Hadroproduction of J/psi and Upsilon in association with a heavy-quark pair

We calculate the cross sections for the direct hadroproduction of J/psi and Upsilon associated with a heavy-quark pair of the same flavour at leading order in alpha_S and v in NRQCD. These processes provide an interesting signature that could be studied at the Tevatron and the LHC and also constitute a gauge-invariant subset of the NLO corrections to the inclusive hadroproduction of J/psi and Upsilon. We find that the fragmentation approximation commonly used to evaluate the contribution of these processes to the inclusive quarkonium production sizeably underestimates the exact calculation in the kinematical region accessible at the Tevatron. Both J/psi and Upsilon are predicted to be unpolarised, independently of their transverse momentum.


Introduction
Since its introduction in 1995 [1,2], NRQCD has become the standard framework to study heavy-quarkonium physics. NRQCD is a non-relativistic effective theory equivalent to QCD, where inclusive cross sections and decay rates can be factorised as the product of short-distance and long-distance parts. The short-distance coefficients are process-dependent but can be computed in perturbative QCD, the strong coupling constant α S being the parameter of the expansion. The long-distance matrix elements are process-independent, i.e., universal, but non-perturbative. They can be classified in terms of their scaling in v, the relative velocity of the heavy quarks in the bound state, and as a result, physical quantities can be expressed as a double series in α S and v.
An innovation of this formalism is the colour-octet mechanism: the heavyquark pair is allowed to be created in a colour-octet state over short distances, the color being neutralised over long distances. It is thanks to this very mechanism that it is possible to account for the CDF data [3,4] on the inclusive J/ψ and ψ ′ cross sections at the Tevatron [5,6,7]. In this case the colour-octet matrix elements fitted from the data roughly scale as the power counting rules of NRQCD predict. On the other hand, the recent data collected at √ s = 1.96 TeV by the CDF collaboration [8] have revealed that the J/ψ is unpolarised, in flagrant disagreement with the expectations of NRQCD. It is fair to say that the mechanisms responsible for the quarkonium production at the Tevatron are not completely understood yet (for a recent review see [9]).
Another challenge to theorists has been provided by the recent measurements at B factories, where rates for inclusive and exclusive J/ψ production in association with a cc quark pair [10] are far larger than the leading order NRQCD theoretical expectations [11]. In this case, keeping only the leading term in α S and v in the NRQCD expansion results in a very rough approximation. In addition, as opposed to the production of the J/ψ in hadron collisions, colour-octet channels cannot be invoked to bring theoretical predictions in agreement with the experimental measurement [12]. Inclusion of the α 3 S (NLO) corrections to this process has been recently found to reduce the discrepancy between theory and measurements [13].
In view of the unexpectedly large measurements for the production of J/ψ associated with a cc quark pair in e + e − annihilation, it is natural to wonder whether the corresponding production pattern could be large in hadroproduction as well. Besides offering a new interesting signature, such as ℓ − ℓ + in association with one or two heavy quark tags, this process contributes to the α 4 S (NLO) corrections to the inclusive colour-singlet hadroproduction of J/ψ and Υ. Historically, these higher-order contributions to the cross section at the Tevatron were considered in the fragmentation approximation as a first attempt to solve the so-called ψ ′ anomaly [14,15].
The purpose of this work is to present and discuss the complete tree-level calculation for the associated production of J/ψ and Υ (commonly noted Q) with a heavy-quark pair of the same flavour. Such associated production has been recently discussed in the k T factorisation formalism [16,17]. Here we consider the leading order term in the NRQCD expansion in the usual collinear factorisation scheme.
This Letter is organised as follows. In Section 2, we briefly decribe the method used to compute the associated hadroproduction of a quarkonium with a heavy-quark pair together with the results for the Tevatron and the LHC. In Section 3, we discuss the validity of the fragmentation approximation Q → QQ at the Tevatron. In Section 4 we discuss the polarisation of the quarkonium produced by colour-singlet transitions. Finally, Section 5 is devoted to the discussion of our results.
2 The calculation of pp → Q + QQ

