Radiative Leptonic $B_c\to \gamma \ell\bar\nu$ Decay in Effective Field Theory beyond Leading Order

We study the radiative leptonic $B_c\to \gamma\ell\bar\nu$ decays in the nonrelativistic QCD effective field theory, and consider a fast-moving photon. As a result the interactions with the heavy quarks can be integrated out, and thus we arrive at a factorization formula for the decay amplitude. We calculate not only the relevant short-distance coefficients at leading order and next-to-leading order in $\alpha_s$, but also the nonrelativistic corrections at the order $|\bold{v}|^2$ in our analysis. We find out that the QCD corrections can sizably decrease the branching ratio and thus is of great importance in extracting the long-distance operator matrix elements of $B_c$. For the phenomenological application, we present our results for the photon energy, lepton energy and lepton-neutrino invariant mass distribution.


I. INTRODUCTION
The search for new degrees of freedom can proceed under two distinctive directions. At the high energy frontier, new particles have different signatures with the standard model (SM) particles, and measurements of their production may provide definitive evidence on their existence. On the other hand, it is likely that low energy processes will be influenced through loop effects. Rare decays of heavy mesons, with tiny decay rates in the SM, are sensitive to the new degrees of freedom and thus can be exploited as indirect searches of these unknown effects, for a recent review see Ref. [1].
The B c meson is the unique pseudo-scalar meson that is long lived and composed of two different heavy flavors. Since this hadron is stable against strong interactions, its weak decays provide a rich phenomena for the study of CKM matrix elements, and also a platform to study the effects of weak interactions in a heavy quarkonium system [2,3]. In the past decades it has received growing attentions since the first observation by the CDF collaboration [4]. This can be particularly witnessed by the recent LHCb measurements of the B c lifetime [5,6], the decay widths of B c → J/ψπ and B c → J/ψ ν [7,8], and various other decay modes [9][10][11][12]. One may expect that more decay channels of B c can be measured by the LHCb, ATLAS and CMS experiments [13][14][15].
On theoretical side, various approaches have been applied to calculate the decay width of B c decays , but most of them are phenomenological. Since both constituents of the B c are heavy and can only be treated nonrelativistically, an effective field theory can be established [53]. Taking the B c → J/ψ ν as the example, one may derive the conjectured non-relativistic QCD (NRQCD) factorization formula for its decay amplitude: where the O f f i,j are constructed by low energy operators. The short-distance, or hard, contributions at the length scale 1/m b,c are encapsulated into the coefficients C ij that can be computed in perturbation theory.
The long-distance, or soft part of, matrix elements have to be extracted in a nonperturbative approach, for instance the Lattice QCD simulation, or constrained by much simpler processes for instance the annihilation modes B c → ν and B c → γ ν. However, the usefulness of the B c → ν is challenged by two aspects. Firstly its decay rate is given by in which the suppression factor m 2 /m 2 Bc arises from the helicity flip. As a result, the B c → µν µ and B c → eν e have tiny branching fractions that may be out of the detector capability at the current experimental facilities. Secondly, there is only one physical observable, namely the decay rate, and thus the B c → ν is not capable to uniquely determine all, typically more than one when relativistic corrections are taken into account, long-distance matrix elements (LDMEs).
On the contrary, the B c → γ ν can provide a wealth of information [54][55][56][57][58], in terms of a number of observables ranging from the decay probabilities, polarizations to an angular analysis. It is interesting to notice that the counterpart in B sector, B → γ ν, has been widely discussed towards the understanding of the B meson light-cone distribution amplitudes [59][60][61][62][63]. The small branching fraction of B c → γ ν can be compensated by the high FIG. 1: Leading order Feynman diagrams for the radiative leptonic B c → γµν µ decay in the SM. The lepton µ can also be e or τ . The photon emission from a virtual W -boson shown in the second panel is suppressed by 1/m 2 W compared to the other contributions.
luminosity at the ongoing hadron colliders and the under-design experimental facilities. The main purpose of this paper is to explore the B c → γ ν at next-to-leading order (NLO) in α s and in |v| 2 , which shall catch up the progress in the B c → ν [55,64]. For the leptonic decay constant, the two-loop calculation is also available in Ref. [65]. The rest of this paper is organized as follows. In Sec. I, we will derive the formulas for various partial decay widths of B c → γ ν. Sec. III is extensively devoted to the nextto-leading order calculation. We will discuss the phenomenological results in Sec. IV. We summarize our findings and conclude in Sec. V. We relegate the calculation details to the Appendix.
In the SM, leading order (LO) Feynman diagrams for the B c → γ ν decay are shown in Fig 1. The photon emission from a virtual W -boson is suppressed by 1/m 2 W compared to other contributions, and thus the second diagram in Fig. 1 can be neglected. Integrating out the off-shell W -boson, we arrive at the effective electro-weak Hamiltonian where V cb is the CKM matrix element. The decay amplitude, matrix element of the abo ve Hamiltonian between the B c and γ ν state, is responsible for the process B c → γ ν.

