Exclusive diffractive bremsstrahlung of one and two photons at forward rapidities: Possibilities for experimental studies in $pp$ collisions at the LHC

We evaluate the cross section for diffractive bremsstrahlung of a single photon in the $pp \to pp \gamma$ reaction at high energies and at forward photon rapidities. Several differential distributions, for instance, in $y$, $k_{\perp}$ and $\omega$, the rapidity, the absolute value of the transverse momentum, and the energy of the photon, respectively, are presented. We compare the results for our standard approach, based on QFT and the tensor-pomeron model, with two versions of soft-photon-approximations, SPA1 and SPA2, where the radiative amplitudes contain only the leading terms proportional to $\omega^{-1}$. The SPA1, which does not have the correct energy-momentum relations, performs surprisingly well in the kinematic range considered. We discuss also azimuthal correlations between outgoing particles. The azimuthal distributions are not isotropic and are different for our standard model and SPAs. We discuss also the possibility of a measurement of two-photon-bremsstrahlung in the $pp \to pp \gamma \gamma$ reaction. In our calculations we impose a cut on the relative energy loss ($0.02<\xi_{i}<0.1$, $i = 1,2$) of the protons where measurements by the ATLAS Forward Proton (AFP) detectors are possible. The AFP requirement for both diffractively scattered protons and one forward photon (measured at LHCf) reduces the cross section for $p p \to p p \gamma$ almost to zero. On the other hand, much less cross-section reduction occurs for $pp \to pp \gamma \gamma$ when photons are emitted in opposite sides of the ATLAS interaction point and can be measured by two different arms of LHCf. For the SPA1 ansatz we find $\sigma(pp \to pp \gamma \gamma) \simeq 0.03$ nb at $\sqrt{s} = 13$ TeV and with the cuts $0.02<\xi_{i}<0.1$, $8.5


I. INTRODUCTION
Bremsstrahlung of photons in nucleon-nucleon collisions is one of the basic processes in physics. It was extensively studied in the pp → ppγ reaction at relatively small c.m. energies where the meson exchanges are responsible for the underlying pp interaction; see e.g. [1,2]. In [1] also the two-photon bremsstrahlung in pp scattering was considered. The virtual-photon bremsstrahlung in the reactions NN → NN(γ * → ℓ + ℓ − ) was discussed in [3,4].
At high energies, the pp → ppγ reaction has not yet been measured. However, some feasibility studies of the measurement of the exclusive diffractive bremsstrahlung cross sections were performed for RHIC energies [5] and for LHC energies using the ATLAS forward detectors [6,7].
In contrast, the inclusive differential production cross-section of forward photons in pp collisions was measured at √ s = 510 GeV with the RHICf detector (see [8]) and at √ s = 0.9, 7 and 13 TeV with the LHCf detector (see [9][10][11]). The LHCf experiment is designed to measure the photons emitted in the very forward rapidity region |y| > 8. 4. In the ATLAS-LHCf combined analysis [12] the forward-photon spectra are measured by the LHCf detector, while the ATLAS inner tracker system is used to suppress nondiffractive events. 1 In addition, several joint analyses with ATLAS-LHCf are on-going; see the discussions in [13,14].
In this Letter, we discuss exclusive diffractive bremsstrahlung of one and two photons in pp collisions for the LHC energy √ s = 13 TeV and at very-forward photon rapidities. We shall work within the tensor-pomeron model as proposed in [15] for soft hadronic high-energy reactions. The theoretical methods which we shall use in our present analysis were developed by us in [16,17]. In [16] we discussed the soft-photon radiation in pion-pion scattering. Our standard, or also called by us "exact", results for diffractive photon-bremsstrahlung were compared to various soft-photon approximations (SPAs).
In [17] we extended these considerations to the pp → ppγ reaction at √ s = 13 TeV, limiting ourselves to |y| < 5 and 1 MeV < k ⊥ < 100 MeV. Here, k ⊥ is the absolute value of the photon transverse momentum. Recently, in [18] we have discussed various centralexclusive production (CEP) processes of single photons. The CEP processes, for instance the photon-pomeron fusion, do not play an important role at forward photon rapidities and can be safely neglected there.
It is also worth noting that exclusive diffractive photon bremsstrahlung in high-energy pp collisions at forward rapidities was discussed earlier in [19][20][21] within somewhat different approaches. In general, the bremsstrahlung is not limited to photon production. The bremsstrahlung-type emission of ω and π 0 mesons in high-energy pp collisions was calculated in [22,23].
