The disappearance searching experiments nu_mu  --->   nu_x  use charged current quasielastic  (CCQE) reaction to detect an arriving neutrino and reconstruct its energy, while the CC1 pi^+ production can mimic the CCQE signal process.  In nu_mu --->nu_e appearance experiments, the NC1 pi^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.Fil: Mariano, Alejandro Edgardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Física; ArgentinaFil: Barbero, César Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Física; ArgentinaFil: López Castro, Gabriel. Instituto Politécnico Nacional. Centro de Investigación y de Estudios Avanzados. Departamento de Física; Méxic

Barbero, César Alberto

López Castro, Gabriel

Mariano, Alejandro Edgardo

Journal of Modern Physics

English

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Journal of Modern Physics, 2013, 4, 1-5 Published Online December 2013 (http://www.scirp.org/journal/jmp) http://dx.doi.org/10.4236/jmp.2013.412A1001  Open Access                                                                                            JMP CC and NC Pion Production Alejandro Mariano1,2*, Cesar Barbero1,2, Gabriel López-Castro3 1Departamento de F sica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina  2Instituto de Fsica La Plata, CONICET, Buenos Aires, Argentina 3Departamento de F sica, Cinvestav, Mexico City, Mexico Email: *mariano@fisica.unlp.edu.ar  Received October 2, 2013; revised November 1, 2013; accepted November 28, 2013  Copyright © 2013 A. Mariano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor-dance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property A. Mariano et al. All Copyright © 2013 are guarded by law and by SCIRP as a guardian. ABSTRACT The disappearance searching experiments x   use charged current quasielastic (CCQE) reaction to detect an ar-riving neutrino and reconstruct its energy, while the CC1π+ production can mimic the CCQE signal process. In e   appearance experiments, the NC1π0 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.  Keywords: Neutrino Scattering; Pion Production; Neutrino Oscillation 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 oscilla-tion probability  2 2 2,sin 2 sin ,2ij i ji jm LPE    and then about the ij  mixing angles and  mass squared differences. New high quality data are becoming from MiniBoone, MINOS, NOMAD, MinerνA and Sci-Boone full dedicated to measure cross sections. 2,i jmDisappearance searching experiments x   use n p   CCQE reaction to detect an arriving neu- trino and reconstruct its energy. E  determination could be wrong for a fraction of CC1   background events (20%) p p    , that can mimic a CCQE one if the pion is absorbed in the target and/or not de- tected. In e   appearance experiment, one detects e  in an (almost)   beam. Here the signal event  is dominated by a NC1π0  background, and the detector can not distinguish between en e p 0πN N  e  and  if one of both photons from the 0π 0π  1232 decay escapes. Then a precise knowledge of cross sec- tions is a prerequisite in order to make simulations in event generators to substract fake 1π events in QE count- ings. 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 inter-mediate resonances, plus a nonresonant (B) term de-scribing other processes (where the cross-  contribu-tion 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 πNπN  resonance and from the consid-eration (or not) of the nonresonant terms and its interfer-ence with the resonant contribution. Nuclear effects and FSI have been introduced by several works, where dif-ferent nuclear models and event generators or simula-tions codes have been implemented in [5] (GiBUU) and [6,7]. *Corresponding author. A. MARIANO  ET  AL. 2 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.  π2. Elementary Amplitude For the πN lN   process the total cross section reads   41 2π 21 0d d2π1                   d cos d ,16lCM CM CMCM lCC NCE ElE EspinFE EE sM E     /    (1) being 22LabCMLabNN Nm EEE m m is the CM-Lab ergy relation,  the merging lepton, l CC NCF /  phase- space factors, and ,B R R .                 (2) the amplitude where the contributions for the B ampli-tude 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 ampli-tude           512         , ,Fi lVi AiGu p i u pu p u p i B R              (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  u -en-   Figure 1. Different contributions to the amplitude. Open Access                                                                                            JMP A. MARIANO  ET  AL. 3 and rescattering of the final  pair, but the effects are not so important as in photoproduction. πN2) Vector amplitude should fulfill electromagnetic gauge invariance (GI) 3 0iu q u   . In the B ampli-tude we must to have same vector FF (   contribution is axial and the   one is self-GI), while for R contribu-tions GI must be fulfilled in presence of finite width ef-fects. 3)  3 2R S    should be invariant under contact transformations (CT), which fixes the Feynman rules to built the amplitude [8], being the corresponding propa-gator (in the momentum space)      3 22 2 21 2 1 2 1 211 12 211 2ˆ,33ˆ ˆ ˆ ,p mG p P p mp m mP P Pm              | |  and a general vertex interaction  , , , . .,g F       h c          (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 kijP3 2k  ,1 2  space. Unstableness of  is included in the  through a self-energy  G p  p  (one loop-corrections), which accounts an energy dependent width and vertex corrections to get GI. Alternatively, we make  2m m i      dressedG G , referred as com-plex 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 consid-ered. 