Charged-current neutrino interactions with nucleons and nuclei at intermediate energies

Charged-current neutrino interactions with nucleons and nuclei at intermediate energies TESIS DOCTORAL Guillermo Daniel Megías Vázquez Departamento de Física Atómica, Molecular y Nuclear Facultad de Física UNIVERSIDAD DE SEVILLA Mayo 2017 Charged-current neutrino interactions with nucleons and nuclei at intermediate energies Memoria de Tesis Doctoral realizada por Guillermo Daniel Megías Vázquez para optar al grado de Doctor en Física Nuclear Dirigida por los Doctores Maria B. Barbaro Juan A. Caballero Carretero Departamento de Física Atómica, Molecular y Nuclear Facultad de Física UNIVERSIDAD DE SEVILLA Mayo 2017 Agradecimientos Más de cuatro años de trabajo decaen a su estado fundamental en estas páginas, las cuales constituyen una ventana abierta hacia un mundo, el de la investigación, donde muchas veces lo último que uno sabe es por donde empezar. Sobre todo, cuando uno se enfrenta a la difícil tarea de sintetizar en unas breves líneas la gratitud hacia todas las personas que, de una u otra forma, han hecho posible este ilusionante proyecto. En primer lugar, quiero tener unas palabras de agradecimiento con mi director de tesis, Juan Antonio Caballero, por la confianza depositada en mí para el desarrollo de este trabajo y su dedicación e interés constante en llevar este proyecto adelante. Un excelente investigador, tanto a nivel profesional como personal, que además ha sido protagonista en mi formación académica desde que tuviera el placer de disfrutar de sus clases hará ya unos años. Recuerdo cuando en la asignatura de Física de Partículas, propuso como trabajo de investigación “adoptar una partícula”. En esos momentos mi duda osciló entre acoger al neutrino o al muón. Lo que yo no imaginaba por aquel entonces era que pasados varios años estaría viendo en su madurez a esas partículas ligeras a las que un día vi nacer. Del mismo modo, agradezco especialmente a mi codirectora de tesis, Maria Barbaro, su apoyo y dedicación sincera durante estos años así como su hospitalidad durante mis estancias en Turín. A todo ello, hay que añadir el innegable esfuerzo de haber corregido toda esta tesis desde la distancia y su ayuda y rapidez siempre que he necesitado algún informe para los distintos enredos burocráticos. Mención especial también merecen Quique Amaro y Nacho Ruiz Simó de la Universidad de Granada con quienes he colaborado en los últimos meses, así como la generosidad y hospitalidad de Carlotta Giusti durante mi breve estancia en la Universidad de Pavia. De todos ellos guardo grandes recuerdos. Si bien no aparece implícitamente como director de esta tesis, gran parte del conocimiento adquirido durante esta etapa y plasmado en estas páginas se lo debo a Bill Donnelly, de quien admiro profundamente su gran sabiduría en el campo de la Física y a quien agradezco sus ideas y su sincero interés durante mi estancia en el MIT. La hospitalidad de Bill así como la de su mujer, Barbara, hicieron de mi experiencia en Boston un recuerdo imborrable. En el plano económico, caben mencionar las ayudas recibidas por los distintos proyectos de investigación de la Universidad de Sevilla y del Ministerio de Economía y Competitividad (anteriormente de Ciencia), así como a los contratos predoctorales de la Fundación Cámara y de los Proyectos de Excelencia de la Junta de Andalucía, pero cabe más aún agradecer honestamente el esfuerzo de los distintos funcionarios y, en general, el de los trabajadores, que con sus impuestos contribuyen a mantener la cada vez más complicada labor científica en este país. Tampoco puedo obviar en estas líneas el indudable valor humano y profesional de todos los investigadores que he conocido en distintos congresos a lo largo del mundo, donde he adquirido una visión más amplia de la investigación en sus distintos ámbitos. Asimismo, en el plano más local, tengo que agradecer al Cluster por aguantar con vida y a todos los compañeros del departamento el buen ambiente en el trabajo y fuera de él, por los almuerzos y los cafés juntos y por los cada vez más olvidados viernes de nachos y futbolín que me hicieron ver de una forma más “realista” los procesos de dispersión. Pero no sólo el mundo de la Física ha contribuido a la realización de este proyecto. Esta memoria no hubiera sido posible sin esas personas que dan un valor medible al observable que es la vida. A los que están incluso cuando no he tenido tiempo para quedar, y a los que en cierto modo me tendré que volver a presentar. Por ello, no puedo terminar estas páginas sin agradecer a quienes están siempre ahí para leer estás líneas (aunque sólo interaccionen débilmente con ellas) y a los que ya no están para verlas desde tan cerca. En especial a mi madre y a mi padre por ayudarme y enseñarme a llegar hasta el día de hoy; y a Bibi por su cariño, sus ánimos y por aguantar todos mis muones. Abstract Nowadays, the interest in neutrinos extends to a large variety of fields in Astrophysics, Nuclear Physics and Particle Physics. One of the open questions in theoretical physics is the description of neutrino oscillations for which an accurate interpretation of neutrino-nucleus reactions is crucial. In this context, recent years have witnessed an intense experimental and theoretical activity to determine the properties of neutrinos and their interaction with matter. This PhD Thesis is thus focused on the analysis of charged-current neutrino-nucleus reactions at kinematics of interest for neutrino oscillation experiments, where the neutrino energy is typically in the GeV region. Additionally, weak interactions in the nuclear medium at intermediate energies are an extraordinary opportunity to study the dynamics of the nuclear many-body system, beyond the information accesible from electron and hadron probes, and to gain a deeper knowledge of the axial structure and the strangeness content of the nucleons. In any accelerator-based neutrino oscillation experiment, neutrinos are produced as the decay products of successive reactions, thus implying a wide-ranged energy beam. Hence, when interacting with the nuclear matter, a large variety of nuclear effects come into play, going from quasielastic scattering to deep inelastic processes, multi-nucleon excitations or meson production via nucleon resonances. Accordingly, robust models that properly describe neutrino-nucleus interactions over the whole experimental range (of the order of 10s of MeV up to 10s of GeV) are required for the experimental analyses. Notice also that the kinematics involved demand a relativistic description of the microscopic nuclear structure. In this thesis, the analysis of these processes are addressed by using realistic models that provide, within a fully relativistic framework, an accurate description of the different reaction mechanisms of relevance for neutrino oscillation measurements. We begin analyzing neutrino scattering off free nucleons and describing the weak hadronic responses together with the inner structure of the nucleons. With the aim of achieveing a consistent analysis of charged-current quasielastic (CCQE) neutrino interactions with nuclei, we present the so-called SuSAv2 model, which is based on the superscaling behavior exhibited by electron scattering data and makes use of the relativistic mean field (RMF) theory to describe the nuclear effects arising in neutrino-nucleus interactions. This prescription accounts for the final-state interactions (FSI) between the outgoing nucleon and the residual nucleus and allows for a description in terms of the different isovector/isoscalar and axial/vector reaction channels that play a role in weak interactions. At very high kinematics, where FSI are negligible, we approach our model to the relativistic plane wave impulse approximation (RPWIA) where no FSI affect the outgoing nucleon. Furthemore, a basic feature in this thesis also concerns the evaluation of multi-nucleon excitations, in particular two-body meson exchange currents (2p-2h MEC) contributions, which are proved to be an essential ingredient to interpret neutrino cross section measurements at interme- 2 1. INTRODUCTION decipherable by using electron or photon scattering. In this sense, the charged-current neutrinonucleon scattering (CC ν-N) is the more efficient way to analyze the nucleon axial form factor. This subject is addressed in detail in this thesis. Moreover, the study of neutral-current neutrinonucleon processes (NC ν-N) shed light on the strange sea quark-gluon effects on the nucleon. Despite its relevance in several fields of Physics, neutrinos are still elusive particles. They only interact through weak forces and are hardly detectable, so they can only be clearly observed through the detection of the secondary particles produced in the process. For this reason, heavy targets are often employed in neutrino experiments in order to increase neutrino-nucleus cross sections. At the same time, a proper theoretical understanding of the weak nuclear response is a prerequisite for the analysis of current and future neutrino oscillation experiments. Therefore, a precise study of neutrino oscillations requires an adequate description of their interactions with nuclei and nucleons. The neutrino-nucleus reaction mechanisms are diverse, going from the quasielastic scattering to deep inelastic scattering, multi-nucleon processes or the excitation of nucleon resonances (see Chapters 3-5 for details). Most of these regimes require both better theoretical and experimental understanding. Accordingly, it would be advantageous to employ the knowledge extracted from electron-nucleus scattering as a solid benchmark to assess the validity of the theoretical neutrino interaction approaches in the different nuclear regimes (see Chapter 6) as well as to improve the understanding of current neutrino oscillation experiments. Theoretical models on (e,e′) reactions can still be improved by taking care of the recent developments in the inelastic structure functions along with the progress on parton density functions (PDFs) through QCD calculations. Furthermore, the employment of theoretical models including multinucleon excitations and realistic mean field theories, in addition to the large amount of (e,e′) data for different nuclei, will likely help us to deepen our understanding of the reaction mechanisms in electromagnetic reactions and its extension to the weak sector, i.e., to the analysis of neutrino interactions. This, makes research on neutrino interactions an attractive ingredient for the future of Particle and Nuclear Physics. 