Helicity form factors for $$D_{(s)} \rightarrow A \ell \nu $$ process in the light-cone QCD sum rules approach

The helicity form factors of the D(s) → Al ν with A = a−1 , a 0 1, b − 1 , b 0 1,K1(1270) and K1(1400) are calculated in the light-cone sum rules approach, up to twist-3 distribution amplitudes of the axial vector meson A. In the helicity form factors parametrization the unitarity constraints are applied to the fitting parameters. In addition, the effects of the low-lying resonances are included in series expansions of aforementioned form factors. The properties of the D(s) → Al ν semileptonic decays are studied by extending the form factors to the whole physical region of q. For a better analysis, a comparison is also made between our results and the predictions obtained using transition form factors via LCSR, 3PSR and CLFQM methods.

In this paper, the helicity form factors for the D (s) decays into axial vectors are calculated with the LCSR. The helicity form factors which can be obtained by contracting the W (or Z) boson polarization vectors and the transition matrix elements, are also functions of q 2 . The relations among the D (s) → A transition matrix elements, transition form factors and the helicity ones are presented in Table I.

Matrix element
Transition form factors Helicity form factors HV,0 HV,1, HV,2 A|ū σ µν γ 5 qν c|D (s) A|ū σ µν qν c|D (s) HT ,0 HT ,1, HT ,2 There are some advantages in using the helicity form factors: * e-mail: samira.momeni@ph.iut.ac.ir 1) Diagonalizable unitarity relations can be imposed on the coefficients of the helicity form factor parameterization. 2) In the helicity form factors, the contributions from the excited states and the spin-parity quantum numbers are considered by relating the dominant poles in the LCSR predictions to low-lying resonances (for more detailed, see [22]).
The masses and quantum numbers J P of low-lying D (s) resonances with the relations among the helicity form factors are provided in Table II. These masses will be used in the helicity form factors parameterizations. Notice that the mass values for D s (1 − ) and none of the (1 + ) states predicted in [23] have been experimentally confirmed yet.
In this study, the branching ratio values are reported for the D (s) → K 1 ℓ + ν decays at θ K = −(34 ± 13) • . Our paper is organized as follows: In Sec. II by using the LCSR, the form factors of D (s) → A decays are derived. Section. III, is devoted to the numerical analysis of the form factors and the branching ratios for semileptonic and decays. A comparison of our results for the branching ratios with the other approaches and existing experimental data is also made in this section and the last section is reserved for summary.
To calculate the helicity form factors of D 0 → a − 1 ℓν decay, the following correlation function is considered: where p i , p f = (p 0 f , 0, 0, | p f |) and q = p i − p f are the four-momentum of the D 0 , a − 1 and W -boson, respectively. Moreover, j int µ =dγ µ (1 − γ 5 )c is the interaction current for D 0 → a − 1 process and j D 0 = iū γ 5 c is the interpolating current for D 0 meson. In Π a1 σ expression, ε α and ε σ denote the polarization for a 1 meson and W -boson, respectively as with Moreover, ε σ=± has similar definition as ε α=± . For off-shell W -boson, ε σ=1 and ε σ=2 are linear combinations of the transverse (±) polarization vectors as In the Light Cone QCD sum rules approach, the correlation function is given in Eq. (3), should be calculated in phenomenological and theoretical representations. Helicity form factors are found to equate both representations of the correlation function through dispersion relation. The phenomenological side can be obtained by inserting a complete series of the intermediate hadronic states with the same quantum numbers as the interpolating current j D 0 . After separating the lowest D 0 meson ground state and applying Fourier transformation, Π a1 σ is obtained as: where, ρ µ is the density of higher states and continuum which can be approximated using the ansatz of the quarkhadron duality as where, ρ QCD is the perturbative QCD spectral density and s 0 is the continuum threshold in D 0 channel. Now, the following definitions are used for the first and second matrix elements in Eq. (9): where H a − 1 σ , f D 0 and m D 0 are the helicity form factor of D 0 → a − 1 ℓν decay, the decay constant and mass of the D 0 meson, respectively. The final result for phenomenological part of correlation function is obtained as: To evaluate the correlation function Π a1 σ in QCD side, the T product of currents should be expanded near the light cone x 2 ≃ 0. After contracting c quark field, is obtained. Where S c (x, 0) is the full propagator of the c quark. In this paper, the contributions from the gluon contributions have been neglected and only the free propagator is considered as: Replacing Eq. (14) in theoretical part of Π a1 σ (p i , p f ) yields: As it is clear from Eq. (15), to calculate the theoretical part of the correlation function, the matrix elements of the nonlocal operators between a − 1 meson and vacuum state are needed. Two-particle distribution amplitudes for the axial vector mesons are given in [31], which are put in the Appendix.
In this step, two-particle LCDAs are inserted in Eq. (15), and then integrals over x and l should be evaluated. To estimate these calculations, the following identities are utilized: Now, to get the LCSR calculations for the D 0 → a − 1 ℓν helicity form factors, the expressions for σ = 0, 1, 2 from both phenomenological and theoretical sides of the correlation function are equated and Borel transform is applied with respect to variable p 2 i as: which eliminates the subtraction term in the dispersion relation and exponentially suppresses the contributions of higher states. Finally, the helicity form factors for D 0 → a − 1 ℓν, transition are obtained in the LCSR as where, Φ , Φ ⊥ are twist-2, g ⊥ , g ⊥ , h (t) and h (p) are twist-3 functions andh are scale-independent scale-dependent decay constants of the a − 1 meson, respectively [31]. We also have: The explicit expressions for twist functions are presented in the Appendix. Following the previous steps in this section, phrases similar to Eqs. (21,22,23) can be obtained for the helicity form For the physical states K 1 (1270) and K 1 (1400) the following relations are used:

III. NUMERICAL ANALYSIS
Our numerical analysis for the helicity form factors and branching ratio values of the semileptonic D (s) → Aℓ + ν, are presented in two subsections. The helicity form factors of the semileptonic D + → a 0  [31]. We can take f A = f ⊥ A at energy scale µ = 1 GeV [31]. The values of Gegenbauer moment for the axial vector mesons, can be found in [31].

