Progress and open questions in the physics of neutrino cross sections

New and more precise measurements of neutrino cross sections have renewed the interest in a better understanding of electroweak interactions on nucleons and nuclei. This effort is crucial to achieve the precision goals of the neutrino oscillation program, making new discoveries, like the CP violation in the leptonic sector, possible. We review the recent progress in the physics of neutrino cross sections, putting emphasis on the open questions that arise in the comparison with new experimental data. Following an overview of recent neutrino experiments and future plans, we present some details about the theoretical development in the description of (anti)neutrino-induced quasielastic scattering and the role of multi-nucleon quasielastic-like mechanisms. We cover not only pion production in nucleons and nuclei but also other inelastic channels including strangeness production and photon emission. Coherent reaction channels on nuclear targets are also discussed. Finally, we briefly describe some of the Monte Carlo event generators, which are at the core of all neutrino oscillation and cross section measurements.

Recent years have witnessed an intense experimental and theoretical activity aimed at a better understanding of neutrino interactions with nucleons and nuclei. Although this activity has been stimulated mostly by the needs of neutrino oscillation experiments in their quest for a precise determination of neutrino properties, the relevance of neutrino interactions with matter extends over a large variety of topics in astrophysics, physics beyond the Standard Model, hadronic and nuclear physics.
Oscillation experiments: At present, the main motivation for neutrino cross section studies comes from oscillation experiments. They aim at a precise determination of mass-squared differences and mixing angles in ν µ disappearance and ν e appearance measurements. The ability to reconstruct the neutrino energy is crucial for this program. Indeed, oscillation probabilities, such as P (ν µ → ν τ ) = sin 2 2θ 23 sin 2 ∆m 2 32 L 4E ν (1) depends on the neutrino energy E ν , which is not known for broad fluxes. A reliable determination of the neutrino energies in nuclear targets requires a good understanding of the reaction mechanisms and a precise simulation of final state interactions. There are also irreducible backgrounds, for example from neutral current π 0 or γ production when these particle produced showers that are misidentified as electrons from ν e n → e − p. Astrophysics: Neutrinos play an important role in astrophysical phenomena and carry information about the emitting sources. In particular, the dynamics of core-collapse supernovae is controlled by neutrino interactions. The neutron rich environment of supernovae is a candidate site for r-process nucleosynthesis because radiated neutrinos convert neutrons into protons. To address these questions a good knowledge of low energy neutrino production and detection cross sections is required [1,2].
Physics beyond the Standard Model: Non-standard neutrino interactions leading, for example, to deviations from universality in the weak couplings or flavor violation in neutral current processes could affect neutrino production, propagation, and detection processes as subleading effects (see Ref. [3] for a recent review). Long and short baseline experiments allow to set bounds on these interactions.
Hadronic physics: Neutrino cross section measurements allow to investigate the axial structure of the nucleon and baryon resonances, enlarging our views of hadron structure beyond what is presently known from experiments with hadronic and electromagnetic probes, not forgetting about lattice QCD. Another fundamental and open question is the strangeness content of the nucleon spin which can be best unraveled in ν p(n) → ν p(n) studies.
Nuclear physics: Modern neutrino experiments are performed with nuclear targets. For nuclear physics this represents a challenge and an opportunity. A challenge because the precise knowledge of neutrino and baryon properties can only be achieved if nuclear effects are under control. An opportunity because neutrino cross sections incorporate richer information than electron-scattering ones, providing an excellent testing ground for nuclear structure, many-body mechanisms and reaction models.
We discuss the progress in the physics of neutrino interactions with nucleons and nuclei at intermediate energies, highlighting some of the open questions and challenges standing ahead. We start with an overview of recent neutrino experiments and future prospects. The theory of neutrino interactions on nucleons is introduced making emphasis on the symmetries, the sources of experimental information and major uncertainties. Pion production is covered, but also other no less interesting inelastic reactions with smaller cross sections. We present different descriptions of neutrino-nucleus scattering from a common perspective, indicating the different approximations and assumptions adopted. The role of multinucleon mechanisms is stressed as well as their impact in the neutrino energy reconstruction. Nuclear medium effects and final state interactions in particle production off nuclear targets are also addressed, paying special attention to the coherent reaction channels and their different theoretical approaches. The last part is devoted to the Monte Carlo models, which are applied to the simulation and analysis of neutrino experiments.
The present review complements other excellent articles of this kind that have recently appeared [4][5][6][7]. In Ref. [7] a very broad range of energies from the eV to EeV is covered, providing an impressive account of the existing experimental data. As we focus on the few-GeV region, we do not discuss the deep inelastic regime, nor questions like shadowing or duality, all of which can be found in Refs. [5,6]. Instead, we address topics like superscaling, quasielastic and inelastic strangeness production, photon emission or Monte Carlo generators, which are at most mentioned in previous reviews.

II. RECENT AND FUTURE OSCILLATION AND CROSS SECTION EXPERIMENTS
The discovery of atmospheric neutrino oscillations [8] was made through the observed distribution of the outgoing charged leptons, produced by the interactions of atmospheric neutrinos with water in the Super-Kamiokande (SK) detector. The signature of the ν µ disappearance was clearly seen as an up-down asymmetry in the zenith-angle distribution of muons, which is expected to be almost symmetric in the absence of neutrino oscillations. This result was quite robust against the uncertainties in the neutrino interactions considering the observed number of events at that time.
Since then, several new accelerator-based long-baseline neutrino oscillation experiments, like K2K [9], MINOS [10], OPERA [11], and T2K [12] have started taking data to confirm the SK results and to measure more precisely the oscillation parameters. All these experiments use high purity ν µ and/orν µ beams and search for the ν µ (ν µ ) oscillation into the other neutrino flavors.
The K2K experiment started operating in 1999 as the first accelerator-based long baseline neutrino experiment. The K2K experiment intended to confirm the neutrino oscillation phenomena observed in atmospheric neutrinos. It also aimed at the search for ν µ → ν e appearance signals. The accelerator and the near detectors were located in KEK, Tsukuba, Japan. The average neutrino energy was 1.3 GeV, and the SK detector, situated 250 km away from KEK, was used as the far detector. K2K reported a deficit in the number of µ-like events observed at SK, and confirmed the ν µ oscillation phenomenon. However, no indication of ν e appearance events was observed. The near detector of the K2K experiment consisted of several components: a 1 kt water Cherenkov detector, a water target scintillator fiber tracker (SciFi), an iron target muon range and a lead glass (LG) detectors. In the later stage of the experiment, the LG dectector was replaced with a fully active scintillator tracking detector (SciBar) (see Fig. 1). With these detectors, various neutrino interactions were studied. One of the interesting observations was a strong suppression of muons in the forward direction [13] and an enhancement of events for large scattering angles. This suppression was observed not only in SciFi but also in the water Cherenkov and the SciBar detectors. These measurements could be well described by increasing the axial vector masses (M A ) entering in the used models for the charged current quasi-elastic (CCQE) scattering and the charged current single pion production (CC1π). Moreover, to achieve a good agreement with data, corrections on the parton distribution functions (PDF's) [14] were also necessary. The CCQE interaction was extensively studied with the SciFi detector and as result an effective value of M A = 1.20 ± 0.12 GeV for the nucleon axial vector mass was extracted from the data [13]. The model dependence in the determination of M A in experiments with nuclear targets, in the light of new theoretical developments, is discussed in Sec. III.
The neutral current π 0 production, which is one of the background sources in the search for ν e appearance signatures, was also studied using the 1kt water Cherenkov detector. The predicted number of interactions from the simulation agreed quite well with the data, but the π 0 momentum distribution turned out to be slightly shifted [15]. Another interesting result from the K2K experiment was the measurement of the charged current coherent pion production cross-section using the SciBar detector. The reported cross-section was almost consistent with zero and much smaller than the predictions from simple PCAC (partial conservation of the axial current) models [16], which however successfully described old high energy experiments. This finding stimulated the theoretical work on coherent reactions induced by (anti)neutrinos described in Sec. V.
The MINOS experiment is also devoted to the study of ν µ andν µ oscillations. The neutrino beam is produced at the Fermi National Laboratory (FNAL) in Illinois, US using 120 GeV protons from the main injector. The mean energy of the neutrino beam is adjustable by changing the configurations of the target and the magnetic horns. Because the neutrino mass difference ∆m 2 32 turned out to be quite small, the nominal energy of the beam was set to 3 GeV. The produced neutrinos are studied in a near detector at FNAL and also in the far detector located in the SOUDAN mine, 750 km away from the target. Both the near and far detectors are steel-scintillator sampling calorimeters with tracking and energy measurement capabilities. These detectors have a magnet field; thus, it is possible to measure the particle momentum and identify its charge. Owing to the intense neutrino beam, the most stringent limit was set on the squared mass difference ∆m 2 32 and the consistency of the oscillation parameters between ν µ andν µ was also discussed [17]. They also studied CCQE scattering in the near detector and obtained a value for the effective axial nucleon mass of around 1.2 GeV [18], which is consistent with the result from K2K.
The OPERA experiment is slightly different from these two previous ones. It aimed to study the ν µ → ν τ oscillation in the appearance channel. The neutrino beam was produced using the 400 GeV SPS proton line in CERN. The far detector is located in the Gran Sasso Laboratory (LNGS) in Italy, which is 732 km away from the target. It consists of an emulsion detector and a high precision tracker, which are used to identify τ leptons by their decay products. The experiment identified 3 τ candidates and confirmed the appearance of τ ′ s from ν µ oscillation [19].
The MiniBooNE experiment [20] was constructed to study the neutrino oscillation pattern observed by the LSND experiment [21], which can not be explained with the standard three neutrino scenario. MiniBooNE used a line of 8 GeV kinetic-energy protons taken from the Booster at FNAL. The averaged energy of the produced neutrino beam was around 800 MeV and the detector is located 541 m away from the target. The MiniBooNE tank contains 800 tons of mineral-oil which is used to detect Cherenkov and scintillation lights. Because the beam intensity was high and the mass of the detector was large, this experiment accumulated more than 100 thousands of neutrino and antineutrino events from 2002 to 2012. MiniBooNE measured CCQE [22,23], CC1π differential cross-sections [24,25], and also the corresponding neutral current (NC) channels [26,27]. Their results corroborated the muon forward suppression observed in K2K and MINOS, and a number of interactions significantly larger than expected from the theoretical predictions with the nominal theoretical parameters, i.e. M A = 1.0 GeV, was also obtained. Besides, the pion momentum distribution of the charged current single π + events is broader and larger than Monte Carlo predictions, which indicates that the effects of π + rescattering in the nucleus are much smaller than expected from the theoretical simulations. On the other hand, the momentum distribution of the NCπ 0 production sample is more consistent with the theoretical predictions. More details in this respect can be found in Sec. IV. MiniBooNE also observed a larger number of ν e −like events than theoretically expected. This could not be explained by the known backgrounds. Various new models including possible "non-standard" oscillation scenarios have been proposed, but no definitive conclusion has yet been reached [28,29].
In 2007 and 2008, another experiment called SciBooNE took data in the Booster neutrino beam-line, which was also used by MiniBooNE. This detector was located at a distance of about 100 m from the neutrino production target. The detector complex consists of a fully active scintillator tracking detector, called SciBar, and a muon tracker. The SciBar detector was used in KEK for the K2K experiment and transported to FNAL to study low energy neutrino and anti-neutrino interactions with much higher statistics. This detector was capable to find low momentum protons and thus, it had power to select CCQE events exclusively [30]. The measured CCQE total cross section was found to be ∼10% smaller than in MiniBooNE [31]. The charged current coherent pion production reaction was also studied and consistent results with the K2K findings were obtained [32]. This time, the neutral current coherent π 0 production cross-section was also measured and found to be consistent with the predictions from PCAC based models [33]. The inclusive CC cross-section was also measured in SciBooNE. In the region, 0.5-1.5 GeV, it turned out to be significantly higher than NEUT and NUANCE Monte Carlo predictions [34].
The ArgoNeuT experiment studies neutrino interactions using a liquid argon (LAr) time projection chamber (TPC) [35]. This LAr TPC has the capability of tracking low energy charged particles, which is difficult for other detectors. For this reason, ArgoNeuT is expected to provide more precise information on various neutrino cross sections. The detector is located just in front of the MINOS near detector, which is used as a muon spectrometer. Despite the small accumulated statistics, because of the limited run period, the inclusive ν µ charged current differential cross sections on argon has been measured [36]. Recently, ArgoNeuT succeeded in tracking a large number of protons after the neutrino interaction, and reported on the observed proton multiplicity. In particular, events with a final charged lepton and two back-to-back protons are quite interesting. These back-to-back protons might have been originated via neutrino interactions with two bound nucleons or from the absorption of a produced pion. This kind of data sets will be useful in the future to improve the performance of the event generators used in neutrino experiments and as a new complement in the study of nuclear structure and the nucleon properties inside the nuclear medium.
The MINERνA experiment placed the detector after ArgoNeuT finished taking data. The setup consists of a fully active scintillator tracking detector, an electromagnetic and a hadron calorimeters, together with various nuclear targets, including helium, carbon, water, iron and lead. This experiment also uses the MINOS near detector as a muon spectrometer. MINERνA can measure cross sections for several nuclei using the same neutrino beam, which will allow to study the nuclear dependence of the different neutrino interactions with a minimized impact of systematic uncer-tainties. Charged current ν µ andν µ quasi-elastic differential cross-sections in carbon have been recently published. The interaction rates are consistent with a value of nucleon axial mass M A ∼ 1.0 GeV [37,38], in agreement with previous results from NOMAD [39], and in contradiction to the other recent experiments. On the other hand, deviations are found between the measured q 2 differential cross section and the expectations from a model of independent nucleons in a relativistic Fermi gas with a small M A value [38].
After the K2K experiment was successfully finished in 2005, the T2K experiment started in 2009 as its natural upgrade. T2K aims to search for ν µ → ν e oscillations and measure or set a limit for the θ 13 mixing angle. This experiment also tries to improve by one order of magnitude the precision on the determination of ∆m 2 32 and θ 23 . These parameters are essential to disentangle the pattern of CP violation in the lepton sector. The T2K experiment reported the first indication of a non-zero value for θ 13 in 2011 at 2.5σ level [40] and the significance reached at 7.5σ in 2013 [41]. After the first result from T2K, θ 13 was also measured in nuclear reactor experiments in the antielectron neutrino disappearance channel, and the precisions are being continuously improved [42][43][44]. The neutrino beam for the T2K experiment is produced by a 30 GeV proton line from the J-PARC proton synchrotron, located in Tokai, Ibaraki, Japan. The direction of the resulting neutrino beam is 2.5 degree shifted from the direction of the far detector, SK located at 295 km from the target, Fig. 2. In this manner, the neutrino spectrum becomes narrower and peaks at lower energies. This configuration is called off-axis. With this method, the average energy of the neutrino beam is adjusted to peak around 700 MeV to match the oscillation maximum. There are several neutrino detectors, located at 280 meters from the target, where the neutrino interactions before oscillations are measured. There are two sets of near detectors. The first one is aligned with the neutrino beam direction and consists of a steel-scintillator sampling tracker that allows to monitor the direction and stability of the neutrino beam. The second set is aligned approximately in the direction to SK. This off-axis detector complex has full active scintillator tracking detectors, TPC's together with calorimeters. These detectors are located in a magnet that allows to identify the charge and momentum of the produced particles. Various studies on neutrino interactions are been carried out using both on-axis and off-axis detectors. The inclusive CC and the CCQE cross-sections have already been measured. Regarding CCQE, the distributions seem to favor a large M A value. Total CC cross-sections have been measured in carbon and iron, which turn out to be similar, once divided by the number of active nucleons, with small differences at the level of 3%.
In 2013, the NOνA experiment has started taking data. This experiment utilizes a fully active liquid scintillator tracking detector to accurately measure θ 13 , study CP violation and the neutrino mass hierarchy. The neutrino beam is produced at FNAL, and is formally used also for MINOS, but upgraded for this new experiment. The far detector is located in the Ash river, 810 km away from the target and 14 mrad off-axis to adjust the maximum neutrino energy. The construction of the far detector is still in process and shall be completed in 2014.
Until now, three mixing angles and two squared mass differences have been measured, but there is no information on whether CP is violated or not in the lepton sector. The mass hierarchy pattern is also unknown. In order to answer these questions, several high precision experiments with much higher statistics and precisions have been proposed. In Japan, a gigantic water Cherenkov detector, called Hyper-Kamiokande (HK) has been proposed. The fiducial volume of HK is more than 20 times larger than SK. It is planned to be located in the Kamioka mine and thus, it will be also possible to study atmospheric neutrinos with much higher precision. Furthermore, the existing neutrino beam line in J-PARC is expected to be upgraded. In the US, a project called LBNE has been launched. The proposed location of the far detector is the Sanford Laboratory, located at 1280 km from FNAL. Two LAr TPC's, with a total fiducial volume of 34 kton, have been proposed as far detectors. This type of detector allows to measure low momentum particles; it is also expected to have a large electron-photon discriminating power. In spite of its relatively small mass compared to HK, the LBNE baseline is rather long, and thus this experiment would be quite sensitive to both the CP δ phase and the neutrino mass hierarchy. In Europe, the LBNO/Laguna project is being discussed. The proposal includes three types of the detectors: a 100 kton LAr TPC called GLACIER, a 500 kton water Cherenkov tank (MEMPHYS) and a 50 kton liquid scintillator detector (LENA). Several locations are under consideration including several detector configurations exploiting different advantages. Further study is underway. This next generation of experiments would reduce the uncertainties of the neutrino interactions down to the few percent level from around the 10% affecting the current experiments. More precise theoretical models will be needed to understand and describe the future wealth of neutrino-nucleus data.

