Renormalization group equation improved analysis of $B \to \pi$ form factors from Light-Cone Sum Rules

Within the framework of $B$-meson light-cone sum rules, we compute the one-loop level QCD corrections to $B\to \pi$ transition form factors at small $ q^{2}$ region, in implement of a complete renormalization group equation evolution. To solve the renormalization group equations, we work at the"dual"space where the anomalous dimensions of the jet function and the light-cone distribution amplitudes are diagonal. With the complete renormalization group equation evolution, the form factors are almost independent of the factorization scale, which is shown numerically. We also extrapolate the results of the form factors to the whole $q^2$ region, and compare their behavior with other studies.


INTRODUCTION
The knowledge of the heavy-to-light transition form factors is of great importance because it is crucial for the determination of parameters of the standard model, and for the understanding of strong interaction dynamics.The B → π transition form factors, which are closely related to the CKM matrix element |V ub |, have been extensively studied in the literature.Because of the appearance of the endpoint singularity in the factorization of the heavy-to-light form factors, the form factors are regarded, in many approaches, as dominated by long-distance QCD dynamics and can be calculated only with non-perturbative methods, such as Lattice QCD, (Light-Cone) QCD Sum Rules, et al.In [1][2][3][4][5][6][7][8], the light-cone sum rules (LCSR) with pion light-cone distribution amplitudes (LCDAs) has been employed to study the B → π form factors, and nextto-leading-order (NLO) corrections to the twist-2 and the twist-3 terms as well as renormalization-group (RG) evolution effects [9] have been considered.In this paper, we use the LCSR with B-meson LCDAs [10,11] to calculate the B → π form factors.The B-meson LCSR has also been established independently in the framework of the soft-collinear effective theory (SCET) [12,13], where jet functions encoding the "hard-collinear" dynamics have been calculated up to O(α s ) [14,15].An alternative approach to analyse NLO corrections to the sum rules is suggested in [16], where the "method of regions" [17] was adopted to compute the vector form factor f + Bπ (q 2 ) and the scalar form factor f 0 Bπ (q 2 ) defined below Results of the B-meson LCSR were shown to be consistent with that of the SCET sum rules.There is another B → π transition form factor which is defined by the tensor current: and this form factor was not computed in [16].In the heavy-quark limit and at leading order in α s , the three independent form factors are proportional to one universal form factor at large recoil.If loop corrections are included, differences between the three form factors appear.The hard-spectator-scattering part can be factorized into a convolution of the perturbative function and the LCDAs of hadrons (B meson and pion) using the QCD factorization approach [18].Both QCD corrections to the universal form factor and to symmetry-breaking hard-spectator interactions have been calculated in [14,15].Using the method of regions, QCD corrections to the symmetry-conserving (universal) form factor and the symmetry-breaking part of the form factors are computed simultaneously.The first step towards using the method of regions is identifying leading regions which are in principle determined by the analytic structure of the Feynman diagram [19].Usually leading regions are closely related to momentum modes of external lines.In this work, there are three momentum modes from external lines, namely the hard (b-quark), the hard-collinear (interpolating current of pion) and the soft (light-spectator quark) modes.Thus, three momentum regions with scaling behaviors can contribute to the correlation function at leading power in λ, where λ ∼ Λ/m b and n µ and nµ are light-cone vectors, satisfying n 2 = n2 = 0 and n • n = 2.In our calculation, only these three regions give leading-power contributions to the correlation function, which is in agreement with the general analysis.The momentum of the fast-moving pion is chosen to be along the n direction.This momentum is also chosen to be hard-collinear and in the Euclidean region (p 2 < 0) to ensure the light-cone operator-product expansion (OPE) of the correlation function [10,11].The method of regions provides us a natural way to perform the factorization of the correlation function because contributions of different momentum regions are considered individually.It has been shown that the correlation function can be factorized into the convolution of the hard function, the jet function and the B-meson LCDA which describe dynamics of the hard, the hardcollinear and the soft regions, respectively.This procedure is equivalent to the two-step matching in the SCET where the hard (jet) function corresponds to the matching coefficient of SCET I (SCET II ) [20,21].
