Neutrino-Nucleus Cross Sections for Oscillation Experiments

# Neutrino-Nucleus Cross Sections for Oscillation Experiments

Teppei Katori and Marco Martini School of Physics and Astronomy, Queen Mary University of London, London, UK ESNT, CEA, IRFU, Service de Physique Nucléaire, Université de Paris-Saclay, F-91191 Gif-sur-Yvette, France Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86, B-9000 Gent, Belgium
###### Abstract

Neutrino oscillations physics is entered in the precision era. In this context accelerator-based neutrino experiments need a reduction of systematic errors to the level of a few percent. Today one of the most important sources of systematic errors are neutrino-nucleus cross sections which in the hundreds-MeV to few-GeV energy region are known with a precision not exceeding 20%. In this article we review the present experimental and theoretical knowledge of the neutrino-nucleus interaction physics. After introducing neutrino oscillation physics and accelerator-based neutrino experiments, we overview general aspects of the neutrino-nucleus cross sections, both theoretical and experimental views. Then, we focus on these quantities in different reaction channels. We start with the quasielastic and quasielastic-like cross section, putting a special emphasis on multinucleon emission channel which attracted a lot of attention in the last few years. We review the main aspects of the different microscopic models for this channel by discussing analogies and differences among them. The discussion is always driven by a comparison with the experimental data. We then consider the one pion production channel where data-theory agreement remains very unsatisfactory. We describe how to interpret pion data, then we analyze in particular the puzzle related to the impossibility of theoretical models and Monte Carlo to simultaneously describe MiniBooNE and MINERvA experimental results. Inclusive cross sections are also discussed, as well as the comparison between the and cross sections, relevant for the CP violation experiments. The impact of the nuclear effects on the reconstruction of neutrino energy and on the determination of the neutrino oscillation parameters is reviewed. A window to the future is finally opened by discussing projects and efforts in future detectors, beams, and analysis.

###### type:
Topical Review
: J. Phys. G: Nucl. Phys.

## 1 Introduction

Neutrino oscillation physics is one of the most flourishing fields in particle physics. The visibility of the active community is further increased by the the 2015 Nobel Prize and the 2016 Breakthrough Prize in Fundamental Physics both given to neutrino oscillations [1, 2, 3, 4, 5, 6].

After the discovery of the neutrino oscillations from the atmospheric neutrinos and solar neutrinos, neutrino oscillations have been further studied in the long baseline accelerator and reactor experiments. Neutrino masses and mixing, a first evidence of a new particle physics beyond the Standard Model (SM), are now well-accommodated in the standard framework of three-neutrino mixing, the so called “Neutrino Standard Model (SM)”, where the three active neutrinos , , are super-positions of three massive neutrinos , , with respective masses , , . With the high precision measurement of the small parameter  [5, 6, 7, 8] we can affirm that neutrino physics is definitely entered the precision era. Beyond a better and better determination of the five known oscillation parameters (two squared-mass differences and three mixing angles) the determination of two unknown parameters, the Dirac CP-violating phase and the neutrino mass ordering (NMO) is motivating the present and future neutrino oscillation experiments. In parallel oscillation experiments aiming at the investigation of the existence of additional massive neutrinos (the sterile neutrinos) are also pursued.

In the present review article we focus on accelerator-based neutrino experiments which, in the precision era, needs a reduction of systematic errors to the level of a few percent. These experiments measure the rate of neutrino interactions, which is the convolution of three factors: the neutrino flux, the interaction cross section and the detector efficiency. Here we discuss all three aspects but we pay particular attention to the neutrino-nucleus cross sections which in the hundreds-MeV to few-GeV energy region are one of the most important sources of systematic errors, being known with a precision not exceeding 20%. Although a majority of interaction systematic errors can be canceled by the internal measurement of oscillation experiments mainly by the near detectors, without improving the interaction models the limitations of internal constrains remain.

After an introduction on neutrino oscillation physics and accelerator-based experiments in Section 1, we discuss in Section 2 some general theoretical and experimental aspects of the neutrino cross sections on nuclei since the detectors of modern neutrino experiments are composed of complex nuclei (C, O, Ar, Fe…) which are more than a simple assembly of protons and neutrons. In accelerator-based experiments the neutrino beams (at difference with respect to electron beams, for example) are not monochromatic but they span a wide range of energies. Several reaction channels can be open and the incomplete lepton kinematic information prevents to compare theories with data in the same way as in the electron scattering. At this moment, the flux-integrated differential cross sections are the golden observables for the theory-experiment comparisons in neutrino scattering.

We focus on these quantities in different reaction channels. We start, in Section 3 with the quasielastic (QE) and quasielastic-like cross section, putting a special emphasis on multinucleon emission channel which attracted a lot of attention in the last few years, after the suggestion [9, 10] of the inclusion of this channel as possible explanation of the MiniBooNE quasielastic cross section on carbon with unexpectedly large normalization [11, 12]. Several theoretical calculations agree today on its crucial role to reproduce MiniBooNE, as well as more recent MINERvA and T2K data. However important quantitative differences remain between the calculations. These differences largely contribute to the systematic error of the neutrino experiments, depending on the way the multinucleon emission channel is inserted in the Monte Carlo generators used for the neutrino experiments. This channel was totally ignored in the generators, and the effort to include these contributions in several Monte Carlo simulations started after 2010 and it is far from conclusion. A treatment of the multinucleon emission channel (related to nucleon-nucleon correlations and meson exchange current contributions) without approximations is particularly difficult and computationally very demanding. Therefore different approximations are employed by the different theoretical approaches and by the Monte Carlo implementations. We review the main aspects of the different microscopic models by discussing analogies and differences among them. The discussion is always driven by a comparison with data (MiniBooNE, T2K, MINERvA, ArgoNeuT) and often respects the chronological order of the theoretical and experimental results, which allows, in our opinion, to better follow the rapid evolution of the field.

The single pion production is the largest misidentified background for both -disappearance and -appearance experiments. However, data-theory agreement remains very unsatisfactory. In particular there is no model which describes MiniBooNE and MINERvA simultaneously, the so called “pion puzzle”. In Section 4, we introduce pion data and describe their interpretations. The complications of pion data analyses lay not only on their primary production models, but also on the fact that all hadronic processes have to be modeled correctly. Combination of data from different channels and different experiments hope to entangle and constrain all processes, however, such an approach has been started very recent and currently we are still struggling against pion puzzle.

Inclusive cross sections are the subject of Section 5 where not only the scattering case, but also the one is presented. The vs comparison, relevant for the CP violation experiments is also discussed.

Since the neutrino beams are not monochromatic but wide-band, the incoming neutrino energy is reconstructed from the final states of the reaction. The determination of the neutrino energy is crucial since it enters the expression of the neutrino oscillation probability. This determination is typically done through the charged current quasielastic events. The reconstructed energy hypothesis used to obtain the neutrino energy from the measured charged lepton variables (energy and scattering angle) via a two-body formula is that the neutrino interaction in the nuclear target takes place on a nucleon at rest. The identification of the reconstructed neutrino energy with the real one is too crude. Several nuclear effects, such as multinucleon emission need to be taken into account. A review on the impact of the nuclear effects on the neutrino energy reconstruction is given in Section 6.

A window to the future is opened in Section 7 where we start the discussion from two flagship future accelerator-based long-baseline neutrino oscillation experiments, DUNE (argon target) [13] and Hyper-Kamiokande (water target) [14]. This clearly shows that argon and oxygen are two of the most important nuclear targets to study. The high precision measurements can be achieved by a number of new approaches mainly focusing on hadronic system information which was previously ignored. Further reductions of systematics could be possible by improving the neutrino beam quality. We discuss the effort to the future both detectors, analyses, and the beam.

We retain that the choices and the emphasis we have put on the different subjects of the present manuscript render the present review a complement of other recent articles [15, 16, 17, 18, 19, 20].

### 1.1 Neutrino oscillation physics

Neutrinos are peculiar particles within the Standard Model (SM), because their flavor states (productions and detections) are superpositions of their Hamiltonian eigenstates, , and they are related with unitary transformation ,

 |να⟩=∑iV∗αi(E)|νi⟩. (1)

In the vacuum, Hamiltonian eigenstates can be identified with mass eigenstates, i.e.,

 |να⟩=∑iU∗αi|νi⟩. (2)

Here, the unitary matrix , so-called PMNS matrix, diagonalize the mass matrix in the flavor basis

 (3)

Over the past years, the neutrino experiment community has tried to understand the structures of neutrino masses and the PMNS matrix. The PMNS matrix is usually written in terms of three Euler angle-like matrices and phase terms

 U=⎛⎜⎝1000c23s230−s23c23⎞⎟⎠⎛⎜⎝c130s13e−iδCP010−s13eiδCP0c13⎞⎟⎠⎛⎜⎝c12s120−s12c120001⎞⎟⎠ ∼⎛⎜⎝c12c13s12c13s13e−iδCP−s12c23−c12s23s13eiδCPc12c23−s12s23s13eiδCPs23c13s12s23−c12c23s13eiδCP−c12s23−s12c23s13eiδCPc23c13⎞⎟⎠ , (4)

where and are the sine and cosine of the angle, respectively. The right matrix represents the rotation between and , or mixing, and this is usually measured by solar neutrino oscillation experiments [21, 22, 23, 24, 25, 26, 27, 28] and long-baseline reactor experiment [4]. Because of the matter effect in the Sun, is known to be heavier than .

