Exclusive photoproduction of B ± c and bottomonia pairs

In this paper we analyze the photoproduction of heavy quarkonia pairs which include b -quarks, such as B + c B − c -mesons or charmonium-bottomonium pairs. Compared to charmonia pair production, these channels get contributions only from some subsets of diagrams, and thus allow for a better theoretical understanding of diﬀerent production mechanisms. In contrast to the production of hidden-ﬂavor quarkonia, for the production of B c -meson pairs there are no restrictions on internal quantum numbers in the suggested mechanisms. Using the Color Glass Condensate approach, we estimated numerically the production cross-sections in the kinematics of the forthcoming Electron-Proton collider and in the kinematics of ultraperipheral collisions at LHC. We found that the production of J/ψ η c and B + c B − c meson pairs are the most promising channels for studies of quarkonia pair production.

from quasi-real photons with small virtuality Q 2 ≈ 0. At central rapidities it is expected that the produced quarkonia will carry a just small fraction of the colliding hadron momenta, x i 1.In this kinematics the saturation effects should play an important role in the dynamics of partons, and thus should be properly accounted for in the theoretical models of interaction.In what follows we'll use the the color dipole framework, also known as Color Glass Condensate or CGC framework [45][46][47][48][49][50][51][52][53], which naturally incorporates the saturation effects and provides a phenomenologically successful description of both hadron-hadron and lepton-hadron collisions [54][55][56][57][58][59][60][61].
The paper is structured as follows.Below in Section II we present the theoretical results for the exclusive photoproduction of heavy quarkonia pairs in the CGC approach.In Section III we present our numerical estimates for meson pairs which include at least one b or b-quark, and analyze the dependence on quantum numbers of produced quarkonia.Finally, in Section IV we draw conclusions.

