Current conservation, screening and the magnetic moment of the $\Delta$
resonance. 111 Supported by the ”Deutsche Forschungsgemeinschaft” under
contract GRK683

1\. Formulation without quark degrees of freedom

A. I. Machavariania b c and Amand Faessler a

a Institute für Theoretische Physik der Univesität Tübingen,
Tübingen D-72076, Germany

b Joint Institute for Nuclear Research, Dubna, Moscow region 141980, Russia

c High Energy Physics Institute of Tbilisi State University, University str.
9, Tbilisi 380086, Georgia

###### Abstract

The pion-nucleon bremsstrahlung
$\pi+N\Longrightarrow\gamma^{\prime}+\pi^{\prime}+N^{\prime}$ is studied in a
new form of current conservation. According to this condition, the internal
and external particle radiation parts of the $\pi N$ radiation amplitude have
opposite signs, i.e., they contain terms which must cancel each other.
Therefore, one has a screening of the internal and external particle radiation
in the $\pi N$ bremsstrahlung. In particular, it is shown that the double
$\Delta$ exchange diagram with the $\Delta-\gamma^{\prime}\Delta^{\prime}$
vertex cancel against the appropriate longitudinal part of the external
particle radiation diagrams. Consequently, a model independent relation
between the magnetic dipole moments of the $\Delta^{+}$ and $\Delta^{++}$
resonances and the anomalous magnetic moment of the proton $\mu_{p}$ is
obtained, where $\mu_{\Delta}$ is expressed by $\mu_{p}$ as
$\mu_{\Delta^{+}}={{M_{\Delta}}\over{m_{p}}}\mu_{p}$ and
$\mu_{\Delta^{++}}={3\over 2}\mu_{\Delta^{+}}$ in agreement with the values
extracted from the fit for the experimental cross section of the
$\pi^{+}p\to\gamma^{\prime}\pi^{+}p$ reaction.

1\. INTRODUCTION

The $\pi N$ bremsstrahlung was extensively investigated in the past in order
to study the electromagnetic properties of the $\Delta$ resonances and their
form factors. The main reason for the determination of the electromagnetic
moments of the $\Delta$ resonances is that on one hand, the $\Delta$’s are
described as a $\pi N$ resonances with the corresponding poles of the $\pi N$
amplitude and, on the other hand $\Delta$’s are often treated as independent
particles in the models of strong interaction. In addition, the quark content
of the proton is the same as for the $\Delta^{+}$, and differs from the quark
content of the $\Delta^{++}$. Therefore, determination of the electromagnetic
moments of the $\Delta$ resonances is important for the definition of the
electromagnetic structure of the nucleons and the $\Delta$ resonances.

In contrast to nucleons, the direct experimental measurement of the
electromagnetic moments of the $\Delta$’s is today impossible. Therefore, the
present experimental electromagnetic moments of the $\Delta$’s are obtained
using a fit to the experimental cross sections of the $\pi N$ bremsstrahlung
[1]-[4]. The analysis of these data by different theoretical models yields
different magnetic moments of the $\Delta$’s. For instance, the magnetic
dipole moment of the $\Delta^{++}$ $\mu_{\Delta^{++}}$ obtained within the
framework of the low energy photon theorem [5]-[12] is
$\mu_{\Delta^{++}}=4.7$-$6.9\mu_{B}$ [2] in nuclear magnetons
$\mu_{B}=e/2m_{N}$, while the potential models yield
$\mu_{\Delta^{++}}=5.6$-$7.5\mu_{B}$ [3] or $\mu_{\Delta^{++}}=4.5\pm
0.95\mu_{B}$ [4]. The theoretical results for different models fitted to the
experimental data [5]-[26] are shown in Table 1 in the conclusions of this
paper. These results indicate substantial discrepancies between the different
predicted values for $\mu_{\Delta^{++}}$ and $\mu_{\Delta^{+}}$.

In this paper an analytic and model-independent relation for the magnetic
moments of the $\Delta$ resonances is suggested. This relation is based on a
new form of current conservation for the total on mass shell and on energy
shell amplitude of the $\pi N$ bremsstrahlung ${\cal
A}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$. The corresponding
current conservation

$k^{\prime}_{\mu}{\cal A}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}=k^{\prime}_{\mu}{\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}+{\cal B}_{\pi^{\prime}N^{\prime}-\pi N}=0$ $None$

consists of the external particle radiation amplitude ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ depicted in Fig. 1 and
the sum of the off shell $\pi N\to\pi^{\prime}N^{\prime}$ scattering
amplitudes ${\cal B}_{\pi^{\prime}N^{\prime}-\pi N}$. This condition is
obtained in the same approach as the Ward-Takahashi identities in the usual
quantum field theory [28, 29]. Using current conservation for the total $\pi
N$ radiation amplitude

$k^{\prime}_{\mu}{\cal A}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}=k^{\prime}_{\mu}{\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}+k^{\prime}_{\mu}{\cal I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}=0.$ $None$

for the on shell external and internal particle radiation amplitudes ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ (Fig.1) and ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ (Fig.2A) one can
represent (1.1a) as

$k^{\prime}_{\mu}{\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}=-k^{\prime}_{\mu}{\cal I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}=-{\cal B}_{\pi^{\prime}N^{\prime}-\pi N}$ $None$

which determines an additional relation between the on shell external and
internal particle radiation amplitudes.

Figure 1: Diagrams describing the external particle radiation amplitude ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ in (1.1a,b). Diagrams
A and C correspond to the radiation of the external nucleons. Diagrams B and D
describe the emission of the photon by the external pions. The hatched circle
indicates the off shell $\pi N$ elastic scattering amplitudes (2.9a,b,c,d).
$N"$ and $\pi"$ denote the intermediate nucleon and pion states.

Thus the problem of the validity of current conservation (1.1a) is reduced to
the determination of the internal particle radiation amplitudes (Fig. 2A)
${\cal I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ which satisfy
the condition (1.1b).

We shall show that the $\Delta$ radiation amplitude ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
in Fig. 2B and the corresponding part of the external particle radiation
amplitude ${\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ (Fig.
1) denoted as $({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ satisfy current conservation

$k^{\prime}_{\mu}\biggl{[}{\cal
A}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}\biggr{]}^{Projection\
on\ spin\ 3/2\ particle\ states}_{2\Delta\ exchange\ with\
\Delta-\gamma^{\prime}\Delta^{\prime}\ vertex}=$ $k^{\prime}_{\mu}({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)+{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ $None$

or

$k^{\prime}_{\mu}({{\cal
E_{L}}^{3/2}})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)=-{k^{\prime}}_{\mu}{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)=-{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta),$ $None$

where the lower index L and the upper index 3/2 denote the longitudinal and
the spin-isospin $(3/2,3/2)$ part of the corresponding amplitudes.
$(\Delta-\gamma\Delta)$ indicates a $\Delta$ radiation vertex with on mass
shell $\Delta$’s in $({{\cal
E_{L}}^{3/2}})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ and in
${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}$.

The intermediate $\Delta$’s in (1.3b) and in the $\Delta-\gamma\Delta$ vertex
are on mass shell, i.e. the four momentum of the $\Delta$ $P_{\Delta}$ is
determined as $P^{o}_{\Delta}=\sqrt{{\sf m}_{\Delta}^{2}+{\bf
P}_{\Delta}^{2}}$, where ${\sf m}_{\Delta}$ denotes the effective complex mass
of the $\Delta$ which is determined by the $\Delta$ pole position of the $\pi
N$ amplitude. In the present approach the intermediate $\Delta$ radiation
amplitude ${\cal I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)$ (Fig. 2B) is constructed unambiguously using
only the on mass shell $\Delta$-pole part of the $\pi N$ amplitude. The
corresponding 3D time-ordered field theoretical construction of the $\Delta$
radiation amplitude was presented by [30, 31, 32]. This approach is
generalized in appendix C for any $s$ depending mass ${\sf m}_{\Delta}(s)$. In
particular, the formulation considered does not use the effective Lagrangian
with the Heisenberg operators of the $\Delta$. Therefore, the off mass shell
$\Delta$ ambiguities does not appear. For the 3D time-ordered representations
of the diagrams in Fig. 1 and Fig. 2B with the on mass shell intermediate
pions, nucleons and $\Delta$’s we shall use the following analytic
decompositions of the amplitudes ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ and ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$ in (1.1a,b) in order to separate current
conservation (1.3a,b):

I. Decomposition over the nucleon and antinucleon exchange parts.

II. Separation of the longitudinal and transverse parts of ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ in (1.1a).

III. Partial wave decomposition of the off shell $\pi N$ amplitudes in ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ and in ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$. This procedure is necessary for separation
of the $\Delta$ resonance $(3/2,3/2)$ spin-isospin states in (1.1a,b). It also
include projections on the intermediate spin $3/2$ states in the $\gamma N-N$
and $\gamma\pi-\pi$ vertices.

IV. Separation of the current conservation conditions with and without
$\Delta$-pole terms in the off mass shell $\pi N$ amplitudes.

V. Reproduction of the double $\Delta$ exchange amplitudes using a sum of the
$\Delta$-pole terms in ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ and in ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$. In the final ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
the $\Delta$ radiation vertex has the same form as the usual
$\Delta-\gamma\Delta$ vertex.

Figure 2: Diagram A presents a symbolic description of the internal particle
radiation amplitude ${\cal I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}$ in (1.1a,b). A special part of the amplitude ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ in diagram A is given
by diagram B which describes the double $\Delta$ exchange amplitude ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
with the photon emission from the intermediate $\Delta$. The
$\Delta-\gamma\Delta$ vertex with on mass shell $\Delta$’s (3.7) in ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
contains the magnetic dipole moment $\mu_{\Delta}$ of the $\Delta$ (see
appendix B). The unambiguous construction of ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
within the 3D time-ordered field theoretical approach is given in appendix C.

An important property of ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
is that it satisfies not the separate current conservation condition

${k^{\prime}}_{\mu}{\cal I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)\neq 0,$ $None$

because the $\Delta-\gamma^{\prime}\Delta^{\prime}$ vertex consists of the
intermediate $\Delta$ four moments $P_{\Delta}^{\mu}$,
${P^{\prime}}_{\Delta}^{\mu}$ and
${k^{\prime}}_{\Delta}^{\mu}=P_{\Delta}^{\mu}-{P^{\prime}}_{\Delta}^{\mu}\neq
k^{\prime}_{\mu}$ (see appendix B).

Current conservation (1.3a,b) can be reinforced if one takes into account that
only $({\cal E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ and ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
have the same double $\Delta$ exchange poles and the same analytical structure
of the $\Delta-\gamma\Delta$ vertex. Therefore,

$({\cal E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)=-{\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta).$
$None$

This relation allows to determine the magnetic dipole moment of the $\Delta$
resonances. Thus, from the equality of the vertex functions in ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
and ${\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ which contain $\mu_{\Delta^{+}}$ and $\mu_{p}$
correspondingly, it follows that $\mu_{\Delta^{+}}$ is analytically defined by
$\mu_{p}$.

The important property of (1.5) is the equality and cancellation of the
intermediate $\Delta$ radiation term ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
(Fig. 2B) and the corresponding longitudinal part of the external particle
radiation amplitudes $({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$. This equality and cancellation are a part of the
general screening of the internal particle terms via the sum of the external
particle radiation diagrams in Fig. 1 because other parts of ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ and ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ in (1.1b) are also
equal and cancel each other.

Current conservation (1.1a,b) has the same form as in the approach based on
the Low theorem (or low energy photon theorem) for the reactions with soft
photons [8]-[14]. Unlike the present formulation, these approaches start from
the external particle radiation amplitude ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (Fig. 1) which
determines the full $\pi N$ bremsstrahlung amplitude in the infrared energy
region of the emitted photon $(k^{\prime}\to 0)$. One can reproduce the double
$\Delta$ exchange amplitude using the sum of the $\Delta$-pole $\pi N$
amplitudes in ${\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$
(Fig. 1) as it was noted in [8]222 The relationship between the external
particle radiation diagrams in Fig. 1 and the double $\Delta$ exchange term in
Fig. 2B were used in [12] based on the Brodsky-Brown identities [36, 37] for
the diagrams in the tree approximation. and was applied in numerous other
papers (see [12]) within the low energy photon approach [5]-[14]. This
approach is based on a approximation ${\cal
A}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}\Longrightarrow{\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}+{\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
which allows to calculate the cross sections of the $\pi N$ bremsstrahlung for
the extraction of the magnetic dipole moment of the $\Delta$. But in this
approach the equality and cancellation of ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
and ${\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ according to (1.5) was not taken into account.
Moreover, the recipe of construction of the bremsstrahlung amplitude in the
low energy photon limit $k^{\prime}\to 0$ is not unique due to the ambiguities
of the low energy photon approximations [12].

Unlike the low energy photon approach [5]-[14], the present approach is not
restricted to the infrared energy region of the emitted photon $(k^{\prime}\to
0)$, i.e., (1.1a,b) and (1.5) are exactly valid for any energy of the final
photon. Moreover, in the present approach the electromagnetic form factors of
the $\Delta$’s are determined through the $\Delta$-pole residues of the off
shell $\pi N$ amplitudes. The suggested formulation can be applied to other
reactions with conserved current like pion photo-production reaction, Compton
scattering, processes with external vector $\rho$ or $\omega$ mesons etc.

This paper consists of four sections and three appendices. Current
conservation (1.1a,b) for the on shell bremsstrahlung amplitudes are derived
in Section 2. In this section the equations (1.1a,b) are decomposed into
independent current conservations with one nucleon and one antinucleon
intermediate states. The following chain of the decompositions ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}\Longrightarrow...\Longrightarrow({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$, ${\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}\Longrightarrow...\Longrightarrow{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ with the final form of current conservation (1.3a,b)
is given in Section 3 and in Appendix A. The derivation of (1.5) with the
extraction of the magnetic dipole moment of the $\Delta^{+}$ and $\Delta^{++}$
resonances is given in Sec. 3. The conclusions and the comparison with the
magnetic dipole moments of other authors (Table 1) are presented in Sec. 4. In
Appendix B construction of the $\Delta-\gamma^{\prime}\Delta^{\prime}$
vertices with the on mass shell $\Delta$’s is considered. Reproduction of the
double $\Delta$ exchange diagram in Fig. 2B within the usual time-ordered
field-theoretical approach for the $\pi N$ bremsstrahlung amplitude is given
in Appendix C.

2\. Ward-Takahashi identities for the pion-nucleon bremsstrahlung amplitude

We consider the radiative pion-nucleon scattering

$\pi(p_{\pi})\ +\ N(p_{N})\Longrightarrow\gamma^{\prime}(k^{\prime})\ +\
\pi^{\prime}(p^{\prime}_{\pi})\ +\ N^{\prime}({p^{\prime}_{N}})$

with the on mass shell momentum of the pions ($p_{\pi}=(\sqrt{{\bf
p}_{\pi}^{2}+m_{\pi}^{2}},{\bf p}_{\pi})$, ${p^{\prime}}_{\pi}=(\sqrt{{\bf
p^{\prime}}_{\pi}^{2}+m_{\pi}^{2}},{\bf p^{\prime}}_{\pi})$, nucleons
($p_{N}=(\sqrt{{\bf p}_{N}^{2}+m_{N}^{2}},{\bf p}_{N})$,
${p^{\prime}}_{N}=(\sqrt{{\bf p^{\prime}}_{N}^{2}+m_{N}^{2}},{\bf
p^{\prime}}_{N})$, and final photon (${k^{\prime}}^{2}=0$). The energy-
momentum of the emitted photon is
$k^{\prime}_{\mu}=(p_{N}+p_{\pi}-p^{\prime}_{\pi}-p^{\prime}_{N})_{\mu}$.

Following the derivation of the Ward-Takahashi identities (see e.g. ch. ${\bf
8.4.1}$ in the book of Itzykson and Zuber[29]) we start with the on shell
amplitude $A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$

${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}({\bf
p^{\prime}_{\pi},p^{\prime}_{N},k^{\prime};p_{\pi},p_{N}})=$
${\overline{u}}({\bf
p^{\prime}}_{N})(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2}){k^{\prime}}_{\mu}{\tau}^{\mu}(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})({p_{\pi}}^{2}-m_{\pi}^{2})u({\bf
p}_{N}),$ $None$

where the Green function ${\tau}^{\mu}$ is expressed via the photon source
operator ${\cal J}^{\mu}(z)$ and the pion and nucleon field operators
$\Phi(x)$ and $\Psi(y)$ as

${k^{\prime}}_{\mu}{\tau}^{\mu}=i\int
d^{4}zd^{4}y^{\prime}d^{4}x^{\prime}d^{4}yd^{4}xe^{ik^{\prime}z+ip^{\prime}_{\pi}x^{\prime}+ip^{\prime}_{N}y^{\prime}-ip_{\pi}x-ip_{N}y}$
${{\partial}\over{\partial z^{\mu}}}<0|{\sf
T}\biggl{(}\Psi(y^{\prime})\Phi(x^{\prime}){\cal
J}^{\mu}(z){\overline{\Psi}}(y)\Phi^{+}(x)\biggr{)}|0>.$ $None$

In this paper we use the same definition and normalization for the Dirac
spinors as in [29]. In particular, $u({\bf p}_{N})$ denotes the spinor of the
nucleon with the three-momentum ${\bf p}_{N}$.

