# Soft-photon radiation in high-energy proton-proton collisions
within the tensor-Pomeron approach: Bremsstrahlung

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 diffractive processes in proton-proton collisions at small momentum
transfers without and with photon radiation. We consider the soft exclusive
reactions $pp\to pp$, $p\bar{p}\to p\bar{p}$, and $pp\to pp\gamma$ within the
tensor-pomeron and vector-odderon approach. We compare our results with the
data for $pp$ and $p\bar{p}$ total cross sections, for the ratio of real to
imaginary part of the forward scattering amplitude, and for the elastic $pp$
cross sections, especially those from TOTEM. To describe the low-energy data
more accurately the secondary reggeons must be included. We write down the
amplitudes for the photon bremsstrahlung in high-energy proton-proton
collisions using the tensor-pomeron model. These results are relevant for the
c.m. energies presently available at the Relativistic Heavy Ion Collider and
at the LHC. We present predictions for the proposed measurements of soft
photons with the planned future upgrade of the ALICE experiment at the LHC. We
investigate the limits of applicability of the soft-photon approximation (SPA)
based on Low’s theorem. The corresponding SPA results are compared to those
obtained from our complete model. The regions of phase space are given
quantitatively where SPA and our complete tensor-pomeron results are close to
each other. As an example, let $k_{\perp}$, $\rm y$, and $\omega$, be the
absolute value of the transverse momentum, the rapidity, and the energy of the
photon, respectively, in the overall c.m. system. For the region $1\;{\rm
MeV}<k_{\perp}<100\;{\rm MeV}$ and $3.5<|\rm y|<5.0$, we find that the SPA
Ansatz with only the pole terms $\propto\omega^{-1}$ agrees at the percent
level with our complete model result up to $\omega\cong 2$ GeV.

##  I Introduction

With this article, we continue our investigations of soft-photon radiation in
diffractive hadronic high-energy reactions. In Ref. Lebiedowicz:2021byo , we
treat, as a first example using our approach, high-energy $\pi\pi$ scattering
without and with photon radiation. In the present paper, we extend these
considerations to $pp$ and $p\bar{p}$ scattering.

The emission of soft photons, that is, photons of energy $\omega$ approaching
zero, was treated in the seminal paper by Low Low:1958sn . In that paper, it
was shown that the term of order $\omega^{-1}$ in the amplitude for the
emission reaction can be obtained from the amplitude without photon emission.
To this order, the emission comes exclusively from the external particles, and
this is a strict consequence of Quantum Field Theory (QFT). Many soft-photon
approximations (SPAs) are based on this result.

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 ; DELPHI:2010cit  have, in many cases, found
large deviations from the SPA calculations. For a review of the experimental
and theoretical situations, see Ref. Wong:2014pY . Clearly, more experimental
and theoretical work is needed in order to clarify this so-called soft-photon
problem.

From the experimental side there is, for instance, the plan for a new
multipurpose detector at the LHC, ALICE 3 Adamova:2019vkf . One physics aim
for this new initiative is a measurement of ultrasoft photons at very low
transverse momentum in $pp$, $pA$ and $AA$ collisions; see, e.g., Refs.
QM2022_PBM ; EMMI_RRTF .

From the theoretical side, many authors have studied soft-photon production
following Ref. Low:1958sn ; see, for instance, Refs. Gribov:1966hs ;
Burnett:1967km ; Bell:1969yw ; Lipatov:1988ii ; DelDuca:1990gz ;
Gervais:2017yxv ; Bern:2014vva ; Lysov:2014csa ; Bonocore:2021cbv . In our
paper Lebiedowicz:2021byo , we have presented two types of soft-photon
studies.

  * (1)

We have studied the amplitude for the reaction $\pi\pi\to\pi\pi\gamma$ in the
limit of the photon c.m. energy $\omega$ going to zero. Using only rigorous
QFT methods, we have calculated the terms of order $\omega^{-1}$ and
$\omega^{0}$. We found agreement with the result of Low Low:1958sn  for the
$\omega^{-1}$ term, but we disagreed with the $\omega^{0}$ term of Ref.
Low:1958sn . We have analyzed the origin of this disagreement, and we give a
critique of the corresponding results of the papers Low:1958sn ;
Burnett:1967km ; Lipatov:1988ii ; DelDuca:1990gz ; Gervais:2017yxv ;
Bern:2014vva ; Lysov:2014csa  in Appendixes A and B of Ref.
Lebiedowicz:2021byo .

  * (2)

We have discussed the reactions $\pi\pi\to\pi\pi$ and $\pi\pi\to\pi\pi\gamma$
at high energies in a specific model, the tensor-pomeron model of Ref.
Ewerz:2013kda . The “standard” results obtained in this approach were then
compared to various soft-photon approximations.

With the present paper, we continue our line (2) of research. Further results
from the line (1) will be presented elsewhere.

One class of hadronic reactions one can study at the LHC is exclusive
diffractive proton-proton collisions. Examples are $pp\to pp$ elastic
scattering and central exclusive production of mesons, for instance, $pp\to
pp\pi^{+}\pi^{-}$. In this paper, we present the results of our investigations
of the following soft reactions at small momentum transfer:

$\displaystyle p+p\to p+p,$ $\displaystyle p+\bar{p}\to p+\bar{p},$
$\displaystyle p+p\to p+p+\gamma\,.$ (1)

We shall work within the tensor-pomeron model as proposed in Ref.
Ewerz:2013kda  for soft hadronic high-energy reactions. There, the soft
pomeron and the charge conjugation $C=+1$ reggeons are described as effective
rank-2 symmetric tensor exchanges, and the odderon and the $C=-1$ reggeons are
described as effective vector exchanges. Now, our task is to construct a soft-
photon diffractive amplitude of the $pp\to pp\gamma$ reaction which satisfies
all theoretical constraints.

Before coming to our present investigations, we make remarks on some related
works. Exclusive diffractive photon bremsstrahlung in $pp$ collisions was
discussed earlier in Refs. Khoze:2010jv ; Lebiedowicz:2013xlb ; Khoze:2017igg
within other approaches. The bremsstrahlung-type emission of $\omega$ and
$\pi^{0}$ mesons was calculated in Refs. Cisek:2011vt ; Lebiedowicz:2013vya .
It is also worth noting that the $pp\to pp\gamma$ reaction has not yet been
measured at high energies; however, feasibility studies were performed for
RHIC energies Chwastowski:2015mua  and for LHC energies Chwastowski:2016jkl ;
Chwastowski:2016zzl .

The theoretical methods which we shall develop in the present paper for soft-
photon production can also be used in a completely different context: for the
production of “dark photons“. Indeed, an interesting proposal of new physics
search was discussed recently in Ref. Foroughi-Abari:2021zbm : to study the
forward production of dark vectors (photons) and scalars via bremsstrahlung in
proton-proton collisions with the proposed Forward Physics Facility (FPF) at
the High-Luminosity LHC Feng:2022inv . At the LHC, such weakly coupled long-
lived particles, with masses $m\sim 10$ MeV–1 GeV, could be produced through
light meson decays and bremsstrahlung in the region that would be covered by
the FPF.

Our present paper is organized as follows. In Sec. II, we discuss the
amplitudes for the reactions listed in (1) within the tensor-pomeron approach.
In Sec. III, we describe two SPAs based on Low’s theorem. The results of our
calculations are presented in Sec. IV. Section IV.1 is devoted to a comparison
of the model results to the available data on the total and elastic $pp$ and
$p\bar{p}$ cross sections. In Sec. IV.2, we present our “exact” model or
“standard” results for the $pp\to pp\gamma$ reaction and a comparison to SPAs.
Subsection IV.3 contains comments on the photon radiation in connection with
diffractive excitation of the proton. Section V contains a summary and our
conclusions. Some details of the present model are given in Appendixes A and
B.

Throughout our paper, we use the metric and $\gamma$-matrix conventions of
Ref. Bjorken:1965 .

##  II Reactions $pp\to pp$, $p\bar{p}\to p\bar{p}$, and $pp\to pp\gamma$

Here we discuss the reactions

$\displaystyle p(p_{a},\lambda_{a})+p(p_{b},\lambda_{b})\to
p(p_{1},\lambda_{1})+p(p_{2},\lambda_{2})\,,$ (2) $\displaystyle
p(p_{a},\lambda_{a})+\bar{p}(p_{b},\lambda_{b})\to
p(p_{1},\lambda_{1})+\bar{p}(p_{2},\lambda_{2})\,,$ (3)

and

$\displaystyle p(p_{a},\lambda_{a})+p(p_{b},\lambda_{b})\to
p(p_{1}^{\prime},\lambda_{1})+p(p_{2}^{\prime},\lambda_{2})+\gamma(k,\epsilon)\,.$
(4)

The momenta are denoted by $p_{a},\ldots,k$; the helicities of the protons are
denoted by $\lambda_{a},\ldots,\lambda_{2}$; and $\epsilon$ is the
polarization vector of the photon. The energy-momentum conservation in (2),
(3), and (4) requires

$\displaystyle p_{a}+p_{b}=p_{1}+p_{2}\,,$ (5) $\displaystyle
p_{a}+p_{b}=p_{1}^{\prime}+p_{2}^{\prime}+k\,.$ (6)

We consider soft hadronic high-energy reactions. We use standard formulas of
the tensor-pomeron and vector-odderon model from Ref. Ewerz:2013kda . In this
model, the assumption is made that the pomeron $\mathbb{P}$ and the charge-
conjugation $C=+1$ reggeons $f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$ couple to
hadrons like symmetric tensors of rank 2, and the odderon $\mathbb{O}$ and the
$C=-1$ reggeons $\omega_{\mathbb{R}}$, $\rho_{\mathbb{R}}$ couple to hadrons
like vectors. We do not treat $\gamma$ exchange in the following.

In Ref. Ewerz:2016onn , it is 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 as shown in Ref. Britzger:2019lvc .

###  II.1 Reaction $pp\to pp$

Figure 1: The diagrams for $pp\to pp$ elastic scattering with $C=+1$ (left)
and $C=-1$ (right) exchanges. There are also the diagrams corresponding to the
exchange $p_{1}\leftrightarrow p_{2}$.

We consider first the reaction (2) for off-shell protons. Also, here, energy-
momentum conservation (5) holds. We have the diagrams of Fig. 1 with $C=+1$
and $C=-1$ exchanges. The kinematic variables are

$\displaystyle s=(p_{a}+p_{b})^{2}=(p_{1}+p_{2})^{2}\,,$ $\displaystyle
t=(p_{a}-p_{1})^{2}=(p_{b}-p_{2})^{2}\,,$ $\displaystyle
u=(p_{a}-p_{2})^{2}=(p_{b}-p_{1})^{2}\,,$ $\displaystyle
m_{a}^{2}=p_{a}^{2}\,,m_{b}^{2}=p_{b}^{2}\,,m_{1}^{2}=p_{1}^{2}\,,m_{2}^{2}=p_{2}^{2}\,.$
(7)

The interchange $p_{1}\leftrightarrow p_{2}$ implies $t\leftrightarrow u$,
where $u=-s-t+m_{a}^{2}+m_{b}^{2}+m_{1}^{2}+m_{2}^{2}$. We are interested in
the kinematic region

$\displaystyle\sqrt{s}\gg m_{p}\,,\quad\sqrt{|t|}\lesssim m_{p}\,,\quad
s\gg|m_{a}^{2}|,\,|m_{b}^{2}|,\,|m_{1}^{2}|,\,|m_{2}^{2}|.$ (8)

There we can neglect the diagrams with $p_{1}\leftrightarrow p_{2}$.

We denote the off-shell $pp$ scattering amplitude by

$\displaystyle{\cal M}^{(0)}(p_{a},p_{b},p_{1},p_{2})\,.$ (9)

In the tensor-pomeron model Ewerz:2013kda , all exchanges at high energies are
assumed to be describable by effective single Regge poles. Some standard
references to Regge theory are Collins:1977 ; Caneschi ; Gribov:2009 . For
further literature, we refer to Ewerz:2013kda ; Lebiedowicz:2021byo . The
special feature of the tensor-pomeron model is, as mentioned above, that all
$C=+1$ exchanges ($\mathbb{P}$, $f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$) are
described as effective rank-2 symmetric tensor exchanges and that all $C=-1$
exchanges ($\mathbb{O}$, $\omega_{\mathbb{R}}$, $\rho_{\mathbb{R}}$) are
described as effective vector exchanges. To calculate the expressions for the
diagrams of Fig. 1, we need the effective propagators for the exchange objects
and their proton-proton coupling vertices. All these quantities are listed in
Chap. 3 of Ref. Ewerz:2013kda .

To write down the amplitude (9) in this model in a convenient way we define
for the $C=+1$ exchanges the functions

$\displaystyle{\cal F}_{\mathbb{P}pp}(s,t)$ $\displaystyle=$
$\displaystyle\big{[}3\beta_{\mathbb{P}pp}F_{1}(t)\big{]}^{2}\frac{1}{4s}(-is\alpha^{\prime}_{\mathbb{P}})^{\alpha_{\mathbb{P}}(t)-1}\,,$
(10) $\displaystyle{\cal F}_{f_{2\mathbb{R}}pp}(s,t)$ $\displaystyle=$
$\displaystyle\big{[}\frac{g_{f_{2\mathbb{R}}pp}}{M_{0}}F_{1}(t)\big{]}^{2}\frac{1}{4s}(-is\alpha^{\prime}_{f_{2\mathbb{R}}})^{\alpha_{f_{2\mathbb{R}}}(t)-1}\,,$
(11) $\displaystyle{\cal F}_{a_{2\mathbb{R}}pp}(s,t)$ $\displaystyle=$
$\displaystyle\big{[}\frac{g_{a_{2\mathbb{R}}pp}}{M_{0}}F_{1}(t)\big{]}^{2}\frac{1}{4s}(-is\alpha^{\prime}_{a_{2\mathbb{R}}})^{\alpha_{a_{2\mathbb{R}}}(t)-1}\,;$
(12)

and for the $C=-1$ exchanges the functions

$\displaystyle{\cal F}_{\mathbb{O}pp}(s,t)$ $\displaystyle=$
$\displaystyle-\eta_{\mathbb{O}}\big{[}3\beta_{\mathbb{O}pp}F_{1}(t)\big{]}^{2}(-is\alpha^{\prime}_{\mathbb{O}})^{\alpha_{\mathbb{O}}(t)-1}\,,$
(13) $\displaystyle{\cal F}_{\omega_{\mathbb{R}}pp}(s,t)$ $\displaystyle=$
$\displaystyle\big{[}\frac{g_{\omega_{\mathbb{R}}pp}}{M_{-}}F_{1}(t)\big{]}^{2}(-is\alpha^{\prime}_{\omega_{\mathbb{R}}})^{\alpha_{\omega_{\mathbb{R}}}(t)-1}\,,$
(14) $\displaystyle{\cal F}_{\rho_{\mathbb{R}}pp}(s,t)$ $\displaystyle=$
$\displaystyle\big{[}\frac{g_{\rho_{\mathbb{R}}pp}}{M_{-}}F_{1}(t)\big{]}^{2}(-is\alpha^{\prime}_{\rho_{\mathbb{R}}})^{\alpha_{\rho_{\mathbb{R}}}(t)-1}\,.$
(15)

We define also the following quantities:

$\displaystyle{\cal F}_{T}(s,t)={\cal F}_{\mathbb{P}pp}(s,t)+{\cal
F}_{f_{2\mathbb{R}}pp}(s,t)+{\cal F}_{a_{2\mathbb{R}}pp}(s,t)\,,$ (16)
$\displaystyle{\cal F}_{V}(s,t)={\cal F}_{\mathbb{O}pp}(s,t)+{\cal
F}_{\omega_{\mathbb{R}}pp}(s,t)+{\cal F}_{\rho_{\mathbb{R}}pp}(s,t)\,.$ (17)

All quantities occurring in (10)–(15) are as defined in Chaps. 3.1 and 3.2 of
Ref. Ewerz:2013kda . To make our article self-contained, we give in Appendix A
a list of the quantities and their values which we use. In considering the
effective propagators and vertices, some additional comments are in order:

  * •

In Ref. Ewerz:2013kda , exchange degeneracy of the reggeons with $C=+1$
($\mathbb{R}_{+}=f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$) and the reggeons with
$C=-1$ ($\mathbb{R}_{-}=\omega_{\mathbb{R}}$, $\rho_{\mathbb{R}}$) and
equality of the $\mathbb{R}_{+}$ and $\mathbb{R}_{-}$ trajectories was
assumed; see also Ref. Donnachie:2002en . This leads for the trajectories to
$\alpha_{\mathbb{R}}(t)=\alpha_{\mathbb{R}_{+}}(t)=\alpha_{\mathbb{R}_{-}}(t)$.
We assume a standard linear form
$\alpha_{\mathbb{R}}(t)=\alpha_{\mathbb{R}}(0)+\alpha_{\mathbb{R}}^{\prime}\,t$
with the intercept and the slope: $\alpha_{\mathbb{R}}(0)=0.5475$,
$\alpha_{\mathbb{R}}^{\prime}=0.9$ GeV-2.

  * •

The pomeron trajectory function in (10) is also taken as linear in $t$,
$\alpha_{\mathbb{P}}(t)=1+\epsilon_{\mathbb{P}}+\alpha_{\mathbb{P}}^{\prime}\,t$.
For the intercept parameter $\epsilon_{\mathbb{P}}$ and the slope parameter
$\alpha_{\mathbb{P}}^{\prime}$, we have the default values from Ref.
Ewerz:2013kda : $\epsilon_{\mathbb{P}}=0.0808$,
$\alpha_{\mathbb{P}}^{\prime}=0.25$ GeV-2. The $\mathbb{P}$ and $\mathbb{R}$
intercept parameters have been determined by Donnachie and Landshoff from a
simultaneous fit to the $pp$ and $p\bar{p}$ scattering data for $\sqrt{s}>10$
GeV Donnachie:1992ny  and should be regarded rather as “effective” parameters.
The parameter $\alpha_{\mathbb{P}}^{\prime}=0.25$ GeV-2 has been determined
from the $pp$ elastic scattering data Donnachie:1983hf  (see also Ref.
Donnachie:2002en ) and is in good agreement with experimental findings on the
exclusive $\rho^{0}$ photoproduction ZEUS:1997rof ; H1:2020lzc . The value for
the coupling constant of the pomeron to protons is
$\beta_{\mathbb{P}pp}=1.87\;{\rm GeV}^{-1}$; see (3.44) and Sec. 6.3 of Ref.
Ewerz:2013kda . For simplicity, in (10)–(15), the Dirac electromagnetic form
factor of the proton is used. It is clear that this cannot be strictly
correct; see the discussion in Chap. 3.2 of Ref. Donnachie:2002en  and in Ref.
Zhou:2006gj . It will be shown in Sec. IV.1 that to effectively describe the
TOTEM data for the elastic $pp$ cross section ($d\sigma/dt$) at $\sqrt{s}=13$
TeV, taking into account only the leading $\mathbb{P}$ contribution, we must
change in Eq. (10) both the $\epsilon_{\mathbb{P}}$ parameter and the pomeron-
proton form factor $F_{1}(t)$.

  * •

For the odderon exchange, we will also consider an alternative Ansatz
corresponding to a double Regge pole (87). Then, we have

$\displaystyle{\cal F}_{\mathbb{O}pp}(s,t)\to\widetilde{\cal
F}_{\mathbb{O}pp}(s,t)$ $\displaystyle=$ $\displaystyle{\cal
F}_{\mathbb{O}pp}(s,t)\left[C_{1}+C_{2}\ln\left(-is\alpha_{\mathbb{O}}^{\prime}\right)\right]\,,$
(18)

where $C_{1}$ and $C_{2}$ are real constants. In (13) and (18), the factor
$\eta_{\mathbb{O}}=\pm 1$ and the odderon trajectory function are unknown. We
assume
$\alpha_{\mathbb{O}}(t)=1+\epsilon_{\mathbb{O}}+\alpha_{\mathbb{O}}^{\prime}\,t$
with the parameters $\epsilon_{\mathbb{O}}$ and $\alpha_{\mathbb{O}}^{\prime}$
that should be determined by experiment. Of course, the pomeron amplitude must
dominate over the odderon one for $s\to\infty$ in order to ensure positive
total $pp$ and $p\bar{p}$ cross sections. Thus, we must require
$\epsilon_{\mathbb{O}}\leqslant\epsilon_{\mathbb{P}}$. For our study here, we
assume $\beta_{\mathbb{O}pp}=0.1\times\beta_{\mathbb{P}pp}\simeq 0.2\;{\rm
GeV}^{-1}$, $\alpha_{\mathbb{O}}^{\prime}=\alpha_{\mathbb{P}}^{\prime}$,
$\epsilon_{\mathbb{O}}=0.0800$, $\eta_{\mathbb{O}}=-1$, and $(C_{1},C_{2})$ as
listed in (88). How these parameters are determined from comparisons between
theory and experiment will be discussed in Sec. IV.1.

