$B \to f_0(980)$ form factors and the width effect from light-cone sum rules

In this paper we calculate the $B \to f_0(980)$ form factors from light-cone sum rules with $B-$meson DAs, the quark-antiquark assignment of $f_0(980)$ is adopted and the three-particle correction from $B-$meson DAs are found to be smaller than $20\%$ in the considered energy regions. We further explore the light-cone sum rules approach to study the $S-$wave $B \to \pi\pi$ form factors, the numerical fitting indicates that the $\sigma$ term should not be included in the resonance model and the $f_0+f'_0+f''_0$ model gives the reasonable prediction. As a by-product, we predict the strong coupling $|g_{f_0\pi\pi}| = 2.42 \pm 0.48$ GeV.

There are many calculations for the heavy-to-light form factor with the final state being a pseudoscalar (P ) or a vector (V ) meson, while to our knowledge the calculation involved a scalar (S) meson still has many controversies, focusing mainly on the underlying structure and the width effect. It has been suggested that the scalar mesons with masses below or near 1 GeV (the isoscalar σ/f 0 (500) and f 0 (980), the isodoublet κ, and the isovector a 0 ) form a SU (3) flavor nonet, and the scalar mesons with masses around 1.5 GeV (f 0 (1370), a 0 (1450), K * 0 (1430) and f 0 (1500)) form another one. The combined analysis with the data [21][22][23] and the orbital angular momentum [24,25] imply that the heavier nonet states are more favourite the quark-antiquark assignment replenished with some possible gluon content, while the scalar mesons in the lighter nonet are more like to be the tetra-quark states. Moreover, the pictures of gluonball [26], hybrid state [27] and molecule state [28] are also discussed for the light scalar mesons. In a word, there is not a general agreement on the contents of f 0 (980).
Actually, there are some attempts to study the B → S form factors from light-cone sum rules with scalar meson DAs [29][30][31][32][33]. We would like to comment that these calculations consid-ered only the scalar mesons in the heavier nonet, and the accuracy is debatable since the input Π 2pSRs (q) = i d 4 xe iqx 0|T {J S (x), J S (0)}|0 . (1) Generally speaking, the correlation function can be studied in twofold ways: the QCD calculation at quark-gluon level in the Euclidean momenta space, and the summing of intermediate states from the view of hadron. The QCD calculation in the negative half plane of q 2 is guaranteed by the operator-product-expansion (OPE) technology, and then the correlation function is written in terms of various quark and gluon condensates. On the other hand, the correlation function can be expressed as the sum of contributions from all possible intermediate states in the positive half-plane, possibly with some subtractions. These two parallel calculations are equated by using the dispersion relation. After taking the quark-hadron duality to eliminate the contributions in the large energy regions, we obtain 1 In fact the width of f 0 is smaller than it of ρ meson. We note that the width effect in B → ρ transition is usually not be considered, because in the experimental analysis the ρ meson is identified with the P-wave ππ signal when the dipion invariant mass locates in the ρ−pole region [38,39]. This makes the narrow-width treatment for B → ρ form factor from LCSRs is consistent with the experimental measurement.