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         33 * *†d 2πe e ,s ts ts tN N s t mm mip x ip xm m s tx p m E u m m ab v m m      ppp ppm(5) being the single particle spectrum      2 2 * *0 0 ,VV B N Vp C m E C E    p p     (6) where   * 2 *2 *, SN N N S NE m m m C    p p *, .m    (7) *N Nm m  is the effective mass acquired by the nucleon [11] through the scalar self-energy . S   SMFT  in-cludes 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). SRHA  includes the same diagram but the full nucleon propagator (which encloses the contribution of the occu-pied negative-energy states) is used in the evaluation of the self-energy. V  and SC  are fixed to reproduce the experimental binding energy per nucleon of 16 MeV at the Fermi momentum  for the normal nuclear matter. For the C11.42 fmFp   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 mat-ter, from which a momentum distribution can be built as    † 30 0 ,  d4A m m AAn a a kn  k kk k    1 2 1 21 2 3 4,h sp p h 1 2 3 41 2 1 221 2 3 4 1 2 3 42,10 02!1,4!p p h hp sp p h h hp s h sc p p h hc p p p p h h h h      where 1 2 1 21 2 1 21 2 1 2ˆ 0,p p h hp p h hp p h h VcE   1 2 3 4 1 2 3 41 2 1 2 1 2 1 2 1 2 3 4 1 2 3 41 2 1 2 1 2 3 4 1 2 3 4ˆ ˆ0.p p p p h h h hp p h h p p p p h h h hcV p p h h p p h h V p p p p h h h hE E 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 sim-plest version [12], that is *   , where      , d* e e , ,optzi v Vi   .    b z zp rr r b z     (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-en- ergy. The dashed lines indicate the propagator of the scalar (S) or vector meson (V) that interacts with a nucleon n (full lines). Open Access                                                                                            JMP A. MARIANO  ET  AL. 4 Assuming a mean distance of trip for  in nucleus π d  , constant nucleon density and the -h model for the  -optical potential, we get    * e e i s di     pp rr ,       2 2 *02 1 2 ,9 N Ns f m m Ts s m    *  22 1 302 3 ,  , ,d R r R r A r cA   1 3 where ,A R  and , T N Z A  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 hy-potesis and a fitting to the differential cross section  π πp πp pn0πp p 2ddQ for the  (ANL) data [15]. The  πp p  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 ( 1) πA A p     process where 12 2A C H   (mineral oil detector) and compare with the corresponding data of the MoniBooNE [16,17] ex-periment. Results corresponding to the   1 0πA A N0π50%  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  30% 0π  Figure 3. CC 1π+ calculation compared with MiniBooNE data [16].  Figure 4. Idem for CC 1π0.   Figure 5. Idem for NC 1π0. Data form [17].  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 GeVE 1 195%exp expACC NCCA     , 0 01 183%exp expACC NCCA    , 0 01 1 92%exp expANC NNCA    . 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. REFERENCES [1] G. L. Fogli and G. Nardulli, Nuclear Physics B, Vol. 160, 1979, pp. 116-150.  http://dx.doi.org/10.1016/0550-3213(79)90233-5  [2] T. Sato, D. Uno and T.-S. H. Lee, Physical Review C, Vol. 67, 2003, Article ID: 065201.  Open Access                                                                                            JMP A. MARIANO  ET  AL. Open Access                                                                                            JMP 5http://dx.doi.org/10.1103/PhysRevC.67.065201 [3] E. Hernandez, J. Nieves and M. Valverde, Physical Re-view D, Vol. 76, 2007, Article ID: 033005.  http://dx.doi.org/10.1103/PhysRevD.76.033005 [4] O. Lalakulich, T. Leitner, O. Buss and U. Mosel, Physical Review D, Vol. 82, 2010, Article ID: 093001.  http://dx.doi.org/10.1103/PhysRevD.82.093001 [5] O. Lalakulich and U. Mosel, Pion Production in the MiniBooNE Experiment, Physical Review C, Vol. 87, 2013, Article ID: 014602.  http://dx.doi.org/10.1103/PhysRevC.87.014602 [6] J. T. Sobczyk and J. Zmuda, Physical Review C, Vol. 87, 2013, Article ID: 065503.  http://dx.doi.org/10.1103/PhysRevC.87.065503 [7] E. Hernandez, J. Nieves and M. J. Vacas, Physical Re-view D, 87, 2013, Article ID: 113009.  http://dx.doi.org/10.1103/PhysRevD.87.113009 [8] M. El-Amiri, G. López Castro and J. Pestieau, Nuclear Physics A, Vol. 543, 1992, pp. 673-684.  http://dx.doi.org/10.1016/0375-9474(92)90553-V  [9] F. de Jong and R. Malfliet, Physical Review C, Vol. 46, 1992, pp. 2567-2581.  http://dx.doi.org/10.1103/PhysRevC.46.2567  [10] C. Barbero, A. Mariano and G. López Castro, Journal of Physics G: Nuclear and Particle Physics, Vol. 39, 2012, Article ID: 085011.  http://dx.doi.org/10.1088/0954-3899/39/8/085011 [11] J. D. Walecka, “Theroretical Nuclear and Subnuclear Physics,” Part I, Oxford University Press, New York, 1995. [12] C. Barbero, A. Mariano and S. B. Duarte, Physical Re-view C, Vol. 82, 2010, Article ID: 067305.  http://dx.doi.org/10.1103/PhysRevC.82.067305 [13] G. López Castro and A. Mariano, Nuclear Physics A, Vol. 697, 2001, pp. 440-468. [14] A. Mariano, Journal of Physics G: Nuclear and Particle Physics, Vol. 34, 2007, p. 1627. [15] C. Barbero, G. López Castro and A. Mariano, Physics Letters B, Vol. 664, 2008, pp. 70-77.  http://dx.doi.org/10.1016/j.physletb.2008.05.011  [16] A. A. Aguilar-Arevalo, et al., Physical Review D, Vol. 83, 2011, Article ID: 052007.  http://dx.doi.org/10.1103/PhysRevD.83.052007 [17] A. A. Aguilar-Arevalo, et al., Physical Review D, Vol. 81, 2010, Article ID: 013005.  http://dx.doi.org/10.1103/PhysRevD.81.013005  

