The quantitative description of the effects of nuclear dynamics on the measured neutrino-nucleus cross sections—needed to reduce the systematic uncertainty of long baseline neutrino oscillation experiments—involves severe difficulties. Owing to the uncertainty on the incoming neutrino energy, different reaction mechanisms contribute to the cross section measured at fixed energy and scattering angle of the outgoing lepton, and must therefore be consistently taken into account within a unified model. We research the theoretical approach based on the impulse approximation and the use of realistic nucleon spectral functions, allowing one to describe a variety of reaction mechanisms active in the broad kinematical range covered by neutrino experiments. The extension of this scheme to include more complex mechanisms involving the two-nucleon currents, which are believed to be important, is also outlined. The impact of nuclear effects on the determination of neutrino oscillation parameters is illustrated by analyzing the problem of neutrino energy reconstruction.
Experimental searches of neutrino oscillations exploit neutrino-nucleus interactions to infer the properties of the beam particles, which are largely unknown. The use of nuclear targets as detectors, while allowing for a substantial increase of the event rate, entails nontrivial problems, as the interpretation of the observed signal requires a quantitative understanding of neutrino-nucleus interactions. Given the present experimental accuracy, the treatment of nuclear effects is in fact regarded as one of the main sources of systematic uncertainty (see, e.g., [
Over the past decade, a growing effort has been made, aimed at making use of the knowledge of the nuclear response acquired from experimental and theoretical studies of electron scattering. Electron-nucleus scattering cross-sections are usually analyzed at fixed beam energy
Schematic representation of the inclusive electron-nucleus cross-section at beam energy around 1 GeV, as a function of energy loss.
It is apparent that the different reaction mechanisms, yielding the dominant contributions to the cross-section at different values of
The bump centered at
The available theoretical models of electron-nucleus scattering provide an overall satisfactory description of the data over a broad kinematical range. In particular, in the region in which quasielastic scattering dominates, the data is generally reproduced with an accuracy of few percentages (for a recent review on electron-nucleus scattering in the quasielastic sector, see [
Because neutrino beams are always produced as secondary decay products, their energy is not sharply defined, but broadly distributed. As a consequence, in charged-current neutrino scattering processes detecting the energy of the outgoing lepton,
(a) Inclusive electron-carbon cross-sections at
The above discussion implies that the understanding of the flux averaged neutrino cross-section requires the development of theoretical models providing a consistent treatment of all reaction mechanisms active in the broad kinematical range corresponding to the relevant neutrino energies.
In Section
Let us consider, for definiteness, charged-current neutrino-nucleus interactions. The formalism discussed in this section can be readily generalized to the case of neutral current interactions [
In the kinematical region corresponding to low momentum transfer, typically
Nonrelativistic nuclear many-body theory, based on dynamical models strongly constrained by phenomenology, provides a fully consistent theoretical approach allowing for an accurate description of the target initial state, independent of momentum transfer. On the other hand, at large
The Impulse Approximation (IA) scheme, extensively employed to analyze electron-nucleus scattering data [
Within the IA picture, the nuclear current of (
Using the above relations, the hadronic tensor can be rewritten in the following form:
Equation (
Using the definition of the tensor describing the interactions of the
It has to be emphasized that the replacement of
Collecting the above results, the nuclear cross-section can be finally written in the following transparent form:
In conclusion, within the IA scheme it is possible to trace back the hadronic tensor corresponding to the nuclear target to the ones describing the elementary interaction with isolated nucleons—which can be, at least in principle, measured using proton and deuteron targets—provided that the four momentum transfer
We emphasize that, as will be discussed below, (
The calculation of the target spectral function requires a model of nuclear dynamics. The simulation codes employed for the analysis of neutrino oscillation experiments are largely based on the relativistic Fermi gas model (RFGM) [
Within the RFGM, the nuclear spectral function, defined in (
