ProbingD_s^*-meson longitudinal twist-2 LCDA

B-meson semileptonic decay is a very important tool for studying the weak interaction. It has great phenomenological implications within the Standard Model (SM) of particle physics. $ B_s\to D_s^*\ell \bar\nu_{\ell} $ decay provides an opportunity for extracting CKM matrix elements and testing the SM [1−4]. Recently, the study of $ B_s\to D_s^*\ell \bar\nu_{\ell} $ has attracted significant interest, driven by advancements in experimental capabilities and theoretical developments. Many experiments have provided increasingly precise data of these decays, which allow the extraction of the CKM matrix element to an increasingly better accuracy. In addition, many theoretical groups have shown great interest in exploring the decay channel.

In 2020, the LHCb Collaboration [5,6] published an article on the experimental measurement of $ B_s\to D_s^*\ell \bar\nu_{\ell} $ decay. They used the experimental data samples collected by the LHCb detector at center-of-mass energies of 7 and 8 TeV and conducted systematic studies on the two decay channels $ B^0_s\to D^-_s\mu^+\nu_\mu $ and $ B^0_s\to D^{*-}_s\mu^+\nu_\mu $ using Caprini-Lellouch-Neubert (CLN) and Boyd-Grinstein-Lebed (BGL) parameterizations. The final measured values of $ |V_{cb}| $ obtained under the two parameterizations were $ (41.4 \pm 0.6 \pm 0.9 \pm 1.2)\times10^{-3} $ and $ (42.3 \pm 0.8\pm 0.9 \pm 1.2)\times 10^{-3} $ , respectively. In Ref. [7], the lattice QCD and HPQCD Collaboration worked together to determine the model independent value of $ V_{cb} = 39.03(56)_{\rm{exp.}} (67)_{\rm{latt.}}\times 10^{-3} $ using the $ B\to D^{*}\ell \bar{\nu}_{\ell} $ data from Belle and the $ B_s\to D_s^*\mu \bar{\nu}_\mu $ data from LHCb, as well as their own transition form factors (TFFs). The TFFs and decay width behaviors for the $ B_s\to D_s^*\ell \bar{\nu}_{\ell} $ process were also given in that article.

Theoretically, there are many methods for making reasonable predictions of $ B_s\to D_s^*\ell \bar\nu_{\ell} $ decay, such as lattice QCD (LQCD) [8−12], soft-collinear effective theory (SCET) [13], covariant confined quark model (CCQM) [14], QCD sum rules (QCDSR) [15], relativistic quark model (RQM) [16], perturbative QCD (pQCD) [17,18], light-front quark model (LFQM) [19], and Bethe-Salpeter method [20]; here, we provides a brief introduction to the above methods. In Ref. [9], the SM semileptonic vector and axial-vector form factors for $ B_s \to D_s^* $ decay are calculated via the LQCD. The relevant calculation methods, the dependence on the heavy quark mass, decay rates and ratios are analysed. Meanwhile, the consistency with LHCb results and the tests on the impact of new physics couplings are also discussed. They give a reasonable reference to study the semileptonic decay of $ B_s \to D_s^* \ell \bar\nu_{\ell} $ . In Ref. [13], the authors utilized the power-counting scheme for the first time to calculate the next-to-leading order (NLO) QCD corrections to the form factors of $ \bar{B}_{(s)}\to D_{(s)}^{(*)}\ell \bar\nu_{\ell} $ transitions in the large recoil region. Within the framework of SCET, they also improved the light-cone sum rules for these form factors and conducted a combined fit for the considered form factors. This significantly refined the previous measurements of the lepton flavor universality ratios. In Ref. [14], the author relied on the SM framework based on the CCQM to calculate TFFs within the entire dynamical category of squared momentum transfer for semileptonic decays. The obtained results, such as decay width ratios, show a degree of consistency with LHCb experiments and LQCD simulations, and the behaviors of differential decay distributions were compared. Other physical observations were also calculated. Through calculation, $ R(D_s) = 0.271\pm0.069 $ , $ R(D^*_s) = 0.240\pm0.038 $ were obtained, and the ratio of decay widths of the $ D_s $ and $ D_s^* $ channels in the muon meson mode $ \Gamma(B_s\to D_s\mu^+\nu_\mu)/ \Gamma(B_s\to D_s^*\mu^+\nu_\mu) = 0.451\pm0.093 $ was determined. In Ref. [15], the phenomenology of theb-flavored strange meson $ B^0_s $ was studied through the QCDSR. Specifically, this included the evaluation of the particle's mass and leptonic constant, as well as the study of the form factors of certain decays (such as $ \bar B^0_s \to D^+_s\ell^-\bar{\nu} $ , $ \bar B^0_s\to D^{*+}_s\ell^-\bar{\nu} $ , $ \bar B^0_s \to K^{*+}\ell^-\bar{\nu} $ ); at the same time, the two-body non-leptonic $ \bar B^0_s $ decays were calculated under the factorization approximation. Finally, the evaluation result of the $ S U(3)_F $ breaking effect in the $ \bar B^0_s $ channel was compared with other estimates.