III. QUASIELASTIC AND QUASIELASTIC-LIKE SCATTERING
A. Quasielastic scattering on the nucleon Let us first consider the processes standing for charge-current (CC) quasielastic (CCQE) and neutral current (NC) elastic (NCE) scattering on nucleons induced by neutrinos. The corresponding reactions with antineutrinos arē At energies low enough that the 4-momentum squared transferred to the nucleon is much smaller than the intermediate , their cross sections can be cast as where E ν ≡ k 0 is the (anti)neutrino energy in the Laboratory frame and M the nucleon mass (isospin symmetry is assumed); G is the Fermi constant while c CC = cos θ C , in terms of the Cabibbo angle, and c NC = 1/4. The leptonic tensor is where the +(−) sign is valid for ν(ν) interactions (ǫ 0123 = +1). The hadronic part is contained in the hadronic tensor H which, in the case of CCQE and NCE processes, takes the form The amputated amplitudes Γ α enter the weak charged and neutral currents. The latter ones can be written in the most general way consistent with the symmetries of the Standard Model, in terms of form factors that contain the information about nucleon properties with the vector and axial currents given by and Vector-current conservation q α V α = 0 and isospin symmetry imply that vector form factors F 1,2 are given in terms of the electromagnetic form factors of protons and neutrons. On the other hand, the PCAC q α A α = i(m u +m d )q u γ 5 q d → 0 in the chiral limit of QCD (m q → 0), together with the approximation of the pion-pole dominance of the pseudoscalar form factor F P , allows to relate F P to the axial form factor F A . Explicit expressions for these relations among form factors can be found, for instance, in Ref. [45] for the CC case, and in Ref. [46] for the NC one.
With these ingredients, a compact expression of the cross section in Eq. (6) as a function of the form factors was obtained [47] and widely utilized [7]. Alternatively, it is instructive to write this cross section as an expansion in small variables 1 q 2 , m 2 l ≪ M 2 , E 2 ν . One has that on proton targets, with κ with κ (n) NC = 1 − µ p + α V µ n − µ s . In the q 2 , m 2 l ≪ M 2 , E 2 ν limit, which is valid for most of the integrated cross section in the few-GeV region, the CCQE cross section is determined by a few experimentally well measured electromagnetic properties of the nucleon: charge, magnetic moments µ p,n and mean squared charge radii where G (p,n) E are the electric form factors, and two axial quantities: coupling g A , also well known from neutron β decay, and radius It is remarkable that r 2 A can be extracted from single pion electroproduction data independently of neutrino experiments which could be distrusted due to low statistics, inaccuracies in the neutrino flux determinations or the influence of nuclear effects. It has been shown [48,49] that, up to O(p 3 ) in a chiral expansion in small momenta and quark masses (Chiral Perturbation Theory) where E (−) 0+ is an s-wave electric dipole amplitude in a specific isospin combination (see Ref. [49] and references therein for more details). The derivative is taken over the virtual photon 4-momentum squared, and f π is the pion decay constant. Using an effective Lagrangian model to extrapolate pion electroproduction data to q 2 = 0 [50] and taking into account the hadronic corrections of Eq. (25) [48] one finds that At higher q 2 , the relation between pion electroproduction amplitudes and the axial form factor becomes more uncertain and model dependent. Therefore, for information on the q 2 dependence of F A one has to rely chiefly on (anti)neutrino experiments off hydrogen and deuterium targets. Although the contribution of the term proportional to F A to the parity-violating asymmetry in electron-proton elastic scattering with polarized beams is typically orders of magnitude smaller than the dominant (magnetic) one, a detailed study at backward angles might also help constraining this form factor [51].
The axial form factor is usually parametrized with a dipole ansatz which corresponds to an exponential shape for the axial charge-density distribution. For such a one parameter function, the so-called axial mass M A is directly related to the axial radius The value of M A extracted from early CCQE measurements on deuterium and, to lesser extent, hydrogen targets is M A = 1.016 ± 0.026 GeV [52]. It is in excellent agreement with the pion electroproduction result, M A = 1.014 ± 0.016 GeV, obtained from the axial radius, Eq. (26), using the relation of Eq. (28). In spite of the fact that deviations from the dipole form have not been observed so far, it is worth stressing that the dipole parametrization is not well justified from a theoretical point of view. In the case of electromagnetic form factor, the dipole behavior arises from cancellations between monopole terms that appear naturally in the vector meson dominance picture [53]. The situation is more uncertain in the axial sector but a similar scenario might be in place from the interplay of two or more axial mesons [54]. The lack of knowledge about the q 2 dependence of F A may result in large uncertainties in the CCQE cross section, specially at large energies as shown in Ref. [55], although the constrains in the axial radius from pion electroproduction would make the bands in Fig. 5 of that paper narrower. In addition to the nucleon electromagnetic and axial properties that define the CCQE hadronic tensor at q 2 , m 2 l ≪ M 2 , E 2 ν , the NCE one depends on the strangeness content of the nucleon via the strange mean squared radius r 2 s , magnetic moment µ s , the strange axial coupling ∆s, which is the strange quark contribution to the nucleon spin, and the corresponding axial radius r 2 As . The impact of r 2 s , µ s , r 2 As , which are not only small (see Ref. [56] for a recent global fit) but also appear in the subleading T (p,n) NC functions, is insignificant. This is not the case of ∆s which, if different from zero, could change the NCE cross section appreciably. This becomes evident by looking at the ratio The MiniBooNE experiment performed a detailed study of NC nucleon knock-out on mineral oil (CH 2 ) but their measurement turned out to be rather insensitive to ∆s because of difficulties to distinguish between protons and neutrons [26,57]. The recently proposed MiniBooNE+ experiment with improved sensitivity to neutrons might allow a better determination of this important nucleon property [58]. The same is true about MicroBooNE [59], where a reliable identification of the low energy protons knocked out of argon should be possible. As a result, the error in the determination of ∆s would be drastically reduced [56]. It should be recalled that for both MiniBooNE+ and MicroBooNE, running with nuclear targets, the presence of multinucleon contributions (discussed in Sec. III C) and other inelastic mechanisms, together with final state interactions [46,60] will certainly affect the extraction of ∆s and should be carefully studied. In fact, we think that dedicated (anti)neutrino-nucleon experiments are needed in order to understand the axial structure of the nucleon in depth.

B. Quasielastic scattering on nuclei
For (anti)neutrinos interacting with nuclear targets, the inclusive cross section per unit volume, which is the proper quantity for an extended system, is given by This formula stands for both CC and NC processes, with c CC = cos θ C and c NC = 1/4. The lepton tensor defined in Eq. (7) as in the nucleon case. By construction, the hadronic tensor can be decomposed as with W s (W a ) being a real symmetric (antisymmetric) tensor. Furthermore, it can be expressed in terms of the polarization propagator (or tensor) This is a classic result, known for the response of many-body systems to external probes [61], that also holds for neutrino interactions. Some examples of different contributions to the polarization propagator are diagrammatically represented in Fig 3. Diagram (c) contributes mostly to pion production 2 discussed in Sec. IV. Diagram (b) is an example of a meson exchange current (MEC) contribution to the interaction mechanisms involving two nucleons from the nucleus, also known as two-particle-two-hole (2p2h) terms. They play a very important role in neutrino scattering in the few-GeV region and will be covered in Sec. III C. Let us now consider diagram (a) of Fig 3. This piece of the polarization tensor contributes mainly to the quasielastic (QE) peak, where the interaction takes place on a single nucleon which is knocked out; these are one-particle-one-hole (1p1h) terms. Nevertheless, through the correlations in the initial or final state, this diagram also contains 2p2h, other multinucleon and even particle emission contributions. It can be shown that for the polarization propagator of diagram (a) in where H αβ (s,a) are the symmetric and antisymmetric components of the QE hadronic tensor for nucleons introduced in Eq. (8). In this way, one disregards off-shell effects on the nucleon current and adopts the same form factors as for the free nucleon.
are particle and hole spectral functions related to the scalar (averaged over spins) nucleon selfenergy Σ present in the full (dressed) in-medium nucleon propagator as a result of nucleon-nucleon (N N ) interactions. Here p 0 ≥ µ and p 0 ≤ µ, with µ being the chemical potential, for A p and A h , respectively. In presence of the nuclear medium, nucleons have a modified dispersion relation and, in addition, become broad states. Practically all approximations employed to calculate neutrino, but also electron, QE scattering with nuclei can be obtained from Eq. (35). The simplest description, present in most event generators used in the analysis of neutrino experiments, is the relativistic global Fermi gas (RgFG) [62]. In this model, the nuclear ground state is a Fermi gas of non-interacting nucleons characterized by a global Fermi momentum p F and a constant binding energy E B . Then, the hole spectral function is where E( p) = M 2 + p 2 − E B . The step function θ (p F − | p|), with p F a constant Fermi momentum, accounts for the Fermi motion of the nucleons in the nucleus. For the particle with E( p) the on-shell energy. The factor 1 − θ (p F − | p|) takes Pauli blocking into account. Such a simple picture with only two parameters (p F , E B ) explains qualitatively inclusive QE electron scattering data but fails in the details. A better description requires a more realistic treatment of nuclear dynamics. An improvement over the RgFG is the so called relativistic local Fermi Gas (RlFG) where the Fermi momentum is fixed according to the local density of protons and neutrons ρ p,n (r) Equations (37) and (38) also hold after replacing the global Fermi momentum by the local one. The binding energy is often neglected but a minimal excitation energy required for the transition to the ground state of the final nucleus has been taken into account in the CCQE model of Ref. [45]. Nuclear spectral functions like the one of Ref. [63], obtained using empirical single particle wave functions and realistic calculations in nuclear matter [adapted for finite nuclei employing the local-density approximation (LDA)], from the convolution model [64] and from other (semi)phenomenological models [65,66] have been applied to electron and neutrino scattering in the QE region [45,[67][68][69][70][71][72]. Sometimes, realistic spectral functions for holes are combined with the use of plane waves for nucleons in the final state, eventually including Pauli blocking with a global Fermi momentum like in Eq. (38). Examples of such an approach, known as plane wave impulse approximation (PWIA), for CCQE scattering can be found in Refs. [69,70,72,73].
To put in perspective the RgFG, RlFG and realistic spectral function descriptions of the nuclear ground state, we consider the nucleon momentum distribution in nuclei, related to the hole spectral function by where V is the nuclear volume, and within the LDA approximation. The momentum distribution is normalized as where A is the nuclear mass number. For simplicity, the same momentum distribution for protons and neutrons, as in symmetric nuclei, has been assumed. In Fig. 4, p 2 n(| p|) on 12 C is plotted for RgFG (neglecting the binding energy), the RlFG (with density profiles taken from Ref. [74]) and the convolution approach of Ref. [64] that provides convenient parametrizations of these distributions. The plot shows the tail at high momenta of the realistic distribution due to short-range correlations, which is absent in the Fermi gas distributions. These correlations are investigated in two-nucleon knockout electron scattering experiments at specific kinematics where they are bound to be relevant [75].
In spite of its simplicity, the RlFG model introduces space-momentum correlations that translate into a considerably more realistic description of the momentum distribution than the RgFG. Besides the short range correlations, the description of the nuclear ground state can be improved with more realistic treatments of the single particle content beyond the non-interacting Fermi-type picture. In the GiBUU model [76], the hole spectral function is modified with respect to the one of the RlFG by the presence of a real position-and momentum-dependent mean-field potential generated by the spectator particles. The hole states remain narrow but the dispersion relation is modified: is the effective mass of the bound nucleon. The initial nucleons have also been treated as shell-model bound states with wave functions obtained as solutions of the Dirac equation in a σ − ω potential [77][78][79], or from a phenomenological energy and target dependent optical potential (OP) [80]. This is known as the relativistic mean field (RMF) approach.
Although the details of the particle spectral functions, and the final state interactions (FSI) in general, do not affect the inclusive integrated cross section, they are important for the differential ones, particularly in the region of low-energy transfer, and are crucial to achieve a realistic description of the reaction final state. It is precisely through the detection of (some of the) final state particles that neutrinos can be detected and its energy, a priori not known in broad beams, identified. The role of final-state correlations is particularly important for NC processes because the outgoing neutrino cannot be detected. In a inclusive CC process of the type of diagram (a) in Fig. 3, the chief effects of the medium in the particle spectral function are an energy shift of the cross section caused by the mean field and a redistribution of the strength caused by N N interactions [69]. Particle spectral functions have been obtained in a variety of approaches: generalizing Glauber theory of multiple scattering [81], also using phenomenological potentials for the real part of the nucleon selfenergy, and the low density limit to relate ImΣ to the N N cross section [71,82]. In Ref. [65], ImΣ is also expressed in terms of the free N N cross section in a LDA framework and taking polarization effects into account; ReΣ is obtained from the imaginary part using dispersion relations. All of them have been employed to study neutrino-induced QE reactions [45,69,71,82].
The wave functions of the outgoing nucleons can be distorted with complex OP in an approach known as distortedwave impulse approximation (DWIA) [72, 77-80, 83, 84] or with the Glauber multiple scattering approximation for knocked out nucleons with more than 1 GeV kinetic energy [83]. DWIA models are successful in describing a large amount of exclusive proton knockout (e, e ′ p) data and are valid when the residual nucleus is left in a given state [85][86][87] but are not appropriate for inclusive scattering. Indeed, the imaginary part of the OP produces an absorption and a reduction of the cross section which accounts for the flux lost towards other channels. This is not correct for an inclusive reaction where all elastic and inelastic channels contribute and the total flux must be conserved. DWIA calculations using the same real potential (RMF) used to describe the initial state, or the real part of a phenomenological OP [78] are more suitable for inclusive processes although one should recall that optical potentials have to be complex owing to the presence of inelastic channels [88]. An alternative is the relativistic Green function (RGF) approach [89][90][91][92] which also starts from a phenomenological complex OP that describes proton-nucleus scattering data, but recovers the flux lost into nonelastic channels in the inclusive case. This in principle renders the model more appropriate for the study of CCQE scattering when nucleons are not detected in the final state, as in the MiniBooNE measurement (see the related discussion in Sec III C). NCQE 3 processes, instead, are identified by the detection of the knocked-out nucleon with no lepton in the final state. The RGF model may include channels which are not present in the measurement but, on the other hand, it contains contributions that are not present in DWIA calculations. DWIA, RMF and RGF approaches have been recently compared for different NCQE kinematics [93]. It is argued that RGF may provide an upper limit to the NCQE cross section while DWIA gives the lower one but one should keep in mind that other mechanisms like MEC not accounted in these approaches can also contribute.
A detailed description of the final state interactions can be achieved using semiclassical Monte-Carlo methods. Moreover, at E ν 500 MeV, this is the only viable way to simulate the final hadronic state. These techniques allow to take into account account rescattering causing energy losses, charge exchange and multiple nucleon emissions [46,60,94]. Furthermore, there are QE-like processes in which a pion produced in the primary interaction and then absorbed, leads only to nucleons in the final state. These mechanisms, which can be very hard, if at all, to disentangle from those originated in an elastic or QE interaction on the nucleon, can be naturally incorporated to the framework. Other approaches as those discussed above have to rely on the subtraction of QE-like events performed by the different experiments, using event generators that do not necessarily incorporate state-of-the art physics input.