1.2 Neutrino Properties and Historical Context As previously mentioned, the “birth” of neutrino dates back to 1930, when Pauli postulated its existence from the analysis of the continuum energy spectrum observed in the beta decay process. This was inconsistent with the image of only electrons being emitted in the nuclear process. The "possible" existence of an extra particle (the neutrino) with negligible mass and electrically neutral, that is, a "silent" partner, was needed in order to preserve energy and momentum conservation. Thus the beta decay process would imply the emission of not only electrons but also of new, unknown, particles that would gather the energy loss in the process. Hence, the description of the β-decay process was defined as: n→p+e−+ν. In the 30’s and 40’s new particles were discovered, specifically, the muon, very similar to the electron but with a much larger mass, and the pion, the particle proposed by Yukawa to explain the strong interaction between nucleons inside the nuclei. A careful study of the pion proved that it decayed into a muon which emerged with an almost perpendicular track in relation to the pion, ensuring the existence of an extraordinary light particle in the process. It was suggested to be a neutrino. A few months later, the experimental proof of the muon decay into an electron led to the conclusion that the latter must be accompanied by two neutrinos. 1.2. NEUTRINO PROPERTIES AND HISTORICAL CONTEXT 3 In the early 1950’s and despite the different hypotheses about the existence of the neutrino, there had no experimental evidence yet. It was in 1956, when Cowan and Reines [4] developed a method to observe the inverse βdecay (ν+p→n+e+), in which this elusive particle was clearly, unambiguously detected. In these experiments, the actual particle taken into consideration was the antineutrino (ν). But how could one distinguish between neutrino and antineutrino? This distinction was explained by Davis and Harmer [5] in the late 1950s, showing that whereas the process ν+n→p+e−was commonly detected, on the contrary, the ν+n→p+e−one never took place. This was the beginning of the lepton number assignment to neutrinos and the rest of leptons as well as its conservation law. Therefore, neutrinos in muon decay should be particle and antiparticle, whose difference comes from its own helicity. This subject will be addressed in detail later. The hypothesis of neutrinos with different flavors came up after realizing that, unlike the process µ→e+ν+ν, the reaction µ→e+γwas not observed. Consequently, it was proposed the existence of two different kinds of neutrinos, one related to the electron and another one to the muon. Therefore, some of the processes mentioned above are really given as: n→p+e−+νe π+→µ++νµ π−→µ−+νµ µ+→e++νe+νµ µ−→e−+νµ+νe Neutrino flavors The Standard Model of Particle Physics contains three different neutrino flavors (νe,νµand ντ), which respectively form a doublet with the corresponding charged leptons (e,µand τ) as shown in Table. 1.1. In the 1990s, the LSND experiment [6] suggested the possibility of further neutrinos to explain neutrino masses and introduced a new type of neutrinos, called “sterile” neutrinos, whose existence is still under investigation. This particle would not interact with any other (except through gravity). The search for sterile neutrinos is an active area of particle physics and it is expected that future neutrino oscillation facilities could shed more light on this issue. Helicity Within the framework of the SM, neutrinos are massless leptons and exhibit purely negative helicity (left-handed), i.e., their spin is antiparallel to their momentum; whereas the reverse occurs for antineutrinos,i.e., their spin projection points out parallel (positive helicity) to the momentum. This reflects the parity violation in electroweak processes. In the SM, only left-handed neutrinos and right-handed antineutrinos exist. Nevertheless, the discovery that neutrinos have mass could modify this description. Moreover, it is not clear yet if neutrinos are Majorana particles, that is, neutrino and antineutrino are the same particles. Neutrinoless double beta decay, which can be viewed as two beta decay events with the produced antineutrinos immediately annihilating with one another, is a possible way to detect if neutrinos are their own antiparticles. Experiments are underway to search for this type of decay. 4 1. INTRODUCTION Charged leptons Neutrinos Name Symbol Charge Mass (MeV) Name Symbol Charge Mass (MeV) 1st gen. Electron e−-1 0.511 Electron neutrino νe0<3·10−6 Positron e++1 Electron antineutrino νe0 2nd gen. Muon µ−-1 105.658 Muon neutrino νµ0<0.19 Antimuon µ++1 Muon antineutrino νµ0 3rd gen. Tauon τ−-1 1766.99 Tauon neutrino ντ0<18.2 Antitauon τ++1 Tauon antineutrino ντ0 Table 1.1: The Protoypical Family of Leptons. Neutrino mass The experimental evidence of solar and atmospheric neutrino oscillations [7–9] led to the conclusion that neutrinos are not massless, even though, at present, we do not know the absolute values of their masses but only the upper limits (see Table 1.1). In addition to the three different flavors, there are also three mass states ν1, ν2and ν3with masses m1,m2and m3respectively, which are associated to the flavor states through the Pontecorvo- Maki-Nakagawa-Sakata (PMNS) matrix [10,11], *., νe νµ ντ+/-=*., Ue1Ue2Ue3 Uµ1Uµ2Uµ3 Uτ1Uτ2Uτ3+/-*., ν1 ν2 ν3+/-.(1.1) The PMNS matrix can be expressed as the product of four sub-matrices [2], which contain three mixing angles (θ12,θ13 and θ23). These parameters determine the degree to which the mass and flavour states are mixed, as well as diverse CP-violation terms. 1.3 Neutrino Oscillations In 1998 the Super-Kamiokande facility [7] revealed the flavor change in atmospheric neutrinos. A few years later, the Sudbury Neutrino Observatory experiment (SNO) [8] concluded that the loss observed in the incoming electron solar neutrino flux was due to neutrino oscillations. These observations were confirmed in subsequent years in different neutrino experiments based on the use of reactors and accelerators [9]. The neutrino oscillation phenomenon, predicted by Bruno Pontecorvo [10], indicates the possibility that a specific neutrino (electronic, muonic or tauonic) transforms into a one with different flavor. This effect is of high experimental and theoretical relevance as it implies neutrinos are not massless. In this context, a proper understanding of cross sections in neutrino-nucleus reactions is essential. In the simple hypothesis that only two families of neutrino exist, the probability that a neutrino of energy Eνchanges its flavor from νito νfafter traveling a distance Lis given by P(νi→νf)=sin22θsin2 ∆m2L 4Eν!,(1.2) where θis the mixing angle related to the combination of the different flavor states into the mass states. The mass square difference parameter ∆m2=m2 1−m2 2is the difference between the two 1.3. NEUTRINO OSCILLATIONS 5 mass states and is the only mass parameter that can be measured in neutrino oscillation experiments, which cannot give information on the absolute masses. The ratio between L/Eνand ∆m2 is optimized in neutrino facilities to maximize the sensitivity to neutrino oscillations. In the threeflavor case the formula for the oscillation probabilities is more complicated and depends upon three mass squares differences (∆m2 12 =m2 1−m2 2,∆m2 13 =m2 1−m2 3,∆m2 23 =m2 2−m2 3), three oscillation angles (θ12,θ13,θ23) and a phase δCP corresponding to the possible violation of CP symmetry in the leptonic sector. The measurements of the latter is one of the most important challenges for future neutrino oscillation experiments. Thus the oscillation probability is given by Pνα→νβ(L,E)=δα β −4X i>j Re(U∗ αiUβiUαjU∗ βj) sin2*, ∆m2 ij 4EL+- +2X i>j Im(U∗ αiUβiUαjU∗ βj) sin *, ∆m2 ij 2EL+-,∆m2 ij ≡m2 i−m2 j.(1.3) The Particle Data Group [2] provides a review of the up-to-date values of the oscillation parameters, shown in Table 1.2. Parameter Value θ12 33.9◦±1.0◦ θ13 9.1◦±0.6◦ θ23 39◦< θ23 <51◦ ∆m2 12 (7.50 ±0.20)10−5eV2 ∆m2 23 (2.32+0.12 −0.08)10−3eV2 Table 1.2: Current world knowledge of neutrino oscillation parameters. ∆m2 13 is determined through ∆m2 13 =∆m2 12 +∆m2 23. There are two methods to determine neutrino oscillations: •Appearance mode: These experiments are focused on the search for a new neutrino flavor, absent in the original beam, or an enhancement of neutrinos of a given flavor in the initial beam. These effects are detected through the corresponding charged lepton produced via charged-current weak interaction: νl+N→l−+X(1.4) with l=e, µ,τ and Xthe final hadron state. These processes will be addressed in following chapters. •Disappearance mode: It is based on the reduction of the expected number of a particular neutrino flavor at the detector, as the case of the SNO experiment [8]. This method requires an accurate understanding of the neutrino beam at the source. Neutrino sources employed in oscillation experiments are diverse, going from nuclear reactors (νe), atmospheric (νe, νe, νµ, νµ), solar (νe) and accelerators (νe, νe, νµ, νµ). In the case of solar and atmospheric neutrinos, there are some constraints on Land Eνwhich prevent from an optimum combination of these parameters for the oscillation measurements. Therefore, the current 6 1. INTRODUCTION efforts focus on accelerator-based neutrino experiments where a better sensitivity to oscillations can be achieved. A detailed analysis of current neutrino experimental facilities will be presented in Sect. 1.6. 1.4 Interaction with Matter The neutrino interaction with matter is purely weak, acting at very short distances with a low intensity, and being able to modify the flavor of the particles involved. Such interaction is described by the electroweak theory, developed by Weinberg [12], Salam [13] and Glashow [14] in the 1970s, by the exchange of the weak bosons W±and Z. The W±boson is connected with charged-current (CC) scattering processes where there is a charge exchange in the interaction vertex, whereas the Z boson is associated to neutral-current scattering where no charge exchange occurs. Moreover, the fact that the electroweak interaction is 5 to 6 orders of magnitude lower than the electromagnetic one, leads to extremely low cross sections which makes experimental procedure highly demanding. In addition to this, the low intensity of weak interactions allows for a perturbative analysis, so that the Born approximation, that is, the 1-boson exchange between the leptonic and hadronic vertices, represents an accurate approach. Unlike electromagnetic scattering, the weak processes do not conserve parity, which provides specific information about the axial structure of the nucleon. In particular, the charged-current neutrino-nucleon (ν-N) scattering contains the most precise information about the nucleon axial form factors (addressed in Chapter 2). On the other side, neutral current processes give essential details about the strange sea quarks in the nucleon structure. In Fig. 1.1, the Feynman diagrams related to the elastic lepton-hadron reactions are shown. In weak interactions, a distinction between charged current processes, where a charged lepton is emitted, and neutral current ones, where the neutrino does not change in the final state, is considered. Figure 1.1: Lepton-hadron elastic scattering processes: a) Electromagnetic interaction. b) Charged-current weak interaction. c) Neutral-current weak interaction. Compared to CC neutrino reactions, the experimental study of NC processes is a fairly demanding task due to the considerable difficulties to obtain information in a process with a reduced cross section and without charged leptons in the final state. Thus, the event recognition lies on the hadron detection. 