aainstitutetext: Institute for Nuclear Research of the Russian Academy of
Sciences,
60th October Anniversary pr-ct 7a, Moscow 117312, Russiabbinstitutetext:
Landau Phystech School of Physics and Research, Moscow Institute of Physics
and Technology,
Institutskiy per. 9, Dolgoprudny 141700, Russiaccinstitutetext: Faculty of
Physics, Lomonosov Moscow State University,
Leninskiye Gory 1-2, Moscow 119991, Russia

#  INR-TH-2023-009 Dark photon production via elastic proton bremsstrahlung
with non-zero momentum transfer

D. Gorbunov a,c E. Kriukova<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore hypothetical vector particles, dark photons $\gamma^{\prime}$,
which mix with the Standard Model photons and thus mediate interactions with
charged particles into the hidden sector. We study the elastic proton
bremsstrahlung of dark photons with masses 0.4–1.8 GeV, relevant for direct
searches with proton accelerators. A key feature of our calculation is that it
explicitly considers the non-zero momentum transfer between protons in the
process $pp\rightarrow pp\gamma^{\prime}$. We compare the obtained
differential and full bremsstrahlung cross sections with the results of other
authors. Our calculation agrees well (up to 3–9% corrections) with the
Weizsacker-Williams approximation that confirms its applicability for proton
beams. Then we refine predictions for the dark photon production with proton
beams of energy 30 GeV, 70 GeV, 120 GeV and 400 GeV relevant for past, present
and future experiments considered in literature.

††arxiv: 2306.15800

##  1 Introduction

A variety of experimental data including observations of galaxy rotation
curves Rubin:1970zza ; deMartino:2020gfi  and neutrino oscillations Super-
Kamiokande:1998kpq ; Ahlers:2018mkf  indicate the need to extend the Standard
Model (SM) of particle physics and explore yet unknown particles of dark
sector singlet with respect to the SM gauge group. One common approach is to
couple the dark sector to the SM via special new particles, mediators, that
are able to interact with both of them Essig:2013lka . There exist only three
ways to write down the renormalizable interaction of the SM fields with a
mediator: scalar, vector and fermion portals Lanfranchi:2020crw . In this
study we concentrate on the vector portal in which mediator is so-called dark
photon.

We consider only the part of the model responsible for the portal-type
interaction, i.e. the SM Lagrangian $\mathcal{L}_{SM}$ is extended as follows
Okun:1982xi ; Galison:1983pa ; Holdom:1985ag ,

$\mathcal{L}=\mathcal{L}_{SM}-\frac{1}{4}F^{\prime}_{\mu\nu}F^{\prime\mu\nu}+\frac{\epsilon}{2}F^{\prime}_{\mu\nu}B^{\mu\nu}+\frac{m_{\gamma^{\prime}}^{2}}{2}A^{\prime}_{\mu}A^{\prime\mu},$
(1)

where $F^{\prime}_{\mu\nu}$ is the dark photon strength tensor, $B_{\mu\nu}$
is the SM hypercharge strength tensor, $\epsilon$ is the kinetic mixing
parameter and $m_{\gamma^{\prime}}$ is the dark photon mass. Due to mixing
with the SM photon the dark photon couples to all SM particles, initially
charged with $q$ under $U(1)_{em}$, with charge $\epsilon q$ Filippi:2020kii .
In what follows we discuss the dark photon production by accelerated protons
incident on target. The corresponding center-of-mass energies are much lower
than the electroweak energy scale, so the $Z$-boson mixing induced by the
portal coupling (1) can be safely neglected.

The vector portal has been intensively studied in experiments with electron
beam dumps (NA64 NA64:2016oww ; NA64:2017vtt ; NA64:2018lsq ; Banerjee:2019pds
; NA64:2019auh , E141 Riordan:1987aw ; Bjorken:2009mm ; Andreas:2012mt ),
proton beam dumps (PS191 Bernardi:1985ny ; Gninenko:2011uv , NOMAD
Gninenko:2011uv ; NOMAD:2001eyx , CHARM CHARM:1985nku ; Gninenko:2012eq ), at
$e^{+}e^{-}$ colliders (BaBar BaBar:2014zli ; BaBar:2017tiz , KLOE
KLOE-2:2011hhj ; KLOE-2:2012lii ; Anastasi:2015qla ; KLOE-2:2016ydq ) and at
$pp$ colliders (LHCb LHCb:2017trq ; LHCb:2019vmc , CMS CMS:2018jid ;
CMS:2019ajt ; CMS:2019buh ; CMS:2020krr ), see Essig:2013lka ; Raggi:2015yfk ;
Agrawal:2021dbo  for the review on dark photon experimental searches. However
for dark photon masses above 0.8 GeV experimental constraints are still quite
weak, $\epsilon<5\cdot 10^{-3}$ Graham:2021ggy . Therefore several experiments
under construction, such as DUNE DUNE:2020fgq ; Breitbach:2021gvv , T2K
deNiverville:2016rqh  and proposals such as SHiP Gorbunov:2014wqa ;
SHiP:2015vad ; SHiP:2020vbd  aim also at searching for dark photons of mass
about 1 GeV produced in $pp$ collisions Batell:2022dpx . In order to estimate
the sensitivity of these projects one needs to study the phenomenology of
${\cal O}(1)$ GeV dark photon, in particular its production modes.

There are three mechanisms of dark photon production in proton-proton
collisions Miller:2021ycl . The dominant one is determined by the value of
dark photon mass $m_{\gamma^{\prime}}$. Dark photons lighter than about 0.4
GeV are mostly produced in meson decays
$\pi^{0}/\eta\rightarrow\gamma^{\prime}\gamma$ due to dark photon mixing with
the SM photon. The QCD Drell-Yan process $q\bar{q}\rightarrow\gamma^{\prime}$
is the most important mode for dark photons with mass
$m_{\gamma^{\prime}}\gtrsim 1.8$ GeV. In the intermediate mass range 0.4–1.8
GeV the dark photon production is mainly due to proton bremsstrahlung
$pp\rightarrow\gamma^{\prime}pp$. In this study we consider dark photon
production in elastic proton bremsstrahlung. Our main result is a new estimate
for its cross section taking into account a realistic non-zero momentum
transfer between the protons.

The paper is organized as follows. In section 2 we give a brief overview of
the existing estimates of bremsstrahlung cross section. Section 3 contains our
calculation of the proton bremsstrahlung cross section with non-zero momentum
transfer. In section 4 we present the numerical results for typical dark
photon masses and compare them with other approximations. Section 5 summarizes
our findings and future prospects.

