The Instantaneous Bethe-Salpeter Equation and Its Analog: the Breit-like Equation

We take ($\mu^\pm e^\mp$) systems and consider the states with quantum number $J^P=0^-$ as examples, to explore the different contents of the instantaneous Bethe-Salpeter (BS) equation and its analog, Breit equation, by solving them exactly. The results show that the two equations are not equivalent, although they are analogous. Furthermore, we point out that the Breit equation contains extra un-physical solutions, so it should be abandoned if one wishes to have an accurate description of the bound states for the instantaneous interacting binding systems.Comment: 5 pages, no figur

Hadronic production of the PP-wave excited BcB_c-states (BcJ,L=1∗B_{cJ,L=1}^*)

Adopting the complete $\alpha_s^4$ approach of the perturbative QCD (pQCD) and updated parton distribution functions, we have estimated the hadronic production of $P$-wave excited $B_c$-states ($B_{cJ,L=1}^*$). In the estimate, special care on the relation of the production amplitude to the derivative of wave function at origin of the potential model is payed. For experimental references, main uncertainties are discussed, and the total cross sections and the distributions of the production with reasonable cuts at the energies of Tevatron and LHC are computed and presented. The results show that $P$-wave production may contribute to the $B_c$-meson production indirectly by a factor about 0.5 of the direct production, and with such a big cross section, it is worth further to study the possibility to observe the $P$-wave production itself experimentally.Comment: 23 pages, 9 figures, to replace for revising the misprints ec

Decays of the Meson BcB_c to a PP-Wave Charmonium State χc\chi_c or hch_c

The semileptonic decays, $B_{c}{\longrightarrow}{\chi_c}(h_c)+{\ell}+{{\nu}}_{\ell}$, and the two-body nonleptonic decays, $B_{c}{\longrightarrow}{\chi_c}(h_c)+h$, (here $\chi_c$ and $h_c$ denote $(c\bar c[^3P_J])$ and $(c\bar c[^1P_1])$ respectively, and $h$ indicates a meson) were computed. All of the form factors appearing in the relevant weak-current matrix elements with $B_c$ as its initial state and a $P$-wave charmonium state as its final state for the decays were precisely formulated in terms of two independent overlapping-integrations of the wave-functions of $B_c$ and the $P$-wave charmonium and with proper kinematics factors being `accompanied'. We found that the decays are quite sizable, so they may be accessible in Run-II at Tevatron and in the foreseen future at LHC, particularly, when BTeV and LHCB, the special detectors for B-physics, are borne in mind. In addition, we also pointed out that the decays $B_c\to h_c+...$ may potentially be used as a fresh window to look for the $h_c$ charmonium state, and the cascade decays, $B_c\to \chi_c[^3P_{1,2}]+l+\nu_l$ ($B_c\to \chi_c[^3P_{1,2}]+h$) with one of the radiative decays $\chi_c[^3P_{1,2}] \to J/\psi+\gamma$ being followed accordingly, may affect the observations of $B_c$ meson through the decays $B_{c}\to {J/\psi}+{l}+\nu_{l}$ ($B_c\to J/\psi+h$) substantially.Comment: 24 pages, 3 figures, the replacement for improving the presentation and adding reference

The meson BcB_c annihilation to leptons and inclusive light hadrons

The annihilation of the $B_c$ meson to leptons and inclusive light hadrons is analyzed in the framework of nonrelativistic QCD (NRQCD) factorization. We find that the decay mode, which escapes from the helicity suppression, contributes a sizable fraction width. According to the analysis, the branching ratio due to the contribution from the color-singlet component of the meson $B_c$ can be of order (10^{-2}). We also estimate the contributions from the color-octet components. With the velocity scaling rule of NRQCD, we find that the color-octet contributions are sizable too, especially, in certain phase space of the annihilation they are greater than (or comparative to) the color-singlet component. A few observables relevant to the spectrum of charged lepton are suggested, that may be used as measurements on the color-octet and color-singlet components in the future $B_c$ experiments. A typical long distance contribution in the annihilation is estimated too.Comment: 26 pages, 5 figures (6 eps-files), submitted to Phys. Rev.

The Color-Octet Contributions to PP-wave BcB_c Meson Hadroproduction

The contributions from the color-octet components $|(c\bar b)_{\bf 8}(^{1}S_{0}) g>$ and $|(c\bar b)_{\bf 8}(^{3}S_{1}) g>$ to the $h_{B_c}$ or $\chi_{B_c}^J$ (the $P$-wave $B_c$ meson) hadroproduction are estimated in terms of the complete ${\cal O}(\alpha_s^4)$ calculation. As necessary inputs in the estimate, we take the values of the octet matrix elements according to the NRQCD scaling rules, and as a result, we have found that the contributions to the $P$-wave production may be the same in order of magnitude as those from the color-singlet ones, $|(c\bar b)_{\bf 1}(^{1}P_{1})>$ and $|(c\bar b)_{\bf 1}(^{3}P_{J})>$ ($J=1,2,3$). Especially, our result indicates that the observation of the color-octet contributions to the $P$-wave production in the low transverse momentum region is not very far from the present experimental capability at Tevatron and LHC.Comment: 14 pages, 4 figures (8 eps-files for figures), add references and correct typo