Numerical study of renormalization group flows of nuclear effective field theory without pions on a lattice

We formulate the next-to-leading-order nuclear effective field theory without pions in the two-nucleon sector on a spatial lattice and investigate nonperturbative renormalization group flows in the strong-coupling region by diagonalizing the Hamiltonian numerically. The cutoff (proportional to the inverse of the lattice constant) dependence of the coupling constants is obtained by changing the lattice constant with the binding energy and the asymptotic normalization constant for the ground state being fixed. We argue that the critical line can be obtained by looking at the finite-size dependence of the ground-state energy. We determine the relevant operator and locate the nontrivial fixed point, as well as the physical flow line corresponding to the deuteron in the two-dimensional plane of dimensionless coupling constants. It turns out that the location of the nontrivial fixed point is very close to the one obtained by the corresponding analytic calculation, but the relevant operator is quite different.

Figure 1

The flow and the fixed points of the NLO NEFT in the $X-Y$ plane obtained by using a sharp momentum cutoff in the continuum formulation. The arrows indicate the directions of the smaller values of the cutoff.

Figure 2

The cutoff dependence of the binding energy (a) and the ANC (b), calculated for a given set of the scattering length and the effective range, $(a_0,r_0)=(5.42,1.75)\mathrm{fm}$, corresponding to deuteron [33].

Figure 3

The flow and the fixed points of the NLO NEFT in the $X-Y$ plane obtained by using a lattice regularization with the three-point formula.

Figure 4

The flow and the fixed points of the NLO NEFT in the $X-Y$ plane obtained by using a lattice regularization with the five-point formula.

Figure 5

The flow line corresponding to the deuteron obtained by using a lattice regularization with the five-point formula.

Figure 6

The flow of the NLO NEFT in the strong-coupling phase in the $X-Y$ plane obtained by numerical diagonalization of the Hamiltonian defined on a lattice with the five-point formula.

Figure 7

The difference between the calculated ground-state energies with $N_s=14$ and $N_s=16$ (upper surface) and between with $N_s=16$ and $N_s=18$ (lower surface) are shown as functions of $X$ and $Y$. This side of the mountain range is the weak-coupling phase; the other side is the strong-coupling phase.

Figure 8

Typical wave functions near the critical line. In the weak-coupling phase (a), the wave function spreads out over the whole space with a small peak at the center of potential. In the strong-coupling phase (b), it is sharply peaked at the center of the potential and decays exponentially.

Figure 9

The ridge line (red) together with the RG flow in Fig. 6 calculated with the five-point formula. From the flow, we infer that the nontrivial fixed point is on the dashed line (green). The nontrivial fixed point is also on the ridge line; it is at the crossing point (blue bullet). The small point (magenta) just above the crossing point is the location of the nontrivial fixed point obtained by the analytic calculation, $(X_{★},Y_{★})=(-0.63338...,-0.098805...)$.

Figure 10

The same as in Fig. 9, but with the three-point formula. The small point (magenta) indicates the location of the nontrivial fixed point obtained by the analytic calculation, $(X_{★},Y_{★})=(-0.76602...,0.17501...)$.

Figure 11

The rotational-symmetry breaking in the asymptotic behavior of the “wave function.” The “wave function” in the $(1,0,0)$, $(1,1,0)$, and $(1,1,1)$ directions are obtained by the diagonalization of the Hamiltonian with the three-point formula (a) and with the five-point formula (b). The gray line indicates the ANC defined as $B/4\pi$ from the coefficient $B$ of the regularized Yukawa term in Eq. (4.17). The calculation is done for the deuteron state, so that ANC is $0.224{\mathrm{fm}}^{-1/2}$.

Figure 12

The flow line corresponding to deuteron is shown as a dotted line (magenta) against the RG flow given in Fig. 9. The calculations are done with the five-point formula. The nontrivial fixed point (blue bullet), phase boundary (red solid line), and the relevant direction (green dashed line) are also shown.

Figure 13

The same as in Fig. 12, but the calculations are done with the three-point formula.