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Effective field theory allows for a systematic and model-independent derivation of the forces between nucleons in harmony with the symmetries of Quantum Chromodynamics. We review the foundations of this approach and discuss its application for light nuclei at various resolution scales. The extension of this approach to many-body systems is briefly sketched. Commissioned article for Reviews of Modern Physics Contents I. QCD and Nuclear Forces 1 A. Chiral symmetry 2 B. Scales in nuclear physics 3 C. Conventional approaches to the nuclear force problem 4 D. Brief introduction to effective field theory 5 E. First results from lattice QCD 6 F. Observables and not-so observable quantities 8 II. EFT for Few-Nucleon Systems: Foundations and Applications 8 A. EFT with contact interactions and universal aspects 8 B. Chiral EFT for few nucleons: foundations 12 C. Chiral EFT for few nucleons: applications 21 D. The role of the ∆-isobar 25 E. Few-nucleon reactions involving pions 30 F. Hyperon-nucleon & hyperon-hyperon interactions 31 G. Nuclear lattice simulations 33 H. Quark mass dependence of nuclear forces and IR limit cycle in QCD 36 III. Towards a Many-Body EFT for Nuclei 39 A. In-medium chiral perturbation theory 39 B. Perturbative chiral nuclear dynamics 41 C. EFT for halo nuclei 42 D. V low k potentials: construction and applications 43 E. Lattice simulations of many-nucleon systems 44

We discuss renormalization of the non-relativistic threebody problem with short-range forces. The problem becomes non-perturbative at momenta of the order of the inverse of the two-body scattering length, and an infinite number of graphs must be summed. This summation leads to a cutoff dependence that does not appear in any order in perturbation theory. We argue that this cutoff dependence can be absorbed in a single three-body counterterm and compute the running of the three-body force with the cutoff. We comment on relevance of this result for the effective field theory program in nuclear and molecular physics.Systems composed of particles with momenta k much smaller than the inverse range 1/R of their interaction are common in nature. This separation of scales can be exploited by the method of effective field theory (EFT) that provides a systematic expansion in powers of the small parameter kR [1]. Generically, the two-body scattering length a 2 is comparable to R, and low-density systems with k ≪ 1/a 2 can be described to any order in kR by a finite number of EFT graphs [2]. However, there are many interesting systems, such as those made out of nucleons or of 4 He atoms, for which a 2 is much larger than R. In this case the expansion becomes non-perturbative at momenta of the order of 1/a 2 , in the sense that an infinite number of diagrams must be resummed. This resummation generates a new expansion in powers of kR where the full dependence in ka 2 is kept. Consequently, the EFT is valid beyond k ∼ 1/a 2 , comprising, in particular, bound states of size ∼ a 2 . There has been enormous progress recently in dealing with this problem in the twobody case [3], where the resummation is equivalent to effective range theory [4]. Ultraviolet (UV) divergences appear in graphs with leading-order interactions and their * bedaque@mocha.phys.washington.edu † hammer@triumf.ca ‡ vankolck@krl.caltech.edu resummation contains arbitrarily high powers of the cutoff. A crucial point is that this cutoff dependence can be absorbed in the coefficients of the leading-order interactions themselves. All our ignorance about the influence of short-distance physics on low-energy phenomena is then embodied in these few coefficients, and EFT retains its predictive power. However, the extension of this program to three-particle systems presents us with a puzzle [5]. Although in some fermionic channels the resummed leading two-body interactions lead to unambiguous and very successful predictions [6,7], amplitudes in bosonic systems and other fermionic channels show sensitivity to the UV cutoff, as evidenced in the Thomas [8] and Efimov [9] effects. This happens even though each leading-order three-body diagram with resummed two-body interactions is individually UV finite. We will argue below that the addition of a one-parameter three-body force counterterm at leading order is necessary and sufficient to eliminate this cutoff dependence. This result extends the EFT program to three-particle systems with large twobody scattering lengths, inc...

We apply the effective field theory approach to the three-nucleon system. In particular, we consider S = 1/2 neutron-deuteron scattering and the triton. We show that in this channel a unique nonperturbative renormalization takes place which requires the introduction of a single three-body force at leading order. With one fitted parameter we find a good description of low-energy data. Invariance under the renormalization group explains some universal features of the three-nucleon system -such as the Thomas and Efimov effects and the Phillips line-and the origin of SU (4) symmetry in nuclei.

We discuss renormalization of the non-relativistic three-body problem with short-range forces. The problem is non-perturbative at momenta of the order of the inverse of the two-body scattering length. An infinite number of graphs must be summed, which leads to a cutoff dependence that does not appear in any order in perturbation theory. We argue that this cutoff dependence can be absorbed in one local three-body force counterterm and compute the running of the three-body force with the cutoff. This allows a calculation of the scattering of a particle and the two-particle bound state if the corresponding scattering length is used as input. We also obtain a model-independent relation between binding energy of a shallow three-body bound state and this scattering length. We comment on the power counting that organizes higherorder corrections and on relevance of this result for the effective field theory program in nuclear and molecular physics.

Dispersion relations provide a powerful tool to analyse the electromagnetic form factors of the nucleon for all momentum transfers. Constraints from meson-nucleon scattering data, unitarity, and perturbative QCD can be included in a straightforward way. In particular, we include the 2pi, rho-pi, and KKbar continua as independent input in our analysis and provide an error band for our results. Moreover, we discuss two different methods to include the asymptotic constraints from perturbative QCD. We simultaneously analyze the world data for all four form factors in both the space-like and time-like regions and generally find good agreement with the data. We also extract the nucleon radii and the omega-NN coupling constants. For the radii, we generally find good agreement with other determinations with the exception of the electric charge radius of the proton which comes out smaller. The omega-NN vector coupling constant is determined relatively well by the fits, but for the tensor coupling constant even the sign can not be determined.Comment: 24 pages, 9 figure

We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two-and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to 4 He atoms and calculate the binding energy of the 4 He tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to 4 He atoms, the correlation is approximately linear.

We discuss the power counting for effective field theories with narrow resonances near a two-body threshold. Close to threshold, the effective field theory is perturbative and only one combination of coupling constants is fine-tuned. In the vicinity of the resonance, a second, "kinematic" fine-tuning requires a nonperturbative resummation. We illustrate our results in the case of nucleon-alpha scattering.

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