Nucleon-nucleon scattering parameters in the limit of SU(3) flavor symmetry

The scattering lengths and effective ranges that describe low-energy nucleon-nucleon scattering are calculated in the limit of SU(3)-flavor symmetry at the physical strange-quark mass with lattice quantum chromodynamics. The calculations are performed with an isotropic clover discretization of the quark action in three volumes with spatial extents of $L\sim 3.4\mathrm{fm}$, $4.5\mathrm{fm}$, and $6.7\mathrm{fm}$, and with a lattice spacing of $b\sim 0.145\mathrm{fm}$. With determinations of the energies of the two-nucleon systems (both of which contain bound states at these up and down quark masses) at rest and moving in the lattice volume, Lüscher's method is used to determine the low-energy phase shifts in each channel, from which the scattering length and effective range are obtained. The scattering parameters, in the ${}^1{S}_0$ channel are found to be $m_{\pi }{a}^{{(}^1{S}_0)}=9.{50}_{-0.69}^{+0.78}{}_{-0.80}^{+1.10}$ and $m_{\pi }{r}^{{(}^1{S}_0)}=4.{61}_{-0.31}^{+0.29}{}_{-0.26}^{+0.24}$, and in the ${}^3{S}_1$ channel are $m_{\pi }{a}^{{(}^3{S}_1)}=7.{45}_{-0.53}^{+0.57}{}_{-0.49}^{+0.71}$ and $m_{\pi }{r}^{{(}^3{S}_1)}=3.{71}_{-0.31}^{+0.28}{}_{-0.35}^{+0.28}$. These values are consistent with the two-nucleon system exhibiting Wigner's supermultiplet symmetry, which becomes exact in the limit of large $N_c$. In both spin channels, the phase shifts change sign at higher momentum, near the start of the $t$-channel cut, indicating that the nuclear interactions have a repulsive core even at the SU(3)-symmetric point.