On Pointlike Interaction between Three Particles: Two Fermions and Another Particle

The problem of construction of self-adjoint Hamiltonian for quantum system consisting of three pointlike interacting particles (two fermions with mass 1 plus a particle of another nature with mass m>0) was studied in many works. In most of these works, a family of one-parametric symmetrical operators {Hε,ε∈ℝ1} is considered as such Hamiltonians. In addition, the question about the self-adjointness of Hε is equivalent to the one concerning the self-adjointness of some auxiliary operators {𝒯l,l=0,1,…} acting in the space L2(ℝ+1,r2dr). In this work, we establish a simple general criterion of self-adjointness for operators 𝒯l and apply it to the cases l=0 and l=1. It turns out that the operator 𝒯l=0 is self-adjoint for any m, while the operator 𝒯l=1 is self-adjoint for m>m0, where the value of m0 is given explicitly in the paper.

THE LONG-RANGE INTERACTION IN MASSLESS (λΦ4)4THEORY

Does massless (λΦ4)4theory exhibit spontaneous symmetry breaking (SSB)? The raw one-loop result implies that it does, but the "RG-improved" result implies the opposite. We argue that the appropriate "low-energy effective theory" is a nonlocal field theory involving an attractive, long-range interaction Φ2(x)Φ2(y)/|x-y|4. RG improvement then requires running couplings for both this interaction and the original pointlike interaction. A crude calculation in this framework yields SSB even after "RG improvement" and closely agrees with the raw one-loop result.