# High-energy $\pi\pi$ scattering without and with photon radiation

Piotr Lebiedowicz<EMAIL_ADDRESS>Institute of Nuclear Physics
Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland  Otto
Nachtmann<EMAIL_ADDRESS>Institut für Theoretische
Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany
Antoni Szczurek 111Also at College of Natural Sciences, Institute of Physics,
University of Rzeszów, ul. Pigonia 1, PL-35310 Rzeszów, Poland.
<EMAIL_ADDRESS>Institute of Nuclear Physics Polish Academy of
Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland

###### Abstract

We discuss the processes $\pi\pi\to\pi\pi$ and $\pi\pi\to\pi\pi\gamma$ from a
general quantum field theory (QFT) point of view. In the soft-photon limit
where the photon energy $\omega\to 0$ we study the theorem due to F.E. Low. We
confirm his result for the $1/\omega$ term of the $\pi\pi\to\pi\pi\gamma$
amplitude but disagree for the $\omega^{0}$ term. We analyse the origin of
this discrepancy. Then we calculate the amplitudes for the above reactions in
the tensor-pomeron model. We identify places where “anomalous” soft photons
could come from. Three soft-photon approximations (SPAs) are introduced. The
corresponding SPA results are compared to those obtained from the full tensor-
pomeron model for c.m. energies $\sqrt{s}=10$ GeV and 100 GeV. The kinematic
regions where the SPAs are a good representation of the full amplitude are
determined. Finally we make some remarks on the type of fundamental
information one could obtain from high-energy exclusive hadronic reactions
without and with soft photon radiation.

##  I Introduction

In this paper we shall be concerned with photon emission in some strong-
interaction processes. In particular, we shall consider soft photon emission,
that is, the emission of photons with energy $\omega$ approaching zero. For
this kinematic region there exists Low’s theorem Low:1958sn  which is based
strictly on Quantum Field Theory (QFT). The theorem states that for $\omega\to
0$ the photons come exclusively from the external hadrons in the process
considered. But this poses immediately the question: how close do we have to
come to $\omega=0$ in order to see the behaviour of the photon-emission
amplitude predicted by Low?

There have been a number of experimental studies trying to verify Low’s
theorem Goshaw:1979kq ; Chliapnikov:1984ed ; Botterweck:1991wf ;
Banerjee:1992ut ; Antos:1993wv ; Tincknell:1996ks ; Belogianni:1997rh ;
Belogianni:2002ib ; Belogianni:2002ic ; Abdallah:2005wn ; Abdallah:2007aa .
For a review of the experimental situation see Wong:2014pY . The result is,
that many experiments see rather large deviations from theoretical
calculations in the soft-photon approximation (SPA) based on Low’s theorem.
Clearly, this situation is unsatisfactory. This has motivated the feasibility
study of measuring soft-photon phenomena in a next-generation experiment in
the framework of the heavy-ion physics programme at the LHC for the 2030’s
Adamova:2019vkf . Clearly, for preparing such soft-photon experiments
accompanying theoretical studies are needed.

One class of hadronic reactions one can study at the LHC are exclusive
diffractive proton-proton collisions. Examples are $pp$ elastic scattering and
central exclusive production (CEP) reactions, for instance $pp\to
p\pi^{+}\pi^{-}p$. In these reactions we can, of course, also have photon
emission:

$\displaystyle p+p\to p+p+\gamma\,,$ $\displaystyle p+p\to
p+\pi^{+}+\pi^{-}+p+\gamma\,,$ (1)

and we can study the soft-photon limit. The advantage of these exclusive
diffractive reactions is that they are “clean” from the experimental side and
that we have reasonable theoretical models for them. We shall work within the
tensor-pomeron model as proposed in Ewerz:2013kda . There, the soft pomeron
and the charge conjugation $C=+1$ reggeons are described as effective rank-2
symmetric tensor exchanges, the odderon and the $C=-1$ reggeons as effective
vector exchanges. The tensor-pomeron model has been applied to quite a number
of CEP reactions Lebiedowicz:2013ika ; Lebiedowicz:2014bea ;
Lebiedowicz:2016ioh ; Lebiedowicz:2016zka ; Lebiedowicz:2018eui ;
Lebiedowicz:2018sdt ; Lebiedowicz:2019jru ; Lebiedowicz:2019boz ;
Lebiedowicz:2020yre ; Lebiedowicz:2021pzd  which can and should all be studied
by the present RHIC and LHC experiments Adam:2020sap ; McNulty:2017ejl ;
Schicker:2019qcn ; Sirunyan:2019nog ; Sirunyan:2020cmr ; Sikora:2020mae . The
next generation LHC experiment Adamova:2019vkf  should be able to study these
reactions in even greater detail, in particular, in the region of low
transverse momenta. Applications of the model of Ewerz:2013kda  have
furthermore been made to photoproduction of $\pi^{+}\pi^{-}$ pairs
Bolz:2014mya , a reaction which is also of interest for the LHC, and to deep-
inelastic lepton-nucleon scattering at low $x$ Britzger:2019lvc . In
Ewerz:2016onn  it was shown that the experimental results Adamczyk:2012kn  on
the spin dependence of high-energy proton-proton elastic scattering exclude a
scalar character of the pomeron couplings but are perfectly compatible with
the tensor-pomeron model. A vector coupling for the pomeron could definitely
be ruled out in Britzger:2019lvc .

With the present paper we want to start the theoretical study of soft photon
emission in exclusive diffractive high-energy reactions in the TeV energy
region in the framework of the tensor-pomeron model. Our first example will
be, for simplicity, pion-pion elastic scattering. This is, of course, not easy
to study for experiments. But, as we shall see, we can in this example compare
our “exact” model results for photon emission to approximations based on Low’s
theorem which gives the photon-emission amplitude to order $\omega^{-1}$ and
$\omega^{0}$ in the photon energy $\omega$ for $\omega\to 0$. We shall show,
as an important result, that the term of order $\omega^{0}$, presented in
Low:1958sn , needs modifications.

Before coming to our present investigations we make remarks on some hadronic
processes where photon emission has been studied, frequently using the soft-
photon approximation.

Direct photons (i.e. photons which originate not from hadronic decays, but
from inelastic scattering processes between partons) are an important
electromagnetic probe of the quark-gluon plasma as created in heavy-ion
collisions. Since pions are the dominant meson species produced in the heavy-
ion collisions, the photon production via bremsstrahlung in pion-pion elastic
collisions was found to be a very important source to interpret the data on
the direct photon spectra and elliptic flow simultaneously Linnyk:2013hta ;
Linnyk:2013wma . In Linnyk:2013hta ; Linnyk:2013wma  the SPA was used and,
therefore, the resulting yield of the bremsstrahlung photons depends on some
model assumptions.

The description of the photon bremsstrahlung in meson-meson scattering beyond
the SPA, within the one-boson exchange (OBE) model, was discussed for the
first time in Eggers:1995jq  and applied to the dilepton bremsstrahlung in
pion-pion collisions. Later on, in Liu:2007zzw , it was applied to the low-
energy photon bremsstrahlung in pion-pion and kaon-kaon collisions. Within the
OBE model the interaction of pions is described by three resonance exchanges
$\sigma$, $\rho$ and $f_{2}(1270)$ in the $t$, $u$ and $s$ channels (the $u$
channel diagrams are needed only in the case of identical pions).

In Linnyk:2015tha ; Linnyk:2015rco  the authors applied the covariant OBE
effective (chiral) model for the pion-pion scattering. The “exact” OBE model
result of the invariant rate of photon bremsstrahlung was compared with that
of the SPA. It was noted there that the accuracy of the SPA approximation can
be significantly improved and the region of its applicability can be extended
by evaluating the on-shell elastic cross section not at the c.m. energy
$\sqrt{s}$ of the $\pi\pi\to\pi\pi\gamma$ process but at a certain smaller
energy. One can see in Fig. 6 of Linnyk:2015tha  (or Fig. 21 of Linnyk:2015rco
) that the “improved SPA model” gives a good approximation to the “exact” OBE
result up to photon energies $\approx 2$ GeV. There the dominant contribution
to the rates comes from low collision energies $\sqrt{s}$. The deviation
between the OBE result and that calculated within the improved SPA is most
pronounced at high $\sqrt{s}$ and high photon energies.

Whereas the examples of photon radiation discussed above concerned low energy
reactions, there have, of course, also been studies of photon radiation for
exclusive reactions at the LHC. Exclusive diffractive photon bremsstrahlung in
proton-proton collisions was discussed in Khoze:2010jv ; Lebiedowicz:2013xlb .
Feasibility studies of the measurement of the exclusive diffractive
bremsstrahlung cross section in proton-proton collisions at the center-of-mass
energy $\sqrt{s}=13$ TeV at the LHC were performed in Chwastowski:2016jkl ;
Chwastowski:2016zzl .

Very interesting general investigations of photon emission in hadronic
reactions have been presented, for instance, in Gribov:1966hs ; Lipatov:1988ii
; DelDuca:1990gz ; Gervais:2017yxv ; Bern:2014vva ; Lysov:2014csa . We shall
comment on results given in these papers below, as far as they have a bearing
on our own investigations.

Now we list the high-energy reactions which we want to study in our present
paper. In Sec. II we discuss the reactions $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$
and $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$ from a general QFT point of view.
Section III deals with the limit of photon-energy $\omega\to 0$ and we discuss
the terms in the amplitude of orders $\omega^{-1}$ and $\omega^{0}$. In Sec.
IV we introduce our model for $\pi^{\mp}\pi^{0}$ and charged-pion scattering
and for the corresponding reactions with photon emission. Section V is devoted
to a comparison of our “exact” model results to various approximations based
Low’s theorem. In Sec. VI we give our conclusions and an outlook on further
work. In Appendix A we compare in detail our results for the terms of order
$\omega^{-1}$ and $\omega^{0}$ in the photon emission amplitude to results
presented in the literature. In Appendix B we discuss the cross section
$d\sigma/d\omega$ for $\omega\to 0$.

##  II General properties of the reactions $\pi\pi\to\pi\pi$ and
$\pi\pi\to\pi\pi\gamma$

Here we study general QFT relations for pion-pion elastic scattering without
and with photon radiation. We shall work to leading order in the
electromagnetic coupling. For simplicity we shall consider $\pi^{-}\pi^{0}$
scattering, that is, the reactions

$\displaystyle\pi^{-}(p_{a})+\pi^{0}(p_{b})\to\pi^{-}(p_{1})+\pi^{0}(p_{2})\,,$
(2)
$\displaystyle\pi^{-}(p_{a})+\pi^{0}(p_{b})\to\pi^{-}(p_{1}^{\prime})+\pi^{0}(p_{2}^{\prime})+\gamma(k,\epsilon)\,.$
(3)

Here $p_{a}$, $p_{b}$, $p_{1}$, $p_{2}$, $p_{1}^{\prime}$, $p_{2}^{\prime}$
and $k$ are the momenta of the particles and $\epsilon$ is the polarisation
vector of the photon, respectively. The energy-momentum conservation in (2)
and (3) requires

$\displaystyle p_{a}+p_{b}=p_{1}+p_{2}\,,$ (4) $\displaystyle
p_{a}+p_{b}=p_{1}^{\prime}+p_{2}^{\prime}+k\,.$ (5)

We denote the amplitude for the reaction (2) by

$\displaystyle{\cal
T}(p_{a},p_{b},p_{1},p_{2})=\braket{\pi^{-}(p_{1}),\pi^{0}(p_{2})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{0}(p_{b})}\,.$ (6)

Since pions have $G$ parity $-1$ all diagrams for (6) are one-particle
irreducible. In QFT we can extend the amplitude (6) for off shell pions (Fig.
1).

Figure 1: Diagram for the off shell $\pi^{-}\pi^{0}$ scattering amplitude.

This off shell scattering amplitude will still satisfy the energy-momentum
conservation (4) and can only depend on the following 6 variables

$\displaystyle s_{L}=p_{a}\cdot p_{b}+p_{1}\cdot p_{2}\,,$ $\displaystyle
t=(p_{a}-p_{1})^{2}=(p_{b}-p_{2})^{2}\,,$ $\displaystyle
m_{a}^{2}=p_{a}^{2}\,,\;\,m_{b}^{2}=p_{b}^{2}\,,\;\,m_{1}^{2}=p_{1}^{2}\,,\;\,m_{2}^{2}=p_{2}^{2}\,.$
(7)

Here we use as squared energy variable $s_{L}$, following Low:1958sn , instead
of the more usual Mandelstam variable $s$. We have

$\displaystyle
s=s_{L}+\frac{1}{2}\left(m_{a}^{2}+m_{b}^{2}+m_{1}^{2}+m_{2}^{2}\right)\,.$
(8)

The on- or off-shell amplitude corresponding to (6) as a function of the
variables (7) will be denoted by

$\displaystyle{\cal
M}^{(0)}(s_{L},t,m_{a}^{2},m_{b}^{2},m_{1}^{2},m_{2}^{2})=\left.{\cal
T}(p_{a},p_{b},p_{1},p_{2})\right|_{\rm on\;shell\;or\;off\;shell}.$ (9)

The on-shell amplitude (6) is then ${\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})$.

Next we study the reaction (3) where we have two one-particle reducible
diagrams (Figs. 2 (a), (b)) and one irreducible diagram (Fig. 2 (c)).

(a) (b)
(c)

Figure 2: One-particle reducible (a, b) and irreducible (c) diagrams for
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$.

For the diagrams (a) and (b) we need the off-shell $\pi\pi$ amplitude (9), the
pion propagator $\Delta(p^{2})$ and the pion-photon vertex function
$\Gamma_{\lambda}(p^{\prime},p)$:

$\displaystyle i\Delta(p^{2})\,,$ (10) $\displaystyle
ie\Gamma_{\lambda}(p^{\prime},p)\,.$ (11)

We denote by $e=\sqrt{4\pi\alpha}>0$ the $\pi^{+}$ charge.

The expressions for the amplitudes of Figs. 2 (a, b) can be written as
follows:

$\displaystyle{\cal M}_{\lambda}^{(a)}=-e\,{\cal
M}^{(0,\,a)}\,\Delta[(p_{a}-k)^{2}]\,\Gamma_{\lambda}(p_{a}-k,p_{a})\,,$
$\displaystyle{\cal M}^{(0,\,a)}={\cal
M}^{(0)}[(p_{a}-k,p_{b})+p_{1}^{\prime}\cdot
p_{2}^{\prime},(p_{b}-p_{2}^{\prime})^{2},(p_{a}-k)^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2}],$
(12) $\displaystyle{\cal
M}_{\lambda}^{(b)}=-e\,\Gamma_{\lambda}(p_{1}^{\prime},p_{1}^{\prime}+k)\,\Delta[(p_{1}^{\prime}+k)^{2}]\,{\cal
M}^{(0,\,b)}\,,$ $\displaystyle{\cal M}^{(0,\,b)}={\cal M}^{(0)}[p_{a}\cdot
p_{b}+(p_{1}^{\prime}+k,p_{2}^{\prime}),(p_{b}-p_{2}^{\prime})^{2},m_{\pi}^{2},m_{\pi}^{2},(p_{1}^{\prime}+k)^{2},m_{\pi}^{2}].$
(13)

The photon-emission amplitude is

$\displaystyle\braket{\gamma(k,\epsilon),\pi^{-}(p_{1}^{\prime}),\pi^{0}(p_{2}^{\prime})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{0}(p_{b})}=(\epsilon^{\lambda})^{*}\,{\cal
M}_{\lambda}$ (14)

where

$\displaystyle{\cal M}_{\lambda}={\cal M}_{\lambda}^{(a)}+{\cal
M}_{\lambda}^{(b)}+{\cal M}_{\lambda}^{(c)}\,.$ (15)

${\cal M}_{\lambda}$ also determines the emission of virtual photons of mass
$k^{2}>0$ which then decay to a lepton pair. For $k^{2}<0$ ${\cal
M}_{\lambda}$ enters the amplitude for the 3-body reaction
$e^{\pm}\pi^{-}\pi^{0}\to e^{\pm}\pi^{-}\pi^{0}$. The amplitude ${\cal
M}_{\lambda}$ must satisfy the gauge-invariance relation, valid for all
$k^{2}$,

$\displaystyle k^{\lambda}{\cal M}_{\lambda}=k^{\lambda}\left({\cal
M}_{\lambda}^{(a)}+{\cal M}_{\lambda}^{(b)}+{\cal
M}_{\lambda}^{(c)}\right)=0\,,$ (16)

that is, we have

$\displaystyle k^{\lambda}{\cal M}_{\lambda}^{(c)}=-k^{\lambda}{\cal
M}_{\lambda}^{(a)}-k^{\lambda}{\cal M}_{\lambda}^{(b)}\,.$ (17)

We shall now use (12), (13), and (17), to get a simple relation between
$k^{\lambda}{\cal M}_{\lambda}^{(c)}$ and ${\cal M}^{(0,\,a)}$, ${\cal
M}^{(0,\,b)}$. For this we recall the normalisation conditions for the pion
propagator and the vertex function. We have

$\displaystyle\left.\Delta^{-1}(p^{2})\right|_{p^{2}=m_{\pi}^{2}}=0\,,\quad\frac{\partial}{\partial
p^{2}}\left.\Delta^{-1}(p^{2})\right|_{p^{2}=m_{\pi}^{2}}=1\,,$
$\displaystyle\left.\Gamma_{\lambda}(p^{\prime},p)\right|_{p^{\prime}=p,p^{2}=m_{\pi}^{2}}=2p_{\lambda}\,.$
(18)

Furthermore we have the Ward-Takahashi identity Ward:1950xp ; Takahashi:1957xn
,

$\displaystyle(p^{\prime}-p)^{\lambda}\Gamma_{\lambda}(p^{\prime},p)=\Delta^{-1}(p^{\prime
2})-\Delta^{-1}(p^{2})\,.$ (19)

From (18) and (19) we obtain for $p_{a}^{2}=m_{\pi}^{2}$

$\displaystyle\Delta[(p_{a}-k)^{2}]\,\Gamma_{\lambda}(p_{a}-k,p_{a})\,k^{\lambda}$
$\displaystyle\qquad\qquad=-\Delta[(p_{a}-k)^{2}]\,\Gamma_{\lambda}(p_{a}-k,p_{a})\,(p_{a}-k-p_{a})^{\lambda}$
$\displaystyle\qquad\qquad=-\Delta[(p_{a}-k)^{2}]\,\\{\Delta^{-1}[(p_{a}-k)^{2}]-\Delta^{-1}[p_{a}^{2}]\\}$
$\displaystyle\qquad\qquad=-1\,.$ (20)

Similarly we get for $p_{1}^{\prime 2}=m_{\pi}^{2}$

$\displaystyle
k^{\lambda}\,\Gamma_{\lambda}(p_{1}^{\prime},p_{1}^{\prime}+k)\,\Delta[(p_{1}^{\prime}+k)^{2}]$
$\displaystyle\qquad\qquad=-[p_{1}^{\prime}-(p_{1}^{\prime}+k)]^{\lambda}\,\Gamma_{\lambda}(p_{1}^{\prime},p_{1}^{\prime}+k)\,\Delta[(p_{1}^{\prime}+k)^{2}]$
$\displaystyle\qquad\qquad=-\\{\Delta^{-1}[p_{1}^{\prime
2}]-\Delta^{-1}[(p_{1}^{\prime}+k)^{2}]\\}\,\Delta[(p_{1}^{\prime}+k)^{2}]$
$\displaystyle\qquad\qquad=1\,.$ (21)

From (12), (13), (17), (20), and (21), we obtain

$\displaystyle k^{\lambda}{\cal M}_{\lambda}^{(a)}=e\,{\cal M}^{(0,\,a)}\,,$
$\displaystyle k^{\lambda}{\cal M}_{\lambda}^{(b)}=-e\,{\cal M}^{(0,\,b)}\,,$
(22) $\displaystyle k^{\lambda}{\cal M}_{\lambda}^{(c)}=-e\,{\cal
M}^{(0,\,a)}+e\,{\cal M}^{(0,\,b)}\,,$ (23)

where ${\cal M}^{(0,\,a)}$ and ${\cal M}^{(0,\,b)}$ are given explicitly in
(12) and (13), respectively.

##  III The expansion of the photon-emission amplitude to the order
$\omega^{-1}$ plus $\omega^{0}$

In this section we discuss the expansion of the amplitude ${\cal M}_{\lambda}$
(15) to the orders $\omega^{-1}$ and $\omega^{0}$. Here $\omega=k^{0}$ and, if
not stated otherwise, we work in the overall c.m. system of the reaction (3).
We shall in the following assume that all components of the photon momentum
are proportional to $\omega$, $k^{\mu}\propto\omega$, with $\omega\to 0$. This
is perfectly alright theoretically, but can this also be realised in nature?
For real photon emission, $k^{2}=0$, this clearly can be realised. It is also
possible for $k^{2}<0$ in the 3-body collision

$\displaystyle e^{\pm}+\pi^{-}+\pi^{0}\to e^{\pm}+\pi^{-}+\pi^{0}\,.$ (24)

For $k^{2}>0$ we can have $e^{+}e^{-}$ production

$\displaystyle\pi^{-}+\pi^{0}\to e^{+}+e^{-}+\pi^{-}+\pi^{0}\,.$ (25)

But here $\omega\geqslant 2m_{e}$ and $k^{2}\geqslant 4m_{e}^{2}$, with
$m_{e}$ the electron mass. Thus, in (25) we cannot reach $\omega=0$. But the
electron mass is very small on a hadronic scale, $m_{e}\simeq 0.5$ MeV, and,
therefore, the limit $\omega\to 0$ should also be of relevance for the
reaction (25).