Superscaling approach
Electron scattering is a major source of information for neutrino interactions studies, providing not only the vector form factors of nucleons but also a testing ground for nuclear scattering models, which can be confronted with a large set of good quality data. Inclusive electron scattering data exhibit interesting systematics that can be used to predict (anti)neutrino-nucleus cross sections. When the experimental (e, e ′ ) differential cross sections are divided by the corresponding single nucleon ones and multiplied by a global Fermi momentum, the resulting function is found to depend on energy and 3-momentum transfers (q 0 , | q|) through a specific combination, the scaling variable ψ ′ , and to be largely independent of the specific nucleus. This property is known as superscaling [95]. Scaling violations reside mainly in the transverse channel [96] and have their origin in the excitation of resonances and meson production in general, 2p2h mechanisms and even the tail of deep inelastic scattering. Therefore, an experimental scaling function f (ψ ′ ) could be reliably extracted by fitting the data for the longitudinal response [97]. The experimental f (ψ ′ ) has an asymmetric shape with a tail at positive ψ ′ (large q 0 ) as can be seen in Fig 5. The requirement of a correct description of the scaling function is a constraint for nuclear models. The RgFG model fulfills superscaling exactly and has a very simple scaling function but with the wrong symmetric shape. It has been observed that models based on the impulse approximation are consistent with the superscaling. However, while PWIA and DWIA approaches with complex OP fail to reproduce the asymmetric shape and the tail of the scaling function, a very good description of it is achieved within the RMF model [99]. In addition, the RMF model predicts a larger scaling function in the transverse mode than in the longitudinal one (f T > f L ) [100], which seems to be supported by data. Extensive studies with a wide class of models reveals the importance of a proper description of the interaction of knocked-out nucleons with the residual nucleus [101] to obtain the tail of the scaling function.
With the superscaling approximation (SuSA) a good representation of the nuclear response can be obtained by embedding nuclear effects in the scaling function: the observables can be calculated with the simple RgFG model followed by the replacement f RgFG → f exp . The same strategy can be used to predict neutrino QE cross section, minimizing in this way the model dependence of the results. It has been found that the SuSA approach predicts a 15 % smaller integrated CCQE cross section compared to the RgFG, close to the result obtained with the RMF model. (Fig. 3 of Ref. [102]). Nevertheless, it should be remembered that scaling fails at ω < 40 MeV and | q| < 400 MeV due to collective effects.

Long-range RPA correlations
The theoretical models discussed so far assume the validity of the impulse approximation according to which the (anti)neutrino interacts with a single nucleon in the nucleus. The influence of the spectator nucleons is only present in the hole and particle spectral function or in FSI. However, when the 3-momentum transferred to the target | q| is small and 1/| q| becomes comparable with the internucleon distance, one should not expect this approximation to hold. The comparison with inclusive electron scattering data shows that at | q| 350 − 400 MeV, systematic discrepancies associated with the breakdown of the impulse approximation start to show up [71,82].
Collective effects can be handled within the RPA (random phase) ring approximation using the bare polarization propagators as input. For CCQE scattering this is illustrated in Fig. 6. These RPA correlations renormalize the different components of the hadronic tensor. One expects them to be relevant for neutrino interactions on the base of the well established quenching of g A in nuclear Gamow-Teller β decay. For CCQE scattering, these longrange correlations were taken into account in Ref. [103] using a RlFG model and an effective N N interaction in the nuclear medium (denoted V in Fig. 6) consisting of pion and rho-meson exchange plus a short range part effectively included in the phenomenological constant g ′ , taken to be the same in the longitudinal and transverse channels. The contributions of the RPA sums to the tensor are expressed in an analytic form in terms of the 1p1h and ∆h polarization propagators. Such a model has been subsequently applied to obtain inclusive and semi-inclusive cross section in different experimental situations [104][105][106][107]. This model was improved in Refs. [45,46] by a more rigorous resummation of the RPA series in the vector-isovector channel, and the introduction of scalar-isoscalar, scalar-isovector and vector-isoscalar contact density dependent terms of the effective interaction (see Eqs. (32)(33)(34)(35)(36) of Ref. [45] for details). The approach set up in Refs [108][109][110] and the one of Ref. [111] are quite similar. In this case the RPA equations are solved numerically. An algebraic solution of these equations was developed in Ref [112] and applied to the study of τ polarizations in CC ν τ ,ν τ -nucleus scattering [113].
The impact of RPA correlations at MiniBooNE energies E ν ∼ 750 MeV is quite significant. This is shown in Fig. 7, where they have been calculated for CCQE on 12 C following Ref. [45]. At low Q 2 ≡ −q 2 < 0.3 GeV 2 these collective effects cause a sizable reduction of the cross section which, as will be discussed in the next section, is crucial to understand CCQE MiniBooNE data. In the dash-dotted lines the nucleons are exposed to the mean field potential while the dashed ones also incorporate spectral functions for the outgoing nucleons [82,94]. The full model with long range (RPA) correlations is denoted by solid lines [115].
In the models outlined in this section, the effective N N interaction contains contact and energy independent interactions with strengths determined from low-energy collective excitations [116]. For this reason, the validity of V , like some of the non-relativistic approximations present in the models [45,110], can be questioned at the rather high energy transferred of a few hundreds of MeV that can be encountered at MiniBooNE and other neutrino experiments. However, it should be remembered, that the inclusion of dynamical pion and rho propagators in the longitudinal and transverse interactions, as well as the presence of ∆h excitations in addition to 1p1h ones , extend the validity of the framework towards higher energies. Furthermore, once q 2 < 0, high q 0 correspond to even higher | q|; one then enters the realm of the impulse approximation, where collective RPA corrections are bound to be negligible. Therefore, even large corrections to the model should have a small impact on the cross sections. A similar situation takes place in inclusive electron scattering on nuclear targets.
Another point of concern is the validity of the RlFG description in the kinematic region of q 0 50 MeV, which can account for a large part of the cross section even at high neutrino energies [55]. It is indeed true that at such low q 0 details of the nuclear structure that are beyond reach for a non-interacting Fermi gas model become important, and continuum RPA [117][118][119][120] or even shell-model calculations [77,79] should be more reliable. Nevertheless, there are indications [121] that the RlFG models with RPA corrections lead to realistic predictions for integrated quantities, for which the details of the excitation spectrum are not so relevant. The success of Ref. [45] describing simultaneously inclusive muon capture on 12 C and the low-energy LSND inclusive CCQE measurements reinforces this conclusion.

C. The role of 2p2h excitations
In May 2009, at NuInt09, the MiniBooNE Collaboration presented a new CCQE cross section measurement using a high-statistics sample of ν µ interactions on 12 C [122]. The subsequent publication [20], describing the first measurement of the double differential CCQE cross section d 2 σ/(dk ′0 d cos θ ′ ) as a function of the energy and angle of the outgoing muon, reaffirmed the surprising result reported earlier: a cross section per nucleon ∼ 20 % higher than expected from bubble chamber low-energy measurements. The size of the cross section was found to be well described with the RgFG model using the dipole parametrization of the axial form factor given in Eq. (27) with an axial mass of M A = 1.35 ± 0.17 GeV obtained with a shape-only fit to data [20]. Such a high value of M A is in contradiction with the previous determinations discussed in Sec. III A.
The origin of this CCQE puzzle has been extensively debated. It was conjectured that nuclear effects on 12 C were influencing the determination of M A , which should be understood as effective. Such a pragmatic attitude that could bring short term benefits in the analysis of neutrino oscillations was dangerous in the long run because the underlying physics was not properly understood. The RgFG model used in the analysis of MiniBooNE data was too simple, but it turned out that more realistic models of the kind reviewed in the previous section also underestimated the data. This situation is illustrated in Fig. 8 for some of the calculations collected in Ref. [123]. Theoretical predictions with M A ∼ 1 GeV lie on a rather narrow band (narrower than the experimental errorbars) clearly below the data. , [124], [70], [94], [78], [110] and [45] in this order, as reported in Ref. [123]. In a CCQE measurement on nuclear targets, there are CCQE-like events for which CC pion production is followed by pion absorption. The subtraction of this background relies partially on the Monte Carlo simulation. In the case of the MiniBooNE CCQE measurement, the event generator has been adjusted to the CCπ + data in order to reduce the uncertainty. In a rather light nucleus like 12 C absorption is not prominent and one should expect a rather small number of these CCQE-like events: it is unlikely that they were so badly underestimated to be responsible for the large CCQE cross section. Martini et al. [110] pointed out that another so far unaccounted source of CCQE-like cross section arises from the contributions of two (or more) interacting nucleons (2p2h excitations), because ejected low energy nucleons are not detected at MiniBooNE.
One should recall that certain 2p2h mechanisms [diagrams (1) and (2) of Fig. 9] had been already taken into account in former CCQE calculations as part of the more general nucleon spectral functions for particles [45,69,71,82] and holes [69,71] and have a small effect on the integrated cross section, although they are known to play an important role in the description of the QE peak in inclusive electron scattering on nuclei [68,69]. Diagram (5) in Fig. 9 has also been considered in calculations of the inclusive neutrino-nucleus cross sections as part of the in-medium ∆(1232) spectral function [69,71,82]. On the other hand, there are many 2p2h terms [for example MEC diagrams (3), (4) and interference diagram (6) in Fig. 9] that are not reduced to particle, hole or resonance spectral functions. They are required for a satisfactory description of the dip region between the QE and the ∆(1232) peaks in inclusive electron scattering [68]. Some of these 2p2h contributions have been taken approximately into account in Ref. [110] using two different parametrizations of the multinucleon terms, from pion absorption [126] and from electron scattering [127], extrapolated to the kinematic region of neutrino interactions. Once they were added to the true CCQE cross section, a very good agreement with the MiniBooNE data was obtained (see the dashed line in Fig. 8). A good description of the data, quite similar to the one obtained with the RgFG and M A = 1.35 GeV, is also obtained with the microscopic model for 2p2h excitations developed in Ref. [128] (dashed double-dotted line in Fig. 8). The importance of MEC has been further stressed in the recent ab initio calculation of the sum rules of the weak NC response functions on 12 C [129]. A significant enhancement (∼ 30%) of the weak response is due to two-nucleon currents. This approach implements non-relativistic currents and treats the ∆ in the static limit but provides a state-of-the-art description of the nuclear ground state and the nuclear correlations. Therefore, it represents a benchmark for more phenomenological methods.
In neutrino experiments, the incident energy is not known a priori and its determination involves some model dependence (see the discussion in Sec. III C 1). For this reason, theoretical models should be directly confronted with the double differential cross section of Ref. [20] for neutrinos and Ref. [22] for antineutrinos. Such a comparison has been performed in Refs. [125,130,131] for neutrinos, and in Refs. [132][133][134] for antineutrinos. In Refs. [130,132] the addition of the vector part of the MEC mechanisms to the result of the SuSa approach led to a better agreement with data. Within the model of Ref. [128] a very good agreement with the ν-12 C double differential cross section has been achieved with M A = 1.077 ± 0.027 GeV (see Table I of Ref. [125]). This value is certainly much lower and closer to the determinations from CCQE on deuterium and pion electroproduction discussed in Sec. III A than the one obtained by the RgFG model.
The role of the different reaction mechanisms according to the model of Refs. [125,128] can be appreciated in Fig 10 where d 2 σ/(dk ′0 d cos θ ′ ) is shown for a single muon-angle bin. Not only multinucleon mechanisms, but also RPA corrections are essential to understand the data. Indeed, RPA strongly decreases the cross section at low muon energies, there where multinucleon contributions accumulate. Therefore, the final picture arises from a delicate balance between the dominant single nucleon scattering, corrected by collective effects, and mechanisms that involve two or more nucleons.
A good description of MiniBooNE data for both d 2 σ/(dk ′0 d cos θ ′ ) and σ(E ν ) with an M A = 1.03 GeV has also been found [90] with the RGF model with empirical OP briefly covered in the previous section. This has been achieved with a model that takes into account those multinucleon contributions that can be ascribed to the particle spectral function (like, for example, diagram (1) in Fig. 9) but does not contain MEC mechanisms or in-medium ∆ modifications. This finding, at odds with the picture outlined above, should be interpreted with care. First of all, the results depend rather strongly on the choice of OP: compare green and red lines in Figs. 1-3 of Ref. [90]. Second and most importantly, the imaginary part of the OP, which adds to the total cross section in the RGF model, is due to inelastic channels. These inelastic channels include pion emission and absorption, and have already been subtracted in the MiniBooNE analysis. Therefore, it would be very interesting to confront the RGF results with fully inclusive CC data.
Ultimately, it would be important to find a more direct experimental signature for the multinucleon processes. Possible observables have been considered in Refs. [135,136]. It is found that 2p2h primary interactions do lead to an increase of multinucleon events. Such an enhancement may indeed be revealed in measurements by looking, for example, at proton pairs in the final state, or the total visible energy which would contain contributions from protons below reconstruction threshold. However, the primary distributions will be heavily distorted by FSI and, therefore, model discrimination would require a high precision and a considerable improvements in the Monte Carlo simulations.

