We apply renormalisation-group methods to two-body scattering by a combination of known long-range and unknown short-range potentials. We impose a cut-off in the basis of distorted waves of the long-range potential and identify possible fixed points of the short-range potential as this cut-off is lowered to zero. The expansions around these fixed points define the power countings for the corresponding effective field theories. Expansions around nontrivial fixed points are shown to correspond to distorted-wave versions of the effective-range expansion. These methods are applied to scattering in the presence of Coulomb, Yukawa and repulsive inverse-square potentials.

A distorted-wave version of the renormalisation group is applied to scattering by an inversesquare potential and to three-body systems. In attractive three-body systems, the short-distance wave function satisfies a Schrödinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalisation of the three-body interactions, with the renormalisation-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.

Abstract. We study the scattering of a particle from a bound pair in an effective field theory using a distorted-wave renormalisation group method to find the power-counting for the three-body force terms. We find that threebody terms appear at lower orders than naively expected. They start with a marginal term that varies as a logarithm rather than a power of the energy scales in the problem. The marginal term has important implications for the three-body problem in nuclear physics.

We apply renormalisation-group methods to two-body scattering by a combination of known long-range and unknown short-range potentials. We impose a cut-off in the basis of distorted waves of the long-range potential and identify possible fixed points of the short-range potential as this cut-off is lowered to zero. The expansions around these fixed points define the power countings for the corresponding effective field theories. Expansions around nontrivial fixed points are shown to correspond to distorted-wave versions of the effective-range expansion. These methods are applied to scattering in the presence of Coulomb, Yukawa and repulsive inverse-square potentials. Close to a fixed point, deviations from it scale as powers of the cut-off. Since the rescaling means that each lowenergy scale appearing in a term of the potential contributes one power of Λ, we can use this to define the power counting for our EFT. A term that behaves like Λ ν is assigned an order d = ν − 1, to match with the Weinberg power counting mentioned above.Perturbations around a fixed point can be classified into three types according to the sign of ν. One with ν > 0 is known as an "irrelevant" perturbation. Its flow is towards the fixed point as Λ → 0. A perturbation with ν = 0 is called "marginal". This is the type of term familiar in conventionally renormalisable field theories. If a marginal perturbation is present, we should expect to find logarithmic flow with Λ. Finally, a perturbation with ν < 0 is called "relevant". It leads to flow away from the fixed point as Λ → 0.If all perturbations about a fixed point are irrelevant, then the fixed point is stable: the couplings of any theory close to that point will flow towards it as Λ → 0. On the other hand, if there are one or more relevant perturbations then the fixed point is unstable.In Ref.[13] these ideas were applied to two-body scattering by short-range forces. Two fixed points were found for s-wave scattering. One is the trivial fixed point, describing systems with no scattering. The terms in the expansion around this point can be organised according to Weinberg power counting. This defines an EFT which is appropriate for describing a system with weak scattering. The second point is a nontrivial one describing systems with a bound state at zero energy. The flow near it includes one unstable direction and the appropriate power counting for expanding around it is KSW counting. The terms of this expansion are in one-to-one correspondence with the terms of the effective-range expansion [15,16,17,18].The expansion around this nontrivial fixed point is appropriate for a system with a large scattering length (or equivalently a bound state or resonance close to zero energy). In the case of nucleon-nucleon scattering, it can be used at very low energies, well below the pion mass, where the whole strong interaction between the nucleons can be treated as short-range. It has been applied successfully to the calculation of various deuteron properties and reactions [19]. The corresponding EFT has...