The left matrix represents mixing, which is measured through long-baseline accelerator neutrino oscillations [29, 30, 31, 32, 33] or atmospheric neutrino oscillations [33, 34, 35, 36, 37]. Although this is the first established neutrino oscillation sector, uncertainty in is the largest among three rotation angles [38], and precise value of is the key for both neutrino mass ordering and measurements.

Finally, the middle matrix is for mixing. Since and are nonzero, nonzero mixing implies nonzero Dirac CP phase, i.e., possible leptonic CP violation in neutrino oscillations. After the discovery of the nonzero both by accelerator neutrino experiments [5, 39, 40, 41] and reactor neutrino experiments [6, 7, 8], the focus of the neutrino oscillation community is mainly the determination of neutrino mass ordering (NMO) and the Dirac CP phase. Note, in the case of Majorana neutrinos, there are two Majorana CP phases, however, they do not contribute to neutrino oscillations so we do not discuss them at here.

The structure of PMNS matrix [38, 42] is strikingly different from CKM matrix [43]. Figure  1 left graphically shows the size of matrix elements. CKM matrix is dominated by diagonal terms, which allows only a small mixing between different flavors. On the other hand, PMNS matrix has large off-diagonal terms, and mixing between different generations are large. However, we do not know which is more “natural”, and the structures of these matrices are big interests of the particle theory community (see for example Ref. [44]).

Another mystery is the structure of neutrino mass ordering (NMO). Figure  1 right shows the order of neutrino masses. Each bar shows fraction of neutrino flavors. Currently, there are two candidates of NMO, so called “normal ordering”, where comes to the top, or “inverted ordering”, where comes to the bottom. The goal of current and future neutrino oscillation experiments is to find the aforementioned unknowns. On top of this, the absolute neutrino mass scale is not known but this is not measurable by oscillation experiments and we do not discuss here.

The completeness of the 3 neutrino mixing paradigm sketched above is challenged by the so called short-baseline anomalies, including reactor [45, 46, 47], Gallium [48, 49, 50, 51, 52], LSND [53], and MiniBooNE results [54, 55] which could indicate that the neutrino mixing framework need an extension in order to accommodate short-baseline oscillations. The additional squared-mass difference required to explain these anomalies with neutrino oscillations necessitates the existence of at least an additional massive neutrino at the 1 eV scale. Since from the LEP measurement of the invisible width of the boson [56], we know that there are only three active neutrinos, in the flavor basis the additional massive neutrinos correspond to sterile neutrinos [57], which do not have standard weak interactions. The search for sterile neutrinos is another strong dynamo of neutrino experimental programs in the world [51, 52, 58, 59, 60, 61].

Next, we take a look on standard neutrino oscillations more closely. The non-trivial part of the vacuum Hamiltonian in the flavor basis can be written as

 H(E)∼12EU⎛⎜ ⎜⎝m21000m22000m23⎞⎟ ⎟⎠U† . (5)

From this Hamiltonian, the evolution of an initial flavor state over a distance is solved. The probability of measuring a flavor state of energy after traveling in vacuum with distance is

 Pνα→νβ(L,E)=δαβ−4∑i>jRe(U∗αiUβiUαjU∗βj)sin2(Δm2ij4EL) (6) + 2∑i>jIm(U∗αiUβiUαjU∗βj)sin(Δm2ij2EL) , Δm2ij≡m2i−m2j .

To discuss further details of neutrino oscillations, we reduce this equation to a simpler form by using the two neutrino oscillation approximation, which is successfully used in past years. We define to be the mixing angle of two mass eigenstates, and . Then, for example, flavor state to oscillation is

 Pνα→νβ(L,E)=sin22θsin2(Δm2214EL) . (7)

First, the neutrino oscillation formula is now reduced to a simple one sinusoidal function with dependence. Super-Kamiokande was first to show the evidence of this dependence of oscillations [34], which is strong evidence that neutrino oscillations are actually caused by neutrino masses within the measured energy region by the Hamiltonian described by Eq. (5), and not any other exotic physics which do not have this dependence.

Second, Since the oscillations generated by a squared-mass difference is observable for , long-baseline neutrino oscillation experiments are characterized by a ratio which make them to be sensitive to . On the other hand, short-baseline neutrino oscillation experiments are characterized by a ratio to explore .

Third, the imaginary part is zero and the oscillation is insensitive to complex CP phases. This is equivalent to the quark sector which requires three flavors for CP violation because there is no Dirac CP phase for two quark flavor mixing. Likewise for neutrino oscillations, leptonic Dirac phase appears only under three neutrino oscillation framework.

Fourth, the mixing angle is involved in terms of , and this causes a degeneracy in , for example and give the same oscillation amplitudes within two neutrino oscillation framework. However, they give different results in full three neutrino oscillation framework where a dependence on also appears. In fact, this degeneracy is the biggest systematic to measure neutrino mass ordering through atmospheric neutrino oscillations [62].

Finally, now there is only one involved in sine square, which means that there is no sensitivity of the sign of . The situation changes if the neutrinos propagate in matter, where the cross section is different between electron neutrinos and others. For the electron neutrinos, both charged and neutral current interactions are possible with electrons in matter, but for other flavors only neutral current interactions are possible. This makes flavor asymmetric potential, the so-called Wolfenstein term [63], in the Hamiltonian

 H(E) = 12EU(m2100m22)U†+(√2GFne000) (8) = V(E)(m21(E)∗00m22(E)∗)V(E)†.

The Wolfenstein term adds different energy dependence in the Hamiltonian and the term cannot be factored out any more, and both mixing matrix elements and effective neutrino mass terms become energy dependent. This makes the oscillation equation sensitive to the sign of . This technique is used to fix the mass ordering of and in solar neutrino mixing, and long-baseline neutrino oscillation experiments are planning to use this to find the mass ordering of  [40, 41, 64, 65, 14, 13, 66]. On the other hand, neutrino mass ordering can be measured through the precise measurement of neutrino oscillations within three neutrino mixing framework. The reactor experiments use that approach to measure neutrino mass ordering [67, 68].

### 1.2 Accelerator-based neutrino experiments

Neutrinos can be generated by natural sources –this is the case of solar neutrinos, atmospheric neutrinos, geo-neutrinos, supernova neutrinos, galactic or extra-galactic neutrinos– or by artificial sources such as reactors and accelerators. In this article we especially focus on accelerator-based neutrino experiments. Here we give a brief overview of these past, present and future experiments which, in parallel with neutrino oscillation program, measure neutrino-nucleus cross sections.

K2K (KEK to Kamioka) experiment was the first long-baseline accelerator-based neutrino oscillation experiment [29], designed to measure disappearance oscillation. The neutrino beam (1.3 GeV) aims 250 km away at the Super-Kamiokande detector. The near detector complex is consisted of multiple detectors, including 1 kton water Cherenkov detector and the vertex detector of the plastic scintillation fiber tracker “SciFi” 111This is still the best name for the particle detector in the world. [69] which was later replaced with the extruded plastic scintillator tracker “SciBar” [70]. Although data from the water Cherenkov detector (HO) was important for the oscillation physics [71], most of the neutrino interaction data are from tracker measurements [72, 73, 74]. Among them, SciBar analysis demonstrates that the measurement of energy deposits around the interaction vertex, the so called “vertex activity”, is a useful variable, especially to select coherent scattering events [75].

MiniBooNE (mini-Booster Neutrino Experiment) was a mineral-oil based Cherenkov detector [76] located on the Booster neutrino beamline (800 MeV, 600 MeV) [77], designed to test the LSND neutrino oscillation results [53], namely the goal of MiniBooNE was to measure () appearance short-baseline oscillation signals [54, 55]. The Cherenkov detector was chosen as a crude way to count neutrino interactions with large fiducial volume, but it turns out that the 4 coverage detector can produce excellent neutrino cross section data. To overcome the disadvantage of not having the near detector unlike other accelerator-based oscillation experiments, MiniBooNE utilized measured () interaction to control systematics of () appearance oscillation analysis. This made MiniBooNE to try to understand detailed kinematics and backgrounds of () interactions [78, 79, 80, 81], and they became the series of first flux-integrated differential cross-section measurements [82, 12, 83, 84, 85, 86, 87].

SciBooNE (SciBar Booster Neutrino Experiment) uses the SciBar detector from K2K at the Booster neutrino beamline at Fermilab 222Extruded scintillators were made by Fermilab. They were shipped to Japan for K2K, and shipped back to Fermilab for SciBooNE, and later shipped to Puebla, Mexico for a solar neutron measurement. Similarly, EC [88] was originally constructed in Italy and used for CHORUS and HARP at CERN, before joining SciBar for K2K, then SciBooNE.. SciBooNE measured many aspects of neutrino interactions which MiniBooNE cannot measure very well, for example it measured charged current quasielastic (CCQE) from 1 and 2-track samples [89], direction reconstruction for protons below Cherenkov threshold [90], and the coherent pion production was measured utilizing the vertex activity [91, 92, 93, 94, 95]. SciBooNE is a nice cross-check of MiniBooNE results, because Cherenkov and tracker detectors are complimentary in neutrino interaction measurements:

• Cherenkov detector, isotropic 4 coverage, but it’s hard to measure more than 1 track.

