Plasma effects on resonant fusion

We investigate the effects of plasma interactions on resonance-enhanced fusion rates in stars. Starting from basic principles we derive an expression for the fusion rate that can serve as a basis for discussion of approximation schemes. The present state-of-the-art correction algorithms, based on the classical correlation function for the fusing particles and the classical energy shift for the resonant state, do not follow from this result, even as an approximation. The results of expanding in a perturbation solution for the case of a weakly coupled plasma are somewhat enlightening. But at this point we are at a loss as to how to do meaningful calculations in systems with even moderate plasma coupling strength. Examples where this can matter are: the effect of a possible low energy $^{12}$ C +$^{12}$ C resonance on X-ray bursts from accreting neutron stars or on supernova 1A simulations; and the calculation of the triple $\alpha$ rate in some of the more strongly coupled regions in which the process enters, such as accretion onto a neutron star.