Real next-to-next-to-leading-order QCD corrections to J/psi and Upsilon hadroproduction in association with a photon

We update the study of the QCD corrections to direct J/psi and Upsilon hadroproduction in association with a photon in the QCD-based approach of the Colour-Singlet (CS) Model. After comparison with the recent full next-to-leading-order (NLO) computation for this process, we provide an independent confirmation to the inclusive case that NLO QCD corrections to quarkonium-production processes whose LO exhibits a non-leading P_T behaviour can be reliably computed at mid and large P_T by considering only the real emission contributions accompanied with a kinematical cut. In turn, we evaluate the leading part of the alpha^4_S alpha contributions, namely those coming from (J/psi,Upsilon)+gamma associated with two light partons. We find that they are dominant at mid and large P_T. This confirms our expectations from the leading P_T scaling of the new topologies appearing at NNLO. We obtain that the yield from the CS becomes one order of magnitude larger than the upper value of the potential Colour-Octet yield. The polarization of the 3S1 quarkonia produced in association with a photon is confirmed to be longitudinal at mid and large P_T.


Introduction
For a long time, the many difficulties to correctly predict quarkonium-production rates at hadron colliders have been attributed to non-perturbative effects associated with channels in which the heavy quark and antiquark are produced in a colour-octet state [1,2,3]. On the basis of leading-order (LO) (α 3 S ) calculations, it had been assumed that coloursinglet production channels give a small contribution at mid and large P T . The confusion most probably came from the fact that quantum-number conservation (J, P, C, and colour) prevents leading P T scaling at LO and, in glue-glue production, at next-to-leading order (NLO) (α 4 S ) 2 . Thus, in contrast with the situation for many other observables, there is still a possibility of unexpectedly large (coloursinglet) contributions at large P T at next-to-next-to-leading order (NNLO) (α 5 S ). In addition, two unexpected features of fragmentation approximations have recently been revealed: heavy-quark fragmentation only dominates over the other topologies at very high P T and gluon fragmentation may not dominate over double t-channel gluon exchanges. As a consequence, it seems that, when leading kinematic contributions are correctly accounted for, colour-singlet production channels will play a more important role -if not the mostthan in past analyses in the explanation of quarkoniumproduction observables and this suggests in turn that colouroctet fragmentation contributions may be less important than had previously been thought.
This became clear thanks to the several recent computations of QCD corrections to quarkonium hadroproduction processes. The NLO corrections to the inclusive yield of J/ψ and Υ were computed [4,5] in the QCD-based approach of the Colour-Singlet (CS) Model 3 [6]. Its polarisation was in turn computed [8] at NLO. These computations were recently complemented [9] by the addition of the real NNLO corrections -thereafter referred to as NNLO -. It was then shown that there may be no need to incorporate Colour-Octet (CO) transition (higher-v corrections of NRQCD) to describe the hadroproduction of Υ at the Tevatron [10,11,12,13]. In the case of the J/ψ and ψ [14,15], the CS contributions are significantly enhanced and brought very near the experimental data of CDF although the large P T direct yield seems not to be fully accounted for in the case of the ψ for instance. As regards the CO channels for J/ψ production, their NLO QCD corrections were recently computed in [16]. It was seen that they minimally affect both the P T dependence and the normalisation of the partonic matrix elements at mid-and large-P T , thus also the value of the 3 S [8] 1 CO Long-Distance Matrix Elements (LDME) fit to the data. Similarly, the polarisation prediction are not modified and remains in disagreement with the latest CDF measurements [17]. For recent reviews, the reader is guided to [1,2,3] along with some perspectives for the LHC [18] and a recent discussion on those aforementioned QCD correction computations [15].
Beside the studies of inclusive production, efforts are Q γ (f) Figure 1: Representative diagrams contributing to the hadroproduction of a J/ψ in association with a photon at orders being made to obtain improved theoretical predictions for complementary observables to the inclusive yield, such as hadroproduction of J/ψ and Υ (thereafter commonly named Q) in association with a photon [19,20,21,22,23,24,25,26]. Recently, the NLO corrections to the hadroproduction of Q + γ in the CS channel were computed by Li and Wang in [27]. As in the inclusive case, NLO corrections are significant in the large P T region since new topologies appear with slower P T falloff (see Fig. 1 (c)), in comparison to LO topologies Fig. 1 (a) and other NLO topologies as the loop corrections Fig. 1 (b).
