Associated production of $\chi_c$ pairs with a gluon in the collinear-factorization approach

We calculate cross section for production of $\chi_c$ pairs in proton-proton collisions. The cross section for the $g g \to \chi_{c J_1} \chi_{c J_2}$ is considerably smaller (especially for $\chi_{c 1} \chi_{c 1}$ final state) than that obtained recently in the $k_T$-factorization approach. We calculate therefore next-to-leading order contributions with $\chi_c$ pair and one extra associated (mini-)jet. We find these contributions to be much larger than those for the $2 \to 2$ contribution. Especially the emission of a leading gluon (carrying a large momentum fraction of one of the incoming gluons) are important. These emissions in the $k_T$-factorization approach are absorbed into the initial state unintegrated gluon distributions. A smaller contribution to the cross section comes from the production of central gluons emitted with rapidities between the $\chi_c$-mesons. They do lead, however, to an enhancement of the $\chi_c$-pair production at large rapidity distance between the mesons. Our present study explains the size of the cross section for the $\chi_c$ pair production obtained previously in the $k_T$-factorization approach. Several differential distributions are presented.


I. INTRODUCTION
The production of quarkonia in the nonrelativistic pQCD approach has a long history. The production of J/ψ is a good example, see for example the review [1]. Using standard parameters of the J/ψ wave functions the lowest-order cross section in the color-singlet model is much below experimental data. Higher order corrections and/or color-octet contributions must be included to get closer to the data [2][3][4]. Furthermore, a large fraction of the prompt production originates from the radiative decays of P-wave χ c quarkonia. Another efficient option is k T -factorization approach [5] where already the lowest-order approach with unintegrated gluon distributions constructed following the prescription in Ref. [6] gives reasonable results (see e.g. [7][8][9][10][11]). In general, the inclusive cross section for J/ψ (the same is true for other quarkonia) grows with energy.
In recent years also the production of J/ψ pairs became accessible experimentally [12][13][14][15][16]. There is no yet sufficient understanding of the measured cross section. An important problem is the understanding of the contribution from single parton scattering (SPS) and double parton scettaring (DPS) mechanisms. Indeed, the importance of charm for the studies of double parton scattering (DPS) has been stressed in [17,18]. Especially production of two J/ψ mesons at large rapidity difference is not well understood. The production of quarkonia with large rapidity distance is often attributed to double parton scattering mechanism for which the two partonic processes are almost uncorrelated, in contrast to single parton scattering mechanism where the correlation is encoded in relevant matrix elements. In this region of phase space the DPS contribution to the cross section for different hard processes is well represented by the factorized ansatz: σ(DPS, J/ψJ/ψ) = 1 2 σ 2 (SPS, J/ψ) σ eff . (1.1) The so-called effective cross section σ eff determines the normalization of the DPS contribution. A value of σ eff ≈ 15 mb was found from several phenomenological studies, see e.g. [19] or a table in Ref. [16]. In the case of J/ψ pair production the cross section for large rapidity distances requires rather small values of σ eff < 5 mb [12][13][14][15][16]. Is the production of J/ψ pairs different than for other partonic processes? We do not see physical arguments to justify such a claim. In Ref. [20] it was found that double χ c production associated with radiative decays of both χ c quarkonia leads to distributions quite similar to those from double parton scattering. A rather sizeable cross section for χ c pair production was obtained from the k T -factorization approach. Can we get a similar result within collinear-factorization approach? The 2 → 2 gg → χ c χ c processes were already calculated long time ago [21]. We intend to calculate both gg → χ c χ c processes (see Fig.1) as well as 2 → 3 processes (see Fig.2). The recent calculation within k T -factorization suggests that the 2 → 3 contributions may be sizeable.
One would expect that the emission of a gluon in the central rapidity region of the parton-level process (see diagram (C) in Fig.2) will enhance the cross section at large rapidity distances between the χ c mesons. The contributions of leading gluons, which carry a large longitudinal momentum fraction of one of the incoming gluons, (see diagrams (A) and (B) in Fig.2) contain a contribution of minijets produced at a large rapidity distance to the χ c -pair. Such contributions-beyond the obvious collinear emissions-are included in the k T -factorization approach already in the lowest order. There these gluons are ab-sorbed into the initial state unintegrated gluon distribution. The 2 → 3 precoesses were studied previously in the context of quarkonium pair production for pp → J/ψJ/ψg reaction [22] and the corresponding cross section turned out to be similar to the leading pp → J/ψJ/ψ contribution and important in order to understand some correlation observables.
We will illustrate our calculations with several examples of χ c χ c pairs. Several differential distributions will be shown. FIG. 2. The lowest-order mechanisms for the χ cJ 1 χ cJ 2 g production in the high-energy kinematics described in the text.
We are interested in three types of configurations in which a final state gluon is produced: firstly, the central production of a gluon gg → χ cJ g χ cJ (diagram (C) in Fig. 2) and secondly the two configurations with leading gluons (diagrams (A) and (B) in Fig.  2), where a gluon carries the largest fraction of momentum of one of the incoming gluons. The leading gluon minijet production is expected of importance for comparison to the k T -factorization approach. This contribution is dominated by a kinematics, where the gluon is emitted at large rapidity distance to the χ c mesons.
A gauge invariant way to organize the calculation in this situation is the use of vertices from the Lipatov effective action [23,24].
Let us introduce the four momenta of incoming protons, neglecting their masses, with the lightlike basis vectors The incoming gluon momenta are The vertex for the "upper" leading gluon reads [23,24] n −ρ Γ µνρ (q a , while for the "lower" leading gluon we have For the vertex of central gluon production (the "Lipatov-vertex") we introduce the momenta of fusing gluons We also need the g * g * → χ cJ vertices. We write them in the form The explicit form of the tensors T µν are found in Ref. [20]. Above a, b are the color indices of incoming gluons, N c = 3 is the number of colors, and M is the mass of the χ c meson. For J = 1 and J = 2 states, the tensors have the form where ε µ (J z , p), ε µν (J z , p) is the polarization vector/tensor for the meson with momentum p. The derivative of the radial wave function at the origin is related to the γγ-decay width as (2.10) We use the value |R ′ (0)| 2 = 0.042 GeV 2 . We can now construct all the 2 → 3 amplitudes of interest from the above tensors. The amplitude for gg → χ cJ 1 gχ cJ 2 with a central gluon reads: The amplitude for the final state with the leading gluons in the fragmentation region of gluon q a or q b can be written in terms of the (half-) off-shell amplitude for the g * g → χ c1 χ c2 process. The 2 → 2 amplitude is obtained from Here the Mandelstam variables arê (2.14) The amplitude of Eq. (2.13) enters the 2 → 3 amplitudes as follows: and likewise We close this section with a brief comment on the gluon exchanges in the crossed channel. The t-channel gluons explicitly depicted in Fig.2 are taken in the respective high-energy limit -they correspond to the reggeized gluons of the effective action [23,24]. For the gluon exchanges in the blobs of diagrams (A) and (B) of Fig.2 we checked that the approximation of reggeized gluon exchange in the gg → χ c χ c subprocess becomes a good approximation at a rapidity distance between χ c 's of ∆y 3. In the numerical calculations, we use the full gluon propagator in Feynman-gauge. We note that the interference between t-and u-channel amplitudes is negligible and confined to a very narrow interval around ∆y ∼ 0.