where Π OPE 2pSRs (s) is the OPE result for the correlation function, s 0 is the threshold with taking into account only the ground state contribution. The mass and the modified decay constant of scalar meson are defined as The neutral scalar meson can not be produced via the vector current because f S is vanished in the SU (3)/isospin limit with the charge conjugation invariance and the conservation of vector current. In order to improve the convergence of OPE calculation and to suppress the contributions from excited states and continuum spectrums, we apply the Borel transformation on both sides of Eq.2, and obtain the result with one-loop perturbative calculation and the vacuum condensate terms up to dimension six [35,41], g sū σ · Gu = g sd σ · Gd = −0.8 qq , g ss σ · Gs = 0.8 g sū σ · Gu , α s /πG a µν G aµν = 0.012 GeV 4 , The light quark masses are estimated as the "current-quark" masses in the MS (µ = 1 GeV) [42], We consider the parameters running with one-loop accuracy [45][46][47], f 0 (980), σ(500) → π + π − 27 ± 13 or 41 ± 11 [48] φ(1020) → f 0 (980)γ → 3γ −48 ± 6 or 86 ± 3 [49] D → f 0 (980)P → ππ(KK)P 34 ± 6 or 46 ± 6 [50] ππ → ππ, KK strong indication for the gluonium nature of σ [51] K → ππeν e 18.7 or 151.3 [52] B → f 0 (980)K * [25,40] The comparable branching ratios of the cascade decays B → f 0 (980) → ππ, KK [42] indicates a quark flavor mixing in f 0 (980), |f 0 (980) = |ss cos θ + |nn sin θ , wherenn = 1 √ 2 (ūu +dd). In Tab.1, we enumerate some constraints for the mixing angle θ, and we preform to take the interval θ ∈ [25,45] (in unit of Degree) in our calculation.   Differentiating both sides of Eq.4 by the Borel mass and taking the ratio of the two equations, one can obtain the mass without the quark flavor mixing dependence, which is actually put into the decay constancef n f 0 ≡ sin θf f 0 . The interval of Borel mass is fixed by the rule of thumb that the contribution from high dimension condensate terms is no larger than ten percents in the truncated OPE, and simultanously the contribution from excited and continuum states is smaller than thirty percents when summing up the hadrons. The threshold s 0 is usually close to the outset of the first excited state with the same quantum number and then a certain vicinity can be expected, we determine it with considering the maximal stability of physical quantities once the Borel mass has been set down. In Fig.1, we displace the mass and the modified decay constant of f 0 (980) predicted from the 2pSRs, where the lightgray band shows the uncertainty associated with the mixing angle θ. Within the intervals M 2 = 1.2 ± 0.1 GeV 2 and s 0 = 1.8 ± 0.2 GeV 2 , we obtain m f 0 (980) = 0.98 ± 0.04 GeV ,f n f 0 (980) = 0.19 ± 0.05 GeV .
The mass agrees well with the PDG average value m f 0 (980) = 0.99 ± 0.02 GeV, which indicates that the quark-antiquark assignment of f 0 (980) is feasible for phenomenology under the QCD sum rules.
3 B → f 0 (980) form factors from the LCSRs The approach of LCSRs with B meson DAs was first proposed to calculate the B → P, V form factors [9], in this section we apply it to calculate the B → f 0 (980) form factors 2 . We start with the correlation function where the scalar current is J u S =ūu, and the weak current is J i ν =ūΓ ν b. The indicator I = A, T correspond to the lorentz structures Γ ν = γ ν γ 5 and σ νµ γ 5 q µ , respectively 3 . After transiting to the heavy quark effective theory (HQET), the heavy-to-light current is reduced to the light-quark current and the correlation function is modified tõ We use the notations p to denote the momentum carried by the scalar current J u S , andq = q − m b v for the effective currentJ I ν =ūΓ I ν h v . In the rest frame the effective b-quark field is defines as where the correlation function does not fluctuate widely and the OPE calculation is applicable.
We contract the same flavor quark fields in the propagator in which the first term is the freedom quark propagator in the QCD limit, and the second term respects the soft one-gluon correction. Two-particle and three-particle B−meson DAs are defined as 4 Two variables ω and ζ are introduced to represent the plus components of light quark momentum and the gluon momentum, respectively. For the sake of simplicity we have omitted the path-ordered gauge factors on the left hand sides. 3 We use the convention σ µν = i 2 (γ µ γ ν − γ ν γ µ ). 4 Recently, the renormalization group equations for three-particle distribution are resolved in the N C limit and the models for higher-twist DAs of B meson are suggested [58], following which the power suppressed correction are supplemented to B decays with energetic final states [59][60][61][62]. In this paper we concern more on the width effect which would be discussed in the next section, and postpone the completely correction from three-particle B−meson DAs for the future improvement.
