Lattice QCD studies on baryon interactions from L\"uscher's finite volume method and HAL QCD method

A comparative study between the L"uscher’s finite volume method and the time-dependent HAL QCD method is given for the $\mathrm{\Xi }\mathrm{\Xi }$(${}^1{\mathrm{S}}_0$) interaction as an illustrative example. By employing the smeared source and the wall source for the interpolating operators, we show that the effective energy shifts $\mathrm{\Delta }{E}_{\mathrm{e}\mathrm{f}\mathrm{f}}(t)$ in L"uscher’s method do not agree between different sources, yet both exhibit fake plateaux. On the other hand, the interaction kernels $V(\vec{r})$ obtained from the two sources in the HAL QCD method agree with each other already for modest values of $t$. We show that the energy eigenvalues $\mathrm{\Delta }E(L)$ in finite lattice volumes ($L^3$) calculated by $V(\vec{r})$ indicate that there is no bound state in the $\mathrm{\Xi }\mathrm{\Xi }{(}^1{\mathrm{S}}_0)$ channel at $m_{\pi }=0.51$ GeV in 2+1 flavor QCD.