Consistency between Luscher's finite volume method and HAL QCD method for two-baryon systems in lattice QCD

There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlation function and the HAL QCD method which extracts observables from the non-local potential associated with the tempo-spatial correlation function. Although the two methods should give the same results theoretically, there have been reported qualitative difference for observables from lattice QCD simulations. Recently, we pointed out in [1, 2] that the separation of the ground state from the excited states is crucial to obtain sensible results in the former, while both states provide useful signals for observables in the latter. In this paper, we identify the contribution of each state in the direct method by decomposing the two-baryon correlation functions into the finite-volume eigenmodes obtained from the HAL QCD method. As in our previous studies, we consider the system in the S-1(0) channel at m = 0.51 GeV in (2+1)-flavor lattice QCD using the wall and smeared quark sources with spatial extents, La = 3.6, 4.3, 5.8 fm. We demonstrate that the pseudo-plateau at early time slices (t = 1 approximate to 2 fm) from the smeared source in the direct method indeed originates from the contamination of the excited states, and the plateau with the ground state saturation is realized only at t > 5 approximate to 15 fm corresponding to the inverse of the lowest excitation energy. We also demonstrate that the two-baryon operator can be optimized by utilizing the finite-volume eigenmodes, so that (i) the finite-volume energy spectra from the HAL QCD method agree with those from the temporal correlation function with the optimized operators and (ii) the correct finite-volume spectra would be accessed in the direct method only if highly optimized operators are employed. Thus we conclude that the long-standing issue on the consistency between Luscher's finite volume method and the HAL QCD method for two baryons is now resolved at least for this particular system considered here: they are consistent with each other quantitatively only if the excited contamination is properly removed in the former.