Self energies of the pion and the delta isobar from the ^3He(e,epi^+)^3H reaction

In a kinematically complete experiment at the Mainz microtron MAMI, pion angular distributions of the $^3$He(e,e$pi^+)^3$H reaction have been measured in the excitation region of the $Delta$ resonance to determine the longitudinal ($L$), transverse ($T$), and the $LT$ interference part of the differential cross section. The data are described only after introducing self-energy modifications of the pion and $Delta$-isobar propagators. Using Chiral Perturbation Theory (ChPT) to extrapolate the pion self energy as inferred from the measurement on the mass shell, we deduce a reduction of the $pi^+$ mass of $Delta m_{pi^+} = (-1.7^{+ 1.7}_{- 2.1})$ MeV/c$^2$ in the neutron-rich nuclear medium at a density of $rho = (0.057^{+ 0.085}_{- 0.057})$ fm$^{-3}$. Our data are consistent with the $Delta$ self energy determined from measurements of $pi^0$ photoproduction from $^4$He and heavier nuclei.