Inclusive prompt $\chi_{c,b}(1^{++})$ production at the LHC

We study the prompt production of the $\chi_c(1^+)$ and $\chi_b(1^+)$ mesons at high energies. Unlike $\chi(0^+,2^+)$ production, $\chi (1^+)$ mesons cannot be created at LO via the fusion of two on-mass-shell gluons, that is $gg\to\chi_{c,b}(1^+)$ are not allowed. However, the available experimental data show that the cross sections for $\chi_c(1^+)$ and $\chi_c(2^+)$ are comparable. We therefore investigate four other $\chi(1^+)$ production mechanisms: namely, (i) the standard NLO process $gg\to\chi_{c,b}(1^+)+g$, (ii) via gluon virtuality, (iii) via gluon reggeization and, finally, (iv) the possibility to form $\chi_{c,b}(1^+)$ by the fusion of three gluons, where one extra gluon comes from another parton cascade, as in the Double Parton Scattering processes.


Introduction
It is well known that according to the generalised Landau-Yang selection rule a spin-1 meson cannot be produced in the fusion of two identical massless spin-1 particles [1]. Therefore the inclusive χ c (1 + ) cross section offers the possibility to probe some non-trivial dynamics of the NLO interaction.
There are two ways to overcome the Landau-Yang selection rule: either to account for different virtualities of incoming gluons (that is to violate their identity) or to consider the formation of χ(1 + ) by three gluons. The third gluon may be emitted as the new secondary particle gg → χ ( 1 + ) + g process, or it may occur in the initial state in the fusion process g + (gg) → χ fusion. Recall that, as shown in [2], the role of such three gluon mechanisms of J/ψ-meson production increases with energy 1 and may even be dominant at the very high that the aim of the paper is not to present a precise quantitative prediction 4 , but rather to compare the role of different mechanisms of χ(1 + ) production in high energy hadron-hadron collisions.
We note that χ b production has advantages over χ c both from the theoretical and experimental viewpoints. Due to the larger mass of χ b (i) the perturbative QCD approach is better justified and (ii) it is easier to detect the more energetic decay products.
In Sect. 2 we first consider the cross section caused by the gg → χ(1 + ) + g subprocess, and secondly due to the different virtualities of two incoming gluons. Then in Sect. 3 we discuss the production via three-gluon fusion where a pair of t-channel gluons represents the reggeization of one incoming gluon. Next, in Sect. 4 we study a most interesting possibility when the two gluons come from two different parton cascades. This mechanism for χ(1 + ) production has not been studied before. At asymptotically high energy this should be the dominating contribution. An interesting fact is that in such a case the cross section in the central region is expected to be smaller than that near the proton fragmentation domain. We give our conclusions in Sect. 5.

Lowest order χ(1 + ) production
In this section we study production by the two-gluon initiated states shown by '1 and 2' in Fig. 1. We show separately the contribution where all three gluons couple to the heavy quark loop (Fig.1a), and the contribution where only two t-channel gluons couple to the heavy quark loop (Fig.1b) which may not vanish in for χ(1 + ) if the virtualities of these two gluons are different (Fig.1b). This contribution may be considered separately and formally called 'production' via two gluon fusion". However actually this is just part of the whole gg → χ(1 + ) + g cross section.
Another point is that it is natural to consider for this part the role of t-channel gluon reggeization, which is the simplest example of χ(1 + ) production via 3-gluon fusion g + (gg) → χ(1 + ). For this reason we will describe the contribution of Fig. 1b in more detail in Section 2.2, and the reggeization in Section 3.
Finally, in Fig.1c we show the process initiated by a highly virtual s-channel gluon.

The gg → χ(1 + ) + g process
The simplest possibility to overcome the Landau-Yang selection rule is to create the χ(1 + ) meson together with an additional gluon. The corresponding 'hard' cross section was calculated in [10] dσ(gg → χ(1 Figure 1: Subprocesses for χ ( 1 + ) production: (a) the standard gg → χ(1 + ) + g process where three on-mass-shell gluons couple to the heavy quark loop; (b,c) where one virtual gluon and one on-mass-shell gluon couple to the heavy quark loop -the virtual gluon has q 2 < 0 for (b) and where s, t, u are the Mandelstam variables for the hard subprocess. The quantities P = st + su + ut and Q = stu; and M is the χ c (1 + ) or χ b (1 + ) meson mass. In eq.(1) the nonrelativistic wave function of the meson is assumed. The derivative of the P -wave wave function at origin is denoted as R . In our calculations for χ c (1 + ) we use the values R 2 /M 2 = 0.006 GeV 3 (as in [10]) and the QCD coupling α s = 0.335 which provide reasonable widths of the χ c (0, 1, 2) mesons calculated accounting for the α s corrections (see e.g. [17]). For χ b (1 + ) we take R 2 = 1 GeV 5 similar to that in [7], which is consistent with the potential model results, and α s = 0.22 corresponding to the higher scale appropriate for χ b production.