The correlation function is factorization-scale independent, thus the scale dependence of the hard function, the B-meson static decay constant, the jet function and the B-meson LCDA must be cancelled, which has been shown at one-loop level [15,16].At present, there is still no complete analysis of RG evolutions of all of the relevant functions.In [16], RG evolutions of the hard function and the B-meson decay constant were performed.The factorization scale in that paper was chosen to be about 1.5GeV, which is a typical hard-collinear scale.This choice is phenomenologically reasonable as the hard-collinear scale is not far from the non-perturbative scale.RG evolutions of the jet function and the B-meson LCDA are non-trivial since anomalous dimensions of these two functions are complicated.But on the conceptual side, a complete RG analysis is necessary.
B-meson LCDAs φ − B (ω) and φ + B (ω) are fundamental inputs of the B-meson LCSR.They are defined by the matrix element [22] 0| where [τ n, 0] is the Wilson line along the n direction, v and fB (µ) are the B-meson velocity vector and the B-meson static decay constant, respectively.The Fourier transformation of φ± B (τ ) leads to The scale dependence of φ ± B (ω) has been studied extensively [23,24].φ ± B (ω) obeys the RG equation with the anomalous dimension being the Lange-Neubert kernel [23].It is difficult to solve the Lange-Neubert equation in the momentum space.With the eigenfunction of the Lange-Neubert kernel found in [24], the RG equation of φ ± B (ω) can be diagonalized and readily solved.The RG equation of the jet function can be simplified in the same way since the scale dependence of the jet function should be partly cancelled by that of the B-meson LCDA.
In this work, we only consider leading-power contributions of the B → π form factors. B-meson threeparticle LCDAs can give subleading-power contributions to the form factors at leading order (LO) in α s and give leading-power contributions at NLO [25].Numerically, the subleading-power correction from threeparticle LCDAs is only a few percent of the leading-power contribution from two-particle LCDAs [10].Feynman diagrams related to three-particle LCDAs can be seen in [16].The calculation of effects of threeparticle LCDAs is expected to be rather complicated, and RG equations of the three-particle LCDAs are not completely available so far.Thus we leave this part for the future work.
This paper is arranged as follows.In Section 2 we briefly review the calculation of the B → π form factors at NLO and emphasize symmetry-breaking effects of these form factors.Then, the RG evolution of the correlation function is shown in details in the following section.In Section 4 we turn to the numerical analysis of the RG improved form factors. Concluding discussions are presented in Section 5.The Appendix includes two parts, the jet function in the "dual" space and dispersion integrals used in the sum rules.

THE B → π FORM FACTORS WITH B-MESON LCSRs
In order to derive the sum rules for the B → π transition form factors, we start with the correlation function where Γ µ = γ µ (σ µν q ν ) denoting the vector (tensor) current.According to the Lorentz-structure analysis, the correlation function is parameterized as Π p) nµ for the vector current, and Π µ (p, q) = Π T (n • p, n • p)ǫ µν q ν for the tensor current, where the anti-symmetric tensor ǫ µν = (n µ nν − n ν nµ )/2.We work in the rest frame of the B meson and the power-counting rule of the pion momentum reads In Euclidean region (n•p < 0) the light-cone OPE is employed to calculate the correlation function.Tree-level results are written as The correlation function can also be expressed in terms of the B → π form factors and the pion decay constant.For instance where ρ h (ω) represents the contribution of excited and continuum states which have the same quantum numbers as pion.The form factors are extracted by matching the partonic and the hadronic representations of the correlation function.After performing the Borel transformation which suppresses the contribution of excited and continuum states, we obtain the sum rules for the tensor form factor at LO Large-recoil symmetry relations [18] indicate: where ξ(n • p) is the Isgur-Wise function.These symmetry relations will be broken by QCD corrections.The method of computing radiative corrections to the correlation function has been introduced in [16].Adopting the diagrammatic factorization method [26], the hard-scattering kernel at NLO is determined by the matching condition where the first and the second terms on the right-hand side of the equation correspond to full-theory diagrams (Fig. 1) and effective diagrams (Fig. 2), respectively.The Lorentz index "µ" is suppressed in this equation.The definition of Φ (0,1) b d can be seen in [16].It has been proved that soft dynamics are completely cancelled between Π (1) b d and Φ (1) b d ⊗ T (0) .Thus the hard-scattering kernel T contains only contributions from hard and hard-collinear regions at leading power in λ, with the hard-region and the hard-collinear-region contributions corresponding to the hard function and the jet function, respectively.The correlation function is factorized as: where a = n, n, T .Results of C n , C n, J n , J n have been given in [16].C T and J T can be calculated following the same method.Nevertheless, using symmetry relations between the form factors, we are able to obtain C T and J T without repeating the calculation.Symmetry relations in Eq.( 11) are broken by loop corrections.
