Non-Perturbative Renormalization Group Equations in Quantum Hadrodynamics

Abstract

Quantum hadrodynamics are studied with non-perturbative renormalization group equations. It is shown that the contributions of one-loop diagrams are important for the effective potential, since effective couplings of lower-order interactions in the effective potential are only slightly affected by quantum effects at low energy. The relation between the localpotential approximation (LPA) and the traditional Hartree approximation is discussed. The LPA includes the contribution of the Fock term to the nucleon self-energy, the contribution in the random phase approximation to the meson self-energy, and vertex corrections in part, as well as the Hartree contribution to the effective potential. §1. IntroductionIn the past quarter century, nuclei and nuclear matter have been studied in the framework of quantum hadrodynamics (QHD). 1), 2) The meson mean-field theory of nuclear matter 1) has produced successful results to account for the saturation properties at normal nuclear density. Following these successes, many studies and modifications have been made of relativistic nuclear models. One of these modifications is the inclusion of vacuum fluctuation effects, which cause divergences in physical quantities when they are naively calculated. Chin 3) estimated the vacuum fluctuation effects in the Hartree approximation by using a renormalization procedure, and found that vacuum fluctuation effects make the incompressibility of nuclear matter smaller and closer to the empirical value than in the original Walecka model.Although the relation between QHD and the underlying fundamental theory, i.e., QCD, is not clear, it is natural that QHD is not valid at very high energy. From this point of view, a cutoff or a form factor should be introduced into the theory of QHD. One may introduce such a cutoff 4) or form factor 5) to avoid the instability of the meson propagators in a random phase approximation (RPA). 6) Cohen 7) introduced a four-dimensional cutoff into the relativistic Hartree approximations (RHA) and found that the vacuum energy contributions may be somewhat different from those in the ordinary renormalization procedures, if the cutoff is not too large.One troublesome problem in the use of a finite cutoff is that physical results depend on the value of the cutoff and the shape of the regulator which are introduced into the theory by hand. It is difficult to determine a suitable value of the cutoff and * )

Effective Hadron Masses and Effective Interactions in Nuclear Matter

“…[33] We regard the Lagrangian (1) as an effective one in which the quantum effects of the vacuum fluctuations have been already included in the higher-order interactions. The Σ s , Σ v and U M may be able to be calculated perturbatively [19,18] or nonperturbatively [34] from the bare Lagrangian which does not include the effects of the vacuum fluctuations. For example, in the one-loop approximation or the relativistic Hartree approximation (RHA), U M is given by [19,18]…”