Single diffractive production of open heavy flavor mesons

In this paper we discuss the single diffractive production of open heavy flavor mesons and non-prompt charmonia in $pp$ collisions. Using the color dipole approach, we found that the single diffractive production constitutes 0.5-2 per cent of the inclusive production of the same mesons. In Tevatron kinematics our theoretical results are in reasonable agreement with the available experimental data. In LHC kinematics we found that the cross-section is sufficiently large and could be accessed experimentally. We also analyzed the dependence on multiplicity of co-produced hadrons and found that it is significantly slower than that of inclusive production of the same heavy mesons.

The recoil proton (lower part) is separated from the heavy hadron by a rapidity gap. The colored vertical and inclined ovals schematically illustrate the contributions of the secondary interactions, whose products might fill the rapidity gap between the recoil proton and the other hadrons (see the text for discussion). Right plot: The leading order contribution to the single diffractive production of prompt charmonia studied in [4][5][6].
correctly the onset of saturation dynamics and thus might be used even for the description of high multiplicity events.
In Section III we present our numerical results and make comparison with experimental data available from the Tevatron, as well as with other theoretical approaches. In Section IV we develop the framework for the description of multiplicity dependence in dipole framework and compare its predictions for multiplicity dependence with that of inclusive production. In Section V we discuss briefly the single diffractive process on nuclei, pA → p + M X. Finally, in Section VI we draw conclusions.

II. SINGLE-DIFFRACTIVE PRODUCTION IN COLOR DIPOLE FRAMEWORK
As was mentioned in the previous section, a defining characteristics of the single-diffractive production is the observation of the recoil proton separated by a large rapidity gap from other hadrons. In LHC kinematics the dominant contribution to such process stems from the diagrams which include the exchange of uncut pomeron between the proton and the other hadrons in the t-channel. The heavy mesons are produced predominantly near the edge of the rapidity gap, and for this reason a pomeron couples directly to the heavy quark loop, as shown in the Figure 1. In this paper we will focus on the production of open heavy-flavor Dand B-mesons, and will also discuss briefly the production of non-prompt charmonia from decays of B-meson. Previously, the single diffractive production for prompt charmonia production has been studied in [4][5][6]. In this last case the dominant contribution differs slightly from that of Dand B-mesons and is shown in the right panel of the Figure 1. In Section III we will use the results of [4][5][6] for comparison with our numerical results for non-prompt charmonia.
The cross-section of the heavy meson production might be related to the cross-section of the heavy quark production as [24][25][26][27].
where y is the rapidity of the heavy meson (D-or B-meson), y * = y − ln z is the rapidity of the heavy quark, p T is the transverse momentum of the produced D-meson, D i (z) is the fragmentation function, which describes the parton i fragmentation into a heavy meson, and dσQ iQi is the cross-section of a heavy quark production with a rapidity y * , discussed below in Subsection II A. The dominant contribution to all heavy mesons stems from the cand b-quarks (prompt and non-prompt mechanisms respectively), so the dσQ iQi might be evaluated in the heavy quark mass limit. The fragmentation functions for the Dand B-mesons, as well as non-prompt J/ψ production, are known from the literature and for the sake of completeness are given in Appendix B.
In Figure 1 we also included colored oval blobs, which stand schematically for the secondary interactions which potentially could fill the large rapidity gap in the final state. The general framework for the evaluation of the rapidity gap survival factors (i.e. the probability that no particles will be produced in a rapidity gap) has been developed in [44][45][46][47][48], and is briefly discussed below in Section II B. The leading order contribution to the amplitude of single diffractive production of heavy quarks separated by a rapidity gap from the recoil proton. Right plot: Illustration indicating how the cross-section of the process is related to the production amplitude from three pomeron fusion. The dashed vertical line stands for the unitarity cut. The diagram includes one cut pomeron (upper gluon ladder) and two uncut pomerons (lower gluon ladders). In both plots a summation over all possible permutations of gluon vertices in the heavy quark line/loop is implied.