CC and NC pion production

The disappearance searching experiments vμ→vx use charged current quasielastic (CCQE) reaction to detect an arriving neutrino and reconstruct its energy, while the CC1π+ production can mimic the CCQE signal process. In vμ→ve appearance experiments, the NC1π0 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.Instituto de Física La Plata (IFLP)Facultad de Ciencias Exacta

López-Castro, Gabriel

CC and NC Pion Production

El Servicio de Difusión de la Creación Intelectual

Journal of Modern Physics, 2013, 4, 1-5 
Published Online December 2013 (http://www.scirp.org/journal/jmp) 
http://dx.doi.org/10.4236/jmp.2013.412A1001  
Open Access                                                                                            JMP 
CC and NC Pion Production 
Alejandro Mariano1,2*, Cesar Barbero1,2, Gabriel López-Castro3 
1Departamento de F sica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina  
2Instituto de Fsica La Plata, CONICET, Buenos Aires, Argentina 
3Departamento de F sica, Cinvestav, Mexico City, Mexico 
Email: *mariano@fisica.unlp.edu.ar 
 
Received October 2, 2013; revised November 1, 2013; accepted November 28, 2013 
 
Copyright © 2013 A. Mariano et al. This is an open access article distributed under the Creative Commons Attribution License, 
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor-
dance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual 
property A. Mariano et al. All Copyright © 2013 are guarded by law and by SCIRP as a guardian. 
ABSTRACT 
The disappearance searching experiments x   use charged current quasielastic (CCQE) reaction to detect an ar-
riving neutrino and reconstruct its energy, while the CC1π+ production can mimic the CCQE signal process. In 
e   appearance experiments, the NC1π0 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. 
 
