The leading twist light-cone distribution amplitudes for the S-wave and P-wave Bc mesons

The light-cone distribution amplitudes (LCDAs) serve as important non-perturbative inputs for the study of hard exclusive processes. In this paper, we calculate ten LCDAs at twist-2 for the S-wave and P-wave $B_c$ mesons up to the next-to-leading order (NLO) of the strong coupling $\alpha_s$ and leading order of the velocity expansion. Each one of these ten LCDAs is expressed as a product of a perturbatively calculable distribution and a universal NRQCD matrix-element. By use of the spin symmetry, only two NRQCD matrix-elements will be involved. The reduction of the number of non-perturbative inputs will improve the predictive power of collinear factorization.

The main difficulty to study the productions and decays of B c mesons is that there are too many energy scales entangled in these processes. However, on the other hand, this entanglement of the many scales provides an excellent platform for testing many fascinating aspects of the perturbative QCD and factorizations. Generally, we think that B c meson is a non-relativistic bound-state of b quark andc quark, similar to quarkonium. Thus, many researches on B c involved processes are done within the framework of non-relativistic QCD (NRQCD) factorization (for reviews of NRQCD, see [9,10]), such as some recent studies on B c decays, B c productions from W and Z boson decays, etc [11][12][13][14][15][16][17][18]. In NRQCD factorization, the cross-section for inclusive process or the amplitude for exclusive process can be factorized into a product of the perturbatively calculable short-distance coefficient and the non-perturbative NRQCD matrix-element. For B c production at large momentum transfer, say Q 2 >> m 2 Bc , the short-distance coefficient in NRQCD factorization involves at least two well-separated scales Q 2 and m 2 Bc . This causes large logarithmic terms like α n s ln n Q 2 /m 2 Bc in perturbation series, which cannot be resummed within the NRQCD method. This is quite similar to the problems in quarkonia production at large momentum transfer. To cure such problems, one needs to either re-factorize the short-distance coefficients in some way, or adopt another factorization scheme, for instance the collinear factorization, to calculate such productions. Taking the exclusive process γ * (q)B − c (p 1 ) → B − c (p 2 ) at large momentum transfer Q 2 ≡ −q 2 = −(p 2 − p 1 ) 2 >> m 2 Bc as an example, the electromagnetic (EM) transition form-factor F (Q 2 ) parameterizing the amplitude can have a similar collinear factorization formula to the EM transition form-factor for γ * π − → π − at large momentum transfer [19,20], which reads as where Φ Bc (x; µ F ) is the light-cone distribution amplitude (LCDA) for B c meson at twist-2, and T H (x, y) represents the perturbatively calculable hard-scattering kernel with x and y being the light-cone momentum fractions. The LCDA Φ Bc (x, µ F ) encodes the hadronization effects, and obeys the celebrated Efremov-Radyushkin-Brodsky-Lepage (ERBL) equation [21,22] where V 0 (x, y) is the Brodsky-Lepage kernel. Different from the LCDAs for light mesons which are totally non-perturbative objects, the LCDA for quarkonium or B c meson can be further factorized into a product of a perturbatively calculable distribution part and a NRQCD matrix-element for the vacuum to quarkonium or B c state transition, at the leading order of non-relativistic expansion parameter v [23,24]. As an examination, it has been shown in [25] that the factorization formula (1) by employing Φ Bc (x; µ) given in [24] do reproduce the asymptotic behavior of the EM transition form-factor calculated directly from the NRQCD computation, up to the next-toleading order (NLO) of the strong coupling α s and leading order of v-expansion. Thus, in this sense, the re-factorization of the LCDAs for B c meson can serve as a bridge to connect the NRQCD factorization and collinear factorization for B c meson exclusive productions at large momentum transfer.
In this paper, we extend the computation of the leading twist LCDA for the S-wave pseudo-scalar B c meson in [24] to all the ten twist-2 LCDAs for S-wave and P-wave B c mesons, up to the NLO of α s and leading order of v. This can also be regarded as a succeeding work of [25,26], in which the LCDAs for quarkonia are calculated in the same manner.
This paper is organized as follows: in section II, we give the notations we use in this paper, and present the definitions of the leading twist LCDAs for the S-wave and P-wave B c mesons in terms of the matrix-elements of a certain class of non-local QCD operators, then we give tree-level results for these LCDAs at the leading order of v within the NRQCD factorization; in section III, we present our results of the LCDAs at the NLO of α s and leading order of v followed by some useful inverse moments of these LCDAs; finally, we discuss some potential applications of our results and summarize our work in section IV.

