Coherent Pions From Neutrino Scattering Off Nuclei
Abstract
We describe a model for pion production off nucleons and coherent pions from nuclei induced by neutrinos in the energy regime. Besides the dominant pole contribution, it takes into account the effect of background terms required by chiral symmetry. Moreover, the model uses a reduced nucleonto resonance axial coupling, which leads to coherent pion production cross sections around a factor two smaller than most of the previous theoretical estimates. Nuclear effects like medium corrections on the propagator and final pion distortion are included.
I Introduction
Neutrinos have been in the forefront of research in particle and nuclear physics for a long time. One of these fields is the study of pion production off nuclei induced by neutrinos. A proper understanding of this process is necessary in the analysis of the present generation of precision neutrino oscillation experiments. For instance, the produced in neutral currents (NC) is the most important induced background to experiments like MiniBooneAguilarArevalo:2007it () that are trying to measure oscillations in the neutrino energy range around . Also of importance is the background that appears from charged current (CC) production in disappearance searches like T2KHiraide:2006zq (). Moreover the pion is strongly coupled to the resonance, and neutrino scattering is presently the best way to access to the axial nucleon transition couplings. The most complete information in this regard comes from the bubble chamber data of ANLBarish:1978pj (); Radecky:1981fn () and BNLKitagaki:1986ct (); Kitagaki:1990vs () where the target was cooled deuterium. However, in present oscillation experiments the target for neutrino interaction is finite nuclei, for instance C (mineral oil in MiniBoone) or O (water target in T2K). This introduces sizable manybody effects that are difficult to disentangle from the genuine single nucleon response to the neutrino probe.
In Sec. II of these paper we describe a phenomenological modelHernandez:2007qq () for pion production induced by neutrino scattering off free nucleons. This model takes into account nonresonant background processes that are usually neglected. These nonresonant processes are determined by chiral symmetry and thus do not introduce free parameters. We then perform a fit of the axial parameters and discuss a possible violation of the nondiagonal GoldbergerTreiman relation. In Sec. III we extend the model to describe the CC coherent process in which the final nucleus is left in its ground state. We will further try to discuss how the coherent reaction can show some light on the values of the axial couplings. We shall focus on the CC process, but the model can be easily extended to antineutrino reactions and NC processes. The interested reader could find further details in Refs. Hernandez:2007qq (); Hernandez:2007ej (); Amaro:2008hd (); Hernandez:2009vm (); Hernandez:2010bx ()
Ii Single Nucleon Pion Production
Here we review the model for the free nucleon reaction
(1) 
as introduced in Ref. Hernandez:2007qq (). This model considers the dominant pole mechanism in which the neutrino excites a resonance that subsequently decays into . In our model we have also included nonresonant background terms as required by chiral symmetry, see Fig. 1. Some previous worksFogli:1979cz (); Fogli:1979qj (); Sato:2003rq () also considered background terms, though they were not consistent with the chiral counting.
The vector part of the interaction can be related to the electromagnetic current by imposing conservation of the vector current. For photon induced reactions extensive experimental data exist and in Ref. Lalakulich:2006sw () they were employed to fit the vector current couplings. We shall use this fit in our work. Unfortunately the axial current is not so well studied. The usual approach is to parameterize the interaction in terms of four form factors . One can assume partial conservation of the axial current (PCAC) and obtain the relation , with the nucleon mass. Furthermore, one can deduceAdler:1964yx () from dispersion relations the following conditions: and . Thus we are left with only a free form factor, the dominant one . A different number of parameterizations have been proposed for this form factor, nevertheless the experimental data are quite limited and thus a simple dipole form
