Single diffractive production of open heavy flavor mesons
SSingle diffractive production of open heavy flavor mesons
Marat Siddikov, Iván Schmidt
Departamento de Física, Universidad Técnica Federico Santa María,y Centro Científico - Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile
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 singlediffractive production constitutes 0.5-2 per cent of the inclusive production of the same mesons. InTevatron kinematics our theoretical results are in reasonable agreement with the available experi-mental data. In LHC kinematics we found that the cross-section is sufficiently large and could beaccessed experimentally. We also analyzed the dependence on multiplicity of co-produced hadronsand found that it is significantly slower than that of inclusive production of the same heavy mesons. Keywords:
I. INTRODUCTION
In the kinematics of the Large Hadronic Collider (LHC), the diffractive events in pp collisions constitute approxi-mately twenty per cent of all inclusive events [1], and for this reason might be used as an additional tool for studiesof the strong interactions. The characteristic feature of the diffractive events is the presence of rapidity gaps betweenhadronic products in the final state. In Quantum Chromodynamics (QCD) such rapidity gaps in high energy kine-matics are explained by the exchange of pomerons in the t -channel. Since the structure of the pomeron is relativelywell understood and largely does not depend on the process, the existence of rapidity gap allows to separate the stronginteractions involving different hadrons. While conventionally diffractive production of mesons has been studied in ep collisions, there are various theoretical suggestions to use pp collisions for studies of the diffractive production ofprompt quarkonia [2–6], dijets [7], gauge bosons [8], Higgs bosons [9], heavy quarks [10, 11], quarkonia pairs [12] andDrell-Yan processes [13]. The possibility to measure diffractive production in pp collisions has been demonstrated atthe Tevatron [14–18]. At the LHC some diffractive processes ( e.g . single diffractive pp → pX ) have been measuredwith very good precision [1], although diffractive production of additional heavy hadrons so far has not been exploredin depth (see however preliminary feasibility study [19]).In this paper we are going to focus on single diffractive production of heavy mesons, pp → p + M X , where M is anopen heavy flavor meson ( D or B ) or a charmonium produced from decay of B -meson; we also assume that the recoilproton in the final state is separated by a rapidity gap from other hadrons. This process deserves special interest bothon its own and because it could help to clarify the role of multipomeron contributions to the production of heavyquarks in general. The role of such mechanisms is not very clear at this moment. Usually it is believed that productionof heavy quarks might be described perturbatively [20, 21] and is dominated by two-gluon (pomeron) fusion [22–30].However, this approach can hardly explain the recently measured dependence of the production cross-sections onthe multiplicity of the charged hadrons co-produced together with a given heavy quarkonia [31–36]. Potentially thisdiscrepancy might indicate sizeable contributions of multigluon production mechanisms. At the same time, for D -and B -mesons such rapidly growing dependence was not observed [31]. On the other hand, theoretical studies [37–39] found that three-pomeron mechanism might give sizeable contribution and can explain the observed multiplicitydependence of quarkonia. For D -mesons it was found in the same framework that the three-pomeron correction isalso pronounced and might constitute up to 40 percent of the result, although in the range of multiplicities availableat present from the LHC it does not contribute to the observed multiplicity dependence due to partial cancellationwith certain interference contributions [40]. Fortunately, it is possible to estimate the role of the three-pomeronfusion directly. The single diffractive production at the partonic level has a similar structure, and thus might provideindependent estimate of the three-pomeron contribution. Since the single diffractive production amplitude includesonly one cut pomeron which might contribute to the observed yields of co-produced hadrons, its cross-section mightbe used as a very clean probe of the multiplicity dependence of individual cut pomerons in high multiplicity events.Earlier the single-diffractive production including heavy quarks has been studied in [10, 11] for the case of promptproduction of quarkonia. As we will see below, the cross-sections of single diffractive production of D - and B -mesonsis larger than that of the prompt charmonia and thus could be easier to study experimentally. The feasibility tomeasure such processes has been discussed in [14, 15, 19]. The study of rare events with large multiplicity requiresbetter statistics, and for this reason we expect that such dependence could be measured during the High LuminosityRun 3 at the LHC (HL-LHC mode) [41–43].The paper is structured as follows. In Section II we develop the general framework for the evaluation of theopen heavy meson production. We will perform our calculations within the color dipole framework, which describes a r X i v : . [ h e p - ph ] A ug MXpS eik S enh pp MXpS eik S enh pp Figure 1: Left plot: The leading order contribution to single diffractive production of open heavy flavor quark mesons. Therecoil proton (lower part) is separated from the heavy hadron by a rapidity gap. The colored vertical and inclined ovalsschematically illustrate the contributions of the secondary interactions, whose products might fill the rapidity gap between therecoil proton and the other hadrons (see the text for discussion). Right plot: The leading order contribution to the singlediffractive production of prompt charmonia studied in [4–6]. 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 theTevatron, as well as with other theoretical approaches. In Section IV we develop the framework for the descriptionof multiplicity dependence in dipole framework and compare its predictions for multiplicity dependence with that ofinclusive 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 theobservation of the recoil proton separated by a large rapidity gap from other hadrons. In LHC kinematics thedominant contribution to such process stems from the diagrams which include the exchange of uncut pomeron betweenthe proton and the other hadrons in the t -channel. The heavy mesons are produced predominantly near the edge ofthe 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 D - and B -mesons, and will also discuss brieflythe production of non-prompt charmonia from decays of B -meson. Previously, the single diffractive production for prompt charmonia production has been studied in [4–6]. In this last case the dominant contribution differs slightlyfrom that of D - and B -mesons and is shown in the right panel of the Figure 1. In Section III we will use the resultsof [4–6] 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 productionas [24–27]. dσ M dy d p T = (cid:88) i ˆ x Q dzz D i (cid:18) x Q ( y ) z (cid:19) dσ ¯ Q i Q i dy ∗ d p ∗ T (1)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 isthe 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 dσ ¯ Q i Q i 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 c - and b -quarks(prompt and non-prompt mechanisms respectively), so the dσ ¯ Q i Q i might be evaluated in the heavy quark mass limit.The fragmentation functions for the D - and B -mesons, as well as non-prompt J/ψ production, are known from theliterature 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 whichpotentially could fill the large rapidity gap in the final state. The general framework for the evaluation of the rapiditygap survival factors ( i.e . the probability that no particles will be produced in a rapidity gap) has been developedin [44–48], and is briefly discussed below in Section II B.
