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\noindent 
\begin{center}
{\bf \Large Electromagnetic corrections to the hadronic phase shifts
in low energy $\pi^+p$ elastic scattering}
\end{center}

\vspace{1.0cm}
\noindent 
\begin{center}
A. Gashi$^a$, E. Matsinos$^a$$^*$, G.C. Oades$^b$, G. Rasche$^a$$^\dagger$, W.S. Woolcock$^c$
\end{center}

\vspace{0.3cm}
\noindent
\begin{center}

  \noindent\textit{{$^{a}$Institut f\"{u}r Theoretische Physik der
  Universit\"{a}t,\\
  	Winterthurerstrasse 190, CH-8057 Zurich, Switzerland \\
  	$^{b}$\ Institute of Physics and Astronomy, Aarhus University, \\
  	DK-8000 Aarhus C, Denmark \\
  	$^{c}$ Department of Theoretical Physics, IAS, \\
  	The Australian National University, Canberra, ACT 0200, Australia}}
  \\
\end{center}

\noindent 
\vspace{0.7cm}


We calculate for the $s$-, $p_{1/2}$- and $p_{3/2}$-waves the electromagnetic 
corrections which must be subtracted from the nuclear phase shifts obtained from the analysis 
of low energy $\pi^+p$ elastic scattering data, in order to obtain hadronic phase shifts.
The calculation uses relativised Schr\"{o}dinger equations containing the sum 
of an electromagnetic potential and an effective hadronic potential. 
We compare our results with those of previous calculations and qualitatively estimate the uncertainties in 
the corrections.


\vspace{0.7cm}
\noindent {\it PACS numbers:} 13.75.Gx,25.80.Dj

\vspace{0.7cm}
\noindent {\it Keywords:} $\pi N$ elastic scattering, $\pi N$ electromagnetic corrections, $\pi N$ phase shifts

\vspace{0.7cm}
\noindent $^\dagger$ Corresponding author. Electronic mail: rasche@physik.unizh.ch;
  Tel: +41 1 635 5810;  Fax: +41 1 635 5604

\noindent $^*$ Present address: The KEY Institute for Brain-Mind Research, 
University Hospital of Psychiatry, Lenggstrasse 31, CH-8029 Zurich, Switzerland.

\newpage
\begin{center}
\section*{I. INTRODUCTION}
\end{center}

The aim of this paper is to present a careful recalculation of the electromagnetic corrections 
which have to be applied in the analysis of low energy $\pi^+p$ elastic scattering data in order to recover 
the hadronic phase shifts.


For these electromagnetic effects in $\pi N$ scattering ($\pi^+p$ elastic and $\pi^-p$ elastic and 
charge exchange) two different approaches were developed: the dispersion theory method 
and a method using potentials for the hadronic and electromagnetic interactions in a 
relativised Schr\"{o}dinger equation (RSE for short).


The dispersion theory approach was initiated by Sauter [1,2]. It was then extended by the 
NORDITA group. The final results for their corrections are given in Tromborg {\it et al.} [3]. 
In Ref.[4] the same authors present a detailed exposition of the underlying ideas and point 
out their limitations.


The potential theory approach was initiated by Oades and Rasche [5,6] and applied in detail 
by Zimmermann [7,8] to an analysis of the then existing data. The final results appear in the 
Landolt-B\"{o}rnstein tables on $\pi N$ scattering [9].


Both methods have their shortcomings. In the dispersion theory approach the only contribution 
from $t$- and $u$-channel exchange which is taken into account is that from $u$-channel nucleon 
exchange. The rest of the unphysical contributions are omitted. Ref.[4] states: ` We are 
not able to determine or even estimate this term, which is of course a serious drawback of 
our method'. For this reason, Ref.[4] makes it clear that the results of an analysis of data 
for the three measured reactions which uses the corrections given there cannot be used 
to draw any conclusion about isospin symmetry violation. It seems that the fact that a dispersion 
theory method cannot give absolute values of the electromagnetic corrections has not been 
sufficiently recognised. This is an unsatisfactory situation, particularly for the analysis 
of the large amount of data at low energies.


In the potential theory approach  of Refs.[7,8] hadronic potentials were constructed via a RSE 
to reproduce the hadronic phase shifts. These potentials then were used in a RSE, containing 
also the electromagnetic potential, to calculate the electromagnetic corrections. The idea 
is that by this addition of an effective short range hadronic potential and the well known long range 
electromagnetic potential we can estimate the corrections reliably. The weak point here is the 
use of a hadronic potential in a RSE. But it should be noted that
\vspace{0.5cm}
\begin{description}
\item[( i )]  the potential model is the only method that can give a well defined separation between 
hadronic quantities and their electromagnetic corrections;

\item[( ii)] the hadronic potential is used only to calculate the corrections, not to calculate the 
hadronic quantities themselves.
\end{description}

It was indeed possible (Ref.[8]) to analyse the best experimental data of that time  
in a charge independent way, 
using energy dependent hadronic potentials.
The main reasons for recalculating the electromagnetic corrections in the potential model approach 
are the following.

\begin{description}
\item[( i )] The results need to be extended to lower energies, where there is now a 
large amount of new data obtained at pion factories.