We shall use the well known relation for the time-ordered product of the
quantum field operators

${{\partial}\over{\partial z^{\mu}}}<0|{\sf
T}\biggl{(}\Psi(y^{\prime})\Phi(x^{\prime}){\cal
J}^{\mu}(z){\overline{\Psi}}(y)\Phi^{+}(x)\biggr{)}|0>=<0|{\sf
T}\biggl{(}\Psi(y^{\prime})\Phi(x^{\prime})\Bigl{(}{{\partial}\over{\partial
z^{\mu}}}{\cal J}^{\mu}(z)\Bigr{)}{\overline{\Psi}}(y)\Phi^{+}(x)\biggr{)}|0>$
$+\delta(z_{o}-x^{\prime}_{o})<0|{\sf
T}\biggl{(}\Psi(y^{\prime})\biggl{[}{\cal
J}^{o}(z),\Phi(x^{\prime})\biggr{]}{\overline{\Psi}}(y)\Phi^{+}(x)\biggr{)}|0>$
$+\delta(z_{o}-y^{\prime}_{o})<0|{\sf
T}\biggl{(}\Phi(x^{\prime})\biggl{[}{\cal
J}^{o}(z),\Psi(y^{\prime})\biggr{]}{\overline{\Psi}}(y)\Phi^{+}(x)\biggr{)}|0>$

.

$+\delta(z_{o}-x_{o})<0|{\sf
T}\biggl{(}\Psi(y^{\prime})\Phi(x^{\prime})\biggl{[}{\cal
J}^{o}(z),\Phi^{+}(x)\biggr{]}{\overline{\Psi}}(y)\biggr{)}|0>$
$+\delta(z_{o}-y_{o})<0|{\sf
T}\biggl{(}(\Psi(y^{\prime})\Phi(x^{\prime})\biggl{[}{\cal
J}^{o}(z),{\overline{\Psi}}(y)\biggr{]}\Phi^{+}(x)\biggr{)}|0>$ $None$

and the equal-time commutation conditions

$\biggl{[}{\cal
J}^{o}(z),\Psi(y^{\prime})\biggr{]}\delta(z_{o}-y^{\prime}_{o})=-e_{N^{\prime}}\delta^{(4)}(z-y^{\prime})\Psi(y^{\prime});\
\ \ \biggl{[}{\cal
J}^{o}(z),{\overline{\Psi}}(y)\biggr{]}\delta(z_{o}-y_{o})=e_{N}\delta^{(4)}(z-y){\overline{\Psi}}(y)$
$None$

$\biggl{[}{\cal
J}^{o}(z),\Phi(x^{\prime})\biggr{]}\delta(z_{o}-x^{\prime}_{o})=-e_{\pi^{\prime}}\delta^{(4)}(z-x^{\prime})\Phi(x^{\prime});\
\ \ \biggl{[}{\cal
J}^{o}(z),{\Phi^{+}}(x)\biggr{]}\delta(z_{o}-x_{o})=e_{\pi}\delta^{(4)}(z-x){\Phi^{+}}(x),$
$None$

where $e_{N^{\prime}}$, $e_{\pi^{\prime}}$, $e_{N}$ and $e_{\pi}$ stand for
the charge of the nucleons and pions in the final and initial states. In
particular, $e_{N}=e,0$ for the proton and the neutron, and $e_{\pi}=\pm e,0$
for pions.

After substitution of (2.3a,b) in (2.2a) and integration over $d^{4}z$ we
obtain

${k^{\prime}}_{\mu}{\tau}^{\mu}=-i\int
d^{4}y^{\prime}d^{4}x^{\prime}d^{4}yd^{4}xe^{ip^{\prime}_{\pi}x^{\prime}+ip^{\prime}_{N}y^{\prime}-ip_{\pi}x-ip_{N}y}\biggl{(}e_{N^{\prime}}e^{ik^{\prime}y^{\prime}}+e_{\pi^{\prime}}e^{ik^{\prime}x^{\prime}}-e_{N}e^{ik^{\prime}y}-e_{\pi}e^{ik^{\prime}x}\biggr{)}$
$<0|T\biggl{(}\Psi(y^{\prime})\Phi(x^{\prime}){\overline{\Psi}}(y)\Phi^{+}(x)\biggr{)}|0>.$
$None$

Equal-time commutators (2.3a,b) are the result of the commutation relations
between the electric charge operator $Q=\int d^{3}x{\cal J}^{o}(x)$ and the
particle field operators with the charge $e$. These conditions express
electric charge conservation for the local fields, i.e., they represent one of
the first principles in quantum field theory.

Substituting (2.2c) into (2.1) we get

${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}({\bf
p^{\prime}_{\pi},p^{\prime}_{N},k^{\prime};p_{\pi},p_{N}})=-i(2\pi)^{4}\
\delta^{(4)}(p^{\prime}_{N}+p^{\prime}_{\pi}+k^{\prime}-p_{\pi}-p_{N})$
$\Biggl{[}{\overline{u}}({\bf
p^{\prime}}_{N})(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N}){{e_{N^{\prime}}}\over{\gamma_{\nu}(p^{\prime}_{N}+k^{\prime})^{\nu}-m_{N}}}<out;{\bf
p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
$+({p^{\prime}}_{\pi}^{2}-m_{\pi}^{2}){{e_{\pi^{\prime}}}\over{(p^{\prime}_{\pi}+k^{\prime})^{2}-m_{\pi}^{2}}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$ $-<out;{\bf
p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|{\overline{J}}(0)|{\bf
p}_{\pi};in>{{e_{N}}\over{\gamma_{\nu}(p_{N}-k^{\prime})^{\nu}-m_{N}}}(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})u({\bf
p}_{N})$ $-<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>{{e_{\pi}}\over{(p_{\pi}-k^{\prime})^{2}-m_{\pi}^{2}}}(p_{\pi}^{2}-m_{\pi}^{2})\Biggr{]}$
$None$

where $J(x)=(i\gamma_{\nu}\partial/\partial x_{\nu}-m_{N})\Psi(x)$ and
$j_{\pi}(x)=(\partial^{2}/\partial x^{\nu}\partial
x_{\nu}+m_{\pi}^{2})\Phi(x)$ denote the source operator of the nucleon and the
pion.

For the on mass shell external particles
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$
(2.4) vanishes. In particular, for $k^{\prime}=0$
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$
disappears due to cancellation of the on shell $\pi N$ amplitudes in (2.4).
Thus expression (2.4) presents current conservation for the on shell
bremsstrahlung amplitude

${k^{\prime}}_{\mu}\Biggl{[}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\bf
p^{\prime}_{\pi},p^{\prime}_{N},k^{\prime};p_{\pi},p_{N}})\Biggr{]}_{on\ mass\
shell\ \pi^{\prime},\ N^{\prime},\ \pi,\ N}=0.$ $None$

It is convenient to extract the full energy-momentum conservation $\delta$
function from the radiative $\pi N$ scattering amplitude
$A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ and introduce the
corresponding amplitude $<out;{\bf p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\cal
J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$

${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-i(2\pi)^{4}\
\delta^{(4)}(p^{\prime}_{N}+p^{\prime}_{\pi}+k^{\prime}-p_{\pi}-p_{N}){k^{\prime}}_{\mu}<out;{\bf
p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>,$ $None$

Afterwards, using the identity $a/(a+b)\equiv 1-b/(a+b)$ in (2.4) $\biggl{(}$
i.e.
$(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})/\Bigl{(}{\gamma_{\nu}(p^{\prime}_{N}+k^{\prime})^{\nu}-m_{N}}\Bigr{)}=1-\gamma_{\mu}k^{\prime\mu}/\Bigl{(}{\gamma_{\nu}(p^{\prime}_{N}+k^{\prime})^{\nu}-m_{N}}\Bigr{)}$;
$({p^{\prime}}_{\pi}^{2}-m_{\pi}^{2})/\Bigl{(}(p^{\prime}_{\pi}+k^{\prime})^{2}-m_{\pi}^{2}\Bigr{)}=1-2{k^{\prime}}^{\mu}(p^{\prime}+k^{\prime})_{\mu}/\Bigl{(}(p^{\prime}_{\pi}+k^{\prime})^{2}-m_{\pi}^{2}\Bigr{)}$
etc.$\biggr{)}$ we obtain

${k^{\prime}}_{\mu}<out;{\bf p^{\prime}}_{N}{\bf p}_{\pi^{\prime}}|{\cal
J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>={\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}+{k^{\prime}}_{\mu}{\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=0,$ $None$

where $p_{N}+p_{\pi}-p_{N^{\prime}}-p_{\pi^{\prime}}-k^{\prime}=0$ and

${\cal B}_{\pi^{\prime}N^{\prime}-\pi N}=e_{N^{\prime}}{\overline{u}}({\bf
p^{\prime}}_{N})<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf
p}_{N};in>+e_{\pi^{\prime}}<out;{\bf p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf
p}_{\pi}{\bf p}_{N};in>$ $-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p}_{N})-e_{\pi}<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>,$ $None$

${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-\Biggl{[}{\overline{u}}({\bf
p^{\prime}}_{N})\gamma^{\mu}{{\gamma_{\nu}(p^{\prime}_{N}+k^{\prime})^{\nu}+m_{N}}\over{2p^{\prime}_{N}k^{\prime}}}e_{N^{\prime}}<out;{\bf
p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
$+(2{p^{\prime}}_{\pi}+k^{\prime})^{\mu}{{e_{\pi^{\prime}}}\over{{2p^{\prime}_{\pi}k^{\prime}}}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
$-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|{\overline{J}}(0)|{\bf
p}_{\pi};in>{{\gamma_{\nu}(p_{N}-k^{\prime})^{\nu}+m_{N}}\over{2p_{N}k^{\prime}}}\gamma^{\mu}u({\bf
p}_{N})$ $-<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>{{e_{\pi}}\over{2p_{\pi}k^{\prime}}}(2p_{\pi}-k^{\prime})^{\mu}\Biggr{]}$
$None$

The identity (2.7) is derived for the on shell total $\pi N$ bremsstrahlung
amplitude (2.6). This identity consists of ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (2.8b), which has the
form of the external particle radiation diagrams in Fig. 1, and ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$ (2.6a), which consists of the sum of the
different off mass shell $\pi N$ amplitudes. (2.7) is derived using the same
technique as for the well-known Ward-Takahashi identity [29]. But the usual
Ward-Takahashi identity connects the off mass shell $n+1$ and $n$ point
vertices and Green functions. For the on mass shell external particles the
usual Ward-Takahashi identity transforms into (2.4), which generates (2.7)
using the simple algebraic identity $a/(a+b)\equiv 1-b/(a+b)$. Therefore, we
designate (2.7) as the modified Ward-Takahashi identity for the on shell $\pi
N$ radiation amplitude $<out;{\bf p^{\prime}}_{N}{\bf p}_{\pi^{\prime}}|{\cal
J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$.

It must be noted that (2.4) and (2.8b) do not contain the full electromagnetic
form factors of the external particles as follows from the equal-time
commutation rules (2.3a,b). In particular, ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (2.8b) consists of the
incomplete $\gamma NN$ and $\gamma\pi\pi$ vertex functions $e_{N}\gamma^{\mu}$
and $e_{\pi}({p^{\prime}}_{\pi}^{\mu}+p_{\pi}^{\mu})$ with the physical
charges and off shell $\pi N$ amplitudes. The amplitude (2.8b) can be
described via a sum of the corresponding Feynman diagrams in Fig.1. ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ with the full
electromagnetic vertices was the basic expression in derivation of the low
energy photon theorem [5]. Various applications of this method are given in
[6, 8, 12, 13, 14]. The external particle radiation amplitudes in Fig. 1 are
responsible for the infrared behavior of the total bremsstrahlung amplitude,
i.e., in the low energy photon limit they represent the leading diagrams. The
present derivation of current conservation (2.7) based on the general
condition (2.2c), i.e., (2.7) is not restricted by the limit $k^{\prime}=|{\bf
k^{\prime}}_{\gamma}|\to 0$.

The off shell $\pi N$ amplitudes in (2.8a,b) are functions of three on mass
shell moments from which one can construct only three independent Lorentz-
invariant (Mandelstam) variables. Therefore, we have

${\overline{u}}({\bf p^{\prime}}_{N})<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf
p}_{\pi}{\bf p}_{N};in>={\cal
T}_{N^{\prime}}\biggl{(}(p^{\prime}_{\pi}-p_{\pi}-p_{N})^{2};s,t_{\pi}\biggr{)}={\cal
T}_{N^{\prime}}\biggl{(}m_{N}^{2}+2k^{\prime}p^{\prime}_{N};s,t_{\pi}\biggr{)}$
$None$

$<out;{\bf p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>={\cal
T}_{\pi^{\prime}}\biggl{(}(p^{\prime}_{N}-p_{\pi}-p_{N})^{2};s,t_{N}\biggr{)}={\cal
T}_{\pi^{\prime}}\biggl{(}m_{\pi}^{2}+2k^{\prime}p^{\prime}_{\pi};s,t_{N}\biggr{)}$
$None$

$<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|{\overline{J}}(0)|{\bf
p}_{\pi};in>u({\bf p}_{N})={\cal
T}_{N}\biggl{(}s^{\prime},t_{\pi};(p^{\prime}_{\pi}+p^{\prime}_{N}-p_{\pi})^{2}\biggr{)}={\cal
T}_{N}\biggl{(}s^{\prime},t_{\pi};m_{N}^{2}-2k^{\prime}p_{N}\biggr{)}$ $None$

$<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>={\cal
T}_{\pi}\biggl{(}s^{\prime},t_{N};(p^{\prime}_{\pi}+p^{\prime}_{N}-p_{\pi})^{2}\biggr{)}={\cal
T}_{\pi}\biggl{(}s^{\prime},t_{N};m_{\pi}^{2}-2k^{\prime}p_{\pi}\biggr{)}.$
$None$

The four moments of the fourth off mass shell particle in the $\pi N$
amplitudes (2.9a,b,c,d) are determined via the energy-momentum conservation
for the bremsstrahlung amplitude $<out;{\bf p^{\prime}}_{N}{\bf
p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$ (2.6) with
$p^{\prime}_{N}+p^{\prime}_{\pi}+k^{\prime}=p_{\pi}+p_{N}$, i.e., the off
shell behavior of these $\pi N$ amplitudes is defined by
$k^{\prime}_{\gamma}$. The related invariant variables are

$s^{\prime}=(p^{\prime}_{N}+p^{\prime}_{\pi})^{2};\ \ \
s=(p_{N}+p_{\pi})^{2}=(p^{\prime}_{N}+p^{\prime}_{\pi}+k^{\prime})^{2}=s^{\prime}+2k^{\prime}(p^{\prime}_{N}+p^{\prime}_{\pi})=s^{\prime}+2k^{\prime}(p_{N}+p_{\pi})$
$None$

$t_{N}=(p^{\prime}_{N}-p_{N})^{2};\ \ \
t_{\pi}=(p^{\prime}_{\pi}-p_{\pi})^{2}$ $None$

with the following relations between them:

$t_{\pi}+(p^{\prime}_{\pi}-p_{N})^{2}+s=m_{\pi}^{2}+2m_{N}^{2}+(p^{\prime}_{\pi}-p_{\pi}-p_{N})^{2},$
$None$

$t_{N}+(p^{\prime}_{N}-p_{\pi})^{2}+s=m_{N}^{2}+2m_{\pi}^{2}+(p^{\prime}_{N}-p_{\pi}-p_{N})^{2},$
$None$

$t_{\pi}+(p_{\pi}-p^{\prime}_{N})^{2}+s^{\prime}=m_{\pi}^{2}+2m_{N}^{2}+(p_{\pi}-p^{\prime}_{\pi}-p^{\prime}_{N})^{2},$
$None$

$t_{N}+(p_{N}-p^{\prime}_{\pi})^{2}+s^{\prime}=m_{N}^{2}+2m_{\pi}^{2}+(p_{N}-p^{\prime}_{\pi}-p^{\prime}_{N})^{2}.$
$None$

Next we rewrite expressions (2.8a,b) in the time-ordered three-dimensional
form, where the particle and antiparticle contributions in the intermediate
states are separated. Using the completeness conditions $u({\bf
p^{\prime}_{N}\pm k^{\prime}}){\overline{u}}({\bf p^{\prime}_{N}\pm
k^{\prime}})+v({\bf p^{\prime}_{N}\pm k^{\prime}}){\overline{v}}({\bf
p^{\prime}_{N}\pm k^{\prime}})={\sf 1}$ for the intermediate one nucleon
state, we obtain

${\cal B}_{\pi^{\prime}N^{\prime}-\pi N}(N)=e_{N^{\prime}}{\overline{u}}({\bf
p^{\prime}}_{N})u({\bf p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$ $+e_{\pi^{\prime}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf p_{N}-k^{\prime}})u({\bf p}_{N})$
$-e_{\pi}<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>,$ $None$

${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(N)=-\Biggl{[}(2{p^{\prime}}_{\pi}+k^{\prime})^{\mu}{{e_{\pi^{\prime}}}\over{{2p^{\prime}_{\pi}k^{\prime}}}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
$+{{{\overline{u}}({\bf
p^{\prime}}_{N})\Bigl{[}(2p^{\prime}_{N}+k^{\prime})^{\mu}-i\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}}\over{2p^{\prime}_{N}k^{\prime}}}u({\bf
p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})e_{N^{\prime}}<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf
p}_{\pi}{\bf p}_{N};in>$ $-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}}){{\Bigl{[}(2p_{N}-k^{\prime})^{\mu}-i\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p}_{N})}\over{2p_{N}k^{\prime}}}$ $-<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>{{e_{\pi}}\over{2p_{\pi}k^{\prime}}}(2p_{\pi}-k^{\prime})^{\mu}\Biggr{]},$
$None$

where $(N)$ indicates the part of ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ with the one-nucleon
propagator. The corresponding part of ${\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}$ is denoted as ${\cal B}_{\pi^{\prime}N^{\prime}-\pi N}(N)$. Similarly, for
the intermediate antinucleon part of (2.8a,b) we have

${\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}({\overline{N}})=-e_{N^{\prime}}{\overline{u}}({\bf p^{\prime}}_{N})v({\bf
p^{\prime}_{N}+k^{\prime}}){\overline{v}}({\bf p^{\prime}_{N}+k^{\prime}})$
$+e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|{\overline{J}}(0)|{\bf
p}_{\pi};in>v({\bf p_{N}-k^{\prime}}){\overline{v}}({\bf
p_{N}-k^{\prime}})u({\bf p}_{N}),$ $None$

${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\overline{N}})={{{\overline{u}}({\bf
p^{\prime}}_{N})\Bigl{[}(2p^{\prime}_{N}+k^{\prime})^{\mu}-i\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}}\over{2p^{\prime}_{N}k^{\prime}}}v({\bf
p^{\prime}_{N}+k^{\prime}}){\overline{v}}({\bf
p^{\prime}_{N}+k^{\prime}})e_{N^{\prime}}<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf
p}_{\pi}{\bf p}_{N};in>$ $-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>v({\bf
p_{N}-k^{\prime}}){\overline{v}}({\bf
p_{N}-k^{\prime}}){{\Bigl{[}(2p_{N}-k^{\prime})^{\mu}-i\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p}_{N})}\over{2p_{N}k^{\prime}}}.$ $None$

For the derivation of (2.12a,b) and (2.13a,b) the simple relations of the
Dirac spinors ${\overline{u}}({\bf
p^{\prime}}_{N})\gamma^{\mu}(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}+m_{N})={\overline{u}}({\bf
p^{\prime}}_{N})2{p^{\prime}_{N}}^{\mu}$, ${\overline{u}}({\bf
p^{\prime}}_{N})\gamma^{\mu}\gamma^{\nu}k^{\prime}_{\nu}={k^{\prime}}^{\mu}{\overline{u}}({\bf
p^{\prime}}_{N})-i{\overline{u}}({\bf
p^{\prime}}_{N})\sigma^{\mu\nu}k^{\prime}_{\nu}$ were used.

