D∗Dρ and B∗Bρ strong couplings in light-cone sum rules

Abstract We present an improved calculation of the strong coupling constants gD∗Dρ and gB∗Bρ in lightcone sum rules including the one-loop QCD corrections of leading power with ρ meson distribution amplitudes. We further compute the subleading-power corrections from two-particle and threeparticle higher-twist contributions at leading order up to twist-4 accuracy. The next-to leading order corrections to leading power contribution offset the subleading-power corrections to some extend numerically, and our numerical results are consistent with previous works from sum rules. The comparisons between our results and the existing model-dependent estimations are also made.


I. INTRODUCTION
This paper aims to give a more precise determination of D * Dρ coupling (g D * Dρ ) and B * Bρ coupling (g B * Bρ ). The couplings, describing the low-energy interaction among heavy mesons and light meson, are of great importance to understand the QCD long-distance dynamics. In the first place, the coupling is a fundamental parameter of the effective Lagrangian of heavy meson chiral perturbative theory (HMχPT) [1][2][3][4]. Phenomenologically, it describes the strength of the final state interactions [5] which are important in the generation of the strong phase within B decays [6,7]. Secondly, the coupling relates the pole residue of D(B) to ρ form factors at large momentum transfer by the dispersion relation. Furthermore, although the couplings have been extensively studied in the literature, the theoretical predictions exhibit a widespread of values.
Various theoretical approaches to determine the coupling have been suggested. With the D(B) to ρ form factors obtained at certain region of the momentum transfer from lightcone sum rules (LCSR) [8][9][10] and the lattice QCD [11], the corresponding pole residue which relates the coupling can be extracted with appropriate extrapolation for the form factors. Phenomenologically, the simplest way is the vector meson dominance (VMD) hypothesis which neglects the continuum spectral. Other modified parameterizations for the form factors have been proposed [12][13][14][15]. The strong couplings are estimated from HMχPT with VMD approximation [16][17][18][19]. However, these estimations have potential theoretical uncertainties.
Another strategy to obtain the strong coupling is calculation from first principles of QCD. We study the strong coupling constants g D * Dρ and g B * Bρ in LCSR by using double dispersion relation. The LCSR was proposed in [20][21][22] based on the light-cone operatorproduct-expansion (OPE) relative to the conventional QCD sum rules (QCDSR) method.
There have been several works regarding the couplings g D * Dρ and g B * Bρ , starting from [23] with including two-particle ρ DAs corrections up to twist-3 at leading order (LO). Years later, [24][25][26][27] improved the formal calculation by considering the two-particle twist-4 corrections [28] at LO. Meanwhile, g D * Dρ is also calculated with three-point QCDSR by taking into account the dimension-5 quark-gluon condensate corrections under flavor SU (3) symmetry [29]. With respect to previous works, we give a calculation including O(α s ) corrections to leading power contribution with the resummation of large logarithm to next-to leading logarithmic accuracy (NLL). For the subleading-power corrections, our results also include the three-particle twist-4 corrections at LO. The paper is organized as follows: in section II we calculate the leading power contributions up to NLO. Following the procedure in [33,34], the hard-collinear factorization is achieved in OPE region with the aid of evanescent operator [35,36] in the frame work of softcollinear effective theory (SCET) [37][38][39] and strategy of regions [40]. Moreover, procedures for analytic continuation and continuum subtraction are similar to [41,42]. Subleadingpower corrections including two-particle and three-particle corrections up to twist-4 at LO are calculated in section III. Section IV provides our numerical results and the phenomenological discussion. We will summarize this work in the last section.