In [9], it was shown that some α 5 S contributions to the inclusive yield coming from three-jet configurations, i.e. Q + j j j, such as those arising from gluon-fragmentation and "high-energy enhanced" (or double t-channel gluon exchange) channels are important and clearly dominate over the other contributions at mid and large P T . In the case which interests us here, Q + γ, those two channels appear at order α 4 S α -see for instance Fig. 1 (d) and (e,f)-and are expected to dominate over the Q + γ yield at mid and large P T .
In this work, we therefore apply the same procedure as in [9] to evaluate their contributions. First, we will check that the NLO computation of Li and Wang [27] can indeed be reproduced by an evaluation of the real α 3 S α contributions (Q + γ + one light partons) complemented by a cutoff on low invariant masses for any pairs of external light partons in the process. The procedure is explained in the next section. We will also check that the sensitivity of our computation on this cut-off dies away when P T grows and that the theoretical uncertainty attached to its choice is typically smaller than the ones attached to the choices of the renormalisation scale µ r for α S , the factorisation scale µ f for the (collinear) parton distribution functions (PDF) and the heavy-quark mass m Q .
Having performed those checks, we will apply the same procedure for the evaluation of the contribution for Q + γ + two light partons -namely the real NNLO contributions to Q + γ production-arguing that they provide with a first reliable estimate of the complete NNLO contributions to Q + γ production at large enough P T . As regards the polarisation, the quarkonia directly produced in association with a photon via those channels are mainly longitudinally polarised, as in the inclusive case.

Inclusive case
As we argued in [9], the NLO contributions to the inclusive yield can be approximated at large enough P T in a relatively simple and reliable manner by computing the α 4 S contributions consisting in the production of a Q with 2 light partons (denoted j thereafter) on which we apply a cutoff on low invariant masses for any light parton pairs in the process. Computations of such cross sections can be done reliably using the automated generator of matrix elements MadOnia [28].
The underlying idea supporting this was twofold: • First, at large enough P T , topologies with the leading P T behaviour will dominate and those are wholly included in this subset of α 4 S contributions (the production of a Q with 2 light partons); • Second, this subset accounts for a physical process at Born level. Its contribution is therefore finite except for soft and collinear divergences. The purpose of the cut-off is to avoid such divergences by imposing a lower bound on the invariant-mass of any light parton pairs (s i j ). For the new channels (with a leading P T scaling) opening up at α 4 S , the dependence on this cut gets smaller for large P T since no collinear or soft divergences can appear there. For other channels, whose Born contribution is at α 3 S , the cut would produce logarithms of s i j /s min i j . Those can be large. Nevertheless, they can be factorised over their corresponding Born contribution, which scales at most as P −8 T . The sensitivity on s min i j is thus expected to vanish at large P T . This insensitivity to the cut and the good agreement between the NLO (Q + j j with a s i j cut) and the full NLO result is recalled in Fig. 2 for the case of the inclusive J/ψ and Υ(1S ) production. The gray band illustrates the sensitivity to the invariant-mass cut s min i j between any pairs of light partons when it is varied from m 2 c to 4m 2 c and 0. (a) J/ψ + γ + X (b) Υ + γ + X Figure 3: (a) full computation at NLO for J/ψ + γ + X (dashed line) [27] vs. J/ψ + X + 1 light parton with a cut on s i j (gray band) at √ s = 14 TeV; (b) full computation at NLO for Υ(1S ) + γ + X (dashed line) [27] vs. Υ(1S ) + γ + 1 light parton with a cut on s i j (gray band) at √ s = 14 TeV. The absolute value of the rapidity of both the Q and the γ is limited to 3.
0218. The parton distribution set used was CTEQ6 M [29]. The yield becomes insensitive to the value of s min i j as P T increases, and it reproduces very accurately the differential cross section at NLO accuracy, both for the J/ψ and Υ(1S ) case.

Production in association with a photon
In the more exclusive case Q + γ, similar topologies are present with the same P T scaling and we also expect to reproduce accurately the yield at NLO accuracy (α 3 S α) computed in [27] by computing the yield from the production of Q + γ with one light parton with the invariant-mass cut s min i j between any pairs of light partons, also referred to as NLO .
