Electroproduction of $D$- and $B$-mesons in high-multiplicity $ep$ collisions

In this paper we study the electroproduction of open heavy flavor $D$- and $B$-mesons in the kinematics of future $ep$ colliders, such as the Electron Ion Collider (EIC), the Large Hadron electron Collider (LHeC) and the Future Circular Collider (FCC-he). We study in detail the dependence of the cross-sections on multiplicity of co-produced hadrons, in view of its possible sensitivity to contributions from multipomeron contributions, and discuss different observables which might be used for its study. According to our theoretical expectations, in $ep$ collisions the multipomeron contributions are small in the EIC kinematics, although they might be sizable at LHeC and FCC-he. We also provide theoretical predictions for the production cross-sections of heavy mesons in the kinematics of all the above-mentioned $ep$ colliders.


I. INTRODUCTION
Due to the high luminosity of the forthcoming LHC upgrade (HL-LHC) and future electron-proton colliders, many rare processes recently got renewed theoretical interest. One of the directions which might benefit from the outstanding luminosity is the production of different hadrons in high-multiplicity events. The development of theoretical framework for the study of such events was initiated more than forty years ago in [1][2][3][4][5][6]. However, for a long time the experimental study of such processes was limited by the insufficient luminosity of existing high-energy experiments (see however the discussion in [7][8][9][10][11][12] related to HERA). At RHIC and LHC, thanks to the very large luminosity, the multiplicity dependence of hadroproduction processes has been studied in great detail, and various elaborate observables have been measured experimentally, extending our understanding of the mechanisms of these processes. For example, the experimental study of yields of light charged hadrons co-produced together with heavier mesons [13][14][15][16][17][18] revealed that the multiplicity dependence is faster than in the absence of heavy mesons, and, as was suggested in [19][20][21][22][23], might be explained by contributions of higher twist multipomeron mechanisms. This finding is important, because it gives possibility to understand better the onset of saturation in high-energy collisions.
It is expected that the future Electron Ion Collider (EIC) [24,25], the Large Hadron electron Collider (LHeC) [26] and the Future Circular Collider (FCC-he) [27][28][29] also will have very large luminosities, which will make possible a study of physics at the intensity frontier in electroproduction processes. The measurement of the multiplicity dependencies at these new colliders might be used for better understanding of the underlying microscopic mechanisms of different electroproduction processes. In what follows we will focus on the production of heavy flavor D-and B-mesons, as well as non-prompt J/ψ mesons. These states might be described approximately in the heavy quark mass limit [30,31], and for this reason have been used since the early days of QCD as a probe for testing the predictions of perturbative Quantum Chromodynamics (QCD) (see e.g. [32][33][34][35][36][37][38][39][40] for an overview). In what follows we will focus on the kinematics of photoproduction, where most of the heavy mesons are produced from quasi-real photons with virtuality Q 2 ≈ 0. In this kinematics the typical values of Bjorken x B are small, x B 1, and the gluon densities significantly exceed the sea quark contributions. In the proton rest frame the interaction might be viewed as a scattering of the color dipole, formed from the photon, in the proton gluonic field. The appropriate description of such process is the color dipole framework (also known as CGC/Sat) [41][42][43][44][45][46][47][48][49]. This approach has been successfully applied to the phenomenological description of both hadron-hadron and lepton-hadron collisions [9,11,[50][51][52][53][54][55], and allows a straightforward extension for the description of high-multiplicity events [10,39,[56][57][58][59][60][61]. The color dipole approach is not valid for larger values of x B 0.1, due to possible contributions of intrinsic quarks (e.g. intrinsic charm). For this reason in what follows we will consider only the variables which do not get significant contributions from that region. We also will analyze explicitly the role of the multipomeron mechanisms, which are usually omitted as higher twist effects. Since such contributions have more pronounced dependence on multiplicity, their presence could be straightforwardly deduced from experimental data on multiplicity dependence.
The paper is structured as follows. In Section II we discuss the framework used for the open-heavy meson production evaluation, taking into account the contributions of the single-and double-pomeron mechanisms, compare the theoretical expectations with experimental data and make predictions for the kinematics of the future electron-proton colliders. In Section III we suggest observables which might help to measure the multiplicity dependence, and make theoretical predictions for them in the dipole framework. Finally, in Section IV we draw conclusions.

