Breakdown of NRQCD Factorization in Processes Involving Two Quarkonia and its Cure

We study inclusive processes involving two heavy quarkonia in nonrelativisitic QCD (NRQCD) and demonstrate that, in the presence of two P-wave Fock states, NRQCD factorization breaks down, leaving uncanceled infrared singularities. As phenomenologically important examples, we consider the decay $\Upsilon \to \chi_{cJ}+X$ via $b\bar{b}({}^3P_{J_b}^{[8]})\to c\bar{c}({}^3P_J^{[1]})+gg$ and the production process $e^+e^-\to J/\psi+\chi_{cJ}+X$ via $e^{+}e^{-}\to c\bar{c}({}^3P_{J_1}^{[8]})+c\bar{c}({}^3P_J^{[1]})+g$. We infer that such singularities will appear for double quarkonium hadroproduction at next-to-leading order. As a solution to this problem, we introduce to NRQCD effective field theory new types of operators whose quantum corrections absorb these singularities.

Because of their large mass scales and nonrelativistic nature, heavy-quarkonium states are ideal probes to study quantum chromodynamics (QCD), which is the fundamental theory to describe the strong interactions between quarks and gluons, from both perturbative and nonperturbative aspects. The traditional color-singlet model (CSM), which restricts the heavy-quark pair, QQ, to form a color singlet and to share the spectroscopic quantum numbers 2S+1 L J of spin S, orbital angular momentum L, and total angular momentum J with the physical quarkonium state, is plagued by infrared (IR) singularities when applied to the production or decay of P-wave quarkonia [1] or quarkonia with L > 1 [2]. Phenomenologically, a cutoff, e.g., on the binding energy of the QQ bound state or the momenta of emitted gluons, has to be introduced to regularize such singularities, which makes the theoretical predictions model dependent and causes the separation of short-and longdistance physics to break down. This problem has been successfully solved by the factorization formalism [3] built on the rigorous effective field theory of nonrelativistic QCD (NRQCD) [4]. In the NRQCD factorization formalism [3], the production and decay rates of heavy quarkonia are separated into short-distance coefficients (SDCs), which can be obtained as expansions in the strong-coupling constant α s through the NRQCD to full-QCD matching of perturbative calculations, and supposedly universal long-distance matrix elements (LDMEs). The sizes of the latter are subject to scaling rules in the velocity v Q of Q andQ in the QQ rest frame [5]. This allows us to calculate heavy-quarkonium production and decay rates systematically as double expansions in α s and v 2 Q . The NRQCD factorization formalism has successfully cured the IR problem of the CSM, as explicitly shown in the literature for inclusive decay and production of P- [6] and D-wave states [7]. Note that in exclusive processes, such as B-meson exclusive decay to χ cJ mesons [8] and exclusive double quarkonium production in e + e − annihilation [9], there are still uncanceled IR divergences at one loop, which disappear in the limits m c /m b → 0 and m c / √ s → 0, respectively [9].
Besides inclusive production and decay processes involving a single quarkonium state, also processes in which two heavy quarkonia participate, such as bottomonium decay to charmonium or double heavy-quarkonium production, are of great phenomenological interest. Measurements of Υ decay to charmonium, like the J/ψ meson, can be traced to the first experiment carried out by CLEO Collaboration about 30 years ago [10], and were then updated by the ARGUS [11] and CLEO [12] Collaborations with much larger data samples. Very recently, more precise results for the branching ratios of Υ → J/ψ(ψ ′ ) + X [13] and Υ → χ c1 + X [14] were obatined by the Belle Collaboration. On the other hand, J/ψ pair and J/ψ + Υ associated production have been a very hot topic at hadron colliders in recent years. In fact, double J/ψ (Υ) and J/ψ + Υ prompt production were extensively measured by the D0 Collaboration [15] at the FNAL Tevatron and by the LHCb [16], CMS [17], and ATLAS [18] Collaborations at the CERN LHC. Interestingly, there are substantial discrepancies between CMS data and NRQCD predictions at leading order (LO) in α s , which are expected to be reduced by the yet unknown next-to-leading-order (NLO) corrections [19]. Moreover, double heavy-quarkonium production serves as a useful laboratory to investigate the double parton scattering mechanism [20] at hadron colliders. On the theoretical side, both bottomonium decay to charmonium and double heavy-quarkonium production have been studied in the NRQCD factorization framework. In the former case, only the color-singlet (CS) 3 S [1] 1 [21] and color-octet (CO) 3 S [8] 1 [22] bb Fock states were considered. In the latter case, the CS contributions are known to NLO in α s [23] and v 2 Q [24], while the CO contributions are only known to LO [19] and partially to NLO in v 2 Q [24]. In all these calculations, IR singularities appearing in intermediate steps were always properly removed by NRQCD factorization, and one might have expected that this is a general rule valid for all QQ Fock states and to all orders in α s and v 2 Q . In this Letter, we change this familiar picture by presenting two counterexamples, suggesting that the wellestablished formalism needs a generalization for the cases at hand. In fact, the violation of NRQCD factorization appears as one includes more QQ Fock states or goes beyond LO in α s . As we demonstrate later on, the leftover IR singularities may, fortunately, be completely absorbed into the QCD corrections to a class of operators introduced here for the first time.
