Calculation of an $A=3$ bound-state matrix element in pionless effective field theory

In this paper, we establish a general framework for calculating pionless matrix elements between $A=3$ bound-states up to next-to-leading-order. This framework is useful for pionless calculations of electroweak observables, such as ${}^3$H,${}^3$He magnetic moments and ${}^3$H $\beta$ decay. Starting from a Bethe-Salpeter equation, we prove that for a bound-state, the three-nucleon wave-function normalization can be expressed diagrammatically in a way that is equivalent to the unit operator between two identical three-nucleon bound-states. This diagrammatic form of the identity matrix element is the foundation for constructing an $A=3$ matrix element of a general operator. We show that this approach can be used to calculate the energy difference between ${}^3$H and ${}^3$He due to the Coulomb interaction, and to calculate the NLO corrections to the ${}^3$H and ${}^3$He scattering amplitudes due to effective range corrections.