Effective Field Theory of 3He

Abstract

He and the triton are studied as three-body bound states in the effective field theory without pions. We study 3 He using the set of integral equations developed by Kok et al. which includes the full off-shell T-matrix for the Coulomb interaction between the protons. To leading order, the theory contains: two-body contact interactions whose renormalized strengths are set by the NN scattering lengths, the Coulomb potential, and a three-body contact interaction. We solve the three coupled integral equations with a sharp momentum cutoff, Λ, and find that a three-body interaction is required in 3 He at leading order, as in the triton. It also exhibits the same limit-cycle behavior as a function of Λ, showing that the Efimov effect remains in the presence of the Coulomb interaction. We also obtain the difference between the strengths of the three-body forces in 3 He and the triton. I. INTRODUCTION Since Weinberg first proposed applying the ideas of effective field theory (EFT) to nuclear forces [1], much effort has gone into this approach. (For reviews, see Refs. [2-4].) Although there is still some debate about how best to implement it at energies where pion-exchange forces are resolved, the picture is clearer at lower energies. Here few-nucleon systems can be described by a "pionless" EFT based on two-and three-body contact interactions [5-8]. The resulting expansion of the two-body force is just that of the effective-range expansion [9], but the EFT framework makes it possible to extend this consistently to other effective operators and three-body forces. This theory has been applied extensively to two-body systems, where it has been extended to include the effects of the Coulomb interaction on proton-proton scattering [10-14]. In that system, it corresponds to a distorted-wave or "modified" version of the effective-range expansion [9].

Investigation of thennΛbound state in pionless effective theory

“…This value is consistent with the values of Λ c found in similar context in the literature. For example, when the triton and He 6 ΛΛ are treated as three-nucleon and ΛΛα bound systems, respectively, the cutoff values are obtained as Λ c ≃ 380 MeV for the case of the triton [36], and Λ c ≃ 400 − 570 MeV for the case of He 6 ΛΛ [12], depending on the adopted value of the S-wave ΛΛ scattering length with the three-body contact interactions turned off. This contrasts with our case as we indeed need a very large cutoff to a form bound state without the three-body contact term.…”