II. EXCLUSIVE PHOTOPRODUCTION OF MESON PAIRS
Nowadays, photoproduction processes might be studied both in electron-proton, proton-proton and proton-nuclear collisions in ultraperipheral kinematics.The corresponding cross-sections of these processes are related to photoproduction cross-section as where in (1) we use standard DIS notation in which y is the inelasticity (fraction of electron energy which passes to the photon), and y a , k ⊥ a , with a = 1, 2, are the rapidities and transverse momenta of the produced quarkonia with respect to electron-proton or hadron-hadron collision axis.The expression dn γ ω ≡ E γ , q ⊥ in (2) is the spectral density of the flux of equivalent photons with energy E γ and transverse momentum q ⊥ with respect to the nucleus, which was found explicitly in [62].The momenta p ⊥ a = k ⊥ a − q ⊥ are the transverse parts of the quarkonia momenta with respect to the produced photon.The nuclear form factors strongly suppress the transverse momenta q ⊥ larger than the inverse nuclear radius R −1 A .For this reason the average values of q ⊥ are quite small, 0.2 GeV/A 1/3 2 , and the p ⊥ -dependence of the cross-sections in the left-hand side of (2) almost coincides with the p ⊥ -dependence of the cross-section in the integrand in the right-hand side.The subscript letter T in the right-hand side of (1) reminds us that the dominant contribution comes from quasireal transversely polarized photons.The corresponding cross-section dσ T is related to the amplitude as where A γ T p→M1M2p is the amplitude of the exclusive process, induced by a transversely polarized photon, and φ is the angle between the vectors p ⊥ 1 and p ⊥ 2 in transverse plane.The variable q + is the light-cone momentum of the photon, and p + 1 , p + 2 are the light-cone momenta of the produced quarkonia.As we will show below, it is possible to express them via quarkonia kinematic variables (y a , p ⊥ a ).For further evaluations of the amplitude A γ T p→M1M2p it is necessary to fix the reference frame and write out explicit light-cone momenta decompositions of the participating hadrons.In what follows we will use the notations: q for the photon momentum, P and P for the momentum of the proton before and after the collision, and p 1 , p 2 for the 4-momenta of produced heavy quarkonia.We will also use the notation ∆ for the momentum transfer to the proton, ∆ = P − P , and t for its square, t ≡ ∆ 2 .The light-cone expansion of the above-mentioned momenta in the lab frame is given by q = q + , 0, 0 ⊥ , q + ≈ 2E γ (4) where m N is the mass of the nucleon, and M 1 , M 2 are the masses of produced quarkonia.In the high-energy collider kinematics, when q + , P − {Q, M a , m N , |t|}, there is an approximate relation between the energy (component q + ) of the photon and the light-cone momenta of the produced quarkonia, which in essence reflects the fact that the change of the light-cone plus-component proton momentum, (P ) + − P + , is negligibly small, in agreement with the eikonal picture expectations.The relations (4-9) allow to express the quarkonia kinematic variables (y a , p ⊥ a ) in terms of conventional DIS variables, such as the Bjorken variable x B or invariant energy W = √ s γp .In what follows we will use these variables (y a , p ⊥ a ), since they allow to keep explicit an symmetry w.r.t.permutation of quarkonia, and are directly measurable in experiments.In terms of these variables, the invariant energy W of the γp collision and the invariant mass M 12 of the produced heavy quarkonia pair are given by and respectively.The photoproduction amplitude A γ T p→M1M2p , which appears in (3), is the central quantity of interest for our study.Since the formation time of quarkonia is larger than the typical size of the proton, the amplitude of the process might be factorized and written as a convolution of the quarkonia wave functions with hard amplitudes which describe photoproduction of two quark-antiquark pairs in the gluonic field of the target.In what follows we will refer to the heavy quarks produced in such hard subprocess as "final state" quarks.The studies of exclusive production are usually performed in the kinematics of small momenta p T , so we may expect that possible contributions of the color octet mechanisms [8,9] are small and might be omitted.While there is no direct experimental check of this assumption, similar studies of single quarkonia photoproduction in exclusive processes [63][64][65] provide indirect evidence that this assumption might be quite reliable.
The amplitude of the double quarkonia photoproduction has been evaluated in [66] using the color dipole (Color Glass Condensate) approach [45,[47][48][49][50][51][52][53].That evaluation was performed for the charm sector, focusing on the production of J/ψ η c pairs.In this paper we are going to extend those results for the case in which the final state quarkonia include b-quarks.For the production of mixed states, such as J/ψ η b , Υ η b and B + c B − c meson pairs, only some subsets of diagrams contribute to the total cross-section, thus providing the possibility to understand the relative contribution of different mechanisms.
In the color dipole approach the hard process is considered in the eikonal picture.Taking into account that the interactions of heavy quarks with the gluonic field of the target are suppressed by the strong coupling α s (m Q ), all the leading order diagrams might be grouped into two main classes, shown schematically in Figure 1.In what follows we will call them "type-A" and "type-B" respectively, and take into account that the amplitude of the whole process might be written as an additive sum, where A (A) and A (B) are the contributions of the respective classes.For the production of all-charm or all-bottom quarkonia pairs, both A (A) and A (B) give nonzero contribution.In this case, C-parity conservation indicates that the produced quarkonia must have opposite C-parities.For the production of mixed B + c B − c meson pairs, the amplitude A (B) ≡ 0, so only the type-A diagrams contribute.The C-parity conservation in this case does not impose any constraints on the produced B c -quarkonia internal quantum numbers, although imposes constraints on the angular momentum L of the relative motion, which should take odd values.This constraint is relaxed if the produced B c mesons have different spins, like e.g.
is the (so far undiscovered) vector state.Finally, for the production of charmonium-bottomonium pairs, such as J/ψ η b or Υ η b , we can see that A (A) ≡ 0, so the amplitude get contributions only of the type-B diagrams.Further analysis of the type-B diagrams allows to reach some conclusions about the relative size of mixed charmonium-bottomonium production cross-sections.Analysis of quantum numbers suggests that a vector particle (J P = 1 − ) might be produced only in the upper loop (with 3 gluon attachments to heavy quark line), whereas scalar particles might originate from the quark loop in lower part of the diagram.This observation allows to understand the behavior of the cross-section under permutation of charm and bottom flavors.Since in the heavy quark mass limit each gluon attachment is suppressed and the natural scale for heavy quark is its mass, we may immediately conclude that in channels with charmonium-bottomonium production, the states with vector bottomonia are suppressed significantly stronger than the states with vector charmonia.In the next section we will corroborate this expectation by explicit comparison of numerical predictions for Υ(1S) η c and J/ψ η b production cross-sections.
In the eikonal picture the impact parameter of the parton is conserved during interaction with the target.The interaction of the colored dipole with the target might be described as a linear combination of the color singlet dipole scattering amplitudes, which are known from Deep Inelastic Scattering.For this reason, it becomes possible to rewrite both types of diagrams as a mere convolution of the four quark component of the photon wave function ψ QQQQ with final state quarkonia wave functions and a linear combination of color singlet dipole scattering amplitudes, where in (13, 14) r ij = x i − x j is the relative distance between partons i and j; α ij = α i / (α i + α j ) is the light-cone fraction carried by the quark in the pair (ij), and b ij = (α i x i + α j x j ) / (α i + α j ) is the (transverse) position of the center of mass of (i, j) pair.The notations Ψ M1 , Ψ M2 are used for the wave functions of the final state quarkonia M 1 and M 2 (for a moment we disregard completely their spin indices), and ψ QQ QQ ({α i , x i } ; q) is the 4-quark lightcone wave function of the virtual photon γ * which is given explicitly in Appendix A. The amplitudes N (A) and N (B)  include resummation over all possible connections of t-channel gluons to quark lines and, as was shown in [66], can be rewritten as a linear superposition of the color singlet dipole amplitudes N (x, r ij , b ij ) The variables Y ij in (13,14) stand for the lab-frame rapidity of quark-antiquark pair made of partons i, j.Explicitly it is given by where α i and α j are light-cone fractions of the heavy quarks which form a given quarkonium.

