Renormalizability of the nuclear many-body problem with the Skyrme interaction beyond mean field

Phenomenological effective interactions like Skyrme forces are currently used in mean-field calculations in nuclear physics. Mean-field models have strong analogies with the first order of the perturbative many-body problem and the currently used effective interactions are adjusted at the mean-field level. In this work, we analyze the renormalizability of the nuclear many-body problem in the case where the effective Skyrme interaction is employed in its standard form and the perturbative problem is solved up to second order. We focus on symmetric nuclear matter and its equation of state, which can be calculated analytically at this order. It is shown that only by applying specific density dependence and constraints to the interaction parameters can renormalizability be guaranteed in principle. This indicates that the standard Skyrme interaction does not in general lead to a renormalizable theory. To achieve renormalizability, other terms should be added to the interaction and employed perturbatively only at first order.

Figure 1

First-order (upper row) and second-order (lower row) contributions to the energy of nuclear matter. Direct (exchange) terms are shown on the left (right). Particles and holes are denoted by oriented solid lines and the interaction is denoted by a dashed line.

Figure 2

EOS for the case $t_1^R=t_2^R=0$ with $\alpha =-1/6$. The fitted second-order EOS, Eq. (18), (red squares) is compared to the benchmark EOS (black solid line). Also shown (blue circles) is a more constrained fit, where $t_0^R$ is not redefined according to Eq. (17), which gives an estimate of the potential size of higher-order effects.

Figure 3

EOS for the case $t_1^R=t_2^R=0$ with $\alpha =1/3$. The fitted second-order EOS, Eq. (21), with $m=939$ MeV (red circles) and with $m$ as a free parameter (blue squares) is compared to the benchmark EOS (black solid line).