Study of various spectroscopic properties of the Ds meson

Spectroscopic parameters of the Ds(cs̄) meson are obtained using phenomenological quark antiquark potential(coulomb plus power) model consisting of O (1/m) correction to the potential. Within Variational scheme Gaussian wave function is employed with a hamiltonian incorporating kinematic relativistic corrections to obtain various properties such as the mass spectra, decay constants, electromagnetic transitions. The results are compared with various experimental measurement as well as other theoretical predictions.


Introduction
The D s meson is a light-heavy quark structure composed of the charm and the strange quark [1].The ground state masses as well as the 1P state masses of the D s meson have been measured quite accurately [1].Mass spectrum as well as many other spectroscopic properties of the D s meson have been extensively studied in various theoretical schemes [2].There exists mutual variation in the masses of 1P states predicted by these models as well as with the experimental measurements.Recently some of the higher excited states such as the D s1 (2710), D sJ (2860) and D sJ (2040) have been experimentally measured.Being a light-heavy quark meson system the D s meson requires a relativistic treatment.In this paper we employ a potential model scheme that incorporates kinematic relativistic corrections to the kinetic energy term as well as O(1/m) corrections to the potential energy term of the hamiltonian.This allows for an opportunity to test the applicability and validity of such a model to a light-heavy quark bound system.
The paper is organized as followingly.Section 2 provides the theoretical background and the calculation of the mass spectrum of the D s meson.In section 3 we obtain the decay constants.In section 4 electromagnetic transition rates are estimated while in section 5 we conclude the present work.

Theoretical formulation
For the study of the D s meson we consider the relativistic Hamiltonian in which motion of the quarks inside the meson is relativistic [3][4][5] a e-mail: nayneshdev@gmail.comb e-mail: vhkher@yahoo.co.in c e-mail: raiajayk@gmail.comwhere p is the relative momentum of the quark-antiquark and m c is the c quark mass and m s is the s quark mass.The Hamiltonian in Eq(1) represents the energy of the meson in the meson rest frame.We expand the kinetic energy(K.E.) part of the Hamiltonian, as and V(r) is the quark-antiquark potential [6,7], where [8][9][10], A is the potential parameter and V 0 is a constant.α c = (4/3)α S M 2 , α S M 2 is the strong running coupling constant.The non-perturbative form of V (1) (r) is not yet known, but leading order perturbation theory yields where C F = 4/3 and C A = 3 are the Casimir charges of the fundamental and adjoint representation, respectively [6].The value of the QCD coupling constant α s (M 2 ) is determined through the simplest model with freezing [11,12], namely where M = 2m c m s/ (m c + m s), M B = 0.95 GeV [11,12], and Λ = 0.413 GeV [13].
We have used the gaussian wave function in the present study.The gaussian wave function in position space has the form and in momentum space has the form Here, μ is the variational parameter and L is Laguerre polynomial.
For the present study, we employ the Ritz variational scheme.We obtain the expectation values of the Hamiltonian as The variational parameter, μ is determined for each state using the Virial theorem [14].Gaussian wavefunction in position space has been employed to obtain the expectation value of the potential energy part in the Virial theorem while momentum space wave function has been used to obtain the kinetic energy part.As the interaction potential assumed here does not contain the spin dependent part, Eq(9) gives the spin averaged masses of the system.The calculated spin averaged mass os the ground state is matched with the experimental spin-averaged mass using the equation [10] ;where M V and M P are the vector and pseudoscalar meson ground state masses taken from ref [1].This fixes the parameter V 0 .Using this value of V 0 we calculate S , P, and D wave spin-averaged masses of D s mesons which are listed in Table 1.For the comparison for the nJ state, we compute the spin-averaged or the center of weight mass from the respective theoretical values as [10] where, M CW,n denotes the spin-averaged mass of the n state and M nJ represents the mass of the meson in the nJ state.The value of the radial wave function R(0) for 0 −+ and 1 −− states would be different due to their spin dependent hyperfine interaction.The spin hyperfine interaction of the heavy-light flavored mesons is small and this can cause a small shift in the value of the wave function at the origin [10,15].
The parameters used to calculate the low lying masses of the D s meson are A = 0.135 GeV −1 , m s = 0.55 GeV, m c = 1.35 GeV and the value of the constant V 0 = −0.123GeV.The spin averaged masses for S, P and D states are tabulated in Table 1.It can be observed that the spin-averaged masses obtained are in good agreement with experimental and other theoretical predictions.

