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\begin{document}
%\vspace*{4cm}
\title{From QCD to Dual Superconductivity to Effective String Theory}

\author{M. Baker}
\address{Department of Physics, University of Washington, P.O. Box 351560, Seattle, WA 98195, USA\\}
%	E-mail: baker@phys.washington.edu, rsteinke@u.washington.edu}

\maketitle\abstracts{
    We show how an effective field theory of long distance QCD,
    describing a dual superconductor,
    can be expressed as an effective string theory of superconducting
    vortices. We use the semiclassical expansion of this effective
    string theory about a classical rotating string solution
    in any spacetime dimension $D$ to obtain the semiclassical meson 
    energy spectrum. We argue that the experimental data on Regge 
    trajectories along with numerical simulations of the heavy quark 
    potentials provide good evidence for an effective string 
    description of long distance QCD.}

%\end{abstract}
%\newpage

% bibliography
%\end{opening}
\section{From QCD to Dual Superconductivity}

In the dual superconductor mechanism of 
confinement~\cite{Nambu,Mandelstam,tHooft} a dual Meissner
effect confines color electric flux 
to narrow tubes~\cite{Nielsen+Olesen} connecting a quark-antiquark pair. 
In the confined
phase, monopole fields $\phi$ condense to a
value $\phi_0$, and dual potentials $C_\mu$
acquire a mass $M = g\phi_0$ via a dual Higgs mechanism.
The dual coupling constant is $g=2\pi/e$, where $e$ is
the Yang--Mills coupling constant. 
Quarks couple to dual potentials via a Dirac string connecting
the quark-antiquark pair along a line L, the ends of which 
are sources and sinks of color electric flux. The color field of the 
pair destroys the dual Meissner effect near L so that $\phi$
vanishes on L. At distances transverse to L greater than $1/M$
the monopole field returns to its bulk value $\phi_0$, so that 
the color field is confined to a tube of radius $a = 1/M$ surrounding 
the line L.
%~\cite{Baker+Ball+Zachariasen:1991,Nora}  
As a result, for quark-antiquark separations $R$ greater than $a$, 
a linear potential develops that confines the quarks in hadrons.
The string tension $\sigma \sim {\phi_0}^2$ , so that $M \sim \sqrt {\sigma/\alpha_s}$.
(The running coupling $\alpha_s$ is evaluated at a scale of order $M$.)

Recent lattice calculations~\cite{Forcrand} and general arguments based on the work of 't Hooft~\cite{tHooft2} show that the confined phase
of non-Abelian gauge
theory is characterized by a dual order parameter (monopole condensate), which
vanishes in regions of space where electric color flux can penetrate
and "dual supercondutivity" 
is destroyed. This provides a basis in QCD for a generic dual 
superconducting effective field theory of long distance Yang Mills 
theory in which the dual gluon mass $M$ serves 
as the ultraviolet cutoff.

To obtain an effective string theory of long distance QCD we do 
not need a specific form for the action 
$S[C_\mu,\phi]$ of the effective dual field theory.
This theory
must have classical $Z_N$ electric vortex 
solutions, and it cannot
have massless particles.
 For $SU(N)$ Yang Mills theory this can be done
by coupling dual non-Abelian potentials $C_\mu$ to $Z_N$
invariant dual Higgs fields $\phi$ . 
(For example there could be Higgs fields in the adjoint representation 
of the gauge group.)
The long distance
 effective dual theory will then have the same symmetries 
as $SU(N)$ Yang Mills theory and
the dual $SU(N)$ gauge symmetry can be "spontaneously broken"
so that the gauge bosons $C_\mu$ all acquire a mass, and 
there are $Z_N$ electric flux tube excitations.~\cite{Baker+Ball+Zachariasen:1991} 


In the classical approximation to the dual theory the axis of the flux 
tube is a straight line between the quark and the antiquark.
The contribution of flux tube fluctuations to the heavy quark potential
is determined by the path integral over all field
configurations $C_\mu,\phi$, for which the monopole fields $\phi$ 
vanish on some line $L$ connecting the quark-antiquark pair.~\cite{Nora}
The fluctuating vortex line $L$ sweeps out a fluctuating 
spacetime surface $\tilde x^\mu$, whose boundary is the loop $\Gamma$
formed by the worldlines of the moving pair.  These surfaces  
$\tilde x^\mu$ determine the location of the vortices, where dual
superconductivity is destroyed and electric color flux can penetrate.

The Wilson loop $W[\Gamma]$ of Yang Mills theory determining the quark-antiquark interaction
is the partition function of the dual theory in the vortex sector.~\cite
{Nora}
\begin{equation}
W[\Gamma] = \int \scrD C_\mu \scrD\phi \scrD\phi^* e^{iS[C_\mu, \phi]}
 \,.
\label{Wilson loop def}
\end{equation}
The path integral \eqnlessref{Wilson loop def} goes over all field
configurations for which the monopole field  $\phi(x)$
vanishes on some sheet $\tilde x^\mu$ bounded by the loop
$\Gamma$.