We start our investigation of the small $\omega$ limit with the pion
propagator (10). We are working to lowest order in the electromagnetic
coupling. Thus, $\Delta^{-1}(p^{2})$ is for us a purely hadronic object. Its
nearest singularity to $p^{2}=0$ is at $p^{2}=(3m_{\pi})^{2}$ as we see from
the Landau conditions (cf. for instance Bjorken:1965 ). Therefore, we can
expand $\Delta^{-1}(p^{2})$ around $p^{2}=m_{\pi}^{2}$ as follows with $c$ a
constant:

$\displaystyle\Delta^{-1}(p^{2})=p^{2}-m_{\pi}^{2}+c(p^{2}-m_{\pi}^{2})^{2}+\ldots\,.$
(26)

This gives for $p_{a}^{2}=m_{\pi}^{2}$ and $p_{1}^{\prime 2}=m_{\pi}^{2}$ the
following

$\displaystyle\Delta^{-1}[(p_{a}-k)^{2}]=(-2p_{a}\cdot
k+k^{2})[1+c(-2p_{a}\cdot k+k^{2})+{\cal O}(\omega^{2})]\,,$
$\displaystyle\Delta[(p_{a}-k)^{2}]=\frac{1}{-2p_{a}\cdot
k+k^{2}}[1-c(-2p_{a}\cdot k+k^{2})+{\cal O}(\omega^{2})]\,,$ (27)
$\displaystyle\Delta^{-1}[(p_{1}^{\prime}+k)^{2}]=(2p_{1}^{\prime}\cdot
k+k^{2})[1+c(2p_{1}^{\prime}\cdot k+k^{2})+{\cal O}(\omega^{2})]\,,$
$\displaystyle\Delta[(p_{1}^{\prime}+k)^{2}]=\frac{1}{2p_{1}^{\prime}\cdot
k+k^{2}}[1-c(2p_{1}^{\prime}\cdot k+k^{2})+{\cal O}(\omega^{2})]\,.$ (28)

From (12), (13) and (15) we see that we must now expand $\Gamma_{\lambda}$,
${\cal M}^{(0,\,a)}$ and ${\cal M}^{(0,\,b)}$ up to order $\omega$ and ${\cal
M}_{\lambda}^{(c)}$ up to order $\omega^{0}$ for getting the total amplitude
${\cal M}_{\lambda}$ expanded up to order $\omega^{0}$.

We start with $\Gamma_{\lambda}(p^{\prime},p)$ which has the general expansion

$\displaystyle\Gamma_{\lambda}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle(p^{\prime}+p)_{\lambda}\,A[p^{\prime
2}-m_{\pi}^{2},p^{2}-m_{\pi}^{2},(p^{\prime}-p)^{2}]$ (29)
$\displaystyle+(p^{\prime}-p)_{\lambda}\,B[p^{\prime
2}-m_{\pi}^{2},p^{2}-m_{\pi}^{2},(p^{\prime}-p)^{2}]\,.$

The functions $A$ and $B$ are analytic in their variables in the region of
interest to us as we see again from the Landau conditions. The Ward-Takahashi
identity gives

$\displaystyle(p^{\prime}-p)^{\lambda}\,\Gamma_{\lambda}(p^{\prime},p)$
$\displaystyle=$ $\displaystyle(p^{\prime 2}-p^{2})\,A[p^{\prime
2}-m_{\pi}^{2},p^{2}-m_{\pi}^{2},(p^{\prime}-p)^{2}]$ (30)
$\displaystyle+(p^{\prime}-p)^{2}\,B[p^{\prime
2}-m_{\pi}^{2},p^{2}-m_{\pi}^{2},(p^{\prime}-p)^{2}]$ $\displaystyle=$
$\displaystyle\Delta^{-1}(p^{\prime 2})-\Delta^{-1}(p^{2})\,.$

Now we set in (30) $p=p_{a}$, $p^{\prime}=p_{a}-k$, $p_{a}^{2}=m_{\pi}^{2}$
and get

$\displaystyle(p^{\prime 2}-m_{\pi}^{2})\,A(p^{\prime
2}-m_{\pi}^{2},0,k^{2})+k^{2}\,B(p^{\prime 2}-m_{\pi}^{2},0,k^{2})$
$\displaystyle\qquad=\Delta^{-1}(p^{\prime 2})=p^{\prime
2}-m_{\pi}^{2}+c(p^{\prime 2}-m_{\pi}^{2})^{2}+\ldots\,.$ (31)

Therefore, we must have

$\displaystyle B(p^{\prime 2}-m_{\pi}^{2},0,k^{2})=(p^{\prime
2}-m_{\pi}^{2})\,\tilde{B}(p^{\prime 2}-m_{\pi}^{2},k^{2})$ (32)

and we get with $p^{\prime 2}-m_{\pi}^{2}=-2p_{a}\cdot k+k^{2}$

$\displaystyle A(-2p_{a}\cdot k+k^{2},0,k^{2})=1+c(-2p_{a}\cdot k+k^{2})+{\cal
O}(\omega^{2})\,,$ $\displaystyle B(-2p_{a}\cdot k+k^{2},0,k^{2})={\cal
O}(\omega)\,.$ (33)

Inserting (33) in (29) we find

$\displaystyle\Gamma_{\lambda}(p_{a}-k,p_{a})=(2p_{a}-k)_{\lambda}[1+c(-2p_{a}\cdot
k+k^{2})]+{\cal O}(\omega^{2})\,.$ (34)

In a completely analogous way we get for $p_{1}^{\prime 2}=m_{\pi}^{2}$

$\displaystyle\Gamma_{\lambda}(p_{1}^{\prime},p_{1}^{\prime}+k)=(2p_{1}^{\prime}+k)_{\lambda}[1+c(2p_{1}^{\prime}\cdot
k+k^{2})]+{\cal O}(\omega^{2})\,.$ (35)

From (27), (28), (34) and (35) we get

$\displaystyle\Delta[(p_{a}-k)^{2}]\,\Gamma_{\lambda}(p_{a}-k,p_{a})=\frac{(2p_{a}-k)_{\lambda}}{-2p_{a}\cdot
k+k^{2}}+{\cal O}(\omega)\,,$ (36)
$\displaystyle\Gamma_{\lambda}(p_{1}^{\prime},p_{1}^{\prime}+k)\,\Delta[(p_{1}^{\prime}+k)^{2}]=\frac{(2p_{1}^{\prime}+k)_{\lambda}}{2p_{1}^{\prime}\cdot
k+k^{2}}+{\cal O}(\omega)\,.$ (37)

Next we investigate the energy-momentum conservation conditions (4) and (5)
for the reactions (2) and (3), respectively. It is clear that for $k\neq 0$ we
cannot have $p_{1}=p_{1}^{\prime}$ and $p_{2}=p_{2}^{\prime}$ since

$\displaystyle p_{a}+p_{b}\neq p_{1}+p_{2}+k\,.$ (38)

This means that when going from (2) to (3) we must have a change of momenta
$p_{1}\to p_{1}^{\prime}\neq p_{1}$ and $p_{2}\to p_{2}^{\prime}\neq p_{2}$.
In fact, choosing for the reaction (3) some $k\neq 0$, even a small momentum
$k$, this does not fix $p_{1}^{\prime}$ and $p_{2}^{\prime}$. This is best
seen in the rest system of the four-vector $p_{a}+p_{b}-k$. There we have
$\mbox{\boldmath$p_{1}^{\prime}$}+\mbox{\boldmath$p_{2}^{\prime}$}=0$,
$|\mbox{\boldmath$p_{1}^{\prime}$}|$ is fixed and thus $p_{1}^{\prime}$ can
still vary on a sphere of radius $|\mbox{\boldmath$p_{1}^{\prime}$}|$. For the
following we work, however, in the overall c.m. system of reaction (3).

We write

$\displaystyle p_{1}^{\prime}=p_{1}-l_{1}\,,\quad
p_{2}^{\prime}=p_{2}-l_{2}\,,$ (39)

and get from (4) and (5) the conditions

$\displaystyle
l_{1}+l_{2}=k\,,\quad(p_{1}-l_{1})^{2}=m_{\pi}^{2}\,,\quad(p_{2}-l_{2})^{2}=m_{\pi}^{2}\,.$
(40)

For given $k$ these are 6 conditions for the 8 unknowns $l_{1}$, $l_{2}$
giving a 2-parameter solution as it should be. Working in the common c.m.
system of the reactions (2) and (3) we set with
$\mbox{\boldmath$\hat{p}_{1}$}=\mbox{\boldmath$p_{1}$}/|\mbox{\boldmath$p_{1}$}|$

$\displaystyle(l_{1}^{\mu})=\left(\begin{array}[]{l}l_{1}^{0}\\\
l_{1\parallel}\mbox{\boldmath$\hat{p}_{1}$}+\mbox{\boldmath$l_{1\perp}$}\\\
\end{array}\right)\,,\quad\mbox{\boldmath$l_{1\perp}$}\cdot\mbox{\boldmath$\hat{p}_{1}$}=0\,,$
(43) $\displaystyle(l_{2}^{\mu})=\left(\begin{array}[]{l}l_{2}^{0}\\\
l_{2\parallel}\mbox{\boldmath$\hat{p}_{1}$}+\mbox{\boldmath$l_{2\perp}$}\\\
\end{array}\right)\,,\quad\mbox{\boldmath$l_{2\perp}$}\cdot\mbox{\boldmath$\hat{p}_{1}$}=0\,,$
(46) $\displaystyle(k^{\mu})=\left(\begin{array}[]{l}\omega\\\
k_{\parallel}\mbox{\boldmath$\hat{p}_{1}$}+\mbox{\boldmath$k_{\perp}$}\\\
\end{array}\right)\,,\quad\mbox{\boldmath$k_{\perp}$}\cdot\mbox{\boldmath$\hat{p}_{1}$}=0\,.$
(49)

Inserting this in (40) we get the system of equations

$\displaystyle l_{2}=k-l_{1}\,,$ $\displaystyle
p_{1}^{0}l_{1}^{0}-|\mbox{\boldmath$p_{1}$}|l_{1\parallel}=\frac{1}{2}l_{1}^{2}\,,$
$\displaystyle
p_{1}^{0}l_{1}^{0}+|\mbox{\boldmath$p_{1}$}|l_{1\parallel}=p_{1}^{0}k^{0}+|\mbox{\boldmath$p_{1}$}|k_{\parallel}-\frac{1}{2}(k-l_{1})^{2}\,.$
(50)

Now we make an important choice for the following. We assume that together
with the soft photon emitted with energy $\omega\to 0$ we consider only slight
changes of the momenta $p_{1}\to p_{1}^{\prime}$ and $p_{2}\to
p_{2}^{\prime}$. That is, we assume

$\displaystyle l_{1}^{\mu}={\cal O}(\omega)\,,\quad l_{2}^{\mu}={\cal
O}(\omega)\,.$ (51)

With this we can neglect the quadratic terms in $l_{1}$, $l_{2}$, $k$ in (50).
The solution of the resulting equations is

$\displaystyle(l_{1}^{\mu})=\left(\begin{array}[]{l}\dfrac{1}{2p_{1}^{0}}(p_{2}\cdot
k)\\\
\dfrac{1}{2|\mbox{\boldmath$p_{1}$}|}\mbox{\boldmath$\hat{p}_{1}$}(p_{2}\cdot
k)+\mbox{\boldmath$l_{1\perp}$}\\\ \end{array}\right),$ (54)
$\displaystyle(l_{2}^{\mu})=\left(\begin{array}[]{l}\dfrac{1}{2p_{1}^{0}}(p_{1}\cdot
k)\\\
\mbox{\boldmath$k$}-\dfrac{1}{2|\mbox{\boldmath$p_{1}$}|}\mbox{\boldmath$\hat{p}_{1}$}(p_{2}\cdot
k)-\mbox{\boldmath$l_{1\perp}$}\\\ \end{array}\right).$ (57)

Here $l_{1\perp}$ stays undetermined, corresponding to the 2-parameter freedom
of the momenta $p_{1}^{\prime}$, $p_{2}^{\prime}$ for given $k$. This is
illustrated in Fig. 3. In the order of $\omega$ considered we get

$\displaystyle p_{1}\cdot l_{1}=0\,,\quad p_{2}\cdot l_{2}=0\,.$ (58)

Figure 3:  Illustration of the momentum configurations for
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ (2) and
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$ (3) in the c.m. system.

Now we can expand ${\cal M}^{(0,\,a)}$ (12) and ${\cal M}^{(0,\,b)}$ (13) up
to order $\omega$. We get with $s_{L}$ and $t$ from (7),

$\displaystyle{\cal M}^{(0,\,a)}$ $\displaystyle=$ $\displaystyle{\cal
M}^{(0)}[(p_{a}-k,p_{b})+p_{1}^{\prime}\cdot
p_{2}^{\prime},(p_{b}-p_{2}^{\prime})^{2},(p_{a}-k)^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2}]$
(59) $\displaystyle=$ $\displaystyle{\cal
M}^{(0)}[s_{L}-(p_{b}+p_{1},k)-(p_{2}\cdot
l_{1}),t-2(p_{a}-p_{1},k-l_{1}),m_{\pi}^{2}-2(p_{a}\cdot
k),m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2}]$ $\displaystyle+{\cal O}(\omega^{2})$
$\displaystyle=$ $\displaystyle\big{\\{}1-\left[(p_{b}+p_{1},k)+(p_{2}\cdot
l_{1})\right]\frac{\partial}{\partial
s_{L}}-\left[2(p_{a}-p_{1},k)-2(p_{a}\cdot
l_{1})\right]\frac{\partial}{\partial t}-2(p_{a}\cdot
k)\frac{\partial}{\partial m_{a}^{2}}\big{\\}}$
$\displaystyle\times\left.{\cal
M}^{(0)}(s_{L},t,m_{a}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\right|_{m_{a}^{2}=m_{\pi}^{2}}+{\cal
O}(\omega^{2})\,,$ $\displaystyle{\cal M}^{(0,\,b)}$ $\displaystyle=$
$\displaystyle{\cal M}^{(0)}[p_{a}\cdot
p_{b}+(p_{1}^{\prime}+k,p_{2}^{\prime}),(p_{b}-p_{2}^{\prime})^{2},m_{\pi}^{2},m_{\pi}^{2},(p_{1}^{\prime}+k)^{2},m_{\pi}^{2}]$
(60) $\displaystyle=$ $\displaystyle{\cal M}^{(0)}[s_{L}-(p_{1}\cdot
k),t-2(p_{a}-p_{1},k)+2(p_{a}\cdot
l_{1}),m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2}+2(p_{1}\cdot k),m_{\pi}^{2}]$
$\displaystyle+{\cal O}(\omega^{2})$ $\displaystyle=$
$\displaystyle\big{\\{}1-(p_{1}\cdot k)\frac{\partial}{\partial
s_{L}}-\left[2(p_{a}-p_{1},k)-2(p_{a}\cdot
l_{1})\right]\frac{\partial}{\partial t}+2(p_{1}\cdot
k)\frac{\partial}{\partial m_{1}^{2}}\big{\\}}$
$\displaystyle\times\left.{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{1}^{2},m_{\pi}^{2})\right|_{m_{1}^{2}=m_{\pi}^{2}}+{\cal
O}(\omega^{2})\,.$

To determine ${\cal M}^{(c)}_{\lambda}$ to order $\omega^{0}$ we use (23). To
order $\omega$ we get, inserting (59) and (60) in (23),

$\displaystyle k^{\lambda}{\cal M}^{(c)}_{\lambda}$ $\displaystyle=$
$\displaystyle e\big{\\{}(p_{b}+p_{2},k)\frac{\partial}{\partial
s_{L}}+2(p_{a}\cdot k)\frac{\partial}{\partial m_{a}^{2}}+2(p_{1}\cdot
k)\frac{\partial}{\partial m_{1}^{2}}\big{\\}}$ (61)
$\displaystyle\times\left.{\cal
M}^{(0)}(s_{L},t,m_{a}^{2},m_{\pi}^{2},m_{1}^{2},m_{\pi}^{2})\right|_{m_{a}^{2}=m_{1}^{2}=m_{\pi}^{2}}+{\cal
O}(\omega^{2})\,.$

From (61) we can read off the term of order $\omega^{0}$ for ${\cal
M}^{(c)}_{\lambda}$:

$\displaystyle{\cal M}^{(c)}_{\lambda}$ $\displaystyle=$ $\displaystyle
e\big{\\{}(p_{b}+p_{2})_{\lambda}\frac{\partial}{\partial
s_{L}}+2p_{a\lambda}\frac{\partial}{\partial
m_{a}^{2}}+2p_{1\lambda}\frac{\partial}{\partial m_{1}^{2}}\big{\\}}$ (62)
$\displaystyle\times\left.{\cal
M}^{(0)}(s_{L},t,m_{a}^{2},m_{\pi}^{2},m_{1}^{2},m_{\pi}^{2})\right|_{m_{a}^{2}=m_{1}^{2}=m_{\pi}^{2}}+{\cal
O}(\omega)\,.$

Now we collect everything together and we obtain from (15), (36), (37), (59),
(60), and (62) the following expansion for the amplitude
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$:

$\displaystyle{\cal M}_{\lambda}$ $\displaystyle=$ $\displaystyle{\cal
M}_{\lambda}^{(a)}+{\cal M}_{\lambda}^{(b)}+{\cal M}_{\lambda}^{(c)}$ (63)
$\displaystyle=$ $\displaystyle e{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\big{[}\frac{(2p_{a}-k)_{\lambda}}{2(p_{a}\cdot
k)-k^{2}}-\frac{(2p_{1}^{\prime}+k)_{\lambda}}{2(p_{1}^{\prime}\cdot
k)+k^{2}}\big{]}$ $\displaystyle+2e\frac{\partial}{\partial s_{L}}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\big{[}-(p_{b}\cdot
k)\frac{p_{a\lambda}}{(p_{a}\cdot k)}+p_{b\lambda}\big{]}$
$\displaystyle-2e\frac{\partial}{\partial t}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\big{[}(p_{a}-p_{1},k)-(p_{a}\cdot
l_{1})\big{]}\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}\big{]}$ $\displaystyle+{\cal
O}(\omega)\,.$

In the first term on the right-hand side (r.h.s.) of (63) we should, for
consistency of the expansion in $\omega$ up to $\omega^{0}$, make the
following replacements:

$\displaystyle\frac{(2p_{a}-k)_{\lambda}}{2(p_{a}\cdot
k)-k^{2}}\to\frac{p_{a\lambda}}{(p_{a}\cdot k)}+\frac{1}{2(p_{a}\cdot
k)^{2}}[p_{a\lambda}k^{2}-k_{\lambda}(p_{a}\cdot k)]\,,$
$\displaystyle\frac{(2p_{1}^{\prime}+k)_{\lambda}}{2(p_{1}^{\prime}\cdot
k)+k^{2}}\to\frac{p_{1\lambda}}{(p_{1}\cdot k)}+\frac{1}{2(p_{1}\cdot
k)^{2}}[p_{1\lambda}(2(l_{1}\cdot
k)-k^{2})-(2l_{1\lambda}-k_{\lambda})(p_{1}\cdot k)]\,.$ (64)

With (63) and (64) we have obtained the terms of order $\omega^{-1}$ and
$\omega^{0}$ in the expansion of the amplitude for the reaction (3). Now we
compare our result with the corresponding one given in Eq. (17) of Low:1958sn
. Using our notation we get for real photons, $k^{2}=0$, from Low’s result an
amplitude $\widetilde{{\cal M}}_{\lambda}$ as follows:

$\displaystyle\widetilde{{\cal M}}_{\lambda}$ $\displaystyle=$ $\displaystyle
e{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}\big{]}$ (65)
$\displaystyle+e\frac{\partial}{\partial s_{L}}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\big{[}-\frac{(p_{b}\cdot
k)}{(p_{a}\cdot k)}p_{a\lambda}-\frac{(p_{2}\cdot k)}{(p_{1}\cdot
k)}p_{1\lambda}+p_{b\lambda}+p_{2\lambda}\big{]}$ $\displaystyle+{\cal
O}(\omega)\,.$

The term of order $\omega^{-1}$ in (65) agrees with that from (63), (64) for
$k^{2}=0$ but the terms of order $\omega^{0}$ from (63), (64) and (65)
disagree. What is the origin of this discrepancy? To elucidate this we have a
look at the derivation of (65). Following Low:1958sn  we consider the
reactions