A. Hard-collinear factorization at LO in QCD
The strong coupling constant g H * Hρ is defined by with pqηε = µναβ p µ q ν η * α ε β . Here we choose H * and H stand for D * − (B * 0 ) meson and D 0 (B + ) meson respectively, and ρ is ρ − . ε µ and η ν are the polarization vectors of the H * and ρ mesons respectively. We use the conventions 0123 = −1 and D µ = ∂ µ − i g s T a A a µ . The couplings of different charge states are related by isospin symmetry, for instance, We construct the following correlation function at the starting point where γ µ ⊥ = γ µ − / n/2n µ − / n/2 n µ , Q is the heavy quark field. The power counting scheme that we take is as follows where we have introduced two light-cone vectors n µ andn µ with n · n =n ·n = 0 , n ·n = 2.
On the hadronic level, taking advantage of the following definitions for decay constants the correlation function (3) can be written as where the second term counts the contributions from higher resonances and continuum states. The ellipses denote the terms that vanish after double Borel transformation.
After double Borel transformation we get the hadronic representation of the correlation For the boundary of the integral Σ, we take s + s = 2 s 0 with s 0 as the threshold of excited and continuum states. The Borel parameters associated with (p+q) 2 and q 2 are quite similar in magnitude, so we set the same value M 2 .
On the quark level, the leading-twist tree diagram is displayed in Fig. 1. The correlation function reads The spinor structure in (8) is the same as the axial-vector part of the correlation function in B → γ l ν in [33]. Following [33] we match the QCD results (8) to SCET. At leading power, we obtain where the tree level SCET operator matrix elements reads with u is the momentum fraction carried by quark. To establish hard-collinear factorization, we decomposed the SCET operator O A,µ in (9) into the light-ray operators where O j, µ (u ) = n · p 2 π dτ e −i u τ n·pξ (τ n) W c (τ n, 0) Γ j ξ(0) , The matching equation including the evanescent operator O E,µ reads at LO the coefficients is Using the definition of the leading-twist ρ DA [28] in appendix A, we get the leading twist tree level factorization formula The result (15) can be written in the form of double dispersion relation where the double dispersion density ρ LT,(0) is defined as Equating (6) and (16) and applying double Borel transformation, then subtracting the continuum states by using quark-hadronic duality, we obtain the master formula as follows where we have chosen the triangle integral region, s + s < 2 s 0 , for performing continuum subtraction [42], and the variables t and v are defined as The expression of wave function φ ⊥ (u, µ) in terms of Gegenbauer polynomials can be written as where b k is the function of Gegenbauer moments a(µ). The spectral density can be obtained as follows [41] Compute the corresponding integral of (18) we obtain the leading-twist strong coupling

B. Hard-collinear factorization at NLO in QCD
The one-loop diagrams are shown in Fig. 2. Only the hard region can generate a nonzero contribution since the collinear region corresponding the scaleless integral in dimensional regularization. With the NDR scheme of the Dirac matrix γ 5 the results of the one-loop where µ is given in (9). The box diagram is order O( ) in dimensional regularization, so has no contribution, which is also confirmed in [33,34]. Adding up the one-loop results, we get the NLO hard amplitude where C (0) i is given by (14). Here we have distinguished the renormalization scale ν from factorization scale µ because of the non-conservation of the pseudoscalar current in QCD.

III. THE SUBLEADING-POWER CORRECTIONS AT LO
In this section, we are going to perform the subleading power corrections to coupling constants, which are involved with two-particle and three-particle corrections up to twist-4.
The Q-quark propagator up to twist-4 in the background field is adopted [44]: A. The two-particle subleading-power corrections Insert the first term of (41) into correlation function (3), the correlation function in Eq. (3) can be written as where we used γ µ γ ν = g µν − i σ µν and σ µν γ 5 = i/2 µναβ σ αβ . Substituting the DAs into (42), we gain the higher twist two-particle corrections The twist-3 g Similar to the leading-twist LO case, after applying double Borel transformations and subtracting the continuum states by using quark-hadronic duality, we obtain the strong coupling constant for two-particle subleading-power corrections at LO The derivation of the spectral density and the corresponding integral are listed in appendix C.
B. The three-particle subleading-power corrections At tree level, the correlation function of three-particle qqg corrections can be written as Employing the definitions of three-particle ρ DAs, we obtain where Here we used the following definition Comparing with the leading-twist corrections (27), the three-particle twist-4 corrections are suppressed by O(Λ 2 QCD /m 2 Q ). The LCSR results for three-particle corrections at LO are as follows where the hat functions of the DAs Φ 3P are defined as Employing the conformal expansion of the three-particle DAs in appendix B, we obtain Collecting (38) (45) and (52), the final LCSR reads where F LT,(1) is defined in (36).