\item[( ii)] The slight effect which a change of the hadronic input has on the electromagnetic 
corrections needs to be controlled. This has not been done by the NORDITA group. 

\item[(iii)] The effect of some fine details of the electromagnetic interaction, which might become important at 
the present precision of the experimental data, needs to be checked. This also is not contained in the NORDITA results.


\item[(iv)] To put the potential model on a firmer basis we want to use energy 
       $independent$ hadronic potentials in an analysis of the experimental data below 100 MeV pion laboratory 
       kinetic energy.
\end{description}

In this paper we treat only the single channel problem of $\pi^+p$ elastic
scattering. It enables us to describe the formalism carefully and to
provide a survey of the relative importance of the contributions to the
corrections. In a second paper we will then apply the corrections
to a phase shift analysis. In further papers the method will be
applied to $\pi^-p$ elastic and charge exchange scattering.


In Sec.II we give a detailed description of our model for the electromagnetic
corrections to $\pi^+p$ scattering. In Sec.III we explain the method of evaluation of
the corrections in this model and in Sec.IV we give the final numerical results
for these corrections. In the next paper we apply these corrections to a detailed
analysis of the existing experimental $\pi^+p$ elastic scattering data below 100 MeV.
\begin{center}
\section*{II. DETAILED DESCRIPTION OF THE MODEL}
\end{center}

We begin this section by writing down the decomposition
of the $\pi^+p$ elastic spin-non-flip and spin-flip scattering amplitudes into
an electromagnetic and a nuclear part:

\begin{equation}
f=f^{em} + \sum_{l=0}^{\infty} \{ (l+1)f_{l+}+lf_{l-} \} P_{l}\, ,
\end{equation}
\begin{equation}
g=g^{em} + i \sum_{l=1}^{\infty} ( f_{l+}-f_{l-} ) P_{l}^{1}\, .
\end{equation}
The bulk of $f^{em}$ comes from the long range point charge Coulomb
interaction. In Born approximation, taking into account
relativistic corrections, this is given by
\[
{2q_c\eta}/{t} \, .
\]
Here $q_c$  is the c.m. momentum; in terms of
the c.m. total energy $W$ and the masses $m_p$  of the proton
and $\mu_c$ of the charged pion it is given by
\begin{equation}
q_c^2=\frac{[W^2-(m_p-\mu_c)^2]\,[W^2-(m_p+\mu_c)^2]}{4W^2} \, .
\end{equation}
The Mandelstam variable $t$ can be expressed in terms of the c.m.
scattering angle $\theta$ by
\[
t=-2q_c^2(1-\cos{\theta})
\]
and the parameter $\eta$  is
\begin{equation}
\eta=\alpha \frac{[W^2-m_p^2-\mu_c^2]}{2q_cW} \, .
\end{equation}
As $q_c\rightarrow 0$, $2q_c\eta /t$ reduces to the standard nonrelativistic result.

From the theory of treating the point charge Coulomb
potential in the presence of short range hadronic interactions
as developed for $pp$ scattering one knows that the most
consistent way to treat the higher order terms in $\eta$ 
is to replace $2q_c\eta/t$ by 
\[
\frac{2q_c\eta}{t}\exp{\{2i\sigma_0-i\eta \ln{(\sin^2{\frac{1}{2}}\theta)}\}} \, ,
\]
where
\[
\sigma_l=\arg{\Gamma(l+1+i\eta)} \, .
\]
This expression is the point charge Coulomb amplitude in
the absence of any short range interaction, to all orders in $\eta$. There is no 
point charge Coulomb contribution to $g^{em}$.


To obtain the complete expressions for $f^{em}$ and $g^{em}$ we take as 
our starting point the one photon exchange amplitudes modified by the 
pion and proton form factors $F^{\pi}$ and $F_1^p$, $F_2^p$ respectively. The most
realistic assumption one can make is that the full amplitudes
$f^{em}$  and $g^{em}$  are obtained from these modified amplitudes
by multiplying them by the phase factor
\[
\exp{\{2i\sigma_0-i\eta \ln (\sin^2\frac{1}{2}\theta)\}} \, .
\]
This procedure has been carefully discussed in Ref.[10]
and differs slightly from the one adopted in Ref.[3]. 
It is then convenient and usual to take out a common factor
\[
\exp{\{2i\sigma_0\}}
\]
from the full amplitudes $f$ and $g$ of Eqs.(1,2), since it does not
appear in any observable and $\sigma_l-\sigma_0$ can be written in the simple form 
\[
\sigma_l-\sigma_0=\sum_{n=1}^{l} \arctan (\frac{\eta}{n}) \, .
\]

