The $B \to \rho $ helicity form factors within the QCD light-cone sum rules

We study the $B \to \rho$ helicity form factors (HFFs) by applying the light-cone sum rules up to twist-4 accuracy. The HFF has some advantages in comparison to the conventionally calculated transition form factors, such as the HFF parameterization can be achieved via diagonalizable unitarity relations and etc. At the large recoil point, only the $\rho$-meson longitudinal component contributes to the HFFs, and we have $\mathcal{H}_{\rho,0}(0)=0.435^{+0.055}_{-0.045}$ and $\mathcal{H}_{\rho,\{1,2\}}(0)\equiv 0$. We extrapolate the HFFs to physically allowable $q^2$-region and apply them to the $B \to \rho$ semileptonic decay. We observe that the $\rho$-meson longitudinal component dominates its differential decay width in low $q^2$-region, and its transverse component dominates the high $q^2$-region. Two ratios $R_{\rm low}$ and $R_{\rm high}$ are used to characterize those properties, and our LCSR calculation gives, $R_{\rm low}=0.967^{+0.305}_{-0.284}$ and $R_{\rm high}=0.219^{+0.058}_{-0.070}$, which agree with the BaBar measurements within errors.


I. INTRODUCTION
The B-meson decays are important for precision test of standard model (SM) and for seeking of new physics beyond the SM. Within the framework of SM, they can be used to fix the masses and couplings of the basic particles, research the CP-violation phenomena, determine more precise values for the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and etc, cf. Refs. [1][2][3][4][5][6][7][8].
For the B-meson decays, one has to deal with the oneparticle, the two-particle, and the three or more particle matrix elements. Those hadronic matrix elements are key components for extracting useful information on the underlying flavor transitions and studying the decay constants, the transition form factors (TFFs), the mixings and decay amplitudes. The γ-structures of those non-perturbative hadronic matrix elements can be decomposed into Lorentz-invariant structures by using covariant decomposition, leading to basic TFFs for various decay channels. Specifically, for the B → light vector meson decays, we need to deal with seven TFFs for the hadronic matrix elements [9,10], which are shown in Table I. For convenience, we also present the relations among the B → vector meson helicity form factors (HFFs) and the hadronic matrix elements in Table I. The B → light vector meson decays have been analyzed by various experimental groups, such as the BaBar collaboration [11,12], the Belle collaboration [13], the LHCb collaboration [14,15], the ATLAS collaboration [16], the CLEO collaboration [17]. On the other hand, the TFFs/HFFs for the B → light vector meson decays have been calculated under various approaches, such as the light-cone sum rules (LCSR) [18][19][20][21][22][23][24][25][26][27][28], the lattice QCD (LQCD) [29][30][31][32][33][34][35][36], the perturbative QCD (pQCD) [37][38][39][40][41], or some Phenomenological model [42,43]. Those approaches are complementary to each other, which are applicable for different q 2 -region. The pQCD approach is valid in low q 2 -region, the LCSR is applicable in small and intermediate q 2 -region around m 2 b − 2m b χ (χ ∼ 500 MeV is the typical hadronic scale of the decay) and the LQCD is applicable in high q 2 -region. Among them, the LCSR prediction can be extrapolated to whole q 2 -region, thus providing an important bridge for connecting various approaches.
There are large differences for the predicted and measured B → ρ decay widths at the large q 2 -region, c.f. Refs. [12,24,32]. In the paper, we shall adopt the LCSR approach to recalculate the B → ρ hadronic matrix elements. In different to previous LCSR treatment [26,27], we shall express the hadronic matrix elements by using the HFF with the help of the covariant helicity projection approach [44]. The HFFs are also Lorentz-invariant functions which can be formally expressed as the linear combination of the usually adopted TFFs.  There are some advantages for the use of HFF [9]: I) Dispersive bounds on the HFF parameterization can be achieved via the diagonalizable unitarity relations; II) There are relations between the HFFs and the spin-parity quantum numbers, especially when taking the heavyquark and/or large-energy limit. Thus, they can be conveniently adopted for considering the contributions from the excited states. The relations among the HFFs and the low-lying states can be obtained by relating the dominant poles in the LCSRs to those low-lying resonances. We present the masses of low-lying B d resonances with explicit quantum numbers J P in Table II, which shall be used in our numerical calculations; III) The LCSRs for the B → V HFFs can be conveniently used for studying the polarized decay widths.