\section{From Dual Superconductivity to Effective String Theory}

We transform the field theory partition function
\eqnlessref{Wilson loop def} 
to a path integral over the vortex sheets $\tilde x^\mu$, so
that $W[\Gamma]$ takes the form of the  partition function
of an effective string theory of these vortices. We do this
in two stages:

\begin{enumerate}
\item We integrate over all field configurations $C_\mu,\phi$,
containing a vortex located on a particular surface
$\tilde x^\mu$, where $\phi(\tilde x^\mu) = 0$. 
This integration determines the action $S_{{\rm eff}}[\tilde x^\mu]$
of the effective string theory.
\item We integrate over all
vortex sheets $\tilde x^\mu(\xi)$, $\xi=\xi^a$, $a=1\,,2$. 
This integration goes over the amplitudes $f^1(\xi)$ and $f^2(\xi)$
of the two transverse vortex fluctuations
in a particular parameterization 
of the world 
sheet,  $\tilde x^\mu(\xi) = x^\mu(f^1(\xi), f^2(\xi), \xi )$ , and
%This gives $W[\Gamma]$ 
gives~\cite{Baker+Steinke2}
\begin{equation}
W[\Gamma] = \int \scrD f^1 \scrD f^2 \Delta_{FP} e^{iS_{{\rm eff}}[\tilde x^\mu]} \,,
\label{Wilson loop eff}
\end{equation}
where
\begin{equation}
\Delta_{FP} = \Det\left[ \frac{\epsilon_{\mu\nu\alpha\beta}}{\sqrt{-g}}
\frac{\partial x^\mu}{\partial f^1} \frac{\partial x^\nu}{\partial f^2}
\frac{\partial \tilde x^\alpha}{\partial \xi^1}
\frac{\partial \tilde x^\beta}{\partial \xi^2}
\right]
\end{equation}
and $\sqrt{-g}$ is the square root
of the determinant of the induced metric 
\begin{equation}
g_{ab} = \frac{\partial \tilde x^\mu}{\partial \xi^a}
\frac{\partial \tilde x_\mu}{\partial \xi^b} \,.
\end{equation}
\end{enumerate}
The partition function \eqnlessref{Wilson loop def}
of an effective field theory has been expressed as the
partition function \eqnlessref{Wilson loop eff} of an effective
string theory.
The presence of the determinant $\Delta_{FP}$ in 
\eqnlessref{Wilson loop eff}  
makes the
path integral invariant under reparameterizations $\tilde x^\mu(\xi)
\to \tilde x^\mu(\xi(\xi^\prime))$ of the vortex worldsheet. 

The parameterization invariant measure in the path 
integral \eqnlessref{Wilson loop eff} is universal and is
independent of the explicit form of the underlying field theory.
On the other hand, the action $S_{{\rm eff}}[\tilde x^\mu]$ 
of the effective string theory is 
not universal, and depends upon parameters in the action 
$S[C_\mu,\phi]$ 
of the effective field theory describing the dual superconductor.
However, for wavelengths $\lambda$ of the string fluctuations greater 
than the flux tube radius $a=1/M$, which are those included 
in \eqnlessref{Wilson loop eff}, the action 
$S_{{\rm eff}}[\tilde x^\mu]$ can be expanded in powers of
the extrinsic curvature tensor $\scrK^A_{ab}$ of the sheet
$\tilde x^\mu$,
\begin{equation}
S_{{\rm eff}}[\tilde x^\mu] = - \int d^4x \sqrt{-g} \left[ \sigma
+ \beta \left(\scrK^A_{ab}\right)^2 + ... \right] \,.
\label{curvature expansion}
\end{equation}
The extrinsic curvature tensor is
\begin{equation}
\scrK^A_{ab} = n^A_\mu(\xi) \frac{\partial^2 \tilde x^\mu}
{\partial\xi^a \partial\xi^b} \,,
\end{equation}
where $n^A_\mu(\xi)$, $A = 1,2$ are vectors normal to
the worldsheet at the point $\tilde x^\mu(\xi)$.
The values of the coefficients in this expansion (
the string tension $\sigma$, the rigidity $\beta$, ...) are
determined by the parameters of the underlying effective field theory,
(For example in a dual superconductor on the border between type I
and type II the rigidity vanishes.)~\cite{Baker+Steinke3}
If these coefficients are taken as parameters, the specific form of the
dual field theory does not enter explicitly in the effective string 
theory of dual superconductivity \eqnlessref{Wilson loop eff}.
The expansion parameter in \eqnlessref{curvature expansion}
is the ratio $(a/\lambda)^2$ of the square of the 
flux tube radius to the square of 
the wave length of the string fluctuations, and the leading term in this
expansion is the Nambu--Goto action.

\section{The Heavy Quark Potentials}

Expanding the action
%$S_{{\rm eff}}[\tilde x^\mu]$
\eqnlessref{curvature expansion} in small fluctuations about a straight 
string connecting two static quarks puts the partition function
\eqnlessref{Wilson loop eff} in the form used by L{\"u}scher, Symanzik
and Weisz~\cite{Luscher1,Luscher2,Luscher3} to calculate
the contribution of string fluctuations to the static heavy quark potentials. The
%potentials.~\cite{Luscher1,Luscher2,Luscher3} The 
leading long distance expression for the energy levels $E_n(R)$
of a fixed string of length $R$, vibrating in D dimensional spacetime
is~\cite{Luscher3}
\begin{equation}
E_n(R) = {\sigma}{R} + (-\frac{D-2}{24} + n)\frac{\pi}{R} \,.
\label{static energy}
\end{equation}
Corrections to \eqnlessref{static energy} arise from the higher order 
terms in $S_{{\rm eff}}[\tilde x^\mu]$ , which give contributions 
proportional to powers of $(a/R)^2$.
For $n=0$, \eqnlessref{static energy} reduces to the L{\"u}scher 
ground state heavy quark potential.

Recent numerical simulations~\cite{Luscher3,Sommer} provide
striking confirmation of the L{\"u}scher potential for 
quark-antiquark separations greater $0.5$ fm. At this distance 
and (with $D=4$) the ratio
$\pi/(12 \sigma R^2)$ of the leading semiclassical correction in eq.
\eqnlessref{static energy} (with $n=0$)
to the classical term $\sigma R$ is already small ($\sim 0.2$). Thus the 
string behavior of the static potential sets in at a distance where
the semiclassical expansion parameter is small.  

