e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, The George Washington University,
Washington, DC 20052 USA 22institutetext: Belarusian State University, 220030
Minsk, Belarus 33institutetext: Institute for Nuclear Problems, Belarusian
State University, 220006 Minsk, Belarus

#  Radiative corrections to the lepton current in unpolarized elastic
$lp$-interaction for fixed $Q^{2}$ and scattering angle

A. Afanaseve1,addr1  A. Ilyicheve2,addr2,addr3

(Received: date / Accepted: date)

###### Abstract

The kinematical difference between the description of radiative effects for
fixed $Q^{2}$ vs a fixed scattering angle in the elastic lepton-proton
($lp$)-scattering is discussed. The technique of calculation as well as
explicit expressions for radiative corrections to the lepton current in
unpolarized elastic $lp$-scattering for these two cases are presented without
using an ultrarelativistic approximation. A comparative numerical analysis
within kinematic conditions of Jefferson Lab measurements and MUSE experiment
in PSI is performed.

††journal: Eur. Phys. J. A

##  1 Introduction

The elastic lepton-proton scattering is a recognized tool for investigation of
the internal proton structure. The observation of the disagreement in
$Q^{2}$-behavior of the proton elastic form factor ratio for unpolarized
Andivahis94 ; Qattan05  and polarized Jones ; Gayou  electron scattering,
along with the proton radius puzzle coming from the different outcomes of the
measurements in electron-proton systems CODATA ; Sick  and in the muonic
hydrogen Pohl  – all of these require understanding of underlying QED
processes that may lead to systematic uncertainties at a per cent level.
Moreover, the results of the recent experiment PRAD PRAD  was in agreement
with muonium spectroscopy experiment that contradicted the previous electron-
proton scattering data. This unexpected result motivates new efforts for the
theoretical and experimental investigations.

One of the important and essential tools for the investigation of the
electromagnetic properties of the proton is an experimental program with high
duty-cycle positron beams at JLab JLabpos . This program with the electron
beams allows to estimate the electromagnetic form factors of the proton
separately as well as to measure a change asymmetry that appears at the lowest
order as an interference of the matrix elements with one- and two-photon
exchanges.

Together with widely discussed two-photon exchange Afanasev_review , the
important source of uncertainties for both lepton and anti-lepton scattering
is from the real photon emission accompanying any process with the charge
particle scattering, as well as the additional virtual particle contributions.
Due to smallness of muon beam momentum at MUSE experiment in PSI MUSE , as
well as scattering by extremely small angles in PRAD-II experiment at
Jefferson Lab PRAD2 , all calculations have to be performed beyond the
ultrarelativistic approximation, $i.e.$ retaining lepton’s mass during the
entire calculation. While for purely elastic scattering at a given beam energy
the four-momentum transfer $Q^{2}$ is in one-to-one correspondence with a
lepton scattering angle, this is not the case for radiative events. It is
therefore of critical importance to understand the role of QED radiative
corrections (RC) in different kinematic scenarios: fixed momentum transfer
$Q^{2}$ vs fixed scattering angle of the detected lepton (as done in MUSE MUSE
or in high-resolution spectrometers with small angular acceptance used in some
of Jefferson Lab experiments).

It should be noted that rather often for estimation of the similar corrections
to the exclusive process the additional particle contributions are calculated
exactly or within ultrarelativistic approximation (with respect to lepton’s
mass) while the real photon emission is considered within the soft photon
approximation. Particularly in the papers Kaiser  and Vanderhaeghen  for
Möller and virtual Compton scattering processes, respectively, the virtual QED
corrections have been calculated beyond the ultrarelativistic limit but only
the soft part of the real photon emission was taken into account.

Mo and Tsai first developed a systemic approach to calculate RC with hard
photon emission in elastic and inelastic electron-proton scattering MoTsai1969
. One limitation in their calculations was the approximate way to consider the
soft-photon contribution, as a result, their final expressions depend on an
artificial parameter $\Delta$ that was introduced to separate the photon
momentum phase space into soft and hard parts.

Here we present the explicit expressions as well as the numerical comparison
of RC to the lepton current both for fixed scattering angle and transferred
momentum squared. Such RC include hard real photon emission from the initial
and final leptons, vacuum polarization and vertex correction. The presented RC
are charge-even, therefore they directly apply to a sum of positron- and
electron scattering cross sections that could be measured by combined
experiments with added positron capabilities at JLab.

For extraction and cancellation of the infrared divergence we use the
covariant approach of Bardin-Shumeiko BSh . One of the important advantages of
this approach over MoTsai1969  consists in the independence of the final
results from the parameter $\Delta$. A similar calculation but for fixed
transferred momentum squared was performed in Ref. AGIM2015 .

Among the other recent results on RC calculations to the lepton current with
the hard photon emission and keeping the lepton mass, we specifically mention
two papers. The first one is by Bucoveanu and Spiesberger Spiesberger  and
includes the second-order RC. The second publication describes FORTRAN code
developed by Banerjee, Engel, Signer, and Ulrich Signer  with a calculation of
the first order RC to several processes in elastic lepton-lepton and lepton-
proton scattering.

The rest of the article is organized as follows. The kinematics of elastic
process and radiative process are discussed in detail in Sec. 2. In
particular, we show that for the description of hard photon emission at fixed
scattering angle the ultrarelativistic approximation is not applicable even
for relativistic electron-proton scattering. The hadronic tensor and Born
cross section are presented in Sec. 3. The additional virtual particle
contributions are given in Sec. 4. For the parameterization of the infrared
and ultraviolet divergences the dimensional regularization is used. In the
next two sections the real photon emission contribution for both fixed $Q^{2}$
and fixed scattering angle is presented. For both cases the infrared
divergence is extracted and cancelled using the Bardin-Shumeiko approach BSh .
The comparative numerical analysis for MUSE MUSE  and Jlab ; Jlab1
experiments can be found in Sec. 7. A brief discussion and conclusions are
presented in the last section. The details of the approach for the infrared
divergence extraction are given in A. The derivation of the compact expression
for the Bardin-Shumeiko function $S_{\phi}$ can be found in B.

##  2 Elastic and inelastic processes

The unpolarized elastic $lp$-scattering

$\displaystyle l(k_{1})+p(p_{1})\to l^{\prime}(k_{2})+p^{\prime}(p_{2}),$ (1)

is considered first. Here $k_{1}$ and $p_{1}$ ($k_{2}$ and $p_{2}$) are the
four-momenta of the initial (final) lepton and proton respectively
($k_{1}^{2}=k_{2}^{2}=m^{2}$, $p_{1}^{2}=p_{2}^{2}=M^{2}$). Although we
consider this process in the target rest frame (${\bf p_{1}}=0$), after
definition of the virtual photon momentum as $q=k_{1}-k_{2}$, it will be
useful to introduce the kinematic invariants:

$\displaystyle\displaystyle S=2p_{1}k_{1},\;Q^{2}=-q^{2},\;X=S-Q^{2},$
$\displaystyle\displaystyle\lambda_{S}=S^{2}-4m^{2}M^{2},\;\lambda_{X}=X^{2}-4m^{2}M^{2},$
$\displaystyle\displaystyle\lambda_{m}=Q^{2}(Q^{2}+4m^{2}),$ (2)

in such a way, that the energies of the initial ($k_{10}$) and final
($k_{20}$) leptons as well as the absolute value of their three-momenta
($|{\bf k_{1}}|$ and $|{\bf k_{2}}|$, respectively) read:

$\displaystyle k_{10}=\frac{S}{2M},\;|{\bf
k_{1}}|=\frac{\sqrt{\lambda_{S}}}{2M},\;k_{20}=\frac{X}{2M},\;|{\bf
k_{2}}|=\frac{\sqrt{\lambda_{X}}}{2M}.\;$ (3)

In the present paper we will consider two types of the cross sections:
$d\sigma/dQ^{2}$ and $d\sigma/d\cos\theta$ where the cosine of the scattering
angle $\theta$ can be expressed through the invariants:

$\displaystyle\cos\theta=\frac{{\bf k_{1}}\cdot{\bf k_{2}}}{|{\bf k_{1}}||{\bf
k_{2}}|}=\frac{SX-2M^{2}(Q^{2}+2m^{2})}{\sqrt{\lambda_{S}\lambda_{X}}}.$ (4)

Taking into account $X=S-Q^{2}$, the quadratic equation over $Q^{2}$ has two
solutions

$\displaystyle Q^{2}_{\pm}=\lambda_{S}\frac{S\sin^{2}\theta+2M^{2}\pm
2M\cos\theta\sqrt{M^{2}-m^{2}\sin^{2}\theta}}{(S+2M^{2})^{2}-\lambda_{S}\cos^{2}\theta},$
(5)

where the direct substitution into (4) shows that $Q^{2}_{-}$ is the correct
expression while $Q^{2}_{+}$ corresponds to the scattering on $180^{o}-\theta$
angle:

$\displaystyle\frac{S(S-Q^{2}_{\pm})-2M^{2}(Q^{2}_{\pm}+2m^{2})}{\sqrt{\lambda_{S}((S-Q^{2}_{\pm})^{2}-4m^{2}M^{2})}}=\mp\cos\theta.$
(6)

The restrictions on the scattering angle $-1<\cos\theta<1$ translate into the
kinematical limits for $Q^{2}$:

$\displaystyle 0<Q^{2}<\frac{\lambda_{S}}{S+m^{2}+M^{2}}.$ (7)

For the description of the inelastic process caused by real photon emission

$\displaystyle l(k_{1})+p(p_{1})\to
l^{\prime}(k_{2})+p^{\prime}(p_{2})+\gamma(k)$ (8)

($k^{2}=0$) three additional variables have to be introduced. We choose the
standard set MASCARAD  of them: inelasticity
$v=(p_{1}+k_{1}-k_{2})^{2}-M^{2}$, $\tau=kq/kp_{1}$ and the azimuthal angle
$\phi_{k}$ between (${\bf k}_{1}$,${\bf k}_{2}$) and (${\bf k}$,${\bf q}$)
planes in the rest frame (${\bf p}_{1}=0$).

Using this set of variables, it is straightforward to show that for real
photon emission the expressions for the energy and the three-momentum of the
scattering lepton have to be modified:

$\displaystyle k_{20}=\frac{X-v}{2M},\;|{\bf
k_{2}}|=\frac{\sqrt{(X-v)^{2}-4m^{2}M^{2}}}{2M}.$ (9)

As a result, $\cos\theta$ can be expressed through the inelasticity value and
$Q^{2}$ in a following way:

$\displaystyle\cos\theta_{R}=\frac{S(X-v)-2M^{2}(Q^{2}+2m^{2})}{\sqrt{\lambda_{S}((X-v)^{2}-4M^{2}m^{2})}},$
(10)

where we introduce the index $R$ to emphasize that at a fixed $Q^{2}$ the
value of $\cos\theta$ depends on the inelasticity of the radiative process.
The restrictions on the scattering angle $-1<\cos\theta_{R}<1$ set the upper
limit for $v$ at fixed $Q^{2}$:

$\displaystyle
v_{q}=\frac{\sqrt{\lambda_{S}}\sqrt{\lambda_{m}}-Q^{2}(S+2m^{2})}{2m^{2}}.$
(11)

Similar to the non-radiative process, there are two possible ways to express
$Q^{2}$ from Eq. (10). After substitution of the obtained expressions for
$Q^{2}$ into the r.h.s. of Eq. (10), the correct solution here is:

$\displaystyle Q_{R}^{2}(v)$ $\displaystyle=$
$\displaystyle\frac{1}{(S+2M^{2})^{2}-\lambda_{S}\cos^{2}\theta}$ (12)
$\displaystyle\times\biggl{[}(S+2M^{2})(\lambda_{S}-vS)-\lambda_{S}(S-v)\cos^{2}\theta$
$\displaystyle-2M\sqrt{\lambda_{S}}\sqrt{{\cal D}}\cos\theta\biggl{]},$

where the index $R$ poses the same meaning as in Eq. (10), namely, at a fixed
$\cos\theta$ the value of $Q^{2}$ depends on the inelasticity of the radiative
process. The quantity

$\displaystyle{\cal
D}=M^{2}(\lambda_{S}+v(v-2S))-m^{2}(\lambda_{S}\sin^{2}\theta+4vM^{2})$

must be positive. It turns out that the upper limit of $v$ for a given
scattering angle follows from that restriction:

$\displaystyle
v_{\theta}=S+2m^{2}-\frac{m}{M}\sqrt{(S+2M^{2})^{2}-\lambda_{S}\cos^{2}\theta}.\;$
(14)

Notice that minimizing ${\cal D}$ maximizes $Q^{2}_{R}$ if $\cos\theta>0$ and
minimizes $Q^{2}_{R}$ if $\cos\theta<0$. The energy and momentum of the
scattering lepton for fixed angle read:

$\displaystyle k_{20}$ $\displaystyle=$
$\displaystyle\frac{S-Q^{2}_{R}(v)-v}{2M},\;$ $\displaystyle|{\bf k_{2}}|$
$\displaystyle=$
$\displaystyle\frac{\lambda_{S}-vS-Q^{2}_{R}(v)(S+2M^{2})}{2M\cos\theta\sqrt{\lambda_{S}}}.$
(15)

From Fig. 1 one can see that when the observable quantity $Q^{2}$ is close to
its kinematical boundaries, the allowed range of the inelasticity reduces to
zero that makes it impossible to emit any real photon. The maximum value of
the inelasticity

$\displaystyle v_{q}^{max}=S-2m(\sqrt{S+m^{2}+M^{2}}-m)$ (16)

comes at the point that can be obtained after substitution (16) into (12),

$\displaystyle
Q^{2}_{R}(v_{q}^{max})=\frac{m(S+2m^{2})}{\sqrt{S+m^{2}+M^{2}}}-2m^{2}.$ (17)

Figure 1:  The dependence of the upper inelasticity limits $v_{q}$ and
$v_{\theta}$ on the observable variables for the muon beam momentum $|{\bf
k}_{1}|=200$ MeV. Lowest: a full kinematic range. Upper: a close-up of the
region near the kinematic boundary. The quantities
$v^{max}_{\theta}=v^{max}_{q}$, $v^{min}_{\theta}$ and
$Q^{2}_{R}(v^{max}_{q})$ are defined by Eqs. (16), (18) and (17) respectively.

From the upper plot of Fig. 1 we can see that for fixed angle the upper
inelasticity limit reaches its maximum value $v_{\theta}^{max}=v_{q}^{max}$ at
the kinematical boundaries $\cos\theta=\pm 1$ and has a minimum

$\displaystyle v_{\theta}^{min}=(S-2mM)\left(1-\frac{m}{M}\right)$ (18)

at $\cos\theta=0$.

The dependence of $Q^{2}_{R}$ on the inelasticity at different fixed angles is
presented in Fig. 2. From this plot one can see that even for $\theta=0^{o}$
real photon emission is not prohibited by any kinematical restrictions.
Opposite to the elastic process, the scattering under zero angle induces non-
zero transferred momentum.

Figure 2:  The dependence of $Q^{2}_{R}$ on $v$, as given by Eq. (12) for the
different fixed scattering angles and the muon beam momentum $|{\bf
k}_{1}|=200$ MeV. Left: a full kinematic range. Right: a close-up of the
region near the kinematic boundary. The joining curves $\theta$ and
$180^{o}-\theta$ line describes by Eq. (19). The two indicated points are
($v_{\theta}^{min}$, $Q^{2}_{R}(v_{\theta}^{min})$) according to Eqs. (18) and
(20), and ($v_{\theta}^{max}$, $Q^{2}_{R}(v_{\theta}^{max})$) according to
Eqs. (16) and (17).