The final result for the amplitudes $f^{em}$, $g^{em}$  is then 
\begin{equation}
f^{em}=f^{pc}+f^{ext}+f^{rel} \, ,
\end{equation}
\begin{equation}
g^{em}=g^{pc}+g^{ext}+g^{rel} \, ,
\end{equation}
with
\[
f^{pc}=\exp{\{i\Phi^c\}}\frac{2q_c\eta}{t} \, ,
\]
\[
f^{ext}=\exp{\{i\Phi^c\}}\frac{2q_c\eta}{t}(F^{\pi}F_1^p-1) \, ,
\]
\[
f^{rel}=\exp{\{i\Phi^c\}}\frac{\alpha}{W}\{ \frac{W+m_p}{E+m_p}F_1^p+2(W-m_p+\frac{t}{4(E+m_p)})F_2^p\}F^{\pi} \, ,
\]
\[
g^{pc}=g^{ext}=0 \, ,
\]
\[
g^{rel}=\exp \{i\Phi^c\}\frac{i\alpha}{2W\tan{\frac{1}{2}\theta}}\{ \frac{W+m_p}{E+m_p}F_1^p+2(W+\frac{t}{4(E+m_p)})F_2^p\}F^{\pi} \, ,
\]
\[
\Phi^c=-\eta \ln{(\sin^2 \frac{1}{2}\theta)} \, ,
\]
where
\[
E=(m_p^2+q_c^2)^{1/2} \, .
\]

In the decomposition (5) of $f^{em}$ the term
$f^{pc}$  represents the relativised point charge Coulomb 
amplitude, correct to all orders in $\eta$. The contribution $f^{ext}$ takes 
account of the finite charge distributions and $f^{rel}$, $g^{rel}$
contain further relativistic corrections and the effect
of the proton anomalous magnetic moment.
We use simple dipole form factors
\[
F_1^p(t)=(1-t/\Lambda_p^2)^{-2}  \, ,
\]
\[
F_2^p(t)=\frac{\kappa_p}{2m_p}F_1^p(t) \, ,
\]
\[
F^{\pi}(t)=(1-t/\Lambda_{\pi}^2)^{-2} \, ,
\]
with
\[
\Lambda_p=805 \, \textnormal{MeV}   \, \, , \, \, \Lambda_{\pi}=1040 \, \textnormal{MeV}.
\]
The parameters $\Lambda_p$ and $\Lambda_{\pi}$ are chosen to correspond 
to the measured charge radii of the proton and charged pion, which can be found in Refs. [11,12].
All numerical constants not given explicitly (e.g. the anomalous magnetic moment $\kappa_p$ 
of the proton) are taken from Ref.[13].



Expanding $f^{pc}$ into partial wave amplitudes gives the point charge
Coulomb phases $\sigma_l$, which we have already defined.
Expanding $f^{ext}$ gives phases $\sigma_l^{ext}$, which to order 
$\alpha$ are
\begin{equation}
\sigma_l^{ext}=\eta q_c^2\int_{-1}^{+1}dz \,t^{-1}P_{l}(z)(F_1^pF^{\pi}-1) \, .
\end{equation}
Expanding $f^{rel}$, $g^{rel}$ gives phases $\sigma_{l\pm}^{rel}$, which to order $\alpha$ 
are
\begin{eqnarray}
\sigma_{l\pm}^{rel}=-\frac{\alpha q_cm_p}{W}\int_{-1}^{+1}dz \, P_{l}(z)F_2^pF^{\pi} \pm \frac{\alpha q_c}{4W(l\pm 1/2+1/2)} \times \nonumber \\
  \int_{-1}^{+1}dz(P_{l}^{'}(z)+P_{l\pm 1}^{'}(z))\{\frac{W+m_p}{E+m_p}F_1^p+(W+\frac{t}{4(E+m_p})2F_2^p\} F^{\pi} \, .
\end{eqnarray}

We now write the
partial wave amplitudes 
\begin{equation}
f_{l\pm}=\exp(2i\Sigma_{l\pm})\frac{\exp {(2i\delta^n_{l\pm})}-1}{2iq_c} \, ,
\end{equation}
where up to terms of higher order in $\alpha$
\[
\Sigma_{l\pm}=(\sigma_l-\sigma_0)+\sigma_l^{ext}+\sigma_{l\pm}^{rel}.
\]
The difference $\sigma_l-\sigma_0$ appears in $\Sigma_{l\pm}$ because the factor 
$\exp(2i\sigma_{0})$ has been removed from the full amplitudes $f$, $g$ of Eqs.(1,2). 
In writing Eq.(9) with real $\delta_{l\pm}^n$ we are ignoring the minute inelasticies due to bremsstrahlung. They are 
however included in the analysis of the experimental data described in the next paper. 