The excited potentials $E_n(R)$, $n>0$, involve wave 
lengths of order $R/n$. When this wave length is of order of the flux 
tube radius, higher order terms in the effective action should become 
important and modify the string behavior of the excited potentials.
Therefore as $n$ increases, the distance $R$ for which 
\eqnlessref{static energy} should be applicable becomes
larger. This expectation 
is in qualitative agreement with the recent lattice measurements of
Juge, Kuti, and Morningstar~\cite{Kuti}
of the excited energy levels. 
They find that, although the excited potentials 
do not have string behavior at $R=0.5$ fm,
there is, for $R \approx 2$ fm, 
a rapid rearrangement of
excited levels towards the string ordering 
\eqnlessref{static energy}. There are questions to be 
clarified, but these measurements of excited  heavy quark 
potentials provide further evidence for an effective string theory 
description of long distance QCD.

\section{Meson Regge Trajectories}

We can also use the effective string theory to calculate the Regge
trajectories of light mesons by attaching massless
scalar quarks to the ends of a rotating string.~\cite{ron}
(We treat neither chiral symmetry breaking nor string breaking.)
Consider a quark-antiquark pair rotating with uniform angular velocity $\omega$
and separated by a distance $R$.
(See Fig.~\ref{rot string fig}).
\begin {figure}
    \begin{center}
	\null \hfill \epsfbox{string_fig.eps} \hfill \null
    \end{center}
    \caption{The string coordinate system}
    \label{rot string fig}
\end {figure}
%The parameters $\xi = (t, r)$ are the time $t$ and a radial 
%coordinate $r$
%running along the length of the string.
Calculating the classical energy and angular momentum of a straight 
rotating 
string gives the usual linear Regge trajectories.
%However with massless quarks the semiclassical 
%corrections contain an infrared divergence so we retain a quark mass 
%as an infrared cutoff which is put equal to zero after renormalization
To obtain the contribution
of string fluctuations to these classical trajectories
we expand the action $S_{{\rm eff}}[\tilde x^\mu]$
in small fluctuations $ f^i $
about a classical rotating straight string solution.
%$\bar x^\mu(r,t)$.
In D spacetime dimensions there are
%dimensions the fluctuation $\theta(r,t)$ is replaced by 
$D-3$ fluctuations perpendicular to the plane of rotation and 
there is 1 fluctuation in the plane of rotation.

The semiclassical calculation of the path
integral \eqnlessref{Wilson loop eff} 
around the classical rotating straight string solution 
with massless quarks
contains an infrared divergence, so we introduce a quark mass 
as an infrared cutoff which can be set equal to zero only after renormalization.  
%similar to the calculation around the static string. However,
%the factor $\Delta_{FP}$ in the measure of the path integral  
%\eqnlessref{Wilson loop eff} plays an important role 
Furthermore, since the classical world sheet is no longer flat,
in addition to the quadratic and linear ultraviolet divergences
present in the calculation of the energy of the string fluctuations
about a static string,
the path integral \eqnlessref{Wilson loop eff} now
contains a term which diverges logarithmically in the ultraviolet 
cutoff $M$. 
This divergence can be absorbed into a renormalization of a term in the
boundary action called
the geodesic curvature \cite{Alvarez:1983}. 
%For a classical rotating string with massless quarks on its ends
%the geodesic curvature is infinite, so that 
%The requirement that the theory is finite in the massless quark limit
%the renormalized coefficient of the geodesic curvature
%must vanish.
%~\cite{ron}

The path integral \eqnlessref{Wilson loop eff}
can then  be evaluated
for any values of the masses of the quarks on the end of the string and
gives the energy levels of the interior of the rotating string for a 
prescribed motion of its ends. The frequency $\omega$ determines the
interquark distance $R$ . These energy levels reduce to the heavy
quark potentials \eqnlessref{static energy} if $\omega$ is set equal 
to zero with $R$ held fixed. 
For massless rotating quarks, these levels can be obtained from
\eqnlessref{static energy} by replacing the length $R$ 
of the string by its proper length $ \pi/\omega $ ,
and by adding the term $ -\omega /2 $, 
which enters because the classical background is not flat.
%Making these modifications in \eqnlessref{static energy}
This gives~\cite{ron} 

\begin{equation}
E_n(\omega) = \frac{\pi\sigma}{\omega} + ( -\frac{D-2}{24} 
+ n )\omega  -\frac{\omega}{2} \,.
\label{hybrid energy ll}
\end{equation}

To obtain the meson energy levels we must also quantize the motion
of the quarks on the ends of the string.~\cite{ron} For massless quarks,
accounting for these boundary fluctuations effectively changes a Dirichlet
boundary condition into a Neumann boundary condition.
The meson energy levels are given by the same expression
\eqnlessref{hybrid energy ll} and the meson angular momentum $J$ is
quantized,
\begin{equation}
J = 
 l + \frac{1}{2} , \kern 1 in l = 0, 1, 2,... \,.
\label{J quant}
\end{equation}

 The value of $\omega$ is given as a function of $J$ through the
classical relation $\omega = \sqrt{\pi\sigma/2J}$. Squaring
both sides of \eqnlessref{hybrid energy ll} and using
the WKB quantization condition \eqnlessref{J quant} yield
the sequence of linear Regge trajectories,
\begin{equation}
E_n^2(l)  = 2\pi\sigma\left(l - \frac{D-2}{24} +n \right) ,
% + O\left( \frac{n^2}{l}\right)\right) ,
 \kern 1 in n =0, 1, 2, ... \,.
\label{E^2 l}
\end{equation}
Corrections to 
\eqnlessref{E^2 l} of order $ n^2/l $ come from the square of the 
term linear in $ \omega $
in \eqnlessref{hybrid energy ll} and from higher order terms 
in the semiclassical expansion.
%For $n^2/l \ll 1$ we can drop these corrections.

