CC and NC Pion Production

The disappearance searching experiments x    use charged current quasielastic (CCQE) reaction to detect an arriving neutrino and reconstruct its energy, while the CC1 π + production can mimic the CCQE signal process. In e    appearance experiments, the NC1 π 0 production process can lead to a fake e  event by the impossibility for the detector of distinguish an arriving electron or a photon. Here we present a consistent model, from the point of view of the construction of the elemental amplitude, for the mentioned pion production background processes including bounding, smearing and final state interaction (FSI) effects in a single fashion. Results are comparable with more evolved approaches based on Monte Carlo simulations.


Introduction
Neutrino oscillation experiments search a distortion in the neutrino flux at a detector positioned far away (L) from the source. By comparing near and far neutrino energy spectra, one gains information about the oscillation probability   decay escapes. Then a precise knowledge of cross sections is a prerequisite in order to make simulations in event generators to substract fake 1π events in QE countings. Several models have been developed over the last thirty years to evaluate these corresponding background elementary cross sections [1][2][3][4]. The scattering amplitude in all these models always contains a resonant term (R) in the system, described by the  -pole contribution and (in some cases) by higher mass intermediate resonances, plus a nonresonant (B) term describing other processes (where the cross- contribution can also be included) leading to final states. The differences between all these models stem mainly from the treatment of the vertexes and the propagator used to describe the πN πN  resonance and from the consideration (or not) of the nonresonant terms and its interference with the resonant contribution. Nuclear effects and FSI have been introduced by several works, where different nuclear models and event generators or simulations codes have been implemented in [5] (GiBUU) and [6,7].
In this paper we reanalyze the elementary amplitude, bounding+ ground state correlations (GSC) effects, and FSI on the emerging nucleon (N) and pion , all what will be developed in the following sections.

Elementary Amplitude
For the π N lN    process the total cross section reads the amplitude where the contributions for the B amplitude are shown in the Figures 1(a)-(g), while for the R contribution is shown in the Figure 1(h).
The requirements on the hadronic part of the amplitude where indicate the lepton and nucleon spinors, are: 1) Unitarity, violated with real B terms. It is possible an unitarization by introduction of experimental phase shifts u  -en-

Bounding + GSC + FSI
The bounding effects in the nucleus are introduced within the relativistic Hartree approximation (RHA) of QHD I where the exchange of ,   mesons is considered. The meson fields are approximated by their vacuum spectation (MFT), i.e. constant values, and within the RHA [11] the vacuum fluctuation corrections are added. The nucleon field is expanded as is the effective mass acquired by the nucleon [11] through the scalar self-energy . S RHA  includes the same diagram but the full nucleon propagator (which encloses the contribution of the occupied negative-energy states) is used in the evaluation of the self-energy. V and S C are fixed to reproduce the experimental binding energy per nucleon of 16 MeV at the Fermi momentum for the normal nuclear matter. For the C 1 1.42 fm  F p   we assume the same scalar and vector self energies that for nucleon, approximation known as "universal couplings". In the structure of the ground state, 2p2h + 4p4h states (p, h ≡ particle, hole) are included through perturbation theory in nuclear matter, from which a momentum distribution can be built as While for pions we use the Eikonal approach in its simplest version [12], that is   presented and should be improved. Nevertheless, it is noted that for example at for Mini-BooNE and ANL or BNL (without cuts), data 1.5 GeV E   This seems to indicate that nuclear effects should be of much minor importance, or that another mechanism coming from nuclear effects should be considered, as 2p2h + 1π configurations generated by FSI added to the 1p1h + 1π considered here, and meson exchange currents contributions that are also capable of generating 2p2h + 1π acting on the nuclear ground state.