(2) 
should be enough. In order to keep the axial transition radius in the range one expect the axial mass to have a value of around . Furthermore one can assume the well known GoldbergerTreiman relation (GTR) for coupling to be also valid for the coupling and thus obtain
(3) 
where is the pion decay constant and , the coupling. Unfortunately, there are no constraints from Chiral Perturbation TheoryGeng:2008bm () and the lattice QCD calculationsAlexandrou:2006mc () are still inconclusive.
Most of the approaches in the literature assume dominance, that is, only include the first two diagrams in Fig. 1. We improve this situation by including nonresonant contributionsHernandez:2007qq () required by chiral symmetry. In addition to the pole () (first row) mechanism the model includes background terms required by chiral symmetry: nucleon (second row) pole terms contact and pion pole contribution (third row) and pioninflight term. We calculate them by using the SU(2) nonlinear model Lagrangian.
The only parameters in the theory are the pion and nucleon masses and the pion decay constant. All other couplings are completely fixed by the theory, so no new parameter is introduced.
We found (see Fig. 2) that these background terms produced significant effects in all channels, namely an enhancement of about % in the cross section that resulted in a disagreement with the ANL data.
As a result we had to readjust the strength of the dominant pole contribution. The least known ingredients of the model are the axial nucleonto transition form factors, of which gives the largest contribution. This strongly suggested a refit of that form factor to the experimental data, which we did by fitting the fluxaveraged ANL differential cross section for pionnucleon invariant masses^{1}^{1}1This cut was introduced in order to avoid the effects of resonances higher than the . . The obtained parameters were
(4) 
with a and a correlation coefficient , that amounts to a % reduction of the GTR prediction. Thus, our full model leads to an overall better description of the data for onepion production reactions off the nucleon. This reduction of the value is consistent with recent results in lattice QCDAlexandrou:2006mc () and quark modelsBarquillaCano:2007yk ().
Recently other fits have been proposed. For instance, in Ref. Leitner:2008ue () they keep the GTR but introduce a nondipole form factor with additional parameters. As in the ANL data the relevant phase space is around , they could keep the GTR at the cost of having a large dependence on for the form factor, that yields a large, somehow unphysical, axial transition radius of around . Furthermore neither statistical errors nor correlation factors are given in that reference. Another analysisGraczyk:2009qm () raised new questions, namely the effect of deuterium wave function on the cross section and the flux uncertainties in the ANL and BNL data. The authors of this latter work took into account both effects, though we believe that their statistical analysis is not quite robust (see discussion in Ref. Hernandez:2010bx ()). Furthermore they only took into account the dominant contribution. The inclusion of deuterium wave function reduces the cross section about an %, so this somehow compensates the neglect of the nonresonant background, and they obtained a best fit of in agreement with the GTR assumption.
RecentlyHernandez:2010bx () we have improved our fit of Ref. Hernandez:2007qq (), improving the lines suggested in Graczyk et al. In summary in this new fit