Electron scattering data have provided overwhelming evidence that the energy-momentum distribution of nucleons in the nucleus is quite different from the one predicted by the RFGM. The differences are to be ascribed to the presence of nucleon-nucleon (NN) correlations, mainly arising from the strongly repulsive nature of the NN interactions at short distances. Dynamical correlations give rise to virtual scattering processes leading to the excitation of the participating nucleons to states of energy larger than the Fermi energy, thus depleting the single particle levels within the Fermi sea. Owing to the contribution of nucleons belonging to a correlated pair, the nuclear spectral function
Highly accurate theoretical calculations of the spectral function can be carried out for uniform nuclear matter, exploiting the simplifications arising from translation invariance [
According to the LDA scheme, the spectral function is written in the following form:
The correlation contribution is given by
The most general expression of the target tensor of (
The contraction of the above tensor with
As already stated, the formalism based on the IA provides a unified framework, suitable to describe neutrino-nucleus interaction in different kinematical regimes. In this Section, we discuss the form of the structure functions
In the CCQE channel, the structure functions involve the energy conserving
The form factors appearing in the vector current,
While more refined parameterizations of the large body of data are available (for a review, see, e.g., [
The generalization of the above formalism to describe the resonance production region only involves minor changes. Unlike the CCQE case, the structure functions depend on both
The CCQE and resonance contributions to the total neutrino-nucleon cross-section reported by the authors of [
QE (solid line) and resonance production (dashed line) contributions to the charged-current neutrino-nucleon scattering cross-section (adapted from [
The decay of the
From the observational point of view, the DIS regime corresponds to hadronic final states with more than one pion.
In principle, the three nucleon structure functions entering the definition of the IA nuclear cross-section, (
An alternative approach, allowing one to obtain the structure functions describing DIS on isolated nucleons, can be developed within the framework of the quark-parton model, exploiting the large database of DIS data collected using charged lepton beams and hydrogen and deuteron targets (see, e.g., [
In addition, the following relation:
Using the above results and the relation
The above procedure rests on the tenet, underlying the IA scheme, that the elementary neutrino-nucleon interaction is
The approach of [
The data set of CCQE events collected by the MiniBooNE collaboration [
As pointed out in the previous Section, the CCQE neutrino-nucleon process is described in terms of three form factors. The proton and neutron electromagnetic form factors, which have been precisely measured up to large values of
It would be tempting to interpret the value of
(a) Inclusive electron-carbon cross-section at beam energy
The authors of [
In MiniBooNE data analysis, an event is labeled as CCQE if no final state pions are detected in addition to the outgoing muon. The simplest reaction mechanism compatible with this definition is single nucleon knockout, induced by the one-nucleon contributions to the nuclear current (see (
It has been suggested that the observed excess of CCQE cross-section may be traced back to the occurrence of events with two particle-two hole final states [
It has to be pointed out, however, that the approaches of [
A fully consistent analysis of the role of two particle-two hole final states within a realistic model of nuclear structure obviously requires that all mechanisms leading to the appearance of these final states be included, using a quantum-mechanical approach that is properly taking into account the interference between the transition amplitudes involving one- and two-nucleon currents. Within the approach of [
Within the IA, ISC are taken into account using realistic spectral functions, which include the contribution of the continuum spectrum associated with unbound states of the residual nucleus. Their main effect is the appearance of a tail of the cross-section, extending to large
In inclusive processes, FSI lead to a shift of the energy loss spectrum, arising from interactions between the knocked out nucleon and the mean field of the recoiling nucleus, and a redistribution of the strength from the quasifree bump to the tails, resulting from rescattering processes. Theoretical studies of electron-nucleus scattering suggest that in the kinematical region relevant to the MiniBooNE analysis, the former mechanism, which does not involve the appearance of two particle-two hole final states, dominates. A recent discussion of the inclusion of FSI within the IA scheme can be found in [
As advocated in [
The role of the two nucleon current in electron scattering is best illustrated by comparing the longitudinal and transverse
