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 (N N ) 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 Δ E 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 N N bound state(s) for heavy quark masses and find that (i) some of the N N data show a clear inconsistency between the behavior of ERE at k²>0 and that at k²<0 , (ii) some of the N N data exhibit a singular behavior of the low-energy parameter (such as the divergent effective range) at k²<0 , (iii) some of the N N data have the unphysical residue for the bound-state pole in the S matrix, and (iv) the rest of the N N 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 N N bound states for pion masses heavier than 300 MeV in the previous studies.