Two-nucleon $S$-wave interactions at the SU(3) flavor-symmetric point with $m_{ud}\approx m_s^{\text{phys}}$: A first lattice QCD calculation with the stochastic Laplacian Heaviside method

We report on the first application of the stochastic Laplacian Heaviside method for computing multiparticle interactions with lattice QCD to the two-nucleon system. Like the Laplacian Heaviside method, this method allows for the construction of interpolating operators which can be used to construct a set of positive-definite two-nucleon correlation functions, unlike nearly all other applications of lattice QCD to two nucleons in the literature. It also allows for a variational analysis in which optimal linear combinations of the interpolating operators are formed that couple predominantly to the eigenstates of the system. Utilizing such methods has become of paramount importance to help resolve the discrepancy in the literature on whether two nucleons in either isospin channel form a bound state at pion masses heavier than physical, with the discrepancy persisting even in the SU(3)-flavor-symmetric point with all quark masses near the physical strange quark mass. This is the first in a series of papers aimed at resolving this discrepancy. In the present work, we employ the stochastic Laplacian Heaviside method without a hexaquark operator in the basis at a lattice spacing of $a\approx 0.086$ fm, lattice volume of $L=48a\approx 4.1$ fm and pion mass $m_{\pi }\approx 714$ MeV. With this setup, the observed spectrum of two-nucleon energy levels strongly disfavors the presence of a bound state in either the deuteron or dineutron channel.

Figure 1

A $t_{\min }$ stability plot of the pion ground-state mass with different $N$ states in the fit function. See text for description. The filled symbol at $t_{\min }=3$ is the chosen fit.

Figure 2

Stability of the single nucleon ground state at zero momentum. The filled (black) circles are the effective mass data from the correlation function. The open squares are the resulting ground-state mass as a function of $t_{\min }$ and the number of states $n$ used in the analysis. The filled square at $t_{\min }=5$ with $n=2$ is the chosen fit. The vertical gray bands indicate time-regions excluded from this fit and the gray filled curve is the effective mass reconstructed from its posteriors and the horizontal band is the value of $m_N$.

Figure 3

Stability plot of the ground-state energy in the T1g irrep with $\mathbf{d}=0$ (a) and the second principal correlator in the $E$ irrep with $\mathbf{d}=1$ (b). The filled (black) circles are the effective mass of the ratio correlation function, Eq. (13). The open squares are the resulting $\mathrm{\Delta }{E}_{\text{g.s.}}$ energy as a function of $t_{\min }$ and the number of “elastic” excited states used, see the text. The filled square is the chosen fit. The vertical gray bands indicate time-regions excluded from this fit, the gray curve is the effective mass reconstructed from its posteriors and the (red) horizontal band is the value of $\mathrm{\Delta }{E}_{\text{g.s.}}$.

Figure 4

Phase shift analysis, ignoring all partial wave mixing, in the irreps that overlap with the S-wave deuteron. The spline/gradient method is shown on the left and the spectrum method on the right. The smaller (magenta) band is the 1-sigma result of the NLO ($q^{*2}$) order fit while the larger (gray) band is the 1-sigma result of the NNLO ($q^{*4}$) order analysis. The solid (cyan) line is the solution of $qcot\delta =iq$ where a bound state would occur if it were in the spectrum.

Figure 5

Same as described in the caption of Fig. 4 except for the $s$-wave dineutron channel.

Figure 6

Values of $qcot\delta$ in the deuteron channel in the present work compared with NPLQCD results at $m_{\pi }\approx 800$ MeV. While there is some communal expectation that discretization effects should be subdominant, one should be cautious to note that these computations have been performed with different lattice actions at only a single lattice spacing each: in the present case, the CLS clover-Wilson action [47] and with NPLQCD, a single-stout smeared [86], tadpole improved [87] clover-Wilson action [13]. Assuming the discretizaton effects are relatively small, and that the phase shift does not have a strong pion mass dependence, these results are in conflict.

Figure 7

Plotting the bootstrap samples within a single irrep, demonstrating the difference between the modified distance vs. the usual regression.