In comparison with the natural parity χ c (0 + ) and χ c (2 + ) mesons, which can be formed via on-mass-shell gluon-gluon fusion with the cross sectionσ ∝ α 2 s , the cross section (1) contains and extra power of QCD coupling (not accompanied by any large logarithm). Therefore gg → χ(1 + ) + g can be considered as a NLO production process.
Expression (1) has to be convoluted with the incoming parton distributions and integrated over t and s. Since we are looking for the inclusive χ(1 + ) production we fix the rapidity, Y , of χ(1 + ) meson and choose to integrate over the variables y and p t , where p t is the transverse momentum of the final gluon while y is the rapidity separation between this gluon and the χ(1 + ) meson. It is easy to check that the corresponding Jacobian J = 1 (see e.g. [18]). Thus the cross section of χ(1 + ) production in proton-proton collisions reads where g(x, µ F ) is the density of gluons which carry a fraction x of the momentum of the incoming proton (measured at factorization scale µ F ). The variables where m 2 t = M 2 + p 2 t . The hard cross section dσ/dt is given by (1) with The expected cross section is shown in Table 1 where the LO MSTW2008 [19] PDFs were used.

Production via
Here we consider the possibility of χ(1 + ) production via 'virtual+real' gluon fusion processes 5 , which are shown in Fig. 1. These processes are not forbidden thanks to the different mass/virtualities of the initial gluons. The corresponding 'hard' cross section can be extracted from (1) using the 'equivalent photon/gluon' approximation [21]. Indeed, in the limit of s M 2 , |t|, the dominant contribution comes from a diagram where first the incoming gluon with momentum, say, p 1 emits the final gluon p 3 and then the virtual gluon, g * , with momentum q = p 1 − p 3 interacts with another incoming (quasi-real) gluon p 2 to produce the χ(1 + ) meson 6 . Thus in the large s limit we may write the cross section (1) as the product of the virtual (The factor dz/z in dN corresponds to the integration over the rapidity separation y and is omitted here.) In this way we get where the virtuality, q 2 , of off-mass-shell gluon g * plays the role of t = q 2 in (1).
Since the elementary cross section (8) vanishes as q 2 → 0 we cannot consider the incoming gluon, with momentum q, as an on-mass-shell parton. The inclusive cross section should therefore be written in terms of the unintegrated gluon density which is defined in such a way that Thus we obtain 5 The production via g * + g fusion was discussed in detail in [20]. 6 Recall that in terms of the unintegrated gluon density f g (x, q t , µ F ) the value of xg(x, µ F ) is given by the That is, each function xg contains a large logarithm. In this logarithmic integration the virtuality of the initial gluon is small, q 2 ∼ q 2 t µ 2 F . So this gluon may be considered as an on-mass-shell particle.
Note that the dq 2 /q 2 integral does not now have a logarithmic structure since the 'hard' cross section (8) contains a factor t = q 2 . In other words expression (10) should be considered as a NLO contribution (in comparison with LO χ(0 + , 2 + ) meson production) at the same level as the cross section (2). Moreover, it should not be considered as a new contribution -it is just a part of the whole cross section (2).
The value of contribution (10) shown in Table 1 is obtained by calculating the unintegrated gluon density starting from the MSTW2008 [19] PDFs based on the KMR/MRW last-step prescription [22,23]. In this prescription we have used the LO splitting functions but kept the more exact NLO kinematics. As it was shown in [23] this provides an accuracy close to that given by the NLO prescription. Note that here we have used the gluons unintegrated over the virtuality, q 2 and not over the transverse momentum squared, q 2 t (see [23] for details). The contributions (2) and (10) were evaulated using the MSTW2008 LO PDFs for the integrated gluons xg. For the very low q 2 < Q 2 0 we assume the saturation-like behaviour

Production via gluon reggeization
An attractive possibility to organize the fusion of one gluon with the pair of incoming gluons, that is the g + (gg) → χ(1 + ) subprocess, is to consider the contribution coming from gluon reggeization, see Fig. 2. On one hand the gluon trajectory is described by the diagrams where the initial t-channel gluon is replaced by the exchange of two t-channel gluons. On the other hand this contribution is enhanced by large logarithms of the proton-proton energy, or to be more precise -by a 1/ω 0 factor, where ω 0 denotes the shift of the position of the BFKL vacuum singularity α BFKL = 1 + ω 0 . Recall that thanks to the bootstrap condition [24] if we account for the gluon trajectory then we include all the contributions (of the antisymmmetric colour-octet states, which are of interest here) enhanced by the 1/ω 0 factor, that is -by the large leading logarithm of the energy.