The quantities $\delta_{l\pm}^n$ are called nuclear phase shifts.
They are decomposed further into the hadronic phase shifts 
$\delta_{l\pm}^h$, which will later be defined precisely, and further electromagnetic corrections $C_{l\pm}$:
\begin{equation}
\delta_{l\pm}^n=\delta_{l\pm}^h+C_{l\pm} \, .
\end{equation}
The quantity $C_{l\pm}$  can, again up to higher terms in $\alpha$, be
split up in the following way:
\begin{equation}
C_{l\pm}=C_{l\pm}^{pc}+C_{l\pm}^{ext}+C_{l\pm}^{rel}+(\sigma_l^{vp}+C_{l\pm}^{vp}) \, .
\end{equation}
The phase shift $\sigma_l^{vp}$  is due to the Uehling potential for
vacuum polarisation. Though this effect is of higher order in $\alpha$
we consider it because the Uehling potential has an extremely long range
compared to the hadronic interaction. It has to be taken
into account in the phases $\delta_{l\pm}^n$  because for this part
of the electromagnetic interaction no amplitude has been
included in $f^{em}$  and $g^{em}$. The quantities $C_{l\pm}^{...}$ arise from
the interference between the hadronic interaction and
those parts of the electromagnetic interaction indicated in
the superscripts.

To calculate the $C_{l\pm}^{...}$  we first of all need for each partial wave potentials which represent
the different contributions to $f^{em}$, $g^{em}$. For $f^{pc}$ (and
the $\sigma_l$) this potential is, omitting relativistic effects to be discussed later,
\begin{equation}
V^{pc}(r)=\alpha /r \, ,
\end{equation}
where $r$ is the distance between $\pi^+$ and $p$ in the c.m.
system.
For $f^{ext}$ (and the $\sigma_l^{ext}$) the dipole form factors determine
a definite charge distribution for $\pi^+$ and  $p$  and therefore a 
definite Coulomb potential for the extended charges. 
The deviation of this potential from the point charge Coulomb potential
we call $V^{ext}$. This can be calculated, but is very complicated.
In practice we find that different charge distributions
(e.g. uniform, gaussian,...) give practically the same $C_{l\pm}^{ext}$,
provided that the charge radii ($<r^2>^{1/2}$) of $\pi^+$ and  $p$
remain always at their experimentally determined values. 
We found it convenient to assume gaussian charge distributions for which 
\begin{equation}
V^{ext}(r)=\alpha /r \{ \textnormal{erf}(r/c)-1 \} \, , 
\end{equation}
where
\[
c^2=2/3 \{ <r^2>_p^{\frac {1}{2}}+<r^2>_{\pi^+}^{\frac {1}{2}} \} \, .
\]
Since the smearing out of the charges is a hadronic effect,
$V^{ext}$ is of short range.


For $f^{rel}$ and $g^{rel}$ (and the $\sigma_{l\pm}^{rel}$) no potential is given {\it a priori}. From
Austen and de Swart [14] we see that $V_{l\pm}^{rel}$ contains
a short range part independent of ($l\pm$) and a part which                                   
behaves like $r^{-3}$ at large $r$ (and is modified at small $r$ because of 
the extended charge distributions)
and depends on ($l\pm$). 
Empirically we find that it is easy  to construct a very short range
energy independent potential $V_0^{rel}$  which reproduces
$\sigma_0^{rel}$ below 100 MeV pion lab kinetic energy. By adding
to $V_0^{rel}$  a function which behaves like $r^{-3}$ for large $r$ and is 
modified for small $r$ to take account of the charge distributions, 
it is possible to fit $\sigma_{1\pm}^{rel}$.


We see from the decomposition of $C_{l\pm}$ that we also need the
Uehling potential for vacuum polarisation. In the notation
of Durand [15] this is given for point charges by
\begin{equation}
V^{vp}(r)=V^{pc}(r)I(2 m_e r) 2\alpha / 3\pi \, ,
\end{equation}
$m_e$  being the electron mass. The explicit integral representation
of $I$ can be taken from Ref.[15]. In our numerical calculation we also
include the tiny short range effect of the extended charges on $V^{vp}$.
Finally we need effective short range hadronic potentials $V_{l\pm}^h$ to represent
the hadronic interaction leading to the hadronic phase shifts $\delta_{l\pm}^h$. These potentials we
adjust so as to reproduce the energy dependence of the $\delta_{l\pm}^h$ over the low energy range.
Details of the construction of these potentials will be given in Sec.III.

