Renormalization group evolution and power counting in nuclear matter

Abstract

In nuclear matter, for interparticle separations larger than the healing distance (a characteristic long-distance scale of finite-density fermionic systems), the in-medium two-body wave function is essentially a free wave function. In terms of the renormalization group (RG), this implies that the running of the effective field theory (EFT) couplings freezes for r-space cutoffs above this distance (or p-space cutoffs below the corresponding healing momentum scale). As a consequence the leading order EFT description of nuclear matter (understood here as the infrared limit of the RG) corresponds to the mean-field approximation and a set of tree-level (i.e. perturbative) leading contact-range couplings. Though the contacts do in principle inherit the power counting they had in the vacuum, their iteration is suppressed in the infrared, explaining why they become perturbative in nuclear matter. In addition to the contact-range potential, RG evolution requires the inclusion of density-dependent terms in the equation of state that can be represented by a pseudo-potential, which (unlike the genuine contacts) should not be iterated. The LO EFT description ends up being a subset of Skyrme forces previously identified by the Orsay group.

Figures (3)

• Figure 1: Vacuum and in-medium zero-energy, S-wave two-body wave functions. The normalization has been set from the condition that both wave functions coincide in the $r \to 0$ limit, where $\Psi(r) \to (1 - {a_0}/r)$ with $a_0$ the scattering length. With this normalization in the asymptotic $r \to \infty$ limit the wave function approaches $\Psi \to 1$ in vacuum and $\Psi \to (1 - 2 a_0 k_F / \pi)$ in medium. The scattering length and Fermi momentum have been taken to be $a_0 = -23.7\,{\rm fm}$ and $k_F = 1.33\,{\rm fm}^{-1}$, which correspond to the singlet neutron-proton scattering length and the saturation density of symmetric nuclear matter.
• Figure 2: Running of the lowest order coupling constant $C_0$ with respect to the r-space cutoff $R_c$ in vacuum and in medium. The scattering length and Fermi momentum are the same as in Fig. \ref{['fig:in-medium-wf']}.
• Figure 3: The function $h_F(x)$, which determines the relative size of the loops in nuclear matter, as calculated for a delta-shell regulator in r-space and a sharp-cutoff in p-space. Here $x = k_F R_c$ or $x = k_F / \Lambda$ for the delta-shell and sharp-cutoff, respectively, with $R_c$ and $\Lambda$ the r-space and p-space cutoffs.