The radiative capture reaction rate from $\Lambda \Lambda$ to H dibaryon in the imaginary time method

Radiative capture rates of thermal $\Lambda\Lambda + \Xi$N states into H dibaryon are calculated in the novel imaginary time method. The H dibaryon is assumed to be a bound state of $\Xi$N with spin $J^{\pi}= 0^+$, isospin $I=0$ and strangeness $-2$. We consider $E1$ transition to H from $\Xi$N $(L=1)$ scattering states which mix with $\Lambda\Lambda (L=1)$. In order to calculate the transition rates, we formulate a coupled-channel imaginary time method by extending the one-channel formula originally proposed by Yabana and Funaki. The imaginary time method allows us to avoid the sum over all the excited thermal initial states, and thus to save computational time significantly. The transition rates are given as a function of temperature and the unknown binding energy of the H dibaryon, which we take as a parameter. It is found that the transition rate is not sensitive to the choices of the H binding energy or the strengths of the channel coupling for temperatures 3 MeV or higher.