Let us first consider the inclusive production of charmonium H = J/ψ, χ cJ , ψ ′ by Υ decay. By NRQCD factorization, the decay width can be expressed as where m and n are bb and cc Fock states, respectively, Γ(bb(m) → cc(n) + X) are the SDCs, and Υ|O(m)|Υ and O H (n) are the LDMEs. According to the velocity scaling rules [5], for Υ (J/ψ, ψ ′ ) the four Fock We have checked explicitly that, in the single P-wave case, where either bb or cc is in a P-wave state, the IR singularities arising in the full QCD calculation can be completely absorbed into the corresponding S-wave LDMEs renderingΓ(bb(m) → cc(n) + X) IR finite. However, when both bb and cc are in P-wave states, there are extra IR singularities that cannot be absorbed into the Swave LDMEs. At LO, there are two such subprocesses, namely bb( 3 P Jc ) + gg for Υ → χ cJ + X. In the following, we focus our attention on the second one. Similar conclusions can be drawn for the first one and eventually be extended to the general case of bottomonium decay to charmonium plus anything.
There are eight Feynman diagrams for bb( 3 P Jc ) + gg, and typical ones are depicted in Fig. 1. For each choice of J b and J c , the corresponding partonic decay rate may be calculated by using appropriate spin and color projectors. Unsurprisingly, they all contain IRdivergent terms. Because of space limitation, we present here only the latter. Furthermore, we sum over J b using heavy-quark spin symmetry, leaving Υ|O( 3 P [8] 0 )|Υ as the overall Υ LMDE. We write the result aŝ whereΓ div 1 (Γ div 2 (J c )) arises from the square of the ampli- tude M 1 (M 2 ) of the diagrams in which the soft gluon is emitted by the c orc (b orb) quarks, andΓ div 3 (J c ) arises from the interference of M 1 and M 2 . We have: where ǫ IR = D/2 − 2 is the infrared regulator of dimensional regularization and r = m c /m b . In Eq. (3), we writeΓ div 1 andΓ div 2 (J c ) as products of the SDCs of bb( 3 P Jc ) + g, whose representative Feynman diagrams are shown in Fig. 1  1 )|Υ , respectively, so as to indicate that they will be canceled after taking into account the contributions of bb( 3 P Jc ) + g. However, in the case ofΓ div 3 (J c ), the NRQCD factorization formalism as we know it simply lacks an operator that could compensate the soft-gluon effects. This rendersΓ div (J c ) IR singular altogether. In other words, we are faced by an IR problem of NRQCD factorization which has gone unnoticed so far! Sincê Γ div 3 (J c ) are due to the interference of diagrams with soft gluons emitted by P-wave bb and cc Fock states, which appears in NRQCD treatments of any inclusive  decay of bottomonium to charmonium at some order of v 2 b and v 2 c , we conclude that the NRQCD factorization formalism, in its familiar and generally accepted form, will break down for any such process. In some cases, this may happen even at relative order v 2 b,c , i.e. at LO, e.g. for the decay χ bJ b → χ cJc + X via the channel bb( 3 P Jc ) + gg. We note in passing that, unlike for the exclusive processes mentioned above [8,9], Γ div 3 (J c ) does not vanish in the limit r → 0. We now turn to our second example, the inclusive production of two heavy quarkonia, e.g. 2J/ψ, J/ψ + Υ, etc. In the NRQCD treatment of J/ψ pair or J/ψ + Υ associated hadroproduction, soft-gluon emission starts from NLO in α s , e.g. gg → cc( 3 P J2 )+g. A complete NLO NRQCD calculation lies beyond the scope of this Letter. Instead, we choose a relatively simple process for illustration, namely e + e − → J/ψ + χ cJ + X proceeding via the LO channel e + e − → cc( 3 P J2 )+g. The IR problem featured here should also appear in double J/ψ production via e + e − annihilation [25].