A. Notations
We adopt the same notations as in [26]: the momentum of B c meson is P µ = m Bc v µ where v µ is the velocity four-vector of B c satisfying v 2 = 1. A 4-vector a can be decomposed as a µ = v ·av µ +a µ ⊤ where v ·a ⊤ ≡ 0. Note that we also use the same notation v to denote the non-relativistic relative velocity between quark and anti-quark inside a rest B c meson, which characterizes the velocity expansion in the NRQCD. We remind that the readers should not confuse these two in the context. Two light-like vectors n ± are also introduced, n 2 ± = 0 and n + n − = 2. A 4-vector a can be decomposed as a µ = n + an µ − /2 + n − an µ + /2 + a µ ⊥ with n ± a ⊥ ≡ 0. For convenience, we set v µ = (n + vn µ − + n − vn µ + )/2 (apparently n + vn − v = 1).

B. Definitions of the LCDAs
The leading twist LCDAs for the S-wave and P-wave B c mesons are defined as the matrixelements of the proper gauge invariant non-local quark bilinear operators where b and c are the fields in QCD for b quark and c quark, respectively. The Wilson-line is a path-ordered exponential with the path along the n + direction where g s is the SU(3) gauge coupling and A µ ≡ A a µ (x)T a (T a are generators of SU(3) group in the fundamental representation). The ten non-vanishing twist-2 LCDAs for the S-wave and P-wave B c mesons are defined as Here f, ε * andφ(x) are decay constants, polarization vectors/tensors, and the distribution parts of twist-2 LCDAs for corresponding B c mesons, respectively. x ∈ [0, 1] denotes the light-cone fraction (we will also adopt the notationx ≡ 1 − x in the rest of this paper), and µ is the renormalization scale . We set the normalization conditions for the LCDAs as following Note that the normalization conditions forφ 1A (x),φ ⊥ 3A (x) andφ S (x) in (6b) are different from the ones for quarkonia [26]. This is due to the fact that we have no C-parity constraints on the LCDAs for B c mesons.
In the calculation below, we actually use the Fourier transformed forms of the definitions given in (5) such that where Here we suppress the dependence of all quantities on the renormalization scale µ. One can see clearly that all the LCDAs are defined in a boost-invariant way, i.e. they are invariant under the Lorentz boost in which a 4-vector a µ transforms as n + a → αn + a, n − a → α −1 n − a and a µ ⊥ → a µ ⊥ .

C. NRQCD factorization for the LCDAs
In [23], the authors show that the LCDA for η c or J/ψ can be factorized into the product of a perturbative function and a NRQCD matrix-element, since quarkonium is a non-relativistic bound-state of heavy quark and anti-quark. Taking advantage of the similar nature of B c meson as a non-relativistic bound-state of b andc quark, we can reduplicate their calculations in the case of B c mesons.
Here, since we are going to deal with two different flavors of heavy quarks, we have to introduce two kinds of heavy quark effective fields in the NRQCD Lagrangian. By use of the four-component notations as in [27], we have the leading order NRQCD Lagrangian Here m Q is the pole mass of the heavy quark Q (Q = b, c), ψ Qv and χ Qv are the effective fields of the heavy-quark and anti-heavy-quark, respectively, satisfying v /ψ Qv = ψ Qv and v /χ Schematically, at operator level, we have the matching equation where n denotes the order of v-expansion, C n Γ (x, µ) is the short-distance coefficient as a distribution over the light-cone fraction x, and O NRQCD Γ,n is the relevant NRQCD operator which scales O(v n ) in the NRQCD power-counting. So, the LCDAs of B c meson can be expressed as Up to the leading order of v, the matrix-elements of the following relevant NRQCD effective operators will be useful in our calculation [27] O Here Using the spin symmetry of heavy quark system to relate the various matrix-elements of S-wave operators and P-wave operators, we have At the leading order of α s and v, these NRQCD matrix-elements can be related to the Schrödinger wave functions at the origin within the color-singlet model [27] Here N c = 3 is the color number, M P and M S are the masses for 1 S 0 and 3 P 0 states of B c mesons, respectively, R nl (r) denotes the radial Schrödinger wave function of the B c meson with radial quantum number n and orbit-angular momentum l, the prime denotes a derivative with respect of r.