Eq. \eqnlessref{E^2 l}, valid in any spacetime dimension , but applicable
only for large angular momentum $l$ gives, for $n=0$, the
contribution of string fluctuations to the leading Regge trajectory.
The ratio $(D-2)/24l$ of the leading semiclassical correction
to the classical term is already small for $l=1$. 
%($1/12$ for $D=4$).
This could provide an explanation for the approximate experimental
linearity of leading light meson Regge trajectories for $l$ 
of the order $1$, in analogy with the expectation from 
\eqnlessref{static energy} that the string behavior of the static
potential should set in at a distance $R \sim 0.5 $ fm.

The energies \eqnlessref{E^2 l} (with $n>0$) of the excited states
of the rotating string give 
rise to daughter Regge trajectories, and the calculation
is applicable when $l$ is much greater than $n^2$ .
According to this picture, linear daughter trajectories should be 
expected only for values of $l$ much greater than 1. This corresponds
to the expected linear behavior of the excited static potentials \eqnlessref{static energy} only at values of $R$ much greater than $0.5$ fm.

%$  Roughly speaking, the experimental data on the light meson leading Regge trajectories are the "$m=0$" analogues of the "$m=$infinity" numerical simulations ~\cite{Sommer, Luscher3} i of the static potential, while experimental data on daughter Regge trajectories would be the "$m=0$" analogue of
%the "$m=$infinity" numerical simulations ~\cite{Kuti}
%of the excited potentials. It would be interesting to
%identify enough particles on daughter trajectories
%to check this expectation.
%in \eqnlessref{hybrid l of E ll}
%$For $D=26$, Eq. \eqnlessref{E^2 l}
%$yields the spectrum
%\begin{equation}
%E^2 = 2\pi\sigma\left(l+n - 1 
%+O\left( \frac{n^2}{l} \right)\right) \, .
%+O\left({n^2}/l \right)\right) \, .
%\label{26 dim spectrum}
%\end{equation}
%The spectrum of energies \eqnlessref{26 dim spectrum}
%, valid in the leading semiclassical approximation, 
%coincides with
%the spectrum of open strings in classical bosonic string theory. 

\section{Summary and Conclusions}

\begin{enumerate}
\item We have found a path, QCD $\rightarrow$  Effective Field Theory of
Dual Superconductivty $\rightarrow$  Effective String Theory, 
 from QCD to effective string theory,
which provides a concrete picture of the QCD string.

%\begin {equation}

%QCD \longrightarrow Effective Field Theory \longrightarrow Effective String Theory 
%\end{equation}
%from QCD to an effective string theory of dual superconductivity.  descibing long distance QCD. This provides a concrete picturie of the QCD string.
\item The derivation of the effective string theory
made no use of the details of the effective dual field theory
from which it was obtained, but the arguments~\cite{Forcrand,tHooft2}
leading from QCD to an effective dual field theory 
description of long distance QCD need to be developed further.
%\item 
%The expression \eqnlessref{E^2 l} for the energy levels of light 
%mesons, for $D=26$, formally coincides with the spectrum
%of the open string in classical bosonic string theory,
%but it is the semiclassical spectrum of an effective string theory 
%valid only for large angular momentum.
\item The effective string theory provides an
understanding of the values of the distances and of
the angular momentum at which string behavior of 
physical quantities sets in.

\end{enumerate}

\section*{Acknowledgements}

I would like to thank Ph. de Forcrand and L. Yaffe for numerous enlightening conversations,
M. L{\"u}scher and P. Weisz for helpful comments, and the organizers 
of this conference for making it so stimulating and pleasant.

%\bibliographystyle{plain}
%\bibliography{newconf}

\begin{thebibliography}{99}


\bibitem{Nambu} Y. Nambu, {\it Phys. Rev.} {\bf D10}, 4262 (1974).

\bibitem{Mandelstam} S. Mandelstam, {\it Phys. Rep.} {\bf 23C}, 245 (1976).

\bibitem{tHooft} G. `t Hooft, in {\it High Energy Physics,
Proceedings of the European Physical Society Conference, Palermo, 1975},
ed. A. Zichichi (Editrice Compositori, Bologna, 1976).

\bibitem{Nielsen+Olesen} H. B. Nielsen and P. Olesen,
{\it Nucl. Phys.} {\bf B61}, 45 (1973).

\bibitem{Forcrand} Ph. de Forcrand and L. von Smekal {\it Nucl. Phys.}
(PS) {\bf 106}, 619 (2002)

\bibitem{tHooft2} G.'t Hooft {\it Nucl. Phys.} {\bf B153}, 141 (1979).
L. Yaffe (private communication)

\bibitem{Baker+Ball+Zachariasen:1991} M. Baker, J. S. Ball and F. Zachariasen,
{\it Phys. Rev.} {\bf D44}, 3328 (1991).

\bibitem{Nora} M. Baker,  J. S. Ball,  N. Brambilla,  G. M. Prosperi and
F. Zachariasen, {\it Phys. Rev.} {\bf D54}, 2829 (1996).

\bibitem{Baker+Steinke2} M. Baker and R. Steinke,
{\it Phys. Rev.} {\bf D63}, 094013 (2001),
.

%\bibitem{ACPZ} E. T. Akhmedov, M. N. Chernodub, M. I. Polikarpov and M. A. Zubkov,
%{\it Phys. Rev.} {\bf D53}, 2087 (1996).

%\bibitem{Polyakov:conformal_paper} A. M. Polyakov, {\it Phys. Lett.}
%{\bf 103B}, 207 (1981).

%\bibitem{Polyakov:book} A. M. Polyakov, {\it Gauge Fields and Strings},
%151--191, (Harwood Academic Publishers, Chur, Switzerland, 1987).