[8]
Jc ) + bb( 3 P [8] J b ) + g, there will be four possible interferences of the four pairings cc( 3 S 1 ), and cc( 3 P 1 ), which yield uncanceled IR singularities. The above two examples clearly demonstrate that NRQCD factorization as we know it is spoiled by uncanceled IR singularities for inclusive production and decay processes involving two (ore more) heavy quarkonia. At this point, we recall that factorization implies a complete separation of perturbative and nonperturbative effects. In the context of the NRQCD factorization framework, one is thus led to find a concept how to separate the problematic IR-singular terms, like Γ div 3 (J c ) and σ div 3 (J 2 ), into contributions pertaining to the hardand soft-scale regimes. The creation and annihilation of heavy-quark pairs clearly take place at short distances. To describe such processes involving two heavy-quark pairs, it is natural to consider products of four heavyquark fields. Since the two heavy-quark pairs cannot be at rest simultaneously, we adopt the covariant form of the NRQCD Lagragian, which at LO reads [26]: where m is the heavy-quark mass, v µ = P µ /(2m) with P µ being the four-momentum of the QQ pair, ψ v and χ v are the nonrelativistic four-component heavy-quark and -antiquark fields, satisfying / vψ v = ψ v and / vχ v = −χ v , D µ is the covariant derivative, and a µ ⊤ = a µ − v µ v · a for any four-vector a µ . The relative momentum q µ between Q andQ corresponds to i∂ µ ⊤ ψ v . Representative diagrams for one-loop corrections to the annihilation or creation of two heavy-quark pairs, Q 1Q1 and Q 2Q2 , are depicted in Fig. 3. In the first two panels, the soft gluon interconnects the same QQ pair. This corresponds to the product of two four-quark operators, which separately receive QCD corrections. The missing link needed to remove the left-over IR singularities is depicted in the last two panels. Here, the two QQ pairs cross talk by exchanging a soft gluon and so form a joint structure, namely an eight-quark operator.
In the following, we refer to bb( 3 P Jc ) + gg as process A and to e + e − → cc( 3 P J2 ) + g as process B and generically denote the total and relative four-momenta of Q iQi as P i and q i , respectively. To generate the appropriate interference parts at one loop for two P-wave QQ states, we need the new operators and their charge conjugatates for process B, where K µν Using the Feynman rules derived from Eq. (6) in connection with the new operators introduced above, we are now in a position to evaluate the last two Feynman diagrams in Fig. 3. Although m c is about three times smaller than m b , we assume that m c ≫ m b v b to ensure that the soft region of bottomonium is sufficiently separated from the hard region of charmonium so that the nonrelativistic approximation still applies to the latter. Working in covariant gauge, we show that the results are gauge independent. The details of our calculation will be presented elsewhere [27]. For space limitation, we here merely explain how to perform the loop integrations, taking the third diagram in Fig. 3 as an example and working in Feynman gauge. The arising loop integral reads Expanding the heavy-quark propagators in 1/m i and dropping terms of order 1/m 2 i and higher, which con-tribute at higher orders in v 2 i , we obtain where ω = v 1 · v 2 and I 0 includes irrelevant terms that will cancel in the sum over all diagrams. The ultraviolet singularities are removed via operator renormalization. Multiplying Eq. (8) with the corresponding SDCs, which is the interference of bb( 3 P Jc ) + g for process A and the interference of e + e − → cc( 3 P  J2 ) for process B, and decomposing the tensor and color structure into the basis of the totalangular-momentum and color states, we find that the IR-singular parts exactly match those in Γ div 3 (J c ) and σ div 3 (J 2 ). We wish to emphasize that the loop integrals are process independent although they depend on v 1 · v 2 .
In summary, we discovered a surprising loophole in the standard formulation of the NRQCD factorization approach which manifests itself in the failure of IR cancelation in the presence of two (or more) P-wave QQ Fock states. This inevitably causes NRQCD factorization to break down for any decay or production process involving two (or more) heavy quarkonia at a certain order of v 2 Q . We illustrated this for two phenomenologically important example processes, Υ → χ cJ + X and e + e − → J/ψ + χ cJ + X, at NLO in α s . As a solution to this problem, we introduced new types of operators and demonstrated that their NLO corrections precisely reproduce the uncanceled IR singularities, which may thus be attributed to the soft regime of NRQCD. This implies that it is possible to generalize the factorization formalism within the very same NRQCD effective field theory so as to allow for the successful theoretical description of processes involving two (or more) heavy quarkonia. The explicit construction of such a generalized NRQCD factorization formalism and its applications to heavy-quarkonium phenomenology are left for future work.
We thank G. T. Bodwin and E. Braaten for very useful comments. This work was supported in part by BMBF Grant No. 05H15GUCC1. The work of X.P.W. was sup-ported in part by CSC Scolarship No. 201404910576.