After substitution of (14) into (12) we find the line with boundary common
points for $\theta$ and $180^{o}-\theta$ curves

$\displaystyle
Q^{2}_{R}(v_{\theta})=\frac{m(S(S+2M^{2})-\lambda_{S}\cos^{2}\theta)}{M\sqrt{(S+2M^{2})^{2}-\lambda_{S}\cos^{2}\theta}}-2m^{2}$
(19)

as it is presented in the right plot of Fig. 2. The quantity
$Q^{2}_{R}(v_{\theta}^{max})$ is defined by Eq. (17) while

$\displaystyle Q^{2}_{R}(v_{\theta}^{min})=m\Biggl{(}\frac{S}{M}-2m\Biggr{)}.$
(20)

From Eq. (19) it can be seen that for the description of hard photon emission
at fixed scattering angle even for high-energy electron-proton scattering the
ultrarelativistic approximation is not applicable.

In practice, however, the contribution of the hard real photon emission to the
cross section can be essentially reduced by applying a cut $v_{cut}$ on the
inelasticity which is also a measured quantity in the single-arm measurement
of the elastically scattered lepton only. Therefore, keeping in mind the
inelasticity maximum values, for an upper limit of this quantity both for the
fixed $Q^{2}$ and scattering angle we will use $v_{cut}$ as an experimentally
observable variable.

The other invariant quantity $\tau$ can be calculated in the rest frame as

$\displaystyle\tau=\frac{1}{M}(q_{0}-|{\bf q}|\cos\theta_{k}),$ (21)

where $q_{0}$ (${\bf q}$) is the energy (three-momentum) of the transfer
momentum $q$ and $\theta_{k}$ is the polar angle between the three-momenta
${\bf q}$ and ${\bf k}$. The range of this variable is defined through
$-1<\cos\theta_{k}<1$ and for fixed $Q^{2}$ and fixed angle $\theta$ it reads:

$\displaystyle\tau^{q}_{max/min}$ $\displaystyle=$
$\displaystyle\frac{Q^{2}+v\pm\sqrt{\lambda_{q}}}{2M^{2}},$
$\displaystyle\tau^{\theta}_{max/min}$ $\displaystyle=$
$\displaystyle\frac{Q^{2}_{R}(v)+v\pm\sqrt{\lambda_{v}}}{2M^{2}}$ (22)

with $\lambda_{q}=(Q^{2}+v)^{2}+4M^{2}Q^{2}$ and
$\lambda_{v}=(Q^{2}_{R}(v)+v)^{2}+4M^{2}Q^{2}_{R}(v)$.

At the end of this Section it is necessary to say about the orientation of the
azimuthal photon angle $\phi_{k}$. It can be defined by choosing a sing in the
expression of $\sin\phi_{k}$ through the pseudoscalar quantity as

$\displaystyle\sin\phi_{k}=\pm\frac{\varepsilon_{\alpha\beta\gamma\delta}p_{1}^{\alpha}q^{\beta}k_{1}^{\gamma}k^{\delta}}{M|{\bf
q}||{\bf k}_{l}^{\bot}|k_{0}\sin\theta_{k}},$ (23)

where ${\bf k}_{l}^{\bot}$ is the transverse three-momenta of the incoming or
scattering lepton with respect to ${\bf q}$, $k_{0}$ is a photon energy.
However, during the estimation of the real photon contribution to elastic or
inclusive lepton-proton scattering even for polarized particles in contrast to
the exclusive or semi-inclusive hadron leptoproduction the sine of $\phi_{k}$
does not appear for any stage of calculations. Therefore, we are not concerned
about this problem and integrate over $\phi_{k}$ without taking into account
its orientation.

##  3 Hadronic tensor and Born contribution

Born contribution to the process depicted by the Feynman graph in Fig. 3(a)
reads:

$\displaystyle d\sigma_{B}=\frac{1}{2\sqrt{\lambda_{S}}}{\cal
M}_{B}^{2}d\Gamma_{2},$ (24)

where the phase space has the form

$\displaystyle d\Gamma_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\delta^{4}(p_{1}+k_{1}-p_{2}-k_{2})\frac{d^{3}k_{2}}{2k_{20}}\frac{d^{3}p_{2}}{2p_{20}}$
(25) $\displaystyle=$
$\displaystyle\frac{dQ^{2}}{8\pi\sqrt{\lambda_{S}}}=\frac{\sqrt{\lambda_{X}}d\cos\theta}{8\pi(S+2M^{2}-\cos\theta
X\sqrt{\lambda_{S}/\lambda_{X}})}..$

a)

b) c)

d) e)

Figure 3: Feynman graphs corresponding to the Born contribution (a), leptonic
vertex correction (b), vacuum polarization (c), and real photon emission from
initial (d) and final (e) leptons.

The matrix element squared is expressed through the convolution of the
leptonic and hadronic tensors

$\displaystyle{\cal M}_{B}^{2}=\frac{e^{4}}{Q^{4}}W_{\mu\nu}(q)L^{\mu\nu}.$
(26)

The leptonic tensor is well known:

$\displaystyle L^{\mu\nu}_{B}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}[\gamma^{\mu}(\hat{k}_{1}+m)\gamma^{\nu}(\hat{k}_{2}+m)],$ (27)

while the hadronic tensor can be defined through the on-shell proton vertex

$\displaystyle\Gamma_{\mu}(q)=\gamma_{\mu}F_{d}(-q^{2})+\frac{i\sigma_{\mu\nu}q^{\nu}}{2M}F_{p}(-q^{2}),$
(28)

where $F_{d}(F_{p})$ is Dirac (Pauli) form factor, in the following way

$\displaystyle W_{\mu\nu}(q)$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}[\Gamma_{\mu}(q)(\hat{p}_{1}+M)\Gamma_{\nu}(-q)(\hat{p}_{1}+\hat{q}+M)]$

and then rearranged into covariant form

$\displaystyle W_{\mu\nu}(q)$ $\displaystyle=$
$\displaystyle-\Biggl{(}g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\Biggr{)}{\cal
F}_{1}(-q^{2})$ (30)
$\displaystyle+\Biggl{(}p_{1\mu}+\frac{q_{\mu}}{2}\Biggr{)}\Biggl{(}p_{1\nu}+\frac{q_{\nu}}{2}\Biggr{)}\frac{{\cal
F}_{2}(-q^{2})}{2M^{2}}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{2}w^{i}_{\mu\nu}(q){\cal F}_{i}(-q^{2}).$

Here

$\displaystyle{\cal F}_{1}(-q^{2})$ $\displaystyle=$
$\displaystyle-q^{2}(F_{d}(-q^{2})+F_{p}(-q^{2}))^{2},\qquad$
$\displaystyle{\cal F}_{2}(-q^{2})$ $\displaystyle=$ $\displaystyle
4M^{2}F_{d}(-q^{2})^{2}-q^{2}F_{p}(-q^{2})^{2}.$ (31)

As a result, after convolution we have

$\displaystyle\frac{d\sigma_{B}}{dQ^{2}}$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha^{2}}{\lambda_{S}Q^{4}}\sum\limits_{i=1}^{2}\theta_{B}^{i}{\cal
F}_{i}(Q^{2}),$ $\displaystyle\frac{d\sigma_{B}}{d\cos\theta}$
$\displaystyle=$ $\displaystyle j_{\theta}\frac{d\sigma_{B}}{dQ^{2}},$ (32)

where

$\displaystyle
j_{\theta}=\frac{\sqrt{\lambda_{S}}\lambda_{X}^{3/2}}{2M^{2}(SX-2m^{2}(Q^{2}+2M^{2}))},$
(33)

and

$\displaystyle\theta_{B}^{1}=Q^{2}-2m^{2},\qquad\theta_{B}^{2}=\frac{SX-M^{2}Q^{2}}{2M^{2}}.$
(34)

##  4 Additional virtual particle contribution

The additional virtual particle contribution can be expressed through Eqs.
(24,26) with replacement of the leptonic tensor (27) by

$\displaystyle L_{V}^{\mu\nu}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}[({\hat{k}}_{2}+m)\Gamma_{V}^{\mu}({\hat{k}}_{1}+m)\gamma^{\nu}]$ (35)
$\displaystyle+\frac{1}{2}{\rm
Tr}[({\hat{k}}_{2}+m)\gamma^{\mu}({\hat{k}}_{1}+m){\bar{\Gamma}}^{\nu}_{V}],$

where the leptonic vertex $\Gamma_{V}$ contains the sum of both the lepton
vertex correction $\Lambda^{\mu}$ and vacuum polarization by lepton
$\Pi^{l\mu}_{\alpha}$ represented by the Feynman graphs in Fig. 3(b) and Fig.
3(c), respectively

$\displaystyle\Gamma_{V}^{\mu}$ $\displaystyle=$
$\displaystyle\Lambda^{\mu}+\Pi^{l\mu}_{\alpha}\gamma^{\alpha},$
$\displaystyle{\bar{\Gamma}}^{\nu}_{V}$ $\displaystyle=$
$\displaystyle\gamma_{0}\Gamma^{\nu\;\dagger}_{V}\gamma_{0}.$ (36)

Similar to AGIM2015  we do not consider the vacuum polarization by the hadron.