Assuming the potentials to be known we can then solve various 
partial wave RSEs and determine the scattering phase shifts. Dropping
the index $l\pm$  for the moment we call $\delta (V)$ the
phase shift for the potential $V$. It is given
by the asymptotic behaviour for $r\rightarrow\infty$ of the regular radial wave function.
Up to a factor this is $\sin (q_cr-l\pi/2+\delta(V))$ for a finite range
potential ( \( \lim_{r\rightarrow\infty} rV(r)=0 \) ) and $\sin (q_cr-\eta \ln (2q_cr)-l\pi/2+\delta(V))$
if $V$ contains $V^{pc}$.
In this notation we have immediately $\delta(V^h)=\delta^h$, $\delta(V^{pc})=\sigma$, $\delta(V^{ext})=\sigma^{ext}$, 
$\delta(V^{rel})=\sigma^{rel}$ and $\delta(V^{vp})=\sigma^{vp}$.
The full potential $V$ is then written as the sum
\begin{equation}
V=V^{em}+V^h \, ,
\end{equation}
where $V^h$ is the effective hadronic potential in the presence of 
the electromagnetic interaction. Moreover $V^{em}$ has the decomposition
\begin{equation}
V^{em}=V^{pc}+V^{ext}+V^{rel}+V^{vp} 
\end{equation}
and the corrections $C^{...}$ are defined by
\[
\delta(V^h+V^{pc})=\delta^h+\sigma+C^{pc}\, ,
\]
\[
\delta(V^h+V^{ext})=\delta^h+\sigma^{ext}+C^{ext}\, ,
\]
\[
\delta(V^h+V^{rel})=\delta^h+\sigma^{rel}+C^{rel}\, ,
\]
\[
\delta(V^h+V^{vp})=\delta^h+\sigma^{vp}+C^{vp}\, .
\]
Neglecting terms of higher order in $\alpha$  we have
\begin{equation}
\delta(V)=\delta^h+\sigma+\sigma^{ext}+\sigma^{rel}+\sigma^{vp}+C^{pc}+C^{ext}+C^{rel}+C^{vp} \, .
\end{equation}
From Eqs.(10,11) we see that the
nuclear phase $\delta^n$ is obtained by subtracting $\sigma+\sigma^{ext}+\sigma^{rel}$ from the last
expression.
\vspace{0.5cm}
{\centering \section*{III. METHOD OF EVALUATION OF THE CORRECTIONS}}

To calculate the corrections $C_{l\pm}$ we have to integrate 
the partial wave RSEs
\begin{equation}
(\,\frac{d^2}{dr^2}-\frac{l(l+1)}{r^2}+q_c^2-2m_cf_{c}V_{l\pm}(r)\,)\,u_{l\pm}(r)=0
\end{equation}
for $l=0$ and $l=1$. 
The full potentials $V_{l\pm}$ are assumed to have the form given in Eqs.(15,16).
Minor relativistic effects are accounted for by $V^{rel}_{l\pm}$.
The main relativistic effects are included through the expression (3) for $q_c^2$ and 
through the relativistic amplification factor $f_c$ which converts the nonrelativistic 
point charge Coulomb amplitude in Born approximation into the Born approximation of $f^{pc}$.
 By comparing the relativistic 
expression for $\eta$ in Eq.(4) with the usual nonrelativistic expression, the factor $f_c$ 
is seen to be
\begin{equation} 
f_c=\frac{W^{2}-m_p^2-\mu_c^2}{2m_cW} \, ,
\end{equation}
where $m_c$ is the reduced mass of the $\pi^{+}p$ system:
\[
m_c=\frac{m_p\mu_c}{m_p+\mu_c} \, .
\]
The factor $f_c$ is of course $1$ for $W=m_p+\mu_c$.

In order to get $f^{pc}$ from the RSEs (18) 
with the potential $V^{pc}$ of Eq.(12), it is clear that the factor $f_c$ is required in 
that case and it is a natural assumption that the same is true for the other parts $V^{ext}$, 
$V^{rel}$ and $V^{vp}$ of the electromagnetic potential. That the same factor applies also for 
$V^{h}$ is less obvious. However, Auvil [16] has given persuasive arguments for treating the 
hadronic potentials in the same way as the Coulomb potential and this is the reason why we use 
the factor $f_c$ in Eq.(18) in order to boost the full potentials $V_{l\pm}$. Zimmerman [8] also 
applies a relativistic boost factor to the full potentials; his choice of the factor is almost the 
same as ours (see Eq.(35) of Ref.[10]).

With the initial conditions $u_{l\pm}(0)=0$, $u^{'}_{l\pm}(0)\neq 0$ (but arbitrary), Eq.(18) 
is integrated outwards from $r=0$ to a distance $R$ where $V^h+V^{ext}+V^{rel}+V^{vp}$ is negligible 
and only $V^{pc}$ remains. Because of the extremely long range of $V^{vp}$ this is around 1000 fm. 
At $r=R$ we match this solution to a linear combination of point charge Coulomb wavefunctions 
$F_l(\eta;q_cr)$ and $G_l(\eta;q_cr)$:
\[
u_{l\pm}(r)=a_{l\pm}F_l(\eta;q_cr)+b_{l\pm}G_l(\eta;q_cr), \; r\geq R \, .
\]
Since $F_l$ and $G_l$ already contain the phase $\sigma_l$  in 
their asymptotic behaviour for large $r$, we
have   
\begin{equation}
\frac{b_{l\pm}}{a_{l\pm}}=\tan(\delta_{l\pm}-\sigma_l) \, ,
\end{equation}
where $\delta_{l\pm}$ (with the index now included) is the same as $\delta (V)$ in Eq.(17).
Eqs.(10,11,17) then show that the nuclear phase shifts $\delta_{l\pm}^n$ are determined by subtracting 
$\sigma_l^{ext}+\sigma_{l\pm}^{rel}$ from $\delta_{l\pm}-\sigma_l$.

