Semileptonic Decays of Charmed Mesons to Light Scalar Mesons

Within the framework of Covariant Confined Quark Model, we compute the transition form factors of $D$ and $D_s$ mesons decaying to light scalar mesons $f_0(980)$ and $a_0(980)$. The transition form factors are then utilised to compute the semileptonic branching fractions. We study the channels namely $D_{(s)}^+ \to f_0(980) \ell^+ \nu_\ell$ and $D \to a_0(980) \ell^+ \nu_\ell$ for $\ell = e$ and $\mu$. For computation of semileptonic branching fractions, we consider the $a_0(980)$ meson to be the conventional quark antiquark structure and the $f_0(980)$ meson as the admixture of $s\bar{s}$ and light quark-antiquark pairs. Our findings are found to support the recent BESIII data.


I. Introduction
Charmed semileptonic decays are important for the study of open flavor hadron spectroscopy in general and heavy quark decay properties in particular. More specifically, the scalar mesons below 1 GeV in final product can provide the key information regarding their internal structure as well as the chiral symmetry in the low energy region of nonperturbative QCD [1]. The internal structure of these mesons is yet to be clearly understood from the theoretical studies attempted so far.
In quark-antiquark picture, several theories have been proposed including the constituent quark model [24]. The semileptonic branching fractions for D s → f 0 (980) were considered in the light front quark model [25] and semileptonic as well as rare decays of the B (s) have also been studied in the formalism of covariant quark model [26]. formalism of linear sigma model [27]. The scalar mesons have been considered to be the bound states of KK molecules in the potential model approach [28,29]. N. N. Achasov et al have considered the scalar meson to be the KK molecule in the radiative decays of φ meson [30]. The assignment of f 0 (980) as the molecular structure of KK have also been used in the phenomenological Lagrangian approach by studying the strong decays of f 0 (980) to ππ and γγ channels [31]. The multiquark structure was also attempted in the unitarized meson model [32], effective field theory [33,34] as well as chiral perturbation theory [35,36]. mesons [41,42]. Light scalar mesons are also studied in the framework of tetraquark mixing [43][44][45].
Lattice quantum chromodynamics investigations of these scalar mesons are reported employing the four quark [48,49] and diquark-diantiquark pictures [50]. The transition form factors for the decays with scalar mesons as daughter products are also computed in the framework of QCD sum rules [51][52][53] and light cone sum rules [54,55] to be the conventional quark antiquark state [56]. The transition form factors are also determined in the light front quark model considering them as a four quark state [20]. R. L.
Jaffe has considered the diquark-diantiquark structure of these mesons in the formalism of MIT bag model [1].
The present work is focused on the semileptonic decay of D and D s mesons to the light scalar mesons namely f 0 (980) and a 0 (980) in the framework of Covariant Confined Quark Model (CCQM) [57][58][59]. The CCQM is the effective field theory approach with the built-in infrared confinement for the hadronic interactions to their constituents. Recently, we have studied the semileptonic decays of D and D s mesons to the pseudoscalar and vector mesons in this formalism in great detail [60][61][62][63][64]. In these papers, we investigated the transition with O µ = γ µ (1 − γ 5 ) and q ∈ d, s. The matrix element in this process is very well parameterized in terms of transition form factors given by Here P = p 1 + p 2 and q = p 1 − p 2 with p 1 and p 2 to be the momentum of D Here, Γ M is the Dirac matrix projecting onto spin of corresponding mesonic state. It should read i.e., Γ M = I, γ 5 , γ µ for scalar, pseudoscalar and vector mesons respectively. g M is the coupling strength of the meson with its constituent quarks. F M , the translation invariant vertex function characterizing the effective physical size of the hadron, is given by with Φ M as the correlation function of two constituent quarks with masses m q 1 and m q 2 and w q i = m q i /(m q 1 + m q 2 ) such that w 1 + w 2 = 1. We choose Gaussian form for the vertex function asΦ The matrix element of self energy diagram and semileptonic decays are constructed from the  diagram is drawn using the convolution of quark propagator and vertex functions ( Fig. 1 and II). The matrix element for self energy diagram for any meson can be written as In Eq. (7),Π ′ M is the derivative of meson mass operator Eq. (8). Similarly, the matrix element for the semileptonic D (s) decays to scalar mesons can be written as The parameters in the double pole approximation for the different decay channels are given in the Tab. II.
where δ ℓ = m 2 ℓ /2q 2 is the helicity flip factor, |p 2 |= λ 1/2 (m 2 D (s) , m 2 S , q 2 )/2m D (s) is the momentum of the daughter (Scalar) meson in the rest frame of the parent (D (s) ) meson and v = 1 − m 2 ℓ /q 2 is the velocity-type parameter. In the above Eq. (11), the bilinear combinations of the helicity structure function are defined in terms of form factors as: This helicity technique is formulated in Ref. [72][73][74] and is also discussed recently in Ref. [70,71]. The computation technique in CCQM is very general and can accommodate hadronic state with any number of constituent quarks.

IV. Conclusion
In this article, we have employed the Covariant Confined Quark Model to study the semileptonic branching fraction of charmed mesons decaying to the light scalar mesons. We  have considered the f 0 (980) meson to be the admixture of ss and light quark component with the mixing angle to be 30 • and a 0 (980) meson to be the conventional quark-antiquark pair.
Our results are found to be consistent with theoretical results as well as available experimental data. We have also provided theoretical calculation for the semileptonic branching fractions of charmed meson to scalar meson in the muon channel for the first time.
The present study can help in understanding the internal structure of the scalar mesons below 1 GeV. As no absolute value of the branching fractions are available in the literature, we expect more accurate data coming from the world wide upgraded experimental facilities to check the validity of our computed results in this study.