Production of $\chi_c$ pairs in $k_T$-factorization

We calculate the production of pairs of $\chi_c(J)$ mesons with all possible combinations of $J=0,1,2$. The leading order production mechanism is the crossed-channel gluon exchange in the gluon-gluon fusion reaction. The building blocks are the vertices $g^* g^* \to \chi_c(J)$ for off shell gluons. We stick to the color-singlet model and calculate the gluon fusion vertices in the limit of heavy quarks with nonrelativistic motion in the bound state. These vertices are used to construct the $g^* g^* \to \chi_c(J_1) \chi_c(J_2)$ amplitudes. We then calculate hadron-level cross sections using the $k_T$-factorization approach. In our numerical predictions, we use the KMR-type unintegrated gluon distributions. Several differential distributions at the $pp$ center of mass energy $\sqrt{s} = 8 \, {\rm TeV}$ are shown. The salient feature of the $t$ and $u$-channel gluon exchange are the broad distributions in rapidity difference $\Delta y$ between $\chi_c$ mesons.


I. INTRODUCTION
Recently, cross sections for the production of J/ψ-pairs were measured at the Tevatron [1] and the LHC [2][3][4][5].There remain a number of puzzles, especially with the CMS and ATLAS data.Here the leading order of O(α 4  S ) (see e.g.[6,7]) is clearly not sufficient.The double parton scattering (DPS) contribution was claimed to be large or even dominant in some corners of the phase space, when the rapidity distance ∆y between two J/ψ mesons is large.However the effective cross sections σ eff found from empirical analyses are about a factor 2.5 smaller than the usually accepted σ eff = 15 mb.It is an open issue at the moment whether this points to a nonuniversality of σ eff or whether there are additional single parton scattering mechanisms which can alleviate the tension.
The production of quarkonium pairs is interesting in a broader context.Here we wish to consider production of pairs of χ c mesons.This process is more difficult to measure experimentally but interesting from the theoretical point of view.A feed down to the double J/ψ channel is interesting in the context of the puzzles mentioned above.
The single-inclusive χ c meson production was a topic of both experimental [8][9][10] and theoretical [11][12][13][14][15] studies.The cross section for single χ c production is rather large.The nonrelativistic perturbative QCD is the standard theoretical approach in this context.In leading order the gluon fusion g * g * → χ c (J), J = 0, 1, 2 is the underlying production mechanism.The k T -factorization approach provides a reasonable description of the experimental data [14,15].
In the present letter we shall include the production of all combinations of χ c meson pair production.The cross section will be calculated in k T -factorization approach using newly derived off-shell matrix elements for the g * g * → χ c (i)χ c (j) process.
A first evaluation of the total cross section will be given.We also show some differential distributions.
A. The pp → χ c (J 1 )χ c (J 2 )X reaction, formalism It was shown in [16,17] that the χ c J/ψ pair production is possible only at O(α 5 s ), while forbidden at O(α 4 s ) due to C parity conservation.In contrast, the production of χ c (J 1 )χ c (J 2 ), see Fig. 1, is possible already at the O(α 4 s ) order.
Of special importance for us is the fact that χ c χ c states are produced by the crossedchannel one-gluon exchange mechanism.This implies that the production amplitudes are flat as a function of g * g * center of mass energy, which implies broad distributions in the rapidity distance ∆y between the produced χ c -mesons.
FIG. 1: A diagrammatic representation of the leading order mechanisms for pp → χ c (J 1 )χ c (J 2 ) → According to our knowledge this contribution was not discussed so far in the literature.There was, however, some calculations for χ c χ b production [18].
In order to calculate the subprocess amplitudes, we first turn to the g * g * → χ c (J) vertices.
FIG. 2: A diagrammatic representation of the g * g * → χ c (λ) vertex being a building block of corresponding g * g * → χ c (J 1 )χ c (J 2 ) and discussed in this section amplitudes.
Here we follow the general rules of NRQCD as explained e.g. in [19][20][21].We restrict ourselves to the color singlet contribution and can write the amplitude for the production of the χ c (J) meson via the fusion of two gluons as: following closely the notation of [11,[22][23][24], where these vertices had been calculated for external reggeized gluons.Below we will need the amplitudes (1.1) for arbitrary offshell momenta of gluons, not only the multiregge kinematics as in [11,[22][23][24].There is however no additional difficulty related with this.As we concentrate on the colorsinglet mechanism, the three-gluon coupling does not enter and we really deal with a QED problem.Consequently the amplitudes (1.1) fulfill the QED-like gauge invariance conditions: q µ 1 V ab µν (J, J z ; q 1 , q 2 ) = 0, q ν 2 V ab µν (J, J z ; q 1 , q 2 ) = 0. (1. 2) The calculation proceeds as follows.
The g * g * → Q Q amplitude is (up to factors) We parametrize In spectroscopic notation, the χ c mesons are 2S+1 L J = 3 P J states, where J = 0, 1, 2. Therefore the spinorial part of the wavefunction is an S = 1 spin triplet state, and the relevant projector can be written as order in k.In fact the Taylor expansion for P-waves starts from the term linear in k: Then, the integration over relative momentum k reduces to the integral Here R ′ (0) is the derivative of the radial wavefunction at the (spatial) origin.
