Semi-inclusive neutrino-nucleus reactions at intermediate energies

University of Seville & University of Turin PhD Thesis SEMI-INCLUSIVE NEUTRINO-NUCLEUS REACTIONS AT INTERMEDIATE ENERGIES by JUAN MANUEL FRANCO PATI ˜ NO B.S., University of Seville, 2017 M.S., University of Seville, 2018 Supervised by: Juan Antonio Caballero and Maria Benedetta Barbaro Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2023 ii © 2023 by JUAN MANUEL FRANCO PATI˜ NO All rights reserved iv Para mi familia, sin quien otra realidad ser´ıa impensable; y esta realidad, intolerable. v vi Acknowledgments Es un placer dedicar unas pocas palabras a todas las personas que me han dado su apoyo y consejo a lo largo de estos ´ultimos cuatro a˜nos. Para empezar, me gustar´ıa expresar mi eterna gratitud a mis directores de tesis, Juan Antonio y Maria, por darme la oportunidad de trabajar en un proyecto tan interesante, su continua orientaci´on, apoyo y por un tener siempre un segundito para charlar. Siempre me he sentido como en casa tanto en Sevilla como en mis visitas a Tur´ın y ha sido gracias a vosotros. Quiero extener mis agradecimientos a todos los colaboradores a los que me he encontrado durante estos a˜nos, tanto en Sevilla y Madrid como en Tur´ın y Ginebra, haciendo especial menci´on a Guille y a Ra´ul, quienes siempre han estado disponibles para discutir ideas y resultados. Tambi´en quiero dar las gracias a Stephen por echarme un cable con las simulaciones de GENIE y por hacerme un hueco para discutir siempre que pod´ıa. Aprovecho tambi´en para agradecer por todo lo que he aprendido en el aspecto docente de todos los compa˜neros que he tenido en las pr´acticas de Cu´antica y a los alumnos por sus preguntas que siempre te hacen mejorar y por su entusiasmo, incluso en los turnos de los viernes por la tarde. Tambi´en quiero dar las gracias a mis amigos del campus por todos esos largos almuerzos, en especial a Gabri y Miguel. Por supuesto, nunca habr´ıa llegado hasta aqu´ı de no ser por mi familia. Gracias a mis padres, a los que jam´as podr´e pagar su sacrificio para cambiar un futuro incierto en Colombia por una oportunidad en Espa˜na para mi y para mi hermana. Espero que al leer esto sint´ais tanto orgullo como he sentido yo al escribirlo. Por ´ultimo, quiero agradecer al instituto Carlos I de F´ısica Te´orica y Computacional de Granada el acceso al cluster de superordenadores PROTEUS en el cual se han hecho gran parte de los c´alculos num´ericos presentados en la tesis y la ayuda econ´omica recibida del Ministerio de Ciencia e Innovaci´on mediante el programa FPI. vii viii Contents 1 Introduction 1 1.1 Neutrinooscillations ............................... 2 1.2 Accelerator neutrino experiments . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Neutrino-nucleus interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 General semi-inclusive neutrino-nucleus scattering formalism 25 2.1 Semi-inclusive neutrino-nucleus kinematics . . . . . . . . . . . . . . . . . . . 27 2.2 Plane-wave impulse approximation . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Nuclearmodels .................................. 34 2.3.1 Relativistic Fermi Gas model . . . . . . . . . . . . . . . . . . . . . . 34 2.3.2 Independent-particle shell model . . . . . . . . . . . . . . . . . . . . . 37 2.3.3 Natural orbitals shell model . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.4 Rome spectral function model . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Semi-inclusive neutrino-nucleus cross section in PWIA . . . . . . . . . . . . 44 3 The relativistic distorted-wave impulse approximation 55 3.1 Initial state within RDWIA: beyond the relativistic mean-field model . . . . 58 3.2 Final state interactions within RDWIA . . . . . . . . . . . . . . . . . . . . . 63 3.2.1 Relativistic optical potential . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.2 Extended relativistic mean-field potential: ED-RMF . . . . . . . . . . 67 4 Neutrino generators approach to semi-inclusive reactions 71 ix 3·1 Parameterization of the missing energy profile ρ(Em) for 12C separated by contributions of the individual shells and the background. The background is normalized so there are 0.8 nucleons up to Em= 300 MeV, i.e. R300 26 dEmF(Em) = 0.8. The remaining nucleons are associated to the shells as summarized in Table3.1. ..................................... 60 3·2 Missing energy profile of neutrons in 40Ar described by the parameterization given in Table 3.2. The red band corresponds to the uncertainties also summarized in Table 3.2. The vertical blue lines show the positions of the seven RMF shells and the black dashed line shows the mean value of the distribution. . . . . . 62 3·3 Energy-dependent A-independent 12C (EDAIC) relativistic optical potential [85] evaluated at different proton kinetic energies TN= 30 MeV (dashed), 200 MeV (dotted) and 800 MeV (dash-dotted). The vector (red lines) and scalar (blue lines) components of the potential are shown separately for both the real (left panel) and imaginary (right panel) parts. . . . . . . . . . . . . . . . . . . . . 65 3·4 Energy-dependent relativistic mean field (ED-RMF) potential [86, 87] for 12C evaluated at different proton kinetic energies TN= 30 MeV (dashed), 200 MeV (dotted) and 800 MeV (dash-dotted). The RMF potential (solid) is also shown for reference. The vector (red lines) and scalar (blue lines) components of the potentials are shown separately. . . . . . . . . . . . . . . . . . . . . . 69 xvi 4·1 Proton kinetic energy (left) and scattering angle (right) distributions for the process e+A→e′+p+B. The electron beam energy is fixed to 1.159 GeV and the outgoing electron phase-space is limited to 17◦< θe′<40◦ and Ee′>0.4 GeV. The red lines show the unfactorized RDWIA predictions obtained with ED-RMF potential for carbon (named “Full”) while the blue lines show the equivalent distributions obtained after applying the GENIE algorithm (Eq. (4.1) and Eq. (4.3)) and before adding GENIE cascade FSI (named “Approx”). Results taken from [108]. . . . . . . . . . . . . . . . . . 76 5·1 T2K 1µCC0π0p semi-inclusive νµ−12C cross section without protons in the final state with momenta above 0.5 GeV as function of the muon kinematics. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from[115]...................................... 82 5·2 T2K 1µCC0πNp semi-inclusive νµ−12C cross section with at least one proton in the final state with momentum above 0.5 GeV as function of the leading proton and muon kinematics. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. . . . . . . . . . . . . . . . . . 85 5·3 T2K 1µCC0πNp semi-inclusive νµ−12C cross section with at least one proton in the final state with momentum above 0.5 GeV as function of the muon scattering angle. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Crosssection measurements taken from [115]. . . . . . . . . . . . . . . . . . . . . . 86 xvii 5·4 T2K 1µCC0πNp semi-inclusive νµ−12C cross section as function of the inferred variable ∆pfor different muon kinematic bins. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. . . . . . . 88 5·5 T2K 1µCC0πNp semi-inclusive νµ−12C cross section as function of the inferred variable ∆θin different muon kinematic bins. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. For readability, the axis range has been reduced to [−30◦,+30◦] hiding an experimental bin above 30◦with very low cross section and centering the [−360◦,−5◦] experimental bin around −20◦. ................................. 89 5·6 T2K 1µCC0πNp semi-inclusive νµ−12C cross section as function of the inferred variable |∆p|in different muon kinematic bins. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. . . . . . . 90 5·7 Scheme showing transverse kinematic imbalances (TKI): δpT,δαTand δϕT. The final lepton and nucleon momenta are projected on the plane perpendicular to the neutrino direction (xy-plane or transverse plane). The transverse component of the transferred momentum (qT) equals −k′ Tand defines the x-axis. ....................................... 91 5·8 T2K 1µCC0πNp semi-inclusive νµ−12C cross sections as function of the transverse kinematic imbalances δpT,δαTand |δϕT|. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. . . . . . . . . . . 95 xviii 5·9 MINERνA 1µCC0πNp semi-inclusive νµ−12C cross section as function of the final muon momentum and scattering angle (top) and as function of the final proton momentum and polar angle (bottom). All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). The original paper from MINERνA was [117] but the cross-section measurements shown here were taken from [116] which corrected a mismodelling in GENIE’s elastic FSI that affected the cross-section measurements presented in the first paper. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. . . . . . . . . . . . . . . . . . . . 97 5·10 MINERνA 1µCC0πNp semi-inclusive νµ−12C cross sections as function of the transverse kinematic imbalances δpT,δpTx,δpT y,δαTand |δϕT|. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). The original paper from MINERνA was [117] but the cross-section measurements shown here were taken from [116] which corrected a mismodeling in GENIE’s elastic FSI that affected the crosssection measurements presented in the first paper. Notice that the convention used in [116] to define the xand yaxis to project δpTon is the opposite to the convention used in this thesis. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 99 xix 5·11 MicroBooNE 1µCC0πNp semi-inclusive νµ−40Ar cross sections as function of the reconstructed muon and proton momenta and scattering angles and the opening angle θreco µp . All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results are from [118]. The bands, drawn for the ED-RMF and ROP models, represent the uncertainties associated with the modeling of the initial nuclear state. The χ2/d.o.f. ratio is given in brackets in the legend of eachdistribution.................................. 102 5·12 MicroBooNE 1eCC0πNp semi-inclusive νe−40Ar cross sections as function of the electron scattering angle and energy and proton kinetic energy and scattering angle. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results are from [125]. The bands, drawn for the ED-RMF and ROP models, represent the uncertainties associated with the modeling of the initial nuclear state. The single white point in the TNdistribution between 0 < TN<50 MeV corresponds to an extra 1eCC0π0p (one electron, no protons above TN= 50 MeV threshold and at least one proton with kinetic energy below the 50 MeV threshold) measurement performed by MicroBooNE [125]. For this single point, additional phase space restrictions on the electron energy (El> 0.5 GeV) and the electron scattering angle (cos θl>0.6) are applied. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. . . . . . 104 xx 5·13 MicroBooNE 1µCC0π1p semi-inclusive νµ−40Ar cross sections as function of the muon and proton momenta and proton polar scattering angle. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results are from [118]. The bands, drawn for the ED-RMF and ROP models, represent the uncertainties associated with the modeling of the initial nuclear state. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. . . . . . . . . . . . . . 106 5·14 MicroBooNE 1µCC0π1p semi-inclusive νµ−40Ar cross sections as function of the muon scattering angle. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results were taken from [119]. The bands drawn for the EDRMF and ROP models are related with the uncertainties associated with the modeling of the initial nuclear state. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. . . . . . . . . . . . . . . . . . . . . . . . . 107 5·15 MicroBooNE 1µCC0π1p semi-inclusive νµ−40Ar cross sections as function of Q2 CCQE and Ecal ν. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results are from [118]. The bands, drawn for the ED-RMF and ROP models, represent the uncertainties associated with the modeling of the initial nuclear state. The covariance matrix is not available for the cross section as function of these variable, therefore it is not possible to calculate the χ2values. . . . 108 B·1 Definition of the coordinate system where the momentum transfer qlies on the z-axis. Notice that the leptons are contained in the x-z plane (scattering plane) and the plane containing the ejected nucleon with momentum pNand polar angle θNforms an angle ϕNwith the scattering plane. . . . . . . . . . 123 xxi D·1 Evolution of χ2between each model and T2K semi-inclusive cross section measurements when the bins contributing the largest χ2are progressively removed. For example, the right-most points of the top left plot correspond to the total χ2in Table D.1 (calculated using all 58 bins reported by T2K), whilst the points just to the left of them show the χ2value once the bin contributing the largest χ2is removed. Points further to the left remove more bins following the same rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 xxii List of Abbreviations xxiii 2p2h . . . . . . . . . . . . . Two-particle-two-hole CC . . . . . . . . . . . . . Charged-current CCQE . . . . . . . . . . . . . Charged-current quasielastic CP . . . . . . . . . . . . . Charge-Parity DIS . . . . . . . . . . . . . Deep inelastic scattering d.o.f. . . . . . . . . . . . . . Degrees of freedom EDAD . . . . . . . . . . . . . Energy-dependent A-dependent EDAI . . . . . . . . . . . . . Energy-dependent A-independent ED-RMF . . . . . . . . . . . . . Energy-dependent relativistic mean field FSI . . . . . . . . . . . . . Final-state interactions IA . . . . . . . . . . . . . Impulse approximation INC . . . . . . . . . . . . . Intranuclear cascade IPSM . . . . . . . . . . . . . Independent-particle shell model IV . . . . . . . . . . . . . Inferred variables LFG . . . . . . . . . . . . . Local Fermi gas MC . . . . . . . . . . . . . Montecarlo MEC . . . . . . . . . . . . . Meson exchange current NC . . . . . . . . . . . . . Neutral current NN . . . . . . . . . . . . . Nucleon-nucleon NO . . . . . . . . . . . . . Natural orbitals non-QE . . . . . . . . . . . . . Non-quasielastic PMNS . . . . . . . . . . . . . Portecorvo-Maki-Nakagawa-Sakata PWIA . . . . . . . . . . . . . Plane-wave impulse approximation QE .. . . . . . . . . . . . Quasielastic RDWIA . . . . . . . . . . . . . Relativistic distorted-wave impulse approximation RFG . . . . . . . . . . . . . Relativistic Fermi gas RMF . . . . . . . . . . . . . Relativistic mean field ROP . . . . . . . . . . . . . Relativistic optical potential RPA . . . . . . . . . . . . . Random phase approximation RPWIA . . . . . . . . . . . . . Relativistic plane-wave impulse approximation rROP . . . . . . . . . . . . . Real part of a relativistic optical potential SF . . . . . . . . . . . . . Spectral function SM . . . . . . . . . . . . . Standard model TKI . . . . . . . . . . . . . Transverse kinematic imbalances xxiv Chapter 1 Introduction Neutrinos are the most abundant massive particles in the universe. When atomic nuclei are created in a star, like our sun, or break apart, like inside a nuclear reactor, neutrinos are created. The existence of neutrinos was proposed by Pauli almost one century ago as an alternative over the violation of energy conservation in βdecay. The confirmation of its existence (or to be more precise, the existence of its antiparticle) came almost three decades after it was postulated [1]. Nowadays, the neutrino holds a distinguished place in the physics landscape, both inside and beyond the Standard Model. The main topic of this thesis is the description of a realistic theoretical model of semi-inclusive neutrino-nucleus reactions, namely, processes where a lepton and one ejected nucleon are detected in coincidence after the interaction of a neutrino with a nucleus. The ultimate goal of the thesis is to perform a systematic analysis of all the semi-inclusive neutrino-nucleus experimental data published until 2023 with a realistic, i.e. relativistic and quantum mechanical, model of the nuclear dynamics. The analysis, proper understanding and modeling of these reactions will help current and future neutrino oscillation experiments by reducing the systematic uncertainties associated to the modeling of neutrino-nucleus interactions. This thesis is structured as follows: in this Chapter, an introduction to neutrino oscillations, accelerator-based neutrino experiments and neutrino-nucleus interactions is presented. Chapter 2 is mainly focused in the general definition of a semi-inclusive neutrino-nucleus reaction and the analysis, from a theoretical point of view, of semi-inclusive results using different 1 8 coming from the neutrino flux and neutrino-nuclei interactions. The near detector, i.e. the one closer to the source of neutrinos, either uses the same technology as the far detector, i.e. the one localized far away from the neutrino source that measures the oscillated neutrino flux, or consists of sub-detectors with complementary functions to obtain detailed information of the neutrino beam and neutrino interactions with complex nuclei. The near detectors are used to characterize the neutrino flux, its energy distribution and the interaction cross sections, which are used as an input to make predictions of observables at the far detector. Deviations between the two energy spectra detected by both detectors may be used to infer the presence of neutrino oscillations, which manifest themselves as the disappearance of the νµbeam, or the appearance of a different neutrino flavor in the νµbeam. However, one should note that the neutrino flux is inevitably different between the near and the far detectors, even without considering neutrino oscillations, due to differences between detector technologies and targets used. The neutrino oscillations change the flavour composition of the neutrino beam quite significantly, as the design of a neutrino oscillation experiment requires. For the precision measurements of neutrino oscillations with accelerator experiments, the understanding of the neutrino-nucleus interaction becomes crucial. Because complex nuclei are used as targets, the nuclear effects complicate the understanding of the neutrino-nucleus interaction as it will be explained in the Section 1.3. Current long-baseline neutrino oscillation experiments measure both the appearance (for instance νµ→νe) and the disappearance νµ→νµchannels, each of them sensitive to some oscillation parameters. We can consider these two specific cases that are relevant for longbaseline experiments. The first one corresponds to the νµ→νeappearance channel with a 9 probability including matter effects [9] given by [10] P(νµ→νe) = 4c2 13s2 13s2 23 sin2∆31 + 8c2 13s12s13s23 (c12c23 cos δCP −s12s13s23) cos ∆32 sin ∆31 sin ∆21 −8c2 13c12c23s12s13s23 sin δCP sin ∆32 sin ∆31 sin ∆21 + 4s2 12c2 13 c2 12c2 23 +s2 12s2 23s2 13 −2c12c23s12s23s13 cos δCP sin2∆21 −8c2 13s2 13s2 23 aL 4Eν1−2s2 13cos ∆32 sin ∆31 + 8c2 13s2 13s2 23 a ∆m2 31 1−2s2 13sin2∆31,(1.10) with ∆ij = ∆m2 ijL/4Eνand a= 2√2GFneEν, where neis the average electron density of the medium and GFis the Fermi constant. The second case is the νµsurvival probability P(νµ→νµ)≈1−sin22θ23 sin2∆32.