Keywords: Neutrino Scattering; Pion Production; Neutrino Oscillation 
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 oscilla-
tion probability 
  2 2 2,sin 2 sin ,
2
ij i j
i j
m L
P
E
     
and then about the ij  mixing angles and  mass 
squared differences. New high quality data are becoming 
from MiniBoone, MINOS, NOMAD, MinerνA and Sci-
Boone full dedicated to measure cross sections. 
2
,i jm
Disappearance searching experiments x   use 
n p   CCQE reaction to detect an arriving neu- 
trino and reconstruct its energy. E  determination 
could be wrong for a fraction of CC1   background 
events (20%) p p    , that can mimic a CCQE 
one if the pion is absorbed in the target and/or not de- 
tected. In e   appearance experiment, one detects 
e  in an (almost)   beam. Here the signal event 
 is dominated by a NC1π0  
background, and the detector can not distinguish between 
en e p  0πN N  
e  and  if one of both photons from the 0π 0π 
 1232
 
decay escapes. Then a precise knowledge of cross sec- 
tions is a prerequisite in order to make simulations in 
event generators to substract fake 1π events in QE count- 
ings. 
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 inter-
mediate resonances, plus a nonresonant (B) term de-
scribing other processes (where the cross-  contribu-
tion 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 consid-
eration (or not) of the nonresonant terms and its interfer-
ence with the resonant contribution. Nuclear effects and 
FSI have been introduced by several works, where dif-
ferent nuclear models and event generators or simula-
tions codes have been implemented in [5] (GiBUU) and 
[6,7]. *Corresponding author. 
A. MARIANO  ET  AL. 2 
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. 
 π
2. Elementary Amplitude 
For the πN lN   process the total cross section 
reads 
   4
1 2π 2
1 0
d d
2π
1                   d cos d ,
16
lCM CM CM
CM l
CC NC E E
lE E
spin
FE E
E s
M
 E
 
 
 





 
 
/
    (1) 
being 
22
Lab
CM
Lab
N
N N
m E
E
E m m





 is the CM-Lab 
ergy relation,  the merging lepton, l CC NCF /  phase- 
space factors, and 
,B R R .                 (2) 
the amplitude where the contributions for the B ampli-
tude 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 ampli-
tude 
      
    
512
         , ,
F
i l
Vi Ai
G u p i u p
u p u p i B R

 
    
  

 

      (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  
u
 -en- 
 
 
Figure 1. Different contributions to the amplitude. 
Open Access                                                                                            JMP 
A. MARIANO  ET  AL. 3
 
and rescattering of the final  pair, but the effects are 
not so important as in photoproduction. 
πN
2) Vector amplitude should fulfill electromagnetic 
gauge invariance (GI) 3 0iu q u

   . In the B ampli-
tude we must to have same vector FF (   contribution is 
axial and the   one is self-GI), while for R contribu-
tions GI must be fulfilled in presence of finite width ef-
fects. 
3)  3 2R S    should be invariant under contact 
transformations (CT), which fixes the Feynman rules to 
built the amplitude [8], being the corresponding propa-
gator (in the momentum space) 
  
   
3 2
2 2 2
1 2 1 2 1 2
11 12 21
1 2ˆ,
3
3ˆ ˆ ˆ ,
p mG p P p m
p m m
P P P
m


 
          
   
| | 
 
and a general vertex interaction 
 , , , . .,g F       h c          (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 
k
ijP
3 2k  ,1 2  space. Unstableness of  is included 
in the  through a self-energy 
 G p  p  (one 
loop-corrections), which accounts an energy dependent 
width and vertex corrections to get GI. Alternatively, we 
make  2m m i      dressedG G , referred as com-
plex 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 consid-
ered. 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 
       
 
33 * *
†
d 2π
e e ,
s t
s t
s t
N N s t m
m m
ip x ip x
m m s t
x p m E u m m a
b v m m

  
 
  
 p
p
p p
p
m
(5) 
being the single particle spectrum 
     2 2 * *0 0 ,VV B N Vp C m E C E    p p     (6) 
where 
  * 2 *2 *, SN N N S NE m m m C    p p *, .m    (7) 
*
N Nm m  is the effective mass acquired by the nucleon 
[11] through the scalar self-energy . S   SMFT  in-
cludes 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). 
S
RHA  includes the same diagram but the full nucleon 
propagator (which encloses the contribution of the occu-
pied negative-energy states) is used in the evaluation of 
the self-energy. V  and SC  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
11.42 fmFp   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 mat-
ter, from which a momentum distribution can be built as 
   † 30 0 ,  d
4A m m A
An a a kn  k kk k  
 