II. PRODUCTION OF OPEN HEAVY FLAVOR MESONS
The cross-section of open heavy-flavor meson production via the fragmentation mechanism is given by [34,35,39,40,[62][63][64] where we use standard DIS notations Q 2 , x B , y for the virtuality of intermediate photon, Bjorken variable x B and elasticity (fraction of electron energy which passes to the photon in the proton rest frame); while η and p T are the rapidity and the transverse momentum of the produced heavy meson. The fragmentation function D i (z) describes the probability of fragmentation of the parton i into a heavy meson. For D-and B-mesons production, as well as for non-prompt J/ψ production, the corresponding fragmentation functions are known from the literature [34,35,65]. While in (1) there is a sum over all parton flavors, the dominant contribution to all the mentioned states stems from the heavy c-and b-quarks. This implies that the cross-section dσ pp→QiQi+X /dη d 2 p T , for heavy quark production might be evaluated in the heavy quark mass limit. It is convenient to separate explicitly the leptonic and hadronic parts of the cross-section, and rewrite it as [64,66] dσ ep→QiQi+X where dσ L and dσ T in the right-hand side of the equation (2) correspond to the cross-sections of heavy quark production by a longitudinally and transversely polarized photon respectively. In the literature the results for leptonic processes are frequently discussed in terms of these photon-proton cross-sections dσ L, T , which have simpler structure. In the dipole approach the cross-sections dσ L,T are given by where η and p T are the rapidity and transverse momenta of the produced heavy meson; Ψ a (r, z) is theQQ component of the light-cone wave function of the photon; r 1,2 are the transverse separation between quarks in the amplitude and its conjugate; while z is the light-cone fraction of the photon momentum carried by the quark. For Ψ a , in the heavy quark mass limit we may use the standard perturbative expressions [67,68] where θ 12 is the azimuthal angle between vectors r 1 and r 1 , m f is the mass of the quark of flavor f , and we used standard shorthand notations The meson production amplitude N M depends on the mechanism of the QQ pair formation. For the case of production on a single-pomeron (see the left panel of the Figure 1), in leading order it is given by by [40,48] N (1)  where N (x, r) is the amplitude of the color singlet dipole scattering. The amplitude (8) has a structure similar to the leading twist result for the hadroproduction of heavy quarks; however, this similarity is no longer valid for higher twist amplitudes. For numerical estimates of this contribution, we need to fix a parametrization of the amplitude N (x, r).
In what follows, for the sake of definiteness we will use the CGC parametrization of the dipole amplitude, which was proposed in [69] (see also [70][71][72][73] for more recent phenomenological analyses). Since we are interested in the p T dependence, we will use the impact parameter dependent fit, taken from [70]. As we can see from Figure 2, the singlepomeron contribution provides a very reasonable description of the available data from HERA. In Figures 3, 4 we have shown the theoretical expectations for the cross-sections of D ± -, B ± -and non-prompt J/ψ meson production, in the kinematics of the future accelerators EIC ( [24][25][26][27][28][29].
It is also interesting to understand the role of the multipomeron mechanisms in electroproduction. While sometimes it is assumed that all such contributions are taken into account by the universal dipole cross-section, in reality the situation is more complicated. The CGC parametrization [70][71][72], used for our numerical estimates, does not take into account such corrections. Another widely used parametrization of the dipole cross-section, the so-called b-Sat [66,73], takes into account such corrections, making additional simplifying assumptions. For this reason our goal is to perform a microscopic evaluation using the CGC model. We understand that a systematic evaluation of all such corrections in high-multiplicity events presents a challenging problem, and for this reason we will focus on the contribution of twopomeron mechanisms, which are shown in the central and right panels of the Figure 1. Formally such contributions are expected to be small, because they are of higher twist. However, it is desired to reassess them for electroproduction, because earlier studies [23] revealed that for hadroproduction such corrections might be pronounced in the charm sector x B 10 7 x B  where and we introduced the shorthand notations The derivation of these expressions is straightforward and follows the procedures described in [23,40,48,49]. Both functions N ± (z, r 1 , r 2 ) are invariant with respect to the permutation r 1 ↔ r 2 . The p T -integrated cross-sections get contributions only from r 1 = r 2 = r, so the cross-sections N ± simplify tõ In Figure (4) we show the ratio of cross-sections, where the numerator and denominator were evaluated using the two-pomeron contribution (9) and the single-pomeron contribution (8) respectively,

III. MULTIPLICITY DEPENDENCE
The theoretical study of multiplicity dependence in high energy collisions was initiated long ago in [1][2][3][4][5][6] in the framework of the Regge approach. Relying on very general properties of particle-reggeon vertices, which are largely independent of the underlying quantum field theory, it was suggested that the enhanced multiplicity of high energy final states could be considered as one of the manifestations of the multiple pomeron contributions. Later it was demonstrated in [7][8][9][10][11][12] that all these findings are also valid in the context of QCD, and thus could be confirmed by experimental evidence. The dependence on multiplicity differs from the dependencies on other kinematic variables, which are sometimes used for the extraction of dipole amplitudes, fragmentation functions or parton distributions from experimental data. This implies that the multiplicity dependence might be used as a litmus test to probe the role of multipomeron contributions.