According to our knowledge the exclusive diffractive photon bremsstrahlung was not yet identified experimentally. In order to answer the question whether this is possible one needs to consider other processes that can be misidentified as bremsstrahlung. A dedicated study is in order but goes beyond the scope of our present article. One of the processes which is potentially important in this context is the pp → ppπ 0 reaction. The decaying neutral pion is a source of unwanted photons that can hinder the identification of bremsstrahlung photons of interest. This reaction was studied by two of us some time ago [23]. In the present letter we wish to briefly discuss the role of this background contribution.
An interesting proposal to study the forward production of "dark photons" via bremsstrahlung in pp collisions with the Forward Physics Facility at the High-Luminosity LHC was discussed recently in [24,25].
Our Letter is organized as follows. In the next section we discuss briefly the theoretical formalism. In Sec. II A we give analytic expressions for radiative amplitudes for the pp → ppγ reaction for our standard and approximate approaches. In Sec. II B we discuss twophoton bremsstrahlung in pp → ppγγ. We present our standard-approach results in Sec. III, along with comparisons to SPAs. Section IV contains our conclusions.

A. pp → ppγ
We consider the reaction at high energies and small momentum transfers. The momenta are indicated in brackets, the helicities of the protons are denoted by λ a , λ b , λ 1 , λ 2 ∈ {1/2, −1/2}, and ǫ is the polarization vector of the photon. The energy-momentum conservation in (2.1) requires The kinematic variables are In the following we work in the overall c.m. system where we choose the 3 axis in the direction of p a (the beam direction). The rapidity of the photon is then where θ is the polar angle of k, cos θ = k 3 /|k|. For the energy k 0 of the photon we use the notation ω. We introduce the variables ξ 1 and ξ 2 which, to a very good approximation, describe the fractional energy losses of the protons p(p a ) and p(p b ) Here the energies of the incoming and outgoing protons, respectively, are and we set M 2 = max(m 2 p , |t 1 |, |t 2 |, k 2 ⊥ ). Alternatively, the proton relative energy-loss parameters can be expressed by the kinematical variables of the photon, The cross section for the photon yield can be calculated as follows see Eqs. (2.33)-(2.35) of [17]. M µ is the radiative amplitude. Our standard photon-bremsstrahlung amplitude, M standard µ , treated in the tensorpomeron approach, see (2.62) and (B3) of [17], includes 6 diagrams shown in Fig. 3(a)-(f) of [17]. The amplitudes (a), (b), (d), and (e), corresponding to photon emission from the external protons, are determined by the off-shell pp elastic scattering amplitude. The contact terms, (c) and (f), are needed in order to satisfy gauge-invariance constraints. For details how to calculate these standard results we refer the reader to Sec. II C and Appendix B of [17].
In the following, we shall compare our standard results to two soft-photon approximations, SPA1 and SPA2, as defined in Sec. III of [17]. In both SPAs we keep only the pole terms ∝ ω −1 . We consider only the pomeron-exchange contribution for the radiative amplitudes, the leading term at high energies.
In SPA1, the radiative amplitude has the form , (2.10) where M (on shell) pp (s, t) is the amplitude for on-shell pp-scattering see (2.19) and (3.1) of [17]. The inclusive photon cross section for the SPA1 case is where we neglect the photon momentum k in the energy-momentum conserving δ (4) (.) function. For SPA1 results we impose restrictions on the proton's relative energy loss variables ξ i by using (2.8) neglecting terms of O(M 2 /s).
In the SPA2 case, we keep the exact energy-momentum relation (2.2). Here we calculate the photon yield using (2.9) replacing the radiative amplitude as follows (2.14) The explicit expressions of these terms are given by (3.4), (B4), and (B15) of [17].

B. pp → ppγγ
Here we consider the reaction We shall study this reaction under specific conditions. We shall require that one photon is emitted at forward and one at backward rapidities, 8.5 < y 3 < 9 and −9 < y 4 < −8.5, respectively, and that 0.02 < ξ 1,2 < 0.1. 2 For the calculation of the radiative amplitudes we use SPA1. Here in the 2 → 4 kinematics for SPA1 we define We shall see below in Sec. III that, indeed, the above cuts on ξ 1 and ξ 2 assure that to a good approximation we can restrict ourselves to the diagrams shown in Fig. 1. Then, the two-photon bremsstrahlung amplitude has the form FIG. 1. Diffractive two-photon bremsstrahlung diagrams for the reaction pp → ppγγ (2.15) with exchange of the pomeron P. These four diagrams (a -d) contribute to the SPA1 amplitude (2.17).