D. Tree-level matching
In this subsection, we explain how to extract the short-distance coefficient at tree-level. This extraction is done through matching the matrix-elements between vacuum and a pair of b quark and anti-c quark state in which heavy quarks move non-relativistically in their center of mass frame. Once the matching at tree-level is done, the extension to NLO is natural.
First we set the momenta for b-quark andc-quark with non-relativistic relative motion as where q µ andq µ are the residual momenta of b quark and anti-c quark, respectively, satisfying . The total momentum of heavy quark pair where M ≡ m b + m c . By use of equations of motion and the power-counting for the residual momenta, the on-shell spinors of quark and anti-quark can be expanded in v as Thus, we have the matrix-element at tree-level where M is the reduced mass of the quark-antiquark pair, and a, b are color indices for the quark and anti-quark, respectively. Note that (19) has an extra term proportional to the spin structure [q /, n / + Γ] compared with (2.43) in [26]. This extra term will make some LCDAs for the P-wave B c mesons more complicated than the ones for quarkonia.
By the matching equation (11) and inserting Γ = 1 , γ 5 , γ α ⊥ and γ α ⊥ γ 5 , we can get the distribution parts of LCDAs and decay constants at tree-level for the S-wave B c mesons, for the B c meson in 1 P 1 state, for the B c meson in 3 P 0 state, for the B c meson in 3 P 1 state, and for the B c meson in 3 P 2 state. Here the superscript (0) denotes the quantities which are calculated at the leading order of α s . By setting m b = m c = m, i.e. x 0 = 1/2, we recover the results fφ(x) for quarkonia at tree-level given in [26].

III. THE CALCULATIONS OF THE LCDAS AT NLO
At the NLO of α s and in the dimensional regularization (DR) scheme, the bare matrixelement of Q[Γ](x) in the Feynman gauge is written as 1 Here α s = g 2 s /(4π) is the running strong coupling, C F = (N 2 c − 1)/2N c with N c = 3 is rank-2 Casimir in the fundamental representation of SU(3) group, and with γ E = 0.5772... being the Euler constant, and d = 4 − 2ε the dimension of the spacetime where the extra dimension ε is used to regulate both of the ultraviolet and infrared divergences appearing in our calculation. The loop momentum k has been decomposed into where we set n ± within 4 dimensions, and k ⊥ runs over the remaining d − 2 dimensions.

A. Matching procedure by method of threshold expansion
The standard procedure to extract the short-distance coefficient C n Γ (x, µ) in (11) at the NLO of α s is to calculate the matrix-elements of Q[Γ](x, µ) and O NRQCD Γ,n at NLO separately, and then match them together, as what done in [23]. However, in this paper, we will perform the matching procedure by adopting the threshold expansion developed in [28], as in [24,26].
In the threshold expansion, the v-expansion of full QCD loop integration is reproduced by a sum of several loop integrations over different loop momentum regions which are characterized by the different power-countings under v-expansion. Since our definitions of the LCDAs are boost-invariant, we choose v µ = (1, 0), i.e. the quark and anti-quark pair is in its rest frame. Thus, the most important integration regions which contribute to the full loop integrals in (25) are [28]: As commonly known, the contributions from soft, ultra-soft and potential regions reproduce the loop corrections to the matrix-element of O NRQCD Γ,n in NRQCD, and the contribution from the hard region gives to the loop corrections to the short-distance coefficient C n Γ (x, µ). Thus, the threshold expansion can simplify the matching procedure greatly. We only need to calculate the hard integration to extract the short-distance coefficients C n Γ (x, µ). In order to get the results up to O(v), we need the following v-expansions of the loop integrands in the hard region. For examples, for the denominators of the integrands, and for the numerators of the integrands. The delta-functions appearing in (25) should be also expanded in v.
After the expansion, a general hard loop integration will look like which can be done straightforwardly by integrating over n − k first by use of residue theorem, and then integrating over k ⊥ . The final result of the integral is an integration over n + k within [−m b n + v, m c n + v], which will be translated to an integration over a light-cone fraction, say y, within [0, 1] as in [26].
With Γ = 1 , γ 5 , γ α ⊥ and γ α ⊥ γ 5 (note that the index α ⊥ runs within 4-dimensions since it is from the definition of the LCDAs), we find that γ ρ n / + Γγ ρ ≡ c n / + Γ n / + Γ where in the NDR scheme and The hard part of the renormalized matrix-element reads where the on-shell renormalization constant for the heavy quark is with Q = b or c, and the renormalization kernels for the operator Q[Γ](x) in the MS scheme are with the Brodsky-Lepage kernel being In total, the hard part of the renormalized matrix-element can be expanded up to O(v) as where S i (i = 0, 1, 2, 3) are both ultraviolet and infrared finite distributions over the lightcone fraction x. Eventually, after substituting Γ with 1, γ 5 , γ α ⊥ and γ α ⊥ γ 5 , one can extract all the three LCDAs for the S-wave states f PφP (x; µ), f Vφ V (x; µ) and f ⊥ Vφ ⊥ V (x; µ) from S 0 , two of the LCDAs for the P-wave states f ⊥ 1Aφ ⊥ 1A (x; µ) and f 3Aφ 3A (x; µ) from S 1 , two LCDAs for the 3 P 2 states f Tφ T (x; µ) and f ⊥ T φ ⊥ T (x; µ) from S 3 , and the rest three LCDAs for the P-wave states, f 1Aφ 1A (x; µ), f SφS (x; µ) and f 3Aφ ⊥ 3A (x; µ) from both S 2 and S 3 .