The difference between C T (J T ) and C n (J n) comes from the symmetry-breaking effect.At one-loop level large-recoil relations can be written as [18]: where r = n • p/m b .The first line of above equations has been confirmed in [14,16].
The first kind of symmetry-breaking effects, which is shown in the square brackets of Eq.( 14), arises from the hard function of weak-vertex correction Fig. 1(a).The weak tensor current ū(0)iσ µν q ν b(0) is not a conserved current, thus there exists operator-renormalization contribution to the correction function of the tensor current.This contribution produces an additional term to the hard function where ν is the renomalization scale.δC µ 2 term (µ should be changed to ν) in Eq.( 14).Inserting the hard function of f + Bπ [16] into Eq.(15), we obtain The second kind of symmetry-breaking effects ∆f T which corresponds to Fig. 1(b,c), comes from the "hard spectator" contribution in the QCD factorization.Since only the hard-collinear region contributes to Fig. 1(b,c) at leading power in λ, ∆f T is related to jet functions.The jet function J n , is the symmetry-conserving term.Only J (+) T corresponds to the symmetry-breaking effect.Comparing ∆f T with ∆f 0 , and employing the result of J (+) n,n , we find where η = −ω/n • p.

RG evolution
The hard and the jet functions contain logarithmic terms such as ln 2 µ n•p and ln µ n•p , which become large when the factorization scale µ is much smaller than n • p.The RG equation approach can be used to resum large logarithms to all orders in α s .In the following, we will present details of the RG evolution of Π T (n • p), and the approach can be easily generalized to Π n (n • p) and Π n(n • p).
Eq.( 16) contains the operator renormalization of the tensor current, so we need to perform evolutions both to the renormalization and the factorization scales.RG equations governing the renormalization-scale and the factorization-scale dependence are given by where Solutions to Eq.( 18) are written by As the hard function has been calculated at one-loop level, evolution functions of the hard function are also required to be expanded to O(α s ).The beta function appearing in Eq.( 20) reads thus anomalous dimensions γ h and γ T need to be expanded to two-loop level: where n ′ f is the number of the active quark flavors.While the cusp anomalous dimension Γ cusp (α s ) should be expanded to three-loop level: since the factor αs αs(µ h ) dα ′ s /β(α ′ s ) starts at O(α −1 s ).Note that the ν dependence of the form factor must be cancelled by that of the Wilson coefficient of tensor current.For phenomenological applications, this renormalization scale can be fixed at ν = m b .Then the corresponding evolution kernel reduces to 1, and the Wilson coefficient should be evolved to m b .We rewrite the second equation of Eq.( 20) as where the specific expression of U 1 (n • p, µ h , µ) at O(α s ) can be found in the appendix of [27].RG equations of the jet function and the B-meson LCDA have following forms These evolution kernels are non-diagonal.It is thus difficult to solve above RG equations in the momentum space.An alternative method was suggested in [24], where the LN kernel can be diagonalized by translating to the "dual" space.It was found in [28] that the basis of the dual space is the eigenfunction of the generator of special conformal transformations.The specific form of the transformation can be written by: The dual-space LCDA ρ − B (ω ′ , µ) satisfies a simpler RG equation where where Using the orthogonality of the Bessel function, we can express Substituting Eq.( 31) into Eq.(13), we obtain the factorization formula of the correlation function in the dual space: where j (−) T (ω ′ , µ) is the jet function in the dual space.In the second term of the RHS, J T (ω) and φ + B (ω) are not transformed to the dual space because it is not necessary to perform the RG evolution to this term, which will be explained in the first comment in the next subsection.We rewrite j − (ω ′ , µ) as j(ω ′ , µ) which satisfies the following RG equation The factorization-scale independence of the correlation function indicates where γ(α s ) is the anomalous dimension of fB (µ).The RG equation of fB (µ) is where and n f is the number of light quark flavors.The evolution factor of fB (µ) is: where z = α s (µ)/α s (µ h2 ).