Multinucleon mechanisms and neutrino energy reconstruction
Neutrino beams are not monochromatic so that the energy of an interaction event is a priori not known. As the oscillation probability is energy dependent, the neutrino energy determination is important for oscillation analyses and also, needless to say, to measure the energy dependence of different cross section, like the CCQE one from MiniBooNE [20] shown in Fig. 8. There are different strategies to reconstruct E ν . In high energy experiments such as MINOS (1 E ν 50 GeV) it is reconstructed as the sum of the muon energy and the hadronic shower energy. As the detection of the final particles is never perfect, the procedure partially relies on the theoretical models contained in the simulation program [137].
At lower energies it is common to rely on a kinematic energy reconstruction based on the event identification as CCQE. In this case, the neutrino energy can be obtained from the measured angle and energy of the outgoing lepton using two-body kinematics This formula, sometimes modified to incorporate the constant binding energy of the RgFG model, is only valid for a target neutron at rest. The Fermi motion of the nucleons in the nucleus causes a smearing of the reconstructed energy around the true value but the procedure remains accurate enough for oscillation analyses, and the energy dependence of the cross section is not affected. On the contrary, CCQE-like events from absorbed pions produce a systematic error in the neutrino energy determination, which is too large to be neglected. This was known and taken into account in the MiniBooNE analysis, by treating these events as a background that must be subtracted [20]. However, the same is true for multinucleon contributions. Once they are sizable, the effect on the E ν determination is significant [138][139][140]. Figure 11 demonstrates the effect of the neutrino-energy reconstruction on the E ν dependence of the CCQE-like (RlFG+RPA+2p2h) cross section according to the model of Refs. [125,128]. The true CCQE (RlFG+RPA) cross section is unaffected by the energy reconstruction. The 2p2h contribution instead is a very different function of the true neutrino energy (green open squares) than as a function of E rec ν (magenta crosses). In other words, E rec ν is a poor estimate of the actual energy for multinucleon mechanism. As a consequence, the total theoretical RlFG+RPA+2p2h cross section, when plotted in terms of E rec ν shows a remarkably good agreement with the MiniBooNE data of Ref. [20] rescaled by a factor 0.9, which is consistent with the experimental normalization error of 10.7%. In conclusion, the The data of Ref. [20] have been rescaled by a factor 0.9 (compatible with flux uncertainties).  [139]. The RlFG+RPA cross section, given by the blue filled squares plotted as a function of E rec , is practically unaffected by the energy reconstruction as can be seen from the comparison with the solid magenta line. The 2p2h parts as a function of Eν and E rec are given by the open green squares and the magenta crosses respectively. The corresponding curves for the total RlFG+RPA+2p2h cross section are the green dotted and the red ones. MiniBooNE data [20] have been rescaled by a factor 0.9.
actual energy dependence of the CCQE-like cross section is not given by the MiniBooNE data but would be steeper and closer to the dotted (green) line in Fig. 11. A similar finding has been made with the model of Refs. [110] as can be seen in Fig. 14 of Ref. [140] The misreconstruction of QE events resulting from many-body dynamics is bound to have an impact on the oscillation analyses of experiments like MiniBooNE, T2K and LBNE [140][141][142]. The bias in the determination of oscillation parameters may remain even after the near detector has been taken into account [143,144].

The high Eν > 1 GeV region
As the neutrino energy increases, so does the range of possible energies that can be transferred to the target. Large energy transfers make possible the excitation of baryon resonances heavier than the ∆(1232) not taken into account in the 2p2h models that have been developed so far. Based on the experience with weak resonance excitation and pion production (see Section IV B) one could expect the D 13 (1520) to play a role. The effective N N interaction used to compute RPA correlations are not realistic at high energies as discussed in Sec. III B 2 although collective effects for this kinematics should be small. In addition, the MEC vertices present in the models come from effective low-energy interactions. Nevertheless, the model of Refs. [125,128] has been applied to neutrino energies of up to 10 GeV, but limiting the 2p2h contribution to | q| < 1.2 GeV [145]. The results obtained in this way, averaged over the neutrino and antineutrino fluxes at MINERvA, have been confronted with data [145] for the reconstructed q 2 distribution obtained using E rec ν . As can be seen in Fig. 12, the agreement is quite good, with a slight overestimation of the data. The q 2 distribution as a function of the true 4-momentum transfer squared is also shown. The impact of the reconstruction procedure in the case of MINERvA flux is small.
According to Ref. [145], as the neutrino energy increases, up to 10 GeV, the 2p2h contribution saturates to ∼ 30% of the QE cross section. A priori, there is no reason for this trend to change drastically at even higher energies. This brings us to a question that remains open: the compatibility of MiniBooNE results with the NOMAD one of M A = 1.05 ± 0.02(stat) ± 0.06(syst) GeV [39]. The answer is not obvious and requires further investigations. The NOMAD measurement includes events with a muon track and one or no knocked out proton track. In principle the events without nucleon tracks should contain at least some 2p2h contributions producing unobservable low energy protons or neutrons but it is possible that, due to the high excitation energies involved, a fraction of the 2p2h events are observed as multi-track ones and removed from the CCQE sample.

D. Quasielastic production of hyperons
The existence of flavor changing CC converting u quarks into s quarks make the QE production of hyperons induced by antineutrinos possible. One has the following ∆S = 1, Cabibbo suppressed, reactions on nucleons which are related to the semileptonic decays of hyperons [146]. The theoretical study of these reactions has been undertaken in Refs. [147][148][149]. The framework outlined in Sec. III A for QE scattering on nucleons remains valid, now with c EW = sin θ C in Eq. (6), where M Y is the hyperon mass and with the vector and axial currents given by and Assuming SU(3) symmetry, the form factors can be related to the electromagnetic and axial form factors of nucleons (see for example Table II of Ref. [147]). In this limit, f 3 = g 2 = 0. SU(3) breaking corrections, which can be systematically studied using chiral perturbation theory [150], are small for the accuracy presently achievable in neutrino experiments. Weak hyperon emission off nuclear target has been addressed in Ref. [147]. Apart from the Fermi motion of the initial nucleon and the mean field potential felt by the hyperons, estimated to be negligible [147], there are important FSI effects. The hyperons produced in the reactions of Eqs. (46) and (47) undergo elastic and charge exchange scattering; Σ 0 can be converted into Λ via radiative decay Σ 0 → Λ γ. These processes, which alter the composition and momentum distributions of the emitted hyperons, have been modeled in Ref. [147] by a Monte Carlo cascade simulation using experimental information on hyperon-nucleon cross sections as input. Another consequence of FSI is that Σ + hyperons, not produced in the primary reactions, can emerge due to processes like Λ p → Σ + n or Σ 0 p → Σ + n although at a small rate (compare Fig. 9 of Ref. [147] with Figs 4-6 of the same article).
An important issue brought up in Ref. [147] and elaborated further in Ref. [151] is that, owing to their weak decays Y → N π, hyperons become a source of pions in experiments with antineutrino beams. As can be seen from Fig. 13, at low incident energies (550 MeV for π − and 650 for π 0 ) the cross section for pion production from hyperons becomes larger than the one from ∆(1232) excitation, which is the dominant mechanism at higher energies (see Sec. IV for more details about weak pion production). Unlike ∆ resonances, hyperons have a large mean life and decay predominantly outside the nucleus. Therefore, the resulting pions are not absorbed in the nucleus. This partially compensates the Cabibbo suppression, particularly for heavy nuclei where absorption is strong. In Ref. [151] it has been shown that for 13. Integrated cross section for π production on 16 O induced by antineutrinos as a function of the antineutrino energy in the laboratory frame. Results for pions produced from ∆ and hyperon (Y) decay are compared. Adapted from Ref. [147].
atmospheric and MiniBooNEν fluxes, a significant fraction of the π − and π 0 originate indeed from hyperon decays.

A. Introduction
Pion production cross section becomes quite relevant for neutrino energies above 400 or 500 MeV, and plays a central role for neutrino energies in the 1 GeV region, of the greatest importance for neutrino oscillation experiments as MiniBooNE or T2K. Recently, the MiniBooNE Collaboration has published one pion production cross sections on mineral oil by ν µ andν µ neutrinos with energies below 2 GeV. The data include ν µ andν µ NC single π 0 production [27], as well as ν µ induced CC charged and neutral pion production [24,25]. These are the first 4 pion production cross sections to be measured since the old deuterium bubble chamber experiments carried out at Argonne National Laboratory (ANL) [153,154] and Brookhaven National Laboratory (BNL) [155].
These new data show interesting deviations from the predictions of present theoretical models that we will briefly discuss in what follows.

