $\deltav$ and $\tau$ be the determinant and trace of  ${\bf g}_l$. 
We have, on one hand, and in the physical region for both channels,    
$$g_l^{(\pm)}=\dfrac{\tau\mp\sqrt{\tau^2-4\deltav}}{2};\quad
{\bf g}_l=\pmatrix{g_l^{(+)}&0\cr
0&g_l^{(-)}\cr};\quad k_1,\,k_2\geq0 
\equn{(5.3.1a)}$$
and, on the other,
$$\eqalign{\deltav=&\det({\bf g}_l)=(k_1k_2)^{-2l-1}\det{\bf
\Phi}_l(s)=(k_1k_2)^{-2l-1}\gamma(s)(s-M^2),\cr 
\tau=&\trace {\bf g}_l=k_1^{-2l+1}\phiv^{(l)}_{11}+k_2^{-2l-1}\phiv^{(l)}_{22}.
\cr}
\equn{(5.3.1b)}$$
The resonating phase is  $\delta_l^{(+)}$ if $\tau$ is positive 
and  $\delta_l^{(-)}$ if $\tau$ is negative 
because, from (5.3.1), it follows that $\deltav$ 
 vanishes for $s=M^2$.

The mixing matrix ${\bf C}_l$ can also be obtained explicitly. 
One has,
$$\eqalign{{\bf C}_l=\pmatrix{\cos\theta&\sin\theta\cr
-\sin\theta&\cos\theta\cr};\quad
\cos\theta=
\left\{\dfrac{k_1^{-2l-1}\phiv^{(l)}_{11}-g_l^{(-)}}{g_l^{(+)}-g_l^{(-)}}\right\}^{1/2}.\cr}
\equn{(5.3.2)}$$

\booksection{5.4. Reduction to a single channel. Weakly coupled channels}
We will now consider the case in which one has two channels, but we are interested 
chiefly on one of them, that we will denote by channel 1. 
We will further assume that this channel opens before channel 2. 
Below the opening of channel 2, the formulas reduce to those of 
one single channel, so we can write (cf. (3.1.5))
$$f^{(l)}_{11}=\dfrac{1}{\pi}
\dfrac{k_1^{2l}}{\phiv^{(l)}_{\rm el}-\dfrac{\ii}{2s^{1/2}}k_1^{2l+1}}.
\equn{(5.4.1)}$$
 $\phiv^{(l)}_{\rm el}$ may be expressed in terms of ${\bf{\Phi}}^{(l)}$ using 
(5.2.2). 
We  define $\kappa_a=\ii k_a$ and get,
$$\phiv^{(l)}_{\rm el}=\,
\dfrac{\vphantom{\Bigg|}\dfrac{(-1)^l}{2s^{1/2}}\kappa_2^{2l+1}\phiv^{(l)}_{11}+\det {\bf{\Phi}}^{(l)}}
{\vphantom{\Bigg|}\dfrac{(-1)^l}{2s^{1/2}}\kappa_2^{2l+1}+\phiv^{(l)}_{22}}.
\equn{(5.4.2)} 
$$
Before the opening of channel 2, 
and above the l.h. cut, $\phiv^{(l)}_{\rm el}$ 
is, as expected, real and analytic.

It is worth noting that \equn{(5.4.2)} is 
still valid above the opening of channel 2, but $\kappa_2$ will now be 
{\sl imaginary}. 
Because of this some care has to be exercised to identify 
the quantity $\delta^{(l)}_{11}$. 
From (2.1.2), which is valid above threshold for channel 1, but  
below  channel 2 threshold we have, using (3.1.5), 
$$\cot\delta^{(l)}_{11}(s)=\dfrac{2s^{1/2}}{k_1^{2l+1}}\,
\phiv^{(l)}_{\rm el}(s),
\equn{(5.4.3)}$$
with $\phiv^{(l)}_{\rm el}$ given by (5.4.2). 
But, because $\kappa_2$ becomes imaginary above the opening of channel 2, 
it follows that $\cot\delta^{(l)}_{11}(s)$ will 
be complex there. 
This is of course to be expected; a real $\delta^{(l)}_{11}(s)$ 
implies strict elastic unitarity.

We next continue with  two channels, but now assume that they are 
weakly coupled. 
This is made transparent by writing
$$\phiv^{(l)}_{12}\equiv \epsilon_{12},$$ 
and we will work to lowest nontrivial order in $\epsilon_{12}$. 
We can write,
$$f^{(l)}_{11}=\dfrac{1}{\pi}
\dfrac{\vphantom{\Bigg|}\phiv^{(l)}_{22}-\dfrac{\ii}{2s^{1/2}}k_2^{2l+1}}
{\vphantom{\Bigg|}\left(\phiv^{(l)}_{11}-\dfrac{\ii}{2s^{1/2}}k_1^{2l+1}\right)
\left(\phiv^{(l)}_{22}-\dfrac{\ii}{2s^{1/2}}k_2^{2l+1}\right)-\epsilon^2_{12}}.
$$
Expanding to lowest order in the mixing, this becomes 
$$f^{(l)}_{11}=\dfrac{1}{\pi}\dfrac{k_1^{2l+1}}{\phiv^{(l)}_{11}-\dfrac{\ii}{2s^{1/2}}k_1^{2l+1}}
\left\{1+\dfrac{\phiv^{(l)}_{11}}{\phiv^{(l)}_{11}-\dfrac{\ii}{2s^{1/2}}k_1^{2l+1}}
\,\dfrac{\epsilon^2_{12}}{\phiv^{(l)}_{22}-\dfrac{\ii}{2s^{1/2}}k_2^{2l+1}}\right\}
\equn{(5.4.4)}$$
i.e., like an effective one-channel amplitude, 
$$\bar{f}^{(l)}_{11}=
\dfrac{1}{\pi}\dfrac{k_1^{2l+1}}{\phiv^{(l)}_{11}-\dfrac{\ii}{2s^{1/2}}k_1^{2l+1}}
\equn{(5.4.5a)}$$
modulated by the factor
$$G^{(l)}_1=
1+\dfrac{\phiv^{(l)}_{11}}{\phiv^{(l)}_{11}-\dfrac{\ii}{2s^{1/2}}k_1^{2l+1}}
\,\dfrac{\epsilon^2_{12}}{\phiv^{(l)}_{22}-\dfrac{\ii}{2s^{1/2}}k_2^{2l+1}}:
\equn{(5.4.5b)}$$
one has,
$$f^{(l)}_{11}=\bar{f}^{(l)}_{11}G^{(l)}_1.
\equn{(5.4.5c)}$$

In the case in which we have a resonance in each channel, we 
write
$$\phiv^{(l)}_{11}(s)\simeq(M_1^2-s)/\gamma_1,\quad
\phiv^{(l)}_{22}(s)\simeq(M_2^2-s)/\gamma_2.
\equn{(5.4.6)}$$
In this case (5.4.4) becomes
$$\eqalign{f^{(l)}_{11}\simeq&\dfrac{1}{\pi}
\dfrac{k_1^{2l+1}\gamma_1}{M_1^2-s-\ii k_1^{2l+1}\gamma_1/2s^{1/2}}\cr
\times&\left\{1+\dfrac{M_1^2-s}
{k_2^{2l+1}\left(M_1^2-s-\ii k_1^{2l+1}\gamma_1/2s^{1/2}\right)}\,
\dfrac{\epsilon^2_{12}\gamma_2k_2^{2l+1}}{M_2^2-s-\ii k_2^{2l+1}\gamma_2/2s^{1/2}}\right\}.\cr
}
\equn{(5.4.7)}$$
It is noteworthy that, if the resonances are narrow, 
and not too near the thresholds, the 
modulation of the first ($M_1$) by the second is negligible 
(of order $\gamma_2\epsilon^2_{12}$) except on top of the 
second, $s\simeq M^2_2$.

The mixing angle also has a simple expression now:
$$\sin\theta=
\dfrac{(k_1k_2)^{l+1/2}|\epsilon_{12}|^2}
{\left|k_2^{2l+1}\phiv^{(l)}_{11}-k_1^{2l+1}\phiv^{(l)}_{22}\right|}.
\equn{(5.4.8)}$$

We note to finish that the coupling of the channels displaces the resonances. 
Defining them as solutions of the equation
$$\det{\bf\Phi}(\widetilde{M}^2_a)=0,\quad a=1,2,
\equn{(5.4.9a)}$$
we see that e.g. for the first we have
$$\widetilde{M}_1^2=M_1^2+\dfrac{\gamma_1\gamma_2\epsilon^2_{12}}{M_2^2-M_1^2}.
\equn{(5.4.9b)}$$

\booksection{5.5. Unitarity for the form factors}

\noindent
The expression for the form factor of scalar particles $A,\, \bar{A}$ 
(which we consider with electric charge $\pm e$) in
the timelike region is defined, for example, in terms of the process 
$$e^+e^-\to A\bar{A}.$$
The corresponding matrix element may be written, to lowest order in the electromagnetic interaction, 
and with the effective photon-hadron interaction 
${\cal L}_{\rm eff}=eJ_\mu(x)A^\mu(x)$, as
$$\eqalign{\langle A(p_1)\bar{A}(p_2)&|S|e^+(k_1)e^-(k_2)\rangle\cr
=\ii& e^2\dfrac{1}{(2\pi)^3}\bar{v}(k_1)\gamma_\mu u(k_2)\dfrac{-\ii}{(p_1+p_2)^2} 
(2\pi)^4\delta(k_1+k_2-p_1-p_2)\langle A(p_1)\bar{A}(p_2)|J^\mu(0)|0\rangle,\cr
}
$$
and we recall that the form factor is defined  (for spinless particles) as
$$\langle A(p_1)\bar{A}(p_2)|J^\mu(0)|0\rangle=
(2\pi)^{-3}(p_1-p_2)^\mu F(s),\quad s=(p_1+p_2)^2.$$

Let us write the $S$ matrix as
$S=1+\ii{\cal T}$ so that 
$$\langle f|{\cal T}|i\rangle=\delta(p_f-p_i)F(i\to f).$$
 Unitarity of $S$ implies the relation
$${\cal T}-{\cal T}^+=\dfrac{1}{\ii}{\cal T}{\cal T}^+.$$
Taking matrix elements, we get
$$\imag \langle  A_a(p_1)\bar{A}_a(p_2)|{\cal T}|e^+e^-\rangle
=\tfrac{1}{2}\langle  A_a(p_1)\bar{A}_a(p_2)|{\cal T}{\cal T}^+|e^+e^-\rangle,$$
and we have assumed that we have several two particle channels, denoted with the 
index $a$. 
Summing now over intermediate states, we find 
$$\imag \langle  A_a(p_1)\bar{A}_a(p_2)|{\cal T}|e^+e^-\rangle
=\sum_b\int\dfrac{\dd^3{\bf q}_1}{2q_{10}}\dfrac{\dd^3{\bf q}_2}{2q_{20}}
\tfrac{1}{2}\langle  A_a(p_1)\bar{A}_a(p_2)|{\cal T}|A_b(q_1)\bar{A}_b(q_2)\rangle
\langle A_b(q_1)\bar{A}_b(q_2)|{\cal T}^+|e^+e^-\rangle.
$$
In terms of the form factors and scattering amplitudes, therefore,
$$\eqalign{
\imag (p_1-p_2)^\mu F_a(s)=&\,
\tfrac{1}{2}\sum_b \int\dfrac{\dd^3{\bf q}_1}{2q_{10}}\,\dfrac{\dd^3{\bf q}_2}{2q_{20}}
(q_1-q_2)^\mu F_b^*(s_q)
\langle  A_a(p_1)\bar{A}_a(p_2)|{\cal T}|A_b(q_1)\bar{A}_b(q_2)\rangle\cr
=&\,\tfrac{1}{2}\sum_b \int\dfrac{\dd^3{\bf q}_1}{2q_{10}}\,\dfrac{\dd^3{\bf q}_2}{2q_{20}}
\delta(q_1+q_2- p_1-p_2)(q_1-q_2)^\mu F_b^*(s_q)F_{ab}(q_1,q_2\to p_1,p_2)\cr
}$$
$s_q=(q_1+q_2)^2$.
In the c.m., $(p_1-p_2)^0=0$, $(p_1-p_2)^i=2k_i$ 
with $\bf k$ the c.m. three-momentum. Considering the spacelike part of above equation (the timelike part
is trivial) we find, after simple manipulations,\fnote{We hope there will be no confusion 
between the form factors, $F_a$, and scattering amplitudes, 
$F_{ab}=F_{ab}(s,t).$}
$$\imag  F_a(s)=
\tfrac{1}{2}\dfrac{1}{2{\bf k}^2}\sum_b  F_b^*(s_q)
\int\dfrac{\dd^3{\bf q}_1}{4q_{10}^2}\,({\bf q}_1{\bf p}_1)\delta(2p_{10}-q_{10}) F_{ab}.
$$
Writing
$$F_{ab}=\sum_l(2l+1)P_l(\cos\theta)f^{(l)}_{ab},$$
$\cos\theta=({\bf q}_1{\bf p}_1)/k_a k_b$, $k_a\equiv|{\bf p}_1|$, 
 $k_b\equiv|{\bf q}_1|$ we finally obtain the expression of unitarity in terms of form factors and 
p.w. amplitudes:
$$\imag  F_a(s)=
\dfrac{3\pi}{8s^{1/2}}\sum_b\dfrac{k^2_b}{k_a} F_b^*(s)f^{(1)}_{ab}(s).
\equn{(5.5.1)}$$

One can diagonalize this. With  the formulas for the $f^{(1)}_{ab}$ 
in terms of the eigenphase shifts, $\widetilde{\delta}^{(l)}_a$, 
and the diagonal p.w. amplitudes, 
we find (matrix notation)
$$\imag {\bf C}^{-1}{\bf k}^{3/2}{\bf F}=
\tfrac{3}{8}\widetilde{\bf f}^{(1)}{\bf C}^{-1}{\bf k}^{3/2}{\bf F}^*.$$ 
It follows that the combination
$$\sum_b C_{ba}k^{3/2}_b F_b$$ 
has a phase equal to $\widetilde{\delta}^{(1)}_a$. 
For the one channel case this proves the equality of the 
phases of form factor and p.w. amplitude (in the P wave).


\booksection{5.6. Unitarity for correlators}

\noindent
We will for definiteness consider a correlator\fnote{We 
write, generally, 
$\langle A\dots B\rangle_0\equiv \langle0|A\dots B|0\rangle$.} of vector currents (not 
necessarily conserved), $J_\mu$:
$$\piv_{\mu\nu}(q)=\ii\int\dd^4x\,\ee^{\ii q\cdot x}\langle{\rm T}J_\mu(x)J^{\dag}_\nu(0)\rangle_0
\equiv(-q^2g_{\mu\nu}+q_\mu q_\nu)\piv_{\rm tr}(q^2)+q_\mu q_\nu \piv_S(q^2),
\equn{(5.6.1)}$$
and we have split it into a transverse component ($\piv_{\rm tr}$) and a scalar one ($ \piv_S$). 
If the current was conserved, $\partial\cdot J=0$, then $\piv_S=0$.

The imaginary part of the correlator is given by the expression

$$\eqalign{
I_{\mu\nu}(q)=&\,\imag \piv_{\mu\nu}(q)=
\tfrac{1}{2}\int\dd^4x\,\ee^{\ii q\cdot x}\langle[J_\mu(x),J^{\dag}_\nu(0)]\rangle_0,
\quad q^2\geq0;\cr
I_{\mu\nu}(q)=&\,0,\quad q^2\leq0.\cr
}
\equn{(5.6.2a)}$$
Inserting a complete sum of states, $\sum_\gammav|\gammav\rangle\langle\gammav|$, this becomes
$$I_{\mu\nu}(q)=\tfrac{1}{2}\int\dd^4x\,\ee^{\ii q\cdot x}\sum_\gammav\langle0|J_\mu(x)|\gammav\rangle
\langle\gammav|J^{\dag}_\nu(0)|0\rangle.$$
Writing also
$$\langle0|J_\mu(x)|\gammav\rangle=\ee^{-\ii p_\gammav\cdot x}\langle0|J_\mu(0)|\gammav\rangle$$
we get the result
$$I_{\mu\nu}(q)=\tfrac{1}{2}(2\pi)^4
\sum_\gammav\delta(q-p_\gammav)\langle0|J_\mu(0)|\gammav\rangle\langle0|J_\nu(0)|\gammav\rangle^*.
\equn{(5.6.2b)}$$
(Of the two terms in the commutator only the first gives a nonzero result, 
because necessarily the momentum of $\gammav$, $p_\gammav$, has to be timelike). 
In particular, (5.6.2b) implies that $I_{\mu\nu}$ is positive definite, i.e., for any $p$,
$p^\mu I_{\mu\nu}p^\nu\geq0$. 
If we write
$$I_{\mu\nu}=(-q^2g_{\mu\nu}+q_\mu q_\nu)\imag\piv_{\rm tr}(q^2)+q_\mu q_\nu \imag\piv_S(q^2)
\equn{(5.6.3a)}$$
then
$$\imag \piv_{\rm tr}\geq0,\quad \imag\piv_S\geq0.
\equn{(5.6.3b)}$$

We will consider two important cases of intermediate states: 
when $|\gammav\rangle$ is a single particle state of mass $m$,  any spin, and when 
it is the state of two spinless particles. 
In the first case,
$$\sum_\gammav\to\sum_\lambda\int\dfrac{\dd^3p}{2p_0}|p,\lambda\rangle\langle p,\lambda|$$
and $\lambda$ is the third component of the spin. 
Then, and working in the c.m. reference system where $q_0=\sqrt{q^2}\equiv s^{1/2}$, ${\bf q}=0$,
$$\eqalign{
I_{\mu\nu}(q)=&\,\tfrac{1}{2}(2\pi)^4\sum_\lambda\int\dfrac{\dd^3p}{2p_0}\,\delta(q-p)
\langle0|J_\mu(0)|p,\lambda\rangle\langle0|J_\nu(0)|p,\lambda\rangle^*\cr
=&\,\dfrac{(2\pi)^4}{4s^{1/2}}\delta(s^{1/2}-m)
\sum_\lambda\dfrac{\sqrt{2}\, F_\mu(q,\lambda)}{(2\pi)^{3/2}}
\dfrac{\sqrt{2}\, F^*_\nu(q,\lambda)}{(2\pi)^{3/2}};
\cr}
$$
 we have defined
$$\langle0|J_\mu(0)|p,\lambda\rangle=\dfrac{\sqrt{2}}{(2\pi)^{3/2}}F_\mu(p,\lambda).
\equn{(5.6.4a)}$$
If the particle is a pion $\pi^-$ and $J_\mu$ is the weak axial current, 
$J_\mu=\bar{u}\gamma_\mu\gamma_5 d$, then $ F_\mu(q,\lambda)$ is related to 
the {\sl pion decay constant}, $f_\pi$:
$$ F_\mu(q,\lambda)=f_\pi p_\mu,\quad f_\pi\simeq 93\,\mev,
\equn{(5.6.4b)}$$
see Chapter~8. 
In general we have
$$I_{\mu\nu}(q)=2\pi\delta(s-m^2)\sum_\lambda F_\mu(q,\lambda)F_\nu(q,\lambda)^*.
\equn{(5.6.5)}$$

For the case of a two-particle intermediate state, with spinless particles,
$$\sum_\gammav\to \int\dfrac{\dd^3 p_1}{2p_{10}}\dfrac{\dd^3 p_2}{2p_{20}}\,
|p_1p_2\rangle\langle p_1p_2|$$
and then
$$\eqalign{
I_{\mu\nu}(q)=&\,\tfrac{1}{2}(2\pi)^4\int\dfrac{\dd^3 p_1}{2p_{10}}\dfrac{\dd^3 p_2}{2p_{20}}\,
\delta(p-p_1-p_2)\langle0|J_\mu(0)|p_1p_2\rangle\langle0|J_\nu(0)|p_1p_2\rangle^*\cr
=&\,\dfrac{(2\pi)^4}{2 s^{1/2}}\int\dd^3k\,
\delta(s-(p_1+p_2)^2)\langle0|J_\mu(0)|p_1p_2\rangle\langle0|J_\nu(0)|p_1p_2\rangle^*;
\cr}
$$
 ${\bf k}={\bf p}_1=-{\bf p}_2$. 
If we assume that the current is conserved, we can express the expectation value of the current 
in terms of a form factor,\fnote{If the current is not conserved we will have terms 
proportional to $p_1+p_2$ in (5.6.6).}
$$\langle0|J_\mu(0)|p_1p_2\rangle=\dfrac{1}{(2\pi)^3}(p_1-p_2)_\mu F(s),
\equn{(5.6.6)}$$
hence
$$I_{\mu\nu}(q)=\dfrac{|F(s)|^2}{2(2\pi)^2s^{1/2}}\int\dd^3k\,
\delta(s-(p_1+p_2)^2)(p_1-p_2)_\mu(p_1-p_2)_\nu.
$$
The integral is easiest calculated in the c.m. reference system. 
Here $(p_1-p_2)_\mu=2k_\mu$, and we have defined $k_0|_{\rm c.m.}=0$. If $\mu$ is the mass of the particles in
the  intermediate state (assumed equal, as they have to be if the current is conserved), then 
$(p_1+p_2)^2=2(\mu^2+{\bf k}^2)$. 
In spherical coordinates,
$$I_{\mu\nu}(q)=\dfrac{\sqrt{s-4\mu^2)}}{4(2\pi)^2s^{1/2}}|F(s)|^2\int\dd\omegav_{\bf k}k_\mu k_\nu.$$
The angular integral, returning to an arbitrary reference system is
$$\int\dd\omegav_{\bf k}k_\mu k_\nu=\dfrac{4\pi}{3s}
\left(\dfrac{s}{4}-\mu^2\right)(-g_{\mu\nu}s+q_\mu q_\nu),$$
so we get the final expression
$$\imag \piv_{\rm tr}(s)=\dfrac{1}{6\pi}\left(\dfrac{s/4-\mu^2}{s}\right)^{3/2}|F(s)|^2.
\equn{(5.6.7)}$$



\bookendchapter
\brochureb{\smallsc chapter 6}{\smallsc p.w. amplitudes for $\pi\pi$ scattering. 
form factors}{31}
\bookchapter{6. Extraction and parametrizations of p.w. amplitudes for $\pi\pi$ scattering.\hb
Form factors}
\vskip-0.5truecm
\booksection{6.1. $\pi\pi$ scattering }

\noindent
There is of course no possibility to arrange collisions of {\sl real} pions. 
One can get information on some phase shifts, at a few energies, 
from processes such as  kaon decays, or from 
the pion electromagnetic or weak form factors (about which more later). 
But a lot of, unfortunately not very precise, information comes from 
peripheral pion production, that we now 
briefly discuss.

What one does is to collide pions with protons and produce two pions and either 
a nucleon, $N$, or a resonance $\Delta$:
$$\pi p\to\pi\pi N;\quad \pi p\to \pi\pi\Delta.$$
One selects events where the momentum $p_\pi$ transferred by the incoming 
pion to the proton
 is small and thus one can assume that the process is mediated 
by  exchange of a virtual pion (\fig~6.1.1). 
The process $\pi p\to \pi\pi\Delta$ is in principle more difficult to analyze than 
$\pi p\to\pi\pi N$; but the last presents a zero for $p_\pi\sim0$, thus suppressing it in the more 
interesting region: 
both processes are, in consequence, equally well (or equally poorly) suited for extracting 
$\pi\pi$ scattering data. 
We then expect that the scattering amplitude for the full process 
will factorize into the $\pi\pi$ scattering amplitude, with one pion off-shell, 
$F(s,t;p_\pi^2)$, and the matrix element $\langle H|\phi_\pi|p_\pi\rangle$.
Here $H=N,\;\Delta$ 
and $\phi_\pi$ is the pion field operator.

It is clear that the method presents a number of drawbacks. 
First of all, a model is necessary for the dependence on $p_\pi$ 
of $F(s,t;p_\pi^2)$ and $\langle H|\phi_\pi|p_\pi\rangle$. 
Indeed, a model is required for  $\langle H|\phi_\pi|p\rangle$ itself. 
Secondly, in factorizing the full processes one is neglecting final state interactions between 
the pions and the $N$ or $\Delta$. These are presumably small, but only rather crude 
models exist for them.
 
\midinsert{
\setbox1=\vbox{\hsize9truecm{\epsfxsize 7.5truecm\epsfbox{pipi_prod.eps}}}
 \setbox6=\vbox{\hsize 5truecm\captiontype\figurasc{Figure 6.1.1 }{\hb 
Diagrams 
for $\pi p\to\pi\pi N,\,\Delta$.\hb
\phantom{X}\hb}} 
\line{\box1\hfil\box6}
\medskip
}\endinsert


Another very important problem is that, as soon as inelastic
 channels become important for $\pi\pi$ scattering, which
occurs for $s^{1/2}\sim1\,\gev$ for the S wave and for $s^{1/2}\gsim 1.4\,\gev$ for 
P, D waves, the analysis becomes impossibly complicated: the errors grow very fast.\fnote{In fact, 
it can be proved (Atkinson, Mahoux and Yndur\'ain, 1973) that, even if one only has two channels, 
say, $\pi\pi$ and $\bar{K}K$,
there is no unique solution (at fixed energy) unless one also measured $\pi\pi\to\bar{K}K$, and
 $\bar{K}K\to\bar{K}K$ as well. 
That inelastic channels are important for $s^{1/2}\gsim1.4\,\gev$ 
is clear by looking at the branching ratios of resonances with higher mass.}
Indeed, above $s^{1/2}\sim 1.5\,\gev$
 it is impossible to disentangle the interesting processes
from 
a number of other ones and, as a consequence, there are 
hardly any reliable  data. 


As a consequence of all these difficulties, it happens that the sets
 of phase shifts one extracts from 
data present unknown biases and, 
in particular,  are dependent on the models used to perform the fits. 
This is very clear in the several sets of solutions presented by Protopopescu et al. (1973), and 
in the large errors of the analysis of Hyams et al. (1973) or Grayer et al.~(1974). 
We could have tried to quantify this 
by introducing systematic error (for example, the difference between 
various determinations) but 
we prefer not to do so in general and 
simply admit that a \chidof\ of up to $2\,\sigma$, with only statistical errors, may 
be  acceptable.

A help out of these difficulties is to use supplementary information from 
processes like 
$$e^+e^-\to\pi^+\pi^-,\quad \tau^+\to\bar{\nu}_\tau\pi^+\pi^0,\quad K\to l{\bar{\nu}_l}\pi\pi,
\quad K\to 2\pi.$$
We will discuss the first three later, but note already that this only provides information on 
the S, P waves at low energy ($s\lsim 1\,\gev^2$). 
Another possibility is to supplement the experimental information with theory; 
in \sects~6.3 to 5 of this chapter we take into account the analyticity
 properties of p.w. amplitudes to write
economical and accurate 
parametrizations of these; the implementation of other constraints, such 
as dispersion relations, is left for next chapter. 

\booksection{6.2. Form factors}
\vskip-0.5truecm
\booksubsection{6.2.1. The pion form factor}

\noindent 
The process $e^+e^-\to\pi^+\pi^-$ (\fig~6.2.1) can, at low energy $t^{1/2}\lsim1\,\gev$, 
be related to the pion form factor.
We can write 
$$\dfrac{\sigma^{(0)}(e^+e^-\to{\rm hadrons})}{\sigma^{(0)}(e^+e^-\to\mu^+\mu^-)}
=12\pi\imag \Piv(t),
$$
where $\Piv$ is hadronic part of the photon polarization function and 
the superindices (0) mean that we evaluate the so tagged quantities to lowest order in 
electromagnetic interactions. 
At low energy this is dominated by the $2\pi$ state and we have
$$\imag\Piv=\imag \piv_{2\pi}(t)=\dfrac{1}{48\pi}\left(1-\dfrac{4\mu^2}{t}\right)^{3/2} 
|F_\pi(t)|^2.\equn{(6.2.1)}$$


\midinsert{
\setbox1=\vbox{\hsize9.5truecm{\epsfxsize 8.5truecm\epsfbox{form_factor.eps}}}
 \setbox6=\vbox{\hsize 5truecm\captiontype\figurasc{Figure 6.2.1 }{\hb
Diagram 
for $e^+e^-\to\pi^-\pi^+$.\hb
\phantom{X}\vskip0.5truecm
\phantom{X}}} 
\line{\box1\hfil\box6}
\medskip
}\endinsert

The evaluation of the pion form factor is slightly complicated by the phenomenon 
of $\omega-\rho$ interference. This can be solved by considering only the 
isospin $I=1$ component, and adding later the $\omega\to2\pi$ 
and interference 
separately; 
that is to say,  in a first approximation we neglect the breaking of isospin 
invariance. We will also neglect for now  electromagnetic corrections. 
In this approximation the properties of $F_\pi(t)$ are the following: 
\item{(i) }{$F_\pi(t)$ is an analytic function of $t$, with a cut 
from $4\mu^2$ to infinity.}
\item{(ii) }{On the cut, the phase of $F_\pi(t)$ is, because of unitarity, identical to 
that of the P wave, $I=1$, $\pi\pi$ scattering, $\delta_1(t)$, and 
this equality 
holds until the opening of the inelastic threshold at $t=s_0$. 
This we showed in \sect~5.4, and the property is known as the
 Fermi--Watson final state interaction 
theorem.}
\item{(iii) }{For large $t$, $F_\pi(t)\sim 1/t$. This follows from 
perturbative QCD.}
\item{(iv) }{$F(0)=1$.}
 
The inelastic threshold occurs, rigorously speaking, at $t=16\mu^2$. 
However, it is an experimental fact that inelasticity is negligible 
until the quasi-two~body channels $\omega\pi,\,a_1\pi\,\dots$ are open. 
In practice we will take
$$s_0\simeq 1\;\gev^2,$$
and fix the best value for $s_0$  empirically. 
It will be $s_0=1.05^2\,\gev^2$, and it so happens that, if we keep close to 
 this value, the dependence of the results of our analysis on  $t_0$ is very slight. 

\booksubsection{6.2.2. Form factor of the pion in $\tau$ decay}

\noindent 
Besides the process $e^+e^-\to\pi^+\pi^-$ one can get  data on the vector pion 
form factor  from the decay
 $\tau^+\to\bar{\nu}_\tau \pi^+\pi^0$ (\fig~6.2.2) 
For this 
 we have to assume isospin invariance, to write the form factor $v_1$ 
for $\tau$ decay 
 in terms of $F_\pi$:
$$v_1=\tfrac{1}{12}\left(1-\dfrac{4\mu^2}{t}\right)^{3/2}|F_\pi(t)|^2,
\equn{(6.2.2a)}$$
where, in terms of the weak vector current $V_\mu=\bar{u}\gamma_\mu d$, 
and in the exact isospin approximation,
$$\Piv^V_{\mu\nu}=\left(-p^2g_{\mu\nu}+p_\mu p_\nu\right)\Piv^V(t)=
\ii\int\dd^4x\,\ee^{\ii p\cdot x}\langle0|{\rm T}V^+_\mu(x)V_\nu(0)|0\rangle;\quad
v_1=2\pi\imag \Piv^V.
\equn{(6.2.2b)}$$
\equn{(6.2.2)} may be verified inserting a complete set of states in the 
expression for $\imag \piv^V$, and 
assuming it to be saturated by the states $|\pi^+\pi^0\rangle$; 
cf.~\equn{(5.6.7)}.



%\topinsert{
\setbox1=\vbox{\hsize7truecm\hfil {\epsfxsize 6.truecm\epsfbox{tau_decay.eps}}\hfil}
 \setbox6=\vbox{\hsize 5truecm\captiontype\figurasc{Figure 6.2.2 }{\hb 
Diagram 
for $\tau\to\bar{\nu}_\tau\pi^0\pi^+$.\hb
\phantom{X}\vskip0.5truecm
\phantom{X}}} 
\line{\box1\hfil\box6}
\smallskip
%}\endinsert

We next make a few remarks concerning  the matter of isospin breaking, due to 
electromagnetic interactions or the  mass difference between $u,\;d$
 quarks, that would spoil the equality (6.2.2a). 
It is not easy to estimate this. A large part of the breaking, the 
$\omega\to2\pi$ contribution and $\omega-\rho$ mixing, may 
 be taken into account explicitly (for the form factor in $\pi^+\pi^-$) 
with the Gounnaris--Sakurai~(1968) method, but this does not exhaust the effects. 
\eqs~(6.2.2) were obtained neglecting 
the mass difference $m_u-m_d$ 
and electromagnetic corrections, 
in particular the $\pi^0 - \pi^+$ mass difference. 
We can take the last partially into 
account by distinguishing between the pion masses in the 
phase space factor in (6.2.2a). 
To do so, we write now (6.2.2b) as
$$\Piv^V_{\mu\nu}=
\ii\int\dd^4x\,\ee^{\ii p\cdot x}\langle0|{\rm T}V^+_\mu(x)V_\nu(0)|0\rangle=
\left(-p^2g_{\mu\nu}+p_\mu p_\nu\right)\Piv^V(t)+p_\mu p_\nu \piv^{S};\quad
v_1\equiv2\pi\imag \Piv^V.
\equn{(6.2.3a)}$$  
We find 
$$v_1=\tfrac{1}{12}
\left\{\left[1-\dfrac{(m_{\pi^+}-m_{\pi^0})^2}{t}\right]
\left[1-\dfrac{(m_{\pi^+}+m_{\pi^0})^2}{t}\right] \right\}^{3/2}|F_\pi(t)|^2.
\equn{(6.2.3b)}$$
To compare with the experimentally measured quantity, 
which involves all of $\imag \piv^V_{\mu\nu}$, we have to 
neglect the scalar component $\piv^S$. This is reasonable, as it is proportional to $(m_d-m_u)^2$, 
and thus likely very small. 
This matter of isospin breaking one thus treats in successive steps. 
First, we neglect isospin breaking. Then we 
take it into account in the masses and widths of the resonances $\rho^0,\,\rho^+$, 
and including $\omega-\rho$ mixing, the
difference in phase space, etc. 
Before doing so, however, we must develop the 
necessary mathematical tools, which we will do in next chapter.

\booksubsection{6.2.3. $K_{l4}$ decay}

\noindent
We now consider the so-called $K_{l4}$ decay, 
$$K\to l\bar{\nu}_l\pi^+\pi^-,$$
with $l$ an electron or a $\mu^-$. 
The effective lagrangian for the decay is
$${\cal L}_{\rm int,eff}=\dfrac{G_F\cos\theta}{\sqrt{2}}\,
\bar{l}\gamma_\mu(1-\gamma_5)\nu_l\,\bar{s}\gamma^\mu(1-\gamma_5)u,
$$
where $G_F$ is Fermi''s constant, $\theta$ the Cabibbo angle, 
and $s$, $u$ the field operators for the 
corresponding quarks. 
The decay amplitude is then
$$F(K\to l\bar{\nu}_l\pi^+\pi^-)=\dfrac{G_F\cos\theta}{\sqrt{2}(2\pi)^2}
\bar{v}_l\gamma_\mu(1-\gamma_5)u_{\nu_l}F^\mu(s);
$$
 the form factor $F^\mu$ is
$$F^\mu(s)=
\langle \pi^+(p_+)\pi^-(p_-)|\bar{s}(0)\gamma^\mu\gamma_5u(0)|K\rangle,
\quad s=(p_++p_-)^2.
$$
If we expand $F^\mu$ into a scalar ($F_S$) and a vector piece, $F_P$,
$$F^\mu=(p_+^\mu+p_-^\mu)F_S+(p_+^\mu-p_-^\mu)F_P;$$
then one can, with an argument like that of \sect~5.5, show that
$${\rm Arg}\,F_S(s)=\delta_0^{(0)}(s),\quad
{\rm Arg}\,F_P(s)=\delta_1(s).$$
It follows that, by measuring the differential decay rate
$$\dfrac{\dd\Gammav(K\to l\bar{\nu}_l\pi^+\pi^-)}{\dd s\dd \Omegav_{\bf p}},$$
with ${\bf p}={\bf p}_+|_{\rm c.m.}$, 
we can separate 
the contributions from $|F_S|^2$, $|F_P|^2$ 
and the interference piece,
 $F_S^*F_P=|F_S|\,|F_P|\cos[\delta_0^{(0)}(s)-\delta_1(s)]$,
and thus get 
 the difference of phases $\delta_0^{(0)}(s)-\delta_1(s)$.
This provides very important information on  low energy 
$\pi\pi$ scattering, particularly since, in this process, both pions 
are on their mass shell.


\booksection{6.3. The P  wave}

\noindent
We present in this and the following two  sections of the present chapter 
parametrizations of the S, P, D and F waves in $\pi\pi$ scattering 
that follow from the theoretical requirements
 we have discussed in previous chapters, and which agree
with  {\sl experimental} data. 
To check that the scattering amplitude that one obtains in this way 
is consistent with dispersion relations or the 
Froissart--Gribov representation 
will be done in the following chapter.

\booksubsection{6.3.1. The P  wave  in the elastic approximation}
We will consider first the P wave for $\pi\pi$ scattering for energies below the 
region were the inelasticity reaches the 2\% level; say, below 
$s_0=1.1\,\gev^2$. 
We will neglect for the moment isospin invariance violations due 
to e.m. interactions or the $u\,-\,d$ quark mass difference. 
This implies, in particular, neglecting the $\omega$ and $\phi$ interference  
effects.
 
We may use the analyticity properties of $\phiv_1(s)$ to write a 
simple parametrization of $\phiv_l(s)$, hence of $\delta_1(s)$. 
An effective range expansion is not enough, as it only converges in the region 
$|s-4\mu^2|<0$ (\fig~3.1.2). To take fully advantage
 of the analyticity domain, shown in \fig~2.1.1, 
the simplest procedure is to make a conformal mapping of the 
cut plane into the unit disk 
(\fig~6.3.1) by means of the transformation\fnote{Out parametrization 
presents a number of advantages with respect to less efficient ones 
used in the literature . The  gain 
obtained by taking into account the correct 
analyticity properties is enormous; see the Appendix here for a discussion and an explicit example, and 
 Pi\u{s}ut~(1970) for other examples and applications to $\pi\pi$ scattering (for discussion of a specific 
example, see the Appendix). Moreover, the physical meaning of,say, (6.3.3) is very clear: 
$b_0$ gives the normalization, and 
$b_1$ 
is related to the average intensity of the l.h. cut and the inelastic cut.}
$$w=
\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}.
\equn{(6.3.1)}$$


One can then expand $\phiv_1(s)$ in powers of $w$, and, reexpressing 
$w$ in terms of $s$, the expansion will be convergent over all the cut $s$-plane. 
Actually, and because we know that the P wave resonates at $s=M^2_\rho$, 
it is more convenient to expand not  $\phiv_1(s)$ 
itself, but ${\psi}(s)$ given by
$$\phiv_1(s)=(s-M^2_\rho){\psi}(s)/4;
\equn{(6.3.2a)}$$
so we write
$${\psi}(s)=\left\{b_0+b_1w+
\cdots\right\}.
\equn{(6.3.2b)}$$

 
\topinsert{
\setbox0=\vbox{\hsize11.5truecm{\epsfxsize 10truecm\epsfbox{map_2.eps}}} 
\setbox6=\vbox{\hsize 3truecm\captiontype\figurasc{Figure 6.3.1 }{The  
mapping $s\to w$.\hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\line{
%\tightboxit
{\box0}\hfil\box6}
\medskip
}\endinsert



In terms of  $\phiv_1(s)$ we find the expression for the phase shift, 
 keeping two terms in the expansion,
$$\delta_1(s)={\rm Arc\; cot}\left\{\dfrac{s^{1/2}}{2k^3}
(M^2_\rho-s)\left[b_0+b_1\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}
\right]\right\};
\equn{(6.3.3)}$$
$M_\rho,\,b_0,\,b_1$ are free parameters to be fitted to experiment.
In terms of $\phiv_1,\,\psi$ we have, for the rho width,
$$\gammav_\rho=\dfrac{2k^3_\rho}{M^2_\rho{\psi}(M^2_\rho)},
\quad
k_\rho=\tfrac{1}{2}\sqrt{M^2_\rho-4\mu^2},
\equn{(6.3.4a)}$$
and the scattering length, $a_1$, is
$$a_1=\dfrac{1}{4\mu\phiv_1(4\mu^2)}.
\equn{(6.3.4b)}$$

The values $b_0=\hbox{const.}$, $b_{i\geq 1}=0$ would 
correspond to a perfect Breit--Wigner. 
Actually, it is known that the $\rho$ deviates from a pure Breit--Wigner 
and for a precision parametrization 
 two terms, $b_0$ and $b_1$, have to be kept in (6.3.3). 
Note that the parametrization holds not only on the physical region 
$4\mu^2\leq s\leq s_0$, but on the unphysical region $0\leq s\leq 4\mu^2$ and 
also over the 
whole region of the complex $s$ plane with $\imag s\neq 0$.
The parametrization given now is the one that has less biases, in the sense 
that no model has been used: we have imposed only the highly safe requirements of 
analyticity and unitarity, depending only on 
causality and conservation of probability.


 
\topinsert{
\setbox0=\vbox{\hsize12.truecm{\epsfxsize11truecm\epsfbox{fit_P_pipi.eps}}} 
\setbox6=\vbox{\hsize 10truecm\captiontype\figurasc{Figure 6.3.2 }{The  
phase shifts of solution 1 from Protopopescu et al.~(1973) (the dots, 
with errors of  the size of the dots) compared with the prediction with 
the parameters (6.3.5a), described by the solid line. 
We emphasize that this solid line is {\sl not} a fit 
to the data of Protopoescu et al., but is obtained 
from the pion form factor.}
} 
\medskip
\centerline{\tightboxit{\box0}}
\bigskip
\centerline{\box6}
\bigskip
}\endinsert 

The best values for our parameters are actually obtained from fits to the 
pion form factor, cf.~\sect~7.2. We have,
$$b_0=1.060\pm0.005,\quad b_1=0.24\pm0.04;\quad M_\rho=772.9\pm0.8\,\mev.
\equn{(6.3.5a)}$$
The corresponding values for the width of the $\rho$ and
 for the scattering length are\fnote{We give the values of the scattering length in terms of
$\mu=138\,\mev$ 
rather than in terms of $m_{\pi^+}=139.6\,\mev$.}
$$\gammav_\rho=147.3\pm0.7\,\mev,\quad a_1=(39.1\pm2.4)\times10^{-3}\mu^{-3}.
\equn{(6.3.5b)}$$
Although the values of the experimental $\pi\pi$ 
phase shifts were {\sl not} included in the fit, the 
phase shifts that (6.3.5a) implies are en very good agreement with them, 
as shown in \fig~6.3.2.