The remaining parts of the paper are organized as follows. In Sec.II, we give the calculation technology for the B → ρ HFFs within the LCSR approach. In Sec.III, we present the numerical results. By extrapolating those HFFs to the whole q 2 -region, we study the properties of the B-meson semileptonic decay B → ρℓν ℓ . Sec.IV is reserved for a summary.

II. CALCULATION TECHNOLOGY FOR THE B → ρ HFFS
As for the B → ρℓν ℓ semileptonic decays, we need to deal with the hadronic matrix element: where k = (k 0 , 0, 0, | k|), ε α (k) are ρ-meson longitudinal (α = 0) and transverse (±) polarization vectors. In the B-meson rest frame with the z axis along the ρ-meson moving direction, and we have where The polarization vectors satisfy k · ε α (k) = 0.
As proposed by Ref. [44], one can adopt the covariant helicity projection approach to study those hadronic ma-trix element (1). The off-shell W -boson has similar polarization vectors as those of ρ-meson, e.g. the off-shell W -boson with momentum q = (q 0 , 0, 0, −| q |) are ε 0 (q) = 1 where | q| = | k|, q 0 = (M 2 B − m 2 ρ + q 2 )/2m ρ , and the extra vector ε t (q) is the time-like polarization vector. The linear combinations of the transverse helicity projection vector ε ± (q) give Using the off-shell W -boson polarization vectors, one can project out the relevant HFFs from the hadronic matrix elements [9] where q = p − k. In the following, we shall not consider the time-like HFF (t), which can be treated by using the same way and has no contribution to semileptonic decay width due to chiral suppression. Following the standard LCSR procedures [7,21,45], we can derive the LCSRs for the B → ρ HFFs. We first define a two-point correlation function as and j † B (0) = im bb (0)γ 5 q(0) which has the same quantum state of the B-meson with J P = 0 − , and σ = (0, 1, 2).
In the time-like q 2 -region, one can insert a complete series of the intermediate hadronic states in the correlator (10) and single out the pole term of the B-meson lowest pseudoscalar, where B|biγ 5 q|0 = m 2 B f B /m b with f B being the Bmeson decay constant. By replacing the contributions from the higher-level resonances and continuum states with the dispersion relations, the invariant amplitudes can be rewritten as where s 0 stands for the continuum threshold parameter and the ellipsis is the subtraction constant or the finite q 2 -polynomial, which has no contribution to the final sum rules. The spectral densities ρ H σ (s) can be approximated by using the ansatz of the quark-hadron duality [46] for the momentum transfer, which correspond to small light-cone distance x 2 ≈ 0, the correlator (10) can be calculated by using the operator product expansion (OPE). By using the b-quark propagator given by Ref. [20], we obtain The nonlocal matrix elements can be expressed in terms of the ρ-meson LCDAs of various twists [21,47], which are put in the Appendix. The LCSRs for the B → ρ HFFs are then ready to be derived by equating the correlator in the time-like and space-like regions due to analytic property of the correlator in different q 2 -regions. After applying the Borel transformation, which removes the subtraction term in the dispersion relation and exponentially suppresses the contributions from unknown excited resonances, we get the required LCSRs for the HFFs: where we have implicitly set the factorization scale as µ.
Up to twist-4 accuracy, the needed ρ-meson light-cone distribution amplitudes (LCDAs) are grouped in Table III, in which δ = m ρ /m b ∼ 0.16. Since the contributions from the twist-4 terms themselves are numerically small, we thus directly adopt the twist-4 LCDA model derived from the conformal expansion of the matrix element to do the numerical calculation [47]. Contribu-  [21].
