$D_{sJ}(2860)$ From The Semileptonic Decays Of $B_s$ Mesons

In the framework of heavy quark effective theory, the leading order Isgur-Wise form factors relevant to semileptonic decays of the ground state $\bar{b}s$ meson $B_{s}$ into orbitally excited $D$-wave $\bar{c}s$ mesons, including the newly observed narrow $D^{*}_{s1}(2860)$ and $D^{*}_{s3}(2860)$ states by the LHCb Collaboration, are calculated with the QCD sum rule method. With these universal form factors, the decay rates and branching ratios are estimated. We find that the decay widths are $\Gamma(B_s\rightarrow D^{*}_{s1}\ell\bar{\nu}) =1.25^{+0.80}_{-0.60}\times10^{-19} \mbox{GeV}$, $\Gamma(B_s\rightarrow D^{'}_{s2}\ell\bar{\nu}) =1.49^{+0.97}_{-0.73}\times10^{-19} \mbox{GeV}$, $\Gamma(B_s\rightarrow D_{s2}\ell\bar{\nu}) =4.48^{+1.05}_{-0.94}\times10^{-17} \mbox{GeV}$, and $\Gamma(B_s\rightarrow D^{*}_{s3}\ell\bar{\nu}) = 1.52^{+0.35}_{-0.31}\times10^{-16} \mbox{GeV}$. The corresponding branching ratios are $\mathcal {B}(B_s\rightarrow D^{*}_{s1}\ell\bar{\nu}) =2.85^{+1.82}_{-1.36}\times 10^{-7}$, $\mathcal {B}(B_s\rightarrow D^{'}_{s2}\ell\bar{\nu}) =3.40^{+2.21}_{-1.66}\times 10^{-7}$, $\mathcal {B}(B_{s}\rightarrow D_{s2}\ell\bar{\nu}) =1.02^{+0.24}_{-0.21}\times 10^{-4}$, and $\mathcal {B}(B_s\rightarrow D^{*}_{s3}\ell\bar{\nu}) = 3.46^{+0.80}_{-0.70}\times 10^{-4}$. The decay widths and branching ratios of corresponding $B^{*}_{s}$ semileptonic processes are also predicted.

(Dated: December 30, 2014) In the framework of the heavy quark effective theory, the leading order Isgur-Wise form factors relevant to semileptonic decays of the groundbs meson B s into the orbitally excited D-wavecs mesons, including the newly observed narrow D * s1 (2860) and D * s3 (2860) states by the LHCb Collaboration, are calculated with the QCD sum rule method. With these universal form factors, the decay rates and branching ratios are estimated. We find that the decay widths are Γ Bs→D
Experimentally, copious samples of charm-strange mesons are, however, available from decays of B 0 s mesons produced at high energy hadron colliders. These have been exploited to study the properties of the orbitally excitedcs mesons, such as D s1 (2536) − and D * s2 (2573) − states, produced in semileptonic B 0 s mesons [10]. The results are important not only from the point-of-view of spectroscopy, but also as they will provide input to future studies of CP violation in B 0 s →D 0 K − π + decays [2]. Actually, the b → c semileptonic processes are the important sources for the the determination of the parameters of the standard model, such as Cabibbo-Kobayashi-Maskawa matrix element |V cb |. They also provide valuable insight in quark dynamics in the nonperturbative domain of QCD. Just because of these reasons, the semileptonic decays of B and B s mesons have been under investigation for many years [11][12][13][14][15][16][17][18][19][20].
In this paper, we assume that the newly observed D * s1 (2860) and D * s3 (2860) mesons are the 1 − and 3 − states which are menbers of the 1D family. Then we use the QCD sum rule method [22] under the framework of heavy quark effective theory (HQET) [13,23] to study the semileptonic decays of groundbs meson doublet H(0 − , 1 − ) into the orbitally Dwave excitedcs meson doublets F (1 − , 2 − ) and X(2 − , 3 − ) containing one heavy anti-quark and one strange quark. The QCD sum rule approach, incorporation with HQET has been proved to be a successful method which was widely applied to investigate the properties and dynamical processes of heavy hadrons containing a single heavy quark [12]. We shall follow the procedure used in Refs. [18,20,24], and study the semileptonic decays mentioned above.