In the following, we use tensor-product notation. The first factors will
always refer to the $p_{a}$-$p_{1}$ line, and the second refer to the
$p_{b}$-$p_{2}$ line of the diagrams shown in Fig. 1.

In our model, the off-shell proton-proton scattering amplitude has the form

$\displaystyle{\cal M}^{(0)}(p_{a},p_{b},p_{1},p_{2})={\cal
M}^{(0)}_{\mathbb{P}}+{\cal M}^{(0)}_{f_{2\mathbb{R}}}+{\cal
M}^{(0)}_{a_{2\mathbb{R}}}+{\cal M}^{(0)}_{\mathbb{O}}+{\cal
M}^{(0)}_{\omega_{\mathbb{R}}}+{\cal M}^{(0)}_{\rho_{\mathbb{R}}}$
$\displaystyle=i{\cal
F}_{T}(s,t)\big{[}\gamma^{\mu}\otimes\gamma_{\mu}(p_{a}+p_{1},p_{b}+p_{2})+(\not\\!p_{b}+\not\\!p_{2})\otimes(\not\\!p_{a}+\not\\!p_{1})-\frac{1}{2}(\not\\!p_{a}+\not\\!p_{1})\otimes(\not\\!p_{b}+\not\\!p_{2})\big{]}$
$\displaystyle\quad-{\cal F}_{V}(s,t)\,\gamma^{\mu}\otimes\gamma_{\mu}\,.$
(19)

With the helicities $\lambda_{a}$, $\lambda_{b}$, $\lambda_{1}$,
$\lambda_{2}\in\\{-1/2,1/2\\}$, we get for the on-shell matrix element

$\displaystyle\braket{p(p_{1},\lambda_{1}),p(p_{2},\lambda_{2})}{{\cal
T}}{p(p_{a},\lambda_{a}),p(p_{b},\lambda_{b})}$ $\displaystyle\equiv{\cal
M}^{({\rm on\;shell})\,pp}(s,t)$
$\displaystyle=\bar{u}_{1}\otimes\bar{u}_{2}{\cal
M}^{(0)}(p_{a},p_{b},p_{1},p_{2})\left.u_{a}\otimes u_{b}\right|_{\rm
on\;shell}$ $\displaystyle=i{\cal
F}_{T}(s,t)\big{[}\bar{u}_{1}\gamma^{\mu}u_{a}\,\bar{u}_{2}\gamma_{\mu}u_{b}\,(p_{a}+p_{1},p_{b}+p_{2})+\bar{u}_{1}\gamma^{\mu}u_{a}(p_{b}+p_{2})_{\mu}\,\bar{u}_{2}\gamma^{\nu}u_{b}(p_{a}+p_{1})_{\nu}$
$\displaystyle\quad-2m_{p}^{2}\,\bar{u}_{1}u_{a}\,\bar{u}_{2}u_{b}\big{]}-{\cal
F}_{V}(s,t)\,\bar{u}_{1}\gamma^{\mu}u_{a}\,\bar{u}_{2}\gamma_{\mu}u_{b}\,.$
(20)

For brevity of notation, we define here and in the following for the spinors
$u_{a}=u(p_{a},\lambda_{a})$, $\bar{u}_{1}=\bar{u}(p_{1},\lambda_{1})$, etc.
We also denote ${\cal M}^{({\rm on\;shell})\,pp}(s,t)$ as the on-shell $pp$
elastic scattering amplitude.

We consider now the high-energy small-angle limit where we have the simple
relations

$\displaystyle\bar{u}(p_{1},\lambda_{1})\gamma^{\mu}u(p_{a},\lambda_{a})\cong(p_{a}+p_{1})^{\mu}\delta_{\lambda_{1}\lambda_{a}}\,,$
$\displaystyle(p_{a}+p_{1},p_{b}+p_{2})\cong 2s\,.$ (21)

From (20) we get, for $s\to\infty$, setting a possible odderon contribution to
zero, the pomeron contribution as leading term in the form

$\displaystyle{\cal M}^{({\rm on\;shell})\,pp}(s,t)\to i8s^{2}{\cal
F}_{\mathbb{P}pp}(s,t)\delta_{\lambda_{1}\lambda_{a}}\delta_{\lambda_{2}\lambda_{b}}\,.$
(22)

In this high-energy small-angle limit the amplitude calculated from the
tensor-pomeron exchange is the same as the standard Donnachie-Landshoff (DL)
amplitude; see the discussion in Chap. 6.1 of Ref. Ewerz:2013kda .

Coming back to the general case, we emphasize that in the following we use the
exact formulas (19) and (20), without the approximations (21).

The total cross section for unpolarized protons, obtained from the forward-
scattering amplitudes using the optical theorem, is

$\displaystyle\sigma_{\rm
tot}(pp)=\frac{1}{\sqrt{s(s-4m_{p}^{2})}}\,\frac{1}{4}\sum_{\lambda_{a},\lambda_{b}}{\rm
Im}\braket{p(p_{a},\lambda_{a}),p(p_{b},\lambda_{b})}{{\cal
T}}{p(p_{a},\lambda_{a}),p(p_{b},\lambda_{b})}$
$\displaystyle\quad=\frac{1}{\sqrt{s(s-4m_{p}^{2})}}\big{\\{}{\rm Re}{\cal
F}_{T}(s,0)\,8\big{[}(s-2m_{p}^{2})^{2}-m_{p}^{4}\big{]}-{\rm Im}{\cal
F}_{V}(s,0)\,2(s-2m_{p}^{2})\big{\\}}\,.\quad$ (23)

With (10)–(18), this gives

$\displaystyle\sigma_{\rm
tot}(pp)=2\big{(}1-\frac{4m_{p}^{2}}{s}\big{)}^{-1/2}\big{\\{}\big{[}\big{(}3\beta_{\mathbb{P}pp}\big{)}^{2}(s\alpha^{\prime}_{\mathbb{P}})^{\alpha_{\mathbb{P}}(0)-1}\cos\big{(}\frac{\pi}{2}(\alpha_{\mathbb{P}}(0)-1)\big{)}$
$\displaystyle\qquad+\big{(}\frac{g_{f_{2\mathbb{R}}pp}}{M_{0}}\big{)}^{2}(s\alpha^{\prime}_{f_{2\mathbb{R}}})^{\alpha_{f_{2\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}(\alpha_{f_{2\mathbb{R}}}(0)-1)\big{)}$
$\displaystyle\qquad+\big{(}\frac{g_{a_{2\mathbb{R}}pp}}{M_{0}}\big{)}^{2}(s\alpha^{\prime}_{a_{2\mathbb{R}}})^{\alpha_{a_{2\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}(\alpha_{a_{2\mathbb{R}}}(0)-1)\big{)}\big{]}\big{(}1-\frac{4m_{p}^{2}}{s}+\frac{3m_{p}^{4}}{s^{2}}\big{)}$
$\displaystyle\qquad-\big{[}-\eta_{\mathbb{O}}\big{(}3\beta_{\mathbb{O}pp}\big{)}^{2}(s\alpha^{\prime}_{\mathbb{O}})^{\alpha_{\mathbb{O}}(0)-1}\big{[}\cos\big{(}\frac{\pi}{2}\alpha_{\mathbb{O}}(0)\big{)}\big{(}C_{1}+C_{2}\ln\left(s\alpha_{\mathbb{O}}^{\prime}\right)\big{)}-C_{2}\frac{\pi}{2}\sin\big{(}\frac{\pi}{2}\alpha_{\mathbb{O}}(0)\big{)}\big{]}$
$\displaystyle\qquad+\big{(}\frac{g_{\omega_{\mathbb{R}}pp}}{M_{-}}\big{)}^{2}(s\alpha^{\prime}_{\omega_{\mathbb{R}}})^{\alpha_{\omega_{\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}\alpha_{\omega_{\mathbb{R}}}(0)\big{)}$
$\displaystyle\qquad+\big{(}\frac{g_{\rho_{\mathbb{R}}pp}}{M_{-}}\big{)}^{2}(s\alpha^{\prime}_{\rho_{\mathbb{R}}})^{\alpha_{\rho_{\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}\alpha_{\rho_{\mathbb{R}}}(0)\big{)}\big{]}\big{(}1-\frac{2m_{p}^{2}}{s}\big{)}\big{\\}}\,.$
(24)

Here, we have used the expression (18) inserted into (17). To get the
expression with (13) for the odderon exchange, we have to set $C_{1}=1$ and
$C_{2}=0$.

###  II.2 Reaction $p\bar{p}\to p\bar{p}$

Here we study the reaction (3), where $p$ and $\bar{p}$ can be on or off
shell. In our model, considering only hadronic exchanges, we have the diagrams
shown in Fig. 2. For high c.m. energies and small momentum transfers (8), the
$s$-channel exchanges should be negligible.

Figure 2: The diagrams for $p\bar{p}\to p\bar{p}$ elastic scattering with the
$t$\- and $s$-channel exchanges.

Now we get for the $p\bar{p}$ off-shell amplitude

$\displaystyle{\cal M}^{(0)}_{p\bar{p}}(p_{a},p_{b},p_{1},p_{2})$
$\displaystyle=i{\cal
F}_{T}(s,t)\big{[}\gamma^{\mu}\otimes\gamma_{\mu}(p_{a}+p_{1},p_{b}+p_{2})+(\not\\!p_{b}+\not\\!p_{2})\otimes(\not\\!p_{a}+\not\\!p_{1})-\frac{1}{2}(\not\\!p_{a}+\not\\!p_{1})\otimes(\not\\!p_{b}+\not\\!p_{2})\big{]}$
$\displaystyle\quad+{\cal F}_{V}(s,t)\,\gamma^{\mu}\otimes\gamma_{\mu}\,.$
(25)

For the on-shell amplitude, we get

$\displaystyle\braket{p(p_{1},\lambda_{1}),\bar{p}(p_{2},\lambda_{2})}{{\cal
T}}{p(p_{a},\lambda_{a}),\bar{p}(p_{b},\lambda_{b})}=\bar{u}_{1}\otimes\bar{v}_{b}{\cal
M}^{(0)}_{p\bar{p}}(p_{a},p_{b},p_{1},p_{2})u_{a}\otimes v_{2}$
$\displaystyle\quad=i{\cal
F}_{T}(s,t)\big{[}\bar{u}_{1}\gamma^{\mu}u_{a}\,\bar{v}_{b}\gamma_{\mu}v_{2}\,(p_{a}+p_{1},p_{b}+p_{2})+\bar{u}_{1}\gamma^{\mu}u_{a}(p_{b}+p_{2})_{\mu}\,\bar{v}_{b}\gamma^{\nu}v_{2}(p_{a}+p_{1})_{\nu}$
$\displaystyle\quad+2m_{p}^{2}\,\bar{u}_{1}u_{a}\,\bar{v}_{b}v_{2}\big{]}+{\cal
F}_{V}(s,t)\,\bar{u}_{1}\gamma^{\mu}u_{a}\,\bar{v}_{b}\gamma_{\mu}v_{2}\,.$
(26)

The total $p\bar{p}$ cross section is

$\displaystyle\sigma_{\rm
tot}(p\bar{p})=\frac{1}{\sqrt{s(s-4m_{p}^{2})}}\,\frac{1}{4}\sum_{\lambda_{a},\lambda_{b}}{\rm
Im}\braket{p(p_{a},\lambda_{a}),\bar{p}(p_{b},\lambda_{b})}{{\cal
T}}{p(p_{a},\lambda_{a}),\bar{p}(p_{b},\lambda_{b})}$
$\displaystyle\quad=\frac{1}{\sqrt{s(s-4m_{p}^{2})}}\big{\\{}{\rm Re}{\cal
F}_{T}(s,0)\,8\big{[}(s-2m_{p}^{2})^{2}-m_{p}^{4}\big{]}+{\rm Im}{\cal
F}_{V}(s,0)2(s-2m_{p}^{2})\big{\\}}\,.\quad$ (27)

With (10)–(18), this gives

$\displaystyle\sigma_{\rm
tot}(p\bar{p})=2\big{(}1-\frac{4m_{p}^{2}}{s}\big{)}^{-1/2}\big{\\{}\big{[}\big{(}3\beta_{\mathbb{P}pp}\big{)}^{2}(s\alpha^{\prime}_{\mathbb{P}})^{\alpha_{\mathbb{P}}(0)-1}\cos\big{(}\frac{\pi}{2}(\alpha_{\mathbb{P}}(0)-1)\big{)}$
$\displaystyle\qquad+\big{(}\frac{g_{f_{2\mathbb{R}}pp}}{M_{0}}\big{)}^{2}(s\alpha^{\prime}_{f_{2\mathbb{R}}})^{\alpha_{f_{2\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}(\alpha_{f_{2\mathbb{R}}}(0)-1)\big{)}$
$\displaystyle\qquad+\big{(}\frac{g_{a_{2\mathbb{R}}pp}}{M_{0}}\big{)}^{2}(s\alpha^{\prime}_{a_{2\mathbb{R}}})^{\alpha_{a_{2\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}(\alpha_{a_{2\mathbb{R}}}(0)-1)\big{)}\big{]}\big{(}1-\frac{4m_{p}^{2}}{s}+\frac{3m_{p}^{4}}{s^{2}}\big{)}$
$\displaystyle\qquad+\big{[}-\eta_{\mathbb{O}}\big{(}3\beta_{\mathbb{O}pp}\big{)}^{2}(s\alpha^{\prime}_{\mathbb{O}})^{\alpha_{\mathbb{O}}(0)-1}\big{[}\cos\big{(}\frac{\pi}{2}\alpha_{\mathbb{O}}(0)\big{)}\big{(}C_{1}+C_{2}\ln\left(s\alpha_{\mathbb{O}}^{\prime}\right)\big{)}-C_{2}\frac{\pi}{2}\sin\big{(}\frac{\pi}{2}\alpha_{\mathbb{O}}(0)\big{)}\big{]}$
$\displaystyle\qquad+\big{(}\frac{g_{\omega_{\mathbb{R}}pp}}{M_{-}}\big{)}^{2}(s\alpha^{\prime}_{\omega_{\mathbb{R}}})^{\alpha_{\omega_{\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}\alpha_{\omega_{\mathbb{R}}}(0)\big{)}$
$\displaystyle\qquad+\big{(}\frac{g_{\rho_{\mathbb{R}}pp}}{M_{-}}\big{)}^{2}(s\alpha^{\prime}_{\rho_{\mathbb{R}}})^{\alpha_{\rho_{\mathbb{R}}}(0)-1}\cos\big{(}\frac{\pi}{2}\alpha_{\rho_{\mathbb{R}}}(0)\big{)}\big{]}\big{(}1-\frac{2m_{p}^{2}}{s}\big{)}\big{\\}}\,.$
(28)

Comparing (19) with (25), (20) with (26), and (24) with (28), we see that for
$pp$ and $p\bar{p}$ scattering the $C=+1$ exchanges contribute with the same
sign and the $C=-1$ exchanges contribute with opposite sign, as it should be.

###  II.3 Reaction $pp\to pp\gamma$

Now, we consider the reaction (4). Here, we have the energy-momentum relation
(6). The kinematic variables are

$\displaystyle s=(p_{a}+p_{b})^{2}=(p_{1}^{\prime}+p_{2}^{\prime}+k)^{2}\,,$
$\displaystyle
s^{\prime}=(p_{a}+p_{b}-k)^{2}=(p_{1}^{\prime}+p_{2}^{\prime})^{2}\,,$
$\displaystyle
t_{1}=(p_{a}-p_{1}^{\prime})^{2}=(p_{b}-p_{2}^{\prime}-k)^{2}\,,$
$\displaystyle
t_{2}=(p_{b}-p_{2}^{\prime})^{2}=(p_{a}-p_{1}^{\prime}-k)^{2}\,.$ (29)

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

Figure 3: Diagrams for the reaction $pp\to pp\gamma$ with exchange of the
pomeron $\mathbb{P}$ (a–f) and the “structure” term (g). The diagrams for
odderon and reggeon exchanges are as in (a–f) with $\mathbb{P}$ replaced by
$f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$, $\mathbb{O}$, $\omega_{\mathbb{R}}$, and
$\rho_{\mathbb{R}}$. In addition there are the diagrams corresponding to the
interchange of the two final protons $p(p_{1}^{\prime})\leftrightarrow
p(p_{2}^{\prime})$. These are not shown here.

For the photon emission process (4), we have seven types of diagrams shown in
Fig. 3. In the diagrams (a), (b), (d), and (e), the photon is emitted from the
external proton lines. The diagrams (c) and (f) correspond to contact terms.
The precise definition of the contact terms and of the “structure” term (g)
will be given below. For high c.m. energies $\sqrt{s}$ and small momentum
transfers $|t_{1,2}|$ [see (29)], the diagrams of Fig. 3 with
$p_{1}^{\prime}\leftrightarrow p_{2}^{\prime}$ are expected to give negligible
contributions. We shall call the diagrams (a), (b), (d), and (e), made gauge
invariant by the addition of (c) and (f), the bremsstrahlung diagrams.

The relevant ${\cal T}$-matrix element is

$\displaystyle\braket{p(p_{1}^{\prime},\lambda_{1}),p(p_{2}^{\prime},\lambda_{2}),\gamma(k,\epsilon)}{{\cal
T}}{p(p_{a},\lambda_{a}),p(p_{b},\lambda_{b})}$
$\displaystyle=(\epsilon^{\mu})^{*}{\cal M}^{(\rm
total)}_{\mu}(p_{a},\lambda_{a};p_{b},\lambda_{b};p_{1}^{\prime},\lambda_{1};p_{2}^{\prime},\lambda_{2};k)\,.$
(30)

We consider ${\cal M}^{(\rm total)}_{\mu}$ for arbitrary $k$. Gauge invariance
requires

$k^{\mu}{\cal M}^{(\rm total)}_{\mu}=0\,.$ (31)

Let the sum of the diagrams of Fig. 3 with $\mathbb{P}$ and all other
exchanges be

${\cal M}_{\mu}(p_{1}^{\prime},p_{2}^{\prime})={\cal M}_{\mu}^{(a)}+{\cal
M}_{\mu}^{(b)}+{\cal M}_{\mu}^{(c)}+{\cal M}_{\mu}^{(d)}+{\cal
M}_{\mu}^{(e)}+{\cal M}_{\mu}^{(f)}+{\cal M}_{\mu}^{(g)}\,.$ (32)

The complete amplitude is then

${\cal M}^{(\rm total)}_{\mu}={\cal
M}_{\mu}(p_{1}^{\prime},p_{2}^{\prime})-{\cal
M}_{\mu}(p_{2}^{\prime},p_{1}^{\prime})\,,$ (33)

where a corresponding exchange of helicities is understood. The relative minus
sign here is due to the Fermi statistics, which requires the amplitude to be
antisymmetric under interchange of the two final protons.