\bibitem{Baker+Steinke3} M. Baker and R. Steinke, {\it Proc. of the
International Symposium on Quantum Chromodynamics and Color Confinement, Osaka, 2000}, ed. H. Suganuma, M. Fukushima and H. Toki (World Scientific,2001), . 

\bibitem{Luscher1} M. L{\"u}scher, K. Symanzik and P. Weisz,
{\it Nucl. Phys.} {\bf B173}, 356 (1980)

\bibitem{Luscher2} M. L{\"u}scher, {\it Nucl. Phys.} {\bf B180}, 317 (1981).

\bibitem{Luscher3} Martin L{\"u}scher and Peter Weisz, .

\bibitem{Sommer} S. Necco and R. Sommer, {\it Nucl. Phys.} {\bf B622},
010 (2002).

\bibitem{Kuti} K. Jimmy Juge, Julius Kuti and Colin Morningstar, .  

\bibitem{ron} M. Baker and R. Steinke, 
 {\it Phys. Rev.} {\bf D65}, 114042 (2002).
.

\bibitem{Alvarez:1983} O. Alvarez, 
{\it Nucl. Phys} {\bf B216}, 125 (1983).


%\bibitem{Pol+Strom} J. Polchinski and A. Strominger, 
%{\it Phys. Rev. Lett.},
%{\bf 67}, 1681 (1991).



%\bibitem{CKNPS} M. N. Chernodub, S. Kato, N. Nakamura, M. I. Polikarpov
%and T. Suzuki,  (1999).

%\bibitem{deVega+Schaposnik} H. J. de Vega and F. A. Schaposnik,
%{\it Phys. Rev. D} {\bf 14}, 1100 (1976).

%\bibitem{Baker+Steinke3} M. Baker and R. Steinke, {\it Proceedings of the

% (2000).



%\bibitem{DHN} R. Dashen, B. Hasslacher and A. Neveu,
%{\it Phys. Rev.} {\bf D10}, 4114 (1974).

%\bibitem{Isgur+Paton} N. Isgur and J. Paton, 
%{\it Phys. Rev. D} {\bf 31}, 2910 (1985).

%\bibitem{Allen+Olsson+Veseli:1998} T. J. Allen, 
%M. G. Olsson and S. Veseli,
%{\it Phys. Lett.} {\bf B434}, 110 (1998).

%\bibitem{GSW} M. B. Green, J. H Schwarz and E. Witten, {\it Superstring Theory}

%\bibitem{LaCourse+Olsson} D. LaCourse and M. G. Olsson, 
%{\it Phys. Rev. D}
%{\bf 39}, 2571 (1989).

%\bibitem{Dubin+Kaidalov+Simonov} A. Yu. Dubin, 
%A. B. Kaidalov and Yu. A. Simonov,
%{\it Phys. Lett.} {\bf 323B}, 41 (1994).


\end{thebibliography}

% appendices

\end{document}
Dear Professors Luscher and  Weisz,

I have read with  with great interest your paper  providing 
quantitative evidence for an effective string theory description of long 
distance QCD . I would like to make a few comments on this paper, and would 
also like to point out its relation to recent work done on effective string 
theory carried out in collaboration with R. Steinke. This letter probably 
won't contain anything that you don't already know, but I thought it might 
be useful to write anyway.

I will first summarize the content of this letter.

1. Your action S_eff (3.2) can be regarded as the action of the effective
string theory of dual superconductivity. The measure in the path integral
of the the effective string theory is determined by the path integral over 
fields of the of the effective field theory of dual superconductivity
describing QCD at distances greater than the flux tube radius. This
interpretation provides a basis for understanding your results as well
as those of Juge, Kuti and Morningstar, (JKM).

2. The string behavior in the static potential sets in at around r = 0.5fm
because, at larger quark-antiquark separations, the semiclassical 
expansion parameter is small.  On the otherhand, the energies of the excited 
states involve shorter wave lengths, and so one must go to larger distances 
before the leading term in the effective string theory dominates. (Higher order
terms in the effective string theory become important when the wave length of 
the string fluctuations are comparable to the flux tube radius.)

3. We have used the effective string theory to calculate, in the leading 
semiclassical approximation, the contribution of string fluctuations to the 
energy levels of a rotating string with massless (scalar) quarks on its ends. 
The measure in the path integral plays an important role since the classical 
world sheet is not flat.  This calculation, valid in any spacetime 
dimension d but applicable only for large angular momentum l, yields,
for d = 26, the spectrum of the open bosonic string in its critical dimension.
For d =4, it gives the leading semiclassical correction to the classical Regge 
formula. The daughter Regge trajectories correspond to the rotational energy 
levels of the the n'th excited mode of the rotating string, and the calculation
is applicable when the angular momentum l is much greater than n^2.

Roughly speaking, the experimental data on the light meson leading Regge
trajectories are the "m =0" analogues of your "m = infinity"  lattice 
calculation of the static potential, while experimental data on daughter 
Regge trajectories would be the "m=0" analogue of the "m = infinity" lattice 
calculation of excited potentials (JKM). The experimental approximate 
linearity of Regge trajectories for l of order 1 corresponds to the 
approximate linear behavior of your lattice calculation of the 
quark-antiquark potential for r greater than 0.5fm.  The semiclassical 
expansion parameter is (d-2)/24l which for d = 4 is already "small" for l =1. 
On the other hand, according to this picture there should be linear daughter 
trajectories only for l greater than n^2. This corresponds to the expected 
linear behavior of excited potentials only at values of r much greater 
than 0.5fm.

The calculations carried out to obtain these results involve just small 
extensions of the calculations in your 1980 paper with Symanzik. 
(Reference [2] of your paper)

The rest of this letter is just an elaboration of the above remarks. 