##  2 Overview of the previously suggested methods

One of the first successful attempts to calculate the electron bremsstrahlung
is given in Kim:1973he . There the modified Weizsacker-Williams approximation
is formulated assuming the one-photon exchange in the process
$e(a)p(P_{i})\rightarrow\gamma^{\prime}(b)e(c)p(P_{f})$. The photon propagator
gives the contribution $1/t^{2}$ to the square of matrix element, where
$t\equiv-q^{2}$ and $q$ is the momentum of virtual photon. Then the main
contribution comes from a phase space region near the minimal value of $t$

$\sqrt{t_{\text{min}}}=\frac{(a\cdot
b)-m^{2}_{\gamma^{\prime}}/2}{a_{0}-b_{0}}.$ (2)

In the lab frame this happens when the difference of 3-dimensional vectors
$\vec{a}-\vec{b}$ and vector $\vec{c}$ are parallel,
$\vec{a}-\vec{b}\parallel\vec{c}$. The differential bremsstrahlung cross
section is approximated then by the differential cross section of the
subprocess $e(a)\gamma(q)\rightarrow\gamma^{\prime}(b)e(c)$ taken at
$t=t_{\text{min}}$

$\left[\frac{\differential^{2}\sigma(ep\rightarrow\gamma^{\prime}ep)}{\differential(a\cdot
b)\differential(b\cdot
P_{i})}\right]_{WW}=\left[\frac{\differential\sigma(e\gamma\rightarrow\gamma^{\prime}e)}{\differential(a\cdot
b)}\right]_{t=t_{\text{min}}}\frac{\alpha}{\pi}\frac{\chi}{(c\cdot P_{i})},$
(3)

where $\alpha$ is the fine-structure constant, $\chi$ is the flux of photons
emitted by the target proton

$\chi=\int\displaylimits_{t_{\text{min}}}^{t_{\text{max}}}\frac{t-t_{\text{min}}}{t^{2}}G_{2}(t)\,\differential
t$ (4)

and $G_{2}(t)$ is the proton electric form factor depending on the energy
transfer as presented in Kim:1973he . This approximation was used to calculate
the dark photon production cross section in electron bremsstrahlung
Bjorken:2009mm . The validity of Weizsacker-Williams approximation for dark
photon production in electron bremsstrahlung was questioned in Liu:2017htz .

Recently the Weizsacker-Williams approximation has been applied to proton
bremsstrahlung
$p(p)p(P_{i})\rightarrow\gamma^{\prime}(k)p(p^{\prime})p(P_{f})$ in the case
of elastic proton scattering by Foroughi-Abari and Ritz Foroughi-Abari:2021zbm
. In this approach the protons exchange virtual effective vector bosons, the
so-called Donnachie-Landshoff pomerons $\mathbb{P}$, which have effective
propagators and proton-pomeron vertices obtained from fits to experimental
data on elastic proton scattering Donnachie:1992ny ; Donnachie:2013xia . The
differential cross section of proton bremsstrahlung is given by

$\left[\frac{\differential^{2}\sigma(pp\rightarrow\gamma^{\prime}pp)}{\differential
z\differential k^{2}_{\perp}}\right]_{WW}=\frac{\epsilon^{2}\alpha
z(1-z)}{16\pi^{2}H^{2}}A_{2\rightarrow
2}\chi_{\mathbb{P}}\absolutevalue{F_{\text{VMD}}}^{2}F_{\text{virt}}^{2}\,.$
(5)

Here $z\equiv k_{z}/p_{z}$ is the ratio of projections of dark photon
3-momentum and incident proton 3-momentum on the beam axis, $k_{\perp}$ is the
transverse momentum of dark photon,

$H\equiv k^{2}_{\perp}+(1-z)m^{2}_{\gamma^{\prime}}+z^{2}M^{2},$ (6)

and $M$ is the proton mass. All particle momenta are assigned below in (14).
In the limit $t\rightarrow 0$ the squared and averaged matrix element of the
subprocess $p(p)\mathbb{P}(q)\rightarrow\gamma^{\prime}(k)p(p^{\prime})$ reads

$A_{2\rightarrow
2}=2\left(\frac{1+\left(1-z\right)^{2}}{1-z}-2\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\frac{k^{2}_{\perp}z^{2}}{H^{2}}\right).$
(7)

The pomeron flux from the target proton is

$\chi_{\mathbb{P}}=\int\displaylimits_{t_{\text{min}}}^{t_{\text{max}}}\left(t-t_{\text{min}}\right)\absolutevalue{\mathcal{M}_{\mathbb{P}}}^{2}\differential
t,$ (8)

where $\mathcal{M}_{\mathbb{P}}$ is the sum of matrix elements describing
single pomeron, double pomeron and triple gluon exchanges. The latter
contribution can be neglected for the interesting here case of $|t|\lesssim 1$
GeV2. The cross section (5) includes two new form factors: the form factor
accounting for possible dark photon mixing with $\rho$\- and $\omega$-family
mesons according to the vector meson dominance (VMD) hypothesis
Faessler:2009tn ,

$F_{\text{VMD}}\equiv\sum_{i}\frac{f_{i}m_{i}^{2}}{m_{i}^{2}-m_{\gamma^{\prime}}^{2}-im_{i}\Gamma_{i}},$
(9)

with the vector meson fit parameters $f_{i}$, masses $m_{i}$ and decay widths
$\Gamma_{i}$ given in table 1,

| $\rho$ | $\omega$ | $\rho^{\prime}$ | $\omega^{\prime}$ | $\rho^{\prime\prime}$ | $\omega^{\prime\prime}$
---|---|---|---|---|---|---
$f_{i}$ | 0.616 | 1.011 | 0.223 | -0.881 | -0.339 | 0.369
$m_{i}$, GeV | 0.770 | 1.25 | 1.45
$\Gamma_{i}$, GeV | 0.150 | 0.0085 | 0.300 | 0.500
Table 1: The parameters of $\rho$\- and $\omega$-family mesons adopted in the
VMD form factor (9).

and the off-shell hadronic form factor with the hard scale $\Lambda=1.5\text{
GeV}$ Feuster:1998cj ,

$F_{\text{virt}}\equiv\frac{\Lambda^{4}}{\Lambda^{4}+\left((p-k)^{2}-M^{2}\right)^{2}},$
(10)

penalising for the proton compositeness.

Another widely used approximation for the proton bremsstrahlung has been
obtained by Blumlein and Brunner in Blumlein:2013cua . Although in the
original paper the authors study the inelastic bremsstrahlung with inclusive
cross section, we find that the similar result can be obtained in the elastic
case along the same lines. The idea of the calculation is the following.
Proton-proton interaction is considered as the exchange of hypothetical
massless vector particles $b$ (in full analogy with virtual photons in the
electron bremsstrahlung Kim:1973he ). In contrast to pomerons in Foroughi-
Abari:2021zbm , these particles have usual massless vector propagator
proportional to $1/q^{2}$. Firstly, using the Weizsacker-Williams
approximation the bremsstrahlung matrix element is related to the amplitude of
the $2\rightarrow 2$ process $pb\rightarrow\gamma^{\prime}p$. Secondly, the
probability of the subprocess $p\rightarrow\gamma^{\prime}p$ is extracted from
the $pb\rightarrow\gamma^{\prime}p$ cross section and the hypothetical boson
momentum $q^{\mu}$ is set exactly to 0 revealing

$\left[\frac{\differential^{2}\sigma(pp\rightarrow\gamma^{\prime}pp)}{\differential
z\differential
k^{2}_{\perp}}\right]_{BB}=w_{p\gamma^{\prime}}(z,k_{\perp}^{2})\sigma_{pp}(\bar{s}),$
(11)

where the splitting function for $p\rightarrow\gamma^{\prime}p$ is given by
Blumlein:2013cua

$w_{p\gamma^{\prime}}(z,k_{\perp}^{2})=\frac{\epsilon^{2}\alpha}{2\pi
H}\left[\frac{1+(1-z)^{2}}{z}-2z(1-z)\left(\frac{2M^{2}+m^{2}_{\gamma^{\prime}}}{H}-z^{2}\frac{2M^{4}}{H^{2}}\right)+\right.\\\
\left.+2z(1-z)(1+(1-z)^{2})\frac{M^{2}m_{\gamma^{\prime}}^{2}}{H^{2}}+2z(1-z)^{2}\frac{m^{4}_{\gamma^{\prime}}}{H^{2}}\right],$
(12)

$\sigma_{pp}(\bar{s})$ is the full elastic proton scattering cross section and
$\sqrt{\bar{s}}$ is the center-of-mass energy of two protons which exchange
the hypothetical boson. Comparing (5)-(8) with (11), (12) one can see that
these two results show completely different behavior as functions of proton
and dark photon masses.