The probability distribution P (N ch , N ch ) of high-multiplicity fluctuations inside each pomeron decreases rapidly as a function of number of produced charged particles N ch , as was found both at ep and pp collisions [77,78]. The theoretical modeling of the essentially nonperturbative probability distribution P (N ch , N ch ) is very challenging. In order to exclude this common suppression factor, it is convenient to analyze the multiplicity dependence of the ratio of two different processes. In pp collisions usually the results are presented for the ratio of cross-sections of heavy meson and inclusive processes in a given multiplicity class, self-normalized to one for n ≡ N ch / N ch = 1 for the sake of convenience [13-15, 18, 79], so effectively such ratios are proportional to a conditional probability to measure a hadron provided N ch charged particles are produced in the final state. It was found that in charm sector such ratios grow with multiplicity, which clearly indicates pronounced multipomeron contributions. For ep collisions this variable is not very convenient, because multiparton configurations might contribute in a similar way both to heavy meson productions and to the inclusive channel. In the latter case a sizable contribution might come from large dipoles, for which multipomeron contributions are even more pronounced than for heavy mesons. Potentially the contribution of large dipoles might be suppressed in some special kinematics (e.g. at large virtualities Q 2 ); however, it will be very challenging to measure multiplicity dependence due to significantly smaller statistics. For this reason below we will consider other variables which might present interest for studies of multiplicity dependence. We need to mention that in contrast to hadroproduction, the multiplicity dependence of electroproduction is simpler at the conceptual level, because there are fewer different mechanisms to produce an enhanced number of charged particles in the final state.
The description of high-multiplicity events in the CGC/Sat framework has been discussed in detail in [10,39,[56][57][58][59][60][61]. It is expected that at high multiplicities the dipole amplitude should satisfy the same Balitsky-Kovchegov equation (and thus maintain its form), although the saturation scale Q s (x, b), which contributes to the dipole amplitude, should be modified as For multipomeron configurations, we should take into account that multiplicity fluctuations occur independently in each pomeron, and the observed multiplicity n might be shared in all possible ways between all cut pomerons in a given rapidity window. However, as was discussed in detail in [19,20,80], with good precision we may assume that the observed multiplicity n is shared equally between all pomerons which participate in ep process. Using this assumption, as well as certain convolution properties of P (N ch , N ch ), it is possible to show that for the ratio of different cross-sections the probability distribution P (N ch , N ch ) cancels altogether. Thus for the evaluation of the cross-sections in a given multiplicity class, we may use a simple prescription (17), properly adjusting the parameter n in each pomeron to take into account equal sharing of total multiplicity. As we can see from the Figure 6, the theoretical estimates suggest that in high-multiplicity events the role of the multipomeron contributions increases. Numerically, in EIC kinematics this contribution becomes pronounced at n 5 for D-mesons, although still remains relatively small for B-mesons. This difference in the size of multipomeron terms suggests that we can study experimentally the multiplicity dependence of the ratio of D-and B-meson cross-section in order to estimate unambiguously the role of the two-pomeron contribution in D-meson production. In order to avoid the effects related to the x B -dependence, we suggest to study the double ratio of cross-sections This ratio equals one in the heavy quark mass limit, yet for finite values of n deviates from this value due to more pronounced higher twist corrections for D-meson (numerator of (18)). In the left panel of Figure 8 we show the dependence of the ratio (18) on n. The dependence on n exists even for the leading twist, due to higher twist corrections, but becomes more pronounced when the multipomeron contributions are taken into account. The growth of the ratio as a function of n agrees with the elevated contribution of multipomeron mechanisms in the large-n kinematics. However, we expect that for asymptotically large values of n the ratio should saturate, because the multipomeron contributions will also become important in the denominator. In the right panel of the same Figure 8, we show a similar self-normalized double ratio (18), in which we replaced B-mesons with non-prompt D-mesons. Since the latter mechanism is dominated by b-meson decays, we can see that qualitatively the ratio has a similar dependence on n. Comparison of the left and right panels of Figure 8 clearly illustrates that the enhancement of the ratio (18) is not related to differences of the D-and B-meson fragmentation functions. We expect that non-prompt charmonia should demonstrate a similar behavior.