1.4. INTERACTION WITH MATTER 7 1.4.1 Neutrino-nucleus scattering At the lowest neutrino energies, the most probable interaction is the elastic scattering where the energy transfer to the nucleus is not enough to release an unbound nucleon so the nucleus recoils intact. The quasielastic (QE) scattering appears once the energy transfer is large enough to scatter off a nucleon from the nucleus. When this process occurs via neutral-current scattering, all neutrinos and anti-neutrinos can scatter off both neutrons and protons in what is referred to as neutral-current quasielastic (NCQE) scattering: νl(νl)+N→νl(νl)+N. Once neutrinos acquire sufficient energy to create the charged lepton’s mass they can also undergo the analogous chargedcurrent quasielastic (CCQE) interaction: νl+n→p+l−and νl+p→n+l+. For Eνµ≈1 GeV, CCQE is the dominant interaction. The interaction of neutrinos with nuclei at intermediate energies, of the order of 100s of MeV up to 10s of GeV, plays an important role in the precise determination of neutrino oscillation parameters. At these energies, nuclear effects are very significant, providing also relevant information on the axial hadronic currents. From the experimental point of view, the data analysis needs to consider a large number of nuclear effects that distort the signals and produce new sources of background that are absent in the elementary neutrino-nucleon processes. In this sense, a complete theoretical model should include, at least, three kinds of contributions: (i) quasielastic (QE) for low-intermediate energy transfers, (ii) nucleon correlations and two-body contributions and (iii) the complete inelastic spectrum, containing pion production arising from the ∆(1232) resonance peak, non-resonant contributions, other meson production as well as higher-energy nucleonic resonances and deep inelastic scattering processes. In this work, we focus on intermediate and high energies covering from QE reactions to deep inelastic scattering processes. The different reaction mechanisms are briefly described in the following lines: •Quasielastic scattering: The most relevant contribution at intermediate energies (Eν∼1 GeV) in which a neutrino scatters off a single bound nucleon being the latter ejected from the nuclear target. We can distinguish between NCQE neutrino interactions, where a neutrino is emitted, and CCQE neutrino interactions with a charged lepton in the final state. Regarding the notation employed in the experimental literature, CCQE-like scattering (or CC0π) is defined as the process where one lepton and no pions are detected in the final state. In this sense, QE-like processes are contaminated with other contributions such as short-range correlations (SRC), multinucleon emissions (np-nh) induced by meson-exchange currents (MEC) and by rescattering processes, pion absorption in the nuclear medium and pionless resonance decay. •2p-2h MEC contributions: This corresponds to a weak boson being exchanged by a pair of nucleons (2-body current) leading to the emission of two nucleons from the primary vertex. This contribution is essential to interpret properly the “dip” region between the QE and ∆ peaks for (e,e′) reactions as well as to reproduce the neutrino QE-like experimental data. •Resonance production: At higher energies, which implies large Q2values, neutrinos gain access to inelastic scattering processes. Here the target nucleon is “knocked” into a baryonic resonance, a ∆or a heavier resonance N∗depending on the energy transfer. These resonances decay into a variety of final states with combinations of nucleons and mesons. Neutrino-induced pion production is the dominant process in the region 0.5 GeV < Eν< 10 GeV. 8 1. INTRODUCTION •Deep inelastic scattering (DIS): At higher energies the neutrino is able to transfer sufficient momentum so the inner structure of the nucleon can be resolved. The neutrino can scatter directly off any of the quarks inside the nucleon, including those which form the “sea” of quarks and anti-quarks that are constantly popping in and out of existence. At lower transfer momentum the nucleons contain mostly up, down and strange quarks, but higher transfer momentum values get access to higher-mass and shorter-lived quarks too. DIS is the dominant process for Eν> 10 GeV and its most visible consequence is the break up of the nucleon containing the struck quark. The different reaction mechanisms contributing to (anti)neutrino-nucleus interactions are shown in Fig. 1.2 as function of the incoming (anti)neutrino energy. In order to give a more complete description of the scattering mechanisms, the nuclear response in terms of the energy transferred to the nucleus is schematically represented in Fig. 1.3 for a fixed momentum transfer q. In the latter, we can observe how small energy transfers of the order of tens of MeV result in elastic scattering off the nucleus followed by a collective excitation of the nucleus. Intermediate energies are related to the process where the lepton scatters off a bound nucleon giving a broad QE peak centered around ∼Q2/2M(being Mthe nucleon mass). At higher energies, multinucleon excitations and nucleonic resonances start to be relevant as well as DIS processes when the incident energy is enough for the nucleon break-up. Notice also the overlap of the different reaction mechanisms in the intermediate kinematical region (1-10 GeV). This effect should be considered carefully when analyzing neutrino-nucleus experimental data. 1.4.2 Elastic neutrino-nucleon interactions and nucleon form factors Doing calculations for the interaction of neutrinos with nuclei presents additional complexity over electrons with nuclei since, in addition to the purely vector electromagnetic responses, new axial and interference vector-axial contributions arising from weak leptonic and hadronic currents come into play. Although essentially composed of only three quarks, nucleons are constantly interacting via exchange of gluons which in turn can produce other temporary quark/anti-quark pairs. The fourmomentum of the weak boson (Q2) determines how much of the nucleon’s internal structure is resolved by a weak interaction. The description of the inner structure of the nucleon can be provided by the use of nucleon form factors, that are given as phenomenological functions expressed in terms of Q2. The type of interaction determines which form factors enter in the process. A more detailed description of hadronic form factors will be addressed in Chapter 2.3. With respect to the nucleon structure, elastic neutrino-nucleon scattering processes are particularly important to neutrino physics for two reasons. First, they provide essential information on the weak nucleon form factors which are difficult or inaccessible for other scattering probes. Second, the previous processes enable the kinematics to be completely reconstructed, and hence the initial neutrino energy can be determined. This is critical for measurements of the oscillation parameters. In spite of this, most of the experiments employ heavy targets in order to get larger cross sections. This makes the analysis of the process much more complex, i.e., a great caution should be drawn on how to reconstruct the kinematics, and how to get information on the weak nucleon form factors. Regarding the weak nucleon form factors in CCQE neutrino scattering we can distinguish vector, F1(Q2) and F2(Q2), axial, FA(Q2), and pseudoscalar, FP(Q2), ones, related to the corre- 1.4. INTERACTION WITH MATTER 9 Figure 1.2: Total neutrino (left) and antineutrino (right) cross sections per nucleon in terms of the incident neutrino energy (Eν). Different reaction channels are shown separately. The figures are taken from [15]. 10 1. INTRODUCTION Figure 1.3: Regimes of the nuclear response in terms of the energy transfered (ω) to the nucleus. sponding components of the hadronic current. The vector form factors, as a consequence of the conserved vector current hypothesis (CVC) [16], can be defined in terms of the electromagnetic ones related to electron-nucleon scattering. Concerning the axial form-factor, FA(Q2), it is common to assume a “dipole” form, where two control parameters must be determined experimentally: the “axial charge” gA=1.26 from beta-decay, and the “axial mass” MA=1.014±0.014 GeV from CCQE neutrino scattering on deuterium and pion production by electrons [17]. It should be noted though that the dipole form is an assumption motivated by the analysis of electromagnetic form factors [18]. A study of the differences between dipole and monopole axial form factor is shown in Appendix D as well as in [19]. Although, to date, neutrino experiments do not have the precision required to test explicitly the Q2-dependence of the weak form factors, a dipole form works nicely in most situations [18]. Finally, the pseudoscalar form factor, FP, is related to the axial one via the “Goldberger-Treiman” relation. The basic features of these functions are analyzed in Chapter 2.3. 