Integrating Eq.(18) with the short range potentials $V_{l\pm}^h$ replacing the full potentials $V_{l\pm}$ 
and matching the solutions $u^h_{l\pm}$ to linear combinations of $q_crj_l(q_cr)$ and $q_crn_l(q_cr)$ at a 
distance outside the range of $V_{l\pm}^h$ leads to the phase shifts $\delta_{l\pm}^h$. This is the precise 
definition of the hadronic phase shifts mentioned just before Eq.(10). The electromagnetic corrections $C_{l\pm}$ 
are just the differences between $\delta_{l\pm}^n$ and $\delta_{l\pm}^h$.

To perform the procedure described above we need to know the various components of the potentials in Eq.(16). The 
explicit forms of $V^{pc}$, $V^{ext}$ and $V^{vp}$ have been given in Eqs.(12,13,14) and the method of construction of 
the potentials $V^{rel}$ has been described. The hadronic parts $V^h_{l\pm}$ of the potentials have to be determined 
so as to reproduce the hadronic phase shifts $\delta_{l\pm}^h$. Since, as emphasised in Sec.I, the hadronic potentials are used 
only to calculate the corrections, we do not give them in detail. They contain a range parameter which we have fixed at 1 fm; 
then for each partial wave we used for $V_{l\pm}^h$ a parametric form with three further parameters.
These were varied in order to get 
the best possible fit to the hadronic phase shifts $\delta_{l\pm}^h$ up to 100 MeV pion lab kinetic energy. As 
discussed in the next paper, the $\delta_{l\pm}^h$ come from the analysis of the experimental data in a parametrised form, 
so the potentials provide nothing else but an alternative parametrisation which is needed for the calculation of the corrections. 
The potentials reproduce the hadronic phase shifts within their experimental errors.
Details can be found in Ref.[17].

The starting point in an iterative procedure to obtain the electromagnetic corrections was to fit the $l=0$ and $l=1$ 
hadronic phase shifts from the analysis of Arndt {\it et al}. [18] using the parametric forms for $V_{l\pm}^h$. Integrating the 
RSE (18) for $l=0$ and $l=1$ with the full potentials $V_{l\pm}$ and with $V_{l\pm}$ replaced by $V_{l\pm}^h$, preliminary 
values of the corrections $C_{l\pm}$ where calculated as described above:
\begin{equation} 
\delta_{l\pm}^n=\delta_{l\pm}-\sigma_l-\sigma_l^{ext}-\sigma_{l\pm}^{rel} \, ,
\end{equation} 
\begin{equation} 
C_{l\pm}=\delta_{l\pm}^n-\delta_{l\pm}^h \, .
\end{equation} 
These preliminary values of $C_{l\pm}$ were then fed into a phase shift analysis of the experimental data on $\pi^+ p$ 
elastic scattering up to 100 MeV pion lab kinetic energy. The full details of this analysis are given in the next paper 
(fitting methods, selection of data, smoothing procedure, $d$- and $f$-wave phase shifts used as input, uncertainties in 
the hadronic phase shifts determined from the data).

The values of the three hadronic phase shifts $\delta_{0+}^h$, $\delta_{1\pm}^h$ obtained from this analysis were then 
used to construct three new hadronic potentials $V_{0+}^h$, $V_{1\pm}^h$, from which new values of the three corrections 
$C_{0+}$, $C_{1\pm}$ were calculated just as before. These corrections were used in a repeat of the phase shift analysis, 
giving new hadronic phase shifts from which further hadronic potentials and electromagnetic corrections were calculated. 
This iterative procedure was continued until the hadronic phase shifts $\delta_{0+}^h$, $\delta_{1\pm}^h$ no longer changed 
within the errors arising from the phase shift analysis of the experimental data. In practice two iteration steps were 
sufficient. 

A technical comment needs to be made already here about the treatment of partial waves with $l\geq 2$. The hadronic phase shifts for 
$l\geq 2$ are known to be very small in our energy region and even for $l=2$ and $l=3$ they are not well determined. The values 
actually used in the analysis are discussed in the next paper. The potentials $V_{l\pm}^h$ are therefore very small 
and the corrections $C_{l\pm}^{...}$ negligible for $l\geq 2$, so that from Eq.(11)
\[
C_{l\pm} \approx \sigma_{l}^{vp} \, \, \, , \, \, l\geq 2 \, .
\]
However, for $l=2$ and $l=3$ the values of $\sigma_l^{vp}$ are much smaller than the uncertainties in the values of 
$\delta_{l\pm}^h$, so that we neglect the corrections $C_{l\pm}$ altogether for these partial waves. 
Partial waves with $l>3$ are omitted from the analysis.