B. Pion production off nucleons
Pion production in weak interactions are window to the poorly know axial properties of baryon resonances. In addition, the first requirement to put neutrino induced pion production on nuclear targets on a firm ground is to have a realistic model at the nucleon level. There have been several theoretical studies of the weak pion production off the nucleon at intermediate energies [47,82,[156][157][158][159][160][161][162][163][164][165][166][167][168][169][170][171][172][173]. Most of them describe the pion production process at intermediate energies by means of the weak excitation of the ∆(1232) resonance and its subsequent decay into N π, and do not incorporate any background terms. The models of Ref. [82, 157-159, 164, 165] include also the weak excitation of several resonance contributions as intermediate states. In these schemes, the vector form factors were fixed from helicity amplitudes extracted in the analysis of pion electroproduction data, while the axial couplings were obtained from PCAC. The most complete model in this respect is the one of Ref. [82], where all 4-star resonances below 1.8 GeV have been included, with vector form factors taken directly over from the MAID analysis ( [174] and [175]). The vector part of the background and its interference with the vector part of the resonant contributions was fixed using the empirical pion electroproduction amplitudes extracted in the MAID analysis [174]. The axial background part (including the vector-axial interference) was taken to be proportional to the vector one. The proportionality constant was adjusted to the old bubble chamber ANL and BNL data, neglecting deuteron effects. According to [82], the D 13 (1520) resonance, besides the ∆(1232), is the only one playing a significant role for neutrino energies below 1-1.5 GeV. However, at E ν > 1 GeV (MINERνA) higher N * in general will become important.
We should pay an special attention to the Rein-Sehgal model [159], because it is used by almost all Monte Carlo (MC) generators. It was an attempt to describe all data available in 1980 on neutrino production of single pions in the resonance region up to πN invariant masses W N π of around 2 GeV. The basic assumption is that single pion production is mediated by all interfering resonances below 2 GeV, supplemented with a simple non-interfering, non-resonant phenomenological background of isospin 1/2. The needed transition matrix elements are calculated using the relativistic quark model of Feynman-Kislinger-Ravndal [176] (formulated in 1970) with SU(6) spin-flavor symmetry, and a total of 18 baryon resonances considered. The original work of Ref. [159] assumes massless leptons. Subsequently, the model for the CC reaction was extended in [177,178] to include finite lepton mass effects in a manner consistent with PCAC. However, we should stress the Rein-Sehgal model provides a really poor description of the pion electroproduction data on protons [169,179]. Actually, it underestimates significantly the electron data, as can be appreciated in the left panel of Fig. 14, and the more accurate predictions from the model of Ref. [82]. The model of Rein and Sehgal also reveals itself unsatisfactory in the axial sector at q 2 = 0 where the divergence of axial current can be related to the πN amplitude by PCAC [180]. This connexion was exploited in Ref. [181] to obtain the forward neutrino cross sections using a dynamical coupled channel model that successfully fits a large set of πN and γN data. The comparison with the Rein-Sehgal model, given in Fig. 3 of Ref. [182], shows a clear disagreement with the more realistic description, underestimating the ∆ peak but overestimating the higher W πN region.  [159] are confronted to data from JLAB [183] (see [179] for details). Right: Model of Refs. [166][167][168] for the W + N → N ′ π amplitude. It consists of direct and crossed ∆(1232), N (1520) and nucleon pole terms, contact and pion pole contribution, and the pion-in-flight term.
Background non-resonant terms, required and totally fixed at threshold by chiral symmetry, were evaluated in [166] 5 . The background terms in [166] are the leading contributions of a SU(2) chiral Lagrangian supplemented with well known form factors in a way that respects both conservation of vector current (CVC) and PCAC. The interference between the ∆ and the background terms produces parity-violating contributions to the pion angular differential cross section, which are intimately linked to T −odd correlations 6 which would be interesting to measure. The model of Refs. [166,168] is diagrammatically shown in the right panel of Fig. 14 (note that the D 13 contributions were not considered originally in [166]). Vector form factors are taken from fits to empirical helicity amplitudes [164]. The axial N ∆ transition is parametrized in terms of four form factors C A 3,4,5,6 (q 2 ), as in [47]. Among the axial form factors the most important contribution comes from C A 5 , whose numerical value is related to the pseudoscalar form factor C A 6 by PCAC. Moreover, in the massless lepton limit, the direct ∆ pole term gives dσ with C A 3,4 (0) contributing to a(q 2 ), i.e. to O(q 2 ), which also gets contributions from vector form factors and terms proportional to dC A 5 /dq 2 q 2 =0 . For the sub-leading C A 3,4 form factors, the Adler's parametrizations [184,185] were adopted (C A 3 = 0, C A 4 = −C A 5 /5), while the dominant axial (C A 5 (q 2 )) form-factor was fitted to the flux averaged ν µ p → µ − pπ + ANL q 2 −differential cross section data [154], finding a correction of the order of 30% to the off diagonal Goldberger-Treiman relation (GTR) prediction C A 5 (0) ∼ 1.15. Considering electroproduction experiments as benchmark, the model provides an accurate description of the data up to pion-nucleon invariant masses of the order of W πN < 1.4 GeV [186].
Deuterium effects and BNL data were not taken into account in the analysis carried out in [166]. It is well known that there exists some tension between ANL and BNL pπ + data samples. It has become a relevant issue that the more than 30-year old ANL and BNL low statistics deuterium pion production data are still nowdays the best source of information about the N ∆ transition matrix element. The authors of [170] made a simultaneous fit, considering only the ∆−mechanism, to both ANL and BNL data including separate overall flux normalization uncertainties for each experiment. The main conclusion was that ANL and BNL data are in fact consistent only when these systematic uncertainties are taken into account. This strategy was followed in [167], where the N ∆ axial transition form-factors within the model derived in [166] were simultaneously fitted to both bubble chamber data sets. Deuterium effects were also taken into account. As a result of this improved analysis, a value of 1.00 ± 0.11 for C A 5 (0), 2σ away from the GTR estimate 7 , was found [167]. This model, which includes a non-resonant chiral background, was further refined in [168] by adding a new resonance (D 13 (1520)) aiming to extend the model to higher energies above the ∆ resonance region for which it was originally developed. Thus, this scheme emerges as one of the most adequate ones, from a theoretical perspective, to analyze pion production data and neutrino energies up to 1 or 1.2 GeV.
Another theoretical description is based on a dynamical model of photo-, electro-and weak pion production [162]. Starting from an effective Hamiltonian with N ∆ couplings obtained with the constituent quark model (30 % below the measured ones), the T matrix is obtained by solving the Lippmann-Schwinger equation in coupled channels. In this way the bare couplings get renormalized by meson clouds. The predicted cross sections are in good agreement with data (Figs. 5-8 of Ref. [162]).
More recently, a Lorentz-covariant effective field theory scheme that contains nucleons, pions, ∆, isoscalar scalar (σ) and vector (ω) fields, and isovector vector (ρ) fields consistent with chiral symmetry has been also employed to study the neutrino-production of pions from nucleons [173]. At low neutrino energies, below 500 MeV, the convergence of the power-counting scheme used in [173] is fast and next-to-leading-order tree-level corrections are found to be small. To go beyond this energy regime, the authors of [173] use phenomenological form factors. Nevertheless, they are mostly concerned with the E ν < 0.5 GeV region, where a satisfactory agreement with ANL data is obtained.
Finally, we just mention the approach of Ref. [180] entirely based on PCAC and valid only at low energies and in the small q 2 region.

C. Pion cross sections in nuclei and the MiniBooNE puzzle
The main contribution to MiniBooNE data comes from 12 C and this poses an extra problem to theoretical calculations because the in-medium modifications of the production mechanisms, and the FSI effects on the produced pions are important. Pions can be absorbed during its way out of the nucleus, but they can also suffer different quasielastic collisions that modify their energy, angle, and charge when they come off the nucleus. For instance, in the case of NC π 0 production, signal events originate mostly from a NC1π 0 primary interaction with a π 0 not being affected by FSI, but also from a NC1π + primary interaction with the π + being transformed into π 0 in a charge exchange FSI reaction. In this particular case, an additional difficulty in interpreting the NCπ 0 production comes from the presence of a coherent contribution. FSI definitely alters the signature of the event and thus the correct simulation of pion production requires a model not only able to describe the elementary reactions (discussed in the previous subsection), but also the final state interactions.
MiniBooNE presented their results as measurements of final states with only one pion, with the appropriate charge, and no other mesons. A variety of flux integrated differential cross sections were reported in final pion momentum or scattering angle. For the case of CC reactions, total cross sections as a function of the reconstructed neutrino energy were provided as well. MiniBooNE data show substantial discrepancies (enhancement) with respect to the Rein-Sehgal interaction [159] prediction with FSI effects as implemented in the NUANCE event generator [187]. Some important discrepancies still remain when these measurements are compared with the most comprehensive approaches [168,188] available in the literature until now, as highlighted in the case of CC 1π 0 in Fig. 15. Both theoretical approaches start from a quite complete microscopic description of the pion production on the nucleon at MiniBooNE energies ( [166][167][168] and [82], respectively) and incorporate a number of standard nuclear medium effects in the initial interaction model (Pauli Blocking, ∆(1232)−spreading potential, etc..).  15. MiniBooNE flux-folded differential dσ/dpπ cross section for CC 1π 0 production by νµ in mineral oil. Data are from Ref. [24]. Left: Predictions from the cascade approach of Ref. [168]. The solid curve corresponds to the full model and the dashed one stands for the results obtained neglecting FSI effects. Right: Predictions from the GiBUU transport model of Ref. [188]. The dashed curves give the results before FSI, the solid curves those with all FSI effects included. Two different form factors C A 5 (q 2 ), tuned to the ANL and BNL data-sets have been employed and give rise to the systematic uncertainty bands displayed in the figure.
The work of Ref. [188] uses the Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) model to account for FSI effects. It is a transport model where FSI are implemented by solving the semi-classical Boltzmann-Uehling-Uhlenbeck equation. It describes the dynamical evolution of the phase space density for each particle species under the influence of the mean field potential, introduced in the description of the initial nucleus state. Equations for various particle species are coupled through this mean field and also through the collision term. GiBUU provides a unified framework for nucleon-, nucleus-, pion-, electron-and neutrino interactions with nuclei, from around a hundred MeV to tens of GeV. It has been extensively and successfully used [76] in the last years, with special attention to pion-and photo-nuclear reactions. The study of [188] is limited only to the incoherent part of the CC induced reaction. The comparison with the MiniBooNE NCπ data was presented and discussed at the NUFACT and NUINT conferences [189,190]. Previous work of the group could be found in Refs. [82,191].
Coherent contributions, when relevant, are included in the work of Ref. [168] from the microscopic approach of Refs. [192,193]. This is based on the same pion production model on nucleons [166,167] employed in [168]. On the other hand [168] makes use of the LDA to evaluate the incoherent production on finite nuclei. This approximation, also part of the GiBUU model, turns out to be quite convenient when using a cascade algorithm, as we will see. The work of Ref. [168] is based on the many body scheme set up in [45,128] to study inclusive neutrino-nucleus reactions. It establishes a systematic many body expansion of the gauge boson absorption modes that includes one, two and even three body mechanisms, as well as the excitation of ∆ resonances and pion production.
To compute the incoherent pion production on a nucleus, one should sum the nucleon cross section over all nucleons in the nucleus. For a neutrino CC process (for antineutrino or NC induced reactions the discussion is similar) within the LDA one gets for initial pion production (prior to any pion FSI) [168] dσ d| k |4πr 2 dr d cos θ π dE π = Φ(| k|) with E N F (r) = M 2 + (p N F (r)) 2 , given in terms of the local Fermi momentum p N F (r) defined in Eq. (39). The step functions implement Pauli blocking. Besides Φ(| k|) is the neutrino flux as a function of the incoming-neutrino energy E ν ≡ | k|; andσ(νN → l − N ′ π) is the cross section at the nucleon level modified by medium effects, which in this case consist in the modification of the ∆(1232) spectral function. The ∆ properties are strongly modified in the nuclear medium [194], and since the direct ∆-mechanism is dominant, a correct treatment is needed for π production inside a nucleus. This is accounted for, both in the works of Refs. [168] and [188], using a realistic spreading potential (∆−selfenergy). In the nuclear medium, on one hand, the width is reduced due to Pauli blocking, but on the other hand, it is increased by the collisions inside the nucleus. For example, via the processes ∆N → N N and ∆N N → N N N , the ∆ can disappear without producing a pion. Secondary pion production is also possible, namely via the process ∆N → πN N . These processes contribute to the in medium ∆ width that generally becomes larger than in the free space. The in medium differential cross section of Eq. (53) is used in a simulation code to generate, at a given point r inside the nucleus and by neutrinos of a given energy, on-shell pions with a certain momentum. These pions are followed through their path across the nucleus. To evaluate these FSI effects, the authors of [168] follow the approach of Ref. [195], where a computer simulation code was developed to describe inclusive pion nucleus reactions (quasielastic, single charge exchange, double charge exchange, and absorption). The πN interaction is dominated by the ∆ resonance excitation, modified in the nuclear medium in the same way as it was modified inσ(νN → l − N ′ π). The different contributions to the imaginary part of its self-energy account for pion, two-and three-nucleon absorption and quasielastic processes. The probabilities for the different processes are evaluated in nuclear matter as a function of the density, then the LDA prescription is used to obtain results in finite nuclei. After a quasielastic event, pions change momentum and may change its electric charge. The probability for charge exchange and the final momentum distribution after a quasielastic interaction were computed in Ref. [195]. That information is used in the simulation program to generate the pion resulting from such a collision.
The model of Ref. [168] provides an overall acceptable description of MiniBooNE data, better for NC than for CC channels, although the theoretical predictions are systematically below data. Differential cross sections, folded with the full neutrino flux, show that most of the missing pions lie in the forward direction and at high energies. An example of this is shown in Fig. 15 for CC1π 0 . FSI effects are clearly visible in the distribution. Because of the FSI some pions are absorbed, but other ones are scattered and loose to nucleons part of their energy. FSI is essential to fill the low momentum part of the distribution. On the other hand, the combined effect of quasielastic scattering and pion absorption through ∆ excitation, depletes the p π = 250 ∼ 450 MeV region producing a distortion of the differential cross section shape that significantly worsens the description of the data. The artificial exclusion of the FSI effects leads to a better description of the high momentum tail of the dσ/dp π distribution.
These findings are fully supported by the results obtained within the GiBUU scheme of Ref. [188]. Actually both approaches produce quite similar results, as can be seen in Fig. 15 for the particular case of CC1π 0 . The authors of [188] also pointed out that MiniBooNE seems to suggest that the higher elementary BNL data for pion production are correct and that the ANL data underestimate the elementary production cross section. Nevertheless, the discrepancy with theory is not only due to an overall normalization factor, but as stressed above, the experimental shape of the pion momentum distributions considerably differ from those predicted by both approaches. However the theoretical model calculations [188,196] turn out to be in agreement with experimental results for the photo-production of pions on nuclei, predicting a suppression in the pion spectra around the ∆ resonance region which is not seen in the MiniBooNE neutrino data.
FSI effects could be reduced by considering the so called formation zone, that among other effects includes the propagation of the ∆ before decaying into a πN pair. The NuWro MC event generator [197,198] includes the concept of formation zone, and its predictions for NC1π 0 have been compared in [199] to MiniBooNE data [27]. The overall agreement is quite satisfactory 8 , better for neutrino induced reactions than for antineutrino ones. Shapes of the distributions of final state π 0 's are affected by an interplay between pion FSI such as absorption and formation time effects, and turn out to agree significantly better with the data than those found in Refs. [168,[188][189][190].
Of course, the formation zone could be adjusted to reproduce data. However, these kind of modifications of the FSI could be difficult to justify, as they might be in conflict with much other phenomenology; it might hide our ignorance on the relevant dynamics. Even when they could help reproducing some observables, if we lack of a correct understanding of the physical mechanisms responsible for them, they might lead to wrong predictions for other observables sensible to other kinematics, dynamical mechanisms or nuclear corrections.
In this respect, the DUET experiment at TRIUMF, that uses the PIAνO detector [201], will provide quite valuable information in the future. Pion FSI models were tuned to previous available pion-nucleus data measured in the 1980's and affected by large uncertainties. The objective of the experiment is to measure π absorption and π charge exchange cross section with ∼ 10% and ∼ 20% accuracy, respectively.
Another useful MiniBooNE measurement was the ratio of CC1π + -like (one pion in the final state) to CCQE-like (no pions in the final state) cross-sections on CH 2 , as a function of the neutrino energy. This ratio was reported in [202] with an accuracy of around 10% (The K2K Collaboration has also measured this ratio [203], but the reported errors are much larger than those in the MiniBooNE data). This ratio becomes quite interesting because of the apparent data/MC normalization discrepancy for the CC1π + production channel at MiniBooNE [25], since it would be free from the overall flux normalization uncertainty. However, it is not a directly observable quantity, because in the experimental analysis it is necessary to reconstruct the neutrino energy. This measurement puts constraints on the theoretical models which include QE, ∆ excitation and MEC/2p2h dynamics. In addition, the theoretical approaches need to include FSI effects as well. There exist three theoretical estimates of this ratio: i) The GiBUU group found a significant discrepancy between the model and the MiniBooNE data, being its prediction smaller than the experimental ratio [191], ii) NuWro MC results [170] are slightly below the MiniBooNE data for larger neutrino energies and iii) the Aligarh group [204] predictions 9 agree with MiniBooNE measurement for E ν < 1 GeV and are below MiniBooNE data for larger neutrino energies.
However, one cannot extract any robust conclusion from these comparisons, since none of the theoretical estimates for the CCQE-like cross section include multinucleon mechanism (2p2h) effects or properly compute RPA corrections. Moreover, as discussed in Sec III C 1, because of the 2p2h effects, the algorithm used to reconstruct the neutrino energy is not adequate when dealing with quasielastic-like events, and a distortion of the total flux-unfolded cross-section shape is produced (redistribution of strength from high to low energies, which gives rise to a sizable excess [deficit] of low [high] energy neutrinos in QE-like distributions) [139].
Forthcoming T2K and MINERνA pion production data will hopefully shed light into the existing puzzle originated by the large discrepancy between MiniBooNE pion production measurements and theoretical model predictions. It is worth noting that the GiBUU group has already studied pion production at the T2K and MINERνA experiments within its model [205,206]. It is found that pion absorption is less pronounced at the MINERνA energies than for MiniBooNE/T2K experiments. This is attributed in [206] to a minimum in the πN cross section at around 0.7 GeV pion kinetic energy.

