Hadronic parity violation in $\vec{n} p \to d γ$ with effective field theory

The parity-violating nucleon-nucleon ($NN$) potential is considered up to next-to-next-to leading order in heavy-baryon chiral perturbation theory. We include the one-pion exchange at the leading order and the two-pion exchange and two-nucleon contact terms at the next-to-next-to-leading order. The effects of intermediate (two-pion exchange) and short-range (two-nucleon contact) terms are probed by calculating the photon asymmetry $A_\gamma$ in $\vec{n} p \to d \gamma$ employing Siegert's theorem and an accurate phenomenological potential for the parity-conserving $NN$ interaction. We explore in detail the uncertainties due to the parameters that control the contribution of the short-range interaction. We obtain about 20% uncertainty in the value of $A_\gamma$ up to the next-to-next-to leading order. We discuss its implication for the determination of the weak pion-nucleon coupling constant and how the uncertainty can be reduced.