1. Interpretation of effective string theory as an effective string 
theory of dual superconducting vortices

Using the work of 't Hooft {Nucl. Phys. B153 141 (1979)} one can show that the 
confined phase of a non-Abelian gauge theory can be characterized by a dual 
order parameter ("monopole condensate") which vanishes in regions of space 
where electric color flux is present in the same way that the presence of 
an external magnetic field destroys ordinary superconductivity.  This generic 
description of the confined phase of QCD  provides a basis for describing long
terms of dual gauge potentials and monopole "Higgs" fields which condense in 
the confined phase. Such a theory has classical solutions, which are vortices
of color electric flux at whose center the monopole field vanishes, and it
describes QCD at distances greater than the radius of the vortex (flux tube).

 In (I), Phys. Rev D63 094013 (2001),  we began with such an 
effective dual field theory and showed how to rewrite the path integral of the 
field theory as a path integral over the surfaces on which the monopole 
condensate field vanishes. These surfaces determine the location of vortices,
where dual superconductivity is destroyed and electric color flux can penetrate.
The measure in the path integral of this effective string theory is determined 
by the measure in the field theory path integral and contains a functional 
determinant which makes the path integral invariant under reparameterizations 
of the vortex world sheet. This measure is universal, independent of the form 
of the underlying dual field theory. 

On the other hand, the action of the effective string theory is not universal 
and depends upon parameters, determined by the action of the effective field 
theory describing the dual superconductor. It can be expanded in powers of the 
extrinsic curvature of the world sheet with coefficients 
(string tension, rigidity,... ) whose values are determined by the underlying 
effective field theory. ( For a superconductor on the border between type I 
and type II the rigidity vanishes.)  

The effective action, Eq. (3.2) of your paper, can then be interpreted as 
the action of the effective string theory of dual superconductivity. This would 
provide a physical origin for the coeffiencts appearing in (3.2). According to
this interpretation, your results should carry over to other non-abelian gauge 
groups because the arguments presented here leading to an effective string
string theory of long distance QCD apply equally well to these theories.

2. Estimate of the range of applicability of the effective string theory 
in leading semiclassical order.

Your observed agreement with effective string theory could be an indication
of the small size of the semiclassical expansion parameter in (3.6),

          E_n = (sigma)r + (pi){-(d-2)/24 + n}/r .     (3.6)

For r= 0.5fm the leading semiclassical correction -[(d-2)/24][(pi)/r]
in Eq.(3.6) (with n=0 ) is of the order of the size of the classical
term (sigma)r. Assume that the ratio of these terms determines the size of 
the semiclassical expansion parameter. (This would mean that, at these
distances, terms in the effective action involving the flux tube radius can be 
neglected in the calculation of the ground state energy.) Then the corrections 
to the leading semiclassical approximation should be small for r much greater 
than 0.5fm.  For d=3 the parameter, (pi)(d-2)/{24(sigma)(r^2)}, is even  
smaller and the agreement should be better, as seen in your data.

On the other hand the excited levels E_n, for n > 0, involve wave lengths of 
order r/n. When this wave length is of order of the flux tube radius higher 
order terms in the effective action should become important and modify the 
string behavior (3.6) of the excited state potentials.  At a given distance r, 
this wave length decreases as n increases.  Therefore as n increases, (3.6) 
should be valid only at a correspondingly larger value of r. This expectation 
is compatible with the "observed" spectrum of excited energy levels found 
by Juge,Kuti,and Morningstar.

3. Application of effective string theory to calculate semiclassical
corrections to meson Regge trajectories and comparison with the
spectrum of the open string.

In the paper (I)  with Steinke mentioned above and in a second paper (II)
Phys. Rev. D65 114042 (2002),  we used the 
effective string theory to calculate semiclassical corrections to meson 
Regge trajectories. In these papers we treated the quarks as massive
scalar particles. (Quark spin and chiral symmytry breaking are different 
questions.) The classical Regge formula is obtained by calculating the 
energy and angular momentum of a rotating quark-antiquark pair connected
by a straight string. Setting the quark mass equal to zero gives the 
usual linear trajectories. However the semiclassical corrections contain 
an infrared divergence when the quark mass is set equal to zero so 
we retain the quark mass as an infrared cutoff which is put equal to zero 
at the end after renormalization.  

In carrying out this calculation we used the regularization proceedure of your 
1980 paper with Symanzik in which you obtained a parameterization invariant, 
regularized formula for the leading semiclassical contribution to the path 
integral of the effective string theory. Your result for the functional 
integral in the background of a straight string of length r with fixed  ends 
is given in Eq.(3.3) of your paper,

                 [-det(-{laplacian}]^-(d-2)/2 .           (A)

Our "new" result was an explicit expression for the functional integral
in the background of a rotating string, which we wrote as a product of 
2 determinants.

The first determinant is the same as (A) except the laplacian is defined with
Dirichlet boundary conditions on a cylinder of height r_p, which is the 
proper length of the rotating string.  (For massless quarks r_p = pi/w)

The second determinant arises from the curvature of the classical world sheet
and contains a logarithmic divergence. We showed that this divergence could be 
absorbed into a renormalization of the coefficient of a term in the 
boundary action ( the geodesic curvature). The requirement that the theory 
has a finite limit as the quark mass approaches zero forces the renormalized 
coefficient of the geodesic curvature to be zero. 

The resulting expression for the contribution of string fluctuations to the 
energy levels of mesons composed of zero mass quarks and rotating with angular 
velocity w can be obtained by replacing in Eq.(3.6) the length r of the string 
by its proper length pi/w, and by adding the quantity -w/2. This latter 
quantity is the contribution of the finite part of the second determinant
in the limit of massless quarks.  Making these modifications in (3.6) gives 

  E_n(w)= (sigma)(pi)/w + w(-(d-2)/24 +n)-w/2,           (B)

applicable to massless rotating quarks. 
 
