Nuclear Potential of Final State Nucleons and Nucleons Plus Pions in Lepton Nucleus Scattering

Within the impulse approximation, the modeling of the energy of final state leptons in electron and neutrino quasielastic and pion production processes on nuclear targets in the region of the $\Delta$ resonance depends on several parameters. These parameters include the removal energy of the initial state nucleon from the nucleus $\epsilon^{P,N}$, the potentials of electrons, nucleons and pions in the Coulomb field of the nucleus $|V_{eff}|$, and the kinetic energy dependent nuclear potential for final state nucleons ($U^{QE}_{opt}$) and "nucleon plus pion" final states in the region of the $\Delta$ resonance which we refer to as $U^{\Delta}_{opt}$. We extract these parameters from electron scattering data. Previous studies have shown that real part of the optical potential for a nucleon bound in $_{6}^{12}C$ at zero kinetic energy $U^{P,N}_{opt}(T=0)\approx$ 44 MeV is larger than that for the $\Delta$(1232) resonance $U^{\Delta}_{opt}(T=0)\approx$ 30 MeV. We find the reverse at higher kinetic energies. For example at T=100 MeV we find a nucleon potential $U^{P,N}_{opt}(T=100 MeV)$=20$\pm$5 MeV and $U^{\Delta}_{opt}(T=100 MeV)$= 30$\pm$5 MeV. The two results are consistent for two reasons. First, theoretically the kinetic energy dependence of the $\Delta$ potential is flatter than that of the nucleon. Secondly, in our analysis the extracted $U^{\Delta}_{opt}$ values are the nuclear potential for {\it"nucleon plus pion"} final states in the region of the $\Delta$ resonances and therefore includes contributions from both resonance and non resonance pion production processes. For Monte Carlo generators that only include the effects of Fermi motion and nuclear potentials, the relevant parameter is the effective nuclear potential for the "nucleon plus pion" final state.