 
1 2 1 2
1 2 3 4
,h s
p p h
 
1 2 3 4
1 2 1 22
1 2 3 4 1 2 3 42
,
10 0
2!
1 ,
4!
p p h h
p s
p p h h h
p s h s
c p p h h
c p p p p h h h h
 
 
 

  
 
where 
1 2 1 2
1 2 1 2
1 2 1 2
ˆ 0
,p p h h
p p h h
p p h h V
c
E
   
1 2 3 4 1 2 3 4
1 2 1 2 1 2 1 2 1 2 3 4 1 2 3 4
1 2 1 2 1 2 3 4 1 2 3 4
ˆ ˆ0
.
p p p p h h h h
p p h h p p p p h h h h
c
V p p h h p p h h V p p p p h h h h
E E

 
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 sim-
plest version [12], that is *   , where 
     , d* e e , ,optzi v Vi   .     b z zp rr r b z     (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-en- 
ergy. The dashed lines indicate the propagator of the scalar 
(S) or vector meson (V) that interacts with a nucleon n (full 
lines). 
Open Access                                                                                            JMP 
A. MARIANO  ET  AL. 4 
Assuming a mean distance of trip for  in nucleus π d  , constant nucleon density and the -h model for 
the 

 -optical potential, we get 
   * e e i s di     pp rr ,  
     2 2 *02 1 2 ,9 N Ns f m m Ts s m     *   
22 1 3
02 3 ,  , ,d R r R r A r cA    1 3  
where ,A R  and , T N Z A  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 hy-
potesis and a fitting to the differential cross section  
π πp 
πp 
p
n
0πp p 

2
d
dQ

 for the  (ANL) data [15]. The  πp p  
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 ( 1) πA A p     process where 
12 2A C H   (mineral oil detector) and compare with 
the corresponding data of the MoniBooNE [16,17] ex-
periment. Results corresponding to the   1 0πA A N
0π
50%
  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  
30% 0π
 
 
Figure 3. CC 1π+ calculation compared with MiniBooNE 
data [16]. 
 
Figure 4. Idem for CC 1π0. 
 
 
Figure 5. Idem for NC 1π0. Data form [17]. 
 
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 GeVE 
1 1
95%exp exp
ACC NCC
A     , 
0 01 1
83%exp exp
ACC NCC
A    , 0 01 1 92%exp expANC NNCA    . 
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. 
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Open Access                                                                                            JMP 
A. MARIANO  ET  AL. 
Open Access                                                                                            JMP 
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Alejandro Mariano