• Tracker detector, relatively smaller angular acceptance, but excellent performance for multi-track events.

Figure 2 shows a comparison of a Cherenkov detector selection and a tracker detector selection. The left figure describes the CCQE sample event selection in MiniBooNE [12]. Isotropic detector contains outgoing muons in 4 direction with relatively constant efficiency. In this analysis a muon decay electron is also tagged, but again the efficiency of this is uniform across the detector. Although kinematic coverage of muons is excellent, the detector has a hard time reconstructing multiple tracks, especially, it can see the low energy protons only through the isotropic scintillation light because most protons are below Cherenkov threshold. The right figure describes the CC inclusive sample event selection in SciBooNE [96]. Notice that the event samples are further classified depending on the event topologies, such as muons stopped inside of the SciBar (“SciBar-stopped”), muons stopped inside of the MRD (“MRD-stopped”), and muons which penetrate the MRD (“muon penetrated”). This is often inevitable because the acceptance of each sub-detector is different. In general, this makes detector efficiency correction more complicated, and it makes it harder to access the true particle kinematics. On the other hand, the detector can analyze more than one particle track, and also the segmented tracker allows measurement of the vertex activity.

MINOS (Main Injector Neutrino Oscillation Search) long-baseline oscillation experiment started shortly after K2K [39, 33]. It utilizes both accelerator-based neutrinos and atmospheric neutrinos, however, in this article we focus on former. The goals of MINOS were to measure both () disappearance oscillation and () appearance oscillation signals [33]. The experiment uses the on-axis NuMI neutrino beam [98, 99], which has the ability to change the energy spectrum by configuring target and horn locations, but most of the data are taken with the low energy configuration, 3 GeV at the flux peak. MINOS has similar near and far detectors, this means the MINOS near detector is also a magnetized iron-scintillator sandwich tracker, and muon momentum is measured by the curvature and the range. A magnetic field allows a charge separation of muon neutrino and muon anti-neutrino interactions, but on the other hand calorimetric reconstruction works fine to measure energies from electromagnetic and hadronic showers. There is a handful of neutrino interaction data published [100, 101, 102]. Currently, NuMI is running with medium energy configuration (7 GeV flux peak on on-axis), and further data are expected from the MINOS extension run, called MINOS+ [103].

ArgoNeuT is the liquid argon time projection chamber (LArTPC), which we further discuss in Sec. 7.1.1. It is a dream detector, because it has almost all the features of all detectors we have discussed, such as 4 coverage, multi-particle tracking, vertex activity measurement, and calorimetric energy reconstruction. Although ArgoNeuT is only 180 L volume, it was located in front of the MINOS near detector to use it as a muon range detector  333Before ArgoNeuT, emulsion detector PEANUT [104] was located in front of the MINOS near detector. Indeed, MINOS near detector is serving as a muon range detector for someone more than 10 years, and contributed a lot of cross section measurements!, and they produced number of interesting results [105, 106, 107, 108, 109, 110].

MINERvA (Main Injector Experiment for v-A) is the tracker detector located in front of the MINOS near detector after ArgoNeuT. MINERvA uses the extruded plastic scintillator for the active material. However, MINERvA also has passive targets which are used to produce a neutrino interaction target dependence results [111, 112]. The charge separation at the down stream MINOS near detector can distinguish neutrino and antineutrino interactions [113, 114, 115, 116, 117, 118, 119, 120, 121], and the high segmentation and timing of the detector allows various particle identifications [122, 123, 124, 125, 126]. Data are taken from MINOS period to NOvA period of NuMI, which means the averaged neutrino energy of earlier MINERvA data are with NuMI low energy configuration (3 GeV), and later data are taken with NuMI medium energy configuration ( 7GeV).

NOMAD (Neutrino Oscillation Magnetized Detector) is a rather higher energy (17 GeV) experiment originally designed for () appearance oscillation measurements. Superior flux systematics and fine grained detector with a magnetic field produced very important data, relevant for context of this review [127, 128].

CHORUS (CERN Hybrid Oscillation Research ApparatUS) and OPERA (Oscillation Project with Emulsion-tRacking Apparatus) are appearance oscillation experiments. Both experiments, as well as DONUT (Direct Observation of NU Tau), use emulsion as an active target material for the main neutrino vertex detector. It has the highest resolution and is useful for the appearance measurements in DONUT and OPERA [129, 31], but it is also useful to measure high charged hadron multiplicity from neutrino interactions [130]. The main chemical elements of the emulsion is the hydrocarbon (60% of mass), however, there are non-negligible amounts of heavy elements (silver, bromine, etc), and this makes interpretation of high precision emulsion data difficult.

T2K (Tokai to Kamioka) experiment is one of two flagship long-baseline neutrino oscillation experiments in the world to date [131, 40]. It uses the J-PARC off-axis neutrino beam (600 MeV) [132]. The Super-Kamiokande detector located 295 km away is the far detector for the oscillation measurement. These designs are for () appearance oscillation measurements to find nonzero and leptonic Dirac CP phase, as well as precise measurement through () disappearance measurements. Although the far detector (Super-Kamiokande) can provide some interesting neutrino cross section results [133], the majority of cross section data are provided from the near detector complex. The ND280 near detector complex is consisted of two tracker-style near detectors. The on-axis INGRID detector [134] is the iron-scintillator sandwich trackers designed to measure the neutrino beam profile. INGRID itself can provide interesting data [135, 136], but the later installed fully active “proton module” (full scintillator tracker) can measure multi-track events and provide further high precision data [137, 138]. The off-axis ND280 detector consists of five sub-detectors: pi-zero detector (P0D) [139], fine-grained detector (FGD) [140], gas argon TPC [141], electromagnetic calorimeter (ECal) [142], and side muon range detector (SMRD) [143]. The analyses combine all of these sub-detectors, and typical analyses are based on either P0D (water target) [144, 145, 146] or FGD (CH and water target) [147, 148, 149, 150, 151, 152, 153] as vertex detectors. However, some analyses also use different targets, such as argon gas (TPC) [154] and lead (ECal) [155].

NOvA (Numi Off-axis Neutrino Appearance) experiment is the other flagship long-baseline neutrino oscillation experiment located at Fermilab. Although NuMI is running with higher energy on on-axis (7 GeV), NOvA near and far detectors are located off-axis from the beam center, making 2 GeV narrow band beam available. This configuration maximizes the sensitivity of () appearance oscillation measurements [41] and () disappearance measurements [32] to find both leptonic Dirac CP phase and NMO as well as  [62]. Like MINOS, identical design is accepted for both near and far detectors. The liquid scintillator tracker detector (CH) may be too coarsely instrumented to perform the precise interaction measurement, however, unique beam profile still allows us to measure useful flux-integrated differential cross sections [156, 157].

### 1.3 Neutrino fluxes

One of the key ingredients in neutrino physics is the neutrino flux which has to be known with the maximal precision. As already mentioned, in this article we focus only on accelerator-based neutrino experiments in the neutrino energy range of 1-10 GeV. Also in this case understanding of the neutrino flux is crucial, as it will be discussed in the following. Below we give a brief overview of current and future neutrino beams.

Modern accelerator-based neutrino beams around 1 to 10 GeV are made in several steps. First, primary protons hit the target to produce secondary meson beams. The target is usually located inside of the magnetic focusing horn, and in the neutrino mode (antineutrino mode) it focuses positive mesons (negative mesons). Then, decay-in-flight (DIF) of mesons make tertiary muon neutrino (neutrino mode) or muon antineutrino (antineutrino mode) dominated beams. These beams are so-called “super beam”, this is contrast with other types of accelerator-based neutrino beam, discussed later (Sec. 7.3).

It is common to place the detector not on the beam center (on-axis), but off from the beam center (off-axis), this makes the neutrino beam narrower and make it more convenient for oscillation analyses, comparing with on-axis configuration [158]. Understandings of production processes of neutrino beams are crucial for accelerator-based neutrino experiments. Detailed descriptions of them are published for major neutrino beamlines, including the Booster Neutrino beamline (BNB) [77] — used for the MiniBooNE/SciBooNE experiments and the host of future Fermilab short baseline programs [159, 160], J-PARC neutrino beamline [132]— being used for the T2K experiment, and the NuMI neutrino beamline [98, 99] — the host neutrino beamline for MINOS, ArgoNeuT, MINERvA, and NOvA. The biggest systematics of the neutrino beam prediction is the correct simulation of the secondary meson kinematics. This is often difficult to simulate, and the most reliable method is to use data by measuring meson production directly from the replica target at hadron measurement facilities. These experiments include HARP at CERN (with BNB and K2K neutrino beam replica target) [161], MIPP at Fermilab (with NuMI replica target) [162], and SHINE at CERN (with J-PARC neutrino beamline replica target) [163]. For more discussions on neutrino beams, see for example Ref. [164].

After incorporating the hadron production information in the neutrino beamline simulation, neutrino flux at the detector site is predicted. In this article we mainly focus on the measurements with neutrino baseline less than  km, since that is the typical baseline for the neutrino cross section experiments and near detectors of long-baseline oscillation experiments. Figure 3 shows flux predictions from current and future accelerator based neutrino experiments. Note, some of them, especially flux predictions of future experiments [14, 165] are preliminary results. The top two plots are for neutrino mode muon neutrino flux predictions, and the bottom two plots are muon antineutrino flux prediction for antineutrino mode. Left two plots are past experiments, and right two plots are current to future experiments.

