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For the attractive interaction, the Luschers finite volume formula gives the phase shift at negative squared moment $k^2<0$ for the ground state in the finite volume, which corresponds to the analytic continuation of the phase shift at $k^2<0$ in the infinite volume. Using this fact, we reexamine behaviors of phase shifts at $k^2 <0$ obtained directly from plateaux of effective energy shifts in previous lattice studies for two nucleon systems on various volumes. We have found that data, based on which existences of the bound states are claimed, show singular behaviors of the phase shift at $k^2<0$, which seem incompatible with smooth behaviors predicted by the effective range expansion. This, together with the fake plateau problem for the determination of the energy shift, brings a serious doubt on existences of the $NN$ bound states claimed in previous lattice studies at pion masses heavier than 300 MeV.
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There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlator and the HAL QCD method which extracts observables from the non-local potential associated with the tempo-spatial correlator. Although the two methods should give the same results theoretically, qualitatively different results have been reported. Recently, we pointed out 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 in the latter. In this paper, we identify the contribution of each state in the direct method by decomposing the two-baryon correlators into the finite-volume eigenmodes obtained from the HAL QCD method. We consider the $XiXi$ system in the $^1$S$_0$ channel at $m_pi = 0.51$ GeV in 2+1 flavor lattice QCD using the wall and smeared quark sources. We demonstrate that the pseudo-plateau at early time slices (t = 1~2 fm) from the smeared source in the direct method indeed originates from the contamination of the excited states, and the true plateau with the ground state saturation is realized only at t > 5~15 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 optimized temporal correlator and (ii) the correct 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 the Luschers finite volume method and the HAL QCD method for two baryons is now resolved: They are consistent with each other quantitatively only if the excited contamination is properly removed in the former.
In this comment, we address a number of erroneous discussions and conclusions presented in a recent preprint by the HALQCD collaboration, arXiv:1703.07210. In particular, we demonstrate that lattice QCD determinations of bound states at quark masses corresponding to a pion mass of $m_pi = 806$ MeV are robust, and that the phases shifts extracted by the NPLQCD collaboration for these systems pass all of the sanity checks introduced in arXiv:1703.07210.
A comparative study between the Luschers finite volume method and the time-dependent HAL QCD method is given for the $XiXi$($^1mathrm{S}_0$) 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 $Delta E_{rm eff} (t)$ in Luschers 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 $Delta E(L)$ in finite lattice volumes ($L^3$) calculated by $V(vec{r})$ indicate that there is no bound state in the $XiXi(^1mathrm{S}_0)$ channel at $m_{pi}=0.51$ GeV in 2+1 flavor QCD.
An extension of the Luschers finite volume method above inelastic thresholds is proposed. It is fulfilled by extendind the procedure recently proposed by HAL-QCD Collaboration for a single channel system. Focusing on the asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave functions (equal-time) near spatial infinity, a coupled channel extension of effective Schrodinger equation is constructed by introducing an energy-independent interaction kernel. Because the NBS wave functions contain the information of T-matrix at long distance, S-matrix can be obtained by solving the coupled channel effective Schrodinger equation in the infinite volume.
A sanity check rules out certain types of obviously false results, but does not catch every possible error. After reviewing such a sanity check for $NN$ bound states with the Luschers finite volume formula[1-3], we give further evidences for the operator dependence of plateaux, a symptom of the fake plateau problem, against the claim in [4]. We then present our critical comments on [5] by NPLQCD: (i) Operator dependences of plateaux in NPL2013[6,7] exist with the $P$-values of 4--5%. (ii) The volume independence of plateaux in NPL2013 does not prove their correctness. (iii) Effective range expansion (ERE) fits in NPL2013 violate the physical pole condition. (iv) Ref.[5] is partly based on new data and analysis different from the original ones[6,7]. (v) A new ERE in Refs.[5,8] does not satisfy the Luschers finite volume formula. [1] T. Iritani et al., JHEP 10 (2016) 101. [2] S. Aoki et al., PoS (LATTICE2016) 109. [3] T. Iritani et al., 1703.0720. [4] T. Yamazaki et al., PoS (LATTICE2017) 108. [5] S.R. Beane et al., 1705.09239. [6] S.R. Beane et al., PRD87 (2013) 034506. [7] S.R. Beane et al., PRC88 (2013) 024003. [8] M.L. Wagman et al., 1706.06550.
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