Besides the QED-like gauge invariance condition, these amplitudes also fulfill the Bose-symmetry 1T µν (J, J z ; q 1 , q 2 ) = T νµ (J, J z ; q 2 , q 1 ) . ( A comment on the J = 1 axial vector is in order.Here the Landau-Yang theorem forbids the decay of the χ c (1) into γγ or gg, and likewise its production through fusion of onshell photons or gluons.Indeed, in the limit q 2 1 → 0, q 2 2 → 0, we have which vanishes, when contracted with the polarization vectors of on-shell photons/gluons as required by the Landau-Yang theorem.
2. The g * g * → χ c (J 1 )χ c (J 2 ) amplitudes Now we wish to discuss the elementary g * g * → χ c (J 1 )χ c (J 2 ) amplitudes, which can be obtained from the building blocks discussed above.
In all cases there are two diagrams (t (left) and u (right) in Fig. 3).
We can write the Feynman amplitudes corresponding to these diagrams as where t = (p These amplitudes are infrared finite and gauge invariant. To obtain the k T -factorization amplitude one should contract (1.18) with the polarization vectors of off-shell gluons Because of the QED-like Ward identities of the gluon fusion vertices, these polarization vectors are equivalent to the more common Gribov's polarizations n + µ , n − ν , for incoming gluons in the high-energy kinematics q 1µ = q + 1 n + µ + q 1Tµ , q 2ν = q + 1 n − ν + q 2Tν .In the nonrelativistic QCD approach the cross section for χ c pair production is proportional to |R ′ (0)| 4 .The result is therefore extremely sensitive to the precise value of the wave function derivative at the origin.In our opinion the best estimate of the parameter can be obtained from: From the experimental value of the diphoton decay width [25] one obtains for the χ c P-wave function squared (1.21) In the following the χ c (J 1 )χ c (J 2 ) cross section is calculated within the k T -factorization approach including off-shell matrix elements for the g * g * → χ c (J 1 )χ c (J 2 ) subprocess and modern unintegrated gluon distributions.
The cross section for pp → χ c (J 1 )χ c (J 2 ) is calculated in the k T -factorization approach.
The corresponding differential cross section for the production of χ c (i)χ c (j) states, where i and j run through 0, 1, 2 can be written as: The unintegrated gluon distribution F (x 1 , q 2 1T , µ 2 F ) is related to the collinear one through and the off-shell matrix element is obtained as (1.24) The longitudinal momentum fractions x 1 and x 2 are calculated from χ c 's transverse masses m Ti = m 2 c + p 2 iT and rapidities: We start presentation of our results by showing integrated cross sections.As an example in Table I we show cross section in a broad range of χ c rapidities.We used an unintegrated gluon distribution constructed from the KMR prescription [26] based on the We used an unintegrated gluon distribution constructed from the KMR prescription [26] based on the MSTW2008 collinear NLO gluon distribution [27].In all cases the gauge invariant matrix elements discussed in the present paper were used.
MSTW2008 collinear NLO gluon distribution [27].For the renormalization scales µ 2 r1 , µ 2 r2 of the running coupling and factorization scales µ 2 F1 , µ 2 F2 entering the unintegrated gluon distribution, we choose .26)where these scales refer to the running coupling/gluon distribution coupling to gluon q 1 or q 2 respectively.We refrain from a detailed analysis of dependence on the factorization scale, the distributions shown below simply serve to get an impression of the salient features of the production mechanism.A more detailed analysis, including theoretical errors will be given in a future work [28], where we will address the feeddown into the J/ψJ/ψ channel.
There are six independent cross sections related to the different spin combinations (see Table I).We see that the cross sections for different spin combinations are of the same order of magnitude.
In Fig. 4 we show rapidity distributions for χ c mesons for different pair combinations.
In the left panel we show: In Fig. 5 we show similar distributions in quarkonia transverse momenta.The distributions for χ c (1) quarkonia are less steep than those for the other mesons.This may have important consequences for large transverse momenta, also for J/ψ pair production (CDF, ATLAS, CMS), but goes beyond the scope of the present letter.
The exchange of gluons leads to broad distributions in the difference of rapidities ∆y of the two quarkonia, as shown in Fig. 6.All final states have in common also a rather deep dip at ∆y = 0. Therefore the χ c pair production will be potentially important rather for experimental setups that cover a large range in rapidities.
In calculations based on collinear gluon distributions, the two χ c mesons are produced back-to-back at the lowest order.This is not so in the k T -factorization approach discussed here.In Fig. 7 we show distributions of the transverse momentum of the meson pair, p T,sum .The distribution for the χ c (1)χ c (1) extends to large pair transverse momenta, which is related to the corresponding vertex structure.The χ c mesons radiatively decay into J/ψ mesons.The double feed down leads to a new contribution to the J/ψJ/ψ channel.The direct J/ψJ/ψ contribution is more than order of magnitude larger than the feed-down contribution.However, the χ c χ c contribution has its own specificity.In Fig. 8 we show distribution in rapidity difference for all χ c χ c contributions weighted by branching fractions into J/ψ channel (solid line) compared to the standard direct J/ψJ/ψ contribution (dashed line).At large rapidity difference the feed-down contribution dominates over the contribution of the standard mechanism.Here we assumed that the J/ψ's from the decay will be collinear to their parent χ c 's.How important is the feed-down contribution for different experimental situations will be discussed elsewhere [28].