1.5 Theoretical approaches to neutrino-nucleus scattering With the aim of achieving a complete theoretical description of neutrino-nucleus interactions, several theoretical groups have developed different models to account for the different ingredients needed for a deep understanding of experimental data. The first MiniBooNE CCQE measurements for neutrino interactions on 12C motivated a large theoretical effort to explain the apparent discrepancy between the experimental cross sections and the theoretical predictions, which seem to underestimate the data [20,21]. A first approach to disentangle this apparent discrepancy was to increase the value of the nucleon axial mass (MA) from the world’s averaged value (1.032 GeV) emerging from deuterium bubble chamber experiments [22, 23] to ∼1.35 GeV. However, while improving the theoretical agreement with the MiniBooNE data, such a high value of MAdoes not match with other experimental data, as NOMAD [24], nor with former elastic neutrino cross sections on light nuclei [25–29]. The key to this issue lays on the definition of the CCQE event in the experiment. Whilst theorist define CCQE as the event where one nucleon is knocked out in the final state along with a charged lepton, MiniBoone defined CCQE as an event with one lepton and no pion observed in the final state, which has been “renormalized” to CCQE-like. Therefore, the definition of CCQE-like event also includes effects beyond the Impulse Approximation, i.e., 1.6. EXPERIMENTAL STATUS 11 the neutrino only interacts with a single-bound nucleon. Thus, nucleon-nucleon correlations and multinucleon emission processes, such as 2p-2h MEC contributions, should be accounted for in the analysis of experimental data. From this baseline, a large number of theoretical groups began to investigate on this topic. In particular, Martini et al. employed a local Fermi gas model [30], taking into account RPA correlations, coherent and incoherent pion production as well as quasielastic excitations. Besides this, the inclusion of contributions arising from np-nh excitations [30] led them to a reasonable agreement with MiniBooNE data without resorting any effective parameter or increasing the MAvalue. A similar approach was taken by Nieves et al. [31], achieving a good agreement with data after a subtraction of ∼10%. Moreover, the Pavia group made use of a relativistic Green’s function approach [32] including the imaginary part of the relativistic optical potential to describe final-state interactions (FSI). Their predictions are in accordance with data for some particular choices of the optical potentials. The contributions of one-body and two-body currents have also been analyzed in [33] using the spectral function formalism. A semiphenomelogical model that considers the quantum-kinetic transport theory (GiBUU) has been also applied for the analysis of electromagnetic and weak interactions [34], using a parametrization of the transverse electromagnetic contributions as a basis for 2p-2h neutrino interactions. In this thesis, the analysis of the electron and neutrino-nucleus interactions in the QE and inelastic regimes will be carried out within the framework of the SuperScaling Approach (SuSA), which assumes the existence of universal scaling functions for both electromagnetic and weak interactions. The analysis of inclusive (e,e′) experimental data [35–38] has proved that scaling is fulfilled with very good accuracy. This implies that the reduced cross section exhibits independence of the momentum transfer (first-kind scaling) and of the nuclear target (second-kind scaling) when expressed as a function of the appropriate scaling variable, itself a function of the energy and momentum transfer. In a recently improved version, called SuSAv2 model, we have employed the Relativistic Mean Field Theory (RMF) and the Relativistic Plane Wave Impulse Approximation (RPWIA) to obtain a complete set of scaling functions that embody all the nuclear dependence of the interaction, being valid for all nuclei. The description of the the many-body physics of the interacting nucleons within the SuSAv2 model, takes into account the different contributions of both longitudinal and transverse nuclear responses, as well as the isovector and isoscalar channels. This is of great interest for charged-current (CC) neutrino reactions, which are purely isovector. Regarding the 2p-2h MEC contributions and contrary to other works on this topic, we employ a microscopic calculation in a fully relativistic framework without further approximations. The SuSAv2-MEC predictions, based on the use of the RMF/RPWIA model plus the 2p-2h MEC fully microscopic calculations, have been succesfully applied to the analysis of (e,e′) and CCQE neutrino-nucleus reactions covering from low to very high energies [38,39]. 1.6 Experimental status Once neutrino oscillations were confirmed in solar and atmospheric neutrinos [7,8], an ample experimental program was developed with the aim of studying these oscillations and determining precisely their masses through the oscillation parameters. This deep research has renewed the interest on neutrino cross sections as an essential ingredient on the analysis of neutrino oscillation experiments as well as on the study of the weak nucleon structure. Nevertheless, due to the extremely reduced cross sections in weak reactions, neutrino experiments are highly complicated. In the words of Haim Harari [40], “Neutrino physics is largely an art of learning a great deal by 18 1. INTRODUCTION collected by MINOS to identify and measure these energetic particles. Figure 1.9: Side-view of the MINERνA experiment. The neutrino beam travels left to right through the various detector components [46]. MINERνA has produced CCQE-like cross sections on CH [46, 47] which are in agreement with theoretical estimations including also multinucleon excitations, without further ingredients or larger axial mass values. Its recent measurements on DIS cross sections for different nuclei [54] could be valuable to assess the relevant ingredients for the neutrino reaction models at very high energies. K2K The K2K experiment [9] was designed in Japan to confirm atmospheric neutrino oscillations using Super-Kamiokande as detector. A schematic diagram of this detector is shown in Fig. 1.10. This consists of a water Cherenkov detector, a water-based liquid scintillator (SciFi, oxygen target) and a muon range detector (MRD). Thus, the impact of 12-GeV protons on an aluminium target produces neutrinos with an energy spread from 1 to 1.5 GeV. Finally, in the SciFi detector the neutrino cross section is measured. Figure 1.10: Schematic view of K2K detector [9]. 1.6. EXPERIMENTAL STATUS 19 T2K The T2K experiment (Tokai to Kamioka) is a long-baseline neutrino experiment in Japan which is focused on neutrino oscillations and can be considered the successor of K2K. It employs a highresolution detector with the aim of determining the neutrino energy spectrum as well as its flavor and the resulting cross sections. T2K has made a search for oscillations from muon neutrinos to electron neutrinos, already published [55] as well as its work on the measurement of oscillations from muon neutrinos to tau ones. A large set of experimental data from forward to backward angles has also been published for the analysis of CC0πproduction processes as well for the inclusive ones [44,45,56]. Figure 1.11: Schematic view of T2K detector [44]. The T2K experiment sends an intense beam of muon neutrinos from Tokai, which is on the east coast of Japan, to Kamioka at a distance of 295 km in western Japan. The neutrino beam is produced from collisions between a proton beam and a graphite target; these collisions produce pions, which quickly decay to muons and muon neutrinos. The muons and any remaining protons and pions are stopped by a second layer of graphite, but the neutrinos pass through it. T2K studies neutrino oscillations with two separate detectors, both of which are 2.5 degrees away from the centre of the neutrino beam. The ND280 near detector is located at 280 metres distance from the target, and measures the number of muon neutrinos in the beam before any oscillation occurs. T2K neutrinos have much higher energies than solar neutrinos, and high-energy neutrinos are more likely to interact. A small number of muon neutrinos interact with scintillator or water in the ND280, and many of these interactions produce a muon. The muon can be detected since it ionises gas which is placed immediately after the interaction points. These ND280 measurements are used to predict the number of muon neutrinos that would be seen in the “far detector” SuperKamiokande if there were no oscillations. The walls of SuperKamiokande are lined with more than 10,000 sensitive photo-multipliers, which detect the cone of Cerenkov light as a ring (see Figure 1.12). This detector system can distinguish muons originated from muon neutrinos (which produce a sharp ring) from electrons arising from electron neutrinos (which produce a more diffuse ring). A next step into the Kamiokande investigations is the Hyper-Kamiokande experiment [57], which will operate in the same beam line as T2K. This detector consists of a megaton scale water tank and ultra high sensitivity photosensors. The Hyper-Kamiokande detector is both a “microscope”, used to observe elementary particles, and also a “telescope” for observing the Sun and su- 20 1. INTRODUCTION Figure 1.12: Experimental view of muon rings in the T2K detector [44]. pernovas, using neutrinos. The Hyper-Kamiokande project has an extremely rich physics portfolio that spans from the study of the CP violation in the leptonic sector and neutrino mixing parameters using accelerator neutrino and anti-neutrino beams to proton decay, atmospheric neutrinos and neutrinos from astronomical origin. NOMAD The NOMAD experiment (Neutrino Oscillation MAgnetic Detector) was designed to search for ντ appearance from neutrino oscillations in the CERN wide-band neutrino beam produced by the 450 GeV proton synchrotron. The single-particle reconstruction and lepton identification capability of the NOMAD detector allowed the search for ντappearance in most of the leptonic and hadronic τ decay channels and also to look for νµ→νeoscillations. A second phase of the NOMAD analysis started after the completion of the oscillation searches, with the aim of exploiting the high quality of the available neutrino data samples for precise measurements of cross sections and particle production. This activity could be also of interest for oscillation studies. The NOMAD detector consisted of a high resolution magnetic detector. In 2000, NOMAD completed its search for νµ→ντthrough their charged-current interactions followed by the τ decay into one or two neutrinos and by the corresponding τdecay daughters. No oscillation signal was found but the limit set was in accordance with the expectations of the proposal, thus demonstrating the validity of the kinematic method used by NOMAD to search for ντcharged-current interactions. This collaboration produced interesting results about the total CCQE muon neutrino cross section on 12C [24] which, unlike MiniBooNE, do not claim for a larger axial mass value of the nucleon. ArgoNeuT The ArgoNeuT (Argon Neutrino Test) Experiment ran on the NuMI beam line at the FermiLab, from