Extended Migdal-Watson formula to evaluate background strength in binary breakup reactions

A new formula to evaluate the nonresonant background for binary breakup is proposed by extending the Migdal-Watson (MW) formula, which was originally developed for $s$-wave breakup in charge neutral systems. The strength of the direct breakup, which is assumed by a one step transition from the initial bound state, is simulated by the complex scaling method (CSM) in binary cluster systems. The extended MW formula is applied to the direct breakup of $^{20}\mathrm{Ne}\phantom{\rule{4pt}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{4pt}{0ex}}\ensuremath{\alpha}+^{16}\mathrm{O}$, and we have found that the formula nicely reproduces the strength of the direct breakup evaluated by CSM. We have demonstrated that new parameters introduced in the extended MW formula have a close connection to the spatial size of the initial wave packet, which is given by the product of the initial wave function and the external field. The application of the new formula to the direct breakup of $^{12}\mathrm{Be}$ into $\ensuremath{\alpha}+^{8}\mathrm{He}$ is also discussed.