A. Input parameters
The masses of quarks in MS scheme and the values of decay constants are listed in Table   I, and the the values of the parameters in ρ DAs are listed in Table II [9,28]. The solution to the two-loop evolution of the Gegenbauer moment a ⊥ n (µ) is where the explicit expressions of E T,n and d T,n can be found in [34]. The scale evolution of other nonperturbative parameters to leading logarithmic accuracy is listed in appendix B.    [45,46] and decay constants [9,[47][48][49], with the scale dependent quantity f T ρ given at µ 0 = 1.0 GeV.

B. Theory predictions
In Table III   contributions up to NLO with resummation at NLL accuracy; g 2P,LL is from sub-leading power two-particle corrections at LO and g 3P,LL is from three-particle corrections at LO. power counting scheme where the parameter χ is of order 1 GeV. We find the scaling of strong couplings from leading-twist contributions in (38) which are independent of m Q , so D * and B * channels have similar values.
The two-particle subleading-power corrections g 2P in (45) can be divided into three parts, the leading-twist corrections at NLP g 2P,LT , the twist-3 DA g (a) ⊥ corrections g 2P,a , the twist-4 DA A T corrections g 2P,A . They have the following scaling behavior the corrections g 2P are dominant by g 2P,a . The suppressed terms g 2P,LT and g 2P,A have opposite sign with g 2P,a and g 2P,LT ∼ 1/m Q , so g 2P for B * channel is larger than D * channel.
Similarly, for the three-particles corrections, from (52) the scaling is It's obvious that g B * Bρ from three-particle corrections is small.

C. Comparison with other approaches
In Table V, we compare our results with other LCSR and QCDSR calculations. Since the NLO effects of leading power cancel with the subleading-power corrections numerically, our results are close to the previous sum rules calculation within errors.
Next we extract the couplings from from factors. The B → ρ form factors V (q 2 ) and T 1 (q 2 ) which relate g B * Bρ are defined as From the dispersion relation of form factor F i (q 2 ) the strong coupling relates to the pole of the form factors at the unphysical point q 2 = m 2 B * . Then we have the relation where f T B * is the tensor coupling of the B * meson and defined as We choose the size f T B * = f B * . Using (62) we extract the coupling g B * Bρ from the recent LCSR works [9,10] in table VI. As it shows, our central value is smaller than the extrapolation of LCSR form factors. When considering the errors, our result is compatible with [10].
The HMχPT effective Lagrangian to parametrize the H * HV coupling can be written as where whereρ is 3 × 3 matrices for light meson nonet, and the heavy H and H * mesons are represented by the doublet field H a with the conventional normalization. The parameter In the chiral and heavy quark limits, we have the following relation for the From the value of g B * Bρ , we find at leading power λ = 0.23 ± 0.03 GeV −1 . In

V. CONCLUSION
We compute the D * Dρ and B * Bρ strong couplings to subleading-power in LCSR. The large-distance dynamics are incorporated in the ρ DAs. We calculated the O(α s ) corrections to leading power of the sum rules. For the γ 5 ambiguity, we take NDR scheme including evanescent SCET operator. Moreover, the evolution of Gegenbauer moments is at two-loop accuracy. The subleading-power corrections are calculated at LO by accounting the twoparticle and three-particle wave functions up to twist-4. The analytical results of double spectral density are obtained, and with respect to the previous work, we also performed a continuum subtraction for the higher-twist corrections. The LO results are independent of the choice of duality region since the special form of the double spectral density.
Numerically, the NLO corrections decrease the tree-level results by 10% and 20% for D * Dρ and B * Bρ respectively. Contrarily, the subleading-power corrections at LO can give rise to the values about 20% and 30% for D * Dρ and B * Bρ respectively. Summing up all the contributions, our values are consistent with the predictions from previous sum rules works.
Moreover, we also predict the coupling λ in HMχPT at leading power. The central value of our result is smaller than the existing model-dependent estimations. A better understanding for this discrepancy is beneficial to shed light on the long-distance QCD dynamics. ACKNOWLEDGMENT We are grateful to Prof. Cai-dian Lü for helpful discussions. This work was supported in part by National Natural Science Foundation of China under Grant No. 11521505.