Precision calculation of $^3$He$(α,γ)^7$Be for solar physics

We calculate the cross section for radiative capture $^3$He$(α,γ)^7$Be at next-to-next-to-leading order (NNLO). At this order of perturbation, momentum dependent two-body currents make their first appearance. We provide a model-independent construction of these currents from gauge and Galilean invariance, where the general framework for constructing higher-order two-body currents in low-energy effective field theories becomes evident. The $^3$He$(α,γ)^7$Be astrophysical S-factor $S_{34}(0)= 0.564^{+0.17}_{-0.015}$ keV b is obtained from a Bayesian analysis at NNLO, with an additional nominal theoretical uncertainty $\pm0.017$ keV b of 3%.