• MiniBooNE used BNB, which also provided the beam for the SciBooNE experiment. Future experiments, such as Fermilab short baseline programs [159, 160] will also use it.

• MINERvA, MINOS, and NOvA use NuMI neutrino beamline. The two important flux configurations are low energy (LE) mode and medium energy (ME) mode. Also, detector configurations can be on-axis or off-axis. Here, MINOS and MINERvA are both LE and ME on-axis experiments, and NOvA is a ME off-axis experiment, and their flux predictions are quite different. Note MINERvA does not provide neutrino flux below 1.5 GeV where flux systematic errors have not been evaluated yet.

• DUNE will use a dedicated beamline, which will have a wide-band beam to measure neutrino oscillations not only the first maximum, but also the second oscillation maximum [165].

• Hyper-Kamiokande uses higher power J-PARC off-axis neutrino beam [14], and here we simply assumed the same shape with current T2K J-PARC off-axis neutrino beam.

The on-axis beam experiments, such as MiniBooNE, MINERvA, and DUNE have a wider beam spectrum, and off-axis beam experiments, such as T2K and NOvA have narrower spectrums. Although spectra are narrower for off-axis beams, they have long tails going to higher energy. This is a standard feature for off-axis beams. Therefore understanding of neutrino interactions are important in all 1-10 GeV spectrum for both on-axis and off-axis beam experiments.

Figure 4 shows more detailed neutrino flux predictions. Here, we use T2K neutrino mode flux prediction as an example to demonstrate the common features of the off-axis super beam. First, we see high energy tails in these fluxes, even though the off-axis beam peak is tuned around 600 MeV. For the neutrino mode flux, positive mesons are focused to enhance muon neutrino () components. However, there are always inevitable contaminations of muon antineutrinos () due to inefficiency to reject negative (wrong sign) mesons and muon decays. Such background ( in neutrino mode beam) is called “wrong sign” (WS) background. There are also tiny amounts of electron neutrinos () and electron antineutrinos () from muon and kaon decays. All of them are considered to be the intrinsic backgrounds of both disappearance and appearance oscillation measurements in neutrino mode. Large far detectors of oscillation experiments often lack magnetic fields to perform a sign separation of positive and negative leptons. Also, resolutions are sacrificed to maximize the fiducial volume, and this makes rather poorer particle ID ability to reject background events. Therefore, for future experiments, understanding of these tiny contaminations is crucial.

These beam contaminations are also major backgrounds of cross section measurements. The experiments often encounter the problem of predicting small contaminations correctly. For example, many WS backgrounds are originated from the forward going mesons which are not rejected by the magnetic horn, and distributions are not measured by hadron production experiments due to the lack of forward direction coverage of detector arrays [81]. Instead, experimentalists often try to measure beam contaminations in neutrino detectors by applying cuts to make a background control sample, and correct their distributions in the simulation. Such a technique was demonstrated by several experiments [81, 166]. This is a powerful way to correct intrinsic beam background distributions and to constrain their errors. However, this technique also needs care. The “backgrounds” are by definition unwanted events in the signal sample. Since measured background events in background control sample never pass the signal selection (if so, they are not the “background” from the beginning!), one needs to be very careful of how to relate measured beam intrinsic background events from background control sample and background events contaminated in the signal sample.

Detailed predictions of the neutrino flux is always difficult, and it is also common to correct flux predictions based on interactions with known cross sections in the neutrino detectors. NOMAD [127] checked the flux normalization in two ways: by utilizing DIS and inverse muon decay (IMD) cross sections. For lower energy neutrino experiments which we focus on this review, neither DIS nor IMD are not very practical for such a purpose. MINERvA [123] measures elastic scattering to constrain the flux. This process also has a theoretically well-known cross section, and distinctive experimental signature (forward going electromagnetic shower) allows them to be selected efficiently. By measuring elastic scattering events, MINERvA can effectively measure the neutrino flux even though the cross section is rather small, and MINERvA found predictions of NuMI flux are consistently higher than the measurement. Later the NuMI flux simulation was improved [99] and this disagreement no longer exists.

## 2 Neutrino cross section generalities

In this Section we discuss some general aspects of the neutrino-nucleus cross sections. We omit to treat the neutrino-nucleon scattering, reviewed for example in Ref. [18]. Our choice is driven by the fact that, as discussed in Sec. 1.2, the detectors of modern neutrino experiments are composed of complex nuclei.

### 2.1 Theory

The double differential cross-section for the reaction is given by

 d2σdΩk′dω=G2Fcos2θC32π2|k′||k|LμνWμν(q,ω). (9)

Here is the differential solid angle in the direction specified by the charged lepton momentum in the laboratory frame, is the energy transferred to the nucleus, the zero component of the four momentum transfer , with and , being the initial and final lepton four momenta. In Eq. (9), is the weak coupling constant, is the Cabbibo angle, and and are the leptonic and hadronic tensors, respectively.

The leptonic tensor is

 Lμν=8(kμk′ν+kνk′μ−gμνk.k′∓iεμναβkαk′β) (10)

where the metric is and the convention for the fully anti-symmetric Levi-Civita tensor is . The sign () before the Levi-Civita tensor refers to neutrino (antineutrino) interaction. This basic asymmetry which follows from the weak interaction theory has important consequences on the differences between neutrino and antineutrino cross sections, as it will be illustrated later.

 Wμν(q,ω)=∑f⟨Ψi|Jμ(q)|Ψf⟩⟨Ψf|Jν(q)|Ψi⟩δ(4)(Pi+q−Pf), (11)

where and are the initial and final hadronic states with four momenta and . is the electroweak nuclear current operator which can be expressed as a sum of one-body and two-body contributions. The sum over final states can be decomposed as the sum of one-particle one-hole (1p-1h) plus two-particle two-hole (2p-2h) excitations plus additional channels

 Wμν(q,ω)=Wμν1p1h(q,ω)+Wμν2p2h(q,ω)+⋯ (12)

According to Eq. (9), the same decomposition holds for the cross section.

The different components of the hadronic tensor can be combined allowing a reformulation of Eq. (9) in terms of projections with respect to the momentum transfer direction. The charged current cross section is a linear combination of five response functions

 d2σdΩk′dω=σ0[LCCRCC+LCLRCL+LLLRLL+LTRT±LT′RT′], (13)

where the kinematical factors come from the contraction with the leptonic tensor and the plus (minus) sign applies to neutrinos (antineutrinos). The letters , and stay for Coulomb, longitudinal and transverse respectively444The notation is often replaced by or .. We omit here to give the explicit expression of the kinematical factors and of the responses, which can be found in many books (e.g. Ref. [167]) and articles (e.g. Refs. [168, 169, 170, 9, 171]).

Below we give instead a simplified expression which ignores the final lepton mass contributions and which is obtained keeping only the leading terms for the hadronic tensor in the development of the hadronic current in [168], where denotes the initial nucleon momentum and the nucleon mass. In this case the response functions entering into the expression of the cross section are reduced to three

 Rα(q,ω)=∑f⟨f|A∑j=1Oα(j)eiq⋅xj|0⟩⟨f|A∑k=1Oα(k)eiq⋅xk|0⟩∗δ(ω−Ef+E0) (14)

with

 Oα(j)=τ±j,(σj⋅ˆq)τ±j,(σj×ˆq)iτ±j, (15)

for , , 555The operators are replaced by the usual 1/2 to 3/2 transition operators in the case of coupling to the .. We have thus the isospin (), the spin-isospin longitudinal () and the spin-isospin transverse () responses (the longitudinal and transverse character of these last two responses refers to the direction of the spin operator with respect to the direction of the transferred momentum ). The explicit expression of the cross section in terms of these three responses is

 d2σdcosθdω = G2Fcos2θcπ|k′|E′lcos2θ2[(q2−ω2)2q4G2ERτ(q,ω) (16) + ω2q2G2ARστ(L)(q,ω) + 2(tan2θ2+q2−ω22q2)(G2Mq24M2N+G2A)Rστ(T)(q,ω) ± 2Eν+E′lMNtan2θ2GAGMRστ(T)(q,ω)].

This expression is particularly useful for illustration since i) the different kinematic variables (related to the leptonic tensor), ii) the nucleon electric, magnetic, and axial form factors (that contain the information about the nucleon properties), and iii) the nuclear response functions (that contain the information about the nuclear dynamics) explicitly appear 666In the generic expression of Eq.(13) the nucleon form factors are implicitly included in the response functions.. It is important to stress that Eqs. (9), (13) and (16) are totally general and apply to different excitations channels: 1p-1h QE, 2p-2h, 1p-1h 1 production, coherent production, etc.

Concerning the form factors, they depend on the square of the 4-momentum transfer . The conserved vector current hypothesis allows to apply the vector (electric and magnetic) form factors measured in electron scattering to neutrino scattering. The axial form factor is usually described by a dipole parameterization

 GA(Q2)=gA(1+Q2/M2A)2. (17)

The coupling is well known from neutron decay, . The value of the axial mass parameter extracted from charged current quasielastic experiments on deuterium bubble chambers [172, 173, 174, 175] is GeV [176]. The value of this axial mass parameter attracted a lot of attention in connection with the MiniBooNE CCQE result, as it will be discussed in Sec. 3.