September 2009 to February 2010. It is the first stage of a project that makes use of Liquid Argon Time Projection Chambers (LArTPCs) as neutrino detectors. ArgoNeuT has collected thou- 1.6. EXPERIMENTAL STATUS 21 sands of beam neutrino events in the 0.1 -10 GeV energy range during its run and some results have been recently published, including the first measurements of the inclusive muon neutrino charged current differential cross sections on argon [48,49]. In a LArTPC experiment, the free electrons are drifted towards the anode in a uniform electric field where they are registered on wires in multiple (at least two) planes, due to electromagnetic induction and collection of the electrons on the last plane of wires. Because the wires in different planes are at an angle with respect to each other, it is possible to get the full 3D reconstruction of an event. The LArTPC, given its ability for simultaneous precise 3D and calorimetric reconstruction of particle interactions, is an extremely interesting detector technology. It connects physics goals, like sterile neutrino search, with technology development milestones that will lead to a multikiloton long-baseline neutrino experiment. This has provided the first ever data for low energy neutrino interactions within a LArTPC, paving the way for construction of larger detectors. Its goals included measurements of the CC inclusive cross sections in the 1-5 GeV range, examining the effects of Final State Interactions (FSI) and testing the Particle ID capabilities of the LArTPC, especially the e/γ separation crucial for future neutrino experiments. νSTORM νSTORM (Neutrinos from STORed Muons) is a proposed storage ring facility [58,59] to deliver beams of muon antineutrinos and electron neutrinos from positive muon decays (muon neutrinos and electron antineutrinos from negative muon decays), with a central muon momentum of 3.8 GeV/c and a momentum acceptance of 10%. The facility will allow searches for eV-scale sterile neutrinos at better than 10 sigma sensitivity. It will be able to provide measurements of neutrino and antineutrino-nucleus scattering cross sections with percent-level precision and will serve as a first step towards developing muon accelerators for particle physics. The flux of the neutrino beam can be determined with percent-level accuracy to perform cross-section measurements for future neutrino oscillation experiments and to resolve the hints for eV-scale sterile neutrinos. νSTORM may be considered as a first step towards a Neutrino Factory and a Muon Collider. In this sense, our studies about the difference between electron neutrino reactions and muon neutrino ones could be valuable for the analysis of its experimental results. DUNE The Deep Underground Neutrino Experiment (DUNE), conducted with the detectors installed in the Long-Baseline Neutrino Facility (LBNF) at FermiLab, is expected to achieve important discoveries in forthcoming years, making definitive determinations of neutrino properties, the dynamics of the supernovae that produced the heavy elements necessary for life, and the possibility of proton decay. With the LBNF facilities and the detectors provided by DUNE [60], the DUNE Collaboration proposes to disentangle the puzzle of neutrinos with broad sensitivity to neutrino oscillation parameters in a single experiment. The focus of the scientific program is the determination of the neutrino mass hierarchy and the analysis of leptonic CP violation by precisely measuring differences between the oscillations of muon-type neutrinos and antineutrinos into electron-type neutrinos and antineutrinos, respectively. Furthermore, the DUNE experiment also focuses on determining the ordering of the neutrino masses as well as on searching for neutrinos beyond the currently known three. 22 1. INTRODUCTION 1.7 Structure and motivation of this thesis As already mentioned in previous sections, most events analyzed by the different collaborations correspond to CCQE processes where a muonic neutrino is scattered by a bound nucleon (νµ+n→ µ−+p). The employment of realistic nuclear models is crucial for the analysis of these and forthcoming experiments, such as NOνA, MINERνA, MiniBooNE or T2K, focused on the search for neutrino oscillations νµ→νe. A proper interpretation of neutrino oscillations implies an accurate description of the CCQE process in a wide range of neutrino energies. It should be noted that the beam neutrino energy is not precisely determined and only an energy distribution of the incoming flux is predicted. Therefore, a control over the uncertainties associated to the nuclear model is indispensable. Likewise, background processes are also important and hard to disentangle from the CCQE signal as the case of the pion production and its subsequent absorption by the nuclear target. Other effects, such as nucleon correlations, meson-exchange currents related to multi-nucleon knock-out or final-state interactions (FSI), can also contribute significantly to the final result and have to be appropriately accounted for in the nuclear models. Moreover, the oscillation probability directly depends on the neutrino energy, which must be reconstructed from the final-state particles. This reinforces the need to dispose of realistic models that provide an accurate description of the mechanisms involved in neutrino-nucleus interactions. These models are in turn used in the simulations with which experiments define their analyses and interpret their data so it is essential to improve the understanding of neutrino interactions in order to improve neutrino oscillation measurements. Therefore, the focus of the following chapters is to study electroweak scattering processes, deepening in neutrino-nucleus reactions along the different nuclear regimes of interest for neutrino oscillations. This will be addressed by using realistic models that treat the process in a fully relativistic way, describing both hadronic and nuclear structure. This is directly connected with the growing interest in neutrino-oscillation experiments where an accurate theoretical description of weak interactions is essential to interpret properly the experimental results. The structure of this thesis will be the following: in Chapter 2 we analyze the case of chargedcurrent neutrino-nucleon elastic interactions where the weak response functions and the nucleon form factors are described; in Chapter 3 we present the formalism of quasielastic neutrino-nucleus interactions within the SuSAv2 model, which is based on the superscaling behavior and the relativistic mean field theory. The Chapter 4 focuses on the 2p-2h MEC contributions as an essential ingredient for the analysis of experimental data, where we make use of an accurate parametrization of fully relativistic microscopic calculations. The extension of the SuSAv2 model to the inelastic regime is detailed in Chapter 5. Afterward, in Chapter 6 we compare the previous theoretical description with electron-nucleus scattering data for all kinematics. This constitutes a solid benchmark to assess the validity of our model for the analysis of the existing charged-current neutrino experimental data, as detailed in Chapter 7. Finally, in Chapter 8 we present a summary and the main conclusions of this PhD thesis. Chapter 2 Charged-current elastic neutrino-nucleon scattering In this chapter we introduce the formalism employed to describe elastic charged-current neutrinonucleon (CC ν-N) scattering processes, which will be of relevance to understand properly the CCQE neutrino-nucleus reactions. Accordingly, we also analyze the relevance of the diverse electromagnetic and weak nucleon form factors and the different single-nucleon responses. This analysis is carried out within the Born Approximation, where a single Wboson is exchanged, thus resulting an accurate approach due to the reduced value of the weak coupling constant and the large W-boson mass. 2.1 General formalism The kinematics of the CC ν-Nscattering process (νl+n→l−+p), where lrepresents e, µ or τ, and the frame of reference employed are schematically shown in Fig. 2.1. In this figure, the elastic neutrino-neutron interaction via the exchange of a W−is displayed, leading to a final state composed of an outgoing lepton and a proton. The antineutrino-nucleon process (νl+p→l++n) is analogous to the previous one. Figure 2.1: Frame of reference and kinematics involved in the CC neutrino-nucleon process where the outgoing lepton is a muon. 23 24 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING Accordingly, the charged-current neutrino (antineutrino) interaction, which is purely isovector, is associated to isospin T=1 as there is a charge exchange where the initial neutron (proton) turns into a proton (neutron) in the final state. This implies an isospin change in the Z-component as |∆Tz|=1. In neutral current interactions (ν+N→ν+N), this Z-component is zero in such a way that isoscalar contributions would also be permitted. The frame of reference selected for these processes is the laboratory frame in which the initial nucleon is at rest. It is taken for simplicity the momentum transfer to the nucleon along the Zaxis and, therefore, the final momentum of the nucleon. Thus, the initial and final momenta of the leptons are contained in the X Z plane. For completeness, the following analysis has also been addressed briefly for the electromagnetic electron-nucleon interaction (e−+N→e−+N) in Appendix A. The notation employed for the kinematics in CC weak processes is summarized as follows: ➲Incident neutrino νl: –4-momentum: kν=(Eν,~ kν) –Mass: mν=0→Eν=|~ kν| ➲Final lepton l−: –4-momentum: kl=(El,~ kl) –Mass: ml –Scattering angle: θl ➲Initial nucleon n: –4-momentum: Pi=(Mn,0) –Mass: Mn ➲Final nucleon p: –4-momentum: Pf=(Ef,~q) –Mass: Mp ➲Exchanged boson W−: –4-momentum: Qν=(ω, ~qν≡qν) The energy-momentum conservation law in the leptonic and hadronic vertexes implies the following relations: Leptonic vertex ➦Energy conservation: Eν−El=ω ➦Momentum conservation: ~ kν−~ kl=~q⇒q2=E2 ν+|~ kl|2−2Eν|~ kl|cos θl Hadronic vertex ➦Energy conservation1:Ef=ω+MN⇒ω2+2MNω=q2⇒−Q2≡ |Q2|=2MNω ➦Momentum conservation: ~q=~ Pi+~ Pf⇒~q=~ Pf As shown in the previous equations, the kinematical variables Eν,Eland θlare not independent so the kinematics of the process is completely defined by determining two of them. 1Notice that a generic nucleon mass MNis employed. This can be either the neutron or proton mass depending on the neutrino or antineutrino case, respectively. 2.1. GENERAL FORMALISM 25 Interaction hamiltonian The transition amplitude Sf i of the neutrino-nucleon scattering process is obtained from the associated interaction hamiltonian (HW), Sf i =−iZd4X HW(X).