A renormalisation-group treatment of two-body scattering

Abstract

Nonrelativistic two-body scattering by a short-ranged potential is studied using the renormalisation group. Two fixed points are identified: a trivial one and one describing systems with a bound state at zero energy. The eigenvalues of the linearised renormalisation group are used to assign a systematic power-counting to terms in the potential near each of these fixed points. The expansion around the nontrivial fixed point is shown to be equivalent to the effective-range expansion.PACS numbers: 03.65. Nk,11.10.Hi,13.75.Cs,12.39.Fe Recently there has been much interest in applying the techniques of effective field theory (EFT) to the scattering of massive particles interacting via short-ranged forces. This has been spurred by Weinberg's powercounting rules for the low-momentum expansion of the nucleon-nucleon potential [1], which raised the possibility of applying the techniques of chiral perturbation theory to nuclear physics [2]. These would provide a systematic method for expanding few-nucleon bound-state properties and scattering observables in powers of nucleon momenta and the pion mass.By focussing on the potential, one avoids contributions where the two intermediate nucleons are almost on-shell, giving small denominators. However, this is the physics responsible for nuclear binding, and so to describe nuclei with an EFT it is not enough to write down a potential; one needs to solve the corresponding Schrödinger or Lippmann-Schwinger equation. At this point one encounters a problem. The EFT is based on a Lagrangian with local couplings between the particles and these include contact interactions between the nucleons. Such interactions correspond to δ-function potentials, and the resulting scattering equations only make sense after a further regularisation and renormalisation.A variety of approaches has been explored for renormalising two-body scattering by such potentials [3][4][5][6][7][8][9][10][11][12][13][14][15]. (For reviews of the various approaches and further references, see Ref. [16].) These have shown that it is difficult to set up a useful and systematic EFT for two-body scattering when the scattering length is unnaturally large, as indeed noted by Weinberg [1]. More recently an alternative to Weinberg's power counting has been suggested by Kaplan, Savage and Wise (KSW) [13], based on dimensional regularisation with a "power divergence subtraction" scheme (PDS). The same counting has also been proposed by van Kolck [10,15] within the framework of a general subtractive renomalisation scheme, using a momentum cut-off set at a large scale, typical of the underlying physics responsible for the short-distance interactions.To examine some of the questions raised by these approaches, we have studied nonrelativistic two-body scattering from the viewpoint of Wilson's continuous (or the "exact") renormalisation group (RG) [17]. In this approach, one imposes a cut-off on the momenta of virtual states at some scale Λ and demands that physical quantities be independent of Λ. A rescaled Hamiltonian is intr...

Partially quenched nucleon-nucleon scattering

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Proton–proton fusion in pionless effective theory

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