The inclusive cross section for the real-photon yield of the reaction (4),
including a statistics factor 1/2 due to identical particles appearing in the
final state, is as follows

$\displaystyle d\sigma({pp\to
pp\gamma})=\frac{1}{2}\frac{1}{2\sqrt{s(s-4m_{p}^{2})}}\,\frac{d^{3}k}{(2\pi)^{3}\,2k^{0}}\int\frac{d^{3}p_{1}^{\prime}}{(2\pi)^{3}\,2p_{1}^{\prime
0}}\frac{d^{3}p_{2}^{\prime}}{(2\pi)^{3}\,2p_{2}^{\prime 0}}$
$\displaystyle\quad\times(2\pi)^{4}\delta^{(4)}(p_{1}^{\prime}+p_{2}^{\prime}+k-p_{a}-p_{b})\frac{1}{4}\sum_{p\,\rm
spins}{\cal M}_{\lambda}^{(\rm total)}\big{(}{\cal M}_{\rho}^{(\rm
total)}\big{)}^{*}(-g^{\lambda\rho})\,.$ (34)

Let, in the c.m. system, $p_{1z}^{\prime}$ and $p_{2z}^{\prime}$ be the
momentum components of $p_{1}^{\prime}$ and $p_{2}^{\prime}$ in the direction
of $p_{a}$. Then we can write (34) with (33) as

$\displaystyle d\sigma({pp\to
pp\gamma})=\frac{1}{2\sqrt{s(s-4m_{p}^{2})}}\,\frac{d^{3}k}{(2\pi)^{3}\,2k^{0}}\int\frac{d^{3}p_{1}^{\prime}}{(2\pi)^{3}\,2p_{1}^{\prime
0}}\frac{d^{3}p_{2}^{\prime}}{(2\pi)^{3}\,2p_{2}^{\prime
0}}\,\theta(p_{1z}^{\prime}-p_{2z}^{\prime})$
$\displaystyle\quad\times(2\pi)^{4}\delta^{(4)}(p_{1}^{\prime}+p_{2}^{\prime}+k-p_{a}-p_{b})$
$\displaystyle\quad\times\frac{1}{4}\sum_{p\,\rm spins}\big{(}{\cal
M}_{\lambda}(p_{1}^{\prime},p_{2}^{\prime})-{\cal
M}_{\lambda}(p_{2}^{\prime},p_{1}^{\prime})\big{)}\big{(}{\cal
M}_{\rho}(p_{1}^{\prime},p_{2}^{\prime})-{\cal
M}_{\rho}(p_{2}^{\prime},p_{1}^{\prime})\big{)}^{*}(-g^{\lambda\rho})\,.\qquad$
(35)

We are interested in high c.m. energies $\sqrt{s}$ and small momentum
transfers. Then, for $p_{1z}^{\prime}>p_{2z}^{\prime}$, the amplitude ${\cal
M}_{\lambda}(p_{2}^{\prime},p_{1}^{\prime})$ is very small and can be
neglected. On the other hand, ${\cal
M}_{\lambda}(p_{1}^{\prime},p_{2}^{\prime})$ is very small for
$p_{1z}^{\prime}<p_{2z}^{\prime}$. Therefore, from (35), we get with very high
accuracy

$\displaystyle d\sigma({pp\to
pp\gamma})=\frac{1}{2\sqrt{s(s-4m_{p}^{2})}}\,\frac{d^{3}k}{(2\pi)^{3}\,2k^{0}}\int\frac{d^{3}p_{1}^{\prime}}{(2\pi)^{3}\,2p_{1}^{\prime
0}}\frac{d^{3}p_{2}^{\prime}}{(2\pi)^{3}\,2p_{2}^{\prime 0}}$
$\displaystyle\quad\times(2\pi)^{4}\delta^{(4)}(p_{1}^{\prime}+p_{2}^{\prime}+k-p_{a}-p_{b})\frac{1}{4}\sum_{p\,\rm
spins}{\cal M}_{\lambda}(p_{1}^{\prime},p_{2}^{\prime})\big{(}{\cal
M}_{\rho}(p_{1}^{\prime},p_{2}^{\prime})\big{)}^{*}(-g^{\lambda\rho})\,.\qquad\;$
(36)

In the following, we shall, for brevity of notation, set ${\cal
M}_{\lambda}\equiv{\cal M}_{\lambda}(p_{1}^{\prime},p_{2}^{\prime})$.

For calculating ${\cal M}_{\lambda}$ from the diagrams of Fig. 3, we use the
following standard proton propagator and $\gamma pp$ vertex:

$\displaystyle
iS_{F}(p)=\frac{i}{\not\\!p-m_{p}+i\epsilon}=i\frac{\not\\!p+m_{p}}{p^{2}-m_{p}^{2}+i\epsilon}\,,$
(37) $\displaystyle\qquad\qquad\qquad\qquad q=p^{\prime}-p\,,$
$\displaystyle\qquad\qquad\qquad\qquad i\Gamma_{\mu}^{(\gamma
pp)}(p^{\prime},p)=-ie\big{[}\gamma_{\mu}+\frac{i}{2m_{p}}\sigma_{\mu\nu}q^{\nu}F_{2}(0)\big{]}\,,$
$\displaystyle\qquad\qquad\qquad\qquad
F_{2}(0)=\big{(}\frac{\mu_{p}}{\mu_{N}}-1\big{)},\;\mu_{N}=\frac{e}{2m_{p}},\;\frac{\mu_{p}}{\mu_{N}}=2.7928\,,$
$\displaystyle\qquad\qquad\qquad\qquad e>0,\;e=\sqrt{4\pi\alpha_{\rm em}}\,.$
(38)

We take the form factors in (38) at $q^{2}=0$ in order to be consistent with
the Ward-Takahashi identity Ward:1950xp ; Takahashi:1957xn :

$(p^{\prime}-p)^{\mu}\Gamma_{\mu}^{(\gamma
pp)}(p^{\prime},p)=-e\big{[}S_{F}^{-1}(p^{\prime})-S_{F}^{-1}(p)\big{]}\,.$
(39)

In any case, we are finally interested in real photon emission where $k=-q$,
$k^{2}=q^{2}=0$.

Now, we list our results for the photon-bremsstrahlung amplitudes ${\cal
M}_{\mu}^{(a)},\ldots,{\cal M}_{\mu}^{(f)}$ corresponding to the pomeron-
exchange diagrams (a)–(f) from Fig. 3 including the other exchanges
$f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$, $\omega_{\mathbb{R}}$,
$\rho_{\mathbb{R}}$, $\mathbb{O}$.

With the off-shell $pp$ scattering amplitude (19), the standard proton
propagator (37), and the $\gamma pp$ vertex function (38), we get the
following radiative amplitudes:

$\displaystyle{\cal M}_{\mu}^{(a)}$ $\displaystyle=$
$\displaystyle-\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\,{\cal
M}^{(0)}(p_{a}-k,p_{b},p_{1}^{\prime},p_{2}^{\prime})\,\big{(}S_{F}(p_{a}-k)\Gamma_{\mu}^{(\gamma
pp)}(p_{a}-k,p_{a})\,u_{a}\big{)}\otimes u_{b}$ (40) $\displaystyle=$
$\displaystyle e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s^{\prime},t_{2})\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}-k+p_{1}^{\prime},p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}-\not\\!k+\not\\!p_{1}^{\prime})$
$\displaystyle-\frac{1}{2}(\not\\!p_{a}-\not\\!k+\not\\!p_{1}^{\prime})\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}-{\cal
F}_{V}(s^{\prime},t_{2})\,\gamma^{\alpha}\otimes\gamma_{\alpha}\big{\\}}$
$\displaystyle\times\big{[}\frac{\not\\!p_{a}-\not\\!k+m_{p}}{(p_{a}-k)^{2}-m_{p}^{2}+i\varepsilon}\big{(}\gamma_{\mu}-\frac{i}{2m_{p}}\sigma_{\mu\nu}k^{\nu}F_{2}(0)\big{)}u_{a}\big{]}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{\mu}^{(b)}$ $\displaystyle=$
$\displaystyle-\big{(}\bar{u}_{1^{\prime}}\Gamma_{\mu}^{(\gamma
pp)}(p_{1}^{\prime},p_{1}^{\prime}+k)S_{F}(p_{1}^{\prime}+k)\big{)}\otimes\bar{u}_{2^{\prime}}\,{\cal
M}^{(0)}(p_{a},p_{b},p_{1}^{\prime}+k,p_{2}^{\prime})\,u_{a}\otimes u_{b}$
(41) $\displaystyle=$ $\displaystyle
e\big{[}\bar{u}_{1^{\prime}}\big{(}\gamma_{\mu}-\frac{i}{2m_{p}}\sigma_{\mu\nu}k^{\nu}F_{2}(0)\big{)}\frac{\not\\!p_{1}^{\prime}+\not\\!k+m_{p}}{(p_{1}^{\prime}+k)^{2}-m_{p}^{2}+i\varepsilon}\big{]}\otimes\bar{u}_{2^{\prime}}$
$\displaystyle\times\big{\\{}i{\cal
F}_{T}(s,t_{2})\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime}+k,p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime}+\not\\!k)$
$\displaystyle-\frac{1}{2}(\not\\!p_{a}+\not\\!p_{1}^{\prime}+\not\\!k)\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}-{\cal
F}_{V}(s,t_{2})\,\gamma^{\alpha}\otimes\gamma_{\alpha}\big{\\}}u_{a}\otimes
u_{b}\,.$

Here, the functions ${\cal F}_{T}$ and ${\cal F}_{V}$ are defined in (16) and
(17), respectively.

Using (39), we find

$\displaystyle k^{\mu}{\cal M}_{\mu}^{(a)}$ $\displaystyle=$
$\displaystyle-e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\,{\cal
M}^{(0)}(p_{a}-k,p_{b},p_{1}^{\prime},p_{2}^{\prime})\,u_{a}\otimes u_{b}\,,$
(42) $\displaystyle k^{\mu}{\cal M}_{\mu}^{(b)}$ $\displaystyle=$
$\displaystyle e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\,{\cal
M}^{(0)}(p_{a},p_{b},p_{1}^{\prime}+k,p_{2}^{\prime})\,u_{a}\otimes u_{b}\,.$
(43)

Now, we impose the gauge-invariance condition, which must hold also for the
photon emission from the $p_{a}$-$p_{1}^{\prime}$ lines in Fig. 3 alone:

$k^{\mu}\big{(}{\cal M}_{\mu}^{(a)}+{\cal M}_{\mu}^{(b)}+{\cal
M}_{\mu}^{(c)}\big{)}=0\,.$ (44)

We obtain then

$\displaystyle k^{\mu}{\cal M}_{\mu}^{(c)}=-k^{\mu}{\cal
M}_{\mu}^{(a)}-k^{\mu}{\cal M}_{\mu}^{(b)}$
$\displaystyle\quad=e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{[}{\cal
M}^{(0)}(p_{a}-k,p_{b},p_{1}^{\prime},p_{2}^{\prime})-{\cal
M}^{(0)}(p_{a},p_{b},p_{1}^{\prime}+k,p_{2}^{\prime})\big{]}u_{a}\otimes
u_{b}$
$\displaystyle\quad=e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s^{\prime},t_{2})\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}-k+p_{1}^{\prime},p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}-\not\\!k+\not\\!p_{1}^{\prime})$
$\displaystyle\quad-\frac{1}{2}(\not\\!p_{a}-\not\\!k+\not\\!p_{1}^{\prime})\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}-{\cal
F}_{V}(s^{\prime},t_{2})\,\gamma^{\alpha}\otimes\gamma_{\alpha}$
$\displaystyle\quad-i{\cal
F}_{T}(s,t_{2})\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime}+k,p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime}+\not\\!k)$
$\displaystyle\quad-\frac{1}{2}(\not\\!p_{a}+\not\\!p_{1}^{\prime}+\not\\!k)\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}+{\cal
F}_{V}(s,t_{2})\,\gamma^{\alpha}\otimes\gamma_{\alpha}\big{\\}}u_{a}\otimes
u_{b}\,.$ (45)

We have from (29)

$\displaystyle s^{\prime}$ $\displaystyle=$ $\displaystyle(p_{a}+p_{b}-k)^{2}$
(46) $\displaystyle=$ $\displaystyle(p_{a}+p_{b})^{2}-2(k,p_{a}+p_{b})+k^{2}$
$\displaystyle=$ $\displaystyle s-(k,2p_{a}+2p_{b}-k)$

and define

$\displaystyle\varkappa=\frac{(k,2p_{a}+2p_{b}-k)}{s}\,.$ (47)

Now, using the expressions given by Eqs. (10)–(15), (46), and (47), we get:

$\displaystyle{\cal F}_{\mathbb{P}pp}(s^{\prime},t_{2})={\cal
F}_{\mathbb{P}pp}(s,t_{2})\big{[}1+\varkappa\big{(}2-\alpha_{\mathbb{P}}(t_{2})\big{)}g_{\mathbb{P}}(\varkappa,t_{2})\big{]}\,,$
(48) $\displaystyle
g_{\mathbb{P}}(\varkappa,t_{2})=\frac{1}{\big{(}2-\alpha_{\mathbb{P}}(t_{2})\big{)}\varkappa}\big{[}(1-\varkappa)^{\alpha_{\mathbb{P}}(t_{2})-2}-1\big{]}\,,$
(49)

and analogously for $f_{2\mathbb{R}}$ and $a_{2\mathbb{R}}$, and

$\displaystyle{\cal F}_{\mathbb{O}pp}(s^{\prime},t_{2})={\cal
F}_{\mathbb{O}pp}(s,t_{2})\big{[}1+\varkappa\big{(}1-\alpha_{\mathbb{O}}(t_{2})\big{)}g_{\mathbb{O}}(\varkappa,t_{2})\big{]}\,,$
(50) $\displaystyle
g_{\mathbb{O}}(\varkappa,t_{2})=\frac{1}{\big{(}1-\alpha_{\mathbb{O}}(t_{2})\big{)}\varkappa}\big{[}(1-\varkappa)^{\alpha_{\mathbb{O}}(t_{2})-1}-1\big{]}\,,$
(51)

and analogously for $\omega_{\mathbb{R}}$ and $\rho_{\mathbb{R}}$.

From (48)–(51) and supplementing with the reggeons we get

$\displaystyle{\cal F}_{T}(s^{\prime},t)={\cal
F}_{T}(s,t)+\varkappa\,\Delta{\cal F}_{T}(s,t,\varkappa)\,,$ (52)
$\displaystyle{\cal F}_{V}(s^{\prime},t)={\cal
F}_{V}(s,t)+\varkappa\,\Delta{\cal F}_{V}(s,t,\varkappa)\,,$ (53)

where

$\displaystyle\Delta{\cal F}_{T}(s,t,\varkappa)$ $\displaystyle=$
$\displaystyle(2-\alpha_{\mathbb{P}}(t))\,g_{\mathbb{P}}(\varkappa,t)\,{\cal
F}_{\mathbb{P}pp}(s,t)+(2-\alpha_{f_{2\mathbb{R}}}(t))\,g_{f_{2\mathbb{R}}}(\varkappa,t)\,{\cal
F}_{f_{2\mathbb{R}}pp}(s,t)$ (54)
$\displaystyle+(2-\alpha_{a_{2\mathbb{R}}}(t))\,g_{a_{2\mathbb{R}}}(\varkappa,t)\,{\cal
F}_{a_{2\mathbb{R}}pp}(s,t)\,,$ $\displaystyle\Delta{\cal
F}_{V}(s,t,\varkappa)$ $\displaystyle=$
$\displaystyle(1-\alpha_{\mathbb{O}}(t))\,g_{\mathbb{O}}(\varkappa,t)\,{\cal
F}_{\mathbb{O}pp}(s,t)+(1-\alpha_{\omega_{\mathbb{R}}}(t))\,g_{\omega_{\mathbb{R}}}(\varkappa,t)\,{\cal
F}_{\omega_{\mathbb{R}}pp}(s,t)$ (55)
$\displaystyle+(1-\alpha_{\rho_{\mathbb{R}}}(t))\,g_{\rho_{\mathbb{R}}}(\varkappa,t)\,{\cal
F}_{\rho_{\mathbb{R}}pp}(s,t)\,.$

The formulas for the odderon in (50), (51), and (55), apply for a single-pole
odderon (13), (85). The corresponding formulas for a double-pole odderon,
(18), (87), are given in Appendix A; see (102)–(104).

Inserting all this in (45), we get

$\displaystyle k^{\mu}{\cal
M}_{\mu}^{(c)}=e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}-i{\cal
F}_{T}(s,t_{2})\big{[}2\gamma^{\alpha}\otimes\gamma_{\alpha}(k,p_{b}+p_{2}^{\prime})+2(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes\not\\!k-\not\\!k\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}$
$\displaystyle\quad+i\frac{(k,2p_{a}+2p_{b}-k)}{s}\Delta{\cal
F}_{T}(s,t_{2},\varkappa)$
$\displaystyle\quad\times\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime}-\not\\!k)-\frac{1}{2}(\not\\!p_{a}+\not\\!p_{1}^{\prime}-\not\\!k)\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}$
$\displaystyle\quad-\frac{(k,2p_{a}+2p_{b}-k)}{s}\Delta{\cal
F}_{V}(s,t_{2},\varkappa)\,\gamma^{\alpha}\otimes\gamma_{\alpha}\big{\\}}u_{a}\otimes
u_{b}\,.$ (56)

Hence, the simplest solution of (56) for ${\cal M}_{\mu}^{(c)}$ has the form

$\displaystyle{\cal
M}_{\mu}^{(c)}=e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}-i{\cal
F}_{T}(s,t_{2})\big{[}2\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{b}+p_{2}^{\prime})_{\mu}+2(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes\gamma_{\mu}-\gamma_{\mu}\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}$
$\displaystyle\quad+i\frac{(2p_{a}+2p_{b}-k)_{\mu}}{s}\Delta{\cal
F}_{T}(s,t_{2},\varkappa)$
$\displaystyle\quad\times\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime}-\not\\!k)-\frac{1}{2}(\not\\!p_{a}+\not\\!p_{1}^{\prime}-\not\\!k)\otimes(\not\\!p_{b}+\not\\!p_{2}^{\prime})\big{]}$
$\displaystyle\quad-\frac{(2p_{a}+2p_{b}-k)_{\mu}}{s}\Delta{\cal
F}_{V}(s,t_{2},\varkappa)\,\gamma^{\alpha}\otimes\gamma_{\alpha}\big{\\}}u_{a}\otimes
u_{b}\,.$ (57)

We define (57) as ${\cal M}_{\mu}^{(c)}$. Possible additions to this solution
of the form

$\displaystyle\widetilde{\cal M}_{\mu}\,,\quad k^{\mu}\widetilde{\cal
M}_{\mu}=0$ (58)

can and will be considered as a part of ${\cal M}_{\mu}^{(g)}$ for which we
require

$k^{\mu}{\cal M}_{\mu}^{(g)}=0\,.$ (59)

For the diagrams (d), (e), and (f), we get

$\displaystyle{\cal M}_{\mu}^{(d)}=\left.{\cal
M}_{\mu}^{(a)}\right|_{\mathop{{}^{(p_{a},\,\lambda_{a})\leftrightarrow(p_{b},\,\lambda_{b})}_{(p_{1}^{\prime},\,\lambda_{1})\leftrightarrow(p_{2}^{\prime},\,\lambda_{2})}}}\,,$
(60) $\displaystyle{\cal M}_{\mu}^{(e)}=\left.{\cal
M}_{\mu}^{(b)}\right|_{\mathop{{}^{(p_{a},\,\lambda_{a})\leftrightarrow(p_{b},\,\lambda_{b})}_{(p_{1}^{\prime},\,\lambda_{1})\leftrightarrow(p_{2}^{\prime},\,\lambda_{2})}}}\,,$
(61) $\displaystyle{\cal M}_{\mu}^{(f)}=\left.{\cal
M}_{\mu}^{(c)}\right|_{\mathop{{}^{(p_{a},\,\lambda_{a})\leftrightarrow(p_{b},\,\lambda_{b})}_{(p_{1}^{\prime},\,\lambda_{1})\leftrightarrow(p_{2}^{\prime},\,\lambda_{2})}}}\,.$
(62)

With the exchanges $(p_{a},\lambda_{a})\leftrightarrow(p_{b},\lambda_{b})$ and
$(p_{1}^{\prime},\lambda_{1})\leftrightarrow(p_{2}^{\prime},\lambda_{2})$, we
shall also exchange the order of the factors in the tensor products. In this
way, the first factors in the tensor products always refer to the
$p_{a}$-$p_{1}^{\prime}$ line, and the second factors refer to the
$p_{b}$-$p_{2}^{\prime}$ line, respectively, in the diagrams of Fig. 3.

We shall call

$\displaystyle{\cal M}_{\mu}^{(\rm standard)}$ $\displaystyle=$
$\displaystyle{\cal M}_{\mu}^{(a)}+{\cal M}_{\mu}^{(b)}+{\cal
M}_{\mu}^{(c)}+{\cal M}_{\mu}^{(d)}+{\cal M}_{\mu}^{(e)}+{\cal M}_{\mu}^{(f)}$
(63)

our standard, or “conventional”, amplitude. We have by construction

$k^{\mu}{\cal M}_{\mu}^{(\rm standard)}=0\,.$ (64)

All “anomalous” terms are then subsumed in ${\cal M}_{\mu}^{(g)}$, which
satisfies (59) and has no singularity for $k_{\mu}\to 0$.

The explicit forms of the amplitudes ${\cal M}_{\mu}^{(a)},\ldots,{\cal
M}_{\mu}^{(f)}$ in (40), (41), (57), and (60)–(62), calculated in the tensor-
pomeron approach, are a main result of our present paper. But it turns out
that these forms are not very convenient for numerical computations.
Therefore, in Appendix B, we rewrite these amplitudes in a different form
which is more easy to handle numerically.

