Investigation of Point-Particle Effective Field Theory for the Inverse-Square Model

Abstract

In this thesis, we study non-relativistic scalar fields in (3+1) space-time subjected to an inverse-square potential. We use a point-particle effective field theory (PPEFT) framework to describe the scalar fields coupling to a point-particle in different cases of interest. In Chapter 3, we encode particle conversion for both Schrodinger and Klein-Gordon fields in a two-species toy model and find that the point-particle couplings all must be renormalized with respect to the radial cut-off near the origin. In addition to this, we find that cross sections have an interesting dependence on the ratio k_{out}/k_{in} of outgoing and incoming momenta. In certain regimes at low energies, we found inelastic behaviour \sigma_{S}^{(in)} ~ O(1) and \sigma_{KG}^{(in)} ~ 1/k_{in} for Schrodinger and Klein-Gordon fields respectively. In Chapter 4, we study the case of a single-particle non-self-adjoint PPEFT whose formulation is taken to next-to-leading order. We find that the point-particle couplings continue to require renormalization and present a series of relevant computations such as field equations, boundary conditions and renormalization runnings, concluding with an exposition of bound state energies, scattering lengths and cross sections. Similar to what was found in Chapter 3, a 1/k_{\text{in}} enhancement is observed in a particular regime of the PPEFT in Chapter 4. In addition, we find that the observables computed therein are modified from what was found in other papers [3][13], where only the leading PPEFT term was kept. These results may provide relevance for future calculations in more complex reactions such as baryon number violation and monopole catalysis.