Finding $B_c(3S)$ States via Their Strong Decays

The experimentally known $B_c$ states are all below open bottom-charm threshold, which experience three main decay modes, and all induced by weak interaction. In this work, we investigate the mass spectrum and strong decays of the $B_c(3S)$ states, which just above the threshold, in the Bethe-Salpeter formalism of and $^3P_0$ model. The numerical estimation gives $M_{B_c(3^1S_0)}=7222\ {\rm MeV}$, $M_{B_c^*(3^3S_1)}=7283\ {\rm MeV}$, $\Gamma\left(B_c(3^1S_0)\to B^*D\right)=6.190^{+0.570}_{-0.540}\ {\rm MeV}$, $\Gamma\left(B_c^*(3^1S_0)\to BD\right)=0.717^{+0.078}_{-0.073}\ {\rm MeV}$ and $\Gamma\left(B_c^*(3^1S_0)\to B^*D\right)=8.764^{+0.801}_{-0.760}\ {\rm MeV}$. Compared with previous studies in non-relativistic approximation, our results indicate that the relativistic effects are notable in $B_c(3S)$ exclusive strong decays. According to the results, we suggest to find the $B_c(3S)$ states in their hadronic decays to $B$ and $D$ mesons in experiment, like the LHCb.

Although there have been many investigations in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] about the properties of B c mesons, the excited B c states, especially above threshold, are rarely explored. The ground state B c meson was first observed by the CDF Collaboration at Fermilab [16] in 1998, while there was no reported evidence of the excited B c state until 2014, the ATLAS Collaboration reported a structure with mass of 6842 ± 9 MeV [17], which is consistent with the value predicted for B c (2S). Recently, the excited B c (2 1 S 0 ) and B * c (2 3 S 1 ) states have been observed in the B + c π + π − invariant mass spectrum by the CMS and LHCb Collaboration, with their masses determined to be 6872.1 ± 2.2 MeV and 6841.2 ± 1.5 MeV [18,19], respectively. Since the low-energy photon in the intermediate decay B * c → B c γ was not reconstructed, the mass of B * c (2 3 S 1 ) meson appears lower than that of B c (2 1 S 0 ).
In quantum field theory, the BS equation provides a basic description for bound states. The BS wave function of a quark-antiquark bound state is defined as where x 1 and x 2 are the coordinates of the quark and antiquark respectively, P is the momentum of the bound state, T denotes the time ordering operator. The wave function in momentum space is where q is the relative momentum between the quark and antiquark. The "center-ofmass coordinate" X and the "relative coordinate" x are defined as: with m 1 and m 2 are the masses of the quark and antiquark respectively. Then the bound state BS equation in momentum space reads Here S i (±p i ) = i ± / p i −m i denotes the fermion propagator; V (P ; q, k) is the interaction kernel; p 1 and p 2 are the momenta of the quark and anti-quark respectively, which can be expressed as where, J = 1 for the quark (i = 1) and J = −1 for the antiquark (i = 2). With the where Under the instantaneous approximation, the interaction kernel in the center of mass frame takes the form V (P ; q, k)| P =0 ≈ V (q ⊥ , k ⊥ ). Then the BS equation can be reduced to with where is the 3-dimensional BS wave function. By introducing the notation ϕ ±± P (q ⊥ ) as: The wave function can be decomposed as And the BS equation (8) can be decomposed into four equations To solve the BS equation, one must have a good command of the potential between two quarks. According to lattice QCD calculations, the potential for a heavy quarkantiquark pair in the static limit is well described by a long-ranged linear confining potential (Lorentz scalar V S ) and a short-ranged one gluon exchange potential (Lorentz vector V V ) [21,22]: Here, the factor e −αr is introduced not only to avoid the infrared divergence but also to incorporate the color screening effects of the dynamical light quark pairs on the "quenched" potential [23]. The potentials in momentum space are Here, α s ( p 2 ) is the running coupling which defined as The constants λ, α, a, and Λ QCD are the parameters that characterize the potential. In the following, we will employ this potential to both the B c system and the heavy-light quark system as an assumption.
In the numerical calculation, following well-fitted parameters [24,25] are used: The numerical results are shown in Table I. For comparison, results obtained from other approaches are also listed.
Our results indicate that the B c (3S) states lie above the threshold for decay into a BD meson pair. The corresponding OZI-allowed two body decay can be depicted by 3 P 0 model, where the additional light quark-antiquark pair is assumed to be created from vacuum, as shown in Fig. 1. The usual 3 P 0 model is a non-relativistic model with a transition operator √ 3g d 3 xψ( x)ψ( x), and it can be extended to a relativistic form [25,26]. The coupling constant g can be parameterized as 2m q γ, where m q is the constitute quark mass and γ is a dimensionless parameter which can  and γ = 0.253 ± 0.010 [27].
The transition amplitude for the OZI-allowed two body decay process (with the momenta assigned as in Fig. 1) can be written as where The Feynman amplitude takes the form [25]: Note, to get the last line of Eq. (19), we have used the fact that the wave function is strongly suppressed when |q ⊥ | is large. With the method developed in Refs. [28,29], the positive energy wave function can be determined by numerically solving Eq. (12).
The strong decay widths of B c (3S) mesons are shown in Table II The total decay widths here are about 5% ∼ 10% the widths of Refs. [13,14], where non-relativistic 3 P 0 model were used.
It's quite a discrepancy but it's also understandable for the following reasons: 1. In Refs. [13,14], the coupling constant g is set to be about 0.264 GeV, while here it's about 0.155 GeV according to Eq. (10) of Ref. [27]. This lead to a 3 times difference in decay width.
2. In the 3 P 0 model with H int = i √ 3gψ(x)ψ(x), the transition operator contains a factor g 2ωq , where ω q is the energy of the created light quark. This factor reduce to g 2mq in the non-relativistic limit. Since the parameter g is extracted from the fit of the non-relativistic 3 P 0 model to the experimental data, our widths are in fact suppressed by a factor m 2 q ω 2 q ∼ 0.270 [26]. By multiplying a compensation factor of  3. The relativistic effect of the wave functions are non-negligible, as discussed in Ref. [30].
According to non-relativistic quantum chromodynamics factorization formalism [31], the production rates of B c (3S) mesons can be estimated through where Ψ H (0) is the wave function at the origin for meson H. With the σ (B c (1S)) predicted in Ref. [32], and the wave functions calculated in Ref. [2], the cross sec-