Analysis of the masses and decay constants of the heavy-light mesons with QCD sum rules

In this article, we calculate the contributions of the vacuum condensates up to dimension-6 including the $\mathcal{O}(\alpha_s)$ corrections to the quark condensates in the operator product expansion, then study the masses and decay constants of the pseudoscalar, scalar, vector and axial-vector heavy-light mesons with the QCD sum rules in a systematic way. The masses of the observed mesons $(D,D^*)$, $(D_s,D_s^*)$, $(D_0^*(2400),D_1(2430))$, $(D_{s0}^*(2317),D_{s1}(2460))$, $(B,B^*)$, $(B_s,B_s^*)$ can be well reproduced, while the predictions for the masses of the $(B^*_{0}, B_{1})$ and $(B^*_{s0}, B_{s1})$ can be confronted with the experimental data in the future. We obtain the decay constants of the pseudoscalar, scalar, vector and axial-vector heavy-light mesons, which have many phenomenological applications in studying the semi-leptonic and leptonic decays of the heavy-light mesons.


Introduction
The charged heavy-light mesons can decay to a charged lepton pair ℓ + ν ℓ through a virtual W + boson. Those leptonic decays are excellent subjects in studying the CKM matrix elements and serve as a powerful probe of new physics beyond the standard model in a complementary way to the direct searches. For example, the decay widths of the pseudoscalar (P) and vector (V) heavy-light mesons can be written as in the lowest order approximation, where the m P/V and f P/V are the masses and decay constants, respectively, the m ℓ is the ℓ mass, the V q1q2 is the CKM matrix element between the constituent quarks q 1q2 , and the G F is the Fermi coupling constant. If we take the CKM matrix element V q1q2 and the branching fractions of the leptonic decays from the CLEO, BaBar, Belle collaborations as input parameters, then the average values f D = (204.6 ± 5.0) MeV, f Ds = (257.5 ± 4.6) MeV and f Ds /f D = 1.258 ± 0.038 are obtained [1]. It is difficult to reproduce the three values consistently in theoretical calculations, such as the QCD sum rules [2,3,4,5] and lattice QCD [6,7,8]. The discrepancies between the theoretical values and experimental data maybe signal some new physics beyond the standard model [9]. In Ref. [10], we observe that if we take into account the O(α 2 s ) corrections to the perturbative terms and the O(α s ) corrections to the quark condensate terms and choose the pole masses, the predictions f D = (211 ± 14) MeV, f Ds = (258 ± 13) MeV and f Ds /f D = 1.22 ± 0.08 are in excellent agreement with the experimental data [1].
In the QCD sum rules for the heavy-light mesons, the Wilson coefficients of the vacuum condensates at the operator product expansion side from different references differ from each other in one way or the other according to the different approximations [2,3,10,11,12]. In this article, we recalculate the contributions of the vacuum condensates up to dimension-6, including the one-loop corrections to the quark condensates, and take into account the terms neglected in previous works, then study the masses and decay constants of the pseudoscalar, scalar, vector and axial-vector heavy-light mesons in a systematic way.
The article is arranged as follows: we derive the QCD sum rules for the masses and decay constants of the heavy-light mesons in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusions.