Eqs.~(6.3.5) above were evaluated with an average of information on the 
two channels that contain the $I=1$ P wave, $\pi^+\pi^-$ (dominated by the 
$\rho^0$) and $\pi^0\pi^+$, dominated by the $\rho^+$. 
The values for a pure $\rho^0$ ($\pi^+\pi^-$) are slightly different; we find
$$\eqalign{b_0=&\,1.070\pm0.006,\quad b_1=0.28\pm0.06,\quad M_{\rho^0}=773.2\pm0.6,\cr 
\gammav_{\rho^0}=&147.4\pm1.0\,\mev,
}
\equn{(6.3.5c)}$$
and $a_1$ does not change appreciably. 
However, this last feature occurs only because 
the fit was made including the constraint 
$a_1=(38\pm3)\times10^{-3}\,\mu^{-3}$; 
see \sect~9.5 for more on this.
(6.3.5)  provide an estimate of the importance of  isospin breaking. 


 

\booksubsection{6.3.2. The $\rho$ and weakly coupled inelastic channels:  $\omega-\rho$ interference}

\noindent
Because of the different masses of the $u,\,d$ quarks, isospin invariance is broken 
and there is a nonzero probability of transition 
between $\pi^+\pi^-$ in isospin 1 and isospin 0
states: hence, a small --but nonzero-- mixing of the $\rho$ and $\omega$ resonances.

To study this phenomenon a popular approximation is that of Gounnaris and 
Sakurai (1968). A  consistent 
treatment requires a two-channel analysis. 
We denote by channel 1 to the P wave isospin 1  $\pi^+\pi^-$  state, and channel 2 will 
be a P wave isospin zero $3\pi$ state. 
To be fully rigorous, we would have to set up a three-body formalism for the last; 
but we will simply take this into account replacing the two body by 
three body phase space for the $\omega$. Using now Eqs.~(5.4.4) to (5.4.7) we write

$$\eqalign{f^{(1)}_{11}=&\dfrac{1}{\pi}
\dfrac{k_1^{3}\gamma_1}{M^2_\rho-s-\ii k_1^{3}\gamma_\rho(s)/2s^{1/2}}\cr
\times&\left\{1+\dfrac{M^2_\rho-s}
{k_\omega^{3}(s)\left(M^2_\rho-s-\ii k_1^{2l+1}\gamma_\rho(s)/2s^{1/2}\right)}\,
\dfrac{\epsilon^2_{12}\gamma_\omega k_2^{3}}{M^2_\omega-s-\ii k_\omega^{3}(s)\gamma_\omega/2s^{1/2}}\right\}.\cr
}
\equn{(6.3.7)}$$
Here we still have
$$k_1=\tfrac{1}{2}\sqrt{s-4\mu^2}
\equn{(6.3.8a)}$$
but for $k_\omega(s)$ 
we have to take the value following from three-body phase space. 
Because the interference effect is only important 
near $s=M^2_\omega$,  a reasonable approximation for it is 
to take $k_\omega$ constant: this is the model of Gounnaris and Sakurai~(1968). 
The model is completed if we
take a constant width for the $\omega$, justified in view of its narrowness, but a full effective range
formula for the 
$\rho$:
$$\gamma_\omega=\gammav_\omega^2/2f_\omega(M^2_\omega),
\quad \gamma_\rho(s)=1/\bar{\phiv}(s),
\equn{(6.3.8b)}$$
with $\bar{\phiv}(s)$ given by a parametrization like (6.3.3). 
The effect of this modulation is a shoulder above the $\rho$ that may be seen in e.g. the 
pion form factor (cf.~\fig~7.2.1).

\booksubsection{6.3.3. The P wave for $1\gev\leq s^{1/2}\leq 1.3\gev$}
In the range $1\,\gev\leq s^{1/2}\leq 1.3\,\gev$ one is sufficiently far away from thresholds to neglect 
their influence (the coupling to $\bar{K}K$ is negligible) and, 
moreover, the inelasticity is reported  small: 
according to Protopopescu et al.~(1973), below the 7\% level. 
A purely empirical parametrization that agrees with the data in this reference 
and Grayer et al.~(1974), up to 
1.2 \gev, within errors, 
is given by a modulated $\rho$ tail,
$$\delta_1(s)={\rm Arc}\,\cot\dfrac{\eta\,(M_\rho-s)}{M_\rho\gammav_\rho}-
\epsilon\,\left(1-\dfrac{4m_K^2}{s}\right)^{3/2},\quad \eta=0.75\pm0.10,\;\epsilon=0.08\pm0.02.
\equn{(6.3.9)}$$
and the second term takes into account the effects of the inelasticity.

For larger $s^{1/2}$, this is incompatible with the properties of the P wave as measured in 
$e^+e^-$ annihilations, where a highly inelastic resonance occurs around 1450 \mev.
An alternate parametrization for the imaginary part of the p.w. 
amplitude  that takes this into account is obtained by adding to the imaginary part produced by 
(6.3.9) the inelastic piece
$$\imag f_{1;{\rm inel}}(s)=\dfrac{2s^{1/2}}{\pi k}\dfrac{{\rm BR}\times
M^2\gammav^2}{(s-M^2)^2+M^2\gammav^2};\quad M=1.45\,\mev, \;\gammav=310\,\mev,\;{\rm BR}\simeq0.15.
\equn{(6.3.10)}$$
The value of BR could vary by 50\%.




\booksection{6.4. The D and F waves}
\vskip-0.5truecm
\booksubsection{6.4.1. Parametrization of the $I=2$ D wave}

\noindent
For isospin equal 2, there are no resonances in the D wave (or, indeed, in any other wave), 
at least at low energies. 
This is an experimental fact that can be understood theoretically by recalling that 
one cannot have $I=2$ with a quark-antiquark state. 
It would seem that we could simply write the parametrization
$$\cot\delta_2^{(2)}(s)=\dfrac{2s^{1/2}}{k^5}\,\dfrac{b_0+b_1w(s)}{4}\,\mu^4,\quad b_i=\hbox{constant},
\equn{(6.4.1a)}$$
but life is more complicated, as we will see presently.
We take
$$w(s)=
\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}
\equn{(6.4.1b)}$$  
but with
$s_0=1.38^2\,\gev^2$ now (see below  for the reason for this choice). 
However, 
a pole term is necessary to get an acceptable fit down to low energy
since we expect $\delta_2^{(2)}$ to change sign near threshold.
Indeed, the measured values (Losty et al.,~1974; Hoogland et al.,~1977) give 
negative and small values for the phase above some $500\,\mev$ while we will 
see that chiral perturbation calculations (\sect~9.4) 
and the Froissart--Gribov representation\fnote{An interesting feature of the Froissart--Gribov 
calculation is that the structure of $\delta_2^{(2)}$, in 
particular the zero near threshold, was 
in fact {\sl predicted} from it (Palou and Yndur\'ain,~1974).}  (\sect~7.5) 
indicate a positive scattering length, $a_2^{(2)}\simeq(1.7\pm0.6)\times10^{-4}\,\mu^{-5}$.
 
Because the phase is so small we expect (comparatively) large, unknown 
systematic errors;  we may then choose to fit only above $650\,\mev$. 
In this case we can  use (6.4.1a), as it stands. 
The best result is obtained  fitting\fnote{If we included the data of 
Losty et al.~(1974) the \chidof\ would jump to a value above 2.}
 the data on $\delta_2^{(2)}(s)$ of
solution A  of Hoogland et al.~(1977) in the 
energy range $650\,\mev\leq s^{1/2}\leq1050\,\mev$. 
Here we get a  $\chidof=10.0/(8-2)$ 
and the values of the parameters
$$b_0=-(2.3\pm0.6)\times10^{3},\quad b_1=-(3.7\pm0.8)\times10^{3};\qquad s^{1/2}\gsim650\;\mev.
\equn{(6.4.1c)}$$
 The errors have been estimated so that the corresponding 
bands for the phase shifts overlap the low energy points of  Hoogland et al.~(1977) 
as well as those of Losty et al.~(1974).



The parametrization (6.3.1) should be considered as purely 
empirical, however, and certainly not valid below the fitted range. 
If we want a parametrization that 
applies down to threshold, we must incorporate the 
zero of the phase shift. So 
we write
$$\cot\delta_2^{(2)}(s)=
\dfrac{2s^{1/2}}{k^5}\,\dfrac{b_0+b_1 w(s)}{4}\,\dfrac{\mu^4 s}{4(\mu^2+\deltav^2)-s}
\equn{(6.4.2a)}$$
with $\deltav$ a free parameter. Moreover, we impose  the 
value $a_2^{(2)}=(1.7\pm1.0)\times10^{-4}\,\mu^{-5}$ for the scattering length 
that follows from the Froissart--Gribov representation, \sect~7.5.  
We get a mediocre fit ($\chidof=17.3/(9-3)$) and the values of the parameters are
$$b_0=(1.96\pm0.06)\times10^3,\quad b_1=(3.03\pm0.03)\times10^3,\quad \deltav=3.4\pm0.4\,\mev.
\equn{(6.4.2b)}$$
  The \chidof\ prefers the higher values of the scattering length, 
$a_2^{(2)}=2.64\times10^{-4}\,\mu^{-5}$; the value of $\deltav$ is small compared to the expected one, 
$\sim30\,\mev$. 
Doubtlessly  unknown biases in the experimental data, 
probably related to those for the $I=2$ S wave, 
precludes a better fit.



\booksubsection{6.4.2. Parametrization of the $I=0$ D wave}

\noindent
The D wave with isospin 0 in $\pi\pi$ scattering presents two resonances 
below $1.7\,\gev$: the $f_2(1270)$ and the $f_2(1525)$, 
that we will denote respectively by $f_2$, $f'_2$. 
Experimentally, 
$\gammav_{f_2}=185\pm4\,\gev$ and $\gammav_{f'_2}=76\pm10\,\gev.$ 
The first, $f_2$,  
couples mostly to $\pi\pi$, with small  couplings to 
$KK$ ($4.6\pm0.5\,\%$), $4\pi$ ($10\pm3\,\%$) and 
$\eta\eta$. The second 
couples mostly to $2K$, with a small coupling to $\eta\eta$ and $2\pi$, 
respectively $10\pm3\,\%$ and $0.8\pm0.2\,\%$. 
This means that the channels $\pi\pi$ and $KK$ are 
essentially decoupled: they only connect indirectly, 
so it is not very profitable to set up a multiple channel calculation.
To a 15\% accuracy we may neglect inelasticity up to $s_0=1.38^2\gev^2$. 
The formulas are like those for the P wave; 
we will discuss them presently. 

There are not many experimental data on the D wave which, at accessible energies, 
 is  small. 
So, the compilation of  $\delta_2^{(0)}$ 
phase shifts of Protopopescu et al. (1973) covers 
only the range $810\leq s^{1/2}\leq 1150\,\mev$. 
In view of this, it is impossible to get accurately the D wave scattering lengths, 
or indeed any other low energy parameter, from this information.
We give here a parametrization whose use lies in that 
it represents with reasonable accuracy the data, something that will be useful later on. 
We write
$$\cot\delta_2^{(0)}(s)=\dfrac{2s^{1/2}}{k^5}\,(M^2_{f_2}-s)\,\dfrac{\mu^2}{4}\,{\psi}(s),\quad
{\psi}(s)=b_0+b_1w(s)+\cdots\,,
\equn{(6.4.3a)}$$
and $w$ is as in (6.4.1b).

We take the data of Protopopescu et al. (1973) and consider the so-called  ``solution 1",
with the two possibilities given in Table~VI and 
Table~XIII (with modified higher moments). 
These data cover the range 
mentioned before, $s^{1/2}=0.810\,\gev$ to $1.150\,\gev$.
The problem with these data points is that they 
are contaminated, for $s\gsim1.1\,\gev^2$, by the 
bias of the S wave with $I=0$ in the same region, whose values there are
quite incompatible with
those of other experiments 
(see \subsect~6.5.3). 
For this reason we perform two fits: either including or excluding the data points for
$s^{1/2}\geq1.075\,\gev$. 
In both cases we present results only for the version with 
modified higher moments (Table~XIII in Protopopescu et al., ~1973) 
as they are the ones that show better compatibility with 
other experiments. 
We also impose the fit to the width of the $f_2$ resonance, 
with the condition $\gammav_{f_2}=185\pm10\,\mev$. 
We find,
$$\eqalign{
\dfrac{\chi^2}{\rm d.o.f.}=&\,\dfrac{46.9}{14-2},\quad b_0=20.16,\quad b_1=19.48,\qquad
\hbox{[All points]};\cr
\gammav_{f_2}&\,=213\,\mev,\quad a_2^{(0)}=17\times10^{-4}\,\mu^{-5}\cr
}
$$
and
$$\eqalign{
\dfrac{\chi^2}{\rm d.o.f.}=&\,\dfrac{20.5}{10-2},\quad b_0=23.95,\quad b_1=18.91,\qquad
\hbox{[Only points for $s^{1/2}<1.075\,\gev$]};\cr
\gammav_{f_2}&\,=187\,\mev,\quad a_2^{(0)}=11\times10^{-4}\,\mu^{-5}.\cr
}
$$
The drastic decrease of the \chidof\ when eliminating the higher
 energy points signals clearly their biased
character.\fnote{We remark again that the \chidof\ is less poor than it looks at first sight, 
as it only takes into account statistical errors, while systematic
 ones are certainly is large as  these.} 
However, the parameters of the fits are rather stable, no doubt because we have imposed the 
correct width of the resonance $f_2$. 
We therefore take as our best result an average of 
the two determinations, with half their 
difference as an estimated error:
$$b_0=22.1\pm1.9,\quad b_1=19.2\pm0.3
\equn{(6.4.3b)}$$
and this corresponds to
$$\gammav_{f_2}=200\pm13\,\mev,\quad a_2^{(0)}=(14\pm3)\times10^{-4}\,\mu^{-5},
$$
 reasonably close to their experimental values (the 
second as deduced from the Froissart--Gribov 
representation, cf.~\sect~7.6). 


\booksubsection{6.4.3. The F wave}

\noindent
The experimental situation for the F wave is somewhat confused. 
According to Protopopescu et al.~(1973) it starts negative
 (but compatible with zero at the $2\,\sigma$ level) and becomes positive around $s^{1/2}=1\,\gev$. 
Hyams et al.~(1973) and Grayer et al.~(1974) report a positive $\delta_3(s)$ 
when it differs from zero (above $s^{1/2}=1\,\gev$). 
In both cases the inelasticity is negligible up to, at least, $s^{1/2}=1.5\,\gev$.

The corresponding scattering length may be calculated with the help of the Froissart--Gribov 
representation and one finds (\sect~7.6)
$$a_3=(6.5\pm0.7)\times10^{-5}\,\mu^{-7}.
\equn{(6.4.4)}$$
It is possible that $\delta_3(s)$ changes sign {\sl twice}, once near threshold 
and once near $s^{1/2}=1\,\gev$.    
However,  we disregard 
this possibility and write, simply,
$$\cot\delta_3(s)=\dfrac{2s^{1/2}}{k^7}\,\dfrac{b_0+b_1w(s)}{4}\,\mu^6,\quad
w(s)=
\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}},\quad s_0=1.5^2\,{\gev}^2,
\equn{(6.4.5a)}$$
and {\sl impose} (6.4.4). It is to be understood
 that this parametrization provides only an empirical representation of the available data, 
and that it may not be reliable except at very low energies, 
where it is dominated by the scattering length, and between 
$1\,\gev\leq s^{1/2}\leq 1.1\,\gev$. 
We find 
$$\dfrac{\chi^2}{\rm d.o.f.}=\dfrac{5.7}{7-2},\quad b_0=(1.07\pm0.03)\times 10^5,\quad
b_1=(1.35\pm0.03)\times 10^5.
\equn{(6.4.5b)}$$
For $s^{1/2}\geq1.1\,\gev$, however, effect of the $\rho_3(1690)$ resonance 
should be included.

\booksection{6.5. The S wave}
\vskip-0.5truecm
\booksubsection{6.5.1. Parametrization of the S wave for  $I=2$}

\noindent
We consider two sets of experimental data.
 The first, that we will denote by
``Hoogland~A", 
 corresponds to solution A in the paper by 
Hoogland et al.~(1977), who use the reaction $\pi^+ p\to\pi^+\pi^+n$; and 
the set denoted by ``Losty,"
 to that from the work of Losty
et al.~(1974), who analyze instead  $\pi^- p\to\pi^-\pi^-\Delta$. 
We will not consider the so-called solution B in the paper of Hoogland et al.~(1977); 
while it produces results similar to the other two, its errors 
are clearly underestimated. These 
results represent a substantial improvement over 
previous ones; since they produce two like charge pions, only isospin 2 
contributes, and one gets rid of the large isospin zero S wave and P wave contamination. 
However, they still present the problem that 
one does not have scattering of real pions.

For isospin 2, there is no low energy resonance, but $f_0^{(2)}(s)$ presents the feature that a zero 
is expected (and, indeed, confirmed by the fits) 
in the region $0<s<4\mu^2$.  If we neglected this and wrote
$$\eqalign{\cot\delta_0^{(2)}(s)=&\,\dfrac{2s^{1/2}}{2k}\,\dfrac{b_0+b_1w(s)}{4};\cr
w=&\,
\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}},\quad s_0=(1.450\;\gev)^2,\cr}
$$
then we could fit the data
with the parameters
$$b_0=-1.87,\quad b_1=5.56.$$
We have a not too bad $\chidof=13.8/(14-2)$ but the expansion has poor convergence 
properties as, in most of the region, $|b_1w|$ is rather larger than $|b_0|$. 
The corresponding value of the scattering length would be $a_0^{(2)}=0.16\,\mu^{-1}$, 
way too large (that a naive fit gives a scattering length of this order 
has been known for a long time; see Prokup et al., 1974). 
Clearly, we have to take the zero of the partial wave into account.


\topinsert{
\setbox0=\vbox{\hsize12.truecm{\epsfxsize 11.2truecm\epsfbox{del_pipi_S2.eps}}} 
\setbox6=\vbox{\hsize 11truecm\captiontype\figurasc{Figure 6.5.1 }{
The  
$I=2$, $S$-wave phase shifts corresponding to (6.5.1), 
with experimental points from Losty et al.~(1974) (open circles) and Hoogland et al.~(1977), solution A 
(black dots).  
}\hb} 
\centerline{\tightboxit{\box0}}
\bigskip
\centerline{\box6}
\medskip
}\endinsert

The zero of $f_0^{(2)}(s)$ is related to the so-called Adler zeros (see Chapter~9) 
and, to lowest order in chiral perturbation theory,  
occurs at $s=2z_2^2$ with $z_2=\mu$. In view 
of this, 
we extract the zero (leaving its value as a {\sl free} 
parameter) and write
$$\cot\delta_0^{(2)}(s)=\dfrac{2s^{1/2}}{k}\,\dfrac{\mu^2}{s-2z_2^2}\dfrac{{b}_0+{b}_1w(s)}{4}.
\equn{(6.5.1a)}$$
The quality of the fit improves substantially: we get $\chidof=8.0/(14-3)$ 
and a second order term such that $|b_1w|<|b_0|$.
 The parameters are now
$${b}_0=-116\pm5.6,\quad {b}_1=-127\pm9,\qquad z_2=145\pm21\,\mev.
\equn{(6.5.1b)}$$
The corresponding scattering length is 
$$a_0^{(2)}=(-0.060\pm0.023)\mu^{-1}.
\equn{(6.5.2)}$$
In the fit (6.5.1) we have not considered experimental data above 0.97 \gev.
The result for the scatering length is compatible (within $\sim1\,\sigma$), as we will see, with the values
suggested by  chiral perturbation theory;\fnote{The 
discrepancy is likely due to 
a systematic bias of the experimental data. 
In fact, if we also include in the fit the data of the CERN-Munich collaboration, in the 
version of Estabrooks and Martin~(1974), the central value of $a_0^{(2)}$ becomes 
$0.46\,\mu^{-1}$.} and this agreement is 
satisfactory also in another respect: 
 the value for $z_2$ which the fit returns, $z_2=145\pm21\,\mev$, comprises the 
value expected from second order chiral perturbation theory that gives $z_2=131\,\mev$.
In \sect~6.9 we will see that our numbers here are also compatible with 
forward dispersion relations.

\booksubsection{6.5.2. Parametrization of the S wave for  $I=0$}

\noindent
The S wave with isospin zero is by far the most difficult to parametrize.
 Here we have a very broad 
enhancement, variously denoted as $\epsilon,\,\sigma,\,
f_0$, around
$s^{1/2}\equiv M_\sigma\simeq700\,\mev$; we will use the name $\sigma$. We will not discuss here whether
 this enhancement is a {\sl bona fides} 
resonance; we merely remark that in all experimental
 phase shift analyses $\delta_0^{(0)}(s)$ crosses 90\degrees\ 
somewhere between 600 and 900 \mev. (This is {\sl not} enough to class 
the object as a resonance. For example, the derivative $\dd \delta_0^{(0)}(s)/\dd s$ is  
more a minimum than a maximum 
 at $M_\sigma$).

There is also a possible resonance (which used to be called
$S^*$ and is now denoted by $f_0(980)$) and another resonance (which was
called
$\epsilon'$ in the seventies), labeled 
as $f_0(1370)$ in the Particle Data Tables, with a mass
around $1.37\,\gev$. 
Moreover,we expect  a zero of $f_0^{(0)}(s)$ (Adler zero), hence a pole of 
the effective range function 
$\phiv_0^{(0)}(s)$, for $s=z_0^2$ with $z_0^2$ in the region $0< s<4\mu^2$. 
In fact, chiral perturbation theory suggests that this zero is located at 
$z_0^2=
\tfrac{1}{2}\mu^2$ but,  as we will discuss in 
\subsect~9.3.5, one cannot trust the accuracy of this prediction, unlike what happened for the 
$I=2$ zero, $z_2$.
 
We can distinguish two energy regions: below $s^{1/2}_0=2m_K$ we are under 
the $\bar{K}K$ threshold. 
Between $s^{1/2}_0$ and $s^{1/2}\sim1.3$ there is a strong coupling between the 
 $\bar{K}K$ and $\pi\pi$ channels and the analysis becomes very unstable, because there 
is little information on the process $\pi\pi\to\bar{K}K$ and even less on 
$\bar{K}K\to\bar{K}K$. 
We will not treat this case here in any detail; the interested reader may find details and references in 
Yndur\'ain~(1975), Aguilar-Ben\'{\i}tez et al.~(1978). 
We will merely present, in the next subsection, an empirical fit in the 
region of energies around and above 1 \gev, and 
 we will now concentrate our efforts in the low energy region.

Below the $\bar{K}K$ threshold we can write a one-channel formula:
$$\cot\delta_0^{(0)}(s)=\dfrac{2s^{1/2}}{k}\,\phiv_0^{(0)}(s).
\equn{(6.5.4)}$$
To parametrize $\phiv_0^{(0)}$ we have, as stated, a difficult situation, from the theoretical 
as well as  from the
experimental point of view.  From the first, 
 and because of the strong coupling
 of the $\bar{K}K$ channel above $s=4m^2_K$, it is
essential to take into account the presence of the associated  cut. 
Moreover, and to reproduce correctly the low energy region data, the Adler zero cannot be neglected:
 we must necessarily use a complicated parametrization.

On the experimental side the situation is a bit confused.
The experimental information we have on this $I=0$ S wave is of three kinds: 
from phase shift analysis in collisions $\pi p\to\pi\pi N,\Delta$; 
from the decay $K_{l4}$; and  from the decay $K_{2\pi}$. 
The last gives the value of the combination 
$\delta_0^{(0)}-\delta_0^{(2)}$ at $s^{1/2}=m_K$; 
the decay $K_{l4}$ gives $\delta_0^{(0)}-\delta_1$ at low energies, $s^{1/2}\lsim380\;\mev$. 
If using the more recent $K_{2\pi}$ information\fnote{I am
grateful to Colangelo, Gasser and Leutwyler for bringing  this to my attention.}
 (Aloisio et al.,~2002) one finds one finds
 the figure 
$\delta_0^{(0)}-\delta_0^{(2)}=58.0\degrees\pm4.6$. 
With the information on the $I=2$ phase obtained in the previous subsection,  
this gives the $I=0$ phase
$$\delta_0^{(0)}(m^2_K)=41.5\degrees\pm4\degrees.
\equn{(6.5.6a)}$$
The change is substantial from the previous experimental values 
that implied (Pascual and
Yndur\'ain,~1974)
 
$$\delta_0^{(0)}(m^2_K)=51\degrees\pm8\degrees.
\equn{(6.5.6b)}$$ 
Unfortunately, both determinations depend crucially on assuming the weak current 
to be a  
pure octet, on isospin inveriance, and on radiative corrections,\fnote{See Cirigliano, Donoghue and 
Golowich~(2000);  Belavin and Navodetsky~(1968)
and  Nachtmann and de~Rafael~(1969).} 
not well defined; and even a small deviations from the values we have taken would cause a
noticeable
 shift in 
$\delta_0^{(0)}(m^2_K)$.
Because of this we will make fits with  both determinations  (6.5.6a,\/b). 

From the various phase shift analyses one concludes that there is {\sl not} a unique solution 
if fitting only $\pi\pi$ data; 
one can get an idea of the uncertainties in old analyses by having a look at 
\fig~3.3.6 in the book by Martin, Morgan and Shaw~(1975) or realizing that 
 the values of the scattering length $a_0^{(0)}$ that the various experimental fits 
(Protopopescu et al., 1973; Hyams et al., 1973; Grayer et al.~1974) gave  vary in the range
$$0.1\leq a_0^{(0)}\leq 0.9\;\mu^{-1}.
$$
Today one can improve substantially on this thanks to the appearance of $K_{l4}$ decay data 
and to  use of consistency conditions, 
but, as we will see, the situation is not as 
satisfactory as for other waves.

We will here  consider here two alternate sets of data to be fitted. 
In both we take the low energy data from $K_{l4}$ decay 
 (Rosselet et al.,~1977; 
Pislak et al.,~2001). [As a technical point, we mention that we have 
increased the error in  the point at highest energy ($s^{1/2}=381.4\,\mev$) 
from the  $K_{e4}$ compilation of Pislak et al.~(2001), whose 
status is dubious (the experimental value represents an average
 over an energy range that extends 
to the edge of phase space)]. To these we add the following points at high energy: 
$$\eqalign{
\delta_0^{(0)}(0.870^2\,\gev^2)=&\,91\pm9\degrees;\quad
\delta_0^{(0)}(0.910^2\,\gev^2)=\,99\pm6\degrees;\cr
\delta_0^{(0)}(0.935^2\,\gev^2)=&109\pm8\degrees;\quad 
\delta_0^{(0)}(0.965^2\,\gev^2)=134\pm14\degrees.\cr
}
\equn{(6.5.7a)}$$
These points are taken from solution 1 of Protopopescu et al.~(1973) 
(both with and without modified moments), with the error 
increased by the difference between this and solution~3 data in the same reference. 
These data points have the rare virtue of agreeing, within errors, 
with the results of  other experimental analyses. 
Their inclusion is essential; if we omit them, the fits would 
produce results at total variance with experimental information above $s^{1/2}=0.5\,\gev$.
We will also include in the fit the data, at similar energies,
of Grayer et al.~(1974):
$$\eqalign{
\delta_0^{(0)}(0.912^2\,\gev^2)=&\,103\pm8\degrees;\quad
\delta_0^{(0)}(0.929^2\,\gev^2)=112.5\pm13\degrees;\cr
\delta_0^{(0)}(0.952^2\,\gev^2)=&\,126\pm16\degrees;\quad 
\delta_0^{(0)}(0.970^2\,\gev^2)=141\pm18\degrees.\cr
}
\equn{(6.5.7b)}$$
The central values are obtained averaging the three solutions given by 
Grayer et al., and the error is calculated adding quadratically
 the statistical error of the highest point, the 
statistical error of the lowest point (for each energy) and the difference 
between the central value and the farthest point.


Then we take two choices, that may be considered to represent two extremes:
 either 
we impose the new or the old value of $\delta_0^{(0)}(m_K^2)$ in (6.5.6a,\/b).
 


For the theoretical formulas we also consider two basic possibilities. 
We  impose the Adler zero at $s=\tfrac{1}{2}\mu^2$ (no attempt is made to vary this),
 and a resonance with mass
$M_\sigma$, a free parameter.
 Then we map the $s$ plane cut along the left hand
cut ($s\leq0$) and  the $\bar{K}K$ cut, writing
$$\cot\delta_0^{(0)}(s)=\dfrac{s^{1/2}}{2k}\,
\dfrac{\mu^2}{s-\tfrac{1}{2}\mu^2}\,\dfrac{M^2_\sigma-s}{M^2_\sigma}\psi(s),
\equn{(6.5.8)}$$
and
$$\psi(s)=\big[b_0+b_1w(s)+b_2w(s)^2\big];\quad
w(s)=\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}},\quad
s_0=4m^2_K$$ (we have taken $m_K=(0.496\,{\gev})$). 
The complicated structure of this wave requires three parameters $b_1$, $b_2$ and 
$b_3$ for an acceptable fit.

This parametrization does not represent fully the 
coupling of the $\bar{K}K$ channel and, indeed, the corresponding phase shift deviates 
somewhat from 
experiment at the upper energy range ($s^{1/2}\geq0.965\,\gev$; see \fig~6.5.2). 
We can, alternatively, try to use the reduction to one channel of the 
two channel formulas (5.4.1,2) and write
$$\cot\delta_0^{(0)}(s)=\dfrac{s^{1/2}}{2k}\psi_{\rm el}(s),\quad
\psi_{\rm el}(s)=\dfrac{\dfrac{\kappa_2}{2s^{1/2}}\,\phiv_{11}(s)+\det {\bf \Phi}\vphantom{\Big|}}
{\dfrac{\kappa_2}{2s^{1/2}}+\phiv_{22}(s)\vphantom{\Big|}}
\equn{(6.5.9a)}$$
where $\kappa_2=\tfrac{1}{2}\sqrt{4m_K^2-s}$. We take a linear 
approximation for the $\phiv_{ii}$, and a constant for $\phiv_{12}$, requiring a zero of 
$\det{\bf \Phi}$ at $s=M_{f_0}^2$, and we allow $M_{f_0}$ to vary between 1 and 1.4 \gev. 
So we write,
$$\eqalign{
\phiv_{11}(s)=&\,\alpha_1+\beta_1s,\quad
\phiv_{22}(s)=\alpha_2+\beta_2s,\cr
\det{\bf \Phi}=&\,(\alpha_1+\beta_1s)(\alpha_2+\beta_2s)-
(\alpha_1+\beta_1M^2_{f_0})(\alpha_2+\beta_2M^2_{f_0}).\cr
}
\equn{(6.5.9b)}$$
This represents correctly the $\bar{K}K$ cut, but does not 
allow for the Adler zero or produce a dynamical 
left hand cut.
Therefore we expect reliability of (6.5.9) near $4m^2_K$, but 
poor description near threshold, which is indeed the case. We do not try to combine the two 
parametrizations as this would lead to a hopeless tangle due to the 
large number of parameters and also to the appearance of 
left hand cut of  $\bar{K}K$, that the $\phiv_{ij}$
inherit (but which must cancel for $\psi_{\rm el}$).


\topinsert{
\setbox0=\vbox{\hsize13.truecm{\epsfxsize 11.3truecm\epsfbox{del_pipi_S.eps}}} 
\setbox6=\vbox{\hsize 11truecm\captiontype\figurasc{Figure 6.5.2 }{
The  
$I=0$, $S$-wave phase shifts corresponding to (6.5.10a) (dashed line) and 
 (6.5.10b), continuous line. 
Also shown are the points, at low energies, from the $K_{l4}$ experiments, 
the point from $K_{2\pi}$ decay (the more recent value),
 and the high energy data of Protopopescu et al. and
Grayer et al.,
 included in
the fits.  }\hb} 
\centerline{\tightboxit{\box0}}
\bigskip
\centerline{\box6}
\medskip
}\endinsert 


We find the following results:\fnote{That, using only 
phase shifts data, there are two alternate 
possibilities for the intermediate energy S wave with 
$I=0$ was recognized already  by, e.g.,  Estabrooks and Martin~(1974); 
see also the textbook of Martin, Morgan and Shaw~(1975) for a discussion.} with (6.5.6a),
$$\eqalign{
\dfrac{\chi^2}{\rm d.o.f.}=&\,\dfrac{20.0}{20-4};\quad 
{b}_0=28.29\pm0.73,\quad {b}_1=39.11\pm1.67,\quad{b}_2=32.37\pm3.48,\quad
M_\sigma=836\pm20\,\mev;\cr
a_0^{(0)}=&\,(0.237\pm0.026)\,\mu^{-1};\quad \delta_0^{(0)}(m_K^2)=38.3\degrees\pm1.4\degrees .
\qquad\hbox{[$\delta_0^{(0)}(m_K^2)$ from (6.5.6a)].}\cr    }
\equn{(6.5.10a)}$$
The \chidof\ is reasonable.

If we impose $\delta_0^{(0)}(m_K^2)$ as given in (6.5.6b) 
we find 
$$\eqalign{
\dfrac{\chi^2}{\rm d.o.f.}=&\,\dfrac{28.1}{20-4};\quad 
{b}_0=15.39\pm0.77,\quad {b}_1=0.58\pm1.71,\quad {b}_2=13.77\pm3.81,\quad
M_\sigma=735\pm31\,\mev;\cr
a_0^{(0)}=&\,(0.212\pm0.029)\,\mu^{-1};\quad \delta_0^{(0)}(m_K^2)=49.5\pm3\degrees .
\qquad\hbox{[$\delta_0^{(0)}(m_K^2)$ from (6.5.6b)].}\cr    
}
\equn{(6.5.10b)}$$

The \chidof\ is not as bad as it looks 
at first sight; a good part of it comes from  points 
above $s^{1/2}=940\,\mev$, so (6.5.10b) 
can be considered an acceptable representation 
of data below that energy. 

It is  to be noted 
that, unfortunately, 
the parameters of the two fits differ a lot among themselves, although, as is clear from 
\fig~6.5.2, one can pass from one to the other 
continuously.  
For example, we may impose for $\delta_0^{(0)}(m_K)$ the average between old and new determinations, 
$\delta_0^{(0)}(m_K)=43.4\degrees\pm2.3\degrees$ and get a solution in the middle of (6.5.10a,\/b), 
$$\eqalign{
\dfrac{\chi^2}{\rm d.o.f.}=&\,\dfrac{22.9}{20-4};\quad 
{b}_0=21.55\pm0.62,\quad {b}_1=16.0\pm1.5,\quad {b}_2=10.5\pm3.5,\quad
M_\sigma=794\pm22\,\mev;\cr
a_0^{(0)}=&\,0.246\pm0.022\,\mu^{-1};\quad \delta_0^{(0)}(m_K^2)=43.2\degrees\pm1.5\degrees.\cr    
}
\equn{(6.5.10c)}$$
This can be considerd as the best solution;\fnote{There exists another solution, which practiaclly overlaps 
with (6.5.10c), with a different set of parameters:
$$\eqalign{
\dfrac{\chi^2}{\rm d.o.f.}=&\,\dfrac{21.9}{20-4};\quad 
{b}_0=21.9\pm0.62,\quad {b}_1=20.3\pm1.55,\quad {b}_2=22.5\pm3.5,\quad
M_\sigma=806\pm21\,\mev;\cr
a_0^{(0)}=&\,0.226\pm0.015\,\mu^{-1};\quad \delta_0^{(0)}(m_K^2)=43.4\degrees\pm1.4\degrees.\cr    
}
$$
We consider (6.5.10c) to be preferred because it exhibits better convergence properties 
(the parameters $b_n$ decrease).
}
however,  we will only make calculations with the two extremes, (6.5.10a,\/b). 

We next say a few words on results using (6.5.9). 
The quality of the fit is substantially lower than all the fits given in (6.5.10). 
Although we expect  (6.5.9) to reproduce better the high energy range, 
the lack of correct left-hand cut structure 
clearly disrupts the lower range. 
Thus, for the fit not imposing $\delta_0^{(0)}(m_K^2)$ we find a \chidof\ of 45/(15-4), 
certainly excessive; so we stick to (6.5.10).

It is difficult to give reasons, at the present stage,
 to prefer any of the three sets of parameters (6.5.10). For 
all the $\chidof\simeq1.2\;{\rm to}\;1.8$, which is a bit large.
We will retain the  
 parametrizations (6.5.10) so long as we cannot find other information that may 
discriminate. 
In particular, it follows that, from the S wave data alone, we cannot 
pin down the scattering length to more than the bounds
$$0.20\,\mu^{-1}\lsim a_0^{(0)}\lsim 0.25\,\mu^{-1}.
$$

In fact, as we will see in \sect~7.4, some forward 
dispersion relations and the Froissart--Gribov representation for $a_1$ 
are well satisfied alternatively by one or the other of the parametrizations (6.5.10a,\/b),
leaving the compromise (6.5.10c) as the best choice. 


\booksubsection{6.5.3. The $I=0$ S wave between $960\,\mev$ and $1300\,\mev$}

\noindent
As we have already commented, the description of 
pion-pion scattering above the $\bar{K}K$ threshold requires a full two-channel 
formalism. To determine the three independent components of the 
effective range matrix $\bf \Phi$, $\phiv_{11}$, 
$\phiv_{22}$ and $\phiv_{12}$, one requires 
measurement of  three cross sections. 
Failing this, one gets an indeterminate set, which is reflected 
very clearly in the wide variations of the effective range matrix parameters in the 
energy-dependent fits of Protopopescu et al.~(1973) and 
Hyams et al.~(1973), Grayer et al.~(1974).
 
The raw data themselves are also incompatible; 
Protopopescu et al. find a phase shift that flattens above $s^{1/2}\simeq 1.04\,\gev$, 
while that of Hyams et al. or Grayer et al. continues to grow. 
This incompatibility is less marked if we choose the 
solution with modified higher moments by 
Protopopescu et al. (Table~XIII there). 
The inelasticities are more compatible among the various determinations, although the errors
 of Protopopescu et al. appear to be underestimated. 

\topinsert{
\setbox0=\vbox{\hsize10.truecm{\epsfxsize 8.2truecm\epsfbox{h_e_del_pipi_S.eps}}} 
\setbox6=\vbox{\hsize 11truecm\captiontype\figurasc{Figure 6.5.3 }{
Fits to the  
$I=0$, $S$-wave phase shift and inelasticity from 960 to 1300 \mev. 
Also shown are the data points from solution~1 of Protopopescu et al.~(1973) (black dots) and 
data  of  Grayer et al.~(1974) (open circles).
}\hb} 
\centerline{\tightboxit{\box0}}
\bigskip
\centerline{\box6}
\medskip
}\endinsert


In spite of this it is possible to give a reasonable semi-phenomenological 
fit to $\delta_0^{(0)}$ and $\eta_0^{(0)}$, defined 
as in (2.1.4). 
We write
$$\cot\delta_0^{(0)}(s)=c_0\,\dfrac{(s-M^2_\sigma)(M^2_{f}-s)}{M^2_{f} s^{1/2}}\,
\dfrac{|k_2|}{k^2_2},\quad k_2=\dfrac{\sqrt{s-4m^2_K}}{2}
\equn{(6.5.12a)}$$
and
$$\eta_0^{(0)}=1-\left(c_1\dfrac{k_2}{s^{1/2}}+c_2\dfrac{k_2^2}{s}\right)
\,\dfrac{M'^2-s}{s}.
\equn{(6.5.12b)}$$
In the first, $c_0$ and $M_\sigma$ are free parameters and we fix $M_{f}=1320\,\mev$. 
In (6.5.12b),  the free parameters are  $c_1,\,c_2$ and we adjust $M'$ 
to get the inelasticity right on the $f_2$. 
We choose to fit the data points of solution~1 of Protopoescu et al. above 
 $\bar{K}K$ threshold, plus two values  at 1.2 and 1.3 \gev\ of Hyams et al. 
for 
the  inelasticity. 
For the phase shift, more conflictive as there is clear  
incompatibility between the two sets of experiments 
we include the seven values of Protopoescu et al. for $s^{1/2}\geq965\,\mev$, and another seven 
points of Grayer et al.~(1974), in the same range. 
The errors of these data have been evaluated as for (6.5.7).
We find,
$$\eqalign{
c_0=&\,1.36\pm0.05,\quad M_\sigma=802\pm11\,\mev;\quad \chi^2/{\rm dof}=36.2/(14-2)
\cr
c_1=&\,6.7\pm0.17,\quad c_2=-17.6\pm0.8;\quad\chi^2/{\rm dof}=7.7/(8-2).\cr
}
\equn{(6.5.13)}$$
The errors for $c_0$, $M_\sigma$ correspond to {\sl three} standard deviations, 
since  we have a $\chidof\simeq3$.
The fit (6.5.13) presents the nice feature that the value of $M_\sigma$ coincides, 
{\sl grosso modo}, with what we found below $\bar{K}K$ threshold. 
The qualitative features of the fits may be seen in \fig~6.5.3, where the incompatibility of the  
data of both sets of experiments is apparent. 



\bookendchapter
\brochureb{\smallsc chapter 7}{\smallsc  analyticity. dispersion relations.
form factors}{47}
\bookchapter{7. Analiticity: dispersion relations 
and the Froissart--Gribov representation.\hb
Form factors: the Omn\`es--Muskhelishvili\hb method}
\vskip-0.5truecm

\booksection{7.1. The Omn\`es--Muskhelishvili method}

\noindent
In the analysis of the pion form factors we have the following situation: we know 
{\sl experimentally} the 
modulus of a quantity, and have information on its phase. 
We would like to translate this into a general parametrization of the quantity. 
This last problem was first solved by Muskhelishvili~(1958) and later applied to the 
physical case by Omn\`es~(1958). 
We turn to this method.

\booksubsection{7.1.1. The full Omn\`es--Muskhelishvili problem}

\noindent
We want to find the most general representation for a function, $F(t)$, 
of which we know that it is analytic in the complex $t$ plane, cut from $t=4\mu^2$ to $\infty$, 
assuming that we know its phase on the cut, 
$${\rm Arg}\;F(t)=\delta(t),\quad 4\mu^2\leq t.
\equn{(7.1.1)}$$
This is the so-called (full) Omn\`es--Muskhelishvili problem.