A. Differential decay widths
Since there is no strong interaction connection between the leptonic and hadronic part, the decay amplitude can be decomposed into two individual sectors: with the matrix elements encoding the hadronic effects: The first one defines the B c decay constant while the B c → γ transition is parametrized by two form factors: with the momentum transfer L = p Bc − k. Here and throughout this work we adopt the convention 0123 = +1. The above equations are similar with the parameterization of the B → γ form factors as given in Ref. [66]. The last term in Eq. (9) that is proportional to the B c decay constant has been added in order to maintain the gauge invariance of the full amplitude [67,68], and see appendix A for a derivation. Substituting Eqs. (7), (8), (9) into Eq. (5), we obtain where s l = L 2 and terms due to lepton mass corrections have been neglected. Apparently, this expression is gauge invariant. For the sake of simplicity, we have defined two abbreviations in the above 1 In terms of the decay constant and form factors, the differential decay width for the B c → γ −ν is given as 1 One shall distinguish the form factor v from the relative velocity v to be defined in the following.
where x k = 2E k /m Bc and y = 2E l /m Bc , and E k and E l is the energy of the photon and charged lepton in the B c rest frame, respectively. One can integrate out the E l and obtain The differential distributions can also be converted to using the relation: The θ l is the polar angle between the lepton flight direction and the opposite direction of the B c meson in the rest frame of the ν pair. Likewise one can integrate out the θ l

B. NRQCD factorization
The factorization properties for the B c → γ ν depend on the kinematics of the photon. In this work, we will not study the soft-photon contribution as discussed in B decays [69], and leave it for future work. In the region where the photon is a collinear (fast-moving) object, its interaction with heavy quarks is highly virtual and thus should be encoded in the short distance coefficients. In the NRQCD scheme, we only need retain those color-singlet operator matrix elements that connect the B c state to the vacuum. To the desired order, one expects the following factorization formula: where v denotes half relative velocity between the charm and bottom quarks in the meson, c f,V,A 0 and c f,V,A 2 are the dimensionless short-distance coefficients that can be expanded in terms of the strong coupling constant 2 . We shall calculate the one-loop corrections to the c f,V,A 0 , but give only the LO results for c f,V,A 2 since the latter ones are already powersuppressed. ψ Q and χ † Q represent Pauli spinor fields that annihilate the heavy quark Q and anti-quarkQ, respectively. Besides, one need note that the state |H(p) in QCD has the standard normalization:

A. Kinematics
Let p 1 and p 2 represent the momenta for the heavy quark Q and anti-quarkQ . Without loss of generality, one may adopt the decomposition: where P Bc is the total momentum of the quark pair. q is a half of the relative momentum between the quark pair with P Bc · q = 0. α and β are the energy fraction for Q andQ in the meson, respectively. The explicit expressions for all the momentum in the rest frame of the B c meson are given by In the rest frame, the meson momentum becomes purely timelike while the relative momentum is spacelike. One can obtain the relations

B. Convariant projection method
In the following calculation, we will adopt the covariant spin-projector method, which can be applied to all orders in v.
The Dirac spinors for the B c system may be written as where ξ λ is the two-component Pauli spinors and λ is the polarization parameters. It is straightforward to derive the covariant form of the spin-singlet combinations of spinor bilinears: with the auxiliary parameter ω = √ Here 1 c is the unit matrix in the fundamental representation of the color SU(3) group.