Since $\Lambda_{\mu}$ and $\Pi^{l\mu}_{\alpha}$ contain the ultraviolet
divergence while $\Lambda_{\mu}$ also includes the infrared divergent terms,
both of these contributions have to be calculated analytically, and we choose
dimensional regularization for this calculation.

After the analytical calculation – detail of which can be found in Appendix D
of HAPRAD3  – $\Lambda_{\mu}$ and $\Pi^{i}_{\alpha\mu}$ read:

$\displaystyle\Lambda_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\biggl{(}\delta_{\rm
vert}^{UV}(Q^{2})\gamma_{\mu}-\frac{1}{2}mL_{m}[{\hat{q}},\gamma_{\mu}]\biggr{)},$
$\displaystyle\Pi^{l}_{\alpha\mu}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\biggl{(}g_{\alpha\mu}+\frac{q_{\alpha}q_{\mu}}{Q^{2}}\biggr{)}\sum_{i=e,\mu,\tau}\delta_{\rm
vac}^{i\;UV}(Q^{2}).$ (37)

The term in $\Lambda_{\mu}$ proportional to

$\displaystyle
L_{m}=\frac{1}{\sqrt{\lambda_{m}}}\log\frac{\sqrt{\lambda_{m}}+Q^{2}}{\sqrt{\lambda_{m}}-Q^{2}}$
(38)

is the anomalous magnetic moment whose contribution reads

$\displaystyle\frac{d\sigma_{AMM}}{dQ^{2}}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{3}m^{2}L_{m}}{2M^{2}Q^{2}\lambda_{S}}$
$\displaystyle\times\Biggl{[}12M^{2}{\cal F}_{1}(Q^{2})-(Q^{2}+4M^{2}){\cal
F}_{2}(Q^{2})\Biggl{]},$ $\displaystyle\frac{d\sigma_{AMM}}{d\cos\theta}$
$\displaystyle=$ $\displaystyle j_{\theta}\frac{d\sigma_{AMM}}{dQ^{2}}.$ (39)

The ultraviolet divergence contained in the remaining terms of Eqs (37) can be
removed by applying the mass-shell renormalization procedure that requires
their vanishing at $Q^{2}\to 0$:

$\displaystyle\delta_{\rm vert}$ $\displaystyle=$ $\displaystyle\delta_{\rm
vert}^{UV}(Q^{2})-\delta_{\rm vert}^{UV}(0),$ $\displaystyle\delta_{\rm
vac}^{i}$ $\displaystyle=$ $\displaystyle\delta_{\rm
vac}^{i\;UV}(Q^{2})-\delta_{\rm vac}^{i\;UV}(0).$ (40)

As a result, we obtain

$\displaystyle\delta_{\rm vert}$ $\displaystyle=$ $\displaystyle-
J_{0}\biggl{(}P_{IR}+\log\frac{m}{\mu}\biggr{)}-2+\biggl{(}\frac{3}{2}Q^{2}+4m^{2}\biggr{)}L_{m}$
$\displaystyle-\frac{Q^{2}+2m^{2}}{\sqrt{\lambda_{m}}}\biggl{(}\frac{1}{2}\lambda_{m}L_{m}^{2}+2{\rm
Li}_{2}\biggl{[}\frac{2\sqrt{\lambda_{m}}}{Q^{2}+\sqrt{\lambda_{m}}}\biggr{]}$
$\displaystyle-\frac{\pi^{2}}{2}\biggr{)},$ $\displaystyle\delta_{\rm
vac}^{l}$ $\displaystyle=$ $\displaystyle\sum_{i=e,\mu,\tau}\delta_{\rm
vac}^{i}=\sum_{i=e,\mu,\tau}\Bigl{[}\frac{2}{3}(Q^{2}+2m^{2}_{i})L_{m}^{i}$
(41)
$\displaystyle-\frac{10}{9}+\frac{8m_{i}^{2}}{3Q^{2}}\Bigl{(}1-2m^{2}_{i}L_{m}^{i}\Bigr{)}\Bigr{]}.$

Here

$\displaystyle J_{0}=2((Q^{2}+2m^{2})L_{m}-1),$ (42)

$\mu$ is an arbitrary parameter of the dimension of a mass,

$\displaystyle
P_{IR}=\frac{1}{n-4}+\frac{1}{2}\gamma_{E}+\log\frac{1}{2\sqrt{\pi}}$ (43)

is the infrared divergent term,

$\displaystyle{\rm Li}_{2}(x)=-\int\limits^{x}_{0}\frac{\log|1-y|}{y}dy$ (44)

is Spence’s dilogarithm, and

$\displaystyle
L_{m}^{i}=\frac{1}{\sqrt{\lambda_{m}^{i}}}\log\frac{\sqrt{\lambda_{m}^{i}}+Q^{2}}{\sqrt{\lambda_{m}^{i}}-Q^{2}},\;\lambda_{m}^{i}=Q^{2}(Q^{2}+4m_{i}^{2}).$
(45)

Finally, the virtual particle contribution reads

$\displaystyle\frac{d\sigma_{V}}{d\zeta}=\frac{d\sigma_{AMM}}{d\zeta}+\frac{\alpha}{\pi}(\delta_{vert}+\delta_{vac}^{l})\frac{d\sigma_{B}}{d\zeta},$
(46)

where $\zeta=Q^{2}$ or $\cos\theta$.

It should be noted that the above obtained expressions for the virtual
particle contributions agree with the results given in Section 3 of Kaiser
and in Appendix A of Vanderhaeghen . Particularly, while the comparison with
Vanderhaeghen  is straightforward, to verify agreement of our results with
Kaiser  we present Eq. (46) in the electron-muon scattering limit: $F_{d}\to
1$, $F_{p}\to 0$, $M\to m_{\mu}$ and $m\to m_{e}$.