B. Parton-level cross sections
Let us now have a look at the parton-level cross section in order to understand better the kinematics and possible singularities in the integration over phase space. The 2 → 3 parton-level cross sections are obtained from where i = A, B, C, and there is no interference between the diagrams of Fig.2. Let us start from the production of a leading gluon along the direction of incoming gluon a, described by amplitude M A . Here, following the rules of the high-energy limit, the fourmomentum q 1 ≡ p g − q a of the exchanged gluon enters the 2 → 2 amplitude in the form We can now use the Ward-identity, to write Then, the 2 → 3 cross section takes the simple form Here one would recognized the factorization in the unintegrated gluon distribution in a gluon and the off-shell cross section for the process g * g → χ c χ c . The off-shell cross section will provide us with a scale µ 2 ∼ M 2 ⊥ , so that for q 2 1⊥ ≪ µ 2 we can neglect the off-shellness of gluon q 1 and only the on shell cross section gg → χ c χ c enters. The parton-level cross section then consists of two parts: Here the first piece contains the infrared divergent integral , which is of course just the collinear logarithm in the g → gg splitting. In a complete NLO calculation of the inclusive χ c χ c the collinear logarithm within some factorization scheme would be absorbed into the evolution of the gluon distribution of one of the protons. The contribution from hard q 2 ⊥1 > µ 2 is a genuine NLO contribution. In our numerical calculations we will simply show the 2 → 3 cross section with a lower cutoff on the transverse momentum of the produced gluon (mini-)jet, p g⊥ > p cut g⊥ ∼ 1 GeV. Let us now come to the contribution from production of a central gluon in the gg → χ c gχ c process. We write the parton-level cross section differential in the gluon rapdity y g and the transverse momenta of χ c mesons p 1,2⊥ . (2.23) The square of the amplitude M C of Eq.(2.11) can be written in the usual impact factor representation Here K r is the real-emission part of the BFKL-kernel [25] K (2.25) Notice that the integral over the gluon rapidity is proportional to Y = log(ŝ/M 2 ), so that the 2 → 3 cross section will be Here we again have an infrared singularity when p g⊥ = − p 1⊥ − p 2⊥ → 0. This is of course just the back-to-back region of the 2 → 2 process. The differential cross section of the Born-level 2 → 2 cross section can be expressed in terms of the same impact factors and reads To the leading order in α S Y, the virtual correction to the 2 → 2 process can be easily calculated using the gluon reggeization property, which amounts to the replacement of the gluon propagator by where (2.29) Expanding the Regge-propagator to the first order, we obtain the 2 → 2 process cross section as Then, the inclusive cross section for the production of χ c -pairs becomes (2.31) (2.32) We cannot absorb the infrared divergencies into the initial state parton distributions in this case. However, for the sufficiently inclusive, say over soft gluon radiation, cross section, the infrared divergencies in the real and virtual part of the BFKL-kernel will cancel. Notice that this mechanism resembles in many respects the Mueller-Navelet dijet production [26], with the χ c playing the role of the jets. However, for this case more involved calculations including a full BFKL-resummation have been performed in recent years [27,28]. As in the case for production of leading gluons, we will in our numerical calculations show the contribution from the χ c χ c g final state with a lower cutoff on the transverse momentum of the gluon p g⊥ = | q 1⊥ + q 2⊥ | > p cut g⊥ ∼ 1 GeV.