Using those constraints, we can obtain the LCDA at the scale of 1 GeV, whose behavior at any other scales can be achieved via the renormalization group evolution [57]. The LCDA at any other scales can be obtained by using the conventional evolution equation. We present the parameters of φ ⊥ 2;ρ and φ 2;ρ in Table IV and V, and the corresponding curves in Fig.1. Those two LCDAs are close in shape, both of which change from a convex behavior to a doubly humped behavior with the increment of the second Gegenbauer moment.   Fig.2 shows how the LCDA φ λ 2;ρ changes with m q . It is drawn by fixing all other input parameters to be their central values, and the LCDA parameters are refitted by fixing the second Gegenbauer moments a ⊥ 2 (1 GeV) = 0.14 and a 2 (1 GeV) = 0.15. As shown by Fig.(2), different choices of light constitute quark m q can make sizable  effects to the LCDA. Thus when discussing the uncertainties, the LCDA uncertainties from different choice of m q shall also be included. As for the LCSRs of the HFFs, we also need to know the continuum threshold s 0 and the allowable range of the Borel parameter M 2 , i.e. the so-called Borel window. The continuum threshold s 0 , being as the demarcation of the B-meson ground state and higher mass contributions, is usually set as the one that is close to the first known resonance of the B-meson ground state. For the purpose, we set s 0 as 34.0 ± 1.0 GeV 2 , which indicates that the excitation energy is around 0.45 GeV to 0.65 GeV. The correlator is expanded over 1/M 2 , when we calculate it to all-power series, it shall be independent to the choice of 1/M 2 . However we only know its first several terms, and we have to set a proper range for M 2 . As a conservative prediction, we require the continuum contribution to be less than 65% of the total LCSR to set the upper limit of M 2 , e.g.
Generally, the net contributions from the highest-twist terms increase with the decrement of M 2 , and the lower limit of M 2 is usually fixed by requiring the highest-twist contributions to be small so as to ensure the convergence of the twist expansion. For the present considered three HFFs H ρ,σ , the twist-4 contributions behave quite differently. As a unified criteria for those HFFs, we adopt the flatness of the HFFs over M 2 to set the lower limit of M 2 , e.g., we require the HFFs to be changed less than 1% within the Borel window. The determined Borel window M 2 are listed in the Table VI. We take the HFFs H ρ,σ (q 2 = 10) as explicit examples to show how the HFFs change with the input parameters. The results are collected in Table VII, where errors from the B-meson decay constant f B , the b-quark pole mass m b , the ρ-meson mass m ρ , the factorization scale µ, the Borel parameter M 2 and the continuum threshold s 0 . Table VII shows that the main errors of those HFFs come from the parameters m b , f B , and s 0 , whose effects could be up to ∼ 10% − 20% accordingly.

B. Extrapolation of the HFFs to all q 2 -region
The LCSR method is only valid for large energy of the final-state vector meson, e.g. E ρ ≫ Λ QCD . It implies a not too large q 2 via the relation q 2 = m 2 B − 2m B E ρ , e.g.
On the other hand, the allowable physical range for q 2 is about [0, 20.3] GeV 2 , in which the upper limit is fixed by q 2 max = (m B −m ρ ) 2 [21]. We adopt the method suggested by Ref. [9] to do the extrapolation of the HFFs, i.e. the where t ∈ [0, 1 2 , · · · , 27 2 , 14] GeV 2 . We put the determined parameters a ρ,σ k in Table VIII, in which all the input parameters are set to be their central values.
We put the extrapolated B → ρ HFFs H ρ,σ (q 2 ) in Fig.(3), where the shaded band stands for the squared average of all the mentioned uncertainties. All the HFFs are monotonically increase with the increment of q 2 , and at the large recoil point, we have H ρ,0 (0) = 0.435 +0.055 In this subsection, we apply the HFFs H ρ,σ (q 2 ) to study the semileptonic decay B → ρℓν ℓ , which is frequently used for precision test the SM and for searching of new physics beyond SM.