The definition of B → S transition form factors is quoted as [63,64] in which the normarlizing factor is κ = sin θ/ √ 2 for the neutral charged meson f 0 (980). Two equivalent definitions in Eq.17 indicate the following relation, Following the Ref. [9], we calculate the correlation function in Eq.13 and obtain the B → f 0 (980) form factors in the narrow width approximation, The dimensionless variable σ ≡ ω/m B is the longitudinal momentum fraction of the light quark inside B−meson, and the virtuality of internal quark is s = m 2 B σ − (q 2 σ − m 2 )/σ. To obtain the above expressions, we have defined an auxiliary distribution with the boundaryφ B ± (0) =φ B ± (∞) = 0. We give several comments timely following on the results: (i) our results shown in Eqs. (20,21) consist with the calculations presented for the B → D * 0 form factors [65] with considering the m c effect, (ii) Eq.22 and Eq.23 are obtained by matching the coefficients associated to p ν and q ν , respectively, and they should be equal to each other with the definition in Eq.18, and (iii) considering the Eqs. (20)(21)(22)(23) at leading power, we get the relation F T ( in the heavy quark limit, and the relations F The dimensionless variable η =σ can be understood as the ratio between the minimal virtuality of the b quark field and the maximal virtuality carried by the internal light quark. The integral over the three-particle DAs is written as here another two auxiliary distributions are introduced as where the red-dashed (black) curves represent the contributions from two-particle (adding the three-particle correction) DAs of B meson.
In Eq.26, the lower indicator N = 1, 2, 3 stands for the power of Borel mass M −2(N −1) premultiplied with the integrals, the coefficients C B→S N,i associated to each three-particle DA are presented in Appendix A.
We take the typical MS for b quark mass m b (m b ) = 4.2 GeV, and use f B = 0.207 GeV obtained from the two-point QCD sum rules [66]. The inverse moment of B−meson DAs is set at λ B = 350 MeV by considering F T,p (0) = F T,q (0), this value is in agreement with the recent estimations [11,67], where λ B = 358 +38 −30 (343 +22 −20 ) MeV is obtained by comparing the B → π(ρ) form factor from LCSRs with pion [68] (rho [8]) and B−meson DAs. In Fig.2, we plot the B → f 0 (980) form factors derived under the narrow f 0 approximation, where the lightgray shadows reveal the total uncertainty came from the quark flavor mixing angle and the LCSRs parameters 5 . It is obvious to check the relations between the form factors in the full recoil energy point. The tensor form factors obtained with Eq.22 and Eq.23 consist very well with each other in the considered energy regions, so if we know these two results with a high accuracy, i.e., with including the NLO QCD corrections, we can determine/restrict the DAs of B−meson. The numerical results show that the three-particle contribution to the tensor form factor F T (Q 2 ) is tiny, while the correction to the form factor F + (Q 2 ) is about 10% in the considered energy regions, this prediction agrees with the result obtained recently in the convariant quark model [69]. For the form factor F − (Q 2 ), the three-particle contribution is opposite in sign to the two-particle contribution, and is dominant in the large momentum transfer regions. If we want to go further accuracy prediction, the width effect of the intermediate scalar states should be considered. 4 The width effect and the B → ππ form factors To discuss the width effect of the intermediate states, let's look back to the dispersion relation of the correlation function in Eq.12 where the imaginary part is obtained rigorously by interpolating a complete set of intermediate states to retain the unitarity, In the above equation, p n is the momentum of each interpolated state |S n , dτ n denotes the integration over the phase space volume of these state. An immediate way to investigate the width effect is to substitute the interpolation of scalar mesons by the stable multi-meson states, such as the ππ, 4π, KK and their continuum states [12].