Expressions (2.13a,b) contain the intermediate
$\pi\to\pi^{\prime}N^{\prime}{\overline{N}}$ transition amplitude. Using the
identity
$k^{\prime}_{\mu}\Bigl{[}(P+P^{\prime})^{\mu}-i\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}=s-s^{\prime}$
it is easy to obtain that

$k^{\prime}_{\mu}{\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\overline{N}})+{\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}({\overline{N}})=0.$ $None$

Consequently, instead of the full Ward-Takahashi identity (2.7) we get

${k^{\prime}}_{\mu}<out;{\bf p^{\prime}}_{N}{\bf p}_{\pi^{\prime}}|{\cal
J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>={\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}(N)+{k^{\prime}}_{\mu}{\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(N)=0.$ $None$

The gauge terms proportional to $k^{\prime}_{\mu}$ in (2.12b) do not
contribute to the $\pi N$ bremsstrahlung amplitude because the product of
polarization vector $\epsilon^{\mu}({\bf k^{\prime}},\lambda)$ of the final
photon and $k^{\prime}_{\mu}$ vanishes $\epsilon^{\mu}({\bf
k^{\prime}},\lambda)k^{\prime}_{\mu}=0.$ The terms proportional to
$k^{\prime}_{\mu}$ modify the Green function $\tau^{\mu}$ in (2.1), but for
the on shell amplitude they can be ignored. In addition, due to
$k^{\prime}_{\mu}k^{\prime\mu}\equiv{k^{\prime}}^{2}=0$ the terms proportional
to $k^{\prime}_{\mu}$ in (2.12b) can also be omitted in the Ward-Takahashi
identity (2.14b).

Current conservation (2.7) and (2.14b) are written for the longitudinal part
of the total $\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N$ amplitude. In
particular, the set of the diagrams which form the first term of the right
side in (2.2b) with ${{\partial}/{\partial z^{\mu}}}\ {\cal J}^{\mu}(z)=0$ are
not included in (2.7). Therefore, the modified Ward-Takahashi identity (2.7)
presents a necessary condition of current conservation which contains only a
longitudinal part of the external particle radiation amplitudes.

One can use the transverse part of the total $\pi N$ bremsstrahlung amplitude
in order to complete ${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(N)$ (2.12b) up to the external particle radiation amplitude with the
anomalous magnetic moment of the external nucleons. For this aim one can pick
out the related transverse terms with $\sigma^{\mu\nu}k^{\prime}_{\nu}$ which
generate the following redefinitions of $\mu_{N^{\prime}}=1$ and $\mu_{N}=1$
with the corresponding anomalous magnetic moments of the nucleon333 Keeping
the identity (2.14b) one can also reproduce the full electromagnetic form
factors of the nucleon ${\em F}_{1}(t)$ and ${\em F}_{2}(t)$ in ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(N)$ (2.12b), where $t$
denotes the corresponding four momentum transfer. Thus, if one picks out the
terms ${\overline{u}}({\bf
p^{\prime}}_{N})\Bigl{[}(2p^{\prime}_{N}+k^{\prime})^{\mu}-i{\widetilde{\mu}}_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}{\em
F}_{2}(t^{\prime}){\em F}_{1}^{-1}(t^{\prime})u({\bf
p_{N}^{\prime}+k^{\prime}}){\cal T}_{N^{\prime}}$ and ${\cal
T}_{N}{\overline{u}}({\bf
p_{N}-k^{\prime}})\Bigl{[}(2p_{N}-k^{\prime})^{\mu}-i{\widetilde{\mu}_{N}}\sigma^{\mu\nu}k^{\prime}_{\nu}{\em
F}_{2}(t){\em F}_{1}^{-1}(t)\Bigr{]}u({\bf p}_{N})$ from the transverse part
of the full $\pi N$ radiation amplitude, one obtains the full $\gamma NN$
vertices in ${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(N)$
(2.12b) with the redefined $\pi N$ amplitudes (2.9a,c) ${\cal
T}_{N^{\prime}}\Longrightarrow{\em F}_{1}^{-1}(t^{\prime}){\cal
T}_{N^{\prime}}$ and ${\cal T}_{N}\Longrightarrow{\em F}_{1}^{-1}(t){\cal
T}_{N}$. Another way to take into account the full electromagnetic form
factors of the external nucleons is to use the modified complete set of the
intermediate Dirac spinors $u({\bf
p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})\Longrightarrow$
$u_{m_{N}\to\sqrt{s_{N}^{\prime}}}({\bf
p_{N}^{\prime}+k^{\prime}}){\overline{u}}_{m_{N}\to\sqrt{s_{N}^{\prime}}}({\bf
p_{N}^{\prime}+k^{\prime}});\ u({\bf p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}})u_{m_{N}\to\sqrt{s_{N}}}({\bf
p_{N}-k^{\prime}}){\overline{u}}_{m_{N}\to\sqrt{s_{N}}}({\bf
p_{N}-k^{\prime}}),$ where $m_{N}$ is replaced by
$s_{N}^{\prime}=\Bigl{(}\sqrt{m_{N}^{2}+({\bf
p_{N}^{\prime}})^{2}}+k^{\prime}\Bigr{)}^{2}-({\bf
p^{\prime}_{N}+k^{\prime}})^{2}$ or $s_{N}=\Bigl{(}\sqrt{m_{N}^{2}+({\bf
p_{N}})^{2}}-k^{\prime}\Bigr{)}^{2}-({\bf p_{N}-k^{\prime}})^{2}$. Then we
obtain the $\gamma^{\prime}NN$ vertex between the one nucleon states with the
four moments $p_{N^{\prime}}+k^{\prime}\to p_{N^{\prime}}$ or
$p_{N}-k^{\prime}\to p_{N}$. In this formulation
$t=t^{\prime}={k^{\prime}}^{2}=0$ and only the threshold values of the
electromagnetic form factors for the external nucleons appear. The large four
momentum transfer of the external nucleons is not important for the
determination of the electromagnetic moments of the $\Delta$’s. Therefore, we
do not include them in the following text.. This procedure implies taking into
account the loop corrections of the $\gamma NN$ vertex. These corrections
reproduces the anomalous magnetic moments within the minimal electromagnetic
coupling scheme (see [27] consideration of (10.81)). Then we obtain444 The
anomalous magnetic moment of the $\Delta$ appears in the $\Delta-\gamma\Delta$
vertex at $\sigma_{\mu\nu}{k^{\prime}}^{\nu}_{\Delta}$ with
${k^{\prime}}^{\nu}_{\Delta}\neq{k^{\prime}}^{\nu}$, i.e., the diagram in Fig.
2B with the anomalous magnetic moment of $\Delta$ is not included in the
transverse part of the $\pi N$ bremsstrahlung amplitude.

${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(N)=-\Biggl{[}(2{p^{\prime}}_{\pi}+k^{\prime})^{\mu}{{e_{\pi^{\prime}}}\over{{2p^{\prime}_{\pi}k^{\prime}}}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
$+{{{\overline{u}}({\bf
p^{\prime}}_{N})\Bigl{[}(2p^{\prime}_{N}+k^{\prime})^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}}\over{2p^{\prime}_{N}k^{\prime}}}u({\bf
p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})e_{N^{\prime}}<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf
p}_{\pi}{\bf p}_{N};in>$ $-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}}){{\Bigl{[}(2p_{N}-k^{\prime})^{\mu}-i\mu_{N}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p}_{N})}\over{2p_{N}k^{\prime}}}$ $-<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>{{e_{\pi}}\over{2p_{\pi}k^{\prime}}}(2p_{\pi}-k^{\prime})^{\mu}\Biggr{]},$
$None$

The external particle $\pi N$ radiation amplitude (2.15) have the fixed
transverse terms $i\mu_{N}\sigma^{\mu\nu}k^{\prime}_{\nu}$ in the vertex
functions of the external nucleons. The full $\gamma NN$ vertices are
necessary for the realistic calculations of the $\pi N$ radiation reactions.
In particular, expression (2.15) automatically satisfies the low energy photon
theorem [5]-[14].

Relation (2.14b) represents the modified Ward-Takahashi identity in the three-
dimensional time-ordered form. This identity establish a relationship between
the external particle $\pi N$ bremsstrahlung amplitude (2.15) and the off mass
shell elastic $\pi N$ scattering amplitudes (2.9a,b,c,d). In order to satisfy
current conservation (2.14b) it is necessary to find the internal particle
radiation diagrams whose four divergence reproduces ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}(N)$. The special case of this problem for
the (3/2,3/2) partial $\pi N$ amplitudes is considered in the next Section.

3\. Internal and external particle radiation parts of the $\pi N$
bremsstrahlung amplitude.

In this Section we show, that the Ward-Takahashi identity (2.14b) after
decompositions of $({\cal E})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(N)$ (2.15) and ${\cal B}_{\pi^{\prime}N^{\prime}-\pi N}(N)$ (2.12a) reduces
to a special identity for the double $\Delta$ exchange amplitude which has the
same structure as on mass shell $\Delta$ radiation diagram in Fig. 2B. For
this aim we separate the longitudinal part of the external particle radiation
amplitude and isolate the $\Delta$ resonance parts of the off shell $\pi N$
amplitudes (2.9a,b,c,d). The symbolic representation of this procedure is
given by the chain of transformations

$({\cal E})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(N)\Longrightarrow({\cal
E_{L}})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}\Longrightarrow({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-
pole)$ $None$

and

${\cal B}_{\pi^{\prime}N^{\prime}-\pi N}(N)\Longrightarrow({\cal
B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi N}(\Delta-pole),$ $None$

where the lower index ${\cal L}$ denotes the longitudinal part of the
amplitude $({\cal E})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$,
the upper index $3/2$ corresponds to the resonance spin-isospin $3/2$ state of
the $\pi N$ amplitudes. The argument $(\Delta-pole)$ indicates the
$\Delta$-pole part of the $\pi N$ amplitudes (2.9a,b,c,d).

Afterwards, using the sum of the $\Delta$-pole terms in the different off
shell $\pi N$ amplitudes in $({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-
pole)$ and $({\cal B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi N}(\Delta-pole)$, one
can separate the double $\Delta$ exchange Ward-Takahashi identities for the
double $\Delta$ exchange amplitudes $({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta\Delta)$ and $({\cal B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta\Delta)$. Moreover, after an algebraic transformation of the Ward-
Takahashi identity for the double $\Delta$ exchange amplitudes one obtains an
independent identity for the amplitude which has the same structure as the
internal $\Delta$ radiation amplitude in Fig. 2B. These decompositions form
the following chain of transformations

$({\cal E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-pole)\Longrightarrow({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta\Delta)\Longrightarrow({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ $None$

$({\cal B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi N}(\Delta-
pole)\Longrightarrow({\cal B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta\Delta)\Longrightarrow{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ $None$

The algebraic decompositions (3.1a,b,c,d) of the Ward-Takahashi identity
(2.14b) are detailed in Appendix A. The resulting identity is

${k^{\prime}}_{\mu}\Biggl{[}<out;{\bf p^{\prime}}_{N}{\bf
p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>\Biggr{]}^{Projection\ on\ spin\ 3/2\ particle\ states}_{2\Delta\
exchange\ with\ a\ \Delta-\gamma^{\prime}\Delta^{\prime}\ vertex}=$
$k^{\prime}_{\mu}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)+{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)=0,$ $None$

where

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)={{<{\bf p^{\prime}}_{N},{\bf
p^{\prime}}_{\pi}|{\sl g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta}>}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}}_{\Delta})g_{bc}\Bigl{[}(P_{\Delta}+P^{\prime}_{\Delta})^{\mu}{\sc
V}_{E}-i\sigma^{\mu\nu}{k^{\prime}_{\Delta}}_{\nu}{\sc
V}_{H}\Bigr{]}u^{c}({\bf P}_{\Delta})\Biggr{\\}}{{<{\bf P}_{\Delta}|{\sl
g}_{\Delta-\pi N}|{\bf p}_{N},{\bf
p}_{\pi}>}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}},$ $None$

${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)={{<{\bf
p^{\prime}}_{N},{\bf p^{\prime}}_{\pi}|{\sl
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta}>}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}}_{\Delta})g_{bc}\Bigl{[}k^{\prime}_{o}(P_{\Delta}+P^{\prime}_{\Delta})^{o}{\sc
V}_{E}-ik^{\prime}_{\mu}\sigma^{\mu o}{k^{\prime}_{\Delta}}_{o}{\sc
V}_{H}\Bigr{]}u^{c}({\bf P}_{\Delta})\Biggr{\\}}{{<{\bf P}_{\Delta}|{\sl
g}_{\Delta-\pi N}|{\bf p}_{N},{\bf
p}_{\pi}>}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}},$ $None$

where $P=p_{\pi}+p_{N}$ and $P^{\prime}=p^{\prime}_{\pi}+p^{\prime}_{N}$ are
the four moments of the $\pi N$ system in the initial and final states,
$P_{\Delta}$ and $P^{\prime}_{\Delta}$ denote the four moments of the $\Delta$

$P_{\Delta}\equiv\Bigl{(}P^{o}_{\Delta}(s),{\bf
P}_{\Delta}\Bigr{)}=\Bigl{(}\sqrt{(M_{\Delta}(s)-{{i\Gamma_{\Delta}(s)}\over
2})^{2}+{\bf P}_{\Delta}^{2}},{\bf P}_{\Delta}\Bigr{)};\ \ \ \ \ \ \ \ {\bf
P}_{\Delta}={\bf P}={\bf p}_{N}+{\bf p}_{\pi}$ $None$
$P^{\prime}_{\Delta}\equiv\Bigl{(}{P^{\prime}}^{o}_{\Delta}(s^{\prime}),{\bf
P^{\prime}}_{\Delta}\Bigr{)}=\Bigl{(}\sqrt{(M_{\Delta}(s^{\prime})-{{i\Gamma_{\Delta}(s^{\prime})}\over
2})^{2}+{\bf P^{\prime}}_{\Delta}^{2}},{\bf P^{\prime}}_{\Delta}\Bigr{)};\ \ \
\ \ \ {\bf P^{\prime}}_{\Delta}={\bf P^{\prime}}={\bf p^{\prime}}_{N}+{\bf
p^{\prime}}_{\pi}.$ $None$

We shall use two models of the $\Delta$ mass [31, 32, 30] ${\sf
m}_{\Delta}(s)=M_{\Delta}(s)-i/2\Gamma_{\Delta}(s)$:

  * $1.$

A model with the fixed mass of the intermediate $\Delta$ resonance

${\sf m}_{\Delta}=M_{\Delta}-{i\over 2}\Gamma_{\Delta}=1232MeV-{i\over
2}120MeV,$ $None$

  * $2.$

and a more general model with an $s$-dependent mass ${\sf m}_{\Delta}(s)$

${\sf m}_{\Delta}(s)=M_{\Delta}(s)-{i\over 2}\Gamma_{\Delta}(s),$ $None$

where ${\sf m}_{\Delta}(s=M_{\Delta}^{2})={\sf m}_{\Delta}$.

$P_{\Delta}$ and $P^{\prime}_{\Delta}$ are on mass shell four moments because
$P_{\Delta}^{2}={\sf m}_{\Delta}^{2}(s)$ and ${P^{\prime}_{\Delta}}^{2}={\sf
m}_{\Delta}^{2}(s^{\prime})$. $u^{b}({\bf P_{\Delta}})$ denotes the Rarita-
Schwinger spinor of the free spin $3/2$ particle with the complex mass ${\sf
m}_{\Delta}(s)$.