First of all, it is clear that, unless we have further information on $F$, the solution to 
this equation is highly nonunique. 
For, if $F_0(t)$ is a solution to (7.1.1), then any
$$\ee^{at}F_0(t),\quad{\rm or}\quad \ee^{a\ee^{bt}}F_0(t),\;\dots$$
would also be a solution. 
Fortunately, in the physically interesting cases we have information on the growth 
of $F(t)$ at large $t$ that precludes such functions. 
For example, the Brodsky--Farrar counting rules imply that
$$F_\pi(t)\lsimsub{t\to\infty}\dfrac{1}{|t|}.
\equn{(7.1.2)}$$
Actually, in QCD one knows the exact behaviour of $F_\pi(t)$ (Farrar and Jackson, 1979), but we will not 
need this here.

We will restrict our analysis to the case where $\delta(t)$ is H\"older continuous 
(Muskhelishvili,~1958). 
We will also, in this subsection, assume
that the phase has a finite, positive limit  as $t\to\infty$:
$$\delta(t)\rightarrowsub_{t\to\infty}\delta(\infty),\quad\delta(\infty)>0. 
\equn{(7.1.3)}$$
In fact, this last is perhaps not a physically reasonable assumption. 
In the real world,  the infinitely many Regge recurrences of the $\rho$
($\rho',\,\rho'',\,\dots$) 
may produce a phase rising linearly with $t$. 
Fortunately, however, it will turn out that (7.1.3) will be the condition that is useful 
for the incomplete Omn\`es--Muskhelishvili problem, which is the one relevant for 
physical applications; so we assume (7.1.3). 

To solve our problem the first step is to form the auxiliary function
$$J(t)=\exp\dfrac{t}{\pi}\int_{4\mu^2}^\infty\dd s\,
\dfrac{\delta(s)}{s(s-t-\ii0)}.
\equn{(7.1.4)}$$ 
We will assume that $F(0)=1$; the modifications 
required if this is not the case are simple, and we leave them to the reader. 
From (7.1.4) two properties of $J$  are immediately obvious: $J(0)=1$ and 
$J$ has no zero in the complex plane (the last because, due to  the continuity of 
$\delta$, the integral in the exponent is finite).


It is easy to verify that the function $J$ has the same analyticity properties and 
 the same phase as $F$. 
For example, using the relation $1/(x\pm\ii0)=\pepe (1/x)\mp\ii\pi\delta(x)$, we have
$$\dfrac{t}{\pi}\int_{4\mu^2}^\infty\dd s\,
\dfrac{\delta(s)}{s(s-t-\ii0)}=\dfrac{t}{\pi}\pepe\int_{4\mu^2}^\infty\dd s\,
\dfrac{\delta(s)}{s(s-t)}+\ii\delta(t).
\equn{(7.1.5a)}$$
At large $t$, the real part of the integral above dominates over its imaginary part and we 
find
$$\dfrac{t}{\pi}\int_{4\mu^2}^\infty\dd s\,
\dfrac{\delta(s)}{s(s-t-\ii0)}\simeqsub_{t\to\infty}-\dfrac{\delta(\infty)}{\pi}\log|t|.
\equn{(7.1.5b)}$$
In view of this last relation we obtain the behaviour,
$$J(t)\simeqsub_{t\to\infty}|t|^{-\delta(\infty)/\pi}.
\equn{(7.1.6)}$$ 

Next step is to form the function $G(t)$, defined by
$$F(t)=G(t)J(t).
$$
Because $J$ never vanishes, $G(t)$ is, at least, analytic in the same domain as $F(t)$. 
Moreover, since $J$ and $F$ have the same phase on the cut, it follows that 
$G(t)$ is real on 
the cut.  
According to the theorem of Painlev\'e, this implies that $G$ is also analytic on the cut, hence $G(t)$ is
analytic in the whole $t$ plane, i.e., is an entire function.

It is now that the growth condition (7.1.2)   enters. 
The only entire functions that do not grow  exponentially (or faster) in some direction are the 
polynomials. Hence, (7.1.2) implies that $G(t)=P_N(t)$, 
where $P_N(t)$ is a polynomial of degree $N$: we have found the general representation
$$F(t)=P_N(t)J(t).
\equn{(7.1.7)}$$

Now, which polynomials are allowed depends on the value of $\delta(\infty)$. 
We will simplify the discussion by assuming that  $\delta(\infty)=n\pi$, 
$n$ an integer; the interested reader may find information on 
other situations in the text of  Muskhelishvili~(1958). 
On comparing (7.1.6) and (7.1.2) it follows 
immediately that $N=n-1$. 
Thus, in the case (that will turn out to be the more interesting one for us here) in which $n=1$, 
the function $J$ is actually the most general solution to the problem:
$$F(t)=J(t)\quad[\delta(\infty)=\pi].
\equn{(7.1.8)}$$

\booksubsection{7.1.2. The incomplete Omn\`es-Muskhelishvili problem}

\noindent
In the physically relevant cases we do not know $\delta(t)$ for all $t$, but only up 
to a certain $s_0$, typically the energy squared at which inelastic channels 
start becoming important. 
We will assume that one has the bounds
$$0<\delta(s_0)\leq\pi;
\equn{(7.1.8)}$$
this is what happens in the interesting cases and, moreover, the 
generalization to other 
situations poses no problem.

The idea for the treatment of this case is to extend $\delta(t)$
 to the full $t$ range, in an appropriate manner, so as to reduce the problem to the previous one. 
Let us call $\delta_{\rm eff}(t)$ to this extension, so that
$\delta_{\rm eff}(t)=\delta(t)$ for $t\leq s_0$. 
We then form 
$$J_{\rm eff}(t)=\exp\dfrac{t}{\pi}\int_{4\mu^2}^\infty\dd s\,
\dfrac{\delta_{\rm eff}(s)}{s(s-t-\ii0)}
\equn{(7.1.9a)}$$
and define $G$ by 
$$F(t)=G(t)J_{\rm eff}(t).
\equn{(7.1.9b)}$$
Because now $\delta_{\rm eff}(t)$ equals $\delta(t)$ only for 
$4\mu^2\leq t\leq s_0$, $G(t)$ will not be analytic on the whole $t$ plane, but will 
retain a cut from $t=s_0$ to $\infty$. $G$ will be an unknown function, that will 
have to be fitted to experiment. 
Because of this, we have interest to have it as smooth as possible, so that a few 
terms will represent it. 
Since discontinuities of $\delta_{\rm eff}(t)$ will generate infinities of $J_{\rm eff}(t)$, 
and  of $G(t)$, we must choose a smooth continuation of $\delta(t)$ above 
$t=s_0$. Moreover, if we do not want to have a $G$ growing without limit  for 
large $t$, we have to construct a $J_{\rm eff}(t)$ that decreases at infinity like 
$F(t)$. 
These conditions are fulfilled if we simply define
$$\delta_{\rm eff}(t)=\cases{\quad
\delta(t),\quad\phantom{-\pi\dfrac{t_0}{t}}\quad t\leq s_0;\cr
\pi+\left[\delta(s_0)-\pi\right]\dfrac{s_0}{t},\quad t\geq s_0.\cr
}
\equn{(7.1.10)}$$
In this case the piece from $s_0$ to $\infty$ in the integral in \equn{(7.1.9a)} 
can be performed explicitely and we get
$$\eqalign{
F(t)&\,=G(t)J_{\rm eff}(t)\cr
=&\,G(t)\ee^{1-\delta(t_0)/\pi}
\left(1-\dfrac{t}{s_0}\right)^{[1-\delta(t_0)/\pi]s_0/t}
\left(1-\dfrac{t}{s_0}\right)^{-1}
\exp\left\{\dfrac{t}{\pi}\int_{4\mu^2}^{s_0} \dd s\;
\dfrac{\delta(s)}{s(s-t-\ii0)}\right\}.\cr
}
\equn{(7.1.11)}$$




It only remains to write a general parametrization of $G(t)$ compatible with 
its known properties. 
To do so, we map the cut $t$ plane into the unit disk in the variable $z$ 
(\fig~7.1.1),
$$z=\dfrac{\tfrac{1}{2}\sqrt{s_0}-\sqrt{s_0-t}}{\tfrac{1}{2}\sqrt{s_0}+\sqrt{s_0-t}}.
\equn{(7.1.12)}$$
The most general $G$ is analytic inside this disk, so we can write
$$G(t)=1+A_0+c_1z(t)+c_2 z(t)^2+c_3 z(t)^3+\cdots
\equn{(7.1.13a)}$$
an expansion that will be convergent for all $t$ inside the cut plane. 
We can implement the condition $G(0)=1$, necessary to ensure 
$F(0)=1$ order by order, by writing
$$A_0=-\left[c_1z_0+c_2 z_0^2+c_3 z_0^3+\cdots\right],\quad
z_0\equiv z(t=0)=-1/3.
\equn{(7.1.13b)}$$
We remark in passing that since, inside the unit circle, $|z|\leq 1$, it follows that 
to every finite order in the expansion (7.1.13a) $G(t)$ is bounded in 
the $t$ plane and hence $F(t)$ and $J_{\rm eff}(t)$ have the same 
asymptotic behaviour, as desired.
 
\topinsert{
\setbox0=\vbox{\hsize11.truecm{\epsfxsize 10truecm\epsfbox{map_1.eps}}} 
\setbox6=\vbox{\hsize 3truecm\captiontype\figurasc{Figure 7.1.1 }{\hb
The  
mapping $t\to z$.\hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\line{
%\tightboxit
{\box0}\hfil\box6}
\smallskip
}\endinsert


We end this section with a simple example that shows clearly the desirability of 
expanding a function which, like $G$, is regular at the frontier of the 
domain of analyticity (which happens because we were careful to extrapolate $\delta$ without introducing 
singularities and keeping the correct asymptotic behaviour). 
Consider the three series
$$\eqalign{
\dfrac{1}{1-z}=&\,\sum_{n=0}^\infty z^n\quad ({\rm A});\cr
\log(1-z)=&\,\sum_{n=1}^\infty\dfrac{1}{n} z^n\quad ({\rm B});\cr
\int_0^z\dd t\,\dfrac{\log(1-t)}{t}=&\,\sum_{n=1}^\infty\dfrac{1}{n^2} z^n\quad ({\rm C}).\cr
}
$$
The first has a pole, the second a logarithmic singularity and the third is regular 
at the edge of the convergence disk. 
The first series diverges there, the second is conditionally convergent at all points except at 
$z=1$, and the third is convergent even at the edge of the disk. 
This pattern is general.



\booksection{7.2. Application 
to the pion form factor of the Omn\`es-Muskhelishvili method}

\noindent
The application to 
the pion form factor 
of the formalism presented in the previous section is straightforward as, indeed, it was
tailored for precisely this case. 
The function $\delta(t)$ is now the  P wave in $\pi\pi$ scattering, 
that we have denoted by $\delta_1(t)$. 
If we consider $\pi^+\pi^-$ scattering, then we have experimental information on 
$|F(t)|$ from $e^+e^-\to\pi^+\pi^-$ and, at $t<0$, we can use 
data on $F(t)$ from $\pi e^-$ scattering. If we take $\pi^+\pi^0$, then 
the information is obtained from the decay $\tau^+\to\bar{\nu}_\tau\pi^+\pi^0$. 
We may parametrize $\delta_1(t)$ as in (6.3.3); 
as for $G(t)$, we take two terms in (7.1.13) and write
$$G(t)=1+
c_1\left[\dfrac{\tfrac{1}{2}\sqrt{s_0}-\sqrt{s_0-t}}{\tfrac{1}{2}\sqrt{s_0}+\sqrt{s_0-t}}
+\tfrac{1}{3}\right]+
c_2\Bigg[\left(\dfrac{\tfrac{1}{2}\sqrt{s_0}-\sqrt{s_0-t}}{\tfrac{1}{2}\sqrt{s_0}+\sqrt{s_0-t}}\right)^2
-\tfrac{1}{9}\Bigg],
\equn{(7.2.1)}$$
$c_1,\,c_2$ free parameters. 
We remark that, although there are only 
 two free parameters, this is because we have imposed the condition
$G(0)=1$; 
the expansion (7.2.1) gives correctly the first three terms.

The quality of the fits, with only five free parameters ($b_0,\,b_1,\,M_\rho;\,c_0,\,c_1$) is remarkable, 
as can be seen in the accompanying figures 7.2.1,\/2; the $\chi^2$ is, including systematic and 
statistical errors, 
$\chidof=213/204$ (the $\omega - \rho$ interference 
effect was treated with the Gounnaris--Sakurai method).



\topinsert{
\setbox0=\vbox{\hsize14.truecm\hfil{\epsfxsize 12.5truecm\epsfbox{2time_11.eps}}\hfil} 
\setbox6=\vbox{\hsize 14truecm\captiontype\figurasc{Figure 7.2.1. }{Plot of 
the fit to $|F_\pi(t)|^2$, timelike and spacelike data. The 
theoretical curve actually drawn is that obtained by fitting also $\tau$ data, but 
the curve obtained fitting only $e^+e^-$ and $\pi e$ data 
 could not be distinguished from that drawn if we plotted it.}\hb
\vskip.1cm} 
\medskip
\centerline{\box0}
\centerline{\box6}
\medskip
}\endinsert 


Because we are interested not only on (relatively) rough 
estimates, but aim at pinning down fine details of isospin breaking as well, 
we will spend some time presenting the results. 
These results have been obtained in the course of the work reported by 
de~Troc\'oniz and Yndur\'ain~(2002), but not all of them have been published.

We consider the following types of fits. 
Firstly, we may take into account $\pi^+\pi^-$ form factor data (in the spacelike 
as well as the timelike regions) and data from  $\tau$ decay into $\nu\pi^+\pi^0$. 
Isospin breaking is incorporated by using the correct phase space for each case,
 and allowing for different masses and
widths for $\rho^0$, $\rho^+$; but 
 the function $G(t)$, whose cut only starts at $t\sim1\,\gev^2$, 
is assumed isospin independent. This produces the best results for 
the hadronic contributions to the $g-2$ of the muon and for the mass and width of the $\rho$: 
$$\eqalign{
M_{\rho^0}=&\;772.6\pm0.5\;\mev,\quad \gammav_{\rho^0}=147.4\pm0.8\,\mev;\cr
M_{\rho^+}=&\;773.8\pm0.6\;\mev,\quad \gammav_{\rho^+}=147.3\pm0.9\,\mev.\cr
}
\equn{(7.2.2)}$$


\topinsert{
\setbox9=\vbox{
\setbox0=\vbox{\hsize16.truecm\line{\hfil{\epsfxsize 7.truecm\epsfbox{31aleph.eps}}\hfil
{\epsfxsize 7.truecm\epsfbox{32opal.eps}}\hfil}}
\setbox6=\vbox{\hsize 13truecm\captiontype\figurasc{Figure 7.2.2 }{Plot 
of the fits to $v_1(t)$ (histograms), and data from $\tau$ decay (black dots).\hb
Left: Aleph data. Right: Opal data.}
Note that the theoretical values (histograms) are results of the {\sl same} 
calculation, with the same parameters, so the differences between the two fits merely 
reflect the slight variations between the two experimental determinations. 
We do not include the result of the fit to the third existing set of data (Anderson et. al,~2000), 
which is much like the ones depicted here.} 
\bigskip
\centerline{\box0}
\centerline{\box6}
\bigskip}
\box9
}\endinsert 

However, for our purposes here it is more interesting to consider two other possibilities. 
We may use only  $\pi^+\pi^-$ data (possibility A) or we may use both $\pi^+\pi^-$
 and $\pi^+\pi^0$  data,\fnote{The experimental numbers are from Barkov et al.~(1985), 
Akhmetsin et al.~(1999), Amendolia et al.~(1986), Anderson et. al,~2000,
 Barate et al.~(1997) and 
Ackerstaff et al.~(1999). There appears to be 
an inconsistency between the old an new versions 
of Akhmetsin et al.~(1999), related to whether the radiative corrections have been correctly incorporated. 
The fit given in the present paper uses the old set of data; we 
have checked that replacing them by the new one leaves the values of the parameters $b_i$, $c_i$, $M_\rho$ 
essentially unchanged.} {\sl neglecting isospin breaking effects}, in particular 
with the same $\rho$ 
parameters (possibility B); this would then represent a 
kind of isospin averaged result. 
The departure of A from B will be a measure of isospin breaking effects.

For the parameters $b_i$, $c_i$ we get, in case A,
$$\eqalign{
c_1=0.19\pm0.04,\quad c _2=-0.15\pm0.10,\cr
b_0=1.070\pm0.006,\quad b_1=0.28\pm0.06\cr
}
\equn{(7.2.3a)}$$
and, in case B, which is the one reported in \subsect~6.3.1,
$$\eqalign{
c_1=0.23\pm0.01,\;c _2=-0.16\pm0.03,\cr
b_0=1.060\pm0.005,\;b_1=0.24\pm0.04.\cr
}
\equn{(7.2.3b)}$$

Another question that has to be taken into account is the relative normalization of the 
various experiments.
This is  particularly important for the 
scattering length and    
 the slope and second derivative of the electromagnetic pion form factor, defined by 
$$F^2_\pi(t)\simeqsub_{t\to0}1+\tfrac{1}{6}\langle r^2_\pi\rangle t+c_\pi t^2.$$
What happens is that, as is clear from \fig~7.2.2, there is
 a small but systematic difference between Opal and Aleph data
and, as shown in de~Troc\'oniz and Yndur\'ain~(2002), 
the spacelike data on $F_\pi(t)$ do not agree well with the 
theoretical curve unless one takes into account systematic normalization effects.

In view of this, we present three sets of values for each quantity. 
In the first, the various experiments are fitted including only 
{\sl statistical} errors. In the second set we repeat the fit, including 
{\sl systematic} normalization effects. 

It is unclear which of the two sets of results is to be preferred; so we add a third set of figures. 
For this, which we consider the best (or, at least, the safest) set,   
we average the two previous ones and add their difference in quadrature to 
the statistical error. The results of all three procedures are 
presented in the following tables:
\bigskip



\setbox0=\vbox{
\setbox1=\vbox{\petit \offinterlineskip\hrule
\halign{
&\vrule#&\strut\hfil\quad#\quad\hfil&\vrule#&\strut\hfil\quad#\quad\hfil&
\vrule#&\strut\hfil\quad#\quad\hfil&\vrule#&\strut\hfil\quad#\quad\hfil\cr
 height2mm&\omit&&\omit&&\omit&&\omit&\cr 
&\hfil $\pi^+\pi^-$ only\hfil&&\hfil Only stat. errors\hfil&
&\hfil With normalization errors\hfil&
&\hfil Best result\hfil& \cr
 height1mm&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule} 
height1mm&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}height1mm&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$10^3\times a_1$ &&$(42\pm2)\;m_{\pi^+}^{-3}$&
&\hfil $(39\pm2)\;m_{\pi^+}^{-3}$ \hfil&&$(40.5\pm2.5)\;m_{\pi^+}^{-3}=(39.1\pm2.4)\;\mu^{-3}$& \cr
\noalign{\hrule}height1mm&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$\langle r^2_\pi\rangle$ (fm)&&$0.433\pm0.002$&
&\hfil $0.426\pm0.003$ \hfil&&$0.430\pm0.0036$& \cr
\noalign{\hrule}height1mm&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$c_\pi\;({\gev}^{-2})$ &&\hfil $3.58\pm0.04$\hfil&
&\hfil $3.49\pm0.06$ \hfil&&$3.54\pm0.06$& \cr
\noalign{\hrule}}}
\centerline{\box1}
\medskip
\centerline{{\sc Table} I{\sc a}}
}\centerline{\box0}

\bigskip
\setbox0=\vbox{
\setbox1=\vbox{\petit \offinterlineskip\hrule
\halign{
&\vrule#&\strut\hfil\quad#\quad\hfil&\vrule#&\strut\hfil\quad#\quad\hfil&
\vrule#&\strut\hfil\quad#\quad\hfil&\vrule#&\strut\hfil\quad#\quad\hfil\cr
 height2mm&\omit&&\omit&&\omit&&\omit&\cr 
&\hfil $\pi^+\pi^-$ \& $\pi^+\pi^0$\hfil&&\hfil Only stat. errors\hfil&
&\hfil With normalization errors\hfil&
&\hfil Best result\hfil& \cr
 height1mm&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule} 
height1mm&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}height1mm&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$10^3\times a_1$ &&$(42\pm2)\;m_{\pi^+}^{-3}$&
&\hfil $(39\pm2)\;m_{\pi^+}^{-3}$ \hfil&&$(40.5\pm2.5)\;m_{\pi^+}^{-3}=(39.1\pm2.4)\;\mu^{-3}$& \cr
\noalign{\hrule}height1mm&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$\langle r^2_\pi\rangle$ (fm)&&$0.438\pm0.003$&
&\hfil $0.435\pm0.003$ \hfil&&$0.436\pm0.0036$& \cr
\noalign{\hrule}height1mm&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$c_\pi\;({\gev}^{-2})$ &&\hfil $3.64\pm0.05$\hfil&
&\hfil $3.56\pm0.04$ \hfil&&$3.60\pm0.06$& \cr
\noalign{\hrule}}}
\centerline{\box1}
\medskip
\centerline{{\sc Table} I{\sc B}}
}\centerline{\box0}
\bigskip
The lack of dependence of $a_1$ on the procedure used is a bit
 fictitious as the fits were obtained including the 
constraint $a_1=(38\pm3)\times10^{-3}\,m_{\pi^+}^{-3}$. 
If we had not included it, the fits would have yielded values as high as 
$43\times10^{-3}\,m_{\pi^+}^{-3}$, difficult to reconcile with $\pi\pi$ scattering data.

The parameters given above are the ones that, in particular, 
 produce the excellent {\sl prediction} of 
P wave phase shifts shown in \fig~6.3.2, as well as the precise values of 
some of  the chiral 
parameters $\bar{l}_i$ that we will give in \sect~9.4.

\booksection{7.3. Dispersion relations and Roy equations}

\noindent A possible way to improve the quality of the analysis of experimental
 data is to use what are known
as {\sl dispersion relations}, either at fixed $t$ or in the form 
of the so-called Roy equations. We start with the first. 

\booksubsection{7.3.1. Fixed $t$ dispersion relations}

\noindent The analyticity properties of $F(s,t)$, as discussed in \sect~2.1,
 imply that we can write a Cauchy
representation for it, fixing $t$ and allowing $s$ to be complex. 
Starting with $s\to s+\ii\epsilon$, $s$ positive and $\epsilon>0,\,\epsilon\to0$, we have
$$F(s+\ii\epsilon,t)=\dfrac{1}{\pi}\int_{4\mu^2}^\infty\dd s'\,
\dfrac{A_s(s',t)}{s'-(s+\ii\epsilon)}+\dfrac{1}{\pi}\int_{4\mu^2}^\infty\dd s'\,\dfrac{A_u(s',t)}{s'-u}.$$
Here $A_s(s',t)=(1/2\ii)\{F(s'+\ii\epsilon,t)-F(s'-\ii\epsilon,t)\}$ 
is the so-called {\sl absorptive} part of the scattering amplitude across the 
right hand cut, which actually equals $\imag F(s',t)$. 
$A_u$ is the corresponding quantity connected  with the left hand cut. 
Taking $\epsilon=0$ above we find a relation between the 
{\sl dispersive} part of $F$, $D(s,t)$, which coincides with its real part, 
and the $A_{s,u}$. For $s$ physical this reads
$$\real F(s,t)=D(s,t)=\dfrac{1}{\pi}\pepe\int_{4\mu^2}^\infty\dd s'\,
\dfrac{A_s(s',t)}{s'-s}+\dfrac{1}{\pi}\int_{4\mu^2}^\infty\dd s'\,\dfrac{A_u(s',t)}{s'-u}
\equn{(7.3.1)}$$
($\pepe$ denotes Cauchy's principal part of the integral).
This is the fixed $t$ dispersion relation.

Actually, and because, in many cases, the $A(s,t)$ grow 
linearly with $s$, (7.3.1) is divergent. 
This is repaired by {\sl subtractions}; that is to say,
 writing the Cauchy representation not for $F$ itself,
but for 
$F(s,t)/(s-s_1)$ where  $s_1$ is a convenient subtraction point, usually taken
 to coincide with a  threshold. 
This introduces a constant in the equations (the value of $F(s,t)$ at $s=s_1$). 
Rewriting our equations with the appropriate subtraction incorporated
 is a technical problem, that we leave for the
reader to take into account; 
for the important case of forward dispersion relations we will 
perform explicitly the subtractions in next subsection.

Let us rewrite the dispersion relation 
in a form such that we separate out 
the unknown high energy contribution. 
We have
$$D(s,t)=\dfrac{1}{\pi}\pepe\int_{4\mu^2}^{s_0}\dd s'\,
\dfrac{A_s(s',t)}{s'-s}+\dfrac{1}{\pi}\int_{4\mu^2}^{s_0}\dd s'\,\dfrac{A_u(s',t)}{s'-u}+
V(s,t;s_0)
\equn{(7.3.2a)}$$
and
$$V(s,t;s_0)=\dfrac{1}{\pi}\int_{s_0}^\infty\dd s'\,
\dfrac{A_s(s',t)}{s'-s}+\int^{\infty}_{s_0}\dd s'\,\dfrac{A_u(s',t)}{s'-u};
\equn{(7.3.2b)}$$
we are assuming $s<s_0$. 
In practice, $s_0\sim1.3^2\,\gev^2$. 
Both $D$ and the $A$ may be written in terms of the {\sl same} set of phase shifts
by expanding them,\fnote{We are actually simplifying a little; 
(7.3.3) should take into account the isospin structure of $s$ and $u$ channels, 
which the reader may find in e.g. the text of Martin, Morgan and Shaw (1976),
or one can consider that we are studying $\pi^0\pi^+$  scattering, for which 
$s$ and $u$ channels are identical.}
$$A(s,t)=\dfrac{2s^{1/2}}{\pi k}\sum_{l=0}^\infty(2l+1)P_l(\cos\theta)\sin^2\delta_l(s),
\equn{(7.3.3a)}$$ 
$$D(s,t)=\dfrac{2s^{1/2}}{\pi k}\sum_{l=0}^\infty(2l+1)P_l(\cos\theta)\cos\delta_l(s)\sin\delta_l(s).
\equn{(7.3.3b)}$$

These equations provide {\sl constraints} for the phase shifts 
provided one knows (or has a reliable model) for the 
high energy term, $V(s,t;s_0)$. 
They enforce analyticity and $s\leftrightarrow u$ crossing symmetry, but not 
 $s\leftrightarrow t$ or  $t\leftrightarrow u$ crossing. 
This is very difficult to implement completely, as it would require analytical continuation, 
but 
a partial verification is possible through the Froissart--Gribov representation 
that we will discuss in \sect~7.5.


\booksubsection{7.3.2. Forward dispersion 
relations}

\noindent By far the more 
important case of dispersion relations is that in which we take $t=0$ ({\sl forward dispersion 
relations}), which we discuss now in some detail.

Let us denote by $F_{\rm o}(s,t)$ to a scattering amplitude which is odd 
under the exchange of $s\leftrightarrow u$, and by $F_{\rm e}(s,t)$ 
to an even one. 
An example of the first is the amplitude corresponding to isospin $I_t=1$ in the $t$ channel,
$$F^{(I_t=1)}=\tfrac{1}{3}F^{(I_s=0)}+ \tfrac{1}{2}F^{(I_s=1)}- \tfrac{5}{6}F^{(I_s=2)}.
\equn{(7.3.4)}$$
Examples of even amplitudes are those for $\pi^0\pi^0\to\pi^0\pi^0$,  
 $\pi^0\pi^+\to\pi^0\pi^+$:
$$F(\pi^0\pi^+\to\pi^0\pi^+)=\tfrac{1}{2}F^{(I_s=1)}+\tfrac{1}{2}F^{(I_s=2)},
\quad
F(\pi^0\pi^0\to\pi^0\pi^0)=\tfrac{1}{3}F^{(I_s=0)}+\tfrac{2}{3}F^{(I_s=2)}.
\equn{(7.3.5)}$$
For odd amplitudes we may profit from the antisymmetry to write a Cauchy representation for  
$F_{\rm o}(s,0)$ and obtain
$$\real F_{\rm o}(s,0)=\dfrac{2s-4\mu^2}{\pi}\,\pepe\int_{4\mu^2}^\infty\dd s'\,
\dfrac{\imag F_{\rm o}(s',0)}{(s'-s)(s'+s-4\mu^2)}.
\equn{(7.3.6)}$$
The integral is convergent; as we will see, from Regge theory we expect, for 
$F_{\rm o}(s,t)=F^{(I_t=1)}(s,t)$, a behaviour 
$$F_{\rm o}(s,t)\simeqsub_{s\to\infty}C s^{0.5+\alpha'_\rho t},\quad \alpha'_\rho\simeq 1/2M^2_\rho;$$
$M_\rho$ is the $\rho$ particle mass. 


For even amplitudes we have to {\sl subtract}, i.e., 
consider combinations $[F_{\rm e}(s,t)-F_{\rm e}(\hat{s},t)]/(s-\hat{s})$, 
where $\hat{s}$ is a convenient energy squared, usually taken in the range $0<\hat{s}\leq4\mu^2$. 
Two popular choices are the $s\leftrightarrow u$ symmetric point, $\hat{s}=2\mu^2$, and 
threshold, 
$\hat{s}=4\mu^2$. 
We then get the equations, respectively,
$$\real F_{\rm e}(s,0)=F_{\rm e}(2\mu^2,0)+\dfrac{(s-2\mu^2)^2}{\pi}
\pepe\int_{4\mu^2}^\infty\dd s'\,
\dfrac{\imag F_{\rm e}(s',0)}{(s'-2\mu^2)(s'-s)(s'+s-4\mu^2)}
\equn{(7.3.7a)}$$
and
$$\real F_{\rm e}(s,0)=F_{\rm e}(4\mu^2,0)+\dfrac{s(s-4\mu^2)}{\pi}
\pepe\int_{4\mu^2}^\infty\dd s'\,
\dfrac{(2s'-4\mu^2)\imag F_{\rm e}(s',0)}{s'(s'-4\mu^2)(s'-s)(s'+s-4\mu^2)}.
\equn{(7.3.7b)}$$
These integrals are  convergent; the behaviour expected from Regge theory is now
$$F_{\rm e}(s,t)\simeqsub_{s\to\infty}C s^{1+\alpha_P t},\quad \alpha_P\simeq 0.12\,\; {\gev}^{-2}.$$
Actually, the convergence of (7.3.7) 
may be proved to follow in a general local field theory.

For a variety of other types of forward dispersion relations, 
see the article of Morgan and Pi\u{s}ut~(1970) or the text of Martin, Morgan and Shaw~(1976).

\booksubsection{7.3.3. The Roy equations}

\noindent
Eqs.~(7.3.2), (7.3.3) look rather cumbersome. Roy~(1971) remarked that they 
appear simpler if we project (7.3.2)  into partial waves: 
one finds the {\sl Roy equations}
$$\cos\delta_l(s)\sin\delta_l(s)=\sum_{l'=0}^\infty\int_{4\mu^2}^{s_0}\dd s'\,
K_{ll'}(s,s')\sin^2\delta_{l'}(s)+V_l(s;s_0).
\equn{(7.3.8)}$$
Here the kernels $K_{ll'}$ are known and the $V_l$ are the (still unknown) 
projections of $V$.

Eq.~(7.3.8) is valid in 
the simplified case we are considering here, i.e., without subtractions. 
If we had subtractions, the fixed $t$ dispersion 
relations would acquire an extra term, a function $g(t)$. 
This may be eliminated, using crossing symmetry, 
in favour of the S wave 
scattering lengths. 
\equn{(7.3.8)} would be modified accordingly.

It should be clear that there is no physics ingredient entering the Roy equations 
that is not present in the fixed $t$ dispersion relation, plus 
partial wave expansions; (7.3.8) is strictly equivalent to the pair (7.3.2b) and (7.3.3). 
(In fact, there is some loss of information when using the Roy equations: in (7.3.2b) we can also require 
agreement between the integral and the real part, at high energies, using also Regge 
theory to evaluate the last).

Roy equations became fashionable in the early seventies, but were soon abandoned; 
not only high energy physicists found other,  more interesting, fish to fry, but it soon became obvious 
that they produced no better results than a straightforward phase shift analysis 
in which one parametrizes the $\delta_l$ in a way compatible with 
analyticity. 
 There are several reasons why this is so. 
First of all (7.3.8) (say) are highly nonlinear integral and matrix equations, 
and it is not clear that a solution to them exists {\sl for a general} set of  $V_l$. 
Solutions are known to exist in some favorable cases; but this constitutes the second 
problem: there are too many of them. 
In fact, Atkinson~(1968) proved a long time ago that, for any arbitrary $V(s,t;s_0)$ 
such that it is sufficiently smooth and decreasing at infinity, 
one can obtain by iteration a solution not 
only of the Roy equations, but of the full Mandelstam representation and 
compatible with inelastic unitarity for {\sl all} $s$ as well.\fnote{As a simple example, 
consider the toy model in Chapter~4. 
It is a field theoretical model, therefore it will satisfy unitarity, Roy's equations,
 Wanders's sum rules and the whole
kit-and-caboodle as accurately as you wish by going to high enough orders in the coupling. 
Moreover, it fits reasonably well the P wave: and yet, it produces totally unrealistic S 
waves. Fit to experiment is essential!} 
Therefore, the solutions to the  equations (7.3.8) are ambiguous in an 
unknown function; only the fact that the phase shifts fit experiment 
really constrains the solution. 
Indeed, it was found in the middle seventies that solutions of the Roy equations
 with suspiciously small 
errors simply reflected the prejudice  
as to what is a {\sl reasonable} $V(s,t;s_0)$ and about which sets of experimental 
phase shifts one ought to fit. 
Recently, the Roy equations have been resuscitated thanks to 
the appearance of new experiemtal data that allow 
more meaningful constraints.

From a practical point of view, the Roy equations present two further drawbacks (with respect to 
the method of parametrizations based on the effective range formalism, plus  
straight dispersion relations). First, 
 they mix various waves and, hence, transmit uncertainties of (say) the S-waves to 
other ones, and they require information on the medium and high energy regions 
($s\gsim1\,\gev^2$) where the 
mixing of $\pi\pi$ with channels such as $\bar{K}K$ is essential. 
Second, in the integrals in the r.h. side in (7.3.8) 
we have to project over partial waves, 
hence integrate with Legendre polynomials 
which, for $l$ larger than 1, 
oscillate and thus create unstabilities, which are difficult to control,
 for the D and higher waves.

Note, however, that this should not be taken as criticism of 
the use of Roy's equations, 
that provide a very useful tool to anlyze $\pi\pi$ scattering. 
In the present review, however, we prefer to concentrate on other 
methods:
 we  leave the implementation of the Roy 
equations (and of fixed $t$ dispersion relations, except 
in a few simple cases) outside the scope of 
these notes. The interested reader may consult the classic papers
 of Basdevant, Froggatt and Petersen~(1972,~1974), Pennington~(1975) or, 
more recently, the very comprehensive articles of 
 Colangelo, Gasser  and Leutwyler (2001) and Ananthanarayan et al.~(2001).

\booksubsection{7.3.4. Reggeology}

\noindent
Although we are here interested only on low and (at times) intermediate energy, it 
is clear that calculations like those 
of the previous subsections 
require a model for high energy $\pi\pi$ scattering. 
Regge pole theory provides such a model and, although outside the scope of this notes, 
we will describe here those of its features that are of interest to us. 

Consider the collision of two hadrons, $A+B\to A+B$. 
According to Regge theory, the high energy scattering amplitude, 
at fixed $t$ and large $s$, 
is governed by the exchange of complex, composite objects (known as {\sl Regge poles}) 
related to the resonances that couple to the $t$ channel. 
The same is true for large $t$, dominated by the resonances in the $s$ channel (this 
is the property originally proved, in potential theory, by T.~Regge).
Thus, for isospin 1 in the $t$ channel, high energy scattering
 is dominated by the exchange of a ``Reggeized" 
$\rho$ resonance.
 If no quantum number is exchanged, we say that the corresponding Regge pole is the vacuum, or
Pomeranchuk Regge pole; this name is often shortened to {\sl Pomeron}. 
In a QCD picture, the Pomeron (for example) will be associated with the excange of 
a gluon ladder between two partons in particles $A$, $B$ (\fig~7.3.1). 
The corresponding formalism has been developed by Gribov, Lipatove and other Russian physicists in 
the 1970s, and is related to the so-called Altarelli--Parisi, or DGLAP mechanism 
in deep inalastic scattering.\fnote{See Barger and Cline~(1969) and Yndur\'ain~(1999) for referencess to the 
original articles.}


One of the useful properties of Regge theory is the so-called {\sl factorization};  
 it can be proved from general properties of Regge theory,\fnote{In potential theory 
the proof can be made mathematically riguous; in relativistic theory, it follows 
from extended unitarity (Gell-Mann, 1962; Gribov and Pomeranchuk, 1962).} or, in QCD, in the DGLAP
formalism,  as is intuitively obvious from \fig~7.3.1.


%\topinsert{
\setbox0=\vbox{\hsize 7cm\epsfxsize 5.6cm\epsfbox{ladder.eps}\hfil}
\setbox1=\vbox{\hsize 5cm\captiontype\figurasc{Fig. 7.3.1. }
 {Cut Pomeron ladder exchanged between the partons $p_A$ and $p_B$ in 
hadrons $A$, $B$. The emitted gluons will materialize into a shower of particles.}\hb
\vskip.1cm
\phantom{x}}
\centerline{\box0\hfil\box1}
%}\endinsert
 
 
Factorization  states that the scattering amplitude $F_{A+B\to A+B}(s,t)$
can be written as a product
$$F_{A+B\to A+B}(s,t)\simeqsub_{{s\to\infty}\atop{t\,{\rm fixed}}}
C_A(t)C_B(t)(s/\hat{s})^{\alpha_R(t)}.
\equn{(7.3.9)}$$ 
$\hat{s}$ is a constant, usually taken to be $1\,\gev^2$ (we will do so here); 
the functions $C_A,\,C_B$ depend on the corresponding particles,  
but the power $(s/\hat{s})^{\alpha_R(t)}$ is universal and
 depends only on the quantum numbers exchanged in 
channel $t$. 
The exponent $\alpha_R(t)$ is called the Regge trajectory associated to the 
quantum numbers in channel $t$ and, for small $t$, 
may be considered linear:
$$\alpha_R(t)\simeqsub_{t\sim0}\alpha_R(0)+\alpha'_Rt.
\equn{(7.3.10)}$$
For the $\rho$ and Pomeron pole, fits to high energy processes give
$$\eqalign{
\alpha_\rho(0)=&\,0.50\pm0.04,\quad\alpha'_\rho\simeq\dfrac{1}{2M^2_\rho}\simeq 1\, {\gev}^{-2},\cr
\alpha_P(0)=&\,1,\quad\alpha'_P=0.11\pm0.05\, {\gev}^{-2},\cr
}
\equn{(7.3.11)}$$
The 
Regge parameters taken here are essentially those of Rarita et al. (1968); 
for $\alpha_\rho(0)$, 
however, we choose the value 0.5
  which is more consistent with determinations based on deep inelastic scattering.
There are indications that the Pomeron is not an ordinary Regge pole but 
we will not discuss this here.

Let us consider the imaginary part of the $\pi N$ or $NN$ scattering amplitudes (here 
by $NN$ we also understand $\bar{N}N$), which we recall are  
normalized so that
$$\sigma_{\rm tot}(s)=\dfrac{4\pi^2}{\lambda^{1/2}(s,m_A^2,m_B^2)} 
\imag F(s,0).$$
We have, letting $f_i$ be the imaginary part of $C_i$,
$$\eqalign{
\imag F_{NN}(s,t)\simeq &\,f^2_N(t)(s/\hat{s})^{\alpha_R(t)},\cr
\imag F_{\pi N}(s,t)\simeq&\,f_\pi(t)f_N(t)(s/\hat{s})^{\alpha_R(t)},\cr
}
\equn{(7.3.12a)}$$
and therefore, using factorization,
$$
\imag F_{\pi \pi}(s,t)\simeq f^2_\pi(t)(s/\hat{s})^{\alpha_R(t)}.
\equn{(7.3.12b)}$$
The functions $f_i(t)$ 
depend exponentially on $t$ for small $t$ and may be written, approximately, as\fnote{Consistency 
requires a more complicated form for the residue functions $f_i(t)$; for 
example, the rho residue should have contained a 
factor $\gammav(1-\alpha_\rho(t))\sin\pi\alpha_\rho(t)$. 
For the small values of $t$ in which we are interested, our expressions are sufficiently accurate.}
$$f_i(t)=\sigma_i\ee^{bt},\quad b=(2.4\pm0.4)\;{\gev}^{-2}.
\equn{(7.3.13)}$$


From (7.3.12) we can deduce the relations among the cross sections
$$\dfrac{\sigma_{\pi\pi\to{\rm all}}}{\sigma_{\pi N\to{\rm all}}}=
\dfrac{\sigma_{\pi N\to{\rm all}}}{\sigma_{N N\to{\rm all}}}.
$$
This relation also holds in the naive 
quark model\fnote{Levin and Frankfurter~(1965). 
For a comprehensive review, see Kokkedee~(1969).}
 in which one considers that scattering of hadrons is given by 
incoherent addition of scattering of their constituent quarks, so we have
$$\sigma_{\pi\pi\to{\rm all}}\;:\;\sigma_{\pi N\to{\rm all}}\;:\;\sigma_{N N\to{\rm all}}=
2\times2\;:\;2\times3\;:\;3\times3.
$$
From any of these relations one can obtain the parameter $\sigma_\pi$ in (7.3.13) in terms of the 
known $\pi N$ and $NN$ cross sections. Using this, we can 
write explicit formulas for 
$\pi\pi$ scattering with exchange of isospin $I_t$ in the $t$ 
channel. For $I_t=0$,
$$\imag F^{(I_t=0)}_{\pi\pi\to\pi\pi}(s,t)\simeqsub_{{s\to\infty}\atop{t\,{\rm fixed}}}
\left\{1+0.24\sqrt{\dfrac{\hat{s}}{s}}\right\}
\sigma_\pi(P)\ee^{bt}(s/\hat{s})^{\alpha_P(0)+\alpha'_P t},
\equn{(7.3.14a)}$$
and we have added empirically the subleading contribution, proportional to $\sqrt{\hat{s}/s}$, 
of the so-called $P'$ pole (associated with the $f_2$ resonance) that is 
necessary at the lowest energy range. 
For $I_t=1$,
$$\imag F^{(I_t=1)}_{\pi\pi\to\pi\pi}(s,t)\simeqsub_{{s\to\infty}\atop{t\,{\rm fixed}}}
\sigma_\pi(\rho)\ee^{bt}(s/\hat{s})^{\alpha_\rho(0)+\alpha'_\rho t}.
\equn{(7.3.14b)}$$
The amplitude for exchange of isospin $I_t=2$ in the $t$ 
channel is considered negligible in Regge theory.
From (7.3.13) and the known cross sections for $\pi N$, $NN$ scattering 
we have\fnote{Our Regge parameters $\sigma(i)$ here are slightly smaller than those 
used in Palou and Yndur\'ain~(1974). 
This is because we now add a $P'$ contribution to the 
Pomeron, and the low-lying resonances to the rho piece.}
$$\sigma_\pi(P)=1.0\pm0.15;\quad \sigma(\rho)=0.8\pm0.12.
\equn{(7.3.14c)}$$
The first holds for $\pi^0\pi^+\to\pi^0\pi^+$, or $\pi^0\pi^0\to\pi^0\pi^0$; 
the second for the amplitude $F^{(I_t=1)}$.