Another observable which might be easily measured experimentally is the dependence of the average momentum p T of heavy mesons on the multiplicity n. This observable has been extensively studied in the context of pp collisions. In Figure 8 we show the dependence of p T on n, for electroproduction of both D-and B-mesons. Since multipomeron contributions are suppressed at large momenta p T , we can see that their inclusion decreases the average p T , compared to what is expected from single-pomeron. Although the expected effect is not very large, we believe that it might be seen in experimental data, since p T might be measured with very good precision.
To summarize, we believe that the multiplicity dependence might reveal information about the contribution of the multipomeron mechanisms. However, in EIC kinematics we do not expect drastic enhancement of the multiplicity dependence, as was observed in pp collisions. This happens because in general multipomeron contributions are small at EIC energies. The situation might be different in the kinematics of future accelerators like LHeC and FCC-he, where the role of the multipomeron contributions is more pronounced. In our analysis we took into account only the first multipomeron correction, namely the production on two pomerons. We could see that its relative contribution is small in EIC kinematics, in agreement with general expectations based on twist counting, and for this reason we do not consider the orrections of even higher order. However, at very small values of x B (significantly smaller than 10 −7 ) we approach the deeply saturated regime, where the expectations based on twist expansion are not reliable, and thus the inclusion of all higher twist might be required.
The mechanism of multiplicity generation suggested in this section introduces dependence on the multiplicity of soft produced particles, and is quite different from other approaches, such as the percolation approach [81] or the modification of the slope of the elastic amplitude [82], suggested earlier in the context of pp studies. We expect that the experimental confirmation of the predicted multiplicity dependence could help to understand better the mechanisms of multiplicity enhancement in high energy collisions.

IV. CONCLUSIONS
In this paper we analyzed the mechanisms of open-heavy flavor meson electroproduction. Motivated by earlier findings in pp collisions, we also analyzed the relative contribution of the first subleading multipomeron correction. We found that for electroproduction this correction is relatively small for EIC kinematics, although it grows with energy and becomes relevant for charm production at LHeC and FCC-he, especially in the small-p T kinematics. This correction is less important for B-mesons and non-prompt charmonia production, and does not exceed ten per cent even at LHeC and FCC-he. The dependence of the correction on p T agrees with general expectations based on large-p T and heavy quark mass limit. Our evaluation is largely parameter-free and describes very well the data from HERA, as well as provides plausible predictions for EIC, LHeC and FCC-he.
We also analyzed the multiplicity dependence, which might be studied experimentally in detail at future EIC, LHeC and FCC-he, due to its outstanding luminosity. The high-multiplicity events present special interest for theoretical studies, because they allow to get better understanding of the production mechanisms at high gluon densities. Since the probability of rare high-multiplicity events is exponentially suppressed, for the analysis of their dynamics it is important to study properly the designed variables. We analyzed in detail the dependence on multiplicity for the average momentum of heavy meson p T and the double ratio defined in (18). The first variable is easier to measure, although it is less sensitive to higher twist effects, due to the smallness of subleading contributions. The double ratio (18) is more interesting, because its deviations from unity allow to quantify directly the size of the higher twist corrections, including multipomeron contributions. Due to the smallness of multipomeron contributions, we do not expect a significant relative enhancement of the cross-sections at large multiplicity in EIC kinematics, and only mild enhancement in the kinematics of LHeC and FCC-he. This expectation differs significantly from what was found experimentally in pp collisions at LHC [13]. We expect that the experimental confirmation of these findings could help to understand better the mechanisms of multiplicity generation in high energy collisions.