Turning to the nuclear responses entering in Eq. (16) an example of these is given in Fig. 5 where the different C spin-isospin transverse responses , calculated in Random Phase Approximation (RPA) according to the approach of Ref. [9], are plotted for fixed values of the momentum transfer , as a function of the energy transfer . One can easily distinguish the quasielastic response, which corresponds to one nucleon knockout. It is peaked around

 ω=√q2+M2N−MN=Q22MN=q2−ω22MN, (18)

the value corresponding to the quasielastic scattering with a free nucleon at rest. RPA collective effects can shift the position of this quasielastic peak [177] with respect to the one given by Eq. (18). The broadening of the quasielastic response is due to the Fermi motion. The quasielastic response can be distorted by Pauli correlations or by the collective nature of the response, as described in RPA.

The curve characterized by a bump at higher corresponds to the resonance excitation, which decays via the pionic channel . It is peaked around

 ω=√q2+M2Δ−MN=Q22MN+M2Δ−M2N2MN. (19)

This second curve is related to the pionic decay in the nuclear medium (hence to the 1 production channel). Pauli blocking of the nucleon and the distortion of the pion are taken into account. In the nuclear medium non pionic decay channels are also possible such as the two-body (2p-2h) and three-body (3p-3h) absorption channels, which leads to np-nh excitations.

The total np-nh excitations channel, which also includes other 2p-2h excitations which are not reducible to a modification of the width is also shown. The part of the np-nh excitations related to the two-body meson exchange currents (MEC) contributions is separately plotted. We postpone a detailed discussion of these np-nh excitations to the Section 3.2. By comparing with data, one can notice that it is crucial to fill the dip between the quasielastic and excitations, as first observed by Van Orden and Donnelly [178] and Alberico et al. [179] in the studies of the electron scattering cross sections and transverse responses.

The different spin-isospin transverse responses calculated in the approach of Ref. [9] are shown in three-dimensional plots in Figs. 8 and 8. These figures well illustrate the response regions i.e. the regions of the and plane where the responses are nonzero. The sum of quasielastic and contributions is given in Fig. 8. This figure allows to easily distinguish the quasielastic and region as well as the position of the quasielastic and peaks, which for a non-interacting system would follow the nucleon and dispersion relations of Eqs. (18) and (19), respectively. The quasielastic response region is delimited by the two lines , where is the Fermi momentum. The spreading of the response region is due to the nucleon Fermi motion and to the decay width which is modified by the interaction of the with surrounding nucleons. The np-nh response function (and region) is separately plotted in Fig. 8. One can observe that it covers mostly the whole and plane, in particular it is non vanishing in the dip region between the nucleon-hole and domains. The two substructures appearing in Fig. 8 reflects the different origins of the np-nh excitations, such as the nucleon-nucleon correlation contributions (lower part) and the non pionic decay contributions (higher part).

For completeness, the main contribution to the coherent pion production response, represented by spin-isospin longitudinal coherent response is also shown in Fig. 8. Other examples of nuclear responses calculated in the same approach can be found in Ref. [9]. For example, Figs. 5 and 6 of Ref. [9] illustrate the reshaping effects of the nuclear responses due to the role of the effective interaction between particle-hole excitations as well as collective features of the spin-isospin longitudinal response, in particular in the coherent channel.

As illustrated, nuclear cross sections are naturally expressed in terms of the nuclear responses, functions of the energy and momentum transferred to the nuclear system. Figure 9 shows a comparison of a typical electron scattering experiment and a typical accelerator-based neutrino experiment [180]. In the electron scattering experiment (left), the beam energy is precisely known, and experimentalists measure energy and angle of scattered electron. In this way, both and are determined from given interactions and kinematics are fully fixed. On the other other hand, modern accelerator-based neutrino experiments (right) are performed with a wide-band beam with a fully active detector to maximize the interaction rate. Since the neutrino energy of a given interaction is not know a priori, experimentalists do not know the neutrino energy of the given interaction, and and cannot be determined from the measured lepton distribution777On the other hand, the fully active detector can records all other tracks and activities from hadrons, and such information can be used to reconstruct the kinematics, as demonstrated by MINERvA [119] and discussed in Sec.3.3.2. The measured variables in neutrino scattering are only the charged lepton energy (or kinetic energy ) and its scattering angle , related to the energy and momentum transfer by

 ω=Eν−El (20)

and

 q2=E2ν+k′2l−2Eν|k′l|cosθ. (21)

The experimental measured quantity is then the flux-integrated double differential cross section in terms of the measurable variables and :

 d2σdTl d cosθ=1∫Φ(Eν) dEν∫ dEν[d2σdω dcosθ]ω=Eν−ElΦ(Eν), (22)

where is the neutrino flux.

The cross section of the r.h.s. of Eq. (22), as expressed in terms of the nuclear responses, according to Eqs. (9) and (16), is non vanishing in the regions of the and plane where the responses are non-zero, regions shown in Figs. 88 and 8. To illustrate how these regions are explored in neutrino reactions we repeat an argument of Refs. [181, 182, 183]. We write the squared four momentum transfer in terms of the lepton observables (for illustration we take the example of an ejected muon)

 Q2=q2−ω2=4(Eμ+ω)Eμ sin2θ2−m2μ+2(Eμ+ω)(Eμ−|k′μ|) cosθ. (23)

For a given set of observables and , this relation defines a hyperbola in the and plane [181]. The asymptotic lines are parallel to the line ( always, the hyperbolas lie entirely in the space-like region) and the intercept of the curves with the axis occurs at a value of the momentum

 q2ω=0=4E2μ sin2θ2−m2μ+2Eμ(Eμ−|k′μ|) cosθ≃4E2μ sin2θ2, (24)

where the second expression is obtained by neglecting the muon mass. With increasing or increasing angle, this point shifts away from the origin. The neutrino cross section for a given and explores the nuclear responses along the corresponding hyperbola. Some examples of hyperbolas are shown in Fig. 10, together with the region of the quasielastic response of a Fermi gas. For simplicity we have omitted to show the , the np-nh and the coherent pion response regions, already illustrated in Figs. 88 and 8. The intercept of the hyperbola with the region of response of the nucleus, whatever its nature (quasielastic, , np-nh, coherent pion), fixes the possible and values for a given value of and .

In the case of a dilute Fermi gas where the region of quasielastic response is reduced to the quasielastic line given by Eq. (18) (the black dashed line of Fig. 10), the intercept values and are completely fixed. Hence the neutrino energy is also determined, , which leads to the well known expression of the reconstructed energy for a fixed set of muon observables, and :

 ¯¯¯¯¯¯Eν=Eμ−m2μ/(2MN)1−(Eμ−|k′μ|cosθ)/MN. (25)

The corresponding value of the reconstructed squared four momentum transfer is

 ¯¯¯¯Q2 = q2int.−ω2int. (26) = 4(Eμ+ωint.)Eμ sin2θ2−m2μ+2(Eμ+ωint.)(Eμ−|k′μ|) cosθ = −m2μ+2¯¯¯¯¯¯Eν(Eμ−|k′μ| cosθ).

The quantities and are often equivalently called and in literature. The set of (,) points satisfying Eqs. (25) and (26) for various values of and is shown in the and plane in Fig. 11 of Sec. 2.3.

A similar procedure could be repeated by considering the intersection of the hyperbolas shown in Fig. 10 with the dispersion relation of Eq. (19) instead of the free nucleon dispersion relation of Eq. (18), when interested in pion production instead of quasielastic.

The complexity of the nuclear physics implies a more subtle and delicate situation. As already discussed, first, the nuclear region of response is not restricted to a line for the QE and , but it spreads around these lines (see Fig. 8) and second, very important, it covers the whole and plane due to multinucleon emission (see Fig. 8). As a consequence, for a given set of values of and , moving along a hyperbola one explores the whole and plane hence, all values of the energy transfer contribute to the cross sections. In other words, for a given set of values of and one explores the full energy spectrum of neutrinos above the muon energy since . This fact has fundamental consequences on the determination of the neutrino energy in the neutrino oscillation experiments, as it will be illustrated in Section 6. Another aspect related to this point is that all the reaction channels (QE, np-nh, pion production,…) are entangled and isolating a primary vertex process from the measurement of neutrino flux-integrated differential cross section is much more difficult than in the cases of monochromatic (such as electron) beams. This is illustrated for example in Ref. [184] where a theoretical model based on the impulse approximation scheme and the nuclear spectral function turns to successfully reproduce the quasielastic peak of electron scattering double differential cross section data on carbon as a function of the transferred energy for fixed scattering angle but not the MiniBooNE neutrino flux integrated quasielastic-like double differential cross section as a function of the muon kinetic energy at the same scattering angle.