III. NUMERICAL RESULTS
The framework presented in the previous section allows to make unambiguous predictions for the cross-sections.We would like to start the presentation of numerical results with a brief discussion of different uncertainties which Left plot: Sensitivity of the J/ψ ηc production cross-section to the choice of the wave function.We compare results with the LC-Gauss parametrization of the wave function [67,68] and the wave functions evaluated in potential models [69][70][71][72].
In the lower panel of the left figure we show the ratio of the cross-sections from the upper panel to the "LC-Gauss" curve.Right plot: Sensitivity of the J/ψ ηc production cross-section to the choice of to the parametrization of the dipole cross-section.In the lower panel of the figure we show the ratio of the cross-sections in b-CGC and b-Sat models.In both plots, for the sake of definiteness, we considered the case when both quarkonia are produced at central rapidities (y1 = y2 = 0) in the lab frame; for other rapidities and quarkonia pairs the pT -dependence has similar shape.
are present in our evaluations.For the sake of definiteness, we'll consider the all-charm sector and focus on J/ψ + η c production, for which the production cross-section is the largest (and thus is easier to study experimentally).
The largest source of uncertainty in our estimates is due to the wave function of the quarkonia, which might be reformulated as uncertainty of the Long Distance Matrix Elements (LDMEs).A popular choice used in phenomenological estimates is the so-called light-cone Gaussian (LC-Gauss) parametrization [67,68].This parametrization depends on unknown parameters, which must be fixed from phenomenology.While for J/ψ and Υ mesons these parameters are known or might be fixed from existing experimental data, for heavier mesons, especially for B c quarkonia, this procedure cannot be applied due to lack of experimental data, thus making it almost impossible to make predictions for heavier mesons.A more systematic approach is to use the wave functions of the quarkonia evaluated in potential models, and using the well-known Brodsky-Huang-Lepage-Terentyev (BHLT) prescription [73][74][75] to convert the rest frame wave function ψ RF into a light-cone wave function Ψ LC .In the small-r region, which is relevant for estimates, the wave functions of the S-wave heavy quarkonia in different schemes are quite close to each other [76][77][78][79], so the uncertainty due to the choice of the potential model should be minimal for physical observables.In order to illustrate this for heavy quarkonia production, in the left panel of Figure 2 we compare predictions for the cross-sections obtained with the LC-Gauss parametrization and various potential models [69][70][71][72].The uncertainty due to the wave function does not exceed 30 per cent, on par with expectations based on α s (m c )-counting.
Another source of uncertainty in our evaluations is the choice of parametrization of the dipole amplitude.In the right panel of the Figure 2 we compare predictions obtained with impact parameter (b) dependent "bCGC" [64,80] and "bSat" [65] parametrizations of the dipole cross-section.In the region of small p T both parametrizations give very close results.In the region of very large p T , the difference between the two models increases due to different small-r behavior implemented in "b-CGC" and "b-Sat" parametrizations: in the former parametrization the dipole amplitude behaves like ∼ r γ , whereas in the latter the dependence is much more complicated due to built-in DGLAP evolution of gluon densities in dipole cross-section.In what follows we will use the impact parameter (b) dependent "b-CGC" parametrization of the dipole cross-section [64,80], In Figure 3 we illustrate the p T -dependence of the cross-section for different quarkonia states (for the sake of definiteness we considered that both quarkonia are produced with the same absolute value of transverse momenta p T ).The strong mass dependence can be understood in the dipole picture: all gluon interactions with dipoles of small size ∼ 1/m Q are strongly suppressed in the heavy quark mass limit, leading to a strong suppression of the cross-sections.As explained in the previous section, the production of B * + c B − c and charmonium-bottomonium pairs EIC get contributions from different classes of diagrams, which explains the significant differences in the cross-sections.The production of bottomonium-bottomonium pairs has significantly smaller cross-sections and does not present any practical interest.For the B + c B − c meson pairs, the C-parity does not impose constraints on internal quantum numbers, and for this reason the suggested mechanism might lead to production of both scalar and vector mesons.In the right panel of Figure 4 we can see that the scalar and vector B c quarkonia should have similar cross-sections at very large p T , although might differ substantially in the region of small momenta p T .Potentially this channel might present special interest for searches of the (so far undiscovered) vector mesons B * ± c .In Figure 4 we study the dependence of the cross-sections on the azimuthal angle φ between the transverse