With the interpolation of the single meson states, the correlation function is rewritten as where the contribution from the ground state is singled out while the rest contributions are retained in the integral, s 0 is again the threshold truncated the excited states. Eq.30 has been used to derive the LCSRs result in sec.3. For the case with the multi-meson states interpolating, Eq.28 is modified to The ellipsis denotes the contributions from other interpolating states with higher thresholds, like 4π, KK, and etc. As we will see later, the threshold of scalar pion form factor is s 2π 0 ∼ 2.4 GeV 2 , then the phase space opens for the 4π and KK states 6 . But in fact, the contribution from the KK state is expected to be suppressed since it is not the main channel coupled to 6 In the case with vector pion form factor [12], the contributions from 4π, KK states are suppressed by the threshold space s 2π 0 1.0 − 1.5 GeV 2 [70].
f 0 , f 0 , f 0 7 . In addition, the Borel exponent e −s/M 2 suppresses the large s contribution with the Borel mass M 2 1.2 GeV 2 , which indicates that, after taking the duality approximation, the contributions from the multi-meson states with the thresholds larger than s 2π 0 are suppressed in a certain degree. That's why we consider the solo dipion contribution in the following calculation.

Formulas
The isoscalar pion form factor in Eq.31 is defined as [71] with the normalization Γ π (0) = m 2 π under the chiral symmetry, the averaged light quark mass ism = (m u + m d )/2. The once-subtracted omenés representation of the form factor is [72] Γ where δ Γπ is defined as the phase of Γ π (t)/P(t). The function P (t) is the polynomial to consider the zero values 8 , in the case of free zeros P (t) = Γ π (0). The Watson theorem makes sure that the identity δ Γπ = δ 0 0 holds in the elastic region s 4m 2 K . Beyond the KK threshold, the generalized Watson theorem with considering the coupled-channel (ππ − KK) is still available to predict the phase up to the energy s = 1.5 2 GeV 2 [76][77][78][79][80]. Unfortunately, the dispersion theory loses the unitarity and the analyticity when going to more higher energies, and one is forced to employ a model [81] or to adopt the perturbative QCD approximation [82]. In our calculation, we would adopt the phase updated from the amplitude analysis [83,84], which can be extrapolated to a high energy ∼ 10 GeV 2 by considering all the measured data, the ππ − KK final state interaction, the mass difference between the charged and neutral kaon, as well as the low energy Roy equation [85].
The remaining matrix element in Eq.31 is expressed in terms of the B → ππ form factors. For the axial-vector current sandwiched between the B-meson and the dipion state, we have [86] −i π + (k 1 )π − (k 2 )|ūγ ν γ 5 b|B − (q + k) Hereafter we take f 0 , f 0 and f 0 to denote the scalar state f 0 (980), f 0 (1370) and f 0 (1700), respectively. 8 The experiment data [73] have made known that in the S−wave isoscalar ππ elastic scattering, the phase δ 0 0 is very close to π and the moduli of amplitude has a sharp dip around the KK threshold, which indicates the corresponding form factor may have a zero in this interval and the phase at there can not be understood without taking into account the KK channel [74,75]. When δ 0 0 = π happens below the threshold s 1 < 4m 2 K , the form factor has a zero at this energy. On the contrast, the zero value does not appear if δ 0 0 = π occurs after opening the KK channel, but a dip close arbitrarily to zero is allowed around the threshold.