A very important observation of a conceptual nature also needs to be made. We have called the quantities $\delta_{l\pm}^h$ 
hadronic phase shifts, but they are obtained from a phase shift analysis which contains the \textit{physical} masses 
of $\pi^+$ and $p$. It is therefore clear that the $\delta_{l\pm}^h$ cannot be considered to be strictly hadronic quantities. 
Such quantities relate to a situation in which the electromagnetic interaction is switched off ($\alpha =0$) and therefore need 
to be obtained using the hadronic masses of $\pi^+$ and $p$. These hadronic masses are not the same as their physical masses
 (indeed it is universally accepted that the hadronic mass of $\pi^+$ is very close to the physical mass 
of $\pi^0$). In the same sense we have noted after Eq.(15) that the quantities $V_{l\pm}^h$ are effective hadronic potentials 
in the presence of the electromagnetic interaction. They would be different in the complete absence of all electromagnetic interactions.
Therefore in our present work, as in all previous work known to us, we avoid speculation 
about the strictly hadronic situation and give quantities that we call hadronic phase shifts and 
electromagnetic corrections which, though they have a precise definition within the framework 
of our potential model, are not the full corrections that would give truly hadronic phase shifts 
by subtraction from the nuclear phase shifts. Any attempt to completely purge the experimental data of all electromagnetic 
effects would be very  speculative. Our aim is the more modest one of taking account of those electromagnetic 
effects that may be calculated with reasonable confidence.

\vspace{0.5cm}
{\centering \section*{IV. NUMERICAL RESULTS FOR THE CORRECTIONS}}


The final results for the three electromagnetic corrections $C_{0+}$, $C_{1\pm}$ are given in Table 1 
from 10 to 100 MeV pion lab kinetic energy.
The accuracy of the final numerical calculation within our potential model as described in the previous 
sections justifies the three decimal places given in the table. In estimating the uncertainties in the 
results of Table 1 we have to distinguish between a) uncertainties coming from applying our model and 
b) uncertainties coming from the choice of the model itself.

One source of a) can be the uncertainty in the hadronic phase shifts, which comes from the analysis 
of the experimental data (experimental errors in the data and uncertainty in the $d$- and $f$-wave phase 
shifts used as input). This uncertainty can be estimated by means of the changes which occur with 
the successive iteration steps described 
in Sec.III. It turns out that the effect on the numbers of Table 1 is negligible. Another source 
of a) can be the use of a particular form of the parametrized hadronic potentials. For the final 
numerical calculations we have fixed the range parameter at 1 fm. Varying it between reasonable limits
also has negligible effect on the corrections. This gives an estimate of the uncertainties coming from 
the fact that some of the fine details of the potentials may have been missed.

The main source of b) is likely to be the assumption that the $V_{l\pm}^h$ are chosen to be energy independent. 
Our experience shows that using energy dependent hadronic potentials can also result in small but 
negligible changes in the corrections. Another source of b) could be the choice of the RSE (18), but as emphasized in 
Ref.[14], it is the only relativistic equation which is well suited for a two-body problem.
In summary we can say that the uncertainties in the corrections are negligible for $C_{0+}$ and $C_{1-}$ and 
for $C_{1+}$ are comparable with the errors in $\delta_{1+}^h$ given in the next paper.

In Table 2 we give (at a smaller set of energies) the various contributions to the corrections (Eq.(11)).
These components are additive to a very good approximation and tiny differences between the sums of the 
numbers in Table 2 and the complete corrections in Table 1 are due to higher order interferences 
between the corrections themselves.

We now compare our results with those of the dispersion theory approach used by NORDITA [3]. It 
is clear from their Ref.[4] that they have not included the vacuum polarisation contribution and thus we must 
compare their results with the sum of the first three numbers in each row of Table 2.
For $C_{0+}^{pc}+C_{0+}^{ext}+C_{0+}^{rel}$ we see that our results agree very well with those of NORDITA, the difference being considerably 
smaller than the error in $\delta_{0+}^h$ given in the next paper. Vacuum polarisation effects are 
negligible compared to these errors in $\delta_{0+}^h$.
Our results for the sum $C_{1+}^{pc}+C_{1+}^{ext}+C_{1+}^{rel}$ differs systematically from those of 
NORDITA by amounts which are comparable with, but rather larger than,  
the errors in $\delta_{1+}^h$. Again vacuum 
polarisation effects can be neglected.
Our results for $C_{1-}^{pc}+C_{1-}^{ext}+C_{1-}^{rel}$ agree with those of NORDITA at low energies, but at 
100 MeV the discrepancy is roughly twice the error in $\delta_{1-}^h$. However, since $\delta_{1-}^h$ is very small, 
the corrections are of minor importance for the results of the analysis. 
Here vacuum polarisation contributions 
are not negligible and make $C_{1-}$ negative at small energies. 