Longitudinal (L) and transverse (T) scaling functions of Carbon at
Figure
The results of highly accurate calculations carried out for light nuclei in the nonrelativistic regime strongly suggest that in the quasielastic region, single nucleon knockout processes are dominant in the longitudinal channel, while both one- and two-nucleon mechanisms provide comparable contributions in the transverse channel [
The authors of [
Comparison between the flux averaged muon energy spectrum at muon scattering angle
Figure
Comparison between the flux averaged muon energy spectra measured by the MiniBooNE collaboration at muon scattering angle
As pointed out above, a fully consistent treatment of processes involving two particle-two hole final states requires a realistic model of nuclear structure, taking into account the effects of NN correlations. Models including MEC within the framework of the IPM, such as those of [
Going beyond this scheme in the kinematical region in which nonrelativistic approximations are not applicable requires an extension of the
The starting point is the generalization of the ansatz of (
It follows that the matrix element of the two nucleon current simplifies to (compare to (
The connection with the spectral function formalism discussed in Section
The two-nucleon spectral function of uniform and isospin symmetric nuclear matter at equilibrium density has been calculated within nuclear many-body theory using a realistic hamiltonian [
Comparison between the two-nucleon relative momentum distribution of nuclear matter computed within nuclear many-body theory using a realistic hamiltonian (solid line) [
Using the
In recent years, nuclear effects in neutrino interactions, while being interesting in their own right, have been mainly studied to appraise their impact on the determination of neutrino oscillations. As an example, in this section, we will discuss the uncertainty on neutrino energy reconstruction arising from the nuclear models employed in data analysis.
Let us consider, for simplicity, two-flavor mixing. The expression of the probability that a neutrino oscillates from flavor
The starting point for neutrino energy reconstruction in
From (
In the analysis of MiniBooNE data [
In general, the neutrino energy distribution,
To gauge the effect of the high momentum and high removal energy tails of the spectral functions obtained within realistic dynamical approaches including NN correlations, the authors of [
(a) Neutrino energy distribution at
The distributions predicted by the RFGM model are more sharply peaked at the neutrino energy given by (
Note that the histograms of Figure
In order to assess the total impact of replacing the RFGM with the approach of [
The differences between the results of the approach obtained from nuclear many-body theory and those of the RFG model turn out to be sizable. The overall shift towards high energies and the tails at large
The reconstruction of neutrino energy in CCQE-like processes is more complex, as the four-momentum transfer is shared between two nucleons. As a consequence, it requires the knowledge of the two-nucleon spectral function of (
The uncertainty associated with the reconstruction of
The results obtained using the MiniBooNE flux and setting
(a) Neutrino energy distributions at
The impact of reaction mechanisms other than single nucleon knock out on neutrino energy reconstruction has been also analyzed in [
The right panels of Figure
Over the past few years, the availability of the double-differential CCQE cross-section measured by the MiniBooNE collaboration and the results of a new generation of theoretical studies have led to a better understanding of neutrino-nucleus interactions in a broad kinematical range, as well as to the identification of a number of outstanding unresolved issues.
In view of the fact that no convincing evidence of medium modifications of the nucleon electromagnetic form factors has yet emerged, the excess of CCQE events in carbon reported by the MiniBooNE collaboration [
The authors of [
The models developed in [
The main problem associated with a fully consistent description of nuclear structure and dynamics in the kinematical region relevant to neutrino experiments such as MiniBooNE and Miner
The dependence of the contribution of processes involving the two-nucleon current on the kinematical conditions should also be carefully investigated. It has been suggested that this analysis may help to shed light on the source of the large disagreement between the values of the nucleon axial mass reported by the MiniBooNE and NOMAD collaborations [
The systematic study of the impact of nuclear effects on the determination of neutrino oscillation parameters is still in its infancy [
The authors declare that some of the results discussed in this paper have been obtained in collaboration with Artur M. Ankowski and Davide Meloni. The authors are also indebted to Camillo Mariani and Makoto Sakuda for many illuminating discussions.