Xppp Q ¯ Q r a p i d i t y g a p pppp Figure 2: Left plot: The leading order contribution to the amplitude of single diffractive production of heavy quarks separatedby a rapidity gap from the recoil proton. Right plot: Illustration indicating how the cross-section of the process is relatedto the production amplitude from three pomeron fusion. The dashed vertical line stands for the unitarity cut. The diagramincludes one cut pomeron (upper gluon ladder) and two uncut pomerons (lower gluon ladders). In both plots a summationover 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 mightbe viewed as a fluctutation of the incoming virtual gluon into a heavy ¯ QQ pair, with subsequent elastic scatteringof the ¯ QQ dipole on the target proton. In perturbative QCD the dominant contribution to such process is givenby the diagram which includes exchange of a single pomeron between Q ¯ Q and a recoil proton, in the spirit of theIngelman-Schlein model [49] (see Figure 2 for details). In LHC kinematics the typical light-cone momentum fractions x , carried by gluons are very small ( (cid:28) ), so the gluon densities are enhanced in this kinematics. This enhancementmodifies some expectations based on the heavy quark mass limit. For example, there could be sizeable correctionsfrom multiple pomeron exchanges between the heavy dipole and the target. For this reason instead of hard processon individual partons it is more appropriate to use the color dipole framework (also known as CGC/Sat) [50–58].At high energies the color dipoles are eigenstates of interaction, and thus can be used as the universal elementarybuilding blocks automatically accumulating both the hard and soft fluctuations [59]. The light-cone color dipoleframework has been developed and successfully applied to phenomenological description of both hadron-hadron andlepton-hadron collisions [60–67]. Another advantage of the CGC/Sat (color dipole) framework is that it allows arelatively straightforward extension for the description of high-multiplicity events, as discussed in [26, 68–74]. Thecross-section of the single diffractive process, shown in Figure 2, in the dipole approach is given by dσ ¯ Q i Q i ( y, √ s ) dy d p T = ˆ d k T x g ( x , p T − k T ) ˆ dz ˆ dz (cid:48) (2) × ˆ d r π ˆ d r π e i ( r − r ) · k T Ψ † ¯ QQ ( r , z, p T ) Ψ ¯ QQ ( r , z, p T ) × N (SD) M ( x ( y ); (cid:126)r , (cid:126)r ) + ( x ↔ x ) ,x , ≈ (cid:112) m M + (cid:104) p ⊥ M (cid:105)√ s e ± y (3)where y and p T are the rapidity and transverse momenta of the produced heavy quark, in the center-of-mass frameof the colliding protons; k T is the transverse momentum of the heavy quark; g ( x , p T ) in the first line of (2) isthe unintegrated gluon PDF; Ψ g → ¯ QQ ( r, z ) is the light-cone wave function of the ¯ QQ pair with transverse separationbetween quarks r and the light-cone fraction of the momentum carried by the quark z . For Ψ g → ¯ QQ ( r, z ) we usestandard perturbative expressions [75] Ψ † T (cid:0) r , z, Q (cid:1) Ψ T (cid:0) r , z, Q (cid:1) = α s N c π (cid:8) (cid:15) f K ( (cid:15) f r ) K ( (cid:15) f r ) (cid:2) e iθ z + e − iθ (1 − z ) (cid:3) (4) + m f K ( (cid:15) f r ) K ( (cid:15) f r ) (cid:9) , Ψ † L (cid:0) r , z, Q (cid:1) Ψ L (cid:0) r , z, Q (cid:1) = α s N c π (cid:8) Q z (1 − z ) K ( (cid:15) f r ) K ( (cid:15) f r ) (cid:9) , (5) (cid:15) f = z (1 − z ) Q + m f (6) (cid:12)(cid:12)(cid:12) Ψ ( f ) (cid:0) r, z, Q (cid:1)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Ψ ( f ) T (cid:0) r, z, Q (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Ψ ( f ) L (cid:0) r, z, Q (cid:1)(cid:12)(cid:12)(cid:12) (7)The meson production amplitude N M depends on the mechanism of the Q ¯ Q pair formation. For the case of thesingle-diffractive production, as we demonstrate in the Appendix A, the contribution to the cross-section is given by N (SD) M ( x, z, (cid:126) r , (cid:126) r ) ≈ ˆ d b (cid:20) N + ( x, z, r , b ) (cid:18) N c (cid:19) + N ( x, r , b ) (cid:18) N c − N c + 16 (cid:19)(cid:21) × (8) × (cid:20) N + ( x, z, r , b ) (cid:18) N c (cid:19) + N ( x, r , b ) (cid:18) N c − N c + 16 (cid:19)(cid:21) . where N + ( x, z, r , b ) ≡ N ( x, z r , b ) + 2 N ( x, ¯ z r , b ) − N ( x, r , b ) , (9)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 (cid:46) m − Q .In widely used phenomenological dipole parametrizations [75–78] it is expected that the b - and r -dependence factorizein this limit, N ( x, r , b ) ≈ N ( x, r ) T ( b ) , (10)where the transverse profile T ( b ) is normalized as ´ d b T ( b ) = 1 , and N ( x, r ) is the dipole cross-section integratedover impact parameter. In this approximation we may rewrite (8) as N (SD) M ( x, z, (cid:126) r , (cid:126) r ) ≈ κ (cid:20) N + ( x, z, r ) (cid:18) N c (cid:19) + N ( x, r ) (cid:18) N c − N c + 16 (cid:19)(cid:21) × (11) × (cid:20) N + ( x, z, r ) (cid:18) N c (cid:19) + N ( x, r ) (cid:18) N c − N c + 16 (cid:19)(cid:21) , where N + ( x, z, r ) ≡ ˆ d b N + ( x, z, r , b ) = 2 N ( x, z r ) + 2 N ( x, ¯ z r ) − N ( x, r ) , (12) κ = ˆ d b T ( b ) . (13)As could be seen from the structure of (8), it is a higher twist ( ∼ O (cid:0) r (cid:1) ) contribution compared to the amplitude ofinclusive production, and thus should have stronger suppression at large p T .The p T -integrated cross-section gets contributions only from dipoles with (cid:126) r = (cid:126) r = (cid:126) r in the integrand. Forthis case it is possible to show that the gluon uPDF x g ( x , p T − k T ) is replaced with the integrated gluon PDF x g G ( x g , µ F ) taken at the scale µ F ≈ m Q . In the LHC kinematics at central rapidities this scale significantlyexceeds 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 thiskinematics the corresponding scale µ F should be taken at the saturation momentum Q s . The gluon PDF x G ( x , µ F ) in this approach is closely related to the