All the described above approaches explicitly put either the momentum of
hypothetical virtual boson or its square to zero assuming that the maximum of
hypothetical boson flux is reached at very small $t$. Although this is indeed
the case for electron bremsstrahlung, the proton bremsstrahlung requires more
detailed treatment. On the base of the analysis performed in this paper, in
figure 1

Figure 1: The differential flux of hypothetical bosons in proton
bremsstrahlung (13) according to the Weizsacker-Williams approximation for
$P=120\text{ GeV}$, $m_{\gamma^{\prime}}=1\text{ GeV}$, $z=0.5$,
$k_{\perp}=0.1\text{ GeV}$.

we plot the differential flux of hypothetical bosons

$\frac{\differential\chi_{b}}{\differential
t}=\left(t-t_{\text{min}}\right)\absolutevalue{\mathcal{M}_{pp}}^{2},$ (13)

where $\absolutevalue{\mathcal{M}_{pp}}^{2}\equiv
4\pi\differential\sigma_{pp}/\differential q^{2}$, the differential cross
section for the elastic proton scattering
$\differential\sigma_{pp}/\differential q^{2}$ is given by (58) and we take
the numerical fit for the amplitude of elastic scattering from
ParticleDataGroup:2016lqr . For the center-of-mass energy of two protons
squared we take its typical value in the proton bremsstrahlung
$\bar{s}=2M^{2}+2MP(1-z)-H/z$ with $P=120\text{ GeV}$,
$m_{\gamma^{\prime}}=1\text{ GeV}$, $z=0.5$ and $k_{\perp}=0.1\text{ GeV}$. It
can be easily seen that the maximum of such hypothetical boson flux is
approached not at zero, but at about QCD scale, $\sqrt{t}\simeq 0.35\text{
GeV}$, which is numerically not a negligible scale in this problem. This
observation suggests a proper refinement of the previous estimates of the dark
photon production via the proton bremsstrahlung.

Thus in the next section we perform the calculation of proton bremsstrahlung
cross section considering non-zero momentum $q$ of the hypothetical virtual
boson. Our main goal is to check whether the Weizsacker-Williams approximation
still gives the reliable result in the case of incident proton beam.

##  3 The bremsstrahlung cross section with non-zero momentum transfer

In this section we calculate the matrix element of bremsstrahlung process
according to the Feynman diagrams in figures 2(a), 2(b). Here we neglect the
dark photon production by the target proton drawing an analogy with the case
of photon emission in electron scattering considered in Altarelli:1964 ;
Baier:1966jf ; LL4 .

(a)

(b)

Figure 2: Feynman diagrams for proton bremsstrahlung.

We extend the approach of ref. Blumlein:2013cua  to the region of the phase
space with non-zero momentum transfer. For the initial particles, $P_{i}$ is
the momentum of the target proton, $p$ is the momentum of the incident proton.
The momenta of outgoing particles are $k$, $p^{\prime}$ and $P_{f}$ for the
dark photon, the outgoing and target protons respectively. The hypothetical
virtual boson responsible for elastic pp scattering has the momentum $q$. In
the lab frame the momenta can be written as

$\displaystyle P_{i}^{\mu}$ $\displaystyle=\\{M,0,0,0\\},$ (14) $\displaystyle
p^{\mu}$ $\displaystyle=\\{P+\frac{M^{2}}{2P},0,0,P\\},$ (15) $\displaystyle
k^{\mu}$
$\displaystyle=\\{zP+\frac{m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}}{2zP},k_{x},k_{y},zP\\},$
(16) $\displaystyle p^{\prime\mu}$
$\displaystyle=\\{p^{\prime}_{0},-k_{x}-q_{x},-k_{y}-q_{y},P(1-z)-q_{z}\\},$
(17) $\displaystyle q^{\mu}$ $\displaystyle=\\{q_{0},q_{x},q_{y},q_{z}\\},$
(18)

where

$p^{\prime}_{0}=P(1-z)+\frac{M^{2}+k^{2}_{\perp}}{2P\left(1-z\right)}+\frac{k_{x}q_{x}+k_{y}q_{y}}{P\left(1-z\right)}-q_{z}+\frac{q^{2}_{\perp}}{2P\left(1-z\right)}$
(19)

and the squares of transverse momenta are $k^{2}_{\perp}\equiv
k_{x}^{2}+k_{y}^{2}$ and $q^{2}_{\perp}\equiv q_{x}^{2}+q_{y}^{2}$. Here we
assume that $M/P\ll 1$ and

$\displaystyle\frac{1}{P}\sqrt{\frac{k^{2}_{\perp}+m^{2}_{\gamma^{\prime}}}{2}}$
$\displaystyle\ll z,$ (20)
$\displaystyle\frac{1}{P}\sqrt{\frac{k^{2}_{\perp}+M^{2}}{2}}$
$\displaystyle\ll 1-z.$ (21)

While constructing the matrix element, for the hypothetical boson we take the
photon propagator to connect currents of the incident ($L^{\mu}$) and the
target ($J^{\mu}$) protons with charge $Q_{b}$ as follows

$i{\cal M}=-i\epsilon
eQ_{b}^{2}L^{\nu}\frac{-i}{q^{2}}g_{\nu\lambda}J^{\lambda}\,.$ (22)

Here the incident proton current reads

$L_{\nu}=\epsilon^{*\mu}_{\gamma^{\prime}}(k)\bar{u}(p^{\prime})\left(\gamma_{\nu}\frac{\hat{p}-\hat{k}+M}{\left(p-k\right)^{2}-M^{2}}\gamma_{\mu}+\gamma_{\mu}\frac{\hat{k}+\hat{p^{\prime}}+M}{\left(k+p^{\prime}\right)^{2}-M^{2}}\gamma_{\nu}\right)u(p)$
(23)

and the target proton current forms the hadronic tensor $W^{\mu\nu}\equiv
J^{\mu}J^{*\nu}$ that is parametrized in the standard way by two form factors
$W_{1}\left(\bar{s},q^{2}\right)$, $W_{2}\left(\bar{s},q^{2}\right)$ as

$W^{\mu\nu}=\left(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}\right)W_{1}+\frac{1}{M^{2}}\left(P_{i}^{\mu}-\frac{q_{\lambda}P_{i}^{\lambda}}{q^{2}}q^{\mu}\right)\left(P_{i}^{\nu}-\frac{q_{\rho}P_{i}^{\rho}}{q^{2}}q^{\nu}\right)W_{2}.$
(24)

Applying the Dirac equation one can easily show that $q^{\mu}L_{\mu}=0$, hence
the absolute square of matrix element averaged over the initial states and
summed over the final ones is

$\overline{|{\cal
M}|^{2}}=\frac{1}{4}\epsilon^{2}e^{2}Q_{b}^{4}\frac{L_{\mu\nu}}{\left(q^{2}\right)^{2}}\left(-g^{\mu\nu}W_{1}+\frac{1}{M^{2}}P_{i}^{\mu}P_{i}^{\nu}W_{2}\right),$
(25)

where $L_{\mu\nu}\equiv L_{\mu}L^{*}_{\nu}$ is the hadronic tensor constructed
of the incident proton current (23).