momenta of J/ψ and η c mesons.For the sake of definiteness, we assumed that transverse momenta p ⊥ J/ψ , p ⊥ η of both quarkonia have equal absolute values.In order to make meaningful comparison of the cross-sections, which differ by orders of magnitude, in the upper row of Figure 4 we plotted the normalized ratio We can see that the ratio has a sharp peak in the back-to-back kinematics (φ = π), which minimizes the momentum transfer to the target |t| = ∆ 2 .In contrast, for the angle φ ≈ 0, which maximizes the variable |t| = ∆ 2 , the ratio has a pronounced dip.The increase of the peak-to-trough ratio with p T is due to the higher values of |t| achievable in φ ≈ 0 kinematics.For p 1 = p 2 the dependence on φ is qualitatively similar, although the maximum and minimum are less pronounced.The dependence on φ has very similar shape for all quarkonia states.Due to smallness of the cross-sections at large p T , it could be challenging to measure the ratio (18).For this reason, we also analyzed the ratio of the p T -integrated cross-sections which should be easier to study experimentally.Its φ-dependence is qualitatively similar to that of (18), although is milder.This happens because the p T -integrated cross-sections get a dominant contribution from the region of relatively small p T , for which the momentum transfer t remains small for all angles φ.We expect that experimental study of the ratios (18,19) could help to understand possible correlations between orientations of the dipole separation vector r and dipole impact parameter b in a color singlet dipole amplitude N (x, r, b).Such dependence is frequently neglected in phenomenological parametrizations, like b-CGC and b-Sat, and for many channels (e.g.DIS, DVCS, DVMP) this simplification is justified, since the corresponding cross-sections are not sensitive to the φ-dependence.However, in different theoretical models it has been demonstarted that such dependence might exist, and its extraction from data becomes possible if the final state includes two hadrons in addition to recoil proton (see [81,82] for more details).While all previous studies of this dependence focused on exclusive dijet production, the exclusive production of heavy quarkonia pairs might be also used for this purpose and presents a lot of interest in view of its very clean final state.In order to illustrate feasibility of such measurement, we analyze the modification of the φ-dependence of the ratios (18,19) due to possible angular dependence of the dipole amplitude.Following [81], temporarily we'll assume that such dependence is given by the dipole amplitude where θ r,b is the angle between vectors r and b, and v 2 is a numerical constant which characterizes the size of angular dependent term.In the left panel of the Figure 5 we illustrate the φ-dependence of the ratio R(φ) for different values of v 2 .Since expected values of v 2 are very small (of order a few percent), the shape of R(φ) experiences only small changes.For this reason, extraction of the constant v 2 from quarkonia pair production requires to use special observables which would enhance sensitivity to v 2 .We suggest to use for this purpose the geometric mean The strong φ-dependence of the cross-sections, which is due to increase of momentum transfer t to the recoil proton, largely cancels in G(φ), and thus extraction of v 2 from this observable might be done wih better precision.Indeed, for small t, the dependence of the cross-sections on t might be approximated with exponent, In the product R(φ)R(π − φ) the exponents with φ-dependence cancel, thus giving possibility to study the "residual" φ-dependence due to O (v 2 )-terms in prefactors.The extension of this proof for the p T -integrated ratios, which appear in (21), is straightforward.As we can see from the right panel of the Figure 5, the observable G(φ) indeed has significantly milder dependence on φ, and thus is much better suited for extraction of v 2 .Finally, in the Figure 6 we show the dependence of the cross-section on the quarkonia rapidities, integrated over the transverse momenta of both heavy mesons.In the left panel we consider the special case when both quarkonia are produced with the same rapidities y 1 = y 2 in the lab frame.The dependence on y i in this setup merely reflects the dependence on the invariant photon-proton energy, as could be seen from (10).In the right panel of the same Figure 6 we show the dependence of the cross-section on the rapidity difference ∆y between the two heavy mesons.For the sake of definiteness we consider that both quarkonia have opposite rapidities in the lab frame, y 1 = −y 2 = ∆y/2.In this setup the variable ∆y might be unambiguously related to the invariant mass of the heavy quarkonia pair using (11).We may observe that in this case the cross-section becomes suppressed as a function of ∆y, which illustrates the fact that the quarkonia are predominantly produced with the same rapidities.Finally, in Figure 7 we show predictions for the pair production in the kinematics of ultraperipheral collisions at LHC.For the sake of definiteness, we consider proton-lead collisions.Qualitatively the behavior of the cross-section is similar to that of ep production.