C. Hadron-level cross sections
We now come to the hadron-level cross sections. Below s is the proton-proton centerof-mass energy squared. The inclusive production of χ c -pairs from the 2 → 2 process is obtained from with p T = | p 1⊥ |, and The cross section for the 2 → 3 processes is calculated from: 36) where m i⊥ = M 2 + p 2 i⊥ and y 1,2 are the cm-rapidities of mesons. We take as the factorization scale µ 2 =ŝ = x 1 x 2 s. For the case of identical χ c -mesons in the final state all of the cross sections must be multiplied by 1/2.

A. Parton level processes
In this subsection we show two examples of rapidity distributions on the parton level for the process (C) in Fig.2.
In Fig.3 we show distribution in rapidity for the gg → χ c0 gχ c0 . Here the center of mass energy has been fixed at W = 50 GeV. The two χ c0 mesons are produced in forward and backward directions while gluon at midrapidity in the partonic center-of-mass system. For comparison we show also rapidity distributions of χ c0 mesons from the gg → χ c0 χ c0 process (solid line).
Similar distribution for the gg → χ c1 gχ c1 process is shown in Fig.4. The situation is similar as for the χ c0 χ c0 pair production. However, the 2 → 3 contribution here is relatively enhanced compared to the 2 → 2 one (solid lines). In each of the gg → χ c1 vertices in the 2 → 2 process only one gluon is off-mass-shell, whereas in the 2 → 3 process in one of the vertices both gluons are off-mass-shell. The vertex g * g * → χ c1 strongly depends on virtualities of the gluons. We remind that when gluons are onmass-shell the vertex vanishes (Landau-Yang theorem [29]).
To ensure validity of the effective Regge action (applicability of the Lipatov-vertex) one should ensure that the gluon is produced at a distance of at least y veto ∼ 1 from the mesons. We therefore show in the left panels of Figs. 3,4 the result obtained for y veto = 1 and in the right panels the result without a rapidity veto. Interestingly, for the χ c0 case, the gluon is automatically produced centrally, while for the case of χ c1 production the rapidity veto is important to exclude contributions from non-central kinematics.