${\sc V}_{E}$ and ${\sc V}_{H}$ in (3.3a,b) are defined through the
$\Delta$-pole residues ${\cal R}_{N^{\prime}}$, ${\cal R}_{\pi^{\prime}}$,
${\cal R}_{N}$ and ${\cal R}_{\pi}$ of the off shell $\pi N$ amplitudes in
equations (A.9a,b,c,d), (A.15a,b) and (A.17a,b) of Appendix A. The $\Delta-\pi
N$ and $\pi N-\Delta$ vertices ${\sl g}_{\pi^{\prime}N^{\prime}-\Delta}$ and
${\sl g}_{\Delta-\pi N}$ are defined as

$<{\bf p^{\prime}}_{N},{\bf p^{\prime}}_{\pi}|{\sl
g}_{\pi^{\prime}N^{\prime}-\Delta}|{\bf P^{\prime}}_{\Delta}>={\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime}){\overline{u}}({\bf
p^{\prime}}_{N})i\gamma_{5}{{(p^{\prime}_{N})_{a}}\over{|{\bf
p^{\prime}}_{N}|}}u^{a}({\bf P^{\prime}}_{\Delta}),$ $None$

$<{\bf P}_{\Delta}|{\sl g}_{\Delta-\pi N}|{\bf p}_{N},{\bf p}_{\pi}>={\rm
g}_{\Delta-\pi N}(s){\overline{u}}^{d}({\bf
P}_{\Delta}){{(p_{N})_{d}}\over{|{\bf p}_{N}|}}i\gamma_{5}u({\bf p}_{N}).$
$None$

The longitudinal part of the external particle radiation amplitude with the
$\Delta$ intermediate states $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ (3.3a) has the same form as the internal
$\Delta$ radiation amplitude ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-\gamma\Delta)$
in Fig. 2B

${\cal I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)=-{{<{\bf p^{\prime}}_{N},{\bf
p^{\prime}}_{\pi}|{\sl g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta}>}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}$
$<{\bf P^{\prime}}_{\Delta},{\sf m}_{\Delta}(s^{\prime})|{\cal
J}^{\mu}(0)|{\bf P}_{\Delta},{\sf m}_{\Delta}(s)>{{<{\bf P}_{\Delta}|{\sl
g}_{\Delta-\pi N}|{\bf p}_{N},{\bf
p}_{\pi}>}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}},$ $None$

where the details of the $\Delta-\gamma\Delta$ vertex $<{\bf
P^{\prime}}_{\Delta},{\sf m}_{\Delta}(s^{\prime})|{\cal J}^{\mu}(0)|{\bf
P}_{\Delta},{\sf m}_{\Delta}(s)>$ with on mass shell $\Delta$’s are given in
Appendix B. In particular, for the low energy photons we have

$<{\bf P^{\prime}}_{\Delta},{\sf m}_{\Delta}(s^{\prime})|J_{\mu}(0)|{\bf
P}_{\Delta},{\sf m}_{\Delta}(s)>=$ ${\overline{u}}^{\sigma}({\bf
P^{\prime}}_{\Delta})g_{\rho\sigma}\biggl{[}{{(P_{\Delta}+P^{\prime}_{\Delta})_{\mu}}\over{2M_{\Delta}}}G_{C0}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})-{{i\sigma_{\mu\nu}{k^{\prime}}_{\Delta}^{\nu}}\over{2M_{\Delta}}}G_{M1}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})\biggr{]}u^{\rho}({\bf
P}_{\Delta})$ $None$

where $G_{C}$ and $G_{M1}$ denote the electric and magnetic dipole form
factors of the $\Delta$’s.

The unambiguous construction of the $\Delta$ radiation amplitude ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma{\Delta})$ (3.6) is given in Appendix C following our
previous papers [31, 32]. Thus ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma{\Delta})$ in Fig. 2B can be determined as a
projection of the complete internal particle radiation amplitude

${\cal I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma{\Delta})=\Biggl{[}{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}\Biggr{]}^{Projection\
on\ spin\ 3/2\ particle\ states}_{2\Delta\ exchange\ with\ a\
\Delta-\gamma^{\prime}\Delta^{\prime}\ vertex},$

Consequently, expressions (3.3a) and (3.6) determine the complete projection
of the $\pi N$ radiation amplitude, and we have

${k^{\prime}}_{\mu}\Biggl{[}<out;{\bf p^{\prime}}_{N}{\bf
p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>\Biggr{]}^{Projection\ on\ spin\ 3/2\ particle\ states}_{2\Delta\
exchange\ with\ a\ \Delta-\gamma^{\prime}\Delta^{\prime}\ vertex}=$
$k^{\prime}_{\mu}\biggl{[}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)+{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)\biggr{]}=0.$ $None$

Combining (3.2) and (3.8) we obtain

$k^{\prime}_{\mu}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)=-k^{\prime}_{\mu}{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)=-{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ $None$

Conditions (3.2) and (3.9) present the four-divergence of the amplitudes
$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ (3.3a) and ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-\gamma\Delta)$
(3.6) with the same double $\Delta$ poles and the corresponding
$\Delta-\gamma\Delta$ vertex. There are no other amplitudes with the same
analytical structure. Moreover, all gauge terms $A_{\mu}$ with the separate
current conservation condition $k^{\prime}_{\mu}A^{\mu}=0$ are included in the
transverse part of the external particle radiation amplitude $({\cal
E_{TR}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (A.4b) and other
transverse parts of the total amplitude which corresponds to the first term in
(2.2b) with $\partial/\partial z^{\mu}\ {\cal J}_{\mu}(z)$. Therefore, current
conservation (3.2) and (3.9) are fulfilled if

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)=-{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-\gamma\Delta)$
$None$

This equation coincides with the final equation (1.5) and allows to
determinate the connection between the form factors of the
$\Delta-\gamma\Delta$ vertices in ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-\gamma\Delta)$
(3.6) and the analogical form factors in $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ (3.3a).

Using the condition (3.10) one easily gets

${k^{\prime}_{\Delta}}_{\mu}\biggl{[}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)+{\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}({\Delta}-\gamma\Delta)\biggr{]}=0$ $None$

which immediately gives

$G_{C0}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})=-2M_{\Delta}{\sc
V}_{E}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf P}_{\Delta})$ $None$

Combining this equation with (3.24) we obtain

$G_{M1}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})=-2M_{\Delta}{\sc
V}_{H}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf P}_{\Delta}).$ $None$

Equations (3.12a,b) determine $G_{C0}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$
and $G_{M1}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$ via the residues of the
$\pi N$ amplitudes ${\cal R}$ (A.9a,b,c,d) which yield $V_{E}$ and $V_{H}$ in
(A.17a,b) and ${\cal V}^{(+)}_{E}$ and ${\cal V}^{(+)}_{H}$ in (A.15a,b).

The threshold values of (3.12a,b) give a relations between $e_{\Delta}$,
$\mu_{\Delta}$ and ${\sc V}_{E}^{k^{\prime}=0}$, ${\sc V}_{H}^{k^{\prime}=0}$
respectively

$e_{\Delta}=G_{C0}({k^{\prime}}_{\Delta}^{2}=0,s^{\prime}=M_{\Delta}^{2},s=M_{\Delta}^{2})=-\biggl{[}2M_{\Delta}{\sc
V}_{E}\biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}}$ $None$

and

$\mu_{\Delta}=G_{M1}({k^{\prime}_{\Delta}}^{2}=0,s^{\prime}=M_{\Delta}^{2},s=M_{\Delta}^{2})=-\biggl{[}{2M_{\Delta}\sc
V}_{H}\biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}}$ $None$

These conditions determine the relations between $e_{\Delta}$, $\mu_{\Delta}$
and residues of the $\pi N$ amplitudes ${\cal R}$ (A.9a,b,c,d).

$e_{\Delta}=-\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}\biggl{(}e_{N}{{{\cal
R}_{N^{\prime}}+{\cal R}_{N}}\over 2}+e_{\pi}{{{\cal R}_{\pi^{\prime}}+{\cal
R}_{\pi}}\over 2}\biggr{)}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s)\Bigr{]}^{-1}\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}},$
$None$

where ${\cal
N}(s)=1/(d{\sqrt{s}}/dk^{\prime})-d{P}^{o}_{\Delta}(s)/d{\sqrt{s}}$ and

$\mu_{\Delta}=-\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}\biggl{(}\mu_{N}{{{\cal
R}_{N^{\prime}}+{\cal R}_{N}}\over 2}\biggr{)}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s)\Bigr{]}^{-1}\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}}.$
$None$

The similarity of the conditions (3.14a) and (3.14b) allows to determine
$\mu_{\Delta}$ using (3.14a) as a normalization condition. For this aim we
consider (3.14b) separately for the
$\pi^{+}n\to\gamma^{\prime}{\pi^{\prime}}^{+}n^{\prime}$ and
$\pi^{o}p\to\gamma^{\prime}{\pi^{\prime}}^{o}p^{\prime}$ reactions. For the
$\pi^{+}n\to\gamma^{\prime}{\pi^{\prime}}^{+}n^{\prime}$ reaction (3.14a)
generates the independent normalization condition

$1=-\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}{{{\cal
R}_{\pi^{\prime}}+{\cal R}_{\pi}}\over 2}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s)\Bigr{]}^{-1}\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}}$
$None$

and for the $\pi^{o}p\to\gamma^{\prime}{\pi^{\prime}}^{o}p^{\prime}$ reaction
we get

$1=-\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}{{{\cal
R}_{p^{\prime}}+{\cal R}_{p}}\over 2}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s)\Bigr{]}^{-1}.\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}}$
$None$

Expressions (3.15) and (3.16) are the normalization conditions for the
residues ${\cal R}_{\pi}$ and ${\cal R}_{p}$ (A.9a,b,c,d) of the $\pi N$
matrices at the $\Delta$ resonance pole position. They show the dependence of
${\cal R}_{\pi}$ and ${\cal R}_{p}$ (A.9a,b,c,d) on the $\Delta$ mass ${\sf
m}_{\Delta}(s)$ (3.4c,d). Therefore, ${\rm g}_{\Delta-\pi N}(s)$ and ${\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})$ form
factors must also include a dependence on a ${\sf m}_{\Delta}(s)$.

The right side of (3.16) differs from the right side of (3.14b) only by the
factor $\mu_{N}$. Therefore, substituting (3.16) into (3.14b) we obtain

$\mu_{\Delta^{+}}=\mu_{p}{{M_{\Delta}}\over{m_{p}}},$ $None$

where the factor ${{M_{\Delta}}/{m_{p}}}$ arises because of the different
units for $\mu_{\Delta^{+}}$ and $\mu_{p}$.

The magnetic dipole moment of $\Delta^{++}$ can be determined from the
relationship between $\mu_{\Delta}$ and
$G_{M1}({k^{\prime}_{\Delta}}^{2}=0,s^{\prime}=M_{\Delta}^{2},s=M_{\Delta}^{2})$
(3.13b) and (3.14b). The difference between $({\cal R}_{N^{\prime}}+{\cal
R}_{N})/2$ in (3.16) for the $\pi^{o}p\to\gamma^{\prime}\pi^{\prime
o}p^{\prime}$ and $\pi^{+}p\to\gamma^{\prime}\pi^{\prime+}p^{\prime}$
reactions is in the isospin factors of the corresponding $\pi N$ amplitudes.
Using the isotospin symmetry between the $\pi^{o}p\to\pi^{o}p$ and
$\pi^{+}p\to\pi^{+}p$ amplitudes we get

$\mu_{\Delta^{++}}={3\over 2}\mu_{\Delta^{+}}={3\over
2}\mu_{p}{{M_{\Delta}}\over{m_{p}}}$ $None$

One cannot use directly the
$\pi^{o}n\to\gamma^{\prime}{\pi^{o}}^{\prime}n^{\prime}$ and
$\pi^{-}n\to\gamma^{\prime}{\pi^{-}}^{\prime}n^{\prime}$ reactions for
determination of the magnetic moments of $\Delta^{o}$ and $\Delta^{-}$,
because the equal-time commutator (2.3a) is zero in this case. This problem is
considered in the next part of the present paper.

The modified Ward-Takahashi identity (3.9) requires equality and cancellation
of the internal $\Delta$ radiation amplitude in Fig. 2B and the corresponding
part of the external particle radiation amplitude according to relation
(3.10). This cancellation is the result of current conservation for the $\pi
N$ radiation amplitude and the special sum of the off shell $\pi N$ amplitudes
in $({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ (3.3a), which have the same analytical structure as
the internal $\Delta$ radiation amplitude (3.6) in Fig. 2B. Therefore, the
amplitude (3.6) in Fig. 2B cancels exactly with $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ (3.3a). Consequently, the internal $\Delta$
radiation diagram in Fig. 2B is screened by the appropriate part of the
external particle radiation diagrams. Generally, screening is built into the
initial Ward-Takahashi identity (2.7), where ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$ must be compensated by the internal
particle radiation diagrams. In other words, the screening corresponds to
equality and cancellation of special parts of the internal and external
particle radiation terms in the total $\pi N$ bremsstrahlung amplitude.

4\. Conclusion

In the present paper $\Delta$’s are considered as resonances of the $\pi N$
system which generate appropriate $\Delta$-poles in the off mass shell $\pi N$
amplitudes (A.9a,b,c,d). The sums of the corresponding residues determine the
$\Delta$ form factors $G_{C}$ and $G_{M1}$ in (3.14a,b). Thus current
conservation (3.2) makes it possible to determine $G_{C}$ and $G_{M1}$ only
using the dynamical information about the residues of the off shell $\pi N$
amplitudes (2.9a,b,c,d). The threshold values of $G_{C}$ and $G_{M1}$ define
the magnetic dipole moments of $\Delta^{+}$ and $\Delta^{++}$ via the
anomalous magnetic moment of the proton. The difference between $\mu_{\Delta}$
and $\mu_{p}$ is formed by different units in the $\Delta-\gamma\Delta$ and
$p-\gamma p$ electromagnetic vertex functions. Another dynamical input for
reproduction of the magnetic dipole moment of $\Delta^{+}$ and $\Delta^{++}$
is the anomalous magnetic moment of the proton, which requires loop
corrections for the $\gamma NN$ vertices and the corresponding redefinition of
the external particle radiation amplitude (2.12b) by expression (2.15). This
redefinition does not change the initial current conservation (2.7) and is
necessary for reproduction of the realistic results for the $\pi N$
bremsstrahlung reactions.

The present investigation of the $\pi N$ radiation is based on current
conservation for the on shell amplitudes (2.7). From the general point of view
only the sum of the external and internal particle radiation parts of the full
bremsstrahlung amplitude satisfies current conservation. The modified Ward-
Takahashi identity (2.7) specifies this statement for the special form of the
external particle radiation amplitude ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ and the appropriate
sum of the off shell $\pi N$ scattering amplitudes ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$. In particular, ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ contains only the
diagrams which are responsible for the infrared behavior of the $\pi N$
radiation amplitude. The sums of the $\Delta$-pole $\pi N$ amplitudes in the
longitudinal part of ${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}$ reproduce the double $\Delta$ exchange poles.

The model-independent properties of current conservation (2.7) can be
generalized for any amplitude of an arbitrary reaction
$a+b\longrightarrow\gamma^{\prime}+f_{1}+...+f_{n}$ $(n=1,2,...)$. Current
conservation requires the existence of the internal particle radiation
amplitude ${\cal I}_{\gamma^{\prime}f_{1}..f_{n}-ab}^{\mu}$ which satisfies
the relation ${k^{\prime}}_{\mu}{\cal
I}_{\gamma^{\prime}f_{1}...f_{n}-ab}^{\mu}=-{k^{\prime}}_{\mu}{\cal
E}_{\gamma^{\prime}f_{1}...f_{n}-ab}^{\mu}={\cal B}_{f_{1}...f_{n}-ab}$, where
${\cal E}_{\gamma^{\prime}f_{1}...f_{n}-ab}^{\mu}$ is the external particle
radiation amplitude. Therefore, the appropriate parts of ${\cal
E}_{\gamma^{\prime}f_{1}..f_{n}-ab}^{\mu}$ and ${\cal
I}_{\gamma^{\prime}f_{1}...f_{n}-ab}^{\mu}$ have a different sign and they
must be subtracted from each other. Consequently, we have a screening of the
internal particle radiation amplitudes by the external one-particle radiation
terms. In the limit $k^{\prime}\to 0$ our approach exactly reproduces the low
energy photon theorems for the bremsstrahlung reactions.

As an example of the screening the identity and cancellation of the double
$\Delta$ exchange amplitude in Fig. 2B and the longitudinal part of the
external particle radiation amplitude is demonstrated in (3.10). This
cancellation allows to determine the magnetic dipole moments
$\mu_{\Delta^{+}}=G_{M1}(0)={{M_{\Delta}}\over{m_{N}}}\mu_{p}$ and
$\mu_{\Delta^{++}}={3\over 2}\mu_{\Delta^{+}}=5.46\mu_{B}$ or
$\mu_{\Delta^{++}}/\mu_{p}\sim 1.95$ of the $\Delta^{+}$ and $\Delta^{++}$
resonances. Our result for $\mu_{\Delta^{++}}$ roughly agrees with the
prediction of the naive $SU(6)$ quark model for
$\mu_{\Delta^{++}}=2\mu_{p}=5.58\mu_{B}$ [15, 16], with the nonrelativistic
potential model [4] $\mu_{\Delta^{++}}=4.52\pm 0.95\mu_{B}$ and with
extraction of $\mu_{\Delta^{++}}$ from the experimental
$\pi^{+}p\to\gamma\pi^{+}p$ cross section within the low energy photon
approach $\mu_{\Delta^{++}}=3.6\pm 2.0\mu_{B}$ [9], $\mu_{\Delta^{++}}=5.6\pm
2.1\mu_{B}$ [10] and $\mu_{\Delta^{++}}=4.7-6.9\mu_{B}$[2]. Our result is
larger than the predictions of the modified $SU(6)$ models [18, 17] and the
low energy photon approximation $\mu_{\Delta^{++}}=3.7\sim 4.9\mu_{B}$ [14].
On the other hand, our result is smaller than the values obtained within the
effective meson-nucleon Lagrangian $\mu_{\Delta^{++}}=6.1\pm 0.5\mu_{B}$ [25],
in the effective quark model $\mu_{\Delta^{++}}=6.17\mu_{B}$ [26], in the
modified bag model $\mu_{\Delta^{++}}=6.54\mu_{B}$ [20], and in the
constituent quark model [21].