##  5 Real photon emission for fixed $Q^{2}$

The contribution of real photon emission from the lepton leg presented in Fig.
3(d, e) has a form:

$\displaystyle d\sigma_{R}=\frac{1}{2\sqrt{\lambda_{S}}}{\cal
M}_{R}^{2}d\Gamma_{3},$ (47)

where the phase space can be expressed through the photonic variables
introduced after (8)

$\displaystyle d\Gamma_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{5}}\delta^{4}(p_{1}+k_{1}-p_{2}-k_{2}-k)\frac{d^{3}k}{2k_{0}}\frac{d^{3}k_{2}}{2k_{20}}\frac{d^{3}p_{2}}{2p_{20}}$
(48) $\displaystyle=$ $\displaystyle\frac{dQ^{2}vdvd\tau
d\phi_{k}}{2^{8}\pi^{4}(1+\tau)^{2}\sqrt{\lambda_{S}\lambda_{q}}}.$

The matrix element squared reads

$\displaystyle{\cal
M}_{R}^{2}=\frac{e^{6}}{t^{2}}W_{\mu\nu}(q-k)L_{R}^{\mu\nu},$ (49)

where $t=-(q-k)^{2}=Q^{2}+\tau R$ and $R=2p_{1}k=v/(1+\tau)$. The leptonic
tensor reads:

$\displaystyle L_{R}^{\mu\nu}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}{\rm
Tr}[\Gamma_{R}^{\mu\alpha}(\hat{k}_{1}+m)\bar{\Gamma}_{R\alpha}^{\nu}(\hat{k}_{2}+m)],$
(50)

with

$\displaystyle\Gamma_{R}^{\mu\alpha}$ $\displaystyle=$
$\displaystyle\Biggl{(}\frac{k_{1}^{\alpha}}{kk_{1}}-\frac{k_{2}^{\alpha}}{kk_{2}}\Biggr{)}\gamma^{\mu}-\frac{\gamma^{\mu}\hat{k}\gamma^{\alpha}}{2kk_{1}}-\frac{\gamma^{\alpha}\hat{k}\gamma^{\mu}}{2kk_{2}},$
$\displaystyle\Gamma_{R\alpha}^{\nu}$ $\displaystyle=$
$\displaystyle\Biggl{(}\frac{k_{1\alpha}}{kk_{1}}-\frac{k_{2\alpha}}{kk_{2}}\Biggr{)}\gamma^{\nu}-\frac{\gamma^{\nu}\hat{k}\gamma_{\alpha}}{2kk_{2}}-\frac{\gamma_{\alpha}\hat{k}\gamma^{\nu}}{2kk_{1}}.$
(51)

It is convenient to introduce the following convolutions integrated over
$\phi_{k}$:

$\displaystyle\int\limits_{0}^{2\pi}d\phi_{k}L_{R}^{\mu\nu}w^{i}_{\mu\nu}(q-k)$
$\displaystyle=$
$\displaystyle-4\pi\sqrt{\lambda_{q}}\sum\limits_{j=1}^{k_{i}}R^{j-3}$ (52)
$\displaystyle\qquad\;\times\theta_{ij}(v,\tau,Q^{2}).$

Here $k_{i}=\\{3,4\\}$, $\theta_{i1}(v,\tau,Q^{2})=4\theta^{i}_{B}F_{IR}$ and
the other components of $\theta_{ij}(v,\tau,Q^{2})$ tensor read:

$\displaystyle\theta_{12}$ $\displaystyle=$ $\displaystyle 4\tau F_{IR},$
$\displaystyle\theta_{13}$ $\displaystyle=$ $\displaystyle-4F-2\tau^{2}F_{d},$
$\displaystyle\theta_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{2M^{2}}\Biggl{[}2(Q^{2}-2\tau M^{2}-2(1+\tau)S)F_{IR}$
$\displaystyle+S_{p}(Q^{2}F_{1+}+2F_{2-}-\tau S_{p}F_{d})\Biggr{]},$
$\displaystyle\theta_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{2M^{2}}\Biggl{[}(\tau(2\tau
M^{2}-Q^{2})+4m^{2})F_{d}-S_{p}F_{1+}$ $\displaystyle+2(1+\tau)(\tau
S_{p}F_{d}+XF_{1+}+F_{IR}-F_{2-})\Biggr{]}+2F,$ $\displaystyle\theta_{24}$
$\displaystyle=$
$\displaystyle-\frac{1}{2M^{2}}\tau(1+\tau)(F_{1+}+(\tau+2)F_{d}),$ (53)

where $S_{p}=S+X=2S-Q^{2}$, and

$\displaystyle F_{d}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau}\biggl{(}\frac{1}{\sqrt{C_{2}}}-\frac{1}{\sqrt{C_{1}}}\biggr{)},\
$ $\displaystyle F_{1+}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{C_{1}}}+\frac{1}{\sqrt{C_{2}}},\;$ $\displaystyle
F_{2\pm}$ $\displaystyle=$ $\displaystyle
m^{2}\biggl{(}\frac{B_{2}}{C_{2}^{3/2}}\pm\frac{B_{1}}{C_{1}^{3/2}}\biggr{)},\;$
$\displaystyle F_{IR}$ $\displaystyle=$ $\displaystyle
F_{2+}-(Q^{2}+2m^{2})F_{d},\;$ $\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\lambda_{q}}}.$ (54)

Here:

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
4m^{2}(Q^{2}+\tau(Q^{2}+v)-\tau^{2}M^{2})$ $\displaystyle+(Q^{2}+\tau S)^{2},$
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle
4m^{2}(Q^{2}+\tau(Q^{2}+v)-\tau^{2}M^{2})$
$\displaystyle+(Q^{2}+\tau(v-X))^{2},$ $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\tau(S(Q^{2}+v)+2M^{2}Q^{2})$ $\displaystyle+Q^{2}(S_{p}-v),$
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\tau((X-v)(Q^{2}+v)-2M^{2}Q^{2})$ (55)
$\displaystyle+Q^{2}(S_{p}-v).$

As a result, we obtain

$\displaystyle d\sigma_{R}=-\frac{\alpha^{3}dQ^{2}d\tau
dv}{2\lambda_{S}(1+\tau)t^{2}}\sum\limits_{i=1}^{2}\sum\limits_{j=1}^{k_{i}}{\cal
F}_{i}(t)R^{j-2}\theta_{ij}(v,\tau,Q^{2}).$

A straightforward integration over the photon phase space is not possible
because of infrared divergence coming from the term with $j=1$ in (52) at the
point $v=0$ (or $R=0$). For the consistent extraction and cancellation of the
infrared divergence we use the Bardin- Shumeiko approach BSh . Following this
method, the identical transformation,

$\displaystyle
d\sigma_{R}=d\sigma_{R}-d\sigma_{R}^{IR}+d\sigma_{R}^{IR}=d\sigma_{R}^{F}+d\sigma_{R}^{IR},$
(57)

allows us to split $d\sigma_{R}$ into the infrared-free $d\sigma_{R}^{F}$ and
infrared-dependent $d\sigma_{R}^{IR}$ parts. The last one can be obtained
before integration over $\phi_{k}$ as a term factorized in front of the Born
cross section:

$\displaystyle d\sigma_{R}^{IR}$ $\displaystyle=$
$\displaystyle\frac{1}{R}\lim_{R\to
0}Rd\sigma_{R}=-\frac{\alpha}{\pi^{2}}d\sigma_{B}\frac{vdvd\tau
d\phi_{k}}{2(1+\tau)^{2}\sqrt{\lambda_{q}}}{\cal F_{IR}},$

where

$\displaystyle{\cal F_{IR}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\biggl{(}\frac{k_{1}}{kk_{1}}-\frac{k_{2}}{kk_{2}}\biggr{)}^{2}.$
(59)

Note that

$\displaystyle F_{IR}$ $\displaystyle=$
$\displaystyle\frac{R^{2}}{2\pi\sqrt{\lambda_{q}}}\int\limits_{0}^{2\pi}d\phi_{k}{\cal
F_{IR}}.$ (60)

The treatment of the infrared divergence by the Bardin-Shumeiko approach
requires to separate $d\sigma_{R}^{IR}$ into the soft $\delta_{S}$ and hard
$\delta_{H}$ parts

$\displaystyle\frac{d\sigma_{R}^{IR}}{dQ^{2}}=\frac{\alpha}{\pi}\delta_{IR}\frac{d\sigma_{B}}{dQ^{2}}=\frac{\alpha}{\pi}(\delta_{S}+\delta_{H})\frac{d\sigma_{B}}{dQ^{2}}$
(61)

by introducing of the infinitesimal inelasticity $\bar{v}$

$\displaystyle\delta_{S}=-\frac{1}{\pi}\int\limits_{0}^{\bar{v}}dv\int\frac{d^{3}k}{k_{0}}\delta((p_{1}+q-k)^{2}-M^{2}){\cal
F_{IR}},$
$\displaystyle\delta_{H}=-\frac{1}{\pi}\int\limits_{\bar{v}}^{v_{cut}}dv\int\frac{d^{3}k}{k_{0}}\delta((p_{1}+q-k)^{2}-M^{2}){\cal
F_{IR}}.$ (62)

This separation allows us to calculate $\delta_{S}$ in the dimensional
regularization by choosing the individual reference systems for each leptonic
propagator $1/kk_{1}$ and $1/kk_{2}$, as well as their combination to make
them independent of the azimuthal angle $\phi_{k}$ while the hard part can be
calculated in straightforward way without any regularization.

