Are two nucleons bound in lattice QCD for heavy quark masses? Consistency check with Lüscher's finite volume formula

On the basis of Lüscher's finite volume formula, a simple test (consistency check or sanity check) is introduced and applied to inspect the recent claims of the existence of the nucleon-nucleon (NN) bound state(s) for heavy quark masses in lattice QCD. We show that the consistency between the scattering phase shifts at k2>0 and/or k2<0 obtained from the lattice data and the behavior of phase shifts from the effective range expansion (ERE) around k2=0 exposes the validity of the original lattice data; otherwise, such information is hidden in the energy shift ΔE of the two nucleons on the lattice. We carry out this consistency check for all the lattice results in the literature claiming the existence of the NN bound state(s) for heavy quark masses and find that (i) some of the NN data show a clear inconsistency between the behavior of ERE at k2>0 and that at k2<0, (ii) some of the NN data exhibit a singular behavior of the low-energy parameter (such as the divergent effective range) at k2<0, (iii) some of the NN data have the unphysical residue for the bound-state pole in the S matrix, and (iv) the rest of the NN data are inconsistent among themselves. Furthermore, we raise a caution of using the ERE in the case of the multiple bound states. Our finding, together with the fake plateau problem previously pointed out by the present authors, brings a serious doubt on the existence of the NN bound states for pion masses heavier than 300 MeV in the previous studies.