Another matter is, when one may apply  formulas like (7.3.14).
From the DGLAP version of the Pomeron (for example) 
we expect the following pattern to occur: in the region $|t|\ll s$, $s\gg\lambdav^2$ (with
$\lambdav\sim0.3\,\gev$ the QCD 
parameter) the ladder exchange mechanism will start to dominate the collision $A+B$. 
We then will have the onset of the Regge regime with, at the same time, 
a large increase of inelasticity (\fig~7.3.1) and a smoothing of the total cross 
section according to the behaviour (7.3.14).  

For $\pi N$, $NN$
scattering this occurs as soon as one is
 beyond the region of elastic resonances; in fact  (as can be seen in the cross section summaries in the 
Particle Data Tables) as soon as the kinetic energy or lab momentum is above 
$1\;-\;1.2$ \gev. For $\pi\pi$ 
we thus expect the Regge description to be valid for 
the corresponding energies, that is to say, for  $s^{1/2}\geq 1.3\,\gev$, which is the region 
where we will use it here. Indeed, and as
 we will see, around $s^{1/2}\sim 1.3\,\gev$ it is possible to calculate the 
$\pi\pi$ scattering 
amplitude from 
experiment and indeed it agrees, within a 15\%, with (7.3.14). 
For example, (7.3.14) implies
 $\sigma_{\pi^0\pi^+}(\infty)=\sigma_{\pi^0\pi^0}(\infty)=40.4\,{\gev}^{-2}$ 
while, from the experimental data, we get the 
experimental value, right above the $f_2$ resonance,
$\sigma_{\pi^0\pi^0}(1.342^2\,{\gev}^2)\simeq46\,{\gev}^{-2}$. 
The ``experimental" cross section $\sigma_{\pi^0\pi^+}$ is a bit smaller, 
but still reasonably close to this.
 
The situation, however,  is less clear for rho 
exchange amplitudes. 
Since these are the difference between much larger pieces, 
it may well be that some important inelastic resonance contribution may coexist with 
the Regge regime. 
We will come back to this later. 
 
As is clear from this minireview, the reliability of the Regge calculation of 
high energy pion-pion scattering cannot go beyond this accuracy of $\sim 15\%$, even 
for small $t$. 
The deviations off simple Regge behaviour are expected to be much larger for large $t$, 
as indeed the counting rules of QCD imply a totally different behaviour for fixed $t/s$. 
This is one of the problems involved in using e.g. the Roy equations that require 
integration up to $-t\sim s\sim 1.7\,\gev^2$, where the Regge picture fails completely 
(we expect instead the Brodsky--Farrar behaviour, $\sigma_{{\rm fixed}\,\cos\theta}\sim s^{-7}$). 
However, for forward dispersion relations or the Froissart--Gribov representation 
 we will work only for $t=0$ or $t=4\mu^2$ for 
which  the largest variation, that of $\ee^{bt}$, is still small, 
$b(t=4\mu^2)\simeq0.19$, and we expect no large error 
due to departure off linearity for the exponent in $f_i(t)$ or for the Regge trajectories, 
$\alpha_R(t)$.


\booksection{7.4. Evaluation of the forward dispersion relation for $\pi\pi$ 
 scattering}
\noindent 
As  examples of application of forward dispersion relations 
we will evaluate here (7.3.7a) for the scattering $\pi^0\pi^+\to\pi^0\pi^+$ 
and  $\pi^0\pi^0\to\pi^0\pi^0$, subtracted at   
 $s=4\mu^2$, and the Olsson sum rule, connected with the $F^{(I_t=1)}$ amplitude.


 
\booksubsection{7.4.1. $\pi^0\pi^+$}

\noindent 
We write $F(s)\equiv F_{\pi^0\pi^+}(s,0)$ and so we have
$$F(4\mu^2)=\dfrac{4\mu}{\pi}a_0^{(2)}=F(2\mu^2)+D,
\equn{(7.4.1a)}$$
where the dispersive integral is
$$D=\dfrac{4\mu^4}{\pi}\int_{4\mu^2}^\infty\dd s\,\dfrac{\imag F(s)}{s(s-2\mu^2)(s-4\mu^2)}.
\equn{(7.4.1b)}$$
Before making a detailed evaluation we will make a quantitative one. 
Because the scattering lengths both for S and P waves are very small, and the 
{\sl imaginary} parts of the amplitudes are (at low energy) proportional to the scattering 
lengths squared, we can, in a first approximation, 
neglect $D$ altogether. Moreover, for $F(2\mu^2)$, 
the S wave is very near its zero. If we therefore  
neglect it we have the approximate sum rule
$$\dfrac{4\mu}{\pi}a_0^{(2)}\simeq3f_1(2\mu^2).
\equn{(7.4.2)}$$
Using the parametrization for the P wave (\subsect~6.3.1) which, it 
will be remembered, converges down to the left hand cut, $s=0$, we find
$3f_1(2\mu^2)=-0.0742$ and thus the scattering length $a_0^{(2)}=-0.058\;\mu^{-1}$, 
a very reasonable number, agreeing with what we deduced from $\pi\pi$ scattering data and, 
to a 20\%, with what is expected in 
chiral perturbation theory.

We next proceed to a more accurate evaluation. 
The contribution to $D$ from energies above $s=1.3^2\,\gev^2$ we evaluate with the 
Regge parametrization of \subsect~7.3.4; the contribution of D and F waves with the parametrizations of
\sect~6.4. Finally, the contributions of S and P 
waves in the region $1\leq s^{1/2}\leq1.3\,\gev$ is obtained also with the 
parametrizations developed in the previous sections. 
All these contributions are small or very small; we get
$$\eqalign{
D({\rm Regge}; s^{1/2}>1.3\;\gev)=(1.5\pm0.2)&\,\times10^{-4},\cr
D({\rm D+F},\;s^{1/2}\leq1.3\,\gev)=1.2&\,\times10^{-5},\cr 
D({\rm S+P},\; 1\leq s^{1/2}\leq1.3\,\gev)=2.3&\,\times10^{-4}.\cr
}
$$
We will write $D_{\rm rest.}$ for the sum of these pieces. 

The subtraction $F(2\mu^2)$ can be very well
 approximated by the contributions of only S and P waves.
 The contribution of 
D and F waves, that may be calculated with the 
formula
$$f_l(s)\simeq\dfrac{2}{\pi}\,s^{1/2}k^{2l}a_l$$
and the values of the scattering lengths obtained from the Froissart--Gribov representation
(see next section), is very small as each is small and, moreover, they partially cancel; 
the result is $(-1.7\pm0.5)\times10^{-5}$.
 We  rewrite
(7.4.1) as
$$\dfrac{4\mu}{\pi}a_0^{(2)}-f_0^{(2)}(2\mu^2)-
\dfrac{1}{\pi}\int_{4\mu^2}^{1\,{\gev}^2}\dd s\,\dfrac{\imag f_0^{(0)}}{s(s-2\mu^2)(s-4\mu^2)}=\deltav,
\equn{(7.4.3a)}$$
and the quantity
$$\deltav=3f_1(2\mu^2)+
\dfrac{3}{\pi}\int_{4\mu^2}^{1\,{\gev}^2}\dd s\,\dfrac{\imag f_1}{s(s-2\mu^2)(s-4\mu^2)}
\equn{(7.4.3b)}$$
may be evaluated using the very precise parametrization for the P wave of \subsect~6.3.1, 
finding
$$\deltav=-0.0698\pm0.0024.
\equn{(7.4.3c)}$$
(We have included also the D and F wave contribution to $F(2\mu^2)$, $(-0.17\pm0.05)\times10^{-4}$, 
in this number for $\deltav$).
Adding this constraint (7.4.3) to the fit of the $I=2$ S wave (\subsect~6.5.1) we find, 
leaving free the $b_i$ and $z_2$, a $\chidof=8.06/(15-3)=0.72$ and 
the parameters
$$b_0=-116.3\pm3.1,\quad b_1=-122.0\pm9.4,\quad z_2=126.5\pm23\,\mev.
\equn{(7.4.4)}$$
The corresponding scattering length is
$$a_0^{(2)}=-(0.0684\pm0.020)\,\mu^{-1}.$$

The central value of $z_2$ in (7.4.4) is in uncanny agreement with the prediction of 
chiral perturbation theory (Chapter~9) that gives $z_2=\mu-9.7\,\mev$. 
That of the scattering length $a_0^{(2)}$ is somewhat higher than the chiral perturbation theory prediction,
$a_0^{(2)}\simeq-0.04\,\mu^{-1}$, 
for both types of fits, 
 but the difference, however, is not sufficient to claim discrepancy. 
In fact, what probably happens is  that the data on $\pi\pi$ scattering have a bias, no doubt 
due to  matters connected with the extrapolation to physical scattering, which 
pushes $a_0^{(2)}$ to values  higher than the physical ones. 
In \sect~7.6 we will report the results of a fit allowing for a systematic 
renormalization of the data; 
for the moment we continue with (7.4.4) as we want to check 
the consistency of our solutions with a few 
theoretical requirements. 



\booksubsection{7.4.2. $\pi^0\pi^0$}

\noindent
The calculation is very similar to that for $\pi^0\pi^+$.
 We have, with $F$ the $\pi^0\pi^0$ amplitude,
$$F(4\mu^2)=F(2\mu^2)+D,\quad D=\dfrac{4\mu^4}{\pi}\int_{4\mu^2}^\infty\dd s\,
\dfrac{\imag F(s)}{s(s-2\mu^2)(s-4\mu^2)}.
\equn{(7.4.5)}$$
In a first approximation we neglect the dispersive integral,
and then get the approximate sum rule
$$\dfrac{8\mu}{3\pi}\left(a_0^{(0)}+2a_0^{(2)}\right)=F(4\mu^2)\simeq F(2\mu^2)
\simeq\tfrac{2}{3}f_0^{(0)}(2\mu^2).
\equn{(7.4.6)}$$
The 
$\pi^0\pi^0$ amplitude contains, in the 
S wave, an $I=2$ component. This we will fix 
as given by (7.4.4). Likewise, we fix the D waves as given by the 
parametrizations of \sect~6.4. Finally, 
 for the S wave with $I=0$, we  take for the moment the parameters of (6.5.10b).
Then $F(4\mu^2)=0.0928$ and $F(2\mu^2)=0.0723$, reasonably close. 


A more precise test requires that we evaluate $D$. 
At high energy, that here we take to mean $s^{1/2}\geq1380\,\mev$ so as to be above the $f_2$ resonance, 
we use the Regge expression for $\imag F$. 
For the D waves we take the parametrizations of \sect~6.4; for the S wave with $I=2$
 we use the parametrization (7.4.4). 
For the S wave with $I=0$ between 960 and 1380 \mev, we take the parametrization of \subsect~6.5.3. 
All of this is very small; the bulk of the contribution to $D$ is that of
 the S wave with $I=0$ below 960 \mev:
$$\eqalign{
D({\rm Regge},\;s^{1/2}\geq1380\,\mev)=&\,0.00031;\cr
D(\hbox{D waves},\;s^{1/2}\leq1380\,\mev)=&\,0.00032;\cr
D(\hbox{S wave},\;I=2,\;s^{1/2}\leq1380\,\mev)=&\,0.0011;\cr
D(\hbox{S wave},\;960\,\mev\leq s^{1/2}\leq1380\,\mev)=&\,0.00013;\cr
D(\hbox{S wave},\;\mev s^{1/2}\leq960\,\mev)=&\,0.0183\pm0.0020,\quad\hbox{(a)};\cr
D(\hbox{S wave},\;\mev s^{1/2}\leq960\,\mev)=&\,0.0193\pm0.0023;\quad\hbox{(b)}.\cr
}
\equn{(7.4.7)}$$
The tags refer to solutions (6.5.6a,\/b), respectively.
We only give the errors in the low energy S wave with $I=0$.
The sum is $D=0.0202\pm0.0020$~(a), $D=0.0212\pm0.0023$~(b)  and, if we add it to 
$F(2\mu^2)=0.0723$, we get $0.0925\pm0.0020$~(a); $0.0935\pm0.0023$~(b), both 
in  agreement (within errors) with the value(s) $F(4\mu^2)=0.0978$~(a),  
$F(4\mu^2)=0.0928$~(b), although solution (b) is slightly 
preferred.  


\booksubsection{7.4.3. The Olsson sum rule}

\noindent
We will now consider the forward dispersion relation for the 
odd amplitude under $s\leftrightarrow u$, $F^{(I_t=1)}$, 
given in (7.3.6). 
At $s=4\mu^2$, 
we have $F^{(I_t=1)}=(8\mu/\pi)(\tfrac{1}{3}a_0^{(0)}-\tfrac{5}{6}a_0^{(2)})$ 
and we find the Olsson~(1967) sum rule,
$$2a_0^{(0)}-5a_0^{(2)}=D_{\rm O},\quad
D_{\rm O}=3\mu\int_{4\mu^2}^\infty \dd s\,
\dfrac{\imag F^{(I_t=1)}(s,0)}{s(s-4\mu^2)}.
\equn{(7.4.8)}$$
The various pieces in the dispersion integral are evaluated as in the 
previous subsections; 
the only difference is that 
we have to use the $\rho$ Regge pole expression\fnote{In fact, 
we have also added the (small) contribution of some large mass resonances; 
see \subsect~7.5.3 for details.} of (7.3.14b,\/c) 
for $\imag F^{(I_t=1)}$ at high energy. 
We find
$$\eqalign{
D_{\rm O}(s^{1/2}\geq1.38\;\gev;\; {\rm Regge})=&\,
(0.07\pm0.01)\mu^{-1}\cr
D_{\rm O}(s^{1/2}\leq1.38\;\gev;\; \hbox{ F wave})=&\,
0.00055\,\mu^{-1}\cr
D_{\rm O}(s^{1/2}\leq1.38\;\gev;\; \hbox{ D wave}\;I=0)=&\,
(0.053\pm0.02)\,\mu^{-1}\cr
D_{\rm O}(s^{1/2}\leq1.38\;\gev;\; \hbox{ D wave}\;I=2)=&\,
-0.0042\,\mu^{-1}.\cr
}
\equn{(7.4.9a)}$$
as happened in the previous cases, the most
 important contributions are those of the P, S waves:
$$\eqalign{
D_{\rm O}(1.0\leq s^{1/2}\leq1.38\;\gev;\; \hbox{ P wave})=&\,
0.0066\,\mu^{-1}\cr
D_{\rm O}(s^{1/2}\leq1.0\;\gev;\; \hbox{ P wave})=&\,
(0.222\pm0.001)\,\mu^{-1}\cr
D_{\rm O}(s^{1/2}\leq1.38\;\gev;\; \hbox{ S wave}\;I=2)=&\,
-(0.070\pm0.015)\,\mu^{-1}\cr
D_{\rm O}(0.965\leq s^{1/2}\leq1.38\;\gev;\; \hbox{ S wave}\;I=0)=&\,
(0.022\pm0.002)\,\mu^{-1},\cr
}
\equn{(7.4.9b)}$$
and, for (respectively) the two choices in 
(6.5.10),
$$\eqalign{
D_{\rm O}(s^{1/2}\leq0.965\;\gev;\; \hbox{ S wave}\;I=0)=&\,
(0.290\pm0.02)\,\mu^{-1}\quad\hbox{(a)}\cr
D_{\rm O}(s^{1/2}\leq0.965\;\gev;\; \hbox{ S wave}\;I=0)=&\,
(0.342\pm0.04)\,\mu^{-1}\quad\hbox{(b)}.\cr
}
\equn{(7.4.10)}$$

Using the values of the scattering lengths $a_0^{(I)}$ given 
previously we find the result (with $D_{\rm O}$ in the l.h. side and $2a_0^{(0)}-5a_0^{(2)}$ 
in the r.h. side)
$$\eqalign{
0.814=0.745\pm0.057\quad \hbox{(a)}\cr
0.609=0.797\pm0.075\quad \hbox{(b)}.\cr
}\equn{(7.4.11)}$$
We have moved all the errors to the r.h. side, as some are correlated.
We get  agreement at the $2\,-\,3\;\sigma$ level (with solution a favoured), which improves to 
 the $1\,-\,2\;\sigma$ level if we take a value at the lower end 
for $a_0^{(2)}$; cf.~\sect~7.6.
 

The fulfillment of the dispersion relations with the values of the 
parameters we found in the previous sections is then,  for $\pi^0\pi^+$,
$\pi^0\pi^0$, and the Olsson sum rule very satisfactory; but  
perhaps the more impressive feature of the calculations is how little 
imposing the fulfillment of the dispersion relation
 affects the values of the parameters.\fnote{This is less surprising
than may look at first  sight, if one realizes that all of our
 solutions are analytic in the correct 
region, so what we are really checking is that the details of 
the discontinuity across the left hand cut are not 
very important.}
Those obtained from the fits to data, respecting the appropriate unitarity and 
analyticity requirements, wave by wave, are  
essentially compatible with the dispersion relations.


\booksection{7.5. The Froissart--Gribov representation and low energy\hb
 P and D wave parameters}
\vskip-0.5truecm
\booksubsection{7.5.1. Generalities}

\noindent
A reliable method to obtain the P and, especially, D and higher
  scattering lengths and effective range parameters,
 which incorporates simultaneously $s,\,u$ and $t$ crossing
symmetry, is the Froissart (1961)--Gribov (1962) representation, to which we now turn.
This method of analysis has been developed long
 ago by Palou and Yndur\'ain~(1974) where, in particular, a rigorous proof of the 
validity of the equations (7.5.3,4) below may be found and, especially, 
by Palou, S\'anchez-G\'omez and Yndur\'ain~(1975), where a complete calculation of
 higher waves and 
effective range parameters is given. Also in the last  
reference  
the method is extended to evaluate the scattering
 lengths for the processes $\bar{K}K\to\pi\pi$.
The interest of the representation is that it ties together $s$, $u$ and $t$ channel 
quantities, without need of singular extrapolations.

Consider a $\pi\pi$ scattering amplitude, $F(s,t)$, symmetric 
or antisymmetric under the exchange $s\leftrightarrow u$, 
such as $\pi^0\pi^0$ or $\pi^0\pi^+$ (symmetric), or the amplitude with isospin 1 in the $t$ channel
(antisymmetric).
  We may project $F(s,t)$  into the $l$th partial wave in the $t-$channel, which is justified for 
$t\leq4\mu^2$. We have,
$$f_l(t)=\tfrac{1}{2}\int^{+1}_{-1}\dd \cos\theta_t\,P_l(\cos\theta_t) F(s,t).
\equn{(7.5.1a)}$$
Here $\cos\theta_t=1+2s/(t-4\mu^2)$ is the $t$ channel scattering angle.
We then write a dispersion relation, in the variable $s$:
$$F(s,t)=\dfrac{1}{\pi}\int_{4\mu^2}^\infty\dd s'\,
\dfrac{\imag F(s',t)}{s'-s}+u\;\hbox{channel}.
\equn{(7.5.1b)}$$
We have not written subtractions that, for $l=1$ and higher do not alter anything, and 
we also have not written explicitly the $u$-channel contribution; 
for the value $t=4\mu^2$, which is the one that interests us here, it simply multiplies
 by 2 the $s$ channel
piece:  for $t=4\mu^2$,  
$u$ and $s$ channel contributions to the final result are identical.

After substituting (7.5.1b) into (7.5.1a), the integral on 
$\dd \cos\theta_t$ can be made with the help of the formula
$$Q_l(x)=\tfrac{1}{2}\int^{+1}_{-1}\dd y\,\dfrac{P_l(y)}{x-y},$$
with $Q_l$ the Legendre function of the second kind. 
This produces the Froissart--Gribov representation,
$$f_l(t)=\dfrac{1}{2k_t^2}\dfrac{1}{\pi}\int_{4\mu^2}^\infty\dd s\,
\imag F(s,t)Q_l\left(\dfrac{s}{2k_t^2}+1\right)+u\;\hbox{channel},\qquad k_t=\dfrac{\sqrt{t-4\mu^2}}{2}.
$$
Taking now the limit $t\to4\mu^2$ in both sides of (6.3.4) and using that 
$$Q_l(z)\simeqsub_{z\to\infty} 2^{-l-1}\sqrt{\pi}\,\dfrac{\gammav(l+1)}{\gammav(l+\tfrac{3}{2})}z^{-l-1}$$
we find, in general,
$$\dfrac{f_l(t)}{k_t^{2l}}\eqsub_{t\to4\mu^2}\dfrac{\gammav(l+1)}{\sqrt{\pi}\,\gammav(l+3/2)}
\int_{4\mu^2}^\infty\dd s\,\dfrac{\imag F(s,t)}{(s+2k_t^2)^{l+1}}+\cdots.
\equn{(7.5.2)}$$
We then use the formula, that can be easily verified for $l\geq1$ from 
the effective range expression,
$$\dfrac{f_l(t)}{k_t^{2l}}\simeqsub_{t\to4\mu^2}\dfrac{4\mu}{\pi}\left\{a_l+k_t^2b_l\right\},
\equn{(7.5.3)}$$
to find the integral representation for $a_l,\,b_l$,
$$\eqalign{a_l=&\,\dfrac{\sqrt{\pi}\,\gammav(l+1)}{4\mu\gammav(l+3/2)}
\int_{4\mu^2}^\infty\dd s\,\dfrac{\imag F(s,4\mu^2)}{s^{l+1}},\cr
b_l=&\,\dfrac{\sqrt{\pi}\,\gammav(l+1)}{2\mu\gammav(l+3/2)}
\int_{4\mu^2}^\infty\dd s\,\left\{\dfrac{4\imag F'_{\cos\theta}(s,4\mu^2)}{(s-4\mu^2)s^{l+1}}-
\dfrac{(l+1)\imag F(s,4\mu^2)}{s^{l+2}}\right\}.\cr
}
\equn{(7.5.4)}$$
Here $F'_{\cos\theta}(s,4\mu^2)=(\partial/\partial\cos\theta)F(s,t)|_{t=4\mu^2}$, 
and an exta factor of 2 should be added to the l.h. 
side for 
identical particles (as if we have a state with well defined isospin).
The method holds, as it is, for waves with 
$l=1$ and higher. For the S wave, the corresponding integrals are divergent; one thus needs subtractions and
the  method becomes much less useful.
We remark that the formulas (7.5.4) are valid, when $l=$ even, {\sl only} 
for amplitudes $F$ symmetric under $s\leftrightarrow u$ crossing, and, 
for $l=$ odd, for amplitudes $F$ which are antisymmetric.

\booksubsection{7.5.2. D waves}

\noindent
For the D wave scattering lengths, the  ones that we will 
calculate now,
$$a_2=\dfrac{4}{15\mu}\int_{4\mu^2}^\infty\dd s\,\dfrac{\imag F(s,4\mu^2)}{s^{3}}.
\equn{(7.5.5)}$$
For these D waves, this method provides the combinations 
$$a_2(\pi^0\pi^0)=2\left[\tfrac{1}{3}a_2^{(0)}+\tfrac{2}{3}a_2^{(2)}\right];
\quad
a_2^{(t)}(\pi^0\pi^+)=2\left[\tfrac{1}{3}a_2^{(0)}-\tfrac{1}{3}a_2^{(2)}\right].
\equn{(7.5.6)}$$
The factor of 2 is due to the identity of the particles. (Note also that, because of 
the use of the Condon--Shortley conventions in e.g.  
the definition of the 
states $|\pi^\pm\rangle$ (\subsect~1.2.2), one has to take that 
the crossed state of  $\pi^\pm$ is $-\pi^\mp$).

We will here  illustrate the method with a detailed evaluation of $a_2^{(t)}(\pi^0\pi^+)$ 
and  $a_2(\pi^0\pi^0)$; we start with the first.
The contribution of the high energy ($s^{1/2}\geq1.3\,\gev$) is obtained integrating 
(7.5.4) in that region with the Regge formula (6.1.14a), finding
$$a_2^{(t)}(\pi^0\pi^+;\;s^{1/2}\geq 1.3\;\gev)=(0.97\pm0.12)\times10^{-4}\;\mu^{-5}
\quad\hbox{[Regge].}
$$
The contribution of D and F waves, up to $s^{1/2}=1.3\,\gev$, is
$$\eqalign{
a_2^{(t)}(\pi^0\pi^+;\;s^{1/2}\leq 1.3\;\gev)=&\,(0.55\pm0.40)\times10^{-4}\;\mu^{-5}
\quad\hbox{[D wave],}\cr
a_2^{(t)}(\pi^0\pi^+;\;s^{1/2}\leq 1.3\;\gev)=&\,(0.11\pm0.05)\times10^{-4}\;\mu^{-5}
\quad\hbox{[F wave].}\cr
}
$$
The contribution of S and P waves in the region $1\,\gev\leq s^{1/2}\leq1.3$ 
is minute; we get $7.3\times10^{-6}\,\mu^{-5}$ and  $1.4\times10^{-8}\,\mu^{-5}$,
 respectively.

The bulk of the contribution comes from the S and, especially, the P waves for $s^{1/2}\leq1\,\gev$:
$$\eqalign{
a_2^{(t)}(\pi^0\pi^+;\;s^{1/2}\leq 1.0\;\gev)=&\,(1.20\pm0.18)\times10^{-4}\;\mu^{-5}
\quad\hbox{[S wave],}\cr
a_2^{(t)}(\pi^0\pi^+;\;s^{1/2}\leq 1.0\;\gev)=&\,(8.34\pm0.06)\times10^{-4}\;\mu^{-5}
\quad\hbox{[P wave].}\cr
}
$$
Putting all of this together we find the very 
precise result
$$a_2^{(t)}(\pi^0\pi^+)=2\left[\tfrac{1}{3}a_2^{(0)}-\tfrac{1}{3}a_2^{(2)}\right]=
(11.24\pm0.48)\times10^{-4}\;\mu^{-5}.
\equn{(7.5.7)}$$
This agrees, within errors, with the value found by Palou, S\'anchez-G\'omez and Yndur\'ain~(1975),
who give 
$(11.07\pm0.52)\times10^{-4}\;\mu^{-5}$  (for the 
range $0.20\,\mu^{-1}\leq a_0^{(0)}\leq0.30\,\mu^{-1}$).

This combination we have calculated is the one that may be evaluated with less ambiguity; 
the values of other low energy  parameters depend 
substantially on the S wave scattering length. 
Thus, from the $\pi^0\pi^0$ scattering amplitude we  calculate the combination 
$$a_2(\pi^0\pi^0)=2\left[\tfrac{1}{3}a_2^{(0)}+\tfrac{2}{3}a_2^{(2)}\right].$$
We find,
$$\eqalign{
a_2(\pi^0\pi^0;\;\hbox{ Regge};\; s^{1/2}\geq1.38\,\gev)=&\,(0.087\pm0.013)\times10^{-4}\,\mu^{-5},\cr
a_2(\pi^0\pi^0;\;\hbox{D waves}, I=0+I=2;\; s^{1/2}\leq1.38\,\gev)=&\,(0.073\pm0.007)\times10^{-4}\,\mu^{-5},\cr
a_2(\pi^0\pi^0;\;\hbox{S waves}, I=0;\;0.96\leq
s^{1/2}\leq1.38\,\gev)=&\,(0.013\pm0.007)\times10^{-4}\,\mu^{-5}.\cr }
$$
As happened for the case of forward dispersion relations, 
the bulk of the contributions come from the 
S wave with $I=2$ (that we take as given by (7.4.4) and, especially, 
the $I=0$ wave below $0.96\,\gev$:
$$\eqalign{
a_2(\pi^0\pi^0;\;\hbox{S wave}, I=2;\; s^{1/2}\leq1.38\,\gev)=&\,(1.7\pm0.3)\times10^{-4}\,\mu^{-5};\cr
\vphantom{x}\cr
a_2(\pi^0\pi^0;\;\hbox{S wave}, I=0;\; s^{1/2}\leq0.96\,\gev)=&\cases{
(12.1\pm0.65)\times10^{-4}\,\mu^{-5}\quad\hbox{(a)}\cr
(14.1\pm1.07)\times10^{-4}\,\mu^{-5}\quad\hbox{(b)}.\cr}\cr
}$$
In the last, the tags (a,\/b,) refer to the fits in \equn{(6.4.10a,\/b,)}. 
Adding all contributions we find
$$2\left[\tfrac{1}{3}a_2^{(0)}+\tfrac{2}{3}a_2^{(2)}\right]=\cases{
(15.4\pm0.7)\times10^{-4}\,\mu^{-5}\quad\hbox{(a)}\cr
(17.4\pm1.1)\times10^{-4}\,\mu^{-5}\quad\hbox{(b)}.\cr}
\equn{(7.5.8)}$$
This can be compared with the value given in Palou, S\'anchez-G\'omez and Yndur\'ain~(1975), 
$(13.6\pm0.6)\times10^{-4}\,\mu^{-5}$ (for $a_0^{(0)}$ in the range $0.2\,\mu^{-1}$ to
 $0.3\,\mu^{-1}$).

\booksubsection{7.5.3. Scattering lengths of P and F waves}

\noindent
We use now (7.5.4), with $\imag F\equiv\imag F^{(I_t=1)}$. 
For the P wave scattering length, the integral is slowly 
convergent; the integrand behaves like $\imag F^{(I_t=1)}/s^2\sim s^{-1.44}$, and 
we do not have the factor $s-4\mu^2$ in the denominator of the Olsson sum rule 
that favoured low energies. 
Because of this, the details of the energy region $1.38\leq s^{1/2}\leq 1.80\,\gev$ 
are not negligible. 
We have represented $\imag F^{(I_t=1)}(s,4\mu^2)$ there by  a Regge 
background, given by the rho 
trajectory, or the same plus  the three 
resonances 
$\rho(1450)$, $\rho(1700)$ and $\rho_3(1690)$, 
whose contribution we have added to the background. 
These resonances we have treated in the 
narrow width approximation,
$$\imag f\simeq \dfrac{2s^{1/2}}{\pi k}\pi M\gammav_{2\pi}\delta(s-M^2),$$
with $M$, $\gammav_{2\pi}$ the mass and two-pion width of the resonance. 
We have taken for these the values of the Particle Data Tables; for the  $2\pi$
branchig ratios  of the $\rho(1450)$, $\rho(1700)$, poorly known,
we have assumed values between 10\% and 30\%. 
The difference between the two choices we have added to the error.

We find the results
$$\eqalign{
a_1({\rm Regge},\;s^{1/2}\geq1.38\;\gev)=&\,(2.24\pm0.4)\times 10^{-3}\;\mu^{-3},\cr
a_1(\rho(1450),\rho(1700),\rho_3)=&\,(1.4\pm0.8)\times 10^{-3}\;\mu^{-3}.\cr
}
$$
For the D, F waves,
$$a_1({\rm D, F};\;s^{1/2}\leq1.38\;\gev)=(3.50\pm0.2)\times 10^{-3}\;\mu^{-3}.
$$
Here most of the contribution comes from the D wave with $I=0$. The 
low energy S and P waves give
$$\eqalign{
a_1({\rm S}, I=0;\;0.965\leq s^{1/2}\leq 1.38\;\gev)\,+\cr
a_1({\rm P}, \;1.0\leq s^{1/2}\leq 1.38\;\gev)=&\,1.30\times 10^{-3}\;\mu^{-3},\cr
\cr
}$$
$$\eqalign{
a_1({\rm S}, I=2;\; s^{1/2}\leq 1.38\;\gev)=&\,(-2.61\pm0.35)\times 10^{-3}\;\mu^{-3},\cr
a_1({\rm P},\; s^{1/2}\leq 1.0\;\gev)=&\,(14.11\pm0.01)\times 10^{-3}\;\mu^{-3}.\cr
}
$$
Finally, and for the  fits in \equn{(6.5.10a,\/b)}, respectively,
$$\eqalign{
a_1({\rm S}, I=0;\;\leq s^{1/2}\leq 0.965\;\gev)=&\,(9.61\pm0.3)\times 10^{-3}\;\mu^{-3},\quad\hbox{(a)}\cr
a_1({\rm S}, I=0;\;\leq s^{1/2}\leq 0.965\;\gev)=&\,(12.26\pm0.5)\times 10^{-3}\;\mu^{-3},\quad\hbox{(b)}.
\cr
}
$$

Adding this we find,
$$\eqalign{
a_1=&\,(34.5\pm2)\times 10^{-3}\;\mu^{-3},\quad\hbox{(a)}\cr
a_1=&\,(36.5\pm2)\times 10^{-3}\;\mu^{-3},\quad\hbox{(b)}.
\cr
}
\equn{(7.5.10)}$$
We note that (7.5.10) deviates from the values given in 
Palou,~S\'anchez-G\'omez and Yndur\'ain~(1975) --the only place where this happens--, 
who give 
$a_1=(39\pm1)\times10^{-3}\,\mu^{-3}$ 
where the errors, however, only include the error due to the variation of $a_0^{(0)}$.

On comparing (7.5.10) with the value found in Chapter~6 from a direct fit,
 $a_1=(39.1\pm2.4)\times10^{-3}\,\mu^{-3}$, we see that 
(7.5.10) is compatible with solution (6.5.10b) at the $1.5\,\sigma$ level, and with (6.5.10a) 
only at  1.9 $\sigma$. 


 
One can also compute from the Froissart--Gribov representation 
the parameter $b_1^{(1)}$, finding (Palou, S\'anchez-G\'omez and Yndur\'ain, 1975)  
$$\eqalign{
a_0^{(0)}=&\,0.20\,\mu^{-1}:\quad b^{(1)}_1=3.44\times10^{-3}\,\mu^{-5};\cr
a_0^{(0)}=&\,0.30\,\mu^{-1}:\quad b^{(1)}_1=2.76\times10^{-3}\,\mu^{-5}.\cr}
\equn{(7.5.11)}$$
The result from the direct fit of the P wave gives $b^{(1)}_1=(4.1\pm0.4)\times10^{-3}\,\mu^{-5}$, 
which  appears to favour a {\sl low} value,
$a_0^{(0)}=(0.10\pm0.06)\,\mu^{-1}$.  The errors, however, are large and, clearly, a recalculation of
the Froissart--Gribov predictions for  these low energy parameters is desirable. 

We conclude the present section with the 
F wave scattering length. 
Here the high energy part is negligible; 
we give only those contributions that are sizeable. 
We have,
$$\eqalign{
a_3({\rm P},\;s^{1/2}\leq1.38\;\gev)=(1.58\pm0.02)\times10^{-5}\;\mu^{-5},\cr
a_3({\rm S},\;I=2,\;\;s^{1/2}\leq1.38\;\gev)=-0.83\times10^{-5}\;\mu^{-5};\cr
\phantom{-}
a_3({\rm S},\;I=0,\;s^{1/2}\leq0.965\;\gev)=\cases{
(4.54\pm0.33)\times10^{-5}\;\mu^{-5},\quad\hbox{(a)},\cr
(5.60\pm0.50)\times10^{-5}\;\mu^{-5},\quad\hbox{(b)}.\cr}\cr
}$$
For the whole $a_3$, adding also the small contributions,
$$\eqalign{
a_3=\cases{
(6.1\pm0.3)\times10^{-5}\;\mu^{-5},\quad\hbox{(a)},\cr
(6.5\pm0.7)\times10^{-5}\;\mu^{-5},\quad\hbox{(b)}.\cr}\cr
}
\equn{(7.5.12)}$$
This agrees, within erors, with both the results of Palou, S\'anchez-G\'omez and Yndur\'ain~(1975), 
$a_3=(6.3\pm0.5)\,\times10^{-5}\,\mu^{-5}$, and 
the compilation of Nagel et al.~(1979) that give 
$a_3=(6\pm2)\,\times10^{-5}\,\mu^{-5}$.

\booksection{7.6. Summary and conclusions}

\noindent
In this section we summarize and discuss the results 
obtained in the two last chapters. In them, 
we have found simple, explicit formulas that satisfy the general requirements of 
analyticity and unitarity and 
which fit well the experimental data; 
then we have verified that our solutions (in the case of the 
S wave, both our solutions) 
are compatible with a few crossing and 
analyticity constraints: forward dispersion relations at threshold and 
the Froissart--Gribov representation. 

 It should be clear that we 
have not made an exhaustive analysis: 
nor was it intended. 
Thus, we have not tried to improve 
our parameters by fully imposing  
 consistency requirements. For example, it is easily verified that a change 
 in the parameters of the P and S wave ($I=0$) to decrease (in the case of the first) 
the scattering lengths by of about \ffrac{1}{2}
$\sigma$ would improve agreement of the Froissart--Gribov and direct determinations of 
$a_1$. However, for the improvements to be more than cosmetic, we should   
 also consider dispersion relations in
a wider range of $s$ and 
$t$ values or, equivalently, 
the Roy equations.
This is the path followed by Ananthanarayan et al.~(2001) and Colangelo, Gasser and Leutwyler~(2001), 
where the interested reader may find details. 
The results found in the first of these papers (which does not impose 
chiral perturbation theory) are compatible with ours, at the 1 to 1.5 $\sigma$ level. 
Also the errors are similar, with theirs generally smaller. 
The price they pay, however, is that all their numbers are correlated, 
whereas ours are not: in this sense, our results are more robust. 
The method of the Roy equations and ours here 
are complementary.

\topinsert{
\setbox0=\vbox{
\setbox1=\vbox{\petit \offinterlineskip\hrule
\halign{
&\vrule#&\strut\hfil\quad#\quad\hfil&\vrule#&\strut\hfil\quad#\quad\hfil&
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 height2mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr 
&\hfil \hfil&&\hfil Nagels\hfil&&PSGY&
&\hfil ACGL\hfil&
&\hfil T.a.\hfil& \cr
 height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule} 
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&$a_1$&&\vphantom{\Big|}$38\pm2$&&$38.5\pm0.6$&
&\hfil$36\pm2$ \hfil&&
$37.3\pm2$ $[\,39.1\pm2.4]^{\,\rm a}$ $\;\times\,10^{-3}$& \cr 
\noalign{\hrule} 
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$b_1$ 
\phantom{\big|}&&\phantom{\Big|}&&$3.2\pm0.3$&
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\noalign{\hrule} 
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&$a_2^{(0)}$&&\vphantom{\Big|}$17\pm3$&
&\hfil $18.5\pm0.6$  \hfil&&$16\pm1$&&$18.05\pm0.7$
$\;\times\,10^{-4}$& \cr
\noalign{\hrule} 
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$a_2^{(2)}$&&$1.3\pm3$&
&\hfil $1.9\pm0.6$ \hfil&&$1.7\pm0.7$&&$1.7\pm0.5$
$\;\times\,10^{-4}$& \cr
\noalign{\hrule}
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$2\big[\tfrac{1}{3}a_2^{(0)}-\tfrac{1}{3}a_2^{(2)}\big]$&&$10.5\pm3$&
&\hfil $11.07\pm0.52$ \hfil&&$9.6\pm0.8$&&$10.90\pm0.45$
$\;\times\,10^{-4}$& \cr
\noalign{\hrule}
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$2\big[\tfrac{1}{3}a_2^{(0)}+\tfrac{2}{3}a_2^{(2)}\big]$&&$13.1\pm5$&
&\hfil $14.9\pm0.8$ \hfil&&$12.9\pm1.2$&&$14.3\pm1.0$
$\;\times\,10^{-4}$& \cr
\noalign{\hrule}
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$b_2^{(0)}$&&$ $&
&\hfil $-3.6\pm0.4$ \hfil&&$-3.2\pm0.5$&&$ $
$\;\times\,10^{-4}$& \cr
\noalign{\hrule}
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$b_2^{(2)}$&&$ $&
&\hfil $-3.5\pm0.6$ \hfil&&$-3.1\pm0.6$&&$ $
$\;\times\,10^{-4}$& \cr
\noalign{\hrule}
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|}$a_3$&&$6\pm2$&
&\hfil $6.2\pm0.9$ \hfil&&$5.4\pm1.1$&&$6.5\pm0.7$
$\;\times\,10^{-4}$& \cr
\noalign{\hrule}
height1mm&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\vphantom{\Big|} $b_3$&&\hfil $ $\hfil&&\hfil $-5.0\pm1.2$ \hfil&
&$-s4.0\pm1.3$&&$ $
$\;\times\,10^{-4}$& \cr
\noalign{\hrule}}
\vskip.05cm}
\centerline{\box1}
\medskip\noindent
Units of $\mu$. Nagel: Nagel et al.,~(1979); ACGL: Anathanarayan et al.~(2001); 
PSGY: Palou, S\'anchez-G\'omez and Yndur\'ain~(1975); T.a.: this article, 
with the parametrizations given below, Eqs.~(7.6.2)~ff. 
(Eq.~(6.5.10b) for the $I=0$ S wave; if we had used (6.5.10c),
 the results would have varied by about
\ffrac{1}{2} $\sigma$). We have reexpressed the figures in ACGL in units of $\mu$, rather than $m_{\pi^+}$, 
and the values quoted here actually correspond to Table~2 in 
Colangelo, Gasser and Leutwyler~(2001).
Note that the errors in PSGY only include the error due to 
variation of $a_0^{(0)}$ between $0.2$ and $0.3$ (in units of $\mu$); 
the full errors could be a factor 2 or so larger.
\medskip
\noindent($^{\rm a}$) The numbers in braces are those following from the 
pion form factor. 
All other numbers in ``T.a." are from the Froissart--Gribov projection.
\medskip
\centerline{\sc Table~II}
\smallskip
\centerrule{5truecm}}
\box0}
\endinsert
\bigskip

In the following Table~II we present a comparison of our results with those of 
earlier calculations.
We think the numbers speak for themselves, but there are  
a few remarks that should be made. 
Our results, although overlapping with,
 are generally above those of Anathanarayan et al.~(2001). 
This is probably due to a number of small effects 
which  go in the same direction. 
To begin with, the high energy behaviour we take  differs from that
 used by these authors.\fnote{In section B.4 of their paper, ACGY  
 explain that they take an asymptotic value  
for the total (Pomeron) cross 
section of 6 mb, corresponding to $\sigma_{\pi\pi}(\infty)\simeq24\,\gev^{-2}$. 
This is a factor of $\sim1.8$ smaller than the value implied 
by factorization, as discussed in our \subsect~7.3.4 here. 
They also take an unorthodox formula for the diffraction peak.}
Secondly,  we have a slight difference with these authors both 
in the P wave and, more markedly, for  the $I=0$ S
wave for $s^{1/2}\leq0.85\,\gev$. 
There is little doubt that a more precise/complicated parametrization 
(perhaps with an extra term in the expansion of $\phiv_0^{(0)}$, or allowing the 
position of the  
Adler zero to vary, or simply using (6.5.10c)) is required to get a  
precise representation for this 
wave. (The reason for the complicated 
shape of the scalar wave is not known with certainty, 
but a likely possibility is that the $\sigma$ structure is not 
 due to a simple 
quark-antiquark state, but has an important glueball component).