dipole scattering amplitude N ( x, r ) = ´ d b N ( x, r , b ) as [68, 79] C F π ¯ α S N ( x, r ) = ˆ d k T k T φ ( x, k T ) (cid:32) − e i k T · r (cid:33) ; x G ( x, µ F ) = ˆ µ F d k T k T φ ( x, k T ) , (14)Eq. (14) can be inverted and gives the gluon uPDF in terms of the dipole amplitude, xG ( x, µ F ) = C F µ F π ¯ α S ˆ d r J ( r µ F ) r ∇ r N ( x, r ) . (15)The corresponding unintegrated gluon PDF can be rewritten as [80] x g (cid:0) x, k (cid:1) = ∂∂µ F xG ( x, µ F ) (cid:12)(cid:12)(cid:12)(cid:12) µ F = k , (16)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 productsof various secondary processes, as shown schematically by the colored vertical and inclined ovals in Figure 1. As wasdemonstrated in [44–47], the effect of these factors is significant at high energies and might decrease the observedyields ( 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 particlesproduced 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 [82–85]). Technically,all these approaches perform evaluations in eikonal approximation, and predict that the observables, which includelarge rapidity gaps, are suppressed by a so-called gap survival factor, (cid:10) S (cid:11) = ´ d b |M ( b, s, ... ) | exp (cid:16) − ˆΩ( b, s ) (cid:17) ´ d b |M ( b, s, ... ) | , (17)where M ( b, s, ... ) is the amplitude of the hard process, b is the impact parameter, and Ω is the opacity or opticaldensity. In a single-channel eikonal model the opacity Ω is directly related to the cross-sections of total, elastic andinelastic processes [83]. It is expected that the energy dependence of the function Ω is controlled by the Pomeronintercept, Ω ∼ s α IP − , 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 inmultichannel extensions of these models, which assume that after interaction with a soft Pomeron the proton mightconvert into additional N D − diffractive states. In this basis, the soft pomeron interaction amplitude ˆΩ should beconsidered as an N D × N D matrix. As was discussed in [82–84], 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 completenessin Appendix C. For the single diffractive scattering the exponent in the expression (17) should be understood as amatrix element between | pp (cid:105) and | p X (cid:105) states [87, 88]. If Φ and Φ are eigenvalues of Ω ik with eigenvalues Ω and Ω , then the matrix exp (cid:16) − ˆΩ( b, s ) (cid:17) reduces in this basis to a linear combination of factors ∼ e − Ω a ( s,b ) , in which thecoefficients can be fixed by projecting the proton and diffractive states onto the eigenstates Φ , Φ of the scatteringmatrix. For the single diffractive production the algorithm for evaluation of the survival factor was introduced earlierfor the pp → pX process in [88], yielding exp (cid:16) − ˆΩ( b, s ) (cid:17) → S ( s pp , b ) ≡ λ ) (cid:16) (1 + λ ) e − (1+ λ ) Ω + (1 − λ ) e − (1 − λ ) Ω + 2 (cid:0) − λ (cid:1) e − ( − λ ) Ω (cid:17) , (18)where parameter Ω is related to eigenvalues Ω , of the matrix Ω ik as Ω = Ω + Ω , (19)and the parameter λ stands for the ratio of the production amplitude of diffractive state X to the amplitude ofelastic proton scattering of the incident proton on a pomeron (see Appendix C for more details). In this paper weare interested only in events without charged particles, produced at pseudorapidity η < y (rapidity gap betweenthe recoil proton and heavy quarks), whereas the evaluation of the survival factor in (17,1821) was performed underthe assumption that there are no co-produced particles in the whole rapidity range η ∈ ( − y max , y max ) , which ismuch stricter than needed in this problem. For this reason we need to correct the estimate (18), using probabilisticconsiderations. In what follows we’ll use notations P A and P B for the probabilities to emit at least one charged particlein the intervals η < y and η > y due to soft interaction of the colliding protons; while ¯ P A ≡ − P A and ¯ P B ≡ − P B are the probabilities not to emit any particles in these intervals (the gap survival factors on these intervals). We willalso use the notation ¯ P A ∪ B for the probability not to produce particles in any of the intervals. The relation betweenthe probabilities ¯ P A ∪ B and ¯ P A , ¯ P B depends crucially on possible correlations between particles from different rapidityintervals. Such correlations have been studied in the literature [89–91], and it is known that they are small whenthe separation between the bins is larger than 1-2 units in rapidity. If we neglect completely such correlations, theprobabilities are related as ¯ P A ∪ B = ¯ P A ¯ P B , which implies that the survival factor should scale with the length of therapidity bin as S (∆ η ) ∼ const ∆ η . For the single diffractive production of heavy mesons we require that no particlesare produced with η < y , although we do not impose any conditions for η > y (so we do not need to introduce thegap survival factor in this region). This implies that the overall survival factor (18) should be adjusted as S → S ( s pp , b ) = (cid:0) S (cid:1) ∆ y y max (cid:38) S , (20)where ∆ y is the width of the rapidity gap interval, and y max = − ln (cid:0) m Q,T /s (cid:1) is the largest possible rapidity ofheavy quarks. This factor S ( 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 (cid:46) m − Q , and we may use a factorized approximation (10).The convolution of S ( s pp , b ) with impact parameter dependent cross-section can be simplified in this limit and yieldsfor the suppression factor a much simpler expression (cid:10) S (cid:11) ≈ ´ d b T ( b ) S ( s pp , b ) ´ d b T ( b ) , (21)which depends only on the energy (Mandelstam variable) s pp of the collision, but does not depend on masses norkinematics 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] N ( x, r , b ) = N (cid:16) r Q s ( x )2 (cid:17) γ eff ( r ) , r ≤ Q s ( x ) − exp ( −A ln ( B r Q s )) , r > Q