(1.11) To obtain the corresponding antineutrino oscillation probabilities one needs to replace δCP → −δCP and a→ −a, which only affects νµ→νeprobability given in Eq. (1.10). In particular, the third term in Eq. (1.10), which contains sin δCP , is the CP violating term that flips sign between νand ¯νand thus introduces CP asymmetry if sin δCP is non-zero. To extract oscillation parameters from neutrino oscillation experiments, experimentalists measure the event rate of reconstructed νµ/νeevents as a function of some observable, for instance the reconstructed neutrino energy. The spectrum for the appearance and disappearance channels is then compared to Monte-Carlo (MC) simulations, in which oscillation parameters are varied until they are in agreement with the data. It is important to note that no single type of neutrino oscillation experiment is sensitive to all oscillation parameters at once, and not with the same precision, but the oscillation probability depends on all parameters at once. Because of this, the results from reactor experiments, which are more 10 sensitive to θ13, or from solar neutrino experiments, that constrain better θ12 and ∆m2 21, are often used by accelerator experiments, and in particular by long-baseline experiments, to improve the measurements of sin2θ23, ∆m2 32 and δCP . The expected event rates at the near and far detectors can be expressed as function of the neutrino energy Eνas follows Nnear να(Eν) = σnear να(Eν)ϕnear να(Eν)ϵnear (Eν) Nfar νβ(Eν) = σfar νβ(Eν)ϕfar νβ(Eν)ϵfar (Eν)Pνα→νβ(Eν).(1.12) They depend on the neutrino flux distributions ϕνα, the neutrino interaction cross-sections with the detector σνα(how probable it is that a neutrino interacts), the detector efficiencies ϵ(how well the detectors will be able to reconstruct the event signaling the neutrino) and, finally, on the oscillation probability Pνα→νβ(for instance how many of the initial νµsurvived or how many νeappeared in the far detector). Since neutrinos interact weakly, the precision on the event rate is strongly limited by statistical errors. To give some idea about how weak the interaction is, we can imagine a neutrino traveling through a material with density ρ∼1023 atoms/cm3. Assuming an associated interaction cross section σ∼10−38 cm2for a 1 GeV neutrino, then it would travel ∆x∼ρσ ∼1010 km before interacting, a distance comparable to 1.5 million Earth radii or 66 times the distance between the Earth and the Sun. Moreover, effects related to the flux, neutrino interaction cross-sections and the detector efficiency are sources of systematic errors. In general, for real experiments we find that near and far detectors can have different acceptances and use different technologies and nuclear targets. Additionally, the initial neutrino energy distribution is unknown because of the special method of producing neutrinos as secondary decay products of hadrons that were produced in a primary nuclear reaction between a beam of protons and a nuclear target. Therefore, the neutrino energy must be reconstructed on an event-by-event basis from the particles present in the final state after the 11 reaction of the neutrino with a nucleus assuming a nuclear model. Under these conditions, what is measured is an averaged oscillation probability as function of the reconstructed neutrino energy that can be expressed as Pνα→νβ(Ereco ν)=RdEνσfar νβ(Eν)ϕfar νβ(Eν)ϵfar (Eν)Pνα→νβ(Eν)S(Ereco ν|Eν) RdEνσnear να(Eν)ϕnear να(Eν)ϵnear (Eν)S(Ereco ν|Eν),(1.13) with S(Ereco ν|Eν) a function that encodes the probability of a specific true neutrino energy Eνto be reconstructed as Ereco ν. In what follows, we briefly summarize some of the past, present and future accelerator neutrino experiments dedicated not only to the measurement of oscillation parameters, but also to the determination of experimental neutrino-nucleus cross sections to test theoretical interaction models which are necessary to extract oscillation parameters. NOMAD The Neutrino Oscillation Magnetic Detector (NOMAD) [11], that used CERN wide-band neutrinos produced by the 450 GeV proton synchrotron, searched for νµ→ντoscillation by detecting τappearance between 1995 and 1998. Its goals were to measure the momenta of charged particles and identify and measure electrons, photons and muons. By the detector design, which used carbon as target, it was also possible to look for νµ→νeoscillation [12]. The analysis of the full NOMAD data sample gave no evidence for ντappearance and set with 90% confidence level (C.L.) that the upper limit on the two-flavour νµ→ντoscillation probability was P(νµ→ντ)<2.2×10−4[13]. The NOMAD experiment applied for the first time a purely kinematical approach for the detection of ντCC interactions. Unfortunately, oscillations were not discovered with the NOMAD experiment. 12 K2K After the success of Super-Kamiokande, that provided the first experimental evidence of atmospheric neutrinos oscillation in 1998, the Japanese National Laboratory for High Energy Accelerator Research Organization (KEK) to Kamiokande (K2K) experiment [14, 15] was created with the main difference of using a well understood muon neutrino beam pointing at the Super-Kamiokande detector at a distance of 250 km. It was the first long-baseline neutrino oscillation experiment to observe the oscillation of νµinto ντand found results of squared mass difference and mixing angle that were consistent with Super-Kamiokande results. The experiment, that ran from 1999 to 2004, used a neutrino beam with a wide energy spectrum ranging from 1 to 1.5 GeV and produced by a 12 GeV proton synchrotron beam interacting with an aluminum target. Since knowing the beam composition is required for looking at νµdisappearance, a 1-kiloton water Cherenkov near detector, a smaller version of Super-Kamiokande, was used to measure the neutrino beam spectrum. Then this distribution was extrapolated using Monte Carlo simulated data to predict the neutrino spectrum at Super-Kamiokande. MINOS The Main Injector Neutrino Oscillation Search (MINOS) was a neutrino oscillation experiment that used a beam from Fermilab and two detectors: one placed a few hundred meters away from the source of the neutrinos and the other 735 km away in Soudan mine in northern Minnesota. The neutrino beam was produced in the NuMI beamline [17] with 120 GeV protons from the Main Injector. Both MINOS detectors were iron-scintillator tracking calorimeters with toroidal magnetic fields [18] designed to be as similar as possible to minimize any systematic errors in comparing the observed neutrino spectra in the two detectors. The NuMI beamline could alter the neutrino energy spectrum by changing the relative position of target and horns. Most of MINOS data were taken with the “low energy” 13 Figure 1·2: First neutrino event detected in June 1999 inside the Super-Kamiokande water Cherenkov detector due to the KEK neutrino beam. The crosses give the reconstructed neutrino interaction vertex and the diamond marks the neutrino beam direction from the vertex. The probability that the event came from an atmospheric neutrino interaction, instead of from a neutrino produced at KEK, was estimated to be 0.01%. Picture taken from [16]. configuration with the peak energy of around 3 GeV. MINOS combined accelerator and atmospheric neutrino data taken from 2005 to 2012 in both disappearance and appearance modes to measure oscillation parameters [19] that were consistent with K2K and Super- Kamiokande measurements of mixing angle and squared mass difference. 14 NOvA The NOvA experiment [20, 21] uses the NuMI beamline with an off-axis configuration. The 14 kt NOvA far detector is located near Ash River, Minnesota, 810 km away from the muon neutrinos source, and uses mineral oil as target. The neutrino energy spectrum at the far detector has a peak around 2 GeV, the energy at which oscillation from muon neutrinos to electron neutrinos is expected to be at a maximum for a 810 km baseline. NOvA aims to measure disappearance νµ→νµand appearance νµ→νein both νµand ¯νµbeams to extract θ23, ∆m2 32 and δCP parameters. The mass ordering affects the rates of νµ→νeand ¯νµ→¯νe oscillations when neutrinos travel through the Earth as compared to the vacuum [9]. These matter effects are important for NOvA due to the large baseline used by the experiment. Depending on the value of δCP and the mass ordering itself, NOvA may be able to exploit the resulting neutrino-antineutrino asymmetry to measure the sign of ∆m2 32 and thus determine the mass ordering. MINERνA The MINERνA (Main INjector ExpeRiment ν-A) experiment [22] studies neutrino-nucleus scattering to improve models describing the process in order to reduce systematic uncertainties in results from oscillation experiments. It measures neutrino reactions covering an energy range from 1 to 20 GeV. The detector, shown in Fig 1·3, is placed on-axis of the NuMI neutrino beam and it is equipped with different targets (C, CH, Fe, H2O, Pb) in order to achieve precise measurements of the nuclear effects [23]. The high energetic muons created in neutrino interactions are capable of escaping the MINERνA detector and entering MINOS, located only 2 meters away from MINERνA. MINERνA uses data collected by MINOS to identify and measure these muons. 15 MINOS Near Detector (Muon Spectrometer) Scintillator Veto Wall Liquid Helium Steel Shield Nuclear Target Region (C, Pb, Fe, H2O) Electromagnetic Calorimeter Hadronic Calorimeter Side ECAL Side HCAL Side ECAL Side HCAL 0.25t 30 tons15 tons 0.6 tons 116 tons 5 m 2 m 2.14 m 3.45 m Elevation View Active Tracker Region 8.3 tons total Figure 1·3: Side view of the MINERνA detector showing the different nuclear targets, the fully-active tracking region and the surrounding calorimeters [22]. T2K After the great success of K2K, the T¯okai-To-Kamioka (T2K) experiment [24, 25] started in 2010 using a newly constructed high-intensity proton synchrotron from the Japanese Proton Accelerator Research Complex (J-PARC) facility at T¯okai. It is the first long-baseline experiment to employ an off-axis neutrino beam. The off-axis angle of 2.5◦was chosen to set the peak of neutrino energy spectrum at 0.6 GeV, matching the first maximum of oscillation probability at the 295 km baseline for ∆m2≈2.5×10−3eV2. T2K employs a set of near detectors at about 280 m from the production target. The on-axis detector, called INGRID, is an array of iron-scintillator sandwich trackers to monitor the beam intensity, direction and profile. The off-axis detector, called ND280, consists of several sub-detectors inside a magnet and is placed in the direction of far detector to measure the neutrino beam properties and to study neutrino interactions on hydrocarbon-based or a hybrid water-hydrocarbon targets. 16 An upgrade of the ND280 is outgoing right now with the aim of reducing the flux and cross-section systematic uncertainties to establish CP violation at 3σlevel for a significant fraction of the possible δCP values [26]. The ND280 measurements are used to predict the number of muon neutrinos that would be seen in Super-Kamiokande if there were no oscillations. Additionally, since it is a magnetized detector, ND280 can measure the wrong sign background in the flux, which is not possible at the far detector. Finally, it is able to precisely reconstruct the exclusive final states of νµand ¯νµinteractions, therefore its data is used as one of the main inputs in the oscillation analysis in order to constrain the neutrino flux and neutrino interaction systematic uncertainties. In the long term, the T¯okai-To-Hyper-Kamiokande (T2HK) experiment has been proposed as successor [27] of the current T2K program. The Hyper-K experiment is a next-generation, large-scale water Cherenkov neutrino detector that will receive neutrinos from J-PARC accelerator facility like T2K. Due to the large size of the Hyper-K detector (60 m in height and 74 m in diameter) and its improved detection techniques, the long-baseline neutrino oscillations program with the Hyper-K detector will benefit from an improved sensitivity to neutrino oscillation parameters, inaccessible to the current T2K experiment, most notably enabling a 5σsensitivity to CP violation discovery for a large number of δCP values. MiniBooNE &MicroBooNE The Mini Booster Neutrino Experiment (MiniBooNE) collected data from 2002 up to 2012 at Fermilab using a muon neutrino beam to search for νeand ¯νeappearance in the same parameter region as the Liquid Scintillator Neutrino Detector (LSND) [28, 29] experiment that ran between 1993 and 1998. The Booster Neutrino Beamline (BNB) with a magnetic horn uses a 8 GeV proton beam from the Fermilab booster to produce a neutrino (antineutrino) beam with energy spectrum peak of 0.6 (0.4) GeV. MiniBooNE, as well as LSND, used a single detector setup. It consisted of a 12.2 m diameter sphere filled with 818 t of mineral 17 oil (CH2) located around half kilometer from the target. The MiniBooNE collaboration reported νeand ¯νeevent excess in both neutrino and antineutrino running modes. In total, 460.5 ±99.0 excess events were observed over the expected backgrounds, corresponding to 4.7σsignificance [30]. The MicroBooNE Collaboration [31], a further step into the BooNE experiment that began in 2015, is currently operating a large Liquid Argon Time Projection Chamber (LArTPC) that is located on the BNB at Fermilab. MicroBooNE will measure neutrino cross sections with reconstructed neutrino energy between 200 and 475 MeV to investigate the low energy excess events observed by the MiniBooNE experiment. The detector also serves as a next step in a phase program towards the construction of massive kiloton scale LArTPC detectors for future long-baseline neutrino physics (DUNE). DUNE DUNE [32] is a new experiment currently under construction, with the goal of discovering CP violation in neutrinos with more than 5σsensitivity, determining the ordering of the neutrino masses as well as searching for neutrinos beyond the currently known three. DUNE, which is planned to start taking data in 2030’s, will be a 1300 km long-baseline experiment based in US with a far detector that will consist of four modules of at least 10 kt fiducial mass LArTPC, located 1.5 km underground at the Sanford Underground Research Facility in South Dakota, and the near detector installed in the Long-Baseline Neutrino Facility (LBNF) at Fermilab. 