Eq.(B) gives the energy levels of the interior of the string
for a prescribed  motion of its ends. The meson energy levels 
are obtained by quantizing the motion of the quarks on the 
ends of the string. The quantum fluctuations of the ends of the string
excite the interior points, which in turn react back on the 
ends, producing an effective quark-antiquark interaction. We have calculated
the propagator for these boundary fluctuations in the leading semiclassical
approximation.

We find, for massless quarks, that there is no contribution to the meson energy
from the fluctuations in the motion of the ends of the string. The energy of 
the meson comes solely from the interior modes of the string, and its energy 
levels are given by (B), with the angular velocity w determined in terms of 
the angular momentum quantum number l by the WKB quantization condition,

   J = (pi)(sigma)/[2(w)^2] = l + 1/2 ,    l=0,1,2...   .   (C)

(Effectively, by making the boundary dynamical the ends of the strings become 
free so that the Dirichlet boundary condition becomes a Neumann boundary 
condition and there is no longitudinal mode. Angular momentum is quantized 
and the meson momentum is zero.)


 Squaring both sides of (B) and using (C) gives

  (E_n)^2 = 2(pi)(sigma)[l +  n - (d-2)/24 + O(n^2/l) ].     (D)
 
The higher order corrections to (D) come from the square of the term linear
in w in (B) as well from the higher order corrections in (B). We
can drop these terms when l >> n^2 . Under such circumstances the
energy levels depend only upon the sum of the vibrational and rotational 
quantum numbers N = n + l. This reflects the conformal invariance of the
effective string theory in the leading semiclassical approximation. This
invariance is broken both by higher order terms in the expansion of the 
effective action and by higher order terms in the semiclassical expansion of 
the path integral.

Comparison with bosonic string theory.

For d = 26, the spectrum of energies (D) coincides with the spectrum of
open strings in classical bosonic string theory, but it is semiclassical 
spectrum applicable only for large values of the angular momentum l.

Meson Regge trajectories.

Solving (D) for l gives the Regge trajectories,

  l = E^2/[(2pi)(sigma)] + (d-2)/24 - n.               (E)

The first semiclassical correction to the leading Regge trajectory, n = 0,
adds the constant (d-2)/24 = 1/12 to the classical Regge formula. 
The small size of this correction could explain why Regge trajectories are 
linear for values of l of order 1. As mentioned earlier this latter 
experimental fact is analogous to your numerical result that (3.6) describes 
the static potential for distances greater than > 0.5fm.


I wrote this letter in response to questions that you raised in the 
conclusion of your article. You mentioned there the possibility that your 
results might indicate the existence of an exact dual formulation of gauge 
theories in terms of a fundamental string theory.  The arguments presented 
here leading to an effective string theory give a different interpretation of 
your results which is also compatible with the results of JKM.

I hope that some of these comments might suggest to you further
lattice or analytic investigations, and of course I would be very interested 
in any of your thoughts.

Best regards,
Marshall Baker

Dear Professors Luscher and Weisz,

Thank you for your reply to my letter.
I would like to make a few comments
on the points you have raised in your letter.

1. We certainly agree that there is no rigorous derivation of the Abelian 
Higgs model as an effective model for QCD . In fact our dervivation
of an effective string theory made no use of the explicit form of the 
Lagrangian of the dual field theory, although in a couple of our 
papers we did write down the Abelian Higgs model as an illustrative example.
This might have led to confusion. The action  of the effective
field theory  determines the values of the coefficients ( string tension, 
rigidity , ...) in the long distance expansion of the action
of the effective string theory , and if these coefficients
are left as parameters, the specific form of the dual field theory does not 
enter explicitly in the effective string theory.
 
The specific form of the dual field theory was  used in  previous calculations
carried out with Ball and Zachariasen. It is an SU(3)
gauge theory coupling non-abelian dual potentials (dual gauge bosons) 
to 3 scalar octets of monopole fields (dual Higgs fields). In the confined
phase the vacuum expectation values of the 3 Higgs fields form j=1 irreducible
representation of an SU(2) subgroup of SU(3). There is then no SU(3) 
transformation which leaves all three dual Higgs fields invariant so that
the dual SU(3) gauge symmetry is completely broken and the eight Goldstone
bosons become the longitudinal components of the now massive dual gauge bosons.
The three octets of adjoint representation dual Higgs fields are also massive so
the theory then contains no massless particles which is a necessary
condition for quark confinement. The particle masses arise by a dual Higgs 
mechanism, so that this theory describes the confined phase rather than an 
ordinary Higgs phase.

 This theory possesses classical solutions which are narrow tubes
of Z_3 flux confined to the neighborhood of the z axis. At large distances
from the flux tube these solutions become a configuration which is a 
singular gauge transformation of the vacuum configuration ( where the Higgs 
fields are constants and the gauge potentials are zero).  
The unit of Z_3 flux arises from the fact that this gauge transformation 
has a discontinuity along some line (plane) perpendicular to the z axis. The
adjoint representation Higgs fields must be continuous across this line.
This forces the gauge transformation on the two sides of this line to be equal 
up to a Z_3  element of the gauge group. This element, exp{i(pi)n/3}, n=0,1,2,
of Z_3 determines the value of the color (electric) flux contained in the flux 
tube, as measured by the path ordered exponential of the integral of the dual 
potential around a loop surrouunding the z axis.  This flux is the ordinary 
(electric) color flux because the gauge potentials are dual (electric). 
The energy per unit length in this Z_3 flux tube is the string tension.