A. Leading order single diffractive contribution
The single diffractive production of onshell heavy quark pair in the reference frame of the recoil proton might be viewed as a fluctutation of the incoming virtual gluon into a heavyQQ pair, with subsequent elastic scattering of theQQ dipole on the target proton. In perturbative QCD the dominant contribution to such process is given by the diagram which includes exchange of a single pomeron between QQ and a recoil proton, in the spirit of the Ingelman-Schlein model [49] (see Figure 2 for details). In LHC kinematics the typical light-cone momentum fractions x 1,2 carried by gluons are very small ( 1), so the gluon densities are enhanced in this kinematics. This enhancement modifies some expectations based on the heavy quark mass limit. For example, there could be sizeable corrections from multiple pomeron exchanges between the heavy dipole and the target. For this reason instead of hard process on individual partons it is more appropriate to use the color dipole framework (also known as CGC/Sat) [50][51][52][53][54][55][56][57][58]. At high energies the color dipoles are eigenstates of interaction, and thus can be used as the universal elementary building blocks automatically accumulating both the hard and soft fluctuations [59]. The light-cone color dipole framework has been developed and successfully applied to phenomenological description of both hadron-hadron and lepton-hadron collisions [60][61][62][63][64][65][66][67]. Another advantage of the CGC/Sat (color dipole) framework is that it allows a relatively straightforward extension for the description of high-multiplicity events, as discussed in [26,[68][69][70][71][72][73][74]. The cross-section of the single diffractive process, shown in Figure 2, in the dipole approach is given by where y and p T are the rapidity and transverse momenta of the produced heavy quark, in the center-of-mass frame of the colliding protons; k T is the transverse momentum of the heavy quark; g (x 1 , p T ) in the first line of (2) is the unintegrated gluon PDF; Ψ g→QQ (r, z) is the light-cone wave function of theQQ pair with transverse separation between quarks r and the light-cone fraction of the momentum carried by the quark z. For Ψ g→QQ (r, z) we use standard perturbative expressions [75] The meson production amplitude N M depends on the mechanism of the QQ pair formation. For the case of the single-diffractive production, as we demonstrate in the Appendix A, the contribution to the cross-section is given by and N (x, r, b) is the color singlet dipole cross-section with explicit dependence on impact parameter b. In the heavy quark mass limit the main contribution to the integrals in (2) comes from small dipoles of size r m −1 Q . In widely used phenomenological dipole parametrizations [75][76][77][78] it is expected that the band r-dependence factorize in this limit, where the transverse profile T (b) is normalized as´d 2 b T (b) = 1, and N (x, r) is the dipole cross-section integrated over impact parameter. In this approximation we may rewrite (8) as As could be seen from the structure of (8), it is a higher twist (∼ O r 2 ) contribution compared to the amplitude of inclusive production, and thus should have stronger suppression at large p T . The p T -integrated cross-section gets contributions only from dipoles with r 1 = r 2 = r in the integrand. For this case it is possible to show that the gluon uPDF x 1 g (x 1 , p T − k T ) is replaced with the integrated gluon PDF x g G (x g , µ F ) taken at the scale µ F ≈ 2 m Q . In the LHC kinematics at central rapidities this scale significantly exceeds the saturation scale Q s (x), which justifies the dominance of the three-pomeron approximation. However, in the small-x kinematics there are sizeable nonlinear corrections to the evolution in the dipole approach. In this kinematics the corresponding scale µ F should be taken at the saturation momentum Q s . The gluon PDF x 1 G (x 1 , µ F ) in this approach is closely related to the dipole scattering amplitude N (x, r) =´d 2 b N (x, r, b) as [68,79] Eq. (14) can be inverted and gives the gluon uPDF in terms of the dipole amplitude, The corresponding unintegrated gluon PDF can be rewritten as [80] x g x, which allows to express the single diffractive cross-section in terms of only the dipole amplitude. The expression (16) will be used below in Section IV for extension of our results to high-multiplicity events.

