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...

Compton scattering from protons and neutrons provides important insight into the structure of the nucleon. For photon energies up to about 300 MeV, the process can be parameterised by six dynamical dipole polarisabilities which characterise the response of the nucleon to a monochromatic photon of fixed frequency and multipolarity. Their zero-energy limit yields the well-known static electric and magnetic dipole polarisabilities α E1 and β M 1 , and the four dipole spin polarisabilities. The emergence of full lattice QCD results and new experiments at MAMI (Mainz), HIγS at TUNL, and MAX-Lab (Lund) makes this an opportune time to review nucleon Compton scattering. Chiral Effective Field Theory (χEFT) provides an ideal analysis tool, since it encodes the well-established low-energy dynamics of QCD while maintaining an appropriately flexible form for the Compton amplitudes of the nucleon. The same χEFT also describes deuteron and 3 He Compton scattering, using consistent nuclear currents, rescattering and wave functions, and respects the low-energy theorems for photon-nucleus scattering. It can thus also be used to extract useful information on the neutron amplitude from Compton scattering on light nuclei. We summarise past work in χEFT on all of these reactions and compare with other theoretical approaches. We also discuss all proton experiments up to about 400 MeV, as well as the three modern elastic deuteron data sets, paying particular attention to the precision and accuracy of each set. Constraining the ∆(1232) parameters from the resonance region, we then perform new fits to the proton data up to ω lab = 170 MeV, and a new fit to the deuteron data. After checking in each case that a two-parameter fit is compatible with the respective Baldin sum rules, we obtain, using the sum-rule constraints in a one-parameter fit, α (p) E1 = 10.7 ± 0.3(stat) ± 0.2(Baldin) ± 0.8(theory), β (p) M 1 = 3.1 ∓ 0.3(stat) ± 0.2(Baldin) ± 0.8(theory), for the proton polarisabilities, and α (s) E1 = 10.9 ± 0.9(stat) ± 0.2(Baldin) ± 0.8(theory), β (s)M 1 = 3.6 ∓ 0.9(stat) ± 0.2(Baldin) ± 0.8(theory), for the isoscalar polarisabilities, each in units of 10 −4 fm 3 . Finally, we discuss plans for polarised Compton scattering on the proton, deuteron, 3 He and heavier targets, their promise as tools to access spin polarisabilities, and other future avenues for theoretical and experimental investigation.

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 analyse the proton Compton-scattering differential cross section for photon energies up to 325 MeV using Chiral Effective Field Theory (χEFT) and extract new values for the electric and magnetic polarisabilities of the proton. Our approach builds in the key physics in two different regimes: photon energies ω < ∼ m π ("low energy"), and the higher energies where the ∆(1232) resonance plays a key role. The Compton amplitude is complete at N 4 LO, O(e 2 δ 4 ), in the low-energy region, and at NLO, O(e 2 δ 0 ), in the resonance region. Throughout, the Delta-pole graphs are dressed with πN loops and γN∆ vertex corrections. A statistically consistent database of proton Compton experiments is used to constrain the free parameters in our amplitude: the M 1 γN∆ transition strength b 1 (which is fixed in the resonance region) and the polarisabilities α E1 and β M 1 (which are fixed from data below 170 MeV). In order to obtain a reasonable fit, we find it necessary to add the spin polarisability γ M 1M 1 as a free parameter, even though it is, strictly speaking, predicted in χEFT at the order to which we work. We show that the fit is consistent with the Baldin sum rule, and then use that sum rule to constrain α E1 +β M 1 . In this way we obtain α E1 = [10.65±0.35(stat)±0.2(Baldin)±0.3(theory)]×10 −4 fm 3 and β M 1 = [3.15 ∓ 0.35(stat) ± 0.2(Baldin) ∓ 0.3(theory)] × 10 −4 fm 3 , with χ 2 = 113.2 for 135 degrees of freedom. A detailed rationale for the theoretical uncertainties assigned to this result is provided. ‡ Permanent address ¶ Permanent address

We update the predictions of the SU(2) baryon chiral perturbation theory for the dipole polarisabilities of the proton, {α E1 , β M 1 } p = {11.2(0.7), 3.9(0.7)} × 10 −4 fm 3 , and obtain the corresponding predictions for the quadrupole, dispersive, and spin polarisabilities:The results for the scalar polarisabilities are in significant disagreement with semiempirical analyses based on dispersion relations, however the results for the spin polarisabilities agree remarkably well. Results for proton Compton-scattering multipoles and polarised observables up to the Delta(1232) resonance region are presented too. The asymmetries Σ 3 and Σ 2x reproduce the experimental data from LEGS and MAMI. Results for Σ 2z agree with a recent sum rule evaluation in the forward kinematics. The asymmetry Σ 1z near the pion production threshold shows a large sensitivity to chiral dynamics, but no data is available for this observable. We also provide the predictions for the polarisabilities of the neutron, the numerical values being {α E1 , β M 1 } n = {13.7(3.1), 4.6(2.7)} × 10 −4 fm 3 , {α E2 , β M 2 } n = {16.2(3.7), −15.8(3.6)} × 10 −4 fm 5 , {α E1ν , β M 1ν } n = {0.1(1.0), 7.2(2.5)} × 10 −4 fm 5 , and {γ E1E1 , γ M 1M 1 , γ E1M 2 , γ M 1E2 } n = {−4.7(1.1), 2.9(1.5), 0.2(0.2), 1.6(0.4)}×10 −4 fm 4 . The neutron dynamical polarisabilities and multipoles are examined too. We also discuss subtleties related to matching dynamical and static polarisabilities.