$\displaystyle\pi^{-}(p_{a})+\pi^{0}(p_{b})\to\pi^{-}(p_{1})+\pi^{0}(p_{2})$
(66)

and

$\displaystyle\pi^{-}(p_{a})+\pi^{0}(p_{b})\to\pi^{-}(p_{1})+\pi^{0}(p_{2})+\gamma(k,\epsilon)\,.$
(67)

But note that requiring energy-momentum conservation for (66),

$\displaystyle p_{a}+p_{b}=p_{1}+p_{2}\,,$ (68)

we cannot have also energy-momentum conservation for (67) if $k\neq 0$:

$\displaystyle p_{a}+p_{b}\neq p_{1}+p_{2}+k\,.$ (69)

Thus, (67) is a fictitious process. We continue, nevertheless, with the
analysis along the same lines as in Sec. II. We get then for (67) setting
$k^{2}=0$:

$\displaystyle\widetilde{{\cal M}}_{\lambda}=\widetilde{{\cal
M}}_{\lambda}^{(a)}+\widetilde{{\cal M}}_{\lambda}^{(b)}+\widetilde{{\cal
M}}_{\lambda}^{(c)}\,,$ (70) $\displaystyle\widetilde{{\cal
M}}_{\lambda}^{(a)}=+e\,\widetilde{{\cal
M}}^{(0,\,a)}\,\frac{p_{a\lambda}}{(p_{a}\cdot k)}\,,$ (71)
$\displaystyle\widetilde{{\cal M}}^{(0,\,a)}={\cal
M}^{(0)}[(p_{a}-k,p_{b})+(p_{1}\cdot
p_{2}),(p_{b}-p_{2})^{2},(p_{a}-k)^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2}]$
$\displaystyle\qquad\quad\,={\cal M}^{(0)}[s_{L}-(p_{b}\cdot
k),t,m_{\pi}^{2}-2(p_{a}\cdot k),m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2}]$
$\displaystyle\qquad\quad\,=\big{\\{}1-(p_{b}\cdot k)\frac{\partial}{\partial
s_{L}}-2(p_{a}\cdot k)\frac{\partial}{\partial m_{a}^{2}}\big{\\}}\left.{\cal
M}^{(0)}(s_{L},t,m_{a}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\right|_{m_{a}^{2}=m_{\pi}^{2}}+{\cal
O}(\omega^{2})\,.$ (72) $\displaystyle\widetilde{{\cal
M}}_{\lambda}^{(b)}=-e\,\widetilde{{\cal
M}}^{(0,\,b)}\,\frac{p_{1\lambda}}{(p_{1}\cdot k)}\,,$ (73)
$\displaystyle\widetilde{{\cal M}}^{(0,\,b)}={\cal M}^{(0)}[(p_{a}\cdot
p_{b})+(p_{1}+k,p_{2}),(p_{b}-p_{2})^{2},m_{\pi}^{2},m_{\pi}^{2},(p_{1}+k)^{2},m_{\pi}^{2}]$
$\displaystyle\qquad\quad\,=\big{\\{}1+(p_{2}\cdot k)\frac{\partial}{\partial
s_{L}}+2(p_{1}\cdot k)\frac{\partial}{\partial m_{1}^{2}}\big{\\}}\left.{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{1}^{2},m_{\pi}^{2})\right|_{m_{1}^{2}=m_{\pi}^{2}}+{\cal
O}(\omega^{2})\,.$ (74)

We determine $\widetilde{{\cal M}}_{\lambda}^{(c)}$ to order $\omega^{0}$
again from the gauge invariance condition

$\displaystyle k^{\lambda}\widetilde{{\cal
M}}_{\lambda}^{(c)}=-k^{\lambda}\widetilde{{\cal
M}}_{\lambda}^{(a)}-k^{\lambda}\widetilde{{\cal M}}_{\lambda}^{(b)}\,.$ (75)

This gives in a way completely analogous to (61), (62)

$\displaystyle\widetilde{{\cal M}}_{\lambda}^{(c)}$ $\displaystyle=$
$\displaystyle e\big{\\{}(p_{b}+p_{2})_{\lambda}\frac{\partial}{\partial
s_{L}}+2p_{a\lambda}\frac{\partial}{\partial
m_{a}^{2}}+2p_{1\lambda}\frac{\partial}{\partial
m_{1}^{2}}\big{\\}}\left.{\cal
M}^{(0)}(s_{L},t,m_{a}^{2},m_{\pi}^{2},m_{1}^{2},m_{\pi}^{2})\right|_{m_{a}^{2}=m_{1}^{2}=m_{\pi}^{2}}$
(76) $\displaystyle+{\cal O}(\omega)\,.$

From (71)–(76) we get, indeed, (65).

Our conclusion is, thus, as follows. The term of order $\omega^{0}$ in the
expansion of the amplitude given in Low:1958sn  corresponds to the fictitious
process (67) which does not respect energy-momentum conservation. The correct
expansion up to order $\omega^{0}$ for the amplitude of the physical process
(3) is given in (63), (64).

In Appendix A we make some remarks on selected papers from the literature
where Low’s theorem is discussed. Our conclusion is that none of the papers
which we have studied gives a derivation of (36) and (37) as we have done it,
using the Ward-Takahashi identity, and none gives a result for the term of
order $\omega^{0}$ equivalent to the one obtained from our Eqs. (63), (64).

##  IV The reactions $\pi\pi\to\pi\pi$ and $\pi\pi\to\pi\pi\gamma$ in the
tensor-pomeron model

In this section we shall discuss elastic $\pi\pi$ scattering, without and with
photon emission, in the tensor-pomeron model Ewerz:2013kda . Let us make some
remarks on the tensor-pomeron model which is a special Regge-type model. Of
course, Regge theory for high-energy reactions has a long history, starting
from papers around the 1960s. Some early papers are Regge:1959mz ; Chew:1962eu
; Gribov:1963gx , for reviews see Collins:1977 ; Collins:1984 ; Caneschi ;
Donnachie:2002en ; Gribov:2009 . In the tensor-pomeron model of Ewerz:2013kda
the assumption is made that the pomeron and the charge-conjugation $C=+1$
reggeons $f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$ couple to hadrons like symmetric
tensors of rank 2, the odderon and the $C=-1$ reggeons $\omega_{\mathbb{R}}$,
$\rho_{\mathbb{R}}$ as vectors. The idea, that the pomeron couplings could be
related to a tensor coupling was, to our knowledge, first proposed in
Freund:1962 . There the pomeron couplings were related to the couplings of the
energy-momentum tensor. For more historical remarks on the tensor pomeron
ideas we refer to Ewerz:2016onn . The tensor-pomeron model Ewerz:2013kda ,
which we shall use in the following, is for soft hadronic high-energy
reactions and has its origin in general investigations of the soft,
nonperturbative, pomeron in QCD using functional-integral techniques
Nachtmann:1991ua . We note that also in holographic QCD a tensor character of
the pomeron couplings is preferred Domokos:2009hm ; Iatrakis:2016rvj . In
Ewerz:2013kda  the constants in the vertex functions describing the pomeron-
hadron couplings were, as far as possible, determined from comparisons of
theory and experiment.

Now we shall first, for simplicity, discuss the reactions
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ and $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$
[see (2), (3)] and then turn to charged-pion scattering.

###  IV.1 The reactions $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ and
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$

We consider the elastic $\pi\pi$ scattering at high c.m. energy $\sqrt{s}$
where pomeron ($\mathbb{P}$) exchange dominates. The amplitude for the
subleading reggeon ($f_{2\mathbb{R}}$, $\rho_{\mathbb{R}}$) exchanges will be
treated in Sec. IV.2. The propagator and the pion couplings of the tensor
pomeron are given in (3.10), (3.11) and (3.34), (3.45), (3.46) of
Ewerz:2013kda , respectively,

$\displaystyle
i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t)=\frac{1}{4s}\left(g_{\mu\kappa}g_{\nu\lambda}+g_{\mu\lambda}g_{\nu\kappa}-\frac{1}{2}g_{\mu\nu}g_{\kappa\lambda}\right)\,(-is\alpha^{\prime}_{\mathbb{P}})^{\alpha_{\mathbb{P}}(t)-1}\,,$
(77)
$\displaystyle\alpha_{\mathbb{P}}(t)=\alpha_{\mathbb{P}}(0)+\alpha^{\prime}_{\mathbb{P}}t\,,\quad\alpha_{\mathbb{P}}(0)=1+\epsilon_{\mathbb{P}}\,,$
$\displaystyle\epsilon_{\mathbb{P}}=0.0808\,,\quad\alpha^{\prime}_{\mathbb{P}}=0.25\;\mbox{GeV}^{-2}\,;$
(78)

$\displaystyle
i\Gamma_{\mu\nu}^{(\mathbb{P}\pi\pi)}(k^{\prime},k)=-i\,2\beta_{\mathbb{P}\pi\pi}F_{M}[(k^{\prime}-k)^{2}]\left[(k^{\prime}+k)_{\mu}(k^{\prime}+k)_{\nu}-\frac{1}{4}\,g_{\mu\nu}(k^{\prime}+k)^{2}\right]\,,\qquad$
(79) $\displaystyle\beta_{\mathbb{P}\pi\pi}=1.76\;\mbox{GeV}^{-1}\,,\quad
F_{M}(t)=\frac{m_{0}^{2}}{m_{0}^{2}-t}\,,\quad m_{0}^{2}=0.50\;{\rm
GeV}^{2}\,.$ (80)

Figure 4: Diagram with pomeron exchange for $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$
in the tensor-pomeron model.

The pomeron-exchange diagram for the reaction (2)
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$, allowing the pions to be off-shell, is
shown in Fig. 4, and easily evaluated. We get with the kinematic variables of
(7) and (8) for (9):

$\displaystyle{\cal
M}^{(0)}_{\mathbb{P}}(s_{L},t,m_{a}^{2},m_{b}^{2},m_{1}^{2},m_{2}^{2})=i{\cal
F}_{\mathbb{P}}(s,t)\left[2(p_{a}+p_{1},p_{b}+p_{2})^{2}-\frac{1}{2}(p_{a}+p_{1})^{2}(p_{b}+p_{2})^{2}\right]$
$\displaystyle\qquad\qquad\qquad=i{\cal
F}_{\mathbb{P}}(s,t)\left[2(2s_{L}+t)^{2}-\frac{1}{2}(-t+2m_{a}^{2}+2m_{1}^{2})(-t+2m_{b}^{2}+2m_{2}^{2})\right]\,.$
(81)

Here we set

$\displaystyle{\cal F}_{\mathbb{P}}(s,t)$ $\displaystyle=$ $\displaystyle{\cal
F}_{\mathbb{P}}\left[s_{L}+\frac{1}{2}(m_{a}^{2}+m_{b}^{2}+m_{1}^{2}+m_{2}^{2}),t\right]$
(82) $\displaystyle=$
$\displaystyle\left[2\beta_{\mathbb{P}\pi\pi}F_{M}(t)\right]^{2}\frac{1}{4s}(-is\alpha^{\prime}_{\mathbb{P}})^{\alpha_{\mathbb{P}}(t)-1}\,.$

For the scattering of $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ with on-shell pions
this gives

$\displaystyle\left.\braket{\pi^{-}(p_{1}),\pi^{0}(p_{2})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{0}(p_{b})}\right|_{\rm on\;shell}={\cal
M}^{(0)}_{\mathbb{P}}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})$
$\displaystyle\qquad\qquad=i{\cal
F}_{\mathbb{P}}(s,t)\left[2(p_{a}+p_{1},p_{b}+p_{2})^{2}-\frac{1}{2}(p_{a}+p_{1})^{2}(p_{b}+p_{2})^{2}\right]$
$\displaystyle\qquad\qquad=8is^{2}{\cal
F}_{\mathbb{P}}(s,t)\Big{[}1-\frac{4m_{\pi}^{2}-t}{s}+\frac{3}{16s^{2}}(4m_{\pi}^{2}-t)^{2}\Big{]}\,,$
(83) $\displaystyle\sigma_{\rm
tot}(\pi^{-}\pi^{0})=\frac{1}{\sqrt{s(s-4m_{\pi}^{2})}}\,{\rm
Im}\braket{\pi^{-}(p_{a}),\pi^{0}(p_{b})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{0}(p_{b})}$
$\displaystyle\qquad=2\left(2\beta_{\mathbb{P}\pi\pi}\right)^{2}(s\alpha^{\prime}_{\mathbb{P}})^{\epsilon_{\mathbb{P}}}\cos\left(\frac{\pi}{2}\epsilon_{\mathbb{P}}\right)\Big{(}1-\frac{4m_{\pi}^{2}}{s}\Big{)}^{-1/2}\Big{[}1-\frac{4m_{\pi}^{2}}{s}+\frac{3}{16}\Big{(}\frac{4m_{\pi}^{2}}{s}\Big{)}^{2}\Big{]}\,.\quad$
(84)

Now we come to the photon-emission process (3)

$\displaystyle\pi^{-}(p_{a})+\pi^{0}(p_{b})\to\pi^{-}(p_{1}^{\prime})+\pi^{0}(p_{2}^{\prime})+\gamma(k,\epsilon)\,.$
(85)

The relevant kinematic variables are here

$\displaystyle s=(p_{a}+p_{b})^{2}=(p_{1}^{\prime}+p_{2}^{\prime}+k)^{2}\,,$
$\displaystyle
t_{1}=(p_{a}-p_{1}^{\prime})^{2}=(p_{b}-p_{2}^{\prime}-k)^{2}\,,$
$\displaystyle
t_{2}=(p_{b}-p_{2}^{\prime})^{2}=(p_{a}-p_{1}^{\prime}-k)^{2}\,.$ (86)

(a) (b) (c)

Figure 5: Pomeron-exchange diagrams for
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$ in the tensor-pomeron model.

We have to calculate ${\cal M}_{\lambda\mathbb{P}}$ (14), (15) from the
diagrams of Fig. 5. First we calculate ${\cal M}^{(a)}_{\lambda\mathbb{P}}$
and ${\cal M}^{(b)}_{\lambda\mathbb{P}}$ from (12) and (13), respectively,
inserting for ${\cal M}^{(0)}$ the tensor-pomeron expression (81).
Furthermore, we use the standard pion propagator and the standard
$\gamma\pi\pi$ vertex function (see e.g. Ewerz:2013kda ; Bolz:2014mya ). This
gives

$\displaystyle\Delta[(p_{a}-k)^{2}]\,\Gamma_{\lambda}(p_{a}-k,p_{a})=\frac{(2p_{a}-k)_{\lambda}}{-2(p_{a}\cdot
k)+k^{2}}\,,$
$\displaystyle\Gamma_{\lambda}(p_{1}^{\prime},p_{1}^{\prime}+k)\,\Delta[(p_{1}^{\prime}+k)^{2}]=\frac{(2p_{1}^{\prime}+k)_{\lambda}}{2(p_{1}^{\prime}\cdot
k)+k^{2}}\,.$ (87)

From (36) and (37) we see that in QFT these relations are exact for $\omega\to
0$ up to corrections of order $\omega$. For us (87) is part of our model
assumptions.

With (81) and (87) we get from (12) the following amplitude ${\cal
M}^{(a)}_{\lambda\mathbb{P}}$ corresponding to the diagram of Fig. 5 (a):

$\displaystyle{\cal M}_{\lambda\mathbb{P}}^{(a)}=-e\,{\cal
M}_{\mathbb{P}}^{(0,\,a)}\,\frac{(2p_{a}-k)_{\lambda}}{-2(p_{a}\cdot
k)+k^{2}}\,,$ $\displaystyle{\cal M}_{\mathbb{P}}^{(0,\,a)}=i{\cal
F}_{\mathbb{P}}[(p_{a}+p_{b}-k)^{2},t_{2}]\left[2(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})^{2}-\frac{1}{2}(p_{a}+p_{1}^{\prime}-k)^{2}(p_{b}+p_{2}^{\prime})^{2}\right].$
(88)

From (13) we get for ${\cal M}^{(b)}_{\lambda\mathbb{P}}$ corresponding to the
diagram of Fig. 5 (b):

$\displaystyle{\cal
M}_{\lambda\mathbb{P}}^{(b)}=-e\,\frac{(2p_{1}^{\prime}+k)_{\lambda}}{2(p_{1}^{\prime}\cdot
k)+k^{2}}\,{\cal M}_{\mathbb{P}}^{(0,\,b)}\,,$ $\displaystyle{\cal
M}_{\mathbb{P}}^{(0,\,b)}=i{\cal
F}_{\mathbb{P}}(s,t_{2})\left[2(p_{a}+p_{1}^{\prime}+k,p_{b}+p_{2}^{\prime})^{2}-\frac{1}{2}(p_{a}+p_{1}^{\prime}+k)^{2}(p_{b}+p_{2}^{\prime})^{2}\right].$
(89)

For ${\cal M}^{(c)}_{\lambda\mathbb{P}}$ we get from (23)

$\displaystyle k^{\lambda}{\cal M}_{\lambda\mathbb{P}}^{(c)}$ $\displaystyle=$
$\displaystyle-e{\cal M}_{\mathbb{P}}^{(0,\,a)}+e{\cal
M}_{\mathbb{P}}^{(0,\,b)}$ (90) $\displaystyle-ie\big{\\{}{\cal
F}_{\mathbb{P}}(s,t_{2})\big{[}-8(k,p_{b}+p_{2}^{\prime})(p_{a}+p_{1}^{\prime},p_{b}+p_{2}^{\prime})+2(k,p_{a}+p_{1}^{\prime})(p_{b}+p_{2}^{\prime})^{2}\big{]}$
$\displaystyle+\big{[}{\cal F}_{\mathbb{P}}[(p_{a}+p_{b}-k)^{2},t_{2})]-{\cal
F}_{\mathbb{P}}(s,t_{2})\big{]}$
$\displaystyle\times\big{[}2(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})^{2}-\frac{1}{2}(p_{a}+p_{1}^{\prime}-k)^{2}(p_{b}+p_{2}^{\prime})^{2}\big{]}\big{\\}}\,.$

Using the explicit expression for ${\cal F}_{\mathbb{P}}(s,t_{2})$ (82) we get

$\displaystyle{\cal F}_{\mathbb{P}}[(p_{a}+p_{b}-k)^{2},t_{2})]-{\cal
F}_{\mathbb{P}}(s,t_{2})={\cal
F}_{\mathbb{P}}(s,t_{2})\,\left(2-\alpha_{\mathbb{P}}(t_{2})\right)\,\frac{2(p_{a}+p_{b},k)-k^{2}}{s}\,g_{\mathbb{P}}(\varkappa,t_{2})\,,$
(91)

where we define

$\displaystyle\varkappa$ $\displaystyle=$
$\displaystyle\frac{2(p_{a}+p_{b},k)-k^{2}}{s}\,,$ (92) $\displaystyle
g_{\mathbb{P}}(\varkappa,t_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\left(2-\alpha_{\mathbb{P}}(t_{2})\right)\varkappa}\big{[}(1-\varkappa)^{\alpha_{\mathbb{P}}(t_{2})-2}-1\big{]}$
(93) $\displaystyle=$ $\displaystyle
1+\frac{\varkappa}{2!}\left(3-\alpha_{\mathbb{P}}(t_{2})\right)+\frac{\varkappa^{2}}{3!}\left(3-\alpha_{\mathbb{P}}(t_{2})\right)\left(4-\alpha_{\mathbb{P}}(t_{2})\right)+\ldots\,.$

The series expansion in (93) is absolutely convergent for $|\varkappa|<1$
which is the only region of interest for us.