The intermediate protons in the diagrams of Fig. 3 (a, b, d, e) are off shell,
and in principle we should take care of that fact in our model. But at
present, we set possible cutoff form factors for off-shell protons in the
pomeron-proton and photon-proton vertices and in the proton propagator to 1.
Note that a naive usage of form factors can violate the Ward-Takahashi
identity (39). We expect, however, that in our regions of interest, in the
small $k_{\perp}$ and $\omega$ ranges and especially in the soft-photon limit,
the off-shell effects should be small.

##  III Soft-photon approximation approach

In the following, we shall compare our exact model results, which we shall
call standard results, for the amplitude (63), using (107)–(120), to two soft-
photon approximations: SPA1 and SPA2. In these SPAs, we consider only the
pomeron-exchange terms for the radiative amplitudes ${\cal
M}_{\mu}^{(a)},\ldots,{\cal M}_{\mu}^{(f)}$ in (63). Below, we list the
explicit expressions for real photon emission in $pp$ scattering.

  1. SPA1:

Here we keep only the pole terms $\propto\omega^{-1}$ for ${\cal
M}_{\mu}^{(a)},\ldots,{\cal M}_{\mu}^{(f)}$ in (63). For real photons
($k^{2}=0$), neglecting gauge terms $\propto k_{\mu}$, and with
$p_{1}^{\prime}\to p_{1}$, $p_{2}^{\prime}\to p_{2}$, we get

$\displaystyle{\cal M}_{\mu}$ $\displaystyle\to$ $\displaystyle{\cal
M}_{\mu,\;{\rm SPA1}}$ (65) $\displaystyle=e{\cal M}^{({\rm
on\;shell})\,pp}(s,t)\Big{[}-\frac{p_{a\mu}}{(p_{a}\cdot
k)}+\frac{p_{1\mu}}{(p_{1}\cdot k)}-\frac{p_{b\mu}}{(p_{b}\cdot
k)}+\frac{p_{2\mu}}{(p_{2}\cdot k)}\Big{]}.\quad$

Here, ${\cal M}^{({\rm on\;shell})\,pp}(s,t)$ is the on-shell $pp$-scattering
amplitude given by (20). The result (65) is easily obtained from (107), (108),
and (119). Inserting (65) into (36), 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:

$\displaystyle d\sigma({pp\to pp\gamma})_{\,\rm SPA1}$ $\displaystyle=$
$\displaystyle\frac{d^{3}k}{(2\pi)^{3}\,2k^{0}}\int
d^{3}p_{1}\,d^{3}p_{2}\,e^{2}\;\frac{d\sigma(pp\to pp)}{d^{3}p_{1}d^{3}p_{2}}$
(66) $\displaystyle\times\Big{[}-\frac{p_{a\mu}}{(p_{a}\cdot
k)}+\frac{p_{1\mu}}{(p_{1}\cdot k)}-\frac{p_{b\mu}}{(p_{b}\cdot
k)}+\frac{p_{2\mu}}{(p_{2}\cdot k)}\Big{]}$
$\displaystyle\times\Big{[}-\frac{p_{a\nu}}{(p_{a}\cdot
k)}+\frac{p_{1\nu}}{(p_{1}\cdot k)}-\frac{p_{b\nu}}{(p_{b}\cdot
k)}+\frac{p_{2\nu}}{(p_{2}\cdot k)}\Big{]}(-g^{\mu\nu})\,.\quad$

Here,

$\displaystyle\frac{d\sigma(pp\to pp)}{d^{3}p_{1}d^{3}p_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s(s-4m_{p}^{2})}}\,\frac{1}{(2\pi)^{3}\,2p_{1}^{0}\,(2\pi)^{3}\,2p_{2}^{0}}$
(67)
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p_{a}-p_{b})\,\frac{1}{4}\sum_{p\,\rm
spins}|{\cal M}^{({\rm on\;shell})\,pp}(s,t)|^{2}\,.\qquad\;$

In (66) and (67), we have a frequently used SPA. One takes the momentum
distribution of the particles without radiation [see (67)] and multiplies it
with the square of the emission factor in the square brackets in (65).

  2. SPA2:

Here we keep the energy-momentum relation (6). We consider again real photon
emission and make in (36) the replacement

$\displaystyle{\cal M}_{\mu}$ $\displaystyle\to$ $\displaystyle{\cal
M}_{\mu,\;{\rm SPA2}}={\cal M}_{{\mathbb{P}},\mu}^{(a+b+c)\,1}+{\cal
M}_{{\mathbb{P}},\mu}^{(d+e+f)\,1}\,.$ (68)

The amplitude (68) corresponds to that given in (108) plus (119), taking only
the pomeron exchange into account. Here, the squared momentum transfer is
$t_{2}$ for the diagrams of Fig. 3 (a–c) and $t_{1}$ for those of Fig. 3
(d–f), where $t_{1}$ and $t_{2}$ are defined in (29). Also, in the calculation
of the photon distributions, we keep the correct energy-momentum conserving
$\delta^{(4)}(.)$ function in (36).

We also consider a slightly modified SPA1 amplitude where we use the high-
energy small-angle limit relations (21). We get then $\widehat{{\cal
M}}_{\mu,\;{\rm SPA1}}$ as in (65) but with (22) inserted for ${\cal M}^{({\rm
on\;shell})\,pp}$:

$\displaystyle\widehat{{\cal M}}_{\mu,\;{\rm SPA1}}=ie8s^{2}{\cal
F}_{\mathbb{P}pp}(s,t)\delta_{\lambda_{1}\lambda_{a}}\delta_{\lambda_{2}\lambda_{b}}\Big{[}-\frac{p_{a\mu}}{(p_{a}\cdot
k)}+\frac{p_{1\mu}}{(p_{1}\cdot k)}-\frac{p_{b\mu}}{(p_{b}\cdot
k)}+\frac{p_{2\mu}}{(p_{2}\cdot k)}\Big{]}.\quad$ (69)

For the SPA2, we examine, furthermore, the approximation

$\displaystyle\widehat{{\cal M}}_{\mu,\;{\rm SPA2}}$ $\displaystyle=$
$\displaystyle ie8s^{2}{\cal
F}_{\mathbb{P}pp}(s,t_{2})\delta_{\lambda_{1}\lambda_{a}}\delta_{\lambda_{2}\lambda_{b}}\Big{[}-\frac{p_{a\mu}}{(p_{a}\cdot
k)}+\frac{p_{1\mu}^{\prime}}{(p_{1}^{\prime}\cdot k)}\Big{]}$ (70)
$\displaystyle+ie8s^{2}{\cal
F}_{\mathbb{P}pp}(s,t_{1})\delta_{\lambda_{1}\lambda_{a}}\delta_{\lambda_{2}\lambda_{b}}\Big{[}-\frac{p_{b\mu}}{(p_{b}\cdot
k)}+\frac{p_{2\mu}^{\prime}}{(p_{2}^{\prime}\cdot k)}\Big{]}\,.$

How these approximations (69) and (70) agree with ${\cal M}_{\mu,\;{\rm
SPA1}}$ from the formula (65) and ${\cal M}_{\mu,\;{\rm SPA2}}$ from the
formula (68) will be discussed at the end of Sec. IV.2.

##  IV Results

Below, we show our results for total and elastic $pp$ and $p\bar{p}$
scattering (Sec. IV.1) and results for the $pp\to pp\gamma$ reaction from the
bremsstrahlung mechanism (Sec. IV.2).

###  IV.1 Comparison of the model results with the total and elastic $pp$ and
$p\bar{p}$ cross section data

Here, we compare our results obtained from Eqs. (20) and (26) to the
experimental data for $\sigma_{\rm tot}$, $d\sigma/dt$, and $\rho$, that is,
the ratio of the real part to imaginary part of the forward scattering
amplitude,

$\displaystyle\rho=\frac{{\rm Re}{\cal T}(s,0)}{{\rm Im}{\cal T}(s,0)}\,.$
(71)

We emphasize that in the present paper it is not our purpose to perform a
precision fit to the available experimental data on these quantities, say,
from ISR to LHC energies. Here ISR means the CERN Intersecting Storage Rings
which operated with a maximum c.m. energy of 63 GeV. Such fits have been, for
instance, given in Refs. Donnachie:2013xia ; Jenkovszky:2017efs ;
Jenkovszky:2018itd ; Khoze:2018kna ; Martynov:2018sga ; Broilo:2020kqg ;
Bence:2021uji ; Godizov:2021ksd . Our aim here is to obtain a reasonable
description of the data mainly at $\sqrt{s}=13$ TeV. We want to fix the
parameters of our model in this way. The main aim of our paper is to give then
predictions for soft-photon radiation in proton-proton elastic collisions for
the LHC energy range; see Sec. IV.2.

In Fig. 4, we present the total $pp$ (see the red points and lines) and
$p\bar{p}$ (see the blue points and lines) cross sections for $\sqrt{s}>10$
GeV. We show our complete theoretical results including the pomeron, the
reggeon, and the odderon exchanges and their separate contributions. First, we
notice that, of course, the high-energy cross sections are dominated by the
pomeron exchange (see the black solid line). The cross sections with the
pomeron contribution alone are $\sigma_{\rm tot}(pp)=102.0$ mb for
$\sqrt{s}=8$ TeV and $\sigma_{\rm tot}(pp)=110.9$ mb for $\sqrt{s}=13$ TeV.
For the odderon exchange, we use the double-pole Ansatz (87) with the
parameters (88) and $(C_{1},C_{2})=(-2.0,0.3)$ adjusted to describe $\rho=0.1$
found by TOTEM at $\sqrt{s}=13$ TeV. See the discussion below in connection
with Fig. 6 which presents $\rho$ as function of $\sqrt{s}$. There we also
show the results for two more values $(C_{1},C_{2})=(-1.0,0.1)$ and
$(-1.5,0.2)$. These lead to values of $\rho>0.1$ at $\sqrt{s}=13$ TeV. Here we
are interested in the total cross sections where the odderon effects are very
small. We get, for instance, for the two cases $(C_{1},C_{2})=(-1.0,0.1)$ and
$(-2.0,0.3)$ the total cross sections, $\sigma_{\rm tot}(pp)=102.2$ and 102.9
mb for $\sqrt{s}=8$ TeV and $\sigma_{\rm tot}(pp)=111.2$ and 111.9 mb for
$\sqrt{s}=13$ TeV, respectively. Our results are in good agreement with the
TOTEM results: $\sigma_{\rm tot}(pp)|_{\rm exp}=(102.9\pm 2.3)$ mb and
$\sigma_{\rm tot}(pp)|_{\rm exp}=(103.0\pm 2.3)$ mb for $\sqrt{s}=8$ TeV
TOTEM:2016lxj , $\sigma_{\rm tot}(pp)|_{\rm exp}=(110.6\pm 3.4)$ mb
TOTEM:2017asr  and $\sigma_{\rm tot}(pp)|_{\rm exp}=(110.3\pm 3.5)$ mb
TOTEM:2017sdy  for $\sqrt{s}=13$ TeV. 222 The cross sections $\sigma_{\rm
tot}(pp)|_{\rm exp}$ from Refs. TOTEM:2017asr ; TOTEM:2017sdy  for
$\sqrt{s}=13$ TeV were determined in a completely independent way and,
therefore, were combined for uncertainty reduction. The result is then
$\sigma_{\rm tot}(pp)|_{\rm exp}=(110.5\pm 2.4)$ mb; see Eq. (21) of Ref.
TOTEM:2017sdy . Note that we get for large energies a total cross section for
$pp$ exceeding that for $p\bar{p}$ collisions, $\sigma_{\rm
tot}(pp)>\sigma_{\rm tot}(p\bar{p})$.

Figure 4:  The total cross sections for $pp$ and $p\bar{p}$ scattering as a
function of the center-of-mass energy $\sqrt{s}$. The red long-dashed line and
the blue dashed line correspond to our complete results for $pp$ and
$p\bar{p}$ scattering, respectively. Here, the pomeron parameters are as in
(79), (80), (89), and (90) with $\epsilon_{\mathbb{P}}=0.0865$. The reggeon
parameters are given in (81)–(84) and we take here for the odderon the
parameters (87), (88), and (101), with $(C_{1},C_{2})=(-2.0,0.3)$. The black
solid line corresponds to the pomeron exchange alone, the dotted lines to the
reggeon exchanges (lower for $pp$, upper for $p\bar{p}$), and the dot-dashed
line corresponds to the odderon contribution increased by a factor of 2 for
visualization. The experimental data are from Ref. TOTEM:2016lxj , from Eq.
(8) and Figs. 4 and 5 of Ref. TOTEM:2017asr , from Ref. TOTEM:2017sdy , and
from the Particle Data Group Zyla:2020zbs  (see Ref. PDG_CrossSections ).

In Fig. 5, we show the $pp$ elastic differential cross sections for our model
and the TOTEM data TOTEM:2017sdy ; TOTEM:2018hki  measured at $\sqrt{s}=13$
TeV. We find a quite good description of the data in the region 0.003 GeV2
$\leqslant-t\leqslant 0.26$ GeV2 taking $\epsilon_{\mathbb{P}}=0.0865$ and
assuming the form factor $F(t)=\exp(-b\,|t|)$ with only one parameter,
$b=2.95$ GeV-2. For comparison, we show also the results for
$\epsilon_{\mathbb{P}}=0.0808$ and the Dirac form factor $F_{1}(t)$. We should
not expect our single-pomeron exchange model to give a precision fit for
$d\sigma/dt$ in the diffractive dip region and beyond. The common lore is that
in order to produce the dip one needs the interference of various terms in the
amplitude, at least three terms: $\mathbb{P}+\mathbb{P}\mathbb{P}+ggg$; see,
e.g., Refs. Donnachie:1983hf ; Donnachie:2013xia . In Refs. Jenkovszky:2017efs
; Jenkovszky:2018itd , the authors discussed the so-called break effect, which
leads to a smooth deflection of the linear exponential behavior of the
diffraction cone observed near $t=-0.1$ GeV2. The effect is related to the
basic analytic properties of the theory ($t$-channel unitarity) and reflects
the presence of the pion cloud around the nucleon. This small, but
interesting, effect could easily be included in our parametrization of the $t$
dependence of the $pp$ elastic cross section. But, as mentioned above, for our
purposes, we only need a reasonable description of the $pp$ scattering data
for $\sqrt{s}=13$ TeV. And, as we show in Fig. 5, we obtain such a description
in the $t$ range of interest for us with our single-exponential form factor
$F(t)$.

Figure 5:  The differential $pp\to pp$ cross sections measured by the TOTEM
experiments at $\sqrt{s}=13$ TeV TOTEM:2017sdy ; TOTEM:2018hki . We show our
theoretical results obtained with two different pomeron parameter sets: 1)
$\epsilon_{\mathbb{P}}=0.0808$ and the Dirac form factor $F_{1}(t)$ (see the
blue dashed lines), and 2) $\epsilon_{\mathbb{P}}=0.0865$ and the exponential
form factor $F(t)=\exp(-b\,|t|)$ with $b=2.95$ GeV-2 (see the black solid
lines). In both cases, we take $\alpha_{\mathbb{P}}^{\prime}=0.25$ GeV-2 from
(80). Here, reggeon and odderon contributions are set to zero.

In Fig. 6, we show our results for $\rho$ for the two elastic scattering
processes $pp$ and $p\bar{p}$ together with the experimental data. The solid
lines represent results of our model without the odderon contribution. The
pomeron and reggeon parameters are the same as in Fig. 4. In the left panel,
the dashed and dotted lines show the results with the odderon exchange for
different values of $(C_{1},C_{2})$ in (18) and with the parameters (88),
(101). The $C=-1$ reggeons lead to a splitting of the $pp$ and $p\bar{p}$
results at smaller energies, while the odderon mainly affects the higher
energies. The odderon parameters $(C_{1},C_{2})$ are in essence determined by
the $\rho$ value at $\sqrt{s}=13$ TeV. The values $\rho=0.103$, 0.116, and
0.129 lead to $(C_{1},C_{2})=(-2.0,0.3)$, $(-1.5,0.2)$, and $(-1.0,0.1)$,
respectively. In the right panel of Fig. 6, we show the results for
$(C_{1},C_{2})=(-2.0,0.3)$ and for $\epsilon_{\mathbb{O}}=0$ and 0.05 to
compare with $\epsilon_{\mathbb{O}}=0.08$. We can see that the $\rho$ values
mentioned can also be obtained by changing $\epsilon_{\mathbb{O}}$.

Figure 6:  Our results for the $\rho$ parameter as function of the c.m. energy
$\sqrt{s}$ for $p\bar{p}$ (the blue upper lines) and $pp$ (the red lower
lines) scattering together with the experimental data from Refs. TOTEM:2016lxj
; TOTEM:2017sdy  and from Ref. PDG_CrossSections . Our complete results
include $\mathbb{P}$, $f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$,
$\omega_{\mathbb{R}}$, $\rho_{\mathbb{R}}$, and $\mathbb{O}$ exchanges. The
solid lines are without the odderon contribution, and the dashed and dotted
lines are with the odderon for different values of the parameters as described
in the figure legends.

The TOTEM Collaboration has reported direct measurements of the $\rho$
parameter (71) through the Coulomb-nuclear interference in the very small
$|t|$ region. The TOTEM results for $\rho$ extracted from the data on the
proton-proton differential cross section are:

  * •

$\rho=0.12\pm 0.03$ at $\sqrt{s}=8$ TeV TOTEM:2016lxj ;

  * •

$\rho=0.09\pm 0.01$ or $\rho=0.10\pm 0.01$ at $\sqrt{s}=13$ TeV TOTEM:2017sdy
, depending on different physics assumptions and mathematical modelings.

We can make the following statement. If we take the above $\rho$ values from
TOTEM as representing the truth, then, in the framework of our model, we need
an odderon contribution at $t=0$. Moreover, we could not find a reasonable
description of the low- and high-energy data for $\rho$ with a single-pole
odderon. For these reasons, we consider here a double-pole odderon. To fit the
TOTEM results for $\rho$, we have to choose the odderon parameters
$(C_{1},C_{2})=(-2.0,0.3)$. These findings are consistent with the results of
Refs. Martynov:2017zjz ; Martynov:2018pye . These authors claim that a small
value of $\rho$ can be considered as the experimental discovery of the odderon
for small $|t|$, namely, in its maximal form. The odderon was first introduced
on theoretical grounds in Ref. Lukaszuk:1973nt . For a review of odderon
physics we refer the reader to Ref. Ewerz:2003xi . However, recent reanalyses
of the TOTEM results Pancheri:2018yhd ; Ezhela:2020hws  have shown that the
values of $\rho$ may be larger than those reported by the TOTEM Collaboration.
For illustration, in Ref. Pancheri:2018yhd , one finds $\rho\simeq 0.135$ at
$\sqrt{s}=8$ TeV and $\rho\simeq 0.133$ at $\sqrt{s}=13$ TeV. Reference
Ezhela:2020hws  gives $\rho=0.123\pm 0.010$ at $\sqrt{s}=13$ TeV. To be
consistent with these values, we should take in our model
$(C_{1},C_{2})=(-1.0,0.1)$.

In view of these ongoing discussions in the literature, we think that the low
value $\rho=0.1$ at $\sqrt{s}=13$ TeV reported by TOTEM probably does not
prove that there is an odderon contribution at $t=0$; see, for instance, Refs.
Donnachie:2019ciz ; Donnachie:2022aiq . However, there are good reasons to
believe that there is an odderon effect at larger $|t|$ (in the dip-bump
region) that leads to a very significant difference between the differential
cross sections of elastic $pp$ and $p\bar{p}$ scattering, as was first seen at
the CERN ISR at $\sqrt{s}=53$ GeV Breakstone:1985pe . Remarkably, such a
difference had been predicted theoretically in Ref. Donnachie:1983hf . The
advantage of corresponding measurements at higher energies, e.g., at the LHC,
is that subleading reggeon exchanges are very small there, and thus an
observation of differences between $pp$ and $p\bar{p}$ would be a clear signal
of the odderon. Indeed, strong evidence for the odderon has been given in Ref.
TOTEM:2020zzr  based on a model-independent analysis of the combined TOTEM and
D0 results in the dip-bump region.

###  IV.2 Our standard results for the $pp\to pp\gamma$ reaction and
comparison with soft-photon approximations

First, we present our exact model or standard bremsstrahlung results for the
$pp\to pp\gamma$ reaction (see Sec. II.3) for the proton-proton collision
energy $\sqrt{s}=13$ TeV. Below, $k_{\perp}$ is the absolute value of the
photon transverse momentum, $\omega=k^{0}$ is the center-of-mass photon
energy, and $\rm{y}$ is the rapidity of the photon. Then, we will compare the
resulting distributions with those obtained via variants of the SPA discussed
in Sec. III.