The method
In this section, we present the numerical method used to compute quarkonium production amplitudes. At leading order in v, the invariant amplitude to create a 3 S 1 quarkonium Q of momentum P and polarisation λ can be expressed as the product of the amplitude to create the corresponding heavy-quark pair, a spin projector N(λ|s 1 , s 2 ), and R(0), the radial wave function at the origin in the configuration space 1 , namely, [18,19] where , s 1 is the quark spin, s 2 the antiquark one and δ ij √ Nc is the projector onto a color-singlet state. In the non-relativistic limit, the spin projector factor can be written as where ε λ µ is the polarisation vector of the quarkonium.
The amplitude for gg → Q λ (P ) + QQ involves 42 Feynman diagrams and an analytical computation is not practical. We have therefore opted for using a custom version of MadGraph [20], to generate the amplitudes and perform the spin projection numerically. The expression in Eq. (2) can be evaluated with the help of the HELAS subroutines [21], as done in Ref. [11]. We validated our algorithm to generate the amplitudes for quarkonium production and decays by comparing with several known analytical results point-by-point in phase space. Finally, we use the techniques introduced in Ref. [22] to perform the phase-space integration. As an exercise, we have reproduced the cross section and the P T distribution for B * c production at the Tevatron and found agreement with the results of Ref. [23].
At hadron colliders a 3 S 1 quarkonium state is observed through its decay into leptons. In order to keep the spin correlations between the vector-like bound state and the leptons, we can replace ε λ µ in Eq. (2) by the leptonic current The polarisation of the quarkonium can therefore be determined by analysing the angular distribution of the leptons. Defining θ as the angle between the ℓ + direction in the quarkonium rest frame and the quarkonium direction in the laboratory frame, the normalised angular distribution I(cos(θ)) is where the relation between α and the polarisation state of the quarkonium is

Cross sections and results
As for the case of open charm and bottom cross sections, heavy-quarkonium hadroproduction is dominated by gluon fusion. We have checked that the lightquark initiated process for Q + QQ production is suppressed by three orders of magnitude, and thus this contribution is neglected in the following.
In our numerical studies we have used: In Fig.2 we show the P T distributions of J/ψ and Υ at the Tevatron and the LHC, for both Q + g and Q + QQ production. The theoretical error bands in the Q + QQ curves correspond to the uncertainties from the renormalisation and factorisation scale ( µ 0 2 ≤ µ f,r ≤ 2µ 0 ) and the heavy-quark masses (m c = 1.5 ± 0.1 GeV or m b = 4.75 ± 0.05 GeV), combined in quadrature.
We start by considering J/ψ and Υ production at the Tevatron, where a rapidity cut |y| < 0.6 is applied. For the charmonium case, we note that the P T distribution peaks at P T ≃ m Q and then it starts a quick descrease, dropping by four orders of magnitude at P T ≃ 20 GeV. In fact, at moderate transverse momentum, P T m Q , Q + QQ production has much milder slope compared to that of Q + g. Even though suppressed by α S , Q + QQ production starts to be comparable to Q + g already for P T ≃ 4m Q . We verified that the topologies where the Q is produced by two different quark lines always dominate. This feature is to be related with the J/ψ production at e + e − colliders, where the presence of two extra charmed mesons seems to indicate that the J/ψ is mostly often created from two different quark lines as well. For the Υ, the curve is less steep in the same P T range due to the larger value of the b-quark mass.
The results for the LHC are displayed in Fig. 2 (c,d). A cut |y| < 0.5 has been applied. Since the rapidity distribution is flat in this range, this is equivalent to consider the quarkonium production at zero rapidity. The P T behaviours for both J/ψ and Υ are very similar to those at the Tevatron while the normalisation is increased by one order of magnitude.

Testing the fragmentation approximation
In this section we discuss the range of applicability of the fragmentation approximation for the process where a QQ pair is produced at high P T and one of the heavy quarks fragments into a quarkonium state, Q → QQ. At sufficiently large P T , it is expected that the quark-fragmentation contributions would dominate the QQQ cross section.

Heavy-quark fragmentation
In the fragmentation approximation, the cross section for the production of a quarkonium Q by gluon fusion via a heavy quark Q i fragmentation is given, at all orders in α S , by where dσ Q i ( P z , µ f rag ) is the differential cross section to produce an on-shell heavy quark Q i with momentum P z and D Q i →Q (z, µ f rag ) is the fragmentation function of Q i into a quarkonium Q.
The fragmentation scale, µ f rag , is usually chosen to avoid large logarithms of P T /µ f rag in σ Q i ( P z , µ f rag ), that is µ f rag ≃ P T . The summation of the corresponding large logarithms of µ f rag /m Q appearing in the fragmentation function can be obtained via an evolution equation [25,26,27].
The perturbative quark fragmentation function into quarkonium Q via a 3 S