The resulting magnetic dipole moments of $\Delta$’s obtained in various
theoretical models differ. Moreover, the results obtained using the same Low
theorem approach for soft photons also differ. This difference can be
explained with the various recipes for the construction of the bremsstrahlung
amplitude in the low energy photon limit $k^{\prime}\to 0$. These ambiguities
are listed in [12]. Our formulation is free off these ambiguities.

Table 1. Magnetic moments of $\Delta^{+}$ and $\Delta^{++}$ in units of the
nuclear magneton $\mu_{B}={e/{2m_{N}}}$. The upper index ∗ at the reference
indicates the theoretical model which is used to fit of the experimental data
and to extract the magnetic moment $\mu_{\Delta}$.

Model | This | $SU(6)$ | Potential and | Skyrme | Low ener. phot. |  Eff. $\pi N$ |
---|---|---|---|---|---|---|---
| work | and Bag | K-matr. app. |  | theorem | Lagran. | quark
|  |  2.79 [15, 16] |  |  |  |  |  3.49[21]
${\mu_{\Delta^{+}}}$ | 3.66 |  2.13[17] |  |  2.0-3.0[24] |  |  |  2.85[22]
|  |  2.20-2.45[18] |  |  |  |  |  2.3-2.7[23]
|  |  3.27[20] |  |  |  |  |  2.79[26]
|  |  5.58 [15, 16] |  6.9-9.7[19]∗ |  |  3.6$\pm$2.0[9]∗ |  |  6.98[21]
${\mu_{\Delta^{++}}}$ | 5.49 |  4.25[17] |  4.52$\pm$0.95[4]∗ |  4.2-7.4[24] |  5.6$\pm$2.1[10]∗ |  6.1$\pm$0.5[25]∗ |  5.33[22]
|  |  4.41-4.89[18] |  5.6-7.5[3]∗ |  |  4.7-6.9[2]∗ |  |  5.1-5.4[23]
|  |  6.54[20] |  |  |  3.7-4.9[14]∗ |  |  6.17[26]

The numerical values of the magnetic moments of the $\Delta^{+}$ and
$\Delta^{++}$ resonances are given in Table 1. In a number of approaches the
magnetic moment of $\Delta$ is treated as an adjustable parameter. The
corresponding results obtained from the experimental cross sections of the
$\pi^{+}p\to\gamma\pi^{+}p$ reaction are indicated in Table 1 with the upper
index ∗. It must be emphasized that only our approach and the naive $SU(6)$
quark model give an analytical form for $\mu_{\Delta^{+}}$ and
$\mu_{\Delta^{++}}$. But our result for $\mu_{\Delta^{+}}$ is
$M_{\Delta}/m_{p}\sim 1.31$ times larger as
$\mu_{\Delta^{+}}=\mu_{p}=2.79\mu_{B}$ in refs. [15, 26].

The $SU(6)$ models [15, 16, 17] and their bag model modifications require
proportionality between the charge and the magnetic dipole moment
$\mu_{\Delta}=e_{\Delta}\mu_{p}$ of the $\Delta$ resonance. Therefore,
$\mu_{\Delta^{+}}=1/2\ \mu_{\Delta^{++}}$ in [15, 16, 17, 18, 20]. This
property is preserved in the constituent quark model [21]. But it is broken in
the Skyrme model [24], chiral quark model [23], chiral quark-soliton model
[22], and effective quark model [26]. Our result for the ratio
$\mu_{\Delta^{++}}/\mu_{\Delta^{+}}$ is determined by the isospin factors of
the $\pi^{+}p$ and $\pi^{o}p$ elastic scattering amplitudes. In addition, we
take into account the difference between units of $\mu_{\Delta}$ and $\mu_{p}$
in the $\Delta-\gamma\Delta$ and $\gamma NN$ vertices. This difference
generates the factor $M_{\Delta}/m_{N}$. Therefore the present value
$\mu_{\Delta^{+}}=3.64\mu_{B}$ is larger than other predictions.

Our approach is based on usual local quantum field theory[28, 29]. This
approach is not dependent on the form of the Lagrangian. Moreover, we have not
used a special representation of the $\pi N$ amplitude and the $\Delta$
propagator. Therefore, the suggested relations between $\mu_{\Delta}$ and the
anomalous magnetic moment of the proton are model independent. But the present
field-theoretical formulation does not include the quark degrees of freedom.
The generalization of the present formulation based on the field-theoretical
approach with the quark-gluon degrees of freedom will be given in the
following paper.

We thank P. Minkowski for his interest in this paper and M. I. Potapov for his
help in preparation of this manuscript.

Appendix A: Projections on the intermediate $\Delta$ states

In this section a set of transformations (3.1a,c) of the external particle
radiation amplitude $({\cal
E})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(N)$(2.15) and the
corresponding Ward-Takahashi identity (2.14b) is performed. The resulting
condition (3.2) as well as other intermediate current conservation conditions
are obtained on the basis of the algebraic identity

$k^{\prime}_{\mu}\Bigl{[}(P+P^{\prime})^{\mu}-i\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}=s-s^{\prime}.$
$None$

The decompositions (3.1a,b,c,d) of current conservation are detailed in the
following subsections.

A. Decomposition over the transverse and longitudinal parts of the external
particle radiation amplitude (2.15), i.e., $({\cal
E})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(N)\Longrightarrow$
$({\cal E_{L}})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(N)$.

In order to separate the transverse part of ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(N)$ (2.15) (Fig. 1) it
is convenient to introduce the total and relative moments

$P=p_{\pi}+p_{N};\ \ \
p={{\alpha_{\pi}p_{N}-\alpha_{N}p_{\pi}}\over{\alpha_{\pi}+\alpha_{N}}};\ \ \
p_{N}={{\alpha_{N}P}\over{\alpha_{\pi}+\alpha_{N}}}+p,\ \ \
p_{\pi}={{\alpha_{\pi}P}\over{\alpha_{\pi}+\alpha_{N}}}-p,$ $None$

$P^{\prime}=p^{\prime}_{\pi}+p^{\prime}_{N};\ \ \
p^{\prime}={{\alpha^{\prime}_{\pi}p^{\prime}_{N}-\alpha^{\prime}_{N}p^{\prime}_{\pi}}\over{\alpha^{\prime}_{\pi}+\alpha^{\prime}_{N}}};\
\ \
p^{\prime}_{N}={{\alpha^{\prime}_{N}P^{\prime}}\over{\alpha^{\prime}_{\pi}+\alpha^{\prime}_{N}}}+p^{\prime},\
\ \
p^{\prime}_{\pi}={{\alpha^{\prime}_{\pi}P^{\prime}}\over{\alpha^{\prime}_{\pi}+\alpha^{\prime}_{N}}}-p^{\prime},$
$None$

where

$\alpha_{N}=k^{\prime}_{\nu}p_{N}^{\nu},\ \ \
\alpha_{\pi}=k^{\prime}_{\nu}p_{\pi}^{\nu};\ \ \
\alpha^{\prime}_{N}=k^{\prime}_{\nu}{p^{\prime}_{N}}^{\nu},\ \ \
\alpha^{\prime}_{\pi}=k^{\prime}_{\nu}{p^{\prime}_{\pi}}^{\nu}.$ $None$

The relative moments $p$ and $p^{\prime}$ are transverse to $k^{\prime}_{\mu}$

$k^{\prime}_{\nu}{p}^{\nu}=0;\ \ \ k^{\prime}_{\nu}{p^{\prime}}^{\nu}=0.$
$None$

Now one can separate the transverse part $({\cal
E_{TR}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ from ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(N)$ (2.15) as

${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(N)=({\sf
E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}+({\cal
E_{TR}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu},$ $None$

where ${k^{\prime}}_{\mu}({\sf
E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}=-{\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}(N)$, ${k^{\prime}}_{\mu}({\cal
E_{TR}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}=0$ and

$({\sf E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-{1\over{s-s^{\prime}}}\Biggl{[}{\overline{u}}({\bf
p^{\prime}_{N}})\Bigl{[}2e_{N^{\prime}}{P^{\prime}}^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p^{\prime}_{N}+k^{\prime}}){\overline{u}}({\bf
p^{\prime}_{N}+k^{\prime}})<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$ $-<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}})\Bigl{[}2e_{N}P^{\mu}-i\mu_{N}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p}_{N})\Biggr{]}$
$-{1\over{s-s^{\prime}}}\Biggl{[}2e_{\pi^{\prime}}{P^{\prime}}^{\mu}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>-2e_{\pi}P^{\mu}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|j_{\pi}(0)|{\bf p}_{N};in>\Biggr{]}$ $None$

$({\cal E_{TR}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-\Biggl{[}e_{N^{\prime}}{{{p^{\prime}}^{\mu}}\over{\alpha^{\prime}_{N}}}{\overline{u}}({\bf
p^{\prime}}_{N})u({\bf p^{\prime}_{N}+k^{\prime}}){\overline{u}}({\bf
p^{\prime}_{N}+k^{\prime}})<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$
$-e_{\pi^{\prime}}{{{p^{\prime}}^{\mu}}\over{\alpha^{\prime}_{\pi}}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>+e_{\pi}{{{p}^{\mu}}\over{\alpha_{\pi}}}<out;{\bf
p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf p}_{N};in>$
$-e_{N}{{{p}^{\mu}}\over{\alpha_{N}}}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf p_{N}-k^{\prime}})u({\bf
p}_{N})\Biggr{]}$ $None$

Other non-longitudinal terms can be obtained using new total four moments
$P_{\pm}$

$P_{\pm}={1\over 2}(P\pm P^{\prime}),\ \ where\ P=P_{+}+P_{-};\ \ \
P^{\prime}=P_{+}-P_{-}\ \ \ and\ \ \ {P_{-}}^{\mu}={1\over
2}{k^{\prime}}^{\mu}.$ $None$

This allows to separate of the term ${\cal
K}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ proportional to
$k^{\prime}_{\mu}$

$({\sf E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}=({\cal
E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}+{\cal
K}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu},$ $None$

where

$({\cal E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-{1\over{s-s^{\prime}}}\Biggl{[}{\overline{u}}({\bf
p^{\prime}_{N}})\Bigl{[}e_{N^{\prime}}(P+P^{\prime})^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p^{\prime}_{N}+k^{\prime}}){\overline{u}}({\bf
p^{\prime}_{N}+k^{\prime}})<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$ $-<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}})\Bigl{[}e_{N}(P^{\prime}+P)^{\mu}-i\mu_{N}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p}_{N})\Biggr{]}$
$-{1\over{s-s^{\prime}}}\Biggl{[}e_{\pi^{\prime}}{(P+P^{\prime})}^{\mu}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>-e_{\pi}{(P+P^{\prime})}^{\mu}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|j_{\pi}(0)|{\bf p}_{N};in>\Biggr{]},$ $None$

${\cal K}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}={{{k^{\prime}}^{\mu}}\over{s-s^{\prime}}}\Biggl{[}{\overline{u}}({\bf
p^{\prime}}_{N})u({\bf p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})e_{N^{\prime}}<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf
p}_{\pi}{\bf p}_{N};in>$ $+e_{\pi^{\prime}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
$+e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|{\overline{J}}(0)|{\bf
p}_{\pi};in>u({\bf p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}})u({\bf p}_{N})+e_{\pi}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|j_{\pi}(0)|{\bf p}_{N};in>\Biggr{]},$ $None$

It is easy to see that ${k^{\prime}}_{\mu}{\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(N)={k^{\prime}}_{\mu}({\cal
E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}=-{\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}(N)$. The resulting expression $({\cal
E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (3.7a) differs
from ${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(N)$ (2.15)
by the $\gamma NN$ and $\gamma\pi\pi$ vertices which have unified factors
$(P+P^{\prime})^{\mu}-i\mu_{N^{\prime}(N)}\sigma^{\mu\nu}k^{\prime}_{\nu}$ and
$(P+P^{\prime})^{\mu}$.

B. Projection on the $\Delta$-pole terms $({\cal
E_{L}})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(N)\Longrightarrow$
$({\cal E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-pole)$ and ${\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}(N)\Longrightarrow$ ${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(\Delta-
pole)$.

In order to separate the $\Delta$-pole parts in $({\cal
E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (A.7a) we shall
use a projection of the $\gamma N-N$ and $\gamma\pi-\pi$ vertex on the spin
$3/2$ intermediate states

$(p^{\prime}_{N}{\bf.}p_{N})\biggl{[}{\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}\Bigl{[}e_{N^{\prime}}{(P+P^{\prime})}^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}\gamma_{5}u({\bf
p^{\prime}_{N}+k^{\prime}})\biggr{]}^{Projection\ on\ spin\ 3/2\ states}=$
$\biggl{(}{\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}{p^{\prime}_{N}}_{a}u^{a}({\bf
P^{\prime}})\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}\biggl{[}e_{N^{\prime}}{(P+P^{\prime})}^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\biggr{]}u^{c}({\bf
P})\Biggr{\\}}{\overline{u}}^{d}({\bf P})(p_{N})_{d}\gamma_{5}u({\bf
p_{N}})\biggl{)}$ ${\overline{u}}({\bf p_{N}})u({\bf
p^{\prime}_{N}+k^{\prime}}),$ $None$

$(p^{\prime}_{N}{\bf.}p_{N})\biggl{[}e_{\pi^{\prime}}{(P+P^{\prime})}^{\mu}\biggr{]}^{Projection\
on\ spin\ 3/2\ states}=\biggl{(}{\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}{p^{\prime}_{N}}_{a}u^{a}({\bf P^{\prime}})$
$\Bigl{[}{\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}e_{\pi^{\prime}}{(P+P^{\prime})}^{\mu}u^{c}({\bf
P})\Bigr{]}{\overline{u}}^{d}({\bf P})(p_{N})_{d}\gamma_{5}u({\bf
p_{N}})\biggl{)}{\overline{u}}({\bf p_{N}})u({\bf p^{\prime}_{N}}).$ $None$

where $(p^{\prime}_{N}{\bf.}p_{N})=(p^{\prime}_{N})_{\sigma}(p_{N})^{\sigma}$
and we omit the spin index $S=\pm 1/2,\pm 3/2$ of the Rarita-Schwinger spinor
$u^{a}({\bf P})$ with the mass $m_{D_{3/2}}^{2}=P_{o}^{2}-{\bf P}^{2}$. The
matrix element (A.8a) corresponds to the transitions $\pi N\to
D_{3/2}\to\gamma^{\prime}D_{3/2}^{\prime}\to\pi^{\prime}N^{\prime}$ with the
intermediate spin $3/2$ particles $D_{3/2}$ and
$D_{3/2}^{\prime}$.55footnotemark: 5

The common factor $\biggl{(}{\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}{p^{\prime}_{N}}_{a}u^{a}({\bf P^{\prime}})$
$\Bigl{[}{\overline{u}}^{b}({\bf P^{\prime}})g_{bc}(...)^{\mu}u^{c}({\bf
P})\Bigr{]}{\overline{u}}^{d}({\bf P})(p_{N})_{d}\gamma_{5}u({\bf
p_{N}})\biggl{)}$ 666 Using the completeness conditions of the spin $3/2$
functions $u^{a}({\bf P},S)$ [38, 39, 40] and $v^{a}({\bf P},S)$ of the free
particle and antiparticle states  $\sum_{S=-3/2}^{3/2}\Biggl{(}u^{a}({\bf
P},S){\overline{u}}^{b}({\bf
P},S)+{{\gamma_{\sigma}P^{\sigma}+s^{1/2}}\over{2s^{1/2}}}\biggl{\\{}\biggl{[}projections\
on\ spin\ 1/2\ states\biggr{]}^{ab}\biggr{\\}}\Biggr{)}+$
$\sum_{S=-3/2}^{3/2}\Biggl{(}v^{a}({\bf P},S){\overline{v}}^{b}({\bf
P},S)+{-{\gamma_{\sigma}P^{\sigma}+s^{1/2}}\over{2s^{1/2}}}\biggl{\\{}\biggl{[}projections\
on\ spin\ 1/2\ states\biggr{]}^{ab}\biggr{\\}}^{\ast}\Biggr{)}=g^{ab}$ one can
rewrite the $\gamma N-N$ vertex as  $g_{bc}{\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}g^{ab}\Bigl{[}e_{N^{\prime}}{(P+P^{\prime})}^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}g^{cd}\gamma_{5}u({\bf
p^{\prime}_{N}+k^{\prime}})={\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}\sum_{S=-3/2}^{3/2}\Biggl{(}u^{a}({\bf
P},S){\overline{u}}^{b}({\bf P},S)+v^{a}({\bf P},S){\overline{v}}^{b}({\bf
P},S)+...\Biggr{)}$
$g_{bc}\Bigl{[}e_{N^{\prime}}{(P+P^{\prime})}^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}\sum_{S^{\prime}=-3/2}^{3/2}\Biggl{(}u^{c}({\bf
P},S){\overline{u}}^{d}({\bf P},S)+v^{c}({\bf P},S){\overline{v}}^{d}({\bf
P},S)+...\Biggr{)}\gamma_{5}u({\bf p^{\prime}_{N}+k^{\prime}}).$ In the
$\Delta$ resonance region one can take into account only the spin $3/2$
intermediate states. In addition, other degrees of freedom with antiparticle
and spin $1/2$ intermediate states form the independent Ward-Takahashi
identities. Then for the final nucleon radiation term we obtain
$\biggl{[}(p^{\prime}_{N})_{a}(p_{N})^{a}{\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}\Bigl{[}e_{N^{\prime}}{(P+P^{\prime})}^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}\gamma_{5}u({\bf
p^{\prime}_{N}+k^{\prime}})\biggr{]}^{Projection\ on\ spin\ 3/2\ states}=$
$\sum_{S,S^{\prime}=-3/2}^{3/2}{\overline{u}}({\bf
p^{\prime}_{N}})\gamma_{5}{p^{\prime}_{N}}_{a}u^{a}({\bf
P^{\prime}},S^{\prime})\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}},S^{\prime})g_{bc}\biggl{[}e_{N^{\prime}}{(P+P^{\prime})}^{\mu}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\biggr{]}u^{c}({\bf
P},S)\Biggr{\\}}{\overline{u}}^{d}({\bf P},S)(p_{N})_{d}\gamma_{5}u({\bf
p^{\prime}_{N}+k^{\prime}}).$ In the same way for the final pion radiation
term we get  $<out;{\bf p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>\Longrightarrow{\overline{u}}({\bf
p^{\prime}}_{N})\gamma_{5}^{2}u({\bf p}_{N}){\overline{u}}({\bf p}_{N})u({\bf
p^{\prime}}_{N})<out;{\bf p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf
p}_{\pi}{\bf p}_{N};in>\Longrightarrow{1\over{(p^{\prime}_{N}{\bf.}p_{N})}}$
$\Biggl{[}{\overline{u}}({\bf
p^{\prime}}_{N})\gamma_{5}(p^{\prime}_{N})_{a}u^{a}({\bf
P^{\prime}})\biggl{\\{}{\overline{u}}^{b}({\bf P^{\prime}})g_{bc}u^{c}({\bf
P})\biggr{\\}}{\overline{u}}^{d}({\bf P})(p_{N})_{d}\gamma_{5}u({\bf
p}_{N})\Biggr{]}{\overline{u}}({\bf p}_{N})u({\bf p^{\prime}}_{N})<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>.$ Hereafter
the spin indexes $S$ and $S^{\prime}$ are omitted for the sake of simplicity.
in (A.8a,b) generates the following redefinition of (A.7a)