Assuming that the functions $W_{1}$ and $W_{2}$ are of the same order of
magnitude like in the case of structureless fermions, one can neglect the
first term in (25) since in the lab frame

$\frac{1}{M^{2}}L_{\mu\nu}P_{i}^{\mu}P_{i}^{\nu}=L_{00}$ (26)

and, as it was shown in Kim:1973he , that cancellations in (25) imply
$g^{\mu\nu}L_{\mu\nu}\ll L_{00}$. Thus one only needs to explicitly calculate
the 00 component of the incident proton hadronic tensor, which may be written
as follows

$L^{\mu\nu}=-\frac{4}{S^{2}U^{2}}\left[g^{\mu\nu}\left(-SU\left(S^{2}+U^{2}\right)+q^{2}\left(2M^{2}(S+U)^{2}+m^{2}_{\gamma^{\prime}}\left(S-U\right)^{2}+\right.\right.\right.\\\
\left.\left.\left.+2SU\left(S+U\right)\right)-2q^{4}SU\right)-\right.\\\
\left.-8M^{2}SUk^{\mu}k^{\nu}+4SU\left(m^{2}_{\gamma^{\prime}}-S\right)p^{\mu}p^{\nu}+4SU\left(m^{2}_{\gamma^{\prime}}-U\right)p^{\prime\mu}p^{\prime\nu}+\right.\\\
\left.+2U\left(k^{\mu}p^{\nu}+p^{\mu}k^{\nu}\right)\left(2M^{2}\left(S+U\right)-m^{2}_{\gamma^{\prime}}\left(S-U\right)+S^{2}-Sq^{2}\right)-\right.\\\
\left.-2S\left(k^{\mu}p^{\prime\nu}+p^{\prime\mu}k^{\nu}\right)\left(2M^{2}\left(S+U\right)+m^{2}_{\gamma^{\prime}}\left(S-U\right)+U^{2}-Uq^{2}\right)+\right.\\\
\left.+2\left(p^{\mu}p^{\prime\nu}+p^{\prime\mu}p^{\nu}\right)\left(2M^{2}\left(S+U\right)^{2}+m^{2}_{\gamma^{\prime}}\left(S^{2}+U^{2}\right)+SU\left(U+S\right)-2SUq^{2}\right)\right],$
(27)

with Lorentz invariants $U\equiv m^{2}_{\gamma^{\prime}}-2p^{\mu}k_{\mu}$ and
$S\equiv m^{2}_{\gamma^{\prime}}+2p^{\prime\mu}k_{\mu}$. These invariants can
be presented in terms of $H$ defined in (6) as

$\displaystyle U$ $\displaystyle=-H/z,$ (28) $\displaystyle S$
$\displaystyle=\frac{H}{z(1-z)}+\frac{2}{1-z}\left(k_{x}q_{x}+k_{y}q_{y}\right)+\frac{q^{2}_{\perp}z}{1-z},$
(29)

where the latter equality is valid up to terms of the second order in $q$.

Using the momentum conservation $p^{\prime}=p-k-q$, we extract in the
numerator of $L^{00}$ terms up to the second order in $q$. Then we arrive at
the approximation for $L^{00}$ (denoted by symbols with a hat)

$\hat{L}^{00}=-\frac{4}{S^{2}U^{2}}\left(\hat{N}_{q^{0}}+\hat{N}_{q^{1}}+\hat{N}_{q^{2}}\right),$
(30)
$\hat{N}_{q^{0}}\equiv\frac{H^{4}}{z^{4}\left(1-z\right)^{3}}\left(1+\left(1-z\right)^{2}\right),$
(31)
$\hat{N}_{q^{1}}\equiv\frac{4q_{0}H^{3}}{z^{3}\left(1-z\right)^{2}}P\left(1+\left(1-z\right)^{2}\right)+\left(k_{x}q_{x}+k_{y}q_{y}\right)\left[\frac{H^{3}}{z^{3}\left(1-z\right)^{3}}\left(6+2\left(1-z\right)^{2}\right)-\right.\\\
\left.-\frac{8H}{z^{2}\left(1-z\right)}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\left(m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}-z^{2}M^{2}\right)\right],$
(32)
$\hat{N}_{q^{2}}\equiv\frac{4q^{2}H^{2}P^{2}}{z^{2}\left(1-z\right)}\left(1+\left(1-z\right)^{2}\right)+16\left(k_{x}q_{x}+k_{y}q_{y}\right)^{2}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)P^{2}+\\\
+\frac{8q_{0}HP}{z\left(1-z\right)^{2}}\left(k_{x}q_{x}+k_{y}q_{y}\right)\left[\frac{H}{z}\left(2+\left(1-z\right)^{2}\right)-\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)z\left(1-z\right)\right].$
(33)

Finally, we substitute (28), (29) into prefactor of (30) and obtain
$\hat{L}^{00}\simeq-4\hat{B}$,

$\hat{B}\equiv
b_{0}+b_{1}q_{0}+b_{2}\left(k_{x}q_{x}+k_{y}q_{y}\right)+b_{3}\left(k_{x}q_{x}+k_{y}q_{y}\right)^{2}+b_{4}q^{2}+b_{5}q_{0}\left(k_{x}q_{x}+k_{y}q_{y}\right),$
(34)

where the coefficients $b_{0},\dots,b_{5}$ take the form

$\displaystyle b_{0}$ $\displaystyle\equiv\frac{1+\left(1-z\right)^{2}}{1-z},$
(35) $\displaystyle b_{1}$
$\displaystyle\equiv\frac{4zP}{H}\left(1+\left(1-z\right)^{2}\right),$ (36)
$\displaystyle b_{2}$
$\displaystyle\equiv-\frac{8\left(1-z\right)z^{2}}{H^{3}}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\left(m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}-z^{2}M^{2}\right)-$
(37)
$\displaystyle\quad\,-\frac{2z\left(-1+\left(1-z\right)^{2}\right)}{H\left(1-z\right)},$
(38) $\displaystyle b_{3}$
$\displaystyle\equiv\frac{16\left(1-z\right)^{2}z^{4}}{H^{4}}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)P^{2}+\frac{z^{2}\left(-12+4\left(1-z\right)^{2}\right)}{H^{2}\left(1-z\right)}+$
(39)
$\displaystyle\quad\,+\frac{32}{H^{4}}\left(1-z\right)z^{3}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\left(m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}-z^{2}M^{2}\right),$
(40) $\displaystyle b_{4}$
$\displaystyle\equiv\frac{4\left(1-z\right)z^{2}P^{2}}{H^{2}}\left(1+\left(1-z\right)^{2}\right),$
(41) $\displaystyle b_{5}$
$\displaystyle\equiv-\frac{8z^{2}P}{H^{2}}\left(\left(1-z\right)^{2}+\frac{z^{2}}{H}\left(1-z\right)\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\right).$
(42)

The differential cross section of the process
$p(p)p(P_{i})\rightarrow\gamma^{\prime}(k)p(p^{\prime})p(P_{f})$ can be cast
as

$\differential{\sigma}=\frac{\epsilon^{2}\alpha
Q_{b}^{4}}{32MP\left(q^{2}\right)^{2}}\left(-\hat{B}\right)W_{2}\frac{\differential{k_{\perp}^{2}\differential{z}\differential{\varphi_{k}}}}{\pi^{2}z}\differential{\Phi_{p^{\prime}P_{f}}},$
(43)

with the two-particle phase space volume

$\differential{\Phi_{p^{\prime}P_{f}}}=\frac{\differential[3]{p^{\prime}}}{\left(2\pi\right)^{3}2p^{\prime}_{0}}\frac{\differential[3]{P_{f}}}{\left(2\pi\right)^{3}2P_{f0}}\left(2\pi\right)^{4}\delta^{(4)}\left(p+P_{i}-k-p^{\prime}-P_{f}\right).$
(44)

One can rewrite (43) by making use of the expression for $W_{2}$ derived in
appendix A as

$\differential\sigma=\frac{\epsilon^{2}\alpha
M\differential{k_{\perp}^{2}}\differential{z}\differential{\varphi_{k}}}{16\pi^{2}zP\tilde{S}^{2}}\int\left(-\hat{B}\right)\absolutevalue{T_{+}(\bar{s},q^{2})+T_{+}^{c}(\bar{s},q^{2})}^{2}\differential{\Phi_{p^{\prime}P_{f}}},$
(45)

where $\sqrt{\bar{s}}$ is the center-of-mass energy of two protons exchanging
the virtual hypothetical boson,
$\tilde{S}^{2}\equiv\left(\bar{s}-2M^{2}\right)^{2}+q^{2}\left(\bar{s}-M^{2}\right)$,
$T_{+}\left(\bar{s},q^{2}\right)$ and $T_{+}^{c}\left(\bar{s},q^{2}\right)$
are the fit functions for the differential cross section of elastic proton
scattering (58) defined as in ParticleDataGroup:2016lqr .