The dot products appeared above are written down by three independent variables, where k 2 is the invariant mass of dipion system, q 2 denotes the squared momentum transfer in the weak decay, and θ π represents the angle between the 3-momentum of π − (k 2 ) and B meson in the dipion rest frame. What's more, β π (k 2 ) = 1 − 4m 2 π /k 2 is the phase factor of dipion system, and is the kinematic Källén function. By the way, the transition matrix element in Eq.31 can also be expressed as, with different helicities of the dipion state (λ = t, 0, +, −). An advantage of the expression in Eq.36 is that the helicity amplitudes can be expanded in terms of the associated Legendre polynomials, then the contributions from different partial waves can be studied. For our calculation the expression in Eq.34 is more convenient with the orthogonal momentum vectors, so we translate the partial wave expansion from the helicity amplitudes H λ to the form factors F i , Substituting Eq.34 and Eq.37 into the numerator in Eq.31, we obtain the S−wave contribution to the imaginary part as In fact, the phase space dτ 2π plays as a S−wave projector for the timelike-helicity form factors F t,0 (q 2 , k 2 , q ·k), which means that only the S−wave component F (l=0) t,0 (q 2 , k 2 ) survives after integrating over the angle θ π . For the form factor F (q 2 , k 2 , q·k), the S−wave projector vanishes and the contribution starts from the D-wave component (l = 2n, n = 1, 2, 3 · · · ), which part is expected tiny in the B → ππ transtion and would not be discussed in this paper. In order to eliminate the contribution from higher power condensate terms and the excited states, we take the semi-local duality with the effective threshold s 2π The LCSRs results for the S−wave B → ππ transition are obtained as, Multiplying both sides of the Eq.17 and Eq.34 by q ν , we obtain the matrix elements deduced by the pseudo-scalar current J P = im bū γ 5 b, Eq.42 suggests an auxiliary LCSRs for the B → f 0 (980) form factor F 0 , Eq.43 hints another LCSRs for the timelike-helicity B → ππ form factor F t , where the phase space integral of the dipion sate is modified to (45) and the corresponding LCSRs is We remark that Eqs. (42,43) are established based on the heavy quark limit, so the auxiliary LCSRs in Eqs. (44,46) can be used to estimate how well does the heavy quark limit work by comparing with the standard calculation in Eqs. (20,21,41). We show in Fig.3 for the B → f 0 (980) form factor F 0 , it is find that the three-particle correction is significant and accounts 20% − 30% of the two-particle contribution in the considered energy regions. With the bluedotted and black curves, we also compare the two independent calculations resulted in Eq. 19 and Eq.44. The prefect agreement of F 0 , as well as the agreement of F T discussed before, implies that the heavy quark expansion holds well up to O(1/m b ) in the low momentum transferred regions.

Models
Eqs. (40,41,46) are the main results in this section. A trouble we encountered immediately is that we can not solve out the B → ππ form factors in terms of the B meson DAs, because they are convoluted with the scalar pion form factor in the integrand. We have to introduce models to parameterise the S−wave B → ππ form factors, and the first candidate coming into mind is the single resonance model written as A condition underlying in Eqs. (40,41,46) is the reality 9 of the imaginary part, We introduce a strong phase φ f 0 to compensate the phase difference between the pion form factor and the f 0 model for B → ππ form factors. Generally speaking, φ f 0 should depend on both the two variables s and q 2 , but the q 2 −dependence does not appear in the single f 0 resonance model, The simple model in Eqs. (47)(48)(49) is inspired by the physics that the sum rules obtained for the B → ππ form factor, in the narrow width approximation, should recover the sum rules for the B → f 0 (980) form factor. To check this point, let's consider the energy-dependent f 0 → ππ width 10 [48]: where Γ tot f 0 is the total width of f 0 (980), and the strong coupling is normalized as with the relation g f 0 π 0 π 0 = g f 0 π + π − / √ 2. The energy-dependence of the width represents the loop effects of two pions coupling to the f 0 state, from this point we take intentionally the f 0 −dominance approximation for the scalar pion form factor 11 , Substituting Eqs. (47,54) into Eq.40 and taking into account Eq.52, the l.h.s of Eq.40 becomes and we recover the result for the form factor F B→f 0 + in the zero-width limit Γ tot f 0 → 0. Similarly, we can recover the LCSRs for the form factors F B→f 0 − and F B→f 0 0 with taking into account the resonance models in Eq.48 and Eq.49, respectively. What's more, the relation defined in Eq.19 also holds well in the resonance models in Eqs. (47,48,49). 9 Here we consider the more strict local reality at each point of invariant mass. 10 We discuss in the general formula without specifying detail expression of Γ f0 (s), i.e., the Flatté model [87]. 11 In fact, the Breit-Winger formula can not be used directly to describe the scalar resonance. The case is different in our previous work about the P −wave B → ππ form factor [12], where the magnitude of timelike pion form factor is well measured [88] and parameterized in the Gounaris-Sakurai model [89].
