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In this paper, we establish a general framework for calculating pionlessmatrix elements between $A=3$ bound-states up to next-to-leading-order. Thisframework is useful for pionless calculations of electroweak observables, suchas $^3$H,$^3$He magnetic moments and $^3$H $\beta$ decay. Starting from aBethe-Salpeter equation, we prove that for a bound-state, the three-nucleonwave-function normalization can be expressed diagrammatically in a way that isequivalent to the unit operator between two identical three-nucleonbound-states. This diagrammatic form of the identity matrix element is thefoundation for constructing an $A=3$ matrix element of a general operator. Weshow 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 NLOcorrections to the $^3$H and $^3$He scattering amplitudes due to effectiverange corrections.

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