### 2.2 Experiment

Let’s now analyze the neutrino cross sections from an experimental perspective. Neutrino experiments measure the rate of neutrino interactions, . The interaction rate is the convolution of three factors: neutrino flux, , interaction cross section, , and the detector efficiency,  [185]

 R∼Φ(Eν)⊗σ(k,k′)⊗ϵ(observed particle kinematics). (27)

Here, neutrino flux is a function of neutrino energy, , and neutrino interaction cross section is a function of initial and final lepton kinematics. The detector efficiency can be a function of any kinematic variables of observables, such as energy deposit, scattering angle, etc. To compare the experimental data with predictions, experimentalists simulate the neutrino flux, the interaction cross section, and the detector model, and convolute them with the Monte Carlo (MC) method. Every experiment strives to understand the detector performance for given kinematics, and this allows experimentalists to unfold detector responses. Then, the measured quantity is the detector effect unfolded rate, and it is proportional to the flux-integrated cross section.

 R′∼Φ(Eν)⊗σ(k,k′) . (28)

Reconstruction of neutrino energy is non-trivial in this energy region as we discuss in Sec. 6. This essentially makes it impossible to unfold neutrino flux term without introducing a model dependence in the final result. Therefore, modern neutrino interaction measurements focus on producing flux-integrated differential cross-sections of direct observable kinematics (lepton energy, lepton scattering angle, total hadron energy deposit, etc.) from topology-based signals. Here, we show how to relate measured event distribution to theoretically calculable quantities.

A histogram of observed neutrino interaction events is distributed in a vector , which may be the functions of lepton kinetic energy and scattering angle, i.e., ,. The index of the data vector emphasizes this vector is a function of observed variables (such as measured muon energy and direction). After subtracting the background contribution , detector bias is corrected. This detector bias unfolding process is often separated into two processes: unsmearing and efficiency correction. A proper unsmearing method transforms the background subtracted data (or sometimes data including backgrounds) from the function of observables to the function of true variables. This corresponds to applying unsmearing matrix to change the index from to , which represents true variables (such as muon energy and direction without any detector bias). Then the estimated detection efficiency is inversely applied to recover the true distribution. Finally, by correcting all normalizations, such as total exposed flux , total target number, T, and bin width and , the double differential cross section is obtained

 (d2σdTlcosθ)i=∑jUij(dj−bj)Φ⋅T⋅ϵi⋅(ΔTl,Δcosθ)i . (29)

As one can see, this is equivalent to Eq. (22), therefore, flux-integrated differential cross section is the point where theorists and experimentalists meet for neutrino interaction physics. Note, details of Eq. (29) may depend on experiments since there are a number of ways to remove backgrounds, unsmear distribution, and correct the efficiency. Furthermore, one could compare experimental data with a theory without unfolding, instead, “fold” detector efficiency and smear and add backgrounds on a theoretical model (forward folding) which may be a favored method from a statistical point of view [186]. However, currently the community standard for experimentalists is to unfolded detector bias to present data.

### 2.3 Matching Theory and Experiment

Experimentalists strive to measure the flux-integrated differential cross section, and theorists calculate that by convoluting their models with flux from each experiment. Because of the flux-dependence of the measured cross sections unlike flux-unfolded total cross sections, every single experiment with different locations or different neutrino beams would measure different differential cross sections. Therefore, comparisons of data sets from different neutrino beams are non-trivial, and they can only be related through the theoretical interaction models. We will come back to this in the next sections, in particular in Sec. 4.1 in connection with the one pion production cross sections. Here we give an example of analysis which can be performed by using a Monte Carlo events generator, the tool bridging theory and experiment.

We have seen in Sec. 2.1 that the and plane of Figs. 10 are useful to appreciate where the nuclear responses lie and which region is explored for fixed values of and . Since the flux-integrated double differential cross sections are function of and , to analyze what happens in the and plane is also very illuminating and allows to bridge theoretical properties of neutrino interactions and nuclear models with the experimental situation. To illustrate this point we consider a simple situation: genuine CCQE events generated by GENIE neutrino interaction generator (version 2.8.0) [187]. This generator uses the relativistic Fermi gas (RFG) model, the simplest model for the nuclear structure and the only one considered in Monte Carlos for many years. Figure 11 shows CCQE flux-integrated double differential cross sections on carbon target for MiniBooNE (top left), T2K ND280 near detector complex (right), and MINERvA (bottom left). For T2K, both muon neutrino in neutrino mode (top right) and muon antineutrino in antineutrino mode (bottom right) are calculated. For MiniBooNE and MINERvA, only muon neutrino CCQE results are shown. Each marker represents an event. The events can be thought of as the intersection between the nuclear response region and the hyperbolas of Figs. 10, weighted by the different neutrino fluxes shown in Sec. 1.3. The differences in these fluxes (in particular the one of MINERvA with respect to the MiniBooNE and T2K cases) are reflected in the different behavior in the and plane. Events are classified by three colored marker types, depending on their three-momentum transfer.

The blue (dark gray) “” markers are events with . This is smaller than roughly the twice of Fermi motion of typical nuclei, and it is the kinematic region where impulse approximation (interaction with one single nucleon in the nucleus) starts to be violated, as illustrated by comparisons with inclusive electron scattering data [188, 189]. Therefore if many events are classified in here, one should consider models beyond the impulse approximation (IA), such as the RPA, or IA-based models with necessary corrections. As one can see in Fig. 11, all experiments considered here include sizable amount of low momentum transfer events (roughly 20% of all CCQE interactions).

The magenta (gray) “” markers have . In this context, this is the “safe region” where most models work fine for the genuine CCQE. We remind however that multinucleon interaction contributes also in this region, so the correct description of the experimental data requires care on models.

Finally, the green (light gray) “” are for . This is the other delicate region. The validity of the nucleon-nucleon effective interaction employed in the RPA-based calculations for genuine quasielastic, such as the ones of Refs. [170, 9], at high momentum transfer is delicate [190]. This is true also for other channels, for example MeV is the limit up to which 2p-2h contributions are included in the model of Nieves et al.  [191] even when they investigate neutrino interactions up to 10 GeV [192].

Several lines are overlaid on Fig. 11 to clarify some kinematics discussions. Four dashed lines are for constant (0.6, 1.0, 1.5, 3 GeV), according to Eq. (25), and four solid lines are for constant (0.2, 0.6, 1.0, 1.4 ), according to Eq. (26). The angular acceptance in the different experiments is represented by a red horizontal line. Below, we summarize what we learn from each plot.

MiniBooNE — There are many events with (27% of all CCQE events) and they are all below . Because the detector has a 4 coverage, angular acceptance is (forward scattering) to (back scattering). This large acceptance helps to understand underlying interaction physics. The K2K experiment [72] showed a deficit of CCQE candidate events at very forward scattering region. It wasn’t understood until MiniBooNE analyzed full 4 kinematic space to show the deficit is not only forward scattering, that could be an inefficiency of the detector, but low region, which must be some physics [78]. As it will be illustrated in Sec. 3, RPA collective effects give a reduction of the CCQE cross section at low .

T2K (neutrino mode) — Narrower J-PARC off-axis beam (see Fig. 3) makes a very small number of interactions below unlike MiniBooNE. The tracker nature of T2K near detectors has small angular acceptance, and here we defined the current acceptance as and draw a red line above where 95% of CCQE candidates are observed by the T2K ND280 CCQE analysis of 2014 [151]. In this limited kinematic space, there is a similar fraction of events as in MiniBooNE (21% of all CCQE interactions). This means that T2K and MiniBooNE should follow similar interaction physics in terms of RFG model. Although current analysis can accept majority of total CCQE events, larger angular events may be also very interesting since they are related to transverse response (the response most affected by the two-body current contributions). We note the potential acceptance of T2K ND280 near detector is larger, because the detector itself has an ability to measure leptons with higher scattering angle thanks to the ECal and the SMRD surrounding the fiducial volume. In principle, can be measured and in this case the analysis can accept more than 95% of CCQE events.

We note the problem of the detector for the total cross section measurements. If the acceptance is small, one needs to “guess” unmeasured number of events to estimate the total cross section, and this often requires the model dependent correction. To overcome this model dependency, neutrino interaction model systematics error must be included and the total error increases, as shown in T2K on-axis CCQE cross section measurements [137]. On top of that, flux-unfolded total cross section requires the reconstruction of neutrino energy (Sec. 6) which adds additional errors. This is a common problem for any cross section measurements. One solution is to present “fiducial cross sections”, that are defined in restricted phase space or limited acceptance. By applying those restrictions in theoretical models, a theory and data are comparable without adding bias in the data.

T2K (antineutrino mode) — The distribution of events is similar with its neutrino mode, except that more events are concentrated (as expected and discussed also in Sec. 3.1 in connection with the MiniBooNE results) in the forward angle region, and lower momentum transfer events () are around 33%, hence higher than in the neutrino mode. This implies physics beyond IA is more important for antineutrino mode than neutrino mode, and models working for neutrino mode, if based on IA, may not work properly in antineutrino mode. Since the antineutrino mode measurement is an important part of CP violation measurement, one should keep in mind this kinematics difference. A quantitative analysis of RPA and multinucleon effects for neutrino and antineutrino can be found for example in Ref. [10]

MINERvA — Higher energy NuMI beam makes very different kinematics with the previous two experiments. The angular acceptance of MINERvA experiment is considerably smaller if the MINOS matching is required. However, in this energy region momentum transfer can be well separated by small angles, and MINERvA also covers a similar kinematic space with MiniBooNE and T2K, for example, the fraction of events with is 21%. Not surprisingly, there are also many events with (18%), because of the higher energy beam. Majority of them are higher energy, forward going events. A similar analysis is performed in Ref. [193] where the Super-Scaling approach is used to evaluate the different and contributions to the MINERvA distribution.