The remainder of this paper is organized as follows. After an introduction, we derive the formulae of the weak current matrix elements in HQET in Sec. II. Then we deduce the three-point sum rules for the relevant universal form factors in Sec. III. In Sec. IV, we give the numerical results and discussions. The decay rates and branching ratios are also estimated in the final section.
The semileptonic decay rate of a B s meson transition into a D s meson is determined by the corresponding matrix elements of the weak vector and axial-vector currents (V µ = cγ µ b and A µ = cγ µ γ 5 b) between them. These hadronic matrix elements can be parametrized in terms of some weak form factors. In HQET, the classification of these form factors has been simplified greatly. At the leading order of the heavy quark expansion, the matrix elements involved in the transitions between the H doublet of thebs mesons and the F or X doublet ofcs mesons can be parametrized in terms of only one Isgur-Wise function.
According to the formalism given in Ref. [25], the heavy-light meson doublets can be expressed as effective operators. For the processes (B s , B * s ) → (D * 1s , D ′ 2s )ℓν, two heavy-light meson doublets H and F are involved. The operators P and P * µ that annihilate members of the H doublet with four-velocity v are, in the form, The fields D * 1ν and D ′µν 2 that annihilate members of the F doublet with four-velocity v are in the representation where / v = v · γ. For the processes (B s , B * s ) → (D 2s , D * 3s )ℓν, the final heavy hadronic states which annihilated by the operators D αβ 2 and D * µνσ 3 are in another doublet X with four-velocity v, namely At the leading order of the heavy quark expansion, the hadronic matrix elements of the weak currents between states in the doublets H v and F v ′ can be calculated from while the corresponding matrix elements between states annihilated by fields in H v and where h (Q) v,v ′ are the heavy quark fields in HQET, and X v ′ = γ 0 X † v ′ γ 0 . v is the velocity of the initial meson and v ′ is the velocity of the final meson in each process. The Isgur-Wise form factors ξ(y) and ζ(y) are universal functions of the product of velocities y(= v · v ′ ).
Here we should notice that each side of Eqs. (4) and (5) is understood to be inserted between the corresponding initialbs and finalcs states. The hadronic matrix elements of )ℓν can be derived directly from the trace formalism (4) and are given as The hadronic matrix elements of B s (B * s ) → D s2 (D * s3 )ℓν are calculated similarly from Eq. (5) as follows: In these matrix elements, ε α (ε ′ α ) is the polarization vector of the initial (final) vector meson and ε αβ and ε ′ σρτ are the polarization tensors of the final tensor mesons. In the derivation of the matrix elements and formulae below, we have used a software called Mathematica with a package called FeynCalc [26]. The only unknown factors in the matrix elements above are the Isgur-Wise form factors ξ(y) and ζ(y) which should be determined through nonperturbative methods. In the following section, we will employ the QCD sum rule approach to estimate them.

III. FORM FACTORS FROM HQET SUM RULES
In order to apply QCD sum rules to study the heavy mesons, we must choose appropriate interpolating currents to represent them. Here we adopt the interpolating currents proposed in Ref. [27] based on the study of Bethe-Salpeter equations for heavy mesons in HQET. Following the remarks given in Ref. [20], we take the interpolating currents that create heavy mesons in the H, F and X doublets as where D α The decay constants f 0,−,1/2 , f 1,−,1/2 , f 1,−,3/2 , f 2,−,3/2 , f 2,−,5/2 , and f 3,−,5/2 are low-energy parameters which are determined by the dynamics of the light degree of freedom.
