Dipion light-cone distribution amplitudes and $B \to \pi\pi$ form factors

We suggest to update the expansion coefficients of 2$\pi$DAs with the distribution amplitudes of light mesons evaluated from lattice QCD, with which we revisit $\overline{B}^0 \to \pi^+\pi^0$ transition form factors from light-cone sum rules approach and extend the predictions from the threshold of dipion invariant mass to high energies with including the resonance intervals. We also derive $B^- \to \pi^0\pi^0$ transition form factors with the isoscalar dipion final state, serving as the supplement to the isovector ones to complete the set of light-cone sum rules prediction of $B \to \pi\pi$ form factors. Our numerics shows that the lowest resonance gives the dominant contribution to $P-$wave form factors, while the resonance contribution in $S-$wave is not so salient.


Introduction
With a large number of angular observables that is sensitive to the spin structure of the underlying short distance operators, the B → ππ ν decays had been suggested to probe the V − A nature of weak interaction [1], whose measurement would provides prolific information to test QCD theoretical approaches and to search physics beyond the SM. At the quark level, it is generated by the semileptonic b → u ν transition, which offers another independent channel to determine the Cabiboo-Kobayashi-Maskawa matrix element V ub if we are able to calculate the B → ππ form factors with an adequate accuracy.
There are some efforts on this topic recently. At large dipion invariant mass, the QCD factorization is available [2,3], when the invariant mass is small and the hadronic recoil is low, the chiral effective theory of heavy meson is proposed to combine with the dispersion theory [4]. The light-cone sum rules (LCSRs) approach is operative in the regions of large hadronic recoil at low invariant mass, i.e., the S−wave generated B → ππ, Kπ form factors has been calculated [5,6] with the combination of the perturbative theory based on the operator production expansion and the low-energy effective theory inspired by the chiral symmetry, corresponding to the effect from the LCDAs and from the scalar form factors, respectively, and recently the B 0 → π + π 0 form factors has been derived from LCSRs approach with the generalized 2πDAs [7][8][9]. In principle, the low invariant mass in LCSRs approach to deal with B → ππ ν decays goes from the threshold 4m 2 π to the resonance intervals, the previous works follow closely around the threshold while the running on the invariant mass is less attended, but actually this part is indispensable to show the resonance contribution. The LCSRs prediction is always influenced by the high power terms, one of which is the non-asymptotic QCD corrections of hadron distribution amplitudes (DAs), in literatures, most of the calculations of B → ππ form factors use the expansion coefficients of 2πDAs obtained 20 years ago from instanton model [10].
In this paper we suggest to update the non-asymptotic coefficients of leading twist 2πDAs with the lattice QCD (LQCD) knowledge of light mesons, which is already quite accurate with the development of the discrete computing technique. We prolong the previous calculation of B 0 → π + π 0 form factors from the threshold of dipion invariant mass to the rather broad range with the energy-dependent 2πDAs, we also derive the new LCSRs prediction for the B − → π 0 π 0 form factors with the isoscalar dipion final state, which is an essential physical quantity in the angular observables of the semileptonic decay. All calculations in this paper are at leading twist and the contributions from high twist 2πDAs are postponed for the future work. The paper is organized as follows. In section.2, we discuss briefly the properties of 2πDAs and update the coefficients of isovector 2πDAs with the LQCD result of light mesons. Section.3 is the main part of this paper, where we present the LCSRs' calculation of B → ππ form factors with both the isovector and isoscalar dipion final states. Section.4 is the numerical result. Our conclusions are presented in Sec.5.

Dipion light-cone distribution amplitudes
The 2πDAs are the most general object to describe the dipion mass spectrum in hard production processes, whose asymptotic formula indicates the information about the deviation of the unstable meson DAs(ρ, f 0 , a 0 etc.), and in further to improve the theoretical accuracy of the nonperturbative information of meson, like the decay width. From the other hand, the crossing symmetry implies a relation between 2πDAs and the skwed parton distributions (SPDs) in the pion, which provide another constraint to determine the SPDs. In this section we brief discuss the properties of 2πDAs, and explain some updates of the nonperturbative inputs. Our precise is still at leading twist.