Let us explain in more detail how the gluon trajectory is built up. Consider the diagrams where the additional t-channel gluon is added to the usual ladder diagram (inside the same parton cascade). The simplest such diagram (in Feynman gauge) is where the additional gluon is between the χ c and the nearest s-channel gluon of the ladder (that is, between the c-quark loop and the gluon 3 in Fig.1b). The contribution of this diagram is enhanced by two logarithms. One logarithm, lnq 2 , comes from the integration over the k t of the new t-channel gluon. The other logarithm comes from the integration over the longitudinal component of the new gluon's momentum, which corresponds to the integration over the mass of the intermediate s-channel gluon which emits the new t-channel gluon. This longitudinal logarithm is actually equal to the rapidity separation between the χ c meson and the nearest s-channel gluon. This rapidity separation is driven by the intercept (the x-dependence) of the parton cascade. In the BFKL approach this logarithm is equal to 1/ω 0 . Moreover, in terms of the BFKL amplitude, the coherent sum of such diagrams (where the new t-channel gluon couples to the different schannel gluons in the ladder) is described by the gluon trajectory Note that in this approach (just as in two gluon fusion g * + g → χ c (1 + )) the corresponding contribution vanishes when the 'reggeized' gluon virtuality q 2 → 0. Indeed, due to gauge invariance, the whole set of diagrams which describe gluon reggeization can be reduced to a one-gluon loop inserted in the place of the original gluon propagator (see, for example, [25]). Thus the result looks like cross section (10) multiplied by the gluon trajectory α G (q 2 ) and by the 1/ω 0 factor. Moreover, contrary to inclusive J/ψ production ( where the gluon pair should be in a symmetric colour-octet state and the analogous contribution is imaginary), in the case of χ(1 + ) we deal with the antisymmetric colour octet -that is, with a true gluon trajectory of negative signature. Therefore the logarithmically enhanced contribution is real and interferes with the lowest α s order g * + g → χ c (1 + ) amplitude considered in the previous section.
Thus, finally, to lowest order in α s , the cross section (10) should be multiplied by a 'double logarithmic' factor A, where Here the q 2 logarithm comes from the integration over the transverse momentum k t in the loop corresponding to the gluon trajectory. In the region of k 2 t q 2 , the integral takes the form Note that the amplitude for unnatural parity 1 + meson production is antisymmetric with respect to the permutation of the two gluons. Therefore in the amplitude (14) we have the logarithm of the ratio q 2 1 /q 2 2 . This result solves the problem of the infrared cutoff at low k t in the integral (13). In the pure symmetric configuration (with q 2 1 = q 2 2 ) we get zero. This fact in some sense is similar to the Landau-Yang selection rule -a spin 1 + particle cannot decay into two identical transverse gluons 7 .
Recall that in the case of χ(1 + ) production the second t-channel gluon (whose distribution is written here in terms of the 'gluon reggeization' trajectory (11)) is not a virtual loop correction to the main amplitude. Rather, this is a particular new channel to produce a spin 1 + meson. For this reason it should not be combined with the real, s-channel gluon emission. The infrared divergence in (12) is cancelled between the diagrams with the upper (q 1 ) and the lower (q 2 ) gluon insertions; and not between the virtual loop (reggeization) and the real gluon emission. 8 The results of the numerical estimate of the corresponding order of α s contribution 9 are presented in the fourth column of Table 1. Here we have used the value of ω 0 = 1/4 which is close to that expected for the BFKL pomeron after the resummation of the next-to-leading log corrections [26]. Recall that at the lowest α s order, the 'reggeized induced' contribution is proportional to α s /ω 0 . We choose reasonable values of ω 0 and α s to indicate the possible size of the effect.