The origin of the differences between our numbers and those of ACGL
 can be seen very clearly if we repeat the
Froissart--Gribov  calculation for, e.g., 
 $a_3$ replacing only our solutions for the S waves (both for isospin 0 and 2), below 800
\mev, by  the ones found in Colangelo, Gasser and Leutwyler~(2001), who
use the Roy equations plus chiral perturbation theory,  but 
keeping the rest as in our calculation. 
In this case  we find
$$a_3=(5.16\pm0.05)\times 10^{-5}\;\mu^{-7}=(5.58\pm0.06)\times 10^{-5}\;m_{\pi^+}^{-7},
\equn{(7.6.1a)}$$
and the error does {\sl not} 
take into account the error from the S wave parametrization of CGL.
This to be compared with the result obtained by those authors, with a direct calculation 
involving also two-loop chiral perturbation theory,\fnote{If we 
had used also their 
value for the P wave below 800 \mev,  one would get (7.6.1a) 
replaced by $a_3=(5.07\pm0.15)\times 10^{-5}\;\mu^{-7}=(5.49\pm0.06)\times 10^{-5}\;m_{\pi^+}^{-7}$.} 
$$a_3=(5.17\pm0.18)\times 10^{-5}\;\mu^{-7}=(5.60\pm0.19)\times 10^{-5}\;m_{\pi^+}^{-7}.
\equn{(7.6.1b)}$$ 
 

As stated above, we find that the accuracy of the results of ACGL and ours is similar; 
an exception is 
 the combination $2\big[\tfrac{1}{3}a_0^{(0)}-\tfrac{1}{3}a_2^{(2)}\big]$ 
where our method produces a clearly more precise 
result, which even compares with what Colangelo, Gasser and Leutwyler~(2001) get 
imposing also chiral perturbation theory to two loops, 
$2\big[\tfrac{1}{3}a_0^{(0)}-\tfrac{1}{3}a_2^{(2)}\big]=(9.94\pm0.3)\times10^{-4}\,\mu^{-5}$.

Finally, we would also like to comment on the agreement of the 
old results of PSGY with 
the more modern determinations, both here and in ACGY. 
This is remarkable and 
lends weight to the 
suitability of the 
Froissart--Gribov method to calculate low energy parameters for 
P, D and higher waves.

We next present the best results that follow from our fits, for ease of reference: 
for the S wave we take th fit with the average value  $\delta_0^{(0)}(m_K)=43.3\pm2.3$,

$$\eqalign{
\cot\delta_0^{(0)}(s)=&\,\dfrac{s^{1/2}}{2k}\,\dfrac{\mu^2}{s-\tfrac{1}{2}\mu^2}\,
\dfrac{M^2_\sigma-s}{M^2_\sigma}\,
\left\{b_0+b_1\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}+
b_2\left[\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}\right]^2\right\};\cr
s_0^{1/2}=2m_K;&\quad\chi^2/{\rm d.o.f.}=21.1/(20-4).\cr
\quad M_\sigma=794\pm22,&\,\quad b_0=21.55\pm0.62,\quad b_1=16.0\pm1.5, \quad b_2=10.5\pm3.5\cr
 [s^{1/2}\leq&\, 0.985\;\gev];\quad
a_0^{(0)}=(0.250\pm0.022)\;\mu^{-1};\quad\delta_0^{(0)}(m_K)=43.2\degrees\pm1.5\degrees.\cr
\cr
}
\equn{(7.6.2)}$$


\smallskip
$$\eqalign{
\cot\delta_0^{(2)}(s)=&\,\dfrac{s^{1/2}}{2k}\,\dfrac{\mu^2}{s-2z_2^2}\,
\left\{b_0+b_1\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}\right\};\cr
s_0^{1/2}=1.45\;\gev;&\quad\chi^2/{\rm d.o.f.}=1.2.\cr
 b_0=&\,-121.3\pm0.6,\quad b_1=-112.6\pm1.6,\quad z_2=133\pm4.5\;\mev.\cr
[s^{1/2}\leq&\, 1.25\;\gev].\cr
}
\equn{(7.6.3)}$$

\smallskip
$$\eqalign{
\cot\delta_1(s)=&\,\dfrac{s^{1/2}}{2k^3}\,(M_\rho-s)\,
\left\{b_0+b_1\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}\right\};\cr
s_0^{1/2}=1.05\;\gev;&\quad\chi^2/{\rm d.o.f.}=1.3.\cr
\quad M_\rho=772.9\pm0.8\;\mev,&\,\quad b_0=1.060\pm0.005,\quad b_1=0.24\pm0.04.\cr
[s^{1/2}\leq&\, 1.0\;\gev].\cr
}
\equn{(7.6.4)}$$
\smallskip

$$\eqalign{
\cot\delta_2^{(0)}(s)=&\,\dfrac{s^{1/2}}{2k^5}\,(M_{f_2}-s)\mu^2\,
\left\{b_0+b_1\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}\right\};\cr
s_0^{1/2}=1.38\;\gev;&\quad\chi^2/{\rm d.o.f.}=2.6.\cr
\quad M_{f_2}=1270\;\mev\;\hbox{(input)},&\,\quad b_0=22.1\pm1.9,\quad b_1=19.2\pm0.3.\cr
[s^{1/2}\leq&\, 1.375\;\gev].\cr
}
\equn{(7.6.5)}$$
\smallskip

$$\eqalign{
\cot\delta_2^{(2)}(s)=&\,\dfrac{s^{1/2}}{2k^5}\,
\dfrac{\mu^4s}{4(\mu^2+\deltav^2)-s}\,
\left\{b_0+b_1\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}\right\};\cr
s_0^{1/2}=1.38\;\gev;&\quad\chi^2/{\rm d.o.f.}=2.9.\cr
\quad b_0=(1.96\pm0.06)\times10^3,&\,\quad b_1=(3.03\pm0.03)\times10^3,\quad \deltav=3.4\pm0.4\;\mev.\cr
[s^{1/2}\leq 1.1\;\gev;&\,\quad \hbox{A 15\% inelasticity has to be included for
$1.1\leq s^{1/2}\leq1.375$}].\cr }
\equn{(7.6.6)}$$
\smallskip

$$\eqalign{
\cot\delta_3(s)=&\,\dfrac{s^{1/2}}{2k^7}\,\mu^6\,
\left\{b_0+b_1\dfrac{\sqrt{s}-\sqrt{s_0-s}}{\sqrt{s}+\sqrt{s_0-s}}\right\};\cr
s_0^{1/2}=1.5\;\gev;&\quad\chi^2/{\rm d.o.f.}=1.4.\cr
a_3=(6.5)\times10^{-5}\;\mu^{-7}\;\hbox{(input)},&\,\quad b_0=(1.07\pm0.03)\times10^5,\quad
b_1=(1.35\pm0.03)\times10^{5}.\cr
 [s^{1/2}\leq&\, 1.2\;\gev; \hbox{empirical fit, see \subsect~6.4.3}].\cr
}
\equn{(7.6.7)}$$
\smallskip
The  S waves require some extra discussion. 
For $I=0$, we have presented the solution (6.5.10c), 
 which joins the advantages of 
both parametrizations  (6.5.6a,\/b).
In what respects the $I=2$ case, (7.6.3) is obtained by fiting
 all the data of Losty et al.~(1974) and Hoogland
et al.~(1977), but including the constraint $a_0^{(2)}=(-0.046\pm0.007)\,\mu^{-1}$ 
suggested by the fit including the CERN-Munich data (cf.~\subsect~6.5.1), 
chiral perturbation theory (\sect~9.3) or the fulfillment of the Olsson sum rule 
(\subsect~7.4.3).

One may remark that, while the S waves are (perhaps) described less precisely in the lower energy
range  than what can be achieved imposing consistency 
requirements systematically (e.g., in the form 
of Roy equations or a complete set of dispersion relations) as in the work of 
Ananthanarayan et al.~(2001) or, to a certain extent,
 Colangelo, Gasser and Leutwyler~(2001), the description of
the remaining waves is as good as anything found in the 
literature, with the added advantage that they are more robust: 
each wave stands on its own. Our P wave, for example, is
somewhat  more accurate at energies around  
and above $M_\rho$ than the solution of Colangelo, Gasser and Leutwyler~(2001). 
However, these authors get a more precise determination at very low 
energy, as is clear comparing their value
 of $a_1=(36.64\pm0.5)\times10^{-3}\,\mu^{-3}$ with ours.\fnote{The author of the present report, however, 
finds it difficult to believe the accuracy claimed 
by these authors; as 
shown in de~Troc\'oniz and Yndur\'ain~(2002), 
the change produced by fitting $\tau$ decay data or $e^+e^-\to\pi^+\pi^-$ 
shifts $a_1$ by an  amount larger than the error given by  Colangelo, Gasser and Leutwyler.} 


\bookendchapter


\brochureb{\smallsc chapter 8}{\smallsc qcd, pcac and chiral symmetry for pions and kaons}{69}
\bookchapter{8. QCD, PCAC and chiral symmetry for pions and kaons}
\vskip-0.5truecm
\booksection{8.1. The QCD Lagrangian. Global symmetries;
conserved currents}

\noindent
In the previous chapters we have discussed {\sl general} properties 
of pion interactions. 
In this  chapter\fnote{This chapter follows, to a large extent, 
the corresponding one in the text of the author, Yndur\'ain~(1999), where we 
send for more details on QCD.} we remember that pions are bound 
states of quark-antiquark, and that we have a theory of strong 
interactions, QCD. 
We will then study properties of 
pion physics which follow, in one way or another, from the QCD 
Lagrangian,
$${\cal L}=-\sum m_l\bar{q}_lq_l+\ii\sum_{l=1}^n\bar{q}_l\slash{D\!}q_l
-\tfrac{1}{4}(D\times B)^2+\hbox{gauge fixing + ghost terms},
\equn{(8.1.1)}$$
where $q_l$ is the quark operator foor flavour $l$, $B$ is the gluon field operator, etc.;
sum over omitted colour indices is generally understood.
In (8.1.1) we assume that we have only {\sl light} 
quarks ($u,\,d$ and, eventually, $s$). The existence of 
heavy quarks has little influence in the physics of small momenta 
in which we are interested here.
 
In the present section we start with  the {\sl global} symmetries
of the QCD Lagrangian.  Since its form is unaltered by renormalization, we 
can neglect the distinction between bare and renormalized $\cal L$.

Clearly, $\cal L$ is invariant under Poincar\'e transformations, $x\rightarrow\Lambdav x+a$. 
The currents corresponding to (homogeneous) Lorentz 
transformations $\Lambdav$ are not of great interest for us here. Space-time 
translations generate the {\sl energy-momentum tensor}. Its form is fixed by Noether's 
theorem, which gives
$$\Thetav^{\mu\nu}=\sum_i \dfrac{\partial {\cal L}}{\partial(\partial_\mu \Phiv_i)}
\,\partial_\nu\Phiv_i-g^{\mu\nu}{\cal L},
\equn{(8.1.2a)}$$
and the sum over $i$ runs over all 
the fields in the QCD Lagrangian. These currents are conserved,
$$\partial_\mu\Thetav^{\mu\nu}=0,$$
and the corresponding ``charges" are the components of the four-momentum
$$P^\mu=\int\dd^3{\bf x}\,\Thetav^{0\mu}(x).$$
The explicit expression for $\Thetav^{\mu\nu}$ in QCD is
$$\eqalign{\Thetav^{\mu\nu}=&\,\ii\sum_q\bar{q}\gamma^\mu D^{\nu}q-
\ii g^{\mu\nu}\sum_q\bar{q}\slash{D\!}q+g^{\mu\nu}\sum_q m_q\bar{q}q\cr
&-g_{\alpha\beta}G^{\mu\alpha}G^{\nu\beta}+\tfrac{1}{4}g^{\mu\nu}G^2+\hbox{gauge fixing}+
\hbox{ghost terms}.\cr}\equn{(8.1.2b)}$$


In the 
quantum version, we understand that products are replaced by Wick ordered products. 
 $\Thetav$ is not unique and, as a matter of fact, direct application of (8.1.2a) does not 
yield a gauge invariant tensor. To obtain the gauge invariant expression (8.1.2b) one may proceed 
by replacing derivatives by covariant derivatives. A more rigorous procedure would be to 
reformulate (8.1.2a) in a way consistent with gauge 
invariance by performing gauge transformations simultaneously to the spacetime 
translation. For an infinitesimal one, $x^\mu\rightarrow x^\mu+\epsilon^\mu$, we then 
define
$$B_a^\mu\rightarrow B_a^\mu+(\epsilon_\alpha\partial^\alpha B_a^\mu\equiv 
D^\mu\epsilon_\alpha B_a^\alpha+\epsilon_\alpha G_a^{\alpha\mu}).$$
The term $D^\mu\epsilon_\alpha B_a^\alpha$ may be absorbed by a gauge transformation, so we may
 write the transformation as $B_a^\mu\rightarrow 
B_a^\mu+\epsilon_\alpha G_a^{\alpha\mu}$. For a discussion of the arbitrariness in the definition 
of the energy--momentum tensor, see Callan, Coleman and Jackiw (1970)
 and Collins, Duncan and Joglekar (1977).

Next, we have the currents and charges associated with
 colour rotations. We leave it to the reader to write them explicitly; 
they are particular cases of colour gauge transformations (with constant parameters). We 
now  pass 
over to a different set of currents {\sl not} associated with interactions 
of quarks and gluons among themselves.

If all the quark  masses vanished, we would have invariance of $\cal L$ under the 
transformations,
$$q_f\rightarrow\sum_{f'=1}^{n_f}W_{ff'}q_{f'},\quad 
q_f\rightarrow\sum_{f'=1}^{n_f}W^5_{ff'}\gamma_5q_{f'}\equn{(8.1.3)} $$
where $f,\,f'$ are flavour indices, and $W,\,W^5$ unitary matrices. This implies that the 
currents
$$\eqalign{V_{qq'}^{\mu}(x)=\bar{q}(x)\gamma^\mu q'(x),\cr
A_{qq'}^{\mu}(x)=\bar{q}(x)\gamma^\mu \gamma_5q'(x)\cr}\equn{(8.1.4)}$$
would be each separately conserved. When mass terms are taken into account, 
only the diagonal $V^{\mu}_{qq}$ are conserved; the others are what is 
called {\sl quasi-conserved} currents, i.e., their divergences are proportional to masses. 
These divergences are easily calculated: since the transformations in (8.1.3) commute 
with the interaction part of $\cal L$, we may evaluate them with free fields, 
in which case use of the free Dirac equation $\ii\slash{\partial}q=m_q q$ gives
$$\partial_\mu V^{\mu}_{qq'}=\ii(m_q-m_{q'})\bar{q}q',\quad
\partial_\mu A^{\mu}_{qq'}=\ii(m_q+m_{q'})\bar{q}\gamma_5q'.\equn{(8.1.5)}$$ 
In fact, there is a subtle point concerning the divergence of 
axial currents. \equ~(8.1.5) is correct as it stands for 
the nondiagonal currents, $q\neq q'$; for 
$q=q'$, however, one has instead
$$\partial_\mu A^{\mu}_{qq}(x)=\ii(m_q+m_q)\bar{q}(x)\gamma_5q(x)+
\dfrac{T_Fg^2}{16\pi^2}\epsilon^{\mu\nu\rho\sigma}G_{\mu\nu}(x)G_{\rho\sigma}(x),
\equn{(8.1.6)}$$
with $T_F=1/2$ a colour factor. This is the so-called Adler--Bell--Jackiw 
anomaly, that we will discuss later.

The {\sl equal time commutation relations} (ETC) of the $V,\,A$ with the fields are 
also easily calculated, for free fields. Using (8.1.4) 
and the ETC of quark fields, one finds,
$$\eqalign{\delta(x^0-y^0)[V^0_{qq'}(x),q''(y)]=&-\delta(x-y)\delta_{qq''}q'(x),\cr 
\delta(x^0-y^0)[A^0_{qq'}(x),q''(y)]=&-\delta(x-y)\delta_{qq''}\gamma_5q'(x),\;\hbox{etc.}\cr}
\equn{(8.1.7)}$$
The $V,\,A$ commute with gluon and ghost fields. The equal time commutation relations of the   
 $V,\,A$ among themselves (again for free fields) are best described for three 
flavours, $f=1,\,2,\,3=u,\,d,\,s$  
by introducing the Gell-Mann $\lambda^a$ matrices in 
flavour space (for two quarks $u,\,d$, replace the $\lambda^a$ by the $\tau^a$ of Pauli, 
and the $f_{abc}$ by $\epsilon_{abc}$ in (8.2.9) below).  So we
let
$$V^{\mu}_{a}(x)=\sum_{ff'}\bar{q}_f(x)\lambda^{a}_{ff'}\gamma^\mu q_{f'}(x),\quad
A^{\mu}_{a}(x)=\sum_{ff'}\bar{q}_f(x)\lambda^{a}_{ff'}\gamma^\mu \gamma_5q_{f'}(x),
\equn{(8.1.8)}$$
and we then obtain the commutation relations
$$\eqalign{\delta(x^0-y^0)[V^0_a(x),V^{\mu}_{b}(y)]=&
\,2\ii\delta(x-y)\sum f_{abc}V^{\mu}_{c}(x),\cr 
\delta(x^0-y^0)[V^0_a(x),A^{\mu}_{b}(y)]=&
\,2\ii\delta(x-y)\sum f_{abc}A^{\mu}_{c}(x),\cr 
\delta(x^0-y^0)[A^0_a(x),A^{\mu}_{b}(y)]=&
\,2\ii\delta(x-y)\sum f_{abc}V^{\mu}_{c}(x),\;\hbox{etc.}\cr} \equn{(8.1.9)}$$
Equations (8.1.7) and (8.1.9) have been derived only for free fields. However, 
they involve short distances; therefore in QCD, and because of asymptotic freedom, 
they will hold as they stand even 
in the presence of interactions.

Equal time commutation relations of conserved or quasi-conserved 
currents with the Hamiltonian may also be easily obtained. If 
$J^\mu$ is conserved, then the corresponding charge 
$$Q_J(t)=\int\dd^3{\bf x}\,J^0(t,{\bf x}),\quad t=x_0,$$
commutes with $\cal H$:
$$[Q_J(t),{\cal H}(t,{\bf y})]=0.$$
Here $\cal H$ is the {\sl Hamiltonian density}, ${\cal H}=\Thetav^{00}$. 
If $J$ is quasi-conserved, let ${\cal H}_m$ be the mass term in $\cal H$,
$${\cal H}_m=\sum_qm_q\bar{q}q.$$
Then,
$$[Q_J(t),{\cal H}_m(t,{\bf y})]=\ii\partial_\mu J^\mu(t,{\bf y}).\equn{(8.1.10)}$$
Of course, $Q_J$ still commutes with the rest of $\cal H$.

\booksection{8.2 Mass terms and invariances: chiral invariance}

\noindent In this section we will consider quarks with masses $m\ll\Lambdav$ 
(with $\lambdav$ the QCD mass parameter), to 
be referred to as {\sl light quarks}.\fnote{It 
is, of course, unclear whether the meaningful parameter in this respect is $\Lambdav$, 
connected to the strong interaction coupling by
$$\alpha_s(t)=\dfrac{12\pi}{(33-2n_f)\log t/\lambdav^2},$$
 or 
$\Lambdav_0$ defined by $\alpha_s(\Lambdav_0)\approx 1$. From considerations of chiral dynamics 
(see later), 
it would appear that the scale for smallness of quark masses is $4\pi f_\pi\sim1\,\gev$,
 where $f_\pi$ is the pion 
decay constant; but even if 
we accept this, it is not obvious at which 
scale $m$ has to be computed. 
 We will see that $m_u,\,m_d\sim4\,\hbox{to}\,10\,\mev$ so there is little doubt
 that $u,\,d$ quarks should be classed as ``light"  with any reasonable definition; but 
the situation is less definite for the $s$ quark with $m_s\sim 180\,\mev$.}  Because 
the only dimensional parameter intrinsic to QCD is, we believe, $\Lambdav$, we may expect 
that to some approximation we may neglect the masses of such quarks, which will yield only 
contributions  of order $m^2/\Lambdav^2$. 

To study this, we consider the QCD Lagrangian,
$${\cal L}=-\sum_{l=1}^n m_l\bar{q}_lq_l+\ii\sum_{l=1}^n\bar{q}_l\slash{D\!}q_l
-\tfrac{1}{4}(D\times B)^2+\hbox{gauge fixing + ghost terms}.
\equn{(8.2.1)}$$
The sum now runs only over {\sl light} quarks;
the presence of heavy quarks will have no practical effect in what follows 
and consequently we neglect them.
We may then split the quark fields into left-handed and right-handed components:
$$q_l=q_{L,l}+q_{R,l};\quad
q_{L,l}\equiv q_{-,l}=\dfrac{1-\gamma_5}{2}q_l,\quad
q_{R,l}\equiv q_{+,l}=\dfrac{1+\gamma_5}{2}q_l.$$
In terms of these, the quark part of the Lagrangian may be written as
$${\cal L}=-\sum_{l=1}^n m_l\left(\bar{q}_{R,l}q_{L,l}+\bar{q}_{L,l}q_{R,l}\right)
+\ii\sum_{l=1}^n\left(\bar{q}_{L,l}\slash{D\!}q_{L,l}+\bar{q}_{R,l}\slash{D\!}q_{R,l}\right)
+\cdots\,.$$

We then consider the set of transformations
 $W^\pm$  
(left-handed times right-handed) given by the independent 
transformations of the $q_{R,l}$, $q_{L,l}$: 
$$q_{R,l}\to\sum_{l'}W^{+}_{ll'}q_{R,l'},\quad
q_{L,l}\to\sum_{l'}W^{-}_{ll'}q_{L,l'};\quad
 W^\pm\;\hbox{unitary}.\equn{(8.2.2)}$$
Clearly, the only term in $\cal L$ that is not invariant under all the transformations 
(8.2.2) is the mass term,
$${\cal M}=\sum_{l=1}^n m_l\bar{q}_lq_l=
\sum_{l=1}^n m_l\left(\bar{q}_{R,l}q_{L,l}+\bar{q}_{L,l}q_{R,l}\right).\equn{(8.2.3)}$$

When written in this form, the mass term is invariant under the set of transformations $[U(1)]^n$,
$$q_l\to \ee^{\ii\theta_l}q_l,\equn{(8.2.4)}$$
but this would not have been the case if we had allowed for nondiagonal terms 
in the mass matrix. To resolve this question of which are the general invariance 
properties of a mass term, we will prove two theorems.\fnote{The theorems are valid for {\sl any}
 quark mass matrix, i.e., also including heavy flavours.}
\smallskip
\hangindent=0.5cm{\noindent\quad{\sc Theorem 1.}  {\sl Any general mass matrix 
can be written in the form (8.2.3) by appropriate redefinition 
of the quark fields. Moreover, we may assume that $m\geq0$. Thus, (8.2.3) is actually the 
most general mass term possible.}}
\smallskip\noindent
 For the proof we consider that 
the most general mass term compatible with hermiticity is
$${\cal M}'=\sum_{ll'}\left\{\bar{q}_{L,l}M_{ll'}q_{R,l'}+
\bar{q}_{R,l}M^*_{ll'}q_{L,l'}\right\}.\equn{(8.2.5)}$$
Let us temporarily denote matrices in flavour space by putting a tilde 
under them. If $\undertilde{M}$ is the matrix with components $M_{ll'}$, then the 
well-known polar decomposition, valid for any matrix, allows us to write
$$\undertilde{M}=\undertilde{m}\undertilde{U},$$
where $\undertilde{m}$ is positive-semidefinite, so all its eigenvalues are $\geq0$, 
and $\undertilde{U}$ is unitary. \equ~(8.2.5) may then be cast in the form 
$${\cal M}'=\sum_{ll'}\left\{\bar{q}_{L,l}m_{ll'}q'_{R,l'}+
\bar{q'}_{R,l}m_{ll'}q_{L,l'}\right\},\quad q'_{R,l}=\sum_{l'}U_{ll'}q_{R,l'},\equn{(8.2.6)}$$
and we have used the fact that $\undertilde{m}$ is Hermitian. Define 
$q'=q_L+q'_R$; because 
$\bar{q}_Rq_R=\bar{q}_Lq_L=0$, (8.2.6) becomes, in terms of $q'$,
$${\cal M}'=\sum\bar{q}'_lm_{ll'}q'_{l'}.$$
It then suffices to transform $q'$ by the matrix that diagonalizes $\undertilde{m}$ 
to obtain (8.2.3) with positive $m_l$. The term $\bar{q}\slash{D\!}q$ in 
the Lagrangian is left invariant by all these transformations, so the theorem is proved.
\smallskip
\hangindent=0.5cm{\noindent\quad{\sc Theorem 2.} {\sl If all the $m_l$ are nonzero 
and different, then the only invariance left is the $[U(1)]^n$ of (8.2.4).}}
\smallskip\noindent
Let us consider the $\undertilde{W}_\pm$ of (8.2.2), and assume that
 $\undertilde{W}_+=\undertilde{W}_-\equiv \undertilde{W}$; to show that this 
must actually be the case is left as an exercise. The condition of invariance of 
$\cal M$ yields the relation
$$\undertilde{W}^{\dag}\undertilde{m}\undertilde{W}=\undertilde{m},\quad {\rm i.e.},
\quad [\undertilde{m},\undertilde{W}]=0.
\equn{(8.2.7)}$$
It is known that any $n\times n$ diagonal matrix can be written as 
$\sum_{k=0}^{n-1}c_k\undertilde{m}^k$ if, as 
occurs in our case, all the eigenvalues of $\undertilde{m}$ are different and nonzero. 
Because of (8.2.7), it then follows that $\undertilde{W}$ commutes with all diagonal 
matrices, and hence it must itself be diagonal: because it is also unitary, it consists of 
diagonal phases, i.e., 
 it may be written as a product of transformations (8.2.4), as 
was to be proved. We leave it to the reader to check that the 
conserved quantity corresponding to the $U(1)$ that acts on 
flavour $q_f$ is the corresponding flavour number.

In the preceding theorems, we have not worried whether the masses $m$
 were bare, running or invariant masses. This is because, 
in the \msbar\ scheme, the mass matrix becomes renormalized as a whole:
$$\undertilde{M}=Z_m^{-1}\undertilde{M}_u,$$
where $Z_m$ is a {\sl number}. The proof of this last property is easy: all we have to do 
is to repeat the standard renormalization of the quark propagator, allowing for the 
matrix character of $M,\,Z_m$. We find, 
for the divergent part and in an arbitrary covariant gauge,
$$\eqalign{\undertilde{S}_R^\xi(p)=&
\dfrac{\ii}{\slash{p}-\undertilde{M}}+
\dfrac{1}{\slash{p}-\undertilde{M}}\Bigg\{-[\undertilde{\Deltav}_F(\slash{p}-\undertilde{M})+
(\slash{p}-\undertilde{M}){\undertilde{\Deltav}}_F^{\dag}]-\delta\undertilde{M}\cr
&-(1-\xi)(\slash{p}-\undertilde{M})N_\epsilon C_F\dfrac{g^2}{16\pi^2}+
3N_\epsilon C_F\dfrac{g^2}{16\pi^2}\undertilde{M}\Bigg\}\dfrac{\ii}{\slash{p}-\undertilde{M}},\cr}$$
and we have defined
$$\undertilde{M}=\undertilde{M}_u+\delta\undertilde{M},\quad 
\undertilde{Z}_F=1-\undertilde{\Deltav}_F.$$
The renormalization conditions then yield
$$\eqalign{{\undertilde{\Deltav}_F}^{\dag}+\undertilde{\Deltav}_F=&\,
-(1-\xi)N_\epsilon C_F\dfrac{g^2}{16\pi^2}={\rm diagonal,}\cr
[\undertilde{\Delta}_F,\undertilde{M}]=&\,0,\quad [\undertilde{M},\delta\undertilde{M}]=0,\cr
\delta\undertilde{M}=&\,3N_\epsilon C_F\dfrac{g^2}{16\pi^2}\undertilde{M}.\cr}$$
Thus, the set of fermion fields and the mass matrix get renormalized as a whole:
$$\undertilde{Z}_F^{-1}=1+N_\epsilon C_F\dfrac{g^2}{16\pi^2},\quad 
\undertilde{Z}_m=1-3N_\epsilon C_F\dfrac{g^2}{16\pi^2},\equn{(8.2.8a)}$$
i.e.,
$$\undertilde{Z}_F=Z_F\,1,\quad\undertilde{Z}_m=Z_m\,1.\equn{(8.2.8b)}$$
We have proved this to lowest order, but the renormalization group 
equations guarantee the result to leading order in $\alpha_s$.

This result can be understood in yet another way. The invariance of 
$\cal L$ under the transformations (8.2.4) implies that we may choose 
the counterterms to satisfy the same invariance, so the mass matrix will remain 
diagonal after renormalization. In fact, this proof shows that in mass independent 
renormalization schemes (such as the \msbar), \equs~(8.2.8b) actually hold to all orders.

The results we have derived show that, if all the $m_i$ are different and nonvanishing,\fnote{As 
seems to be the case in nature. As we will see, one finds ${m}_d/{m}_u\sim2$, 
 ${m}_s/{m}_d\sim20$, ${m}_u\sim 5\,\mev$.} the 
only global symmetries of the Lagrangian are those 
associated with flavour conservation, (8.2.4). As stated above, however, under 
certain conditions it may be a good approximation to neglect the $m_l$. In this case, all 
the transformations of \equ~(8.2.2) become symmetries of the Lagrangian. The measure of the accuracy of 
the symmetry is given by, for example, the divergences of the corresponding currents 
or, equivalently, the conservation of the charges. This has been 
discussed in \sect~7.1, and we now present some extra details.

Let us parametrize the $W$ as $\exp\{(\ii/2)\sum\theta_a\lambda^a\}$, where the $\lambda$ are the 
Gell-Mann matrices. (We consider the case $n=3$; for $n=2$, replace the $\lambda$ by the 
$\tau$ of Pauli). We may denote by $U_\pm(\theta)$  the operators that implement (8.2.2):
$$U_\pm(\theta)\dfrac{1\pm\gamma_5}{2}q_lU_\pm^{-1}(\theta)=
\sum_{l'}\left(\ee^{(\ii/2)\sum\theta_a\lambda^a}\right)_{ll'}\dfrac{1\pm\gamma_5}{2}q_{l'}.
\equn{(8.2.9)}$$
For infinitesimal $\theta$, we write
$$U_\pm(\theta)\simeq1-\dfrac{\ii}{2}\sum \theta_aL^a_\pm,\quad
(L^a_\pm)^{\dag}=L^a_\pm,$$
so that (8.2.9) yields
$$[L^a_\pm,q_{\pm,l}(x)]=-\sum_{l'}\lambda^a_{ll'}q_{\pm,l'}(x),\quad
q_{\pm,l}\equiv \dfrac{1\pm\gamma_5}{2}q_l.\equn{(8.2.10)}$$
Because the $U_\pm$ leave the interaction part of the Lagrangian invariant, and since QCD 
is a free field theory at zero distance, we may solve (8.2.10) using free-field 
commutation relations. The result is
$$L^a_\pm(t)=:\int\dd^3{\bf x}\sum_{ll'}\bar{q}_{\pm,l}(x)\gamma^0\lambda^a_{ll'}
q_{\pm,l'}(x):,
\quad t=x^0.\equn{(8.2.11)}$$
These will be recognized as the charges corresponding to the currents
$$J^{a\mu}_{\pm}(x)=
:\sum_{ll'}\bar{q}_l(x)\lambda^a_{ll'}\gamma^\mu\dfrac{1\pm\gamma_5}{2}q_{l'}(x):.
\equn{(8.2.12)}$$
If the symmetry is exact, $\partial_\mu J_{\pm}^{a\mu}=0$, and a standard calculation 
shows that the $L^a_\pm(t)$ are actually independent of $t$. Otherwise, we 
have to define {\sl equal time} transformations and modify (8.2.9, 10) writing, for 
example,
$$[L^a_\pm(t),q_{\pm,l}(x)]=-\sum_{l'}\lambda^a_{ll'}q_{\pm,l'}(x),\quad
t=x^0.\equn{(8.2.13)}$$
The set of transformations
$$U_\pm(\theta,t)=\exp\left\{-\dfrac{\ii}{2}\sum L^a_\pm(t)\theta_a\right\},$$
builds up the group of {\sl chiral transformations} generated 
by the currents (8.2.12). In our present case we find the 
chiral $SU_F^+(3)\times SU_F^-(3)$ group. Its generators may be rearranged
 in terms of the set of vector and axial currents $V^{\mu}_{ll'}(x)$, $A^{\mu}_{ll'}(x)$ 
introduced in \sect~8.2. (Actually, not
 all diagonal elements are in $SU_F^+(3)\times SU_F^-(3)$, but they are in the group 
 $U_F^+(3)\times U_F^-(3)$).  An important subgroup of 
 $SU_F^+(3)\times SU_F^-(3)$ is that generated by the vector currents, 
which is simply the flavour $SU(3)$ of Gell-Mann and 
Ne'eman.

The exactness of the symmetries is related to the time independence of the 
charges $L_\pm$, which in turn is linked to the divergence of the currents. 
These divergences are proportional to differences of
 masses, $m_l-m_{l'}$ for the vector, and sums $m_l+m_{l'}$ for the 
axial currents.
Thus, we conjecture that $SU_F(3)$ will be good to the extent that 
$|m_l-m_{l'}|^2\ll \Lambdav^2$ 
and chiral $SU_F^+(3)\times SU_F^-(3)$ to the extent that $m_l\ll \Lambdav$. In 
the real world, it appears that mass differences are of the same order as the 
masses themselves, so we expect chiral symmetries to be almost as good 
as flavour symmetries. This seems to be the case experimentally.

\booksection{8.3 Wigner--Weyl and Nambu--Goldstone realizations of symmetries}

\noindent The fact that flavour and chiral $SU(3)$ (or $SU(2)$) appear to 
be valid to similar orders of approximation does not mean that these symmetries are 
realized in the same manner. In fact, we will see that there are good 
theoretical and experimental reasons why they are very different.

Let us begin by introducing the charges with definite parity,
$$Q^a=L^a_++L^a_-,\quad Q^a_5=L^a_+-L^a_-.\equn{(8.3.1)}$$
Their {\sl equal time commutation relations} are
$$\eqalign{[Q^a(t),Q^b(t)]=&\,2\ii\sum f^{abc}Q^c(t),\cr 
[Q^a(t),Q^b_5(t)]=&\,2\ii\sum f^{abc}Q^c_5(t),\cr 
[Q^a_5(t),Q^b_5(t)]=&\,2\ii\sum f^{abc}Q^c(t).\cr }\equn{(8.3.2)}$$
The set $Q^a$ builds the group $SU_F(3)$. In the limit $m_l\to0$, all 
$Q$, $Q_5$ are $t$-independent and
$$[Q^a,{\cal L}]=[Q^a_5,{\cal L}]=0.\equn{(8.3.3)}$$
The difference between $Q^a$, $Q^a_5$, however, lies in the vacuum. 
In general, given a set of generators $L^j$ of symmetry
 transformations of $\cal L$, we have two possibilities:
$$L^j|0\rangle=0,\equn{(8.3.4)}$$ 
which is called a {\sl Wigner--Weyl} symmetry, or
$$L^j|0\rangle\neq0,\equn{(8.3.5)}$$
or {\sl Nambu--Goldstone} symmetry. Obviously,
 we will in general have a mixture of the two symmetries, 
with some $L^i$, $i=1,\dots,r$, verifying (8.3.4) and the 
rest, $L^k$, $k=r+1,\dots,n$, satisfying (8.3.5). Since the commutator
 of two operators that annihilate the 
vacuum also annihilates the 
vacuum, it follows that the subset of Wigner--Weyl symmetries forms a subgroup.  

Two theorems are especially relevant with respect to these questions.
 The first, due to Coleman (1966), asserts that ``the 
invariance of the vacuum is the invariance of the world", or, in more transparent terms, 
that the physical states (including bound states) 
are invariant under the transformations of a Wigner--Weyl group of symmetries. It follows that, 
if we assumed that chiral symmetry was all of it realized in the Wigner--Weyl mode, 
we could conclude that the masses of all mesons in a flavour multiplet would be degenerate,
 up to corrections of order $m_q^2/M_h^2$, with $M_h$ the (average) hadron mass. 
This is true of the $\omega$, $\rho$, $K^*$, $\phi$, but if we include parity doublets 
this is no longer the case. Thus, for example, there is no scalar 
meson with a mass anywhere near that of the pion, and the axial vector meson masses are 
more than half a \gev\ larger than the masses of $\omega$ or $\rho$. Thus 
it is strongly suggested that $SU_F(3)$ is a Wigner--Weyl symmetry, but chiral  
 $SU_F^+(3)\times SU_F^-(3)$ contains generators of the Goldstone--Nambu type. We assume, 
therefore,
$$Q^a(t)|0\rangle=0,\quad Q^a_5(t)|0\rangle\neq0.\equn{(8.3.6)}$$

The second relevant theorem is Goldstone's (1961). It states that, for 
each generator that fails to annihilate the vacuum, there must exist a massless boson with the 
quantum numbers of that generator. Therefore, we ``understand" the smallness of the 
masses of the pion or kaon\fnote{The particles with zero flavour quantum numbers 
present problems of their own (the so-called $U(1)$ problem) 
that will be discussed later.} because, in the limit $m_u,\,m_d,\,m_s\to0$, 
we would also have $\mu\to0$, $m_K\to0$. Indeed, we will later show that
$$\mu^2\sim m_u+m_d,\quad m^2_K\sim m_{u,d}+m_s.\equn{(8.3.7)}$$

We will not prove either theorem here, but we note that (8.3.7) affords a 
quantitative criterion for the validity of chiral  symmetries; they hold to 
corrections of order $\mu^2/m^2_\rho$ for $SU(2)$ and of $m^2_K/m^2_{K^*}$ for 
$SU(3)$.

We also note that a Nambu--Goldstone realization (Nambu, 1960; Nambu and 
Jona--Lasinio, 1961a,b) is never possible in perturbation theory.
 Since the symmetry generators are Wick-ordered products of 
field operators, it is clear that  
to all orders of perturbation theory  $Q^a_5(t)|0\rangle=0$. 
This means that the physical vacuum is different from the vacuum 
of perturbation theory in the 
limit $m\to0$. We emphasize this by writing $|0\rangle$ for 
the perturbation-theoretic vacuum and $|{\rm vac}\rangle$ 
for the physical one when there is danger of confusion.
 So we rewrite (8.3.6) as
$$Q^a(t)|{\rm vac}\rangle=0,\quad Q^a_5(t)|{\rm vac}\rangle\neq0.
\equn{(8.3.8)}$$
It is not difficult to see how this may come about in QCD. Let $a^{\dag}_P({\bf k})$ 
be the creation operator for a particle P with three-momentum $\bf k$. The states
$$a^{\dag}_P({\bf 0})\overbrace{\mathstrut\dots}^n a^{\dag}_P({\bf 0})|0\rangle=
|n\rangle$$
are all degenerate in the limit $m_P\to 0$. Therefore, the physical vacuum will be, 
in this limit, 
$$|{\rm vac}\rangle=\sum_nc_n|n\rangle,$$
i.e., it will contain zero-frequency massless particles (Bogoliubov's 
model). In QCD we have the gluons which 
are massless, and so will the light quarks be, to a good 
approximation, in the chiral limit.
\goodbreak
\booksection{8.4. PCAC, $\pi^+$ decay, the pion propagator and light quark mass ratios}

\noindent We are now in a position to obtain quantitative results on the masses of the light quarks, 
relating them to the masses of pions and kaons.
 To do so, consider the current
$$A^{\mu}_{ud}(x)=\bar{u}\gamma^\mu\gamma_5d(x),$$
and its divergence
$$\partial_\mu A^{\mu}_{ud}(x)=\ii(m_u+m_d)\bar{u}\gamma_5d(x).$$
The latter has the quantum numbers of the $\pi^+$, so we can use 
it as a composite pion field operator. We thus write
$$\partial_\mu A^{\mu}_{ud}(x)=\sqrt{2}f_\pi \mu^2 \phi_\pi(x).\equn{(8.4.1a)}$$
The factors in (8.4.1a) are chosen for historical reasons (our convention is  
not universal, however). $\phi_\pi(x)$ is the pion field normalized to
$$\langle0|\phi_\pi(x)|\pi(p)\rangle=\dfrac{1}{(2\pi)^{3/2}}\ee^{-\ii p\cdot x},
\equn{(8.4.1b)}$$
with $|\pi(p)\rangle$ the state of a pion with momentum $p$. The constant $f_\pi$ 
may be obtained from experiment as follows. Consider the weak decay 
$\pi^+\to\mu^+\nu$. With the effective Fermi Lagrangian for weak interactions 
(see, e.g., Marshak, Riazzudin and Ryan, 1969)
$${\cal L}^{\rm Fermi}_{\rm int}=
\dfrac{G_F}{\sqrt{2}}\bar{\mu}\gamma_\lambda(1-\gamma_5)\nu_\mu
\bar{u}\gamma^\lambda(1-\gamma_5)d+\cdots,$$
we find the decay amplitude
$$F(\pi\to\mu\nu)=
\dfrac{2\pi G_F}{\sqrt{2}}\bar{u}_{(\nu)}(p_2)\gamma_\lambda(1-\gamma_5)v_{(\mu)}(p_1,\sigma)
\langle0|A^{\lambda}_{ud}(0)|\pi(p)\rangle.\equn{(8.4.2a)}$$
Now, on invariance grounds,
$$\langle0|A^{\lambda}_{ud}(0)|\pi(p)\rangle=\ii p^\lambda C_\pi;\equn{(8.4.2b)}$$
contracting with $p_\mu$ we find $C_\pi=f_\pi\sqrt{2}/(2\pi)^{3/2}$ and hence
$$\mu^2 C_\pi=\langle0|\partial_\lambda A^{\lambda}_{ud}(0)|\pi(p)\rangle=
\sqrt{2}f_\pi \mu^2\dfrac{1}{(2\pi)^{3/2}}.\equn{(8.4.2c)}$$
Therefore
$$\tau(\pi\to\mu\nu)=\dfrac{4\pi}{(1-m^2_\mu/\mu^2)^2G^2_Ff^2_\pi \mu m^2_\mu},$$
and we obtain $f_\pi$ from the decay rate. 
An accurate evaluation of $f_\pi$ requires taking into account 
the Cabibbo rotation and electromagnetic radiative corrections.
One gets $f_\pi\simeq 93\,\mev$. A 
remarkable fact is that, if we repeat the analysis for kaons,
$$\partial_\mu A^{\mu}_{us}(x)=\sqrt{2}f_Km^2_K\phi_K(x),\equn{(8.4.3)}$$
we find that, experimentally, $f_K\approx110\,\mev$: it agrees with 
$f_\pi$ to 20\%. Actually, this is to be expected because, in the limit 
$m_{u,d,s}\to0$, there is no difference 
between pions and kaons, and we would find strict equality. That $f_\pi$, $f_K$ are 
so similar in the real world is a good point in favour of $SU_F(3)$ chiral 
ideas.