The next step is to integrate over the phase space volume. Since in the lab
frame $\vec{P_{f}}=\vec{q}$ we replace the second integration variable in (44)
with $\differential^{3}q$ and integrate over $\differential^{3}p^{\prime}$
using the $\delta$-function. Then we change the basis of coordinate system
from the old one ($\vec{e}_{x}$, $\vec{e}_{y}$, $\vec{e}_{z}$) to the new
basis ($\vec{n}_{x}$, $\vec{n}_{y}$, $\vec{n}_{z}$) of coordinate system with
$z$ axis oriented along the vector $\vec{p}-\vec{k}$. The coordinates of
vectors $\vec{n}_{i}$ in the basis $\vec{e}_{j}$ read

$\displaystyle\vec{n}_{x}$ $\displaystyle=N_{x}^{-1}\\{P(1-z),0,k_{x}\\},$
(46) $\displaystyle\vec{n}_{y}$
$\displaystyle=N_{y}^{-1}\\{-k_{x}k_{y},k_{x}^{2}+P^{2}(1-z)^{2},k_{y}P(1-z)\\},$
(47) $\displaystyle\vec{n}_{z}$
$\displaystyle=N_{z}^{-1}\\{-k_{x},-k_{y},P(1-z)\\},$ (48)

with the normalization factors

$N_{x}^{2}\equiv P^{2}(1-z)^{2}+k_{x}^{2},\hskip 14.22636ptN_{z}^{2}\equiv
P^{2}(1-z)^{2}+k_{\perp}^{2},\hskip 14.22636ptN_{y}^{2}\equiv
N_{x}^{2}N_{z}^{2}.$ (49)

Then integration over $\varphi_{q}$ gives

$\displaystyle\int\displaylimits_{0}^{2\pi}\left(k_{x}q_{x}+k_{y}q_{y}\right)\differential\varphi_{q}$
$\displaystyle=-2\pi N_{z}^{-1}|\vec{q}|k^{2}_{\perp}\cos\theta_{q},$ (50)
$\displaystyle\int\displaylimits_{0}^{2\pi}\left(k_{x}q_{x}+k_{y}q_{y}\right)^{2}\differential\varphi_{q}$
$\displaystyle=\pi
N_{z}^{-2}|\vec{q}|^{2}k_{\perp}^{2}\left(P^{2}(1-z)^{2}\sin^{2}\theta_{q}+2k^{2}_{\perp}\cos^{2}\theta_{q}\right).$
(51)

The remaining $\delta$-function fixes the value of $\cos\theta_{q}$

$\cos\hat{\theta}_{q}=\frac{1}{2|\vec{p}-\vec{k}||\vec{q}|}\left(\frac{H}{z}+\frac{t}{M}\left(P(1-z)+M+\frac{M^{2}}{2P}-\frac{m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}}{2zP}\right)\right),$
(52)

where $t\equiv-q^{2}$. Since the integrand does not depend on the azimuthal
angle $\varphi_{k}$, we integrate over it and finally obtain

$\frac{\differential[2]{\sigma}}{\differential{k_{\perp}^{2}}\differential{z}}=\frac{\epsilon^{2}\alpha}{32\left(2\pi\right)^{2}zP\tilde{S}^{2}\sqrt{P^{2}(1-z)^{2}+k_{\perp}^{2}}}\int\displaylimits_{t_{\text{min}}}^{t_{\text{max}}}\differential
t\absolutevalue{T_{+}+T_{+}^{c}}^{2}\times\left[-b_{0}-\right.\\\
\left.-\frac{b_{1}t}{2M}+b_{4}t+\left(b_{2}+\frac{b_{5}t}{2M}\right)\frac{k^{2}_{\perp}|\vec{q}|\cos\hat{\theta}_{q}}{\sqrt{P^{2}(1-z)^{2}+k_{\perp}^{2}}}-\frac{b_{3}k_{\perp}^{2}}{P^{2}(1-z)^{2}+k_{\perp}^{2}}\times\right.\\\
\left.\times\left(\frac{t}{2}\left(\frac{t}{4M^{2}}+1\right)P^{2}(1-z)^{2}+|\vec{q}|^{2}\cos^{2}\hat{\theta}_{q}\left(k_{\perp}^{2}-\frac{P^{2}}{2}(1-z)^{2}\right)\right)\right],$
(53)

where the limits of integration are taken as in Kim:1973he :
$\sqrt{t_{\text{max}}}\sim m_{\gamma^{\prime}}$ and

$\sqrt{t_{\text{min}}}=\frac{H}{2z(1-z)P}.$ (54)

In order to reproduce the cross section in the Weizsacker-Williams
approximation one needs to keep in the integrand of (53) only terms
proportional to $b_{0}$, $b_{4}$, the leading in $P^{2}$ part of $b_{3}$
multiplied by the leading and a subleading in $P^{2}$ part of the
corresponding factor in (53),

$I=\int\displaylimits_{t_{\text{min}}}^{t_{\text{max}}}\differential
t\absolutevalue{T_{+}+T_{+}^{c}}^{2}\left(-b_{0}+b_{4}t-\frac{b_{3}}{2}k^{2}_{\perp}\left(t-t_{\text{min}}\right)\right)\,.$
(55)

Then, using (35), (40) and (41) it is straightforward to show that

$I\simeq\int\displaylimits_{t_{\text{min}}}^{t_{\text{max}}}\differential
t\absolutevalue{T_{+}+T_{+}^{c}}^{2}A_{2\rightarrow
2}\frac{t-t_{\text{min}}}{2t_{\text{min}}},$ (56)

with $A_{2\rightarrow 2}$ defined in (7). Thus if one omits relatively small
terms dependent on the direction of transferred momentum, (53) will coincide
with the cross section in the Weizsacker-Williams approximation (5) without
form factors $F_{\text{VMD}}$ and $F_{\text{virt}}$.

##  4 Numerical estimates and comparison

First, we present the numerical results obtained for the incident proton beam
with the momentum in the lab frame $P=120\text{ GeV}$ as in the DUNE
experiment under construction. The dependence of differential cross section
(53) integrated over the momentum ratio $z$ or over the square of dark photon
momentum $k^{2}_{\perp}$ on dark photon mass is shown in figures 3(a), 3(b).

(a)

(b)

Figure 3: The differential cross section of proton bremsstrahlung for
different values of dark photon mass $m_{\gamma^{\prime}}$ as a function of
(a) ratio $z$ of the dark photon 3-momentum to the incident proton 3-momentum,
$z\equiv k_{z}/p_{z}$, (b) the square of dark photon transverse momentum
$k^{2}_{\perp}$. The incident proton momentum in the lab frame $P=120\text{
GeV}$, dark photon masses are listed in the legend.