$\biggl{[}({\cal E_{L}})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}\biggr{]}^{Projection\ on\ spin\ 3/2\ states}\equiv({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}=$
${{(p^{\prime}_{N})_{a}(p_{N})_{d}{(P+P^{\prime})}^{\mu}}\over{|{\bf
p^{\prime}}_{N}||{\bf p}_{N}|(s-s^{\prime})}}{\overline{u}}({\bf
p^{\prime}_{N}})i\gamma_{5}u^{a}({\bf
P^{\prime}})\Bigl{\\{}{\overline{u}}^{b}({\bf P^{\prime}})g_{bc}u^{c}({\bf
P})\Bigr{\\}}{\overline{u}}^{d}({\bf P})i\gamma_{5}u({\bf p_{N}})$
$\Biggl{[}{{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}\over{(p^{\prime}_{N}{\bf.}p_{N})}}\biggl{(}{\overline{u}}({\bf
p_{N}})u({\bf p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})e_{N^{\prime}}<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf
p}_{\pi}{\bf p}_{N};in>$ $-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf p_{N}-k^{\prime}})u({\bf
p^{\prime}_{N}})$ $+{\overline{u}}({\bf p_{N}})u({\bf
p^{\prime}_{N}})e_{\pi^{\prime}}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
$-e_{\pi}<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>{\overline{u}}({\bf p_{N}})u({\bf
p^{\prime}_{N}})\biggr{)}\Biggr{]}^{Projection\ on\ spin\ 3/2\ states}$
$+{{(p^{\prime}_{N})_{a}(p_{N})_{d}}\over{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|(s-s^{\prime})}}{\overline{u}}({\bf
p^{\prime}_{N}})i\gamma_{5}u^{a}({\bf
P^{\prime}})\Bigl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}(-i\sigma^{\mu\nu}k^{\prime}_{\nu})u^{c}({\bf
P})\Bigr{\\}}{\overline{u}}^{d}({\bf P})i\gamma_{5}u({\bf p_{N}})$
$\Biggl{[}{{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}\over{(p^{\prime}_{N}{\bf.}p_{N})}}\biggl{(}{\overline{u}}({\bf
p_{N}})u({\bf p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})\mu_{N^{\prime}}<out;{\bf
p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf p}_{N};in>$ $-\mu_{N}<out;{\bf
p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|{\overline{J}}(0)|{\bf
p}_{\pi};in>u({\bf p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}})u({\bf p^{\prime}_{N}})\biggr{)}\Biggr{]}^{Projection\ on\
spin\ 3/2\ states}.$ $None$

The $\pi N$ amplitudes (2.9a,b,c,d) consist of the resonant and non-resonant
parts 777 For the on energy shell $\pi N$ and $\pi^{\prime}N^{\prime}$ states
with $s=s^{\prime}$, i.e., $k^{\prime}=0$, the operator  ${\cal Q}({\bf
p^{\prime}}_{N},{\bf p}_{N},{\bf P^{\prime}}_{\Delta},{\bf
P}_{\Delta})={1\over{(p^{\prime}_{N}{\bf.}p_{N})}}{\overline{u}}({\bf
p^{\prime}}_{N})i\gamma_{5}({{p^{\prime}}_{N}})_{a}u^{a}({\bf
P^{\prime}}_{\Delta}){\overline{u}}^{d}({\bf
P}_{\Delta})({{p}_{N}})_{d}i\gamma_{5}u({\bf p}_{N})$ is transformed into the
projection operator on the $\pi N$ state with the orbital momentum $L=1$ and
the total momentum $J=3/2$ ${\cal P}_{1}^{3/2}({\bf p^{\prime}}_{N},{\bf
p}_{N})$ [32] ${\cal P}_{1}^{3/2}({\bf p^{\prime}}_{N},{\bf
p}_{N})={{6m_{N}}\over{4\pi{\bf p}{\bf p^{\prime}}(m_{N}+\sqrt{m_{N}^{2}+{\bf
p}^{2}})}}{\overline{u}}({\bf
p^{\prime}}_{N})i\gamma_{5}{{p^{\prime}}_{N}}_{a}u^{a}({\bf
P^{\prime}}_{\Delta}){\overline{u}}^{d}({\bf
P}_{\Delta}){{p}_{N}}_{d}i\gamma_{5}u({\bf p}_{N}).$ Therefore, we have
$\Biggl{[}{\cal Q}({\bf p^{\prime}}_{N},{\bf p}_{N},{\bf
P^{\prime}}_{\Delta},{\bf P}_{\Delta})\Biggr{]}^{|{\bf
k^{\prime}}|=0}={{4\pi{\bf p}{\bf p^{\prime}}(m_{N}+\sqrt{m_{N}^{2}+{\bf
p}^{2}})}\over{6m_{N}(p^{\prime}_{N}{\bf.}p_{N})}}{\cal P}_{1}^{3/2}({\bf
p^{\prime}}_{N},{\bf p}_{N})$

$\biggl{[}{{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}\over{(p^{\prime}_{N}{\bf.}p_{N})}}{\overline{u}}({\bf
p_{N}}){{(p^{\prime}_{N}+k^{\prime})_{\sigma}\gamma^{\sigma}+m_{N}}\over{2m_{N}}}{\overline{u}}({\bf
p^{\prime}}_{N})<out;{\bf p^{\prime}}_{\pi}|J(0)|{\bf p}_{\pi}{\bf
p}_{N};in>)\biggr{]}^{Projection\ on\ spin\ 3/2\ states}=$ ${{{\cal
R}_{N^{\prime}}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-P^{o}_{\Delta}(s)}}+{r}_{N^{\prime}}(s^{\prime}{\bf
P^{\prime}}_{\Delta};s{\bf P}_{\Delta});$ $None$

$\biggl{[}{{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}\over{(p^{\prime}_{N}{\bf.}p_{N})}}{\overline{u}}({\bf p_{N}})u({\bf
p^{\prime}_{N}})<out;{\bf p^{\prime}}_{N}|j_{\pi^{\prime}}(0)|{\bf
p}_{\pi}{\bf p}_{N};in>\biggr{]}^{Projection\ on\ spin\ 3/2\ states}=$
${{{\cal R}_{\pi^{\prime}}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}}+{r}_{\pi^{\prime}}(s^{\prime}{\bf
P^{\prime}}_{\Delta};s{\bf P}_{\Delta});$ $None$

$\bigg{[}{{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}\over{(p^{\prime}_{N}{\bf.}p_{N})}}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}(0)|{\bf p}_{\pi};in>u({\bf
p}_{N}){{(p_{N}-k^{\prime})_{\sigma}\gamma^{\sigma}+m_{N}}\over{2m_{N}}}u({\bf
p^{\prime}_{N}})\biggr{]}^{Projection\ on\ spin\ 3/2\ states}=$ ${{{\cal
R}_{N}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}_{N}}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}+{r}_{N}(s^{\prime}{\bf
P^{\prime}}_{\Delta};s{\bf P}_{\Delta});$ $None$ $\biggl{[}{{|{\bf
p^{\prime}}_{N}||{\bf p}_{N}|}\over{(p^{\prime}_{N}{\bf.}p_{N})}}<out;{\bf
p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>{\overline{u}}({\bf p_{N}})u({\bf
p^{\prime}_{N}})\biggr{]}^{Projection\ on\ spin\ 3/2\ states}=$ ${{{\cal
R}_{\pi}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}+{r}_{\pi}(s^{\prime}{\bf
P^{\prime}}_{\Delta};s{\bf P}_{\Delta}).$ $None$

The linear relativistic $\Delta$ propagator in (A.9a,b,c,d) is most similar to
non-relativistic quantum mechanical $\Delta$ propagator.

77footnotetext:  Equations (A.9a,b,c,d) are also valid in the models, where
the $\Delta$’s are considered as the intermediate one-particle states which
decays into the asymptotic $\pi N$ states (see for example ref. [42]). The
${\cal R}$ and $r$ functions and the $\Delta-\pi N$ vertices in this case are
defined in the one-particle approach. Nevertheless, intermediate one-particle
$\Delta$’s are not equivalent to real particle degrees of freedom which have
appropriate one-particle asymptotic states. Therefore, the introduction of the
effective $\pi N\Delta$ Lagrangian with the one-particle Heisenberg operator
of $\Delta$ requires an additional assumption.

Projection on the spin $3/2$ states and separation of the $\Delta$ pole terms
modifies Ward-Takahashi identity (A.7c) as

$\Biggl{[}{k^{\prime}}_{\mu}<out;{\bf p^{\prime}}_{N}{\bf
p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>\Biggr{]}^{Projection\ on\ spin\ 3/2\ particle\
states}={k^{\prime}}_{\mu}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}+{\cal
B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}$ $={k^{\prime}}_{\mu}\Bigl{(}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-pole)+({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(non-
pole)\Bigr{)}+{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}({\Delta}-pole)+{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(non-pole)=0.$
$None$

With identity (A.1), it is easy to see, that the $\Delta$-pole and $non-pole$
parts of $({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}$ separately satisfy the independent Ward-Takahashi identities. In
particular,

$\Biggl{[}{k^{\prime}}_{\mu}<out;{\bf p^{\prime}}_{N}{\bf
p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>\Biggr{]}^{Projection\ on\ spin\ 3/2\ particle\ states}_{\Delta-
pole}$ ${k^{\prime}}_{\mu}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-pole)+{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}({\Delta}-pole)=0,$ $None$

where

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-pole)={1\over{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}}{\overline{u}}({\bf
p^{\prime}_{N}})(p^{\prime}_{N})_{a}i\gamma_{5}u^{a}({\bf P^{\prime}})$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}\Bigl{[}(P+P^{\prime})^{\mu}{\cal
V}_{E}-i\sigma^{\mu\nu}k^{\prime}_{\nu}{\cal V}_{H}\Bigr{]}u^{c}({\bf
P})\Biggr{\\}}{\overline{u}}^{d}({\bf P})(p_{N})_{d}i\gamma_{5}u({\bf
p_{N}}),$ $None$

${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}({\Delta}-pole)={{(p^{\prime}_{N})_{a}(p_{N})_{d}}\over{|{\bf
p^{\prime}}_{N}||{\bf p}_{N}|}}{\overline{u}}({\bf
p^{\prime}_{N}})i\gamma_{5}u^{a}({\bf
P^{\prime}})\Bigl{\\{}{\overline{u}}^{b}({\bf P^{\prime}})g_{bc}u^{c}({\bf
P})\Bigr{\\}}{\overline{u}}^{d}({\bf P})i\gamma_{5}u({\bf p_{N}})$
$\Biggl{[}{{{\cal R}_{N^{\prime}}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}}+{{{\cal
R}_{\pi^{\prime}}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}}-{{{\cal
R}_{N}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}-{{{\cal
R}_{\pi}(s^{\prime}{\bf P^{\prime}}_{\Delta};s{\bf
P}_{\Delta})}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}\Biggr{]},$
$None$

where

${\cal V}_{E}={{e_{N^{\prime}}{\cal
R}_{N^{\prime}}}\over{(s-s^{\prime})\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}+{{e_{\pi^{\prime}}{\cal
R}_{\pi^{\prime}}}\over{(s-s^{\prime})\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}$
$-{{e_{N}{\cal
R}_{N}}\over{(s-s^{\prime})\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}}}-{{e_{\pi}{\cal
R}_{\pi}}\over{(s-s^{\prime})\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}}},$
$None$

${\cal V}_{H}={{\mu_{N^{\prime}}{\cal
R}_{N^{\prime}}}\over{(s-s^{\prime})\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}-{{\mu_{N}{\cal
R}_{N}}\over{(s-s^{\prime})\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}}},$
$None$

Hereafter we omit the variables of ${\cal R}$ functions for the sake of
simplicity .

The resulting expressions (A.10c,d) have the common factors
${\overline{u}}({\bf p^{\prime}_{N}})(p^{\prime}_{N})_{a}i\gamma_{5}u^{a}({\bf
P^{\prime}})$, ${\overline{u}}^{d}({\bf P})(p_{N})_{d}i\gamma_{5}u({\bf
p_{N}})$, ${\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}(P+P^{\prime})^{\mu}u^{c}({\bf P})$,
${\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}\Bigl{[}-i\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u^{c}({\bf
P})$ and ${\overline{u}}^{b}({\bf P^{\prime}})g_{bc}u^{c}({\bf P})$ which are
needed for the $\Delta-\gamma\Delta$-type vertex in (3.6) depicted in Fig. 2B.

C. The double $\Delta$ exchange amplitude and transitions $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-
pole)\Longrightarrow$ $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)$ and ${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-pole)\Longrightarrow$ ${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta\Delta)$.

Next we have to extract from ${\cal E_{L}}^{3/2})_{\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-pole)$ (A10b) the amplitude which has the same analytical
properties as the double $\Delta$ exchange term in Fig. 2B. Using a simple
algebra we rewrite (A.11b) as

${\cal
V}_{H}={1\over{\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}\Biggl{\\{}R_{+}\biggl{[}{{|{\bf
k^{\prime}}|}\over{s-s^{\prime}}}-{{{P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}\over{s-s^{\prime}}}\biggr{]}$
$+{{R_{-}}\over{s-s^{\prime}}}\biggl{[}{p}_{\pi}^{o}+{p_{N}}^{o}+{p^{\prime}}_{\pi}^{o}+{p^{\prime}_{N}}^{o}-{P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\biggr{]}\Biggr{\\}},$
$None$

where

$R_{\pm}={1\over 2}\Bigl{[}\mu_{N^{\prime}}{\cal
R}_{N^{\prime}}\pm\mu_{N}{\cal R}_{N}\Bigr{]}$ $None$

and we use the identities $1/a\pm 1/b=1/a\ (b\pm a)\ 1/b$ with
$a=\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}$
and $b=\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}$. These
transformations play a central role in connection between the amplitudes in
Fig. 1 and in Fig. 2B.

The first part in (A.12a) is regular at $|{\bf k^{\prime}}|=0$, where
$s=s^{\prime}$, because
$\Bigl{(}{P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}/(s-s^{\prime})$
is finite at $s=s^{\prime}$.

The second part of ${\cal V}_{H}$ (A.12a) can describe only one $\Delta$
exchange because
${p}_{\pi}^{o}+{p_{N}}^{o}+{p^{\prime}}_{\pi}^{o}+{p^{\prime}_{N}}^{o}-{P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime})$
cancels out one of the $\Delta$ propagators. This expression has different
behavior in two different cases:

${\bf 1.}$ For charge exchange reactions $R_{-}/(s-s^{\prime})$ can be
singular at the threshold $|{\bf k^{\prime}}|=0$. This case needs a special
investigation which is out of the scope of this article.