B. Gap survival factors
The rapidity gap between the recoil proton and the produced heavy meson might be filled potentially by products of various secondary processes, as shown schematically by the colored vertical and inclined ovals in Figure 1. As was demonstrated in [44][45][46][47], the effect of these factors is significant at high energies and might decrease the observed yields (i.e. probability of non-observation of particles in the gap) by more than an order of magnitude [47,48]. This suppression is due to soft interactions between the colliding protons and thus is not related to the particles produced due to hard interactions. The evaluation of this suppression conventionally follows the ideas of Good-Walker [81], which are usually implemented in the context of different models (see for review [82][83][84][85]). Technically, all these approaches perform evaluations in eikonal approximation, and predict that the observables, which include large rapidity gaps, are suppressed by a so-called gap survival factor, where M(b, s, ...) is the amplitude of the hard process, b is the impact parameter, and Ω is the opacity or optical density. In a single-channel eikonal model the opacity Ω is directly related to the cross-sections of total, elastic and inelastic processes [83]. It is expected that the energy dependence of the function Ω is controlled by the Pomeron intercept, Ω ∼ s α IP −1 , so the factor (17) decreases as a function of energy. The single-channel model is very simple, yet its predictions are at tension with experimental data [48]. More accurate description of data is achieved in multichannel extensions of these models, which assume that after interaction with a soft Pomeron the proton might convert into additional N D − 1 diffractive states. In this basis, the soft pomeron interaction amplitudeΩ should be considered as an N D ×N D matrix. As was discussed in [82][83][84], for a good description it is sufficient to choose N D = 2, with the common parametrization for the matrix Ω ik given in [86] and briefly summarized for the sake of completeness in Appendix C. For the single diffractive scattering the exponent in the expression (17) should be understood as a matrix element between |pp and |p X states [87,88]. If Φ 1 and Φ 2 are eigenvalues of Ω ik with eigenvalues Ω 1 and Ω 2 , then the matrix exp −Ω(b, s) reduces in this basis to a linear combination of factors ∼ e −Ωa(s,b) , in which the coefficients can be fixed by projecting the proton and diffractive states onto the eigenstates Φ 1 , Φ 2 of the scattering matrix. For the single diffractive production the algorithm for evaluation of the survival factor was introduced earlier for the pp → pX process in [88], yielding where parameter Ω is related to eigenvalues Ω 1,2 of the matrix Ω ik as and the parameter λ stands for the ratio of the production amplitude of diffractive state X to the amplitude of elastic proton scattering of the incident proton on a pomeron (see Appendix C for more details). In this paper we are interested only in events without charged particles, produced at pseudorapidity η < y (rapidity gap between the recoil proton and heavy quarks), whereas the evaluation of the survival factor in (17,1821) was performed under the assumption that there are no co-produced particles in the whole rapidity range η ∈ (−y max , y max ), which is much stricter than needed in this problem. For this reason we need to correct the estimate (18), using probabilistic considerations. In what follows we'll use notations P A and P B for the probabilities to emit at least one charged particle in the intervals η < y and η > y due to soft interaction of the colliding protons; whileP A ≡ 1 − P A andP B ≡ 1 − P B are the probabilities not to emit any particles in these intervals (the gap survival factors on these intervals). We will also use the notationP A∪B for the probability not to produce particles in any of the intervals. The relation between the probabilitiesP A∪B andP A ,P B depends crucially on possible correlations between particles from different rapidity intervals. Such correlations have been studied in the literature [89][90][91], and it is known that they are small when the separation between the bins is larger than 1-2 units in rapidity. If we neglect completely such correlations, the probabilities are related asP A∪B =P APB , which implies that the survival factor should scale with the length of the rapidity bin as S 2 (∆η) ∼ const ∆η . For the single diffractive production of heavy mesons we require that no particles are produced with η < y, although we do not impose any conditions for η > y (so we do not need to introduce the gap survival factor in this region). This implies that the overall survival factor (18) should be adjusted as where ∆y is the width of the rapidity gap interval, and y max = − 1 2 ln m 2 Q,T /s is the largest possible rapidity of heavy quarks. This factor S 2 (s pp , b) should be included into the expressions (2,8) from the previous Section (II A).
In the heavy quark mass limit the dipoles are small, r m −1 Q , and we may use a factorized approximation (10). The convolution of S 2 (s pp , b) with impact parameter dependent cross-section can be simplified in this limit and yields for the suppression factor a much simpler expression which depends only on the energy (Mandelstam variable) s pp of the collision, but does not depend on masses nor kinematics of the produced heavy quarks.