The anomalous dimension of the jet function can be expressed as where hc = 0.There is no calculation about the two-loop anomalous dimension γ + −γ (1) ), since γ + is unknown yet.However, it has been checked numerically that the form factors are insensitive to γ (1) hc .In addition, NLO corrections should not be very large for the convergence of the α s expansion.We choose γ (1) hc = 0 in our calculation.The solution of Eq.( 33) can be obtained straightforwardly with j(ω ′ , µ hc ) given by where η′ = e −2γ E η ′ = −ω ′ /n • p.The detailed derivation of j(ω ′ , µ hc ) is given in the Appendix A. Collecting evolution factors of the hard function, the jet function, the B-meson LCDA and the B-meson decay constant together, we obtain the RG improved correlation function where We are now in the position to construct the sum rules for the three B → π form factors at NLO. Useful dispersion integrals are collected in the Appendix B. Performing the Borel transformation, we obtain the sum rules for the B → π form factors.The tensor form factor reads where with p F q (a 1 ...a p ; b 1 ...b q ; z) being the generalized hypergeometric function.Vector and scalar form factors f +,0 Bπ have similar forms where r ′ = n • p/m B .For above results, several comments are as follows: • We perform the complete RG evolution of φ − B terms.For φ + B terms, we only apply the RG evolution to the hard function and the B-meson decay constant not to J (+) i and φ + B since firstly the anomalous dimension of J (+) i is unknown yet and secondly evolution effects of the jet function and the B-meson LCDA will partially cancel each other.In principle we should resum logarithmic terms in J (+) i and φ + B to leading-logarithm level because the jet function starts at O(α s ).While due to the factorizationscale independence of the correlation function, we set µ to be a hard-collinear scale in the numerical analysis, which means that there are no large logarithms in J (+) i and φ + B .Also uncertainties, arising from the variation of µ, of these φ + B terms should be small since these terms are suppressed by α s .As we can see from the numerical analysis, the factorization-scale dependence of these terms is negligible.
• Dispersion integrals of the correlation function in the momentum space is non-trivial for the appearance of both pole and branch-cut singularities, which can be seen in the appendix of [16].While in the dual space, the imaginary part of the jet function in Eq.( 44) can be obtained much more easily, i.e., we can simply make the replacement n • p → Ωe iπ .