C. Perturbative matching
Due to the simplicity of the final state, one can directly match the QCD currents onto the NRQCD ones. To determine the values of c 0 and c 2 , we follow the spirit that those shortdistance coefficients are insensitive to the long-distance hadronic dynamics. As a convenient choice, one can replace the physical B − c meson by a freecb pair of the quantum number 1 S [1] 0 , so that both the full amplitude, A[cb( 1 S [1] 0 ) → γ ν], and the NRQCD operator matrix elements can be directly accessed in perturbation theory. The short-distance coefficients c i can then be solved by equating the QCD amplitude A and the corresponding NRQCD amplitude, order by order in α s . For this purpose, we introduce a decay constant and two form factors at the free quark level: Analogous to (18,19,20), one can write down the matching formula: where we have adopted the nonrelativistic normalization.
One can organize the full amplitudes defined in Eqs. (30,31,32) in powers of the relative momentum betweenc and b, denoted by q. To the desired accuracy, one can truncate the series at O(q 2 ), with the first two Taylor coefficients. We will compute both amplitudes at LO in α s in subsection III D, and the calculation at NLO in α s will be conducted in subsection III E.
The NRQCD matrix elements encountered in the above equations are particularly simple at LO in α s : where the factor √ 2N c is due to the spin and color factors of the normalizedcb( 1 S [1] 0 ) state. The computation of these matrix elements to O(α s ) will be addressed in subsection III F.

D. Tree-level amplitude
Adopting the above notation, one can easily obtain the tree-level amplitude for the decay constant where the q µ terms have been omitted and is defined as the reduced mass of thecb system. The vector current is similarly evaluated as: We have introduced the abbreviation Here e c = 2/3 and e b = −1/3 is the electric charge of the c and b quark, respectively. One can perform the Taylor expansion of the amplitudes in powers of q µ : Those terms linear in q should be dropped since this auxiliary momentum introduced at the quark level has no correspondence at the hadron level. In this paper, the O(|q| 2 ) contributions will be retained. In order to simplify the calculation in the covariant derivation, one shall use the following replacement: The result for the axial-vector current is a bit lengthy: In order to extract the A form factor, we only need to keep the µ term which corresponds to Feynman gauge · p Bc = 0, but we have explicitly checked the gauge invariance up to v 2 order. The tree-level NRQCD matrix elements for thecb have been given in Eq. (36), and thus the above results in Eqs. (37,39,42) lead to the tree-level Wilson coefficients In the above results, we have defined z = m c /m b andz = 1 + z. c f,0 i means the LO of Wilson coefficient c f i . It is interesting to notice that the Wilson coefficients c A,0 2 depends on the energy of the emitted photon, which will induce nontrivial behaviors as will be demonstrated later.

E. NLO amplitudes in QCD
Typical one-loop diagrams for the QCD corrections to the B c → γ ν decay are shown in Fig. 2. In calculating the one-loop amplitudes, we use the dimensional regularization to regulate the ultraviolet (UV) and infrared (IR) divergence.
The diagram (a) in Fig. 2 contributes to the NLO decay constant: with We have introduced the abbreviation 1 The heavy quark field renormalization and mass term are given as For the vector current form factor, the sub-diagram in Fig. 2 gives out the corresponding contribution where the auxiliary functions b i , c i , and d i are defined in Appendix B.
The counter-mass terms and wave function renormalization corrections give: For the axial-vector current form factor, the sub-diagram has gauge-dependent contributions, however, the summed result is gauge-invariant. We will show the detail in Appendix C.

