Speaker
Description
As a recent topic of heavy flavor systems, exotic charmonia called $X,Y,Z$ have been observed experimentally above the meson-meson threshold. Masses of $X,Y,Z$, however, are not reproduced by the Cornell potential only with the degrees of freedom of $\bar{c}c$. This indicates that the $X,Y,Z$ states have coupled channel effects of $\bar{c}c$ and meson-meson states strongly.
Because of the color confinement of quarks, the $\bar{c}c$ potentials diverge at large distance. On the other hand, the meson-meson potentials vanish at large distance, because the interaction range is limited by the inverse pion mass. What then is the effect of the coupling to the two-hadron channels in the $\bar{c}c$ potentials and vice versa? It is expected that the coupling to the meson-meson channel affect $\bar{c}c$ potentials and the coupling of mesons with $\bar{c}c$ affect meson-meson potentials.
In this talk, we consider the channel couplings between the $\bar{c}c$ and meson-meson potentials, and investigate the properties of the effective potentials which are obtained by eliminating one of the channels.[1]
We show that these effective potentials $V_{\rm eff}(E)$ at energy $E$ are written as follows
$$
\langle \boldsymbol{r'}_{\bar{D}D} | V^{\bar{D}D}_{\rm eff}(E)|\boldsymbol{r}_{\bar{D}D} \rangle =V^{\bar{D}D}(\boldsymbol{r})\delta( \boldsymbol{r'}- \boldsymbol{r}) +\sum_n \frac{\langle \boldsymbol{ r'}_{\bar{D}D} | V^{t}|\phi_n \rangle \langle \phi_n| V^{t} |\boldsymbol{r}_{\bar{D}D} \rangle}{E-E_n}, \\
\langle \boldsymbol{r'}_{\bar{c}c} | V^{\bar{c}c}_{\rm eff}(E)|\boldsymbol{r}_{\bar{c}c} \rangle = V^{\bar{c}c}(\boldsymbol{r})\delta(\boldsymbol{r'}-\boldsymbol{r})+\int d \boldsymbol{p} \frac{\langle \boldsymbol{r'}_{\bar{c}c} | V^{t}|\boldsymbol{p}_{\rm full} \rangle \langle
\boldsymbol{p}_{\rm full} |V^{t} |\boldsymbol{r}_{\bar{c}c} \rangle}{E -E_{\boldsymbol{p}}+i0^+},
$$
where $\boldsymbol{r}$ and $\boldsymbol{r'}$ are coordinates before and after interactions, $V$ is potentials of internal interactions, $V^t$ is the transition potential of channel couplings, $|\phi_n \rangle$ is the eigenstate of the $\bar{c}c$ Hamiltonian with energy $E_n$, and
$|{\boldsymbol p}^{\rm full} \rangle$ is the meson-meson eigenstate with energy $E_{\boldsymbol{p}}$.
We discuss that the coupling to the eliminated channel induces the non-local and energy dependent effective potential, irrespective of the behavior of the transition potential. In addition, when the hadron channel having continuous scattering eigenstates is eliminated, the resulting $\bar{c}c$ potential contains an imaginary part. The physical property which stems from imaginary part, however, may be lost by the finite terms of the derivative expansion.
[1] H. Feshbach, Ann. Phys. 5, 357 (1958); ibid., 19, 287 (1962).