Inserting (91) in (90) we get

$\displaystyle k^{\lambda}{\cal M}^{(c)}_{\lambda\mathbb{P}}$ $\displaystyle=$
$\displaystyle-ie{\cal
F}_{\mathbb{P}}(s,t_{2})\big{\\{}-8(k,p_{b}+p_{2}^{\prime})(p_{a}+p_{1}^{\prime},p_{b}+p_{2}^{\prime})+2(k,p_{a}+p_{1}^{\prime})(p_{b}+p_{2}^{\prime})^{2}$
(94)
$\displaystyle+\frac{2(p_{a}+p_{b},k)-k^{2}}{s}\,\left(2-\alpha_{\mathbb{P}}(t_{2})\right)\,g_{\mathbb{P}}(\varkappa,t_{2})$
$\displaystyle\times\big{[}2(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})^{2}-\frac{1}{2}(p_{a}+p_{1}^{\prime}-k)^{2}(p_{b}+p_{2}^{\prime})^{2}\big{]}\big{\\}}\,.$

From this we see that a simple solution of (94) for ${\cal
M}^{(c)}_{\lambda\mathbb{P}}$ is

$\displaystyle{\cal M}^{(c)}_{\lambda\mathbb{P}}$ $\displaystyle=$
$\displaystyle-ie{\cal
F}_{\mathbb{P}}(s,t_{2})\big{\\{}-8(p_{b}+p_{2}^{\prime})_{\lambda}(p_{a}+p_{1}^{\prime},p_{b}+p_{2}^{\prime})+2(p_{a}+p_{1}^{\prime})_{\lambda}(p_{b}+p_{2}^{\prime})^{2}$
(95)
$\displaystyle+(2p_{a}+2p_{b}-k)_{\lambda}\,\left(2-\alpha_{\mathbb{P}}(t_{2})\right)\,g_{\mathbb{P}}(\varkappa,t_{2})$
$\displaystyle\times\frac{1}{s}\big{[}2(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})^{2}-\frac{1}{2}(p_{a}+p_{1}^{\prime}-k)^{2}(p_{b}+p_{2}^{\prime})^{2}\big{]}\big{\\}}\,.$

However, we could add to ${\cal M}^{(c)}_{\lambda\mathbb{P}}$ from (95), for
instance, terms proportional to

$\displaystyle p_{a\lambda}(p_{1}^{\prime}\cdot
k)-p^{\prime}_{1\lambda}(p_{a}\cdot k)\,,$ (96)

or

$\displaystyle\varepsilon_{\lambda\mu\nu\rho}p_{a}^{\mu}p_{b}^{\nu}k^{\rho}\big{(}\varepsilon_{\alpha\beta\gamma\delta}p_{a}^{\alpha}p_{b}^{\beta}p_{1}^{\prime\gamma}p_{2}^{\prime\delta}\big{)}\,,$
(97)

and still have a solution of (94). Thus, the solution (95) for ${\cal
M}^{(c)}_{\lambda\mathbb{P}}$ is in general not unique as it is to order
$\omega^{0}$; see (62). This fact is well known in the literature; see for
instance Bern:2014vva .

Collecting now everything together we get for the amplitude of reaction (85)
in our model

$\displaystyle{\cal
M}_{\lambda\mathbb{P}}^{(\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma)}={\cal
M}_{\lambda\mathbb{P}}^{(a)}+{\cal M}_{\lambda\mathbb{P}}^{(b)}+{\cal
M}_{\lambda\mathbb{P}}^{(c)}$ (98)

with ${\cal M}_{\lambda\mathbb{P}}^{(c)}$ given in (95) and ${\cal
M}_{\lambda\mathbb{P}}^{(a)}$ and ${\cal M}_{\lambda\mathbb{P}}^{(b)}$
obtained from (88), (89) and (91), as follows:

$\displaystyle{\cal M}_{\lambda\mathbb{P}}^{(a)}=ie{\cal
F}_{\mathbb{P}}(s,t_{2})\Big{[}1+\left(2-\alpha_{\mathbb{P}}(t_{2})\right)\,\frac{2(p_{a}+p_{b},k)-k^{2}}{s}\,g_{\mathbb{P}}(\varkappa,t_{2})\Big{]}$
$\displaystyle\qquad\quad\;\;\times\left[2(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})^{2}-\frac{1}{2}(p_{a}+p_{1}^{\prime}-k)^{2}(p_{b}+p_{2}^{\prime})^{2}\right]\frac{(2p_{a}-k)_{\lambda}}{2(p_{a}\cdot
k)-k^{2}}\,,\qquad$ (99) $\displaystyle{\cal
M}_{\lambda\mathbb{P}}^{(b)}=-ie{\cal
F}_{\mathbb{P}}(s,t_{2})\left[2(p_{a}+p_{1}^{\prime}+k,p_{b}+p_{2}^{\prime})^{2}-\frac{1}{2}(p_{a}+p_{1}^{\prime}+k)^{2}(p_{b}+p_{2}^{\prime})^{2}\right]\frac{(2p_{1}^{\prime}+k)_{\lambda}}{2(p_{1}^{\prime}\cdot
k)+k^{2}}.$ (100)

These results hold for arbitrary $k$. Below in Sec. V we shall consider only
real photon emission where we have $k^{2}=0$.

Some comments on these results are in order. We are interested in soft photon
emission where $\omega\ll\sqrt{s}$. We have then from (92) and (93)
$|\varkappa|={\cal O}(\omega/\sqrt{s})$ and
$g_{\mathbb{P}}(\varkappa,t_{2})\approx 1$. Looking at ${\cal
M}_{\lambda\mathbb{P}}^{(a)}$ we see that there the term proportional to
$g_{\mathbb{P}}(\varkappa,t_{2})$ is a correction of order $\omega/\sqrt{s}$
relative to the leading term. On the other hand, in ${\cal
M}_{\lambda\mathbb{P}}^{(c)}$ the term proportional to
$g_{\mathbb{P}}(\varkappa,t_{2})$ is not suppressed relative to the first term
in the wavy brackets of (95). But in the soft photon region ${\cal
M}_{\lambda\mathbb{P}}^{(c)}$ is, anyway, only of order $\omega/\sqrt{s}$
relative to ${\cal M}_{\lambda\mathbb{P}}^{(a)}$ and ${\cal
M}_{\lambda\mathbb{P}}^{(b)}$. Thus, in the soft-photon region our model
should give reliable results. But the question arises how high we can go in
$\omega$ and still trust the model. We have, as basis of the model, used the
high-energy approximation, given by the pomeron-exchange term, for the
$\pi\pi$ scattering amplitude. Therefore, in ${\cal M}^{(0)}$ (81), (82) the
c.m. energy squared $s$ should be large enough, above the resonance region,
say

$s\geqslant s_{0}=(5\;{\rm GeV})^{2}\,.$ (101)

But in the reaction $\pi\pi\to\pi\pi\gamma$ we need the off-shell amplitudes
${\cal M}^{(0,a)}$ (88) and ${\cal M}^{(0,b)}$ (89) where the squared c.m.
energies are, respectively,

$\displaystyle
s_{a}=(p_{a}+p_{b}-k)^{2}=(p_{1}^{\prime}+p_{2}^{\prime})^{2}\,,$ (102)
$\displaystyle s_{b}=s\,.$ (103)

Surely, in order to apply our Regge model also for ${\cal M}^{(0,a)}$ we
should require

$\displaystyle s_{a}=(p_{a}+p_{b}-k)^{2}=s-2(p_{a}+p_{b},k)+k^{2}\geqslant
s_{0}\,.$ (104)

In the overall c.m. system this means

$\displaystyle\omega\leqslant\frac{1}{2\sqrt{s}}\,\left(s-s_{0}+k^{2}\right)\,.$
(105)

Below, in Sec. V, we shall take this constraint into account.

In Bolz:2014mya  vertices for the coupling of $\gamma\pi\pi$ and
$\mathbb{P}\gamma\pi\pi$ were derived from a Lagrangian; see (B.66)–(B.71)
there. Using these vertices for evaluating the diagrams of Fig. 5 and using in
all three diagrams the pomeron propagator
$\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t_{2})$ with the common value
$s=(p_{a}+p_{b})^{2}$ gives ${\cal M}_{\lambda\mathbb{P}}^{(a)}$, ${\cal
M}_{\lambda\mathbb{P}}^{(b)}$ and ${\cal M}_{\lambda\mathbb{P}}^{(c)}$ as in
(99), (100), and (95), respectively, but setting
$g_{\mathbb{P}}(\varkappa,t_{2})=0$. Thus, our full results for ${\cal
M}_{\lambda\mathbb{P}}^{(a)}$, ${\cal M}_{\lambda\mathbb{P}}^{(b)}$, ${\cal
M}_{\lambda\mathbb{P}}^{(c)}$ above are an improvement of the simple results,
as we respect now the general QFT structure of the amplitudes shown in Fig. 2.
As discussed above, for soft photons the improvement amounts to suitable
additions of non-leading terms of relative order $\omega/\sqrt{s}$.

What about anomalous soft photons in this framework? Given the amplitude for
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ we have constructed ${\cal
M}_{\lambda\mathbb{P}}^{(a)}$ and ${\cal M}_{\lambda\mathbb{P}}^{(b)}$ in a
straightforward way. Of course, we had to extrapolate to off-shell pions and
to assume (87) to hold not only for $\omega\to 0$. But by and large we think
that ${\cal M}_{\lambda\mathbb{P}}^{(a)}$ and ${\cal
M}_{\lambda\mathbb{P}}^{(b)}$ leave little room for anomalous soft photons.
This is quite different for ${\cal M}_{\lambda\mathbb{P}}^{(c)}$ which we
determined here as the simplest solution of the gauge-invariance condition
(94). Clearly, other solutions of (94) for ${\cal
M}_{\lambda\mathbb{P}}^{(c)}$ are possible which could describe “anomalous”
production of soft photons. One of the present authors has been involved in a
suggestion for the origin of such anomalous soft photons: “synchrotron
radiation from the vacuum” Nachtmann:1983uz ; Botz:1994bg ; Nachtmann:ELFE ;
Nachtmann:Lectures ; Nachtmann:2014qta . For a list of suggestions by other
authors we refer to Wong:2014pY .

###  IV.2 Charged-pion scattering without and with photon radiation

In this section we consider the following reactions at high energies in the
tensor-pomeron model:

$\displaystyle\pi^{-}(p_{a})+\pi^{+}(p_{b})\to\pi^{-}(p_{1})+\pi^{+}(p_{2})\,,$
(106)
$\displaystyle\pi^{-}(p_{a})+\pi^{+}(p_{b})\to\pi^{-}(p_{1}^{\prime})+\pi^{+}(p_{2}^{\prime})+\gamma(k,\epsilon)\,,$
(107)

and

$\displaystyle\pi^{\pm}(p_{a})+\pi^{\pm}(p_{b})\to\pi^{\pm}(p_{1})+\pi^{\pm}(p_{2})\,,$
(108)
$\displaystyle\pi^{\pm}(p_{a})+\pi^{\pm}(p_{b})\to\pi^{\pm}(p_{1}^{\prime})+\pi^{\pm}(p_{2}^{\prime})+\gamma(k,\epsilon)\,.$
(109)

Again we leave $k$ arbitrary and do not require $k^{2}=0$.

Figure 6: The diagram for $\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}$ elastic scattering
with exchange of the pomeron, the $f_{2\mathbb{R}}$, and the
$\rho_{\mathbb{R}}$ reggeons.

The diagrams for the elastic scattering processes (106) and (108) are
analogous to the one in Fig. 4 but now we include the subleading
$f_{2\mathbb{R}}$ and $\rho_{\mathbb{R}}$ reggeon exchanges; see Fig. 6. To
evaluate these diagrams we need the effective $f_{2\mathbb{R}}$ and
$\rho_{\mathbb{R}}$ propagators and their couplings to pions. In our model
these are given in (3.12)–(3.15) and (3.53), (3.54), (3.63), (3.64) of
Ewerz:2013kda , respectively. The $f_{2\mathbb{R}}$ propagator and the
$f_{2\mathbb{R}}\pi\pi$ couplings are as in (77)–(79) with the replacements

$\displaystyle\alpha_{\mathbb{P}}(t)\to\alpha_{f_{2\mathbb{R}}}(t)=\alpha_{f_{2\mathbb{R}}}(0)+\alpha^{\prime}_{f_{2\mathbb{R}}}t\,,$
$\displaystyle\alpha_{f_{2\mathbb{R}}}(0)=0.5475\,,\quad\alpha^{\prime}_{f_{2\mathbb{R}}}=0.9\;\mbox{GeV}^{-2}\,,$
$\displaystyle
2\beta_{\mathbb{P}\pi\pi}\to\frac{g_{f_{2\mathbb{R}}\pi\pi}}{2M_{0}}\,,$
$\displaystyle g_{f_{2\mathbb{R}}\pi\pi}=9.30\,,\quad M_{0}=1\;\mbox{GeV}\,.$
(110)

For the effective $\rho_{\mathbb{R}}$ propagator and the
$\rho_{\mathbb{R}}\pi\pi$ coupling we have

$\displaystyle
i\Delta^{(\rho_{\mathbb{R}})}_{\mu\nu}(s,t)=ig_{\mu\nu}\frac{1}{M_{-}^{2}}\,(-is\alpha^{\prime}_{\rho_{\mathbb{R}}})^{\alpha_{\rho_{\mathbb{R}}}(t)-1}\,,$
(111)
$\displaystyle\alpha_{\rho_{\mathbb{R}}}(t)=\alpha_{\rho_{\mathbb{R}}}(0)+\alpha^{\prime}_{\rho_{\mathbb{R}}}t\,,$
$\displaystyle\alpha_{\rho_{\mathbb{R}}}(0)=0.5475\,,\quad\alpha^{\prime}_{\rho_{\mathbb{R}}}=0.9\;\mbox{GeV}^{-2}\,,$
$\displaystyle M_{-}=1.41\;\mbox{GeV}\,.$ (112) $\displaystyle
i\Gamma_{\mu}^{(\rho_{\mathbb{R}}\pi^{-}\pi^{-})}(k^{\prime},k)=-i\Gamma_{\mu}^{(\rho_{\mathbb{R}}\pi^{+}\pi^{+})}(k^{\prime},k)=\frac{i}{2}\,g_{\rho_{\mathbb{R}}\pi\pi}F_{M}[(k^{\prime}-k)^{2}]\,(k^{\prime}+k)_{\mu}\,,$
$\displaystyle g_{\rho_{\mathbb{R}}\pi\pi}=15.63\,,$ (113)

where $F_{M}(t)$ is defined in (80).

Now everything is prepared to evaluate the diagram of Fig. 6 for the general
off-shell $\pi^{-}\pi^{+}$ scattering amplitude. We get [cf. (9) and (81)]

$\displaystyle\left.\braket{\pi^{-}(p_{1}),\pi^{+}(p_{2})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{+}(p_{b})}\right|_{\rm off\;shell}$ $\displaystyle=$
$\displaystyle{\cal
M}^{(0)\pi^{-}\pi^{+}}(s_{L},t,m_{a}^{2},m_{b}^{2},m_{1}^{2},m_{2}^{2})$ (114)
$\displaystyle=$ $\displaystyle{\cal M}^{(0)}_{\mathbb{P}}+{\cal
M}^{(0)}_{f_{2\mathbb{R}}}+{\cal M}^{(0)}_{\rho_{\mathbb{R}}}\,,$

where

$\displaystyle{\cal M}^{(0)}_{\mathbb{P}}$ $\displaystyle=$ $\displaystyle
i{\cal
F}_{\mathbb{P}}(s,t)\left[2(p_{a}+p_{1},p_{b}+p_{2})^{2}-\frac{1}{2}(p_{a}+p_{1})^{2}(p_{b}+p_{2})^{2}\right]$
(115) $\displaystyle=$ $\displaystyle i{\cal
F}_{\mathbb{P}}(s,t)\left[2(2s_{L}+t)^{2}-\frac{1}{2}(-t+2m_{a}^{2}+2m_{1}^{2})(-t+2m_{b}^{2}+2m_{2}^{2})\right]\,,$
$\displaystyle{\cal M}^{(0)}_{f_{2\mathbb{R}}}$ $\displaystyle=$
$\displaystyle i{\cal
F}_{f_{2\mathbb{R}}}(s,t)\left[2(p_{a}+p_{1},p_{b}+p_{2})^{2}-\frac{1}{2}(p_{a}+p_{1})^{2}(p_{b}+p_{2})^{2}\right]$
(116) $\displaystyle=$ $\displaystyle i{\cal
F}_{f_{2\mathbb{R}}}(s,t)\left[2(2s_{L}+t)^{2}-\frac{1}{2}(-t+2m_{a}^{2}+2m_{1}^{2})(-t+2m_{b}^{2}+2m_{2}^{2})\right]\,,$
$\displaystyle{\cal M}^{(0)}_{\rho_{\mathbb{R}}}$ $\displaystyle=$
$\displaystyle{\cal F}_{\rho_{\mathbb{R}}}(s,t)(p_{a}+p_{1},p_{b}+p_{2})$
(117) $\displaystyle=$ $\displaystyle{\cal
F}_{\rho_{\mathbb{R}}}(s,t)(2s_{L}+t)\,.$

Here ${\cal F}_{\mathbb{P}}(s,t)$ is defined in (82) and we have set

$\displaystyle{\cal
F}_{f_{2\mathbb{R}}}(s,t)=\left[\frac{g_{f_{2\mathbb{R}}\pi\pi}}{2M_{0}}F_{M}(t)\right]^{2}\frac{1}{4s}(-is\alpha^{\prime}_{f_{2\mathbb{R}}})^{\alpha_{f_{2\mathbb{R}}}(t)-1}\,,$
(118) $\displaystyle{\cal
F}_{\rho_{\mathbb{R}}}(s,t)=\left[\frac{g_{\rho_{\mathbb{R}}\pi\pi}}{2M_{-}}F_{M}(t)\right]^{2}(-is\alpha^{\prime}_{\rho_{\mathbb{R}}})^{\alpha_{\rho_{\mathbb{R}}}(t)-1}\,.$
(119)

For the on-shell elastic $\pi^{-}\pi^{+}$ scattering we get, setting
$m_{a}^{2}=m_{b}^{2}=m_{1}^{2}=m_{2}^{2}=m_{\pi}^{2}$ in (7), (8) and
(115)–(117)

$\displaystyle\braket{\pi^{-}(p_{1}),\pi^{+}(p_{2})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{+}(p_{b})}={\cal
M}^{(0)\pi^{-}\pi^{+}}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})$
$\displaystyle\qquad\equiv{\cal M}^{(0)\pi^{-}\pi^{+}}(s,t)$
$\displaystyle\qquad=i\Big{[}{\cal F}_{\mathbb{P}}(s,t)+{\cal
F}_{f_{2\mathbb{R}}}(s,t)\Big{]}\Big{[}2(p_{a}+p_{1},p_{b}+p_{2})^{2}-\frac{1}{2}(p_{a}+p_{1})^{2}(p_{b}+p_{2})^{2}\Big{]}$
$\displaystyle\qquad\quad+{\cal
F}_{\rho_{\mathbb{R}}}(s,t)(p_{a}+p_{1},p_{b}+p_{2})$
$\displaystyle\qquad=8is^{2}\Big{[}{\cal F}_{\mathbb{P}}(s,t)+{\cal
F}_{f_{2\mathbb{R}}}(s,t)\Big{]}\Big{[}1-\frac{4m_{\pi}^{2}-t}{s}+\frac{3}{16s^{2}}(4m_{\pi}^{2}-t)^{2}\Big{]}$
$\displaystyle\qquad\quad+2s{\cal
F}_{\rho_{\mathbb{R}}}(s,t)\Big{[}1-\frac{4m_{\pi}^{2}-t}{2s}\Big{]}\,.$ (120)

For brevity of notation we use in the following the notation ${\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t)$ for the on-shell pion-pion elastic scattering
amplitude.

Turning now to the reactions (108) of like sign $\pi\pi$ scattering we get
from the diagrams analogous to Fig. 6 the following for on-shell pions

$\displaystyle\braket{\pi^{+}(p_{1}),\pi^{+}(p_{2})}{{\cal
T}}{\pi^{+}(p_{a}),\pi^{+}(p_{b})}=\braket{\pi^{-}(p_{1}),\pi^{-}(p_{2})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{-}(p_{b})}$ $\displaystyle\qquad=8is^{2}\Big{[}{\cal
F}_{\mathbb{P}}(s,t)+{\cal
F}_{f_{2\mathbb{R}}}(s,t)\Big{]}\Big{[}1-\frac{4m_{\pi}^{2}-t}{s}+\frac{3}{16s^{2}}(4m_{\pi}^{2}-t)^{2}\Big{]}$
$\displaystyle\qquad\quad-2s{\cal
F}_{\rho_{\mathbb{R}}}(s,t)\Big{[}1-\frac{4m_{\pi}^{2}-t}{2s}\Big{]}+(p_{1}\leftrightarrow
p_{2})\,.$ (121)

The exchange $p_{1}\leftrightarrow p_{2}$ implies $t\leftrightarrow u$ where
$u=-s-t+4m_{\pi}^{2}$.

The total cross sections for $\pi\pi$ scattering are obtained from the
forward-scattering amplitudes using the optical theorem. In this way we get
from (120) for $\pi^{-}\pi^{+}$ scattering

$\displaystyle\sigma_{{\rm
tot}\,,\pi^{-}\pi^{+}}(s)=\frac{1}{\sqrt{s(s-4m_{\pi}^{2})}}\,{\rm
Im}\braket{\pi^{-}(p_{a}),\pi^{+}(p_{b})}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{+}(p_{b})}$
$\displaystyle\qquad=2\Big{(}1-\frac{4m_{\pi}^{2}}{s}\Big{)}^{-1/2}\Big{\\{}\Big{[}\left(2\beta_{\mathbb{P}\pi\pi}\right)^{2}(s\alpha^{\prime}_{\mathbb{P}})^{\alpha_{\mathbb{P}}(0)-1}\cos\left(\frac{\pi}{2}(\alpha_{\mathbb{P}}(0)-1)\right)$
$\displaystyle\qquad+\left(\frac{g_{f_{2\mathbb{R}}\pi\pi}}{2M_{0}}\right)^{2}(s\alpha^{\prime}_{f_{2\mathbb{R}}})^{\alpha_{f_{2\mathbb{R}}}(0)-1}\cos\left(\frac{\pi}{2}(\alpha_{f_{2\mathbb{R}}}(0)-1)\right)\Big{]}\Big{(}1-\frac{4m_{\pi}^{2}}{s}+\frac{3m_{\pi}^{4}}{s^{2}}\Big{)}$
$\displaystyle\qquad+\left(\frac{g_{\rho_{\mathbb{R}}\pi\pi}}{2M_{-}}\right)^{2}(s\alpha^{\prime}_{\rho_{\mathbb{R}}})^{\alpha_{\rho_{\mathbb{R}}}(0)-1}\sin\left(\frac{\pi}{2}(1-\alpha_{\rho_{\mathbb{R}}}(0))\right)\Big{(}1-\frac{2m_{\pi}^{2}}{s}\Big{)}\Big{\\}}\,.$
(122)

The total cross sections for $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ scattering
are obtained from (121) for $t=0$. Here for $s\gg 4m_{\pi}^{2}$ and $t=0$ the
term $(p_{1}\leftrightarrow p_{2})$ is highly suppressed and, thus, very
small. Neglecting the term $(p_{1}\leftrightarrow p_{2})$ for $t=0$ we get the
total cross sections for $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ scattering as
in (122) but with a sign change in the $\rho_{\mathbb{R}}$ term.