## 3 Quasielastic

### 3.1 CCQE, CCQE-like, and CC0π

In the discussion of the CCQE cross section, the MiniBooNE measurement, obtained using a high-statistics sample of CCQE events on C, plays a central role. The results were presented for the first time at the NuInt09 conference [11] and then published in Ref. [12]. In this work the quasielastic cross section is defined as the one for processes in which only a muon is detected in the final state, but no charged pions. However it is possible that in the neutrino interaction, a pion produced via the excitation of the resonance escapes detection, for instance because it is reabsorbed in the nucleus. In this case it imitates a quasielastic process. The MiniBooNE analysis of the data corrected for this possibility via a data driven correction based on the simultaneous measurement of CC1 sample. The net effect amounted to a reduction of the observed quasielastic cross section. After applying this correction, the quasielastic cross section thus defined still displayed an anomaly. The comparison of these results with a prediction based on the relativistic Fermi gas model using in the axial form factor (cf. Eq. (17)) the standard value of the axial cut-off mass GeV, consistent with the one extracted from bubble chamber experiments, reveals a substantial discrepancy. The introduction of more realistic theoretical nuclear models, assuming the validity of the hypothesis that the neutrino interacts with a single nucleon in the nucleus, does not alter this conclusion. This is illustrated in Fig. 13. This figure, published in Ref. [194] shows the CCQE -C cross section as a function of neutrino energy calculated within several models already applied in electron scattering studies where they provided satisfactory agreement with data. These models are the spectral function approaches of Refs. [195, 196], the local Fermi gas plus RPA approaches of Refs. [170, 9, 197], the relativistic mean field of Ref. [198] and GiBUU [199, 189]. All these theoretical results were collected for the NuInt09 conference and published in Ref. [200] where a synthetic description of these different theoretical models is also given. From Fig. 13 it clearly appears that the mentioned theoretical predictions, all using the standard value for the axial mass, underestimate the MiniBooNE data. In Fig. 13 is also shown that in the relativistic Fermi gas model an increase of the axial mass from GeV to the larger value of GeV can reproduce the MiniBooNE data, as discussed in Refs. [11, 12].

A possible solution of this apparent puzzle was suggested by Martini et al. [9] which drew the attention to the existence of additional mechanisms beyond the interaction of the neutrino with a single nucleon in the nucleus, which are susceptible to produce an increase of the quasielastic cross section. The absorption of the boson by a single nucleon, which is knocked out, leading to 1 particle - 1 hole (1p-1h) excitations, is only one possibility. In addition one must consider coupling to nucleons belonging to correlated pairs (NN correlations) and two-nucleon currents arising from meson exchange (MEC). This leads to the excitation of two particle -two hole (2p-2h) states. 3p-3h excitations are also possible. Together they are called np-nh (or multinucleon) excitations. A schematic and pictorial representation of the 1p-1h and 2p-2h excitations is shown in Fig. 14. As shown in Ref. [9] and in Fig. 13 the addition of the np-nh excitations to the genuine quasielastic (1p-1h) contribution leads to an agreement with the MiniBooNE data without any increase of the axial mass. Isolating a genuine quasielastic event in electron scattering experiments where the kinematics are fixed by the knowledge of the energy and momentum of incoming and outgoing electron beams is relatively easy. In the double differential cross sections, or in the nuclear responses, one can isolate the bump centered at corresponding to single nucleon knockout, shown for example in Fig. 5. This is not the case in neutrino scattering experiments. Due to the broadening of the incoming neutrino flux as illustrated in Sec. 2.1, one explores the whole energy- and momentum-transfer plane, hence the multinucleon excitations are strictly entangled with the single nucleon knockout events. This is particularly true for Cherenkov detectors. The importance of 2p-2h excitations in neutrino scattering processes was suggested for the first time by Delorme and Ericson in Ref. [181]. The confusion between one nucleon knock out processes and multinucleon excitations in the Cherenkov detectors was stressed as first by Marteau et al. [201, 202] in connection with the atmospheric neutrino measurements at Super-Kamiokande. Today one generally refers to single nucleon knockout processes as true or genuine quasielastic. Processes in which only a final charged lepton is detected, hence including multinucleon excitations, but pion absorption contribution is subtracted, are usually called quasielastic-like, or QE-like. Thus, what MiniBooNE published was not CCQE data, but CCQE-like data. To avoid the confusion of the signal definition, it is increasingly more popular to present the data in terms of the final state particle, such as “1 muon and 0 pion, with any number of protons”. This corresponds to the CCQE-like data without subtracting any intrinsic backgrounds (except beam and detector related effects) and it is called CC0. We will discuss the advantage of such topology-based signal definition in Sec. 4.1.

The results presented in Figs.13 and 13 relate to cross sections as a function of the neutrino energy. Nevertheless the experimental points shown in these figures are affected by the energy reconstruction problem, see Section 6. For a comparison between theory and experiment, the most significant quantities are the neutrino flux-integrated double differential cross sections, as defined in Eq. (22) (theory) and Eq. (29) (experiment), which are functions of two measured variables: the muon energy and the scattering angle. The comparison between the experimental MiniBooNE results [12] and the theoretical calculations of Martini et al., as published in Ref. [182] is given in Fig. 15. A very good agreement with data is obtained once the multinucleon component is included. Similar conclusions have been obtained by Nieves et al. in Ref. [203].

In 2013 the MiniBooNE collaboration published the measurements of the antineutrino CCQE-like cross section on carbon [86]. Similar agreements between theory and experiments for the flux-integrated double differential cross sections and similar conclusions on the crucial role of np-nh excitations have been obtained by Nieves et al. [205] and by Martini and Ericson [204]. A full calculation of the MiniBooNE flux-integrated neutrino and antineutrino double differential cross section was also given by Amaro et al. in Ref. [206, 207] in the super-scaling analysis (SuSA) approach. In this context for the neutrino scattering the inclusion of the vector MEC gives a relatively small contribution which reduces the discrepancy between the MiniBooNE results and the theoretical predictions, but it is not enough to reproduce data. The situation is different for antineutrino cross section where the vector MEC contribution turns to be large. These results have been updated by Megias et al. in Ref. [208] and further updated in Ref. [209] by considering the SuSAv2 approach and by including the axial MEC contributions. We postpone the discussion on the comparison among models and on the relative role of np-nh excitation in neutrino and antineutrino scattering to the next subsection. For the moment we show in Fig. 16 the neutrino and antineutrino differential cross section and , as measured in Refs. [12] and [86] and as calculated in Refs. [182] and  [204]. From the panels, one can observe that the antineutrino cross sections falls more rapidly with angle than the neutrino one. This also reflects in the distributions. It is a consequence of the difference of sign in front of the vector-axial interference term of the cross section, cf. Eq. (16). This different and behavior between neutrino and antineutrino cross section is due first to the kinematic factor in front of the nuclear responses. It would survive even if one considered, as in the Fermi gas model, that all the nuclear responses would be the same. In reality, due to the nuclear interaction, the nuclear responses are different in the different spin isospin channels. As a consequence, the vector-axial interference term introduce an additional asymmetry between neutrino and antineutrino since the various nuclear responses weigh differently in the neutrino and antineutrino cross sections. This point was analyzed in details in Refs. [10, 207]. The asymmetry of the nuclear effects for neutrino and antineutrino is important for CP violation studies. The nuclear cross-section difference for neutrinos and antineutrinos stands as a potential obstacle in the interpretation of experiments aimed at the measurement of the CP violation angle, hence has to be fully mastered. It will be further discussed in subsection 3.2. In subsections 3.2 and 3.3, other experiments following the MiniBooNE one will be also discussed.

### 3.2 np-nh excitations: theory vs experimental dσ

After the suggestion [9] of the inclusion of np-nh excitations mechanism as the likely explanation of the MiniBooNE anomaly, the interest of the neutrino scattering and oscillation communities on the multinucleon emission channel rapidly increased. Indeed this channel was not included in the generators used for the analyses of the neutrino cross sections and oscillations experiments. It can be inferred also from the large values of the axial mass deduced from other neutrino scattering experiments on nuclei: K2K on oxygen GeV [72]; K2K on carbon GeV [210]; MINOS on iron GeV [101]. The only exception is NOMAD, a higher-energy experiment on carbon who obtains a value of GeV [127]. The high values of the axial mass indicate the presence of a np-nh component not considered in the analysis. Today there is an effort to include this np-nh channel in several Monte Carlo [211, 212, 213, 214].

Concerning the theoretical situation, nowadays several calculations agree on the crucial role of the multinucleon emission in order to explain the MiniBooNE neutrino [12] and antineutrino [86] data as well as the SciBooNE [96] and T2K inclusive  [148, 150] and CC  [215] cross sections. Nevertheless there are some differences on the results obtained for this np-nh channel by the different theoretical approaches. The aim of this section is to review the actual theoretical status on this subject.

The theoretical calculations of np-nh excitations contributions to neutrino-nucleus cross sections are actually performed essentially by three groups. There are the works of Martini et al. [9, 10, 182, 183, 216, 204, 217, 218, 219], the ones of Nieves et al. [191, 203, 220, 205, 192] and the ones of Amaro et al. [206, 221, 207, 222, 223, 208, 224, 225, 226, 209].