QCD sum rules for the heavy-light mesons
In the following, we write down the two-point correlation functions Π 0/5 (p) and Π µν V /A (p) in the QCD sum rules, where the currents J 5 (x), J 0 (x), J µ V (x) and J µ A (x) interpolate the pseudoscalar, scalar, vector and axial-vector heavy-light mesons, respectively, Q = c, b and q = u, d, s. We can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J 5 (x), J 0 (x), J µ V (x) and J µ A (x) into the correlation functions Π 0/5 (p) and Π µν V /A (p) to obtain the hadronic representation [38,39]. After isolating the ground state contributions from the pseudoscalar, scalar, vector and axial-vector heavy-light mesons, we get the following results, where the decay constants f S/P/V /A are defined by the ǫ µ are the polarization vectors of the vector and axial-vector mesons.  Now we carry out the operator product expansion at large space-like region P 2 = −p 2 . The analytical expressions of the perturbative O(α s ) corrections to the perturbative terms for all the correlation functions [17,18] and the semi-analytical expressions of the perturbative O(α 2 s ) corrections to the perturbative terms for the pseudoscalar current's correlation functions [19] are available now. We take into account those analytical and semi-analytical expressions directly [17,18,19]; and recalculate the contributions of the vacuum condensates, i.e. we calculate the Feynman diagrams shown in Figs.1-5, where the solid and dashed lines denote the light and heavy quark lines, respectively, the wave line denotes the gluon line. In calculating the diagrams in Fig.2, we correct the minor errors in Ref. [10], where the quark condensate qq 12 in the full light-quark propagators is replaced with qq 3D , the D is the dimension of the space-time. A minor error occurs when there exist divergences, such a step should be deleted, i.e. the quark condensate qq 12 survives in the D-dimension. In Ref. [40], we correct the minor errors and improve the calculations, and obtain the correct expressions. Furthermore, we obtain the perturbative O(α s ) corrections to the quark condensate terms for the vector and axial-vector currents.   Once analytical expressions of the QCD spectral densities are obtained, then we can take the quark-hadron duality below the continuum thresholds and perform the Borel transforms with respect to the variable P 2 = −p 2 to obtain the QCD sum rules, where and the s 0 are the continuum threshold parameters. The perturbative O(α s ) corrections R 5 (x) and R V (x) are taken from Refs. [17,18]. We can also take into account the semi-analytical perturbative O(α 2 s ) corrections to the perturbative terms for the B T Π 0 5 , are mathematical functions defined at the energy-scale of the pole mass µ = m c , here the n l counts the number of massless quarks [19]. We can derive Eqs. (9)(10)(11)(12) with respect to 1/T 2 , then eliminate the decay constants f S/P/V /A to obtain the QCD sum rules for the masses.
Once the masses m S/P/V /A are obtained, we can take them as input parameters and obtain the decay constants from the QCD sum rules in Eqs. (9)(10)(11)(12).
In the case of the light-quark currents, the perturbative O(α s ) corrections to the perturbative terms amount to multiplying the factors 1 + 11 3 αs π ≈ 1 + 3.67 αs π and 1 + αs π to the perturbative terms in the correlation functions for the pseudoscalar (scalar) and vector (axial-vector) currents, respectively [39]. In the present case, if we take the approximation µ 2 = m 2 c = T 2 , the perturbative O(α s ) corrections to the quark condensate terms amount to multiplying the factors 1 + 3.47 αs π and 1 + 0.94 αs π to the quark condensate terms in the correlation functions for the pseudoscalar (scalar) and vector (axial-vector) currents, respectively. The analogous O(α s ) corrections indicate that the present calculations are reliable.

Numerical results and discussions
In the heavy quark limit, the heavy-light mesons Qq can be classified in doublets according to the total angular momentum of the light antiquark s ℓ , s ℓ = sq + L, where the sq and L are the spin and orbital angular momentum of the light antiquark, respectively. The spin doublets (D,  [41]. We take the values √ s 0 = m gr + (0.4 − 0.8) GeV as guides, here the gr denotes the ground states, and search for the optimal threshold parameters s 0 and Borel parameters T 2 to satisfy the following criteria: • Pole dominance at the phenomenological side; • Convergence of the operator product expansion; • Appearance of the Borel platforms; • Reappearance of experimental values of the ground state heavy meson masses. The contributions of the ground states can be fully taken into account by choosing the threshold parameters The contaminations of the excited states are very small if there are some contaminations, we expect that the couplings of the currents to the excited states are more weak than that to the ground states. For example, the decay constants of the pseudoscalar mesons π(140) and π(1800) have the hierarchy f π(1300) ≪ f π(140) from the lattice QCD [42], the QCD sum rules [43], or from the experimental data [44].
The vacuum condensates are taken to be the standard values s GGG = 0.045 GeV 6 at the energy scale µ = 1 GeV [38,39]. The quark condensates and mixed quark condensates evolve with the renormalization group equation, qq (µ) = qq (Q) αs(Q) αs(µ) 4 9 and qg s σGq (µ) = qg s σGq (Q) αs(Q) αs(µ)   Table 1. From Table 1, we can see that the pole dominance can be satisfied. On the other hand, the dominant contributions come from the perturbative terms and the quark condensate terms, so we expect to obtain reliable predictions.
After taking into account the uncertainties of the input parameters, we obtain the values of the masses and decay constants of the heavy-light mesons, which are shown in Figs.6-9 and Table 1. From the figures, we can see that the masses and decays constants are rather stable with variations of the Boral parameters T 2 , the predictions are reasonable.
