Techniques developed for handing inverse-power-law potentials in atomic physics are applied to the tensor one-pion exchange potential to determine the regions in which it can be treated perturbatively. In S-, P -and D-waves the critical values of the relative momentum are less than or of the order of 400 MeV. The RG is then used to determine the power counting for short-range interaction in the presence of this potential. In the P -and D-waves, where there are no low-energy bound or virtual states, these interactions have half-integer RG eigenvalues and are substantially promoted relative to naive expectations. These results are independent of whether the tensor force is attractive or repulsive. In the 3 S1 channel the leading term is relevant, but it is demoted by half an order compared to the counting for the effective-range expansion with only a short-range potential. The tensor force can be treated perturbatively in those F -waves and above that do not couple to P -or D-waves. The corresponding power counting is the usual one given by naive dimensional analysis.

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...

We solve a nonlocal generalisation of the NJL model in the Hartree approximation. This model has a separable interaction, as suggested by instanton models of the QCD vacuum. The choice of form factor in this interaction is motivated by the confining nature of the vacuum. A conserved axial current is constructed in the chiral limit of the model and the pion properties are shown to satisfy the Gell-Mann-Oakes-Renner relation. For reasonable values of the parameters the model exhibits quark confinement.

We calculate the amplitude T1 for forward doubly-virtual Compton scattering in heavy-baryon chiral perturbation theory, to fourth order in the chiral expansion and with the leading contribution of the γN∆ form factor. This provides a model-independent expression for the amplitude in the low-momentum region, which is the dominant one for its contribution to the Lamb shift. It allows us to significantly reduce the theoretical uncertainty in the proton polarisability contributions to the Lamb shift in muonic hydrogen. We also stress the importance of consistency between the definitions of the Born and structure parts of the amplitude. Our result leaves no room for any effect large enough to explain the discrepancy between proton charge radii as determined from muonic and normal hydrogen.

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.