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Register a new userThe weak decay $B_c$ to $Z(3930)$ and $X(4160)$ by Bethe-Salpeter method
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Considering $Z(3930)$ and $X(4160)$ as $chi_{c2}(2P)$ and $chi_{c2}(3P)$ states, the semileptonic and nonleptonic of $B_c$ decays to $Z(3930)$ and $X(4160)$ are studied by the improved Bethe-Salpeter(B-S) Method. The form factors of decay are calculated through the overlap integrals of the meson wave functions in the whole accessible kinematical range. The influence of relativistic corrections are considered in the exclusive decays. Branching ratios of $B_c$ weak decays to $Z(3930)$ and $X(4160)$ are predicted. Some of the branching ratios are: $Br(B_c^+to Z(3930)e^+ u_e)$$=(3.03^{+0.09}_{-0.16})times 10^{-4}$ and $Br(B_c^+to X(4160)e^+ u_e)$$=(3.55^{+0.83}_{-0.35})times 10^{-6}$. These results may provide useful information to discover $Z(3930)$ and $X(4160)$ and the necessary information for the phenomenological study of $B_c$ physics.
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Considering $X(3940)$ and $X(4160)$ as $eta_c(3S)$ and $eta_c(4S)$, we study the productions of $X(3940)$ and $X(4160)$ in exclusive weak decays of $B_c$ meson by the improved Bethe-Salpeter(B-S) Method. Using the relativistic B-S equation and Mandelstam formalism, we calculate the corresponding decay form factors. The predictions of the corresponding branching ratios are: $Br(B_c^+to X(3940)e^+ u_e)$$=1.0times10^{-4}$ and $Br(B_c^+to X(4160)e^+ u_e)=2.4times10^{-5}$. That will provide us a new way to observe the $X(3940)$ and $X(4160)$ in the future, as well as to improve the knowledge of $B_c$ meson decay.
Inspired by the newly observed state $X^{*}(3860)$, we analyze the strong decay behaviors of some charmonium-like states $X^{*}(3860)$,$X(3872)$, $X(3915)$, $X(3930)$ and $X(3940)$ by the $^{3}P_{0}$ model. We carry out our work based on the hypothesis that these states are all being the charmonium systems. Our analysis indicates that $0^{++}$ charmonium state can be a good candidate for $X^{*}(3860)$ and $1^{++}$ state is the possible assignment for $X(3872)$. Considering as the $3^{1}S_{0}$ state, the decay behavior of $X(3940)$ is inconsistent with the experimental data. So, we can not assign $X(3940)$ as the $3^{1}S_{0}$ charmonium state by present work. Besides, our analysis imply that it is reasonable to assign $X(3915)$ and $X(3930)$ to be the same state, $2^{++}$. However, combining our analysis with that of Zhou~cite{ZhouZY}, we speculate that $X(3915)$/$X(3930)$ might not be a pure $coverline{c}$ systems.
The new mesons $X(3940)$ and $X(4160)$ have been found by Belle Collaboration in the processes $e^+e^-to J/psi D^{(*)}bar D^{(*)}$. Considering $X(3940)$ and $X(4160)$ as $eta_c(3S)$ and $eta_c(4S)$ states, the two-body open charm OZI-allowed strong decay of $eta_c(3S)$ and $eta_c(4S)$ are studied by the improved Bethe-Salpeter method combine with the $^3P_0$ model. The strong decay width of $eta_c(3S)$ is $Gamma_{eta_c(3S)}=(33.5^{+18.4}_{-15.3})$ MeV, which is closed to the result of $X(3940)$, therefore, $eta_c(3S)$ is a good candidate of $X(3940)$. The strong decay width of $eta_c(4S)$ is $Gamma_{eta_c(4S)}=(69.9^{+22.4}_{-21.1})$ MeV, considering the errors of the results, its closed to the lower limit of $X(4160)$. But the ratio of the decay width $frac{Gamma(Dbar D^*)}{Gamma (D^*bar D^*)}$ of $eta_c(4S)$ is larger than the experimental data of $X(4160)$. According to the above analysis, $eta_c(4S)$ is not the candidate of $X(4160)$, and more investigations of $X(4160)$ is needed.
The challenge to obtain from the Euclidean Bethe--Salpeter amplitude the amplitude in Minkowski is solved by resorting to un-Wick rotating the Euclidean homogeneous integral equation. The results obtained with this new practical method for the amputated Bethe--Salpeter amplitude for a two-boson bound state reveals a rich analytic structure of this amplitude, which can be traced back to the Minkowski space Bethe--Salpeter equation using the Nakanishi integral representation. The method can be extended to small rotation angles bringing the Euclidean solution closer to the Minkowski one and could allow in principle the extraction of the longitudinal parton density functions and momentum distribution amplitude, for example.
We construct weak axial one-boson exchange currents for the Bethe-Salpeter equation, starting from chiral Lagrangians of the N-Delta(1236)-pi-rho-a_1-omega system. The currents fulfil the Ward-Takahashi identities and the matrix element of the full current between the two-body solutions of the Bethe-Salpeter equation satisfies the PCAC constraint exactly.
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