We see from Fig. 4 that at $\sqrt{s}=13$ TeV the $pp$-scattering cross section
is completely dominated by the pomeron exchange. Therefore, in the following,
we shall take into account only the pomeron-exchange contribution as shown in
the diagrams of Fig. 3 (a)–(f).

In Fig. 7, we show the distributions in $|t_{1,2}|$, where $t_{1,2}$ is either
$t_{1}$ or $t_{2}$ defined in (29), and in transverse momentum of the proton
(here, $p_{t,p}=|\mbox{\boldmath$p_{t,1}^{\prime}$}|$ or
$|\mbox{\boldmath$p_{t,2}^{\prime}$}|$). At $t_{1,2}=0$ and $p_{t,p}=0$, all
contributions vanish. We can see how the $t_{1,2}$ dependence is sensitive to
the form factor in the $\mathbb{P}pp$ vertex (89) and $\epsilon_{\mathbb{P}}$
the pomeron intercept parameter; see the discussion after Eq. (80). In the
following, we take in our calculations $\epsilon_{\mathbb{P}}=0.0865$ and the
exponential pomeron-proton form factor $F(t)=\exp(-b\,|t|)$ with $b=2.95$
GeV-2 adjusted to the TOTEM data (see Fig. 5). We see from Fig. 7 that photons
with $1\,{\rm MeV}<k_{\perp}<100\,{\rm MeV}$ and $|\rm y|<5$ come
predominantly from $pp$ collisions with momentum transfers between the protons
of order $p_{t,p}\sim\sqrt{|t_{1,2}|}\sim 0.3$ GeV. Very small values of
$p_{t,p}$ and/or $|t_{1,2}|$ hardly contribute.

Figure 7:  The distributions in 4-momentum transfer squared $|t_{1,2}|$ and in
the transverse momentum of the outgoing protons for the $pp\to pp\gamma$
reaction calculated for $\sqrt{s}=13$ TeV, $|\rm y|<5$, and $1\,{\rm
MeV}<k_{\perp}<100\,{\rm MeV}$. The meaning of the lines is the same as in
Fig. 5.

In Fig. 8, we show the results obtained for $\sqrt{s}=13$ TeV, $|\rm y|<5$,
and for two $k_{\perp}$ intervals: $1\,{\rm MeV}<k_{\perp}<400\,{\rm MeV}$
(top panels) and $1\,{\rm MeV}<k_{\perp}<100\,{\rm MeV}$ (bottom panels). We
see that in the small $k_{\perp}$ and $\omega$ regions the Dirac term from the
$\gamma pp$ vertex function (38) dominates, while for larger values, the
anomalous magnetic moment of the proton (Pauli term) plays an important role.
Of course, for the complete result, all contributions to ${\cal M}_{\mu}^{(\rm
standard)}$ from the diagrams of Fig. 3 with Dirac and Pauli terms have to be
added coherently. For more details on the size of various contributions to
${\cal M}_{\mu}^{(\rm standard)}$, we refer to the discussions in Appendix B.
We get the integrated cross sections for $\sqrt{s}=13$ TeV and in the
$k_{\perp}$ range $1\,{\rm MeV}<k_{\perp}<100\,{\rm MeV}$: $\sigma(pp\to
pp\gamma)=0.21$ nb for $|\rm y|<3.5$ and $\sigma(pp\to pp\gamma)=4.01$ nb for
$3.5<|\rm y|<5$.

Figure 8:  The differential cross sections for the $pp\to pp\gamma$ reaction
calculated for $\sqrt{s}=13$ TeV, $|\rm y|<5$, and for the $k_{\perp}$
intervals as specified in the figure legends. Shown are our standard results
(total) and the results for the Dirac and Pauli terms alone.

In Fig. 9 (left panels), we show the two-dimensional differential cross
sections in the $\omega$-$k_{\perp}$ plane calculated for $\sqrt{s}=13$ TeV,
$1\,{\rm MeV}<k_{\perp}<100\,{\rm MeV}$, $|\rm y|<3.5$ (the top panel) and
$3.5<|\rm y|<5$ (the bottom panel). The latter rapidity range is relevant for
the planned ALICE 3 measurements Adamova:2019vkf ; QM2022_PBM ; EMMI_RRTF  of
ultrasoft photons. Large ${\rm y}$ is near the $\omega$ axis, and ${\rm y}=0$
corresponds to the line $\omega=k_{\perp}$, both in accordance with
$\omega=k_{\perp}\cosh{\rm y}$. The phase-space region where
$\omega<k_{\perp}$ is forbidden. We see that, due to our cuts on ${\rm y}$ and
$k_{\perp}$ applied in phase space, there are additional regions that are not
accessible kinematically.

In the right panels of Fig. 9, we show the ratio

$\displaystyle{\rm R}(\omega,k_{\perp})=\frac{d^{2}\sigma_{\rm SPA2}/d\omega
dk_{\perp}}{d^{2}\sigma_{\rm standard}/d\omega dk_{\perp}}\,.$ (72)

One can see that the SPA2 given by (68) stays within 1$\,\%$ accuracy for
$k_{\perp}\lesssim 22$ MeV and $\omega\lesssim 0.35$ GeV considering $|\rm
y|<3.5$ and up to $\omega\cong 1.7$ GeV for $3.5<|\rm y|<5.0$. It is difficult
to draw the ratio ${\rm R}(\omega,k_{\perp})$ for SPA1/standard due to
different integration procedures in two different codes: one for the exact
three-body phase space (standard approach) and one for the two-body phase
space supplemented by additional integration over photon 3-momentum (SPA1).

Figure 9:  The left panels show the cross section $d^{2}\sigma/d\omega
dk_{\perp}$ as a function of ($\omega$, $k_{\perp}$) calculated for
$\sqrt{s}=13$ TeV. The top panels are for $1\,{\rm MeV}<k_{\perp}<100\,{\rm
MeV}$, $|\rm y|<3.5$, and the bottom panels are for $3.5<|\rm y|<5$. The right
panels show the ratio ${\rm R}(\omega,k_{\perp})$ as defined in (72).

In Fig. 10, we compare our standard result to various SPAs on a
semilogarithmic scale. We see that both, the SPA1 (65) and SPA2 (68), follow
the standard results very well. Surprisingly, the SPA1, which does not have
the correct energy-momentum relations, fares somewhat better than SPA2. But
let us now have a closer look at these kinematic regions on a linear scale.

Figure 10:  The differential distributions in transverse momentum of the
photon and in the energy of the photon for the $pp\to pp\gamma$ reaction. The
calculations were done for $\sqrt{s}=13$ TeV, $1\,{\rm
MeV}<k_{\perp}<100\,{\rm MeV}$, $|\rm y|<3.5$ (the top panels) and $3.5<|\rm
y|<5$ (the bottom panels). The black solid line corresponds to the standard
result (63), the red dotted line corresponds to SPA1 (65), and the blue long-
dashed line corresponds to SPA2 (68).

Figure 11 shows the ratios of the SPAs to the standard cross sections,

$\displaystyle\frac{d\sigma_{\rm SPA}/dk_{\perp}}{d\sigma_{\rm
standard}/dk_{\perp}}\,\,,$ (73) $\displaystyle\frac{d\sigma_{\rm
SPA}/d\omega}{d\sigma_{\rm standard}/d\omega}\,,$ (74)

as functions of $k_{\perp}$ and $\omega$, respectively. The fluctuations of
the ratio SPA1/standard (see the red dashed lines) are due to a different
organization of integration in the two codes. One can see that the deviations
of the SPA1 from the standard results are up to around 1$\,\%$ for $|\rm
y|<3.5$, $1\,{\rm MeV}<k_{\perp}<100\,{\rm MeV}$, and $\omega\lesssim 0.7$
GeV. For the forward region, $3.5<|\rm y|<5$, this accuracy occurs even up to
larger $\omega\simeq 2$ GeV. For the SPA2, the deviations increase rapidly
with growing $k_{\perp}$ and $\omega$ (see also the right panels of Fig. 9).

Figure 11:  The ratios $\sigma_{\rm SPA}/\sigma_{\rm standard}$ given by (73)
and (74), respectively. The red dotted line corresponds to SPA1 (65), and the
blue long-dashed line corresponds to SPA2 (68). The oscillations in the SPA1
results are of numerical origin.

In Fig. 12, we show the results for the SPA1 (69) and SPA2 (70) Ansätze using
the high-energy small-angle approximation. The calculations were done for
$\sqrt{s}=13$ TeV, $1\,{\rm MeV}<k_{\perp}<400\,{\rm MeV}$, and $|\rm y|<5$.
From the left panel, we see that the SPA1 is in very good agreement with our
standard result if we include there only the Dirac terms. We have checked
numerically that the results from (65) and (69) overlap. From the right panel,
we see that the SPA2 results (68) and (70) are very close to each other.

Figure 12:  Comparison of the cross sections $d\sigma/dk_{\perp}$ for our
standard bremsstrahlung model and the SPA1 (left panel) and for the two
versions of the SPA2 (right panel). In the left panel, we show the standard
complete result and the result for the Dirac term alone from Fig. 8 together
with the SPA1 result (69). In the right panel, the blue long-dashed line
corresponds to (68), while the red solid line corresponds to (70).

###  IV.3 Comments on the photon radiation in connection with diffractive
excitation of the proton

One may ask about the role of diffractive excitations of the protons in
connection with photon radiation. There are two types of exclusive processes
to be considered,

$\displaystyle p+p\to p+\gamma+N^{*}\,,$ (75)

with the $N^{*}$ charged nucleon resonance decaying hadronically, typically
$N^{*}\to p\pi^{0}$, $p\pi^{0}\pi^{0}$, $p\pi^{+}\pi^{-}$, and

$\displaystyle p+p\to(N^{*}\to p+\gamma)+p\,,$ (76) $\displaystyle
p+p\to(N^{*}\to p+\gamma)+(N^{*}\to{\rm hadrons})\,.$ (77)

Relevant examples of diagrams for these processes with pomeron exchange are
shown in Fig. 13 for the reaction (75) and in Fig. 14 for (76) and (77).

Figure 13: A diagram for the reaction (75) with photon emission from the
proton line and hadronic decay of the $N^{*}$. The diagrams where the photon
is attached in all other possible ways to the charged particle lines are not
shown.

(a) (b)

Figure 14: Examples of diagrams: (a) for the reaction (76) and (b) for the
reaction (77).

The reactions (75) and (77) lead to a final state $p+\gamma+p+{\rm pion(s)}$.
Thus, there is no interference with the process $pp\to pp\gamma$ which we
study in this paper. Of course, the question of whether experimentally the
processes (75) and (77) can be separated from the elastic process (4) is a
very relevant one. However, this will depend on the experiment and must be
studied by the experimentalists themselves. The reaction (76), on the other
hand, leads to the final state $pp\gamma$ and, thus, should be discussed here.
Let us first note that the diagram of Fig. 14 (a) will not lead in the
amplitude to a bremsstrahlung term $\propto\omega^{-1}$. Indeed, this diagram
will lead in the amplitude to a factor, for real photons where $k^{2}=0$,

$\displaystyle\left[(p_{1}^{\prime}+k)^{2}-m_{N^{*}}^{2}+im_{N^{*}}\Gamma_{N^{*}}\right]^{-1}=\left[2(p_{1}^{\prime}\cdot
k)-(m_{N^{*}}^{2}-m_{p}^{2})+im_{N^{*}}\Gamma_{N^{*}}\right]^{-1}\,.$ (78)

Here, $m_{N^{*}}$ and $\Gamma_{N^{*}}$ are the mass and width of the $N^{*}$
resonance considered. Clearly, Eq. (78) will not lead to a singular
contribution as $k\to 0$. Thus, in our terminology, it is part of the
structure term (g) of Fig. 3. In the following, we give a short discussion of
$N^{*}$ resonances which can contribute to the photon flux via the diagrams of
the type of Fig. 14 (a).

The $N(1440)$ with $J^{P}=1/2^{+}$, $N(1520)$ with $J^{P}=3/2^{-}$, and
$N(1680)$ with $J^{P}=5/2^{+}$ states are potential candidates for the role of
$N^{*}$ in (76). These resonances satisfy the Gribov-Morrison rule; see, e.g.,
Chap. 3.9 of Ref. Donnachie:2002en . The Roper resonance $N(1440)$ is
doubtful. This poorly known state was considered, however, in Refs.
Jenkovszky:2012hf ; Jenkovszky:2018itd  as an important contribution to
forward low-mass single diffraction dissociation. In Ref. Lebiedowicz:2013vya
, it was shown for the $pp\to pp\pi^{0}$ reaction that the nonresonant
diffractive processes (Drell-Hiida-Deck type model) lead to an enhancement in
the invariant mass $M_{\pi p}\approx 1.4$ GeV. Thus, one cannot be sure
whether the Roper resonance plays an important role there. Coming back to the
$pp\to(N^{*}\to p\gamma)p$ reaction, the Roper resonance has much smaller
branching ratio Zyla:2020zbs  ${\cal BR}(N(1440)\to p\gamma)=0.035-0.048\,\%$
than the $N(1520)$, ${\cal BR}(N(1520)\to p\gamma)=0.31-0.52\,\%$, and the
$N(1680)$, ${\cal BR}(N(1680)\to p\gamma)=0.21-0.32\,\%$. For the $N(1680)$
resonance, a sizeable cross section $\sigma(pp\to pN(1680))=170\pm 60$ $\mu$b
was estimated at the CERN ISR energy of $\sqrt{s}=45$ GeV deKerret:1976ze .
One should, however, bear in mind that at ISR energies secondary reggeon
exchanges are still important. For the bremsstrahlung-type processes at the
LHC energies, the reggeon-exchange contributions are very small; thus,
diffractively excited $N^{*}$ resonances are there due to pomeron exchange and
should be produced preferentially in forward/backward rapidity region. All
these processes require dedicated studies if the low $\gamma p$ invariant-mass
region can be measured in the forward rapidity range.

In Fig. 15, we show the differential cross sections $d\sigma/dM_{\gamma
p,\,{\rm low}}$ (the same sign rapidity of $\gamma$ and $p$) and
$d\sigma/dM_{\gamma p,\,{\rm high}}$ (the opposite sign rapidity of $\gamma$
and $p$) for our standard bremsstrahlung (nonresonant) model. The calculations
were done for $\sqrt{s}=13$ TeV, $3.5<{\rm y}<5$, and $1\;{\rm
MeV}<k_{\perp}<100\;{\rm MeV}$. Figure 16 shows the two-dimensional
distributions in ($M_{\gamma p,\,{\rm low}},{\rm y}$) and in ($M_{\gamma
p,\,{\rm low}},k_{\perp}$). In the present paper we consider soft-photon
emission only by protons. As already mentioned, photons may also be produced
from the radiative decays of diffractively excited nucleon resonances.
Candidates are $N(1440)$, $N(1520)$, and $N(1680)$. If these processes
contribute significantly to our $pp\to pp\gamma$ reaction, then we should see
them in the $M_{\gamma p,\,{\rm low}}$ distribution (possibly distorted by
interference effects) as a resonance enhancement at $M_{\gamma p}=m_{N^{*}}$
with a width $\Gamma_{N^{*}}$ over our nonresonant term. However, we expect
that the decay photons will be emitted at rapidities larger than covered by
the dedicated ALICE 3 detector. Other background contributions, for instance,
soft photons from central-exclusive production via the fusion processes
$\gamma\mathbb{P}\to\gamma$, $\mathbb{O}\mathbb{P}\to\gamma$, etc., should be
taken into account. Possible interference effects between various mechanisms
should be also included. This goes beyond the scope of the present paper. We
leave a detailed analysis of other contributions for future studies. Finally,
we conclude that the measurement of forward/backward protons would be crucial
for a better understanding of the mechanisms of the $pp\to pp\gamma$ reaction.

Figure 15:  The distributions in $M_{\gamma p,\,{\rm low}}$ (left panel) and
in $M_{\gamma p,\,{\rm high}}$ (right panel) for our standard bremsstrahlung
(nonresonant) model.

Figure 16:  The two-dimensional distributions in ($M_{\gamma p,\,{\rm
low}},{\rm y}$) and in ($M_{\gamma p,\,{\rm low}},k_{\perp}$) for the $pp\to
pp\gamma$ reaction for our standard bremsstrahlung (nonresonant) model.

##  V Conclusions

In this paper, we have presented a detailed discussion of the reactions $pp\to
pp$, $p\bar{p}\to p\bar{p}$, and $pp\to pp\gamma$ at high c.m. energies using
the framework of the tensor-pomeron model Ewerz:2013kda . To calculate the
amplitudes for these reactions we have considered pomeron ($\mathbb{P}$),
odderon ($\mathbb{O}$), and reggeon ($f_{2\mathbb{R}}$, $a_{2\mathbb{R}}$,
$\omega_{\mathbb{R}}$, $\rho_{\mathbb{R}}$) exchanges.

Our theoretical results for elastic $pp$ and $p\bar{p}$ scattering are given
in Secs. II.1 and II.2 and compared to experimental data in Secs. IV.1 and
IV.2. With our model, adjusting the pomeron and odderon parameters, we get a
reasonable fit to the TOTEM data on $pp$ elastic scattering for $\sqrt{s}=13$
TeV and $|t|\lesssim 0.3$ GeV2; see Fig. 5. We emphasize that we are not out
to produce a precision fit to all $pp$ and $p\bar{p}$ elastic scattering data.
We only need a reasonable description of the data for $\sqrt{s}=13$ TeV as a
prerequisite for the calculation of photon radiation in $pp$ collisions at
this energy. Nevertheless, we also had a look at the data for $\rho$, the
ratio of real and imaginary parts of the forward-scattering amplitudes. We
found that, taking the TOTEM data for granted, we need an odderon contribution
in the framework of our model, and a double Regge pole structure of this
odderon with intercept slightly above 1 seems to be preferred; see Fig. 6.
Clearly, further experimental studies of reactions where odderon effects could
be present would be very welcome. Examples of such reactions are
photoproduction of $f_{2}$ mesons, $\gamma p\to f_{2}p$, and central exclusive
production of single and double $\phi$ mesons, $pp\to p\phi p$ and $pp\to
p\phi\phi p$; see, for instance, Refs. Bolz:2014mya ; Lebiedowicz:2019boz ;
Lebiedowicz:2019jru . All these reactions can be studied at ALICE3.

Turning now to the main topic of our paper, the calculation of soft-photon
radiation in $pp$ collisions, we have given the explicit formulas for this
process in the framework of our tensor-pomeron model. The amplitudes
corresponding to photon emission from the external proton lines [see the
diagrams of Fig. 3 (a), (b), (d), and (e)] are determined by the off-shell
$pp\to pp$ scattering amplitude. By construction, the amplitudes (c) and (f),
the contact terms, have to satisfy gauge-invariance constraints involving the
previous amplitudes. In this way, we obtained our standard results for the
$pp\to pp\gamma$ reaction. The relevant distributions are shown in Figs. 7–11.

We have compared our standard results to two soft-photon approximations SPA1
and SPA2; see Sec. III. In the SPA1, we considered only the pole terms
$\propto\omega^{-1}$ in the radiative amplitudes. This SPA1 agrees rather
well, at the percent level, with our exact model (or standard result) in the
kinematic range considered; see Figs. 10 and 11. For $1\;{\rm
MeV}<k_{\perp}<100\;{\rm MeV}$ and $3.5<|\rm y|<5.0$, for instance, we find
agreement of SPA1 with our standard result at the percent level up to
$\omega\cong 2$ GeV. The SPA2 is a good approximation to the standard result,
within 1$\,\%$ accuracy, for $k_{\perp}\lesssim 22$ MeV and $\omega\lesssim
0.35$ GeV considering $|\rm y|<3.5$ and up to $\omega\cong 1.7$ GeV for
$3.5<|\rm y|<5.0$; see Fig. 9.

