Radiative transitions and the mixing parameters of the D meson

Spectroscopic parameters of heavy-light flavoured D meson are obtained within the framework of phenomenological quark-antiquark potential (Coulomb plus linear confinement) model using the Gaussian wave function. We incorporated O (1/m) corrections to the potential energy term and relativistic corrections to the kinetic energy term of the hamiltonian. We obtain the radiative (electric and magnetic) transitions and the mixing parameters of the DâĹŠ D¯ oscillation. The results are compared with various experimental measurement as well as other theoretical predictions.

We employ Gaussian wave function for the calculation of the overlap integrals. The form of the wave function is given by here µ is the variational parameter, n, l are the quantum numbers of the bound state and L are the Laguerre polynomials. The variational parameter µ was fitted using virial theorem by using a relativized Hamiltonian which includes O (1/m) corrections to the potential energy term and relativistic corrections to the kinetic energy term. For details see ref [5].

M1 radiative transitions
The rates for magnetic dipole transitions correspond to triplet-singlet between S-wave states of the same n quantum number as well as either triplet-singlet or singlet-triplet transitions between S-wave states of different n quantum numbers given by [9] Here L = 0 for S-waves and j 0 (x) is the spherical Bessel function. Our calculated radiative magnetic dipole transitions widths for states of D meson are tabulated in Table 2 using the masses estimated by our model [5].

Meson transition
This work

Mixing parameters
The neutral D meson, exhibit particle-antiparticles mixing, leading to oscillations between the light (L) and heavy (H) mass eigenstates [1]. With assuming CPT conservation throughout, in each system, the light (L) and heavy (H) mass eigenstates are, have a mass difference ∆m q = m H −m L > 0, and a total decay width difference ∆Γ q = Γ L −Γ H . The time evolution of the neutral D meson doublet is described by the Schrodinger equation [1] i d dt where M 12 and Γ 12 are the off-diagnol elements of the mass and decay matrices. The formula of predictions for the off-diagonal elements of the mass and the decay matrices are [1] The known function S 0 (x t ) can be approximated very well by 0.784x 0.76 t [10], and V ij are the elements of the CKM matrix. The parameters η D and η D correspond to the gluonic corrections for respective meson.
In the absence of CP violation, the time -integrated mixing probability (χ q ) is given by where, x q = ∆m q τ Dq , y q = ∆Γ q τ Dq 2 The mass difference ∆m is a measure of the frequency of the change from a (D) 0 into aD 0 or vice versa. We have the time-integrated mixing rate for semi-leptonic decays as [1] For the estimation of the mixing parameters x q , y q , χ q and R M , we use η D = 0.86, η D = 0.21 and the gluonic correction to the oscillation is given by Ref. [11,12]. The bag parameter B Dq = 1.34 is taken from the lattice result of [13], while the pseudoscalar mass (M D , and the pseudoscalar decay constant (f D ) of the charmed mesons are taken from our previous study in the article [5]. The calculated mixing parameters x q , y q , χ q and R M are tabulated in table 3.  [16] 0.13±0.22±0.20 Ref. [17] 0.04−0.6+0.7 Ref. [18] 0.02±0.47±0.14 Ref. [19] 1.6±2.9±2.9

Conclusion
From comparison of our estimated radiative (E1 and M1 dipole) transitions widths with other theoretical estimations, we conclude that the various models have different predictions of E1 and M1 dipole transitions may be due to different parameters and treatments are used in the relativistic corrections in the model. We have compared our results of the mixing parameters to PDG [1] as well as other theoretical results and they are in agreement with our results.