IV. CONCLUSIONS
In this manuscript we have studied in detail the exclusive photoproduction of heavy quarkonia pairs, which include bottom mesons.We focused on the leading order contribution, which leads to production of charmonia and bottomonia pairs with opposite C-parities.For B + c B − c pairs, the C-parity does not impose any constraints on the internal quantum numbers of quarkonia, so the suggested mechanism might be used as a clean channel for studies of (so far undiscovered) B c mesons with different internal quantum numbers.The analysis of mixed charm-bottom pairs (e.g.B + c B − c or J/ψ η b , Υη c pairs) allows to single out contributions of two main classes of diagrams in the suggested production mechanism.In all cases the quarkonia are produced with relatively small opposite transverse momenta p T , and small separation in rapidity: the kinematic which minimizes the momentum transfer to the recoil proton and the invariant mass of the produced pair.The dependence of the cross-section on azimuthal angle between transverse momenta of produced quarkonia might present special interest, since it allows to test the dependence of the dipole amplitude N (x, r, b) on the relative angle between the dipole separation r and impact parameter b.We estimated numerically the crosssections in the kinematics of ultraperipheral collisions at LHC and the kinematics of the forthcoming Electron-Ion Collider.We found that J/ψ η c and B + c B − c might be studied with reasonable precision in forthcoming experiments.The production cross-sections of other quarkonia pairs, especially from the all-bottom states (like e.g.Υη b ) are numerically significantly smaller due to extra suppression by the heavy mass and a different production mechanism.