In Sec. II.3, we have written the amplitude for $pp\to pp\gamma$ in the
standard, straightforward, way with the photon coupling to the protons via the
Dirac and Pauli terms. It turned out that the corresponding formulas (40),
(41), (57), and (60)–(62) were not convenient for numerical work. Therefore,
we have rewritten these amplitudes in Appendix B as a sum of seven terms,
labeled by $j=1,\ldots,7$, for the tensor exchanges and of four terms, labeled
by $j^{\prime}=1,\ldots,4$, for the vector exchanges [see (107)–(120)]. Each
of these 11 terms is separately gauge invariant. The pole terms are only
contained in the $j=1$ tensor term and $j^{\prime}=1$ vector term. The other
terms have no singularity for $\omega\to 0$. Thus, one expects that for soft-
photon emission the terms with $j=1$ and $j^{\prime}=1$ will be dominant. We
found that indeed this is correct but there is a very interesting effect
guaranteeing this. Considering for pomeron exchange the gauge-invariant and
nonsingular terms with $j=2$ and $j=4$ alone, we find that the pole term only
dominates over these $j=2$ and $j=4$ terms individually for very small
$k_{\perp}\approx\omega\lesssim 2m_{p}^{2}/\sqrt{s}\cong 0.15$ MeV [see (126),
(127), and Fig. 18]. This effect is, in essence, due to the Pauli coupling of
the photon. We note that in the literature such small values for $\omega$ as a
limit for the dominance of the $\omega^{-1}$ term are mentioned. For instance,
in Ref. DelDuca:1990gz , it is argued that for hard high-energy elastic
processes Low’s original result gives a reliable representation of the
radiative amplitude only in the vanishingly small region $\omega\lesssim
m^{2}/Q$ in the limit $Q\to\infty$. Here, $Q$ is the scale of the hard
process, and $m$ is the charged particle mass. But since in this work only
hard processes with photon emission are considered, these arguments do not
apply to our case. We consider the exclusive soft process $pp\to pp\gamma$
with soft-photon emission. We have, of course, to take all contributions with
different labels $j$ into account and add them coherently; see (107). We find
then large cancellations between the $j=2$ and $j=4$ terms; see Fig. 17. This
leads to a much larger region in $k_{\perp}$ and $\omega$ where the pole terms
alone give a good representation of the radiative amplitude; see Fig. 9. Our
conclusion is that simple order of magnitude estimates for the $\omega$
regions where the pole term dominates, using only parts of the amplitudes, may
give completely wrong results. It is essential to add coherently all the
various parts of the amplitude for soft-photon emission in order not to miss
important interference effects.

In this article, we have only discussed the bremsstrahlung-type emission of
soft photons in $pp$ collisions. Anomalous emission terms have been subsumed
in the amplitude ${\cal M}_{\mu}^{(g)}$, which must satisfy (59) and can have
no singularity for $k\to 0$. There are, however, quite conventional
contributions to ${\cal M}_{\mu}^{(g)}$, for instance, soft photons from
central-exclusive production via the fusion processes
$\gamma\mathbb{P}\to\gamma$, $\mathbb{O}\mathbb{P}\to\gamma$. The
contributions of such processes are expected to be important in the
midrapidity region, around ${\rm y}=0$. In a future paper, we plan to study
central-exclusive production of photons within the tensor-pomeron approach.

Finally, we emphasize that we have taken care to write the formulas for the
$pp\to pp\gamma$ amplitude in such a way that they also apply to soft virtual
photon production, for instance, $pp\to pp(\gamma^{*}\to e^{+}e^{-})$. Thus,
our Eqs. (40), (41), (57), and (60)–(62), as well as (105)–(120), can be
directly used for soft virtual photon production. But further investigations
of this interesting topic go beyond the scope of our present article.

We hope that our theoretical studies of the exclusive $pp\to pp\gamma$
reaction will find experimental counterparts with measurements of soft photons
at the Relativistic Heavy Ion Collider and at the LHC, for instance, with the
planned ALICE3 detector Adamova:2019vkf ; QM2022_PBM ; EMMI_RRTF .

###### Acknowledgements.

We thank Johanna Stachel, Peter Braun-Munzinger, and Carlo Ewerz for very
useful discussions. We are grateful to Carlo Ewerz for reading the manuscript.
This work and the stay of Piotr Lebiedowicz in Heidelberg were supported by
the Bekker Program of the Polish National Agency for Academic Exchange under
Contract No. BPN/BEK/2021/2/00009/U/00001. This study was also partially
supported by the Polish National Science Centre under Grant No.
2018/31/B/ST2/03537.

##  Appendix A List of quantities and their values used in the paper

The quantities listed in the following are in essence taken from Chap. 3 of
Ref. Ewerz:2013kda . Here, the propagators and vertices involving the pomeron,
the odderon, and the reggeons are to be understood as effective propagators
and vertices.

The effective propagators for $\mathbb{P}$, reggeons, and $\mathbb{O}$ are as
follows:

  * •

pomeron $\mathbb{P}$ (see (3.10), (3.11), and Sec. 6.1 of Ref. Ewerz:2013kda
):

$\displaystyle
i\Delta^{(\mathbb{P})}_{\mu\nu,\kappa\lambda}(s,t)=\frac{1}{4s}\left(g_{\mu\kappa}g_{\nu\lambda}+g_{\mu\lambda}g_{\nu\kappa}-\frac{1}{2}g_{\mu\nu}g_{\kappa\lambda}\right)(-is\alpha^{\prime}_{\mathbb{P}})^{\alpha_{\mathbb{P}}(t)-1}\,,$
(79)
$\displaystyle\alpha_{\mathbb{P}}(t)=\alpha_{\mathbb{P}}(0)+\alpha^{\prime}_{\mathbb{P}}\,t\,,\quad\alpha_{\mathbb{P}}(0)=1+\epsilon_{\mathbb{P}}\,,$
$\displaystyle\alpha^{\prime}_{\mathbb{P}}=0.25\;\mathrm{GeV}^{-2}\,.$ (80)

The default value for $\epsilon_{\mathbb{P}}$ from Ref. Ewerz:2013kda  is
$\epsilon_{\mathbb{P}}=0.0808$. In our present paper we find from a comparison
to the data from TOTEM measurements at $\sqrt{s}=13$ TeV TOTEM:2017sdy ;
TOTEM:2018hki  a slightly higher value: $\epsilon_{\mathbb{P}}=0.0865$; see
Fig. 5. This value for the soft pomeron intercept is in agreement with those
obtained in Refs. Cudell:1996sh ; Luna:2003kw ; Luna:2004gr . Our value above
also agrees, within practically 1 standard deviation, with that obtained in
Ref. Britzger:2019lvc  from a fit to photoproduction and low-x deep inelastic
scattering which gave $\epsilon_{\mathbb{P}}=0.0935(^{+76}_{-64})$.

  * •

reggeons $\mathbb{R}_{+}=f_{2\mathbb{R}},a_{2\mathbb{R}}$ (see (3.12), (3.13),
and Sec. 6.3 of Ref. Ewerz:2013kda ):

$\displaystyle
i\Delta^{(\mathbb{R}_{+})}_{\mu\nu,\kappa\lambda}(s,t)=\frac{1}{4s}\left(g_{\mu\kappa}g_{\nu\lambda}+g_{\mu\lambda}g_{\nu\kappa}-\frac{1}{2}g_{\mu\nu}g_{\kappa\lambda}\right)(-is\alpha^{\prime}_{\mathbb{R}_{+}})^{\alpha_{\mathbb{R}_{+}}(t)-1}\,,$
(81)
$\displaystyle\alpha_{\mathbb{R}_{+}}(t)=\alpha_{\mathbb{R}_{+}}(0)+\alpha^{\prime}_{\mathbb{R}_{+}}\,t\,,$
$\displaystyle\alpha_{\mathbb{R}_{+}}(0)=0.5475\,,$
$\displaystyle\alpha^{\prime}_{\mathbb{R}_{+}}=0.9\;\mathrm{GeV}^{-2}\,.$ (82)

  * •

reggeons $\mathbb{R}_{-}=\omega_{\mathbb{R}},\rho_{\mathbb{R}}$ (see (3.14),
(3.15), and Sec. 6.3 of Ref. Ewerz:2013kda ):

$\displaystyle
i\Delta^{(\mathbb{R}_{-})}_{\mu\nu}(s,t)=ig_{\mu\nu}\frac{1}{M_{-}^{2}}(-is\alpha^{\prime}_{\mathbb{R}_{-}})^{\alpha_{\mathbb{R}_{-}}(t)-1}\,,$
(83)
$\displaystyle\alpha_{\mathbb{R}_{-}}(t)=\alpha_{\mathbb{R}_{-}}(0)+\alpha^{\prime}_{\mathbb{R}_{-}}\,t\,,$
$\displaystyle\alpha_{\mathbb{R}_{-}}(0)=0.5475\,,$
$\displaystyle\alpha^{\prime}_{\mathbb{R}_{-}}=0.9\;\mathrm{GeV}^{-2}\,,$
$\displaystyle M_{-}=1.41\;\mathrm{GeV}\,.$ (84)

The numbers for the reggeon parameters in (82) and (84) are taken from Ref.
Donnachie:1992ny  and Figs. 3.1 and 3.2 of Ref. Donnachie:2002en , except for
$M_{-}$, which is discussed in Sec. 6.3 of Ref. Ewerz:2013kda .

  * •

odderon $\mathbb{O}$

Our Ansatz for a single-pole odderon is as in Ref. Ewerz:2013kda  [see (3.16),
(3.17), and Sec. 6.2 therein]:

$\displaystyle
i\Delta^{(\mathbb{O})}_{\mu\nu}(s,t)=-ig_{\mu\nu}\frac{\eta_{\mathbb{O}}}{M_{0}^{2}}(-is\alpha^{\prime}_{\mathbb{O}})^{\alpha_{\mathbb{O}}(t)-1}\,,$
(85)
$\displaystyle\alpha_{\mathbb{O}}(t)=\alpha_{\mathbb{O}}(0)+\alpha^{\prime}_{\mathbb{O}}\,t\,,\quad\alpha_{\mathbb{O}}(0)=1+\epsilon_{\mathbb{O}}\,,$
$\displaystyle M_{0}=1\;\mathrm{GeV}\,,$ $\displaystyle\eta_{\mathbb{O}}=\pm
1\,,$ $\displaystyle\alpha^{\prime}_{\mathbb{O}}=0.25\;\mathrm{GeV}^{-2}\,.$
(86)

Here, $M_{0}=1$ GeV is introduced for dimensional reasons, and
$\alpha^{\prime}_{\mathbb{O}}=\alpha^{\prime}_{\mathbb{P}}$ (80) is set as
default.

For a double-pole odderon, we set

$i\widetilde{\Delta}^{(\mathbb{O})}_{\mu\nu}(s,t)=i\Delta^{(\mathbb{O})}_{\mu\nu}(s,t)\left[C_{1}+C_{2}\ln\left(-is\alpha_{\mathbb{O}}^{\prime}\right)\right]\,.$
(87)

Here, $C_{1}$ and $C_{2}$ are real constants.

From the comparison with the $\rho$ values of $pp$ and $p\bar{p}$ scattering,
we find that the following values give a reasonable description of the TOTEM
data (see Fig. 6):

$\displaystyle\eta_{\mathbb{O}}=-1\,,\quad\alpha^{\prime}_{\mathbb{O}}=0.25\;\mathrm{GeV}^{-2}\,,\quad\epsilon_{\mathbb{O}}=0.0800\,,$
$\displaystyle(C_{1},C_{2})=(-1.0,0.1),(-1.5,0.2),(-2.0,0.3)\,.$ (88)

The effective proton-pomeron, proton-reggeon, and proton-odderon vertices are
taken as in Ref. Ewerz:2013kda , but at least for the $\mathbb{P}pp$ vertex,
we use a different form factor in order to fit the TOTEM data TOTEM:2017sdy ;
TOTEM:2018hki  at $\sqrt{s}=13$ TeV and in the low $|t|$ region.

Now here is the list of effective vertices.

  * •

$\mathbb{P}pp$ vertex (see (3.43), (3.44), and Sec. 6.1 of Ref. Ewerz:2013kda
):

$\displaystyle i\Gamma_{\mu\nu}^{(\mathbb{P}pp)}(p^{\prime},p)$
$\displaystyle=$ $\displaystyle
i\Gamma_{\mu\nu}^{(\mathbb{P}\bar{p}\bar{p})}(p^{\prime},p)$ (89)
$\displaystyle=$
$\displaystyle-i3\beta_{\mathbb{P}pp}F_{1}[(p^{\prime}-p)^{2}]$
$\displaystyle\times\left\\{\frac{1}{2}\left[\gamma_{\mu}(p^{\prime}+p)_{\nu}+\gamma_{\nu}(p^{\prime}+p)_{\mu}\right]-\frac{1}{4}g_{\mu\nu}(\not\\!p^{\prime}+\not\\!p)\right\\}\,,\quad$
$\displaystyle\beta_{\mathbb{P}pp}$ $\displaystyle=$ $\displaystyle
1.87\;\mathrm{GeV}^{-1}\,.$ (90)

In Ref. Ewerz:2013kda , the form factor $F_{1}(t)$ was taken as the
electromagnetic form factor of the proton,

$\displaystyle
F_{1}(t)=\big{(}1-\frac{t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\big{)}\big{(}1-\frac{t}{4m_{p}^{2}}\big{)}^{-1}\,G_{D}(t)\,,$
$\displaystyle\mu_{N}=\frac{e}{2m_{p}},\quad\frac{\mu_{p}}{\mu_{N}}=2.7928\,,$
$\displaystyle G_{D}(t)=\big{(}1-\frac{t}{m_{D}^{2}}\big{)}^{-2},\quad
m_{D}^{2}=0.71\;{\rm GeV}^{2}\,.$ (91)

In the present paper, we use $F_{1}(t)\to F(t)=\exp(-b\,|t|)$ with $b=2.95$
GeV-2 adjusted to the TOTEM data (see Fig. 5).

  * •

$\mathbb{R}_{+}pp$ vertex, where
$\mathbb{R}_{+}=f_{2\mathbb{R}},a_{2\mathbb{R}}$ (see (3.49)–(3.52) and Sec.
6.3 of Ref. Ewerz:2013kda ):

$\displaystyle i\Gamma_{\mu\nu}^{(f_{2\mathbb{R}}pp)}(p^{\prime},p)$
$\displaystyle=$ $\displaystyle
i\Gamma_{\mu\nu}^{(f_{2\mathbb{R}}\bar{p}\bar{p})}(p^{\prime},p)$ (92)
$\displaystyle=$ $\displaystyle-
ig_{f_{2\mathbb{R}}pp}\frac{1}{M_{0}}F_{1}[(p^{\prime}-p)^{2}]$
$\displaystyle\times\left\\{\frac{1}{2}\left[\gamma_{\mu}(p^{\prime}+p)_{\nu}+\gamma_{\nu}(p^{\prime}+p)_{\mu}\right]-\frac{1}{4}g_{\mu\nu}(\not\\!p^{\prime}+\not\\!p)\right\\}\,,\quad$
$\displaystyle g_{f_{2\mathbb{R}}pp}$ $\displaystyle=$ $\displaystyle
11.04\,,\quad M_{0}=1\;\mathrm{GeV}\,;$ (93) $\displaystyle
i\Gamma_{\mu\nu}^{(a_{2\mathbb{R}}pp)}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle
i\Gamma_{\mu\nu}^{(a_{2\mathbb{R}}\bar{p}\bar{p})}(p^{\prime},p)$ (94)
$\displaystyle=$ $\displaystyle-
ig_{a_{2\mathbb{R}}pp}\frac{1}{M_{0}}F_{1}[(p^{\prime}-p)^{2}]$
$\displaystyle\times\left\\{\frac{1}{2}\left[\gamma_{\mu}(p^{\prime}+p)_{\nu}+\gamma_{\nu}(p^{\prime}+p)_{\mu}\right]-\frac{1}{4}g_{\mu\nu}(\not\\!p^{\prime}+\not\\!p)\right\\}\,,\quad$
$\displaystyle g_{a_{2\mathbb{R}}pp}$ $\displaystyle=$ $\displaystyle
1.68\,,\quad M_{0}=1\;\mathrm{GeV}\,.$ (95)

  * •

$\mathbb{R}_{-}pp$ vertex, where
$\mathbb{R}_{-}=\omega_{\mathbb{R}},\rho_{\mathbb{R}}$ (see (3.59)–(3.62) and
Sec. 6.3 of Ewerz:2013kda ):

$\displaystyle i\Gamma_{\mu}^{(\omega_{\mathbb{R}}pp)}(p^{\prime},p)$
$\displaystyle=$
$\displaystyle-i\Gamma_{\mu}^{(\omega_{\mathbb{R}}\bar{p}\bar{p})}(p^{\prime},p)$
(96) $\displaystyle=$ $\displaystyle-
ig_{\omega_{\mathbb{R}}pp}F_{1}[(p^{\prime}-p)^{2}]\gamma_{\mu}\,,$
$\displaystyle g_{\omega_{\mathbb{R}}pp}$ $\displaystyle=$ $\displaystyle
8.65\,;$ (97) $\displaystyle
i\Gamma_{\mu}^{(\rho_{\mathbb{R}}pp)}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle-i\Gamma_{\mu}^{(\rho_{\mathbb{R}}\bar{p}\bar{p})}(p^{\prime},p)$
(98) $\displaystyle=$ $\displaystyle-
ig_{\rho_{\mathbb{R}}pp}F_{1}[(p^{\prime}-p)^{2}]\gamma_{\mu}\,,$
$\displaystyle g_{\rho_{\mathbb{R}}pp}$ $\displaystyle=$ $\displaystyle
2.02\,.$ (99)

  * •

$\mathbb{O}pp$ vertex (see (3.68), (3.69), and Sec. 6.2 of Ref. Ewerz:2013kda
):

$\displaystyle i\Gamma_{\mu}^{(\mathbb{O}pp)}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle-i\Gamma_{\mu}^{(\mathbb{O}\bar{p}\bar{p})}(p^{\prime},p)$ (100)
$\displaystyle=$
$\displaystyle-i3\beta_{\mathbb{O}pp}M_{0}\,F_{1}[(p^{\prime}-p)^{2}]\gamma_{\mu}\,,$
$\displaystyle\beta_{\mathbb{O}pp}$ $\displaystyle=$ $\displaystyle
0.2\;\mathrm{GeV}^{-1}\,,\quad M_{0}=1\;\mathrm{GeV}\,.$ (101)

The coupling constant $\beta_{\mathbb{O}pp}$ is taken as free parameter to be
determined by experiment. In the present study, we assume (101):
$\beta_{\mathbb{O}pp}=0.1\times\beta_{\mathbb{P}pp}\simeq 0.2\;{\rm
GeV}^{-1}$.

At the end of this Appendix, we give the formulas corresponding to (50), (51),
and (55) for the case of a double-pole odderon (87). We have here to replace
${\cal F}_{\mathbb{O}pp}(s,t)$ by $\widetilde{\cal F}_{\mathbb{O}pp}(s,t)$
according to (18). In this way, we get instead of (50)

$\displaystyle\widetilde{\cal
F}_{\mathbb{O}pp}(s^{\prime},t_{2})=\widetilde{\cal
F}_{\mathbb{O}pp}(s,t_{2})+\varkappa\,\widetilde{\Delta\cal
F}_{\mathbb{O}pp}(s,t_{2},\varkappa)\,,$ (102)

where

$\displaystyle\widetilde{\Delta\cal F}_{\mathbb{O}pp}(s,t_{2},\varkappa)$
$\displaystyle=$ $\displaystyle{\cal
F}_{\mathbb{O}pp}(s,t_{2})\big{\\{}C_{1}\big{(}1-\alpha_{\mathbb{O}}(t_{2})\big{)}g_{\mathbb{O}}(\varkappa,t_{2})$
$\displaystyle+C_{2}\big{[}\big{(}1-\alpha_{\mathbb{O}}(t_{2})\big{)}g_{\mathbb{O}}(\varkappa,t_{2})\ln\big{(}-is(1-\varkappa)\alpha_{\mathbb{O}}^{\prime}\big{)}+\frac{1}{\varkappa}\ln(1-\varkappa)\big{]}\big{\\}}\,.$

For $\Delta{\cal F}_{V}(s,t,\varkappa)$ in (55), this gives the replacement

$\displaystyle\widetilde{\Delta\cal F}_{V}(s,t,\varkappa)$ $\displaystyle=$
$\displaystyle\widetilde{\Delta\cal
F}_{\mathbb{O}pp}(s,t,\varkappa)+(1-\alpha_{\omega_{\mathbb{R}}}(t))\,g_{\omega_{\mathbb{R}}}(\varkappa,t)\,{\cal
F}_{\omega_{\mathbb{R}}pp}(s,t)$ (104)
$\displaystyle+(1-\alpha_{\rho_{\mathbb{R}}}(t))\,g_{\rho_{\mathbb{R}}}(\varkappa,t)\,{\cal
F}_{\rho_{\mathbb{R}}pp}(s,t)\,.$

##  Appendix B The amplitudes with photon emission

Here, we rewrite the amplitude ${\cal M}_{\mu}^{(a+b+c)}={\cal
M}_{\mu}^{(a)}+{\cal M}_{\mu}^{(b)}+{\cal M}_{\mu}^{(c)}$, see (40), (41), and
(57), in a way that is more suitable for numerical computations.