Also, as a well-established theory that can be effectively applied to the exclusive decay process, the QCD light-cone sum rule (LCSR) [21,22] incorporates both the hard and soft contributions in the computation of hadron transitions. In the LCSR method, the two-point correlation function of vacuum to meson is constructed for calculating heavy-to-light TFFs, where the matrix elements of nonlocal operators are considered in the light-cone region $ x^2 \to 0 $ . The difference between this method and traditional SVZ sum rule [23] is that all non-perturbative dynamics are parameterized according to the light-cone distribution amplitudes (LCDAs) with progressively higher twists instead of quark and gluon condensates [24]. Currently, the LCSR has been widely applied to the study ofBto light-flavor meson decays [25−30]. In this work, we adopt LCSR to calculate the $ B_s\to D^*_s $ TFFs. Specifically, due to the presence of both longitudinal $ (\|) $ and transverse $ (\perp) $ polarization states in vector mesons, employing traditional currents to construct correlation functions poses a challenge involving fifteen LCDAs. Therefore, to simplify the calculation process, we adopt chiral currents instead of traditional currents, allowing the contributions of twist-2 LCDA to dominate and eliminating contributions from other LCDAs. The specific procedure will be elaborated in the next section. Therefore, an accurate prediction of the twist-2 $ \phi_{2;D_s^*}^\|(x,\mu) $ is of great importance. In recent relevant studies, breakthrough progress has been achieved in theoretical calculations of heavy meson distribution amplitudes. For instance, in Refs. [31−33], heavy meson distribution amplitudes were determined based on the Large Momentum Effective Theory and by utilizing the LQCD method. Furthermore, the amplitudes of $ D_s^* $ -mesons were studied in these articles [34,35], indicating that the research on $ D_s^* $ -meson amplitudes has attracted considerable attention. Currently, the LCDA of many mesons depends on the Gegenbauer moment, which can be calculated using QCD sum rules. As a mature theoretical method, the background field theory (BFT) decomposes the quark field into a classical background field that describes non-perturbative effects and a quantum field that describes perturbative effects. This can provide a clean physical picture for separating the perturbative and non-perturbative properties of the QCD theory and provide a systematic way to derive the QCDSR for hadron phenomenology. Meanwhile, due to the ability to adopt different gauges for quantum fluctuations and the background field in BFT, the calculations can be greatly simplified. In our previous work, the longitudinal twist-2 LCDA ofρ-meson was successfully studied in BFT and constructed by the light cone harmonic oscillator (LCHO) model [36]. Motivated by this, in this work, we employ the BFT method to study the twist-2 LCDA of the $ D_s^* $ -meson and attempt to integrate the phenomenological LCHO model to provide another perspective to understand the momentum distribution of quarks and gluons inside $ D_s^* $ -mesons.

The remainder of this article is organized as follows. In Sec. II, we derive the summation rules for theξ-moments of the $ D_s^* $ -meson longitudinal leading-twist $ \phi_{2;D_s^*}^\|(x,\mu) $ LCDAs and $ B_s\to D_s^* $ TFFs $ A_1(q^2) $ , $ A_2(q^2) $ , and $ V(q^2) $ , as well as establish the LCHO for the $ D_s^* $ -meson leading-twist LCDAs. In Sec. III, we provide relevant numerical results and a detailed discussion. Section IV summarizes the article.

The differential decay width of semileptonic decays $ B_s\to D_s^*\ell\bar\nu_\ell $ can be written in terms of the helicity components basis [37−39]:

where the Fermi coupling constant $ G_F = 1.1663787(6)\times 10^{-5}\; {\rm{GeV}}^{-2} $ , $ |V_{cb}| $ is the CKM matrix element, $ \lambda\equiv \lambda(m_{B_s}^2,m_{m_{D_s^*}}^2,q^2) = m_{B_s}^4+m_{D_s^*}^4+q^4-2(m_{B_s}^{2}m_{D_s^*}^{2}+m_{B_s}^{2}q^{2}+m_{D_s^*}^{2}q^{2}) $ is the phase-space factor, $ m_\ell $ is the lepton mass $ (\ell = e,\mu,\tau) $ , and $ {\cal{H}}_{\rm{total}} $ represents the overall helicity structure:

The symbols $ {\cal{H}}_{I}(I = U,L,S) $ are the bilinear combinations of the helicity components of the hadronic tensor. In this study, the leptonic mass $ m_\ell $ is very small in case of $ \ell = (e,\mu) $ when compared with the squared transition momentum $ q^2 $ , which can be safely neglected. Thus, only two helicity structures remain,i.e., $ {\cal{H}}_U $ and $ {\cal{H}}_L $ , which have the following formulas:

The helicity amplitudes $ H_i $ with index $ i = (\pm,0,t) $ are denoted as functions of invariant mass $ q^2 $ , which are formed from the $ B_s\to D_s^* $ TFFs with different combinations:

As we know, the three $ B_s\to D_s^* $ TFFs $ A_1(q^2) $ , $ A_2(q^2) $ , and $ V(q^2) $ are the important hadronic inputs for studying the relevant implications of semileptonic decays $ B_s\to D_s^*\ell\bar\nu_\ell $ . Therefore, to derive their analytic expressions within the LCSR approach, we construct the following chiral current correlation function (correlator):

where $ J_\mu(x) = \bar c(x)\gamma_\mu (1-\gamma_5)b(x) $ and $ J_{B_s}^\dagger(0) = $ $\bar b(0){\rm i} (1- \gamma_5)s(0) $ represent the left-handed current. As the $ D_s^* $ -meson DAs are relatively complex structures, there are both chiral-even and chiral-odd DAs for the $ D_s^* $ -meson, which has longitudinal and transverse polarization states. The adopted left-handed current can effectively highlight the contributions from the chiral-even DAs such as $ \phi_{2;D_s^*}^\|(x,\mu) $ , $ \phi_{3;D_s^*}^{\bot}(x,\mu) $ , $ \psi_{3;D_s^*}^{\bot}(x,\mu) $ , $ \Phi_{3;D_s^*}^\|(x,\mu) $ , $ \tilde{\Phi}_{3;D_s^*}^\|(x,\mu) $ , $ \phi_{4;D_s^*}^\|(x,\mu) $ , and $ \psi_{4;D_s^*}^\|(x,\mu) $ , while the chiral-odd DAs provide zero contributions. For the remaining chiral-even DAs, only $ \phi_{2;D_s^*}^\|, \phi_{3;D_s^*}^\bot(x,\mu) $ , and $ \psi_{3;D_s^*}^\bot(x,\mu) $ account for dominant contributions to the LCSR, while other chiral-even DAs offer negligible contributions. Moreover, the twist-3 DAs $ \phi_{3;D_s^*}^\bot(x,\mu) $ and $ \psi_{3;D_s^*}^\bot(x,\mu) $ can be related to $ \phi_{2;D_s^*}^\|(x,\mu) $ under the Wandzura-Wilczek (WW) approximation [40,41]:

Therefore, the longitudinal leading-twist $ \phi_{2;D_s^*}^\|(x,\mu) $ may provide a dominant contribution, either directly or indirectly.

The correlation function (5) is defined at both the time-like and space-like $ q^2 $ -regions. According to the basic steps of LCSR, the correlation function can first be treated in the timelike $ q^2 $ -region by inserting a complete set of intermediate hadronic states with the same quantum numbers as the current operator $ \bar{c} i(1+\gamma_5)s $ . By separating the pole term of the lowest $ B_s $ -meson, the hadronic representation can be obtained as

Among them, $ \langle B_s|\bar b i \gamma_5 s|0\rangle = m_{B_s}^2 f_{B_s}/m_b $ , where $ f_{B_s} $ represents the decay constant of the $ B_s $ meson. Moreover, the $ B_s\to D_s^* $ transition matrix elements have the following expressions:

Then, by substituting Eq. (8) into Eq. (7), the invariant amplitudes can be written as

with $ i = (1,2,3) $ . The reduced function $ \widetilde{A}_{i}(q^2) $ take the forms $ \widetilde{A}_{1}(q^2) = (m_{B_s}+m_{D_s^*})A_1(q^2) $ , $ \widetilde{A}_{2}(q^2) = -A_2(q^2)/ (m_{B_s}+ m_{D_s^*}) $ , and $ \widetilde{A}_{3}(q^2) = -2{\rm i}V(q^2)/(m_{B_s}+m_{D_s^*}) $ . Under the general dispersion relation in the momentum squared $ (p+q)^2 $ of the $ B_s $ -meson [42], we can get

with $ t_{\min} = (m_b+m_s)^2 $ . The spectral density is given by

The contributions from the high resonances and continuum states is replaced by $ \rho _{i}^{\mathrm{H}}(q^2,s) $ , which can be approximated by quark-hadron duality in the QCD sum rule applications [23]:

where $ \mathrm{Im}\Pi _{i}^{\mathrm{QCD}} $ can be obtained by calculating the correlation function via QCD theory in the spacelike region. In this region, the correlation function can be treated by performing OPE near the light-cone $ x^2\rightsquigarrow 0 $ , where the $ s_0 $ is the threshold parameter. After applying the Borel transformation to suppress contributions from the higher excited states and continuum states, Eq. (10) can be further written as

Meanwhile, the invariant amplitude in QCD can be written using the same concept:

Finally, equating the results of correlation functions in different regions, the resultant TFFs under the LCSR can be obtained:

The three simplified $ D_s^* $ -meson LCDAs $ A_{D_s^*}^\|(u), B_{D_s^*}^\|(u) $ , and $ C_{D_s^*}^\|(u) $ are represented as follows:

Furthermore, $ s(\varrho) = [m_b^2 -\bar \varrho(q^2-\varrho m_{D_s^2}^2)]/\varrho $ with $ \bar \varrho = (1-\varrho) $ and $ \varrho $