D. Final results for the LCDAs at NLO
Here we present the final results for the LCDAs at the NLO of α s and leading order of v. The three LCDAs for the S-wave B c mesons arê and the corresponding decay constants are Here, ∆ = 0 for the NDR scheme, and ∆ = 1 for the HV scheme. Similarly, the seven LCDAs for the P-wave B c mesons arê and the decay constants The +++ , ++ , and + functions used in the above expressions are defined as

IV. DISCUSSIONS AND SUMMARY
In this paper, we have calculated all ten twist-2 LCDAs for the S-wave and P-wave B c mesons, up to the NLO of α s and the leading order of v. These LCDAs are re-factorized into the products of perturbatively calculable distribution parts and universal NRQCD matrixelements for vacuum to B c state transition. And by use of the spin symmetry, the number of the non-perturbative NRQCD matrix-elements is reduced to two. Thus, such reduction of the non-perturbative inputs will potentially improve theoretical predictive power in hard exclusive productions of B c mesons within the framework of collinear factorization.
On the other hand, as we mentioned before, our results of LCDAs for B c mesons can be used to check the asymptotic behavior of the NRQCD predictions for the hard exclusive B c meson productions as in [25], and resum the large α n s ln n Q 2 /m 2 Bc terms by use of the ERBL equations as what done for the hard exclusive quarkonia productions in [32].
With the excellent running of the LHC experiments, a great deal of data of W , Z and the Higgs bosons has been accumulated. This makes it possible to study many rare and even very rare decays of W , Z and Higgs bosons experimentally in the near future. Recently, there are a number of theoretical researches on exclusive radiative decays of W , Z and Higgs bosons, including decays to a photon and a quarkonium or a B c meson, within the collinear factorization, in order to test the QCD precisely or probe the Yukawa couplings of light quarks [33][34][35][36][37][38][39]. Thus, our results can be applied to such processes. However, we should mention, that the study of B c exclusive production through weak charge-current is almost infeasible in experiments since the production rates is greatly suppressed by the CKM factor |V cb | 2 ∼ 10 −3 [39] . We think, that the study on B c exclusive productions through flavor-conserving processes, for instance Z 0 → B + c B − c , is more plausible experimentally. Besides the exclusive processes, our results can also be applied to study the B c semiinclusive productions in hadron colliders at relatively low p T region. Some researchers have pointed out that the double parton fragmentation mechanism may become as important as the single-parton fragmentation mechanism for quarkonia productions at colliders at relatively low p T region [40][41][42][43][44]. In the corresponding factorization formula, the double-parton fragmentation functions serve as very important non-perturbative quantities. The colorsinglet parts of these double-parton fragmentation functions for quarkonia can be related to the LCDAs of quarkonia [45,46]. Therefore, with the same reasoning, our results of LCDAs for B c mesons can deduce the color-singlet parts of the double-parton fragmentation functions for B c mesons.
Although we have calculated the LCDAs up to NLO of α s and leading order of v, the corresponding relativistic corrections may be as important as, or even more important than, the radiative corrections. Thus, in order to test the applications of LCDAs in B c meson productions precisely, one should consider the relativistic corrections too. We leave this task to future work.