This is indeed the case. For instance, the differential cross section for Q + γ at NLO accuracy from Li and Wang [27] is displayed in Fig. 3 and is very well reproduced by the NLO computed for different values of s min i j . When P T grows, the latter becomes completely insensitive to the value chosen for s min i j . The same parameter values were used for Fig. 3 as for Fig. 2 and α was set to 1/137. This result is a clear and completely independent confirmation of the validity of the reasoning initially given in [9] that NLO QCD corrections to quarkonium-production processes whose LO shows a non-leading P T behaviour can be reliably computed at mid and large P T by considering only the real emission contributions accompanied with a kinematical cut. In turn, this reinforces our confidence that the impact of NNLO contributions can evaluated likewise by 3 computing the NNLO contributions, as done in the following section.
This also gives us confidence that much better Monte Carlo simulations of inclusive production at mid and large P T could be achieved using NLO and NNLO partonic matrix elements, which can be interfaced [30] with eventgenerators such as PYTHIA [31]. In any case, they would give results much more reliable than simulations based on matrix elements for CS channels at LO only.

Cross section
Among the contributions appearing at α 4 S α, we find the topologies of Fig. 1 (d) (gluon fragmentation) and Fig. 1 (e,f) ("high-energy enhanced" or double t-channel gluon exchange), those exhibit new kinematical enhancements appearing in higher-order QCD corrections. In other words, those provide us with new mechanisms to produce a high-P T Q with a γ with a lower kinematic suppression, still via CS transitions. They are therefore expected to dominate the differential cross section at NNLO accuracy in the region of large transverse momentum.
Those are also entirely contained in the contributions to pp → Q + γ + j j, namely the real α 4 S α corrections, and we can follow the procedure validated in the previous section, by "simply" adding one light partons in the final state.
The computation of pp → Q+γ+ j j at tree level is in principle systematic, but technically quite challenging: a dozen parton-level subprocesses contribute, most involving a few hundred Feynman diagrams. As done in [9], we follow the approach described in Ref. [28], which allows the automatic generation of both the subprocesses and the corresponding scattering amplitudes.
The differential cross-sections for J/ψ + γ and Υ(1S ) + γ are shown in Fig. 4. The gray band (referred to as NLO ) corresponds to the sum of the LO and the real α 3 S α contributions. The red (or dark) band (referred to as NNLO ) corresponds to the sum of the LO, the real α 3 S α and the real α 4 S α contributions. The α 4 S α contributions in both case dominate over the yield at large P T . The uncertainty bands are obtained from the combined variations 0.5µ 0 ≤ µ r, f ≤ 2µ 0 with for the J/ψ, m c = 1.5 ± 0.1 GeV and 1 ≤ s min i j /(1.5 GeV) 2 ≤ 2 and, for the Υ(1S ), m b = 4.75 ± 0.25 GeV and 0.5 ≤ s min i j /(4.5 GeV) 2 ≤ 2. At the leading order in the heavy-quark velocity (v), the results for the radially excited states ψ(2S ), Υ(2S ) and Υ(3S ) are readily obtained by changing |R Q (0)| 2 and the branching ratio into dileptons.

Polarisation
As regards the polarisation parameter λ, it is computed by analysing the angular distribution (θ) between the + direction in the quarkonium rest frame and the quarkonium from which we can extract λ bin by bin in P T . Our results for the J/ψ are shown in Fig. 5 along with the curves for the NLO . Our predictions for the polarisation parameter λ for the NLO are in qualitative agreement with those of [27], i.e. the J/ψ's produced in association with a photon are dominantly longitudinal. We did not go further in the comparison since, for both numerical computations, 5% precision in λ could only be reached at a high cost of computing time and since the NNLO is anyhow larger. As regards the latter, it confirms the trends of the NLO (and thus NLO) results. This is not a surprise knowing the NNLO results for the inclusive yield, differing essentially 4 4 at least as far as the dominant channels are concerned. in the replacement of the photon by a gluon. This replacement is indeed not expected to change results concerning the polarisations of the particles produced.

Discussion of the results at NNLO
First, let us stress that although the uncertainty associated with the choice of the cut s min i j is somewhat larger than at NLO , it is nevertheless smaller than the one attached to the mass, the renormalisation scale and the factorisation scale 5 . The latter dependence is expected: on the one hand, we miss the virtual part at low P T where it is sizable and where it is expected to reduce the renormalisation-scale dependence; on the other hand, the contributions dominating at large P T are directly sensitive to the fourth power of α S . Yet, the dependence is smaller than in the inclusive case [9] where five powers of α S are involved.