In addition there are the diagrams where the two photons are emitted from the p a -p ′ 1 line or from the p b -p ′ 2 line, and various contact terms. These diagrams are not shown here. and the inclusive two-photon cross section is (2.18)

A. Single photon emission
In the following we consider explicitly only photon emission in very forward direction (y ≫ 1), where the photon is emitted predominantly from p(p a ) plus p(p ′ 1 ); see (2.1). As we see from (2.8) we have here ξ 1 sizeable but ξ 2 very small. In fact, then the proton p(p ′ 2 ), having nearly the same energy as the incoming proton p(p b ), cannot be measured by the present AFP detectors. Therefore, requiring for single emission that both final state protons are measured by the AFP detectors reduces the cross section essentially to zero. Thus, for photon emission at y ≫ 1 we consider only detection of p(p ′ 1 ) in the AFP detector. Of course, for y ≪ −1 the roles of p(p ′ 1 ) and p(p ′ 2 ) are interchanged and all distributions shown below for y ≫ 1 are easily transferred to y ≪ −1. . Shown are the results in the forward rapidity region for our standard approach (standard bremsstrahlung results) for the pp → ppγ reaction together with the results obtained via SPA1 and SPA2 discussed in Sec. II A. We see from the panels (a) and (b) of Fig. 2 that bremsstrahlung photons are emitted predominantly in very forward-rapidity region 9 < y < 10 and with small values of k ⊥ ; see also the left panel of Fig. 4. Forward photons will be measured by the LHCf experiment in the regions 8.5 y 9 and y 11. Due to the cut 0.02 < ξ 1 < 0.1 the energy of the photons is limited to 130 GeV < ω < 650 GeV; see the panels (c) and (d) of Fig. 2. Note, that due to the cuts specified in the figure legend the distributions in k ⊥ and ω have no singularity for k ⊥ → 0, respectively ω → 0. The k ⊥ distribution reaches a maximum at k ⊥ ∼ 0.014 GeV, and then it quickly decreases with increasing k ⊥ .
In the SPA1, the photon momentum k was, on purpose, omitted in the energymomentum conserving δ function in the evaluation of the cross section [see (2.12) and (2.13)]. Here, the cross section is integrated over k ⊥ from k ⊥min to a maximal value k ⊥max which we set to 1 GeV. In the SPA2, the correct 2 → 3 kinematics is used. Recall that in both SPAs we keep only the pole terms ∝ ω −1 in the radiative amplitudes. We see from Fig. 2 that the SPA1, which does not have the correct energy-momentum relations, performs surprisingly well in the kinematic range considered. For the SPA2, the deviations from our standard result increase rapidly with growing k ⊥ and ω. From this comparison we see the importance of the interference between the pole term and the non-leading, but numerically large, terms occurring in the radiative amplitudes. It is essential to add coherently all the various parts of the amplitude for the bremsstrahlung-type emission of photons in order not to miss important interference effects. For more details on the size of various contributions we refer to the discussions in [17] (see, e.g., Fig. 17 there).
In the standard result, all contributions to the radiative amplitude with Dirac and Pauli terms are included. Figure 3 shows the complete (total) standard results and the results for Dirac and Pauli terms individually. The anomalous magnetic moment of the proton (Pauli term) plays an important role for larger values of k ⊥ and ω.
In Fig. 4 we show the two-dimensional differential cross sections in the k ⊥ -y plane (left panel) and in the ω-y plane (right panel) imposing a typical cut 0.02 < ξ 1 < 0.1 on the fractional energy loss of the 'emitting' proton.
In Fig. 5 we show the results for dσ/dk ⊥ and dσ/dω for √ s = 13 TeV, 0.02 < ξ 1 < 0.1, as in Figs. 2(b) and 2(d) but now for a more restrictive y cut, 8.5 < y < 9. We see that the k ⊥ and ω distributions are reduced by a factor of order 10 compared to their counterparts in Figs. 2 (b) and (d). The kink at k ⊥ ≈ 0.17 GeV is due to the cut on y.
In Fig. 6 we show the distributions in absolute value of the transverse momenta, p t,1 and p t,2 , of the outgoing protons p(p ′ 1 ) and p(p ′ 2 ), see the solid and dashed lines, respectively. Results for two y intervals are shown: 6 < y < 13 (left panel) and 8.5 < y < 9 (right panel). One can see that the cross sections reach a maximum at p t,p ∼ |t 1,2 | ∼ 0.25 GeV and that dσ/dp t,1 = dσ/dp t,2 . We find that for p t,1 > 0.8 GeV, the Pauli component gives a sizeable contribution.