${\bf 2.}$ For the $\pi N$ bremsstrahlung reactions without charge exchange
(e.g. $\pi^{\pm}p\longrightarrow\gamma\pi^{\pm}p$ or
$\pi^{o}p\longrightarrow\gamma\pi^{o}p$) we have $e_{N^{\prime}}=e_{N}$ and
$\mu_{N^{\prime}}=\mu_{N}$. In this case $R_{-}/(s-s^{\prime})$ is finite at
the threshold $|{\bf k^{\prime}}|=0$ and this part corresponds to the one
$\Delta$ exchange diagrams of the $\pi
N\to\gamma^{\prime}\pi^{\prime}N^{\prime}$ reaction. Using identity (A.1) one
can separate one and double $\Delta$ exchange contributions through the
different current conservation conditions, i.e., one can split $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-pole)$ and ${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}({\Delta}-pole)$ into two parts

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-pole)=({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)+({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta)$
${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}({\Delta}-pole)={\cal
B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(\Delta\Delta)+{\cal
B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(\Delta),$ $None$

where

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}\Delta)={1\over{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}}{\overline{u}}({\bf
p^{\prime}_{N}})(p^{\prime}_{N})_{a}i\gamma_{5}u^{a}({\bf P^{\prime}})$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}\Bigl{[}(P+P^{\prime})^{\mu}{\cal
V}_{E}^{(+)}-i\sigma^{\mu\nu}k^{\prime}_{\nu}{\cal
V}_{H}^{(+)}\Bigr{]}u^{c}({\bf P})\Biggr{\\}}{\overline{u}}^{d}({\bf
P})(p_{N})_{d}i\gamma_{5}u({\bf p_{N}}),$ $None$

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta})={1\over{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}}{\overline{u}}({\bf
p^{\prime}_{N}})(p^{\prime}_{N})_{a}i\gamma_{5}u^{a}({\bf P^{\prime}})$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}})g_{bc}\Bigl{[}(P+P^{\prime})^{\mu}{\cal
V}_{E}^{(-)}-i\sigma^{\mu\nu}k^{\prime}_{\nu}{\cal
V}_{H}^{(-)}\Bigr{]}u^{c}({\bf P})\Biggr{\\}}{\overline{u}}^{d}({\bf
P})(p_{N})_{d}i\gamma_{5}u({\bf p_{N}}),$ $None$

${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}({\Delta}\Delta)={{s-s^{\prime}}\over{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}}{\overline{u}}({\bf
p^{\prime}_{N}})(p^{\prime}_{N})_{a}i\gamma_{5}u^{a}({\bf
P^{\prime}})\Biggl{\\{}{\overline{u}}^{b}({\bf P^{\prime}})g_{bc}{\cal
V}_{E}^{(+)}u^{c}({\bf P})\Biggr{\\}}{\overline{u}}^{d}({\bf
P})(p_{N})_{d}i\gamma_{5}u({\bf p_{N}}),$ $None$

${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}({\Delta})={{s-s^{\prime}}\over{|{\bf p^{\prime}}_{N}||{\bf
p}_{N}|}}{\overline{u}}({\bf
p^{\prime}_{N}})(p^{\prime}_{N})_{a}i\gamma_{5}u^{a}({\bf
P^{\prime}})\Biggl{\\{}{\overline{u}}^{b}({\bf P^{\prime}})g_{bc}{\cal
V}_{E}^{(-)}u^{c}({\bf P})\Biggr{\\}}{\overline{u}}^{d}({\bf
P})(p_{N})_{d}i\gamma_{5}u({\bf p_{N}}),$ $None$

where

${\cal V}_{E}^{(+)}={{e_{N}({\cal R}_{N^{\prime}}+{\cal R}_{N})+e_{\pi}({\cal
R}_{\pi^{\prime}}+{\cal R}_{\pi})}\over{2(s-s^{\prime})}}{{|{\bf
k^{\prime}}|-({P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime}))}\over{\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}$
$None$

${\cal V}_{H}^{(+)}={{\mu_{N}({\cal R}_{N^{\prime}}+{\cal
R}_{N})}\over{2(s-s^{\prime})}}{{|{\bf
k^{\prime}}|-({P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime}))}\over{\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}$
$None$

${\cal V}_{E}^{(-)}={{e_{N}({\cal R}_{N^{\prime}}-{\cal R}_{N})+e_{\pi}({\cal
R}_{\pi^{\prime}}-{\cal
R}_{\pi})}\over{2(s-s^{\prime})}}{{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})+{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}\over{\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}$
$None$

${\cal V}_{H}^{(-)}={{\mu_{N}({\cal R}_{N^{\prime}}-{\cal
R}_{N})}\over{2(s-s^{\prime})}}{{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})+{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}\over{\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}$
$None$

With (A.1) it easy to check that

${k^{\prime}}_{\mu}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}{\Delta})=-{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}({\Delta}{\Delta});\ \ \ {k^{\prime}}_{\mu}({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta})=-{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}({\Delta}).$
$None$

Thus we have extracted the double $\Delta$ exchange part from the $\Delta-
pole$ amplitudes (A.10b,c). Afterwards, the Ward-Takahashi identity (A.10a) is
divided into two independent identities (A.16). The resulting Ward-Takahashi
identity (A.16) contains the double $\Delta$ exchange terms.

D. An alternative form of the double $\Delta$ exchange amplitude $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)$.

Hereafter it is convenient to represent $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)$ (A.14a) and ${\cal
B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}({\Delta}\Delta)$ (A.14c) through the
$\pi N\to\Delta$, $\Delta\to\gamma^{\prime}\Delta^{\prime}$,
$\Delta^{\prime}\to\pi^{\prime}N^{\prime}$ vertices (3.5a,b) and the
intermediate $\Delta$ propagators. Therefore we rewrite (A.15a,b) as

${\cal
V}^{(+)}_{E}={1\over{\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime}){\sc
V}_{E}{\rm g}_{\Delta-\pi N}(s),$ $None$

${\cal
V}^{(+)}_{H}={1\over{\Bigl{(}{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})\Bigr{)}\Bigl{(}{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)\Bigr{)}}}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime}){\sc
V}_{H}{\rm g}_{\Delta-\pi N}(s).$ $None$

Relations (A.17a,b) allows to rewrite (A.14a,c) as

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)={{<{\bf p^{\prime}}_{N},{\bf p^{\prime}}_{\pi}|{\sl
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta}>}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}}_{\Delta})g_{bc}\Bigl{[}(P+P^{\prime})^{\mu}{\sc
V}_{E}-i\sigma^{\mu\nu}k^{\prime}_{\nu}{\sc V}_{H}\Bigr{]}u^{c}({\bf
P}_{\Delta})\Biggr{\\}}{{<{\bf P}_{\Delta}|{\sl g}_{\Delta-\pi N}|{\bf
p}_{N},{\bf p}_{\pi}>}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}},$
$None$

${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(\Delta\Delta)=-{{<{\bf
p^{\prime}}_{N},{\bf p^{\prime}}_{\pi}|{\sl
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta}>}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}}_{\Delta})g_{bc}\Bigl{[}(s-s^{\prime}){\sc V}_{E}\Bigr{]}u^{c}({\bf
P}_{\Delta})\Biggr{\\}}{{<{\bf P}_{\Delta}|{\sl g}_{\Delta-\pi N}|{\bf
p}_{N},{\bf p}_{\pi}>}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}}.$
$None$

The amplitude (A.18a) has the form of the usual $\Delta$ radiation diagram in
Fig. 2B with the $\Delta-\gamma\Delta$ vertex function
$(P+P^{\prime})^{\mu}{\sc V}_{E}-i\sigma^{\mu\nu}k^{\prime}_{\nu}{\sc V}_{H}$
instead of the $\Delta-\gamma\Delta$ vertex (B.3a,b) in Appendix B.

E. Transitions $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)\Longrightarrow$ $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)$ and ${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta\Delta)\Longrightarrow$ ${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$

It is important to note that ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-\gamma\Delta)$
(3.6) and $({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)$ (A.18a) contain different $\Delta-\gamma\Delta$
vertices. In order to unify these vertex functions we extract from $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)$ (3.15d) the part with the $\Delta-\gamma\Delta$ vertex
from (3.6)

$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}\Delta)=({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)+({E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta),$ $None$

$({\cal B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi N}^{\mu}({\Delta}\Delta)=({\cal
B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)+({B}^{3/2})_{\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta).$ $None$

where $({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ and $({\cal
B}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ are defined by (3.3a,b) and

$({E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta)={{<{\bf p^{\prime}}_{N},{\bf p^{\prime}}_{\pi}|{\sl
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta}>}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}}_{\Delta})g_{bc}\Bigl{[}(P+P^{\prime}-P_{\Delta}-P^{\prime}_{\Delta})^{\mu}{\sc
V}_{E}-\ i\sigma^{\mu\nu}({k^{\prime}-k^{\prime}_{\Delta}})_{\nu}{\sc
V}_{H}\Bigr{]}u^{c}({\bf P}_{\Delta}){{<{\bf P}_{\Delta}|{\sl g}_{\Delta-\pi
N}|{\bf p}_{N},{\bf
p}_{\pi}>}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}}.$ $None$

${B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(\Delta)={{<{\bf p^{\prime}}_{N},{\bf
p^{\prime}}_{\pi}|{\sl g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta}>}\over{{p^{\prime}}_{\pi}^{o}+{p^{\prime}}_{N}^{o}-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}}$
$\Biggl{\\{}{\overline{u}}^{b}({\bf
P^{\prime}}_{\Delta})g_{bc}\Bigl{[}k^{\prime}_{o}(P+P^{\prime}-P_{\Delta}-P^{\prime}_{\Delta})^{o}{\sc
V}_{E}-ik^{\prime}_{\mu}\sigma^{\mu o}(k^{\prime}-k^{\prime}_{\Delta})_{o}{\sc
V}_{H}\Bigr{]}u^{c}({\bf P}_{\Delta})\Biggr{\\}}{{<{\bf P}_{\Delta}|{\sl
g}_{\Delta-\pi N}|{\bf p}_{N},{\bf
p}_{\pi}>}\over{{p}_{\pi}^{o}+{p}_{N}^{o}-{P}^{o}_{\Delta}(s)}}.$ $None$

The difference between $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)$ (A.18a) and $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ (3.3a) makes the zero components of the
kinematic factors
$(P+P^{\prime})^{\mu}-(P_{\Delta}+P^{\prime}_{\Delta})^{\mu}=\delta^{\mu
0}\Bigl{[}P^{o}+{P^{\prime}}^{o}-P_{\Delta}^{o}-{P^{\prime}}^{o}_{\Delta}\Bigr{]}$;
$(k^{\prime}-k^{\prime}_{\Delta})_{\nu}=\delta_{\nu
0}\Bigl{[}P^{o}-{P^{\prime}}^{o}-P_{\Delta}^{o}+{P^{\prime}}^{o}_{\Delta}\Bigr{]}$.
These kinematic factors cancel out one of the $\Delta$ propagators
$1/({P^{\prime}}^{o}-{P^{\prime}}^{o}_{\Delta})$ or
$1/({P}^{o}-{P}^{o}_{\Delta})$ in
$({{E_{L}}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta)$ (3.20c). Therefore
$({{E_{L}}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta)$ (A.20b) corresponds to the one $\Delta$ exchange term.

Modification of $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta\Delta)$ (3.3a) and ${\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta\Delta)$ (3.3b) according to (3.19a,b) generates two new Ward-
Takahashi identities: identity (3.2) for the amplitudes (3.3a,b) and
$k^{\prime}_{\mu}({E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta)=-{B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi N}(\Delta)$ for the
amplitudes (A.20a,b).

An additional expression like
$({E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta)$
(A.20a) was also used in other papers [8, 19] in order to ensure current
conservation for the total $\pi N$ bremsstrahlung amplitude. Unlike the case
in those papers, $({E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta)$ corresponds to the one-$\Delta$ exchange amplitude which
satisfies the independent Ward-Takahashi identity.

Appendix B: $\gamma^{\prime}\Delta^{\prime}-\Delta$ vertex function with on
mass shell $\Delta$’s

The $\Delta-\gamma\Delta$ vertices can be constructed using the analytical
continuation of the spin $3/2$ particle electromagnetic vertex function
$<out;{\bf P^{\prime}}|{\cal J}^{\mu}(0)|{\bf P};in>$ in the complex region.
The electromagnetic vertex of the spin $3\over 2$ particles with the real mass
$m_{3/2}$ is

$<out;{\bf P^{\prime}}|{\cal J}^{\mu}(0)|{\bf
P};in>=(P+P^{\prime})^{\mu}\Biggl{(}{\overline{u}}^{\sigma}({\bf
P^{\prime}})\Bigl{[}g_{\rho\sigma}G_{1}({k^{\prime}}^{2})+{{{k^{\prime}}_{\sigma}{k^{\prime}}_{\rho}}\over{m_{3\over
2}^{2}}}G_{3}({k^{\prime}}^{2})\Bigr{]}\Biggr{)}u^{\rho}({\bf P})$
$+{\overline{u}}^{\sigma}({\bf
P^{\prime}})\Biggl{(}-i\sigma^{\mu\nu}{k^{\prime}}_{\nu}\Bigl{[}g_{\rho\sigma}G_{2}({k^{\prime}}^{2})+{{{k^{\prime}}_{\sigma}{k^{\prime}}_{\rho}}\over{m_{3\over
2}^{2}}}G_{4}({k^{\prime}}^{2})\Bigr{]}\Biggr{)}u^{\rho}({\bf P}),$ $None$

where ${k^{\prime}}_{\mu}=(P-P^{\prime})_{\mu}$ denotes the four momentum of
the emitted photon, ${\bf P}={\bf p}_{N}+{\bf p}_{\pi}={\bf P}_{\Delta}$,
${P}^{o}=\sqrt{m_{3\over 2}^{2}+{\bf P}^{2}}$; ${\bf P^{\prime}}={\bf
p}^{\prime}_{N}+{\bf p^{\prime}}_{\pi}={\bf P^{\prime}}_{\Delta}$,
${P^{\prime}}^{o}=\sqrt{m_{3\over 2}^{2}+{\bf P^{\prime}}^{2}}$ are the four
moments of spin $3/2$ particles with a mass $m_{3\over 2}$ in the initial and
final states.

Expression (B.1) can be analytically continued for the unequal masses of the
particles in the “in” ($m_{3/2}(in)=m_{3/2}$) and in the ”out”
($m_{3/2}(out)=m^{\prime}_{3/2}$) states

$<out;{\bf P^{\prime}}m^{\prime}_{3\over 2}|{\cal J}^{\mu}(0)|{\bf P}m_{3\over
2};in>=(P+P^{\prime})^{\mu}\Biggl{(}{\overline{u}}^{\sigma}({\bf
P^{\prime}})\Bigl{[}g_{\rho\sigma}G_{1}({k^{\prime}}^{2},m^{2}_{3\over
2},m^{\prime 2}_{3\over
2})+{{{k^{\prime}}_{\sigma}{k^{\prime}}_{\rho}}\over{m^{\prime}_{3\over
2}m_{3\over 2}}}G_{3}({k^{\prime}}^{2},m^{2}_{3\over 2},m^{\prime 2}_{3\over
2})\Bigr{]}\Biggr{)}u^{\rho}({\bf P})$ $+{\overline{u}}^{\sigma}({\bf
P^{\prime}})\Biggl{(}-i\sigma^{\mu\nu}{k^{\prime}}_{\nu}\Bigl{[}g_{\rho\sigma}G_{2}({k^{\prime}}^{2},m^{2}_{3\over
2},m^{\prime 2}_{3\over
2})+{{{k^{\prime}}_{\sigma}{k^{\prime}}_{\rho}}\over{m^{\prime}_{3\over
2}m_{3\over 2}}}G_{4}({k^{\prime}}^{2},m^{2}_{3\over 2},m^{\prime 2}_{3\over
2})\Bigr{]}\Biggr{)}u^{\rho}({\bf P}),$ $None$

where for $m^{\prime}_{3/2}=m_{3/2}$ expression (B.2) coincides with (B.1).

The extension of (B.2) in the complex energy and mass region of the $\Delta$
resonance implies the replacements $m_{3/2}\Longrightarrow{\sf m}_{\Delta}(s)$
and $m^{\prime}_{3/2}\Longrightarrow{\sf m}_{\Delta}(s^{\prime})$, where ${\sf
m}_{\Delta}$ is given in (3.9b). Correspondingly, we obtain the complex
energies (3.9c,d) and the complex zero component of the four-vector of the
four momentum transfer ${\bf k^{\prime}}={\bf k^{\prime}}_{\Delta}$,
$k^{\prime}_{o}\Longrightarrow(k^{\prime}_{\Delta})_{o}=\sqrt{{\sf
m}^{2}(s)+{\bf P}^{2}_{\Delta}}-\sqrt{{\sf m}^{2}(s^{\prime})+{\bf
P^{\prime}}^{2}_{\Delta}}$.

The general double $\Delta$ exchange term ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma{\Delta})$ (3.18) contains the following full
$\Delta-\gamma^{\prime}\Delta^{\prime}$ vertex function $<{\bf
P^{\prime}}_{\Delta},s^{\prime}|J_{\mu}(0)|{\bf P}_{\Delta},s>$ with on mass
shell $\Delta$’s

77footnotetext:  Another double $\Delta$ exchange term contains the
$\Delta-\pi^{\prime}\Delta^{\prime}$ vertex function. But this term in
negligible small[8].

$<{\bf P^{\prime}}_{\Delta},{\sf m}_{\Delta}(s^{\prime})|{\cal
J}^{\mu}(0)|{\bf P}_{\Delta},{\sf
m}_{\Delta}(s)>=(P_{\Delta}+P^{\prime}_{\Delta})^{\mu}\Biggl{(}{\overline{u}}^{\sigma}({\bf
P^{\prime}}_{\Delta})\Bigl{[}g_{\rho\sigma}G_{1}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$
$+{{{k^{\prime}_{\Delta}}_{\sigma}{k^{\prime}_{\Delta}}_{\rho}}\over{M^{2}_{\Delta}}}G_{3}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})\Bigr{]}u^{\rho}({\bf
P}_{\Delta})\Biggr{)}$ $+{\overline{u}}^{\sigma}({\bf
P^{\prime}}_{\Delta})\Biggl{(}-i\sigma^{\mu\nu}{k^{\prime}_{\Delta}}_{\nu}\Bigl{[}g_{\rho\sigma}G_{2}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})+{{{k^{\prime}_{\Delta}}_{\sigma}{k^{\prime}_{\Delta}}_{\rho}}\over{M^{2}_{\Delta}}}G_{4}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})\Bigr{]}\Biggr{)}u^{\rho}({\bf
P}_{\Delta}),$ $None$

where we introduced the auxiliary four-vector
${k^{\prime}_{\Delta}}_{\mu}=(P_{\Delta}-P^{\prime}_{\Delta})_{\mu}$ for the
momentum transfer and $g_{\mu\nu}$ is the metric tensor. An additional
dependence of the form factors $G_{i}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$
on the variables $s$ and $s^{\prime}$ is generated by ${\sf m}_{\Delta}(s)$
and ${\sf m}_{\Delta}(s^{\prime})$ (3.4d).