Weak K andK production
Although in the few-GeV region, the attention has been focused on pion production because it is the inelastic process with the largest cross section, (anti)kaon, and strangeness production in general, are also relevant. In particular, kaon production induced by atmospheric neutrinos is a potential background for the proton decay mode with a kaon in the final state (p →ν K + ), which has large branching ratios in different theories beyond the Standard Model. Several neutrino oscillation experiments like MINOS, NOνA, T2K or LBNE have or will have fluxes that extend to energies where strange particles can be produced. Therefore, a better understanding of these reactions is important to reduce systematic errors, which will be the dominant ones in the era of precise neutrino oscillation measurements. Such a progress is even more urging for experiments running in theν mode as no measurements of strange-particle production cross sections exist withν fluxes.
The main reaction channels at these energies are single K production (∆S = 1) singleK production (∆S = −1)ν and associated strangeness production (∆S = 0) Associated strangeness production is the dominant process but it has a high threshold because both a kaon and a hyperon are produced. This leaves single kaon production as the main source of kaons at low energies, in spite of being Cabibbo suppressed. Owing to the absence of flavor-changing NC in the Standard Model, ∆S = ±1 reactions proceed via CC interactions. Channels (54), (55), (57) and (58) make coherent K (K) production in nuclei induced by ν(ν) possible. These are discussed in Sec. V C.
After the early work of Ref. [207], new theoretical developments on ∆S = 1 reactions (54-56) have become available only recently [208]. In the description of Ref. [208], the reaction mechanisms are derived from a Lagrangian that implements the QCD chiral symmetry breaking pattern. Although the vertices are SU(3) symmetric, this flavor symmetry is broken in the amplitudes by the physical hadron masses. The resulting set of diagrams for the hadronic currents are shown in Fig. 16 and are refereed as contact (CT), kaon pole (KP), u-channel crossed Σ (CrΣ) and Λ (CrΛ), pion in flight (πP) and eta in flight (ηP) terms. The absence of S = 1 baryons implies that there are no s-channel amplitudes. The structure of these currents and the corresponding cross sections at threshold are fully determined by chiral symmetry, with couplings fixed from nucleon and hyperon semileptonic decays and pion decay. Some corrections of next order, whose couplings are constrained by measured values of nucleon magnetic moments, have also been included. PCAC is implemented for the axial part of the currents. As the dependence of the different terms of the hadronic current on the momentum transferred to the nucleon is poorly known, if at all, the authors of Ref. [208] adopt a global dipole form factor assuming M F = 1 GeV. The results obtained in Ref. [208] for the integrated cross sections are shown in Fig. 16 for the reaction of Eq. (54). In the validity region assumed for the model (E ν ≤ 2 GeV, the CT amplitude, not included in Ref. [207], is dominant and interferes destructively with the rest. The KP term is negligible, while the CrΣ contribution is much smaller than the CrΛ one because of the much smaller coupling in the strong vertex. The cross section of the reaction of Eq. (56) has similar size and exhibits much the same features. Instead, the reaction of Eq. (55) has a 4-5 times smaller cross section almost completely determined by the CT.
For the ∆S = −1 reactions (57-59) close to threshold, the relevant mechanisms can also be obtained from chiral SU(3) Lagrangians (see Fig 17). The CT, KP, πP and ηP contributions to the hadronic current are present but now the Λ and Σ hyperons appear in the s-channel. As for K + production, the structure of these amplitudes close to threshold is fully defined by chiral symmetry, with the couplings determined from semileptonic decays. In Ref. [209], the q 2 dependence of Eq. (70) was also adopted. In pion production reactions, the excitation of the spin-3/2 ∆(1232) plays a dominant role at relatively low excitation energies (∼ 200 MeV). Therefore, the corresponding state of the baryon decuplet Σ * (1385) that couples to NK should be considered here. The vector and axial N − Σ * form factors, which are not known, were related to the better known N − ∆(1232) ones using SU(3) rotations [209].  As can be seen in Fig. 17, the CT provides the largest contribution to the cross section. The small impact of the Σ * resonance, contrasting with the dominance of the ∆ in the pion case, can be explained by the fact that the Σ * is below the kaon production threshold [209]. The other two channels (58) and (59) have similar cross sections [209]. It should be mentioned that while the weak K production model described above represents a theoretically solid prediction at threshold, the situation is different for theK channels. The presence of a baryon resonance, Λ(1405), just below theKN threshold might have a non negligible influence on the weakK production cross sections that needs to be investigated.
The same formalism has been applied to the study of CC associated strangeness production reactions, Eqs. (60)(61)(62)(63)(64)(65), at low (anti)neutrino energies. Again, it is found that the CT vertices, present in the leading order chiral Lagrangian and ignored in previous calculations [210,211] are responsible for most of the cross section [212]. Channels (61) and (63) where CTs are not allowed (at leading order) show relatively smaller cross sections. However, we must point out there may be important resonant contributions to the ∆S = 0 processes, such as the N * (1535) below the KΛ threshold, which have not been considered. This is known to be the case for associated strangeness photoproduction. Recently, a model that describes associated strangeness production induced by pions and photons has been used to predict the corresponding reactions induced by neutrinos in the forward direction by applying PCAC [181].
The dynamics of strange-particle production in nuclear targets is considerably more involved. The interaction of kaons with the nuclear medium is not so strong due to the absence of baryon resonances but quasielastic and charge-exchange scattering are present. Furthermore, as the energy increases, inelastic processes like secondary kaon production K N → K ′ N ′ π become sizable. Instead, theK interaction with the nucleons is strong starting from very low energies due to the presence of the Λ(1405);K can disappear leading to hyperons viaK N → π Y , η Y with Y = Λ, Σ. Due to FSI, (anti)kaons can be produced in secondary collisions such as π N → Y K, KK N ′ and N N → N Y K. Indeed, the number of low-energy K produced in neutrino-nucleus collisions is actually enhanced by FSI; in the case ofK, secondary interactions tend to compensate their absorption [213]. For this reason, from the ongoing exclusive strangeness production measurements on nuclear targets at MINERνA it will be very hard to extract the corresponding reactions on the nucleon, unless a very reliable modeling of the nuclear dynamics is implemented [213].

NC photon emission
One of the possible inelastic reaction channels is photon emission induced by NC interactions (NCγ), which can take place on single nucleons or on nuclear targets via incoherent [Eq. (72)] or coherent [Eq. (73)] scattering. It is also possible that, after nucleon knockout, the residual excited nucleus decays emitting γ rays [Eq. (74)]. This mechanism has been identified as an important source of low (∼ 10 MeV) photons for neutrinos of intermediate energies, whose main reaction mechanism is QE scattering [214].
Weak photon emission has a small cross section compared, for example, with pion production. Indeed, while pion production involves predominantly two weak vertices followed by a strong (resonant) decay, in NCγ one has a much weaker electromagnetic vertex instead of the strong one. In spite of this, NCγ turns out to be one of the largest backgrounds in appearance ν µ → ν e (ν µ →ν e ) experiments when photons with hundreds of MeV energies are misidentified as e ∓ from CCQE scattering of ν e (ν e ) in the detector. This was the case at MiniBooNE, where the photon background is estimated from the measured NCπ 0 rate assuming that it comes form radiative decay of weakly produced resonances, mainly ∆ → N γ. The experiment has found an excess of events with respect to the predicted background in both ν andν modes. In theν mode, the data are consistent withν µ →ν e oscillations and have some overlap with a previous LSND result [29]. In contrast, the reconstructed neutrino-energy distribution (E QE ν ) of e-like events in the ν mode is only marginally compatible with a two-neutrino oscillation model, showing an unexplained excess of events for 200 < E QE ν < 475 MeV [29,215]. This puzzle triggered a theoretical interest in the NCγ processes as an important background to the MiniBooNE measurement, which was not directly constrained by data. Theoretical models for the reaction of Eq. (71) in the few-GeV region have been developed in Refs. [216][217][218]. These calculations incorporate s-and u-channel amplitudes with nucleons and ∆(1232) in the intermediate state (see Fig. 18). The structure of nucleon pole terms, N P and CN P , at threshold is fully determined by the symmetries of the Standard Model. They are infrared divergent when the photon energy E γ → 0 but this becomes irrelevant when the experimental detection threshold (E γ > 140 MeV in the case of MiniBooNE) is taken into account. The extension towards higher energy transfers required to make predictions for the neutrino cross sections is performed by the introduction of phenomenological parametrization of the weak and electromagnetic form factors. The same strategy has been followed for the ∆P and C∆P terms. As can be seen in Fig. 19, where the NCγ cross sections for the different mechanisms according to the model of Ref. [218] are displayed, the ∆(1232) excitation followed by its radiative decay is the dominant mechanism. Heavier N * resonances, P 11 (1440), D 13 (1520) and S 11 (1535), were included as intermediate states in Ref. [218]. The contribution from the D 13 (1520) on proton targets is sizable above E ν ∼ 1.5 GeV. Instead the other two N * are negligible (see Fig. 19).  [218]. A cut of Eγ ≥ 140 MeV in the phase space integrals has been applied. The error bands in the full-model results (solid lines) represent the uncertainty in the axial N ∆ coupling C A 5 (0) = 1.00 ± 0.11 according to the determination of Ref. [167]. The curves labeled N , ∆, D13, P11 and S11 stand for the partial contributions of the BP and CBP mechanisms of Fig 18; the label π corresponds to the πEx one. The lines labeled as "no N * " display the predictions without the N * contributions.
The pion pole (πEx) mechanism originates from the Z 0 γπ vertex, which is fixed by the axial anomaly of QCD. It is nominally of higher order [173] and, indeed, gives a very small contribution to the cross section as shown in the lower panels of Fig. 19. Other t-channel mechanisms from the exchange of vector (ρEx) and pseudoscalar (ωEx) mesons [216] arise from the anomaly-mediated Z 0 γρ and Z 0 γω interactions [219]. Among them, the ωEx contribution is favored by the size of the couplings. In addition, the isoscalar nature of the ω meson makes the ωEx mechanism potentially interesting for the coherent scattering reaction of Eq. (73). This is discussed in Sec. V D. The Z 0 γω vertex has been revisited using a framework which incorporates vector mesons as composite gauge bosons of the spontaneously broken hidden local symmetry [220]. It is shown that this vertex arises from the homogeneous part of the general solution to the anomaly equation and is not fully determined by the anomaly; the corresponding free parameters are related to the ω → π 0 γ decay. Reference [173] assumes that the ρEx and ωEx mechanisms, taken from Ref. [216], saturate the low-energy constants in the contact terms, although it is emphasized that other sources are possible. The contribution of these contact terms, and of the ωEx in particular, to the NCγ cross section on the nucleon is very small at E ν ≤ 550 MeV [173,216], as expected from power counting arguments. The extension to higher energies requires the introduction of poorly understood form factors [216,217]. The cross section from these mechanisms increases fast with energy. This rapid growth might be a concern for experiments at higher energies, or with a high-energy tail in the neutrino flux (like T2K), as a source of NCγ events and, therefore, unconstrained background. Moreover, one should recall that this trend will be limited by unitarity bounds: in a realistic framework, these amplitudes will be modified by loop contributions and partially canceled by contact terms of even higher orders.
The incoherent NCγ reaction, Eq. (72), on nuclear targets has been studied in Refs. [218,221] using the relativistic local Fermi gas approximation. The broadening of the ∆ resonance in the medium has also been incorporated to the models using a spreading potential in Refs. [217,221], while Ref. [218] uses the parametrization of the imaginary part of the in-medium ∆ selfenergy as a function of the local nuclear density derived in Ref. [194]. In the left panel of Fig. 20, the cross sections on 12 C obtained with the model of Ref. [218] are shown. It is clear that the neglect of nuclear medium corrections is a poor approximation. By taking into account Fermi motion and Pauli blocking, the cross section already goes down by more than 10%. With the full model the reduction is of the order of 30%. A similar net effect is obtained in Ref. [217]. However, the reduction quoted for the direct ∆ mechanism is substantially larger for neutrinos (∼ 50%) but not for antineutrinos, which is hard to understand (see the comparison in Fig. (9) of Ref. [218] and the related discussion). The cross section shows an approximate scaling with the target mass (A), as can be seen in the right panel of Fig. 20. Nevertheless, the cross section is smaller for heavier nuclei, particularly 208 Pb. The theoretical models outlined above have been used to calculate the NCγ events at MiniBooNE (including the coherent contribution described in Sec V D). With the model of Ref. [216] the number of these events were calculated to be twice as many as expected from the MiniBooNE in situ estimate [222]. The conclusion was that NCγ events give a significant contribution to the low-energy excess of e-like events. However, in Ref. [222] the nuclear target ( 12 C) was treated as an ensemble of nucleons, neglecting the important nuclear-medium corrections. Furthermore, an energy independent and rather high efficiency correction compared with the presently available figures [223] was assumed in the analysis. In contrast, the predictions based on the models of Refs [217,218] are compatible with the MiniBooNE determination in spite of the quantitative differences in these approaches [217,224,225]. One would then conclude that the NCγ reactions cannot explain the excess of e-like events at low E QE ν observed at MiniBooNE, which remains an open question.