s ( x ) , (22) A = − N γ s (1 − N ) ln (1 − N ) , B = 12 (1 − N ) − − N N γs , (23) Q s ( x, b ) = (cid:16) x x (cid:17) λ/ T G ( b ) , γ eff ( r ) = γ s + 1 κλY ln (cid:18) r Q s ( x ) (cid:19) , (24) γ s = 0 . , λ = 0 . , x = 1 . × − , T G ( b ) = exp (cid:18) − b γ s B CGC (cid:19) . (25)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 productionconstitutes approximately one per cent of the inclusive cross-section. In the large- p T region the contribution from thesingle 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 resultsare only available for the ratio of the integrated cross-sections of diffractive and inclusive processes, R (diff . )¯ bb ( s ) ≡ σ diff B + ( s ) σ incl B + ( s ) . (26) Single diffr. vs inclusive, pp → D + | y |< s = - prompt2 - pomeron inclusive3 - pomeron inclusive - - - p T [ GeV ] d σ / d p T [ μ b / G e V ] pp → p + D + X | y |< s = s =
13 TeV s =
100 TeV - - - - p T [ GeV ] d σ / d p T [ μ b / G e V ] Figure 3: The cross-section dσ/dp T of the single diffractive production of D + -mesons. Integration over the rapidity bin | y | < . 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 yieldsfrom 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 overviewin Appendix A 2 and more detailed discussion in [40]). The experimental data are for inclusive production from [92]. Rightplot: √ s -dependence of the data in the kinematics of LHC and the planned Future Cicular Collider (FCC) [93]. For other D -mesons the p T -dependence has a very similar shape, yet differs numerically by a factor of two. | y |< | y |< Single diffr. vs inclusive, p p → B ± X s = y = - pomeron, incl.3 - pomeron, incl. - - - - - p T [ GeV ] d σ / d p T d y [ μ b / G e V ] p p → p + B ± X y = s = s = s =
13 TeV s =
100 TeV - - - - - p T [ GeV ] d σ / d p T d y [ μ b / G e V ] Figure 4: Cross-section for the single diffractive B ± -mesons production. Left plot: Comparison of single diffractive predictionswith inclusive cross-sections (experimental and theoretical results). 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 ashort discussion in Appendix A 2).The experimental data are for inclusive production from CMS [94](“ | y | < . “ data points)and ATLAS [95](“ | y | < . ” data points). For some experimentally measured results bin-integrated cross-sections dσ/dp T wasconverted into dσ/dp T dy dividing by the width of the rapidity bin (this is justified since in LHC kinematics at central rapidities y ≈ the cross-section is flat). Right plot: The p T -dependence of the cross-section dσ/dy dp T for several energies √ s . For energy √ s = 1 . it was found that R (diff . )¯ bb ( √ s = 1 . . ± . ± .
16) % . (27)In the Table I we present our theoretical expectations for this value. For Tevatron kinematics the model prediction R (diff . )¯ bb ≈ . agrees with (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 dueto the fact that the production of heavy quark in single diffraction events is formally a higher twist effect, and thushas an additional suppression by the factor ∼ (Λ QCD /m Q ) . While the absolute cross-sections of single diffractiveand inclusive production increase as a function of energy, the ratio (27) slowly decreases due to energy dependence of ◆ ◆ ◆ ◆ ◆ ◆ Single diffr. vs inclusive, pp → B → J / ψ y = s = - pomeron inclusive3 - pomeron inclusive - - - - - p T [ GeV ] d σ / ⅆ p T ⅆ y [ μ b / G e V ] Single diffractive, pp → B → J / ψ y = s = s = s =
13 TeV s =
100 TeV - - - - - p T [ GeV ] d σ / ⅆ p T ⅆ y [ μ b / G e V ] Figure 5: Cross-section for the single diffractive non-prompt
J/ψ -mesons production. Left plot: Comparison of single diffractivepredictions with inclusive cross-sections (experimental and theoretical results). The theoretical curves marked “2-pomeroninclusive” 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]. Rightplot: The p T -dependence of the cross-section dσ/dy dp T for several energies √ s . √ s R (diff)¯ cc R (diff)¯ bb R (diff) J/ψ c - and b -quarks ( R (diff)¯ cc and R (diff . )¯ bb respectively). Thelast column R (diff) J/ψ is for the non-prompt
J/ψ production. the gap survival factor in single-diffractive cross-section.We extended the definition (26) and analyzed the ratio of differential cross-sections, R (diff . ) M ( s, y, p T ) ≡ dσ diff M /dy dp T dσ incl M /dy dp T , M = D ± , B ± , ..., (28)which presents a novel observable. In Figure 6 we show this ratio as a function of p T for D -mesons, both for promptand non-prompt mechanisms. For the sake of definiteness we considered D + mesons, although the results for theratio (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 quitefast at large p T . This behavior agrees with our earlier observation that the single-diffractive mechanism is formallya higher twist effect compared to the dominant two-gluon fusion mechanism, in the case of inclusive production. Asexpected, at small p T the ratios are similar for B -mesons and non-prompt J/ψ ; for larger p T the results differ due todifferences 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 productionfrom [5, 6] (color octet contributions + gluon fragmentation, dominant at large p T ) and from [4] (color evapora-tion model). As we can expect, the non-prompt mechanism is smaller than the prompt contribution, although thequalitative behavior is similar in both cases.In Figure 9 we compare our predictions with earlier results from [11] obtained in the framework of Ingelman-Schleinmodel. We can see that in the region p T (cid:46) 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 pre-dicted in [6] that for the prompt contributions R (diff , prompt) J/ψ ≈ . ± . , although later the CDF collaboration [14]found a value twice larger R (diff , CDF)
J/ψ ≈ . ± . (29) y = prompt s = s = s =
13 TeV - - - p T [ GeV ] R D + ( d i ff ) ( y = , p T ) y = non - prompt s = s = s =