1.3 Neutrino-nucleus interactions Neutrino interaction with matter occurs via the weak and gravitational forces, the latter being negligible due to the very small neutrino mass. The corresponding cross sections are very small, of the order of 10−38 cm2/GeV. The type of interaction can be identified exclusively through the particles produced after the reaction. Neutrino weak interactions can 24 25 Chapter 2 General semi-inclusive neutrino-nucleus scattering formalism As discussed in Secs. 1.2 and 1.3, accelerator neutrino experiments use large amount of target material to increase the probability of interaction of the incoming neutrinos. The targets commonly used by the experimental collaborations are hydrocarbons and water hence neutrinos (or antineutrinos) typically interact with hydrogen, carbon or oxygen nuclei. However, other heavier nuclei such as iron or argon are also used. In particular, argon is used in many present programs, like MicroBooNE, and future experiments, including DUNE, which employ LArTPC detectors. In order to extract the oscillation parameters it is necessary to reconstruct the incident neutrino energy from the reaction products because the neutrino production mechanism typically results in neutrino beams with a very broad flux distribution (for instance see Fig. 2·4). Moreover, this reconstruction method requires basic understanding of nuclear reactions and the nuclear structure of the target nucleus. With the aim of increasing the number of events that are detected, experimental collaborations usually measure inclusive CC or CC0πcross sections where only a final-state charged lepton (i.e. an electron or a muon) is detected in presence or not of pions in the final state, respectively. Since these inclusive cross sections are completely integrated over the hadronic degrees of freedom, they are relatively insensitive to the model chosen for the description of the nuclear state. This means that simple models of the nucleus can provide CC inclusive cross section results that are not very different from those found in more elaborated models [49, 50, 51, 26 52]. However, comparison with these neutrino scattering data is not sufficient to test the validity of the different models. The wide nature of the neutrino beam makes the separation of true QE events from contributions of other elemental processes a very complicated task and the cross section uncertainties are usually too large to discriminate between different nuclear models. Therefore, it is important to compare the different model predictions to the much more accurate electron scattering data for which the incoming electron energy is known. The connection between electron and neutrino scattering has been analyzed in the past (see [53] and references therein) and it is commonly accepted within the neutrino community that any nuclear model that aims to describe the neutrino-nucleus interaction for oscillation experiments must be first validated against electron scattering data. In addition, in the last years a number of cross section measurements have been performed by different neutrino collaborations where not only the final charged lepton is detected but also some hadron as well. This is driven by the goal of constraining the incident neutrino energy much better compared with the case where only the final charged lepton is detected. One of these reactions corresponds to the interaction of an incoming neutrino with a bound neutron of the target nucleus and the detection in coincidence of a charged lepton and a proton in the final state. This reaction, which will be the focus of this thesis, is not an inclusive one but a semi-inclusive reaction [49, 50, 51, 54]. Although the theoretical description of semi-inclusive reactions is more complicated than modeling inclusive processes, the richer structure of the cross sections allows one to better discriminate between different nuclear models and improve the reconstruction of the neutrino energy. In this chapter, the focus will be in the theoretical description of the semi-inclusive CC neutrino-nucleus reaction where a charged lepton and an ejected nucleon are detected in the final state. We will start by describing in general the kinematics of the reaction and later the analytic expressions of the semi-inclusive CC cross section in the plane-wave impulse approximation for different models of the initial nuclear state will be given. 27 2.1 Semi-inclusive neutrino-nucleus kinematics The general semi-inclusive CC neutrino1-nucleus reaction [54], schematically shown in Fig. 2·1, involves an incident neutrino Kµ= (ϵ, k) that interacts with a target nucleus Pµ A= (MA,0), considered at rest in the laboratory system with rest mass MA, by exchanging energy ωand momentum qvia an intermediate boson Qµ= (ω, q). We assume that the final state is composed by a charged lepton K′µ= (El,k′), an ejected nucleon Pµ N= (EN,pN) and the residual nucleus Pµ A−1= (EA−1,pA−1). Generally, the residual nucleus could be in its ground state or in some discrete excited state with invariant mass WA−1. Given that the energies involved in the typical accelerator neutrino experiments are of the order of GeV, in this chapter the treatment of the kinematics, and later of the nuclear models, will be completely relativistic. The energy and momentum conservation in the hadronic vertex yields MA+ω=EN+EA−1 pA−1=q−pN=−pm,(2.1) 1This thesis is focused in neutrino-nucleus semi inclusive cross sections. Of course the formalism is also valid for reactions with antineutrinos. However there is not semi-inclusive antineutrino-nucleus experimental data available at the time of writing because the additional difficulty of detecting neutrons in the final state. Q k k0 PA pN PA−1 Figure 2·1: Schematic representation of the reaction A(ν, l N)A-1 in the Born approximation, i.e.one boson exchange. 28 where we have defined the missing momentum pmas minus the momentum of the residual nucleus, hence the total energy of the residual nucleus can be expressed as EA−1=qp2 m+W2 A−1. It is convenient to introduce a variable that measures the degree of internal excitation of the residual nucleus. The excitation energy of the residual nucleus Eis defined as E=EA−1−E0 A−1=qp2 m+W2 A−1−qp2 m+M2 A−1≥0 (2.2) with MA−1the rest mass of the residual nucleus. We can also define the so-called missing energy Emas Em=WA−1+mN−MA(2.3) and easily relate it with the excitation energy Ein the limit pm<< MA−1 Em=E+Es,(2.4) where Es=MA−1+mN−MAis the separation energy, namely the minimum energy necessary to remove a nucleon from a nucleus A. Using the conservation of energy and momentum in Eq. (2.1) and the definition of Ewe have E=MA−1−Es+mN+ω−qm2 N+p2 m+q2+ 2pmqcos θ−qp2 m+M2 A−1,(2.5) with θthe angle between the missing and transfer momenta. This relation, for some fixed values of qand ω, gives a curve E(pm) for each value of θ. Since the value of θis limited to the range −1≤cos θ≤1, not all the plane (E, pm) is accessible and the maximum and minimum values of pmthat are allowed will depend on the specific state of the residual nucleus, qand ω[55, 56]. The advantage of using (E, pm) or (Em, pm) as dynamic variables of the process, in addition to qand ω, instead of experimental-accessible variables like the ejected proton 29 momentum and polar angle respect to the neutrino beam is that these variables are more appropriate to describe the nuclear dynamics. The most relevant contributions of the semiinclusive cross section are localized at relatively small values of E, where one typically finds distributions as functions of pmthat reflect the shell structure of the target nucleus. For instance, in a simple shell model of the nucleus one sees features that reflect the knockout of nucleons from the valence shell, the next-to-valence shell and so on. These momentum distributions fall relatively fast with increasing pm. Nevertheless for the analysis of exclusive (e, e′p) measurements these simple models have been proven not to be adequate and one requires overall suppression of these momentum distributions by factors of typically 30% via the so-called spectroscopic factors [57]. Additionally, from past (e, e′p) studies one knows that part of this missing strength is moved to higher values of E, partially through nuclear interactions that make both initial and final nuclear states more complex than the simple shell model. For instance, the NN interaction has both long- and short-range contributions, and especially the latter can promote strength to higher values of Eand pm[57, 58]. Around 20%−30% of the strength is known to occupy this high Eand pmregion, although the actual contributions are not very well determined. 2.2 Plane-wave impulse approximation In the previous section we have described in a generic way the semi-inclusive CC neutrinonucleus reaction and its kinematics. However due to the complexity of the interaction of the exchanged boson with a many-body system, i.e.the initial nucleus, we will make use of the impulse approximation (IA), only valid for high values of q, to describe the hadronic vertex. In the IA, the boson exchanged in the process is absorbed by a single bound nucleon of the initial nucleus, therefore the hadronic current is constructed only by one-body operators and two-body currents are not considered. If one neglects FSI, the nucleon ejected from the nucleus and detected in the final state is the one that interacted directly with the 30 Q pm k0 k PA−1 PA pN Figure 2·2: Semi-inclusive neutrino-nucleus interaction in the IA. exchanged boson, while the rest of the nucleons inside the nucleus participate in the reaction as spectators. The semi-inclusive reaction in this approximation is schematically represented in Fig. 2·2 and two new variables are introduced, the momentum pand energy Eof the bound nucleon, which can be related with the rest through conservation E=EN−ω=MA−EA−1 p=pN−q=pm.(2.6) Notice that the momentum of the bound nucleon is equal to the missing momentum and, since a nucleon bound inside a nucleus is off-shell, in general its energy and momentum are not related through the usual on-shell relativistic expression of the energy: E=E=pp2 m+m2 N. Regarding the description of the ejected nucleon, to start we assume a further simplification by considering the ejected nucleon as a plane-wave, i.e., the distortion caused by the residual nucleus is neglected. This corresponds to the plane-wave impulse approximation (PWIA) and allows us to greatly simplify the description of the hadronic vertex by factorizing the nuclear hadronic tensor Hµν in the product of two objects. On of them is the single-nucleon hadronic tensor Wµν that is constructed by considering that the neutrino interacts with a single off-shell nucleon described by Dirac spinors with positive energy. And the other is the so-called spectral function S(pm, Em) that depends on the specific nuclear model used 31 to describe the initial nuclear state and is defined as the probability of finding a nucleon in the initial nucleus with energy Emand momentum pmcompatible with the kinematics of the process. This exact factorization can also be found in the analysis of exclusive (e, e′p) processes [59] under similar approximations. Although PWIA is a severe oversimplification of the semi-inclusive neutrino-nucleus scattering and will certainly fail to describe precisely the experimental data, this approximation allow us to get an analytic expression of the hadronic tensor for some simple nuclear models and reduce the numerical complexity of the calculation for more realistic nuclear models. Hence PWIA calculations will allow us later to check the impact on the semi-inclusive cross section of the different models of the initial nuclear state without complications introduced by FSI, that will be included in a fully quantum and relativistic fashion in later chapters when the comparison with data from different accelerator neutrino experiments is presented. Since in accelerator-based neutrino experiments the direction of the neutrino is known, we will work in a coordinate system where the neutrino direction is fixed along the z-axis as shown in Fig. 2·3. The sixth-differential CC neutrino-nucleus cross section as function of the neutrino energy and the detected lepton and nucleon kinematics is given by [49, 54] dσ dk′dΩk′dpNdΩL N =(GFcos θck′pN)2mN 8kε′EN(2π)6 ∞ Z 0 dEZd3pmυ0F2 χSpm, Em(E, pm) ×δ(MA+k−ε′−EN−qp2 m+M2 A−1−E)δ(k−k′−pN+pm) (2.7) with the final lepton and nucleon solid angles Ωk′= (θl, ϕl) and ΩL N=θL N, ϕL N,GFand θcthe Fermi constant and the Cabibbo angle respectively, F2 χ=2 υ0LµνWµν the Lorentz invariant resulting from the contraction of the leptonic tensor Lµν and the single-nucleon hadronic tensor Wµν,υ0= (k+El)2−q2a kinematical factor and S(pm, Em) the spectral function. The two δ−functions guarantee the energy and momentum conservation (see Eq. (2.1)). Notice that in Eq. (2.7) an integration over unmeasured quantities related with the residual 32 nucleus is done and that, as pointed out before, in the PWIA there is a clear factorization of the cross section into a part that describes the elementary neutrino-nucleon interaction, contained in F2 χ, and the spectral function S(pm, Em), that depends on the nuclear model used for the description of the initial state. The single-nucleon hadronic tensor Wµν in PWIA can be constructed supposing that the initial and final nucleons are described by free positive-energy Dirac spinors. The detailed calculation of Wµν can be found in Appendix A. Additionally, in Appendix B we provide the analytic expression of F2 χand the nuclear responses it depends on. The spectral function S(Em, pm) in Eq. (2.7) can be related to the nuclear momentum distribution n(pm) using its definition n(pm) = ∞ Z 0 dEmS(Em, pm),(2.8) Scattering plane Reaction plane z x y x x0 k pN k0 θl θL N φL N Figure 2·3: Definition of the coordinate system where the direction of the incoming neutrino kis fixed along the z-axis. The leptons are contained in the denominated scattering plane, while the ejected nucleon with momentum pNand polar angle with respect to the neutrino direction θL Nis contained in the so-called reaction plane, which forms an angle ϕL Nwith the scattering plane. 33 and taking into account that the momentum distribution is normalized as follows N=1 (2π)3 ∞ Z 0 dpmp2 mn(pm) (2.9) with Nthe number of nucleons that participate in the scattering. In our case, since we will focus in the analysis of CCνreactions, this is the number of neutrons in the target nucleus. The Eq. (2.7) depends on the kinematics of the final lepton and nucleon detected and assumes that the neutrino energy kis known, hence ωand qare also known and fixed. However, to compare theoretical semi-inclusive predictions with experimental measurements it is necessary to take into account that in accelerator-based neutrino experiments the neutrino beam does not have a well-defined energy. In Fig. 2·4 we show νµflux predictions for different neutrino experiments, which can be separated in low-energy, i.e.T2K, MiniBooNE, or NINJA, and high-energy, i.e.MINERνA or DUNE, experiments. As a consequence of these broad neutrino fluxes, one needs to integrate the cross section over all the possible energies weighted by the normalized experimental neutrino flux P(k) in order to compare with the experimental data. Using the energy and momentum conservation to integrate over Eand pmin Eq. (2.7), the flux-averaged semi-inclusive cross section can be expressed as dσ dk′dΩk′dpNdΩL N=(GFcos θck′pN)2mN 8ε′EN(2π)6 ∞ Z 0 dkP(k) kυ0F2 χSpm, Em(E, pm),(2.10) where pm=k′+pN−k, E=MA+k−El−EN−pp2 m+MA−1 ≈k−El−EN+mN−Es,(2.11) 40 Assuming that the discrete states occupied by the nucleons, commonly known as shells, are characterized by the usual quantum numbers n, l, j with an associated bound energy −Enlj, the spectral function will be given simply by SIPSM (pm,E) = X n,l,j (2j+ 1) nnlj (pm)δ(E+Es−Enlj),(2.34) where nnlj(pm) is the positive-energy RMF momentum distribution given in Eq. (2.32), the sum is extended to all the occupied shells (n, l, j), i.e.,Pj(2j+ 1) = N, and the energy conservation states E=Enlj −Es; hence the separation energy in the IPSM is the energy of the highest shell. Lastly, one can notice that the normalization of the relativistic wave function in Eq. (2.31) is broken when the negative-energy components are dropped to construct the positive-energy momentum distribution in Eq. (2.32). However, the overall contribution of the negative-energy component and its interference term with the positiveenergy component is rather small (2% −3%) [59] and in our case a renormalization of Eq. (2.32) is performed to compensate the lack of negative-energy contributions and also to satisfy the normalization adopted in Eq. (2.9). Introducing Eq. (2.34) into Eq. (2.10) one can get the flux-averaged semi-inclusive cross section for the IPSM dσ dk′dΩk′dpNdΩL N=(GFcos θck′pN)2mN 8(2π)6ε′ENX n,l,j (2j+ 1)P(knlj) knlj υ0F2 χnnlj(pm),(2.35) where the neutrino energy knlj =El+EN−mN+Enlj (2.36) and the missing momentum p2 m=k2 0nlj +k′2+p2 N−2k0nljk′cos θl−2k0nljpNcos θL N + 2k′pN(cos θlcos θL N+ sin θlsin θL Ncos ϕL N) (2.37) 41 are fixed by energy conservation. 2.3.3 Natural orbitals shell model The Natural orbitals (NO) shell model is presented as an extension of the IPSM specially constructed to create a realistic spectral function that leads to a good agreement with the scaling function extracted from the analysis of inclusive electron-nucleus scattering data [64, 65]. The model starts from the simple IPSM spectral function given in Eq. (2.34), which can be rewrite as SIPSM (pm,E) = X i (2ji+ 1)ni(pm)δ(E −Ei),(2.38) where the sum iruns up to the last shell occupied with eigenvalue Ei. Then, effects beyond the mean field approximation are included by replacing the δ-function in Eq. (2.38) by a function with finite width in energy like a Lorentzian function LΓi(E −Ei) = 1 2π Γi (E −Ei)2+ (Γi/2)2,(2.39) where Γiis the width of the isingle-particle state, and by using single-particle wave functions that diagonalize the one-body density matrix [66] ρ(r,r′) = X a Naψ∗ a(r)ψa(r′).(2.40) The NO single-particle wave functions in Eq. (2.40), that include short-range NN correlations, are used to obtain the natural occupation numbers Nαand the wave functions in momentum space, i.e., the momentum distributions. The NO spectral function is then given by [67] SNO(pm,E) = 1 2πA X i (2ji+ 1)Niψi(pm) 2LΓi(E −Ei),(2.41) 42 where the single-particle NO momentum distributions ψi(p) 2are normalized according to Eq. (2.9). The flux-averaged semi-inclusive cross section associated to this model is dσ dk′dΩk′dpNdΩL N=ZdE(GFcos θck′pN)2mNP(k) 8kε′EN(2π)7A ×X i (2ji+ 1)Ni|ψi(pm)|2LΓi(E −Ei)υ0F2 χ,(2.42) where the neutrino and missing momenta are fixed by conservation k=Es+EN+El−mN+E, p2 m=k2+k′2+p2 N−2kk′cos θl−2kpNcos θL N2k′pN(cos θlcos θL N+ sin θlsin θL Ncos ϕL N). (2.43) Notice that in the NO model, the substitution of the δ-function by LΓi(E −Ei) in Eq. (2.41) does not allow us to solve the integral over Eanalytically, like for the previous nuclear models, hence the integral has to be performed numerically. 