A. Introduction
As discussed in Sec. IV, neutrino-induced pion, photon and (anti)kaon production off nucleons and nuclei in the intermediate energy region are not only important for neutrino oscillation experiments but are also sources of relevant data on the structure of hadrons and specially of their axial properties. At intermediate energies, pions, antikaons or photons are mainly produced through resonance excitation and these reactions can be used to extract information on nucleon-to-resonance axial transition form factors. In reactions on nuclei, these outgoing particles can be produced incoherently or coherently. In the latter case the nucleus is left in its ground state in contrast with the incoherent case where it is either broken or left in some excited state. For instance, the charge current (CC) coherent pion production (COHπ) reaction reads The same is true for the rest of neutrino induced coherent channels reviewed in this section. A proper understanding of the coherent processes is very important in the analysis of neutrino oscillation experiments. For instance, coherent π 0 or γ production by NC are among the most important ν µ -induced background to experiments that measure ν µ → ν e oscillations in the neutrino energy range around 1 GeV [215]. This is because NC γ or π 0 events can mimic ν e signal events, since the electromagnetic showers instigated by electrons or positrons and photons are not distinguishable in the large Cherenkov tanks used as far detectors 10 . Similarly, coherent CC π + production is a background in ν µ disappearance searches [226].
Coherent reactions have smaller cross sections and are clearly more forward peaked than incoherent ones. Indeed, large momentum transfers to the nucleus are suppressed by the nuclear form factor (Fourier transform of the nuclear density), which takes its largest values when the lepton transferred ( q ) and produced particle ( k coh = k π , k γ , k K · · · ) momenta are similar. Because the kinetic energy of the final nucleus is negligible, the energy of the outgoing particle, k 0 coh , coincides, in a very good approximation, with the lepton transferred energy q 0 . The limiting case q = k coh would correspond to q 2 = m 2 , being m the mass of the produced particle (photon, pion, kaon,...), which is not kinematically accessible (except for photon emission). Instead, q 2 = 0 can be reached for massless leptons in the strictly forward kinematics. In this case, the lepton tensor is such that and, because of CVC, its contraction with the vector part of the hadron tensor vanishes. This is the reason why the COHπ part of electron and photon induced reactions turned out to be a quite small fraction of the total inclusive nuclear absorption cross section [227,228]. The largest differential cross sections in coherent particle production with electromagnetic probes arise in kinematics that optimize the product of the amplitude squared of the elementary process times the nuclear form factor. However, in weak processes, there is an axial part which is not suppressed for kinematics where q and k coh are almost equal (the exception being coherent photon emission, as discussed in Sec V D). Thus, the reduction induced by the nuclear form factor is less significant, and the relative contribution of the coherent production channel to the total cross section, larger. A similar scenario is encountered in some hadronic reactions such as ( 3 He, 3 H π + ) in nuclei [229]. It is worth stressing the important role played by nuclear effects. The ∆(1232) resonance, which is the dominant intermediate state in coherent photon and pion production, is strongly modified in the nuclear medium. In addition, the pion and (anti)kaon outgoing wave functions are distorted inside the nuclei. This distortion is particularly strong in the case of pions, owing to the presence of the ∆(1232) resonance in the pion-nucleus optical potential and rather mild for kaons due to the absence of KN resonances. Thus, these processes are quite sensitive to the pion or (anti-)kaon dynamics in nuclei. We should also mention the nonlocality in the ∆ propagation which is often neglected in the microscopic models. The effects of this approximation were discussed in Ref. [230] in the context of COHπ reactions, without considering the modification of ∆ properties in the nucleus and the distortion of the outgoing pion. Sizable effects (reductions as large as a factor of two at E ν ∼ 500 MeV) were claimed in [230], which might question the validity of the local ∆ propagation. In the more realistic description of Ref. [231], the nonlocality is preserved for the ∆ kinetic term in a linearized version of the ∆ propagator but, at the same time, a local approximation for vertices and ∆ selfenergy have been adopted. Nevertheless, the mismatch between the non-local recoil effects and the local approximation is likely to be minimized by the fact that the parameters in the ∆ selfenergy are adjusted to describe pion-nucleus scattering data with the same model. The problem of nonlocality should be further investigated, performing a realistic calculation including full nuclear effects, and disentangling up to what extent some of the non-local effects are effectively accounted for in the empirical ∆−nucleus optical-potential, usually tested/fitted in pion-nucleus reactions, employed to describe the dynamics of the ∆ resonance in the nuclear environment.
B. CC and NC coherent π production reactions Models for COHπ can be classified as PCAC or microscopic. The dominance of the axial contributions at q 2 = 0 has been extensively exploited, through the PCAC hypothesis, to relate 11 the neutrino COHπ cross sections σ COH (ν + A Z | gs → ν + A Z | gs + π 0 ), σ COH (ν + A Z | gs → ℓ − + A Z | gs + π + ) with the pion-nucleus elastic ones σ(π 0 + A Z | gs → π 0 + A Z | gs ), σ(π + + A Z | gs → π + + A Z | gs ) [177,200,[232][233][234], and similarly for antineutrino induced reactions. On the other hand, microscopic approaches [110,192,193,231,[235][236][237][238] start from a model for weak pion production on the nucleon, and perform a coherent sum over all nucleon contributions, taking into account modifications of the elementary amplitudes in the nuclear medium. Since the nucleus remains in its ground state, a quantum treatment of pion distortion becomes possible. In microscopic COHπ models, the hadronic and nuclear physics input is the same employed to describe the related incoherent pion production process. The main drawbacks of this kind of approaches are: i) the available descriptions are restricted to the kinematic region where pion production is dominated by the excitation of the ∆(1232) resonance and cannot be easily extended to higher energies, and ii) these models are also technically more involved than PCAC ones and difficult to implement in MC simulations.
High neutrino energy (E ν ≥ 2 GeV) COHπ production data (including the recent NOMAD measurement of the NC COHπ 0 cross section [239]) were successfully explained with the PCAC based model of Ref. [200]. The experimental investigation at E ν ∼ 1 GeV started only recently. Contrary to PCAC based models expectations, the K2K Collaboration obtained only an upper bound for CC COHπ at E ν = 1.2 GeV [16]. This unexpected result triggered a renewed theoretical interest in this process. The present experimental situation is puzzling because the upper limits obtained by K2K [16] and SciBooNE [32] coexist with measurements of NC COHπ 0 (MiniBooNE [27] and SciBooNE [33]) with a presumably larger cross section. Indeed, SciBooNE reported [33] a value σ CC−COHπ + σ NC−COHπ 0 = 0.14 +0.30 −0.28 (78) for carbon and an average neutrino energy in the 0.8 GeV region. This result is difficult to accommodate with the relation σ CC−COHπ + /σ NC−COHπ 0 = 2 which, up to kinematic corrections, follows from PCAC and isospin invariance in the case of isoscalar nuclei. There exists a general consensus within the theory community, that though this ratio in carbon at these energies could deviate from 2, it is in any case expected to be close to 1.5. This is more than 4σ far from the SciBooNE measurement. PCAC was used by Rein and Sehgal [200] to study the COHπ 0 reaction, extending it to q 2 = 0 by means of a phenomenological form factor. Subsequently, the model for the CC reaction, has been upgraded [177] to include lepton mass effects important for low E ν studies. This simple model is a reference in the field of neutrino interactions and has been adopted by the MC simulations employed in neutrino experiments 12 . However in the E ν < 2 GeV region, and for light nuclei, there are drawbacks that render the model inaccurate [192,240]. Far from the q 2 = 0 kinematic point, PCAC models cannot be safely used to determine the angular distribution of the outgoing pions. Terms that vanish at q 2 = 0, and that are not considered in PCAC based models, provide much more forward peaked pion angular distributions [240]. (We will discuss this in certain detail below, Fig. 21). For neutrino energies above 2 GeV, the involved momenta, in most of the available phase space, are sufficiently large to guaranty that only kinematics close to the q = k coh configuration lead to non totally negligible values of the nuclear form factor, making the q 2 = 0 approximation quite appropriate. Hence, PCAC models provide accurate predictions of neutrino COHπ cross sections for high neutrino energies. However, for lower neutrino energies, these models overestimate both the CC and NC total coherent cross sections as compared to the results obtained in more realistic microscopical models.
Within microscopic models there are still in principle various approaches e.g due to differences in the treatment of the non-resonant background. A common assumption is that the reaction proceeds solely via the ∆ excitation [110,[235][236][237]. Non-resonant contributions required by chiral symmetry at threshold have been also included in Refs. [192,193] using the model for the weak pion production off the nucleon of Refs. [166,167], though they turn out to be very small because of large cancellations, some of them exact for isospin symmetric nuclei. The absolute normalization of the predicted cross-section depends on the adopted value of the dominant axial N → ∆ form factor 11 At q 2 = 0 only the axial current survives, being its contribution proportional to its divergence (qµA µ ). The relation with the pion-nucleus elastic differential cross section follows from PCAC ∂µA µ (x) ∼ fπm 2 π π(x) with fπ the pion decay constant and π(x) a pion field. In addition, one should assume dominance of the q = kπ kinematics (thanks to the nuclear form factor) and neglect some off-shell effects since q 2 = 0 = m 2 π . 12 An important improvement [240][241][242] in these models is the use of a better input for the elastic π + A Z |gs → π + A Z |gs angular differential cross section than that employed in the Rein and Sehgal model [200]. There, this distribution was approximated by the forward cross section modulated by the nuclear form factor and an angular-independent attenuation factor that should take into account effects of the outgoing pion absorption in the nucleus. As a result in [241,242], and using experimental information on the angular dependence of the elastic pion-nucleus cross section, the predicted COHπ cross section became reduced by a factor of 2-3 for neutrino energies around 1 GeV and for light nuclei in the carbon region. Apart from the obvious limitation coming from the lack of experimental data for many pion energies and nuclei, it could be also argued that because of the strong distortion of the incoming pion in the on-shell elastic pion-nucleus process, one cannot directly relate the amplitude of the latter reaction to that of pion production induced by a weak current (more details can be found in Ref. [240]).
C A 5 (0), as the process is dominated in large extent by the axial part of the weak current (central values for the total cross sections may suffer by some 20%30% uncertainty), while differential cross sections do not appreciably change their shape [193].

PWIA ∆ in medium
Full Model  [192] and [200], respectively. The variable η is defined as Eπ (1 − cos θπ), with the pion angle defined with respect the incoming neutrino momentum. In the right top panel and in the bottom panels, the microscopic ∆ contribution is calculated following [192], but using C A 5 (0) = 1.2.
In Fig. 21, we show some results at low and intermediate neutrino energies from the microscopic ∆−hole and the PCAC Rein and Sehgal models of Refs. [192] and [200], respectively. The scheme of Refs. [236,237] is quite similar to that employed in [192] and leads to qualitative similar results, while the approach of [235], although less sophisticated in the treatment of the distortion of the outgoing pion waves, makes use of the same model to account for the ∆ properties in the nuclear medium. The importance of the nuclear medium effects is firstly stressed in the top left panel of Fig. 21. To this end, different predictions for the pion momentum differential cross section are displayed [192]. The long-dashed line (in red) has been calculated using planes waves for the outgoing pion and without including any in-medium correction for the ∆. Results with ∆ nuclear medium effects are shown by the dotted line (in blue). The full model calculation of Ref. [192] including medium effects on the ∆ and the distortion of the outgoing pion wave function 13 is shown by the solid line (in black). We can see the in-medium modifications of the ∆ properties produce a strong reduction and broadening of the peak. Pion distortion further decreases the cross section and moves the maximum to lower energies.
Pion angular and q 2 distributions are displayed in the other three panels of Fig. 21, where the results from a microscopic calculation following [192], but neglecting non-resonant background terms and using C A 5 (0) = 1.2, is compared with the predictions of the PCAC Rein-Sehgal model [200]. Outgoing pion distortion is only included in the right top panel, where the modifications of the ∆ properties in the nuclear medium are also taken into account within the microscopical approach. In the bottom panels, plane waves for the outgoing pion and a free ∆ resonance were employed. From the figure, we can conclude: 1. Rein-Sehgal model predicts much larger and wider q 2 differential cross sections. As mentioned above, this is mainly due to the poor approximation assumed in this model for the the elastic π + A Z | gs → π + A Z | gs angular differential cross section. The effects of this approximation decrease with the neutrino energy and atomic number, but they are still important for E ν ∼ 1.5 − 2 GeV in a medium sized nucleus like calcium [240].
2. Regarding the outgoing pion distributions, the microscopic calculation is much more peaked around η = E π (1 − cos θ π ) = 0 than the Rein-Sehgal model results, specially far from q 2 = 0. Terms that vanish at q 2 = 0, and that are not considered in PCAC based models, provide much more forward peaked pion angular distributions. This is also true for distorted results [240].
Finally we briefly review other approaches. The model of [238] starts from Lorentz-covariant effective field theories with nucleon, pion, ∆(1232) but also scalar (σ) and vector (ρ, ω) mesons as the relevant degrees of freedom, and exhibit a nonlinear realization of (approximate) SU (2) L ⊗SU (2) R chiral symmetry. This is the same scheme as that employed in [173] for the neutrino-production of pions from nucleons. An special attention is paid to the power counting, which is shown to be valid only for quite low neutrino energies below 550 MeV. On the other hand, the approach of Ref. [110] shares many of the ingredients of previous models (in-medium ∆ modification, effective ∆−hole interactions) but there, the coherent cross section is related to the coherent part of the nuclear response (mostly longitudinal). This response is obtained within the RPA approximation when the intermediate pion is placed on the mass shell and thus, some RPA corrections driven by short distance dynamics in the spin-isospin longitudinal channel are neglected. In the forward direction, the authors of Ref. [110] explicitly relate their forward neutrino coherent cross section to the elastic cross section of physical pions, as predicted by Adler's theorem [243], thus making contact with the PCAC based models mentioned above (actually the results of [110] agree quite well, see [123], with those found in [241,242] that use the elastic differential pion-carbon data.). This infinite nuclear matter approach derived in Ref. [110] uses plain (undistorted)pion wave-functions. In spite of this, the obtained COHπ cross sections do not differ much [123] from those obtained with the microscopic models addressed above.
A different microscopic COHπ model is derived in [231] starting from a dynamical model in coupled channels, where the bare N ∆ transition from a constituent quark model is renormalized by meson clouds. Pion distortion and ∆ spreading potential effects are also taken into account. The free parameters in the spreading potential and pionnucleus optical potential are fitted to pion-nucleus elastic scattering data. The results of this approach reasonably agree with those found in [192,193] (see Figs. 8 and 9 of Ref. [123]).
To conclude, we stress once more that all microscopical theoretical models, though lead to different cross sections, however, predict a value for the ratio σ CC−COHπ + /σ NC−COHπ 0 in carbon always around 1.5 for E ν ∼ 0.8 GeV, which is in clear contradiction with the SciBooNE measurement (0.14 +0. 30 −0.28 ). This becomes an open problem and a challenge for both, theory and experimental communities.

