Strong Decays of the Radial Excited States $B(2S)$ and $D(2S)$

The strong OZI allowed decays of the first radial excited states $B(2S)$ and $D(2S)$ are studied in the instantaneous Bethe-Salpeter method, and by using these OZI allowed channels we estimate the full decay widths: $\Gamma_{B^0(2S)}=24.4$ MeV, $\Gamma_{B^+(2S)}=23.7$ MeV, $\Gamma_{D^0(2S)}=11.3$ MeV and $\Gamma_{D^+(2S)}=11.9$ MeV. We also predict the masses of them: $M_{B^0(2S)}=5.777$ GeV, $M_{B^+(2S)}=5.774$ GeV, $M_{D^0(2S)}=2.390$ GeV and $M_{D^+(2S)}=2.393$ GeV.

In the past few years, there are many new states observed in experiments. Among them, the new states D * s0 (2317), D s1 (2460) [1], B s1 (5830) and B s2 (5840) [2] are orbitally excited states, which are also called P wave states. So far, great progress has been made on the physics of orbital excited states D * s0 (2317) and D s1 (2460) [3], and there are already exist some investigations of B s1 (5830) and B s2 (5840) [4]. Around the energy of these hadrons, according to constitute quark model, there may be the radial excited S wave states B(2S) and D(2S). But due to their absence, the experimental and theoretical studies for the radial excited 2S states B(2S) and D(2S) are still missing in the literature.
We know that the first radial excited 2S state has a node structure in its wave function, which means relativistic correction of 2S state is much larger than the one of corresponding basic state, even the 2S state is a heavy meson, so to consider the physics of radial excited state a relativistic method is needed. Bethe-Salpeter equation [5] and its instantaneous one, Salpeter equation [6], are famous relativistic methods to describe the dynamics of a bound state. In a previous letter [7], we have solved the full Salpeter equations for pseudoscalar mesons, the masses of first radial excited 2S states are obtained, The mass of B(2S) is 310 MeV higher than the threshold of mass scale of B * π, but lower than the threshold of B * s K, and the mass of D(2S) is 240 MeV higher than the threshold of mass scale of D * π, but lower than the threshold of D * s K, so the strong decays B(2S) → B * + π and D(2S) → D * + π are OZI allowed strong decays, and they are dominate decay channels of B(2S) and D(2S), respectively. In this letter, we calculate the strong decay widths of B(2S) → B * + π and D(2S) → D * + π in the framework of Bethe-Salpeter method.
Since one of the final state is π meson in the OZI allowed B(2S) or D(2S) strong decay, we use the reduction formula, PCAC relation and low energy theorem, so for the strong decays (considering the B 0 (2S) → B * + π − as an example) shown in Fig. 1, the transition matrix element can be written as [8]: where P , P f 1 and P f 2 are the momenta of the initial state B 0 (2S), final states B * + and π − , respectively, and f P f 2 is the decay constant of π − meson.
To evaluate Eq. (1), we need to calculate the hadron matrix element B * + (P f 1 )|ūγ µ γ 5 d|B 0 (P ) . It is well known that the Mandelstam formalism [9] is one of proper approaches to compute the hadron matrix elements sandwiched by the Bethe-Salpeter or Salpeter wave functions of two bound-state. With the help of this method, in leading order, the hadron matrix elements in the center of mass system of initial meson can be written as [8,10]: where q is the relative three-momentum of the quark-anti-quark in the initial meson B 0 (2S) and r, M is the mass of B 0 (2S), r is the three dimensional momentum of the final meson B * + , ϕ ++ P is the positive energy B.S. wave function for the relevant mesons andφ ++ For the initial state pseudoscalar meson B 0 (2S) (J P = 0 − ), the positive energy wave function takes the general form [7]: where q ⊥ = (0, q) and ω i = m 2 i + q 2 , f i ( q) are eigenvalue wave functions which can be obtained by solving the full 0 − state Salpeter equations. For the final state vector meson B * + (J P = 1 − ), the positive energy wave function takes the general form [11]: where ǫ is the polarization vector of meson, and A, B, C, D, E, F, G, H are defined as: where M ′ is the mass of B * + , eigenvalue wave functions f i ( q ′ ) can be obtained by solving the full 1 − state Salpeter equations.
In calculation of transition matrix element and solving the full Salpeter equation, there are some parameters have to be fixed, the input parameters are chosen as follows [7]:  Table 1.
In our results only the 1 − 0 − final states are calculated (B * π and D * π), in our estimate of mass spectra, there are no other OZI allowed strong decay channels. For example, from the analysis of quantum number, there may be the decay channels with P wave in the final states, for example, the final state can be 0 + 0 − B(1P )π states, but due to our estimate the mass of lightest P wave 0 + state m B(1P ) = 5.665 GeV [13], which is larger than the threshold of B(2S) (the same results for D(2S) cases), so there is no phase space for this channel, if later experimental discovery of mass of this state is lower than theoretical estimate like happened to D s0 (2317), which has been hoped much higher than 2317 MeV, this channel become a OZI allowed one, but because the phase space is very small, and it is a P wave, the transition decay width should be smaller than the case when it is S wave, so we can ignore the contributions of these channels and other electroweak channels, and we use these OZI allowed decay widths to estimate the full decay width of this 2S state.