A. Analysis of helicity form factors
The formulas of helicity from factors, Eqs. (21,22,23), contain two free parameters s 0 and M 2 , which are the continuum threshold and Borel mass-square, respectively. In this paper the values of continuum threshold are chosen as s 0 = (7 ± 0.2) GeV 2 [21] and working region for M 2 is provided that the contribution of higher states as well as higher twist contributions, be small. Fig. 1 shows the dependence of the D 0 → a − 1 helicity form factors with respect to M 2 . Since H σ=1,2 vanish at q 2 = 0, these two form factors are plotted at q 2 = 0.01 GeV 2 . It is easily seen from Fig. 1, that the form factors H The contributions of twist-2 and twist-3 distribution amplitudes and higher states in the D 0 → a − 1 helicity form factors, with respect to M 2 , are displaced in Figs. 2 and 3. It can be observed that at the above-mentioned interval from Borel mass, the higher twist contributions as well as higher states, are suppressed. Our numerical analysis shows, that the contribution of the higher states is smaller than about 8% of the total value.
Using all the input values and parameters, the helicity form factors can be evaluated as a function of q 2 . The values of H 0 for aforementioned decays at the zero transferred momentum square q 2 = 0 are presented in Table III. In this table, the contributions of twist-2 distribution amplitudes are also reported. The main uncertainty in H 0 (q 2 = 0) comes from c quark mass m c and Φ ⊥ light cone distribution amplitude. For H a 1 0 we take q 2 = 0 while, for H a 1 1,2 the results are plotted at q 2 = 0.01 GeV 2 . The threshold parameter is taken s0 = (7 ± 0.2) GeV 2 for every plot. In order to extend LCSR prediction to the whole physical region, m 2 ℓ ≤ q 2 ≤ (m D (s) − m A ) 2 , we use the series process H0(q 2 = 0) Twist-2 process H0(q 2 = 0) Twist-2 and Ds → K1A(K1B) ℓ + ν decays at q 2 = 0. expansion given in [22] as: where z(q 2 , t) = √ where t = t 0 , t − , m D (s) with t ± = (m D (s) ± m A ) 2 and t 0 = t + (1 − 1 − t − /t + ). Moreover, D r (s) shows the resonance states are given in Table II. The function φ(q 2 ) is given by [33]: where χ 0 has been calculated using OPE and is given by [22]: It should be noted that for the functions z(q 2 , t − ) and φ(q 2 ) the replacement m D (s) → m D r (s) must be made. For the series expansion parameterizations 25, 26 and 27, the unitarity constraints are obtained as [22]: We use parameter ∆ defined as: where 0 ≤ q 2 ≤ (m D (s) − m A ) 2 /2 to estimate quality of fit for each helicity form factor. Table IV includes the values of a σ 1 , a σ 2 and ∆ for the helicity form factors of the semileptonic decays. For these results all the input parameters are    Now, we are ready to estimate the branching ratio values for the semileptonic D (s) → Aℓν decays. The differential decay width of considered semileptonic decays is evaluated in SM as: where V cq ′ = V cd (V cs ) is used for c → d(s) ℓν transition. To calculate the branching ratios, the total mean life time τ D 0 = 0.41, τ D + = 1.04 and τ D + s = 0.50 ps [24] are used for the D (s) states. The differential branching ratios of D 0 → a − 1 (b − 1 )ℓν with their uncertainly regions, are plotted with respect to q 2 in Fig. 5. Moreover, our results for the branching ratio values of the semileptonic decays D 0 → a − 1 (b − 1 )ℓν and D + → a 0 1 (b 0 1 )ℓν decays as well as the estimations of the other approaches are presented in Fig. 6. The predictions of LCSR, 3PSR and CLFQM are calculated by using transition form factors.  The θ K dependence of the branching ratio values of D (s) decays into the physical states K 1 (1270) and K 1 (1400), are displaced in Fig. 7; and comparison between our results and other theoretical technics at θ K = −(34 ± 13) • are given in Fig. 8. The D + → K 0 1 (1270) e + ν e decay is searched at the BEPCII collider and its decay branching fraction is determined to be B(D + → K 0 1 (1270) e + ν e ) = (2.30 ± 0.69) [37]. Our branching ratio of D + → K 0 1 (1270) e + ν e agrees with the experimental measurement when θ K = −(36.68 ± 6.30) • .
In summary, we calculate the D (s) to axial vector mesons a − 1 , a 0 1 , b − 1 , b 0 1 , K 1 (1270) and K 1 (1400) helicity form factors using the light cone QCD sum rules. The uncertainties of the helicity form factors come from the borel parameter M 2 , the charm quark mass m c and Φ ⊥ twist-2 light cone distribution amplitude of the axial vector meson. To extend the LCSR calculations to the full physical region, the extrapolated series expansions are used and the low-lying D meson resonances with 1 + and 1 − quantum numbers were utilized as the dominant poles. Based on the fitted form factors, predictions for the branching ratios of relevant semileptonic decays were reported and a comparison was made between our results and other method estimations. Our calculation for branching ratio of D + → K 0 1 (1270) e + ν e decay is in good agreement with the BEPCII collider measurement within errors at the mixing angle θ k = −(36.68 ± 6.30) • This work LCSR [34] 3PSR [21] Br (D