We use the following relations:

$\displaystyle\frac{\not\\!p_{a}-\not\\!k+m_{p}}{(p_{a}-k)^{2}-m_{p}^{2}+i\varepsilon}\big{(}\gamma_{\mu}-\frac{i}{2m_{p}}\sigma_{\mu\nu}k^{\nu}F_{2}(0)\big{)}u_{a}$
$\displaystyle\qquad=\frac{1}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}\big{\\{}2p_{a\mu}-k_{\mu}+(k_{\mu}-\not\\!k\gamma_{\mu})+\frac{F_{2}(0)}{2m_{p}}\big{[}2(p_{a\mu}\not\\!k-(p_{a}\cdot
k)\gamma_{\mu})$
$\displaystyle\qquad\quad+2m_{p}(k_{\mu}-\not\\!k\gamma_{\mu})-(\not\\!kk_{\mu}-k^{2}\gamma_{\mu})\big{]}\big{\\}}u_{a}\,,$
(105)
$\displaystyle\bar{u}_{1^{\prime}}\big{(}\gamma_{\mu}-\frac{i}{2m_{p}}\sigma_{\mu\nu}k^{\nu}F_{2}(0)\big{)}\frac{\not\\!p_{1}^{\prime}+\not\\!k+m_{p}}{(p_{1}^{\prime}+k)^{2}-m_{p}^{2}+i\varepsilon}$
$\displaystyle\qquad=\bar{u}_{1^{\prime}}\frac{1}{2p_{1}^{\prime}\cdot
k+k^{2}+i\varepsilon}\big{\\{}2p_{1\mu}^{\prime}+k_{\mu}-(k_{\mu}-\gamma_{\mu}\not\\!k)+\frac{F_{2}(0)}{2m_{p}}\big{[}-2(p_{1\mu}^{\prime}\not\\!k-(p_{1}^{\prime}\cdot
k)\gamma_{\mu})$
$\displaystyle\qquad\quad-2m_{p}(k_{\mu}-\gamma_{\mu}\not\\!k)-(k_{\mu}\not\\!k-k^{2}\gamma_{\mu})\big{]}\big{\\}}\,.$
(106)

Using (105) and (106) and exploiting the properties of the Dirac spinors,
$\not\\!p_{a}u_{a}=m_{p}u_{a}$,
$\bar{u}_{1^{\prime}}\not\\!p_{1^{\prime}}=\bar{u}_{1^{\prime}}m_{p}$ etc., we
can write our ${\cal M}_{\mu}^{(\rm standard)}$ from (63) as follows:

$\displaystyle{\cal M}_{\mu}^{(\rm standard)}=\sum_{j=1}^{7}\big{(}{\cal
M}_{{\rm T},\mu}^{(a+b+c)\,j}+{\cal M}_{{\rm
T},\mu}^{(d+e+f)\,j}\big{)}+\sum_{j^{\prime}=1}^{4}\big{(}{\cal M}_{{\rm
V},\mu}^{(a+b+c)\,j^{\prime}}+{\cal M}_{{\rm
V},\mu}^{(d+e+f)\,j^{\prime}}\big{)}\,.$ (107)

Here, ${\rm T}$ and ${\rm V}$ stand for the tensor- and vector-exchange
diagrams, respectively, and $j$ and $j^{\prime}$ are just labels for the
subamplitudes in the sums on the rhs of (107). We have

$\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,1}$ $\displaystyle=$
$\displaystyle e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s,t_{2})\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime},p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime})$
(108)
$\displaystyle-2m_{p}^{2}\mathbb{1}\otimes\mathbb{1}\big{]}\big{[}\frac{2p_{a\mu}-k_{\mu}}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}+\frac{2p_{1\mu}^{\prime}+k_{\mu}}{2p_{1}^{\prime}\cdot
k+k^{2}+i\varepsilon}\big{]}\big{\\}}u_{a}\otimes u_{b}\,,$
$\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,2}$ $\displaystyle=$
$\displaystyle e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s^{\prime},t_{2})\frac{1}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})$
(109)
$\displaystyle+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime}-\not\\!k)\big{]}\big{[}k_{\mu}-\not\\!k\gamma_{\mu}+\frac{F_{2}(0)}{2m_{p}}\big{(}2p_{a\mu}\not\\!k-2(p_{a}\cdot
k)\gamma_{\mu}$
$\displaystyle+2m_{p}(k_{\mu}-\not\\!k\gamma_{\mu})-(\not\\!kk_{\mu}-k^{2}\gamma_{\mu})\big{)}\big{]}\otimes\mathbb{1}\big{\\}}u_{a}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,3}$ $\displaystyle=$
$\displaystyle-e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s^{\prime},t_{2})\frac{m_{p}}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}\big{[}2p_{a\mu}\not\\!k-2(p_{a}\cdot
k)\gamma_{\mu}+2m_{p}(k_{\mu}-\not\\!k\gamma_{\mu})$ (110)
$\displaystyle-(\not\\!kk_{\mu}-k^{2}\gamma_{\mu})+\frac{F_{2}(0)}{2m_{p}}\big{(}(-2(p_{a}\cdot
k)+k^{2}+4m_{p}^{2})(k_{\mu}-\not\\!k\gamma_{\mu})$
$\displaystyle+2m_{p}(2p_{a\mu}\not\\!k-2(p_{a}\cdot
k)\gamma_{\mu})-2m_{p}(\not\\!kk_{\mu}-k^{2}\gamma_{\mu})\big{)}\big{]}\otimes\mathbb{1}\big{\\}}u_{a}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,4}$ $\displaystyle=$
$\displaystyle e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s,t_{2})\frac{1}{2p_{1}^{\prime}\cdot
k+k^{2}+i\varepsilon}\big{[}-(k_{\mu}-\gamma_{\mu}\not\\!k)$ (111)
$\displaystyle+\frac{F_{2}(0)}{2m_{p}}\big{(}-2p_{1\mu}^{\prime}\not\\!k+2(p_{1}^{\prime}\cdot
k)\gamma_{\mu}-2m_{p}(k_{\mu}-\gamma_{\mu}\not\\!k)-(k_{\mu}\not\\!k-k^{2}\gamma_{\mu})\big{)}\big{]}\otimes\mathbb{1}$
$\displaystyle\times\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime}+k,p_{b}+p_{2}^{\prime})+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime}+\not\\!k)\big{]}\big{\\}}u_{a}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,5}$ $\displaystyle=$
$\displaystyle-e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s,t_{2})\frac{m_{p}}{2p_{1}^{\prime}\cdot
k+k^{2}+i\varepsilon}\big{[}-2p_{1\mu}^{\prime}\not\\!k+2(p_{1}^{\prime}\cdot
k)\gamma_{\mu}-2m_{p}(k_{\mu}-\gamma_{\mu}\not\\!k)$ (112)
$\displaystyle-(\not\\!kk_{\mu}-k^{2}\gamma_{\mu})-\frac{F_{2}(0)}{2m_{p}}\big{(}(2(p_{1}^{\prime}\cdot
k)+k^{2}+4m_{p}^{2})(k_{\mu}-\gamma_{\mu}\not\\!k)$
$\displaystyle+2m_{p}(2p_{1\mu}^{\prime}\not\\!k-2(p_{1}^{\prime}\cdot
k)\gamma_{\mu})+2m_{p}(\not\\!kk_{\mu}-k^{2}\gamma_{\mu})\big{)}\big{]}\otimes\mathbb{1}\big{\\}}u_{a}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,6}$ $\displaystyle=$
$\displaystyle
e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i\Delta{\cal
F}_{T}(s,t_{2},\varkappa)\big{[}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{a}+p_{1}^{\prime}-k,p_{b}+p_{2}^{\prime})$
(113)
$\displaystyle+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes(\not\\!p_{a}+\not\\!p_{1}^{\prime}-\not\\!k)-m_{p}(2m_{p}-\not\\!k)\otimes\mathbb{1}\big{]}$
$\displaystyle\times\big{[}\frac{(2p_{a}+2p_{b}-k,k)}{s}\frac{2p_{a\mu}-k_{\mu}}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}+\frac{(2p_{a}+2p_{b}-k)_{\mu}}{s}\big{]}\big{\\}}u_{a}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,7}$ $\displaystyle=$
$\displaystyle e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}i{\cal
F}_{T}(s,t_{2})\big{[}\big{(}\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{b}+p_{2}^{\prime},k)+(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes\not\\!k-m_{p}\not\\!k\otimes\mathbb{1}\big{)}$
(114)
$\displaystyle\times\big{(}\frac{2p_{1\mu}^{\prime}+k_{\mu}}{2p_{1}^{\prime}\cdot
k+k^{2}+i\varepsilon}-\frac{2p_{a\mu}-k_{\mu}}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}\big{)}$
$\displaystyle-2\gamma^{\alpha}\otimes\gamma_{\alpha}(p_{b}+p_{2}^{\prime})_{\mu}-2(\not\\!p_{b}+\not\\!p_{2}^{\prime})\otimes\gamma_{\mu}+2m_{p}\gamma_{\mu}\otimes\mathbb{1}\big{]}\big{\\}}u_{a}\otimes
u_{b}\,.$

For the subamplitudes with the vector exchanges we get

$\displaystyle{\cal M}_{{\rm V},\mu}^{(a+b+c)\,1}$ $\displaystyle=$
$\displaystyle-e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}{\cal
F}_{V}(s,t_{2})\gamma^{\alpha}\otimes\gamma_{\alpha}\big{[}\frac{2p_{a\mu}-k_{\mu}}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}+\frac{2p_{1\mu}^{\prime}+k_{\mu}}{2p_{1}^{\prime}\cdot
k+k^{2}+i\varepsilon}\big{]}\big{\\}}$ (115) $\displaystyle\times u_{a}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{{\rm V},\mu}^{(a+b+c)\,2}$ $\displaystyle=$
$\displaystyle-e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}\Delta{\cal
F}_{V}(s,t_{2},\varkappa)\gamma^{\alpha}\otimes\gamma_{\alpha}\big{[}\frac{(2p_{a}+2p_{b}-k,k)}{s}\frac{2p_{a\mu}-k_{\mu}}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}$ (116)
$\displaystyle+\frac{(2p_{a}+2p_{b}-k)_{\mu}}{s}\big{]}\big{\\}}u_{a}\otimes
u_{b}\,,$ $\displaystyle{\cal M}_{{\rm V},\mu}^{(a+b+c)\,3}$ $\displaystyle=$
$\displaystyle-e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}{\cal
F}_{V}(s^{\prime},t_{2})\frac{1}{-2p_{a}\cdot
k+k^{2}+i\varepsilon}\gamma^{\alpha}\big{[}k_{\mu}-\not\\!k\gamma_{\mu}$ (117)
$\displaystyle+\frac{F_{2}(0)}{2m_{p}}\big{(}2p_{a\mu}\not\\!k-2(p_{a}\cdot
k)\gamma_{\mu}+2m_{p}(k_{\mu}-\not\\!k\gamma_{\mu})-(\not\\!kk_{\mu}-k^{2}\gamma_{\mu})\big{)}\big{]}\otimes\gamma_{\alpha}\big{\\}}$
$\displaystyle\times u_{a}\otimes u_{b}\,,$ $\displaystyle{\cal M}_{{\rm
V},\mu}^{(a+b+c)\,4}$ $\displaystyle=$
$\displaystyle-e\bar{u}_{1^{\prime}}\otimes\bar{u}_{2^{\prime}}\big{\\{}{\cal
F}_{V}(s,t_{2})\frac{1}{2p_{1}^{\prime}\cdot
k+k^{2}+i\varepsilon}\big{[}-(k_{\mu}-\gamma_{\mu}\not\\!k)$ (118)
$\displaystyle+\frac{F_{2}(0)}{2m_{p}}\big{(}-2p_{1\mu}^{\prime}\not\\!k+2(p_{1}^{\prime}\cdot
k)\gamma_{\mu}-2m_{p}(k_{\mu}-\gamma_{\mu}\not\\!k)-(k_{\mu}\not\\!k-k^{2}\gamma_{\mu})\big{)}\big{]}\gamma^{\alpha}\otimes\gamma_{\alpha}\big{\\}}$
$\displaystyle\times u_{a}\otimes u_{b}\,.$

According to (60)–(62), we have

$\displaystyle{\cal M}_{{\rm T},\mu}^{(d+e+f)\,j}$ $\displaystyle=$
$\displaystyle\left.{\cal M}_{{\rm
T},\mu}^{(a+b+c)\,j}\right|_{\mathop{{}^{(p_{a},\,\lambda_{a})\leftrightarrow(p_{b},\,\lambda_{b})}_{(p_{1}^{\prime},\,\lambda_{1})\leftrightarrow(p_{2}^{\prime},\,\lambda_{2})}}}\quad{\rm
for}\;\;j=1,\ldots,7\,,$ (119) $\displaystyle{\cal M}_{{\rm
V},\mu}^{(d+e+f)\,j^{\prime}}$ $\displaystyle=$ $\displaystyle\left.{\cal
M}_{{\rm
V},\mu}^{(a+b+c)\,j^{\prime}}\right|_{\mathop{{}^{(p_{a},\,\lambda_{a})\leftrightarrow(p_{b},\,\lambda_{b})}_{(p_{1}^{\prime},\,\lambda_{1})\leftrightarrow(p_{2}^{\prime},\,\lambda_{2})}}}\quad{\rm
for}\;\;j^{\prime}=1,\ldots,4\,,$ (120)

where we also exchange the order of the tensor products in (108)–(118). Note
that all subamplitudes ${\cal M}_{{\rm T},\mu}^{(a+b+c)\,j}$ and ${\cal
M}_{{\rm V},\mu}^{(a+b+c)\,j^{\prime}}$ are separately gauge invariant, as is
easy to check,

$\displaystyle k^{\mu}{\cal M}_{{\rm T},\mu}^{(a+b+c)\,j}$ $\displaystyle=$
$\displaystyle 0\,,\quad j=1,\ldots,7\,,$ $\displaystyle k^{\mu}{\cal M}_{{\rm
V},\mu}^{(a+b+c)\,j^{\prime}}$ $\displaystyle=$ $\displaystyle 0\,,\quad
j^{\prime}=1,\ldots,4\,.$ (121)

The same holds for ${\cal M}_{{\rm T},\mu}^{(d+e+f)\,j}$ and ${\cal M}_{{\rm
V},\mu}^{(d+e+f)\,j^{\prime}}$.

Note that only the $j=1$ terms in (108) and (119), and the $j^{\prime}=1$
terms in (115) and (120), contain the pole terms proportional to $\omega^{-1}$
for $\omega\to 0$. The SPA1 and SPA2 results are derived from these terms; see
Sec. III.

In Fig. 17, we show differential distributions for the $pp\to pp\gamma$
reaction for $\sqrt{s}=13$ TeV, $|\rm y|<4$, and for two $k_{\perp}$
intervals: $1\,{\rm MeV}<k_{\perp}<100\,{\rm MeV}$ (the left panels),
$100\,{\rm MeV}<k_{\perp}<400\,{\rm MeV}$ (the right panels). In these
calculations, we limit ourselves to the leading pomeron-exchange contribution.
We show the complete result (“total”) including interference effects and the
results for individual $j$ terms (108)–(114) plus (119), except for $j=3$ and
5, which are very small and can be safely neglected. Note that there is
significant cancellation among the terms $j=2$ and $4$, due to destructive
interference. The coherent sum of the amplitudes $(a+b+c)$ plus $(d+e+f)$ with
$j=2$ and $j=4$ is denoted by “$2+4$”.

Figure 17:  The differential distributions for the $pp\to pp\gamma$ reaction
calculated for $\sqrt{s}=13$ TeV, $|\rm y|<4$, and for two $k_{\perp}$
intervals as specified in the figure legends. The results for individual $j$
terms in (107) and their coherent sum (total) are shown. The coherent sum of
the amplitudes $(a+b+c)$ plus $(d+e+f)$ with $j=2$ and $j=4$ is denoted by
“$2+4$”.

How can we understand these results? First, we remark that the destructive
interference of the 2 and 4 terms above is not due to a gauge cancellation.
These terms are separately gauge invariant; see (121). Let us have a closer
look at the terms 1, 2, and 4 of (108), (109), and (111), respectively, for
real photons, $k^{2}=0$, and with transverse momentum only:

$\displaystyle(k^{\mu})=\left(\begin{array}[]{c}\omega\\\
\omega\,\mbox{\boldmath$\hat{k}_{\perp}$}\\\
\end{array}\right)\,,\quad|\mbox{\boldmath$\hat{k}_{\perp}$}|=1\,.$ (124)

We work in the c.m. system with $|\mbox{\boldmath$p_{a}$}|$ defining the z
axis. Then, disregarding terms on the rhs of (108) and (109), which are of the
same order in $|\mbox{\boldmath$p_{a}$}|$, we get very roughly

$\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,1}$ $\displaystyle\propto$
$\displaystyle\frac{p_{a\mu}}{-p_{a}\cdot
k}+\frac{p_{1\mu}^{\prime}}{p_{1}^{\prime}\cdot k}$ (125)
$\displaystyle\propto$ $\displaystyle\frac{\Delta
p}{|\mbox{\boldmath$p_{a}$}|\omega}\,.$

Here $\Delta p={\cal O}(m_{p})$ is a measure of the transverse momentum change
from $p_{a}$ to $p_{1}^{\prime}$.

The term ${\cal M}_{{\rm T},\mu}^{(a+b+c)\,2}$ has no singularity for
$\omega\to 0$. The main term here comes from the anomalous magnetic moment
$F_{2}(0)$ and can be estimated as

$\displaystyle{\cal M}_{{\rm T},\mu}^{(a+b+c)\,2}$ $\displaystyle\propto$
$\displaystyle\frac{1}{|\mbox{\boldmath$p_{a}$}|\omega}\frac{|\mbox{\boldmath$p_{a}$}|\omega}{m_{p}}=\frac{1}{m_{p}}$
(126)

and similarly for ${\cal M}_{{\rm T},\mu}^{(a+b+c)\,4}$. Thus, the term ${\cal
M}_{{\rm T},\mu}^{(a+b+c)\,1}$ will win over the 2 and 4 terms individually
for $\omega\to 0$. However, for this to happen, we must require

$\displaystyle\frac{\Delta
p}{|\mbox{\boldmath$p_{a}$}|\omega}\approx\frac{m_{p}}{|\mbox{\boldmath$p_{a}$}|\omega}\gtrsim\frac{1}{m_{p}}\,,$
$\displaystyle\omega\lesssim\frac{m_{p}^{2}}{|\mbox{\boldmath$p_{a}$}|}\,.$
(127)

For our case with $2|\mbox{\boldmath$p_{a}$}|=13$ TeV, this requires

$\displaystyle k_{\perp}\approx\omega\lesssim 0.15\;{\rm MeV}\,.$ (128)

Explicit calculations confirm the order of magnitude of this estimate. Indeed,
from Fig. 18, we see that the $j=1$ term exceeds the $j=2$ term only for
$k_{\perp}\lesssim 0.35$ MeV and $\omega\lesssim 0.7$ MeV. The crossing place
of these terms depends on the ${\rm y}$ range since the $j=1$ term has a
minimum at ${\rm y}=0$ and grows rapidly with $|{\rm y}|$ increasing, while
the terms 2 and 4 are flat in the midrapidity region (see the top panels of
Fig. 17). When going with ${\rm y}\to 0$, the crossing place of the terms 1
and 2 shifts to lower values of $k_{\perp}$ and $\omega$. In reality, however,
there is destructive interference between the terms 2 and 4, and their sum is
harmless, well below the term 1, at least for $k_{\perp}<100$ MeV and
$\omega<2$ GeV, as we see from the left panels of Fig. 17. In the right panels
of Fig. 17 we show the results for larger $k_{\perp}$ and $\omega$ ranges. The
destructive interference of the terms 2 and 4 is again a salient feature.

Figure 18:  The cross sections $d\sigma/dk_{\perp}$ (left) and
$d\sigma/d\omega$ (right) for the terms ${\cal M}_{{\rm
T},\mu}^{(a+b+c)\,j}+{\cal M}_{{\rm T},\mu}^{(d+e+f)\,j}$ for pomeron exchange
for $j=1,2$, and 4. See (108), (109), (111), and (119). The crossing of the
term $j=1$ containing the $\omega^{-1}$ pole with the nonpole terms $j=2$ and
4 occurs for $k_{\perp}\cong 0.35$ MeV and $\omega\cong 0.7$ MeV.