Cesar Barbero

Gabriel López-Castro

Crossref

CONICET Digital

Journal of Modern Physics, 2013, 4, 1-5 Published Online December 2013 (http://www.scirp.org/journal/jmp) http://dx.doi.org/10.4236/jmp.2013.412A1001  Open Access                                                                                            JMP CC and NC Pion Production Alejandro Mariano1,2*, Cesar Barbero1,2, Gabriel López-Castro3 1Departamento de F sica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina  2Instituto de Fsica La Plata, CONICET, Buenos Aires, Argentina 3Departamento de F sica, Cinvestav, Mexico City, Mexico Email: *mariano@fisica.unlp.edu.ar  Received October 2, 2013; revised November 1, 2013; accepted November 28, 2013  Copyright © 2013 A. Mariano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor-dance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property A. Mariano et al. All Copyright © 2013 are guarded by law and by SCIRP as a guardian. ABSTRACT The disappearance searching experiments x   use charged current quasielastic (CCQE) reaction to detect an ar-riving neutrino and reconstruct its energy, while the CC1π+ production can mimic the CCQE signal process. In e   appearance experiments, the NC1π0 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.  Keywords: Neutrino Scattering; Pion Production; Neutrino Oscillation 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 oscilla-tion probability   2 2 2,sin 2 sin ,2ij i ji jm LPE     and then about the ij  mixing angles and  mass squared differences. New high quality data are becoming from MiniBoone, MINOS, NOMAD, MinerνA and Sci-Boone full dedicated to measure cross sections. 2,i jmDisappearance searching experiments x   use n p   CCQE reaction to detect an arriving neu- trino and reconstruct its energy. E  determination could be wrong for a fraction of CC1   background events (20%) p p    , that can mimic a CCQE one if the pion is absorbed in the target and/or not de- tected. In e   appearance experiment, one detects e  in an (almost)   beam. Here the signal event  is dominated by a NC1π0  background, and the detector can not distinguish between en e p  0πN N  e  and  if one of both photons from the 0π 0π  1232 decay escapes. Then a precise knowledge of cross sec- tions is a prerequisite in order to make simulations in event generators to substract fake 1π events in QE count- ings. 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 inter-mediate resonances, plus a nonresonant (B) term de-scribing other processes (where the cross-  contribu-tion 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 πNπN  resonance and from the consid-eration (or not) of the nonresonant terms and its interfer-ence with the resonant contribution. Nuclear effects and FSI have been introduced by several works, where dif-ferent nuclear models and event generators or simula-tions codes have been implemented in [5] (GiBUU) and [6,7]. *Corresponding author. A. MARIANO  ET  AL. 2 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.  π2. Elementary Amplitude For the πN lN   process the total cross section reads    41 2π 21 0d d2π1                   d cos d ,16lCM CM CMCM lCC NC E ElE EspinFE EE sM E     /    (1) being 22LabCMLabNN Nm EEE m m is the CM-Lab ergy relation,  the merging lepton, l CC NCF /  phase- space factors, and ,B R R .                 (2) the amplitude where the contributions for the B ampli-tude 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 ampli-tude           512         , ,Fi lVi AiG u p i u pu p u p i B R              (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  u -en-   Figure 1. Different contributions to the amplitude. Open Access                                                                                            JMP A. MARIANO  ET  AL. 3 and rescattering of the final  pair, but the effects are not so important as in photoproduction. πN2) Vector amplitude should fulfill electromagnetic gauge invariance (GI) 3 0iu q u   . In the B ampli-tude we must to have same vector FF (   contribution is axial and the   one is self-GI), while for R contribu-tions GI must be fulfilled in presence of finite width ef-fects. 3)  3 2R S    should be invariant under contact transformations (CT), which fixes the Feynman rules to built the amplitude [8], being the corresponding propa-gator (in the momentum space)      3 22 2 21 2 1 2 1 211 12 211 2ˆ,33ˆ ˆ ˆ ,p mG p P p mp m mP P Pm              | |  and a general vertex interaction  , , , . .,g F       h c          (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 kijP3 2k  ,1 2  space. Unstableness of  is included in the  through a self-energy  G p  p  (one loop-corrections), which accounts an energy dependent width and vertex corrections to get GI. Alternatively, we make  2m m i      dressedG G , referred as com-plex 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 consid-ered. 