It can be seen from the explicit expressions for $\delta_{S}$ and $\delta_{H}$
– details of their calculation can be found in A – that for $Q^{2}\to 0$ both
of the m tend to zero and their sum

$\displaystyle\delta_{IR}$ $\displaystyle=$ $\displaystyle
J_{0}\biggl{[}P_{IR}+\log\frac{v_{cut}}{\mu
M}\biggr{]}+\frac{1}{2}SL_{S}+\frac{1}{2}XL_{X}$ (63)
$\displaystyle+S_{\phi}(k_{1},k_{2},p_{2})$

does not depend on the separated inelasticity $\bar{v}$ and contains the
infrared term $P_{IR}$ as well as a parameter $\mu$ that have to be cancelled
against corresponding terms in $\delta_{\rm vert}$.

Therefore RC for fixed $Q^{2}$ read:

$\displaystyle\frac{d\sigma_{RC}}{dQ^{2}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\pi}(\delta_{VR}+\delta_{\rm
vac}^{l})\frac{d\sigma_{B}}{dQ^{2}}+\frac{d\sigma_{AMM}}{dQ^{2}}+\frac{d\sigma_{F}}{dQ^{2}}.$
(64)

Here the expression for $\delta_{\rm vac}^{l}$ is defined in Eq. (4),
$\delta_{VR}$ is an infrared-free sum $\delta_{\rm vert}$ and $\delta_{IR}$:

$\displaystyle\delta_{VR}$ $\displaystyle=$
$\displaystyle\delta_{IR}+\delta_{\rm
vert}=J_{0}\log\frac{v_{cut}}{mM}+\frac{1}{2}SL_{S}+\frac{1}{2}XL_{X}$ (65)
$\displaystyle+S_{\phi}(k_{1},k_{2},p_{2})-2+\biggl{(}\frac{3}{2}Q^{2}+4m^{2}\biggr{)}L_{m}$
$\displaystyle-\frac{Q^{2}+2m^{2}}{\sqrt{\lambda_{m}}}\biggl{(}\frac{1}{2}\lambda_{m}L_{m}^{2}+2{\rm
Li}_{2}\biggl{[}\frac{2\sqrt{\lambda_{m}}}{Q^{2}+\sqrt{\lambda_{m}}}\biggr{]}$
$\displaystyle-\frac{\pi^{2}}{2}\biggr{)}.$

The general expression for $S_{\phi}(k_{1},k_{2},p_{2})$ is reproduced in B
and for our case

$\displaystyle S_{\phi}(k_{1},k_{2},p_{2})$ $\displaystyle=$
$\displaystyle\frac{Q^{2}+2m^{2}}{\sqrt{\lambda_{m}}}\Biggl{(}\frac{1}{4}\lambda_{X}L_{X}^{2}-\frac{1}{4}\lambda_{S}L_{S}^{2}$
$\displaystyle+$ $\displaystyle{\rm
Li}_{2}\biggl{[}1-\frac{(X+\sqrt{\lambda_{X}})\rho}{8m^{2}M^{2}}\biggr{]}$
$\displaystyle+$ $\displaystyle{\rm
Li}_{2}\biggl{[}1-\frac{\rho}{2(X+\sqrt{\lambda_{X}})}\biggr{]}$
$\displaystyle-$ $\displaystyle{\rm
Li}_{2}\biggl{[}1-\frac{Q^{2}(S+\sqrt{\lambda_{S}})\rho}{2M^{2}(Q^{2}+\sqrt{\lambda_{m}})^{2}}\biggr{]}$
$\displaystyle-$ $\displaystyle{\rm
Li}_{2}\biggl{[}1-\frac{2m^{2}Q^{2}\rho}{(Q^{2}+\sqrt{\lambda_{m}})^{2}(S+\sqrt{\lambda_{S}})}\biggr{]}\Biggr{)},$

where
$\rho=(Q^{2}+\sqrt{\lambda_{m}})(S_{p}-\sqrt{\lambda_{m}})/\sqrt{\lambda_{m}}$.

The anomalous magnetic moment contribution is represented by Eqs. (39). At
last, the finite part of the cross section reads:

$\displaystyle\frac{d\sigma_{F}}{dQ^{2}}$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{3}}{2\lambda_{S}}\int\limits_{0}^{v_{cut}}dv\sum\limits_{i=1}^{2}\Biggl{[}4\frac{J_{0}\theta_{B}^{i}{\cal
F}_{i}(Q^{2})}{vQ^{4}}$ (67) $\displaystyle+$
$\displaystyle\int\limits_{\tau_{min}^{q}}^{\tau_{max}^{q}}\frac{d\tau}{(1+\tau)t^{2}}\sum\limits_{j=1}^{k_{i}}{\cal
F}_{i}(t)R^{j-2}\theta_{ij}(v,\tau,Q^{2})\Biggr{]},$

where the integration limits over $\tau$ are defined by Eqs. (22).

##  6 Real photon emission for fixed scattering angle

The phase space for this case reads:

$\displaystyle d\Gamma_{3}$ $\displaystyle=$ $\displaystyle
J_{\theta}(v)\frac{vdvd\cos\theta d\tau
d\phi_{k}}{2^{8}\pi^{4}(1+\tau)^{2}\sqrt{\lambda_{S}\lambda_{v}}},$ (68)

where

$\displaystyle J_{\theta}(v)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{S}-vS-Q^{2}_{R}(v)(S+2M^{2})}{(S+2M^{2})^{2}-\lambda_{S}\cos^{2}\theta}$
(69) $\displaystyle\times$
$\displaystyle\Biggl{(}\frac{S+2M^{2}}{\cos\theta}+M\sqrt{\frac{\lambda_{S}}{\cal
D}}(S-v+2m^{2})\Biggr{)},$

and $J_{\theta}(0)=j_{\theta}$.

After some algebra similar to the previous section, we have:

$\displaystyle\frac{d\sigma_{RC}}{d\cos\theta}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\pi}(\delta_{VR}+\delta_{\rm
vac}^{l})\frac{d\sigma_{B}}{d\cos\theta}+\frac{d\sigma_{AMM}}{d\cos\theta}+\frac{d\sigma_{F}}{d\cos\theta},$

while the finite part has the following structure:

$\displaystyle\frac{d\sigma_{F}}{d\cos\theta}=-\frac{\alpha^{3}}{2\lambda_{S}}\int\limits_{0}^{v_{cut}}dv\sum\limits_{i=1}^{2}\Biggl{[}4j_{\theta}\frac{J_{0}\theta_{B}^{i}{\cal
F}_{i}(Q^{2})}{vQ^{4}}+J_{\theta}(v)$
$\displaystyle\times\int\limits_{\tau_{min}^{\theta}}^{\tau_{max}^{\theta}}\frac{d\tau}{(1+\tau)t^{2}}\sum\limits_{j=1}^{k_{i}}{\cal
F}_{i}(t)R^{j-2}\theta_{ij}(v,\tau,Q^{2}_{R}(v))\Biggr{]}.$

##  7 Numerical results

Here we present the relative RC which is defined as a ratio of RC to the Born
cross section

$\displaystyle\delta_{RC}=\frac{d\sigma_{RC}/d\zeta}{d\sigma_{B}/d\zeta}$ (72)

both for fixed $Q^{2}$ ($\zeta=Q^{2}$) and the scattering angle
($\zeta=\cos\theta$) presented in Eqs. (64) and (LABEL:rcth), respectively.
Corresponding Born contributions are defined by Eqs. (32).