We employ the z−series expansion [90] to parameterise the B → f 0 form factors, where j = +, −, 0, F B→f 0 i (0) is the values at the full recoil energy point, and the parameters b F i , c F i indicate the coefficients associated with the ζ−functions,

Numerics
With the single f 0 model in Eqs. (47,48,49), we rewrite the LCSRs predictions for the S−wave B → ππ form factors in a more general formula, where for the sake of brevity we introduce the following notations: The integral coefficient on the l.h.s reads as I OPE j represent the OPE calculations on the r.h.s of Eqs. (40,41,46), There is no physics requirement that the threshold s 2π 0 should be equal to s 0 . We fixed it in an independent way by considering again the correlation function in Eq.1, but with interpolating the multi-meson states with the same quantum number. The 2pSRs is then written in terms of the S−wave isoscalar pion form factor as With the form factor expressed in Eq.33 and displaced in Fig.4, we obtain the threshold s 2π 0 = 2.4 GeV 2 , which is larger than the threshold s 0 = 1.8 GeV 2 in the case of single meson interpolating. This is easy to be understood by the broad structure of the S−wave isosclalar ππ state. It is apparent from Fig.4 that there are two smooth peaks appear in the S−wave isoscalar pion form factor, so it is nature to question what's the roles of σ and the excited states, like f 0 , f 0 , in the B → ππ transition. To convince ourselves, we suggest additionally the f 0 + σ model and the f 0 + f 0 + f 0 model.
We consider the first one by appending σ to the single f 0 model, and modify the Eq.(49) to We again tacitly assume that the strong phases φ S associated to each resonance are only dependent on the invariant mass of dipion state, the simplest way to achieve it is to take the same q 2 −evolution for both the two resonances: F B→σ j (q 2 ) = γ σ F j F B→f 0 j (q 2 ). The dimensionless parameter γ σ 0 F j indicates the relative contribution from σ comparing to the contribution from f 0 which is normalized as unit. In this way, the general formula in Eq.59 is rewritten in the f 0 + σ model, Inspired by the Eq.19, we also apply the relation 2F B→S    [91]. In that work, the phase-shift of ππ scattering and the twice-subtracted Omnés solution are adopted to express the asymptotic coefficient B 10 (s). As mentioned above for the isoscalar pion form factor, with the aim to take into account the contributions from the ππ − KK scattering, it is more reasonable to take the phase of ππ scattering and the once-subtracted Omnés solution. Moreover, the normarlizations are different 12 in the isoscalar dipion DAs and the isoscalar pion form factor: B 10 (0) = −5/9 and Γ π (0)/m = 2m π 0 . We revisit the work in Ref. [91] with the same inputs (formulas) adopted here and update the asymptotic result from the dipion LCSRs: t,asy (4m 2 π , 0)/m B = 5.40 ± 1.00. As depicted in Fig.5, the result obtained here under the f 0 + σ 0 model is √ q 2 F (l=0) t (4m 2 π , 0)/m B = 36.0 ± 10.0, which is impossible to reconcile with the asymptotic prediction. We conclude temporarily that the σ−term should not been included in the resonance model to describe the S−wave B → ππ transition with the small 12 In the case of the P −wave B → ππ from factor, the expansion coefficient of the asymptotic isovector dipion DAs and the timelike pion form factor have the same normalization: B 01 (0) = F π (0) = 1, then the results obtained in these two ways are consistent [12,92,93].