We integrate over $k^{2}_{\perp}$ in the region 0.01-1 GeV2 and over $z$ in
the region obeying the assumptions (20), (21). One can observe that the
differential cross section in figure 3(a) exhibits a maximum in the $z$ range
0.2-0.4 for dark photons lighter than $0.8$ GeV. For heavier dark photons the
behavior of the cross section changes and the maximum is reached at the
maximal $z$. Figure 3(b) shows the rapid decrease in the differential cross
section with growing transverse dark photon momentum $k_{\perp}$. Note that
the dark photon distribution over the transverse momentum is important, since
it defines the outgoing trajectories which show which part of the produced
dark photons can reach the downstream detector, that is a typical setup
adopted e.g. in SHiP, DUNE and other experiments planning to search for the
dark photon.

Second, the dependence of the full integrated bremsstrahlung cross section on
dark photon mass for several phenomenologically interesting values of $P$ is
outlined in figures 4(a), 4(b).

(a)

(b)

Figure 4: Full proton bremsstrahlung cross section as a function of dark
photon mass for the values of incident proton momenta in the lab frame $P$ at
J-PARC (green line), U-70 (red line), Fermilab (light blue line), CERN SPS
(dark blue line). In figure 4(a) the cross section is shown _without_ a non-
trivial contribution of VMD form factor, i.e. we set $F_{\text{VMD}}=1$. The
pronounced peak in figure 4(b) at $m_{\gamma^{\prime}}\approx 0.8$ GeV and the
knee at $m_{\gamma^{\prime}}\approx 1.3$ GeV are induced by the form factor
$F_{\text{VMD}}$ (9) accounting for the dark photon mixing with vector meson
states, see table 1.

Figure 4(a) shows the differential cross section (53) integrated over
$k_{\perp}^{2}$ and $z$. In figure 4(b) we further multiply the result by the
absolute square of VMD form factor (9). The bremsstrahlung cross section for
proton beam with $P=30$ GeV in the lab frame produced by the Main Ring of
J-PARC facility is shown in green. Red line demonstrates the bremsstrahlung
cross section of the proton beam with $P=70$ GeV at proton synchrotron U-70.
The full cross section for Fermilab proton beam with $P=120$ GeV is shown in
light blue. Dark blue line corresponds to the CERN SPS proton beam with
$P=400$ GeV. The differential cross section (53) feebly depends on $P$, since
the $1/P^{2}$ integral prefactor behaviour gets compensated by the integral
$I$ proportional to $P^{2}$ which is evident from (56). Thus the dependence of
the cross section on $P$ is mostly due to varying limits of integration (20),
(21). This dependence vanishes at large $P$, as is demonstrated by light and
dark blue lines in figure 4(a).

In figures 5(a), 5(b)

(a)

(b)

Figure 5: At the top: the differential cross section of proton bremsstrahlung
according to the result of this work (dark blue line), Blumlein and Brunner
(red line) Blumlein:2013cua , Foroughi-Abari and Ritz (green line, includes
the off-shell hadronic form factor $F_{\text{virt}}$ (10)) Foroughi-
Abari:2021zbm  and in the Weizsacker-Williams approximation (light blue line)
Kim:1973he ; at the bottom: the relative deviation of the differential cross
section obtained in this work from the WW approximation as a function of (a)
the ratio $z$ of the dark photon momentum to the incident proton momentum, (b)
the square of dark photon transverse momentum $k^{2}_{\perp}$. The incident
proton momentum in the lab frame $P=120\text{ GeV}$, dark photon mass
$m_{\gamma^{\prime}}=1\text{ GeV}$.

we fix $m_{\gamma^{\prime}}=1\text{ GeV}$ and compare the bremsstrahlung
differential cross sections: the result of this work, (53) integrated over $z$
or $k^{2}_{\perp}$ (dark blue line), agrees well (within 3–9%) with the
Weizsacker-Williams (WW) approximation that is shown in light blue and is
given by (5) with unity form factors $F_{\text{VMD}}$ and $F_{\text{virt}}$
Kim:1973he . The red line presents the answer (11)-(12) of Blumlein and
Brunner (BB) Blumlein:2013cua , where the splitting function (12) is
multiplied by the _elastic_ $pp$-cross section (58) integrated over $q^{2}$.
The result of Foroughi-Abari and Ritz (FAR) Foroughi-Abari:2021zbm , where we
include in (5) the off-shell hadronic form factor $F_{\text{virt}}$ (10) and
still put $F_{\text{VMD}}=1$, is shown in green. If one puts
$F_{\text{virt}}=1$, the result of FAR (5) coincides with the WW
approximation. While the result of BB exceeds our result in figure 5(a) by 1-3
orders of magnitude, the result of FAR is smaller than the latter. Since from
(56) it follows that our result should agree well with the WW approximation,
in the bottom of figures 5(a), 5(b) we plot the relative deviation of the
differential cross section, obtained in this work from the WW answer

$\delta\left(\frac{\differential\sigma}{\differential
z}\right)\equiv\left.\left(\frac{\differential\sigma}{\differential
z}-\left(\frac{\differential\sigma}{\differential
z}\right)_{\text{WW}}\right)\middle/{\left(\frac{\differential\sigma}{\differential
z}\right)_{\text{WW}}}\right.,$ (57)

and the analogous one for the differential cross section dependent on
$k^{2}_{\perp}$, $\delta\left({\differential\sigma}/{\differential
k^{2}_{\perp}}\right)$.

It is important to note that these three answers show completely different
behavior at small $z$: the growth of BB cross section is due to the term
proportional to $1/z$ in (12), and the off-shell hadronic form factor
$F_{\text{virt}}$, see eq. (10) suppresses the FAR cross section in the limit
$z\rightarrow 0$. Although this discrepancy still needs to be resolved, the
$1/z$ singularity in the BB cross section looks very unnatural in comparison
with WW approximation and our result which is obtained considering the diagram
of $2\rightarrow 3$ bremsstrahlung process as a whole, without any splitting
into auxiliary subprocesses.

The differential cross section obtained in this work agrees not only with the
WW approximation, but also with the result of FAR, if one does not multiply
the latter by the absolute square of the off-shell hadronic form factor
$F_{\text{virt}}$ (10). We should also note that Foroughi-Abari and Ritz used
the pomeron propagator with the non-integer power of $q^{2}$ and the proton-
pomeron couplings obtained within the framework of Donnachie-Landshoff pomeron
model. In contrast, we utilize the photon-type propagator and derive the value
of hypothetical boson-proton-proton coupling from the experimental fit for
differential cross section of elastic $pp$-scattering. Despite that our
approaches are, obviously, different, the results obtained in the WW
approximation, our work and by FAR coincide numerically within about 10%. In
addition, Foroughi-Ababi and Ritz performed Foroughi-Abari:2021zbm  another
calculation of the elastic bremsstrahlung with the one-pomeron exchange, that
also exhibits a close agreement with the WW approximation.

Figures 6(a), 6(b)

(a)

(b)

Figure 6: Full proton bremsstrahlung cross sections as functions of dark
photon mass (a) _without_ the VMD form factor, (b) with the VMD form factor
$F_{\text{VMD}}$ (9). The color scheme is the same as in figures 5(a), 5(b).
The incident proton momentum in the lab frame is $P=120\text{ GeV}$.

show the full integrated bremsstrahlung cross sections as functions of the
dark photon mass. In figure 6(a) we omit the VMD form factor putting
$F_{\text{VMD}}=1$, while in figure 6(b) we multiply all the results by the
absolute square of the same form factor $F_{\text{VMD}}$ (9). The result of
this work is shown in dark blue, the red line represents the answer of BB, the
answer of FAR is indicated by the green line. Our result with non-zero
momentum transfer agrees very well with the bremsstrahlung cross section in
the WW approximation (light blue line) and deviates from the FAR answer by a
factor 2-5, while the BB computation greatly exceeds both of them.