2.3.4 Rome spectral function model One of the most realistic and complete descriptions of nuclear targets commonly used in accelerator-neutrino experiments is given by the so-called Rome spectral function model [58, 68], usually referred just as Rome or spectral function (SF) model within the neutrino community. In the SF model, the nuclear spectral function is split in two parts SRome (pm, Em) = SSP (pm, Em) + Scorr (pm, Em),(2.44) the single-particle contribution SSP (pm, Em) that gives the bulk contribution to the spectral function for missing energies up to ∼40 MeV and missing momenta up to ∼250 − 300 MeV and the correlated contribution Scorr (pm, Em). This correlated part shows a 43 widespread background extending up to large values of both Emand pm. This is in contrast to SSP (pm, Em), which has a tail that is typically extended just above the Fermi level and is sharply peaked in Emaround the bound energy values of the single-particle states in the nuclear shell model. The correlated part of the spectral function arises from contributions of configurations having pairs of strongly correlated nucleons, which are mostly sensitive to the short-range NN interaction. Since it is expected that the correlated part of the spectral is only mildly affected by finite-size effects, this part is extracted from a calculation in nuclear matter which is recalculated for finite nuclei employing the local-density approximation (LDA) [68]. The SF model employs (e, e′p) experimental data (e.g. Saclay [69] for 16O) to parametrize SSP (pm, Em) in the following way SSP (pm, Em) = X n Znϕn(pm) 2Fn(Em−En),(2.45) where ϕn(pm) is the wave function describing the single-particle states in momentum space, which is usually fitted to experimental data with Woods-Saxon wave functions, Fn(Em−En) is the width of the single-particle states (e.g. Lorentzian- or Gaussian-shaped function) and Znis the spectroscopic factor that gives the normalization of the nshell. From the analysis of (e, e′p) data it has been shown that Zn<1 and that the single-particle part of the spectral function accounts for 70 −80% of the total normalization, while the remaining 20 −30% is associated to the correlated part. The 12C and 16O spectral functions calculated using this approach are shown in Fig. 2·5 and the flux-averaged semi-inclusive cross section associated to this model is obtained by simply introducing Eq. (2.44) in Eq. (2.10) dσ dk′dΩk′dpNdΩL N=(GFcos θck′pN)2mN 8ε′EN(2π)6 ∞ Z 0 dkP(k) kυ0F2 χSRome (pm, Em).(2.46) To see more clearly the differences between the nuclear models, Fig. 2·6 shows the momentum distribution n(pm) of 12C for all the models described in this section. As already 44 100 200 300 400 500 (MeV) m p 110 2 10 (MeV) m E 0 2 4 6 8 10 12 14 16 18 9− 10× ) -4 ) (MeV m , p m S(E C 12 100 200 300 400 500 (MeV) m p 110 2 10 (MeV) m E 0 5 10 15 20 25 30 35 40 9− 10× ) -4 ) (MeV m , p m S(E O 16 Figure 2·5: 3-D plot of the carbon (left) and oxygen (right) spectral functions calculated using the Rome approach [58, 68], i.e.the single-particle contribution extracted from the analysis of (e, e′p) experimental data and using the LDA for the correlated part. mentioned above, the RFG momentum distribution is constant up to kFand then drops to zero, whereas the other models display a more complex shape characterized by tails that extend beyond kF. Also it can be seen that the NN correlations included in the Rome momentum distribution significantly increase the strength of the distribution for large values of pm. This effect is observed also for the NO model when it is compared to IPSM distribution, although this additional contribution is smaller than in the Rome case. 2.4 Semi-inclusive neutrino-nucleus cross section in PWIA In the previous sections the flux-averaged semi-inclusive neutrino-nucleus cross section in PWIA for different models of the initial nuclear state was introduced. Since we are trying to understand the differences between the models, in what follows we show semi-inclusive cross sections for some specific kinematics where the dependence of the momentum distribution on the missing momentum is particularly relevant. We begin considering the CC νµsemi-inclusive cross sections for IPSM and RFG model 45 0 100 200 300 400 500 0 2 4 6 8 pm (MeV) n(pm) (105GeV-3) RFG IPSM-Total IPSM-1s1/2 IPSM-1p3/2 NO Rome 0 100 200 300 400 500 10-10 10-5 100 pm (MeV) n(pm) (105GeV-3) RFG IPSM-Total IPSM-1s1/2 IPSM-1p3/2 NO Rome 0 100 200 300 400 500 0 2 4 6 8 pm (MeV) n(pm) (105GeV-3) RFG IPSM-Total IPSM-1s1/2 IPSM-1p3/2 NO Rome 0 100 200 300 400 500 10-10 10-5 100 pm (MeV) n(pm) (105GeV-3) RFG IPSM-Total IPSM-1s1/2 IPSM-1p3/2 NO Rome Figure 2·6: Momentum distribution n(pm) calculated by integrating the spectral function of different nuclear models of 12C. The value of the Fermi momentum for 12C is kF= 0.228 GeV and the individual shells that are included in the IPSM are shown separately. The distributions are shown in linear (left) and semi-logarithmic (right) scales to expose the differences between the models in the low- and high-missing momentum zones. with muon momentum k′= 1.5 GeV, muon scattering angle θl= 30◦and two different values for the final proton azimuthal angle, namely ϕL N= 180◦and ϕL N= 165◦, as function of the ejected proton momentum pNand the angle θL N. We consider 40Ar as the target and use the neutrino flux corresponding to DUNE. Results for the RFG model are presented in Fig. 2·7 using kF= 0.241 GeV for two different points of view, namely “side” (top panels) and “hawk” (bottom) views. The graphs on the left correspond to ϕL N= 180◦, while the ones on the right to ϕL N= 165◦. In both cases the shape of the cross section is simple being only different from zero in a very well-defined area given by the Fermi condition pm≤kF. Notably, compared to the ϕL N= 180◦scenario, the region where the cross section exists is significantly reduced when ϕL N= 165◦. In the latter kinematics, only a few points in the (pN, θL N) plane satisfy the condition that the corresponding missing momentum, as defined in Eq. (2.24), is smaller than kF. Fig. 2·8 shows the semi-inclusive cross sections for the IPSM. For ϕL N= 180◦(left panels), the shape is more complex compared to the RFG model. The contour appears more diffuse, and the maximum is approximately positioned at the center 46 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ 0 1 2 3 4 5 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ 0 0.5 1 1.5 2 2.5 3 3.5 4 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0 0.5 1 1.5 2 2.5 3 3.5 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ Figure 2·7: Semi-inclusive cross section for 40Ar and DUNE flux with k′= 1.5 GeV, θl= 30◦,ϕL N= 180◦(left panels) and ϕL N= 165◦(right panels) using the RFG model. of the existing cross-section region. However, its distribution clearly differs from the results observed for ϕL N= 165◦(right panels). Here, the cross section displays a symmetric shape with a well-defined maximum at the center of the projected contour. As one moves away from the center, the magnitude uniformly decreases in all directions. It is noteworthy that the maximum value of the cross section decreases by approximately ∼70% when ϕL N= 165◦, in contrast to the corresponding value at ϕL N= 180◦. On the contrary, note that for the RFG model (Fig. 2·7) the reduction is only ∼15%. This is clearly illustrated in Table 2.1 where the specific values, denoted by (˜pN,˜ θL N), for which the semi-inclusive cross section reaches its maximum for the two ϕL N-angles considered and both nuclear models are summarized. The origin of these features is further investigated in the next plots. 47 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ 0 1 2 3 4 5 6 7 8 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0 1 2 3 4 5 6 7 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ 0 0.5 1 1.5 2 2.5 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 ) -2 GeV 2 cm -37 (10 L N Ωd N dp k' Ωdk'd σd 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (GeV) N p 0 10 20 30 40 50 60 70 80 90 (deg) L N θ Figure 2·8: Semi-inclusive cross section for 40Ar and DUNE flux with k′= 1.5 GeV, θl= 30◦,ϕL N= 180◦(left panels) and ϕL N= 165◦(right panels) using the IPSM. ϕL N= 180◦ϕL N= 165◦ ˜pN˜ θL NCross section ˜pN˜ θL NCross section RFG 1.43 GeV 43.50◦4.62 ×10−37 1.24 GeV 42.89◦3.95 ×10−37 IPSM 1.00 GeV 49.54◦7.56 ×10−37 1.00 GeV 48.33◦2.32 ×10−37 Table 2.1: Values of pNand θL Nthat give the maximum cross sections in Fig. 2·7 and Fig. 2·8, i.e., (˜pN,˜ θN). Cross sections in cm2GeV−2. Fig. 2·9 shows the semi-inclusive cross sections for the IPSM (red dashed) and the RFG (blue solid) models and the two values of ϕL N. The graphs on the left present the semiinclusive cross section as a function of the ejected nucleon momentum pNevaluated at the values of ˜ θL Nwhere the maximum in the cross section occurs in each model. The panels on the right show the corresponding cross sections as function of θL Nat fixed ˜pN. Not only the 48 shapes in the two models completely disagree but also the region in (pN,θL N) where the cross section is defined differs very significantly. 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 ϕNL = 180º, θNL = θNL pN(GeV) dσ/dk'dΩk'dpNdΩNL(10-37cm2GeV-2) � RFG IPSM 20 40 60 80 0 2 4 6 8 ϕNL = 180º, pN = pN θNL(deg) dσ/dk'dΩk'dpNdΩNL(10-37cm2GeV-2) � 0.0 0.5 1.0 1.5 2.0 0 1 2 3 4 5 ϕNL = 180º, θNL = θNL pN(GeV) dσ/dk'dΩk'dpNdΩNL(10-37cm2GeV-2) � 20 40 60 80 0 1 2 3 4 5 ϕNL = 180º, pN = pN θNL(deg) dσ/dk'dΩk'dpNdΩNL(10-37cm2GeV-2) � Figure 2·9: Semi-inclusive cross sections as function of pN(θL N) for the two values considered of the azimuthal angle ϕL N(see text for details). In each case the cross section is evaluated at the corresponding values ˜ θL N(˜pN) that give the maximum cross section in Figs. 2·7 and 2·8. The values of ˜pNand ˜ θL Nare summarized in Table 2.1. The specific contribution of the different shells in the IPSM to the semi-inclusive cross section in the case of 40Ar is shown in Fig. 2·10, where a cut in the plane θL N=˜ θL Nof the semi-inclusive cross section shown in Fig. 2·8 is presented. The left (right) panel in Fig. 2·10 corresponds to ϕL N= 180◦(ϕL N= 165◦). In both graphs the behavior and allowed values of the missing momentum pm(solid black line) are shown. According to the 40Ar momentum distribution (see Fig. 2·11), the s-shell contribution is dominant for very low 49 0.5 1.0 1.5 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 � ϕNL = 180º, θNL = θNL pN(GeV) dσ/dk'dΩk'dpNdΩNL(10-37cm2GeV-2) pm(GeV) f-shell d-shells p-shells s-shells All shells pm 0.5 1.0 1.5 0 1 2 3 0.0 0.2 0.4 0.6 0.8 � ϕNL = 165º, θNL = θNL pN(GeV) dσ/dk'dΩk'dpNdΩNL(10-37cm2GeV-2) pm(GeV) f-shell d-shells p-shells s-shells All shells pm Figure 2·10: (Left axes) Contributions to the semi-inclusive cross section by each shell in the IPSM for 40Ar. The value of ˜ θL Nis given in Table 2.1 and the lepton variables are fixed to k′= 1.5 GeV and θl= 30◦. (Right axes) Missing momentum pmdefined in Eq. (2.37) for the IPSM. Since its value is different for each shell, here is shown the average over all the 40Ar shells. missing momentum. For the kinematics considered in the left panel of Fig. 2·10 this very low-pmregion corresponds to values of the ejected nucleon momentum in the vicinity of pN = 1.0 GeV (pm∼40−50 MeV). The other shells give a smaller contribution, generating the secondary peaks observed in Fig. 2·8 at ϕL N= 180◦. These peaks disappear at ϕL N= 165◦ being also the cross section significantly smaller. As shown in the right panel of Fig. 2·10, the missing momentum gets also its minimum value in the region of pNclose to 1 GeV. However, here pm∼200 MeV, i.e., much larger than the corresponding value in the previous case. As clearly illustrated by the 40Ar momentum distribution (Fig. 2·11), at pm≃200 MeV the shells that contribute the most are the dand p-ones. This is consistent with the more symmetric shape of the semi-inclusive cross section shown in Fig. 2·8 for ϕL N= 165◦(right panels) with only one visible peak. According to the general energy and momentum conservation given by Eqs. (2.1, 2.5), it is possible to obtain an expression for E, or equivalently Em, as function of pmfor a selected 56 averaged semi-inclusive neutrino-nucleus cross section can be written as [73, 74, 75] dσ dk′dΩk′dpNdΩL N=G2 Fcos2θck′2p2 N 64π5Zdk WA−1 EA−1frec LµνHµν P(k),(3.1) where the recoil factor is given by frec = 1−pm·ˆz EA−1 .(3.2) Notice that the integral over the neutrino momentum kin Eq. (3.1) is equivalent to the integral over the missing energy Embecause they are related through the energy-momentum conservation: Em=WA−1+mN−MA=k−El+mN−EN−TA−1(3.3) with TA−1the kinetic energy of the recoiling system. The hadronic tensor, which contains all the information about the nuclear dynamics of the process including FSI, is constructed in a generic way as Hµν =X iX f Jµ fiJν fi†=DA−1, pN ˆ JµAEDA−1, pN ˆ JνAE∗,(3.4) where one averages over initial and sums over final states, |A⟩is the initial nuclear state given by the target nucleus A,|A−1, pN⟩is the final hadronic state composed of the ejected proton with asymptotic momentum pNand the undetected final nucleus A−1, and ˆ Jµis the weak current operator. Assuming that the IA is valid, then the current operator ˆ Jµis reduced to an incoherent sum of relativistic one-body current operators. Additionally, we are assuming that the initial nucleus is described by the independent-particle shell model. As already explained in Sec. 2.3.2, within the IPSM the initial state is made of a finite number of nucleons characterized by the quantum numbers κand µ, and described by a relativistic 57 wave function solution of the Dirac equation using the RMF potentials. Then, taking into account the IA and IPSM assumptions, the hadronic tensor is given by Hµν κ=ρκ(Em)X µ,sN Jµ κ,µ,sNJν∗ κ,µ,sN(3.5) with ρκ(Em) the missing energy density or profile of the initial nucleus, which are Dirac deltas δ(Em−Eκ) at the values of the bound energies if one uses a pure IPSM to describe the initial state, and Jµ κ,µ,sN=Zdreir·qΨsN(r, pN)ˆ Jµ NΦµ κ(r),(3.6) where µis the third component of the total angular momentum j=|κ| − 1/2 of the bound nucleon, sNthe spin projection of the final nucleon and ˆ Jµ Nis the CC2 one-body current operator given in Eq. (A.4). The wave functions ΨsNand Φµ κare four-dimensional spinors which describe, respectively, the scattered and bound nucleons. Within RDWIA, the scattered proton wave function ΨsNwith asymptotic momentum pNis obtained as a partial wave expansion ΨsN(r, pN) = 4πrEN+mN 2ENX κ,µ,m′ e−iδ∗ κil< l m′1 2sN|j µ > Y ∗ lm(ˆ r)ψµ κ(r, pN) (3.7) with Ylm(ˆ r) the spherical harmonics that describe the nucleon direction and lthe orbital angular momentum. The relativistic wave function ψµ κ(r, pN) for a certain κcan be expressed as in Eq. (2.25) where the radial functions gκ(r) and fκ(r) are solutions of the Dirac coupled differential equations (see Eq. (2.27)) with some scalar and vector potentials, in addition to the Coulomb potential. However, as we will see in Sec. 3.2, the scalar and vector potentials can be complex for some FSI models, therefore the outgoing proton will be described in that case by the complex conjugates of the radial functions. Finally, the phase shifts δ∗ κand the 58 normalization are calculated by matching the solutions of Eq. (2.27) with its asymptotic wave function in the limit r→ ∞. In the special case of the long-range Coulomb potential, the solutions are matched with the analytic expressions of the Dirac-Coulomb wave functions [77]. The bound wave functions Φµ κ(r) in Eq. (3.6) will be the complete bound-state solutions of the Dirac equation with the RMF potentials (see Eq. (2.25)). In the next two sections, we describe more in detail the specific model of the initial state used in this thesis and the possible choices of potentials to model FSI within RDWIA. 