1. 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
and then about the 
mixing angles and 
mass squared differences. New high quality data are becoming from MiniBoone, MINOS, NOMAD, MinerνA and SciBoone full dedicated to measure cross sections.
Disappearance searching experiments 
use 
CCQE reaction to detect an arriving neutrino and reconstruct its energy. 
determination could be wrong for a fraction of CC1
background events (20%)
, that can mimic a CCQE one if the pion is absorbed in the target and/or not detected. In 
appearance experiment, one detects 
in an (almost) 
beam. Here the signal event 
is dominated by a NC1π0 
background, and the detector can not distinguish between 
and 
if one of both photons from the 
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 [1-4]. 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 
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.
2. Elementary Amplitude
For the 
process the total cross section reads
(1)
being 
is the CM-Lab 
-energy relation, 
the merging lepton, 
phasespace factors, and
(2)
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
(3)
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
Figure 1. Different contributions to the amplitude.
and rescattering of the final 
pair, but the effects are not so important as in photoproduction.
2) Vector amplitude should fulfill electromagnetic gauge invariance (GI)
. In the B amplitude we must to have same vector FF (
contribution is axial and the 
one is self-GI), while for R contributions GI must be fulfilled in presence of finite width effects.
3) 
should be invariant under contact transformations (CT), which fixes the Feynman rules to built the amplitude [8], being the corresponding propagator (in the momentum space)
and a general vertex interaction
(4)
depending on the fields (nucleon, pion, photon, W-Z bosons, etc.) interacting with the 
field
. We have introduced 
(defined in Ref. [9]) which projects on the 
space. Unstableness of 
is included in the 
through a self-energy 
(one loop-corrections), which accounts an energy dependent width and vertex corrections to get GI. Alternatively, we make
, referred as complex mass scheme that results to be appropriated along the resonance region [10].
3. 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
(5)
being the single particle spectrum
(6)
where
(7)
is the effective mass acquired by the nucleon [11] through the scalar self-energy
. 
includes the tadpole diagram from Figure 2, retaining in its evaluation only the contribution from nucleons in the filled Fermi sea in the nucleon propagator (tick full lines). 
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. 
and 
are fixed to reproduce the experimental binding energy per nucleon of -16 MeV at the Fermi momentum 
for the normal nuclear matter. For the 
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
where
FSI on nucleons are taken (Toy model!) through the used effective fields within the RHA also for final N. While for pions we use the Eikonal approach in its simplest version [12], that is
, where
(8)
(a) (b)
Figure 2. (a) Tadpole diagram included in the MFT and RHA self-energies. (b) Tadpole exchange diagram that is added in order to get the relativistic Hartree Fock self-energy. The dashed lines indicate the propagator of the scalar (S) or vector meson (V) that interacts with a nucleon n (full lines).
Assuming a mean distance of trip for 
in nucleus
, constant nucleon density and the 
-h model for the 
-optical potential, we get
where 
and
, 
are the mass number, radii and isospin factor of the nucleus respectively.
4. Results and Conclusions
The different coupling constants presented in the B and R terms have been fixed by the fitting to the elastic 
cross section data [13], to the 
and 
processes [14], and using the CVC hypotesis and a fitting to the differential cross section
for the 
(ANL) data [15]. The results obtained for the background processes mentioned at the introduction are shown in Figures 3-5, where the different effects, bounding, smearing (GSC) and FSI are added gradually. In Figure 3 we show the total cross section for the 
process where 
(mineral oil detector) and compare with the corresponding data of the MoniBooNE [16,17] experiment. Results corresponding to the 
reaction are shown in Figure 4, and those for NC1
per nucleon in Figure 5.
Calculations are 
below MoniBooNE for CC 1π (comparable to the GiBUU Monte Carlo results) and 
for NC 
production. The FSI inclusion is very primitive, and perhaps an overvaluation of them is
Figure 3. CC 1π+ calculation compared with MiniBooNE data [16].
Figure 5. Idem for NC 1π0. Data form [17].
presented and should be improved. Nevertheless, it is noted that for example at 
for MiniBooNE and ANL or BNL (without cuts), data 
,
,
.
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.
NOTES