General review of 2πDAs
The concept of wave function of a single meson has been generalized to a multi-hadron system [11], and the perturbative behavior of meson pairs is calculated at large invariant mass [12,13], while the factorized form of exclusive electroproduction process [14,15] involves the 2πDAs at small invariant mass [16,17]. In this work we quote the chirally even and odd two quark 2πDAs defined in Refs. [10,18] as, respectively, where the index f, f respects the (anti-)quark flavor, a, b indicates the electro charge of each pion, the coefficient κ +−/00 = 1 and κ +0 = √ 2, k = k 1 + k 2 is the invariant mass of dipion state, n 2 = 0, τ = 1/2, τ 3 /2 corresponds to the isoscalar and isovector 2πDAs, respectively, the chirally odd constant f ⊥ 2π is defined by the local matrix element, The generalized 2πDAs depend on three independent kinematic variables, the momentum fraction z carried by anti-quark with respecting to the total momentum of dipion state, the longitudinal momentum fraction carried by one of the pions ζ = k + 1 /k + , and the invariant mass squared k 2 . The normalization conditions of the distribution amplitudes read as: where F π (k 2 ) is the timelike pion form factor, F t (k 2 ) is the tensor pion form factor, normalized by F π (0) = 1 and F t (0) = 1, M (π) 2 is the momentum fraction carried by quarks in the pion associated to the usual quark distribution [19], F EMT (k 2 ) is the form factor of the quark part of the energy momentum tensor with the normalization F EMT π (0) = 1 [20]. The 2πDAs can be decomposed, respecting to flavor/isospin, as 1 : As is well known, 2πDAs can be double decomposed in terms of Gegenbauer polynomials C 3/2 n (2z−1) (eigenfunction of the evolution equation) and the Legendre polynomials C 1/2 (2ζ −1) (partial wave expansion), The coefficients B n (k 2 , µ) have the similar scale dependence as the gegenbauer moments of pion and rho mesons [21,22], where β 0 = 11 − 2N f /3 and the one-loop anomalous dimension are [23] γ ,(0) We note that for isovector (isoscalar) 2πDAs, the gegenbauer index n goes over even (odd) and the partial-wave index l goes over odd (even) numbers, which is guaranteed by the C-parity. Several comments are in time for the expansion coefficients B n (k 2 ): ‡ When the four momentum of one of the pions goes to zero, soft pion theorem relates the chirally even coefficients and the gegenbauer moments of pion meson, The 2πDAs are also related to the skewed parton distributions (SPDs) in the pion by the crossing, which support us to express the moments of SPDs in terms of B nl (k 2 ) in the forward limit as ‡ The Watson theorem of pion-pion scattering amplitudes implies a intuitive way to express the imaginary part of 2πDAs, which subsequently deduces the Omnés solution of N −subtracted dispersion relation for the coefficients, .
With two subtraction, this expression gives an excellent description of the experimental data of pion form factor not only below the inelastic threshold k 2 < 16m 2 π , but also in the resonance region up to k 2 ∼ 2.5 GeV 2 . In this way, 2πDAs in a wide range energies is given by the ππ phase shift δ I and a few subtraction constants.  (14).
- ‡ Taking the vanishing width limit in the vicinity of resonance, 2πDAs reduce to the distribution amplitudes of resonance (ρ), which implies another relation between the gegenbauer moments of rho meson and the coefficients B n , The decay constants of resonance are related to the imaginary part of B nl (m 2 ρ ) as with the strong coupling defining in π(k 1 )π(k 2 )|ρ = g ρππ (k 1 − k 2 ) α α .