3.1 Initial state within RDWIA: beyond the relativistic mean-field model In Sec. 2.3.2 we briefly introduced the independent-particle shell model within the context of semi-inclusive neutrino-nucleus reactions in PWIA. There, the initial nuclear state was described only by the positive-energy bound solutions of the Dirac equation using the RMF potentials. Within the RDWIA not only the positive-energy but also negative-energy solutions using the RMF potentials are going to be used for the initial nuclear state Φµ κ(r) in Eq. (3.6). Also, it is known from exclusive (e, e′p) experiments that effects beyond the mean-field approximation, such as short- and long-range correlations, modify the missing energy distribution predicted by a pure shell model like the RMF model. Hence the RMF description of the initial state will be extended to include effects beyond the mean-field approximation taking as reference the spectral function formalism (as described in Sec. 2.3.4), that simultaneously includes the depletion of the occupation of the shell-model states and the appearance of nucleons at higher missing-energies due to nucleon-nucleon correlations. However since the SF formalism assumes the factorization of the semi-inclusive cross section in to the elemental single neutrino-nucleon cross section times the spectral function, while the RDWIA formalism is an unfactorized calculation, we follow [73, 75] to include, in an effective way, the improvements of the SF formalism without imposing factorization in the 59 distorted-wave calculation for 16O and 12C. For both nuclei, the missing energy profile ρ(Em) that appears in Eq. (3.5) can be divided in different regions. In the low missing-energy region, the p-shell states (both 1p3 2and 1p1 2for 16O and 1p3 2for 12C) will have a Gaussian energy dependence rather than δ-functions as in the IPSM. The same is applied to the 1s1 2shell in the intermediate missing-energy region. The high Emand pmregion of the spectral function, that accounts for correlations, will be reproduced in the RDWIA calculation by introducing an additional swave, denominated background, that is broad in momentum space and that is fitted to reproduce the momentum distribution of the Rome spectral function [58, 68] for both nuclei. In the intermediate missing-energy region, where the s-shell and the background coexist, and in the high missing-energy zone, where only the background contributes, the background is parametrized as follows [73]: F(Em) = Sbexp (−100 b) exp [−(Em−c)/w]+1 (3.8) if 26 < Em<100 MeV, and F(Em) = Sbexp (−b Em) (3.9) if Em>100 MeV. The values of the parameters are obtained by performing a fit of the Rome spectral function, which yields Sb= 0.0177 MeV−1,b= 0.011237 MeV−1,c= 40 MeV and w= 5 MeV. Since we are more interested in the context of this thesis in 12C, the specific parameterization of ρ(Em) for this nucleus is shown in Fig. 3·1. Also, the contribution of two shells of 12C to the missing energy profile is summarized in Table 3.1. In later chapters the comparison with semi-inclusive cross sections measurements and the RDWIA predictions for 40Ar will be presented, however an equivalent Rome spectral function does not exist for this nucleus. Therefore for 40Ar a missing energy profile ρ(Em) will be proposed [76] combining RMF results, phenomenology and exclusive (e, e′p) measurements on 40Ar by JLAB collaboration [78]. As for 12C, the wave functions of the 22 neutrons 60 50 100 150 200 250 300 10-3 10-2 10-1 100 Em (MeV) ρ(Em) (MeV-1) 1p3/2 1s1/2 Background Total Figure 3·1: Parameterization of the missing energy profile ρ(Em) for 12C separated by contributions of the individual shells and the background. The background is normalized so there are 0.8 nucleons up to Em= 300 MeV, i.e. R300 26 dEmF(Em) = 0.8. The remaining nucleons are associated to the shells as summarized in Table 3.1. κ Eκ(MeV) σκ(MeV) nκ 1s1/237.0 10.0 1.9 1p3/217.8 2.0 3.3 Table 3.1: Parameterization of the missing energy distributions for the two shells of 12C. The contribution to the missing energy density ρκ(Em) of each shell is given by ρκ(Em) = nκ √2πσκexp(−Em−Eκ 2σκ2 ), with Eκposition of the peak, σκthe width and nκthe occupation number, i.e. the number of nucleons in each shell. in 40Ar will be given by the bound states solutions of the Dirac equation with the RMF potentials. However the realistic missing energy profile ρ(Em) for 40Ar will be modeled as a Maxwell-Boltzmann distribution [78] for each shell κ ρκ(Em) = 4Sκ √πσκEm−Eκ+σκ σκ2 exp −Em−Eκ−σκ σκ2(2jκ+ 1) ,(3.10) with Eκthe position of the peak, σκthe width, Sκthe occupancy of the shell, i.e. the ratio 61 between the number of nucleons in a shell and the number of nucleons predicted by the IPSM, and jκthe total angular momentum of the shell κ. A summary with the parameterization for the seven shells in 40Ar can be found in Table 3.2. There, the position of the peaks are given by the RMF predictions except for the valence shell (1f7 2) which is fixed to the experimental neutron separation energy [79]. The widths of the shells used in this thesis to describe neutrons in argon are inspired from (e, e′p) exclusive measurements in 40Ar by JLAB [78]. Finally, the spectroscopic factors assigned to each shell are based on phenomenology: the internal shells (1s1 2, 1p3 2and 1p1 2) are affected by short-range correlations while external shells (1d5 2, 2s1 2, 1d3 2and 1f7 2) do depend on both short- and long-range correlations [80, 81], therefore their spectroscopic factors are lower. In addition to the value of the parameters, Table 3.2 also shows the uncertainty associated to each parameter. Proceeding in this way we provide, for the first time, an uncertainty band in theoretical semi-inclusive neutrinonucleus predictions due to the modeling of the initial nuclear state. This uncertainty band will be generated by randomly sampling the values of the missing energy profile parameters within their uncertainties with an uniform probability distribution and running the RDWIA calculation for up to one hundred different combinations of the ρ(Em) parameterization. The only parameter missing for this model of neutrons in 40Ar is the background normalization Sbin Eqs. (3.8) and (3.9). This parameter, related with the number of neutrons in the background, is constrained by imposing that the total number of neutrons is fixed to 22 for each different parameterization of ρ(Em). However, using this method, some random combinations of the parameters could give up to 40-50% of the neutrons in background for argon which is clearly unrealistic. To avoid this, the RDWIA calculation is done only if the background contains between 15-25% of the 22 neutrons, which is consistent with previous studies [82, 83, 84]. The corresponding missing energy profile ρ(Em) for argon is shown in Fig. 3·2 compared with the RMF prediction. 62 α Eκ(MeV) σκ(MeV) Sκ 1s1/255 ±6 30 ±15 0.9 ±0.15 1p3/239 ±4 12 ±6 0.9 ±0.15 1p1/234 ±3 12 ±6 0.9 ±0.15 1d5/223 ±2 5 ±3 0.75 ±0.15 2s1/216.1 ±1.6 5 ±3 0.75 ±0.15 1d3/216.0 ±1.6 5 ±3 0.75 ±0.15 1f7/29.869 ±0.005 5 ±3 0.75 ±0.15 Table 3.2: Parameterization of the missing energy distribution for the 22 neutrons in 40Ar adopted in this work. The missing energy distributions are modeled as Maxwell-Boltzmann distributions (see the text). The spectroscopic factors or occupancies of the shells Sκgive the relative occupancy of the shell respect to the pure shell model prediction. The position of the 1f7 2shell was set to the experimental neutron separation energy [79], and the others were set to the RMF values. The widths used in this model were inspired from the proton results obtained by JLAB (e, e′p) experimental data on 40Ar [78]. 0 50 100 0.0 0.2 0.4 0.6 Em (MeV) ρ(Em) (MeV-1) average ρ(Em) RMF ρ(Em) uncertainty Figure 3·2: Missing energy profile of neutrons in 40Ar described by the parameterization given in Table 3.2. The red band corresponds to the uncertainties also summarized in Table 3.2. The vertical blue lines show the positions of the seven RMF shells and the black dashed line shows the mean value of the distribution. 63 3.2 Final state interactions within RDWIA In this section two approaches to FSI within RDWIA, denoted as Relativistic optical potential (ROP) and Energy-dependent relativistic mean field (ED-RMF), are discussed. In general, FSI models can either use complex potentials or only real potentials. Complex potentials, given that the imaginary part of the potential produces a loss of flux to inelastic channels, should be used when the kinematic constrains of the experiment being analyzed isolates the elastic channel, for instance exclusive (e, e′p) experiments. This is equivalent to constrain the values of the missing energy Embelow the two nucleons knockout energy, which guarantees that only the elastic channel is being measured. Instead, pure real potentials are more appropriate for inclusive-like measurements where all the channels contribute to the experimental signal, including the inelastic ones. 3.2.1 Relativistic optical potential The ejected nucleon wave function is calculated as the scattering solution of the Dirac equation in the presence of a complex relativistic optical potential. In this thesis, we use phenomenological relativistic optical potentials that are not derived from a specific microscopic theory but have been fitted to certain elastic proton-nucleus scattering observables. Specifically, we will use global scalar-vector (S—V) potentials that can be applied to protons with different kinetic energies. Cooper et at [85] published different parametrizations for nuclei of interest for accelerator neutrino experiments, such as 12C, 16O or 40Ca, for protons kinetic energies in the range 20 ≤TN≤1040 MeV. Although 40Ca is not used as target in accelerator neutrino experiments, its parametrized relativistic optical potential will be employed to generate the predictions for 40Ar in later chapters. The scalar or vector global 64 S—Vrelativistic optical potentials take the general form [85] U(r, E, A) = VV(E, A)fV(r, E, A) + VS(E, A)fS(r, E, A) (3.11) +iWV(E, A)gV(r, E, A) + WS(E, A)gS(r, E, A), where Eand rare the proton center of mass energy and position, Ais the atomic mass number of the target and the superscripts Vand Srefer to the volume and surface peaked terms respectively. The geometric functions fiand gi, with i=V,S, are fV(r, E, A) = cosh R(E, A)/a (E, A)−1 cosh R(E, A)/a (E, A)+ cosh r/a (E, A)−2, fS(r, E, A) = cosh R(E, A)/a (E, A)−1cosh r/a (E, A)−1 cosh R(E, A)/a (E, A)+ cosh r/a (E, A)−22.(3.12) The same functional forms are used for gi. As mentioned above, the optical potential consists of scalar and vector terms each having both real and imaginary parts. In the parametrization the eight strength parameters, namely Viand Wifor the scalar and vector terms, have the same polynomial dependence upon Eand A. For instance VV(E, A) = v0+ 4 X m=1 vmxm+ 3 X n=1 vn+4yn+v8xy +v9x2y+v10xy2,(3.13) where x= 1000/E, with Ein MeV, and y=A/ (A+ 20). Generally, both scalar and vector potentials, as well as their real and imaginary parts, may have different geometry parameter values. The functions R(E, A) and a(E, A) appearing in Eq. (3.12) are also represented by 65 012345 -400 -200 0 200 400 Real part r (fm) Potentials (MeV) Vector Scalar EDAIC (800 MeV) EDAIC (200 MeV) EDAIC (30 MeV) 012345 -200 -100 0 100 200 Imaginary part r (fm) Potentials (MeV) Scalar Vector EDAIC (800 MeV) EDAIC (200 MeV) EDAIC (30 MeV) Figure 3·3: Energy-dependent A-independent 12C (EDAIC) relativistic optical potential [85] evaluated at different proton kinetic energies TN= 30 MeV (dashed), 200 MeV (dotted) and 800 MeV (dash-dotted). The vector (red lines) and scalar (blue lines) components of the potential are shown separately for both the real (left panel) and imaginary (right panel) parts. polynomial expansions, as follows R(E, A) = A1/3r0+ 4 X m=1 rmxm+ 3 X n=1 rn+4yn+r8xy +r9x2y+r10xy2,(3.14) a(E, A) = a0+ 4 X m=1 amxm+ 3 X n=1 an+4yn+a8xy +a9x2y+a10xy2.(3.15) The optical potentials that will be used in this thesis are the ones denominated energydependent A-independent (EDAI) potentials where the dependence on Aof all the parameters is dropped. Further simplifications are performed, compared with the general energydependent A-dependent (EDAD) potentials, to reduce the total number of parameters for each target nucleus to 70 [85]. The number of data points used in the fits for each target is well over 1000. Fig. 3·3 shows the radial dependence of the EDAIC potential (the one for 12C) evaluated at different proton kinetic energies. As it was said previously, the ROP contains a real and an imaginary term, where the 72 nuclear model implemented in the generator. (IV) The interaction mode is selected from the probabilities for the corresponding incident neutrino energy. For instance, if a large neutrino energy was initially sampled, it is more probable that the selected mode is inelastic scattering rather than QE. (V) The neutrino-nucleon interaction is simulated according to the interaction mode selected and the differential cross section of the interaction is used to simulate the kinematics of the outgoing lepton, nucleon(s) and other particles (for instance, pions). (VI) The possible re-interactions of the particles within the nucleus are simulated, according to the FSI model. The semi-classical intranuclear cascade model (INC) is the most commonly used FSI model in neutrino generators. It considers each particle produced individually, starting from the interaction vertex and continuing until the particle exits the nucleus. During this propagation, the particle is moved in small steps (around 0.1 fm, typically), and at each step, the probability of an interaction is evaluated. This probability is based on the particle’s mean free path, which is influenced by the density of the surrounding medium (determined according to the nuclear model) and the probability of interacting with nucleons. Various types of interactions with the nucleus are possible, and their individual cross sections depend on the particles’ energies. The only quantum effect considered is Pauli blocking. Apart from that, the model is classical in nature, treating the propagated particles as free and assuming they travel in straight lines between consecutive scatterings. The scatterings themselves are evaluated using free-space cross sections. Pauli blocking can be included as an average effect by reducing the nucleon interaction cross section through a global factor or as a local effect by ensuring that the momentum of the final-state particles is above some value of the Fermi momentum at each interaction point. The implementation of a specific neutrino interaction model in a neutrino event generator 73 usually requires a fast method of calculating the differential cross section. However the RDWIA calculations demand significant computational resources to generate predictions for comparison with experimental measurements, although the computation can be accelerated by, for instance, precalculating and storing in tables the hadronic tensor components Hµν κ as function of four dynamic variables (for instance q, ω, pmand Em). Until this date few microscopic models capable of exclusive predictions are implemented in neutrino event generators. Some work is currently being done to implement models based in RDWIA approach in MC generators [74]. Models like SuSAv2 [91, 93, 94] and Hartree-Fock continuum random phase approximation (HF-CRPA) [104, 105] have been implemented in some neutrino event generators like GENIE, but these models yield inclusive predictions as function of the outgoing lepton kinematics only. Later on, we will elaborate on the implementation of these models [106, 107] in generators using a factorization ansatz [106]. In this approach, lepton kinematics is directly computed from a microscopic model calculation before the hadronic system is incorporated using approximate methods. As a result, the model accurately reflects the microscopic model’s predictions for lepton kinematics, but it provides only a broad estimation for outgoing hadron kinematics due to the limited information available. Summarizing, models that are in principle inclusive can be used in combination with neutrino event generators to provide semi-inclusive predictions by means of different approximations. Exploiting this approach, in this thesis we will compute the semi-inclusive predictions using the SuSAv2 implementation in GENIE [106] and directly compare them with the RDWIA results and experimental semi-inclusive measurements by different international collaborations. Since all the available semi-inclusive experimental data correspond to CC0π, we will also make use of the SuSAv2 2p2h model implemented in GENIE as well as the GENIE’s pion absorption model to generate the non-quasielastic (non-QE) contributions to semi-inclusive cross sections. In what follows, we discuss in detail the implementation approach of the SuSAv2 (QE and MEC) model in GENIE. Additionally, a brief description of GENIE’s 74 pion absorption model is also presented before we move on to the comparison with the experimental data in the next chapter. 4.1 SuSAv2 implementation in GENIE At present, only a few neutrino event generators are capable of calculating exclusive predictions. Most theoretical models implemented in the generators can predict the kinematics of the outgoing leptons but are not directly applicable to describe hadrons in the final state. Nevertheless, it is feasible to generate semi-inclusive predictions, involving both a lepton and a hadron in the final state, by employing inclusive models implemented in generators with the aid of approximations like the factorization approach [106]. However, given the impact of these semi-inclusive reactions on oscillation analyses, it is necessary to validate the accuracy of these approximations against experimental measurements of cross sections and also against microscopic neutrino interaction models capable of predicting final lepton and hadron kinematics. Among the different nuclear models for neutrino interactions, those based on the RMF theory are promising candidates to be implemented in event generators due to their accurate description of the nuclear dynamics and their good agreement with both inclusive and semiinclusive electron- and neutrino-nucleus scattering data without relying on any factorization approach. As a first attempt in this direction, the SuSAv2-MEC model [91, 93, 94], a purely inclusive approach based on the RMF theory which has proven to successfully predict inclusive cross sections for electrons and neutrinos in a wide range of kinematics, was implemented in the neutrino event generator GENIE [106] for both 1p1h and 2p2h channels. This implementation has been carried out via SuSAv2 1p1h and 2p2h inclusive hadron tensor tables, Hµν(q, ω), using a binning of 5 MeV in the transferred momentum and energy which is combined with GENIE’s interpolation methods between adjacent bins. Since the SuSAv2 is an inclusive model and all the information about hadrons in the final state has been 75 integrated, a factorization between the leptonic and hadronic vertices is used to generate semi-inclusive predictions. Under this approach the outgoing hadronic state is generated starting with an initial nucleon state sampled from a local Fermi gas (LFG) nuclear model for the 1p1h channel. A missing momentum pmis sampled from the LFG momentum distribution and the outgoing nucleon energy ENis calculated as [108] EN=qp2 m+m2 N−ω−Eb(q),(4.1) where Eb(q) is a q-dependent binding energy inspired by the phenomenological energy shift from the SuSAv2 model Eb= max (5,−17.687 + 0.0564q) [MeV] (q < 827MeV) (4.2) and Eb(q) = Eb(q= 827 MeV) ≃29 MeV for q > 827 MeV. The outgoing nucleon momentum calculated according to Eq. (4.1) will not agree in general with momentum conservation Eq. (2.1), i.e. pE2 N−m2 N=|pm+q|. To partially solve this issue, the magnitude of the outgoing nucleon momentum is taken from Eq. (4.1) but the direction is constructed according to the momentum conservation, i.e. pN=qE2 N−m2 Npm+q |pm+q|.