III. NUMERICAL RESULTS
For our numerical evaluations here and in what follows we will we use the impact parameter (b) dependent "bCGC" parametrization of the dipole cross-section [77,78] In Figures 3, 4 and 5 we show the production cross-sections of the D-mesons, B-mesons and non-prompt J/ψ mesons.
We can see that in the small-p T region, which encompasses most of the events, the single diffraction production constitutes approximately one per cent of the inclusive cross-section. In the large-p T region the contribution from the single diffractive production is strongly suppressed since it is formally a higher twist effect.
To the best of our knowledge there is no direct experimental data for the cross-sections of the suggested process. The diffractive production of B-mesons has been studied earlier by the CDF collaboration in [15], although the results are only available for the ratio of the integrated cross-sections of diffractive and inclusive processes, Single diffr. vs inclusive, pp→D + dσ/dp dσ/dp T [μb/GeV] Figure 3: The cross-section dσ/dpT of the single diffractive production of D + -mesons. Integration over the rapidity bin |y| < 0.5 is implied. Left plot: Comparison with inclusive production in the LHC kinematics for √ s = 7 TeV (theory and experiment). The curves with labels "SD, prompt" and "SD, non-prompt" correspond to single diffractive contributions to D-meson yields from the fragmentation of the c and b quarks respectively. The curves marked "2-pomeron inclusive" and "3-pomeron inclusive" stand for the contributions of 2-and 3-pomeron fusion mechanisms to inclusive D-meson yields respectively (see a short overview in Appendix A 2 and more detailed discussion in [40]). The experimental data are for inclusive production from [92]. Right plot: √ s-dependence of the data in the kinematics of LHC and the planned Future Cicular Collider (FCC) [93]. For other D-mesons the pT -dependence has a very similar shape, yet differs numerically by a factor of two.
|y|<2.1 |y|<0.5 Single diffr. vs inclusive, p p→B ± X s =7 TeV    The theoretical curves marked "2-pomeron incl." and "3-pomeron incl." stand for the additive contributions from 2-and 3-pomeron fusion mechanisms respectively (see [40] and a short discussion in Appendix A 2).The experimental data are for inclusive production from CMS [94]("|y| < 2.1" data points) and ATLAS [95]("|y| < 0.5" data points). For some experimentally measured results bin-integrated cross-sections dσ/dpT was converted into dσ/dpT dy dividing by the width of the rapidity bin (this is justified since in LHC kinematics at central rapidities y ≈ 0 the cross-section is flat). Right plot: The pT -dependence of the cross-section dσ/dy dpT for several energies √ s.
For energy √ s = 1.8 TeV it was found that In the Table I (27), within uncertainty of experimental data (27). As we can see from the same Table I, in LHC kinematics the ratio (26) is approximately of the same order. The smallness of the values in the Table I is due to the fact that the production of heavy quark in single diffraction events is formally a higher twist effect, and thus has an additional suppression by the factor ∼ (Λ QCD /m Q ) 2 . While the absolute cross-sections of single diffractive and inclusive production increase as a function of energy, the ratio (27)  The theoretical curves marked "2-pomeron inclusive" and "3-pomeron inclusive" stand for the additive contributions from 2-and 3-pomeron fusion mechanisms respectively (see [40] and a short discussion in Appendix A 2).The experimental data are for inclusive production from CMS [96]. Right plot: The pT -dependence of the cross-section dσ/dy dpT for several energies √ s.   the gap survival factor in single-diffractive cross-section. We extended the definition (26) and analyzed the ratio of differential cross-sections, which presents a novel observable. In Figure 6 we show this ratio as a function of p T for D-mesons, both for prompt and non-prompt mechanisms. For the sake of definiteness we considered D + mesons, although the results for the ratio (28) are almost the same for other choices of D-mesons. In Figure 7 we show the same ratio for the B-mesons (B + for definiteness) and non-prompt J/ψ. We can see that the ratio is smaller than for D-mesons, and decreases quite fast at large p T . This behavior agrees with our earlier observation that the single-diffractive mechanism is formally a higher twist effect compared to the dominant two-gluon fusion mechanism, in the case of inclusive production. As expected, at small p T the ratios are similar for B-mesons and non-prompt J/ψ; for larger p T the results differ due to differences in fragmentation functions (see Appendix B for details).
