Three-body scattering from nonperturbative flow equations

Abstract

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.

Functional renormalization for the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation crossover

“…The accuracy of this result is somewhat surprising as no momentum dependence of l fj has been taken into account. The latter has turned out to be important for the atom-dimer scattering [49]. On the other hand, the general leading-order effect of fermionic momentum dependence is captured by the wave function renormalization Z j , the effect of which is largest in the strongly interacting regime, when |c −1 | < 1 (cf.…”

“…The additional inclusion of the atom-dimer couplingl fj closes the truncation on the level of interaction terms quartic in the fields and describes threebody scattering [49]. It leads to quantitative modifications for the many-body problem.…”

Functional renormalization group and conformal invariance in cold atoms

“…above the state with zero density n = 0 and zero temperature T = 0. As was shown in [9,10], numerous mathematical simplifications appear in the nonrelativistic vacuum compared with the many-body state. For instance, all Feynman diagrams with loop lines pointing in the same direction vanish.…”

“…As advocated in [9,10], a systematic vertex expansion of Γ k is especially suitable for studies of few-body problems. The vertex expansion is an expansion in powers of fields, i.e.…”

“…3. The flow equation for P ϕ (Q) can be solved analytically [9,10], and the solution for bosons and fermions differs only by a simple multiplicative factor. Keeping in mind the question asked in the end of Sec.…”