(4.3) After the nucleon is extracted, the propagation is carried out through the nucleus using GENIE’s semi-classical cascade FSI model. In [108] the distributions as function of the nucleon kinetic energy and scattering angle were obtained for exclusive (e, e′p) events on carbon using the ED-RMF model and GENIE’s factorization approach described above. The distributions are shown in Fig. 4·1 where a shift in the nucleon kinetic energy distribution is observed between the exact calculation (“Full”) and the GENIE implementation (“Approx”). 76 Regarding the angular distribution, the differences found are larger than for the energy distribution with a slight displacement of the maximum and significant shape differences. As pointed out in [108] these differences could be washed out after applying the GENIE cascade FSI model, however it was found that the effects of the factorization approach remain visible. In the case of the SuSAv2 2p2h channel, based on the relativistic Fermi 0.0 0.1 0.2 0.3 0.4 0.5 0 2 4 Tp (GeV) P(Tp) (GeV-1) Full Approx -1.0 -0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 cos(θp) P(cos(θp)) Full Approx Figure 4·1: Proton kinetic energy (left) and scattering angle (right) distributions for the process e+A→e′+p+B. The electron beam energy is fixed to 1.159 GeV and the outgoing electron phase-space is limited to 17◦< θe′<40◦and Ee′>0.4 GeV. The red lines show the unfactorized RDWIA predictions obtained with ED-RMF potential for carbon (named “Full”) while the blue lines show the equivalent distributions obtained after applying the GENIE algorithm (Eq. (4.1) and Eq. (4.3)) and before adding GENIE cascade FSI (named “Approx”). Results taken from [108]. gas (RFG) calculation [109, 110], the momentum and energy transfer are assigned to a pair of nucleons drawn from a LFG nuclear model after taking into account the removal energy, which is a constant in this case. The probability of having neutron-neutron (protonproton for antineutrinos) or proton-neutron pairs as initial cluster is chosen based on the kinematics of the inclusive reaction using the SuSAv2-MEC 2p2h theoretical model [111]. The transferred momentum and energy are shared equally between the cluster components, one neutron (proton) is turned into a proton (neutron) for neutrinos (antineutrinos) and the cluster breaks up into two nucleons. The two nucleons are then propagated through the 77 nucleus via GENIE’s semi-classical cascade model. 4.2 GENIE pion absorption model GENIE can further be used to model the pion absorption contribution to semi-inclusive CC0π measurements. The predominant contribution stems from GENIE’s simulation of single pion production using an extension of the Rein-Sehgal model [112] to account for nonzero lepton masses [113], which produces pions and nucleons which are propagated through the nucleus via the same semi-classical cascade FSI model used for other interaction channels. In a portion of the events, the outgoing pions are absorbed within the nuclear medium, often leading to the ejection of additional nucleons in the process. The small additional contributions, where all mesons are absorbed by FSI, arise from more inelastic channels. These channels are evaluated using an effective leading-order model that incorporates the modifications proposed by Bodek and Yang [114] to accurately describe scattering at low momentum-transfers. 78 79 Chapter 5 Comparison with semi-inclusive experimental measurements In this chapter all the available CC semi-inclusive neutrino-nucleus experimental measurements from T2K [115], MINERνA [116, 117] and MicroBooNE [118, 119] are compared with the theoretical RDWIA QE predictions using different FSI models described in Chapter 3: ROP, rROP, ED-RMF and RPWIA. These unfactorized predictions will be compared to those obtained by the SuSAv2 model implemented in GENIE neutrino event generator as explained in Sec. 4.1. For a fair comparison with the experimental measurements, which are all CC0π, the SuSAv2-2p2h MEC and pion absorption contributions calculated with GENIE will be added to the QE predictions. 5.1 T2K The T2K collaboration measured [115] νµ−12C CC0πsemi-inclusive cross sections with two different topologies: i) one muon and no protons in the final state (1µCC0π0p) with momenta above 0.5 GeV, and ii) one muon and at least one proton in the final state (1µCC0πNp) where the leading (i.e.the fastest) proton has momentum above 0.5 GeV. 80 5.1.1 1µCC0π0p By the definition of this topology one could think that this corresponds to the inclusive CC0πchannel, where only the final lepton is detected while no information about hadrons in the final state does exist. However, in the definition of the inclusive cross section one explicitly integrates over the ejected hadron variables in the entire phase-space allowed by conservation and the kinematics of the reaction. For the 1µCC0π0p signal, the integration over the ejected proton kinematics is performed as in the inclusive case except for the integral over pNwhich is limited by the T2K experimental proton detection threshold of 0.5 GeV. Because of this limitation in the proton momentum, no inclusive model can be compared with these data because it would necessarily include contributions of protons with momenta above 0.5 GeV. Therefore these cross sections are indeed semi-inclusive and only models capable of predicting both the final muon and proton kinematics, like the RDWIA, can be compared against them. In Fig. 5·1 the microscopic RDWIA and the GENIE-SuSAv2 predictions are compared with T2K 1µCC0π0p measurements as function of the muon kinematics. For backward angles the microscopic calculation predicts a rather small difference between RPWIA and the models with FSI (rROP, ROP and ED-RMF), all of them underestimating the experimental measurements in contrast with the better agreement achieved with GENIE-SuSAv2. As one moves to more forward muon angles GENIE-SuSAv2 predictions start to overestimate some of the experimental points, an outcome probably due to scaling or factorization violations and poor treatment of low-energy effects which are accounted for more consistently by the ED-RMF model. The two RDWIA models that use real potentials, namely ED-RMF and rROP, give similar predictions for backward muon angles. However, some differences appear as one moves to more forward muon angles and the maximum discrepancy can be observed in the peak of the 0.98 <cos θl<1.0 distribution. In this zone, where qand ωare small, the ED-RMF prediction is closer to the data due to the better treatment of orthogonalization 81 between the initial and final nucleon wave functions. All the RDWIA models except the ED-RMF have at some degree lack of orthogonalization in this region, but in particular the overlap between the initial bound wave functions and the plane waves used in RPWIA is very large yielding big spurious contributions to the cross section for this model which largely overestimates the data at low values of k′. For all the angular bins the 2p2h contribution, although non-negligible, is limited to a few percent of the total cross section. Interestingly, the final state proton kinematic restriction (pN<0.5 GeV) leaves the pion absorption contribution negligible. No model is able to reproduce the sharp oscillation shown by the data just after the maximum in the last two bins (0.94 <cos θl<0.98 and 0.98 <cos θl< 1.0), but it should be noted that, once the reported correlations in the measured cross section are accounted for, the measurement shows no significant preference for an oscillation in the cross section. To quantify the agreement of the different models with the data, in Appendix D a χ2 analysis using the covariance matrices provided with the cross sections measurements is discussed in detail. All the results for T2K are summarized in Table D.1. For CC0π measurements without protons in the final state and momenta above 0.5 GeV, the ROP and ED-RMF models have associated a smaller χ2compared with GENIE-SuSAv2 results, although still much larger than the number of degrees of freedom (d.o.f.), indicating that low momentum protons are not quantitatively described by these models. 5.1.2 1µCC0πNp: Muon and proton kinematics In Fig. 5·2 and Fig. 5·3 the different models are compared with T2K 1µCC0πNp semiinclusive cross-section measurements with at least one proton with momentum above 0.5 GeV as function of the leading proton’s kinematics and the muon scattering angle, respectively. In general, the 2p2h channel seems to be more relevant for this case compared with the previous analysis, especially at forward scattering angles. This is not surprising, since the 88 -0.5 0.0 0.5 1.0 0 2×10-40 4×10-40 6×10-40 -1.0 < cos(θl) < -0.6 Δp (GeV) dσ/dΔp (cm2GeV-1nucleon-1) 2p2h GENIE-SuSAv2 ED-RMF ROP RPWIA rROP Other -0.5 0.0 0.5 1.0 0 1×10-40 2×10-40 3×10-40 4×10-40 -0.6 < cos(θl) < 0.0 ; k' < 0.25 GeV Δp (GeV) dσ/dΔp (cm2GeV-1nucleon-1) -0.5 0.0 0.5 1.0 0 2×10-40 4×10-40 6×10-40 8×10-40 -0.6 < cos(θl) < 0.0 ; k' > 0.25 GeV Δp (GeV) dσ/dΔp (cm2GeV-1nucleon-1) -0.5 0.0 0.5 1.0 0 1×10-40 2×10-40 3×10-40 4×10-40 0.0 < cos(θl) < 0.8 ; k' < 0.25 GeV Δp (GeV) dσ/dΔp (cm2GeV-1nucleon-1) -0.5 0.0 0.5 1.0 0 1×10-39 2×10-39 3×10-39 0.0 < cos(θl) < 0.8 ; k' > 0.25 GeV Δp (GeV) dσ/dΔp (cm2GeV-1nucleon-1) -0.5 0.0 0.5 1.0 0 2×10-40 4×10-40 6×10-40 0.8 < cos(θl) < 1.0 ; 0.25 GeV < k' < 0.75 GeV Δp (GeV) dσ/dΔp (cm2GeV-1nucleon-1) -0.5 0.0 0.5 1.0 0 2×10-40 4×10-40 6×10-40 8×10-40 1×10-39 0.8 < cos(θl) < 1.0 ; k' > 0.75 GeV Δp (GeV) dσ/dΔp (cm2GeV-1nucleon-1) Figure 5·4: T2K 1µCC0πNp semi-inclusive νµ−12C cross section as function of the inferred variable ∆pfor different muon kinematic bins. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. 89 -20 0 20 0 5×10-42 1×10-41 1.5×10-41 2×10-41 2.5×10-41 -1.0 < cos(θl) < -0.6 Δθ (degrees) dσ/dΔθ (cm2degree-1nucleon-1) 2p2h RPWIA ROP rROP GENIE-SuSAv2 ED-RMF Other -20 0 20 0 2×10-42 4×10-42 6×10-42 8×10-42 -0.6 < cos(θl) < 0.0 ; k' < 0.25 GeV Δθ (degrees) dσ/dΔθ (cm2degree-1nucleon-1) -20 0 20 0 5×10-42 1×10-41 1.5×10-41 2×10-41 -0.6 < cos(θl) < 0.0 ; k' > 0.25 GeV Δθ (degrees) dσ/dΔθ (cm2degree-1nucleon-1) -20 0 20 0 2×10-42 4×10-42 6×10-42 8×10-42 0.0 < cos(θl) < 0.8 ; k' < 0.25 GeV Δθ (degrees) dσ/dΔθ (cm2degree-1nucleon-1) -20 0 20 0 1×10-41 2×10-41 3×10-41 4×10-41 5×10-41 0.0 < cos(θl) < 0.8 ; k' > 0.25 GeV Δθ (degrees) dσ/dΔθ (cm2degree-1nucleon-1) -20 0 20 0 1×10-42 2×10-42 3×10-42 4×10-42 5×10-42 0.8 < cos(θl) < 1.0 ; 0.25 GeV < k' < 0.75 GeV Δθ (degrees) dσ/dΔθ (cm2degree-1nucleon-1) -20 0 20 0 5×10-42 1×10-41 1.5×10-41 0.8 < cos(θl) < 1.0 ; k' > 0.75 GeV Δθ (degrees) dσ/dΔθ (cm2degree-1nucleon-1) Figure 5·5: T2K 1µCC0πNp semi-inclusive νµ−12C cross section as function of the inferred variable ∆θin different muon kinematic bins. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. For readability, the axis range has been reduced to [−30◦,+30◦] hiding an experimental bin above 30◦with very low cross section and centering the [−360◦,−5◦] experimental bin around −20◦. 90 0.0 0.5 1.0 0 2×10-40 4×10-40 6×10-40 8×10-40 1×10-39 -1.0 < cos(θl) < -0.6 |Δp| (GeV) dσ/d|Δp|(cm2GeV-1nucleon-1) 2p2h GENIE-SuSAv2 ED-RMF ROP RPWIA rROP Other 0.0 0.5 1.0 0 1×10-40 2×10-40 3×10-40 -0.6 < cos(θl) < 0.0 ; k' < 0.25 GeV |Δp| (GeV) dσ/d|Δp|(cm2GeV-1nucleon-1) 0.0 0.5 1.0 0 2×10-40 4×10-40 6×10-40 8×10-40 -0.6 < cos(θl) < 0.0 ; k' > 0.25 GeV |Δp| (GeV) dσ/d|Δp|(cm2GeV-1nucleon-1) 0.0 0.5 1.0 0 1×10-40 2×10-40 3×10-40 0.0 < cos(θl) < 0.8 ; k' < 0.25 GeV |Δp| (GeV) dσ/d|Δp|(cm2GeV-1nucleon-1) 0.0 0.5 1.0 0 1×10-39 2×10-39 3×10-39 0.0 < cos(θl) < 0.8 ; k' > 0.25 GeV |Δp| (GeV) dσ/d|Δp|(cm2GeV-1nucleon-1) 0.0 0.5 1.0 0 1×10-40 2×10-40 3×10-40 4×10-40 0.8 < cos(θl) < 1.0 ; 0.25 GeV < k' < 0.75 GeV |Δp| (GeV) dσ/d|Δp|(cm2GeV-1nucleon-1) 0.0 0.5 1.0 0 2×10-40 4×10-40 6×10-40 8×10-40 0.8 < cos(θl) < 1.0 ; k' > 0.75 GeV |Δp| (GeV) dσ/d|Δp|(cm2GeV-1nucleon-1) Figure 5·6: T2K 1µCC0πNp semi-inclusive νµ−12C cross section as function of the inferred variable |∆p|in different muon kinematic bins. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. 91 5.1.4 1µCC0πNp: Transverse kinematic imbalances Another type of variables that can be used to analyze correlations between the particles detected in the final in semi-inclusive reactions are the so-called transverse kinematic imbalances (TKI) [120]. TKI are designed to enhance experimental sensitivity to nuclear effects, and therefore discriminate between different models, with minimal dependence on the neutrino energy. In particular, the use of TKI can help in disentangling effects linked to FSI, initial state correlations and/or multi-nucleon excitations (2p2h). They are defined by projecting the final lepton and the ejected nucleon momenta on the plane perpendicular to the neutrino direction (transverse plane) as can be seen in Fig. 5·7. More specifically, the vector magnitude Transverse plane z x y k pN pN T k0 k0 T qT δφT δpT δαT Figure 5·7: Scheme showing transverse kinematic imbalances (TKI): δpT,δαTand δϕT. The final lepton and nucleon momenta are projected on the plane perpendicular to the neutrino direction (xy-plane or transverse plane). The transverse component of the transferred momentum (qT) equals −k′ Tand defines the x-axis. 92 of the momentum imbalance (δpT) and the two angles (δαTand δϕT) are: δpT=|δpT|=|k′ T+pN,T|,(5.3) δαT= arccos −k′ T·δpT |k′ T| |δpT|,(5.4) δϕT= arccos −k′ T·pN,T |k′ T| |pN,T|,(5.5) where k′ Tand pN,Tare, respectively, the projections of the final lepton and nucleon momentum on the transverse plane (if the neutrino direction is taken as the z-axis, then the projections only have components in the xy-plane as it is shown in Fig. 5·7). In the absence of FSI and assuming a pure QE event, the momentum imbalance is generated entirely by the description of the initial nuclear state dynamics [120, 121]. In this approximation δpT is a direct measurement of the transverse component of the bound nucleon momentum distribution, therefore the RFG model, widely used in neutrino event generators, would be at a disadvantage compared to more realistic nuclear models like the independent-particle shell model or the spectral function model [115, 121]. This was explicitly shown in Ref. [51], where the RFG was found to give a much poorer description of the δpTdistribution than the IPSM in PWIA. Also in the PWIA limit, the δαTdistribution is expected to be flat due to the isotropy property shown by the nucleon momentum distribution, although presence of FSI and other effects beyond the impulse approximation break this behavior. The comparison of the cross sections as function of the transverse kinematic imbalances for the different models with T2K measurements is presented in Fig. 5·8. For these data, the following phase-space restrictions are applied: k′>0.25 GeV, cos θl>−0.6, 0.45 < pN<1.0 GeV and cos θL N>0.4. The δpTdistribution shown in Fig. 5·8 favours the ED-RMF and rROP calculations over the GENIE-SuSAv2 predictions in the low δpTregion, which is mainly dominated by initial-state effects with negligible contribution from the 2p2h and pion absorption channels. This could be caused by the inconsistencies of the implementation 93 of the SuSAv2 model, which is based on the RMF theory, in GENIE, that generates the initial state nucleon using a local Fermi gas model. For imbalances above the Fermi level, nucleonnucleon correlations become more important and the microscopic calculation predicts small FSI effects. In this region all the microscopic models except the ROP model overestimate the cross-section measurements after including the non-QE contributions calculated with GENIE. Note that this comparison between theory and experiment does present some inconsistency because the 2p2h contribution is calculated with a Fermi gas while the microscopic calculations for the QE process use the RMF model with corrections to include nucleonnucleon correlations. In any case, it is clear that the QE contribution with nucleon-nucleon correlations included is not enough to describe the region of high-momentum imbalance and additional contributions are essential to describe the experimental results. For the distribution as function of δpTshown in Fig. 5·8 there is an additional model called “RFG”, which corresponds to the predictions based on the plane-wave approximation using the RFG model for the initial state as described in Sec. 2.3.1. The RFG prediction greatly overestimates the measurements, while RPWIA with a realistic description of the initial state (see Sec. 3.1) yields a lower and shifted to the left cross section. Of course some kind of FSI and non-QE contributions must be added to improve the agreement with the data, but qualitatively one can see that, even in the plane-wave limit, this imbalance is largely affected by the description of the initial nuclear state. Regarding the angular TKI, δϕTis more dependent on the neutrino energy and less sensitive to nuclear effects than δpT[120]. The variable δαTmeasures with good approximation the angle between the initial nucleon momentum and the transferred momentum [120]. All the model predictions except ROP as function of δϕTshown in Fig. 5·8 overestimate the cross-section measurements, although the overestimation is less severe in the case of the EDRMF and rROP models for low values of δϕTcompared with GENIE-SuSAv2. In the case of δαT, it is expected to have a rather flat distribution due to the isotropy of the momentum 94 distribution of the bound nucleon which is broken mainly by non-QE effects and FSI, although this deviation from flatness is partially washed out by the constraint on the outgoing proton kinematics in current experimentally accessible signal definitions. However, it is interesting to note that all of the microscopic models, including RPWIA, predict a significant QE-driven rise in δαTwhen there is no or low proton momentum threshold (although this is not shown here), in contrast to what is often predicted by neutrino event generators. In the results presented in Fig. 5·8 there is an overestimation of the cross-section measurements by all the models except the ROP, which is less significant for the microscopic calculations using the ED-RMF and rROP models compared with GENIE-SuSAv2 results specially at high δαTvalues. The non-QE contributions become more relevant for higher values of δαT where the biggest differences between the microscopic and the GENIE-SuSAv2 calculations also appear, with the former in better agreement with the cross-section measurements. The differences between both kinds of calculations could be explained by the different treatment of FSI. Simulations performed with the neutrino event generator NuWro [122], which also uses a semi-classical cascade model for FSI with tuned parameters, are in better agreement with the microscopic calculation than with GENIE-SuSAv2. Quantitatively, χ2values summarized in Table D.1 correspond to values of χ2/d.o.f. close to 1 for the ROP model for the three TKI variables, with the best agreement found for the δαTdistribution. 