From Table 1 and B s1 also lie below the corresponding BK and B * K thresholds, respectively. The strong decays B * s0 → BK and B s1 → B * K are kinematically forbidden, the P-wave heavy mesons B * s0 and B s1 can decay through the isospin violation precesses B * s0 → B s η → B s π 0 and B s1 → B * s η → B * s π 0 respectively or through the radiative decays [46]. The η and π 0 transition matrix is very small according to Dashen        t ηπ = π 0 |H|η = −0.003 GeV 2 , the P-wave bottomed mesons B * s0 and B s1 , just like their charmed cousins D * s0 (2317) and D s1 (2460), maybe very narrow [48]. The present predictions are consistent with our previous work [13], but the analysis is refined by including more terms in the operator product expansion.
The values of the decay constants of the pseudoscalar mesons are slightly different from the ones in our previous work [10]. In Table 2, we compare the present predictions to the experimental data and other theoretical calculations, such the QCD sum rules (QCDSR) [2,3,4,5,14] and lattice QCD (LQCD) [6,7,8]. The present predictions f D = (208 ± 10) MeV and f B = (194 ± 15) MeV are consistent with the experimental data within uncertainties, while the prediction f Ds = (240 ± 10) MeV is lies below the lower bound of the experimental value f Ds = (257.5 ± 4.6) MeV [1]. We take the M S mass m c (µ) and truncate the perturbative corrections to the order O(α s ), the experimental values of the f D , f Ds and f Ds /f D cannot be reproduced consistently by the QCD sum rules. The existence of a charged Higgs boson or any other charged object beyond the standard model would modify the decay rates, see Eq.(1), therefore modify the values of the decay constants, for example, the leptonic decay widths are modified in two-Higgs-doublet models [49]. If the predictions of the f D , f Ds and f Ds /f D based on the QCD sum rules are close to the true values, new physics beyond the standard model are favored so as to smear the discrepancies between the theoretical calculations and experimental data. The Borel parameters, continuum threshold parameters, pole contributions, and the resulting masses and decay constants of the heavy pseudoscalar mesons are shown in Table 3, the values are slightly different from the ones in our previous work [10]. From Table 1 and Table 3, we can see that the present predictions f D = (210 ± 11) MeV, f Ds = (259 ± 10) MeV and f B = (192 ± 13) MeV are in excellent agreement with the experimental data within uncertainties [1]. The ratio f Ds /f D = 1.23 ± 0.07 is also in excellent agreement with the experimental data f Ds /f D = 1.258 ± 0.038 [1], which indicates that the perturbative O(α 2 s ) corrections should be taken into account. However, the pole masses m Q and energy scales µ = m Q have be chosen, as the semi-analytical expressions are obtained at such conditions. In this case, new physics beyond the standard model are not favored, as the agreements between the experimental data and present theoretical calculations are already excellent.
In Table 4, we compare the present predictions for the decay constants of the heavy vector mesons to other theoretical calculations, such as the QCD sum rules [5,15,21], lattice QCD [22,23,24,25], the relativistic potential model (RPM) [30], the field-correlator method (FCM) [32], and the light-front quark model [34]. From the table, we can see that the predictions differ from each other in one way or the other. In Table 5, we compare the present predictions for the decay constants of the heavy scalar mesons to the ones from the QCD sum rules [16] and lattice QCD [26]. From the table, we can see that the predictions are consistent with the ones from lattice calculations but differ greatly from the ones from the QCD sum rules.
If we turn off the perturbative O(α s ) corrections to the quark condensates and choose the same parameters, such as the M S masses, Borel parameters and continuum threshold parameters, etc, the masses and decay constants undergo reduction or increment in a definite way according to the spin and parity, see Table 6. From the table, we can see that the mass-shifts of the D-mesons with J P = 0 ± are larger than 40 MeV, while the shifts of the masses and decay constants of all the B-mesons are small and can be neglected. We can re-choose the Borel windows to warrant the mass-shifts δm S/P/V /A = 0, and account for the net effects by the shifts of the decay constants δf S/P/V /A , which are shown the bracket in Table 6. From the table, we can see that the largest shift of the decay constant δf D = −11 MeV, which exceeds the total uncertainty of the decay constant δf D = ±10 MeV (see Table 1), the shifts of the decay constants of the D-mesons with J P = 0 ± , 1 − are larger than 5 MeV, while for other mesons, the shifts of the decay constants |δf | ≤ 4 MeV. All in all, we should take into account the perturbative O(α s ) corrections to the quark condensates in a comprehensive study.