Implementing the three-neutron quantization condition

Abstract

We describe in detail the implementation of the relativistic three-neutron finite-volume quantization condition derived in Ref. [1]. In particular, we show how the complications due to Wigner rotations acting on spins are included, and present concrete formulas for the case when the angular momenta within pairs is restricted to be less than 2. We describe the symmetries of the matrices appearing in the quantization condition, and decompose solutions into irreducible representations of the appropriate doubled finite-volume symmetry groups. We present an implementation of the three-particle K matrix, keeping the two lowest-order terms in the threshold expansion. We provide numerical predictions for the finite-volume spectrum for a setup with nearly physical parameters, including two-particle interactions that are based on experimental results. This exploratory study shows the how lattice QCD calculations of the three-neutron spectrum with sufficient precision can provide detailed information on both two- and three-particle interactions.

Figures (8)

• Figure 1: Form of the scattering phase shifts for each of the $nn$ channels that we consider. Data points are from Ref. Stoks:1994wp; the blue lines are our chosen forms. All quantities are expressed in units of the neutron mass. For further discussion, see text.
• Figure 2: Three-neutron spectrum in the rest frame using ${\mathcal{K}_{\mathrm{df},3}}=0$, broken down by irrep. Energies are in units of $M_N$; numerical values of the energies are given in \ref{['tab:nP0noKdf']}. Noninteracting energy levels are shown in the left-most column (denoted "free"), along with their degeneracies (which include the degeneracies within the irrep). Results are for $M_N L=20$ and $M_\pi/M_N=0.15$. The inelastic threshold is shown by the dashed horizontal line.
• Figure 3: As for \ref{['fig:nP0noKdf']}, but for (a) $n_P^2=1$ and (b) $n_P^2=3$. In the right panel, the dotted (orange) line in the $F_1/F_2$ irrep indicates a level with unphysical residue, as discussed in the text. The inelastic threshold in the right panel lies above the plot range at $E=3.1967 M_N$.
• Figure 4: Eigenvalue of $(\mathcal{K}_{2,L}^{-1}+F+G)M_N$ for $n_P^2=3$ in the $F_1$ irrep with the smallest absolute value plotted against $(E - E_0)/M_N$, where $E_0$ is the lowest noninteracting energy in this irrep. Blue points are for $M_N L=20$ ($M_\pi L=3$), for which $E_0/M_N=3.1385$; red points are for $M_N L = 80/3$ ($M_\pi L=4$), for which $E_0/M_N=3.0801$. A physical crossing corresponds to the eigenvalue passing through zero from above.
• Figure 5: Shift in the energy of the lowest $G_1(1)$ level when either the $\mathcal{K}_A$ (solid blue curve) or $\mathcal{K}_B$ term (dashed red curve) from ${\mathcal{K}_{\mathrm{df},3}}$ is included. The horizontal axis is $\tanh(c_i/100)$, for $i=A,B$; the vertical axis is $10^3 (E-E_0)/M_N$, where $E_0=3.0430 M_N$ is the energy of the state when ${\mathcal{K}_{\mathrm{df},3}}=0$. Parameters are $M_N L=20$ and $M_\pi/M_N=0.15$. The curves have been cut off at the maximal values of $|c_i|$ for which no unphysical solutions are present. The blue and red dotted horizontal lines (barely distinguishable by eye) are explained in the text.