The relations (8.4.1) and (8.4.3) are at times called 
PCAC\fnote{Partially conserved axial current. In fact, in the limit 
$\mu^2\to0$, the right hand side of 
(8.4.1a) vanishes.} but this is not very meaningful, for these 
equations are really {\sl identities}. One may use any pion field operator 
one wishes, in particular (8.4.1), provided that it has the right quantum numbers and its 
vacuum-one pion matrix element is not zero. The nontrivial part of PCAC 
will be described below.

The next step is to consider the two-point function, or {\sl correlator} (we 
drop the $ud$ index from $A_{ud}$)
$$\piv^{\mu\nu}(q)=
\ii\int\dd^4x\,\ee^{\ii q\cdot x}\langle{\rm T}A^\mu(x)A^\nu(0)^{\dag}\rangle_{\rm vac},$$
and contract with $q_\mu,\,q_\nu$:
$$\eqalign{q_\nu q_\mu \piv^{\mu\nu}(q)=&
-q_\nu\int\dd^4x\,\ee^{\ii q\cdot x}\partial_\mu
\langle{\rm T}\,A^\mu(x)A^\nu(0)^{\dag}\rangle_{\rm vac}\cr
=&-q_\nu\int\dd^4x\,\ee^{\ii q\cdot x}\delta(x^0)
\langle[A^0(x),A^\nu(0)^{\dag}]\rangle_{\rm vac}\cr
&-q_\nu\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}\,\partial\cdot A(x)A^\nu(0)^{\dag}\rangle_{\rm vac}\cr
=&\,2\ii\int\dd^4x\,\ee^{\ii q\cdot x}\delta(x^0)
\langle[A^0(x),\partial\cdot A(0)^{\dag}]\rangle_{\rm vac}\cr
&+\ii\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}\,\partial\cdot A(x)\partial\cdot A(0)^{\dag}\rangle_{\rm vac}.\cr}$$
Using \equs~(8.4.1, 2) and evaluating the 
commutator, we find
$$\eqalign{q_\nu q_\mu \piv^{\mu\nu}(q)=&\,2(m_u+m_d)\int\dd^4x\,\ee^{\ii q\cdot x}
\delta(x)\langle\bar{u}(x)u(x)+\bar{d}(x)d(x)\rangle_{\rm vac}\cr
&+2\ii f^2_\pi \mu^4\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}\,\phi_\pi(x)\phi_\pi(0)^{\dag}\rangle_{\rm vac},\cr}$$
or, in the limit $q\to0$,
$$\eqalign{2(m_u+m_d)\langle:\bar{u}(0)u(0)+\bar{d}(0)d(0):\rangle_{\rm vac}\cr
=-2\ii f^2_\pi \mu^4\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}\,\phi_\pi(x)\phi_\pi(0)^{\dag}\rangle_{\rm vac}\big|_{q\to0},\cr}$$
and we have reinstated explicitly the colons of normal ordering.
The right hand side of this equality has  contributions from 
the pion pole and from the continuum; by writing 
a dispersion relation (Cauchy representation) for $\Piv(t)$, defined by
$$\Piv(q^2)=\ii\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}\,\phi_\pi(x)\phi_\pi(0)^{\dag}\rangle_{\rm vac},$$
they can be expressed  as\fnote{The equation 
below should have  been written with subtractions, to compensate for the growth 
of $\Piv(q^2)$ for large $q^2$; but these do not alter the conclusions.}
$$\eqalign{\ii\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}\,\phi_\pi(x)\phi_\pi(0)^{\dag}\rangle_{\rm vac}\big|_{q\to0}&
=\left\{\dfrac{1}{\mu^2-q^2}+
\dfrac{1}{\pi}\int\dd t\,\dfrac{\imag \Piv(t)}{t-q^2}\right\}_{q\to0}\cr
&=\dfrac{1}{\mu^2}+\dfrac{1}{\pi}\int\dd t\dfrac{\imag \Piv(t)}{t}.\cr
}$$
The order of the limits is essential; we first must take $q\to0$ and the chiral limit 
afterwards. In the limit $\mu^2\to0$, the first term on the right hand side 
above {\sl diverges}, and the second remains finite.\fnote{Properly speaking, 
this is the PCAC limit, for in this limit 
the axial current is conserved.} We then get
$$\eqalign{(m_u+m_d)\langle\bar{u}u+\bar{d}d\rangle=
-2f^2_\pi \mu^2\left\{1+O(\mu^2)\right\},\cr
\langle\bar{q}q\rangle\equiv\langle:\bar{q}(0)q(0):\rangle_{\rm vac},\quad q=u,d,s,\dots.\cr}
\equn{(8.4.4)}$$
This is a strong indication 
that $\langle\bar{q}q\rangle\neq 0$ because, in order to ensure that it vanishes, we would 
require $f_\pi=0$ (or very large $O(\mu^2)$ corrections). We also note that we have not 
distinguished in e.g. (8.4.4), between bare ($u$) or renormalized 
($R$) quark masses and operators; 
the distinction is not necessary because $m_q$ and 
 $\langle\bar{q}q\rangle$ acquire opposite renormalization, so that $m_u\langle\bar{q}q\rangle_u=
m_R \langle\bar{q}q\rangle_R$.

We may repeat the derivation of (8.4.4) for 
kaons. We find, to leading order in $m_K^2$,
$$\eqalign{(m_s+m_u)\langle\bar{s}s+\bar{u}u\rangle\simeq&-2f_{K^+}^2m_{K^+}^2,\cr
(m_s+m_d)\langle\bar{s}s+\bar{d}d\rangle\simeq&-2f_{K^0}^2m_{K^0}^2.\cr}\equn{(8.4.5)}$$
We may assume $f_{K^+}=f_{K^0}$ since, in the limit $m_{u,d}^2\ll \Lambdav^2$ they should 
be strictly equal. For the 
same reason, one can take it that the VEVs $\langle\bar{q}q\rangle$ are equal 
for all light quarks. Under these circumstances, we may eliminate the VEVs and 
obtain
$$\dfrac{m_s+m_u}{m_d+m_u}\simeq\dfrac{f_K^2m^2_{K^+}}{f^2_\pi \mu^2},\quad
\dfrac{m_d-m_u}{m_d+m_u}\simeq\dfrac{f_K^2(m^2_{K^0}-m^2_{K^+})}{f^2_\pi \mu^2}.$$
A more careful evaluation requires consideration of electromagnetic contributions 
to the observed $\pi$, $K$ masses (Bijnens, 1993; Donoghue, Holsten and Wyler, 1993) 
and higher order chiral corrections (Kaplan and Manohar, 1986;
 Bijnens, Prades and de~Rafael, 1995).\fnote{The method  originates
 in the work of Glashow and Weinberg (1968a,\/b) and Gell-Mann, Oakes and Renner (1968). In QCD, see 
Weinberg (1978a), Dom\'\i nguez (1978) and Zepeda (1978). Estimates of the 
quark masses essentially agreeing with (8.4.6, 7) below had been obtained even before QCD by
 e.g. Okubo (1969), but nobody knew what to do with them. The first 
evaluation in the context of QCD is due 
to Leutwyler (1974).} In this way we find
$$\dfrac{m_s}{m_d}=18\pm5,\quad \dfrac{m_d}{m_u}=2.0\pm0.4.\equn{(8.4.6)}$$
If we couple this with the phenomenological estimate (from meson and baryon 
spectroscopy) $m_s-m_d\approx 100$ to $200$ \mev, $m_d-m_u\approx 4\,\mev$, we obtain the 
masses (in \mev)
$$\bar{m}_u(Q^2\sim m^2_\rho)\approx 5,\quad
\bar{m}_d(Q^2\sim m^2_\rho)\approx 9,\quad
\bar{m}_s(Q^2\sim m^2_\rho)\approx 190,\equn{(8.4.7)}$$
where the symbol $\approx$ here means that a 50\% error 
would not be very surprising.

This method for obtaining light quark masses is admittedly very rough; 
in the next section we will describe more sophisticated ones.

To conclude this section we make a few comments concerning light quark condensates, 
$\langle\bar{q}q\rangle$. The fact that these do not vanish implies spontaneous breaking of 
chiral symmetry because, under $q\to\gamma_5q$, $\langle\bar{q}q\rangle\to-\langle\bar{q}q\rangle$. 
One may thus wonder whether chiral symmetry would not be restored in 
the limit $m_q\to0$, which would 
imply
$$\langle\bar{q}q\rangle\rightarrowsub_{m_q\to0}0.\equn{(8.4.8)}$$
This possibility is discussed for example by Gasser and Leutwyler (1982). 
The equation (8.4.8) 
is highly unlikely. If it held, one would expect in particular the ratios,
$$\langle\bar{s}s\rangle:\langle\bar{d}d\rangle:\langle\bar{u}u\rangle\sim
190:9:5,$$
which runs contrary to all evidence, from hadron spectroscopy 
to SVZ sum rules which suggest
$$\langle\bar{s}s\rangle\sim\langle\bar{d}d\rangle\sim\langle\bar{u}u\rangle$$
to a few percent. Thus we obtain an extra indication that chiral symmetry is indeed 
spontaneously broken in QCD.

\booksection{8.5. Bounds and estimates of light quark masses in terms of the pion and kaon masses}

\noindent In this section we describe a method for obtaining bounds and estimates 
of light quark masses. The method was first used (to get rough estimates)
 by Vainshtein et al. (1978) and 
further refined by Becchi, Narison, de Rafael and Yndur\'ain (1981),
 Gasser and Leutwyler (1982), etc. One
 starts with the correlator,
$$\eqalign{\Psiv^5_{ij}(q^2)=&\;
\ii\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}\partial\cdot A_{ij}(x)\partial\cdot A_{ij}(0)^{\dag}\rangle_{\rm vac}\cr
=&\;\ii(m_i+m_j)^2\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm T}J^5_{ij}(x)J^5_{ij}(0)^{\dag}\rangle_{\rm vac},\cr}\equn{(8.5.1)}$$
where $A_{ij}^\mu=\bar{q}_i\gamma^\mu\gamma_5q_j$, 
$J^5_{ij}=\bar{q}_i\gamma_5q_j$, $i,j=u,d,s$.

To all orders of perturbation theory, the function
$$F_{ij}(Q^2)=\dfrac{\partial^2}{\partial(q^2)^2}\Psiv^5_{ij}(q^2), \quad Q^2=-q^2,$$
vanishes as $Q^2\to\infty$. Hence, we may write a dispersion relation of the form
$$F_{ij}(Q^2)=
\dfrac{2}{\pi}\int_{m_P^2}^{\infty}\dd t\,\dfrac{\imag\Psiv^5_{ij}(t)}{(t+Q^2)^3},
\quad P=\pi,\,K. 
\equn{(8.5.2)}$$

For large values of $Q^2$, $t$ we may calculate $F_{ij}(Q^2)$, 
$\imag\Psiv^5_{ij}(t)$. The calculation has been improved along the 
years due to increasing precision of the QCD evaluations of
 these quantities.\fnote{Broadhurst (1981) and Chetyrkin et al. (1995) 
for subleading mass corrections; Becchi, Narison, de Rafael and Yndur\'ain (1981), 
 Generalis (1990), Sugurladze and Tkachov (1990), Chetyrkin, Gorishnii and Tkachov (1982), 
Groshny, Kataev, Larin and Sugurladze (1991) and Pascual and de~Rafael (1982) for 
radiative corrections to various terms.} Here, however, we will consider only
 leading effects and first order subleading corrections. We then have, 
$$\eqalign{F_{ij}(Q^2)=&
\,\dfrac{3}{8\pi^2}\,\dfrac{[\bar{m}_i(Q^2)+\bar{m}_j(Q^2)]^2}{Q^2}\cr
&\times\Bigg\{1+\tfrac{11}{3}\dfrac{\alpha_s(Q^2)}{\pi}+
\dfrac{m_i^2+m_j^2+(m_i-m_j)^2}{Q^2}+
\dfrac{2\pi}{3}\,\dfrac{\langle\alpha_sG^2\rangle}{Q^4}\cr
&-\dfrac{16\pi^2}{3Q^4}
\left[\left(m_j-\dfrac{m_i}{2}\right)\langle\bar{q}_iq_i\rangle+
\left(m_i-\dfrac{m_j}{2}\right)\langle\bar{q}_jq_j\rangle\right]\Bigg\}\cr}
\equn{(8.5.3a)}$$
and
$$\imag \Psiv^5_{ij}(t)
=\dfrac{3[\bar{m}_i(t)+\bar{m}_j(t)]^2}{8\pi}
\left\{\left[1+\tfrac{17}{3}\,\dfrac{\alpha_s(t)}{\pi}\right]t-
(m_i-m_j)^2\right\}.\equn{(8.5.3b)}$$
The contributions containing the condensates are easily evaluated 
taking into account the nonperturbative parts of the quark and gluon 
propagators. The quantities $m_i\langle\bar{q}_jq_j\rangle$ 
may be reexpressed in terms of experimentally known quantities, $f_{K,\pi}$, $m_{K,\pi}$ 
as in (8.4.4, 5). For the case 
$ij=ud$, which is the one we will consider in more detail, their 
contribution is negligible, as are the terms of order $m^2/Q^2$ in \equs~(8.5.3).
 We will henceforth 
neglect these quantities. 
Because one can write the imaginary part of the spectral function as
$$\imag\Psiv_{ij}^5(t)=\tfrac{1}{2}
\sum_{\Gammav}\left|\langle{\rm vac}|\partial^{\mu}A^{ij}_{\mu}(0)|\Gammav\rangle\right|^2
(2\pi)^4\delta_4(q-p_{\Gammav})$$
it follows that $\imag\Psiv_{ij}^5(t)\geq 0$: it is this
 positivity that will allow us to derive 
quite general bounds. To obtain tight ones it is important to use the information contained in 
both \equs~(8.5.3a,b); to this end, we define the function
$$\eqalign{\varphi_{ij}(Q^2)=&\;F_{ij}(Q^2)-\int_{Q^2}^{\infty}\dd t\,
\dfrac{1}{(t+Q^2)^3}\,\dfrac{2\imag\Psiv^5_{ij}(t)}{\pi}\cr
=&\int_{m_P^2}^{Q^2}\dd t\,\dfrac{1}{(t+Q^2)^3}\,\dfrac{2\imag\Psiv^5_{ij}(t)}{\pi}.\cr}
\equn{(8.5.4)}$$
For sufficiently large $Q^2$ we may use (8.5.3) and integrate the imaginary part
 to obtain, for $ij=ud$,
$$\eqalign{\varphi_{ud}(Q^2)=\dfrac{3}{8\pi^2}
\left\{\dfrac{[\bar{m}_u(Q^2)+\bar{m}_d(Q^2)]^2}{Q^2}
\left[\tfrac{1}{4}+\left(\tfrac{5}{12}+2\log 2\right)\dfrac{\alpha_s}{\pi}\right]\right.\cr
\left. +\,\dfrac{1}{3Q^6}\left[8\pi^2f^2_\pi \mu^2+
2\pi\langle\alpha_sG^2\rangle\right]\right\},\cr}\equn{(8.5.5a)}$$
and to this accuracy the two loop expression  for the running masses is to be used.
For the $ij=us,ds$ cases, and neglecting $m_{u,d}/m_s$,
$$\eqalign{\varphi_{us,ds}(Q^2)=&\,\dfrac{3}{8\pi^2}
\Bigg\{\dfrac{\bar{m}_s^2}{Q^2}
\left[\tfrac{1}{4}+\left(\tfrac{5}{12}+2\log 2\right)\dfrac{\alpha_s}{\pi}\right]\cr
-&\,\dfrac{2\bar{m}_s^4}{Q^4}\left[\tfrac{3}{4}+\left(6+4\log 2\right)\dfrac{\alpha_s}{\pi}\right]
 +\dfrac{1}{3Q^6}\left[8\pi^2f_K^2m_K^2+
2\pi\langle\alpha_sG^2\rangle\right]\Bigg\}.\cr
}\equn{(8.5.5b)}$$
We can extract the pion (or kaon, as the case may be) pole explicitly from 
the low energy dispersive integral in (8.5.4) thus getting 
 for e.g., $\varphi_{ud}$
$$\varphi_{ud}(Q^2)=\dfrac{4f^2_\pi \mu^4}{(\mu^2+Q^2)^3}+
\int_{t_0}^{Q^2}\dd t\,\dfrac{1}{(t+Q^2)^3}\,\dfrac{2\imag\Psiv^5_{ij}(t)}{\pi};
\equn{(8.5.6)}$$
the continuum threshold $t_0$ is  $3\mu^2$ for
 $ij=ud$ or $(m_K+2\mu)^2$ for $ij=(u,d)s$. Because of the positivity of 
$\imag \Psiv$ this immediately gives bounds on $m_i(Q^2)+m_j(Q^2)$ as soon 
as $Q^2\geq Q_0^2$, where $Q_0^2$ is a 
momentum  large enough for the QCD estimates (8.5.5) to be valid: thus,
to leading order,
$$\bar{m}_u(Q_0^2)+\bar{m}_d(Q_0^2)\geq\left\{\dfrac{2^7\pi^2f^2_\pi \mu^4}{3}\,
\dfrac{Q_0^2}{(Q_0^2+\mu^2)^3}\right\}^{\frac{1}{2}};\equn{(8.5.7a)}$$
for the combination $us$,
$$\bar{m}_s(Q_0^2)\geq\left\{\dfrac{2^7\pi^2f^2_K m^4_K}{3}\,
\dfrac{Q_0^2}{(Q_0^2+m^2_K)^3}\right\}^{\frac{1}{2}}.\equn{(8.5.7b)}$$
The bound depends a lot on the value of $Q_0^2$. We find, for example, the bounds
$$\eqalign{\bar{m}_u(1\,\gev^2)+\bar{m}_d(1\,\gev^2)\geq&\, 13\,\mev,\quad Q^2_0=1.75\;\gev^2,\cr
\bar{m}_u(1\,\gev^2)+\bar{m}_d(1\,\gev^2)\geq&\, 7\;\mev,\quad Q^2_0=3.5\;\gev^2\cr}
\equn{(8.5.8a)}$$
and
$$\eqalign{\bar{m}_s(1\,\gev^2)\geq&\, 245\;\mev,\quad Q^2_0=1.75\;\gev^2,\cr
\bar{m}_s(1\,\gev^2)\geq&\, 150\;\mev,\quad Q^2_0=3.5\;\gev^2.\cr}
\equn{(8.5.8b)}$$
As is customary, we have translated the 
bounds (as we will also do for the estimates later on) 
to bounds on the running masses defined at 1 \gev.
The bounds can be stabilized somewhat by considering derivatives of $F^5_{ij}$, 
but (8.5.8) do not change much.

To get {\sl estimates} for the masses, a model is necessary for the 
low energy piece of the dispersive integral (8.5.6). At very low energy, one can calculate 
$\imag \Psiv^5$ using chiral perturbation theory (see for example Pagels and Zepeda, 1972; 
Gasser and Leutwyler, 1982); the contribution is minute. The important region is that where 
the quasi-two body channels are open, the $(\rho,\omega)-\pi$ channels for the 
$ud$ case. This is expected to be dominated by the $\pi'$ resonance, with a mass of 
$1.2\,\gev$. One can take the residue of the resonance as a free 
parameter, and fit the QCD expression (8.5.5). This is the procedure 
followed by Narison and de~Rafael (1981), Hubschmid and Mallik (1981), 
Gasser and Leutwyler (1982), Kataev, Krasnikov and 
Pivovarov (1983),  Dom\'\i nguez and de~Rafael (1987), 
Chetyrkin, Pirjol and Schilcher (1997), etc. The errors 
one finds in the literature are many times {\sl overoptimistic} because they do not 
take into account the important matter of the value $Q_0^2$ at which the 
perturbative QCD evaluation is supposed to be valid (Yndur\'ain, 1998).
 Now, as is clear from 
\equ~(8.5.5), the radiative corrections feature a large coefficient, so 
it is difficult to estimate reliably a  figure  for $Q_0^2$. Both bounds (as shown above) and 
estimates will depend on this. As reasonably safe estimates we may quote the values
$$\eqalign{\bar{m}_u(Q^2=1\,\gev^2)=&\,4.2\pm 2\;\mev,\cr
\bar{m}_d(Q^2=1\,\gev^2)=&\,8.9\pm 4.3\;\mev,\cr
\bar{m}_s(Q^2=1\,\gev^2)=&\,200\pm 50\;\mev,\cr}\equn{(8.5.9a)}$$
and we have, to reduce  the errors a bit, taken also into 
account the chiral theory estimates of the 
mass ratios given in the previous section, \equ~(8.4.6).

For the $s$ quark,  independent estimates (Chen et al., 2001, Narison, 1995) following from 
$$\tau\to\nu_\tau+\;\hbox{strange 
particles},\quad e^+e^-\to\;\hbox{strange 
particles}$$
give slightly smaller numbers. 
Taking them into account we may write
$$\bar{m}_s(Q^2=1\,\gev^2)=183\pm 30\;\mev.\equn{(8.5.9b)}$$


\booksection{8.6. The triangle anomaly; $\pi^0$ decay. The gluon anomaly. The $U(1)$ problem}
\vskip-0.5truecm
\booksubsection{8.6.3. The triange anomaly and the $\pi^0$ decay}

\noindent
 Historically, one of the first motivations for the
 colour degree of freedom came from the study of the 
decay $\pi^0\to \gamma\gamma$, which we now consider in  some detail.

The amplitude for the process $\pi^0\to\gamma\gamma$ may be written, using the 
reduction formulas, as
$$\eqalign{\langle\gamma(k_1,\lambda_1),\gamma(k_2,\lambda_2)|S|\pi^0(q)\rangle
=\dfrac{-\ii e^2}{(2\pi)^{9/2}}\,
\epsilon^*_{\mu}(k_1,\lambda_1)\epsilon^*_{\nu}(k_2,\lambda_2)\cr
\times\int\dd^4x_1\,\dd^4x_2\,\dd^4z\,\ee^{\ii(x_1\cdot k_1+x_2\cdot k_2-z\cdot q)}
(\dal_z+\mu^2)\langle{\rm T}J^{\mu}_{\rm em}(x_1) J^{\nu}_{\rm em}(x_2)\phi_{\pi^0}(z)\rangle_0,\cr}
\equn{(8.6.1)}$$
and we have used the relation $\dal A^{\mu}_{\rm ph}(x)=J^{\mu}_{\rm em}(x)$,
 with $ A^{\mu}_{\rm ph}$ the photon field. We leave it as an exercise for the 
reader to check this, as well as  to verify that, in our particular case, one can replace
$$\dal_{x_1}\dal_{x_2}{\rm T}\{ A^{\mu}_{\rm ph}(x_1) A^{\nu}_{\rm ph}(x_2)\phi_{\pi^0}(z)\}
\to{\rm T}\{(\dal A^{\mu}_{\rm ph}(x_1))(\dal A^{\nu}_{\rm ph}(x_2))\phi_{\pi^0}(z)\},
$$
i.e., that potential delta function terms that appear 
when the derivatives in the d'Alembertians act on the 
theta functions $\theta(x_1-z),\dots$ implicit in the T-product make no 
contribution. Separating off the delta of four-momentum 
conservation, we then find
$$F\left(\pi^0\to\gamma(k_1,\lambda_1),\gamma(k_2,\lambda_2)\right)=
\dfrac{e^2(q^2-\mu^2)}{\sqrt{2\pi}}\epsilon^*_{\mu}(k_1,\lambda_1)\epsilon^*_{\nu}(k_2,\lambda_2)
F^{\mu\nu}(k_1,k_2),\quad q=k_1+k_2,\equn{(8.6.2a)}$$
where we have defined the VEV
$$F^{\mu\nu}(k_1,k_2)=\int\dd^4x\,\dd^4y\,\ee^{\ii(x\cdot k_1+y\cdot k_2)}
\langle{\rm T}J^{\mu}(x)J^{\nu}(y)\phi(0)\rangle_0.\equn{(8.6.2b)}$$
We henceforth suppress the indices ``em" and ``$\pi^0$" in $J$ and $\phi$ respectively.

We next use the equation (8.4.3), generalized to include the $\pi^0$:
$$\eqalign{\partial_\mu A_0^\mu(x)=\sqrt{2}f_\pi \mu^2\phi(x),\quad
\phi\equiv\phi_{\pi^0},\cr    
A_0^{\mu}(x)=\dfrac{1}{\sqrt{2}}\left\{\bar{u}(x)\gamma^\mu\gamma_5u(x)-
\bar{d}(x)\gamma^\mu\gamma_5d(x)\right\}.\cr}\equn{(8.6.3a)}$$
It will prove convenient to use, instead of $A_0$, the current $A_3$, defined as
$$A_3^{\mu}(x)=\left\{\bar{u}(x)\gamma^\mu\gamma_5u(x)-
\bar{d}(x)\gamma^\mu\gamma_5d(x)\right\};\equn{(8.6.3b)}$$
with  it, we write
$$\eqalign{F^{\mu\nu}(k_1,k_2)=&\,\dfrac{1}{f_\pi \mu^2}T^{\mu\nu}(k_1,k_2),\cr  
T^{\mu\nu}(k_1,k_2)=&\,\tfrac{1}{2}
\int\dd^4x\,\dd^4y\,\ee^{\ii(x\cdot k_1+y\cdot k_2)}
\langle{\rm T}J^{\mu}(x)J^{\nu}(y)\partial\cdot A_3(0)\rangle_0.\cr}
\equn{(8.6.4)}$$

Up to this point, everything has been exact. The next step involves using the PCAC 
hypothesis in the following form: we assume that $F(\pi\to\gamma\gamma)$ can 
be approximated by its leading term in the limit $q\to0$. On purely 
kinematic grounds, this is seen to imply that also $k_1,k_2\to0$. One may write
$$T^{\mu\nu}(k_1,k_2)=\epsilon^{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta}\Phiv+O(k^3).
\equn{(8.6.5)}$$
The PCAC hypothesis means that we retain only the first
 term in \equ~(8.6.5). As will be seen presently, this 
will lead us to a contradiction, the resolution of which will involve introducing the so-called 
{\sl axial}, or {\sl triangle anomaly},
 and will allow us actually to calculate $T^{\mu\nu}$ exactly to all orders 
of perturbation theory (in the PCAC approximation).

The first step is to consider the quantity
$$R^{\lambda\mu\nu}(k_1,k_2)=
\ii\int\dd^4x\,\dd^4y\,\ee^{\ii(x\cdot k_1+y\cdot k_2)}
\langle{\rm T}J^\mu(x)J^\nu(y)A_3^\lambda(0)\rangle_0.\equn{(8.6.6)}$$
On invariance grounds, we may write the general decomposition,
$$R^{\lambda\mu\nu}(k_1,k_2)=\epsilon^{\mu\nu\lambda\alpha}k_{1\alpha}\Phiv_1+
\epsilon^{\mu\nu\lambda\alpha}k_{2\alpha}\Phiv_2+O(k^3),\equn{(8.6.7)}$$
where the $O(k^3)$ terms are of the form
$$\epsilon^{\mu\lambda\alpha\beta}k_{i\alpha}k_{j\beta}k_{l\lambda}\Phiv_{ijl}+
\hbox{ permutations of}\;i,j,l=1,2,3,$$
and, for quarks with nonzero mass, the $\Phiv$ are regular as $k_i\to0$. 

The conservation of the e.m. current, $\partial\cdot J=0$, yields two equations:
$$k_{1\mu}R^{\mu\nu\lambda}=k_{2\nu}R^{\mu\nu\lambda}=0.\equn{(8.6.8)}$$
The first implies
$$\Phiv_2=O(k^2);\equn{(8.6.9a)}$$
the second gives
$$\Phiv_1=O(k^2).\equn{(8.6.9b)}$$
Now we have, from (8.6.4) and (8.6.6),
$$q_\lambda R^{\lambda\mu\nu}(k_1,k_2)=T^{\mu\nu}(k_1,k_2),\quad {\rm i.e.},\;\Phiv=\Phiv_2-\Phiv_1,
\equn{(8.6.10)}$$
and hence we find the result of Veltman (1967) and Sutherland (1967),
$$\Phiv=O(k^2).\equn{(8.6.11)}$$
Because the scale for $k$ is $\mu$, this means that $\Phiv$ should be of order $\mu^2/M^2$, 
where $M$ is a typical hadronic mass. Thus, we expect that $\Phiv$ would be 
vanishing in the chiral limit, and hence very small in the real world. Now, this is in 
disagreement with experiment, as the decay $\pi^0\to2\gamma$
 is in no way suppressed; but worse still, 
(8.6.11) contradicts a direct calculation. In fact, we may use the equations of motion and write
$$\partial_\mu A_3^\mu(x)=
2\ii\left\{m_u\bar{u}(x)\gamma_5u(x)-m_d\bar{d}(x)\gamma_5d(x)\right\}.\equn{(8.6.12)}$$
We will calculate first neglecting strong interactions; (8.6.11) should certainly 
be valid in this approximation. This involves the diagrams of \fig~8.6.1 with a $\gamma_5$ 
vertex. The result, as first obtained by Steinberger (1949) is, in the limit $k_1,k_2\to0$,  
and defining $\delta_u=1$, $\delta_d=-1$,

\topinsert{
\setbox0=\vbox{\hsize 12.4cm\epsfxsize 10.6cm\epsfbox{7_7_1.eps}\hfil}
\centerline{{\box0}}
\setbox1=\vbox{\hsize 11.1cm\captiontype\figurasc{Fig. 8.6.1. }
 {Diagrams connected with the anomaly ($\pi^0\to\gamma\gamma$ decay).}\hb
\vskip.1cm}
\centerline{\box1}
}\endinsert

$$\eqalign{T^{\mu\nu}(k_1,k_2)&=2N_c\sum_{f=u,d}\delta_fQ^2_fm_f\cr
\times&\int\dfrac{\dd^4p}{(2\pi)^4}\,
\dfrac{\trace\gamma_5(\slash{p}+\slash{k}_1+m_f)\gamma^\mu
(\slash{p}+m_f)\gamma^\nu(\slash{p}-\slash{k}_2+m_f)}{[(p+k_1)^2-m_f^2][(p-k_2)^2-m_f^2]
(p^2-m_f^2)}\cr
=&-\dfrac{1}{4\pi^2}
\epsilon^{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta}\left\{3(Q_u^2-Q_d^2)\right\}+O(k^4)\cr
=&-\dfrac{1}{4\pi^2}
\epsilon^{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta}+O(k^4).
\cr}$$
The factor $N_c=3$ comes from the sum over the three colours of the 
quarks and the factor 2
 from the two diagrams in \fig~8.6.1 (which in fact contribute equally to the amplitude). We 
thus find that
$$\Phiv=-\dfrac{1}{4\pi},\equn{(8.6.13)}$$
which contradicts (8.6.11). This is the {\sl triangle anomaly} (Bell and Jackiw, 1969; Adler, 1969).

What is wrong here? Clearly, we cannot maintain (8.6.12), which was 
obtained with free-field equations of motion, $\ii\slash{\partial}q=m_qq$; 
we must admit that in the presence of interactions with vector fields (the 
photon field in our case), \equ~(8.6.12) is no longer valid. To 
obtain agreement with (8.6.13) we have to write (Adler, 1969)
$$\eqalign{\partial_\mu A_3^\mu(x)=&\,
2\ii\left\{m_u\bar{u}(x)\gamma_5u(x)-m_d\bar{d}(x)\gamma_5d(x)\right\}\cr
&+N_c(Q_u^2-Q_d^2)\dfrac{e^2}{16\pi^2}F_{\mu\nu}(x)\widetilde{F}^{\mu\nu}(x),\cr}\equn{(8.6.14)}$$
where the {\sl dual} $\widetilde{F}$ has been defined as
$$\widetilde{F}^{\mu\nu}=\tfrac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta},\quad 
F^{\mu\nu}=\partial^\mu A^{\nu}_{\rm ph}-\partial^\nu A^{\mu}_{\rm ph}.$$
More generally, for fermion fields interacting with vector fields with strength $h$, 
we find
$$\partial_\mu\bar{f}\gamma^\mu\gamma_5f=
2\ii m_f\bar{f}\gamma_5f+\dfrac{T_Fh^2}{8\pi^2}H^{\mu\nu}\widetilde{H}_{\mu\nu};\equn{(8.6.15)}$$
$H^{\mu\nu}$ is the vector field strength tensor.

Let us return to the decay $\pi^0\to 2\gamma$. From (8.6.13) we calculate
 the amplitude, in the PCAC limit 
$\mu\sim0$,
$$F(\pi^0\to2\gamma)=\dfrac{\alpha}{\pi}\,
\dfrac{\epsilon^{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta}\epsilon^*_\mu(k_1,\lambda_1)
\epsilon^*_\mu(k_2,\lambda_2)}{\sqrt{2\pi}},
\equn{(8.6.16)}$$
and the decay rate
$$\Gammav(\pi^0\to2\gamma)=\left(\dfrac{\alpha}{\pi}\right)^2\dfrac{m^3_\pi}{64\pi f^2_\pi}
=7.25\times10^{-6}\,\mev,$$
to be compared with the experimental figure,
$$\Gammav_{\rm exp}(\pi^0\to2\gamma)=7.95\times10^{-6}\,\mev.$$
Actually, the sign of the decay amplitude can also be measured (from the Primakoff effect) 
and it agrees with the theory. It is important to note that, if we had no colour,
our result would have decreased by a factor $1/N_c^2$, i.e., it would have been 
off experiment by a full order of magnitude.

One may wonder what credibility to attach to this calculation: after all,
 it was made to zero order in $\alpha_s$. In fact, the calculation is exact to all 
orders in QCD;\fnote{The proof is essentially contained in the original paper of 
Adler and Bardeen (1969). See also Wilson (1969), 
Crewther (1972) and Bardeen (1974).} the only approximation is the PCAC one 
$\mu\approx0$. To show this we will give an alternate
 derivation of the basic result, \equ~(8.6.13). We then return to (8.6.6). To zero 
order in $\alpha_s$,
$$\eqalign{R^{\mu\nu\lambda}=&\sum\delta_fQ^2_f\cr
\times&\int\dfrac{\dd^4p}{(2\pi)^4}\,
\dfrac{\trace\gamma^\lambda\gamma_5(\slash{p}+\slash{k}_1+m_f)\gamma^\mu
(\slash{p}+m_f)\gamma^\nu(\slash{p}-\slash{k}_2+m_f)}{[(p+k_1)^2-m_f^2][(p-k_2)^2-m_f^2]
(p^2-m_f^2)}\cr
+&\;\hbox{crossed term}\cr}$$
(\fig~8.6.1 with the $\gamma^\lambda\gamma_5$ vertices). More generally, we 
regulate the integral by working in dimension $D$, and consider 
an arbitrary axial triangle with
$$R^{\mu\nu\lambda}_{ijl}=2\int\dfrac{\dd^Dp}{(2\pi)^D}\,
\trace\gamma^\lambda\gamma_5\dfrac{1}{\slash{p}+\slash{k}_1-m_i}\gamma^\mu
\dfrac{1}{\slash{p}-m_j}\gamma^\nu\dfrac{1}{\slash{p}-\slash{k}_2-m_l}.\equn{(8.6.17)}$$
We would like to calculate $q_\lambda R^{\mu\nu\lambda}_{ijl}$. Writing identically
$$(\slash{k}_1+\slash{k}_2)\gamma_5=
-(\slash{p}-\slash{k}_2-m_l)\gamma_5+(\slash{p}+\slash{k}_1+m_i)\gamma_5-
(m_i+m_l)\gamma_5,$$
we have
$$\eqalign{q_\lambda R^{\mu\nu\lambda}_{ijl}=&\,-2(m_i+m_l)\cr
\times&\int\dfrac{\dd^Dp}{(2\pi)^D}\,
\dfrac{\trace\gamma_5(\slash{p}+\slash{k}_1+m_i)\gamma^\mu
(\slash{p}+m_j)\gamma^\nu(\slash{p}-\slash{k}_2+m_l)}{[(p+k_1)^2-m_i^2][(p-k_2)^2-m_l^2]
(p^2-m_j^2)}\cr
&+a^{\mu\nu\lambda}_{ijl},\cr}\equn{(8.6.18a)}$$
$$\eqalign{a^{\mu\nu\lambda}_{ijl}=-2&\int\dd \hat{p}\,
\trace\left\{(\slash{p}-\slash{k}_2-m_l)\gamma_5-(\slash{p}+\slash{k}_1+m_i)\gamma_5\right\}\cr
&\times\dfrac{1}{\slash{p}+\slash{k}_1-m_i}\gamma^\mu\,\dfrac{1}{\slash{p}-m_j}\gamma^\nu
\dfrac{1}{\slash{p}-\slash{k}_2-m_l}.\cr}\equn{(8.6.18b)}$$
The first term on the right hand side of (8.6.18a) is what 
we would have obtained by naive use of the equations of motion, 
$\partial_\mu\bar{q}_i\gamma^\mu\gamma_5q_l=\ii(m_i+m_l)\bar{q}_i\gamma_5q_l$; 
$a^{\mu\nu\lambda}_{ijl}$ is the anomaly. If we accepted the commutation
 relations $\{\gamma^\mu,\gamma_5\}=0$ also for dimension $D\neq4$, we could rewrite it as
$$\eqalign{a^{\mu\nu\lambda}_{ijl}=-2\int\dd\hat{p}\,\Bigg\{&\trace\gamma_5
\dfrac{1}{\slash{p}+\slash{k}_1-m_i}\gamma^\mu\dfrac{1}{\slash{p}-m_l}\gamma^\nu\cr
+&\trace\gamma_5\gamma^\mu\dfrac{1}{\slash{p}-m_j}\gamma^\mu
\dfrac{1}{\slash{p}-\slash{k}_2-m_l}\Bigg\}.\cr}\equn{(8.6.18c)}$$
Then we could conclude that $a^{\mu\nu\lambda}_{ijl}$ vanishes 
because each of the terms in (8.6.18c) consists of an
 antisymmetric tensor that depends on a single vector ($k_1$ for 
the first term, $k_2$ for the second) and this is zero. It is thus 
clear that the nonvanishing of $a^{\mu\nu\lambda}_{ijl}$ is due to the fact that 
it is given by an ultraviolet divergent integral: if it was convergent, 
one could take $D\to4$ and $a^{\mu\nu\lambda}_{ijl}$ would vanish. Incidentally, 
this shows that $a^{\mu\nu\lambda}_{ijl}$ is actually independent of the masses 
because $(\partial/\partial m)a^{\mu\nu\lambda}_{ijl}$ is convergent, 
and thus the former argument applies. We may therefore write $a^{\mu\nu\lambda}_{ijl}=a^{\mu\nu}$, 
where $a^{\mu\nu}$ is obtained by setting all masses to zero. A similar 
argument shows that $a^{\mu\nu}$ has to be of the form
$$a^{\mu\nu}(k_1,k_2)=a\epsilon^{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta},\quad a={\rm constant},
\equn{(8.6.19a)}$$
and thus we may obtain $a$ as
$$a\epsilon^{\mu\nu\alpha\beta}=
\dfrac{\partial^2}{\partial k_{1\alpha}\partial k_{2\beta}}a^{\mu\nu}(k_1,k_2)\Big|_{k_i=0}.
\equn{(8.6.19b)}$$ 
If we could write the formula (8.6.18c) for $a$, we would immediately conclude from 
(8.6.19b) that $a=0$, in contradiction with the Veltman--Sutherland theorem.
 But this is easily seen to be inconsistent: if we would have shifted 
variables in (8.6.18c), say $p\to p-\xi k_2$, we would have found a finite but 
nonzero value, actually $\xi$-dependent  for $a$, $a=-\xi/2\pi^2$. This shows
 that the commutation relations\fnote{These commutation relations are 
actually self-contradictory. For example, using only the commutation relations of 
the $\gamma_\mu$, $\mu=0,\dots,D-1$ for $D\neq4$, we have
$$\trace \gamma_5\gamma^\alpha\gamma^\mu\gamma^\nu\gamma^\rho\gamma_\alpha\gamma^\sigma=
(6-D)\trace\gamma_5\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma,$$
while, if we allow $\gamma_5$ anticommutation, we can obtain
$$\trace\gamma_5\gamma^\alpha\gamma^\mu\gamma^\nu\gamma^\rho\gamma_\alpha\gamma^\sigma=
-\trace\gamma_5\gamma^\mu\gamma^\nu\gamma^\rho\gamma_\alpha\gamma^\sigma\gamma^\alpha=(D-2)
\trace\gamma_5\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma,$$
which differs from the former by a term $O(D-4)$. These problems, however, only arise for 
arrays with an odd number of $\gamma_5$ and at least four other 
gammas.} 
$\{\gamma^\mu,\gamma_5\}=0$ cannot be accepted for $D\neq0$, for 
they lead to an undefined value for the anomaly. If, however,
 we start from (8.6.18b) and refrain from commuting $\gamma_5$ and $\gamma^\mu$s, 
$$a\epsilon^{\mu\nu\alpha\beta}=-2\int\dd\hat{p}\trace\gamma_5
\left\{\dfrac{1}{\slash{p}}\gamma^\alpha
\dfrac{1}{\slash{p}}\gamma^\mu
\dfrac{1}{\slash{p}}\gamma^\nu
\dfrac{1}{\slash{p}}\gamma^\beta-
\dfrac{1}{\slash{p}}\gamma^\mu
\dfrac{1}{\slash{p}}\gamma^\nu
\dfrac{1}{\slash{p}}\gamma^\beta
\dfrac{1}{\slash{p}}\gamma^\alpha\right\}.$$
Performing symmetric integration and using only the rules 
 for $D\neq4$, we obtain an unambiguous result:
$$\eqalign{a\epsilon^{\mu\nu\alpha\beta}=&\,
\dfrac{8(D-1)(4-D)}{D(D+2)}\,\dfrac{\ii}{16\pi^2}\dfrac{2}{4-D}\,
\trace\gamma_5\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta+O(4-D)\cr
&\rightarrowsub_{D\to4}-\dfrac{1}{2\pi^2}\epsilon^{\mu\nu\alpha\beta}.\cr}$$
This is one of the peculiarities of the anomaly: a {\sl finite} Feynman 
integral whose value depends on the regularization prescription. Fortunately,
 we may eschew the problem by using the Veltman--Sutherland theorem to conclude 
that, at any rate, there is a {\sl unique} value of $a^{\mu\nu}$ 
compatible with gauge invariance for the e.m. current, viz.,
$$a^{\mu\nu}_{ijl}=a^{\mu\nu}=
-\dfrac{1}{2\pi^2}\epsilon^{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta}.\equn{(8.6.20)}$$
We have explicitly checked that our regularization leads to 
precisely this value; to verify that it also respects gauge invariance is left as as simple exercise.