As expected, the result is negative since at q 2 < 0 the gluon trajectory is shifted to lower values of α G (q 2 ) < 1. However due to the cancellation between the q 1 and q 2 terms the whole contribution is not too large inspite of the 1/ω 0 enhancement.
Recall however that at the present stage we account for the order of α s reggeized correction only. When the "correction" becomes large the higher order α s terms become important replacing effectively the first and negative α s contribution (let denote it as −δ R ) by the positive exponential factor like exp(−δ R ).

Production via two parton cascades
Here we consider the situation when all three (quasi-real) gluons couple to the heavy quark loop directly. The most interesting possibility is to form the incoming (gg) pair taking the two gluons 7 Recall that as we are looking for the leading logarithm the gluons may be considered as quasi-real, transverse particles. 8 Just as in the case of the colourles Higgs boson production, the infrared divergence caused by real emission is cancelled by the true reggeization diagram where (in Feynman gauge) the additional t-channel gluon couples to the s-channel gluons above and below the colourless boson. 9 The contribution of the terms ∝ ln q 2 1 and ∝ ln q 2 2 are calculated separately. In the first case the gluon q 1 is written in terms of the unintegrated distribution, f g , while for gluon q 2 we may use the integrated gluon distribution x 2 g(x 2 , µ F ), and vice versa. from two different parton cascades, see Fig. 3. The probability to find the corresponding pair is given by the product of the gluon densities, x 1 g(x 1 ) × x 2 g(x 2 ), multiplied by the probability that the two cascades overlap in transverse space. Since in the low x region (which is relevant at high energies) the parton density grows as a power of (1/x) -that is xg(x) ∝ x −λ -this contribution will dominate asymptotically at very high energies.
In Double Parton Scattering (DPS) the probability of cascade overlap in transverse space is given by a factor 1/σ DPS eff (see e.g. [27]), where Actually this factor results from the integration (15) over the ('pomeron') loop formed by the two gluon cascades. The integral is driven by the proton form factor F (t), that is by the t q 2 dependence of the proton-pomeron vertex. However, while for DPS the loop contains four form factors (leading to F 4 in (15)) in our case we have only two form factors (we need two cascades on one side of Fig. 3 only). Therefore (assuming exponential dependence) we have We emphasize that the typical value of |t| in the pomeron loop integration simultaneously plays the role of the lower limit for the factorization scale, µ 2 F . In terms of σ DPS eff (and assuming the exponential t-behaviour) this limit is µ 2 F > |t| = 4π/σ DPS eff . Let us denote the momenta of the three gluons as k i (i = 1, 2, 3). In terms of the incoming proton's momenta, p A and p B we may write To calculate the corresponding matrix element we use the gauge invariance condition (M µ k µ = 0) and replace the proton's momenta p A , p B transferred through the spin part (or numerator) of gluon's propagators by k 1t /α and k it /β i respectively. Note that in order to keep the leading (DGLAP) logarithms in k t integrals we retain the lowest power of k it in the matrix element 10 .
Thus the matrix element for ggg → χ(1 + ), corresponding to the heavy quark loop in Fig. 3, takes a rather simple form where f abc is the antisymmetric colour tensor, N c = 3 is the number of colours, and g is the QCD coupling (α s = g 2 /4π). The basic amplitude A s contains a spin part (given by the trace around the quark loop) and a part corresponding to the propagator poles. It is of the form where the trace gives and the poles are where the denominators include the contribution of the longitudinal (αβ 23 s(1 − 2z)) and transverse (−| k 1t − k 2t | 2 ) components of the square of the momentum. Here e χ and e a are the χ(1 + ) and k 1 -gluon polarization vectors. Also we have introduced the relative momentum fraction z = β 2 /(β 2 + β 3 ) where The value of β 23 is fixed by the χ(1 + ) meson mass and its rapidity. Finally, the normalization constant B in (19) is given by Recall that in the diagram for the cross section, Fig. 3, we have four t-channel gluons from the proton p B side 11 . These 4 gluons form three loops of integration. The integrals over 10 Together with the denominators (1/k 2 i ) of the gluon propagators this will give in the cross section the logarithm k 2 it d 2 k it /k 4 i . 