In Figure 8 we compare our results for non-prompt production of J/ψ with the predictions for prompt production from [5,6] (color octet contributions + gluon fragmentation, dominant at large p T ) and from [4] (color evaporation model). As we can expect, the non-prompt mechanism is smaller than the prompt contribution, although the qualitative behavior is similar in both cases.
In Figure 9 we compare our predictions with earlier results from [11] obtained in the framework of Ingelman-Schlein model. We can see that in the region p T 5 GeV, where a majority of heavy mesons are produced, both approaches give comparable contributions. At larger p T the discrepancy between the two approaches increases.
Finally, we would like to stop briefly on the ratio R (diff) J/ψ of single diffractive and inclusive contributions. It was predicted in [6] that for the prompt contributions R (diff, prompt) J/ψ ≈ 0.65 ± 0.15%, although later the CDF collaboration [14] found a value twice larger The ratio of single diffractive to inclusive production cross-sections, as defined in (28). The left plot is for the B mesons, the right panel is for non-prompt production of J/ψ-mesons.
This mismatch might be explained by sizeable non-prompt contributions: combining R (diff, prompt) J/ψ with R (diff, non−prompt) J/ψ from the first line in Table I, we get R (diff, prompt+nonprompt) J/ψ ≈ 1.22%, in reasonable agreement with the experimental value (29).

IV. MULTIPLICITY DEPENDENCE
According to the Local Parton Hadron Duality (LPHD) hypothesis [97][98][99], the multiplicity of produced hadrons in a given event is directly related to the number of partons produced in a collision. For this reason the study of multiplicity dependence of different processes presents an interesting extension, which allows to understand better the onset of the saturation regime in high energy collisions. A feasibility to measure such processes was demonstrated for inclusive channels by the STAR [32,100] and ALICE [31,101] collaborations. The extension of these experimental measurements to single diffractive production is quite straightforward, since their detectors have the capability to detect simultaneously both the rapidity gaps and the charged particles outside of the rapidity window. Since the crosssection of single diffractive production is significantly smaller than that of inclusive production, and the probability of events with large multiplicity is exponentially suppressed [101], each measurement will require larger integrated luminosity.
In order to get rid of a common exponential suppression at large multiplicities, for a comparison of the multiplicity pT -dependence of differential cross-sections of prompt and non-prompt mechanisms for single diffractive production of J/ψ mesons. The results for the prompt mechanism are taken from [5,6], and the width of the green band reflects the uncertainty due to one of the model parameters (gluon fraction of pomeron fg). The results for the non-prompt mechanism (blue solid curve) are results of this paper. Right plot: Energy dependence of total cross-sections of prompt and non-prompt single diffractive production mechanisms of J/ψ-mesons. The prompt contribution (green dashed line) is taken from [4]. dσ/ⅆp T [μb/GeV] Figure 9: Comparison of color dipole approach predictions (this paper) with results of [11] obtained in the framework of Ingleman-Schlein model [49]. The left plot corresponds to single-diffractive charm production, the right plot is for bottom quarks.
dependence in different channels it is widely accepted accepted to use a self-normalized ratio [102] dN M /dy where N ch = ∆η dN ch /dη is the average number of particles detected in a given pseudorapidity window (η−∆η/2, η+ ∆η/2), n = N ch / N ch is the relative enhancement of the number of charged particles in the same pseudorapidity window, w (N M ) / w (N M ) and w (N ch ) / w (N ch ) are the self-normalized yields of heavy meson M (M = D, B) and charged particles (minimal bias events) in a given multiplicity class; dσ M (y, √ s, n) is the production crosssections for heavy meson M with rapidity y and N ch = n N ch charged particles in the pseudorapidity window (η − ∆η/2, η + ∆η/2), whereas dσ ch (y, √ s, n) is the production cross-sections for N ch = n N ch charged particles in the same pseudorapidity window. Mathematically the ratio (30) gives a conditional probability to produce a meson M in a single diffractive collision in which N ch charged particles are produced.