We also show in the appendix B how the sensitivity of the SHiP experiment
($P=400\text{\leavevmode\nobreak\ GeV}$) depends on the approach chosen for
estimating the dark photon production cross section. There we revisit the
contours of the potentially excluded regions obtained earlier in
Gorbunov:2014wqa  and SHiP:2020vbd . Following these works we consider only
the proton bremsstrahlung and assume nearly null background conditions
provided by the SHiP muon shield. Our results are summarized in figure 7. For
the given dark photon mass $m_{\gamma^{\prime}}$ the excluded value of kinetic
mixing $\epsilon$ can vary up to 1 order of magnitude. We stress that this is
not the final answer for the sensitivity of the SHiP experiment, since
following the original work of SHiP collaboration SHiP:2020vbd  we did not
combine these results with the production via meson decays and Drell-Yan
process. It will be also important to consider the inelastic bremsstrahlung
(for further discussion see section 5). We would like to draw the attention of
the researchers planning searches for dark photons at the fixed-target
experiments to this issue: the applicability of BB approximation should be
first studied thoroughly.

##  5 Discussion

To summarize, we have obtained the elastic bremsstrahlung cross section
considering the non-zero momentum transfer between incident and target
protons. Our result agrees well with the Weizsacker-Williams approximation.
That is the Weizsacker-Williams approximation is applicable not only to
structureless fermions, but also to protons and can give reliable results.
With our method we present the refined predictions for the dark photon
production cross section to be used for the data analysis with proton beams of
30 GeV, 70 GeV, 120 GeV and 400 GeV.

Following the well-known study of the proton bremsstrahlung Blumlein:2013cua ,
in this paper we considered _elastic_ proton scattering as the protons
exchanging a hypothetical boson coupled to them like a massless vector, with
the photon-type propagator. Generally speaking, the elastic proton scattering
should be described as the exchange of effective particle, pomeron, with a
completely different propagator dependence on 4-momentum squared $q^{2}$ (see
Donnachie:2002en  for a review). The pomeron model, closest to our rough
consideration, is the Donnachie-Landshoff (DL) model Donnachie:1983hf ;
Donnachie:1985iz ; Donnachie:1987gu . The DL pomeron is a vector with the
charge parity $C=+1$. For a long time the DL model was successfully used to
describe the elastic proton scatterings as it was one of the first proposed
models, being simple, easy to use and adequately describing the experimental
data available at the time Donnachie:2002en . Later it has been criticised for
poor grounding in QFT and has been argued to be ruled out by experiment
Britzger:2019lvc . However, there are alternative pomeron models considering
it as an effective spin-0 or spin-2 particle Ewerz:2013kda . The scalar nature
of pomeron is argued to be excluded by the STAR experiment and tensor pomeron
model is suggested to be the most suitable Ewerz:2016onn . Recent works
Lebiedowicz:2022nnn ; Lebiedowicz:2023mhe  contain the calculations of the SM
bremsstrahlung and the central-exclusive production cross sections within the
tensor pomeron model. One of the crucial future tasks is to revisit the dark
photon production cross section in the framework of contemporary pomeron
models.

In order to estimate the dark photon production cross section in the
bremsstrahlung taking place in a real fixed-target experiment, one needs to
consider the process $pA\rightarrow pA\gamma^{\prime}$, where $A$ is the
target nuclei. Following Watanabe:2023rgp , we assume that the hypothetical
vector particle considered in our study has equal coupling constants with
proton and neutron ($Q_{b}$ charge in our calculation) just like a pomeron.
Then the roughest estimate for the cross section of proton bremsstrahlung on a
target composed of atoms with the number of nucleons $A$ is
$\sigma\left(pA\rightarrow
pA\gamma^{\prime}\right)=A\times\sigma\left(pp\rightarrow
pp\gamma^{\prime}\right)$.

The cross section of the inelastic bremsstrahlung is expected to be an order
of magnitude greater than the elastic bremsstrahlung cross section that we
studied in this paper. It is the inelastic process that mostly contributes to
dark photon production. Therefore it is also important to explore the
inelastic dark photon production. From general considerations there should be
the possibility to break the bremsstrahlung cross section into the proton
(in)elastic scattering cross section and the splitting probability as it has
been done by Blumlein and Brunner in Blumlein:2013cua . But then it is not
clear why the result for elastic cross section obtained in this splitting
approach is so different from the Weizsacker-Williams approximation and our
non-zero momentum transfer calculation. One of the questions for future
research is whether the inelastic bremsstrahlung cross section calculated
taking into account non-zero momentum transfer indeed rapidly grows at small
$z$ as is stated in Blumlein:2013cua  or not. The task to resolve the
differences between existing approaches requires further consideration.

###### Acknowledgements.

We are indebted to V. Kim, I. Timiryasov, O. Teryaev and V. Vechernin for
valuable discussion and correspondence. The work is partially supported by the
Russian Science Foundation RSF grant 21-12-00379. The work of EK is supported
by the grant of “BASIS” Foundation no. 21-2-10-37-1.

##  Appendix A Elastic pp scattering

In this appendix we consider the elastic proton scattering
$p\left(p^{\prime}+q\right)p\left(P_{i}\right)\rightarrow
p\left(p^{\prime}\right)p\left(P_{f}\right)$. The fit of differential cross
section to experimental data is usually parameterized as follows

$\derivative{\sigma_{pp}}{q^{2}}=\frac{\absolutevalue{T_{+}(\bar{s},q^{2})+T_{+}^{c}(\bar{s},q^{2})}^{2}}{16\pi\bar{s}\left(\bar{s}-4M^{2}\right)},$
(58)

where the Mandelstam variable is
$\overline{s}\equiv\left(p^{\prime}+q+P_{i}\right)^{2}$, functions
$T_{+}(\bar{s},q^{2})$ and $T_{+}^{c}(\bar{s},q^{2})$ are defined as in
ParticleDataGroup:2016lqr  and the numerator is the absolute square of matrix
element averaged over initial and summed over final spins,

$\overline{\absolutevalue{\mathcal{M}}^{2}}(\bar{s},q^{2})\equiv\absolutevalue{T_{+}(\bar{s},q^{2})+T_{+}^{c}(\bar{s},q^{2})}^{2}.$
(59)

The matrix element constructed along the same lines as for the proton
bremsstrahlung reads

$i\mathcal{M}=\frac{iQ_{b}^{2}}{q^{2}}\bar{u}\left(p^{\prime}\right)\gamma_{\mu}u\left(p^{\prime}+q\right)J^{\mu},$
(60)

where $J^{\mu}$ is the target proton current. Then the averaged matrix element
square is

$\overline{\absolutevalue{\mathcal{M}}^{2}}=\frac{1}{4}\frac{Q_{b}^{4}}{\left(q^{2}\right)^{2}}\widetilde{L}_{\mu\nu}W^{\mu\nu},$
(61)

with the hadronic tensor of incident proton

$\widetilde{L}_{\mu\nu}=4\left(2p^{\prime}_{\mu}p^{\prime}_{\nu}+p^{\prime}_{\mu}q_{\nu}+p^{\prime}_{\nu}q_{\mu}-g_{\mu\nu}p^{\prime\lambda}q_{\lambda}\right)$
(62)

and target proton hadronic tensor $W^{\mu\nu}$ defined in the main text by eq.
(24). Since the incident proton hadronic tensor is transversal,
$q^{\mu}\widetilde{L}_{\mu\nu}=0$, for the product of hadronic tensors one
obtains

