Diagrammatic Approach to Four-Boson Systems in Effective Field Theory

Abstract

The diagrammatic approach is used to study four-boson systems using two- and three-boson contact interactions from a leading-order effective field theory (EFT). In order to determine at what order a four-boson force is needed in an EFT, an renormalization group (RG) analysis, e.g., a study of the cutoff dependence, must be performed for four-boson observables. While existing four-boson calculations are limited below a threshold cutoff, $\Lambda_{t}$, above which deeply bound trimers appear, the method used in this calculation works at higher cutoffs by systematically incorporating the trimer poles in the four-boson calculation. Tetramer binding energies for cold $^4$He atoms are computed numerically and it is necessary to go above $\Lambda_{t}$ in order to obtain convergence. No four-boson force is needed at LO, as argued by previous studies, but more rigorously demonstrated here by the convergence at much higher cutoffs. Tetramer binding energies for cold $^4$He atoms agree with existing calculations. Tetramer binding energies and decay widths are also computed near the unitary limit. These results in the unitary limit also agree with existing calculations. The method used in this paper may be extended to study four-nucleon systems and the cutoff dependence of four-nucleon observables at high cutoffs.Comment: 42 pages, 14 page