This is the non-abelian version of dual superconductivity.
In Phys.Rev.D 56, 1400, (1997)  with Ball and Zachariasen,
using this model, we had calculuated all the heavy quark potentials in the 
classical approximation to the dual theory. The quark-antiquark pair were 
coupled to the dual potentials via a Dirac string along the z axis connecting 
the pair. This coupling was proportional to the color hypercharge matrix so 
that one unit of Z_3 flux flowed between the quark and the antiquark.
The requirement that the flux tube configuration has finite energy
forces the dual Higgs fields to vanish along the axis of the flux tube
between the pair. These fields were proportional to the same color matrices
as in the vacuum , and the classical equations of motion reduced to a form 
similar to that of the Abelian Higgs model with one unit of Abelian flux.
(However the dual Abelian Higgs model has flux tube solutions carrying n 
units of flux for any integer n.) In our solution all the dual
Higgs mesons had the same mass which was also about the same as the mass of the 
dual gauge boson (about 950 MEV.) This equality corresponds to a dual
superconductor on the border between typeI and typeII. Because of our special 
color ansatz we could not be sure that we had found the lowest energy solution
to the non-abelian equations. so this was an additional assumption.

The details of the potentials depended upon the specific Lagrangian,
the classical approximation, and the particular color ansatz, although  the 
qualitative features were determined by basically 2 parameters , the string
tension and the flux tube radius , which is the inverse of the dual gluon 
(gauge boson) mass M. The dual Higgs mechanism relates M to the string tension 
and (alpha)_s, $M = \sim \sqrt {(sigma)/(alpha)_s}$. The dual gluon mass  
serves as the ultraviolet cutoff in the dual theory, and (alpha)_s is evaluated 
at a scale of order M.

In the classical approximation the axis of the flux tube is a straight line 
between the quark and the antiquark.

The path integral of the dual theory over all 
configurations containing any vortex line connecting the quark-antiquark pair
determines the contribution of flux tube fluctuations to the quark-antiquark
interaction. The fluctuating vortex line sweeps out a world sheet whose
boundary is the loop Gamma formed from the world lines of the moving 
quark-antiquark pair. The monopole fields vanish on this sheet so that the 
path integral goes over all field configurations for which the monopole fields 
vanish on some sheet bounded by the loop Gamma. This path integral is the
Wilson loop evaluated in the dual theory. 

The surfaces of zeros of the monopole fields locate the positions of the 
vortex sheets in the path integral. To obtain the effective string theory we
 evaluate the Wilson loop by first integrating over all field 
configurations in which the vortex is located on a particular world sheet. This
integration determines the action of the effective string theory. The remaining 
integration over all world sheets gives the path integral of the effective 
field theory  the form of a path integral of an effective string theory 
with short distance string fluctuations cut off at M. 
The details of the dual field theory have been integrated out and do not 
appear in the effective string theory. Th effective string theory derived
from the dual Abelian Higgs model would have the same form , but the values
of the parameters appearing in the expansion of the effective action would 
be different.


This is our effective model for long distance QCD, and now after this lengthy 
introduction I will try to use the work of 't Hooft to connect this model
to QCD. The essential feature of the dual field theory was the presence of 
Z_3 flux tubes, resulting from coupling dual potentials to 3 sets of monopole 
fields in the adjoint representation in a manner so that the dual gauge 
symmetry is completely broken. 't Hooft begins with pure Yang Mills theory,
notes that the usual Wilson loop operator A creates electric flux and measures 
magnetic flux, and then defines an operator B which creates magnetic
flux and measures electric flux.  He begins with a path integral over field 
configurations in a toroidal box with periodic boundary conditions
He then defines Non-Abelian magnetic flux by first choosing a stack 
of parallel planes in the box, for example, the set of all xt planes,
each one of which is labeled by a fixed value of coordinate pair (y,z). 
The boundary of each of these planes is a rectangular loop which winds around 
the box.  The usual periodic boundary conditions require that the gauge 
potentials at the two edges of this loop lying along the t direction be equal 
for all values of x, with the corresponding periodicity conditions on the 
other two edges.

He then considers a more general class of periodic boundary condition,
for which the gauge potentials on the two edges in the t direction are related
by a gauge transformation depending on x, and  for which the potentials
on the two edges in x direction are related by a gauge transformation
depending on t. The values of these  gauge tranformations at the 
edges of the loop  determine 
(by going around the loop in opposite directions) two
gauge two transformations relating the potentials
on diagonally opposite edges. These two gauge transformations
must then be equal up to a Z_3 element of  the gauge group in order that the
potentials obtained by going around the loop in the two direction are equal.
This  twisted boundary condition on the gauge potentials in all xt  planes
forces a unit if Z_3 flux to wind around the box passing through this
stack of xt planes. This can be seen by evaluating the Wilson loop operator 
in a configuration which differs from the vacuum configuration by such a
gauge transformation. This Wilson loop operator then has a value given
by the  Z_3 element determining the twisted boundary condition. That
is, the Wilson loop measures gauge invariant magnetic flux (Z_3 flux).
just as the dual Wilson loop measures electric flux , as seen
in dual field theory where we had an explicit construction of the 
dual Wilson loop in terms of dual potentials. Of course we don't where
where the unit of Z_3 magnetic flux crosses each xt plane.

We now want to construct the dual Wilson loop in term of operators 
defined in the original  Yang Mills theory. to construct a line of Z_3 
magnetic flux along the z axis, we use the same method we used to construct 
a line of Z_3 electric flux in the dual theory. We consider a configuration
which at large distances from the z axis is a gauge transformation of 
the vacuum configuration and which has a discontinuity along some plane
perpendicular to the z axis, for example, the xz half-plane for x<0 , y=0.
We take this gauge transformation to be independent of z and we take the gauge
transformation on the the two sides of this half plane to be equal up to a Z_3
elemnt of the gauge group , y=0.