Before continuing, a few words on the Veltman--Sutherland theorem for zero quark 
masses are necessary. In this case, the first term on the 
right hand side of (8.6.18a) is absent: it would appear that we could not maintain our result for the anomaly, 
\equ~(8.6.20), because this would imply
$$q_\lambda R^{\mu\nu\lambda}_{ijl} =
-\dfrac{1}{2\pi^2}\epsilon^{\mu\nu\alpha\beta}k_{1\alpha}k_{2\beta}\neq0,$$
thus contradicting the Veltman--Sutherland conclusion, $q_\lambda R^{\mu\nu\lambda}_{ijl}=0$. 
This is not so. The relation $q_\lambda R^{\mu\nu\lambda}_{ijl}=a^{\mu\nu}$ 
and the value of $a^{\mu\nu}$ {\sl are} correct. What occurs is that 
for vanishing masses the functions $\Phiv_i$ in (8.6.7) possess singularities of the type 
$1/k_1\cdot k_2$, singularities coming from the denominators in, 
for example, \equ~(8.6.17) 
when $m_i=0$. Therefore, the 
Veltman--Sutherland theorem is {\sl not} applicable. This is yet another 
peculiarity of the anomalous triangle: we have the relation 
$$\lim_{m\to0}q_\lambda R^{\mu\nu\lambda}_{ijl}=0$$
 but, 
if we begin with $m=0$,
$$q_\lambda R^{\mu\nu\lambda}_{m\equiv0}=a^{\mu\nu}\neq0.$$
\topinsert{
\setbox0=\vbox{\hsize 12.1truecm{\epsfxsize11.truecm\epsfbox{7_7_2.eps}}\hfil}
\centerline{{\box0}}
\setbox1=\vbox{\hsize 11.1truecm\captiontype\figurasc{Fig. 8.6.3. }
 {(a) A nonanomalous diagram. (b) ``Opened" diagram\hb 
corresponding to (a).}\hb
\vskip.4cm}
\centerline{\box1}
\vskip0.4cm
}\endinsert

Let us return to our original discussion, in particular for $m\neq0$. 
The present method shows how one can prove that the result does not get renormalized. 
The Veltman--Sutherland theorem is exact; so we have actually shown that 
it is sufficient to prove that (8.6.20) is not altered by higher orders in $\alpha_s$.
 Now, consider a typical 
higher order contribution (\fig~8.6.2a). It may be written as an integral 
over the gluon momenta and an integral over the quark momenta. But for the latter, the 
triangle has become an hexagon (\fig~8.6.2b) for 
which the quark integral is convergent and here the limit $D\to4$ 
may be taken: it vanishes identically. In addition, the above  
arguments have shown that the anomaly is in fact related 
to the large momentum behaviour of the theory and thus we 
expect that the exactness of (8.6.13) will not be spoiled by nonperturbative effects.
We will not make the proof more precise, but refer to the literature.\fnote{For 
a detailed discussion, see the reviews of Adler (1971) and Ellis (1976). 
The triangle graph is the only one that has {\sl primitive} anomalies; it does, 
however, induce secondary anomalies 
in square and pentagon graphs. The triangle with three axial currents has
 an anomaly closely related to the one we have discussed, cf. the text of Taylor (1976).
 An elegant discussion 
of currents with anomalies for arbitrary interaction may be found in Wess and Zumino (1971).
The derivation of the anomaly in the context of the path integral formulation of field 
theory, where it is connected with the {\sl divergence} of the measure, 
may be found in  Fujikawa (1980,~1984,~1985).}



\booksubsection{8.6.2. The $U(1)$ problem and the gluon anomaly}

\noindent In the previous section, we discussed the triangle anomaly
 in connection with the decay $\pi^0\to\gamma\gamma$. 
As remarked there, the anomaly is not restricted to photons; in particular, we 
have a gluon anomaly. 
Although this lies outside the scope of the present review, we will say a few words on the subject.
 Defining the current
$$A_0^\mu=\sum_{f=1}^n\bar{q}_f\gamma^\mu\gamma_5q_f,\equn{(8.6.23)}$$
we find that it has an anomaly
$$\partial^\mu A_0^\mu=\ii\sum_{f=1}^n2m_f\bar{q}_f\gamma_5q_f+
\dfrac{ng^2}{16\pi^2}\widetilde{G}G,\equn{(8.6.24)}$$
where
$$\widetilde{G}_a^{\mu\nu}\equiv \tfrac{1}{2}\epsilon^{\mu\nu\alpha\beta}G_{a\alpha\beta},
\quad \widetilde{G}G=\sum_a\widetilde{G}_a^{\mu\nu}G_{a\mu\nu}.$$
The current (8.6.23) is the so-called $U(1)$ current (pure flavour singlet) and is 
atypical in more respects than one. In particular, it is associated with the 
$U(1)$ problem, to which we now turn.

Assume that we have $n$ light quarks; we only consider these and will  neglect 
(as irrelevant to the problem at hand) the existence of heavy flavours.
 We may take $n=2$ ($u,d$) and then we speak of 
``the $U(1)$ problem of $SU(2)$" or $n=3$ ($u,d,s$), which is the $SU(3)$ $U(1)$ problem. 
 Consider now the $n^2-1$ matrices in flavour space $\lambda_1,\dots,\lambda_{n^2-1}$;
for $SU(3)$ they coincide 
with the Gell-Mann matrices, and for $SU(2)$ with the Pauli matrices. 
 Define further $\lambda_0\equiv1$. Any 
$n\times n$ Hermitian matrix may be written as a linear combination of the $n^2$ matrices
 $\lambda_\alpha,\;\alpha=0,1,\dots,n^2-1$.
 Because of this completeness, it is sufficient to consider 
the currents
$$A_{\alpha}^{\mu}=\sum_{ff'}\bar{q}_f\gamma^\mu\gamma_5\lambda^{\alpha}_{ff'}q_{f'};
\quad \alpha=0,1,\dots,n^2-1.$$
Of course, only $A_0$ has an anomaly. 

Now let $N_1(x),\dots,N_k(x)$ 
denote local operators (simple or composite) and 
consider the quantity
$$\langle{\rm vac}|{\rm T}A_{\alpha}^{\mu}(x)\prod_jN_j(x_j)|{\rm vac}\rangle.
\equn{(8.6.25)}$$
For $\alpha=a\neq0$, the Goldstone theorem implies that the masses of the pseudoscalar particles 
$P_a$ with the quantum numbers of the $A_a$ vanish in the chiral limit; introducing a common 
parameter $\epsilon$ for all the quark masses by letting $m_f=\epsilon r_f$, 
$f=1,\dots,n$, where the 
$r_f$ remain fixed in the chiral limit, we have
$$m^2_{P_a}\approx \epsilon.\equn{(8.6.26)}$$
Therefore, 
in this limit, the quantity (8.6.25) develops a pole at $q^2=0$, for $\alpha=a\neq0$. 
To be precise, what this means is that in the chiral limit (zero quark masses),
$$\lim_{q\to0}\int\dd^4x\,\ee^{\ii q\cdot x}
\langle{\rm vac}|{\rm T}A_{\alpha}^{\mu}(x)\prod_jN_j(x_j)|{\rm vac}\rangle
\approx \hbox{(const.)}\times\,q^\mu \dfrac{1}{q^2}.$$
If we neglect anomalies, the derivation of (8.6.26) can be repeated for the case 
$\alpha=0$ and we would thus find that the $U(1)$ (flavour singlet)
 particle $P_0$ would also have vanishing mass in the chiral limit (Glashow, 1968). This statement 
was made more precise by Weinberg (1975) who 
proved the bound $m_{P_0}\leq\sqrt{n}\times({\rm average}\;m_{P_a})$.
 Now, this is a catastrophe since, for the $SU(2)$ case,
$m_\eta\gg\sqrt{2}\,\mu$ and, for 
$SU(3)$, the mass of the $\eta'$ particle also violates the bound. This is the 
$U(1)$ problem. In addition, 
Brandt and Preparata (1970) proved that under these conditions the decay 
$\eta\to3\pi$ is forbidden, which is also in contradiction with experiment. We are thus led to 
{\sl assume} that (8.6.25) remains regular as 
$\epsilon\to0$ for 
$\alpha=0$. If we could {\sl prove} that this is so, we would have solved the 
$U(1)$ problem. We will not discuss this matter 
any more here, sending to the standard references.\fnote{Adler (1969); Bardeen
(1974); Crewther (1979b), etc.}

\bookendchapter

\brochureb{\smallsc chapter 9}{\smallsc  chiral perturbation theory}{89}
\bookchapter{9. Chiral perturbation theory}
\vskip-0.5truecm
\booksection{9.1 Chiral Lagrangians}
\vskip-0.5cm
\booksubsection{9.1.1. The $\sigma$ Model}

\noindent In this and the following sections we will  describe a method
 that has been devised to explore 
{\sl systematically} the consequences of the chiral symmetries of QCD, in the limit of 
small momenta and neglecting the light quark masses (or to leading order in 
 these). 
The method consists in writing Lagrangians consistent with chiral symmetry 
for pion field operators. 
These Lagrangians are not unique but, on the mass shell and for momenta $p^2$ 
much smaller than $\Lambdav^2$, all produce the {\sl same} results 
(Coleman, Wess and Zumino, 1969; Weinberg, 1968a). 
The Lagrangians are not renormalizable, but 
this is not important as they are to be used only at tree level
 (actually, it turns out to be possible to go beyond tree level,  at the 
cost of introducing a number of phenomenological constants, as we will discuss later). 
One can then use these Lagrangians to calculate low energy quantities 
involving pions, if the symmetry we consider is chiral $SU(2)$, 
reproducing the results obtained in a more artisanal way with the help of 
current algebra and soft pion (PCAC) techniques. 
This general formulation of chiral dynamics was first proposed by 
Weinberg~(1979) and later developed  in much greater detail by 
Gasser and Leutwyler~(1984,~1985a,\/b). 
We will begin in this section with a few examples, to proceed in next section to 
contact with PCAC and present a first application; the general formulation of 
chiral perturbation theory will be left for \sect~9.3.

The starting point to formulate the effective chiral Lagrangian theories is to write the chiral 
transformation properties of pions,\fnote{We will consider here explicitly only
 chiral $SU(2)$; the extension to  chiral $SU(3)$, 
that is to say, to processes involving also kaons and the 
$\eta$, is straightforward.}  whose field we denote by $\vec{\varphi}$, with the 
the vector representing an isospin index, 
and a fictitious, scalar particle that we will 
denote by $\sigma$. 
This is the so-called {\sl sigma model} for spontaneous  symmetry breaking, devised by 
Gell-Mann and L\'evy~(1960). For infinitesimal chiral (i.e., parity changing) transformations we write,
$$\eqalign{
\sigma\to& \sigma+\delta\sigma,\quad \delta\sigma=-\vec{\alpha}\vec{\varphi}\cr
\vec{\varphi}\to&\vec{\varphi}+\delta\vec{\varphi},\quad \delta\vec{\varphi}=\vec{\alpha}\sigma.\cr
}
\equn{(9.1.1a)}$$
The $\vec{\alpha}$  are the parameters of the chiral transformations in $SU(2)\times SU(2)$, 
which would correspond, in a quark formulation, to the transformations 
involving $\gamma_5$. 

For ordinary isospin transformations, with parameters $\vec{\theta}$, we have
$$\delta\sigma=0,\qquad\delta\vec{\varphi}=\vec{\theta}\times\vec{\varphi}.
\equn{(9.1.1b)}$$
Because we suppose invariance under the full  $SU(2)\times SU(2)$ transformations 
it follows that $\sigma$ and $\vec{\varphi}$ fields 
should have the same mass, that (in a first approximation) we take to be zero. 

We now assume that the interaction is such that the field $\sigma$ acquires a vacuum expectation value, 
$\langle\sigma\rangle=k\neq0$; this will provide a large (i.e., of order $\Lambdav$) mass 
for the sigma field, which will then disappear from the
 low energy effective theory.\fnote{Alternatively, 
we could interpret it as the enhancement experimentally observed in 
the isospin zero S-wave in 
pion-pion scattering at an energy around 750 \mev. 
The key point, of course, is that 
at low energies only the pions give appreciable contributions; 
those from other particles are suppressed by powers $p^2/M_{\sigma}^2$.} 
To formulate the last, we want to redefine fields which do no more mix 
under chiral transformations. It happens that this is not possible if using {\sl linear} transformations; 
but can be achieved if  
nonlinear ones are allowed ({\sl nonlinear sigma models}).
 A simple choice is to set $\sigma'=\sigma-k$ 
(so the VEV of $\sigma'$ vanishes) 
and then  define
$$\eqalign{R=&\sqrt{(\sigma'+k)^2+\vec{\varphi}^2}-k,\cr
\vec{\pi}=&\dfrac{k}{\sqrt{(\sigma'+k)^2+\vec{\varphi}^2}}\,\vec{\varphi}.\cr}
\equn{(9.1.2)}$$
For small energies we can expand the new fields in terms of the old 
(in effect, this is an expansion in powers of $k^{-1}$), 
$$R\simeq\sigma'+\cdots,\qquad\vec{\pi}\simeq \vec{\varphi}+\cdots$$
so that the $R$, $\vec{\pi}$ coincide, at leading order, with the old fields. 
However, the new fields do not mix under chiral transformations: 
we get
$$\delta
R=0,\qquad\delta\vec{\pi}=\vec{\alpha}\,\dfrac{k\sigma}{\sqrt{\sigma^2+\vec{\varphi}^2}}=
\vec{\alpha}\sqrt{k^2-\vec{\pi}^2}.
\equn{(9.1.3a)}$$
Under ordinary isospin we still have,
$$\delta R=0,\qquad\delta\vec{\pi}=\vec{\theta}\times\vec{\pi}.
\equn{(9.1.3b)}$$

Because of these properties we can write a Lagrangian, invariant under chiral 
transformations, using 
only the field $\vec{\pi}$: we have 
succeeded in decoupling the sigma field. 
The Lagrangian is not unique; a choice, suggested by 
Coleman, Wess and Zumino (1969) is to take
$${\cal L}=\tfrac{1}{2}\dfrac{1}{(1+a^2\vec{\Piv}^2)^2}\left(\partial\vec{\Piv}\right)^2,\quad 
(\partial\vec{\Piv})^2\equiv (\partial_\mu\vec{\Piv}) (\partial^\mu\vec{\Piv});\qquad
a=1/2k,
\equn{(9.1.4a)}$$
with $\vec{\Piv}$ a reparametrization of $\vec{\pi}$:
$$\vec{\Piv}=\dfrac{2\vec{\pi}}{1+\sqrt{1-\vec{\pi}^2/k^2}};$$
it transforms chirally as
$$\delta\vec{\Piv}=\dfrac{1}{a}
\left[\vec{\alpha}\left(1-a^2\vec{\Piv}^2\right)
+2a^2\vec{\Piv}(\vec{\alpha}-\vec{\Piv})\right].$$

We may expand $\cal{L}$ in powers of $a$ getting
$${\cal L}=\tfrac{1}{2}\left(\partial\vec{\Piv}\right)^2-
\dfrac{a^2}{2}\vec{\Piv}^2\left(\partial\vec{\Piv}\right)^2
+\dfrac{a^4}{2}\vec{\Piv}^4\left(\partial\vec{\Piv}\right)^2+\cdots
\equn{(9.1.5)}$$


 
\topinsert{
\setbox0=\vbox{\hsize7.truecm{\epsfxsize 6.truecm\epsfbox{feynman.eps}}} 
\setbox6=\vbox{\hsize 6.5truecm\captiontype\figurasc{Figure 9.1.1 }{\hb
The  four 
pion graph.\hb
\phantom{XX}}\hb
\vskip.1cm} 
\medskip
\line{
%\tightboxit
{\box0}\hfil\box6}
\medskip
}\endinsert


To show the usefulness  
of the effective Lagrangian formulation, we
 calculate $\pi\pi$ scattering to lowest order in $a$. 
Denote by $i,\,j,\,k,\,l$ to the isospin indices, 
varying from 1 to 3. 
The Feynman rule corresponding to (9.1.5) is, for a four-pion vertex with 
momenta $p_1,\,p_2,\,p_3,\,p_4$, all incoming (\fig~9.1.1),
$$\eqalign{\ii a^2g_{\mu\nu}\Big[
&\delta_{ij}\delta_{kl}\left(p_3^\mu p_4^\nu+p_1^\mu p_2^\nu\right)\cr
+&\delta_{ik}\delta_{jl}\left(p_2^\mu p_4^\nu+p_1^\mu p_3^\nu\right)\cr
+&\delta_{il}\delta_{jk}\left(p_2^\mu p_3^\nu+p_1^\mu p_4^\nu\right)
\Big].\cr}
\equn{(9.1.6)}$$
In terms of the Mandelstam variables
$$s=(p_1+p_2)^2,\quad t=(p_2+p_4)^2,\quad u=(p_2+p_3)^2,$$
 we can write the scattering amplitude that follows from (9.1.6) to lowest order as
$$F(i+j\to k+l)=
\dfrac{a^2}{(2\pi)^2}
\left\{\delta_{ij}\delta_{kl}s+\delta_{ik}\delta_{jl}t+\delta_{il}\delta_{jk}u\right\}.
\equn{(9.1.7)}$$
We will later identify $a$ with $1/f_\pi$, the inverse of the pion decay constant, so 
(9.1.7) gives the low energy ($s,\,t,\,u\ll \Lambdav^2$) pion-pion collision amplitude.
The simplicity of this evaluation contrasts with that based on ``old fashioned" 
PCAC, current algebra and soft pion techniques (Weinberg, 1966). 

\booksubsection{9.1.2. Exponential formulation}

\noindent A more elegant, but equivalent formulation uses a matrix representation of the pion field. 
Letting $\vec{\tau}$ be the Pauli matrices, for isospin space, 
we construct the $2\times2$ matrix
$$\pi=\vec{\tau}\vec{\varphi}
\equn{(9.1.8a)}$$
with $\vec{\varphi}$ the pion field. 
We then exponentiate $\pi$ and set the matrix 
$$\Sigmav=\exp2\ii\pi/F.
\equn{(9.1.8b)}$$ 
The chiral $SU(2)\times SU(2)$ transformations are defined in terms of the 
unitary matrices $W_L,\,W_R$:
$$\Sigmav\to\Sigmav'\equiv W_L\sigmav W_R^{\dag}.
\equn{(9.1.8c)}$$
The symmetry breaking condition is implemented by assuming a nonzero VEV for 
$\Sigmav$:
$$\langle \Sigmav\rangle=
\pmatrix{F&0\cr
0&F\cr}.$$
The advantage of the present method is that we only work with the pion field from the 
beginning. 

It is convenient to parametrize the $W_{R,L}$ as 
$$\eqalign{
W_L=&\ee^{\vec{\alpha}\vec{\tau}}\ee^{\vec{\theta}\vec{\tau}},\cr
W_R=&\ee^{-\vec{\alpha}\vec{\tau}}\ee^{\vec{\theta}\vec{\tau}}.\cr}
\equn{(9.1.9)}$$
For ordinary isospin transformations we simply set $\vec{\alpha}=0$ so that $W_L$ and $W_R$ coincide 
and (9.1.8) is equivalent to $\pi'=W(\vec{\theta}\/)\pi W^{-1}(\vec{\theta}\/)$. 
Then, 
for the pion field itself we have
$\vec{\pi}'=R(\vec{\theta}\/)\vec{\pi}$ with $R(\vec{\theta}\/)$ the three-dimensional rotation 
corresponding to the $SU(2)$ matrix $W(\vec{\theta}\/)$ given by the relation
$$W(\vec{\theta}\/)\tau_i W^{-1}(\vec{\theta}\/)=\sum_j R^{-1}_{ij}(\vec{\theta}\/)\tau_j.$$ 
Under an infinitesimal chiral transformation, (9.1.8) gives, after expanding,
$$\vec{\pi}'=\vec{\pi}+F\vec{\alpha}+\cdots.$$

Next we construct a Lagrangian invariant under 
(9.1.8). The one which contains {\sl less} derivatives is
$${\cal L}=\dfrac{F^2}{4}\trace \left(\partial_\mu\Sigmav^+\right)\partial^\mu\Sigmav,
\equn{(9.1.10)}$$
and the overall constant is chosen so that, after expanding, the kinetic 
term is $\tfrac{1}{2}(\partial_\mu\vec{\pi})\partial^\mu\vec{\pi}$.
This shows clearly the arbitrariness of the method: we can add 
extra terms with higher derivatives to (9.1.10). 
However, they will, on dimensional grounds, contribute to higher orders in the momenta. 
But it is important to realize that the effective Lagrangian methods are only 
valid to give the first orders in the expansion in powers of the momenta, $p^2/\Lambda^2$. 
The theory says nothing {\sl a priori} about higher corrections, which involve more and more 
arbitrary parameters. 

In this formalism we can introduce in a natural manner leading order 
symmetry breaking by considering that it is due to a quark mass 
matrix,
$$M=\pmatrix{m_u&0\cr0&m_d\cr}.$$
This is not invariant under chiral (or even ordinary isospin) 
transformations. 
We may couple $M$ and $\Sigmav$; the lowest dimensionality scalar that can be formed is the 
function
$$v^3\trace (\Sigmav^+M+M\Sigmav).$$
$v$ is a constant with dimensions of mass, that we will identify later.
Expanding in powers of $\pi$, we find that the first 
nonzero term is the quadratic one,
$$-\dfrac{4v^3}{F^2}\trace M\pi^2=-\dfrac{4v^3}{F^2}(m_u+m_d)\vec{\pi}^2,
\equn{(9.1.11)}$$
and we have used that $(\vec{\lambda}\vec{\tau})^2=\vec{\lambda}^2$ for any $\vec\lambda$.
\equn{(9.1.11)} provides the lowest order mass term for the pions; it has the nice feature 
that it reproduces (as it should) the result we had obtained with the help of PCAC and 
current algebra in (8.2.4). This allows us to 
realize that $v^3$ is proportional to the quark condensate.
Applications of this to calculate some hadronic corrections 
to low energy weak interactions may be found in the book of Georgi (1984).

An alternate to the exponential formulation presented here will be given in \sect~9.3.


\booksection{9.2. Connection with PCAC, and a first application}

\noindent Before starting to calculate with the chiral Lagrangians described in the 
previous section we have to interpret the constant ($F$ or $a$) that 
appears there. 
For this we have to introduce the {\sl axial current} in the present formalism, 
which we choose to do in the original Coleman--Wess--Zumino version.\fnote{For 
the derivation in the exponential version, somewhat messier, see the text of Georgi (1984).} 
To do so we use a method which is a variant of Noether's method, due to Adler (for 
details on it, see Georgi, 1984 or Adler, 1971). 
Let us consider a general Lagrangian ${\cal L}(\phi)$ depending on the field $\phi$, 
and make 
an infinitesimal transformation on the fields, characterized by the 
infinitesimal parameters
$\epsilon_i$:
$$\delta\phi=\sum_i\epsilon_i\xi_i(\phi).$$
The corresponding variation of the Lagrangian is then
$$\delta{\cal L}= K_i(\phi)\epsilon_i+L_i^\mu(\phi)\partial_\mu\epsilon_i+
M_i^{\mu\nu}(\phi)\partial^\mu\partial^\nu\epsilon_i+\hbox{higher derivatives}.$$
(sum over repeated indices understood). 
The variation of the action can then be written, after integrating by parts, as
$$\delta{\cal A}=\int\dd^4x\,\left\{K_i+\partial_\mu J_i^\mu\right\}\epsilon_i$$
and we have defined the current $J$ by
$$J_i^\mu=-L_i^\mu+\partial_\nu M_i^{\mu\nu}+\cdots.$$
For a symmetry of the system, the change must leave 
the action unchanged, hence $\partial_\mu J_i^\mu=-K_i$.
Moreover, choosing $\epsilon$ constant, $\cal L$ will be invariant only if 
$K_i=0$. 
In this case, $J_i^\mu$ is obtained simply as the coefficient of 
$\partial_\mu\epsilon_i$ in the variation of $\cal L$. 
It is interesting to note that, if $\cal L$ only contains 
first order derivatives of the field $\phi$, then all the terms $M,$ etc. above vanish 
so  $J_i^\mu$  coincides with $-L_i^\mu$. 

This can be immediately applied to the Lagrangian (9.1.4a). 
Working to lowest order in $\Piv$, we find immediately the 
axial current to be
$$\vec{A}_\mu=-\dfrac{1}{a}\partial_\mu\vec{\Piv}+\hbox{higher orders}=
-\dfrac{1}{a}\partial_\mu\vec{\varphi}+\hbox{higher orders}.$$
Taking derivatives of both sides and using the equations of motion this gives
$$\partial^\mu\vec{A}_\mu=\dfrac{1}{a}\mu^2\vec{\varphi}.$$
On comparing with the definitions in \sect~7.3, we identify
$$\dfrac{1}{a}=f_\pi,$$
$f_\pi$ the pion decay constant, $f_\pi\simeq 93\;\mev$. 
(The factor $\sqrt{2}$ in the definitions of \sect~7.3 has disappeared because 
the physical pion states are related to the ones used now, $\vec{\pi}$, by 
$\pi^\pm=\mp2^{-1/2}(\pi_1\pm\ii \pi_2)$).    

With this identification we get the pion-pion scattering amplitude, 
given in  
\equn{(9.1.7)}, as
$$F(i+j\to k+l)=
\dfrac{1}{4\pi^2f^2_\pi}
\left\{\delta_{ij}\delta_{kl}s+\delta_{ik}\delta_{jl}t+\delta_{il}\delta_{jk}u\right\}.
\equn{(9.2.1)}$$
From this one can evaluate the low energy parameters for $\pi\pi$ scattering. 
For example, the isospin 1, P wave scattering length is 
calculated as follows. First, we identify the physical pion states in terms
 of the $i,\,j,\,\dots=1,\,2,\,3$ ones
 as 
$$|\pi^0\rangle=
|3\rangle,\qquad|\pi^{\pm}\rangle=\mp2^{-1/2}\left\{|1\rangle\pm\ii|2\rangle\right\};$$
the isospin 1 state will appear in particular in the combination $|\pi^0\pi^+\rangle$ as 
$$|\pi^0\pi^+\rangle=2^{-1/2}|I=1\rangle+2^{-1/2}|I=2\rangle.$$ 
Moreover, we have the partial wave expansion, for states with well defined isospin $I$, 
$$F^{(I)}=2\sum_l(2l+1)P_l(\cos\theta)f_l^{(I)};\qquad f^{(I)}_l=
\dfrac{2s^{1/2}}{\pi k}\sin \delta_l^{(I)}\ee^{\ii \delta^{(I)}_l},
\equn{(9.2.2)}$$
 with $\delta_l^{(I)}$ the phase shifts.\fnote{Recall
 that the factor 2 in the partial wave expansion is due to
the identity 
of the particles, in states with well-defined isospin.}

At small energy we write 
the {\sl partial wave amplitudes}, $f_l^{(I)}$, in terms of the 
{\sl scattering lengths}, $a_l^{(I)}$:
$$f_l^{(I)}(s)\simeqsub_{s\to 4\mu^2}\;\dfrac{4\mu k^{2l}}{\pi}a_l^{(I)}.$$
$k$ is the center of mass momentum; for massless pions, we can take $k^2=s/4$. 
With all this we find, for the P wave
$$a_1=\dfrac{1}{24\pi f_\pi^2\mu}\simeq 0.029\, \mu^{-3}.
\equn{(9.2.3)}$$
Experimentally, and from the analysis of \sect~6.8, we know that 
$$a_1({\rm exp.})=(0.0391\pm0.0024)\,\mu^{-3}.$$ 
The agreement between theory and experiment improves if including pion mass corrections, 
and higher order 
chiral perturbative theory terms 
(to be discussed later).

The S-wave phase shifts are similarly calculated, and 
 we get,
$$\eqalign{
a_0^{(0)}=&\dfrac{7\mu}{32\pi f^2_\pi}\simeq0.155\; \mu^{-1};\cr
a_0^{(2)}=&-\dfrac{\mu}{16\pi f^2_\pi}\simeq-0.044\; \mu^{-1}.
\cr}$$
The agreement of these with experiment is less good than before, although 
it is difficult to tell since the experimental data 
are not unambiguous.  Including corrections, the predicted 
value for $a_0^{(0)}$ (for example) could go up 
to $0.22\,\mu^{-1}$, while  experiment gives 
values in the  range $0.20\, \mu^{-1}$ to $0.30\,\mu^{-1}$, 
as we saw in \sect~6.5.

\booksection{9.3 Chiral perturbation theory: general formulation}
There is a large number of further applications of chiral perturbation theory  
(at times also denoted by the name of {\sl $\chi$PT}\/), to leading order, which the interested reader 
may find in the text of Georgi (1984). But one may ask if 
it is possible to go beyond. 
In fact, an enormous amount of work has been devoted to the matter in recent years,
 particularly following the basic 
papers of Gasser and Leutwyler (1984, 1985a,\/b).\fnote{We will not be able 
to give an amount of information comparable to that
 presented in these papers; we urge the reader to consult
them for a more detailed treatment and further applications. 
The subject has had an enormous growth in the 
last years; a recent review, with references, 
is that by Scherer~(2002). An introductory one is 
the text by Dobado et al.~(1997).} 
In the present section we will indeed describe  the general formalism of chiral
 perturbation theory, following, precisely, 
the excellent expos\'e of these authors. 
We will restrict ourselves to chiral isospin; the extension to chiral $SU(3)$ may be found in 
Gasser and Leutwyler (1985a).

The idea is the following: we will first extend the chiral symmetry in QCD 
to a gauge symmetry. Then we will construct the more general Lagrangians involving 
pions (for chiral $SU(2)$), first to 
leading order and then to higher orders, consistent with the 
PCAC definition $\partial\cdot A=\sqrt{2} f_\pi \mu \phi_\pi$ 
and verifying the gauge chiral symmetry. 
Because these Lagrangians share the symmetry with the QCD one, it will follow 
that the theory based on pions will satisfy identical Ward identities and commutation relations 
as QCD; 
therefore they will show the same low energy properties.


\booksubsection{9.3.1. Gauge extension of chiral invariance}
As stated, we start by extending the $SU(2)\times SU(2)$ symmetry to a gauge symmetry. 
We do so by introducing {\sl sources} in the QCD Lagrangian. 
We denote by ${\cal L}_{{\rm QCD}0}$ to the QCD Lagrangian for massless $u,\,d$ quarks,
$${\cal L}_{{\rm QCD}0}=\sum_{\alpha=u,d}\bar{q}_\alpha\ii \Slash{D}q_\alpha-\tfrac{1}{4}G^2.
\equn{(9.3.1a)}$$
Then we consider ${\cal L}(v_\mu,a_\mu,s,p)$ where 
$v_\mu,\,a_\mu,\,s,\,p$ are, respectively, vector, axial, scalar and pseudoscalar sources, 
and we define
$$\eqalign{
{\cal L}(v_\mu,a_\mu,s,p)=&\,{\cal L}_{{\rm QCD}0}\cr
+&\,\sum_{\alpha,\beta}
\bar{q}_\alpha\gamma_\mu \left(v^\mu_{\alpha\beta}+a^\mu_{\alpha\beta}\gamma_5\right)q_\beta
+\sum_{\alpha,\beta}
\bar{q}_\alpha \left(-s_{\alpha\beta}+\ii p_{\alpha\beta}\gamma_5\right)q_\beta.\cr
}
\equn{(9.3.1b)}$$
We include the mass matrix in $s_{\alpha\beta}$ so that 
$$s_{\alpha\beta}=m_\alpha\delta_{\alpha\beta}+\widetilde{s}_{\alpha\beta}.
\equn{(9.3.1c)}$$
$\alpha,\,\beta$ are flavour indices that run over the values $u,\,d$, in our case.

The Lagrangian (9.3.1b) is invariant under independent local gauge transformations of the left and 
right components of the $q$, {\sl provided} we at the same time transform the sources: 
$$\eqalign{q\to q'=&\,\left\{\tfrac{1}{2}(1+\gamma_5)W_R(x)+\tfrac{1}{2}(1-\gamma_5)W_L(x)\right\}q;\cr
v^\mu\pm a^\mu\to&\, v'^\mu\pm a'^\mu=W_{R,L}\left(v^\mu\pm a^\mu\right)W^{\dag}_{R,L}
+\ii W_{R,L}\partial^\mu W^{\dag}_{R,L},\cr
s+\ii p\to&\, s'+\ii p'=W_R(s+\ii p)W^{\dag}_L.\cr
}
\equn{(9.3.2)}$$
Here the $W_{R,L}$ are independent $SU(2)$ matrices. 
The symmetry may be extended to a $U(2)\times U(2)$ symmetry; however, the current associated 
with the diagonal piece presents an anomaly, as we know. 
We will not study this piece here, but refer to Gasser and Leutwyler (1985a). 
To avoid it we will restrict the $v^\mu$, $a^\mu$ to be traceless. 
This is automatic if we parametrize them in terms of the three-vectors $v_i^\mu$, $a_i^\mu$ writing
$$v^\mu=\tfrac{1}{2}\sum_iv_i^\mu \tau_i,\quad
a^\mu=\tfrac{1}{2}\sum_ia_i^\mu \tau_i
\equn{(9.3.3)}$$
and the $\tau_i$ are the Pauli matrices in flavour space. 
The $s,\,p$ may likewise be parametrized in terms of the ({\sl Euclidean}) four dimensional vectors 
$s_A$, $p_A$ with
$$s=\sum_As_A\tau_A,\quad
p=\sum_Ap_A\tau_A;\qquad \tau_0\equiv1.
\equn{(9.3.4)}$$

At low energy the only degrees of freedom are those associated with the pions; 
moreover, we have to take also into account that, in QCD, the 
scalar densities have a nonzero expectation value in the ground state (the physical vacuum). 
We will use the quantity $B$ defined as
$$B=-\dfrac{\langle\bar{q}q\rangle}{f^2}.
\equn{(9.3.5)}$$
We write $f$ for the pion decay constant in the chiral limit 
($m_{u,d}\to0$). 
In \subsect~9.3.3 we will see the connection with the 
physical decay constant, whose value we take to be $f_\pi\simeq93\,\mev.$
In the chiral limit, $B$ is independent of which $q$ ($u$ or $d$) we take. 
Comparing with (9.2.4) we have
$$B=\mu^2/(m_u+m_d).$$

\booksubsection{9.3.2. Effective Lagrangians in the chiral limit}

\noindent
We will start by working in the chiral limit, $m_{u,d}=0$. 
At low energies an effective Lagrangian should only include 
pion fields and, apart from the nonzero value of the 
condensate, should respect chiral gauge invariance.\fnote{This is, of course, a limitation 
of the chiral dynamics approach; 
it must fail at distances where the {\sl composite} character 
of the pions becomes relevant; thus certainly at energies 
of the order of the $\rho$ mass, as this particle is 
a quark-antiquark bound state, and decays into two pions.}
To construct this Lagrangian we proceed as for the 
nonlinear $\sigma$-model of \sect~9.1. 
We define a chiral four-dimensional vector $\varphi_A$, $A=0,\,1,\,2,\,3$ such that  
$\vec{\varphi}=\vec{\pi}$ (the pion field) 
and $\varphi_0=\sigma$ (the $\sigma$ field). 
We get rid of the last by imposing the invariant constraint
$$\sum_A\varphi_A\varphi_A=f^2.
\equn{(9.3.6a)}$$
We could include this into the Lagrangian, using a multiplier, or simply by admitting that 
$\varphi_0$ is not an independent field, but one has
$$\varphi_0=\sqrt{f^2-\vec{\varphi}^2}.
\equn{(9.3.6b)}$$

The transformation properties of $\varphi$ under  $SU(2)\times SU(2)$ 
imply the following values for the chiral covariant derivative, 
that we denote by $\nabla^\mu$:
$$\eqalign{
\nabla^\mu\varphi_0=&\,\partial^\mu\varphi_0+\vec{a}^\mu(x)\vec{\varphi},\cr
\nabla_\mu\vec{\varphi}=&\,\partial^\mu\vec{\varphi}+\vec{v}^\mu(x)\times\vec{\varphi}-
\vec{a}^\mu(x)\varphi_0.\cr
}
\equn{(9.3.7)}$$

We then construct the more general Lagrangians which are  
compatible with \equn{(9.3.2)}, and involve only $\varphi_A$. 
We start at lowest order in the momenta, $O(p^2)$. 
If we only allow two powers of the momenta at tree level, then only two derivatives can occur and the more
general form 
of this first order Lagrangian is, simply,
$${\cal L}_{{\rm ch.}1}=\tfrac{1}{2}\sum_A(\nabla_\mu\varphi_A)\nabla^\mu\varphi_A.
\equn{(9.3.8)}$$
The index ``ch." reminds us that this is valid in the chiral limit, 
and the factor $1/2$ is included so that the kinetic energy term 
agrees with that for three real, (pseudo-)scalar fields. 
One can evaluate the axial current from (9.3.8) and identify 
$f$ with the value of the pion decay constant, $f_\pi$, in the chiral limit. 
(In this case the identification of the axial current is simpler than before, as it is 
the current coupled to the
axial source, $\vec{a}^\mu$).

In particular, to lowest order and replacing $\varphi_0$ in terms of 
$\vec{\varphi}$, this gives
$${\cal L}_{{\rm ch.}1}\simeq\tfrac{1}{2}(\partial_\mu\vec{\varphi})\partial^\mu\vec{\varphi}
+\dfrac{1}{2f^2}\left(\vec{\varphi}\partial_\mu\vec{\varphi}\right)
\left(\vec{\varphi}\partial^\mu\vec{\varphi}\right)+\;\hbox{source terms}\;+\;\hbox{higher orders}.
\equn{(9.3.9)}$$
To order $p^2$, this is equivalent to (9.1.5).

Let us next consider $O(p^4)$. 
Simple power counting shows that the loop corrections generated by (9.3.8) 
are of relative order $p^2$ for each new loop; hence, one loop corrections induced by 
${\cal L}_{{\rm ch.}1}$ will be of order $p^4$. 
These corrections (which are necessary in order to respect unitarity of the 
effective theory) are, generally speaking, divergent. 
However, if we use a regularization that respects gauge invariance (such as 
dimensional regularization in the absence of anomalies) these divergences will multiply 
chiral gauge invariant polynomials of degree $p^4$. 
They can thus be absorbed into suitable counterterms.

This leads us to construct all possible terms of order $p^4$ which will build the 
second order effective Lagrangian, ${\cal L}_{{\rm ch.}2}$. 
After use of the equations of motion it can be seen (Gasser and Leutwyler, 1984) 
that its most general form will be (sum over repeated indices $A,\,B,\,C$ understood)
$$\eqalign{
{\cal L}_{{\rm ch.}2}=&\,\dfrac{1}{f^4}\Big\{l_1\left(\nabla^\mu\varphi_A \nabla_\mu\varphi_A\right)^2
+l_2\left(\nabla^\mu\varphi_A\nabla^\mu\varphi_A\right)\left(\nabla_\mu\varphi_B\nabla_\mu\varphi_B\right)\cr
+&\,l_5\varphi_A F^{\mu\nu}_{AB}F_{BC,\mu\nu}+l_6\nabla_\mu\varphi_A F^{\mu\nu}_{AB}\nabla_\nu\varphi_B\cr
+&\,h_2\trace F_{\mu\nu}F^{\mu\nu}\Big\}.\cr
}
\equn{(9.3.10a)}$$
Here $F$ is defined by
$$\left(\nabla^\mu\nabla^\nu-\nabla^\nu\nabla^\mu\right)\varphi_A=F^{\mu\nu}_{AB}\varphi_B
\equn{(9.3.10b)}$$
and the reason for the
numbering of the constants $l_1,\,\dots,\,h_2$ (that agrees with the definitions of Gasser and
Leutwyler, 1984) will be seen below.

The constants $l_1,\,\dots,\,h_2$ will be divergent: 
their divergence is to be adjusted so that it cancels the one loop divergences generated by 
${\cal L}_{{\rm ch.}1}$. The theory will, therefore, {\sl predict} 
the coefficients of terms of type $p^4\log p^2/\nu^2$, with $\nu$ a renormalization scale  
(and, when we take into account leading symmetry breaking by the pion mass, also terms in $p^4$, 
$\mu^4$ and $p^2\mu^2$ multiplied by either 
$\log p^2/\nu^2$ or $\log \mu^2/\nu^2$).
However, the finite parts of the constants $l_1,\,\dots,\,h_2$ are {\sl not} 
 given by the theory. 
In fact, what one does is to {\sl fix} these constants  
by requiring agreement of the predictions using ${\cal L}_{{\rm ch.}1}$, 
${\cal L}_{{\rm ch.}2}$ with experiment. 
Chiral dynamics does {\sl not} allow an evaluation from first principles of 
corrections of order $p^4$. 
What it does is to {\sl correlate} these corrections to all processes in terms of a finite number of
constants, the  $l_1,\,\dots,\,h_2$.