C. CC coherent kaon and antikaon production reactions
Here we briefly discuss the coherent production of charged kaons. Namely, we consider the ∆S = ±1, Cabibbo suppressed, weak strangeness production reactions . (79) These reactions have been theoretically studied in the recent work of Ref. [244] using a microscopic approach along the lines of the COHπ study of Ref. [192] described above. At the nucleon level, the model of Ref. [208,245], presented in Sec. IV D 1, was implemented. The distortion of the outgoing kaons is treated in a quantum-mechanical way by solving pion-nucleus optical potential, based on the ∆−hole model plus some other low energy terms, that successfully describes the interaction of pions with nuclei at low and intermediate energies. This is a improvement with respect to the simpler eikonal approximation used for example in Ref. [235]. the Klein-Gordon equation with realistic optical potentials accounting for the (very different) K andK interactions in the nuclear medium.
The resulting cross sections for incident muon neutrinos of 1-2 GeV are small, with cross sections per nucleon much smaller than the corresponding ones on free nucleons. This can be explained by the rather large momentum transferred to the nucleus (due to the large value of the kaon mass compared to the typical kaon momenta) which reduces drastically the nuclear form factors. At these large momentum transfers, the nuclear form factors depend strongly on the details of the proton and neutron density distributions. Angular kaon and lepton momentum distributions are forward peaked, as it is normally the case in coherent processes. The cross section dependence on the atomic (Z) and mass numbers (A) of the target nuclei is shown in Fig. 22 for 1 GeV (anti)neutrinos. The isospin factors of the dominant CT mechanisms (see Figs 16 and 17) suggest a quadratic dependence of the cross section with (A+Z) but no significant enhancement for heavy nuclei is observed, even if the distortion is neglected. To understand this, one should recall that heavier nuclei have narrower form factors, which causes a larger suppression at high momentum transfers. The error-bars in the full-model results represent the uncertainties in the model. The errors in the proton and neutron density distributions, as well as a 10 % one in M F (Eq. (70)), accounting for the uncertainty in the elementary production model, have been propagated to the final results. In the case ofK production, the uncertainty in the K − distortion, which turns out to be the major error source, has also been estimated [244].
At higher energies, where the present model is not directly applicable, larger kaon momenta are present so that the suppressing role of the kaon mass is less important. A fast increase of the cross section is therefore expected. In view of this, measuring this reaction at MINERνA would be quite interesting.

D. NC coherent γ production reactions
The coherent contribution (NC COHγ), Eq. (73), is an important ingredient in a realistic description of NC photon emission on nuclear targets. At high energies, it was firstly studied in the early eighties [246,247]. A discussion about these works can be found in Section V.E of Ref. [216]. Here, we focus on the intermediate energy region, of relevance for the MiniBooNE and T2K experiments. At these energies, there exist three recent theoretical calculations of this reaction channel. The model of Ref [216] mostly ignores nuclear corrections. The nucleus is treated as a scalar particle, including a nuclear form factor to ensure that the coherence is restricted to low-momentum transfers. The microscopical approaches of Refs. [238] and [218] are more robust and rely on the same formalism and approximations used to study COHπ processes [192,238]. In Ref. [218], the NCγ model on the nucleon is directly applied by summing the different amplitudes coherently. The total cross section is clearly dominated by the ∆(1232), with small corrections from D 13 (1520) excitation (see Fig. 11 of Ref. [218]). Nucleon-pole contributions are negligible because the coherent kinematics favors a strong cancellation between the direct and crossed terms. The πEx terms vanish exactly for isospin symmetric nuclei because amplitudes for protons and neutrons cancel with each other. The results from the ∆P + C∆P + N P + CN P part of the model of Refs. [217,238] are within the uncertainty band of those in Ref. [218] up to (anti)neutrino energies of 1.4-1.5 GeV.
Unlike π-and ρ t-channel terms, the coherent contribution of the ωEx does not vanish for symmetric nuclei because amplitudes on protons and neutrons add up rather than cancel. In Ref. [216] it was found that the COH ωEx mechanism plays a sub-dominant role at E ν ∼ 1 GeV, compared to naive estimates, being suppressed by form factors and recoil. On the other hand, because of the strong energy dependence of the contact terms in Ref. [217], with couplings derived from the ρEx and ωEx amplitudes of Ref. [216], the NC COHγ cross section above E ν,ν = 0.65 GeV is dominated by these contact terms and not by the ∆. However, as discussed in Sec. IV D 2 in the nucleon case, the results are not only highly sensitive to unknown form factors but should also be constrained by unitarity. There are two features that make the dynamics of NC COHγ processes substantially different to that governing the COHπ ones. First, the outgoing particle (γ) does not suffer from strong distortion effects during its way out of the nucleus, and, second, the axial contribution turns out to be purely transverse ∼ ( k γ × q ) and also vanishes when q = k γ (optimal configuration for the nuclear form factor), which corresponds to q 2 = 0. Therefore, the largest differential cross sections arise in kinematics that optimize the product of the amplitude squared of the elementary process times the nuclear form factor, as in pion coherent production reactions induced by electrons and photons [227,228]. The photon angular dependence exhibited in the left panels of Fig. 23, obtained with the model of Ref. [218], should be understood from this perspective. Notice that θ γ is the photon angle with respect to the direction of the incoming (anti)neutrino beam and not the angle formed by q and k γ , which is not observable. Indeed, for each θ γ , and integration over all possible q is performed. The interference pattern between the direct dominant ∆ mechanism and the cross ∆ and the N (1520)−pole (weak excitation of this resonance and its subsequent decay into N γ) terms, also included in the elementary model employed in [218], strongly influences the angular distributions shown in Fig. 23. Neutrino (antineutrino) COHγ cross sections are about a factor 15 (10) smaller than the incoherent ones [218]. These proportions are similar to those found for pion production reactions [168,193] in spite of the suppression of the axial current for forward kinematics in COHγ reactions. This suppression is however partially compensated by the reduction of about a factor of two due to the strong distortion of the outgoing pion, which is not present in the photon case.
Predictions [218] for the outgoing photon energy distributions in various nuclei are displayed in the right panels of Fig. 23. The pronounced peak is produced by the dominant ∆ resonance. The peak position does not appreciably change from nucleus to nucleus, but it gets wider as A increases. The second, smaller and broader peak that can be seen for neutrinos but not for antineutrinos corresponds to the excitation of the D 13 (1520) resonance. For the integrated cross sections it is found [218] that these neither scale with A, like the incoherent one approximately does, nor with A 2 as one would expect from the coherence of the dominant ∆ mechanism.

VI. MONTECARLO GENERATORS
Neutrino beams are not mono-energetic and usually contain several flavors. In the experiments, it is necessary to identify the flavor of the interacting neutrino and reconstruct its direction and energy from the particles observed in the detectors. Therefore, a detailed knowledge of the final state after a neutrino interaction is essential to analyze and interpret the data. In order to evaluate the efficiency, resolution and purity of the selected event samples, or to establish the method to analyze the data, simulation program libraries, developed specifically for neutrino interactions, are extensively used. These libraries, which provide all the information on the produced particles for the different neutrino interaction mechanisms, are called Monte-Carlo (MC) neutrino event generators.
There are several generators available, such as ANIS [248], GENIE [249], GiBUU [76], NEGN [250], NEUT [251], NUANCE [187], FLUKA [252] or NuWRO [197]. Some of them have been developed by different experimental groups to be used in their analysis. There are several reasons why the different collaborations developed their own generators. The first reason is that the neutrino beam energy spectrum is specific for each experiment and thus, the dominant/relevant interaction modes could be quite different. Unfortunately, there is no unified framework able to describe all neutrino interactions in a broad energy range, so that each generator needs to combine various models. Sometimes rather simplified schemes are used to describe some interactions, which are not too relevant for a given experiment. The second reason is that the target nuclei and detectors used in each experiment are different as well. Furthermore, the detection efficiencies for each type of particle are detector dependent and can become quite different. A widespread strategy to deal with these challenges is to choose the underlying models according to the specific needs of the experiment and tune them from the data measured at the (near) detectors. These specialized generators are useful within a single experiment but, sometimes, their diversity makes the comparison of results from different experiments difficult. This also obscures the physical interpretation of the cross section data and slows down the development of more precise approaches. In order to avoid this kind of problems, the GENIE collaboration undertook the development of a new MC aiming to cover all the relevant neutrino energy ranges. On the other hand, some theory groups have also developed generators to interpret the experimental data sets that have become available in the last years.
In general, an event generator is required to provide, for any neutrino flavor, energy and target nucleus: • the total cross section, • the partial contributions to the cross section from each interaction mode, • the energy and direction of all the produced particles, simulating particle re-interactions inside the target nucleus when necessary.
The simulation programs are carefully designed to deal with the different dynamical models, paying special attention to prevent that some important interactions could be missed and to avoid unwanted overlaps between different energy regimes. Some event generators incorporate rather old theoretical models to simulate neutrino interactions, even though more sophisticate schemes have recently become available. A first reason for this is that the generators need to simulate several nuclear targets, but sometimes the models are not applicable for all nuclei. Another reason is that the models often have limitations in the kinematic ranges where they can be trustfully used, although this is no less true for the old approaches. Because the event generators have to cover the entire allowed phase space and be capable of simulating all the relevant exclusive channels, some models cannot be accommodated in them. Finally, to be able to produce millions of events, the generators have speed limitations that are not complied by all theoretical descriptions.
As mentioned in Sec. III B, to simulate CCQE and NCQE scattering, several generators still use the simple RgFG. This is because it is easy to implement and allows to change the target nucleus by modifying just a few parameters. Recently, some of the generators have started to adopt some of the more advanced approaches, with spectral functions or the RlFG with RPA corrections, discussed in Sec. III B.
For single meson production via resonances, many generators implement the relativistic harmonic oscillator model of Rein-Sehgal [159]. Even though this approach underestimates significantly pion electroproduction data (see Fig. 14 in Sec. IV B and the related discussion), it is still extensively used. Indeed, the code to calculate the helicity amplitudes was provided by the authors, it is easy to implement and includes higher energy resonances, which allows the extension of the model to simulate the resonant production of kaon, eta and multiple pions. Some authors [169] have improved the vector form factors for the dominant ∆(1232) resonance, taking advantage of the helicity amplitudes obtained from recent pion electroproduction data. Some modern generators, like GiBUU [76] and NuWRO [253], are using better models, which rely on pion electroproduction data for the vector part of the amplitudes, and on the neutrino pion production cross sections from the old ANL and BNL bubble chamber neutrino experiments to constrain the resonance axial-vector couplings.
For the simulation of multi-hadron production, deep inelastic (neutrino-quark) scattering (DIS) is assumed. DIS differential cross sections can be expressed in terms of PDF's. These PDF's are fitted to various experimental data sets and provided as a library. These PDF's are not applicable in the low q 2 0.8 GeV 2 , and low resonance mass (W 2 GeV) regions. Therefore, some correction functions have been proposed in these kinematic regions [254]. There could also be some overlaps with the single meson production contributions depending on the implementation. These overlaps are usually avoided by limiting W and the multiplicity. The kinematics of hadrons produced in DIS, the so-called hadronization, is determined with the help of PYTHIA [255,256], the standard library used in the high energy experiments. PYTHIA is also designed for large values of W , at least bigger than 2 GeV. Thus, most generators use a scaling function, known as KNO [257], whose parameters are also extracted from data.
The particles produced inside the nucleus may interact with the nucleons on their way out. These are the FSI addressed in different parts of this review. Most generators, except GiBUU and NUNDIS/NUNRES, simulate these rescatterings with a semi-classical cascade. In particular, the pions are carefully treated because their interaction probability is rather high, once many of these are produced from resonance decay. Although the basic idea is the same in several generators, the actual implementations of the particle mean free paths, the determination of the rescattered particle kinematics, or the particle production multiplicities are substantially different in the various generators. GiBUU takes FSI into account using a semi-classical transport model in coupled channels, which was briefly described in Sec. IV C. Finally, NUNDIS/NURES simulates the rescattering in the framework of the FLUKA simulation package.

VII. CONCLUDING REMARKS
Neutrino interactions offer unique opportunities for exploring fundamental questions in astrophysics, nuclear and particle physics. One of these questions regards the neutrino oscillation phenomenon, which has been established over the last 15 years. Neutrino oscillation experiments are currently evolving from the discovery to the precision stage. Unavoidably, any oscillation experiment faces a major difficulty: the elusive nature of the neutrinos. The presence of neutrinos, being chargeless particles, can only be inferred by detecting the secondary particles created when the neutrinos interact with the nuclear targets used as detectors. A better understanding of the neutrino-nucleus interactions is then crucial to minimize systematic uncertainties in neutrino oscillation experiments. For nuclear physics this represents not only a challenge, but also an opportunity. Indeed, with the recent intense experimental activity a wealth of new more precise neutrino-nucleus cross section data have become available, and more are awaited in the future.
Most of the relevant event sample falls inside the poorly understood region of resonance excitation. Experiments like MiniBooNE and SciBooNE have produced good quality data for quasi-elastic scattering and pion production at intermediate energies. These data show interesting deviations from theoretical predictions. Some of them have found an explanation (CCQE, in terms of 2p2h excitations), but other still await for a proper interpretation.
We have reviewed the recent progress in the physics of neutrino cross sections, highlighting the open questions revealed by the comparison with new experimental data. Among others, we have identified both incoherent and coherent pion production on nuclei as topical problems, where significant discrepancies between data and theoretical predictions exist and have not yet been understood. In the first case, state of the art microscopic approaches fail to describe the CC1π MiniBooNE differential and total cross sections: theoretical estimates are systematically below the data. Microscopic models predict a suppression in the pion spectra around the ∆(1232) resonance region, which is not seen in the experimental distribution of events. This points out either to an incorrect theoretical description of pion FSI effects or to a misunderstanding on the actual (dynamical) origin of the events measured by MiniBooNE. Regarding coherent pion production, the SciBooNE measurement of the ratio σ CC−COHπ + /σ NC−COHπ 0 on carbon is more than 4σ away from the bulk of the theoretical results, which are consistent with what is expected from the combination of PCAC and isospin invariance.
So far, more precise nuclear cross sections have not allowed to gain insight into the axial hadron properties because of the intrinsic difficulties in the treatment of nuclear and FSI effects. The more than 30-year old ANL and BNL low statistics deuterium pion production data are still nowadays the best source of information about the N ∆ transition matrix element. New measurements of neutrino cross sections on hydrogen and deuterium are thus absolutely necessary, even more when there is tension between ANL and BNL pπ + data samples.
We should also stress that, to achieve the precision goals in neutrino oscillation measurements and to reliably extract information about the axial properties of the nucleon and baryon resonances, it is crucial that the current theoretical developments are implemented in the event generators used in the experimental data analysis. The strategy, adopted by some oscillation experiments, of fitting the new data with the parameters available in the MC event generators is dangerous in the long term.
The MINERνA experiment at FermiLab, fully dedicated to the measurement of neutrino cross sections on nuclear targets, has started to take data. This experiment shall provide valuable information complementary to the findings of Jefferson Lab with electrons, and stimulate a productive activity in the field of neutrino cross sections in the next years.