Spectra and decay rates of b ¯ b meson using Gaussian wave function

. Using the Gaussian wave function mass spectra and decay rates of b ¯ b meson are investigated in the framework of phenomenological quark anti-quark potential (coulomb plus power) model consisting of relativistic corrections to the kinetic energy term. The spin-spin, spin-orbit and tensor interactions are employed to obtain the pseudoscalar and vector meson masses. The decay constants (cid:2) f P / V (cid:3) are computed using the wave function at the origin. The di-gamma and di-leptonic decays of the b ¯ b meson are investigated using Van-Rayan Weisskopf formula as well as in the NRQCD formalism.

The new role of the heavy flavour studies as the testing ground for the non-perturbative aspects of QCD, demands extension of earlier phenomenological potential model studies on quarkonium masses to their predictions of decay widths with the non-perturbative approaches like NRQCD [24].
We present details of the semi-relativistic treatment of the heavy quarks along with the computed results in section-2. The decay constants f P,V of bb meson incorporating QCD correction is presented in section-3. In section-4 we present the details of the computations of the di-gamma decays of pseudoscalar states and the di-leptonic decay widths of the vector states of the bb quarkonia in the frame work of the NRQCD formalism as well as in the conventional Van For the study of the bb meson we consider the relativistic Hamiltonian in which motion of the quarks inside the meson is relativistic [25][26][27][28] where p is the relative momentum of the quark-antiquark and m b is the b 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 up to O p 4 and V(r) is the quark-antiquark potential [15,16,18], A is the potential parameter, ν is a general power index, 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 value of the QCD coupling constant α s (M 2 ) is determined through the simplest model with freezing [10,11], namely where M = 2m b m¯b/ m b + m¯b , M B = 0.95 GeV [10,11], 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 R nl (μ, r) = μ 3/2 2 (n − 1)! Γ (n + l + 1/2) 1/2 (μr) l e −μ 2 r 2 /2 L l+1/2 n−1 (μ 2 r 2 ); (4) 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 For a chosen value of ν between 0.5 to 2.0 the variational parameter, μ is determined for each state using the Virial theorem [29]. As the interaction potential assumed here does not contain the spin dependent part, Eq(5) gives the spin averaged masses of the system. The calculated spin averaged mass of the ground state is matched with the experimental spin-averaged mass using the equation 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 bb mesons which are listed in Tables 1 and 2. For the comparison for the nJ state, we compute the spin-averaged or the center of weight mass from the respective theoretical values as [18] 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-heavy flavored mesons is small and this can cause a small shift in the value of the wave function at the origin [18,31].  Tables 1 and 2. It can be observed that the spin-averaged masses obtained are in good agreement with experimental and other theoretical predictions at ν = 1, but some excited states are over estimated by 100 MeV.

Excited states
In the case of quarkonia the quark-antiquark bound states are represented by n 2S +1 L J , identified with the J PC values, with SQ, parity P = (−1) L+1 and the charge conjuation C = (−1) L+S with (n, L) being the radial quantum numbers [32]. For computing the mass differences between different degenerate QQ meson states, the spin dependent part of the potential employed is given by where The spin dependent part Eq(8) is added to the Hamiltonian Eq(5) to calculate the excited state masses. Masses are tabulated in Table-3.

Decay constants
The decay constants of mesons are important parameters in the study of leptonic or non-leptonic weak decay processes. In the non-relativistic limit, we compute the decay constants using the Van-Royen-Weisskopf formula [33], EPJ Web of Conferences 04054-p.4 Table 3. Masses of the bb mesons(in GeV).

Di-gamma and di-leptonic decay rates
The di-gamma decay of 1 S 0 state and the di-leptonic decay of 3 S 1 state using the conventional Van-Royen Weisskopf formula [33] is Omitting O v 4 Γ the NRQCD formula for the decays can be written as [24] The short distance coefficients f 's and g's computed in the order of α 2 as [24] Im f γγ ( 1 S 0 ) = πQ 4 α 2 (20) Img ee where color factor C F = N 2 c − 1 / (2N c ) = 4/. Details of NRQCD is nicely given in ref [24].

Conclusion
In present work, bb meson is studied in the general frame work of potential model. The mass spectra, decay constants and decay rates are calculated using the Gaussian wave function. The potential model parameters and the masses of the bb meson are employed to study their decay properties in the frame work of NRQCD formalism as well as using the conventional Van-Royen-Weisskopf formula.
The mass spectra obtained in the present scheme is reasonably close to experimental values as well other theoretical results. The some excited states like 3S, 4S and 5S are little overestimated. The mass predicted for η b (2S ) is very close to the recent measurements of ref [4]. The decay constants of the pseudoscalar ( f P ) and the vector meson ( f V ) are computed with and without QCD corrections, is found to be 100 MeV off with the recent predictions [31,35,36,42]. The departure from the predicted f P and f V need more refined mechanism related to their wave functions incorporating the confinement and hyperfine splitting. Using the predicted masses and redial wave functions at the origin, the digamma and di-leptonic, decay rates of bottomonium are studied and tabulated in Table- (5). The digamma decay rate calculated using Van-Rayan Weisskopf formula close to the other predicted values, but NRQCD results are well off from others. In the case of the di-leptonic decay both the results are underestimated. For the di-gamma prediction results calculated with potentials [37,38,40] are in good agreement with other theoretical predictions. For the di-leptonic decay rates, only Log potential is giving good result [40]. Finally, predicted results suggested that it required relativistic corrections to the potential, which will improve the present results.