13 TeV - - - p T [ GeV ] R D + ( d i ff ) ( y = , p T ) Figure 6: The ratio of single diffractive to inclusive production cross-sections, as defined in (28). The left plot correspondsto the prompt production (from c → D fragmentation), and the right plot is for the non-prompt mechanism (from b → D fragmentation). For the sake of definiteness we considered D + mesons; for other D -mesons the results have a very similarshape. s = s = s =
13 TeV - - - p T [ GeV ] R B + ( d i ff ) ( y = , p T ) s = s = s =
13 TeV - - - p T [ GeV ] R J / ψ ( d i ff ) ( y = , p T ) Figure 7: 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 ≈ . , in reasonable agreementwith the experimental value (29). IV. MULTIPLICITY DEPENDENCE
According to the Local Parton Hadron Duality (LPHD) hypothesis [97–99], the multiplicity of produced hadronsin a given event is directly related to the number of partons produced in a collision. For this reason the study ofmultiplicity dependence of different processes presents an interesting extension, which allows to understand better theonset of the saturation regime in high energy collisions. A feasibility to measure such processes was demonstrated forinclusive channels by the STAR [32, 100] and ALICE [31, 101] collaborations. The extension of these experimentalmeasurements to single diffractive production is quite straightforward, since their detectors have the capability todetect simultaneously both the rapidity gaps and the charged particles outside of the rapidity window. Since the cross-section of single diffractive production is significantly smaller than that of inclusive production, and the probabilityof events with large multiplicity is exponentially suppressed [101], each measurement will require larger integratedluminosity.In order to get rid of a common exponential suppression at large multiplicities, for a comparison of the multiplicity0 pp → p + J / ψ + X s = - - - - - p T [ GeV ] d σ / ⅆ p T [ n b / G e V ] pp → p + J / ψ + X promptnon - prompt
100 500 1000 5000 10 s [ GeV ] σ t o t [ n b ] Figure 8: Left plot: p T -dependence of differential cross-sections of prompt and non-prompt mechanisms for single diffractiveproduction of J/ψ mesons. The results for the prompt mechanism are taken from [5, 6], and the width of the green bandreflects the uncertainty due to one of the model parameters (gluon fraction of pomeron f g ). The results for the non-promptmechanism (blue solid curve) are results of this paper. Right plot: Energy dependence of total cross-sections of prompt andnon-prompt single diffractive production mechanisms of J/ψ -mesons. The prompt contribution (green dashed line) is takenfrom [4]. pp → p + D + + X s =
14 TeVDipole approachIngelman - Schlein model - - - p T [ GeV ] d σ / ⅆ p T [ μ b / G e V ] pp → p + B + + X s =
14 TeVDipole approachIngelman - Schlein model - - - p T [ GeV ] d σ / ⅆ p T [ μ b / G e V ] Figure 9: Comparison of color dipole approach predictions (this paper) with results of [11] obtained in the framework ofIngleman-Schlein model [49]. The left plot corresponds to single-diffractive charm production, the right plot is for bottomquarks. dependence in different channels it is widely accepted accepted to use a self-normalized ratio [102] dN M /dy (cid:104) dN M /dy (cid:105) = w ( N M ) (cid:104) w ( N M ) (cid:105) (cid:104) w ( N ch ) (cid:105) w ( N ch ) = dσ M ( y, η, √ s, n ) /dydσ M ( y, η, √ s, (cid:104) n (cid:105) = 1) /dy (cid:30) dσ ch (cid:0) η, √ s, Q , n (cid:1) /dηdσ ch ( η, √ s, Q , (cid:104) n (cid:105) = 1) /dη (30)where (cid:104) N ch (cid:105) = ∆ η dN ch /dη is the average number of particles detected in a given pseudorapidity window ( η − ∆ η/ , η +∆ η/ , n = N ch / (cid:104) N ch (cid:105) is the relative enhancement of the number of charged particles in the same pseudorapiditywindow, w ( N M ) / (cid:104) w ( N M ) (cid:105) and w ( N ch ) / (cid:104) w ( N ch ) (cid:105) 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 cross-sections for heavy meson M with rapidity y and N ch = n (cid:104) N ch (cid:105) charged particles in the pseudorapidity window ( η − ∆ η/ , η + ∆ η/ , whereas dσ ch ( y, √ s, n ) is the production cross-sections for N ch = n (cid:104) N ch (cid:105) charged particles inthe 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 beendeveloped in [26, 68–74]. In this picture the observation of enhanced multiplicity signals that a larger than averagenumber of partons is produced in a given event. Nevertheless, we still expect that each pomeron should satisfy the1nonlinear Balitsky-Kovchegov equation. The bCGC dipole amplitude (22) was constructed as an approximate solutionof the latter, and for this reason it should maintain its form, although the value of the saturation scale Q s might bemodified. As was demonstrated in [68–70], the observed number of charged multiplicity dN ch /dy of soft hadrons in pp collisions is proportional to the saturation scale Q s (modulo logarithmic corrections), for this reason the eventswith large multiplicity might be described in dipole framework by simply rescaling Q s as a function of n [68–74], Q s ( x, b ; n ) = n Q ( x, b ) . (31)It was demonstrated in [26] that the error of the approximation (31) is less than 10% in the region of interest ( n (cid:46) ),and for this reason we will use it for our estimates. While at LHC energies it is expected that the typical values ofsaturation scale Q s ( x, b ) fall into the range 0.5-1 GeV , from (31) we can see that in events with enhanced multiplicitythis 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 newregime, 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 uninte-grated gluon density g ( x, k ⊥ , n ) in (2) is also modified. This change might be found taking into account the relationof gluon density with the dipole amplitude N ( x, r, b ) given by (16). For the sake of simplicity below we’ll focus on themultiplicity