F. NLO amplitudes in NRQCD
The NRQCD Lagrangian can be derived by integrating out the degrees of freedom of order heavy quark mass [53]: The replacement in the last line implies that the corresponding heavy anti-quark bilinear sector can be obtained through the charge conjugation transformation. L light represents the Lagrangian for the light quarks and gluons. The coefficients c D , c F , and c S have perturbative expansions in powers of α s , which can be written as c i = 1 + O(α s ). The matrix element of thecb to vacuum at NLO can be written as This is in agreement with the results in Ref. [70].

G. Determination of c i : Matching QCD to NRQCD
Up to α s and v 2 , one can expand the decay constant and form factors as Matching the QCD results onto the NRQCD, one can obtain the UV and IR finite shortdistance coefficient Note that the scale dependent term in the brace of Eqs. (61) and (62) will be cancelled each other, the residual dependence only lies in the strong coupling constant.
We first present numerical results for the decay constant f Bc : The strong coupling constant at the Z-boson peak is [71] α s (m Z ) = 0.1185 ± 0.0006, which corresponds to With these values, one can see the α s corrections can reduce the decay constant by approximately 9.5% − 16.2%.
To estimate the size of O(|v| 2 ) effects, one requests the size of non-perturbative LDMEs, for which we use Buchmüller-Tye (B-T) potential model [72]: For an estimate of q 2 , one may make use of the relative velocity. Using the heavy quarks kinetic and potential energy approximation [53], we have |v| α s (2m red |v| ) .
Choosing m b = 4.8 GeV and m c = 1.5 GeV, and using two-loop strong coupling constant, we get For a value v 2 Bc 0.186, we have As a result, the decay constant will be further reduced by about 9%. For the short-distance coefficients for B c → γ transition form factors V and A, our results are shown in Fig. 3. The solid line denotes the leading-order coefficient c  (18,19,20) is valid only for a hard photon, while the soft-photon contribution needs special treatment [69]. Thus a cut-off on the photon energy should be introduced, however we have checked that the cut-off will not affect the results significantly in Tabs. I and II. With the estimated long-distance matrix elements, results for differential distributions are given in Figs. 4 and 5, where the QCD and relativistic corrections are shown respectively. The integrated branching ratios of B c → γ ν and B c → ν are presented in Tabs. I and II. Ignoring the lepton mass, the branching ratio of B c → γeν e is identical to that of B c → γµν µ . The LO results are in agreement with Ref. [54][55][56][57][58] with the same input parameters. From the calculation, one can see that both the QCD and relativistic corrections give destructive contributions to the process B c → ν. However, relativistic corrections produce a constructive contribution to the B c → γ ν. Our results have demonstrated that the QCD and relativistic corrections are mandatory towards a more accurate extraction of the value of LDMEs for B c system.  In this work, we have analyzed the radiative leptonic B c → γ ν decays in the NRQCD effective field theory. NRQCD factorization ensures the separation of short-distance and long-distance effects of B c → γ ν into all order of α s . Treating the photon as a collinear object whose interactions with the heavy quarks can be integrated out, we arrive at a factorization formula for the decay amplitude.
We have calculated not only the short-distance coefficients at leading order and next-toleading order in α s , but also the nonrelativistic corrections at the order |v| 2 in our analysis. We found that the QCD corrections can sizably decrease the branching ratio, which has very important impact on extracting the long-distance operator matrix elements of B c . For phenomenological applications, we have estimated the long-distance matrix elements, which are further used to explore the photon energy, lepton energy and lepton-neutrino invariant mass distribution. These results can be examined at the LHCb experiment.

Appendix B: Passarino-Veltman integrals
The coefficients b i , c i and d i are related to the scalar Passarino-Veltman integrals defined in Ref. [76,77], and we have split the finite pieces Here we give the the results of divergence integrals.