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arXiv:0911.4608 · Physical Review A

Renormalization group, dimer-dimer scattering, and three-body forces

Boris Krippa, Niels R. Walet, Michael C. Birse

Published Apr 26, 2010• 16 citations
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Abstract

We study the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for systems of nonrelativistic fermions, using the same functional renormalization technique as previously applied to fermionic matter. We find a strong dependence on the cutoff function used in the renormalization flow for a two-body truncation of the action. Adding a simple three-body term substantially reduces this dependence.

DOI
10.1103/physreva.81.043628

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We study the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for systems of nonrelativistic fermions, using the same functional renormalization technique as previously applied to fermionic matter. We find a strong dependence on the cutoff function used in the renormalization flow for a two-body truncation of the action. Adding a simple three-body term substantially reduces this dependence.
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Abstract. We consider the applications of functional renormalisation group to few and many-body systems. As an application to the few-body dynamics we study the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for systems of few nonrelativistic fermions. We find a strong dependence on the cutoff function used in the renormalisation flow for a two-body truncation of the action. Adding a simple three-body term substantially reduces this dependence. In the context of many-body physics we study the dynamics of both symmetric and asymmetric many-fermion systems using the same functional renormalisation technique. It is demonstrated that functional renormalisation group gives sensible and reliable results and provides a solid theoretical ground for the future studies. Open questions as well as lines of further developments are discussed. Nonperturbative or Exact Renormalisation Group (ERG) has become a popular tool to study strongly interacting dy-namics [1]. It has been used to study a variety of physical systems, from nonrelativistic two-body [2] and many-body [3] systems to effective quark models [4] and gauge theo-ries [5]. The main object of the ERG is the scale-dependent average effective action Γk, where k is an auxiliary running
We consider fermion-dimer scattering in the presence of a large positive scattering length in the frame of functional renormalization group equations. A flow equation for the momentum dependent fermion-dimer scattering amplitude is derived from first principles in a systematic vertex expansion of the exact flow equation for the effective action. The resummation obtained from the nonperturbative flow is shown to be equivalent to the one performed by the integral equation by Skorniakov and Ter-Martirosian (STM). The flow equation approach allows to integrate out fermions and bosons simultaneously, in line with the fact that the bosons are not fundamental but build up gradually as fluctuation induced bound states of fermions. In particular, the STM result for atom-dimer scattering is obtained by choosing the relative cutoff scales of fermions and bosons such that the fermion fluctuations are integrated out already at the initial stage of the RG evolution.
The functional renormalization group is applied to the effective action for scattering of two nonrelativistic fermions. The resulting physical effective action is shown to contain the correct threshold singularity. The corresponding ``bare'' action respects Galilean invariance up to second order in momenta. Beyond that order it contains terms that violate this symmetry and, for the particular regulator considered, nonanalytic third-order terms. The corresponding potential can be expanded around a nontrivial fixed point using the power counting appropriate to a system with large scattering length.