Figure 4:  Relative RC vs the value of the scattering lepton kinetic energy
for elastic $ep$ and $\mu p$ scattering, beam momenta is equal to 115 MeV, 153
MeV and 210 MeV for $\theta=20^{o}$ (1), $60^{o}$ (2), $100^{o}$ (3). Solid
(dashed) line corresponds to fixed $Q^{2}$ ($\cos\theta$).

As mentioned above, a cut applied on the upper integration limit over
inelasticity allows to reduce the contribution of hard photon emission. On the
other hand, for the radiative process the energy of the scattering lepton
depends on the inelasticity as it is presented in Eqs. (9) for the fixed
$Q^{2}$ and (15) for the fixed scattering angle. Therefore instead of the
upper limit over inelasticity, we can set a cut on the lower limit of the
scattered-lepton energy.

The result of these cuts under MUSE kinematic conditions MUSE  is presented in
Fig. 4. As we can see, the situation for the scattering electron for fixed
$Q^{2}$ and scattering angle for soft photon emission is almost identical
while for hard photon emission it is dramatically different: for the fixed
scattering angle RC increase to 80% while for the fixed $Q^{2}$ RC do not
exceed 5% . This is a key observation for both electron and positron
scattering in the experimental analysis.

Another interesting issue consists in the $\varepsilon$-behavior at JLab
kinematic conditions Jlab ; Jlab1 . Following our previous work chaslet  we
can define

$\displaystyle\varepsilon_{q}=\Biggl{[}1+2\biggl{(}1+\frac{Q^{2}}{4M^{2}}\biggr{)}\frac{M^{2}(Q^{2}-2m^{2})}{S(X-v_{cut})-M^{2}Q^{2}}\Biggr{]}^{-1}$
(73)

for fixed $Q^{2}$ and in a similar way

$\displaystyle\varepsilon_{\theta}=\varepsilon_{q}\big{|}_{Q^{2}\to
Q^{2}_{R}(v_{cut})}$ (74)

for fixed scattering angle.

The numerical result presented in Fig. 5 shows almost identical values of RC
for the soft photon emission and different behavior of RC with the hard real
photons for the fixed $Q^{2}$ vs. a fixed scattering angle. In the first case
with growing $v_{cut}$ the value of the variable $\varepsilon$ decreases and
RC for the hard photon (when $v_{cut}=v_{q}$) does not exceed 1.6 times the
Born contribution, while for the fixed scattering angle $\varepsilon$ goes a
little bit up but the absolute value of the relative RC rapidly increases
reaching the values up to 45 times (when $v_{cut}=v_{\theta}$). Such a rapid
change of RC near the kinematic limit of fixed-angle measurements sets more
stringent requirements on energy resolution for lepton detection in the fixed-
angle kinematic setting, as opposed to fixed $Q^{2}$ analysis.

Figure 5:  Relative RC vs value of the scattering particle kinetic energy as a
function of $\varepsilon$ at $Q^{2}=0.85$ GeV2 and $Q^{2}=1.45$ GeV2. The
solid (dashed) correspond fixed $Q^{2}$ ($\cos\theta$).

##  8 Conclusion

We discussed essential differences between the kinematic description of
radiative effects for fixed $Q^{2}$ vs. fixed scattering angle in the elastic
lepton-proton scattering. In particular, it was shown that for the description
of hard-photon emission at fixed scattering angle, even for the high-energy
electron-proton scattering the ultrarelativistic approximation is not
applicable in the considered kinematics as we approach the limits of phase
space. The technique of Bardin-Shumeiko for the covariant extraction and
cancellation of the infrared divergence as well as the explicit expressions
for RC to the lepton current in unpolarized elastic $lp$-scattering within
these two cases were presented. The numerical analysis within kinematic
conditions of Jefferson Lab measurements and MUSE experiment in PSI has shown
the almost identical values of RC for the soft photon emission and
significantly different behavior of RC with the hard real photon for the fixed
$Q^{2}$ compared with fixing a lepton scattering angle. The presented
formalism may be of use also for the high-energy muon scattering case of the
AMBER proposal AMBER .

Based on our recent work chaslet , in the nearest future we intend to
generalize the numerical comparison of RC calculation for the fixed $Q^{2}$
and scattering angle with an electron/positron and muon/antimuon charge
asymmetry. We also intend to include simulations of radiative events for the
fixed scattering angle into Monte Carlo generator ELRADGEN ELRADGEN1 ;
ELRADGEN2 , that is used for the hard photon generation in the elastic
$lp$-scattering.

###### Acknowledgements.

The authors thank the anonymous referee of this paper for insightful comments.
Data Availability Statement This manuscript has no associated data or the data
will not be deposited. [Authors’ comment: The discussion presented in this
article develops from already existing and published data which are duly
referenced.]

##  Appendix A Calculation of $\delta_{S}$ and $\delta_{H}$

For calculation of $\delta_{S}$ in the dimensional regularization

$\displaystyle\frac{d^{3}k^{\prime}}{k^{\prime}_{0}}$ $\displaystyle\to$
$\displaystyle\frac{d^{n-1}k^{\prime}}{(2\pi\mu)^{n-4}k_{0}}$ (75)
$\displaystyle=$ $\displaystyle\frac{2\pi^{n/2-1}k_{0}^{\prime
n-3}dk_{0}(1-x^{2})^{n/2-2}dx}{(2\pi\mu)^{n-4}\Gamma(n/2-1)},$

where $x=\cos\theta$ ($\theta$ is defined as the spatial angle between the
photon three-momentum and ${\bf k}^{\prime}_{i}$ ($i=1-3$) that are introduced
below) and $\mu$ is an arbitrary parameter of the dimension of a mass the
reference system ${\bf p}_{1}{\bf+q=0}$ is used.

The Feynman parameterization of (59) gives

$\displaystyle{\cal F_{IR}}$ $\displaystyle=$
$\displaystyle\frac{1}{4k_{0}^{\prime
2}}\int\limits_{0}^{1}dy\Biggl{[}\frac{m^{2}}{k_{10}^{\prime
2}(1-x\beta_{1})^{2}}+\frac{m^{2}}{k_{20}^{\prime 2}(1-x\beta_{2})^{2}}$ (76)
$\displaystyle-\frac{Q^{2}+2m^{2}}{k_{30}^{\prime
2}(1-x\beta_{3})^{2}}\Biggr{]}=\frac{1}{4k_{0}^{\prime
2}}\int\limits_{0}^{1}dy{\cal F}(x,y).$

Here $\beta_{i}=|{\bf k^{\prime}}_{i}|/k^{\prime}_{i0}$ for $i=1,2,3$ and
$k_{3}=yk_{1}+(1-y)k_{2}$.

After the substitution of Eqs. (75) and (76) into the definition of
$\delta_{S}$ by Eq. (62) and, using $\delta$-function, integrated over the
photon energy $k_{0}$ one can find that

$\displaystyle\delta_{S}$ $\displaystyle=$
$\displaystyle-\frac{1}{2(4\mu\sqrt{\pi})^{n-4}\Gamma(n/2-2)}\int\limits_{-1}^{1}dx(1-x^{2})^{n/2-2}$
(77) $\displaystyle\times\int\limits_{0}^{1}dy{\cal
F}(x,y)\int\limits_{0}^{\bar{v}}\frac{dv}{v}\left(\frac{v}{M}\right)^{n-4}.$

The integration over $v$ and the expansion of the obtained expression into the
Laurent series around $n=4$ result in

$\displaystyle\delta_{S}$ $\displaystyle=$
$\displaystyle\delta_{S}^{IR}+\delta_{S}^{1},$ (78)

where

$\displaystyle\delta_{S}^{IR}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\biggl{[}P_{IR}+\log\frac{\bar{v}}{\mu
M}\biggr{]}\int\limits_{0}^{1}dy\int\limits_{-1}^{1}dx{\mathcal{F}}(x,y)$ (79)

and

$\displaystyle\delta_{S}^{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\int\limits_{0}^{1}dy\int\limits_{-1}^{1}dx\log\biggl{[}\frac{1}{4}(1-x^{2})\biggr{]}{\mathcal{F}}(x,y).$
(80)

Here $P_{IR}$ is the infrared divergent term defined by Eq. (43). Taking into
account that $k_{3}^{2}=y(1-y)Q^{2}+m^{2}$ the integration over $x$ and $y$
variables in $\delta_{S}^{IR}$ is simple:

$\displaystyle\delta_{S}^{IR}$ $\displaystyle=$ $\displaystyle
J_{0}\biggl{[}P_{IR}+\log\frac{\bar{v}}{\mu M}\biggr{]},$ (81)

where $J_{0}$ is defined by Eq. (42).