(a) (b) (c) (d) (e) (f)

Figure 7: Diagrams for the reaction $\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$
with exchange of the pomeron, the $f_{2\mathbb{R}}$, and the
$\rho_{\mathbb{R}}$ reggeons.

For the photon emission process (107) we have 6 diagrams shown in Fig. 7. The
diagrams for (109) are analogous but in addition we have the diagrams with
$p_{1}^{\prime}$ and $p_{2}^{\prime}$ interchanged. The kinematic variables
for these reactions are as in (86). We have

$\displaystyle\braket{\pi^{-}(p_{1}^{\prime}),\pi^{+}(p_{2}^{\prime}),\gamma(k,\epsilon)}{{\cal
T}}{\pi^{-}(p_{a}),\pi^{+}(p_{b})}=(\epsilon^{\lambda})^{*}{\cal
M}_{\lambda}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}\,,$ (123)
$\displaystyle\braket{\pi^{+}(p_{1}^{\prime}),\pi^{+}(p_{2}^{\prime}),\gamma(k,\epsilon)}{{\cal
T}}{\pi^{+}(p_{a}),\pi^{+}(p_{b})}=(\epsilon^{\lambda})^{*}{\cal
M}_{\lambda}^{(\pi^{+}\pi^{+}\to\pi^{+}\pi^{+}\gamma)}\,.$ (124)

Our building blocks for these ${\cal M}_{\lambda}$ amplitudes are ${\cal
M}_{\lambda}^{(a)},\ldots,{\cal M}_{\lambda}^{(f)}$ corresponding to the
diagrams (a)–(f) from Fig. 7. We have here

$\displaystyle{\cal M}_{\lambda}^{(a)}={\cal
M}_{\lambda\mathbb{P}}^{(a)}+{\cal M}_{\lambda f_{2\mathbb{R}}}^{(a)}+{\cal
M}_{\lambda\rho_{\mathbb{R}}}^{(a)}$ (125)

and similarly for ${\cal M}_{\lambda}^{(b)},\ldots,{\cal M}_{\lambda}^{(f)}$.
The amplitudes ${\cal M}_{\lambda\mathbb{P}}^{(a)}$, ${\cal
M}_{\lambda\mathbb{P}}^{(b)}$, and ${\cal M}_{\lambda\mathbb{P}}^{(c)}$ are as
in (99), (100), and (95), respectively. From these we obtain the amplitudes
${\cal M}_{\lambda f_{2\mathbb{R}}}^{(a)}$, ${\cal M}_{\lambda
f_{2\mathbb{R}}}^{(b)}$, and ${\cal M}_{\lambda f_{2\mathbb{R}}}^{(c)}$ with
the replacements (110). For $\rho_{\mathbb{R}}$ exchange we get

$\displaystyle{\cal M}_{\lambda\rho_{\mathbb{R}}}^{(a)}$ $\displaystyle=$
$\displaystyle e{\cal
M}_{\rho_{\mathbb{R}}}^{(0,a)}\frac{(2p_{a}-k)_{\lambda}}{2(p_{a}\cdot
k)-k^{2}}\,,$ $\displaystyle{\cal M}_{\rho_{\mathbb{R}}}^{(0,a)}$
$\displaystyle=$ $\displaystyle{\cal
F}_{\rho_{\mathbb{R}}}(s,t_{2})\Big{[}1+\left(1-\alpha_{\rho_{\mathbb{R}}}(t_{2})\right)\,\frac{2(p_{a}+p_{b},k)-k^{2}}{s}\,g_{\rho_{\mathbb{R}}}(\varkappa,t_{2})\Big{]}$
(126) $\displaystyle\times(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})\,,$
$\displaystyle{\cal M}_{\lambda\rho_{\mathbb{R}}}^{(b)}$ $\displaystyle=$
$\displaystyle-e\frac{(2p_{1}^{\prime}+k)_{\lambda}}{2(p_{1}^{\prime}\cdot
k)+k^{2}}{\cal M}_{\rho_{\mathbb{R}}}^{(0,b)}\,,$ $\displaystyle{\cal
M}_{\rho_{\mathbb{R}}}^{(0,b)}$ $\displaystyle=$ $\displaystyle{\cal
F}_{\rho_{\mathbb{R}}}(s,t_{2})\,(p_{a}+p_{1}^{\prime}+k,p_{b}+p_{2}^{\prime})\,,$
(127) $\displaystyle{\cal M}_{\lambda\rho_{\mathbb{R}}}^{(c)}$
$\displaystyle=$ $\displaystyle e{\cal
F}_{\rho_{\mathbb{R}}}(s,t_{2})\Big{\\{}2(p_{b}+p_{2}^{\prime})_{\lambda}-\frac{(2p_{a}+2p_{b}-k)_{\lambda}}{s}\left(1-\alpha_{\rho_{\mathbb{R}}}(t_{2})\right)g_{\rho_{\mathbb{R}}}(\varkappa,t_{2})$
(128)
$\displaystyle\times(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})\Big{\\}}\,.$

Here $\varkappa$ and $g_{\mathbb{P}}(\varkappa,t)$ are defined in (92) and
(93), respectively, $g_{f_{2\mathbb{R}}}(\varkappa,t)$ is defined analogously

$\displaystyle
g_{f_{2\mathbb{R}}}(\varkappa,t)=\frac{1}{\left(2-\alpha_{f_{2\mathbb{R}}}(t)\right)\varkappa}\big{[}(1-\varkappa)^{\alpha_{f_{2\mathbb{R}}}(t)-2}-1\big{]}\,,$
(129)

and $g_{\rho_{\mathbb{R}}}(\varkappa,t)$ is defined as

$\displaystyle g_{\rho_{\mathbb{R}}}(\varkappa,t)$ $\displaystyle=$
$\displaystyle\frac{1}{\left(1-\alpha_{\rho_{\mathbb{R}}}(t)\right)\varkappa}\big{[}(1-\varkappa)^{\alpha_{\rho_{\mathbb{R}}}(t)-1}-1\big{]}\,,$
(130) $\displaystyle=$ $\displaystyle
1+\frac{\varkappa}{2!}\left(2-\alpha_{\rho_{\mathbb{R}}}(t)\right)+\frac{\varkappa^{2}}{3!}\left(2-\alpha_{\rho_{\mathbb{R}}}(t)\right)\left(3-\alpha_{\rho_{\mathbb{R}}}(t)\right)+\ldots\,.$

We emphasize that ${\cal M}_{\lambda\mathbb{P}}^{(c)}$, ${\cal M}_{\lambda
f_{2\mathbb{R}}}^{(c)}$ and ${\cal M}_{\lambda\rho_{\mathbb{R}}}^{(c)}$ are
obtained as the simplest solution of the gauge-invariance relation

$\displaystyle k^{\lambda}\left({\cal M}_{\lambda}^{(a)}+{\cal
M}_{\lambda}^{(b)}+{\cal M}_{\lambda}^{(c)}\right)=0\,.$ (131)

For the diagrams of Fig. 7 (d)–(f) we find

$\displaystyle{\cal M}_{\lambda}^{(d)}=\left.-{\cal
M}_{\lambda}^{(a)}\right|_{p_{a},p_{1}^{\prime}\leftrightarrow
p_{b},p_{2}^{\prime}}\,,$ (132) $\displaystyle{\cal
M}_{\lambda}^{(e)}=\left.-{\cal
M}_{\lambda}^{(b)}\right|_{p_{a},p_{1}^{\prime}\leftrightarrow
p_{b},p_{2}^{\prime}}\,,$ (133) $\displaystyle{\cal
M}_{\lambda}^{(f)}=\left.-{\cal
M}_{\lambda}^{(c)}\right|_{p_{a},p_{1}^{\prime}\leftrightarrow
p_{b},p_{2}^{\prime}}\,.$ (134)

Note that $(p_{a},p_{1}^{\prime})\leftrightarrow(p_{b},p_{2}^{\prime})$
implies $t_{1}\leftrightarrow t_{2}$; see (86). We have also here

$\displaystyle k^{\lambda}\left({\cal M}_{\lambda}^{(d)}+{\cal
M}_{\lambda}^{(e)}+{\cal M}_{\lambda}^{(f)}\right)=0\,.$ (135)

For the amplitudes (123) and (124) we get finally

$\displaystyle{\cal M}_{\lambda}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}$
$\displaystyle=$ $\displaystyle{\cal M}_{\lambda}^{(a)}+{\cal
M}_{\lambda}^{(b)}+{\cal M}_{\lambda}^{(c)}+{\cal M}_{\lambda}^{(d)}+{\cal
M}_{\lambda}^{(e)}+{\cal M}_{\lambda}^{(f)}\,,$ (136) $\displaystyle{\cal
M}_{\lambda}^{(\pi^{+}\pi^{+}\to\pi^{+}\pi^{+}\gamma)}$ $\displaystyle=$
$\displaystyle-\left(\hat{\cal M}_{\lambda}^{(a)}+\hat{\cal
M}_{\lambda}^{(b)}+\hat{\cal M}_{\lambda}^{(c)}\right)+\hat{\cal
M}_{\lambda}^{(d)}+\hat{\cal M}_{\lambda}^{(e)}+\hat{\cal M}_{\lambda}^{(f)}$
(137) $\displaystyle+(p_{1}^{\prime}\leftrightarrow p_{2}^{\prime})\,.$

Here we define

$\displaystyle\hat{\cal M}_{\lambda}^{(a)}={\cal
M}_{\lambda\mathbb{P}}^{(a)}+{\cal M}_{\lambda f_{2\mathbb{R}}}^{(a)}-{\cal
M}_{\lambda\rho_{\mathbb{R}}}^{(a)}$ (138)

and similarly for $\hat{\cal M}_{\lambda}^{(b)},\ldots,\hat{\cal
M}_{\lambda}^{(f)}$.

The inclusive cross section for the real-photon yield of the reaction (107) is
as follows

$\displaystyle
d\sigma({\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma(k)})=\frac{1}{2\sqrt{s(s-4m_{\pi}^{2})}}\,\frac{d^{3}k}{(2\pi)^{3}\,2k^{0}}\int\frac{d^{3}p_{1}^{\prime}}{(2\pi)^{3}\,2p_{1}^{\prime
0}}\frac{d^{3}p_{2}^{\prime}}{(2\pi)^{3}\,2p_{2}^{\prime 0}}$
$\displaystyle\qquad\times(2\pi)^{4}\delta^{(4)}(p_{1}^{\prime}+p_{2}^{\prime}+k-p_{a}-p_{b}){\cal
M}_{\lambda}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}\big{(}{\cal
M}_{\rho}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}\big{)}^{*}(-g^{\lambda\rho})$
(139)

and similarly for $\pi^{+}\pi^{+}\to\pi^{+}\pi^{+}\gamma$, including a
statistic factor 1/2.

In the following we shall compare our “exact” model results, which we shall
call “standard” results, for (136) and (137), using (99), (100), (95),
(125)–(128), and (132)–(134), to various soft-photon approximations (SPAs).
Below we list the explicit expressions for photon emission in $\pi^{-}\pi^{+}$
scattering.

  1. SPA1:

Here we keep only the pole terms $\propto\omega^{-1}$ for ${\cal
M}_{\lambda}^{(a)}\cdots{\cal M}_{\lambda}^{(f)}$ in (136). From (95), (99),
(100), (120), (125)–(128), and (132)–(134) we see that this amounts to the
following replacements, using $k^{2}=0$, and $p_{1}^{\prime}\to p_{1}$,
$p_{2}^{\prime}\to p_{2}$:

$\displaystyle{\cal M}_{\lambda}^{(a)}\to e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t)\frac{p_{a\lambda}}{(p_{a}\cdot k)}\,,$
$\displaystyle{\cal M}_{\lambda}^{(b)}\to-e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t)\frac{p_{1\lambda}}{(p_{1}\cdot k)}\,,$
$\displaystyle{\cal M}_{\lambda}^{(c)}\to 0\,,$ $\displaystyle{\cal
M}_{\lambda}^{(d)}\to-e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t)\frac{p_{b\lambda}}{(p_{b}\cdot k)}\,,$
$\displaystyle{\cal M}_{\lambda}^{(e)}\to e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t)\frac{p_{2\lambda}}{(p_{2}\cdot k)}\,,$
$\displaystyle{\cal M}_{\lambda}^{(f)}\to 0\,.$ (140)

From (136) and (140) we get then

$\displaystyle{\cal
M}_{\lambda}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}\to{\cal
M}_{\lambda,\;{\rm SPA1}}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}$
$\displaystyle\qquad\qquad=e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t)\Big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}-\frac{p_{b\lambda}}{(p_{b}\cdot
k)}+\frac{p_{2\lambda}}{(p_{2}\cdot k)}\Big{]}.$ (141)

Inserting this in (139) we get the following SPA1 result for the inclusive
photon cross section where, for consistency, we neglect the photon momentum
$k$ in the energy-momentum conserving $\delta^{(4)}(.)$ function:

$\displaystyle d\sigma({\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma(k)})_{\,\rm
SPA1}=\frac{d^{3}k}{(2\pi)^{3}\,2k^{0}}\int d^{3}p_{1}\,d^{3}p_{2}\,e^{2}$
$\displaystyle\qquad\times\Big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}-\frac{p_{b\lambda}}{(p_{b}\cdot
k)}+\frac{p_{2\lambda}}{(p_{2}\cdot k)}\Big{]}$
$\displaystyle\qquad\times\Big{[}\frac{p_{a\rho}}{(p_{a}\cdot
k)}-\frac{p_{1\rho}}{(p_{1}\cdot k)}-\frac{p_{b\rho}}{(p_{b}\cdot
k)}+\frac{p_{2\rho}}{(p_{2}\cdot k)}\Big{]}(-g^{\lambda\rho})$
$\displaystyle\qquad\times\frac{d\sigma(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+})}{d^{3}p_{1}d^{3}p_{2}}\,,$
(142)

where

$\displaystyle\frac{d\sigma(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+})}{d^{3}p_{1}d^{3}p_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s(s-4m_{\pi}^{2})}}\,\frac{1}{(2\pi)^{3}\,2p_{1}^{0}\,(2\pi)^{3}\,2p_{2}^{0}}$
(143)
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p_{a}-p_{b})\,|{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t)|^{2}\,.\qquad$

In (142), (143) we have a frequently used SPA. One takes the distribution of
the particles without radiation [see (143)] and multiplies with the square of
the emission factor in the square brackets in (141).

  2. SPA2:

Here we take into account that the squared momentum transfer is $t_{2}$ for
the diagrams of Fig. 7 (a)–(c) and $t_{1}$ for those of Fig. 7 (d)–(f), where
$t_{1,2}$ are defined in (86). We make in (139) the replacement:

$\displaystyle{\cal
M}_{\lambda}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}\to{\cal
M}_{\lambda,\;{\rm SPA2}}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}$
$\displaystyle\qquad\qquad=e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t_{2})\Big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}^{\prime}}{(p_{1}^{\prime}\cdot k)}\Big{]}$
$\displaystyle\qquad\qquad\quad+e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t_{1})\Big{[}-\frac{p_{b\lambda}}{(p_{b}\cdot
k)}+\frac{p_{2\lambda}^{\prime}}{(p_{2}^{\prime}\cdot k)}\Big{]}.$ (144)

In the calculation of the photon distribution we keep the correct energy-
momentum conserving $\delta^{(4)}(.)$ function in (139).

  3. SPA3:

In our third example we make in (139) the replacement

$\displaystyle{\cal
M}_{\lambda}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}\to{\cal
M}_{\lambda,\;{\rm SPA3}}^{(\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma)}$
$\displaystyle\qquad\qquad=e{\cal
M}^{(0)\pi^{-}\pi^{+}}(s,t^{\prime})\Big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}^{\prime}}{(p_{1}^{\prime}\cdot
k)}-\frac{p_{b\lambda}}{(p_{b}\cdot
k)}+\frac{p_{2\lambda}^{\prime}}{(p_{2}^{\prime}\cdot k)}\Big{]},$ (145)

where we choose

$\displaystyle t^{\prime}={\rm min}(t_{1},t_{2})\,.$ (146)

Also here we keep the correct energy-momentum conserving $\delta^{(4)}(.)$
function in the evaluation of (139).

We shall also consider approximations which we shall call “improved SPA1” and
“improved SPA2”, respectively. For this we consider Fig. 7. In the diagrams
(a) and (d) the squared c.m. energy of the off-shell $\pi\pi\to\pi\pi$
amplitude is $s_{a}=(p_{a}+p_{b}-k)^{2}$, in the diagrams (b) and (e) it is
$s=(p_{a}+p_{b})^{2}$; see (102), (103). For real photons, $k^{2}=0$, and
working in the overall c.m. system we have

$\displaystyle s_{a}=s-2\omega\sqrt{s}\,.$ (147)

Now we take, as a compromise, the average value $\tilde{s}$ of $s_{a}$ and
$s$,

$\displaystyle\tilde{s}=s-\omega\sqrt{s}\,,$ (148)

as squared c.m. energy of the $\pi\pi\to\pi\pi$ amplitudes. With the
amplitudes ${\cal M}^{(0)\pi^{-}\pi^{+}}(\tilde{s},t)$ and ${\cal
M}^{(0)\pi^{-}\pi^{+}}(\tilde{s},t_{1,2})$ in (141) and (144), respectively,
we get what we call the improved SPA1 and SPA2 results. The prescription to
replace $s$ by $\tilde{s}$ (148) has been advocated by Linnyk et al.
Linnyk:2015tha ; Linnyk:2015rco .

##  V Results

Below we show our results for elastic $\pi\pi\to\pi\pi$ scattering (subsection
V.1) and results for the $\pi\pi\to\pi\pi\gamma$ reaction (subsection V.2).

###  V.1 Comparison with the total and elastic $\pi\pi$ cross sections

Here we compare our model results with the $\pi^{-}\pi^{+}$ and
$\pi^{\pm}\pi^{\pm}$ total and total elastic cross section data.

First we briefly review the experimental results for the $\pi\pi$ total and
elastic cross sections. There are no direct measurements of total and elastic
$\pi\pi$ cross sections at present. However, indirect data at low and
intermediate $\sqrt{s}$, the pion-pion center-of-mass energy, have been
extracted from reactions like $\pi^{-}p\to\pi^{+}\pi^{-}n$,
$\pi^{-}\pi^{-}\Delta^{++}$ Biswas:1967mpl ; Cohen:1973yx ; Losty:1973et ;
Robertson:1973tk  and $\pi^{\pm}p\to\Delta^{++}X$ and $\pi^{\pm}n\to pX$
Hanlon:1976ct ; Abramowicz:1979ca . They are compared with our predictions in
Fig. 8. In the left panel the experimental data are from Biswas:1967mpl ;
Cohen:1973yx ; Losty:1973et ; Robertson:1973tk ; Hanlon:1976ct ;
Hoogland:1977kt ; Abramowicz:1979ca 222There are also the data of the total
$\pi^{-}\pi^{-}$ cross section from Biswas:1967mpl ; Robertson:1973tk  (see,
e.g., Fig. 3 of Pelaez:2003ky  or Fig. 2 of Caprini:2011ky ). It was stated in
Pelaez:2003ky  that these results are not consistent with other data at lower
energies probably due to incorrect treatment of final state interactions. The
uncertainties of these data are therefore very large and hence we do not show
them in Fig. 8. while in the right panel from Srinivasan:1975tj ;
Alekseeva:1982uy .