(2.1) with the hamiltonian expressed as HW(X)="g 2√2#2 J(l)† µ(X)Aµ (N)(X)="g 2√2#2 J(l)† µ(X)Dµν W(Q)J(N) ν(X),(2.2) where gis the dimensionless weak coupling constant, related with the Fermi constant, GF, as GF √2 =g2 8M2 W ;MW=80.401(38) GeV .(2.3) The 4-potential Aµ (N)(X), i.e., the field generated by the nucleon in the interaction with the corresponding lepton, can be described in terms of the hadronic current, J(N) ν, and the weak propagator, Dµν W, Aµ (N)(X)=Zd4Y DW z }| { Zd4Q (2π)4*,−gµν +QµQν/M2 W Q2−M2 W+iε+-eiQ·(X−Y)J(N) ν(Y) ≈Zd4YZd4Q (2π)4eiQ·(X−Y)*, 1 M2 W+-Jµ (N)(Y),(2.4) where the approximation |Q2| ≪ M2 W(MW=80.401 GeV/c2), valid for energies corresponding to the elastic and QE regimes, has been considered. Next we introduce the leptonic and hadronic currents, used to obtain the double differential cross section. Leptonic and hadronic currents The leptonic j(l) µcurrent is defined as a sum of vector (γµ) and axial (γµγ5) terms associated to the vector and axial structure of the nucleon. The axial term makes possible the parity violation in weak interactions. The previous definition is general for all weak processes and, in particular, for the charged current ones. On the contrary, the analysis of the leptonic current (j(e) µ) in electromagnetic (e,e′) processes only involves the vector component. Thus, the weak leptonic current is defined as j(l) µ=ψl(kl,sl)γµ(1 ∓γ5)ψνl(kνl,sνl),(2.5) where the -(+) sign refer to neutrino (antineutrino) processes. Likewise, the hadronic currents for charged-current weak processes, Jµ (N), are also composed of vector and axial terms associated to the vector and axial structure of the nucleon. On the contrary, only the vector part remains for electromagnetic interactions (see Appendix A for details), Jµ (N)=ψPf(Pf,Sf)H ΓµψPi(Pi,Si),(2.6) 26 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING where H Γµ=FV 1γµ+ iFV 2 2MN σµνQν | {z } H Γµ V +GAγµγ5+FPQµγ5 | {z } H Γµ A .(2.7) At first order in perturbation theory and considering plane waves, the wave functions related to the hadronic and leptonic particles can be defined as ψα(X)=rmα EαVuα(pα,sα)e−ipα·X,(2.8) where u(pα,sα) are the Dirac spinors and αthe corresponding single particle state. In the previous expresion for the hadronic current (2.6), the inner structure of the nucleon is described by means of the purely-isovector nucleon form factors (FV 1,FV 2,GAand FP), arising from the CC weak interaction and dependent on Q2. The isospin symmetry allows for a description of the isovector form factors FV 1and FV 2in terms of the electromagnetic Dirac, Fp,n 1and Pauli, Fp,n 2, form factors for protons and neutrons. Conserved vector current (CVC) When comparing both electron-nucleon and charged-current neutrino-nucleon scattering, it can be observed that the electromagnetic current (see Appendix A) and the vector component of the weak one are related through the conserved flavor current. This is knwon as the CVC (conserved vector current) hypothesis and implies that the weak vector component is conserved. In the words of J.D. Walecka [16], the CVC implies that the vector part of the single nucleon matrix element of the charge changing weak current, whatever the detailed dynamic structure of the nucleon, can be obtained from electron scattering through the electromagnetic interaction. The CVC hypothesis assumes that the vector part of the hadronic current and the electromagnetic current are components of the same conseved-current isospin multiplet and thus their form factors are interrelated, FV 1(Q2)= Fp 1(Q2)−Fn 1(Q2) 2,FV 2(Q2)= µpFp 2(Q2)−µnFn 2(Q2) 2,(2.9) where µp=2.793 and µn=−1.913 are the corresponding proton and neutron magnetic moments. For completeness, the isoscalar form factors are defined as FS 1(Q2)= Fp 1(Q2)+Fn 1(Q2) 2,FS 2(Q2)= µpFp 2(Q2)+µnFn 2(Q2) 2.(2.10) Finally, the axial form factor, GA, describes the axial-pseudovector structure of the nucleon whereas the FPone englobes the pseudoscalar structure. All these form factors have been studied in detail in the literature and will be carefully analyzed in Section 2.3. Additional information on the electroweak form factors can also be found in [61–63]. 2.1. GENERAL FORMALISM 27 Neutrino-nucleon elastic cross section The double differential cross section for the elastic neutrino-nucleon interaction can be expressed, in terms of the outgoing lepton energy (El) and the solid scattering angle (Ωl), as: d2σ dEldΩl =|~ kl| |~ kνl| G2 F 4π2δ ω−|Q2| 2MN!Hηµν H Wµν ,(2.11) where Hηµν and H Wµν are the leptonic and hadronic tensors, respectively, which arise from the contraction of the leptonic and hadronic currents. The leptonic tensor reads, Hηµν =j(l)† µj(l) ν=kf,µki,ν −kf·kigµν +kf,ν ki,µ ±iεµναλ kα fkλ i,(2.12) being the axial contribution negative for neutrinos and positive for antineutrinos as a consequence of their different helicity. Moreover, the hadronic tensor H Wµν is given by: H Wµν =Jµ† (N)Jν (N)=H Wµν V+H Wµν A+H Wµν V A =−W1(Q2)gµν +W2(Q2)Pµ iPν i M2 N +iW3(Q2) 2M2 N εµν ρσPi,ρQσ +W4(Q2) M2 N QµQν+W5(Q2)Pµ iQν+QµPν i 2M2 N ,(2.13) where Wi(Q2) are the hadronic structure functions defined in terms of the nucleon form factors [63], W1=|Q2| 4M2 NFV 1+FV 22 +(GA)2+G2 A(2.14) W2=(FV 1)2+|Q2| 4M2 N (FV 2)2+(GA)2(2.15) W3=2GA(FV 1+FV 2) (2.16) W4=|Q2| −4M2 N (4M2 N)2(FV 2)2−FPGA MN +|Q2| 4M2 N (FP)2(2.17) W5=W2.(2.18) In the previous expression (2.13), the antisymmetric term corresponds to the axial contributions and, in particular, to the interference vector-axial term associated to W3. The contraction of the leptonic and hadronic tensors Hηµν H Wµν results in the following expression, d2σ dEldΩ =|~ kl| Eνl G2 F 4π2δ ω−|Q2| 2MN!Hηµν H Wµν =|~ kl|El G2 F 4π2δ ω−|Q2| 2MN!(2W1sin2θ 2+W2Elcos2θ 2±W3 Eνl+El MN Elsin2θ 2 + m2 l El(El+|~ kl|)"W1cos θ−W2 2cos θ±W3 2*, Eνl+El MN cos θ−El+|~ kl| MN+- +W4 2*, m2 l M2 N cos θ+2El(El+|~ kl|) M2 N sin2θ 2+-−W5El+|~ kl| 2MN#).(2.19) 34 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING Galster dipolar parametrization This parametrization, firstly introduced in 1971 [18], is widely used by the theoretical and experimental community and gives a reasonable description of the proton experimental data for |Q2| ≤ 1 GeV2(∼5%). In the neutron case, the description is not very accurate due to the data uncertainty. It presents a high simplicity based on a dipolar functional form, Gp E(Q2)=GV D(Q2) (2.78) Gn E(Q2)=−µnτGV Dξn(2.79) Gp M(Q2)=µpGV D(Q2) (2.80) Gn M(Q2)=µnGV D(Q2),(2.81) where ξn=(1 +λnτ)−1with λn=5.6 and τ=|Q2|/4M2 N, and µp=2.793 and µn=−1.913 are the proton and neutron magnetic moments. The dipolar form factor, GV D, is described as: GV D(Q2)=1 1+|Q2| M2 V2≡1 1+λV Dτ2;GV D(0) =1 ; GV D(|Q2| → ∞)=0,(2.82) where the parameter values are λV D=4M2 N/M2 V=4.97 and MV=0.843 GeV. For Q2→0, the form factors are fixed by the electric charge and the nucleon magnetic moments. Typically this parametrization is constrained to be consistent with experimental data for the neutron charge radius. However, recent and more precise experimental results [67] have shown the limitation of this parametrization to fit all these data. This has motivated the analysis of other approaches. Kelly parametrization Recently, a new parametrization of the electromagnetic form factors has been developed by J.J. Kelly [68]. This is basically an extension of the Galster one, providing a reasonable description of all experimental data. In this approach, the electric and magnetic form factors are defined as, G(Q2)∝ n X k=0 akτk 1+ n+2 X k=1 bkτk ,(2.83) where both numerator and denominator are polynomials in τ. With n=1 and a0=1, this parametrization provides excellent fits to Gp E,Gp M/µpand Gn M/µnusing only four parameters. However, this approach is less successful for Gn E. More specific details can be found in [68]. VMD models: GKeX parametrization A more accurate description of the nucleon form factors can be obtained from the VMD (vector meson dominance) models. Within this phenomenological description, the form factors are expressed in terms of mesonic propagators and meson-nucleon form factors. The most representative VMD models are the Gari-Krumpelmann parametrization [69], which incorporates the description 2.3. HADRONIC STRUCTURE 35 for high |Q2|introduced by the pQCD; and the recently-developed GKeX parametrization, also known as Lomon prescription [70]. The validity of the GKeX parametrization extends over all range of Q2where electron data exist (0.1-10 GeV). This description represents an extension of the Gari-Krumpelmann one, including also the effects arising from the vector mesons: ρ,ρ′,ω,ω′ and φ. 0.2 0.4 0.6 0.8 1 1.2 GE p/GD Galster GKeX Data 0.95 1 1.05 1.1 GM p/(µpGD) 0.001 0.01 0.1 1 10 |Q2| (GeV2) -0.2 0 0.2 0.4 0.6 GE n/GD 0.01 0.1 1 10 |Q2| (GeV2) 0.6 0.7 0.8 0.9 1 1.1 GM n/(µnGD) Figure 2.2: Electromagnetic Sachs form factors as a function of |Q2|for the Galster and GKeX parametrizations and divided by the appropriate dipole factor. Experimental data taken from [71]. In Fig. 2.2 we compare the electromagnetic form factors, Gn,p EyGn,p M, divided by the appropriate dipole term for the Galster and GKeX parametrizations. Both are very similar for low |Q2| values, whereas their differences increase substantially with |Q2|. Furthermore, the GKeX model reproduces the experimental results accurately, due to its more sophisticated description of the nucleon structure. For completeness, we also show in Fig. 2.3 the isovector form factors, G(1) EyG(1) M, associated to the weak current for both parametrizations. This analysis reinforces the idea of using the GKex parametrization in the nucleon form factors employed in the study of QE processes. Additionally, a detailed analysis of various nucleon form factors’ parametrizations and their relevance on the charged-current quasielastic neutrino-nucleus cross section will be analyzed in Chapter 7. 36 2. CHARGED-CURRENT ELASTIC NEUTRINO-NUCLEON SCATTERING 0.001 0.01 0.1 1 10 |Q2| (GeV2) -0.5 0 0.5 1 GE V/GD Galster GKeX 0.001 0.01 0.1 1 10 |Q2| (GeV2) 0.85 0.9 0.95 1 1.05 1.1 GM V/(µV GD) Figure 2.3: Isovector form factors as a function of |Q2|for the Galster and GKeX paramterizations where µVmakes reference to the isovector magnetic moment. 