dependence of the p T -integrated cross-section, which is easier to measure experimentally. For this casethe 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) reducesto a common factor dN M /dy (cid:104) dN M /dy (cid:105) = ´ d r J ( r µ F ) r ∇ r N ( y, r , n ) ´ d r J ( r µ F ) r ∇ r N ( y, r , , (32)the same for all mesons. In Figure 10 we show the multiplicity dependence of the ratio (32). At very small n , whensaturation effects are small, the size of the dipole is controlled by the mass of heavy quark ∼ /m Q , and thus the dipoleamplitude N ( y, r , n ) might be approximated as N ( y, r , n ) ∼ ( r Q s ( y, n )) γ , where γ ≈ . − . is a numericalparameter. In view of (31) this translates into the multiplicity dependence dN M /dy (cid:104) dN M /dy (cid:105) ∼ n γ , (33)as shown in the same Figure 10 with red dotted line. At larger values of n , due to saturation effects, the curve deviatesfrom the small- n asymptotic behavior. As we can see from the right panel of the same Figure 10, this behavior isdifferent from the dependence seen by ALICE for inclusive the production [31], as well as from our theoretical resultfor inclusive production from [40]. This happens because in single diffractive production the co-produced hadronsstem from only one cut pomeron, whereas in inclusive production, in the setup studied in [31], at least two pomeronscan contribute to the observed multiplicity enhancement. Since each cut pomeron gives a factor ∼ n γ in multiplicitydependence, 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 growsrapidly with atomic number A , and thus is easier to measure experimentally. The AA collisions are not suitable forthis purpose due to formation of hot Quark-Gluon Plasma at later stages [103–109]. For this reason we will focuson pA collisions and in the kinematics when the scattered proton in the final state is separated by large rapidity gapfrom the produced heavy meson and nuclear debris.In CGC framework the nucleus differs from the proton by larger size R A = A / R p and larger values of thesaturation scale Q sA . As was found in [110, 111] from analysis of the experimental data, the dependence of Q sA onatomic number A might be approximated by Q sA ( x ) ≈ Q s ( x ) A / δ δ ≈ . ± . . (34)The value δ < indicates that the saturation scale grows faster than ∼ A / 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 dipole2 p T > y = s = s =
13 TeV s =
100 TeV n γ - dN ch / d η〈 dN ch / d η〉 d N D / d y 〈 d N D / d y 〉 pp , s = p T > y = - dN ch / d η〈 dN ch / d η〉 d N J / ψ non - p r o m p t d y d N J / ψ non - p r o m p t d y Figure 10: Left plot: Multiplicity dependence of open heavy flavor meson production cross-sections with single diffractivemechanism (the same for all mesons, see the text for explanation). The red dotted line corresponds to the asymptotic expressionfor small multiplicities, as explained in the text. Right plot: comparison of multiplicity dependence for inclusive and singlediffractive production for non-prompt
J/ψ mesons. The experimental points are from ALICE [31] for inclusive production, thetheoretical curve for inclusive production is from [40]. amplitude using (15,16). The magnitude of nuclear effects is conventionally expressed in terms of the normalized ratioof the cross-sections on the nucleus and proton, R A ( y ) = dσ pA → p M X /dyA dσ pp → p M X /dy . (35)For the sake of simplicity we’ll focus on the p T -integrated cross-section. In this case the dependence on the gluonPDF factorizes, and thus the ratio (35) reduces to a common prefactor R A ( y ) ≈ g A ( x ( y ) , µ F ) g N ( x ( y ) , µ F ) = 1 A ´ d b ´ d r J ( r µ F ) r ∇ r N A (cid:0) y, r , b /A / (cid:1) ´ d b ´ d r J ( r µ F ) r ∇ r N ( y, r , b ) , (36)where N A ( y, r , b ) is a nuclear dipole amplitude with adjusted saturation scale (34), and the rescaling of the impactparameter 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 decreasesby up to a factor of two for very heavy nuclei. This finding is in agreement with expected suppression of nuclear gluondensities 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), A R A ( y, A ) = dN M /dy (cid:104) dN M /dy (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) n = ( Q sA /Q s ) , which might be checked experimentally. VI. CONCLUSIONS
In this paper we studied single diffractive production of open heavy-flavor mesons. We analyzed in detail theproduction of D - and B -mesons, as well as non-prompt production of J/ψ mesons. While in general diffractiveevents constitute up to 20 per cent of inclusive cross-section [1], we found that for heavy mesons production thesingle diffractive events constitutes only 0.4-2 per cent of all inclusively produced heavy mesons. This happensbecause the leading order contribution to single diffractive production is formally a higher twist effect (compared toleading order inclusive diagrams) and thus includes additional suppression ∼ (Λ QCD /m Q ) . Similarly, the observed3 y = s = s =
13 TeV s =
100 TeV
10 20 50 100 2000.30.50.71. A R A Figure 11: The nuclear suppression factor R A defined in (35) as a function of the atomic number A for the p T -integratedcross-section (the same for all mesons, see the text for explanation). ( a ) ( b ) ( c ) Figure 12: The diagrams which contribute to the heavy meson production cross-section in the leading order perturbativeQCD. 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” isreplaced with reggeized gluon (BK pomeron), which satisfies the Balitsky-Kovchegov equation and corresponds to a fan-likeshower of soft particles. suppression at large transverse momentum p T of the produced heavy meson agrees with expected pattern of highertwist suppression. Nevertheless, we believe that the cross-sections are sufficiently large and thus could be measuredwith reasonable precision at the LHC.We also analyzed the dependence on multiplicity of co-produced hadrons, assuming that these are produced onlyon one side of the heavy meson. We found that the dependence on multiplicity is mild, in contrast to the vigorouslygrowing multiplicity seen by ALICE [31] for inclusive production. Our evaluation is largely parameter-free and reliesonly 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. Acknowldgements