FIG. 2. The differential distributions in the rapidity of the photon (a), in the transverse momentum of the photon (b), in the energy-loss variable ξ 1 (c), and in the energy of the photon (d) for the pp → ppγ reaction via bremsstrahlung. The calculations were done for √ s = 13 TeV, 6 < y < 13, and with the cut 0.02 < ξ 1 < 0.1. For SPA1 an additional cut k ⊥ < 1 GeV was imposed. The solid line corresponds to our standard bremsstrahlung model, the black long-dashed line corresponds to SPA2 (2.14), and the red dotted line corresponds to SPA1 (2.10). The oscillations in the SPA1 results are of numerical origin p(p ′ i ) and the photon γ(k) p ′ i⊥ = p t,i e iφ i , (i = 1, 2) , k ⊥ = k ⊥ e iφ 3 , 0 φ i < 2π , (i = 1, 2, 3) . (3.1) Here the azimuth φ = 0 corresponds to some fixed transverse direction in the LHC system which is also the c.m. system for our reactions. Transverse-momentum conservation requires     Therefore, a measurement of p ′ 1⊥ and k ⊥ determines also p ′ 2⊥ . Figure 7 shows the distributions inφ ij defined as where we require In the top panels of Fig. 7 we show the results inφ 12 , the angle between the transverse momenta of the outgoing protons, for our standard and SPA2 calculations. For SPA1 (not shown here) the outgoing protons are back-to-back (φ 12 = π). We see that also for our standard approachφ 12 ≈ π gives the main contribution. That is, very roughly the outgoing protons and the beam are in one plane S 0 . The width of the distribution inφ 12 depends on the y cut with the more restrictive y cut (right panel) giving a wider distribution. The bottom panels of Fig. 7 show the distributions inφ 13 (φ 23 ), the azimuthal angles between the proton p(p ′ 1 ) (p(p ′ 2 )) and γ(k); see Eq. (3.3). The SPA1 and our standard results show maxima forφ 13 andφ 23 around π/2 and 3π/2. This corresponds to emission of the photon in a plane S 1 which is orthogonal to the plane S 0 defined above. The SPA2 results for theφ 13 andφ 23 distributions deviate very significantly from our standard results.   √ s = 13 TeV, 0.02 < ξ 1 < 0.1, 6 < y < 13 (left panel), and 8.5 < y < 9 (right panel). The solid (dashed) line, denoted as p t,1 (p t,2 ), corresponds to the proton p(p ′ 1 ) (p(p ′ 2 )).
In Fig. 8 we present the result of a study of the pp → pp(π 0 → γγ) background. For this we take the upper estimate of the cross section for the pp → ppπ 0 reaction (Drell-Hiida-Deck type model) from [23] which corresponds to Λ N = Λ π = 1 GeV, and without taking into account the absorptive corrections. 3 The red lines represent the distributions of our (signal) standard bremsstrahlung of a single photon associated with a proton for which the fractional energy loss ξ 1 is in the intervals specified in the figure legend. We vary the upper limit of ξ 1 . The signal distribution moderately depends on the upper limit; see Fig. 2. In the left panel of Fig. 8, for the background contribution we assume that one photon is measured (called "first" in the following) by the LHCf in the rapidity interval 8.5 < y < 9, ω > 130 GeV, and with ω max ≈ √ s 2 ξ 1,max . The distributions of the measured photon correspond to the black lines (within the LHCf acceptance), while the distributions of the second photon correspond to the blue lines. In the latter case, the maximum of the cross section corresponding to the background is located outside the LHCf acceptance region. The percentage of measured/unmeasured photons depends on the upper limit of ξ 1 . For ξ 1,max = 0.06 (the solid lines) the second photon practically cannot be measured. The signal-to-background ratio for ξ 1,max = 0.06 is somewhat larger than 1. In the right panel of Fig. 8, we present the contribution of the background for the y > 10.5 acceptance range. In this case the second photon cannot be measured. Here, the signal-to-background ratio is of order of 4 for ξ 1,max = 0.1, and about 10 for ξ 1,max = 0.08.

B. Results for emission of two photons
In Fig. 9 we show the results for the pp → ppγγ reaction, discussed in Sec. II B, calculated for √ s = 13 TeV and in the kinematic region specified in the figure caption. The shapes of distributions in the first four panels are analogous to the ones for single photon emission. The lowest panel shows azimuthal correlations between forward proton and forward photon (solid line) and between backward proton and backward photon (dashed line). The fact of seemingly different distributions for forward and backward emissions is due to the way how the azimuthal angles are defined in (3.1) and (3.3) which leads to the symmetry relation This explains the observed differences between the solid and dashed lines.