With these currents, we can now estimate the Isgur-Wise functions ξ(y) and ζ(y) from QCD sum rules. First comes the ξ(y). The jumping-off point is the following three-point correlation function: where J are the weak currents, J 0,−,1/2 and J α 1,−,3/2 are the interpolating currents defined in Eq. (14) and Eq. (16). Here it is worth noting that ξ(y) can also be estimated by choosing the interpolating current (15) for the initial state and the current (17) for the final state because of the heavy quark symmetry. Ξ 1 (ω, ω ′ , y) is analytic functions in ω = 2v · k and ω ′ = 2v ′ · k ′ , and are not continual when where "· · · " denotes contributions from higher resonances and continuum states, f 1,−,3/2 is the decay constant defined in Eq. (22). As we can see from the Eqs. (28) and (29), the pole contribution to Ξ 1 (ω, ω ′ , y) is proportional to the universal function ξ(y). The QCD sum rule then can be constructed directly from Ξ 1 (ω, ω ′ , y) by isolating the Lorentz structures.
The theoretical side of the correlator is calculated by means of the operator product expansion. The perturbative part can be expressed as a double dispersion integral in ν and ν ′ plus possible subtraction terms. Therefore the theoretical expression for the correlation function in (28) is of the form The perturbative spectral density ρ pert can be calculated straightforwardly from HQET Feynman rules. At the leading order of perturbation and heavy quark expansion, we obtain the perturbative spectral density of the sum rule for ξ(y) as Assuming quark-hadron duality, the contribution from higher resonances is usually approximated by the integration of the perturbative spectral density above some threshold. Equating the phenomenological and theoretical representations, the contribution of higher resonances in the phenomenological expression (29) can be eliminated. Following the arguments in Refs. [12,28], we can not directly assume local duality between the perturbative and the hadronic spectral densities, but first integrate the spectral density over the "off-diagonal" variable ν − = ν − ν ′ , keeping the "diagonal" variable ν + = ν+ν ′ 2 fixed. Then the quark-hadron duality is assumed for the integration of the spectral density in ν + . The integration region is restricted by the Θ functions above in terms of the variables ν − and ν + , and usually the triangular region defined by the bounds: 0 ≤ ν + ≤ ω c , −2 y−1 y+1 ν + ≤ ν − ≤ 2 y−1 y+1 ν + is chosen. A double Borel transformation in ω and ω ′ is performed on both sides of the sum rule, in which for simplicity we take the Borel parameters equal [12,16,17]: T 1 = T 2 = 2T . It eliminates the subtraction terms in the dispersion integral (30) and improves the convergence of the operator product expansion series. Our calculations are confined at the leading order of perturbation. Among the operators in the operator product expansion series, only those with dimension D ≤ 5 are included. For the condensates of higher dimension (D > 5), their values are negligibly small and their contributions are suppressed by the double Borel transformation. So they can be safely omitted. Finally, we obtain the sum rule for the form factor ξ(y) as follows: The derivation of the sum rule for ζ(y) is totally similar. Only now the correlation function we need to consider is where J are also the weak currents, J 0,−,1/2 and J αβ 2,−,3/2 are the interpolating currents defined in Eqs. (14) and (18). By repeating the above procedure, we reach the perturbative spectrum density as below: Then the sum rule for ζ(y) appears as below: = 2810 MeV [6], M D s2 = 2820 MeV [6], and M D * s3 = 2860.5 MeV [1]. In order to obtain information of Isgur-Wise function ξ(y) and ζ(y) with less systematic uncertainty, we can divide the three-point sum rules (32) and (35) with the square roots of relevant two-point sum rules for the decay constants, as many authors did [12,16,17]. This can not only reduce the number of input parameters but also improve stabilities of the three-point sum rules. In the calculation of ξ(y), the two-point QCD sum rule we need here are [30] and [6]  Here the cutoff parameter µ is fixed at 1GeV and the Euler parameter γ E = 0.577. In order to calculate ζ(y), we need the two-point QCD sum rule (36) and [6] f 2 2,−,5/2 e −2Λ 