The form factors $G_{i}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$ are simply
connected with the charge monopole $G_{C0}$, magnetic dipole $G_{M1}$,
electric quadrupole $G_{E2}$ and magnetic octupole $G_{M3}$ form factors of
the $\Delta$ resonance. 888 Other choices of $G_{i}$ form factors are
considered in ref. [35]

The terms $\sim{k^{\prime}}_{\Delta}^{2}/4M_{\Delta}^{2}$ in the
$\Delta-\gamma^{\prime}\Delta^{\prime}$ vertex for the low energy photons can
be neglected and only terms $\sim 1/M_{\Delta}$ can be taken into account.
Then (B.3) reduces to (3.7).

An important property of the electromagnetic $\Delta$ vertices (B.3) and (3.7)
is that at the threshold ($k^{\prime}=0$ and $k^{\prime}_{\Delta}=0$) they
coincide with $G_{i}(0,m^{2}_{3\over 2},m^{\prime 2}_{3\over 2})$ in (B.2).
But the form factors $G_{i}({k^{\prime}}^{2},m^{2}_{3\over 2},m^{\prime
2}_{3\over 2})$ in (B.2) are real according to the $C,P,T$ invariance.
Consequently, the form factors $G_{i}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$
at the threshold are also real. Therefore,
$G_{C0}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$ and
$G_{M1}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$ satisfy the following
normalization conditions

$G_{C0}({k^{\prime}}_{\Delta}^{2}=0,s,s)=e_{\Delta};\ \ \
G_{M1}({k^{\prime}}_{\Delta}^{2}=0,s,s)=\mu_{\Delta},$ $None$

where $e_{\Delta}$ and $\mu_{\Delta}$ denote the charge and magnetic dipole
moment of the $\Delta$’s.

The exact form of vertex functions (B.3a,b) ensure the validity of the special
one-body current conservation condition for the $\Delta$ vertex function

${k^{\prime}}_{\Delta}^{\mu}<{\bf P^{\prime}}_{\Delta},{\sf
m}_{\Delta}(s^{\prime})|J_{\mu}(0)|{\bf P}_{\Delta},{\sf m}_{\Delta}(s)>=$
$\left\\{\begin{array}[]{llllllllllllll}0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ for\ constant\ {\sf m}_{\Delta}\ (3.4c)\\\ {{{\sf m}_{\Delta}^{2}(s)-{\sf
m}_{\Delta}^{2}(s^{\prime})}\over{2M_{\Delta}}}G_{C0}({k^{\prime}}_{\Delta}^{2},s,s^{\prime}){\overline{u}}^{\sigma}({\bf
P^{\prime}}_{\Delta})g_{\rho\sigma}u^{\rho}({\bf P}_{\Delta})\ \ \ for\ {\sf
m}_{\Delta}(s)\ (3.4d)\end{array}\right.$ $None$

Equation (B.4b) expresses the analytical continuation of usual current
conservation for the real spin $3/2$ particle vertex function in the complex
energy-mass region of the $\Delta$’s. Certainly, this “one-body intermediate
$\Delta$ current” conservation is not necessary for real current conservation
of the full $\pi N$ radiation amplitude.

It must be emphasized that the present formulation of the $\Delta$ degrees of
freedom does not use a Heisenberg local field operator of the $\Delta$
resonance or a Lagrangian with the local $\Delta$ field operators. This
simplifies the formulation because it is not possible to construct a Fock
space for a “free” resonance state with a complex mass and a complex energy. A
renormalization procedure for real spin $3/2$ states can generate intermediate
$\Delta$ complex states. This renormalization is equivalent to the extension
of the vertex functions (B.1) or (B.2) into the complex $\Delta$ vertex (B.3).
In the present approach we use the vertices only with the on mass shell
$\Delta$’s. Therefore, ambiguities generated by unphysical gauge
transformations of the $\Delta$-particle field operator
$\Psi_{\Delta}^{a}\longrightarrow\Psi_{\Delta}^{a}+C\gamma^{a}\gamma_{b}\Psi_{\Delta}^{b}$
[40] with an arbitrary parameter $C$ do not appear in the present formulation.
Sensitivity of the $\gamma p\to\gamma^{\prime}\pi^{\prime}p^{\prime}$
observable to the choice of the form of the intermediate $\Delta$ propagator
is demonstrated in [32].

In the off mass shell region, where ${P^{\prime}}^{2}\neq m^{2}_{\Delta}$ and
${P}^{2}\neq m^{2}_{\Delta}$, the $\Delta-\gamma^{\prime}\Delta^{\prime}$
vertex is a function of two independent four moments of each $\Delta$.
Therefore, for the off mass shell $\Delta$’s (B.3) and (3.7) take a much more
complicated forms with the increasing number of the form factors $G_{i}$,
because each of the conditions $P^{2}_{\Delta}\neq m_{\Delta}^{2}$ and
$(i\gamma_{\sigma}P_{\Delta}^{\sigma}\neq m_{\Delta}^{2})$ reduplicates the
number of the form factors. Therefore, instead of two form factors in (3.7) we
get $8$ form factors for the off mass shell
$\Delta-\gamma^{\prime}\Delta^{\prime}$ vertices. The role of these six
additional form factors is as important as the contribution of the off shell
effects like the mass and charge renormalization. In addition, these form
factors of the $\Delta-\gamma^{\prime}\Delta^{\prime}$ vertex with off mass
shell $\Delta$’s depend on three complex variables
${k^{\prime}}^{2}_{\Delta}$, $P^{2}_{\Delta}$ and ${P^{\prime}}^{2}_{\Delta}$.
Therefore, the use of the off mass shell $\Delta$ propagators together with
the on mass shell $\Delta-\gamma^{\prime}\Delta^{\prime}$, as is done in refs.
[33, 34, 35], is inconsistent.

Appendix C: On mass shell $\Delta$ degrees of freedom and construction of the
double $\Delta$ exchange term in Fig. 2B.

The on mass shell intermediate $\Delta$ states are usually introduced via the
$\Delta$ resonance pole position in the $\pi N$ amplitude. We shall shortly
consider the corresponding formulation within the time-ordered field-
theoretical approach [31, 32, 30]. In this formulation the off mass shell $\pi
N$ amplitudes (2.9a,b,c,d) are simply connected with the $\pi N$ $t$-matrix
${\sc T}({\bf p^{\prime}_{N}p^{\prime}_{\pi},p_{N}p_{\pi}};E)$, which
satisfies the relativistic Lippmann-Schwinger-type equation in the c.m. frame

${\sc T}({\bf p^{\prime},p};E_{\bf p})={\sc U}({\bf p^{\prime},p};E_{\bf
p})-\int{\sc U}({\bf p^{\prime},q};E_{\bf p}){{d^{3}{\bf q}}\over{E_{\bf
p}-E_{\bf q}-i\epsilon}}{\sc T}({\bf q,p};E_{\bf p}),$ $None$

where $E_{\bf p}\equiv P_{o}=\sqrt{{\bf p}^{2}+m_{\pi}^{2}}+\sqrt{{\bf
p}^{2}+m_{N}^{2}}$ and ${\bf p}$ are the $\pi N$ energy and the relative
three-momentum in c.m. frame. Equation (C.1) can be symbolically represented
as

${\sc T}(E_{\bf p})={\sc U}(E_{\bf p})+{\sc U}(E_{\bf p}){\sc G_{o}}(E_{\bf
p}){\sc T}(E_{\bf p})={\sc U}(E_{\bf p})+{\sc U}(E_{\bf p}){\cal G}_{\pi
N}(E_{\bf p}){\sc U}(E_{\bf p}),$ $None$

where ${\sc G_{o}}(E_{\bf p})$ and ${\cal G}_{\pi N}(E_{\bf p})$ are the free
and total Green functions of the $\pi N$ system and ${\sc U}(E)\equiv{\sc
U}({\bf p^{\prime},p};E)={\sc A}({\bf p^{\prime},p})+E{\sc B}({\bf
p^{\prime},p})$ is the linear energy depending on the field-theoretical
potential with a Hermitian ${\sc A}({\bf p^{\prime},p})$ and ${\sc B}({\bf
p^{\prime},p})$ matrices. The full $\pi N$ Green function satisfies the
completeness condition

${\cal G}_{\pi N}(E)=\sum_{\pi N}{{|{\Psi}_{\pi N}({\bf q})>d^{3}{\bf
q}<\Psi_{\pi N}({\bf q})|(1-{\sc B})}\over{E-E_{\bf q}-i\epsilon}},$ $None$

where ${\Psi}_{\pi N}({\bf q})$ denotes the $\pi N$ wave function which can be
determined via the solution of (C.1).

The $\Delta$ resonance pole in the complex energy region generates the
following representation of the full $\pi N$ wave function

${\cal G}_{\pi N}(E)=\sum_{\Delta}{{|\Psi_{\Delta}><\Psi_{\Delta}|(1-{\sc
B})}\over{E-E_{\Delta}}}+\ non-pole\ part,$ $None$

where $m_{\Delta}=1232MeV-{i\over 2}120MeV$ and $E_{\Delta}\equiv
P^{o}_{\Delta}=\sqrt{m_{\Delta}^{2}+{\bf P}^{2}}$ according to (3.9a) and
(3.9c). $m_{\Delta}$ indicates the $\Delta$-resonance pole position of the
full $\pi N$ Green function or the total $\pi N$ amplitude.

Using (C.4) and (C.2) one can extract the $\Delta$ exchange part of the $\pi
N$ $t$-matrix

$\biggl{[}{\sc T}(E)\biggr{]}_{one\ \Delta\ exchange\
part}=\sum_{\Delta}{{{\sc U}(E)|\Psi_{\Delta}><\Psi_{\Delta}|(1-{\sc B}){\sc
U}(E)}\over{E-E_{\Delta}}}$ $None$

This expression can be reproduced in the separable model of the $\pi N$
$t$-matrix [31, 32]

$T({\bf p^{\prime},p};E)=\lambda g({\bf p^{\prime}})g({\bf
p})(p_{\Delta}-p^{\prime}_{N})_{\alpha}{{u^{\alpha}({\bf
p}_{\Delta}){\overline{u}}^{\beta}({\bf
p}_{\Delta})}\over{{d}_{\Delta}(E)}}(p_{\Delta}-p_{N})_{\beta}$ $None$

where

${d}_{\Delta}(E)=1-\lambda\int{{d^{3}{\bf
q}}\over{(2\pi)^{3}}}{m_{N}\over{{2E_{\bf q}}_{\pi}{E_{\bf q}}_{N}}}{{{\bf
q}^{2}g^{2}({\bf q})}\over{E+io-{E_{\bf q}}_{\pi}-{E_{\bf q}}_{N}}}$ $None$

in the usual separable potential model and

${d}_{\Delta}(E)=E-E_{\Delta}(bare)-\Sigma_{\pi N}(E)$ $None$

in the more complicated microscopic models with the bare energy
$E_{\Delta}(bare)$ and the mass operator of the $\Delta$ resonance
$\Sigma_{\pi N}(E)$.

Using the normalization condition [19], we get

${d}_{\Delta}\Bigl{(}E=\sqrt{m_{\Delta}^{2}+{\bf P}_{\Delta}^{2}}\
\Bigr{)}=0.$ $None$

Equations (C.4c) and (C.4d) can be represented in the form of (C.4b) with the
corresponding redefinition of the form factors of the $\Delta$ resonances

$|{g}_{\Delta}(E)>\equiv\biggl{(}{{{d}_{\Delta}(E)}\over{E-E_{\Delta}}}\biggr{)}^{{1\over
2}}{\sc U}(E)|\Psi_{\Delta}>$ $None$

In this way the expression $E-E_{\Delta}(s)$ can be replaced by the propagator
${d}_{\Delta}(E)$ which is constructed in the separable model.

$E-E_{\Delta}(s)\Longrightarrow{d}_{\Delta}(E).$ $None$

This allows to separate the $\Delta$ pole and non-pole parts of the $\pi N$
amplitude in accordance with the (A.9a,b,c,d).

The double $\Delta$ exchange term with the
$\Delta-\gamma^{\prime}\Delta^{\prime}$ vertex (3.21c) (Fig. 2B) can be
extracted from the $\pi N$ bremsstrahlung amplitude $<out;{\bf
p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$ (2.6) in the same way as in our previous papers [31, 32, 30]. Thus
the $s$-channel part of the full $\pi N$ bremsstrahlung amplitude with the
double $\pi N$ intermediate states is

$<out;{\bf p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf
p}_{\pi}{\bf
p}_{N};in>\Longrightarrow\sum_{{\pi^{\prime\prime}N^{\prime\prime}},{\pi^{\prime\prime\prime}N^{\prime\prime\prime}}}\int
d^{4}x^{\prime}e^{ip^{\prime}_{\pi}x^{\prime}}d^{4}xe^{-ip_{\pi}x}<out;{\bf
p^{\prime}}_{N}|j_{\pi^{\prime}}(x^{\prime})\Bigr{]}|{\pi}^{\prime\prime\prime}N^{\prime\prime\prime};out>$
$\theta(x^{\prime}_{o})<out;\pi^{\prime\prime\prime}N^{\prime\prime\prime}|{\cal
J}^{\mu}(0)|{\pi^{\prime\prime}N^{\prime\prime}};in>\theta(-x_{o})<in;{\pi^{\prime\prime}N^{\prime\prime}}|j_{\pi}(x)|{\bf
p}_{N};in>$ $None$

which after integration is transformed to

$<out;{\bf p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf
p}_{\pi}{\bf
p}_{N};in>\Longrightarrow(2\pi)^{6}\sum_{{\pi^{\prime\prime}N^{\prime\prime}},{\pi^{\prime\prime\prime}N^{\prime\prime\prime}}}{{<out;{\bf
p^{\prime}}_{N}|{\sc U}(E_{\bf
p^{\prime}})|{\Psi}_{\pi^{\prime\prime}N^{\prime\prime}}({\bf
p^{\prime\prime}})>}\over{{p^{\prime}}^{o}_{N}+{p^{\prime}}^{o}_{\pi}-{p^{\prime\prime\prime}}^{o}_{N}+{p^{\prime\prime\prime}}^{o}_{\pi}-i\epsilon}}$
$<out;\pi^{\prime\prime\prime}N^{\prime\prime\prime}|{\cal
J}^{\mu}(0)|{\pi^{\prime\prime}N^{\prime\prime}};in>{{<{\Psi}_{\pi^{\prime\prime}N^{\prime\prime}}({\bf
p^{\prime\prime}})|{\sc U}(E_{\bf p})|{\bf
p}_{N};in>}\over{{p}^{o}_{N}+{p}^{o}_{\pi}-{p^{\prime\prime}}^{o}_{N}+{p^{\prime\prime}}^{o}_{\pi}-i\epsilon}}$
$None$

where we used a connection between the $\pi N$ amplitude and the $\pi N$ wave
function [31, 30]

$<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\pi}(0)|{\bf
p}_{N};in>=<out;{\bf p^{\prime}}_{N}|{\sc U}(E_{\bf p^{\prime}})|{\Psi}_{\pi
N^{\prime}}({\bf p})>,$ $None$

Next we separate the $\pi N$ irreducible part
$\biggl{[}<out;\pi^{\prime\prime\prime}N^{\prime\prime\prime}|{\cal
J}^{\mu}(0)|{\pi^{\prime\prime}N^{\prime\prime}};in>\biggr{]}_{\pi\ N\
irreducible}$ of the full $\pi N$ bremsstrahlung amplitude
$<out;\pi^{\prime}N^{\prime}|{\cal J}^{\mu}(0)|{\pi N};in>$ as

$<out;\pi^{\prime}N^{\prime}|{\cal J}^{\mu}(0)|{\pi
N};in>=\sum_{{\pi^{\prime\prime}N^{\prime\prime}},{\pi^{\prime\prime\prime}N^{\prime\prime\prime}}}<{\Psi}_{\pi^{\prime}N^{\prime}}|(1-{\sc
B})|\pi^{\prime\prime\prime}N^{\prime\prime\prime};out>$
$\biggl{[}<out;\pi^{\prime\prime\prime}N^{\prime\prime\prime}|{\cal
J}^{\mu}(0)|{\pi^{\prime\prime}N^{\prime\prime}};in>\biggr{]}_{\pi\ N\
irreducible}<in;{\pi^{\prime\prime}N^{\prime\prime}}|(1-{\sc B})|{\Psi}_{\pi
N}>,$ $None$

Substituting (C.7) into (C.5b) and using (C.3) and (C.4a) we get (3.6), where

$<{\bf P^{\prime}}_{\Delta},{\sf m}_{\Delta}(s^{\prime})|{\cal
J}^{\mu}(0)|{\bf P}_{\Delta},{\sf
m}_{\Delta}(s)>=\sum_{{\pi^{\prime\prime}N^{\prime\prime}},{\pi^{\prime\prime\prime}N^{\prime\prime\prime}}}<{\Psi}_{\Delta^{\prime}}|(1-{\sc
B})|\pi^{\prime\prime\prime}N^{\prime\prime\prime};out>$
$\biggl{[}<out;\pi^{\prime\prime\prime}N^{\prime\prime\prime}|{\cal
J}^{\mu}(0)|{\pi^{\prime\prime}N^{\prime\prime}};in>\biggr{]}_{\pi\ N\
irreducible}<in;{\pi^{\prime\prime}N^{\prime\prime}}|(1-{\sc
B})|{\Psi}_{\Delta}>,$ $None$

and

$<{\bf p^{\prime}}_{N},{\bf p^{\prime}}_{\pi}|{\sl
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}|{\bf
P^{\prime}}_{\Delta^{\prime}}>=<{\bf p^{\prime}}_{N},{\bf
p^{\prime}}_{\pi}|{\sc U}(E_{\bf p^{\prime}})|{\Psi}_{\Delta^{\prime}}({\bf
P^{\prime}})>,$ $None$

$<{\bf P}_{\Delta}|{\sl g}_{\Delta-\pi N}|{\bf p}_{N},{\bf
p}_{\pi}>=<{\Psi}_{\Delta}({\bf P})|{\sc U}(E_{\bf p})|{\bf p}_{N},{\bf
p}_{\pi}>$ $None$

## References

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