Second, we find that the subprocess gg → Q+γ+gg dominates, providing with more than two thirds of the whole yield in the J/ψ case. In addition, we have checked that this fraction is slightly increasing with P T and only weakly dependent on the value of the invariant mass cut-off of light partons s min i j , removing the collinear and infrared divergences of gg → Q + γ + gg. Another important observation is that the size of the yield from gg → Q + γ + gg does not vary much when s min i j is changed from 2.25 to 9 GeV 2 in the J/ψ case for instance. This indicates that those divergences are not -after being cut-artificially responsible for a large part of the NNLO yield.
Third, it is likely that the largest part of this contribution is not from gluon fragmentation topologies, but rather from 5 Note that the uncertainty associated the choice of µ f has a negligible impact on the final results compared to the other theoretical uncertainties. double t-channel gluon exchange ones 6 , keeping in mind that such a decomposition in terms of the corresponding Feynman graphs is not gauge invariant. There are a couple of indications supporting this: • Such fragmentation contributions would be expected to provide transverse quarkonia if there were neither corrections due to the off-shellness of the fragmenting gluons nor possible spin-flip contributions when radiated gluon energies (in Q rest frame) are not small compared to the heavy-quark mass. While a depolarisation (in the helicity frame) is possible, it would be still far from the longitudinal polarisation computed here for Q produced via gg → Q + γ + gg.
• Processes such as qq → Q + γ + qq , which proceed uniquely via double t-channel gluon exchange, have the same P T dependence as the process gg → Q + γ + gg and the difference in normalisation is naturally accounted for by colour factors and the smaller value of the quark PDF compared to the gluon one at low x. Another similarity with gg → Q + γ + gg is that the polarisation of the yield from qq → Q + γ + qq is strongly longitudinal, for P T larger than 5 GeV, as observed for the full NNLO yield dominated by gg → Q + γ + gg.
The previous discussion can in fact be extended to the inclusive case studied in [9]. Indeed, the results then obtained at NNLO appeared to be significantly higher than the ones using the fragmentation approximation. Further, the polarisation of the yield from gg → Q + ggg processes happened to become more and more longitudinal at large P T in contradiction with the expected polarisation from a fragmentation channel. This lets us think that, for both processes Q + γ and Q + X at NNLO , the yield may mainly come from double t-gluon channel exchanges appearing for the first time at this order.
A careful kinematical analysis of the yield from gg → Q + γ + gg would certainly be helpful. Yet, it would be highly computer-time demanding, especially to obtain a distribution 7 of the relative momentum between the J/ψ and its closest gluon precise enough to unequivocally attribute the most part of the yield to the double t-channel gluon exchange channels rather than to the fragmentation ones. Indeed, it concerns the most complicated process with a couple of hundreds of diagrams. This work is left for a future analysis and would certainly be expediently done along with the one for the inclusive case. 6 In the inclusive case, those have been previously discussed [32] in the k t factorisation formalism. See [33] for a recent application to Υ hadroproduction. 7 The evolution of the distribution for different P T would be even better. 5

Photon detectability
In order to detect the photon, we evidently have to impose that it possesses a finite transverse momentum for it not to go in the beam pipe and that it is isolated to avoid misidentifications with Bremsstrahlung radiations. Yet, isolation criteria highly depend on the detector potentialities. The determination of optimum values for P γ T,min and for an isolation criterion for the γ improving the signal over background ratio is beyond the scope of this theoretical analysis. We have therefore decided to apply minimal values for those constraints, which would be further increased after a full detector simulation. Along the same lines, it is worth noting here that isolation cut for CO mediated signal (a priori suppressed, as discussed below) can only be imposed in such simulation. Indeed, such a cut requires a simulation of the hadronic activity following the discolouration of the CO pairs as done e.g. in [34].
At LO, the P T requirement is trivially satisfied for a Q with a finite P Q T since P γ T is balancing P Q T . As discussed by Li and Wang [27], this is not automatically the case at NLO and a minimum P γ T cut has to be applied and it affects the yield vs P Q T up to roughly three times the value of this cut. Typically the cut P γ T,min = 3 GeV applied here affects the yield up to P Q T = 7 − 10 GeV. As regards the isolation cut, we excluded any events with partons (q,g) within a cone ∆R = ∆η 2 + ∆φ 2 = 0.1 from the photon [35] 8 . As noted above, those are minimal cuts.
In our case an analysis of different P γ T -cut values is less relevant than in [27] for two reasons: first, our computation cannot be reliably extended to low P T and, second, the main contributions -the double t-channel gluon exchange and gluon-fragmentation channels-create a γ with a similar momentum as that of the Q (see the graphs of Fig. 1 (d) & (e)). Overall, such a kinematical cut insuring its detectability does not affect the NNLO yield for P T larger than 10 GeV.