Excited states
We add separately (in Eq.( 9)) the spin-dependent part of the usual one gluon exchange potential (OGEP) between the quark anti quark for computing the hyperfine and spin-orbit shifting of the lowlying S , P and D-states.Thus to take into account the spin dependent and spin-orbit interaction, causing the splitting of the nL levels one introduces additional term in the Hamiltonian [19][20][21] 05005-p.3

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where V (0) (r) is the phenomenological potential, the first terms takes into account the relativistic corrections to the potential V(r), the second term accounts spin orbital interaction, third term is usual spin-spin interaction part which is responsible for pseudoscalar and vector meson splitting(Eq.( 15) & ( 16)) and fourth term stands for tensor interaction.
In the case of quark and antiquark of unequal mass charge-conjugation parity is no longer a good quantum number and so the states with J = L, are mixtures of spin-triplet 3 L L and spin-singlet 1 L L states: J = L = 1, 2, 3, . . .
where φ is the mixing angle and the primed state has the heavier mass.Such mixing occurs due to the nondiagonal spin-orbit and tensor terms in Eq (9).The masses of the physical states were obtained by diagonalizing the mixing matrix.The calculated values of the mass spectra of D s meson are listed in Table 2.We are following spectroscopic notation n 2S +1 L J in Table 2. Overall the mass spectrum is in satisfactory agreement with others.

Electromagnetic transition widths 4.1 Electric Dipole Transition
The radiative widths are calculated in the dipole approximation.The E1 matrix elements are determined by using the variational radial wave functions of the initial and the final state and explicitly performing the angular integration given by [34] Γ for 1 P 1 → 1 S 0 (2J + 1)/3 for 3 S 1 → 3 P J 3 for 1 S 0 → 1 P 1 (18) 05005-p.5

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Here, α is the fine structure constant, k is the photon energy, e q and e Q are the quark charges in units of the proton charge, E f is the energy of the final meson state, M i is the mass of the initial meson state, and m q and m Q are the quark masses employed within the present work.The E1 radiative transition widths are listed in tables (4).

Magnetic Dipole Transitions
The M1 rate for transitions between S -wave levels is given by [24,37] is where the magnetic dipole moment is and k is the photon energy.Rates for the allowed transitions between the spin-triplet and the spinsinglet states are given in Table (5).

Conclusion
In this paper we have obtained the mass spectra and various decay properties of the D s meson within the framework of potential models with kinematic relativistic correction to the kinetic energy term 05005-p.6 EPJ Web of Conferences as well as O (1/m) correction to potential energy term.Looking at table 1 it can be seen that the spin averaged masses for the various s, p and d wave states are in good agreement with the experimental measurements as well as other theoretical estimates.From table 2 it can be observed that the entire mass spectrum is also in fair agreement with various models and experimental results.The pseudoscalar and vector decay constants without QCD correction are in satisfactory agreement with other theoretical predictions as can be seen from table 3. The e1 transition rates listed in table 4 as well as M1 transition rates listed in table 5 obtained with the present framework show significant disagreement with other theoretical estimates.In conclusion we find that the present model is adequately predicts the mass spectrum however the decay properties are in disagreement.

Table 1 .
Spin averaged masses of D s meson.

Table 2 .
Masses of the D s mesons(in GeV).

Table 3 .
Decay constants of the D s meson(in GeV).

Table 4 .
E1 transition widths in the D s meson.

Table 5 .
M1 transition widths in the D s meson.