We present for the scattering of $\pi^{-}\pi^{+}$ (opposite-sign pions) and
$\pi^{\pm}\pi^{\pm}$ (same-sign pions) the total (left panel) and total
elastic (right panel) cross sections versus $\sqrt{s}$. The results for the
single pomeron exchange ($\mathbb{P}$), for the pomeron and $f_{2\mathbb{R}}$
reggeon exchanges ($\mathbb{P}+f_{2\mathbb{R}}$), and the complete results
($\mathbb{P}+f_{2\mathbb{R}}+\rho_{\mathbb{R}}$) are shown. The corresponding
theoretical expressions are given in (110)–(122). According to our model we
treat the $\rho_{\mathbb{R}}$ reggeon as effective vector exchange and the
pomeron and $f_{2\mathbb{R}}$ reggeon as effective tensor exchanges. Thus, in
the Regge parametrization of the $\pi^{\pm}\pi^{\pm}$ cross section, the
$\rho_{\mathbb{R}}$ contributes with a sign opposite to $\mathbb{P}$ and
$f_{2\mathbb{R}}$.

We find good agreement with the experimental data taking into account the
default values from Ewerz:2013kda  for the parameters of the propagators and
vertices. One has to keep in mind that for the subleading exchanges the errors
of the coupling constants are quite large, in particular for the coupling
$g_{\rho_{\mathbb{R}}\pi\pi}$, as was discussed in Sec. 7.1 of Ewerz:2013kda .
In addition one also has to keep in mind that there should be a smooth
transition from reggeon to particle exchanges when going to very low energies.
Note that the same-sign-pions channels do not contain $s$ channel resonances
in contrast to the opposite-sign-pions channel. Thus, our theoretical results,
which include only $t$-channel exchanges, are in better agreement with the
experimental data for $\sigma^{\pi^{\pm}\pi^{\pm}}$ than for
$\sigma^{\pi^{-}\pi^{+}}$. Moreover, such effects as absorption corrections
and multiple soft and hard exchanges, discussed in Szczurek:2001py , were not
included in our calculation. Clearly, all these topics deserve careful
analyses, but this goes beyond the scope of the present paper.

There are also the data of $\pi^{\pm}\pi^{-}$ total cross sections from the
analysis performed in Zakharov_data . In that work, a triple reggeon model
with absorption was used to extract $\sigma_{\rm tot}^{\pi^{\pm}\pi^{-}}$ from
the $\pi^{\pm}p\to\Delta^{++}X$ and $\pi^{\pm}n\to pX$ processes. The authors
of Zakharov_data  found that the inclusion of absorptive corrections in these
two reactions decreases the results by about 10$\,\%$ to 15$\,\%$. The
uncertainty of these results is large and therefore we do not show these data
in Fig. 8 and instead we refer to Szczurek:2001py ; Pelaez:2004ab . In
Szczurek:2001py  the effect of absorption corrections (double-scattering
effect) on the total cross section for $\pi\pi$ scattering as a function of
$\sqrt{s}$ was discussed. The $t$-dependence of the elastic $\pi\pi$ cross
sections was also discussed there. The authors of Szczurek:2001py  found that
the absorption is much weaker for the same-sign pions than for the opposite-
sign pions; see, e.g., Figs. 5, 9 and Table 2 of Szczurek:2001py .

The total $\pi^{+}\pi^{-}$ and $\pi^{\pm}\pi^{\pm}$ cross sections including
subleading reggeon exchanges were also discussed in Pelaez:2003ky ;
Halzen:2011xc ; Caprini:2011ky . There is the question of the reliability of
the Regge model down to low energies and whether in the region of low
$\sqrt{s}$ but not low $|t|$ the Regge parametrization can be properly
applied. On general grounds, one expects Regge theory to work when $s\gg|t|$,
$s_{0}$ [see (101)] and $|t|\lesssim 1$ GeV2 and, in fact, the Regge
parametrization for $\pi\pi$ becomes unreliable at large $|t|$. The interested
reader may consult Refs. Pelaez:2003ky ; Caprini:2011ky  for the detailed
discussion of this and other related issues.

Figure 8:  Results for $\pi^{-}\pi^{+}$ and $\pi^{+}\pi^{+}$ or
$\pi^{-}\pi^{-}$ total (left panel) and total elastic (right panel) cross
sections as a function of $\sqrt{s}$ together with the experimental data. The
single pomeron exchange is given by the blue solid line, the pomeron plus
$f_{2\mathbb{R}}$ reggeon exchanges by the blue dotted line, the complete
result ($\mathbb{P}+f_{2\mathbb{R}}+\rho_{\mathbb{R}}$) for the opposite-sign
pions and the same-sign pions is given by the black long-dashed line and the
red short-dashed line, respectively. In the right panel we show also the
$f_{2\mathbb{R}}$ reggeon and the $\rho_{\mathbb{R}}$ reggeon terms
separately.

In the next subsection we shall discuss soft-photon emission in $\pi\pi$
scattering for c.m. energies $\sqrt{s}=10$ GeV and 100 GeV. We see from Fig. 8
that at $\sqrt{s}\gtrsim 100$ GeV the $\pi\pi$ cross sections are completely
dominated by the pomeron-exchange contribution. At least, this is the result
of our model. Therefore, in Sec. V.2 we shall take into account only the
pomeron-exchange term for the reactions $\pi\pi\to\pi\pi\gamma$ at
$\sqrt{s}\geqslant 100$ GeV. At $\sqrt{s}\simeq 10$ GeV we will show results
including the pomeron exchange alone and in addition the $\rho_{\mathbb{R}}$
and $f_{2\mathbb{R}}$ reggeon exchanges. As we will show below in Fig. 14, the
secondary reggeon exchanges play a significant role there.

###  V.2 Comparison of our “exact” model or “standard” results for the
$\pi\pi\to\pi\pi\gamma$ reactions with various soft-photon approximations

First, in Fig. 9, we present the two-dimensional distributions in ($\omega$,
$k_{\perp}$), ($\omega$, $\rm{y}$), and ($k_{\perp}$, $\rm{y}$), for the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction for our “standard” result
(136), (139), including only the pomeron exchange. Calculations were done for
the pion-pion collision energy $\sqrt{s}=10$ GeV. Here, $\omega=k^{0}$ is the
center-of-mass photon energy, $k_{\perp}$ is the absolute value of the photon
transverse momentum, and $\rm{y}$ is the rapidity of the photon. We must
remember here, that in order to stay with all amplitudes in the Regge regime
we certainly have to require (105) which reads here, with $k^{2}=0$ and
$s_{0}=25\;{\rm GeV}^{2}$,

$\displaystyle\omega\leqslant\frac{1}{2\sqrt{s}}\,\left(s-s_{0}\right)=3.75\;{\rm
GeV}\,.$ (149)

To be on the safe side, we shall in the following only show results for
$\omega<3$ GeV. In the panel (a) we show the lines corresponding to the
absolute value of the rapidity of the photon ${\rm y}=1,2,\ldots,6$. Large
${\rm y}$ is near the $\omega$ axis and ${\rm y}=0$ on the $k_{\perp}$ axis.
There are in all three plots also regions that are not accessible
kinematically. From the panel (b) we see that an upper cut on $\omega$ is
effecting the upper limit of the allowed ${\rm y}$ range.

(a)
(b) (c)

Figure 9:  The two-dimensional distributions in ($\omega$, $k_{\perp}$),
($\omega$, $\rm{y}$), and ($k_{\perp}$, $\rm{y}$), for the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction including only the pomeron
exchange. Calculations were done for $\sqrt{s}=10$ GeV, $0.001\;{\rm
GeV}<k_{\perp}<0.2\;{\rm GeV}$, $\omega<3\;{\rm GeV}$, and $|\rm{y}|<8$. The
lines plotted in the panel (a) correspond to the photon rapidities ${\rm
y}=1,2,\ldots,6$. In panels (b) and (c) we show the results only for
$0<\rm{y}<8$ since these distributions are symmetric under $\rm{y}\to-\rm{y}$.

Now we compare our “exact” model or “standard” result for the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction to various soft-photon
approximations (SPAs) discussed in Sec. IV.2. We consider $\sqrt{s}=10$ GeV
and include only the pomeron exchange.

A quantity of great interest is the ratio of the cross section calculated in
one of the SPAs to the “standard” result. This ratio will now be studied as a
function of $\omega=k^{0}$ and $k_{\perp}$ in the $\omega$-$k_{\perp}$ plane.
In Fig. 10 we show, in two-dimensional plots, the ratio

$\displaystyle{\rm R}(\omega,k_{\perp})=\frac{d^{2}\sigma_{\rm SPA}/d\omega
dk_{\perp}}{d^{2}\sigma_{\rm standard}/d\omega dk_{\perp}}\,.$ (150)

The results for the three scenarios of the SPA amplitudes are presented. The
result on the panels (a) corresponds to SPA1 (141), the result on the panels
(b) corresponds to SPA2 (144), and the result on the panels (c) corresponds to
SPA3 (145). We also show the lines corresponding to ${\rm y}=1,2,\ldots,6$.

(a) (b) (c)

Figure 10:  The ratios ${\rm R}(\omega,k_{\perp})$ (150) for the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction for $\sqrt{s}=10$ GeV for the
three soft-photon approximations SPA1 (141), SPA2 (144), and SPA3 (145). The
lines corresponding to the photon rapidities ${\rm y}=1,2,\ldots,6$ are also
plotted. The right panels show the region of small $k_{\perp}$ and small
$\omega$ in greater detail.

Now we discuss results separately for the three $k_{\perp}$ intervals of
photon transverse momenta: $0.1\;{\rm MeV}<k_{\perp}<1\;{\rm MeV}$, $1\;{\rm
MeV}<k_{\perp}<10\;{\rm MeV}$, $10\;{\rm MeV}<k_{\perp}<100\;{\rm MeV}$. We do
so due to difficulties in the numerical evaluation of integrals. In Fig. 11 we
show the distributions in ${\rm y}$ for the standard results [see Eq. (136)]
including only the pomeron exchange. Calculations were done for $\sqrt{s}=10$
GeV and 100 GeV. When going from $\sqrt{s}=10$ GeV to $\sqrt{s}=100$ GeV the
maximum of the ${\rm y}$ distribution shifts from ${\rm y_{max}}\simeq 3.4$ to
${\rm y_{max}}\simeq 5.8$.

Figure 11:  The distributions in rapidity of the photon in the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction calculated for $\sqrt{s}=10$
GeV, $\omega<3$ GeV (left panel) and for $\sqrt{s}=100$ GeV, $\omega<40$ GeV
(right panel) for different $k_{\perp}$ intervals. Plotted are the results
only for positive $\rm{y}$ since the distributions are symmetric under
$\rm{y}\to-\rm{y}$.

In Fig. 12 we present the distributions in $k_{\perp}$ and $\omega$ for the
reaction $\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ calculated for $\sqrt{s}=10$
GeV including only the pomeron exchange. Results are shown for three
$k_{\perp}$ intervals for our model and for the various SPAs. From the semi-
logarithmic plots of Fig. 12 we see that the three SPAs follow the general
trend of our standard results quite well for $k_{\perp}\lesssim 20$ MeV and
for $\omega\lesssim 1$ GeV. But let us now have a closer look at these
kinematic regions at a linear scale.

Figure 12:  The differential distributions in transverse momentum of the
photon and in the energy of the photon for the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction. The calculations were done
for $\sqrt{s}=10$ GeV and $\omega<3$ GeV. The black solid line corresponds to
the standard result, the red long-dashed line corresponds to SPA1 (141), the
green dashed line corresponds to SPA2 (144), and the blue dotted line
corresponds to SPA3 (145).

Figure 13 shows the ratios of the SPAs to the standard cross section:

$\displaystyle\frac{d\sigma_{\rm SPA}/dk_{\perp}}{d\sigma_{\rm
standard}/dk_{\perp}}\,\,,$ (151) $\displaystyle\frac{d\sigma_{\rm
SPA}/d\omega}{d\sigma_{\rm standard}/d\omega}\,,$ (152)

as functions of $k_{\perp}$ and $\omega$, respectively. The rapid fluctuations
of the ratio as a function of $k_{\perp}$ are due to different organization of
integration in the two codes: one for the full three-body phase space
(standard approach, SPA2, SPA3) and one for the two-body phase space
supplemented by additional integration over photon three momentum (SPA1). The
SPAs which we consider deviate from the standard results only at the percent
level for $0.1\;{\rm MeV}<k_{\perp}<1\;{\rm MeV}$ but at the 10$\,\%$ to
50$\,\%$ level for $k_{\perp}\cong 50$ MeV; see the left panels of Fig. 13.
From the right panels of Fig. 13 we see that the deviations of the SPAs from
the standard results are up to around 50$\,\%$ for $\omega<1.5$ GeV. We also
note that the discrepancies of the SPAs to the standard results typically
increase rapidly with growing $k_{\perp}$ and $\omega$. For the SPA1
approximation we have on purpose set $k=0$ in the energy-momentum conserving
delta function in (139), since this corresponds to a frequently used procedure
in the literature. Thus, the SPA1 approach does not respect the upper
kinematic limit for $\omega$. But this is no problem for us since we are
interested here only in soft-photon production. But we note that the accuracy
of the SPA1 can be significantly improved and the region of its applicability
can be extended by keeping the correct energy-momentum conservation as in the
SPA2 and SPA3.

Figure 13:  The ratios $\sigma_{\rm SPA}/\sigma_{\rm standard}$ given by (151)
and (152), respectively. The oscillations in the SPA1 results are of numerical
origin.

Now we wish to illustrate the effect of inclusion of reggeon exchanges
($\rho_{\mathbb{R}}$ and $f_{2\mathbb{R}}$) in addition to the pomeron
exchange. In Fig. 14 we show the ratio $\sigma_{\rm
standard}^{(\mathbb{P}+\mathbb{R})}/\sigma_{\rm standard}^{(\mathbb{P})}$ for
our model as a function of $k_{\perp}$, $\omega$, and $\rm{y}$. Inclusion of
the subleading reggeon exchanges in the calculations leads to a sizable
increase of the cross section. We get for the ratio of the total cross
sections with the cuts $1\;{\rm MeV}<k_{\perp}<10\;{\rm MeV}$ and $\omega<3$
GeV

$\displaystyle\frac{\sigma_{\rm
standard}^{(\mathbb{P}+\mathbb{R})}}{\sigma_{\rm
standard}^{(\mathbb{P})}}=\frac{29.50\;\mu{\rm b}}{21.76\;\mu{\rm b}}\simeq
1.36\,,$ (153)

that is, an about 36$\,\%$ increase due to the reggeon exchanges. From the
ratios of differential distributions in $\omega$ and in $\rm{y}$ we see that
these ratios vary from 1.25 to 1.55 depending on kinematics.

Figure 14:  The ratios $\sigma_{\rm
standard}^{(\mathbb{P}+\mathbb{R})}/\sigma_{\rm standard}^{(\mathbb{P})}$ in
the $\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction calculated for
$\sqrt{s}=10$ GeV, $1\;{\rm MeV}<k_{\perp}<10\;{\rm MeV}$, and $\omega<3$ GeV.

Now we turn to the results at c.m. energy $\sqrt{s}=100$ GeV. Here we include
in the calculations only the pomeron-exchange contributions. As we see already
from Fig. 8 the nonleading exchanges are negligible there.

In Fig. 15 we show the distributions in ($\omega$, $k_{\perp}$), ($\omega$,
$\rm{y}$), and ($k_{\perp}$, $\rm{y}$), for our standard results. Here we
consider only c.m. photon energies $\omega<10$ GeV. The constraint (105),
setting $k^{2}=0$, is then always well satisfied. That is, we are in the Regge
regime for all relevant amplitudes. These distributions are the analogs of
those shown in Fig. 9 for $\sqrt{s}=10$ GeV.

(a)
(b) (c)

Figure 15:  The two-dimensional distributions in ($\omega$, $k_{\perp}$),
($\omega$, $\rm{y}$), and ($k_{\perp}$, $\rm{y}$), for the reaction
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ at $\sqrt{s}=100$ GeV. This is the
same as in Fig. 9 but for $\sqrt{s}=100$ GeV, $\omega<10$ GeV, and
$|\rm{y}|<11$. The lines plotted in the panel (a) correspond to the photon
rapidities ${\rm y}=4,6,8$.

Figure 16 shows the ratios ${\rm R}(\omega,k_{\perp})$ (150) for the reaction
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ at $\sqrt{s}=100$ GeV for the
approximations SPA1 (141), SPA2 (144) and SPA3 (145).

(a) (b) (c)

Figure 16:  The ratios ${\rm R}(\omega,k_{\perp})$ (150) for the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction for $\sqrt{s}=100$ GeV for
the three SPAs. The lines corresponding to the photon rapidities ${\rm
y}=4,6,8$ are also plotted.

In Figs. 17 and 18 we show the results for $\sqrt{s}=100$ GeV which are
analogs of those shown in Figs. 12 and 13 for $\sqrt{s}=10$ GeV. The
calculations were done with cuts on $\omega$ specified in the figure legends.
In all cases the constraint on $\omega$ from (105) is well satisfied. We see
that at $\sqrt{s}=100$ GeV the three SPAs are all close to our standard
results in the region of small $k_{\perp}$ and $\omega$. For $0.1\;{\rm
MeV}<k_{\perp}<1\;{\rm MeV}$ the SPA1 result deviates strongly from the
standard result for $\omega\gtrsim 4$ GeV; see the upper most right panel of
Fig. 17. This is due to the incorrect energy-momentum $\delta$ function used,
on purpose, there; see (140)–(143). Figure 18 shows that for
$k_{\perp}\lesssim 10\;{\rm MeV}$ the deviations of the SPAs from the standard
results are only at the percent level. For the $\omega$ distributions these
differences are up to around 10$\,\%$ for $\omega\lesssim 3$ GeV.

We also note that in Fig. 18 the SPA results are in most cases above the
standard results (ratio > 1) but in some cases also below (ratio < 1). Thus,
the ratios SPA/standard depend strongly on the kinematics.

As for $\sqrt{s}=10$ GeV, the rapid oscillations of the ratios for SPA1 in
Fig. 18 are a numerical artefact caused by different integration procedures in
two different codes.

Figure 17:  The same as in Fig. 12 but for $\sqrt{s}=100$ GeV. Shown are
results for three $k_{\perp}$ intervals and with cuts on $\omega$ specified in
the figure legends.

Figure 18:  The same as in Fig. 13 but for $\sqrt{s}=100$ GeV. Shown are
results for three $k_{\perp}$ intervals and with cuts on $\omega$ specified in
the figure legends. The oscillations in the SPA1 results are of numerical
origin.

At this point we can discuss the qualitative accuracy estimates of the SPA1
approximation as given a long time ago in Gribov:1966hs . For the scattering
reaction this estimate is given following Eq. (19) of Gribov:1966hs  and reads
for our case

$\displaystyle p_{a}\cdot k\approx p_{1}\cdot k\ll m_{\pi}^{2}\,,$ (154)
$\displaystyle p_{b}\cdot k\approx p_{2}\cdot k\ll m_{\pi}^{2}\,.$ (155)

In the c.m. system, choosing $p_{a}$ in $z$ direction, $p_{a}\cdot k$
($p_{b}\cdot k$) can only become small for the longitudinal component $k_{L}$
of $k$ being positive (negative). We have then

$\displaystyle p_{a}\cdot k=p_{a}^{0}\omega-|\mbox{\boldmath$p_{a}$}|k_{L}$
$\displaystyle=$ $\displaystyle
p_{a}^{0}\omega-|\mbox{\boldmath$p_{a}$}|\sqrt{\omega^{2}-k_{\perp}^{2}}$
(156) $\displaystyle=$
$\displaystyle\frac{2m_{\pi}^{2}\omega^{2}+\frac{1}{2}(s-4m_{\pi}^{2})\,k_{\perp}^{2}}{\sqrt{s}\omega+\sqrt{s-4m_{\pi}^{2}}\sqrt{\omega^{2}-k_{\perp}^{2}}}\quad{\rm
for}\;k_{L}>0\,,$

and

$\displaystyle p_{b}\cdot
k=\frac{2m_{\pi}^{2}\omega^{2}+\frac{1}{2}(s-4m_{\pi}^{2})\,k_{\perp}^{2}}{\sqrt{s}\omega+\sqrt{s-4m_{\pi}^{2}}\sqrt{\omega^{2}-k_{\perp}^{2}}}\quad{\rm
for}\;k_{L}<0\,.$ (157)

In Fig. 19 we show again the ratio SPA1/standard in the $\omega$-$k_{\perp}$
plane together with the lines

$\displaystyle\frac{2m_{\pi}^{2}\omega^{2}+\frac{1}{2}(s-4m_{\pi}^{2})\,k_{\perp}^{2}}{\sqrt{s}\omega+\sqrt{s-4m_{\pi}^{2}}\sqrt{\omega^{2}-k_{\perp}^{2}}}=c\,m_{\pi}^{2}\,,\quad
c=1.0,\;0.5,\;0.1,\;0.01\,.$ (158)

Figure 19:  The ratio SPA1/standard (150) for $\sqrt{s}=10$ GeV (top panel)
and for $\sqrt{s}=100$ GeV (bottom panels). The lines corresponding to
$c=1.0,0.5,0.1$, and 0.01 in (158) are plotted. All other curves in the bottom
right panel are numerical artefacts.