B. Hadron level cross sections
The integrated cross sections (full phase space) for different components are shown in Table 1 for √ s = 8 TeV. We restrict ourselves to the case of "identical" pairs, i.e. χ c0 χ c0 , χ c1 χ c1 , or χ c2 χ c2 . We see, that the cross sections for the 2 → 2 processes are consistently lower than the ones obtained in the k T -factorization approach in Ref. [20].
0.16 nb pp → χ c1 gχ c1 1.78 · 10 −2 nb pp → χ c0 gχ c0 0.03 nb In Fig.5 we show transverse momentum distribution of one of the χ c mesons for the 2 → 2 and 2 → 3 processes. The 2 → 3 contributions were calculated with p Tg > 1 GeV. The factorization scale is chosen as µ f = √ŝ and the energy in the center of mass of two protons is 8 TeV. We use the MSTW2008nlo [30] parton distribution functions. For illustration in a few plots we present only diagram (B) from Fig.2, since the behaviour of the process described by diagram (A) is exactly opposite. We discuss pairs, where two of χ ′ c s have the same spin, it is good example to show all characteristics of these mesons. Notice, that in high transverse momentum region, the pp → [χ c χ c ]g process dominates for each χ cJ , though for χ c1 it is not big effect.
In Fig.6 we show rapidity distributions of χ c mesons for the different 2 → 2 and 2 → 3 mechanisms discussed above. One can see that in the rapidity range |y| > 3 the pp → [χ c1 χ c1 ]g process is not negligible (the middle plot in Fig.6). While the 2 → 2 sub-processes lead to the production of χ c mesons at midrapidities the 2 → 3 processes generate χ c mesons also at large |y|. Such mesons are then suppressed in the midrapidity experiments as ATLAS or CMS. The same may be true in the case of forward LHCb experiment. When the forward emitted meson is measured the second meson is emitted preferentially at midrapidities (diagrams (A) and (B)) or even in opposite directions (diagram (C)). We leave detailed studies relevant for a given experiment for the future.
In Fig.7 we compare rapidity distributions of χ c mesons and the associated gluon (see diagram (C) in Fig.2). In this case, while the χ c quarkonia are produced preferentially in forward or backward directions, gluons are emitted preferentially at midrapidities. For comparison we show also distributions of χ c quarkonia from the 2 → 2 sub-processes.
In Fig.8 we show similar distributions for diagrams (A) and (B) in Fig.2 and for reference also the distributions from the 2 → 2 sub-processes.
In general, there is rapidity ordering of final state particles for the considered 2 → 3 processes. To see it even better let us present now distributions in rapidity differences between final state objects.
The distribution in rapidity distance between two χ c mesons is shown in Fig.9 for different components discussed in the present paper: χ c0 χ c0 , χ c1 χ c1 and χ c2 χ c2 . Indeed, as expected, the largest distances between the χ c quarkonia are populated by processes with the gluon emitted among both χ c mesons. Then also a sizeable gap at small rapidity distances can be observed.
In Fig.10 we show similar distributions, this time for rapidity distance between one of the χ c mesons and the associated gluon for χ c0 χ c0 , χ c1 χ c1 and χ c2 χ c2 . The considered mechanisms prefer large distances also in this variable.
Let us discuss now some correlation observables. In Fig.11 -13 we show two-dimensional distributions in transverse momenta of both χ c quarkonia for separate ((A) or (B) in Fig. 2) diagrams. Such separation is possible due  Fig.2 for χ c0 χ c0 (left) and χ c1 χ c1 (middle) and χ c2 χ c2 (right). Here µ 2 f =ŝ was used. tum conservation it equals the transverse-momentum distribution of the emitted gluon. A significant difference between diagram (A) and (B) or (C) appears for χ c1 . Emission of the gluon is suppressed in diagram (A) at small p T region. While the distributions for χ c0 χ c0 and χ c2 χ c2 are similar, the distributions for χ c1 χ c1 are clearly less steep. Similar observation was already made in the k T -factorization study in [20]. This is particularly spectacular for the central emission diagram (diagram (C) in Fig.2) when both gluons are off-mass-shell.   Here µ 2 f =ŝ.

IV. CONCLUSIONS
In the present paper we have calculated differential cross sections for χ c pair production in the collinear approach including next-to-leading order corrections (2 → 3 processes). Here we have considered only symmetric pairs (identical χ c mesons). The present results can be compared to previously calculated cross sections in the k Tfactorization approach with the KMR unintegrated gluon distributions. We have found that the leading-order 2 → 2 processes give much smaller cross sections than those in the k T -factorization approach. Therefore we have calculated higher-order corrections including 2 → 3 processes. There are three typical diagrams with emission of leading and central gluons (see Fig.2). The cross section for leading gluon emission is much larger.
When adding the leading and (real emission part of the) next-to-leading order contribution we have obtained results that are similar to the k T -factorization results for the production of χ c0 χ c0 and χ c2 χ c2 but still considerably less than in the k T -factorization approach for the χ c1 χ c1 . The latter disagreement is likely due to even higher-order (NNLO) contributions (involving 2 → 4 processes) contained effectively in the k T -factorisation which may be crucial to include for the χ c1 χ c1 channel as here the vertices vanish for on-shell gluons. In general, the larger numerical value of deviation from the on-shell situation the larger the vertex. We expect that consistent inclusion of the NNLO corrections may be important in this particular case and much less important for other cases. A detailed study will be done elsewhere.
The central gluon emission is interesting in that it enhances production of χ c 's at large rapidity distances. This is similar to the Mueller-Navelet production of large rapidity distance dijets and one may think of a larger enhancement from resummation.
We have calculated several single-particle differential distributions in rapidity and transverse momentum of χ c mesons as well as some correlation observables such as twodimensional distribution in transverse momenta of both χ c quarkonia or in transverse momentum of the quarkonium pair.