5.2 MINERνA The MINERνA collaboration measured [116, 117] semi-inclusive 1µCC0πNp events on 12C and presented the differential cross section results as function of the muon and leading proton kinematics, and also as function of TKI. The experimental constrains applied in this case are summarized in Table 5.1. 95 0.0 0.2 0.4 0.6 0.8 0 5×10-39 1×10-38 δpT(GeV) dσ/dδpT(cm2GeV-1nucleon-1) ED-RMF RPWIA ROP GENIE-SuSAv2 2p2h Other RFG rROP 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2×10-40 4×10-40 6×10-40 8×10-40 1×10-39 δαT (rad) dσ/dδαT(cm2rad-1nucleon-1) 0.0 0.5 1.0 0 1×10-39 2×10-39 3×10-39 4×10-39 5×10-39 |δϕT| (rad) dσ/d|δϕT|(cm2rad-1nucleon-1) Figure 5·8: T2K 1µCC0πNp semi-inclusive νµ−12C cross sections as function of the transverse kinematic imbalances δpT,δαTand |δϕT|. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Cross-section measurements taken from [115]. MINERνAk′(GeV) cos θlpN(GeV) cos θL NϕL N(◦) 1.5-10 >0.939 0.45-1.2 >0.342 - Table 5.1: Phase-space restrictions applied to the semi-inclusive CC0πcross-section measurements with one muon and at least one proton in the final state shown by MINERνA collaboration in [116, 117]. 5.2.1 1µCC0πNp: Muon and proton kinematics Despite a larger contribution from non-quasielastic channels, due to the higher energy neutrino beam compared with T2K (see Fig. 2·4), the semi-inclusive cross sections predicted by ROP, shown in Fig. 5·9 as function of the muon and proton kinematics together with the 96 other models, seem qualitatively in reasonable agreement with experimental measurements, except for the θL Ncross section where there is an overestimation in the high-θL Nregion. This is partially confirmed by the χ2-values presented in the legend of Fig. 5·9, which shows ROP as the favored model that best matches the measurements of muon and proton kinematic variables. Whilst all models other than ROP overestimate the cross section data, the agreement between the ED-RMF, rROP and the GENIE-SuSAv2 predictions is very good except for the pNdistribution where differences can be seen in the whole interval of proton momentum. It should be noted that the apparent overestimation of the non-ROP models (and the outstanding agreement with the data of the ROP model) may be due to a mismodeling of the strength of the 2p2h or pion absorption contributions calculated with GENIE and may therefore not suggest an issue in the modeling of the QE channel. 5.2.2 1µCC0πNp: Transverse kinematic imbalances In Fig. 5·10 the results of the different FSI models with the MINERνA cross-section measurements of the TKI distributions are presented. Even without adding the non-QE contributions, all the models except ROP overestimate the data in the peak of the δpTdistribution. In the high-momentum imbalance tail the contribution from the non-QE channels are sufficient and clearly necessary to match any prediction with the experimental cross section. A similar situation is found for the δϕTcross section where all the models except ROP overestimate the cross-section measurements near zero imbalance and non-QE contributions are clearly required to describe the tail of the distribution. Regarding the δαTresults, it is interesting to point out the appearance of a clear peak at large values of the imbalance in the MINERνA cross-section data that is not present in the T2K cross-sections shown in Fig. 5·8, which might be caused by additional non-quasielastic contributions present in MINERνA due to the higher energy of the neutrinos. In case of T2K results shown in Fig. 5·8, it has been shown [123] that the T2K restriction of the proton momentum to be above 0.45 97 2345678910 0 5×10-40 1×10-39 1.5×10-39 k' (GeV) dσ/dk' (cm2GeV-1nucleon-1) 2p2h RPWIA (79/32) ROP (30/32) ED-RMF (64/32) GENIE-SuSAv2 (71/32) Other rROP (61/32) 0 5 10 15 20 0 1×10-40 2×10-40 3×10-40 θl(degree) dσ/dθl(cm2degree-1nucleon-1) ED-RMF (52/19) GENIE-SuSAv2 (50/19) ROP (35/19) RPWIA (61/19) rROP (50/19) Other 2p2h 0.5 0.7 0.9 1.1 0 2×10-39 4×10-39 6×10-39 8×10-39 pN(GeV) dσ/dpN(cm2GeV-1nucleon-1) RPWIA (120/25) ROP (61/25) rROP (106/25) ED-RMF (115/25) GENIE-SuSAv2 (116/25) Other 2p2h 10 20 30 40 50 60 70 0 2×10-41 4×10-41 6×10-41 8×10-41 1×10-40 θNL(degree) dσ/dθNL(cm2degree-1nucleon-1) GENIE-SuSAv2 (62/26) ED-RMF (76/26) RPWIA (93/26) rROP (74/26) ROP (40/26) Other 2p2h Figure 5·9: MINERνA 1µCC0πNp semi-inclusive νµ−12C cross section as function of the final muon momentum and scattering angle (top) and as function of the final proton momentum and polar angle (bottom). All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). The original paper from MINERνA was [117] but the cross-section measurements shown here were taken from [116] which corrected a mismodelling in GENIE’s elastic FSI that affected the crosssection measurements presented in the first paper. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. GeV removes most of the interactions in which FSI plays an important role eliminating the peak at large δαT. The GENIE-SuSAv2 prediction and all the RDWIA results except the ROP overestimate the cross-section data, although the shape of the rise in δαTseems to be well described by the combination of FSI and non-QE contributions except the last bin. Additional projections on the plane perpendicular to the neutrino direction of the momentum imbalance δpTare also presented in Fig. 5·10. If the interaction occurred on a free nucleon at rest, then we would expect a delta-function at δpT= 0 because the muon 104 -1.0 -0.5 0.0 0.5 1.0 0 5×10-39 1×10-38 1.5×10-38 2×10-38 cos(θl) dσ/dcos(θl) (cm2nucleon-1) 2p2h Other GENIE-SuSAv2 (7.0/4) ED-RMF (7.0/4) ROP (0.2/4) 0 1 2 3 4 0 1×10-39 2×10-39 3×10-39 4×10-39 5×10-39 El (GeV) dσ/dEl(cm2GeV-1nucleon-1) GENIE-SuSAv2 (9/4) ROP (2/4) ED-RMF (9/4) 2p2h Other -1.0 -0.5 0.0 0.5 1.0 0 2×10-39 4×10-39 6×10-39 8×10-39 1×10-38 cos(θNL) dσ/dcos(θNL) (cm2nucleon-1) GENIE-SuSAv2 (13/4) ROP (7/4) ED-RMF (14/4) 2p2h Other 0.00 0.05 0 1×10-38 2×10-38 3×10-38 0.2 0.4 0.6 0.8 TN (GeV) dσ/dTN(cm2GeV-1nucleon-1) GENIE-SuSAv2 (9/5) Other 2p2h ROP (3/5) ED-RMF (9/5) Figure 5·12: MicroBooNE 1eCC0πNp semi-inclusive νe−40Ar cross sections as function of the electron scattering angle and energy and proton kinetic energy and scattering angle. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results are from [125]. The bands, drawn for the ED-RMF and ROP models, represent the uncertainties associated with the modeling of the initial nuclear state. The single white point in the TNdistribution between 0 < TN<50 MeV corresponds to an extra 1eCC0π0p (one electron, no protons above TN= 50 MeV threshold and at least one proton with kinetic energy below the 50 MeV threshold) measurement performed by MicroBooNE [125]. For this single point, additional phase space restrictions on the electron energy (El>0.5 GeV) and the electron scattering angle (cos θl>0.6) are applied. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. 105 the 1µCC0π1p topology, leaving aside the non-quasielastic contributions, is closer to the picture drawn by the ROP model, in which the imaginary part of the optical potential subtracts all the inelastic nuclear FSI, leaving only the elastic channel (i.e. the outgoing proton interacting elastically with the residual system). The inclusion of the correlated part of the spectral function introduces, in an effective way, states with multiple nucleons being knocked out. Notice that this contribution is very minor in the cross sections shown in Figs. 5·13 and 5·14, in fact, the result obtained after its subtraction is contained within the uncertainty for both RDWIA predictions. The results presented as function of the proton kinematics show good agreement between the ROP and data, while the predictions by the other models overestimate the measured cross sections, especially the ED-RMF model. Regarding the lepton kinematics, the bins around the peak of the k′distributions are slightly underestimated by the ROP but the ED-RMF and GENIE-SuSAv2 overestimate the data in the rest of the bins. Finally, the cos θldistribution is shown in Fig. 5·14. The ROP and GENIE-SuSAv2 predictions are within the experimental uncertainty except for the forward angle bin that is overestimated by the GENIE-SuSAv2 and ED-RMF models. However, recent work by the MicroBooNE collaboration [126] shows a cos θldistribution that is reproduced correctly by different neutrino generators. This suggests that the discrepancy observed in Fig. 5·14 at small muon scattering angles might be due to the use of an old version of the GENIE configuration that accounts for efficiency corrections and beam-induced backgrounds. 106 0.5 1.0 1.5 0 1×10-39 2×10-39 3×10-39 4×10-39 -0.65 < cos(θl) < 0.95 k' (GeV) dσ/dk' (cm2GeV-1nucleon-1) 2p2h Other ED-RMF (21/7) ROP (7/7) GENIE-SuSAv2 (10/7) 0.5 1.0 1.5 0 1×10-39 2×10-39 3×10-39 4×10-39 -0.65 < cos(θl) < 0.8 k' (GeV) dσ/dk' (cm2GeV-1nucleon-1) Other GENIE-SuSAv2 (4/5) ROP (5/5) ED-RMF (6/5) 2p2h 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1×10-39 2×10-39 3×10-39 4×10-39 5×10-39 6×10-39 -0.65 < cos(θl) < 0.95 cos(θNL) dσ/dcos(θNL) (cm2nucleon-1) GENIE-SuSAv2 (14/7) ED-RMF (27/7) ROP (8/7) Other 2p2h 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1×10-39 2×10-39 3×10-39 4×10-39 5×10-39 -0.65 < cos(θl) < 0.8 cos(θNL) dσ/dcos(θNL) (cm2nucleon-1) GENIE-SuSAv2 (10/7) Other 2p2h ED-RMF (21/7) ROP (11/7) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 1×10-39 2×10-39 3×10-39 4×10-39 5×10-39 6×10-39 7×10-39 -0.65 < cos(θl) < 0.95 pN(GeV) dσ/dpN(cm2GeV-1nucleon-1) 2p2h Other GENIE-SuSAv2 (6/7) ED-RMF (25/7) ROP (2/7) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 1×10-39 2×10-39 3×10-39 4×10-39 5×10-39 -0.65 < cos(θl) < 0.8 pN(GeV) dσ/dpN(cm2GeV-1nucleon-1) GENIE-SuSAv2 (9/7) ED-RMF (20/7) ROP (5/7) Other 2p2h Figure 5·13: MicroBooNE 1µCC0π1p semi-inclusive νµ−40Ar cross sections as function of the muon and proton momenta and proton polar scattering angle. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results are from [118]. The bands, drawn for the ED-RMF and ROP models, represent the uncertainties associated with the modeling of the initial nuclear state. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. 107 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0 2×10-39 4×10-39 6×10-39 cos(θl) dσ/dcos(θl) (cm2nucleon-1) GENIE-SuSAv2 (28/7) 2p2h Other ED-RMF (53/7) ROP (10/7) Figure 5·14: MicroBooNE 1µCC0π1p semi-inclusive νµ−40Ar cross sections as function of the muon scattering angle. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results were taken from [119]. The bands drawn for the ED-RMF and ROP models are related with the uncertainties associated with the modeling of the initial nuclear state. The χ2/d.o.f. ratio is given in brackets in the legend of each distribution. 5.3.4 1µCC0π1p: Reconstructed Eνand Q2 In Fig. 5·15, we also present the 1µCC0π1p predictions as function of the reconstructed neutrino energy and Q2 CCQE, which are defined as follows [119] Ecal ν=El+TN+ 40 MeV , Q2 CCQE =Ecal ν−El2−(k−k′)2,(5.6) where the argon binding energy is assumed to be 40 MeV. Both RDWIA calculations tend to underestimate (ROP) or overestimate (ED-RMF) the measurements as function of Q2 CCQE, being the ED-RMF prediction closer to data in the bin that excludes forward muon angles, i.e. −0.65 <cos θl<0.8. In the case of the cross section as function of Ecal ν, all the models overpredict the data in the tail of the distribution (large Ecal ν-values). Given that a good 108 description of Ecal νrequires a description of the fully exclusive final state (including very low momentum hadrons below detection threshold), the poor agreement is unsurprising. 0.0 0.5 1.0 0 2×10-39 4×10-39 6×10-39 -0.65 < cos(θl) < 0.95 Q2CCQE (GeV2) dσ/dQ2CCQE (cm2GeV-2nucleon-1) 2p2h Other ED-RMF ROP GENIE-SuSAv2 0.0 0.5 1.0 0 1×10-39 2×10-39 3×10-39 4×10-39 5×10-39 -0.65 < cos(θl) < 0.8 Q2CCQE (GeV2) dσ/dQ2CCQE (cm2GeV-2nucleon-1) GENIE-SuSAv2 Other 2p2h ED-RMF ROP 0.3 0.6 0.9 1.2 0 1×10-39 2×10-39 3×10-39 4×10-39 -0.65 < cos(θl) < 0.95 Eνcal (GeV) dσ/dEνcal (cm2GeV-1nucleon-1) Other 2p2h GENIE-SuSAv2 ED-RMF ROP 0.3 0.6 0.9 1.2 0 1×10-39 2×10-39 3×10-39 -0.65 < cos(θl) < 0.8 Eνcal (GeV) dσ/dEνcal (cm2GeV-1nucleon-1) Other 2p2h GENIE-SuSAv2 ED-RMF ROP Figure 5·15: MicroBooNE 1µCC0π1p semi-inclusive νµ−40Ar cross sections as function of Q2 CCQE and Ecal ν. All curves include the 2p2h and pion absorption (denoted “other”) contributions evaluated using GENIE (shown separately). Experimental results are from [118]. The bands, drawn for the ED-RMF and ROP models, represent the uncertainties associated with the modeling of the initial nuclear state. The covariance matrix is not available for the cross section as function of these variable, therefore it is not possible to calculate the χ2values. 