11 In general, there may be a diagram where the gluon pair (gg) goes in the p B direction in one amplitude, A, but goes in the other, p A , direction in the amplitude A * . Such a contribution either corresponds to the three gluon singularity (in the ω plane) which has a lower intercept (ω 0 ∼ 0 or even negative -this contribution is small): or, dealing with the BFKL pomeron, we have to consider this (gg) pair as gluon reggeization which (in the lowest α s order) was already considered in Sect. 3. the transverse momenta components have already discussed: two loops gives the logarithmic integrals k 2 it d 2 k it /k 4 i corresponding to the gluon PDF given by the parton cascade i, while the third integral (over t) is limited by the proton form factor. The longitudinal momentum component in the central loop is fixed by the condition that the χ(1 + ) meson (and other secondary partons) should be on-mass-shell. Finally, we have the loop integrations over the gluon longitudinal momentum in each amplitude, A and A * . These integrals can be written in terms of z and can be closed on the quark pole 12 ; that is, in our non-relativistic approximation (for the χ(1 + ) wave function) we get an intermediate state with both quarks on-mass-shell. The pole position is So, finally, the matrix element (19) reads To obtain the cross section we have to sum over the χ(1 + ) polarizations, e χ , average over e a and integrate over the gluon transverse momenta. A problem is that according to (27) the value of z depends on these momenta and we cannot use the usual, integrated PDFs; each value of k it corresponds to its own z. Therefore we have to use the unintegrated distribution in the same way as described in Sect. 2.2. Moreover in our calculation we have to keep only the poles with the positive z < 1. Otherwise one of the gluons (z or 1 − z) will have negative energy and so must be considered as an outgoing and not an incoming particle. Such a configuration describes the gg → χ(1 + ) + g subprocess which has already been discussed in Sect. 2.
Thus the 'DPS' component of inclusive χ(1 + ) cross section (which we call σ DPS ) reads where z is given by (27) and where we take the factorization scale µ F = M . Since, within the Leading Logarithm approximation, both k 2 1t , k 2 2t M 2 , we may replace in the first argument of f g (β 23 (1 − z), ...) the value of (1 − z) by 1, that is f g (β 23 (1 − z), ...) becomes f g (β 23 , ...). The origin of cos 2 φ in the cross section comes from the convolution of a 1 and a 2 .
The predicted 'DPS' part of the χ(1 + ) production cross section is presented in the fifth column of Table 1. In the numerical calculation we have used MSTW2008LO partons [19] and σ DPS eff = 10 mb [29].  Table 1: The cross section dσ/dY in µb for producing χ c (1 + ) by the various mechanisms at 7 (13) TeV. Note that the g * g → χ is already included in the gg → χ + g contribution, and has only been considered separately to facilitate the study of the Regge contribution.

Discussion
As seen from Tables 1 and 2, in the central rapidity region (Y = 0) the main contribution to χ(1 + ) production comes from the most trivial gg → χ(1 + ) + g subprocess, even at 13 TeV. This contribution decreases with |Y |, that is, towards the edges of the available rapidity space.
The expected correction due to gluon reggeization vanishes at Y = 0, in accord with the generalized Landau-Yang selection rule, which is valid for the symmetric configuration. Moreover, for Y = 0 it is negative, and at large rapidity Y it becomes quite large in comparison with the original (without reggeization) contribution. The absolute value of the Regge contribution increases with |Y | (up to Y ∼ 5) making the overall rapidity (Y ) distribution a bit narrower. We emphasize, however that the results of Tables 1 and 2 are presented just for illustration purposes, and include the O(α s ) reggeized correction only. When the 'correction' increases, the higher-order α s terms become important replacing effectively the first and negative α s contribution (which we denote as −δ R ) by a positive exponential factor like exp(−δ R ).
The DPS contribution, which originates from the fusion of gluons from two parton cascades, reveals quite a different behaviour. Due to the growth of the low x gluon density this part of cross section increases with the initial energy and/or with |Y |. The DPS mechanism for χ c (1 + ) production is seen to provide about 30% of the cross section for Y = 4 to 5. At very high energies (asymptotically) this contribution starts to dominate.
The values of the χ b (1 + ) cross sections are much smaller, but due to the larger χ b mass the perturbative predictions are better justified. Note also the strong correction caused by gluon reggeization, which arises since the virtuality of the gluon is larger. Y gg → χ + g g * g → χ G-regge DPS 0.  Table 2: The cross section dσ/dY in nb for producing χ b (1 + ) by the various mechanisms at 13 TeV. Note that the g * g → χ is already included in the gg → χ + g contribution, and has only been considered separately to facilitate the study of the Regge contribution.