B → ππ form factors
We present the result of B → ππ from factors with the updated subtraction constants in this section, in which the evolution of F I=1 ,⊥ on invariant mass and the form factors with isoscalar dipion state F I=0 are the new results, as the supplement to the previous work [7,8].
B → ππ transition matrix element is defined in terms of the form factors as [1], where we use the same notations for the kinematics of dipion state as in Eqs. (1,2) indicates the momentum transfer in the decay, the vector products is expressed in terms of the independent variables as with the phase factor β π (k 2 ) = 1 − 4m 2 π /k 2 , and θ π is the angle between the pions in their c.m. frame. The derivation of B → ππ from factors from LCSRs approach starts from defining an approximate correlation function, which is written down as the non-local matrix element with the B meson interpolating current j This correlation function is only valid for calculating the form factors F ⊥ and F , but is failed to derive the timelike-helicity ones (F t and F 0 ) due to the kinematic singularity, to overcome this problem we introduce an modified correlation with replacing the V − A weak current by the pseudoscalar current j (P ) (x) = im b q f (x)γ 5 b(x) [8].
For the sake of simplicity, we focus on the form factors contributed at tree level 4 in semileptonic decays B → ππ ν , where the flavor of light antiquark in weak current is identified by f=u, and the quark in internal interpolating current is f = d and u for {ab} = {+0} and {00/ + −}, respectively. The QCD calculations and hadron analysis of the correlation functions are the same as in Refs. [7,8], we here quote the result: where s(u) = (m 2 B − q 2 u + k 2 uu)/u and u 0 is the solution to s(u 0 ) = s B 0 , M 2 and s B 0 are the Borel mass and threshold parameter introduced in LCSRs approach. Eqs. (24)(25)(26) collect the total contributions from all partial wave components, to obtain the contribution from each partial wave, we use the following expansion,

Form factors with isovector dipion state
For the isovector dipion state, the partial wave contribution to the form factors is gained by multiplying both sides of Eqs.(23,24) (Eq.25) by sin θ π P (1) (cos θ π ) (P (0) (cos θ π )) and integrating over cos θ π , where we introduce the short-hand notation for Legender integration, and for the functions integrated over the momentum fraction, To derive these expressions we use the orthogonality relation of the Legender polynomials In the pervious work [7,8], these form factors are considered only around the threshold k 2 ∼ 4m 2 π , we will replenish their evolution on k 2 to high energies in the next section with Eq.14 for the numerical computing.

Form factors with isoscalar dipion state
When the final two pions forms an isoscalar state, we multiply Eqs. (23,24) by cos θ π P (0) (cos θ π ) 5 and Eq.25 by P (0) (cos θ π ), then the B − → π 0 π 0 form factors are arranged as the angular integration in this case is we reveal here that I I=0 = 0 when goes over odd numbers, I I=0
neglecting the contributions from higher partial waves, we arrive at the final expression for B − → π 0 π 0 form factors which had not been studied before, the comparison of Eq.41 and Eq.42 indicates the relation 2B ⊥ 10 (k 2 ) = B ⊥ 12 (k 2 ) if we acquiesce in the convergence of gegenbauer expansion and keep only the first term with n = 1, this relation leads to another constraint to check ππ phase shifts, As presented in Tab.1, the coeffficent B 10 (B 12 ) is also studied in instanton model, which allows us to predict the timelike-helicity form factor q 2 F ( =0),I=0 t (q 2 , k 2 ), while the chirally odd coefficients are still missing for the other form factors in B − → π 0 π 0 ν decay.