$\widetilde{L}_{\mu\nu}W^{\mu\nu}=-g^{\mu\nu}\widetilde{L}_{\mu\nu}W_{1}+\frac{1}{M^{2}}P_{i}^{\mu}P_{i}^{\nu}\widetilde{L}_{\mu\nu}W_{2}.$
(63)

Let us assume that the functions $W_{1}$ and $W_{2}$ are of the same order of
magnitude as in the case of structureless fermions. Then for typical values of
$M$, $q^{2}$ and $\bar{s}$ the first term in (63) is numerically much less
than the second one, and hence can be neglected, which yields from eqs. (59)
and (61)

$W_{2}\left(\bar{s},q^{2}\right)=\frac{2\left(q^{2}\right)^{2}M^{2}}{Q_{b}^{4}}\frac{\absolutevalue{T_{+}(\bar{s},q^{2})+T_{+}^{c}(\bar{s},q^{2})}^{2}}{\left(\bar{s}-2M^{2}\right)^{2}+q^{2}\left(\bar{s}-M^{2}\right)}\,.$
(64)

##  Appendix B The sensitivity of SHiP experiment to dark photons

Due to the kinetic mixing with the SM photons in the Lagrangian (1), the dark
photon can decay into pairs of charged leptons with the decay width
Miller:2021ycl

$\Gamma(\gamma^{\prime}\rightarrow
l^{+}l^{-})=\frac{\alpha\epsilon^{2}}{3}m_{\gamma^{\prime}}\sqrt{1-\frac{4m_{l}^{2}}{m_{\gamma^{\prime}}^{2}}}\left(1+\frac{2m_{l}^{2}}{m^{2}_{\gamma^{\prime}}}\right),$
(65)

where $m_{l}$ is the lepton mass. The decay width to hadrons,

$\Gamma(\gamma^{\prime}\rightarrow\text{hadrons})=\frac{\alpha\epsilon^{2}}{3}m_{\gamma^{\prime}}R(m_{\gamma^{\prime}})\sqrt{1-\frac{4m_{\mu}^{2}}{m_{\gamma^{\prime}}^{2}}}\left(1+\frac{2m_{\mu}^{2}}{m^{2}_{\gamma^{\prime}}}\right)\,,$
(66)

explicitly depends on the experimentally-determined R-ratio
ParticleDataGroup:2022pth

$R(\sqrt{s})\equiv\frac{\sigma(e^{+}e^{-}\rightarrow\text{hadrons})}{\sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-})}.$
(67)

In this appendix we consider the dark photon which can decay only to visible
SM final states while its decay into invisible final states with dark matter
is, e.g., forbidden by the kinematics. Then the total dark photon decay width
reads

$\Gamma_{\text{tot}}=\Gamma(\gamma^{\prime}\rightarrow
e^{+}e^{-})+\Gamma(\gamma^{\prime}\rightarrow\mu^{+}\mu^{-})+\Gamma(\gamma^{\prime}\rightarrow\text{hadrons}).$
(68)

The decay length for the flux of dark photons moving with the relativistic
factor $\gamma$ and velocity $\beta\equiv k_{z}/k_{0}$ is

$L(z,k^{2}_{\perp})=\frac{\gamma\beta}{\Gamma_{\text{tot}}}.$ (69)

So the probability for a dark photon to decay inside the fiducial volume of
SHiP detector is

$w_{\text{det}}(z,k^{2}_{\perp})=e^{-l_{\text{sh}}/L(z,k^{2}_{\perp})}\left(1-e^{-l_{\text{det}}/L(z,k^{2}_{\perp})}\right),$
(70)

where we took the conservative estimates from Gorbunov:2014wqa  for shielding
length $l_{\text{sh}}=60\text{\leavevmode\nobreak\ m}$ and the length of
detector fiducial volume $l_{\text{det}}=50\text{\leavevmode\nobreak\ m}$. It
allows us to illustrate the impact of the cross section on the estimate of the
SHiP sensitivity to the model parameters. Namely, we compare the sensitivity
of SHiP experiment based on our result of the hidden photon production with
the estimates previously obtained in Gorbunov:2014wqa  using the BB answer.
Following Gorbunov:2014wqa  we select only dark photons which have the momenta
lying inside the cone with the angle

$\theta_{\text{cr}}=\frac{r}{l_{\text{sh}}+l_{\text{det}}},$ (71)

where the cone radius is $r=2.5\text{\leavevmode\nobreak\ m}$.

Finally, we take the estimate for the luminosity of the SHiP experiment from
SHiP:2020vbd

$\mathcal{L}_{\text{SHiP}}=\frac{N_{\text{POT}}}{\sigma^{\text{inel}}_{\text{SHiP}}}$
(72)

with $N_{\text{POT}}=10^{20}$ protons on target and the inelastic proton-
nucleon cross section
$\sigma^{\text{inel}}_{\text{SHiP}}=10.7\text{\leavevmode\nobreak\ mb}$.

Combining the expressions (65)–(72) gives the total number of signal events

$N_{\gamma^{\prime}}=\mathcal{L}_{\text{SHiP}}|F_{\text{VMD}}|^{2}\int\differential
k^{2}_{\perp}\differential z\frac{\differential^{2}\sigma}{\differential
k^{2}_{\perp}\differential
z}w_{\text{det}}(z,k^{2}_{\perp})H\left(\theta_{\text{cr}}-\frac{k_{\perp}}{zP}\right),$
(73)

where $H(x)$ is the Heaviside step-function equal to 1 for $x>0$ and to 0 for
$x<0$.

Figure 7

Figure 7: Contours of the regions in the dark photon parameter space expected
to be explored by the SHiP experiment for the elastic cross section obtained
in this work with the VMD form factor $F_{\text{VMD}}$ (9) (blue), the BB
inelastic cross section (11) with the Dirac form factor $F_{D}$ (74) (red) and
the VMD form factor $F_{\text{VMD}}$ (9) (green).

shows the contours in the parameter space $(m_{\gamma^{\prime}},\epsilon)$
corresponding to 3 signal events inside the SHiP detector. The blue curve
corresponds to the number of events estimated using the elastic bremsstrahlung
cross section obtained in this work (53) and VMD form factor $F_{\text{VMD}}$
(9). To compare with the results of SHiP:2020vbd , we also show the contour of
potentially excluded regions evaluated with the BB cross section (11) and the
VMD form factor $F_{\text{VMD}}$ (9) (green line). The conservative expected
exclusion limits analogous to the obtained in Gorbunov:2014wqa  are shown in
red and were computed using the BB cross section (11) and the Dirac form
factor SHiP:2020vbd

$F_{D}(m_{\gamma^{\prime}})=\begin{cases}\frac{\Lambda^{4}}{m_{\gamma^{\prime}}^{4}},&m_{\gamma^{\prime}}>\Lambda,\\\
1,&m_{\gamma^{\prime}}<\Lambda,\end{cases}$ (74)

where $\Lambda^{2}=0.71\leavevmode\nobreak\ \text{GeV}^{2}$. The non-
observation of signal events for parameter values inside the contours means
the exclusion of these regions at the 95% CL. The proposed construction of
muon shield in the SHiP experiment allows one to neglect the possible
background SHiP:2020vbd . One can see that changes in the predicted
experimental sensitivity, depending on the model of dark photon production,
are quite serious, that warns the experimental community.

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