In the color dipole (CGC/Sat) approach, the framework for description of the high-multiplicity events has been developed in [26,[68][69][70][71][72][73][74]. In this picture the observation of enhanced multiplicity signals that a larger than average number of partons is produced in a given event. Nevertheless, we still expect that each pomeron should satisfy the nonlinear Balitsky-Kovchegov equation. The bCGC dipole amplitude (22) was constructed as an approximate solution of the latter, and for this reason it should maintain its form, although the value of the saturation scale Q s might be modified. As was demonstrated in [68][69][70], the observed number of charged multiplicity dN ch /dy of soft hadrons in pp collisions is proportional to the saturation scale Q 2 s (modulo logarithmic corrections), for this reason the events with large multiplicity might be described in dipole framework by simply rescaling Q 2 s as a function of n [68][69][70][71][72][73][74], It was demonstrated in [26] that the error of the approximation (31) is less than 10% in the region of interest (n 10), and for this reason we will use it for our estimates. While at LHC energies it is expected that the typical values of saturation scale Q s (x, b) fall into the range 0.5-1 GeV, from (31) we can see that in events with enhanced multiplicity this parameter might exceed the values of heavy quark mass m Q and lead to an interplay of large-Q s and large-m Q limits. The expression (31) explicitly illustrates that the study of the high-multiplicity events gives us access to a new regime, which otherwise would require significantly higher energies. The observation of enhanced multiplicity in the process shown in the left diagram of Figure 1 implies that unintegrated gluon density g (x, k ⊥ , n) in (2) is also modified. This change might be found taking into account the relation of gluon density with the dipole amplitude N (x, r, b) given by (16). For the sake of simplicity below we'll focus on the multiplicity dependence of the p T -integrated cross-section, which is easier to measure experimentally. For this case the cross-section (2) simplifies considerably, since, after integration over p T , the multiplicity dependent (integrated) gluon density factorizes and contributes to the result as a multiplicative factor. For this reason the ratio (30) reduces to a common factor the same for all mesons. In Figure 10 we show the multiplicity dependence of the ratio (32). At very small n, when saturation effects are small, the size of the dipole is controlled by the mass of heavy quark ∼ 1/m Q , and thus the dipole amplitude N (y, r, n) might be approximated as N (y, r, n) ∼ (r Q s (y, n)) γ , where γ ≈ 0.63 − 0.76 is a numerical parameter. In view of (31) this translates into the multiplicity dependence as shown in the same Figure 10 with red dotted line. At larger values of n, due to saturation effects, the curve deviates from the small-n asymptotic behavior. As we can see from the right panel of the same Figure 10, this behavior is different from the dependence seen by ALICE for inclusive the production [31], as well as from our theoretical result for inclusive production from [40]. This happens because in single diffractive production the co-produced hadrons stem from only one cut pomeron, whereas in inclusive production, in the setup studied in [31], at least two pomerons can contribute to the observed multiplicity enhancement. Since each cut pomeron gives a factor ∼ n γ in multiplicity dependence, this explains the predicted difference between the single diffractive and inclusive processes.

V. NUCLEAR EFFECTS
The study of the single diffractive production on nuclear collisions is appealing because its cross-section grows rapidly with atomic number A, and thus is easier to measure experimentally. The AA collisions are not suitable for this purpose due to formation of hot Quark-Gluon Plasma at later stages [103][104][105][106][107][108][109]. For this reason we will focus on pA collisions and in the kinematics when the scattered proton in the final state is separated by large rapidity gap from the produced heavy meson and nuclear debris.