As we mentioned already, the terms with labels $j=2$ and 4 contain the Dirac
and Pauli couplings, and the latter one dominates. In the result “$2+4$” shown
in Fig. 17, we find destructive interference of the Pauli parts of $j=2$ and
$j=4$ in the term $(a+b+c)$ and in the term $(d+e+f)$ individually. But for
the Pauli parts, we find numerically that there is practically no interference
between the $(a+b+c)$ and the $(d+e+f)$ terms. In the “$2+4$” term, the Dirac
part wins over the Pauli part. The visible dip for $d\sigma/d{\rm y}$ at ${\rm
y}\approx 0$ in the “$2+4$” result (see the left upper panel of Fig. 17) is
due to destructive interference of the Dirac parts between the $(a+b+c)$ and
the $(d+e+f)$ terms.

For the vector-exchange contributions, (115)–(118) and (120), the situation is
similar to that for the tensor exchanges. The terms containing the
$\omega^{-1}$ pole are the $j^{\prime}=1$ terms. Individually, the vector-
exchange terms with $j^{\prime}=3$ and $j^{\prime}=4$ are much larger than the
$j^{\prime}=1$ term, except for very small $k_{\perp}$ and $\omega$. But again
there is destructive interference of the $j^{\prime}=3$ and $j^{\prime}=4$
terms and the sum $3+4$ is well below the $j^{\prime}=1$ term in the same
kinematic regions as shown in Fig. 17. In this kinematic region, the
$j^{\prime}=2$ term is very small.

## References

  * (1) P. Lebiedowicz, O. Nachtmann, and A. Szczurek, High-energy $\pi\pi$ scattering without and with photon radiation, Phys. Rev. D 105 no. 1, (2022) 014022, arXiv:2107.10829 [hep-ph].
  * (2) F. E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974.
  * (3) A. T. Goshaw, J. R. Elliott, L. E. Evans, L. R. Fortney, P. W. Lucas, W. J. Robertson, W. D. Walker, I. J. Kim, and C.-R. Sun, Direct Photon Production from $\pi^{+}p$ Interactions at 10.5 GeV/$c$, Phys. Rev. Lett. 43 (1979) 1065.
  * (4) P. V. Chliapnikov, E. A. De Wolf, A. B. Fenyuk, L. N. Gerdyukov, Y. Goldschmidt-Clermont, V. M. Ronzhin, and A. Weigend, (Brussels-CERN-Genoa-Mons-Nijmegen-Serpukhov Collaboration), Observation of direct soft photon production in $K^{+}p$ interactions at 70 GeV/$c$, Phys. Lett. B 141 (1984) 276.
  * (5) F. Botterweck et al., (EHS/NA22 Collaboration), Direct soft photon production in $K^{+}p$ and $\pi^{+}p$ interactions at 250 GeV/$c$, Z. Phys. C 51 (1991) 541.
  * (6) S. Banerjee et al., (SOPHIE/WA83 Collaboration), Observation of direct soft photon production in $\pi^{-}p$ interactions at 280 GeV/$c$, Phys. Lett. B 305 (1993) 182.
  * (7) J. Antos et al., Soft photon production in 400 GeV/$c$ $p$-Be collisions, Z. Phys. C 59 (1993) 547.
  * (8) M. L. Tincknell et al., Low transverse momentum photon production in proton-nucleus collisions at 18 GeV/$c$, Phys. Rev. C 54 (1996) 1918.
  * (9) A. Belogianni et al., (WA91 Collaboration), Confirmation of a soft photon signal in excess of Q.E.D. expectations in $\pi^{-}p$ interactions at 280 GeV/$c$, Phys. Lett. B 408 (1997) 487, arXiv:hep-ex/9710006.
  * (10) A. Belogianni et al., Further analysis of a direct soft photon excess in $\pi^{-}p$ interactions at 280 GeV/$c$, Phys. Lett. B 548 (2002) 122.
  * (11) A. Belogianni et al., Observation of a soft photon signal in excess of QED expectations in $pp$ interactions, Phys. Lett. B 548 (2002) 129.
  * (12) J. Abdallah et al., (DELPHI Collaboration), Evidence for an excess of soft photons in hadronic decays of $Z^{0}$, Eur. Phys. J. C 47 (2006) 273, arXiv:hep-ex/0604038.
  * (13) J. Abdallah et al., (DELPHI Collaboration), Observation of the muon inner bremsstrahlung at LEP1, Eur. Phys. J. C 57 (2008) 499, arXiv:0901.4488 [hep-ex].
  * (14) J. Abdallah et al., (DELPHI Collaboration), Study of the dependence of direct soft photon production on the jet characteristics in hadronic $Z^{0}$ decays, Eur. Phys. J. C 67 (2010) 343, arXiv:1004.1587 [hep-ex].
  * (15) C.-Y. Wong, An Overview of the Anomalous Soft Photons in Hadron Production, PoS Photon 2013 (2014) 002, arXiv:1404.0040 [hep-ph].
  * (16) D. Adamová et al., A next-generation LHC heavy-ion experiment, arXiv:1902.01211 [physics.ins-det].
  * (17) P. Braun-Munzinger.  A talk “Experimental opportunities for new physics at low transverse momentum” presented at Quark Matter 2022, 4-10 April 2022, Kraków, Poland.
  * (18) Workshop EMMI Rapid Reaction Task Force on “Real and virtual photon production at ultra-low transverse momentum and low mass at LHC”, organised by K. Schweda, R. Bailhache and S. Flörchinger, https://indico.gsi.de/event/11529/overview.
  * (19) V. N. Gribov, Bremsstrahlung of Hadrons at High Energies, Sov. J. Nucl. Phys. 5 (1967) 280.
  * (20) T. H. Burnett and N. M. Kroll, Extension of the Low Soft-Photon Theorem, Phys. Rev. Lett. 20 (1968) 86.
  * (21) J. S. Bell and R. Van Royen, On the Low-Burnett-Kroll theorem for soft-photon emission, Nuovo Cim. A 60 (1969) 62.
  * (22) L. N. Lipatov, Massless particle bremsstrahlung theorems for high-energy hadron interactions, Nucl. Phys. B 307 (1988) 705.
  * (23) V. Del Duca, High-energy bremsstrahlung theorems for soft photons, Nucl. Phys. B 345 (1990) 369.
  * (24) H. Gervais, Soft photon theorem for high energy amplitudes in Yukawa and scalar theories, Phys. Rev. D 95 no. 12, (2017) 125009, arXiv:1704.00806 [hep-th].
  * (25) Z. Bern, S. Davies, P. Di Vecchia, and J. Nohle, Low-energy behavior of gluons and gravitons from gauge invariance, Phys. Rev. D 90 no. 8, (2014) 084035, arXiv:1406.6987 [hep-th].
  * (26) V. Lysov, S. Pasterski, and A. Strominger, Low’s Subleading Soft Theorem as a Symmetry of QED, Phys. Rev. Lett. 113 no. 11, (2014) 111601, arXiv:1407.3814 [hep-th].
  * (27) D. Bonocore and A. Kulesza, Soft photon bremsstrahlung at next-to-leading power, arXiv:2112.08329 [hep-ph].
  * (28) C. Ewerz, M. Maniatis, and O. Nachtmann, A Model for Soft High-Energy Scattering: Tensor Pomeron and Vector Odderon, Annals Phys. 342 (2014) 31, arXiv:1309.3478 [hep-ph].
  * (29) V. A. Khoze, J. W. Lämsä, R. Orava, and M. G. Ryskin, Forward physics at the LHC: detecting elastic $pp$ scattering by radiative photons, JINST 6 (2011) P01005, arXiv:1007.3721 [hep-ph].
  * (30) P. Lebiedowicz and A. Szczurek, Exclusive diffractive photon bremsstrahlung at the LHC, Phys.Rev. D87 (2013) 114013, arXiv:1302.4346 [hep-ph].
  * (31) V. A. Khoze, A. D. Martin, and M. G. Ryskin, Can invisible objects be ’seen’ via forward proton detectors at the LHC?, J. Phys. G 44 no. 5, (2017) 055002, arXiv:1702.05023 [hep-ph].
  * (32) A. Cisek, P. Lebiedowicz, W. Schäfer, and A. Szczurek, Exclusive production of $\omega$ meson in proton-proton collisions at high energies, Phys.Rev. D83 (2011) 114004, arXiv:1101.4874 [hep-ph].
  * (33) P. Lebiedowicz and A. Szczurek, Exclusive $pp\to pp\pi^{0}$ reaction at high energies, Phys. Rev. D87 (2013) 074037, arXiv:1303.2882 [hep-ph].
  * (34) J. Chwastowski, L. Fulek, R. Kycia, R. Sikora, J. Turnau, A. Cyz, and B. Pawlik, Feasibility Studies of Exclusive Diffractive Bremsstrahlung Measurement at RHIC Energies, Acta Phys. Polon. B 46 no. 10, (2015) 1979, arXiv:1501.06264 [hep-ex].
  * (35) J. J. Chwastowski, S. Czekierda, R. Kycia, R. Staszewski, J. Turnau, and M. Trzebiński, Feasibility studies of the diffractive bremsstrahlung measurement at the LHC, Eur. Phys. J. C 76 no. 6, (2016) 354, arXiv:1603.06449 [hep-ex].
  * (36) J. J. Chwastowski, S. Czekierda, R. Staszewski, and M. Trzebiński, Diffractive bremsstrahlung at high-$\beta^{\star}$ LHC, Eur. Phys. J. C 77 no. 4, (2017) 216, arXiv:1612.06066 [hep-ex].
  * (37) S. Foroughi-Abari and A. Ritz, Dark sector production via proton bremsstrahlung, Phys. Rev. D 105 no. 9, (2022) 095045, arXiv:2108.05900 [hep-ph].
  * (38) J. L. Feng et al., The Forward Physics Facility at the High-Luminosity LHC, arXiv:2203.05090 [hep-ex].
  * (39) J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields.  McGraw-Hill, Inc., New York, 1965.
  * (40) C. Ewerz, P. Lebiedowicz, O. Nachtmann, and A. Szczurek, Helicity in Proton-Proton Elastic Scattering and the Spin Structure of the Pomeron, Phys. Lett. B763 (2016) 382, arXiv:1606.08067 [hep-ph].
  * (41) L. Adamczyk et al., (STAR Collaboration), Single spin asymmetry $A_{N}$ in polarized proton-proton elastic scattering at $\sqrt{s}=200$ GeV, Phys. Lett. B719 (2013) 62, arXiv:1206.1928 [nucl-ex].
  * (42) D. Britzger, C. Ewerz, S. Glazov, O. Nachtmann, and S. Schmitt, Tensor Pomeron and low-$x$ deep inelastic scattering, Phys. Rev. D100 no. 11, (2019) 114007, arXiv:1901.08524 [hep-ph].
  * (43) P. D. B. Collins, An Introduction to Regge Theory and High-Energy Physics.  Cambridge University Press, Cambridge, England, 1977.
  * (44) L. Caneschi, ed., Regge Theory of Low-$p_{t}$ Hadronic Interactions.  Elsevier Science Publishers B.V., Amsterdam, 1989.
  * (45) V. N. Gribov, Strong Interactions of Hadrons at High Energies: Gribov Lectures on Theoretical Physics.  Cambridge University Press, Cambridge, England, 2009.  edited by Y. L. Dokshitzer and J. Nyiri; Published in: Cambridge Monogr. Part. Phys., Nucl. Phys., Cosmol. 27 (2012).
  * (46) A. Donnachie, H. G. Dosch, P. V. Landshoff, and O. Nachtmann, Pomeron physics and QCD, Cambridge Monogr. Part. Phys., Nucl. Phys., Cosmol. 19 (2002) 1.
  * (47) A. Donnachie and P. V. Landshoff, Total cross sections, Phys.Lett. B296 (1992) 227, arXiv:hep-ph/9209205 [hep-ph].
  * (48) A. Donnachie and P. V. Landshoff, $pp$ and $\bar{p}p$ elastic scattering, Nucl.Phys. B231 (1984) 189.
  * (49) J. Breitweg et al., (ZEUS Collaboration), Elastic and proton-dissociative $\rho^{0}$ photoproduction at HERA, Eur. Phys. J. C 2 (1998) 247, arXiv:hep-ex/9712020.
  * (50) V. Andreev et al., (H1 Collaboration), Measurement of exclusive $\pi^{+}\pi^{-}$ and $\rho^{0}$ meson photoproduction at HERA, Eur. Phys. J. C 80 no. 12, (2020) 1189, arXiv:2005.14471 [hep-ex].
  * (51) L.-J. Zhou, Q. Wu, W.-X. Ma, and Y.-T. Gu, Pomeron-Quark Coupling from Charge Conjugation Invariance, Commun. Theor. Phys. 46 (2006) 287.
  * (52) J. C. Ward, An Identity in Quantum Electrodynamics, Phys. Rev. 78 (1950) 182.
  * (53) Y. Takahashi, On the generalized Ward identity, Nuovo Cim. 6 (1957) 371.
  * (54) A. Donnachie and P. V. Landshoff, $pp$ and $\bar{p}p$ total cross sections and elastic scattering, Phys. Lett. B 727 (2013) 500, arXiv:1309.1292 [hep-ph]. [Erratum: Phys.Lett.B 750, 669–669 (2015)].
  * (55) L. Jenkovszky, I. Szanyi, and C.-I. Tan, Shape of proton and the pion cloud, Eur. Phys. J. A 54 no. 7, (2018) 116, arXiv:1710.10594 [hep-ph].
  * (56) L. Jenkovszky, R. Schicker, and I. Szanyi, Elastic and diffractive scatterings in the LHC era, Int. J. Mod. Phys. E 27 no. 08, (2018) 1830005, arXiv:1902.05614 [hep-ph].
  * (57) V. A. Khoze, A. D. Martin, and M. G. Ryskin, Elastic and diffractive scattering at the LHC, Phys. Lett. B 784 (2018) 192, arXiv:1806.05970 [hep-ph].
  * (58) E. Martynov and B. Nicolescu, Odderon effects in the differential cross-sections at Tevatron and LHC energies, Eur. Phys. J. C 79 no. 6, (2019) 461, arXiv:1808.08580 [hep-ph].
  * (59) M. Broilo, D. A. Fagundes, E. G. S. Luna, and M. Peláez, Soft Pomeron in light of the LHC correlated data, Phys. Rev. D 103 no. 1, (2021) 014019, arXiv:2012.08664 [hep-ph].
  * (60) N. Bence, A. Lengyel, Z. Tarics, E. Martynov, and G. Tersimonov, Froissaron and Maximal Odderon with spin-flip in $pp$ and $\bar{p}p$ high energy elastic scattering, Eur. Phys. J. A 57 no. 9, (2021) 265.
  * (61) A. A. Godizov, High-energy elastic diffractive scattering of nucleons in the framework of the two-Reggeon eikonal approximation (from U-70 to LHC), Eur. Phys. J. C 82 no. 1, (2022) 56, arXiv:2110.11895 [hep-ph].
  * (62) G. Antchev et al., (TOTEM Collaboration), Measurement of elastic $pp$ scattering at $\sqrt{s}=8$ TeV in the Coulomb–nuclear interference region: determination of the $\rho$-parameter and the total cross-section, Eur. Phys. J. C 76 no. 12, (2016) 661, arXiv:1610.00603 [nucl-ex].
  * (63) G. Antchev et al., (TOTEM Collaboration), First measurement of elastic, inelastic and total cross-section at $\sqrt{s}=13$ TeV by TOTEM and overview of cross-section data at LHC energies, Eur. Phys. J. C 79 no. 2, (2019) 103, arXiv:1712.06153 [hep-ex].
  * (64) G. Antchev et al., (TOTEM Collaboration), First determination of the $\rho$ parameter at $\sqrt{s}=13$ TeV: probing the existence of a colourless C-odd three-gluon compound state, Eur. Phys. J. C 79 no. 9, (2019) 785, arXiv:1812.04732 [hep-ex].
  * (65) P. Zyla et al., (Particle Data Group), Review of Particle Physics, PTEP 2020 no. 8, (2020) 083C01.
  * (66) Data files and plots of cross-sections and related quantities. https://pdg.lbl.gov/2022/hadronic-xsections/hadron.html.
  * (67) G. Antchev et al., (TOTEM Collaboration), Elastic differential cross-section measurement at $\sqrt{s}=13$ TeV by TOTEM, Eur. Phys. J. C 79 no. 10, (2019) 861, arXiv:1812.08283 [hep-ex].
  * (68) E. Martynov and B. Nicolescu, Did TOTEM experiment discover the Odderon?, Phys. Lett. B778 (2018) 414, arXiv:1711.03288 [hep-ph].
  * (69) E. Martynov and B. Nicolescu, Odderon: models vs experimental data - a short review of recent papers, arXiv:1811.07635 [hep-ph].
  * (70) L. Łukaszuk and B. Nicolescu, A Possible interpretation of $pp$ rising total cross-sections, Lett. Nuovo Cim. 8 (1973) 405.
  * (71) C. Ewerz, The Odderon in Quantum Chromodynamics, arXiv:hep-ph/0306137 [hep-ph].
  * (72) G. Pancheri, S. Pacetti, and Y. Srivastava, Analysis and implications of precision near-forward TOTEM data, Phys. Rev. D 99 no. 3, (2019) 034014, arXiv:1811.00499 [hep-ph].
  * (73) V. V. Ezhela, V. A. Petrov, N. P. Tkachenko, and A. A. Logunov, On the $\rho$ and $\sigma_{tot}$ measurement by the TOTEM Collaboration: in the wake of recent discoveries, arXiv:2003.03817 [hep-ph].
  * (74) A. Donnachie and P. V. Landshoff, Small $t$ elastic scattering and the $\rho$ parameter, Phys. Lett. B 798 (2019) 135008, arXiv:1904.11218 [hep-ph].
  * (75) A. Donnachie and P. V. Landshoff, Lack of evidence for an odderon at small $t$, Phys. Lett. B 831 (2022) 137199, arXiv:2203.00290 [hep-ph].
  * (76) A. Breakstone et al., Measurement of $\bar{p}p$ and $pp$ Elastic Scattering in the Dip Region at $\sqrt{s}=53$ GeV, Phys. Rev. Lett. 54 (1985) 2180.
  * (77) V. M. Abazov et al., (TOTEM and D0 Collaborations), Odderon Exchange from Elastic Scattering Differences between $pp$ and $p\bar{p}$ Data at 1.96 TeV and from $pp$ Forward Scattering Measurements, Phys. Rev. Lett. 127 no. 6, (2021) 062003, arXiv:2012.03981 [hep-ex].
  * (78) L. Jenkovszky, O. Kuprash, R. Orava, and A. Salii, Low missing mass, single- and double diffraction dissociation at the LHC, Phys. Atom. Nucl. 77 no. 12, (2014) 1463, arXiv:1211.5841 [hep-ph]. [Odessa Astron. Pub. 25 (2012) 102].
  * (79) H. de Kerret, E. Nagy, M. Regler, W. Schmidt-Parzefall, K. R. Schubert, et al., Experimental Results on Diffractive One-Pion Production at the CERN ISR, Phys.Lett. B63 (1976) 477.
  * (80) A. Bolz, C. Ewerz, M. Maniatis, O. Nachtmann, M. Sauter, and A. Schöning, Photoproduction of $\pi^{+}\pi^{-}$ pairs in a model with tensor-pomeron and vector-odderon exchange, JHEP 1501 (2015) 151, arXiv:1409.8483 [hep-ph].
  * (81) P. Lebiedowicz, O. Nachtmann, and A. Szczurek, Searching for the odderon in $pp\to ppK^{+}K^{-}$ and $pp\to pp\mu^{+}\mu^{-}$ reactions in the $\phi(1020)$ resonance region at the LHC, Phys. Rev. D 101 (2020) 094012, arXiv:1911.01909 [hep-ph].
  * (82) P. Lebiedowicz, O. Nachtmann, and A. Szczurek, Central exclusive diffractive production of $K^{+}K^{-}K^{+}K^{-}$ via the intermediate $\phi\phi$ state in proton-proton collisions, Phys. Rev. D99 no. 9, (2019) 094034, arXiv:1901.11490 [hep-ph].
  * (83) J.-R. Cudell, K. Kang, and S. K. Kim, Bounds on the soft pomeron intercept, Phys. Lett. B 395 (1997) 311, arXiv:hep-ph/9601336.
  * (84) E. G. S. Luna and M. J. Menon, Extrema bounds for the soft Pomeron intercept, Phys. Lett. B 565 (2003) 123, arXiv:hep-ph/0305280.
  * (85) E. G. S. Luna, M. J. Menon, and J. Montanha, An analysis on extrema and constrained bounds for the soft pomeron intercept, Nucl. Phys. A 745 (2004) 104, arXiv:hep-ph/0408211.