all diagrams in Fig. 1 are included;

the fitted data are the full ANL data set and the BNL total cross sections at the three lowest neutrino energies (we neglect higher energies where the effect of higher resonances beyond the must be addressed; the BNL differential cross sections were not taken into account as they are not normalized);

deuterium wave function effects were introduced following the prescription of Ref. AlvarezRuso:1998hi ();

the form factors and were tentatively included in one fit, though the data were found to be quite insensitive to their values so we decided to stick to the Adler’s assumption; and

the uncertainties in ANL and BNL flux normalization are introduced as fully correlated systematic errors.
In this way we obtain a best fit of and with a goodness of fit value of . Thus we observe a violations of the off diagonal GoldbergerTreiman relation at the level of .
Iii The Coherent Reaction
Here we describe our modelAmaro:2008hd () for the coherent reaction
(5) 
where the target nucleus is left in the ground state (). To calculate the amplitude of this process we sum over all individual nucleon wave functions, which are modelled by a Fermi gas in local density approximation. The individual nucleon amplitudes are modelled following the model of the previous section, using the fit of Eq. 4 for the form factor. On top of that a number of manybody effects are introduced. In first place we take into account the inmedium modificationsOset:1987re () of the properties. This implies a shift in the pole mass towards lower energies and most importantly a net broadening of the width (the opening of new decay channels in the nuclear medium compensates the Pauli blocking of the decay channel). Also important is the distortion of the outgoing pion by strong interaction with the nucleus. Thus we consider the wave function of the pion to be the outgoing solution to the KleinGordon equation with a microscopic optical potentialNieves:1991ye () whose imaginary part takes into account the inelastic interactions of the pion with the nucleus, that thus disappear from the coherent channel. We must emphasize here that solving the KleinGordon equation is the correct way of describing the distortion of the outgoing pion. Other approaches use either a Monte Carlo simulationAguilarArevalo:2008xs () or include an attenuation factor fitted to the pion nuclei scattering cross sectionNakamura:2009iq (). The first procedure, though physically sound, can be a bit misleading as it includes in its cross section processes (like quasielastic scattered pions) that do not leave the nucleus in it ground state, thus are not coherent. This kind of models are used in the analysis of MiniBoone experiment, thus making a bit messy the direct comparison between theoretical models and experimental results (see discussion in Ref. Amaro:2008hd ()). The second approach is an oversimplification, as the pionnucleus scattering is quite a different process from the neutrino induced pion production. As pion nucleus interaction is governed by the strong interaction, the incoming pion interacts strongly with the nuclear surface, thus the pion nucleus cross section is quite insensitive to the details of the nuclear core. On the other hand, neutrino scattering is a weak process, dominated by nuclear density, so pions are mostly produced in the deep, high density regions of the nucleus. The physics of pion interaction is thus quite different in pion scattering off nuclei and pion production by electroweak probes.
In left panel of Fig. 3 we show the pion momentum distribution for CC coherent pion production, in the peak energy region of the T2K experiment.
Including inmedium selfenergy (longdashed line) reduces the PWIA results (shortdashed line). Further inclusion of pion distortion (full model, solid line) reduces the cross section, and the peak is shifted towards lower energies, reflecting the strong absorption and the higher probability of a quasielastic collisions of the pion in the kinematical region. The total cross section reduction is around . Similar nuclear effects were already studied in Refs. AlvarezRuso:2007tt (); AlvarezRuso:2007it (). However, the authors of these references neglected the nucleon momenta in the Dirac spinors. The effect of this approximation (nucleons at rest, dotted line) results in a % decrease of the total cross section. In the right panel of Fig. 3 we show the pion angular distribution with respect to the incoming neutrino direction. The reaction is very forward peaked, as expected due to the nucleus form factor. The angular distribution profile keeps its forward peaked behavior after introduction of nuclear medium effects. Furthermore we corrected some numerical errors in the mentioned papers. However one must be aware that this model does not take into account the nonlocalitiesLeitner:2009ph () in the propagation. We believe this effect is partially taken into account in an effective fashion by our treatment of the in nuclear medium; nevertheless further studies would be interesting.
Reaction  Experiment  Exp  

CC C  K2K  
NC C  MiniBoone  
CC C  MiniBoone  
CC O  T2K  
CC C  T2K 
In Table 1 we compare our model with present results of K2K experimentHasegawa:2005td () and show our predicitions for MiniBoone and T2K. Our prediction, subject to sizable uncertainties, lies well below the K2K upper bound, due to the use of a low value for , while our prediction for the NC MiniBoone cross section is notably smaller than that given in the PhD thesis of J. L. RaafRaaf:2005up (). However, we believe (see discussion in Refs. Amaro:2008hd (); Hernandez:2009vm () that the MiniBoone analysis might importantly overestimate this cross section, not only because some of the s which undergo FSI collisions are accounted for instead of being removed, but also because a possible mismatch between the absolute normalisation of the background and coherent yields. Note that the K2K and MiniBoone results seems somehow incompatible with the approximate relation , which would be expected from dominance and neglecting finite muon mass effects.
Acknowledgments
M. V. acknowledges a Postdoctoral Fellowship from the Japanese Society for the Promotion of Science (JSPS). Research supported by DGI contracts FIS200801143, FIS200603438, FPA200765748 and CSD200700042, JCyL contracts SA016A07 and GR12, Generalitat Valenciana contract PROMETEO/2009/0090 and by EU HadronPhysics2 contract 227431.
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