Beside the problem of the photon detectability, we have to make sure that other processes will not contribute to the yield of Q + γ. In the J/ψ case, we expect the nonprompt background to be properly subtracted (by vertexdisplacement method for instance). However, in this case, the ratio "non-prompt over prompt" at very large P T is expected to be lower than in the inclusive case since the photon emission is required for both processes, whereas in the inclusive case the B feed-down can proceed at a lower order in α S . Similarly, the χ Q feed-down will not be significant (except in the region where the invariant mass of the pair Q + γ is close to the mass of the χ Q ) since suppressed by v 2 and by the branching while having the same α s suppression than for the J/ψ for similar topologies. As regards the feed-down of the radially excited states 3 S 1 , they are readily accounted for by constant multiplicative factors to the cross section of the state fed in : ∼ 1.4 for the ψ(2S ) into J/ψ, ∼ 1.1 for both the Υ(2S ) into Υ(1S ) and Υ(3S ) into Υ(2S ) 9 .

Colour-octet yield
Now, let us discuss the possible colour-octet contributions, ignoring first the modifications, induced by the QCD corrections to CO and CS contributions, of the CO LDME extracted from the Tevatron data. As discussed in [27], a quick comparison with the results obtained in [26] shows that the NLO CS and the LO CO yields are of the same order at the LHC, with roughly the same P T dependence.
Yet, contrary to what is claimed in [27], higher-QCD corrections are expected to impact more on the CO partonic matrix elements in Q + γ than in the inclusive case [16], for which NLO corrections seem unimportant at large P T . Indeed, the difference between the two cases lies in the fact that in Q + γ P −4 T topologies, such as fragmentation ones, are not opened at the order αα 2 S as studied in [26] (only one initiated by quark, and thus sub-dominant at low x, qq → γ + (QQ) [8] , with a quark in the t-channel shows a P −4 T scaling).
In the light of the NNLO results, we know that when the real α 4 S α contributions are taken into account, the CS contributions will be increased by about one order of magnitude compared to the CS at NLO. To be sure that CS contributions are then dominant over CO ones at higher orders, we have to check that fragmentation contributions from CO opening at αα 3 S are small. This is indeed the case since they are mediated by either 1 S [8] 0 or 3 P [8] 0 C = +1 CO states and their corresponding LDMEs are know to be severely constrained when higher-QCD corrections are taken into account for the analysis of the inclusive data at low-and mid-P T at the Tevatron [16,36], at various fixed-target experiments [37] and more recently at the B factories [38,39]. Finally, a further suppression is expected from the increase of the CS yield from NLO and NNLO contributions [9]. Using an upper conservative value of the LDME combination M Jψ k = 0|O J/ψ [ 1 S [8] 0 ]|0 + k , we analysed the two extreme cases ( 0|O J/ψ [ 3 P [8] 0 ]|0 = 0 (case I) and 0|O J/ψ [ 1 S [8] 0 ]|0 = 0 (case II)) to evaluate those CO αα 3 S contributions via pp → j + (cc) 8,C=+1 + γ using the same parameter values as the central NLO curve and s min i j = m 2 c . At P T 10 GeV, the case I gives a contribution about 5 times smaller than the central αα 3 S (NLO) CS values while the case II is further below 10 by another factor of 5. At P T 40 GeV, case I is less than a factor of 2 larger than the central NLO values, thus still clearly below the NNLO yield, while case II is still a bit suppressed and comparable to the central NLO values. A dedicated analysis including a global fit of the C = +1 CO LDMEs and taking into account DGLAP radiation of the fragmenting gluon is beyond the scope of this work but would confirm that αα 3 S CO corrections to Q + γ matter only at large P T and are most likely not large enough to challenge the dominance of CS contributions to this process.
This dominance is expected since the photon has to be emitted by quarks. If the hard scattering part is gg → gg, the photon can only be emitted by the heavy quarks with topologies similar to CS channels. At α 2 S α, the gluon-fusion CO channels do not scale like P −4 T . One has to go to α 3 S α to have the first gluon fragmentation channels initiated by gluon fusion. In this case, as discussed above, the gluon fragments into a photon and a CO C = +1 (whose LDMEs are severely constrained) 11 . Last but not least, contrary to the double t-channel gluon exchanges dominating the CS yield, the latter fragmentation CO topologies will systematically produce nearly collinear γ and hadrons from the discolouration of the CO state. Those events are therefore likely to be rejected by experimental photon-isolation cuts.