2.3.3 Axial structure The weak hadronic current shown in Eq. (2.6) also depends on the axial GA(Q2) and pseudoscalar GP(Q2) form factors. The axial one, GA, has been widely studied in last years [71, 72] in order to determine its functional structure. On the contrary, the pseudoscalar GP≡FP/2MN, is harder to analyze, being its contribution very small in most of the kinematical situations addressed and negligible for the momentum transferred involved in β-decay processes. Moreover, current data are not precise enough to determine the functional form of GA, which is usually parametrized using a dipole form, analogously to the vector form factors, GA(Q2)=gA 1+Q2 M2 A2,(2.84) where gA=−1.267 is the axial-vector coupling constant and MA=1.032(36) GeV is the nucleon axial mass. The gAparameter is determined through β-decay processes with neutrons in the elastic limit Q2→0 [73], and the axial mass MAvalue has been extracted from deuterium-filled bubble chamber experiments [17]. More information regarding the functional form of GAand the possible quenching of gAcan be found, respectively, in Appendices D and E. The axial and pseudoscalar form factors can be connected making use of the PCAC (partially conserved axial current) hypothesis (see [74] for details), through the Goldberger-Treiman relation, GP(Q2)= 4M2 N Q2+m2 π GA(Q2) ; mπ: pion mass .(2.85) In current neutrino-nucleus investigations, one of the main sources of uncertainty comes from the axial form factor, and specifically, its dependence with Q2and MA. In this sense, recent CCQE neutrino experiments on 12C, such as MiniBooNE or MINERνA, have estimated higher MAvalues (MA≈1.35 GeV) in disagreement with the standard estimations. It must be taken into account that the standard value, MA=1.03 GeV, is consistent with deuterium experimental data where nuclear effects are negligible as well as with weak pion production data at low |Q2|. Hence the increase of 2.3. HADRONIC STRUCTURE 37 the axial mass value in these experiments must be interpreted as the lack of some ingredients in the model employed for the data analysis. In particular, the role played by multinucleon effects, such as 2p-2h MEC in CCQE neutrino-nucleus experiments will be explored in Chapter 7 as a possible explanation of the “apparent” increase of MA. In Fig. 2.4, we analyze the axial form factors, GAand GP, using the world average axial mass value as well as an increase value of 1.35 GeV. Whilst small differences emerge at very low |Q2|, we can observe how GAincreases with MAfor large values of |Q2|. On the contrary, this effect is not visible for GPdue to the factor 1 1+|Q2|/m2 π where mπ≪MA. Moreover, the pseudoscalar form factor falls to zero faster than the axial one which turns into a less relevant contribution of pseudoscalar effects in the region of elastic, QE scattering and beyond. ✵✵✵✁ ✵✵✁ ✵✁ ✁ ✁✵ ⑤✂ ✷ ⑤ ✄☎✆✝ ✷ ✮ ✵ ✞ ✵✟ ✵✠ ✵ ✡ ✁ ✁ ✞ ● ❆ ✴ ☛ ❆ ▼ ☞ ❂ ✁✌✵✍☎✆✝ ▼ ☞ ❂ ✁✌✍ ✎☎✆✝ ✵✵✵✵✁ ✵✵✵✁ ✵✵✁ ✵✁ ✁ ✁✵ ⑤✂ ✷ ⑤ ✄☎✆✝ ✷ ✮ ✵ ✵ ✞ ✵✟ ✵✠ ✵✡ ✁ ✁ ✞ ● P ✯ ✏ ♣ ✴ ✑ ☛ ❆ ✭ ✒ ✓ ✔ ✕ ❪ ▼ ☞ ❂ ✁✌✵✍☎✆✝ ▼ ☞ ❂ ✁✌✍✎ ☎✆✝ Figure 2.4: Axial (GA) and pseudoscalar (GP) form factors for two MAvalues as a function of |Q2|. Finally, it is worth mentioning that the differences between neutrino and antineutrino cross sections, which arise basically from the V-A interference term (2.21), start to disappear as |Q2| increases. Notice that the RV A T′response (see Eq. 2.68) depends on GAthat approaches zero for large |Q2|. Chapter 3 Charged-current quasielastic neutrino-nucleus scattering After presenting the formalism of neutrino-nucleon elastic scattering, in this chapter we extend our theoretical description to charged-current neutrino-nucleus interaction in the quasielastic regime and the study of nuclear effects. This is of interest for most recent neutrino experiments that employ different nuclear targets to measure the oscillation parameters at energies where the quasielastic regime dominates. Unlike the elastic scattering where we only consider the inner structure of the nucleon, quasielastic reactions require a description of the nuclear structure. In this PhD thesis, the description of the lepton-nucleus interaction and, in particular, the nuclear dynamics, is addressed within the context of the SuperScaling Approach (SuSA), which assumes the existence of universal scaling functions for both electromagnetic and weak interactions. For a proper understanding of this approach, we give a brief description of the relativistic Fermi gas model that will help to introduce the basis of the SuSA model as a semiphenomelogical approach based on the analysis of inclusive (e,e′) data. Next, we extend our description to the relativistic mean field (RMF) theory and the relativistic plane wave impulse approximation (RPWIA) models as a more sophisticated procedure to include final-state interaction (FSI) effects as well as the mean field generated by the nuclear constituents in the neutrino-nucleus interaction. The RMF has the merit of treating final-state interactions in a relativistic framework, which is of relevance for the analysis of neutrino experiments. In neutrinonucleus interactions, after the final-state particles have been created, they propagate out through the nucleus, undergoing strong interactions with the other nucleons inside the nucleus. These “finalstate interactions” can significantly alter the momentum and direction of the final-state particles as well as the type and number of particles. Accordingly, pions and nucleons can be absorbed within the nuclear medium or their collisions with other nucleons can generate additional particles. Thus a consistent and relativistic treatment of these interactions is essential to determine the contribution of the different reaction mechanisms to neutrino-nucleus cross section. The description of the the many-body physics of the interacting nucleons within the RMF and RPWIA models is therefore included in our framework, in the so-called SuSAv2 (SuperScaling Approach version 2) model. This approach has been recently applied to the analysis of QE electron scattering data [38] for several nuclear targets as well as to charged-current quasielastic (CCQE) neutrino-nucleus experiments [39], yielding an accurate description of the experimental data in both cases. 39 40 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 3.1 General formalism of the CCQE process In this section we show schematically the kinematics involved in studies of lepton scattering from nuclei, focusing on charge-changing neutrino reactions. Most kinematic variables have been previously defined in Chapter 2 for elastic neutrino-nucleon scattering. Here we employ that notation for (νl,l−) and (νl,l+) neutrino-nucleus reactions. We specifically analyze the CCQE neutrino scattering process in which an incident beam of neutrinos with 4-momentum kµ=(Eν,kν) interacts with a nucleus. In the final state, a charged lepton with 4-momentum k′µ=(E′ l,k′ l) emerges as well as an outgoing nucleon. This process is mediated by a weak boson (W) and can be described as, νµ(νµ)+A→µ−(µ+)+p(n)+(A−1) (3.1) with Athe nuclear target and (A−1) the residual nucleus after the interaction. Compared to the elastic case, the complexity of the nuclear dynamics introduces some uncertainties in the description, related to the inner structure of the nucleus or final-state interactions described above. Many theoretical approaches evaluate these processes in the Impulse Approximation (IA), which assumes the incident lepton to only interact with a single bound nucleon as shown in Fig. 3.1. The influence of the remaining nucleons over the entire process is taken into account in different ways depending on the nuclear model employed. Hence, the IA describes the nuclear many-body matrix element as a sum of single-nucleon current matrix elements. Figure 3.1: Schematic view of the charged-current neutrino-nucleus scattering process in the Impulse Approximation (IA). 3.1. GENERAL FORMALISM OF THE CCQE PROCESS 41 3.1.1 Lepton scattering kinematics Going into detail to the description of lepton kinematics, the neutrino and final-lepton energies are given as Eνl=qm2 ν+k2 ν(3.2) El=qm2 l+k′2 l,(3.3) where mνand mlare the masses of the incident neutrino and outgoing lepton, respectively. The 4-momentum transfer to the nucleus Qµ=(ω,q) is given by ω=Eν−El(3.4) q=kν−k′ l,(3.5) being ωand qthe energy transfer and 3-momentum transfer, respectively. Given an excitation from target rest mass Mito some final rest mass Mf≥Mi(that is, the final hadronic rest frame total energy is W=Mf), we define a sort of excitation energy ω0≡1 2MiM2 f−M2 i≥0,(3.6) related to the mass excess of the residual nucleus. This value sets a minimum neutrino energy for the QE neutrino interaction to take place. In the case of NCQE scattering, this would be the minimum energy required for the process whereas in the CCQE one we must also consider the energy needed to create the outgoing-lepton mass. Then from energy-momentum conservation one has ω=ω0+|Q2| 2Mi .(3.7) Solving Eqs. (3.4) and (3.7), one gets expressions for the scattered lepton’s energy and 3-momentum. Defining ǫ1≡qM2 i+2MiEν+m2 ν+k2 νsin2θl(3.8) ǫ2≡qMi(Eν−ω0)+H M2,(3.9) where θlis the lepton scattering angle (the angle between kνand k′ l) and H M=q(m2 l+m2 ν)/2, it can be shown that k′ l=1 ǫ2 1ǫ2 2(kνcos θl)+(Mi+Eν)qǫ4 2−m2 lǫ2 1(3.10) El=1 ǫ2 1ǫ2 2(Mi+Eν)+(kνcos θl)qǫ4 2−m2 lǫ2 1,(3.11) where for the results to be real for all scattering angles, the beam energy must be greater than Eν,min, Eν,min =ml+ω0+ mlω0+(m2 l−m2)/2 Mi−ml .(3.12) This result represents the minimum energy required for the CCQE process in terms of the lepton mass and the mass excess of the residual nucleus. In addition, the contribution of the ω0parameter 42 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING is generally smaller than the lepton mass. In the particular case of (ν, µ−) reaction on 12C, we have mµ≫ω0as the mass excess of the residual nitrogen, ∼15 MeV, is much smaller than the muon mass, mµ≈105 MeV. Taking into account the mass difference between leptons and nucleus, the third term in Eq. (3.12) gives a negligible contribution in most of the cases. Hence, for given values of the excitation energy ω0, beam energy Eνand the scattering angle θl, the quantities ǫ1,2can be computed and from them the final lepton’s energy and 3-momentum are fixed. The 4-momentum transfer is then given as well. 3.2 Relativistic Fermi Gas Model and SuperScaling Approach Next, in order to describe the structure of the nuclear responses in our theoretical prescription, we introduce the concept of scaling and the main features of the relativistic Fermi gas model [75,76] that has been used as a basis for the study of scaling and superscaling behaviors. 