Finally we note that emission of the two photons either from the p a -p ′ 1 line or from the p b -p ′ 2 line in Fig. 1 should essentially not contribute to the distributions shown in Fig. 9. Such processes will give two photons on one side of the interaction point and the final proton on the opposite side will miss the ξ cut which we impose. FIG. 9. The distributions for the two-photon bremsstrahlung in the pp → ppγγ reaction. The calculations were done for √ s = 13 TeV, 8.5 < y 3 < 9, −9 < y 4 < −8.5, and 0.02 < ξ 1,2 < 0.1. Shown are results for SPA1.

IV. CONCLUSIONS
In this paper we have studied single-and double-photon bremsstrahlung at veryforward and backward photon rapidities in proton-proton collisions at high c.m. energies. To calculate the amplitudes of the reactions pp → ppγ and pp → ppγγ the framework of the tensor-pomeron model was used.
We have started our analysis with the reaction pp → ppγ, single photon production. We have compared our standard bremsstrahlung results and the results using the approximations SPA1 and SPA2. These approaches were discussed in our previous paper [17] but in a different kinematic region. In the present paper, we have shown that in the forward-rapidity region and for 0.02 < ξ 1 < 0.1 (that corresponds to 130 GeV < ω < 650 GeV) the standard results and the SPA1 results for purely photonic distributions (dσ/dk ⊥ , dσ/dω, dσ/dy) are very close to each other, while the SPA2 overestimates differential cross sections; see Fig. 2 and Fig. 5.
We have studied the azimuthal angle correlations between outgoing particles (see Fig. 7). We observe very interesting correlations between protons and photons. Moreover, these correlations are significantly different for our standard approach and for the SPA1 and SPA2 approaches. Therefore, we emphasize that for detailed comparisons of our predictions with experiment and in order to distinguish our standard and the approximate approaches measurement of the outgoing protons would be most welcome, if not indispensable.
We emphasize that for ω → 0 our bremsstrahlungs distributions are an exact result of QCD plus lowest order electromagnetism. This follows from Low's theorem [27] and the fact that our tensor-pomeron model describes proton-proton elastic scattering at √ s = 13 TeV quite well; see [17]. But is ω in the range 130-650 GeV small enough? In order to answer this question we have to consider, for forward emission of the photon, the scalar products p a · k and p ′ 1 · k. We get As we see from Fig. 2(b) the main part of the k ⊥ distribution is for k ⊥ < 0.2 GeV. With 0.02 < ξ 1 < 0.1 we have then always p a · k ≈ p ′ 1 · k < 1 GeV 2 . Thus, we think that this is well in the region where Low's theorem should be applicable.
We have estimated the coincidence cross section for two-photon bremsstrahlung in the pp → ppγγ reaction within the SPA1 approach. We have required that the final state protons and photons can be measured by the ATLAS forward proton spectrometers (AFP) and LHCf detectors, respectively. We have imposed the kinematical cuts 8.5 < y 3 < 9, −9 < y 4 < −8.5, 0.02 < ξ 1,2 < 0.1, and obtained the corresponding cross section σ ≃ 0.03 nb for √ s = 13 TeV. Our predictions can be verified by the ATLAS-LHCf measurement.
We have also briefly estimated the background contribution due to the pp → ppπ 0 diffractive process for single photon bremsstrahlung. We have compared the signal and background contributions in two LHCf acceptance regions, 8.5 < y < 9 and y > 10.5. One can increase the signal-to-background ratio to about 1 for the first acceptance region. For the second acceptance region the ratio is bigger than 3.5. We conclude that there is a chance to measure single photon bremsstrahlung with the present experimental configuration discussed here. For the two-photon bremsstrahlung a background estimate is much more complicated and goes beyond the scope of this Letter.
Let us finally comment on experimental signatures for our processes. The single photon bremsstrahlung mechanism should be identifiable by the measurement of proton and photon on one side and by checking the exclusivity condition (no particles in the main detector) without explicit measurement of the opposite side proton by AFP. Whether this is sufficient requires further studies, since such a measurement will probably include one-side diffractive dissociation, which can be of the order of 20-30 %. Let us note that in order to isolate our signal reaction pp → ppγ it would be very helpful if the transverse momenta of the outgoing photon and protons could be measured. For the signal reaction these transverse momenta must add up to zero; see (3.2). The background reactions discussed in Sec. III A and above will not satisfy (3.2). Thus, a cut on the quantity (3.2) could be used to eliminate background to a good part. The cross section for two photons on different sides is rather small but should be measurable. Here a study of the background contributions should be done.