The np-nh channel is taken into account through more phenomenological approaches by Lalakulich, Mosel et al. [227, 228, 229, 230] in GiBUU and by Bodek et al. [231] in the so called Transverse Enhancement Model (TEM). In the case of GiBUU in Refs. [227, 228, 229] the size of the squared matrix element of the neutrino-induced two-nucleon knock-out cross section is obtained by fitting the neutrino charged current quasielastic MiniBooNE cross section on carbon. This is a pure two-nucleon phase-space model. Recently a more realistic np-nh contribution, where an empirical response function deduced from electron scattering data is used as a basis, has been implemented in GiBUU [230]. It allows a simultaneous description of neutrino and antineutrino MiniBooNE CCQE-like data, as well as and T2K CC inclusive cross sections. In the TEM model [231] the magnetic form factor is enhanced with respect to the standard dipole parameterization according to the formula . The parameters and are fitted to reproduce inclusive electron scattering data on carbon. This is is the effective way to include the meson exchange currents contribution in electron and neutrino scattering. In the same spirit of the modification of the axial mass, instead to modify the nuclear responses contribution, which requires elaborated many-body calculations, a simple modification of the magnetic form factor, easy to implement in the Monte Carlo is proposed. Two-body current contributions to the axial part of the cross sections are not taken into account in this TEM model. Recent ab initio many-body calculations of neutral-weak responses and sum rules performed by Lovato et al. [232, 233] have separated these axial contributions, showing their relevance. Also very recently, fully relativistic calculations of MEC contributions to the weak nuclear responses and neutrino cross sections, performed by Simo et al.  [225] and Megias et al.  [209] respectively, show the role of the axial contribution. This important point will be analyzed in more detail later.

Beyond all the theoretical approaches and models mentioned above, other interesting calculations discussing the 2p-2h excitations in connection with the neutrino scattering appeared in 2015 and 2016  [234, 235, 236]. Since, for the moment no comparison with neutrino flux-integrated differential cross sections are shown, in the following we will focus essentially on the results obtained by the three theoretical approaches which calculate these quantities: the ones of Amaro et al., Martini et al. and Nieves et al.

Considering these three different models, it is important to remind that there exist some differences already at level of genuine quasielastic, which can be particularly important when one compares the double differential cross sections. Amaro et al. considered the relativistic super-scaling approach (SuSA) [237] based on the super-scaling behavior exhibited by electron scattering data. It has been recently extended by Gonzalez-Jimenez et al. [238] in order to take into account the different behavior of the longitudinal and transverse nuclear responses due to relativistic mean field effects. This new version of the model is called SuSAv2. The models of Martini et al. and Nieves et al. are more similar: they start from a local Fermi gas picture of the nucleus. They consider medium polarization and collective effects through the random phase approximation (RPA) including -hole degrees of freedom, and meson exchange and Landau-Migdal parameters in the effective interaction.

Turning to the np-nh sector, let’s remind, as first, the sources and the references of the calculations of the three groups. The np-nh contributions in the papers of Martini et al. are obtained starting from the microscopic calculations of the transverse response in electron scattering performed by Alberico et al. [179], from the results of pion and photon absorption of Oset and Salcedo [239] and from the results of pion absorption at threshold of Shimizu and Faessler [240]. The 2p-2h contributions considered by Amaro et al. are taken from the full relativistic model of De Pace et al. [241] related to the electromagnetic transverse response. The extension of this full relativistic model to the weak sector by the addition of the axial MEC has been given very recently by Simo et al.  [225]. The results of Amaro et al. of Refs.  [206, 221, 207, 208, 224] do not include these last contributions, while the very recent results of Refs. [226, 209] do it. The approach of Nieves et al. can be considered as a generalization of the work of Gil et al. [242], developed for the electron scattering, to the neutrino scattering. The contributions related to the non-pionic decay are taken, as in the case of Martini  et al. from Oset and Salcedo [239].

We remind that there exist several contributions to two-body currents , see for example Refs. [179, 241, 191, 225]. In the electromagnetic case, there are the so called pion-in-flight term , the contact term and the -intermediate state or -MEC term . At level of terminology, in the past some authors refer just to the first two terms as Meson Exchange Currents contributions (like in [9]) but actually the most current convention consists of including the -term into MEC. Here we follow this convention. In the electroweak case another contribution, the pion-pole term , appears. It has only the axial component and therefore it is absent in the electromagnetic case.

In the 2p-2h sector, the three microscopic models that we are discussing now are based on the Fermi gas, which is the simplest independent particle model. In other words, the calculations are performed in a basis of uncorrelated nucleons. If also in the 1p-1h sector a basis of uncorrelated nucleons is used, one needs to consider also the nucleon-nucleon (NN) correlations contributions since the protons and the neutrons in the nucleus are correlated and the short range correlated (SRC) pairs act as a unique entity in the nuclear response to an external field. In the framework of independent particle models, like Fermi gas based models or mean field based models, these NN correlation are included by considering an additional two-body current, the correlation current . Detailed calculations and results for these NN correlation current contributions are given for example in Refs. [179, 243, 244, 236]. In other approaches, like the one of Lovato et al. [232, 233] the NN correlations are included in the description of the nuclear wave functions. With the introduction of the NN correlation contributions, also the NN correlations-MEC interference contributions to the 2p-2h excitations (the terms called in the works of Martini et al.) naturally appear. In the correlated-basis based approach, these contributions are referred as one nucleon-two nucleon currents interference. This point will be further analyzed later.

Focusing now on the Fermi gas based models, it is important to stress that even in this simple model an exact relativistic calculation is difficult for several reasons. Let’s start from the general expression of the 2p-2h hadronic tensor

 Wμν2p−2h(q,ω) = V(2π)9∫d3p′1d3p′2d3h1d3h2m4NE1E2E′1E′2 (30) θ(p′2−kF)θ(p′1−kF)θ(kF−h1)θ(kF−h2) ⟨0|Jμ|h1h2p′1p′2⟩⟨h1h2p′1p′2|Jν|0⟩ δ(E′1+E′2−E1−E2−ω)δ(p′1+p′2−h1−h2−q),

where and are the momenta of the two nucleons ejected out of the Fermi sea, leaving two hole states in the daughter nucleus with moments and . Using energy and momentum conservation, for fixed values of and , the 2p-2h calculations involves the computation of 7-dimensional integrals . The first difficulty is that one needs to perform these 7-dimensional integrals for a huge number of 2p-2h response Feynman diagrams. Second, divergences in the NN correlations sector and in the angular distribution of the ejected nucleons [222, 223] may appear and need to be regularized. Furthermore, as illustrated in Sec. 2.1 the neutrino cross section calculations should be performed for all the kinematics compatible with the experimental neutrino flux (and not only for some fixed values of the momentum- or energy-transfer, as in the case of the electron scattering where the incoming and outgoing electron energies and momenta are known). For these reasons an exact relativistic calculation is very demanding with respect to computing, and as a consequence different approximations are employed by the different groups in order to reduce the dimension of the integrals, and to regularize the divergences. The choice of subsets of diagrams and terms to be calculated also presents important differences. In this connection Amaro et al. only explicitly add the MEC contributions and not the NN correlations-MEC interference terms (these last terms were analyzed for electron scattering in Ref.[244]) to the genuine quasi-elastic. MEC contributions, NN correlations and NN correlations-MEC interference are present both in Martini et al. and Nieves et al. Martini et al. consider only the -MEC 888The main reason for Martini et al. to discard the other contributions from the explicit calculation of MEC is that they are peculiar to the external probe. They want a “universal” spin-isospin 2p-2h response, to use in different processes, like in Ref. [179] where this response was used to study electron scattering and pion absorption. However MEC contributions to the time component of the axial current (due to the seagull and pion-pole terms) are taken into account in an effective way by introducing in the time component of the axial current a renormalization factor , see Appendix A of Ref. [9]. The impact of this renormalization on the cross section is small.. This is the dominant contribution, as shown for example by De Pace et al. (Ref. [241], Fig. 9). The interference between direct and exchange diagrams is neglected by Martini et al. and Nieves et al. The treatment of Amaro et al. is fully relativistic as well as the one of Nieves et al. (even if the non pionic decay contribution of -MEC are taken from the non-relativistic work [239], as in the case of Martini et al.) while the results of Martini et al. are related to a non-relativistic reduction of the two-body currents. Interestingly, Simo et al. have shown (Ref. [222], Fig.12) that for the 2p-2h phase-space integral (obtained from Eq.(30) by setting to one the current matrix elements) the differences between a non relativistic and a fully relativistic calculation are relatively small. The two results are close to each other in particular if compared with the calculation implementing only relativistic kinematics and not the Lorentz-contraction factor .

Beyond these differences, there are also other differences related to the terms of the neutrino-nucleus cross sections affected by 2p-2h contribution. Amaro et al. in Refs.  [206, 221, 207, 208, 224] consider the 2p-2h contribution only in the vector sector while Martini et al. and Nieves et al. also consider the axial vector. Fully relativistic calculations of Amaro et al. for the axial sector have been recently presented in the paper of Simo et al. [225] focusing on the nuclear responses and in the in the paper of Megias et al.  [209] focusing on the neutrino flux integrated differential cross sections.<