NUMERICAL ANALYSIS
In this section, we perform the numerical analysis of the B → π form factors. B-meson LCDAs serve as fundamental ingredients of the B-meson LCSR approach.Nevertheless there is a very limited knowledge about these LCDAs so far.Several phenomenological models of the B-meson LCDA φ + B are suggested.We employ three typical models [22,29,30]: Neglecting the contribution from the three-particle Fock state, φ − B (ω, µ 0 ) is determined by the Wandzura-Wilczek approximation [18] φ The parameter ω 0 equals the inverse moment of B-meson LCDAs, i.e., λ B is closely related to exclusive B-meson decays.The experimental data can only give a very rough constraint on this parameter.In the previous work [16], the inverse moment is determined by fixing f + Bπ (0) = 0.28 ± 0.03 which is the prediction of the pion LCSR [8].In this paper we adopt the same value as in [16] for comparison, i.e., ω 0 = 0.354 +0.038 −0.030 (GeV).Because the RG equation of φ − B is evaluated in the dual space, the corresponding dual-space expression of this LCDA is required.ρ − B has a similar meaning with Gegenbauer moments of the light-meson LCDAs.Explicit forms of ρ − B are To give a more intuitive picture of LCDAs in the dual space, we plot the ω ′ dependence of them in Fig. 3.The dual-space LCDAs are factorization-scale dependent.The RG evolution effect can modify the behavior of the original model [31], and this effect has been considered in our calculation of the form factors. Before presenting numerical results of the form factors, we first show behaviors of evolution factors of the hard function (U 1 (µ, µ h1 )), the jet function (U j (µ, µ hc )), the B-meson LCDA (Uρ(µ, µ 0 )) and the B-meson decay constant (U 2 (µ, µ h2 )), in Fig. 4. ω ′ is fixed at 1.0GeV when plotting this figure.This choice leads to large logarithmic terms in evolution kernels of the LCDA and the jet function, hence evolution effects Figure 5: The factorization-scale dependence of f + Bπ (0).Solid, dotted, dot-dashed and dashed lines stand for values of the form factor with full RG evolution at NLL level, at LL level, with RG evolution only respect to the hard coefficient and the B-meson decay constant and without RG evolution, respectively. of these two functions are significant.But there is a strong cancellation between these two effects due to different signs of slopes of their curves.This cancellation is important to guarantee the scale invariance of the form factors.To illustrate the effect of the RG evolution, we plot in Fig. 5 the scale dependence of f (+) Bπ (0), where the first type of B-meson LCDAs φ ± B1 (ω) and ρ − B1 (ω ′ ) are employed.It is obvious that after the complete RG evolution, the theoretical prediction of the form factor is almost independent of the factorization scale as expected.Results of f (+) Bπ (0) with leading logarithm (LL) resummation and next-toleading logarithm (NLL) resummation are both displayed for a comparison.It can be seen that the scale dependence is mild in both cases, but the NLL resummation reduces the value of the form factor about 3% compared to the LL-resummation value.For terms contain φ + B , the RG evolution has not been performed (suppressed by the coupling constant).This figure also indicates that φ + B terms of the form factors are almost factorization-scale independent.In the numerical analysis, we set the factorization scale to be a hard-collinear scale (µ = 1.5 ± 0.5GeV), as there are no large logarithmic terms in φ + B terms at this scale [16].
Since B-meson LCDAs are most important inputs of the B-meson LCSR, we need to test the LCDA- model dependence of the form factors.In Fig. 6, the vector form factor computed with three different B-meson LCDA models is displayed.Central values of the inverse moment are fitted as 0.392 in ρ − B2 and 0.382 in ρ − B3 .From this figure, we can see that model of the B-meson LCDA has a tiny influence on the shape of the form factor. Hereafter we will take ρ − B1 as the default model.Bπ (0).Solid, dot-dashed and dotted lines in the left (right) figure correspond to s 0 = 0.7GeV 2 , 0.75GeV 2 and 0.65GeV 2 ( M 2 = 1.25GeV 2 , 1.0GeV 2 and 1.5GeV 2 ), respectively.
In the LCSR approach, the form factors should be insensitive to the Borel parameter and the effective threshold.These parameters are constrained following conditions in [16], where the contribution from excited and continuum states should be less than 40% and the rate of change We fix q 2 = 0 to study ω M and s 0 dependence of the form factors. Above constraints lead to a region 0.24 ≤ ω M ≤ 0.36 (corresponding to 1.0 ≤ M 2 /GeV 2 ≤ 1.5) for all of the three form factors.We plot the Borel mass dependence of the form factor f + Bπ (0) in Fig. (7), a manifest platform at M 2 ∈ [1.0GeV 2 , 1.5GeV 2 ] guarantees that our calculation is insensitive to this unphysical parameter.The form factors are also almost independent on the effective threshold s 0 when it is adopted as s 0 = (0.7 ± 0.05)GeV 2 .