The corrections $C_{0+}^{pc}+C_{0+}^{ext}$ and $C_{1+}^{pc}+C_{1+}^{ext}$ at $T_{\pi}=90$ MeV, $100$ MeV can also 
be compared to the results of Zimmermann, who uses a slightly different version of the potential model. 
The agreement is quite good, which is an indication of the stability of the results obtained with the potential model.

As we discussed in Sec.1, the calculation using dispersion relations omits what could be important 
medium range effects due to $t$-and $u$-channel exhanges, which the potential model includes quite reliably. 
The differences of our results from those of NORDITA are probably due to such medium range effects.
In particular the NORDITA results for $C_{1+}^{pc}+C_{1+}^{ext}+C_{1+}^{rel}$ 
show a somewhat stronger energy dependence than ours and 
this is likely to be a medium range effect, perhaps due to the omission of $t$-channel $\pi\pi$ 
($T=0$, $J=0$) exchange. We therefore claim a higher degree of reliability for our present 
calculation of the electromagnetic corrections, compared with that of NORDITA.


{\bf Acknowledgements.} 
We thank the Swiss National Foundation and PSI (`Paul Scherrer Institut') for financial support. We acknowledge very interesting discussions with W. R. Gibbs.

\newpage
\noindent
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\end{thebibliography}


\newpage
%\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
 $T_{\pi}$ & $C_{0+}$ & $C_{1+}$ &$C_{1-}$ \\\hline
 10 & 0.063 & -0.034 &  -0.006 \\
 20 & 0.071 & -0.056 &  -0.001 \\
 30 & 0.078 & -0.083 &   0.002 \\
 40 & 0.084 & -0.115 & 0.005  \\
 50& 0.089 & -0.154 & 0.007  \\
 60 & 0.094 & -0.203 & 0.008  \\
 70& 0.098 & -0.264 & 0.009  \\
 80& 0.101 & -0.341 & 0.009  \\
 90 & 0.104 & -0.438 & 0.010  \\
100& 0.106 & -0.556 & 0.010 \\ \hline
\end{tabular}
\end{center}
\vspace{0.5cm}
{\large \bf TABLE 1:}
{Values in degrees of the electromagnetic corrections $C_{0+}$, $C_{1-}$ and $C_{1+}$ 
as functions of the pion lab kinetic energy $T_{\pi}$ (in MeV).}
%\end{table}
\vspace{1cm}
%\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$T_{\pi}$ & $C_{0+}^{pc}$ & $C_{0+}^{ext}$ & $C_{0+}^{rel}$ & $C_{0+}^{vp}$+$\sigma_0^{vp}$ \\\hline
10 & 0.092  & -0.009 & -0.002 & -0.018 \\
20 & 0.103  & -0.014 & -0.004 & -0.016 \\
40 & 0.124  & -0.022 & -0.005 & -0.014 \\
60 & 0.144  & -0.031 & -0.007 & -0.014 \\
80 & 0.162  & -0.040 & -0.009 & -0.014 \\
100 & 0.176  & -0.050 & -0.010 & -0.014 \\ \hline
$T_{\pi}$ & $C_{1+}^{pc}$ & $C_{1+}^{ext}$ & $C_{1+}^{rel}$ & $C_{1+}^{vp}$+$\sigma_1^{vp}$ \\\hline
10 & -0.022 & 0.000 &  0.000 & -0.011 \\
20 & -0.047 & 0.001 &  0.000 & -0.010 \\
40 & -0.104 & 0.002 & -0.004 & -0.010 \\
60 & -0.184 & 0.007 & -0.016 & -0.010 \\
80 & -0.300 & 0.016 & -0.048 & -0.011 \\
100 & -0.465 & 0.035 & -0.118 & -0.012 \\ \hline
$T_{\pi}$ & $C_{1-}^{pc}$ & $C_{1-}^{ext}$ & $C_{1-}^{rel}$ & $C_{1-}^{vp}$+$\sigma_1^{vp}$ \\\hline
10 & 0.004 & 0.000 &  0.000 & -0.011 \\
20 & 0.008 & 0.000 &  0.000 & -0.010 \\
40 & 0.013 & 0.000 &  0.000 & -0.009 \\
60 & 0.017 & 0.000 &  0.000 & -0.009 \\
80 & 0.020 & -0.001 & -0.002 & -0.009 \\
100 & 0.023 & -0.001 & -0.004 & -0.009 \\ 
\hline
\end{tabular}
\end{center}
\vspace{0.5cm}
{\large \bf TABLE 2:}
{Values in degrees of the various contributions to the electromagnetic corrections 
$C_{0+}$, $C_{1-}$ and $C_{1+}$ 
as functions of the pion lab kinetic energy $T_{\pi}$ (in MeV).}
%\end{table}
\end{document}