We thank our colleagues at UTFSM university for encouraging discussions. This research was partially supportedby the project Proyecto Basal FB 0821 (Chile) and Fondecyt (Chile) grant 1180232. Also, we thank Yuri Ivanov fortechnical support of the USM HPC cluster, where some evaluations were performed.4 ( a ) ( b ) ( c ) Figure 13: The diagrams which contribute to the single diffractive heavy meson production in the leading order in perturbativeQCD ( O ( α s ) -correction). In diagrams ( a ) and ( c ) all possible attachments of the gluon to the quarks and antiquarks are implied.In QCD the interaction of the color dipole with a pomeron might be understood as a gluon ladder (BFKL pomeron), for thisreason 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 amplitudes1. Single diffractive production
In this Appendix, for the sake of completeness, we explain the main technical steps and assumptions used for thederivation of the single diffractive cross-section (2, 8). The general rules which allow to express the cross-sectionsof hard processes in terms of the color singlet dipole cross-section might be found in [50–58]. In the heavy quarkmass limit the strong coupling α s ( m Q ) is small, which allows to consider the interaction of a heavy ¯ QQ dipole withgluons perturbatively and discuss them similar to the treatment of the k T -factorization approach. At the same timewe 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 describedby a factor ± ig t a γ ( x ⊥ ) , where x ⊥ is the transverse coordinate of the quark, and the function γ ( x ⊥ ) is related to adistribution of gluons in the target. This function is related to a dipole cross-section σ ( x, r ) as ∆ σ ( x, r ) ≡ σ ( x, ∞ ) − σ ( x, r ) = 18 ˆ d b | γ ( x, b − z r ) − γ ( x, b + ¯ z r ) | (A1)where r is the transverse size of the dipole, and z is the light-cone fraction of the dipole momentum carried by thequarks. The equation (A1) might be rewritten in the form ˆ d b γ ( x, b ) γ ( x, b + r ) = 12 σ ( x, r ) + ˆ d b | γ ( x, b ) | − σ ( x, ∞ ) (cid:124) (cid:123)(cid:122) (cid:125) =const . (A2)For very small dipoles, the dipole cross-section is related to the gluon uPDF as [115] σ ( x, (cid:126) r ) = 4 πα s ˆ d k ⊥ k ⊥ F ( x, k ⊥ ) (cid:0) − e ik · r (cid:1) + O (cid:18) Λ QCD m c (cid:19) , (A3)so the functions γ ( x, r ) might be also related to the unintegrated gluon densities. With the help of (A2), for manyhigh energy processes it is possible to express the exclusive amplitude or inclusive cross-section as a linear combinationof the color singlet dipole cross-sections σ ( x, r ) with different arguments. While in the deeply saturated regime we canno longer speak about individual gluons (or pomerons), we expect that the relations between the dipole amplitudesand 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 diagramsshown in the Figure (13). As was explained at the beginning of this appendix, in the heavy quark mass limit theinteractions of ¯ QQ with gluons become perturbative, which implies that the t -channel pomeron might be consideredas 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 A (3) (cid:0) x, (cid:126) r Q , (cid:126) r ¯ Q (cid:1) = (cid:20) N c γ (cid:0) x, (cid:126) r Q , (cid:126) r ¯ Q (cid:1) + (cid:18) N c − N c + 16 (cid:19) γ − (cid:0) x, (cid:126) r Q , (cid:126) r ¯ Q (cid:1)(cid:21) t a ≡ a (cid:0) x, (cid:126) r Q , (cid:126) r ¯ Q (cid:1) t a . ( a ) ( b ) Figure 14: (color online) The three-pomeron contributions (diagram ( a )) contribute at the same order in α s as the interferenceof LO and NNLO diagrams (diagram ( b )). In both plots the vertical dashed line is a unitary cut, lower blob is a target (proton),and all possible connections of pomerons (thick wavy lines) to the heavy Q, ¯ Q quark lines are implied. Note that in diagram(a) both pomerons are cut, whereas in case of the interference contribution one of the pomerons is uncut. where γ + ( x, (cid:126) r , (cid:126) r ) = γ ( x, (cid:126) r ) + γ ( x, (cid:126) r ) − γ (cid:18) x, (cid:126) r + (cid:126) r (cid:19) ,γ − ( x, (cid:126) r , (cid:126) r ) = γ ( x, (cid:126) r ) − γ ( x, (cid:126) r ) ,a is the color index of the incident (projectile) gluon, and (cid:126) r Q , r ¯ Q are the coordinates of the quarks. For evaluation ofthe p T -dependent cross-section we need to project the coordinate space quark distribution onto the state with definitetransverse momentum p T , so we have for the evaluate the additional convolution ∼ ´ d r d r e ip T · ( r − r ) , where (cid:126) r , are the coordinates of the quark in the amplitude and its conjugate, viz: (cid:12)(cid:12)(cid:12) A (3) ( p T ) (cid:12)(cid:12)(cid:12) = (cid:0) η (cid:1) ˆ d x ¯ Q ˆ d x Q ˆ d y Q e i p T · ( x Q − y Q ) (cid:16) A (3) ( (cid:126) x i ) (cid:17) ∗ A (3) ( (cid:126) y i ) (cid:12)(cid:12)(cid:12) (cid:126) x ¯ Q = (cid:126) y ¯ Q (A4) = (cid:18) η (cid:19) ˆ d x ¯ Q ˆ d x Q ˆ d y Q e i p T · ( x Q − y Q ) a ∗ (cid:0) x, (cid:126) x Q , (cid:126) x ¯ Q (cid:1) a (cid:0) x, (cid:126) y Q , (cid:126) x ¯ Q (cid:1) . As discussed earlier, at high energies we may apply iteratively the relation (A1) and express the three-pomeron dipoleamplitude in terms of the color singlet dipole cross-sections, as given in (8). In the frame where the momentum of theprimordial gluon is not zero, we should take into account an additional convolution with the momentum distributionof the incident (“primordial”) gluons, as shown in (2), and was demonstrated in [27].