It has been argued that the B → π form factors calculated using the B-meson LCSR can be trusted at q 2 ≤ q 2 max = 8 GeV 2 (see [11] for more detailed discussions).To extrapolate the form factors calculated with the B-meson LCSR at large recoil toward large momentum transfer, we follow the same vein with [16], where the z-series parameterization was employed.In this parameterization, the cut q 2 -plane (the branch cut is the q 2 > t + region of the real axis) is mapped onto the unit disk |z(q 2 , t 0 )| < 1 via the conformal transformation where t + = (m B + m π ) 2 denotes the threshold of continuum states in the B * -meson channel.The free parameter t 0 ∈ (−∞ , t + ) determines the value of q 2 mapped onto the origin in the z plane.One can adjust the value of t 0 to minimize the corresponding z interval in the region q 2 min ≤ q 2 ≤ q 2 max , in order that the z-series expansion converges rapidly.Here we choose the same value as that in [8] where q 2 min = −6.0GeV 2 and t − ≡ (m B − m π ) 2 .Using the z-series expansion and taking into account the threshold t + behavior, one can obtain the parametrization of each form factor.
Parametrizations of the vector and the scalar form factors have been given in [8,32].The parametrization of the tensor form factor is similar with that of f + Bπ (q 2 ) [33] where the expansion coefficient(s) b T k is (are) determined by matching the large-recoil f T Bπ (q 2 ) onto Eq.( 52).As the interval in the z plane is constrained in a small region, it is reasonable to truncate the z-series at N = 2 in the practical calculation.Slop parameters b 1 , b1 , b T 1 are collected in Table 1.Uncertainties from different sources, including the inverse moment, the model of B-meson LCDA, the Borel parameter, the effective threshold, quark masses, et al, are taken into account in our numerical analysis.In Table 1, we collect parameters which arise large uncertainties.
In Fig. (8) the q 2 dependence of form factors f 0,+ Bπ (q 2 ) are shown, where Lattice results are from HPQCD collaboration [34], Fermilab/MILC collaboration [35] and RBC/UKQCD collaboration [36].Our results of f 0 Bπ within errors are in agreement with the Lattice data.While our results of f + Bπ is larger than the Lattice data.Results of f 0,+ Bπ (q 2 ) from the pion LCSR are also shown.It is manifest that slopes of these two form factors with the B-meson LCSR are greater than that of the pion LCSR.The difference between the B-meson and other approaches can be understood through following points.(1) As we can see from Table 1, the parameter λ B brings huge uncertainty to results of the form factors. Values of the form factors at q 2 = 0 GeV 2 are significantly influenced by the changing of λ B .But the value of this parameter is not determined yet.(2) We only calculate leading-power contributions of the form factors in this work.While power-suppressed contributions, which are induced by higher-twist pion LCDAs, are taken into account in calculations of the pion LCSR.The subleading-power effect in the B-meson LCSR may influence both values of the form factors at q 2 = 0 GeV 2 and slopes of the form factors.  Bπ (q 2 ), and the re-scaled form factor f + Bπ (q 2 ).Black and red curves are results of the B-meson and the pion LCSRs, respectively.Lattice QCD results are taken from HPQCD collaboration [34] (blue squres), Fermilab/MILC collaboration [35] (blue band) and RBC/UKQCD collaboration [36] (green and blue triangles).