ACKNOWLDGEMENTS
We thank our colleagues at UTFSM university for encouraging discussions.This research was partially supported by Proyecto ANID PIA/APOYO AFB180002 (Chile) and Fondecyt (Chile) grant 1220242.The research of S. Andrade

EIC
Figure 4. Upper row: Dependence of the normalized ratio R(φ), defined in (18), on the angle φ (difference between azimuthal angles of both quarkonia).The left plot corresponds to J/ψ ηc pair production, but with different transverse momenta.The right plot corresponds to different quarkonia states, and fixed absolute values of the transverse momentum pT .Lower row: Similar dependence of pT -integrated ratio R(φ), defined in (19), on the angle φ.In the left plot we compare the predictions for J/ψ ηc with different rapidities; in the right plot we compare the predictions for different quarkonia pairs, at fixed central rapidity in the lab frame.The appearance of the sharp peak in back-to-back kinematics is explained in the text.For other rapidities the pT -dependence has similar shape.

EIC
Figure 5. Left plot: The dependence of the pT -integrated ratio R(φ) defined in (19) on the angle φ (azimuthal angle between quarkonia).The curves differ by choice of the parameter v2 defined in (20).Right plot: Similar dependence for the geometric mean G(φ) defined in (21).Both plots correspond to J/ψ ηc pair production at central rapidities (y1 = y2 = 0).For other quarkonia pairs the φ-dependence has similar shape.line connected to a photon, and m 2 for the current masses of the quark-antiquark pair produced from the virtual gluon.The evaluation of the diagrams follows the standard rules of the light-cone perturbation theory [16,83] where the first and the second terms correspond to contributions of non-instantaneous and instantaneous parts of propagators of all virtual particles, and for the sake of brevity we omitted color and helicity indices of heavy quarks (c i and a i respectively).The non-instantaneous contribution is given by the sum where A ({α i , r i }) = − 2e q α s (µ) (t a ) c1c2 ⊗ (t a ) c3c4 The variable b j1...jn corresponds to the center of mass position of the n partons j 1 , ...j n .The variable n i,j1...jn = (x i − b j1...jn ) / |x i − b j1...jn | is a unit vector pointing from quark i towards the center-of-mass of the system of quarks j 1 ...j n .The tree-like structure of the leading order diagrams 1, 2 in Fig. 8 and iterative evaluation of the coordinate of the center of mass of two partons b ij = (α i r i + α j r j ) / (α i + α j ), allows to reconstruct the transverse coordinates

Figure 1 .
Figure 1.Main classes of diagrams which contribute in the leading order over αs (mQ) to exclusive photoproduction of quarkonia pairs (type-A and type-B diagrams).The eikonal interactions are shown schematically as exchanges of t-channel gluons, indicated by the red wavy lines.In both plots it is implied: (a) summation over all possible attachments of t-channel gluons to partons inside blue dashed rectangle in upper part of diagram (b) inclusion of diagrams with inverted direction of heavy quark lines ("charge conjugation").In the right diagram the t-channel gluons must be connected to different quark loops in order to guarantee production of color singlet QQ in final states.The blue dashed rectangle schematically shows part of the diagrams which (in absence of eikonal interactions) would contribute to the QQ QQ-component of the photon wave function ψ (γ) QQ QQ .

y 1 =y 2 =0
Figure2.Left plot: Sensitivity of the J/ψ ηc production cross-section to the choice of the wave function.We compare results with the LC-Gauss parametrization of the wave function[67,68] and the wave functions evaluated in potential models[69][70][71][72].In the lower panel of the left figure we show the ratio of the cross-sections from the upper panel to the "LC-Gauss" curve.Right plot: Sensitivity of the J/ψ ηc production cross-section to the choice of to the parametrization of the dipole cross-section.In the lower panel of the figure we show the ratio of the cross-sections in b-CGC and b-Sat models.In both plots, for the sake of definiteness, we considered the case when both quarkonia are produced at central rapidities (y1 = y2 = 0) in the lab frame; for other rapidities and quarkonia pairs the pT -dependence has similar shape.

Figure 3 .
Figure 3. Left plot: Production cross-sections of different quarkonia pairs with spin-parity 1 − , 0 − .The cross-sections differ significantly, due to the heavy constituents masses and wave functions, as well as the classes of diagrams which might contribute.Right plot: Comparison of the cross-sections for BcBc meson pairs with different spin-parity: 1 − , 0 − vs. 0 − , 0 − .For the sake of definiteness we considered the case when both quarkonia are produced at central rapidities (y1 = y2 = 0) in the lab frame; for other rapidities the pT -dependence has a similar shape.

Figure 8 .
Figure 8.The leading order contribution to the wave function ψ (γ) QQ QQ defined in the text.The momenta ki shown in the right-hand side are Fourier conjugates of the coordinates xi.It is implied that both diagrams should be supplemented by all possible permutations of final state quarks (see the text for more details).
. The wave function might be represented as a sum ψ