Comparison
We now compare the results of the full LO cross sections for pp → Q + QQ with those calculated in the fragmentation approximation. The same set of parameters of Section 2.2 is employed 2 . The comparison is shown in Fig. 3. The approximation is lower than the full computation in the P T range accessible at the Tevatron. We have verified that for J/ψ production, the two curves still differ by a little bit less than 10% at P T = 80 GeV. For the Υ, much larger values of P T are required (Fig. 3 (b)).
Therefore, contrary to the common wisdom, in the kinematical region accessible at the Tevatron, pp → J/ψ + cc, which is an NLO subset of pp → J/ψ + X, is not dominated by the fragmentation contributions. Idem for pp → Υ + bb.
4 Polarisation: gg → Q + QQ vs. gg → Q + g Let us first review the analytical result for gg → Q + g, which we used as a check of our numerical procedure. To compute the polarisation parameter α discussed in Section 2.1, it is sufficient to consider the unpolarised (or total) cross section and the longitudinal one in the laboratory frame. The parameter α introduced in Section 2.1 can then be written as The total cross section is calculated from the squared amplitude summed over the quarkonium polarisation which reads [18,29] where M Q = 2m Q andŝ,t andû are the usual Mandelstam variables for parton quantities. To compute the squared amplitude for longitudinally polarised quarkonia, we need to define the longitudinal polarisation vector in the laboratory frame: with where x 1 and x 2 are the usual momentum fractions of the incoming gluons. From that definition and the Eqs. (10.505) of Ref. [30], we can then easily compute If x 1 and x 2 are approximately equal and the masses can be neglected, it is possible to see from Eq. (9) and Eq. (12) that the quarkonia will be transversally polarised. Using the same parameters as for the previous plots, we find that α gets rapidly close to one for P T > m Q , see Fig. 4. Our results agree with those of Ref. [31]. Note that this behaviour of α is not expected to be seen in the data, since the yield from pp → Q+g is not dominant, even when only colour-singlet channels are considered. For the process gg → Q + QQ, α remains close to zero in the whole P T range. Since the latter is the one which dominates over the other colour-singlet contributions, we conclude that the quarkonia not produced by a colour-octet mechanism will be unpolarised at high P T .

Conclusion
In this Letter, we have presented the tree-level calculation for the associated production of J/ψ and Υ with a heavy-quark pair. The motivation was twofold. First, these processes offer a new interesting signature, that could be tested experimentally by measuring the fraction of quarkonium produced with at least one heavy-light quark meson. Second, they contribute to the α 4 S (NLO) corrections to the inclusive colour-singlet hadroproduction of J/ψ and Υ.
We have found that the quark-fragmentation approximation employed to describe quarkonium production should not be applied in the range of transverse momenta reached at the Tevatron and analysed by the CDF collaboration. The fragmentation approximation actually underestimates the full contribution of pp → J/ψ + cc by more than a factor four in the region P T ≃ 10 GeV. This approximation starts to be justified (within 10% of error) only for P T ≥ 80 GeV. The same conclusion applies for the ψ ′ . For pp → Υ + bb the fragmentation approximation is relevant at even higher P T , as expected. It is interesting to note that for J/ψ production from e + e − annihilation, the fragmentation approximation is only 6% away from the full contribution at √ s = 50 GeV [11]. One may try to gather an intuitive explanation for such glaring difference by considering the Feynman diagrams contributing to the two processes. In e + e − annihilation, there are only four diagrams at leading order, two of which contribute to the fragmentation topologies. On the other hand, the dynamics underlying the 3 S 1 -quarkonia production at the Tevatron is much more involved. The number of Feynman diagrams is significantly larger (42 vs. 4) and only a few of them give rise to fragmentation topologies. As a result, the fragmentation approximation starts to be relevant at a much higher regime in P T . A similar situation holds for the process γγ → J/ψcc [32] and also for the B * c hadroproduction, where the fragmentation approximation is not accurate in the P T range explored at the Tevatron [33,23]. We have also shown that the J/ψ or Υ produced in association with a heavy-quark pair of the same flavour are unpolarised. Further analysis, including combination with the results of Ref. [34] for pp → Q + X at NLO and Ref. [35], to compare with the available data from the Tevatron, is ongoing.
In conclusion, we look forward to the measurement of the fraction of events in the J/ψ sample at the Tevatron, with at least one charmed meson in the final state. Such a measurement could also provide further insight to the mechanism responsible for inclusive heavy-quarkonium production at hadron colliders.