2,−,5/2 /T = 1 640π 2 After the divisions have been done, the Isgur-Wise functions ξ(y) and ζ(y) depend only on the Borel parameter T and the continuum thresholds. The determination of the Borel parameter is an important step of sum rules. After a careful analysis, we find the sum rule for ξ(y) works well in a sum rule "window": 0.4 GeV < T < 0.6 GeV, which overlaps with that of the two-point sum rule (36) [30]. For the sum rule of ζ(y), we choose the "window" as 0.5 GeV < T < 0.7 GeV. Note that the Borel parameters in the three-point sum rules are twice of those in the two-point sum rules. In the evaluation, we have taken 2.0 GeV < ω 0 < 2.4 GeV, 2.8 GeV < ω 1 < 3.2 GeV, and 3.2 GeV < ω 2 < 3.6 GeV [20]. The regions of these continuum thresholds are fixed by analyzing the corresponding two-point sum rules [30]. Following the discussions in Refs. [12,28], the upper limit ω c1 for ν + in (32) and ω c2 in (35) should be evaluated in the regions 1 2 [(y +1)− y 2 − 1]ω 0 ω c1 1 2 (ω 0 +ω 1 ) and 1 2 . So they can be fixed in the regions 2.4 GeV < ω c1 < 2.6 GeV and 2.5 GeV < ω c2 < 2.7 GeV . After all these parameters are putted into the calculation, we get the results that are shown in Fig. 1 and Fig. 2 The errors mainly come from the uncertainty due to ω c 's and T . It is difficult to estimate the systematic errors which are brought in by the quark-hadron duality. Using the linear approximates for the universal form factors above, we can calculate the semileptonic decay rates of processes B s (B * s ) → D * s1 (D ′ s2 )ℓν and B s (B * s ) → D s2 (D * s3 )ℓν. For this purpose, we have to derive firstly the formulae for the differential decay rates of these processes in terms of the Isgur-Wise functions ξ(y) and ζ(y) from the matrix elements (6)-(13) given in Sec. II. After some derivation, the formulae of the differential decay rates of the processes while for the processes B s (B * s ) → D s2 (D * s3 )ℓν, they can be found to be 1000π 3 |ζ(y)| 2 (y − 1) 5/2 (y + 1) 7/2 [(r 2 5 + 1)(7y − 3) − 2r 5 4y 2 − 3y + 3 ], 3s |ζ(y)| 2 (y − 1) 5/2 (y + 1) 7/2 [(r 2 6 + 1)(11y + 3) − 2r 6 8y 2 + 3y + 3 ], where r i (i = 1, · · · , 8) is the ratio between the mass of the finalcs meson and that of the initialbs meson in each process, e.g., The maximal values of y for these semileptonic processes are given in Table I. In addition, we need the input parameters V cb = 0.04 and G F = 1.166 × 10 −5 GeV −2 . By integrating the differential decay rates over the kinematic region 1.0 ≤ y ≤ y max , we get the decay widths of these semileptonic decay modes which are listed in Table II. Notice that the lifetime of B 0 s meson is τ B 0 s = 1.5ps   , which means the total decay width is about Γ B 0 s = 4.388 × 10 −13 GeV. There is no experimental result for the total width of the B * s meson by now, but we know that its dominant decay mode is the radiative decay B * s → B s γ [29], the width of which is calculated theoretically to be about Γ B * s = 0.07keV [31,32]. We can take it as the total width of B * s meson for a rough estimation for the branching rations of its semileptonic decays. Taking all these into account, we get the final branching ratios of the semileptonic decays mentioned above (see Table II). It worth noting that the large errors which come from the systematical uncertainty of the QCD sum rule approach can be reduced by taking higher order corrections into account. As we can see in Table II, the branching ratio of B * s semileptonic decays into the D * s1 (2860) and D * s3 (2860) are too small to be observed, while the branching ratios of B 0 s semileptonic decays into these states are large enough to be measured by the LHCb experiments.
In summary, we have studied the semileptonic decays of the ground statebs meson doublet (0 − , 1 − ) into the 1D excited family ofcs meson, including the newly observed D * s1 (2860) and D * s3 (2860) mesons by the LHCb collaboration. Under the framework of HQET, we have employed the QCD sum rule approach to estimate the leading-order universal form factors describing these weak transitions. With these universal form factors, the decay rates and branching ratios are estimated. We find that the decay widths are Γ Bs→D * s1 ℓν = (3.2 ± 1.1) × 10 −19 GeV, Γ Bs→D ′ s2 ℓν = (3.8 ± 1.3) × 10 −19 GeV, Γ Bs→D s2 ℓν =