109 Chapter 6 Summary and conclusions The discovery of neutrino oscillations opened a new frontier in particle physics. Past, ongoing, and forthcoming experiments are dedicated to unraveling the mechanisms responsible for neutrino oscillations by measuring the physical parameters governing this phenomenon. As for now, the neutrino mass hierarchy remains unknown, yet it is an important physical information. Additionally, neutrino oscillations have hinted at the existence of charge-parity violation within the lepton sector. To determine the presence of such violation, a fundamental parameter of the PMNS paradigm needs to be measured. Presently, accelerator-based neutrino experiments are actively gathering data, while new experiments are in development, all geared towards quantifying the degree of CP violation present in neutrino oscillations. One of the main limiting systematic errors in neutrino oscillation physics comes from our limited knowledge of neutrino-nucleus interactions. To help to constrain nuclear effects for the modeling of neutrino-nucleus interactions and improve the reconstruction of the neutrino energy for oscillation experiments, in this thesis we have studied semi-inclusive neutrino interactions with complex nuclei at intermediate lepton energies, i.e., neutrino beam energies ranging from 0.5 up to 10 GeV. Due to the kinematics of the particles involved and the complexity of the interaction, we have developed a fully relativistic and quantum mechanical model able to describe not only inclusive but also semi-inclusive electron and neutrino reactions with complex nuclei. Although there are multiple neutrino-nucleus interaction mechanisms, the focus of this 110 thesis has been the description of quasielastic scattering of an incoming neutrino by nuclei and the detection in coincidence of a lepton and an ejected proton, also known as oneproton knockout process. The study of this process can be used, with the right selection of experimental observables, to identify relevant nuclear effects related to both the initial state dynamics and to final-state interactions (FSI), as well as to the influence of twoparticle-two-hole excitations. After describing the general kinematics of the reaction, we have provided the expressions of the flux-averaged sixth-differential CC neutrino-nucleus cross sections in the plane-wave impulse approximation (PWIA) for different nuclear models, namely the relativistic Fermi gas (RFG), the independent-particle shell model (IPSM), the natural orbitals (NO) shell model and the spectral function model (SF). The results in PWIA show that RFG, the simplest approach considered, differs completely from the two shell-based models, IPSM and NO. Not only the shape of the semi-inclusive cross section is totally different, without any sub-shell structure, but also its magnitude and behavior with the kinematic variables, particularly, with the azimuthal angle of the outgoing nucleon ϕL N. On the contrary, IPSM and NO lead to rather similar results showing only some discrepancies in the low missing momentum region because of the different wave-functions used to construct the momentum distributions. Aiming to give a state-of-the-art description of the quasielastic contribution of the oneproton knockout process, we have included FSI using the relativistic distorted-wave impulse approximation (RDWIA) framework. Additionally, for the description of the initial nuclear state, we have used the relativistic mean field (RMF) model extended to include effects beyond the mean-field approximation, like the depletion of the occupation of the shell-model states and the appearance of nucleons at higher missing-energies due to nucleon-nucleon correlations, using the SF model as reference. Within RDWIA, two different approaches to describe FSI were discussed: on one hand, the relativistic optical potential (ROP) model which uses complex phenomenological potentials fitted to reproduce elastic proton-nucleus 111 scattering data; and on the other hand, an extension of the RMF model called energydependent relativistic mean field (ED-RMF) model that preserves orthogonality for small values of the proton momentum and improves the agreement with inclusive electron and neutrino scattering data. The RDWIA approach was used to generate theoretical predictions to compare with experimental semi-inclusive measurements performed by T2K, MINERνA and MicroBooNE collaborations using 12C and 40Ar nuclei as targets. In the specific case of MicroBooNE, the cross-section measurements were collected using both νµand νebeams. Although the quasielastic scattering is the dominant contribution in the energy range that has been studied, additional contributions beyond the quasielastic need to be included to compare with the cross-section measurements. These contributions, the two-particle-two-hole (2p2h) and pion production and reabsorption, have been included based on the predictions of GENIE neutrino event generator. Additionally, the quasielastic predictions from the inclusive model SuSAv2 implemented in GENIE were also included in the comparison with the data. Overall, we have seen that the predictions based on the RDWIA approach match or improve the agreement of the GENIE-SuSAv2 model with the 12C semi-inclusive data measured by T2K and MINERνA collaborations. In particular, the microscopic calculation based on the ED-RMF model improves the agreement with the experimental results at forward lepton angles, i.e. low momentum and energy transfer, where scaling violations and low-energy effects not included in the SuSAv2 model are relevant. The detection in coincidence of two particles in the final state, a lepton and the ejected proton, opens the possibility of extracting cross sections as function of variables that measure correlations between both detected particles. These can help to explore better different nuclear effects like initial-state physics, FSI or 2p2h. For instance, the transverse kinematic imbalance (TKI) δpTis particularly sensitive to the model used for the initial nuclear state, therefore realistic models of the nuclei, like the RMF model, are needed in order to describe 112 correctly this distribution. The tails of the distributions as function of δpTas well as the inferred variable |∆p|, exclusively measured by T2K, can help to explore the size and shape of the 2p2h contribution. The |∆p|distribution is especially interesting for forward muon scattering angles because of the clear differences between the high and low muon momentum cases, with the former enhancing the relative 2p2h contribution over the latter. It is important to point out the limitations in the predictive power for nucleon kinematics of the implementation of the SuSAv2 2p2h model within GENIE, as well as of the pion absorption contribution, that are included in all the comparisons with experimental data presented in this thesis. It is shown that different models of the 2p2h channel can yield very different semi-inclusive cross-section predictions for this channel, therefore the impact of the factorization approximation used to generate semi-inclusive predictions from inclusive models implemented in GENIE, or any other neutrino event generator that uses the same approximations, and the inconsistencies introduced by using different nuclear models need to be studied carefully. In addition to the 12C cross sections, we have also analyzed the semi-inclusive νµ−40Ar and νe−40Ar cross section data measured by the MicroBooNE collaboration. We have constructed a SF model for neutrons on argon based on RMF results, phenomenology and exclusive (e, e′p) measurements. Also, the RDWIA calculation is performed taking into account conservative uncertainties associated with the modeling of the SF used for the description of the initial state. Among the two RDWIA approaches considered in this work, the ROP model provides the best overall agreement with data for both CC0πNp and CC0π1p topologies, although the former presents large uncertainties. It is worth mentioning in the CC0π1p case the accordance between the ROP predictions and data as function of the muon and proton kinematics, except for the forward muon scattering angles where the reported very low data point appears not to be present in subsequent analyses. Finally, there are ongoing efforts from different experimental collaborations to improve 113 the measurements of more neutrino-nucleus exclusive channels, like the one-proton knockout process. For example, the upgraded version of the T2K near detector ND280 with improved reconstruction capabilities will be able to detect protons with much lower momentum, hence we can expect much more precise measurements of interactions with proton production that will require models able to predict hadrons kinematics like the RDWIA. Also, the majority of the neutrino event generators simulate FSI using semi-classical cascade models. However, recent studies suggest that this framework only provides reliable predictions for nucleons with momentum above 200 MeV, while the optical potential approach produces more sensible results. This could justify investigating in the future how to implement in neutrino event generators more exclusive models and more sophisticated descriptions of the initial state that go beyond a simple Fermi gas, which would improve the treatment of semi-inclusive reactions by neutrino event generators. 120 which is reduced to m2 NWµν AA =G2 A"Pµ NPν+Pν NPµ−gµν2m2 N−Q2 2#−G2 PQ2 8m2 N QµQν−GAGP 2QµQν+QνQµ. (A.7) Finally, the vector-axial component of the hadronic tensor is 4m2 NWµν V A =Tr / PN+mNF1γµ+iF2 2mN σµαQα/ P+mNGAγνγ5−GP 2mN γ5Qν Aµν + Tr / PN+mNGAγµγ5+GP 2mN γ5Qµ/ P+mNF1γν−iF2 2mN σνβQβ Bµν , with the first term further divided into Aµν =F1Tr / PN+mNγµ/ P+mNGAγνγ5−GP 2mN γ5Qν Aµν 1 + iF2 2mN Tr / PN+mNσµαQα/ P+mNGAγνγ5−GP 2mN γ5Qν Aµν 2 . Knowing that Tr γ5γκγλγµγν=−4iϵκλµν, where ϵκλµν is the totally antisymmetric rank - 4 Levi-Civita tensor, Tr (γ5γµγν) = 0 and that the trace of γ5multiplied by any odd number of γµmatrices is zero, then to the Aµν 1term defined above only contributes one trace Tr / PNγµ/ Pγνγ5proportional to GAand Aµν 2is given by Aµν 2=imNGATr / PNγµγαγνγ5+Tr γµγα/ Pγνγ5−GP 2mN Tr γ5/ PNγµγα/ PQνQα, 121 where we have used that σµα =i(γµγα−gµν) to reduce the number of traces that are not zero. Then we have that Aµν =−4iF1GAϵαµβνPNαPβ+F2 2mN−GAmNϵµναβ P+PNβ+GP 2mN ϵβµασPNβPσQνQα. Equivalently, the Bµν term is given by Bµν =−4iF1GAϵαµβνPNαPβ+F2 2mN−GAmNϵµναβ P+PNβ−GP 2mN ϵβνασPNβPσQµQα. Therefore, the final expression of the vector-axial component of the CC2 hadronic tensor is m2 NWµν V A =iGAϵµναβ 2F1PNαPβ−F2PN+Pα+GPF2 4m2 N−Qµϵναβσ +QνϵµαβσPNαPσQβ. (A.8) For completeness, we also present here the expression of the single nucleon hadronic tensor calculated using the CC1 current. Following a similar procedure to the one used above for CC2, for the CC1 case the vector-vector and vector-axial components of the hadronic tensor are given by m2 NWµν V V =F1+F22PµPν N+PνPµ N+¯ Q2 2gµν −F2F1+F2−F2 2 21−Q2 4m2 NP+PNµP+PNν(A.9) and m2 NWµν V A = 2iGAF1+F2ϵαβµνPNαPβ.(A.10) 122 Appendix B Single nucleon responses in PWIA As in the case of exclusive (e, e′p) scattering in PWIA [59, 131, 132], the semi-inclusive neutrino-nucleus cross section in PWIA can be reduced to the neutrino-nucleon cross section times a spectral function, describing the probability of finding a nucleon in the target nucleus with certain energy and momentum which should be compatible with the kinematics of the process. This factorization property is broken in general when FSI effects or other ingredients beyond the impulse approximation are introduced. Moreover, even in RPWIA, i.e., the plane-wave approach but the bound nucleons wave functions are the full relativistic solutions of the Dirac equation in presence of potentials (positive and negative energy components included), factorization is lost. In general, the elementary neutrino-nucleon cross section is proportional to the contraction of the leptonic Lµν and the hadronic Wµν tensors. In Appendix A we have obtained in detail the single nucleon hadronic tensor under PWIA for both CC1 and CC2 prescriptions of the weak nucleon current. The weak leptonic current for CC processes is defined as Jl µ=ul(k′, s′) (γµ−χγµγ5)uνl(k, s), where uland uνlare the free Dirac spinors that describe the final lepton and the initial neutrino and χ= 1(−1) corresponds to neutrino (antineutrino) processes. The leptonic tensor constructed from this current reads, Lµν =KµK′ ν+KνK′ µ−K·K′gµν −iχϵµναβKαK′β,(B.1) 123 with Kµand K′ µthe 4-momenta of the neutrino and the final lepton. Defining a coordinate system where the three momentum qis along the z-axis (see Fig. B·1), then the 4-momenta of the initial and final nucleons and leptons can be expressed as Pµ N= (EN, pNsin θNcos ϕN, pNsin θNsin ϕN, pNcos θN), Pµ=E, pNsin θNcos ϕN, pNsin θNsin ϕN, pNcos θN−q, Kµ=k, −kk′sin θl q,0,−k(k−k′cos θl) q, K′ µ=El,−kk′sin θl q,0,k′(k′−kcos θl) q.(B.2) Also, the transferred 4-momenta take trivial forms in this coordinate system: Qµ= (ω, 0,0, q) and Qµ= (ω, 0,0, q), with ω=EN−E. The main advantage of using this coordinate system with the momentum transfer fixed on the z-axis is the possibility of expressing the contraction of the leptonic and hadronic tensors as the linear combination of 10 nuclear response functions, defined as specific components of the hadronic tensor, and factorizing the ϕNdependence, which is simply reduced to cos ϕNor cos 2ϕNterms. The contraction of Scattering plane Reaction plane z x y x x′ k k′ θl q pN θN ϕN Figure B·1: Definition of the coordinate system where the momentum transfer qlies on the z-axis. Notice that the leptons are contained in the x-z plane (scattering plane) and the plane containing the ejected nucleon with momentum pNand polar angle θNforms an angle ϕNwith the scattering plane. 124 the hadronic and leptonic tensors, a Lorentz invariant that has the same value in any system of reference, can be expressed as [49] F2 χ=2 v0 LµνWµν =VCC(RCC V V +RCC AA)+2VCL(RCL V V +RCL AA) + VLL(RLL V V +RLL AA) +VT(RT V V +RT AA) + VT T (RT T V V +RTT AA) cos 2ϕN+VT C(RT C V V +RTC AA) cos ϕN +VTL(RT L V V +RTL AA) cos ϕN+χVT′RT′ V A +VTC′RT C′ V A cos ϕN+VT L′RT L′ V A cos ϕN,(B.3) where υ0= (k+El)2−q2. The different weak hadronic responses are given by specific components of the single nucleon hadronic tensor Wµν RCC =W00, RCL =W03, RLL =W33, RT=W11 +W22, RTT =W22 −W11, RTC = 2√2W01, RTL = 2√2W31, RT′= 2iW12, RTC′= 2√2iW02, RTL′= 2√2iW32,(B.4) 125 and the leptonic factors are specific components of the leptonic tensor given in Eq. (B.1) VCC =2 υ0 L00, VCL =2 υ0 L03, VLL =2 υ0 L33, VT=L11 +L22 υ0 , VTT =L22 −L11 υ0 , VTC =2 √2υ0 L01, VTL =2 √2υ0 L31, VT′=−2i υ0 L12, VTC′=−2i √2υ0 L02, VTL′=−2i √2υ0 L32.(B.5) Notice that due to momentum conservation q=pN−pthe xand ycomponents of the 3-momenta of the initial and ejected nucleon are equal because qonly has zcomponent. Exploiting this, we can derive the ϕNdependence of the invariant F2 χas follows: the responses that contain both 0 and 3 indices (RCC,RCL and RLL) cannot have ϕNdependence because they depend on the first and last components of the 4-momenta (the energy and the projection on the z-axis) that do not have any ϕNdependence. The response that has both 1 and 2 indices, i.e. RT′, is calculated only with the vector-axial component of the hadronic tensor. As it can be seen in Eq. (A.8) and Eq. (A.10), this response is proportional to ϵ12αβPNαPβor ϵ12αβ PN+PαQβfor both CC1 and CC2 currents, which again involve components of the 4-momenta that are ϕN−independent. The RTresponse is constructed by the sum of the 11 and 22 components of the hadronic tensor and the 126 only dependence on ϕNis introduced by terms proportional to P1P1 N+P2P2 N, or using Eq. (B.2), proportional to p2 Nsin2θNcos2ϕN+ sin2ϕN=p2 Nsin2θN, hence RTdoes not present any ϕNdependence. The same argument can be used for RT T =W22 −W11, where the difference, instead of the sum as in the previous case, leads the response to be proportional to p2 Nsin2θNcos2ϕN−sin2ϕN=p2 Nsin2θNcos 2ϕN. The terms that do not contain this angular dependence are zero or are eliminated when the subtraction is performed. The responses RT C and RT L have ϕNdependence introduced by terms proportional to P1 Nor P1 thus, they are proportional to pNsin θNcos ϕN. Finally, for the RT C′and RTL′responses only the terms proportional to pNsin θNcos ϕNsurvive, with the cross terms that depend on cos ϕNsin ϕNeliminated due to the antisymmetry properties of the Levi-Civita tensor. Although we have used the special properties found by considering a coordinate system with direction of qfixed in the z-axis, from the point of view of an accelerator-based neutrino experiment it is common to have fixed the direction of the incoming neutrino kinstead. This coordinate system, presented in Fig. 2·3 and used to generate cross sections that can be directly compared with experimental measurements, can be related with the one with fixed direction of q, shown in Fig. B·1, by a rotation in the scattering plane of an angle cos θq=k−k′cos θl q,(B.6) which is the angle between qand k. The angular variables identifying the ejected nucleon in both coordinate systems can be related using the following expressions cos θN= cos θL Ncos θq−cos ϕL Nsin θL Nsin θq,(B.7) sin θN=p1−cos2θN,(B.8) cos ϕN=cos ϕL Nsin θL Ncos θq+ cos θL Nsin θq sin θN ,(B.9) sin ϕN=sin ϕL Nsin θL N sin θN .(B.10) 127 To conclude, since we already found out the complete expression of the single nucleon hadronic current in Appendix A, in what follows we present the exact analytic expressions, for the CC2 current, of all the hadronic and leptonic responses calculated as defined in Eq. (B.4) and Eq. (B.5). Since we have already derived the entire ϕNdependence of the invariant, the 4-momenta defined in Eq. (B.2) will be evaluated at ϕN= 0 to simplify the calculations. Taking into account the following definitions p⊥=pNsin θN, p∥=pNcos θN−q , δ=EN−E−ω , (B.11) and that the relevant Lorentz invariants are P·Q=−Q2 2−δ2 2−δ(E+ω), PN·Q=Q2 2−δ2 2−δE , Q2=Q2+δ2+ 2ωδ , (B.12) the single nucleon hadronic responses are 8m4 NRCC V V = 4E2(4F2 1m2 N+F2 2|Q2|)+4Eω(4F2 1m2 N+F2 2|Q2|)−4F2 1m2 N|Q2| −8F1F2m2 N(ω2+|Q2|) + F2 2(ω2|Q2|−4m2 N(ω2+|Q2|)) −2δ(2E+ω)(F2 2(2Eω +ω2−|Q2|)−4F2 1m2 N) + δ2(−4F2 2E2 −12F2 2ωE + 4F2 1m2 N+F2 2(|Q2|−5ω2)) −4δ3F2 2(E+ω)−δ4F2 2,(B.13) 128 8m4 NRCC AA = 16E2G2 Am2 N+ 16EG2 Am2 Nω−8GAGPω2m2 N −4G2 Am2 N(4m2 N+|Q2|) + G2 Pω2|Q2|+δ(16EG2 Am2 N + 8G2 Am2 Nω−8GAGPm2 Nω−2G2 Pω3) + δ2(4G2 Am2 N−G2 Pω2),(B.14) 8m4 NRCL V V = 2E(2p∥+q)(4F2 1m2 N+F2 2|Q2|) + ω(8F2 1m2 Np∥ −8F1F2qm2 N+F2 2(−4m2 Nq+ 2p∥|Q2|+q|Q2|)) +δF2 2h−(4E2q+Eω(4p∥+ 6q)+2ω2(p∥+q)−|Q2|(2p∥+q))i + 8F2 1m2 Np∥−4F1F2qm2 N−δ2F2 2(4Eq + 2ωp∥+ 3ωq)−δ3F2 2q , (B.15) 8m4 NRCL AA = 8EG2 Am2 N(2p∥+q) + ω(8G2 Am2 Np∥−8GAGPqm2 N+G2 Pq|Q2|) +δ(8G2 Am2 Np∥−4GAGPm2 Nq−2G2 Pω2q)−δ2G2 Pωq , (B.16) 8m4 NRLL V V =F2 1(16m2 Np∥(p∥+q)+4m2 N|Q2|)−8F1F2m2 Nω2+F2 24p∥q(ω(E+ω) −q(p∥+q)) + |Q2|((2p∥+q)2+ 4m2 N−|Q2|+ 2Eω)−4m2 Nq2 +δω(−8F2 1m2 N−8F1F2m2 N+F2 2h2q(2p∥+q)−4ω2−8ωE −4E2i) −δ2(4F2 1m2 N+F2 26ω(E+ω)−ω2)−2F2 2δ3ω , (B.17) 8m4 NRLL AA =G2 A(16m2 Np∥(p∥+q) + 16m4 N+ 4m2 N|Q2|) + G2 Pq2|Q2|−δ(2G2 Pq2ω +G2 A8m2 Nω)−δ2(4G2 Am2 N+G2 Pq2)−8m2 NGAGPq2,(B.18) 129 8m4 NRT V V = 16F1F2m2 N|Q2|+ 4F2 2(2m2 N+p2 ⊥)|Q2|+ 8F2 1m2 N(2p2 ⊥+|Q2|) −16δF1(F1+F2)m2 Nω+δ2(8F2 2E2+ 8F2 2ωE −8F2 1m2 N−2F2 2|Q2|)+4δ3F2 2(2E+ω)+2δ4F2 2,(B.19) 8m4 NRT AA = 8G2 Am2 N(4m2 N+ 2p2 ⊥+|Q2|)−16δG2 Am2 Nω−8δ2G2 Am2 N,(B.20) 8m4 NRTT V V =−4p2 ⊥(4F2 1m2 N+F2 2|Q2|),(B.21) 8m4 NRTT AA =−16G2 Am2 Np2 ⊥,(B.22) 8m4 NRTC V V = 4√2p⊥(2E+ω)(4F2 1m2 N+F2 2|Q2|)+4√2δp⊥(F2 2(−2Eω −ω2+|Q2|)+4F2 1m2 N)−4δ2√2F2 2ωp⊥,(B.23) 8m4 NRTC AA = 16√2G2 Am2 Np⊥(2E+ω) + 16√2δG2 Am2 Np⊥,(B.24) 8m4 NRTL V V = 4√2p⊥(2p∥+q)(4F2 1m2 N+F2 2|Q2|)−4√2δF2 2p⊥q(2E+ω)−4√2δ2F2 2p⊥q , (B.25) 8m4 NRTL AA = 16√2G2 Am2 Np⊥(2p∥+q),(B.26)