Within the SM, the total differential decay width of B → ρℓν ℓ can be written as where the terms proportional m 2 ℓ have been suppressed due to the large chiral suppression for the light leptons with negligible masses, the parameter G = G 2 F /(192π 3 m 3 B ) with the fermi coupling constant G F = 1.166 × 10 −5 GeV −2 [48], and the phase-space factor λ(q 2 ) = (m 2 B +m 2 ρ −q 2 ) 2 −4m 2 B m 2 ρ . Our LCSR prediction for the differential decay width 1/|V ub | 2 × dΓ/dq 2 is presented in Fig.(4), where the uncertainties from all error sources are added in quadrature. As a comparison, the UKQCD group LQCD prediction [29] and their extrapolated LQCD prediction (with the help of the heavy quark symmetry, kinematic constraints and the LCSR scaling relations) [32] are presented as a comparison. Our LCSR prediction is consistent with the LQCD prediction within the intermediate q 2 -region; however our LCSR prediction prefer a larger 1/|V ub | 2 × dΓ/dq 2 in low q 2 -region and a smaller 1/|V ub | 2 × dΓ/dq 2 in high q 2 -region.
As a minor point, we pick out the uncertainty caused FIG. 4. The LCSR prediction for the differential decay width 1/|V ub | 2 × dΓ/dq 2 . The LQCD prediction [29] and the extrapolated prediction of UKQCD group by using of the LQCD result [32] are presented as a comparison. The shaded bands are their theoretical errors. by varying m q ∈ [0.2, 0.4] GeV from the above uncertainty, and present the LCSR prediction for the differential decay width 1/|V ub | 2 ×dΓ/dq 2 in Fig.(5). It shows the uncertainty caused by m q is small, which agree with the observation of Table VII that   We present the total decay width Γ/|V ub | 2 in Table  IX, in which we also present the ratio Γ /Γ ⊥ as a useful reference. The total decay width, Γ = Γ +Γ ⊥ , where the decay width for the ρ-meson longitudinal components Γ is defined as and the decay width for the ρ-meson transverse components Γ ⊥ is defined as Table IX shows that, due to the large cancelation of the differences among different q 2 -regions, the difference for the total decay width Γ between the integrated LCSR and LQCD predictions shall be greatly suppressed.
We present the LCSR predictions for the polarized differential decay widths 1/|V ub | 2 × dΓ /dq 2 and 1/|V ub | 2 × dΓ ⊥ /dq 2 in Fig.(6), in which all the input parameters are set to be their central values. Fig.(6) shows that the differential decay widths for the final-state ρ-meson transverse and longitudinal components behave quite differently. The longitudinal differential decay width dΓ /dq 2 monotonously deceases with the increment of q 2 , and the transverse differential decay width dΓ ⊥ /dq 2 shall first increase and then decrease with the increment of q 2 . Both of them tend to zero for q 2 → q 2 max due to the phasespace suppression. As a result, the ρ-meson longitudinal component dominates low q 2 -region, and its transverse component dominates high q 2 -region 1 .
FIG. 7. The LCSR predictions for the ratios R low and R high . The BaBar [12] results and the values by using extrapolated LQCD predictions [32] are also presented.

V. SUMMARY
We have studied the HFFs for the B-meson semileptonic decay B → ρℓν ℓ within the LCSR approach. Fig.(3) shows that the extrapolated HFFs within the whole q 2region. At the large recoil point, only the ρ-meson longitudinal component contributes, e.g. H ρ,0 (0) = 0.435 +0.055 −0.045 and H ρ,{1,2} (0) ≡ 0, where the errors are squared averages of the considered error sources. By applying the extrapolated HFFs to the semileptonic decay B → ρℓν ℓ , we observe that the differential decay width 1/|V ub | 2 × dΓ/dq 2 , as shown by Fig.(4), is consistent with the Lattice QCD prediction within the intermediate q 2 -region. However our LCSR prediction prefer a larger 1/|V ub | 2 × dΓ/dq 2 in low q 2 -region and a smaller 1/|V ub | 2 × dΓ/dq 2 in high q 2 -region. More explicitly, Fig.(6) shows that the longitudinal decay width dominates the lower q 2 -region and the transverse one dominates the higher q 2 -region. Two typical ratios R low and R high can be used to test those properties. Our LCSR calculation shows that R low = 0.967 +0.308 −0.285 and R high = 0.219 +0.058 −0.070 . Fig.(7) shows that those predictions agree with the BaBar measurements within errors. Thus by using the HFFs with definite polarizations, some useful information can be achieved. A more precise measurement of those ratios shall be helpful for testing various calculation approaches.