In principle one can extend this procedure to higher orders and, indeed, the 
$O(p^6)$ corrections have been considered in the literature,\fnote{Akhoury and Alfakih (1991); 
 Fearing and Scherer (1996); Knecht et al.~(1995,~1996); Bijnens et al.~(1996);
 Bijnens, Colangelo and Eder
(2000).}  but we will not discuss this in any detail here.
  Not only the number of constants to be fitted to experiment
 grows out of hand, but it is practically impossible
to separate the 
$O(p^4)$ and $O(p^6)$ pieces of the   $l_1,\,\dots$, as we will see in two examples later. 
More interesting is to take into account the corrections due to 
the nonzero masses of the $u,\,d$ quarks (or, equivalently, of the pions) to which we now turn.

\booksubsection{9.3.3. Finite pion mass corrections}
Because the mass of the pion will appear in pion propagator denominators, $1/(p^2-\mu^2)$, 
a consistent way to treat the finiteness of the pion mass requires that we consider $p^2$ and $\mu^2$ to be
of the same order of magnitude, and calculate to all orders in their ratio; 
otherwise we would be replacing
$$\dfrac{1}{p^2-\mu^2}\quad\hbox{by}\quad\dfrac{-1}{\mu^2}\left\{1+\dfrac{p^2}{\mu^2}+
\dfrac{p^4}{\mu^4}+\cdots\right\},$$
 not  a very accurate procedure.

To leading order we have to find the lowest order  terms that can be added to 
${\cal L}_{{\rm ch.}1}$ and which contain $s_0$; we recall that $s_0$ included the quark masses. 
There is only one such term that also preserves parity,
${\rm Constant}\times (s_0\varphi_0+\vec{p}\vec{\varphi})$. 
The constant may be identified requiring that the new term reproduce the 
equality (9.2.4) for the pion propagator. 
We then have the full ${\cal L}_{1}$, correct to $O(p^2)$, $O(\mu^2)$,
$${\cal L}_{1}={\cal L}_{{\rm ch.}1}+2Bf\left(s_0\varphi_0+\vec{p}\vec{\varphi}\right),
\equn{(9.3.11a)}$$
which corresponds to the pion mass
$$\mu^2=(m_u+m_d)B.
\equn{(9.3.11b)}$$
To next order, 
$${\cal L}_{2}={\cal L}_{{\rm ch.}2}+\dfrac{1}{f^4}\Big\{l_3(\xi_A\varphi_A)^2
+l_4\nabla^\mu\xi_A\nabla_\mu\varphi_A+l_7(\eta_A\varphi_A)^2+h_1\xi_A\xi_A+h_3\eta_A\eta_A\Big\}.
\equn{(9.3.12a)}$$
We have defined
$$\xi_0=2Bs_0,\quad\vec{\xi}=2Bp;\qquad
\eta_0=2Bp_0,\quad\vec{\eta}=-2B\vec{s}
\equn{(9.3.12b)}$$
and ${\cal L}_{{\rm ch.}1}$, ${\cal L}_{{\rm ch.}2}$ are as given in (9.3.8), (9.3.10).

For reference, we note the correspondence between our definitions and those of 
Gasser and Leutwyler (1984):
$$U_A=\dfrac{1}{f}\,\varphi_A,\quad \chi_A=\dfrac{1}{f}\,\xi_A,\quad 
\widetilde{\chi}_A=\dfrac{1}{f}\,\eta_A.
\equn{(9.3.13)}$$

\booksubsection{9.3.4. Renormalized effective theory}
Renormalization for the one loop graphs generated by ${\cal L}_1$ 
proceeds  in the usual manner. 
The divergences, as stated in the previous subsection, can be 
canceled by divergent pieces in the $l_i,\,h_j$. 
One finds (Gasser and Leutwyler, 1984, where the $c_i$ are denoted by $\gamma_i$ and the 
$d_j$ by $\delta_j$)
$$\eqalign{
l_i=&\,l_i^{\rm loop.}(\nu)=\dfrac{c_i}{32\pi^2}\left\{\dfrac{2}{D-4}+\log\nu^2-
(\log4\pi-\gammae+1)\right\},\cr
 h_j=&\,h_j^{\rm loop.}(\nu)=
\dfrac{d_j}{32\pi^2}\left\{\dfrac{2}{D-4}+\log\nu^2-(\log
4\pi-\gammae+1)\right\};\cr
}
\equn{(9.3.14a)}$$
$\nu$ is the renormalization point and
$$\eqalign{
c_1=&\,\tfrac{1}{3},\quad c_2=\tfrac{2}{3},\quad c_3=-\tfrac{1}{2},\quad c_4=2,\quad
c_5=-\tfrac{1}{6},\quad c_6=\tfrac{1}{3},\quad c_7=0;\cr
d_1=&\,2,\quad d_2=\tfrac{1}{12},\quad  d_3=0.\cr}
\equn{(9.3.14b)}$$
The renormalized constants $l_i^{\rm ren.}$ may be obtained 
by comparing with experimental quantities.\fnote{The $h_j$ 
depend on the renormalization scheme and, in fact, 
do not intervene in any physical observable. 
This is discussed in Gasser and Leutwyler, 1984.} 
They depend on the renormalization point, $\nu$. Alternatively, one may replace them by the 
quantities $\bar{l}_i$, defined as (proportional to) the $l_i^{\rm ren.}(\nu)$ 
with $\nu=\mu_{\rm ch.}$. (Here we denote by $\mu_{\rm ch.}$ to the pion mass in the leading order in
chiral symmetry breaking, 
that is to say, using (9.3.11b) but evaluating $B=-\langle\bar{q}q\rangle/f$ 
in the chiral limit).  Then, we have
$$l_i^{\rm ren.}(\nu)=\dfrac{c_i}{32\pi^2}\left\{\bar{l}_i+\log\dfrac{\mu_{\rm ch.}}{\nu^2}\right\}.
\equn{(9.3.14c)}$$
We remark that this implies that the 
$\bar{l}_i$ are divergent in the chiral limit, as we are renormalizing at 
$\nu=\mu_{\rm ch.}$ which vanishes in this limit:
$$\bar{l}_i\;\simeqsub_{m_{u,d}\to0}\;-\log \mu_{\rm ch.}.$$

We can now compare the results of calculations made 
with ${\cal L}_1$ and ${\cal L}_2$  with experimental quantities, and obtain the $\bar{l}_i$. 
As an example we consider $\pi\pi$ scattering. 
If we use the full ${\cal L}_1$ and ${\cal L}_2$ we obtain, after a
 straightforward but tedious calculation
(Gasser and Leutwyler, 1984)
$$F(i+j\to k+l)=
\dfrac{1}{4\pi^2}
\left\{\delta_{ij}\delta_{kl}A(s,t,u)+\delta_{ik}\delta_{jl}A(t,s,u)+\delta_{il}\delta_{jk}A(u,t,s)\right\}
\equn{(9.3.15a)}$$
where now
$$A(s,t,u)=\dfrac{s-\mu^2_{\rm ch.}}{f^2}+B(s,t,u)+C(s,t,u).
\equn{(9.3.15b)}$$
Here $B$, $C$ are, respectively, the logarithmic and polynomial fourth order corrections:
$$\eqalign{
B(s,t,u)=&\,\dfrac{1}{96\pi^2f^4_\pi}\Bigg\{3(s^2-\mu^2)I(s)\cr
+&\,
\left[t(t-u)-2\mu^2 t+4\mu^2 u-2\mu^4\right]I(t)\cr
+&\,\left[u(u-t)-2\mu^2 u+4\mu^2 t-2\mu^4\right]I(u)\Bigg\};\cr
I(s)=&\,\beta\log\dfrac{\beta-1}{\beta+1}+2,\quad 
\beta=\sqrt{1-4\mu^2_{\rm ch.}/s};
\cr}
\equn{(9.3.16a)}$$
$$\eqalign{
C(s,t,u)=&\,\dfrac{1}{96\pi^2f^4_\pi}\Bigg\{2\left(\bar{l}_1-\tfrac{4}{3}\right)(s-2\mu^2)^2\cr
+&\,\left(\bar{l}_2-\tfrac{5}{6}\right)\left[s^2+(t-u)^2\right]-12 \mu^2 s+15 \mu^4\Bigg\}.
\cr}
\equn{(9.3.16b)}$$
The expression for $B$ in the chiral limit ($\mu=0$) has been known for a long time 
(Lehmann, 1972).
 To leading order $B,\,C\to0$, $\mu\to0$, and (9.3.15) of course reproduces (9.2.1).

The extension to $SU(3)$ (i.e., including kaons and $\eta$) may be found, for 
chiral perturbation theory, in 
Gasser and Leutwyler~(1985a); 
for $\pi\pi$ scattering  
in Bernard, Kaiser and Meissner~(1991) for some cases and, in general, in 
the paper of G\'omez-Nicola and Pel\'aez~(2002). 


\booksubsection{9.3.5. The parameters of chiral perturbation theory}

\noindent
To one loop, the $\pi\pi$ scattering amplitude depends on the two unknown constants $\bar{l}_1,\,\bar{l}_2$ 
(besides, of course, $f_\pi$ and $\mu$). 
A technical point to be cleared is that, in $A$, we have the quantities 
$f$ and $\mu_{\rm ch.}$ which we have to 
relate to the physical ones. 
The details may be found again in the paper of Gasser and Leutwyler (1984); we have
$$\mu^2=\mu^2_{\rm ch.}\left\{1-\dfrac{\mu^2}{32\pi^2f^2_\pi}\bar{l}_3\right\},\quad
f_\pi=f\left\{1+\dfrac{\mu^2}{16\pi^2f^2_\pi}\bar{l}_4\right\}. 
\equn{(9.3.17)}$$
Thus, $F$ in (9.3.15) depends also indirectly on the constants $\bar{l}_3,\,\bar{l}_4$. 
We can, however, obtain directly  $\bar{l}_1,\,\bar{l}_2$ by selecting an 
observable that depends only on second order effects. 
Such observables are the D waves  at low energy:
$$f_2^{(I)}\simeq \dfrac{(s-4\mu^2)^2}{4\pi}\mu a_2^{(I)},$$
and $I=0,\,2$ is the isospin index. 
Because this vanishes (for $\mu=0$)
  as $s^2=p^4$, the contributions to them  start at second order 
and we find
$$\eqalign{
a_2^{(0)}=&\,\dfrac{\mu^{-1}}{1440\pi^3f^4_\pi}\left\{\bar{l}_1+4\bar{l}_2-\tfrac{53}{8}\right\},\cr
a_2^{(2)}=&\,\dfrac{\mu^{-1}}{1440\pi^3f^4_\pi}\left\{\bar{l}_1+\bar{l}_2-\tfrac{103}{40}\right\}.\cr
}
\equn{(9.3.18)}$$


We can also improve our previous determination
 of the scattering lengths; for example, for the S and  P waves, 
including pion mass and $O(p^4)$ terms gives
$$\eqalign{
a_0^{(0)}=&\,\dfrac{7\mu}{32\pi f^2_\pi}\left\{1+\dfrac{5\mu^2}{84\pi^2 f^2_\pi}\left[
\bar{l}_1+2\bar{l}_2-\tfrac{3}{8}\bar{l}_3+\tfrac{21}{10}\bar{l}_4+\tfrac{21}{8}\right]\right\},\cr
a_0^{(2)}=&\,-\dfrac{\mu}{16\pi f^2_\pi}\left\{1-\dfrac{\mu^2}{12\pi^2f^2_\pi}\left[
\bar{l}_1+2\bar{l}_2+\tfrac{3}{8}\right]+\dfrac{\mu^2}{32\pi^2f^2_\pi}\left[\bar{l}_3+4\bar{l}_4\right]
\right\};\cr}
\equn{(9.3.19a)}$$
$$a_1=\dfrac{\mu^{-1}}{24\pi f^2_\pi}
\left\{1-\dfrac{1}{12\pi^2}\,\dfrac{\mu^2}{f_\pi^2}\left[\bar{l}_1-\bar{l}_2+\tfrac{65}{48}\right]
+\dfrac{\mu^2}{8\pi^2f^2_\pi}\bar{l}_4\right\}.
\equn{(9.3.19b)}$$
Note that we 
here use the  definition (cf.~(7.5.3))
$$\dfrac{\pi}{2k^{2l}}\real f_l^{(I)}(s)=a_l^{(I)}+b_l^{(I)}k^2+\cdots\,.
$$ 
This may be compared with the standard effective range expansion: 
$$k^{2l+1}\cot\delta_l^I(s)\,\simeqsub_{k\to 0}\,\dfrac{1}{a_l^{(I)}}+\tfrac{1}{2}r_0k^2+O(k^{4}).
$$
The connection between the parameters $a^I_l|_{\rm G.\&L.},\,b_l^{I}|_{\rm G.\&L.}$
 of Gasser and Leutwyler (1984) and our $a_l^{(I)},\,b_l^{(I)}$, 
and also with $r_0$ and with
 the parameters $b_l^{I}|_{\rm P.S-G.Y.}$ of Palou, S\'anchez-G\'omez and
Yndur\'ain~(1975) is
$$
a^I_l|_{\rm G.\&L.}=\mu a^{(I)}_l,\quad 4b_l^{I}|_{\rm P.S-G.Y.}=b_l^{I}|_{\rm G.\&L.}= 
b_1^{(I)}=a_l^{(I)}\dfrac{1-\mu^2 a^{(I)}_lr_0}{2\mu}.$$

For the parameters $b_l^{(I)}$ one loop chiral perturbation theory gives
$$\eqalign{
b_0^{(0)}=&\,\dfrac{\mu^{-1}}{4\pi f^2_\pi}
\left\{1+\dfrac{\mu^2}{12\pi^2f^2_\pi}\left[2\bar{l}_1+3\bar{l}_2-\tfrac{13}{16}\right]
+\dfrac{\mu^2}{8\pi^2f^2_\pi}\bar{l}_4\right\},\cr
b_0^{(2)}=&\,-\dfrac{\mu^{-1}}{8\pi f^2_\pi}
\left\{1-\dfrac{\mu^2}{12\pi^2f^2_\pi}\left[\bar{l}_1+3\bar{l}_2-\tfrac{5}{16}\right]
+\dfrac{\mu^2}{8\pi^2f^2_\pi}\bar{l}_4\right\};\cr
b_1^{(1)}=&\,\dfrac{\mu^{-1}}{288\pi^3f^4_\pi}\left\{-\bar{l}_1+\bar{l}_2+\tfrac{97}{120}\right\}.
\cr}
\equn{(9.3.20)}$$

The values of the $a_0,\,b_0$ given above imply that the S waves with isospin $I$ 
have a zero each, for
$s=z_I^2$ in the range
$0<s<4\mu^2$, located at
$$\eqalign{z_0^2=&\,4\mu^2-\dfrac{7\mu^2}{2}\left\{1+\dfrac{5\mu^2}{84\pi^2f^2_\pi}
\left[\bar{l}_1+2\bar{l}_2-\tfrac{3}{8}\bar{l}_4+\tfrac{21}{8}\right]
-\dfrac{\mu^2}{12\pi^2f^2_\pi}
\left[2\bar{l}_1+3\bar{l}_2-\tfrac{13}{16}\right]+
\dfrac{\mu^2}{8\pi^2f^2_\pi}\bar{l}_4\right\};\cr
z_2^2=&\,4\mu^2-2\mu^2\left\{1+\dfrac{\mu^2}{32\pi^2f^2_\pi}\bar{l}_3+
\dfrac{\mu^2}{12\pi^2f^2_\pi}\left[\bar{l}_2+\tfrac{1}{16}\right]\right\}.
\cr
}\equn{(9.3.21)}$$
These zeros are often called 
{\sl Adler zeros}, after the work of Adler~(1965)
 on zeros of scattering amplitudes implied by PCAC.
It should be clear, however, that while the location of $z_2$ 
is probably well described by (9.3.21), 
there is no reason why the same should be the case for $z_0$. Indeed, to get this last, 
we have used the expansion of $f_l^{(I)}$ for 
$s=z_0^2\simeq\tfrac{1}{2}\mu^2$ were, due to the vicinity of the 
left hand cut of $f_0^{(0)}(s)$, starting at $s=0$, we would 
expect it to give a poor approximation. 
Actually, while fits to data do confirm $z_2$ (\subsect~6.5.1), the 
situation is less clear for $z_0$; recall \subsect~6.5.2.


\booksection{9.4. Comparison of chiral perturbation theory to one loop 
with experiment}

\noindent
Using the experimental values of the quantities evaluated in Chapters~6,\/7,
 and others as well, we can find
the  constants $\bar{l}_i$. In fact, there are many more observables than constants; 
for example, $a^{(2)}_2,\,a^{(0)}_2$ and $b^{(1)}_1$ depend only on the two $\bar{l}_1,\,\bar{l}_2$.
 So the agreement of 
various determinations among themselves is a nontrivial 
check of second order chiral perturbation theory.\fnote{The tests are less impressive 
than what they may look at first sight. 
In fact, chiral perturbation theory is just a (very convenient) way to 
summarize properties that hold in any local field theory: 
analyticity, crossing and unitarity, 
plus the dynamical properties embodied in the 
constants $f_\pi$, $\mu$ and 
the $\bar{l}_i$. 
Thus for example, by comparing the r.h. sides of the 
Olsson sum rule (7.4.8) and the 
Froissart-Gribov representation for $a_1$, (7.5.4), we discover that, 
in any local field theory we must have the 
equality
$2a_0^{(0)}-5a_0^{(2)}=18\mu^2a_1+O(\mu^4)$.}

We collect here the more recent values of the constants $\bar{l}_i$; the
reader interested in  the details of the calculations should
 consult the original papers. 
We have, from Bijnens, Colangelo and Talavera~(1998); Colangelo, Gasser and Leutwyler~(2001),
$$\eqalign{
\bar{l}_1=&\,-0.4\pm0.6,\quad \bar{l}_2=6.0\pm1.3,\quad \bar{l}_3=2.9\pm2.4,\cr
\bar{l}_4=&\,4.4\pm0.2,\quad \bar{l}_5=13.0\pm1.0,\quad \bar{l}_6=16\pm1.\cr
}
\equn{(9.4.1)}$$
Actually, these determinations include estimates of 
two loop effects.
The value of $\bar{l}_7$ is not known with any accuracy; an estimate for it is 
$\bar{l}_7\sim5\times10^{-3}$ (Gasser and Leutwyler, 1984).

Some of this constants can be calculated independently with   comparable accuracy 
(but  only at the one loop level)  using the 
results reported in \sect~7.6 (Table~II) here. So, 
from the combination $a_2^{(0)}-a_2^{(2)}$,
that  we evaluated very precisely, there follows the value
$$\bar{l}_2=6.38\pm0.22.
\equn{(9.4.2a)}$$
Likewise, the constant $\bar{l}_1$ can be deduced from 
the combination $a_2^{(0)}+2a_2^{(2)}$ that follows from the Froissart--Gribov 
representation for $\pi^0\pi^0$
 and above value of $\bar{l}_2$. 
 We find
$$\bar{l}_1=-2.2\pm0.7,
\equn{(9.4.2b)}$$
somewhat larger in magnitude than the value given in (9.4.1). 
The difference may easily be due to two loop corrections, 
taken into account in (9.4.1) but not by us. 

Use of the  quadratic charge radius of the pion as input 
(see \equn{(9.4.5)} below) allows us 
also an accurate evaluation of $\bar{l}_6$:
$$\bar{l}_6=16.35\pm0.14.
\equn{(9.4.2d)}$$
Our improvements, however, are somewhat illusory. In fact, 
the accuracy of {\sl all} the calculations may be challenged 
because of the possible contributions of two loop corrections, 
that we have not taken into account, and of 
electromagnetic corrections, that may be 
important; see next section for a discussion of a few examples 

In what respects S and P waves scattering lengths, substituting (9.4.2) into (9.3.19)
 we find the chiral perturbation 
theory predictions 
$$\eqalign{
a_0^{(0)}=&\,(0.198\pm0.009)\,\mu^{-1},\quad
a_0^{(2)}=(-0.041\pm0.002)\,\mu^{-1},\quad
a_1=(36.7\pm0.6)\times10^{-3}\,\mu^{-3};\quad\hbox{(c)}.\cr
}
\equn{(9.4.3)}$$
The predictions for the $a_0^{(I)}$ are   in agreement with the experimental values we 
found before.  
The value 
of the P wave scattering length  in (9.4.3) is also  compatible with the result of 
the direct fit, $a_1=(39.1\pm2.4)\times10^{-3}\,\mu^{-3}$, within errors. 
It is also compatible with the results of other authors:
$$
a_1=\cases{
(36.64\pm0.5)\times10^{-3}\;\mu^{-3}\quad\hbox{(Colangelo, Gasser and Leutwyler, 2001)}\cr
(35.96\pm2)\times10^{-3}\;\mu^{-3}\quad\hbox{(Ananthanarayan et al., 2001)}\cr
(36.7\pm2)\times10^{-3}\;\mu^{-3}\quad\hbox{(Amor\'os, Bijnens and Talavera, 2000).}\cr
}
$$

We also note, as  consistency tests, that the value of $\bar{l}_4$ 
that follows from  $a_1$, via \equn{(9.3.19b)}, is compatible, 
within the rather large error, with (9.4.1a), 
as one gets $\bar{l}_4=5.5\pm2.0$, and that $b_1^{(1)}$ would yield 
a number for  $\bar{l}_2-\bar{l}_1$ compatible 
(also within errors) with what we found before.

The very precise calculation of the pion form factor possible with the 
Omn\`es--Muskhelishvili techniques also allows a direct 
determination of a second order (two loop) 
parameter. 
According to Gasser and Meissner~(1991), Colangelo, Finkelmeir and Urech~(1996),  and 
Fearing and Scherer~(1966), 
one has
$$c_\pi=\dfrac{1}{16\pi^2f^2_\pi}\left\{\dfrac{1}{60\mu^2}+\dfrac{1}{16\pi^2f^2_\pi}\bar{f}_2\right\}.
\equn{(9.4.4a)}$$
With the value  $c_\pi=3.60\pm0.03\,\gev^{-4}$ 
given in de~Troc\'oniz and Yndur\'ain~(2002), this 
implies
$$\bar{f}_2=5.520\pm0.056.
\equn{(9.4.4b)}$$
Note, however, that this result is purely formal; indeed, the nominally leading term ($1/60\mu^2$) 
is in fact {\sl smaller} than the nominally second order one, $\bar{f}_2/(16\pi^2f^2_\pi)$.

This shows clearly the limitations of chiral perturbation theory. 
Another example is the charge radius of the pion, for which one has, to 
second order (Gasser and Meissner,~1991, and Colangelo, Finkelmeir and Urech,~1996),
$$\langle
r^2_\pi\rangle=\dfrac{1}{16\pi^2f^2_\pi}\left[\bar{l}_6-1+\dfrac{\mu^2}{16\pi^2f^2_\pi}\bar{f}_1\right].
\equn{(9.4.5)}$$
Here the two loop term is much smaller than the leading one, for reasonable values 
of $\bar{f}_1$. The value of $\bar{l}_6$ given above was obtained 
neglecting $\bar{f}_1$; but a value of this quantity of the order of 
that of  $\bar{f}_2$  would alter $\bar{l}_6$  by 14\%.



It is also possible to give a prediction, based on chiral perturbation theory and 
the Froissart--Gribov representation, for 
scattering lengths for large $l$. 
We will give some details for the amplitude for isospin 1 in the $t$ channel,
$$F^{(I_t=1)}=\tfrac{1}{3}F^{(I_s=0)}+\tfrac{1}{2}F^{(I_s=1)}-\tfrac{5}{6}F^{(I_s=2)}.$$
The corresponding scattering length is given by \equn{(7.5.4)},
$$2a_l^{(1)}=\dfrac{\sqrt{\pi}\,\gammav(l+1)}{4\mu\gammav(l+3/2)}
\int_{4\mu^2}^\infty\dd s\,\dfrac{\imag F^{(I_t=1)}(s,4\mu^2)}{s^{l+1}}
\equn{(9.4.6)}$$
and the factor 2 in the l.h. side is due to the identity of the pions. 
As $l\to\infty$, only the behaviour of $\imag F^{(I_t=1)}(s,4\mu^2)$ 
near threshold matters; hence we can replace
$$\eqalign{
\imag F^{(I_t=1)}(s,4\mu^2)\simeq&\,\tfrac{1}{3}\imag F^{(I_s=0)}-\tfrac{5}{6} \imag F^{(I_s=2)}\cr
\simeq&\, 2\times\dfrac{2s^{1/2}}{\pi k}
\left\{\tfrac{1}{3}\sin^2\delta_0^{(0)}(s)-\tfrac{5}{6}\sin^2\delta_0^{(2)}(s)\right\}\cr
\simeq&\,2\times\dfrac{2s^{1/2}}{\pi k}
\left\{\tfrac{1}{3}\left[a_0^{(0)}\right]^2-\tfrac{5}{6}\left[a_0^{(2)}\right]^2\right\}.\cr
}
\equn{(9.4.7)}$$
The factor 2 in the r.h. side also comes from the identity of the pions. 
Replacing the $a_0^{(I)}$ by their values at leading order in chiral perturbation theory, 
\equn{(9.3.19)}, we find that we can approximate
$$\imag F^{(I_t=1)}(s,4\mu^2)\simeq 2\dfrac{13\mu^2s^{1/2}k}{512\pi^3 f^4_\pi}.
$$
Substituting into (9.4.6) and performing the integral we get the result 
$$a_l^{(1)}\,\simeqsub_{l\to\infty}\,\dfrac{13\mu^{3-2l}}{2\times4^{l+5}\pi^2f^4_\pi}
\dfrac{\gammav(l+1)\gammav(l-1)}{\gammav(l+3/2)\gammav(l+1/2)}
\,\simeqsub_{l\to\infty}\,\dfrac{13\mu^{3-2l}}{2\times4^{l+5}\pi^2l^2f^4_\pi}
\equn{(9.4.8a)}$$
and, in the last step, we have used the asymptotic properties of the gamma function.
The same method gives a prediction for even waves; for example, for $I=0$ we have,
$$a_l^{(0)}\simeqsub_{l\to\infty}\dfrac{23\mu^{3-2l}}{2\times4^{l+5}\pi^2l^2f^4_\pi}.
\equn{(9.4.8b)}$$


Gasser and Leutwyler~(1983) have produced a formula for all 
scattering lengths with $I=1$ to leading order in chiral 
perturbation theory that is exact (and not only 
 valid for $l\to\infty$), based on a direct calculation.
They give the expression, valid for $l\geq3$,
$$a_l^{(1)}({\rm G.- L.})=\dfrac{\mu^{3-2l}}{512\pi^3 f^4_\pi}\,\dfrac{l!(l-3)!}{[(2l+1)!!]^2}
\,\big(13l^2+5l-22\big).
\equn{(9.4.9)}$$
If, in  (9.4.9), 
we replace $(2l+1)!!=\gammav(2l+2)/2^l\gammav(l+1)$ and use 
the duplication formula of the gamma function we find
$$a_l^{(1)}({\rm G.- L.})=
\dfrac{13\mu^{3-2l}}{2\times4^{l+5}\pi^2f^4_\pi}\,
\dfrac{\gammav(l+1)\gammav(l-1)}{\gammav(l+3/2)\gammav(l+1/2)}\,
\dfrac{l^2}{(l-2)(l+1/2)}\,\left(1+\dfrac{5}{13l}-\dfrac{22}{13l^2}\right)$$
that, as $l\to\infty$, agrees with (9.4.7a). 

The calculation  using leading order chiral perturbation theory 
yields the figure $a_3=1.8\times10^{-5}\,\mu^{-7}$. 
From the Froissart--Gribov representation 
we found in  \sect~7.6 results ranging between 
5.4 and 6.5, in units of $10^{-5}\,\mu^{-7}$. 
 A large disagreement (a factor of 3 to 4) is thus found between the 
leading chiral perturbation result and the ``experimental" results.

For $l=4$, our expression (9.4.7b) gives $a_4^{(0)}=1.3\times10^{-5}\,\mu^{-9}$ 
while the ``experimental" value (from the 
Froissart--Gribov representation) is $(0.9\pm0.1)\times10^{-5}\,\mu^{-9}$.
The disagreement for $a_3$, and the difference between the two values for $a_4^{(0)}$, 
 show that, in some cases, 
the corrections  due to subleading effects in chiral perturbation 
theory may be very large: 
for the quantity $a_3$, two to three times the nominally leading term. 
This is not surprising; as is clear in our derivation, the value we obtain depends on 
the {\sl square} of the S wave scattering lengths, for which 
one loop corrections are at least of 25\%. 

\booksection{9.5. Weak and electromagnetic interactions. 
The accuracy of chiral\hb perturbative calculations}
Weak and electromagnetic interactions, at tree level, can be introduced by 
making the standard minimal replacement in the covariant derivatives; 
for e.g., electromagnetic interactions,  
$\nabla_\mu\to\nabla_\mu-eA_\mu$. In this way one can 
calculate chiral dynamics values of quantities like the 
pion electromagnetic form factor, or strong interactions corrections to 
weak decays. 
Another matter are virtual electromagnetic corrections. 
These break chiral invariance, and are large. 
For example, the $\pi^+-\pi^0$ mass difference is of order $(m_d-m_u)^2$ 
in chiral perturbation theory; the corresponding calculation yields a very small number,
$$m^2_{\pi^+}-m^2_{\pi^0}=(m_d-m_u)^2\dfrac{2B^2l_7}{f^2_\pi}$$
which would give $m_{\pi^+}-m_{\pi^0}\sim0.2\,\mev.$
However, the experimental value is $m_{\pi^+}-m_{\pi^0}=4.6 \mev$. 
In this case one can use current algebra techniques to estimate the electromagnetic 
contribution, which is indeed of the right order of magnitude (Das, Mathur and Okubo, 1967), 
but in general this is not possible; we expect (generally unknown) 
electromagnetic corrections of something up to this order of magnitude, 
$\sim3.4\%$, to chiral perturbation theory calculations.

A case in which the electromagnetic corrections to the constants $\bar{l}_i$ is 
known is that of  
$\bar{l}_6$. The value reported in (9.4.2d) above is actually an average of those 
obtained from the charge radii of the pion with $\pi^0\pi^+$ and $\pi^+\pi^-$. 
If we use only the last (associated with the $\rho^0$), hence the parameters of (6.3.5c), we find 
instead 
$$\bar{l}_6=16.07\pm0.18.
\equn{(9.5.1)}$$
The difference between the two, a 1.5\%, is the minimum
 extra error due to electromagnetic corrections 
that we should append to all the determinations of chiral perturbation theory parameters.

A place where isospin violation corrections are 
potentially large are the scattering lengths. 
If we repeat the fits of de~Troc\'oniz and Yndur\'ain~(2002) without 
imposing the constraint
$a_1=(38\pm3)\times10^{-3}\,\mu^{-3}$, 
and fit separately $\pi^+\pi^-$ and 
$\tau$ decay data we find the numbers,
$$\eqalign{
a_1(\pi^+\pi^-)=&\,(36\pm3)\times10^{-3}\,\mu^{-3},\cr
a_1(\pi^+\pi^0)=&\,(42\pm3)\times10^{-3}\,\mu^{-3}.\cr
}$$
The two values overlap, but only barely; a difference of the order of 
$3\times10^{-3}$ (in units of $\mu$) 
cannot be excluded. 
This is perhaps the reason for the discrepancy between the low energy P wave of 
Colangelo, Gasser and Leutwyler~(2001) and what one finds directly from 
the pion form factor.


Another question is the scale of higher corrections in chiral perturbation theory. 
For the {\sl logarithmic} corrections 
we know that this scale is $1/(4\pi f_\pi)^2$, so for energies of the order 
of $\mu^2$ we expect corrections $O(\mu^2/(4\pi f_\pi)^2)\simeq1.4\%$. 
However, this estimate forgets the {\sl constant} contributions to the 
$l_i$. There is no reason why 
they should be suppressed by powers of $1/(4\pi f_\pi)^2$; all we can expect  
is a suppression of order $O(\mu^2/\lambdav^2_0)$, with 
$\lambdav^2_0$ proportional to the QCD parameter $\lambdav\sim400\,\mev$ (for 2 flavours). 
In some cases the coefficients of these terms will be small; in other they may be large. 
This last situation occurs for example for the isospin zero S wave in $\pi\pi$ scattering, 
where the correction necessary to get agreement between the leading 
value obtained from chiral dynamics, $a_0^{(0)}=0.16\,\mu^{-1}$, with the experimental 
values which  vary between 
$a_0^{(0)}\simeq0.26\,\mu^{-1}$ and $a_0^{(0)}\simeq0.20\,\mu^{-1}$
 is at least a third of the leading one.\fnote{
This possibility is particularly relevant in view of the doubts expressed by other researches 
on some aspects of chiral perturbation theory; see, 
for example, Fuchs, Sazdjian and Stern (1991); Knecht et al.~(1996).} 

In this context, we would like to emphasize that the situation is 
even worse for the quantity $a_3$; here, and as mentioned, 
the leading order calculation gives $a_3=1.8\times10^{-5}\,\mu^{-7}$ 
while, according to Colangelo, Gasser and Leutwyler~(2001), including one and 
two loop 
corrections changes this to $a_3=5.2\times10^{-5}\,\mu^{-7}$, 
in agreement with the experimental value (obtained from e.g. the Froissart--Gribov representation).

In some cases the size of the corrections may be gleaned from external arguments. 
For example, for the isospin zero S-wave in $\pi\pi$ scattering, 
chiral dynamics implies that its imaginary part should be suppressed with respect to 
the real part, at energy squared $s$, by powers $s/\lambdav^2_0$. 
However, already at $s^{1/2}=500\,\mev$, i.e., only $200\,\mev$ above threshold, 
real and imaginary part are of the same order of 
magnitude; so, we would expect poor convergence in this case, 
as indeed happens.  

  

\bookendchapter

\bookchapter{Appendix: the conformal mapping method}
\brochureb{\smallsc the conformal mapping method}{\smallsc the conformal mapping method}{105}

\noindent Let us consider a function, $f(z)$, 
analytic in a domain, $\cal{D}$; for example, this domain may be a plane with two cuts, as for 
the partial wave amplitudes. 
According to general theorems (see, e.g., Ahlfors,~1953), 
it is always possible to map the interior of this domain into the interior 
of the  disk $\deltav(0,1)$, with center at the origin 
and unit radius. 
Let us call $w=w(z)$ to the corresponding variable. 
Then, in this variable, $f$ is analytic inside  $\deltav(0,1)$ 
and thus the ordinary Taylor expansion in terms of $w$ is absolutely and uniformly convergent 
in  $\deltav(0,1)$. 
Therefore, undoing the mapping, it follows that we can write
$$f(z)=\sum_{n=0}^\infty c_nw(z)^n,
\equn{(A.1)}$$
and this expansion is absolutely and uniformly convergent inside all of  $\cal{D}$.

It is important to realize that the representation (A.1) does not imply any 
supplementary assumption on $f(z)$ besides its analyticity properties; 
the convergence of (A.1) and the analyticity of $f$ in $\cal{D}$ are strictly 
equivalent statements.

We next say a few words about the specific situations we encountered in the 
main text.\fnote{ Further discussion
(with references) of the present method, and also of other 
similar ones (for example, mapping into an ellipse and expanding in 
a Legendre series there),  applied in particular to $\pi\pi$
scattering, may be found in the reviews of Pi\u{s}ut~(1970) and Ciulli~(1973). 
The question of the stability of extrapolations is also discussed there.} 
In some cases we have a function $f$ analytic inside $\cal{D}$ 
except for a pole at $z_0$. 
Then the function
$$\varphi(z)=(z-z_0)f(z)$$
is analytic inside  $\cal{D}$ and it is $\varphi$ that can be expanded as in (A.1). 
In some other cases, we have a function $f(z)$ analytic inside  $\cal{D}$, with a zero at $z_0$. 
Of course, this zero does not spoil the analyticity, so we could expand $f$ itself. 
But, because the expansion of a function converges best if the function varies little, 
we have interest in extracting this zero and write
$$f(z)=(z-z_0)\psi(z),$$ 
expanding then $\psi$, which has the same analyticity properties as $f$.


The gain in convergence and stability obtained by expanding in the conformally transformed variable 
is enormous. 
The reader may verify this with
 the simple example of the function $\log(1+x)$. 
Here the region $\cal{D}$ is the complex plane cut from $-\infty$ to $-1$.
 If we expand in powers of
$x$,
$$\log(1+x)=-\sum_{n=1}^\infty\dfrac{(-x)^n}{n}
\equn{(A.2)}$$
 then for e.g. 
$x\sim1/2$ we need five terms for a 1\% percent accuracy.




\topinsert{
\setbox0=\vbox{\hsize13.truecm{\epsfxsize 11.3truecm\epsfbox{del_pipi_SS.eps}}} 
\setbox6=\vbox{\hsize 11truecm\captiontype\figurasc{Figure}{
The  
$I=0$, $S$-wave phase shifts corresponding to (6.5.10a) (continuous line) and 
 Colangelo, Gasser and Leutwyler~(2001), dashed line.}\hb} 
\centerline{\tightboxit{\box0}}
\bigskip
\centerline{\box6}
\medskip
}\endinsert
 

In this case the expansion in the conformal variable can be made explicitly. 
The transformation that maps $\cal{D}$ into $\deltav(0,1)$ is 
$$w(x)= \dfrac{(1+x)^{1/2}-1}{(1+x)^{1/2}+1},$$
with inverse $x=2w/(1-w)$. 
Substituting this into $\log(1+x)$, we get the expansion
 in the conformally transformed variable 
$$\log(1+x)=4\sum_{n={\rm odd}}^\infty\dfrac{1}{n}\left[\dfrac{(1+x)^{1/2}-1}{(1+x)^{1/2}+1}\right]^n.
\equn{(A.3)}$$
If using this expansion, only two terms are necessary for an accuracy of a part in a thousand 
for $x\sim1/2$. 
Even for $x=25$, very far from the region of convergence of (A.2), (A.3) 
still represents the function closely: only three terms 
in (A.3) are necessary to get  a precision better than  2\%.\fnote{In this example 
we compare the virtues of (A.1), (A.3) as {\sl expansions}, for simplicity; in 
the main text, they are, however, used to {\sl fit}. 
Thus, we should really give ourselves the values of $\log(1+x)$ at a series of 
points, $x_1,\,x_2,\,\dots\,x_n$ and fit 
with (A.1), (A.3). 
The improvement is less spectacular than before, but 
it is still substantial.} 


This economy is also apparent in our parametrizations of 
the partial waves or the Omn\`es auxiliary function $G(t)$, 
where only two, or in one case three terms, are necessary.
Indeed, 
the simplicity and economy of our parametrization contrasts 
with some of the complicated ones found in the literature. 
Thus, for example, Colangelo, Gasser and Leutwyler~(2001), who take 
it from Schenk~(1991), write
$$\cot\delta=\dfrac{2s^{1/2}}{k^{2l+1}}\dfrac{s-s_R}{4\mu^2-s_R}
\,\dfrac{1}{A+B k^2+C k^4+Dk^6}.
\equn{(A.4)}$$
For the P wave, they need these four parameters $A,\,B,\,C,\,D$ (apart from the 
squared mass of the resonance, $s_R$) when we only require two. 
Moreover, (A.3) only converges in the shaded disk in \fig~3.1.2 
(but it is used in the whole range, which is a recognized cause
 of unstability)
 and, in general, 
presents complex singularities, hence violating 
causality. 

It s true that (A.4) is only used by Schenk and by Colangelo {\sl et alii} in the 
physical region; 
this, in fact, is one of its disadvantages: our parametrization can be used in all the 
cut complex plane and is therefore suited to discuss effects such as location of the 
poles asociated with the resonances or the Adler zeros for the 
S waves. 
As a graphical example, consider the phase $\delta_0^{(0)}$ as given
 by Colangelo, Gasser and Leutwyler~(2001); for $s^{1/2}\leq0.8\,\gev$ 
it agrees, at the percent level, with our solution (6.5.10a). 
However, the CGL phase goes bererk above $0.9\,\gev$, while 
(6.5.10a) continues to represent it fairly well up to the $\bar{K}K$ threshold: 
see the accompanying figure. 
It is also possible that at least some of the wiggles that te 
CGL solution presents (for example, the one below and around 0.8 \gev, 
clearly seen in the figure) are due to 
the unstability of the Schenk expansion.


It may be argued 
that, even if using Schenk's parametrization, 
one can get at $f_l(s)$ outside the physical region 
indirectly 
via Roy's equations. 
Using ours, however, you can get that {\sl both} 
directly 
and via Roy's 
equations, which would 
provide useful consistency tests. 
As an example, we mention that the value we obtain for the 
Adler zero in \equn{(7.6.3)}, with a simple fit to data and only three 
parameters, namely $z_2=133\pm4.5\;\mev$, is consistent with (and the central 
value even slightly more 
accurate than) what Colangelo, Gasser and Leutwyler~(2001) get with 
the parametrization of Schenk, with five parameters,
 after imposing fulfillment of the Roy equations and a large number of 
crossing and analyticity sum rules: $z_2=136\,\mev$.

The fact that we manage with a smaller number of parameters is important 
not only as a matter of economy or consistency, but also in that 
we avoid  spureous minima which are liable to occur when 
large number of parameters are present. 
 

\bookendchapter

\brochureb{\smallsc references}{\smallsc references}{109}
\booksection{Acknowledgments}

\noindent
I am grateful to J.~Gasser for information concernig, in particular, the 
best values for the $\bar{l}_i$ parameters, and to the same, to G.~Colangelo and 
to H.~Leutwyler for a number of comments
and  criticisms --constructive as well as destructive,  
but always of interest. 
A number of remarks by A.~Pich have been also very useful. 
 I am also thankful to J.~Pel\'aez for helpful discussions and 
suggestions.

Finally, I am grateful to the hospitality of CERN (Geneva) and NIKHEF (Amsterdam) 
where part of this work was done.

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}

\bookendchapter
\bye






 
 
 
 
 