Numerics
To obtain the numerical result, we fix b quark mass at m b (3 GeV) = 4.47 GeV [53], and the decay constant of B meson at f B = 0.207 GeV [54,55], with neglecting the small uncertainty from renormalization scale. For the LCSRs parameters we adopt the same inputs as in Refs. [7,8]: Other parameters entered in the numerical computing are explained in Sec.2.
In Figs.(1-3) we present the LCSRs prediction for B → ππ form factors, in which we adopt the notations F ,⊥ (q 2 ) = F ,⊥ (q 2 , k 2 = 4m 2 π ), F t,0 (q 2 ) = F t,0 (q 2 , k 2 = 0.1 GeV 2 ) and F (k 2 ) = F (q 2 = 0, k 2 ). For the transition form factors with isovector dipion state, the P −wave contributions to F (q 2 ) are calculated at low k 2 with the few first expansion coefficients of B nl (k 2 ) = B nl (0) + k 2 d/dk 2 ln B nl (0) listed in Tab.1, while the curves of F asy (k 2 ) are acquired by using the Omnés solution of B 01 (k 2 ) expressed in Eq.14. When the final dipion is isoscalar state, we plot only the asymptotic shapes of the form factors because we do not have any information so far beyond the asymptotic coefficient B 10 (k 2 ) from experiment measurement or from effective low energy theory. The resonance information in the invariant mass spectrum is carried by the pion-pion phase shift δ I (k 2 ), which is well measured and described by the Regge parameterization in the range from the threshold to 1100 MeV [56]. This is not enough because the coefficients B nl (k 2 ) in Eq.14 integrate over the whole region of invariant mass suqared, we adopt the result from amplitude analysis with marriage of dispersion relations with unitarity [57,58], which it is able to extrapolate the phase shifts to a high energy ∼ 5 GeV for both S−wave and P −wave, with considering all the well measured data, the ππ − KK final state interaction, the mass difference between charged and neutral Kaon and also the low energy Roy-Equation.
The upper panels of Figs. (1,2) show the P −wave contribution to q 2 −dependence form factors of B 0 → π + π 0 ν decay around the dipion threshold, consisting with our previous work [7,8] with in the uncertainty analysis. The lower panels are the new result for the form factors in a wide range of invariant mass squared at the full recoil point, the shape of timelike-helicity form factor F t (k 2 ) is compatible with it obtained [8] by using the normalization condition B 01 (k 2 ) = F π (k 2 ) and the data of pion form factor measured up to 1.78 GeV [59], which in turn supports that Eq.14 is powerful at least in the few low resonances interval. Comparing with the result obtained from LCSRs with the B−meson distribution amplitudes [9], the ∼ 25% differences is regarded as the contributions from high twist 2πDAs. Fig.3 depicts the first calculation of B − → π 0 π 0 form factors, where the S−wave contribution is shown. It is obvious to see that the q 2 −dependence of F I=0 t is decreasing, in contrasting to F I=1 t , which is originated from the different gegenbauer polynomials C 3/2 n (2u − 1) in function J t n . In the right panel for the k 2 −dependence, the uncertainties from LCSRs parameters cannel between the exponential e m 2 B /M 2 and the function J t 0 (M 2 , s B 0 ).

Conclusion and outlook
In this paper, we discuss and update the isovector 2πDAs by their relations to the distribution amplitudes of pion and rho mesons, with which we revisit the B 0 → π + π 0 form factors from LCSRs approach. With the Omnés solution of expansion coefficients of 2πDAs in terms of the ππ phase shifts and a few subtraction constants, the LCSRs predictions for these form factors are extended from the threshold of dipion invariant mass to a wide range energies with including resonances. We also study the B − → π 0 π 0 form factors with the isoscalar 2πDAs calculated from effective low energy theory based on instanton vacuum. For the form factors with isovector dipion state, the updated calculation does not bring noticeable deviation from the previous work, the lowest intermediate resonance ρ dominates in P −wave contribution (∼ 80%), the high partial (l = 3) contribution is tiny and not exceed a few percents of the P −wave contribution, and the contribution from high gegenbauer term (n = 2) is barely too. For the form factors with isoscalar dipion state, our calculation is asymptotic with neglecting the contributions from higher partial waves and lacking of the information for high gegenbauer terms, the contribution from resonance in S−wave is not apparent as in the P −wave, indicating a more complicated inner structure in isoscalar dipion system.
Further improvements on this project include: (a) Developing and promoting the effective low energy theory and/or other approaches to calculate the chirally odd coefficients of isoscalar 2πDAs, also for the higher power terms of chirally even coefficients. (b) Finding the relation between isoscalar dipion state and intermediate meson state (f 0 ) to restrict the subtraction constants in isoscalar 2πDAs. (c) Forwarding the calculation to include the contributions from 2πDAs at twist-3 and from the next-to-leading-order QCD correction, to meet the precision requirement for extracting the CKM matrix element V ub . (d) Revisiting these form factors from another LCSRs with B meson DAs as input, in order to improve the theoretical accuracy.