) are adopted to simplify the phase elimination, and the particular relation γ S F 0 = γ S F + is complemented to each resonance 13 . The integral coefficients of the excited states f 0 and f 0 are to obtain it we have taken the total widths as Γ tot f 0 = 0.35±0.15 GeV and Γ tot f 0 = 0.11 GeV [42]. The fitting result within the f 0 + f 0 + f 0 model are presented in Tab.3. At the bottom of Tab.3, for the comparison we supplement the B → f 0 form factors calculated directly from the LCSRs, whose uncertainty comes only from the LCSRs parameters. From the result list in the second column, we can extract the strong coupling |g f 0 π + π − | = 1.98 ± 0.38 GeV and obtain subsequently |g f 0 ππ | = 2.42±0.48 GeV, which is consistent with the result |g f 0 ππ | = 1.60±0.80 GeV calculated directly from Eq.52 within the uncertainty. Because the dependences on the quark flavor mixing angle cancel between κ f 0 F i and F B→f 0 j (0), the coupling extracted in this way has the smaller uncertainty.
The contribution from each resonance to the OPE result is listplotted in Fig.6, from which we can see that f 0 plays as the leading role as expected. We also find that the contribution from f 0 is at least one times larger than it from f 0 , the underlying physics is that f 0 couples only to the ρρ state while f 0 can couples directly to the ππ state. We plot in Fig.7 for the S−wave 13 In fact, the Eq.48 generalized with multi-resonances implies a strict relation at the full recoil energy point which is further reduced in terms of the strength parameters γ S Fi : In the f 0 + σ model, γ σ F0 = γ σ F+ is the exactly solution. And in the f 0 + f 0 + f 0 model discussed following, γ S F0 = γ S F+ is one set of the particular solution. (4m 2 π , 0) = 11.0 ± 5.0 is close to the asymptotic result obtained from the LCSRs with dipion DAs, on the basis of this we suggest that the f 0 + f 0 + f 0 model is the more reasonable one to describe the S−wave B → ππ from factors.

Conclusion
In this paper we calculate the B → f 0 (980) form factor as the first time from the light-cone sum rules with B−meson DAs, and explore the calculation to study the S−wave B → ππ form factors. To ensure the usefulness of the quark-antiquark assignment for f 0 (980), we revisit the 2pSRs with scalar current and predict the mass of f 0 (980) with considering the quark flavor mixing. For the B → f 0 (980) form factor, we find that the three-particle B−meson DAs gives no more than 20% correction to the leading twist contribution. In order to investigate the width effect, we suggest two resonance models to parameterize the S−wave B → ππ from factors, and the fitting results show that (i) the f 0 + σ model is not apposite because the σ resonance gives one times enhancement to the form factor at the full recoil energy point, which pulls the result far away from the asymptotic calculation processed in the LCSRs with dipion DAs, (ii) the f 0 + f 0 + f 0 model gives the compatible result to the asymptotic calculation, and as a by-product, provides a new way to determine the strong coupling |g f 0 ππ | = 2.42 ± 0.48 GeV.
Our predictions still have large uncertainty, mainly comes from the freedom to choose the widths of f 0 and f 0 states. Further improvements on this project include mainly the follows: (i) completing the rest corrections from the three-particle DAs of B−meson, which part is indispensable to provide the final result at subleading power, (ii) pushing the OPE calculation to the next-to-leading-order to improve the theoretical accuracy, (iii) studying the high power terms in dipion DAs and their evolution on the invariant mass, with which the B → ππ form factors can be calculated beyond the asymptotic contribution, (iv) and last but not least, forwarding this approach to calculate the B → Kπ form factors, which is more interested in phenomenology.
A Coefficients in the three-particle correction We present here the coefficients appeared in the three-particle correction for B → S transition 14 . 14 We omit the upper indicator B → S for the simplicity, and use the character m to denote the light quark mass of the internal propagator.