In CGC framework the nucleus differs from the proton by larger size R A = A 1/3 R p and larger values of the saturation scale Q 2 sA . As was found in [110,111] from analysis of the experimental data, the dependence of Q 2 sA on atomic number A might be approximated by The value δ < 1 indicates that the saturation scale grows faster than ∼ A 1/3 expected from naive geometric estimates. In single diffractive process the nucleus contributes in (2) only through the unintegrated gluon density g(x, k). Currently the latter is poorly defined experimentally [112], for this reason we will estimate it from the dipole production cross-sections with single diffractive mechanism (the same for all mesons, see the text for explanation). The red dotted line corresponds to the asymptotic expression for small multiplicities, as explained in the text. Right plot: comparison of multiplicity dependence for inclusive and single diffractive production for non-prompt J/ψ mesons. The experimental points are from ALICE [31] for inclusive production, the theoretical curve for inclusive production is from [40]. amplitude using (15,16). The magnitude of nuclear effects is conventionally expressed in terms of the normalized ratio of the cross-sections on the nucleus and proton, For the sake of simplicity we'll focus on the p T -integrated cross-section. In this case the dependence on the gluon PDF factorizes, and thus the ratio (35) reduces to a common prefactor where N A (y, r, b) is a nuclear dipole amplitude with adjusted saturation scale (34), and the rescaling of the impact parameter b in the numerator reflects the increase of the nuclear radius. In the Figure 11 we have shown the ratio (35) as a function of the atomic number A. We can see that due to nuclear (saturation) effects the cross-section decreases by up to a factor of two for very heavy nuclei. This finding is in agreement with expected suppression of nuclear gluon densities found in [112] from global fits of experimental data. Finally, from comparison of (32) and (36) we may obtain the relation between the nuclear suppression factor R A and the multiplicity dependence of the proton cross-section (32), , which might be checked experimentally.

VI. CONCLUSIONS
In this paper we studied single diffractive production of open heavy-flavor mesons. We analyzed in detail the production of Dand B-mesons, as well as non-prompt production of J/ψ mesons. While in general diffractive events constitute up to 20 per cent of inclusive cross-section [1], we found that for heavy mesons production the single diffractive events constitutes only 0.4-2 per cent of all inclusively produced heavy mesons. This happens because the leading order contribution to single diffractive production is formally a higher twist effect (compared to leading order inclusive diagrams) and thus includes additional suppression ∼ (Λ QCD /m Q ) 2 . Similarly, the observed (a) (b) (c) Figure 12: The diagrams which contribute to the heavy meson production cross-section in the leading order perturbative QCD. The contribution of the last diagram (c) to the meson formation might be also viewed as gluon-gluon fusion gg → g with subsequent gluon fragmentation g →QQ. In CGC parametrization of the dipole cross-section approach each "gluon" is replaced with reggeized gluon (BK pomeron), which satisfies the Balitsky-Kovchegov equation and corresponds to a fan-like shower of soft particles.
suppression at large transverse momentum p T of the produced heavy meson agrees with expected pattern of higher twist suppression. Nevertheless, we believe that the cross-sections are sufficiently large and thus could be measured with reasonable precision at the LHC. We also analyzed the dependence on multiplicity of co-produced hadrons, assuming that these are produced only on one side of the heavy meson. We found that the dependence on multiplicity is mild, in contrast to the vigorously growing multiplicity seen by ALICE [31] for inclusive production. Our evaluation is largely parameter-free and relies only on the choice of the parametrization for the dipole cross-section (22).
We expect that suggested processes might be studied by the CMS (see their recent feasibility study in [19]), ALICE [31,101] and STAR collaborations. In QCD the interaction of the color dipole with a pomeron might be understood as a gluon ladder (BFKL pomeron), for this reason its interaction with a dipole is described as with a pair of gluons in a color singlet state (see the text for explanation).
Appendix A: Evaluation of the dipole amplitudes

Single diffractive production
In this Appendix, for the sake of completeness, we explain the main technical steps and assumptions used for the derivation of the single diffractive cross-section (2,8). The general rules which allow to express the cross-sections of hard processes in terms of the color singlet dipole cross-section might be found in [50][51][52][53][54][55][56][57][58]. In the heavy quark mass limit the strong coupling α s (m Q ) is small, which allows to consider the interaction of a heavyQQ dipole with gluons perturbatively and discuss them similar to the treatment of the k T -factorization approach. At the same time we tacitly assume that each such gluon should be understood as a parton shower ("pomeron").