For the calculation of $\delta_{S}^{1}$ we note that in the system ${\bf
p}_{1}{\bf+q=0}$ the energies of the initial and scattering lepton through the
invariants:

$\displaystyle k_{10}^{\prime}=\frac{X}{2M^{2}},\qquad
k_{20}^{\prime}=\frac{S}{2M^{2}}.$ (82)

As a result,

$\displaystyle\delta_{S}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}SL_{S}+\frac{1}{2}XL_{X}+S_{\phi}(k_{1},k_{2},p_{2}),$
(83)

where

$\displaystyle
L_{S}=\frac{1}{\sqrt{\lambda_{S}}}\log\frac{S+\sqrt{\lambda_{S}}}{S-\sqrt{\lambda_{S}}},$
$\displaystyle
L_{X}=\frac{1}{\sqrt{\lambda_{X}}}\log\frac{X+\sqrt{\lambda_{X}}}{X-\sqrt{\lambda_{X}}},$
(84)

and

$\displaystyle S_{\phi}(k_{1},k_{2},p_{2})$ $\displaystyle=$
$\displaystyle\frac{Q^{2}+2m^{2}}{4}\int\limits_{-1}^{1}dx\int\limits_{0}^{1}dy\frac{\log[(1-x^{2})/4]}{k^{\prime
2}_{30}(1-x\beta_{3})^{2}}.$

Notice that the standard expressions for $S_{\phi}$ are rather cumbersome, see
for example Eqs. (35) and (A.14) of work AGIM2015 . In B we present a more
compact analytical expression for this quantity.

For the calculation of $\delta_{H}$ the straightforward integration is used.
Taking into account (60), one can find that

$\displaystyle\delta_{H}$ $\displaystyle=$
$\displaystyle-\int\limits_{\bar{v}}^{v_{cut}}\frac{dv}{v}\int\limits_{\tau_{min}}^{\tau_{max}}d\tau
F_{IR}=J_{0}\log\frac{v_{cut}}{\bar{v}}.$ (86)

##  Appendix B Calculation of $S_{\phi}$

Here we present a general approach suggested by ’t Hooft and Veltman in their
work tHooft  for a compact representation of the $S_{\phi}$-function
introduced by Bardin and Shumeiko in BSh . Let us consider a real photon with
a momentum $k$ and three other time-like four-momenta $a_{i}$ ($i=1,2,3$) with
masses $m_{i}^{2}=a_{i}^{2}$. The basic idea consists in Feynman
parameterization. Instead of usual approach used in the standard Bardin-
Shumeiko technique with two fermionic propagator presented in previous
appendix, taken in the system ${\bf a}_{3}=0$:

$\displaystyle\frac{1}{a_{1}k}\frac{1}{a_{2}k}=\gamma\frac{1}{a_{1}k}\frac{1}{\gamma
a_{2}k}=\frac{\gamma}{k_{0}^{2}}\int\limits_{0}^{1}\frac{dy}{a_{40}^{2}(1-x\beta)^{2}}.$
(87)

Here, as in the previous appendix $x=\cos\theta$, a new four-vector
$a_{4}=ya_{1}+(1-y)\gamma a_{2}$, and $\beta=|{\bf a}_{4}|/a_{40}$. The
quantity $\gamma$ is choosing in such a way, that $(a_{1}-\gamma
a_{2})^{2}=0$, $i.e.$ $a_{1}-\gamma a_{2}$ is lightlike vector.

Now introduce the following invariants:

$\displaystyle s_{1}=2a_{1}a_{3},\;\lambda_{1}=s_{1}^{2}-4m_{1}^{2}m_{3}^{2},$
$\displaystyle s_{2}=2a_{2}a_{3},\;\lambda_{2}=s_{2}^{2}-4m_{2}^{2}m_{3}^{2},$
$\displaystyle s_{3}=2a_{1}a_{2},\;\lambda_{3}=s_{3}^{2}-4m_{1}^{2}m_{2}^{2}.$
(88)

Then equation $(a_{1}-\gamma a_{2})^{2}=0$ has the following two solutions:

$\displaystyle\gamma_{1}=\frac{2m_{1}^{2}}{s_{3}+\sqrt{\lambda_{3}}},\qquad\gamma_{2}=\frac{s_{3}+\sqrt{\lambda_{3}}}{2m_{2}^{2}},$
(89)

and the generalized form of $S_{\phi}$ looks as (LABEL:sf):

$\displaystyle S_{\phi}=\frac{1}{4}\gamma
s_{3}\int\limits_{0}^{1}\frac{dy}{a_{40}^{2}}\int\limits_{-1}^{1}dx\frac{\log[(1-x^{2})/4]}{(1-x\beta)^{2}}.$
(90)

The first integration over $x$ is straightforward

$\displaystyle S_{\phi}=\frac{1}{2}\gamma
s_{3}\int\limits_{0}^{1}\frac{dy}{m_{4}^{2}\beta}\log\frac{1-\beta}{1+\beta},$
(91)

where $m_{4}^{2}=a_{4}^{2}=ym_{1}^{2}+(1-y)\gamma^{2}m_{2}^{2}$. The second
integration has to be performed after the standard substitutions, while taking
into account that for the first two momenta $a_{i0}=s_{i}/(2m_{3})$.

Finally, we can find that for the general case $S_{\phi}$ depends on six
variables and for $\gamma=\gamma_{1}$ it has the following structure:

$\displaystyle S_{\phi}(a_{1},a_{2},a_{3})$ $\displaystyle=$
$\displaystyle\frac{s_{3}}{\sqrt{\lambda_{3}}}\Biggl{(}\log^{2}\frac{s_{1}+\sqrt{\lambda_{1}}}{2m_{1}m_{3}}-\log^{2}\frac{s_{2}+\sqrt{\lambda_{2}}}{2m_{2}m_{3}}$
$\displaystyle+{\rm
Li}_{2}\biggl{[}1-\frac{(s_{1}+\sqrt{\lambda_{1}})\rho}{8m_{1}^{2}m_{3}^{2}}\biggr{]}$
$\displaystyle+{\rm
Li}_{2}\biggl{[}1-\frac{\rho}{2(s_{1}+\sqrt{\lambda_{1}})}\biggr{]}$
$\displaystyle-{\rm
Li}_{2}\biggl{[}1-\frac{(s_{2}+\sqrt{\lambda_{2}})\rho}{4m_{3}^{2}(s_{3}+\sqrt{\lambda_{3}})}\biggr{]}$
$\displaystyle-{\rm
Li}_{2}\biggl{[}1-\frac{m_{2}^{2}\rho}{(s_{2}+\sqrt{\lambda_{2}})(s_{3}+\sqrt{\lambda_{3}})}\biggr{]},$

where
$\rho=(2s_{1}(s_{3}+\sqrt{\lambda_{3}})-4m_{1}^{2}s_{2})/\sqrt{\lambda_{3}}$.

It should be noted that

$\displaystyle S_{\phi}(a_{1},a_{2},a_{3})=S_{\phi}(a_{2},a_{1},a_{3})$ (93)

The r.h.s. of this equation corresponds $\gamma=\gamma_{2}$.

In our case $a_{1}=k_{1}$, $a_{2}=k_{2}$, $a_{3}=p_{2}$, and $s_{1}=X$,
$s_{2}=S$, $s_{3}=Q^{2}+2m^{2}$, $m_{1}=m_{2}=m$, $m_{3}=M$.

## References

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