Finally, while contributions from s-channel cut to J/ψ production could appear in the low P T region as in the inclusive case [40] (the final state gluon of the gg → J/ψg being simply replaced by a photon), in the large P T region, the colour-transfer-enhancement mechanism discussed in [41] is not expected to matter for the present process. In the inclusive case, gg → J/ψ + cc may account for a significant part of the yield at very large P T . This indicates that topologies for which colour transfer could occur are not much suppressed and those could impact on the inclusive yield. In the present case, we have checked that the contribution to J/ψ + γ from the partonic process gg → J/ψ + cc + γ is sub-dominant and found that it is one order of magnitude smaller than the process from gg → J/ψ + γ + gg with the same P T dependence between 20 and 50 GeV. It is therefore quite unlikely that colour transfers, acting on topologies with at least 3 heavy quarks, be visible in J/ψ+γ yields.

Conclusion
In conclusion, we have computed the real next-to-nextto-leading order QCD contributions to the hadroproduction of a J/ψ + γ and Υ + γ via colour singlet transitions along the same lines as [9] for the inclusive case and argued that 11 If one required that P γ T exactly balance P Q T in order to select the LO CSM, aiming at the extraction of the gluon distributions for instance, one would have to refine the analysis to take into account NNLO corrections in the CS channels and the CO yield [25] which would then be dominated by gg → g → (QQ) [8] + γ via 1 S [8] 0 or 3 P [8] J if the C = +1 CO channels are not too much suppressed.
it provided a first reliable evaluation of the corresponding yield at NNLO accuracy.
Indeed, prior to that, we have shown that the full NLO evaluation of Li and Wang [27] is very accurately reproduced for P T > 5 GeV by the sole evaluation of the real emission contributions up to order α 3 S α, namely the NLO . In the inclusive case [9], a similar observation was made and motivated the study of the NNLO contributions to evaluate the yield at NNLO accuracy. We have thus reach a completely independent, but similar, conclusion for the production of a quarkonium in association with a photon.
A priori, integrating the amplitudes for such real emission contributions leads to divergences in some phase-space regions. Those can be avoided by imposing a minimal invariant mass between the external light partons, while in a complete calculation those divergences are canceled by the virtual corrections. Yet, at NLO, the latter scale as P −8 T and, at NNLO, as P −6 T , hence are suppressed compared to respectively the real NLO part scaling as P −6 T and the real NNLO part scaling as P −4 T . This explains why we can neglect the virtual corrections at mid and large P T and use a cut-off on which the results become insensitive when P T grows, and also why the sole NLO reproduces the full NLO accurately.
We have also shown that the differential cross section in P T at NNLO is one order of magnitude larger than at NLO for large P T -as expected from their P T scaling-. We have also identified the process responsible for the most part of it, i.e. gg → Q + γ + gg. Although, the distinction between the gluon-fragmentation and the double t-channel gluon exchange graphs cannot be carried out in gauge invariant way, we provided some hints that the second type of topology is dominating, similar to the inclusive case [9]. One of this hint is the polarisation of the quarkonia produced in association with a photon.
Indeed, we have computed the polarisation parameter λ of the NNLO yield which is negative, indicating a longitudinally polarised yield. This confirms the trend observed at NLO and hints at the dominance of double t-channel gluon exchange contributions.
When the NNLO contributions are incorporated in the CS yield, it becomes one order of magnitude larger than the potential CO yield, which would mainly produce transversally polarised Q. The measurements of the cross section for the production of Q + γ would directly measure the size of the CS without being sensitive to the non-perturbative CO parameters. This is a complementary case to the study of J/ψ + cc and Υ + bb as discussed in [5,42,15].
Similarly to the analysis of the inclusive case [9], this analysis cannot be extended to too low P T , where the approximations on which it is based no longer hold. One way to improve the predictions could be achieved by merging the matrix elements with parton showers using one of the approaches available in the literature [43], or by performing an analytic resummation [44].
Finally, the results presented here strongly support the procedure used for the very first evaluation of the inclu-7 sive yield at NNLO accuracy [9] and which showed an agreement with the Tevatron measurements. This in turn confirms that much better Monte Carlo simulations than the ones based on matrix elements for CS channels at LO only are now possible at mid and large P T . This could be achieved using NLO and NNLO partonic matrix elements generated by MadOnia [28,30], for the process studied here pp → Q + γ + X, but also for the inclusive measurements to be performed at the LHC.