2. 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 brieflythe main expressions used for evaluation of the cross-sections for the latter case. A detailed discussion of inclusiveproduction, as well as comparison with experimental data might be found in [40]. The evaluation of the cross-sectionfollows the steps outlined in the previous Appendix A 1. The leading order contribution in the inclusive case is dueto a standard fusion of two gluons (pomerons). In the evaluation of the three-pomeron we should take into accountthat there are two complementary mechanisms, shown schematically in Figure 14. In what follows we’ll refer to thecontribution 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 adifferent multiplicity dependence. As we discussed in [40], both twist-three corrections give sizeable contributions atsmall p T (cid:46) GeV. For D -mesons the two corrections together contribute up to 40-50 per cent of the leading orderresult, whereas for B -mesons these contributions are of order 10% even for p T ∼ , in agreement with the heavy masslimit.6Both the leading order cross-section and the higher twist correction might be written as dσ pp → ¯ Q i Q i + X ( y, √ s ) dy d p T = ˆ d k T x g ( x , p T − k T ) ˆ dz ˆ dz (cid:48) (A5) × ˆ d r π ˆ d r π e i ( r − r ) · k T Ψ † ¯ QQ ( r , z, p T ) Ψ † ¯ QQ ( r , z, p T ) × N M ( x ( y ); (cid:126)r , (cid:126)r ) + ( x ↔ x ) , (see the Section II for notations and definitions). For the leading order contribution, the amplitude N M is givenby [27, 40] N (2) M ( x, (cid:126)r , (cid:126)r ) = (A6) = − N ( x, (cid:126)r − (cid:126)r ) −
116 [ N ( x, (cid:126)r ) + N ( x, (cid:126)r )] − N ( x, ¯ z ( (cid:126)r − (cid:126)r ))+ 916 [ N ( x, ¯ z(cid:126)r − (cid:126)r ) + N ( x, ¯ z(cid:126)r − (cid:126)r ) + N ( x, ¯ z(cid:126)r ) + N ( x, ¯ z(cid:126)r )] . Similarly, the three-pomeron contribution shown in the diagram ( a ) of the Figure 14 may be rewritten as N (3) M ( x, z, (cid:126) r , (cid:126) r ) ≈ σ eff (cid:34) N ( x, z, (cid:126) r , (cid:126) r ) (cid:18) N c (cid:19) + N − ( x, (cid:126) r , (cid:126) r ) (cid:32) (cid:0) N c − N c + 720 (cid:1) N c (cid:33) (A7) + (cid:0) N c − (cid:1) N + ( x, z, (cid:126) r , (cid:126) r ) N − ( x, (cid:126) r , (cid:126) r ) (cid:35) where N − ( x, (cid:126) r , (cid:126) r ) ≡ −
12 [ N ( x, (cid:126) r − (cid:126) r ) − N ( x, (cid:126) r ) − N ( x, (cid:126) r )] (A8) N + ( x, z, (cid:126) r , (cid:126) r ) ≡ −
12 [ N ( x, (cid:126) r − (cid:126) r ) + N ( x, (cid:126) r ) + N ( x, (cid:126) r )] + N ( x, ¯ z(cid:126) r − (cid:126) r ) + N ( x, ¯ z(cid:126) r ) (A9) + N ( x, − ¯ z(cid:126) r + (cid:126) r ) + N ( x, − ¯ z(cid:126) r ) − N ( x, ¯ z ( (cid:126) r − (cid:126) r )) and σ eff ≈
20 mb is a numerical parameter. Finally, for the interference term shown in the diagram ( b ) of the Figure 14we may get in a similar way N (int) M ( x, z, (cid:126) r , (cid:126) r ) = − σ eff (cid:20) N + ( x, z, (cid:126) r , (cid:126) r ) ˜ N + ( x, z, (cid:126) r ) (cid:18) N c (cid:19) + (A10) − N − ( z, (cid:126) r , (cid:126) r ) ˜ N − ( x, (cid:126) r ) (cid:32) (cid:0) N c − N c + 720 (cid:1) N c (cid:33) ++ (cid:0) N c − (cid:1) (cid:16) N + ( z, (cid:126) r , (cid:126) r ) ˜ N − ( x, (cid:126) r ) + ˜ N + ( x, (cid:126) r ) N − ( z, (cid:126) r , (cid:126) r ) (cid:17)(cid:35) . Appendix B: Fragmentation functions
For the sake of completeness, in this appendix we briefly summarize the fragmentation functions used in ourevaluations. Since the fragmentation functions are essentially nonperturbative and cannot be evaluated from firstprinciples, 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 fragmentationfunction of this process we used the parametrization from [24] D b → B ( z, µ ) = N z α (1 − z ) β , (B1)7 D ˜ b → B ( z ) D ˜ b → J / ψ ( z, P B = ) D ˜ b → J / ψ ( z, P B =
10 GeV ) D ˜ b → J / ψ ( z, P B =
20 GeV ) z D ˜ i → M ( z ) Figure 15: The fragmentation function of B -quarks and non-prompt J/ψ mesons. To facilitate comparison of the shapes,we normalized all the fragmentation functions to unity (so we use the notation ˜ D i → M instead of D i → M ). The normalizationcoefficients for b → B and b → J/ψ cases differ by the branching fraction Br B → J/ψ ≈ . . N c a c γ c N b a b γ b D . × .
54 3 .
58 78 . .
76 1 . D + . × .
16 3 .
39 185 7 .
08 1 . Table II: The values of parameters of D -meson fragmentation function with parametrization (B4), as found in [114]. where N = 56 . , α = 8 . , β = 1 . . The shape of parametrization (B1) is close to another widely used parametriza-tion from [113] D b → B ( z, µ ) = Nz (cid:16) − z − (cid:15) − z (cid:17) , (B2) (cid:15) ≈ . (B3)The production of non-prompt charmonia which stem from decays of the B -mesons might also be described using afragmentation function, which is related to that of B -mesons as [25] D b → J/ψ ( z, µ ) = ˆ z dx D b → B (cid:16) xz , µ (cid:17) × B d Γ dz ( z, P B ) where Γ B ≡ /τ B is the total decay width of the B -meson, and the function d Γ ( z, P B ) /dz was evaluated in detailin [25]. In the Figure 15 we compare the fragmentation functions D b → B and D b → J/ψ . These two functions differby the branching fraction Br B → J/ψ ≈ . , and for this reason in order to facilitate comparison, we plotted thefragmentation functions normalized to unity, ˜ D ( z ) = D ( z ) / ´ dz D ( z ) . As we can see, the distribution D b → J/ψ issignificantly wider than D b → B and has a peak near smaller values of z ≈ . .The D -mesons might be produced either from fragmentation of c -quarks (prompt mechanism) or from b -quarks(non-prompt mechanism). The fragmentation functions for both cases are available from [114], D i → D ( z, µ ) = N i z − ( γ i ) (1 − z ) a exp (cid:0) − γ i /z (cid:1) , i = b, c (B4)with parameters given in the Table II. Though the parameters for D + and D in the table differ significantly, theirfragmentation functions have very similar shapes and differ only by a factor of two in normalization. Appendix C: Parametrization for the matrix Ω ik In this appendix we briefly summarize the parametrization of the soft pomeron scattering amplitude Ω ik used inSection II B. In the two-channel model it is assumed that in addition to proton there is another diffractive state X ,8which might be produced instead of proton in inelastic processes ( e.g . single diffractive, double diffractive). Thematrix Ω ik is thus a × 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 ) = ˆ d q π e iq · b ˜Ω ik (cid:0) t = − q , s (cid:1) (C1) ˜Ω ik ( t, s ) = v i F i ( t ) F k ( t ) (cid:18) ss (cid:19) α IP − , (C2) F i ( t ) = exp (cid:16) b i (cid:16) c d i i − ( c i − t ) d i (cid:17)(cid:17) , (C3) s ≈ , v , = √ σ (1 ± λ ) , (C4) σ ≈
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10 GeV − , b ≈ . − , (C6) c ≈ .
233 GeV , c ≈ .
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