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         33 * *†d 2πe e ,s ts ts tN N s t mm mip x ip xm m s tx p m E u m m ab v m m      ppp ppm(5) being the single particle spectrum      2 2 * *0 0 ,VV B N Vp C m E C E    p p     (6) where   * 2 *2 *, SN N N S NE m m m C    p p *, .m    (7) *N Nm m  is the effective mass acquired by the nucleon [11] through the scalar self-energy . S   SMFT  in-cludes 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). SRHA  includes the same diagram but the full nucleon propagator (which encloses the contribution of the occu-pied negative-energy states) is used in the evaluation of the self-energy. V  and SC  are fixed to reproduce the experimental binding energy per nucleon of 16 MeV at the Fermi momentum  for the normal nuclear matter. For the C11.42 fmFp   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 mat-ter, from which a momentum distribution can be built as    † 30 0 ,  d4A m m AAn a a kn  k kk k    1 2 1 21 2 3 4,h sp p h 1 2 3 41 2 1 221 2 3 4 1 2 3 42,10 02!1 ,4!p p h hp sp p h h hp s h sc p p h hc p p p p h h h h      where 1 2 1 21 2 1 21 2 1 2ˆ 0,p p h hp p h hp p h h VcE   1 2 3 4 1 2 3 41 2 1 2 1 2 1 2 1 2 3 4 1 2 3 41 2 1 2 1 2 3 4 1 2 3 4ˆ ˆ0.p p p p h h h hp p h h p p p p h h h hcV p p h h p p h h V p p p p h h h hE E 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 sim-plest version [12], that is *   , where      , d* e e , ,optzi v Vi   .     b z zp rr r b z     (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-en- ergy. The dashed lines indicate the propagator of the scalar (S) or vector meson (V) that interacts with a nucleon n (full lines). Open Access                                                                                            JMP A. MARIANO  ET  AL. 4 Assuming a mean distance of trip for  in nucleus π d  , constant nucleon density and the -h model for the  -optical potential, we get    * e e i s di     pp rr ,       2 2 *02 1 2 ,9 N Ns f m m Ts s m     *   22 1 302 3 ,  , ,d R r R r A r cA    1 3  where ,A R  and , T N Z A  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 hy-potesis and a fitting to the differential cross section  π πp πp pn0πp p 2ddQ for the  (ANL) data [15]. The  πp p  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 ( 1) πA A p     process where 12 2A C H   (mineral oil detector) and compare with the corresponding data of the MoniBooNE [16,17] ex-periment. Results corresponding to the   1 0πA A N0π50%  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  30% 0π  Figure 3. CC 1π+ calculation compared with MiniBooNE data [16].  Figure 4. Idem for CC 1π0.   Figure 5. Idem for NC 1π0. Data form [17].  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 GeVE 1 195%exp expACC NCCA     , 0 01 183%exp expACC NCCA    , 0 01 1 92%exp expANC NNCA    . 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. REFERENCES [1] G. L. Fogli and G. Nardulli, Nuclear Physics B, Vol. 160, 1979, pp. 116-150.  http://dx.doi.org/10.1016/0550-3213(79)90233-5  [2] T. Sato, D. Uno and T.-S. H. Lee, Physical Review C, Vol. 67, 2003, Article ID: 065201.  Open Access                                                                                            JMP A. MARIANO  ET  AL. Open Access                                                                                            JMP 5http://dx.doi.org/10.1103/PhysRevC.67.065201 [3] E. Hernandez, J. Nieves and M. Valverde, Physical Re-view D, Vol. 76, 2007, Article ID: 033005.  http://dx.doi.org/10.1103/PhysRevD.76.033005 [4] O. Lalakulich, T. Leitner, O. Buss and U. Mosel, Physical Review D, Vol. 82, 2010, Article ID: 093001.  http://dx.doi.org/10.1103/PhysRevD.82.093001 [5] O. Lalakulich and U. Mosel, Pion Production in the MiniBooNE Experiment, Physical Review C, Vol. 87, 2013, Article ID: 014602.  http://dx.doi.org/10.1103/PhysRevC.87.014602 [6] J. T. Sobczyk and J. Zmuda, Physical Review C, Vol. 87, 2013, Article ID: 065503.  http://dx.doi.org/10.1103/PhysRevC.87.065503 [7] E. Hernandez, J. Nieves and M. J. Vacas, Physical Re-view D, 87, 2013, Article ID: 113009.  http://dx.doi.org/10.1103/PhysRevD.87.113009 [8] M. El-Amiri, G. López Castro and J. Pestieau, Nuclear Physics A, Vol. 543, 1992, pp. 673-684.  http://dx.doi.org/10.1016/0375-9474(92)90553-V  [9] F. de Jong and R. Malfliet, Physical Review C, Vol. 46, 1992, pp. 2567-2581.  http://dx.doi.org/10.1103/PhysRevC.46.2567  [10] C. Barbero, A. Mariano and G. López Castro, Journal of Physics G: Nuclear and Particle Physics, Vol. 39, 2012, Article ID: 085011.  http://dx.doi.org/10.1088/0954-3899/39/8/085011 [11] J. D. Walecka, “Theroretical Nuclear and Subnuclear Physics,” Part I, Oxford University Press, New York, 1995. [12] C. Barbero, A. Mariano and S. B. Duarte, Physical Re-view C, Vol. 82, 2010, Article ID: 067305.  http://dx.doi.org/10.1103/PhysRevC.82.067305 [13] G. López Castro and A. Mariano, Nuclear Physics A, Vol. 697, 2001, pp. 440-468. [14] A. Mariano, Journal of Physics G: Nuclear and Particle Physics, Vol. 34, 2007, p. 1627. [15] C. Barbero, G. López Castro and A. Mariano, Physics Letters B, Vol. 664, 2008, pp. 70-77.  http://dx.doi.org/10.1016/j.physletb.2008.05.011  [16] A. A. Aguilar-Arevalo, et al., Physical Review D, Vol. 83, 2011, Article ID: 052007.  http://dx.doi.org/10.1103/PhysRevD.83.052007 [17] A. A. Aguilar-Arevalo, et al., Physical Review D, Vol. 81, 2010, Article ID: 013005.  http://dx.doi.org/10.1103/PhysRevD.81.013005  

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CC and NC pion production

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