If the accuracy estimate (154), (155) would be valid, say with $p_{a}\cdot
k\leqslant cm_{\pi}^{2}$ and $p_{b}\cdot k\leqslant cm_{\pi}^{2}$ and $c=0.1$
or 0.01, the area below the corresponding curves in Fig. 19 should be coloured
green, indicating that there the SPA1 is a good representation of our standard
result. But we see from Fig. 19 that the green areas, which we obtained from
explicit calculations, seem to have only very little to do with the
qualitative conditions (154), (155). The areas below the curves of constant
$c$ have regions where the SPA1 is not a good representation of the standard
results. On the other hand, there are large green areas above these curves
where the SPA1 gives a good representation of the standard results. Of course,
all these statements refer to a comparison of SPA1 to our standard results and
things could be different for possible other calculations including, for
instance, “anomalous” terms.

Figure 20 shows the ratios ${\rm R}(\omega,k_{\perp})$ for the two “improved
SPA” scenarios calculated for $\sqrt{s}=10$ GeV and 100 GeV. Comparing these
results to the corresponding results from Figs. 10 and 16 we observe that the
“improved SPAs” greatly reduce the discrepancies to our standard results. This
is particularly the case for the improved SPA1 case. One can see from Figs. 20
(a) and (e) that the improved SPA1 is a good approximation with the accuracy
up to 10$\,\%$ for $k_{\perp}\lesssim 0.2$ GeV and $\omega\lesssim 2$ GeV for
$\sqrt{s}=10$ GeV, and for $k_{\perp}\lesssim 0.2$ GeV and $\omega\lesssim 20$
GeV for $\sqrt{s}=100$ GeV.

(a) (b) (c) (d) (e)

Figure 20:  The ratios ${\rm R}(\omega,k_{\perp})$ (150) for the
$\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ reaction for our “improved SPA”
scenarios for $\sqrt{s}=10$ GeV and 100 GeV.

##  VI Conclusions

In this paper we have studied elastic pion-pion scattering without and with
photon radiation. In Sec. II we have given a detailed analysis, from a QFT
point of view, of the reactions $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ and
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$. We have used this analysis in Sec.
III to derive the expansion of the amplitude for
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$ in powers of $\omega$, the photon
energy in the overall c.m. system, for $\omega\to 0$. The term of order
$\omega^{-1}$ agrees with that given by F.E. Low in Low:1958sn  but, to our
great surprise, our term of order $\omega^{0}$ disagrees with that given in
Low:1958sn . We have analysed this important discrepancy and we have shown
that our expansion is for the photon-emission amplitude satisfying energy-
momentum conservation. In contrast, we find that the term of order
$\omega^{0}$ from Low:1958sn  corresponds to the expansion of an amplitude
violating energy-momentum conservation for photon-momentum $k\neq 0$. In
Appendix A we compare our findings for this term of order $\omega^{0}$ to
further results from the literature. We find that our result is new compared
to the results of the papers which we study in Appendix A. We emphasize that
our result for the $\omega^{0}$ term is a strict consequence of QFT.
Therefore, absolutely no model dependence is contained there.

In Sec. IV we have calculated the amplitudes for $\pi\pi\to\pi\pi$ and
$\pi\pi\to\pi\pi\gamma$ in the tensor-pomeron model. The diagrams for the
latter process where the photon is emitted from the external pion lines [Fig.
7 (a), (b), (d), (e)] are determined completely by the (off-shell)
$\pi\pi\to\pi\pi$ scattering amplitude. The amplitudes corresponding to the
diagrams of Fig. 7 (c) and Fig. 7 (f), the “structure terms”, have to satisfy
gauge-invariance constraints involving the previous amplitudes. We have given
a solution of these constraints which involves again only the (off-shell)
$\pi\pi\to\pi\pi$ scattering amplitude. But we have emphasized that this
solution is not unique (as is well known in the literature, see e.g.
Bern:2014vva ) and there “anomalous” terms in the $\pi\pi\to\pi\pi\gamma$
amplitudes, not directly related to the $\pi\pi\to\pi\pi$ amplitude, could
come up. We considered then as “standard”, or “exact” model, our
$\pi\pi\to\pi\pi\gamma$ amplitudes without such “anomalous” terms. Clearly, in
Sec. IV we used a model. We summarize here our main model assumptions.

  1. (1)

We used the tensor-pomeron model, both for the on-shell and off-shell
$\pi\pi\to\pi\pi$ amplitudes. For the high-energy reactions which are our main
interest we needed the effective pomeron propagator and the ${\rm
pomeron}\,\pi\pi$ vertex. These quantities were taken from Ewerz:2013kda
where they were derived from comparison of theory to data, in particular for
nucleon-(anti)nucleon and pion-nucleon scattering.

  2. (2)

We used the standard pion propagator and $\gamma\pi\pi$ vertex; see (87).
Possible off-shell form factors in the pomeron- and photon-pion vertices and
the pion propagator are set to one.

  3. (3)

To determine the structure terms [Figs. 5 (c), 7 (c), 7 (f)] we used the
simplest solutions of the respective gauge-invariance relations; see (94),
(95), (128). Throughout our paper we denote as “anomalous” terms possible
additional structure terms, which then have to satisfy the gauge-invariance
relations by themselves. In our model we excluded such “anomalous” terms in
the radiative amplitudes.

We consider point (3) as our main model assumption.

We have defined three soft-photon approximations to our above “exact model”:
SPA1, SPA2, and SPA3; see Sec. IV.2. In the SPA1 the photon momentum $k$ was,
on purpose, omitted in the energy-momentum conserving $\delta^{(4)}(.)$
function in the evaluation of the cross section. In the SPA2 and SPA3 the
correct energy-momentum conservation was required.

In Sec. V we have presented quantitative calculations for the elastic $\pi\pi$
scattering without and with photon radiation within the tensor-pomeron model.
We have shown results for our “standard model” and for the three SPAs for two
different collision energies $\sqrt{s}=10$ GeV and 100 GeV. We have shown, for
instance, the results of our model for the two-dimensional distributions in
photon transverse momentum $k_{\perp}$ and rapidity ${\rm y}$ [Figs. 9 (c) and
15 (c)]. For $\sqrt{s}=10$ GeV the distribution is largest for
$k_{\perp}\lesssim 0.1$ GeV and $2\lesssim{\rm y}\lesssim 5$. For
$\sqrt{s}=100$ GeV the distribution is largest for $k_{\perp}\lesssim 0.004$
GeV and $3\lesssim{\rm y}\lesssim 7$. These are the results of our
calculations in the framework of our “standard” model where we have listed the
assumptions in (1), (2), (3) above. We note that the distributions in
$k_{\perp}$ and ${\rm y}$ give very small values for ${\rm y}\approx 0$, that
is, in the mid-rapidity region. Clearly, this is then a region where
“anomalous” contributions to the radiative amplitude could be large compared
to our “standard”. But we note that such anomalous contributions cannot come
directly from the high-energy exchange object, the pomeron $\mathbb{P}$.
Charge-conjugation ($C$) invariance of the strong and electromagnetic
interactions forbids a $\mathbb{P}\mathbb{P}\gamma$ vertex. The pomeron has
$C=+1$ and the photon $C=-1$. But in $\pi\pi$ scattering there can be central
exclusive production (CEP) of single photons by the fusion reactions
$\mathbb{P}+\rho_{\mathbb{R}}\to\gamma$ and
$f_{2\mathbb{R}}+\rho_{\mathbb{R}}\to\gamma$. In the terminology used in our
paper these would be called “anomalous” photon contributions even if their
origin is quite conventional. These photons, indeed, can be expected to
populate preferentially the mid-rapidity region; see Lebiedowicz:2013xlb .

Another main purpose of our paper was a study of the various soft-photon
approximations (SPAs). How close or far away are they from our standard
results? As expected, the SPAs are good approximations to the standard results
for low $k_{\perp}$ and low $\omega$. To be concrete: this means
$k_{\perp}\lesssim 10$ MeV and $\omega\lesssim 50$ MeV for $\sqrt{s}=10$ GeV
(see Fig. 10) and $k_{\perp}\lesssim 10$ MeV and $\omega\lesssim 0.5$ GeV for
$\sqrt{s}=100$ GeV (see Fig. 16). For larger values of $k_{\perp}$ and/or
$\omega$ the discrepancies between the standard and SPA results increase
rapidly. But these discrepancies also depend on the detailed kinematics. The
“improved SPA” approaches with the variable $\tilde{s}$, defined in (148), in
the $\pi\pi\to\pi\pi$ amplitudes greatly reduce the discrepancies to our
standard results, especially in the case of SPA1 (see Fig. 20). For these
numerical studies we have considered only the leading exchange at high
energies, the pomeron. This should be a very good approximation for
$\sqrt{s}=100$ GeV. For $\sqrt{s}=10$ GeV we have also considered the
subleading reggeon exchanges and we found that they increase the cross
sections for $\pi^{-}\pi^{+}\to\pi^{-}\pi^{+}\gamma$ by about 20$\,\%$ to
40$\,\%$.

As already mentioned in the Introduction there are plans for a new detector
for the LHC, ALICE 3. One physics aim for this new initiative is an
experimental study of soft-photon emission in hadronic reactions. What can we
say in this context from our investigation of $\pi\pi$ scattering without and
with photon radiation? From the theory side we have a good model for the basic
process $\pi\pi\to\pi\pi$. This allowed us to construct our standard amplitude
for $\pi\pi\to\pi\pi\gamma$ but we have excluded anomalous terms, as described
above. Suppose now that we have experimental measurements at all photon
energies $\omega$. Then we could study, as an example, the ratio

$\displaystyle R_{\rm exp}(\omega)=\frac{d\sigma_{\rm
exp}/d\omega}{d\sigma_{\rm standard}/d\omega}\,.$ (159)

From the results of our present paper we know that the terms of order
$1/\omega$ and $\omega^{0}$ in the expansion of the standard amplitude are
strict results from QFT without approximations, given the on-shell
$\pi\pi\to\pi\pi$ amplitudes. Therefore, if QFT describes experiment we must
have (see Appendix B for a detailed discussion)

$\displaystyle\lim_{\omega\to 0}R_{\rm exp}(\omega)=1\,,\quad\lim_{\omega\to
0}\frac{dR_{\rm exp}(\omega)}{d\omega}=0\,.$ (160)

A violation of these relations would mean a terrible crisis for QFT! For
higher $\omega$ a value $R_{\rm exp}(\omega)\neq 1$ would mean that there are
soft photons from “anomalous” terms (in the sense defined above) present in
experiment. From our point of view the origin of such “anomalous” terms should
be searched for in nonperturbative QCD. One will have to first consider
carefully all conventional sources of “anomalous” photons like photons from
central exclusive production reactions (see above) and then more
unconventional sources; see for instance Wong:2014pY  and Nachtmann:1983uz ;
Botz:1994bg ; Nachtmann:ELFE ; Nachtmann:Lectures ; Nachtmann:2014qta . Let us
note that for very small $\omega$ one has to take care of infrared divergences
and multiple soft photon emission. But these effects can be calculated with
the methods originally developed by Bloch and Nordsieck Bloch:1937pw .

What can we do if we do not have a good model for the amplitude of the basic
process, e.g. for multi-particle production? Typically one has then the
experimental or theoretical distributions of particles and one uses the analog
of our SPA1 approximation (140)–(143) instead of $d\sigma_{\rm
standard}/d\omega$ in (159):

$\displaystyle\tilde{R}_{\rm exp}(\omega)=\frac{d\sigma_{\rm
exp}/d\omega}{d\sigma_{\rm SPA1}/d\omega}\,.$ (161)

Then, the firm prediction from QFT is only

$\displaystyle\lim_{\omega\to 0}\tilde{R}_{\rm exp}(\omega)=1\,.$ (162)

Note that the ratios $R(\omega)$ for SPA1 shown in the right panels of Fig. 13
and Fig. 18 do not satisfy

$\displaystyle\lim_{\omega\to 0}\frac{dR(\omega)}{d\omega}=0\,,$ (163)

and this must be expected to be the case in general. If then $\tilde{R}_{\rm
exp}(\omega)$ turns out $\neq 1$ for larger $\omega$ the conclusions for
“anomalous” terms in the photon-emission process will not be so
straightforward, since it will depend on an estimate of the accuracy of the
SPA used. For our $\pi\pi$ scattering reaction these accuracies can be read
off, as function of the kinematic region considered, from the figures shown in
Sec. V. But, in general, such accuracy estimates are a difficult task.

In the future we plan to study proton-proton elastic scattering and central
exclusive production (CEP) reactions like $pp\to p\pi^{+}\pi^{-}p$ without and
with soft photon production using the methods which we have developed here for
the $\pi\pi$ scattering case. We hope that with the planned ALICE 3 detector
at the LHC our theoretical studies of soft photon emission in exclusive
reactions will find their experimental counterparts. The goals will be to
establish if QFT has a crisis there in the sense of a violation of relations
of the type (160) and if “anomalous” soft photons, compatible with QFT, are
present.

###### Acknowledgements.

We thank Johanna Stachel, Peter Braun-Munzinger, Carlo Ewerz, and Stefan
Flörchinger for very useful discussions and for providing us information on
relevant literature. We thank Charles Gale and Heinrich Leutwyler for
correspondence on the topics of our paper. This work is partially supported by
the Polish National Science Centre under Grant No. 2018/31/B/ST2/03537 and by
the Center for Innovation and Transfer of Natural Sciences and Engineering
Knowledge in Rzeszów (Poland).

##  Appendix A Some remarks on the literature concerning Low’s theorem

Here we compare our findings concerning the revision of the order $\omega^{0}$
term in the soft-photon expansion from Sec. III to results from a number of
papers from the literature. We find it surprising that so many versions of
“Low’s theorem” can be found in the literature. This, clearly, poses the
question if they are all equivalent. A genuine theorem of QFT should give a
unique result. We think that a clarification of this question is important
especially for experimentalist trying to check this theorem. They should know
precisely what they are supposed to check. In this spirit, as a service to
experimentalists, and in order to answer to various points raised by the
referee of our paper, we shall in the following compare results from the
literature to our findings.

For these comparisons we shall mainly restrict ourselves to the simple pion-
pion scattering reactions (2) and (3).

We start by recalling our Eqs. (63) and (64) which we specialise here for real
photons, $k^{2}=0$. We get then for $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$,
dropping gauge terms proportional to $k_{\lambda}$,

$\displaystyle{\cal M}_{\lambda}$ $\displaystyle=$ $\displaystyle
e\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot k)}-\frac{p_{1\lambda}}{(p_{1}\cdot
k)}\big{]}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})$ (164)
$\displaystyle+e\big{\\{}-\frac{1}{(p_{1}\cdot
k)^{2}}\big{[}p_{1\lambda}(l_{1}\cdot k)-l_{1\lambda}(p_{1}\cdot
k)\big{]}+2\big{[}-\frac{(p_{b}\cdot k)}{(p_{a}\cdot
k)}p_{a\lambda}+p_{b\lambda}\big{]}\frac{\partial}{\partial s_{L}}$
$\displaystyle-2\big{[}(p_{a}-p_{1},k)-(p_{a}\cdot
l_{1})\big{]}\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}\big{]}\frac{\partial}{\partial
t}\big{\\}}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})$
$\displaystyle+{\cal O}(\omega)\,.$

Now we have a look at Gribov’s paper Gribov:1966hs . As far as we can see the
emphasis there is on the question of determining the kinematic region of
validity of the $\omega^{-1}$ term in Low’s theorem. The $\omega^{0}$ term is
only mentioned in the context of the cancellation of the off-mass-shell
effects. This happens also in our calculation when adding the amplitudes
${\cal M}_{\lambda}^{(a)}$, ${\cal M}_{\lambda}^{(b)}$, and ${\cal
M}_{\lambda}^{(c)}$ in (63). The question where the $\omega^{-1}$ term gives a
reliable result is discussed in detail in our Sec. V where we also compare to
Gribov:1966hs .

Next we study Lipatov’s paper Lipatov:1988ii . From the many interesting
considerations presented in this paper we are only concerned with the form of
Low’s theorem given there for the photon case. From Eq. (11) of Lipatov:1988ii
we get for our process (3), using our notation,

$\displaystyle\left.{\cal M}_{\lambda}\right|_{\rm Lipatov}$ $\displaystyle=$
$\displaystyle e\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p^{\prime}_{1\lambda}}{(p^{\prime}_{1}\cdot k)}\big{]}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})$ (165)
$\displaystyle-e(p_{a}-p_{1},k)\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}\big{]}\frac{\partial}{\partial t}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\,.\qquad$

This result clearly is different from our Eq. (164). There is in (165) no term
$\partial{\cal M}^{(0)}/\partial s_{L}$ and the term proportional to
$\partial{\cal M}^{(0)}/\partial t$ is different from our result.

Concerning the paper DelDuca:1990gz  we see no overlap and, therefore, no
conflict with our results. In DelDuca:1990gz  hard processes with photon
emission are considered. Let $Q$ be the scale of a hard process. Then,
including photon radiation, the region of $\omega$ mainly discussed in
DelDuca:1990gz  is $m^{2}/Q\leq\omega\lesssim m$, with $m$ being some mass
scale. But we are considering a soft process and we are interested in the
strict limit $\omega\to 0$.

Concerning the papers which we want to discuss next we would like to make a
general remark. In many papers the results for the soft-photon expansion of
the amplitude, say for $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$, contains
derivatives with respect to the momenta of the basic amplitude, here for
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$. Let us consider as in (9) the, in general
off-shell, amplitude

$\displaystyle\widetilde{\cal T}(p_{a},p_{b},p_{1},p_{2})=\left.{\cal
T}(p_{a},p_{b},p_{1},p_{2})\right|_{\rm off\;shell\;or\;on\;shell}.$ (166)

In order to calculate derivatives like $\partial\widetilde{\cal T}/\partial
p_{a}^{\mu}$ we have to consider

$\displaystyle\widetilde{\cal T}(p_{a}+\delta p_{a},p_{b},p_{1},p_{2})={\cal
T}(p_{a},p_{b},p_{1},p_{2})+\delta p_{a}^{\mu}\,\frac{\partial\widetilde{\cal
T}}{\partial p_{a}^{\mu}}(p_{a},p_{b},p_{1},p_{2})+{\cal O}((\delta
p_{a})^{2})\,.\qquad$ (167)

But clearly, with (167) we have to go outside the physical region for
$\widetilde{\cal T}$ which requires always $p_{a}+p_{b}-p_{1}-p_{2}=0$. Thus,
it is our opinion that all expressions for Low’s theorem which contain
derivatives like $\partial\widetilde{\cal T}/\partial p_{a}^{\mu}$ etc. have
to be considered as potentially problematic.