B → π form factors are very important phenomenologically.Here we briefly discuss two applications of our result.The CKM matrix element |V ub | can be determined from the (partial) branching fraction of B → πℓν ℓ .If we neglect mass of leptons, the integrated decay width is written by where | p π | is the magnitude of the pion three-momentum in the B-meson rest frame, l = e, µ and ∆ζ(0, q A straightforward extraction of |V ub | can be performed using the relation where ∆BR(0, q 2 0 ) is the integrated branching ratio and the mean lifetime of the B 0 meson τ B 0 = (1.519± 0.005) ps [37].Experimental measurements of ∆BR(0, q 2 0 ) of the semi-leptonic B0 → π + µ ν µ decay [38,39] Utilizing the result of the form factor f + Bπ (q 2 ) which is computed with the B-meson LCSR and extrapolated with the z-series parametrization we can obtain ∆ζ(0, 12 GeV 2 ) = 4.93 +0.
Then the extracted CKM matrix element where the theoretical uncertainty comes from uncertainties of ∆ζ(0, 12 GeV 2 ) as displayed in (57).This |V ub | is larger compared to [16], since the RG evolution reduces the value of f + Bπ (q 2 ).In Fig. 9, we display the normalized differential q 2 distribution of B → πµν µ .Black curves represent the prediction of this work, where the solid line is the central value and dashed curves correspond to uncertainties.Due to the cancellation of the uncertainty of f + Bπ (q 2 ), the uncertainty of the normalized differential distribution of B → πµν µ is small.Our prediction is in agreement with the experimental data from BarBar [38,40,41] and Belle [39,42].

CONCLUSION AND DISCUSSION
We reviewed the method of calculating the B → π tensor form factor with the B-meson LCSR.In this framework, the method of regions was employed and contributions from different momentum regions are separated naturally.Precise soft cancellation guarantees the factorization theorem.The correlation function was factorized into the convolution of the hard function, the jet function and the B-meson LCDA which correspond to contributions from hard, hard-collinear and soft regions, respectively.We obtained one-looplevel hard and jet functions through the analysis of symmetry-breaking effects.
To resum large logarithmic terms in the form factors, we carried out the complete RG evolution of the factorized correlation function, including evolutions of the jet function and the B-meson LCDA.The B-meson LCDA, defined via the HQET, obey the Lange-Neubert equation which contains non-diagonal anomalous dimension.Following the approach in [24], we diagonalized the RG equation of the B-meson LCDA in the dual space and solved the diagonalized RG equation.The same method was also applied to the evolution of the jet function.Combining the evolution of each part together, we obtained the RG improved B → π form factors.
On the numerical side, we checked behaviors of the four evolution kernels (U 1 , U 2 , U j and U ρ ) and illustrated cancellation effects among the kernels.We examined the factorization-scale dependence of the RG improved form factors and compared our predictions with previous results.We extrapolated the q 2 dependence of the form factors to the whole physical region using the z-series expansion.Then we compared values of the form factors in this work with that in the LQCD and the pion LCSR.Phenomenologically we extracted the CKM matrix element |V ub | and analysed the normalized differential q 2 dependence of B → πµν µ .The B-meson form factors have many other phenomenological applications, such as the tensor form factor can give important contributions to FCNC processes B → (π, K)l + l − .Of course a complete study of phenomenological applications are far more complicated, and we left it for the future work.This work supplements the framework proposed in [16], and can be applied to various transition processes.

B Dispersion integrals
To obtain final expressions of the form factors, we need to extrapolate n • p to physical region.For the consistency of our derivation, we must have The above equation indicates that the branch cut of the root and logarithmic function be along negative real axis, and −Ω − iǫ = Ωe −iπ .The following equation can be derived from Eq. (61) From which we obtain following useful results: Following a similar way, another useful result is also obtained

Figure 1 :Figure 2 :
Figure 1: Diagrammatic representation of the correlation function Π µ (n • p, n • p) at next-to-leading order in α s .

T 2 b
(n • p, µ, ν) corresponds to the ln m

( 1 )
hc till now.This parameter could not be determined by the factorizationscale independence of the correlation function (γ(1)

Figure 4 :
Figure 4: Evolution factors of the hard function, the B-meson decay constant, the jet function and the B-meson LCDA, which are denoted using dotted, solid, dashed and dot-dashed lines, respectively.

Table 1 :
z-parameter fitted values of f + Bπ