Two-baryon systems with twisted boundary conditions

We explore the use of twisted boundary conditions in extracting the nucleon mass and the binding energy of two-baryon systems, such as the deuteron, from lattice QCD calculations. Averaging the results of calculations performed with periodic and antiperiodic boundary conditions imposed upon the light-quark fields, or other pairwise averages, improves the volume dependence of the deuteron binding energy from $\sim e^{-\kappa \mathrm{L}}/\mathrm{L}$ to $\sim e^{-\sqrt{2}\kappa \mathrm{L}}/\mathrm{L}$. However, a twist angle of $\pi /2$ in each of the spatial directions improves the volume dependence from $\sim e^{-\kappa \mathrm{L}}/\mathrm{L}$ to $\sim e^{-2\kappa \mathrm{L}}/\mathrm{L}$. Twist averaging the binding energy with a random sampling of twist angles improves the volume dependence from $\sim e^{-\kappa \mathrm{L}}/\mathrm{L}$ to $\sim e^{-2\kappa \mathrm{L}}/\mathrm{L}$, but with a standard deviation of $\sim e^{-\kappa \mathrm{L}}/\mathrm{L}$, introducing a signal-to-noise issue in modest lattice volumes. Using the experimentally determined phase shifts and mixing angles, we determine the expected energies of the deuteron states over a range of cubic lattice volumes for a selection of twisted boundary conditions.

Figure 1

Leading loop contributions to the mass of the nucleon. The solid line, solid-double line and dashed line denote a nucleon, a $\mathrm{\Delta }$ resonance and a pion, respectively. The black disks denote axial couplings.

Figure 2

The deuteron binding energy as a function of $\mathrm{L}$ using i-PBCs [${\mathit{\varphi }}^p=-{\mathit{\varphi }}^n\equiv \mathit{\varphi }=(\frac{\pi }{2},\frac{\pi }{2},\frac{\pi }{2})$]. The blue curve corresponds to the ${\mathbb{A}}_2$ irrep of the $C_{3v}$ group, while the red curve corresponds to the $\mathbb{E}$ irrep. The brown dashed curve corresponds to the weighted average of the ${\mathbb{A}}_2$ and $\mathbb{E}$ irreps, $-\frac{1}{3}(2B_d^{(\mathbb{E})}+B_d^{({\mathbb{A}}_2)})$, while the black solid curve corresponds to the $S$-wave limit. The infinite-volume deuteron binding energy is shown by the black dotted line.

Figure 3

(a) The deuteron binding energy as a function of L from PBCs (green curve) and from APBCs (purple curve). The black solid curve represents the average of these energies. (b) A closer look at the average in part (a) compared with energies obtained with i-PBCs, ${\mathbb{A}}_2$ (blue curve) and $\mathbb{E}$ (red curve).