Keeping this in mind we shall now discuss the aspects relevant for Low’s
theorem in the paper by H. Gervais Gervais:2017yxv . In Gervais:2017yxv  the
reactions considered are fermion-scalar ($f$-$s$) elastic scattering and the
corresponding photon-emission process. In our notation we have then

$\displaystyle f(p_{a})+s(p_{b})\to f(p_{1})+s(p_{2})\,,$ (168) $\displaystyle
f(p_{a})+s(p_{b})\to
f(p_{1}^{\prime})+s(p_{2}^{\prime})+\gamma(k,\epsilon)\,.$ (169)

The masses of $f$ and $s$ are $m_{f}$ and $m_{s}$. Now it is correctly stated
in Gervais:2017yxv  that the momenta of $f$ and $s$ cannot all stay fixed when
going from (168) to (169); see Eq. (9) of Gervais:2017yxv . The four variables
$\xi_{i}$, $\eta_{i}$ $(i=1,2)$ introduced there correspond to our $l_{1,2}$
variables; see (39). We have only two such variables since we keep $p_{a}$ and
$p_{b}$ the same in (168) and (169) which is, of course, legitimate. Let
$\widetilde{\cal T}(p_{a},p_{b},p_{1},p_{2})$ be the elastic amplitude,
stripped from the spinors, and, in general, for off-shell particles. The
relevant formula giving the terms of order $\omega^{-1}$ and $\omega^{0}$ for
the amplitude from (169) is then Eq. (20) of Gervais:2017yxv . There, the
following expressions, using our notation, occur:

$\displaystyle I_{1}=\big{[}-l_{1}^{\alpha}\frac{\partial}{\partial
p_{1}^{\alpha}}-l_{2}^{\alpha}\frac{\partial}{\partial
p_{2}^{\alpha}}-k^{\alpha}\frac{\partial}{\partial
p_{a}^{\alpha}}\big{]}\widetilde{\cal T}(p_{a},p_{b},p_{1},p_{2})\,,$
$\displaystyle I_{2}=\big{[}-l_{1}^{\alpha}\frac{\partial}{\partial
p_{1}^{\alpha}}-l_{2}^{\alpha}\frac{\partial}{\partial
p_{2}^{\alpha}}+k^{\alpha}\frac{\partial}{\partial
p_{1}^{\alpha}}\big{]}\widetilde{\cal T}(p_{a},p_{b},p_{1},p_{2})\,,$
$\displaystyle I_{3\mu}=\frac{\partial}{\partial p_{a\mu}}\widetilde{\cal
T}(p_{a},p_{b},p_{1},p_{2})\,,$ $\displaystyle
I_{4\mu}=\frac{\partial}{\partial p_{1\mu}}\widetilde{\cal
T}(p_{a},p_{b},p_{1},p_{2})\,.$ (170)

We shall now choose a simple trial function for $\widetilde{\cal T}$:

$\displaystyle\widetilde{\cal
T}(p_{a},p_{b},p_{1},p_{2})=h\big{[}p_{a}^{2}+p_{1}^{2}-p_{b}^{2}-p_{2}^{2}\big{]}\,.$
(171)

On shell we have

$\displaystyle\widetilde{\cal T}(p_{a},p_{b},p_{1},p_{2})_{\rm on\;shell}$
$\displaystyle=$ $\displaystyle h\big{[}2m_{f}^{2}-2m_{s}^{2}\big{]}$ (172)
$\displaystyle=$ $\displaystyle{\rm const}\,.$

We shall assume that $h(2m_{f}^{2}-2m_{s}^{2})\neq 0$ and that also the
derivative

$\displaystyle h^{\prime}\big{[}2m_{f}^{2}-2m_{s}^{2}\big{]}\neq 0\,,$ (173)

but otherwise arbitrary. Evaluating $I_{1},\ldots,I_{4}$ from (170) we get,
using (58),

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle[-(l_{1}\cdot
p_{1})+(l_{2}\cdot p_{2})-(k\cdot p_{a})]\,2h^{\prime}(2m_{f}^{2}-2m_{s}^{2})$
$\displaystyle=$ $\displaystyle-(k\cdot
p_{a})2h^{\prime}(2m_{f}^{2}-2m_{s}^{2})\,,$ $\displaystyle I_{2}$
$\displaystyle=$ $\displaystyle(k\cdot
p_{1})2h^{\prime}(2m_{f}^{2}-2m_{s}^{2})\,,$ $\displaystyle I_{3\mu}$
$\displaystyle=$ $\displaystyle
2p_{a\mu}\,h^{\prime}(2m_{f}^{2}-2m_{s}^{2})\,,$ $\displaystyle I_{4\mu}$
$\displaystyle=$ $\displaystyle
2p_{1\mu}\,h^{\prime}(2m_{f}^{2}-2m_{s}^{2})\,.$ (174)

We conclude, that the expression (20) of Gervais:2017yxv  which is supposed to
give Low’s theorem up to order $\omega^{0}$ contains, in our simple example,
the arbitrary quantity $h^{\prime}(2m_{f}^{2}-2m_{s}^{2})$. Thus, in our
opinion, this equation has a problem. On the other hand, inserting
$\widetilde{\cal T}$ from (171) in our Eq. (164) we get a sensible result.
Here, the on shell $\widetilde{\cal T}$ equals the constant (172) and on the
right-hand side (r.h.s.) of (164) only the terms proportional to ${\cal
M}^{(0)}$ survive, $\partial{\cal M}^{(0)}/\partial s_{L}$ and $\partial{\cal
M}^{(0)}/\partial t$ being zero.

Finally we want to discuss the form of Low’s theorem presented in Eqs. (2.8),
(2.9) of Bern:2014vva  and (25), (26) of Lysov:2014csa . For our reactions (2)
and (3) these give for ${\cal M}_{\lambda}$ from our Eq. (14)

$\displaystyle{\cal M}_{\lambda}$ $\displaystyle=$
$\displaystyle\big{\\{}\frac{e}{(p_{a}\cdot
k)}\big{(}p_{a\lambda}-i\eta_{a}k^{\nu}J_{a\lambda\nu}\big{)}-\frac{e}{(p_{1}\cdot
k)}\big{(}p_{1\lambda}-i\eta_{1}k^{\nu}J_{1\lambda\nu}\big{)}\big{\\}}\widetilde{\cal
T}(p_{a},p_{b},p_{1},p_{2})$ (175) $\displaystyle+{\cal O}(\omega)\,.$

Here

$\displaystyle J_{a\lambda\nu}=i\big{(}p_{a\lambda}\frac{\partial}{\partial
p_{a}^{\nu}}-p_{a\nu}\frac{\partial}{\partial p_{a}^{\lambda}}\big{)}\,,$
$\displaystyle J_{1\lambda\nu}=i\big{(}p_{1\lambda}\frac{\partial}{\partial
p_{1}^{\nu}}-p_{1\nu}\frac{\partial}{\partial p_{1}^{\lambda}}\big{)}\,,$
(176)

and we have inserted factors $\eta_{a}=\pm 1$ and $\eta_{1}=\pm 1$ in (175)
because we could not always find out the precise momentum orientations used in
Bern:2014vva  and Lysov:2014csa . But this will play no role in the following.

Now we shall use as trial function in (175)

$\displaystyle\widetilde{\cal
T}(p_{a},p_{b},p_{1},p_{2})=\tilde{h}\big{[}(p_{a}+p_{b})^{2}-(p_{1}+p_{2})^{2}\big{]}\,,$
(177)

where $\tilde{h}$ is a function satisfying

$\displaystyle\tilde{h}(0)\neq 0\quad{\rm and}\quad\tilde{h}^{\prime}(0)\neq
0$ (178)

but otherwise arbitrary. In the physical region we have, on shell and off
shell, $p_{a}+p_{b}=p_{1}+p_{2}$, and therefore our ${\cal M}^{(0)}$ from (9)
is given by

$\displaystyle{\cal M}^{(0)}=\tilde{h}(0)={\rm const}\,.$ (179)

Thus, from (164) we get our result as

$\displaystyle{\cal M}_{\lambda}=e\big{\\{}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}-\frac{1}{(p_{1}\cdot
k)^{2}}\big{[}p_{1\lambda}(l_{1}\cdot k)-l_{1\lambda}(p_{1}\cdot
k)\big{]}\big{\\}}\,\tilde{h}(0)+{\cal O}(\omega)\,.\qquad$ (180)

On the other hand, from (175) we find

$\displaystyle{\cal M}_{\lambda}$ $\displaystyle=$ $\displaystyle
e\big{\\{}\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}\big{]}\,\tilde{h}(0)$ (181)
$\displaystyle+\big{[}\eta_{a}\big{(}p_{a\lambda}\frac{(p_{b}\cdot
k)}{(p_{a}\cdot
k)}-p_{b\lambda}\big{)}+\eta_{1}\big{(}p_{1\lambda}\frac{(p_{2}\cdot
k)}{(p_{1}\cdot
k)}-p_{2\lambda}\big{)}\big{]}2\tilde{h}^{\prime}(0)\big{\\}}+{\cal
O}(\omega)\,.$

Clearly, the results (180) and (181) differ. In (181) we also see the
completely arbitrary quantity $\tilde{h}^{\prime}(0)$ occurring. That is, at
least for this example (164) and (175) are not equivalent.

The reader may wonder if our trial function (177) is reasonable since in
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ we have the same particles in the initial
and the final state. Should there be some symmetry requirement for
$\tilde{h}(\cdot)$? We can counter such an argumentation by considering
instead of $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ the reaction $\pi^{-}\pi^{0}\to
K^{-}K^{0}$ where there is no symmetry between the initial and the final
state. The results (180) and (181) stay the same.

With this we close our remarks on some papers from the literature.

##  Appendix B The cross section $d\sigma/d\omega$ for $\omega\to 0$

In this appendix we shall discuss the cross section $d\sigma/d\omega$ for
$\omega\to 0$ for the reaction $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$ for
real photon emission. The results for charged-pion scattering are analogous.

We consider, thus, the reaction (3) where the scattering amplitude is defined
in (14). The cross section is given as in (139). We work in the overall c.m.
system, setting $k^{0}\equiv\omega$ and

$\displaystyle{\cal M}_{\lambda}\equiv{\cal
M}_{\lambda}^{(\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma)}\,.$ (182)
$\displaystyle d\sigma({\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma(k)})$
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s(s-4m_{\pi}^{2})}}\,\frac{d^{3}k}{(2\pi)^{3}\,2k^{0}}\int\frac{d^{3}p_{1}^{\prime}}{(2\pi)^{3}\,2p_{1}^{\prime
0}}\frac{d^{3}p_{2}^{\prime}}{(2\pi)^{3}\,2p_{2}^{\prime 0}}$ (183)
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{1}^{\prime}+p_{2}^{\prime}+k-p_{a}-p_{b})(-1){\cal
M}_{\lambda}({\cal M}^{\lambda})^{*}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{s(s-4m_{\pi}^{2})}}\frac{1}{2^{4}(2\pi)^{5}}\,\omega\,d\omega\,d\Omega_{\hat{k}}\int
d\Omega_{\hat{p}_{1}^{\prime}}\int_{0}^{\infty}d|\mbox{\boldmath$p_{1}^{\prime}$}|\frac{|\mbox{\boldmath$p_{1}^{\prime}$}|^{2}}{p_{1}^{\prime
0}p_{2}^{\prime 0}}$ $\displaystyle\times\delta(p_{1}^{\prime 0}+p_{2}^{\prime
0}+\omega-\sqrt{s})(-{\cal M}_{\lambda}{\cal M}^{\lambda*})\,.$

Here we have

$\displaystyle p_{1}^{\prime 0}$ $\displaystyle=$
$\displaystyle\sqrt{|\mbox{\boldmath$p_{1}^{\prime}$}|^{2}+m_{\pi}^{2}}\,,$
$p_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-\mbox{\boldmath$p_{1}^{\prime}$}-\mbox{\boldmath$k$}\,,$
$\displaystyle p_{2}^{\prime 0}$ $\displaystyle=$
$\displaystyle\sqrt{|\mbox{\boldmath$p_{2}^{\prime}$}|^{2}+m_{\pi}^{2}}$ (184)
$\displaystyle=$
$\displaystyle\sqrt{|\mbox{\boldmath$p_{1}^{\prime}$}|^{2}+2|\mbox{\boldmath$p_{1}^{\prime}$}|\omega(\mbox{\boldmath$\hat{p}_{1}^{\prime}$}\cdot\mbox{\boldmath$\hat{k}$})+\omega^{2}+m_{\pi}^{2}}\,,$

where
$\mbox{\boldmath$\hat{p}_{1}^{\prime}$}=\mbox{\boldmath$p_{1}^{\prime}$}/|\mbox{\boldmath$p_{1}^{\prime}$}|$
and
$\mbox{\boldmath$\hat{k}$}=\mbox{\boldmath$k$}/|\mbox{\boldmath$k$}|=\mbox{\boldmath$k$}/\omega$.
The $\delta$ function of the energies in (183) requires

$\displaystyle p_{1}^{\prime 0}+p_{2}^{\prime 0}=\sqrt{s}-\omega$ (185)

which allows us to calculate $|\mbox{\boldmath$p_{1}^{\prime}$}|$ for given
$\sqrt{s}$, $\omega$, $\hat{p}_{1}^{\prime}$ and $\hat{k}$. We get as solution

$\displaystyle|\mbox{\boldmath$p_{1}^{\prime}$}|$ $\displaystyle=$
$\displaystyle\sqrt{\frac{s(\sqrt{s}-2\omega)^{2}-4(\sqrt{s}-\omega)^{2}m_{\pi}^{2}}{4\left[(\sqrt{s}-\omega)^{2}-\omega^{2}(\mbox{\boldmath$\hat{p}_{1}^{\prime}$}\cdot\mbox{\boldmath$\hat{k}$})^{2}\right]}+\frac{1}{4}\left[\frac{\sqrt{s}(\sqrt{s}-2\omega)\omega(\mbox{\boldmath$\hat{p}_{1}^{\prime}$}\cdot\mbox{\boldmath$\hat{k}$})}{(\sqrt{s}-\omega)^{2}-\omega^{2}(\mbox{\boldmath$\hat{p}_{1}^{\prime}$}\cdot\mbox{\boldmath$\hat{k}$})^{2}}\right]^{2}}$
(186)
$\displaystyle-\frac{1}{2}\frac{\sqrt{s}(\sqrt{s}-2\omega)\omega(\mbox{\boldmath$\hat{p}_{1}^{\prime}$}\cdot\mbox{\boldmath$\hat{k}$})}{(\sqrt{s}-\omega)^{2}-\omega^{2}(\mbox{\boldmath$\hat{p}_{1}^{\prime}$}\cdot\mbox{\boldmath$\hat{k}$})^{2}}\,.$

For $\omega=0$ this gives, as it must be,

$\displaystyle\left.|\mbox{\boldmath$p_{1}^{\prime}$}|\right|_{\omega=0}=\frac{1}{2}\sqrt{s-4m_{\pi}^{2}}\,.$
(187)

Now we define the phase space function

$\displaystyle{\rm
J}(s,\omega,\mbox{\boldmath$\hat{p}_{1}^{\prime}$},\mbox{\boldmath$\hat{k}$})$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}d|\mbox{\boldmath$p_{1}^{\prime}$}|\frac{|\mbox{\boldmath$p_{1}^{\prime}$}|^{2}}{p_{1}^{\prime
0}p_{2}^{\prime 0}}\,\delta(p_{1}^{\prime 0}+p_{2}^{\prime
0}+\omega-\sqrt{s})$ (188) $\displaystyle=$
$\displaystyle\frac{|\mbox{\boldmath$p_{1}^{\prime}$}|^{2}}{|\mbox{\boldmath$p_{1}^{\prime}$}|p_{2}^{\prime
0}+p_{1}^{\prime
0}\left[|\mbox{\boldmath$p_{1}^{\prime}$}|+\omega(\mbox{\boldmath$\hat{p}_{1}^{\prime}$}\cdot\mbox{\boldmath$\hat{k}$})\right]}\,.$

Here $p_{1}^{\prime 0}$ and $p_{2}^{\prime 0}$ have to be substituted
according to (184) and finally everywhere $|\mbox{\boldmath$p_{1}^{\prime}$}|$
from (186) has to be inserted. For $\omega=0$ we get

$\displaystyle{\rm
J}(s,0,\mbox{\boldmath$\hat{p}_{1}^{\prime}$},\mbox{\boldmath$\hat{k}$})=\frac{1}{2}\sqrt{1-\frac{4m_{\pi}^{2}}{s}}\,.$
(189)

Collecting everything together we find

$\displaystyle\frac{d\sigma}{d\omega}(\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma)=\frac{1}{\sqrt{s(s-4m_{\pi}^{2})}}\frac{1}{2^{4}(2\pi)^{5}}\,\omega\int
d\Omega_{\hat{k}}\,d\Omega_{\hat{p}_{1}^{\prime}}\,{\rm
J}(s,\omega,\mbox{\boldmath$\hat{p}_{1}^{\prime}$},\mbox{\boldmath$\hat{k}$})(-{\cal
M}_{\lambda}{\cal M}^{\lambda*}).$ (190)

We are interested in the behaviour of $d\sigma/d\omega$ for $\omega\to 0$.
Therefore, we shall now consider the expansion of ${\cal M}_{\lambda}$ in
powers of $\omega$ as given in (164). We have used in (190) $\hat{k}$ and
$\hat{p}_{1}^{\prime}$ as phase-space variables. Therefore, we should choose
the expansion of ${\cal M}_{\lambda}$ keeping $\hat{p}_{1}^{\prime}$ fixed,
independent of $\omega$:

$\displaystyle\mbox{\boldmath$\hat{p}_{1}^{\prime}$}=\mbox{\boldmath$\hat{p}_{1}$}\,.$
(191)

This means that we choose in the expansion (164)

$\displaystyle\mbox{\boldmath$l_{1\perp}$}=0\,;$ (192)

see (57). Then $k^{\mu}$, $l_{1}^{\mu}$ and $l_{2}^{\mu}$ are, for fixed
$\hat{k}$, strictly proportional to $\omega$. Therefore, we can write from
(164):

$\displaystyle{\cal M}_{\lambda}=\frac{1}{\omega}\widehat{{\cal
M}}_{\lambda}^{(0)}+\widehat{{\cal M}}_{\lambda}^{(1)}+{\cal
O}(\omega^{2})\,,$ (193)

where

$\displaystyle\widehat{{\cal M}}_{\lambda}^{(0)}$ $\displaystyle=$
$\displaystyle\widehat{{\cal
M}}_{\lambda}^{(0)}(s,\mbox{\boldmath$\hat{p}_{a}$},\mbox{\boldmath$\hat{p}_{1}$},\mbox{\boldmath$\hat{k}$})$
(194) $\displaystyle=$ $\displaystyle
e\omega\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}\big{]}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\,,$
$\displaystyle\widehat{{\cal M}}_{\lambda}^{(1)}$ $\displaystyle=$
$\displaystyle\widehat{{\cal
M}}_{\lambda}^{(1)}(s,\mbox{\boldmath$\hat{p}_{a}$},\mbox{\boldmath$\hat{p}_{1}$},\mbox{\boldmath$\hat{k}$})$
$\displaystyle=$ $\displaystyle e\big{\\{}-\frac{1}{(p_{1}\cdot
k)^{2}}\big{[}p_{1\lambda}(l_{1}\cdot k)-l_{1\lambda}(p_{1}\cdot
k)\big{]}+2\big{[}-\frac{(p_{b}\cdot k)}{(p_{a}\cdot
k)}p_{a\lambda}+p_{b\lambda}\big{]}\frac{\partial}{\partial s_{L}}$
$\displaystyle-2\big{[}(p_{a}-p_{1},k)-(p_{a}\cdot
l_{1})\big{]}\big{[}\frac{p_{a\lambda}}{(p_{a}\cdot
k)}-\frac{p_{1\lambda}}{(p_{1}\cdot k)}\big{]}\frac{\partial}{\partial
t}\big{\\}}{\cal
M}^{(0)}(s_{L},t,m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2},m_{\pi}^{2})\,.$

Inserting (193) in (190) we find

$\displaystyle\frac{d\sigma}{d\omega}(\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma)=\frac{1}{\omega}\big{[}{\cal
A}^{(0)}(s,\omega)+\omega{\cal A}^{(1)}(s,\omega)+{\cal
O}(\omega^{2})\big{]}\,,$ (196)

where we have with (191)

$\displaystyle{\cal
A}^{(0)}(s,\omega)=\frac{1}{\sqrt{s(s-4m_{\pi}^{2})}}\frac{1}{2^{4}(2\pi)^{5}}\int
d\Omega_{\hat{k}}\,d\Omega_{\hat{p}_{1}}\,{\rm
J}(s,\omega,\mbox{\boldmath$\hat{p}_{1}$},\mbox{\boldmath$\hat{k}$})(-\widehat{{\cal
M}}^{(0)}_{\lambda}\widehat{{\cal M}}^{(0)\lambda*})\,,$ (197)
$\displaystyle{\cal
A}^{(1)}(s,\omega)=\frac{1}{\sqrt{s(s-4m_{\pi}^{2})}}\frac{1}{2^{4}(2\pi)^{5}}\int
d\Omega_{\hat{k}}\,d\Omega_{\hat{p}_{1}}\,{\rm
J}(s,\omega,\mbox{\boldmath$\hat{p}_{1}$},\mbox{\boldmath$\hat{k}$})$
$\displaystyle\qquad\qquad\qquad\times\big{[}-\widehat{{\cal
M}}^{(1)}_{\lambda}(\widehat{{\cal M}}^{(0)\lambda})^{*}-\widehat{{\cal
M}}^{(0)}_{\lambda}(\widehat{{\cal M}}^{(1)\lambda})^{*}\big{]}\,.$ (198)

We clearly have from (197)

$\displaystyle{\cal A}^{(0)}(s,\omega)\neq 0\,.$ (199)

Furthermore, ${\cal A}^{(0)}(s,\omega)$ and ${\cal A}^{(1)}(s,\omega)$ are
continuous functions of $\omega$ in the region of interest for us.

Given the amplitude ${\cal M}^{(0)}$ for the basic process
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ the result for $d\sigma/d\omega$ in (196) is
a strict consequence of QFT. Thus, the experimental cross section should also
be of the form

$\displaystyle\left.\frac{d\sigma}{d\omega}(\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma)\right|_{\rm
exp}=\frac{1}{\omega}\big{[}{\cal A}^{(0)}(s,\omega)+\omega{\cal
A}^{(1)}(s,\omega)+{\cal O}(\omega^{2})\big{]}\,.$ (200)

Therefore, if we have a good “standard” representation of the basic
$\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}$ process, compatible with experiment, and if
the corresponding calculation of $d\sigma/d\omega$ respects the rules of QFT,
in particular the relation (164), this standard result must also give the
expansion (196) for $\omega\to 0$ and we shall then have

$\displaystyle R_{\rm exp}(\omega)=\frac{d\sigma_{\rm
exp}/d\omega}{d\sigma_{\rm standard}/d\omega}=1+{\cal O}(\omega^{2})\,.$ (201)

Equation (201) should, in particular, be true for our standard result as
discussed in Sec. IV.1 for $\pi^{-}\pi^{0}\to\pi^{-}\pi^{0}\gamma$ and in Sec.
IV.2 for charged-pion scattering. Of course, there we must stay in the