In the high-energy eikonal picture, the interaction of the quarks and antiquark with a t-channel gluon are described by a factor ±ig t a γ (x ⊥ ), where x ⊥ is the transverse coordinate of the quark, and the function γ (x ⊥ ) is related to a distribution of gluons in the target. This function is related to a dipole cross-section σ(x, r) as where r is the transverse size of the dipole, and z is the light-cone fraction of the dipole momentum carried by the quarks. The equation (A1) might be rewritten in the form For very small dipoles, the dipole cross-section is related to the gluon uPDF as [115] σ (x, r) = 4πα s 3ˆd so the functions γ (x, r) might be also related to the unintegrated gluon densities. With the help of (A2), for many high energy processes it is possible to express the exclusive amplitude or inclusive cross-section as a linear combination of the color singlet dipole cross-sections σ(x, r) with different arguments. While in the deeply saturated regime we can no longer speak about individual gluons (or pomerons), we expect that the relations between the dipole amplitudes and color singlet cross-sections should be valid even in this case. For the case of single-diffractive heavy quark pair production, the leading-order contribution is given by the diagrams shown in the Figure (13). As was explained at the beginning of this appendix, in the heavy quark mass limit the interactions ofQQ with gluons become perturbative, which implies that the t-channel pomeron might be considered as a color singlet pair of gluons. Taking into account all the diagrams shown in the Figure 13 and properties of the SU (N c ) structure constants, we may express the amplitude of the single diffractive process as where a is the color index of the incident (projectile) gluon, and r Q , rQ are the coordinates of the quarks. For evaluation of the p T -dependent cross-section we need to project the coordinate space quark distribution onto the state with definite transverse momentum p T , so we have for the evaluate the additional convolution ∼´d 2 r 1 d 2 r 2 e ip T ·(r1−r2) , where r 1,2 are the coordinates of the quark in the amplitude and its conjugate, viz: As discussed earlier, at high energies we may apply iteratively the relation (A1) and express the three-pomeron dipole amplitude in terms of the color singlet dipole cross-sections, as given in (8). In the frame where the momentum of the primordial gluon is not zero, we should take into account an additional convolution with the momentum distribution of the incident ("primordial") gluons, as shown in (2), and was demonstrated in [27].

Inclusive production
In Section III we compared predictions for single-diffractive production of heavy quarks with those of the inclusive production of the same mesons. For the sake of completeness, in this Appendix we would like to mention briefly the main expressions used for evaluation of the cross-sections for the latter case. A detailed discussion of inclusive production, as well as comparison with experimental data might be found in [40]. The evaluation of the cross-section follows the steps outlined in the previous Appendix A 1. The leading order contribution in the inclusive case is due to a standard fusion of two gluons (pomerons). In the evaluation of the three-pomeron we should take into account that there are two complementary mechanisms, shown schematically in Figure 14. In what follows we'll refer to the contribution shown in the diagram (a) as genuine three-pomeron corrections, whereas the contribution of the diagram (b) is the interference term. The two diagrams differ by number of cut pomerons, and for this reason they have a different multiplicity dependence. As we discussed in [40], both twist-three corrections give sizeable contributions at small p T 5 GeV. For D-mesons the two corrections together contribute up to 40-50 per cent of the leading order result, whereas for B-mesons these contributions are of order 10% even for p T ∼ 0, in agreement with the heavy mass limit.

Appendix B: Fragmentation functions
For the sake of completeness, in this appendix we briefly summarize the fragmentation functions used in our evaluations. Since the fragmentation functions are essentially nonperturbative and cannot be evaluated from first principles, currently their parametrization is extracted from the phenomenological fits of e + e − annihilation data. For the B-mesons the dominant contribution comes from the fragmentation of b-quarks, and for the fragmentation function of this process we used the parametrization from [24] D b→B (z, µ 0 ) = N z α (1 − z) β ,  Table II: The values of parameters of D-meson fragmentation function with parametrization (B4), as found in [114].
where N = 56.4, α = 8.39, β = 1.16. The shape of parametrization (B1) is close to another widely used parametrization from [113] D b→B (z, µ 0 ) = N z 1 − 1 z − 1−z which might be produced instead of proton in inelastic processes (e.g. single diffractive, double diffractive). The matrix Ω ik is thus a 2 × 2 matrix in the subspace which includes a proton and the diffractive state X. For our evaluations we used a parametrization from [45], which has a form Ω ik (b, s) =ˆd 2 q 4π 2 e iq·bΩ ik t = −q 2 , s (C1) and has been fitted using recent LHC data on elastic, single diffractive and double diffractive scattering.