II. CONCLUSIONS
We have made a first exploratory study of χ c pair production in proton-proton collisions.The g * g * → χ c (J i )χ c (J j ) amplitudes for off-shell gluons and different spin combinations J i , J j = 0, 1, 2 are calculated based on g * g * → χ c (J) verticies calculated within the color-singlet nonrelativistic pQCD approach.In this approach the vertices are proportional to the derivative of the spatial wave function at the origin |R ′ (0)|.The value of this quantity can be obtained from models of the quarkonia states.Here it has been obtained from the χ c (0) → γγ branching fraction which was measured experimentally.
We have performed calculations within the k T -factorization approach for the pp → χ c χ c X process at √ s = 8 TeV using Kimber-Martin-Ryskin [26] type unintegrated gluon distribution based on the MSTW2008 [27] collinear gluons.
We have found that the cross sections for different combinations of χ c quarkonia are of a similar size.The integrated cross sections for different channels are of the order of a few nb.This is of the same order of magnitude as the cross section for J/ψ pair production.This means that a feedown from the double χ c decays χ c → J/ψγ leads to extra nonnegligible contribution which has to be included in the total prompt production of two J/ψ mesons.Due to specific branching fractions the χ c (1)χ c (1), χ c (1)χ c (2) and χ c (2)χ c (2) channels are the dominant ones.The other three contributions can be safly neglected.
The χ c χ c contribution to the J/ψJ/ψ final state is interesting but goes beyond the scope of the present analysis and will be studied in detail in future dedicated analyses.
The salient feature of the t and u-channel gluon exchange mechanism are the broad distributions in rapidity difference ∆y between χ c mesons.This is to be contrasted with the narrow ∆y distribution of J/ψ pairs at leading order.A feed-down from double χ c production to the double J/ψ channel is therefore expected to be important at large ∆y and may mimic the kinematical behaviour of double parton scattering mechanisms.

FIG. 7 :
FIG. 7: Distributions in the transverse momentum of quarkonium pairs for different spin combinations.

FIG. 8 :
FIG.8: Distributions in the rapidity difference between two J/ψ (dashed line) and for the sum over all χ c χ c combinations multiplied by combined branching fractions.

TABLE I :
Cross sections in nb for production of different combinations of χ c (J 1 )χ c (J 2 ) dimeson states for -8 < y 1 , y 2 < 8 at √ s = 8 TeV.The numbers are obtained in the k T -factorization approach.