3.2.1 The concept of scaling and superscaling Scaling is a phenomenon observed in several areas of Physics and is a fundamental part of the scientific methodology. This is not new; we can think of Galileo’s observations of the oscillations of a pendulum, Kepler’s discovery of the equal area law for planetary motion and Newton’s inverse square law of gravitation. The establishment of a scaling relationship between physical quantities reveals an underlying driving mechanism, and it is the task of Physics to understand and to provide a formalism for that mechanism. The concept of yscaling in lepton-nucleus reactions was firstly introduced by West [77] as an analog of xscaling in high-energy physics. There it was shown that a collection of nonrelativistic point-like charged “nucleons” with negligible final-state interactions leads to an inclusive quasielastic electron scattering cross section that can be written in terms of a factor containing a single-nucleon cross section times a specific function. In the limit of large momentum transfers, this function scales; that is, it becomes a function of only one variable, usually represented by y, and largely independent on the momentum transfer. Further studies focused on the analysis of this scaling behavior on inclusive electron-nucleus scattering data in a relativistic framework [77] show that this behaviour can also be observed for different nuclei [36,78]. Concerning the difficulties for this analysis, it should be noticed that the explicit energy dependence of the cross sections complicates the attempt to factorize the electronnucleus cross section by extraction of a unique single-nucleon cross section. In addition to this, there also exists the question of the off-mass-shell extrapolation of the single-nucleon current as well as the fact that FSI and multi-nucleon interactions can also affect the scaling analysis. In the particular case of QE electron-nucleus scattering processes, in most of the models based on IA, the inclusive (e,e′) cross section can be approximated by a single-nucleon cross section times a specific function of (q,ω). In this case, the lepton interacts with a many-body system in such a way that the energy ωand momentum qare transferred only to individual constituents of the complex system. Scaling occurs at some specific kinematics, where the specific function scales, that is, it becomes dependent on only a single quantity, namely, the scaling variable ψ. This quantity, whose definition is discussed later, is in turn a function of qand ω:ψ=ψ(q,ω). The function that results once the single-nucleon cross section has been divided out is called the scaling function 3.2. RELATIVISTIC FERMI GAS MODEL AND SUPERSCALING APPROACH 43 f=f(q,ψ). In other words, to the extent that at some kinematics this function depends on ψ, but not on q, one says that ψ-scaling occurs. The study of the scaling function can shed light on the dynamics of the nuclear system. Indeed, within some specific approaches, the scaling function is related to the momentum distribution of the nucleons in the nucleus (or, more generally, with the spectral function) [79,80]. When studying (e,e′) processes it is useful to introduce the following concepts: •Scaling of first kind. This is related to the concept of y-scaling: it is satisfied when the scaling function does not explicitly depend on the transferred momentum, but only on ψ including its implicit dependence on qand ω. •Scaling of second kind. It is observed when the scaling function is independent of the nuclear species. •Scaling of zeroth kind. It occurs when the scaling functions linked to the different channels that make up the cross section, longitudinal (L) and transverse (T), are equal. Hence in inclusive electron scattering, zeroth-kind scaling means that the electromagnetic (EM) scaling functions satisfy f=fL=fT, where frepresents the scaling function associated to the total cross section whereas fL,Trefer to the scaling functions obtained from the separate longitudinal and transverse responses. •Superscaling. Finally, when scaling of both the first and second kinds occurs simultaneously one has superscaling [35,36]. With these ingredients it is clear that the natural starting point for such an examination is the relativistic Fermi gas (RFG) model in which the nucleus is described as a noninteracting gas of nucleons. Indeed, this model fulfills exactly all of the kinds of scaling previously defined. The RFG has the appeal of simplicity while maintaining important aspects in the problem such as Lorentz covariance and gauge invariance in a fully relativistic way. Naturally it ignores potentially important effects such as those stemming from strong final-state interactions or two-body MEC and employs an oversimplified initial-state spectral function; nevertheless, such ingredients can be added to the basic model and appear not to invalidate it as a basic starting point for analyses of scaling. The general procedure used to define scaling functions consists of constructing the inclusive cross section, or nuclear response functions, within a particular theoretical model (or experimental data) and divide them by the appropriate single-nucleon quantity computed within the RFG model. Here the word “appropriate” entails two aspects to be considered. First, the usual analysis in the region of the quasielastic (QE) peak assumes that the dominant process is elastic scattering from nucleons in the nuclear ground state followed by quasifree ejection of the nucleons from the nucleus, and hence the appropriate single-nucleon form factors are the elastic ones. Second, the nucleons in the nuclear ground state are moving (Fermi motion) and accordingly the single-nucleon cross section used must take this into account. 3.2.2 Nuclear effects and dynamical parameters in the RFG model The RFG model can be taken as a guide for incorporating relativistic ingredients to more sophisticated models and, particularly, as a first approximation to the nuclear dynamics involved in the QE 50 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING Figure 3.2: Scaling function f(ψ′) as a function of ψ′for different nuclei (A≥12) and several kinematics covering from forward to very backward angles as well as from low to very high incident energies. The values of Acorresponding to different symbols are shown in the figure. Data taken from [36]. Nevertheless, the scaling behavior becomes particularly clear if one studies the experimental cross section separated into its longitudinal fL(ψ′) and transverse fT(ψ′) contributions, as shown in Fig. 3.3. The separate longitudinal and transverse contributions lead to the conclusion that the longitudinal experimental data superscale throughout the whole region of the QE peak, whereas the transverse data do not scale, being scaling violations more prominent in the region above the QE peak (ω > ωQEP,i.e.,ψ′>0). Scaling violations at high ωoccur because of other non-QE processes, such as meson production and resonance excitations, which are predominantly transverse, come into play. At very high transfer energies (i.,e. high ψ′-values) deep inelastic scattering starts to be relevant. Likewise, 2p-2h states induced by meson-exchange currents are known to have a relevant contribution in the “dip” region between the QE and the ∆peaks. As we detail in Chapter 4, 2p-2h MEC contributions are predominantly transverse for electromagnetic interactions. All these contributions beyond the IA are responsible for scaling violations, mainly observable in the transverse channel. Other mechanisms, not included in the present work, can introduce some effects in the general discussion. This is the case of RPA nucleon-nucleon correlation effects [30] that can modify the longitudinal and transverse responses because of the very different isospin character of the two channels. However, RPA is only relevant for very low energy/momentum transfers and, in overall, they are expected to be very small. The prescription adopted in the first version of the SuperScaling Approach has been to employ the experimental longitudinal responses to define a general scaling function for both electromagnetic longitudinal and transverse channels. This implies that both Land Tscaling functions are roughly the same after subtracting the non-scaling contributions related to processes beyond the QE regime, that is, zeroth kind scaling is fulfilled. As we shall illustrate, the most modern version 3.3. SUPERSCALING APPROACH:A SEMIPHENOMELOGICAL MODEL 51 of the model (SuSAv2) contains corrections to this assumption based on Relativistic Mean Field theory. Figure 3.3: Scaling function, fL(ψ′) and fT(ψ′), from the longitudinal and transverse response, respectively, as a function of ψ′for different nuclei (A≥12) and for different values of q(in MeV/c). Data taken from [82]. From the analysis of the separate longitudinal data shown in Fig. 3.3 (left panel), a “universal” phenomenological Lsuperscaling function fLhas been extracted. In Fig. 3.4, we show the data analysis. The solid curve refers to a fit of the longitudinal data given by the following parametrization, which corresponds to the SuSA scaling function, fSuSA(ψ′)≡fL(ψ′)=p1 [1 +p2 2(ψ′−p3)2](1 +ep4ψ′)(3.69) where p1=2.9883, p2=1.9438, p3=0.67310 and p4=−3.8538. The RFG scaling function is also shown for reference (dashed line). Note that fSuSA(ψ′) presents an asymmetric shape and a tail that extends towards positive values of ψ′,i.e.,ω > ωQEP. Its maximum reaches about ∼0.6. In contrast, the RFG scaling function is symmetric in the scaling variable ψ′, is limited strictly to the region −1≤ψ′≤+1 and has a maximum value of 3/4. This behavior clearly differs from data and, hence, the RFG does not reproduce the scaling behavior shown by the incusive electron scattering data. The previous superscaling behavior shown in Fig. 3.4 has been evaluated with other models based on the harmonic oscillation shell model (HO+FSI) and the relativistic Fermi gas (RFG+FSI) that incorporate different descriptions of FSI (see [83] for details). Some asymmetry in the corresponding scaling functions emerges from these calculations, mainly due to the FSI, but the theoretical predictions still differ significantly from the data. In subsequent sections we provide a detailed study of the longitudinal and transverse scaling function within the framework of the RMF theory, analyzind the specific role played by FSI and the relativistic nuclear dynamics. 52 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING -1 0 1 2 3 ψ/ 0 0,2 0,4 0,6 0,8 fL (ψ/) RFG SuSA Figure 3.4: Averaged experimental fSuSA(ψ)versus ψ′in the quasielastic region together with a phenomenological parameterization of the selected (e,e′) longitudinal scaling data obtained from the RLdata on [82]. The integral of the curve has been normalized to unity. The RFG scaling function is also shown as reference. . Moreover, in spite of the difficulty in analyzing the transverse scaling function, previous studies [84] based on the modeling of the QE longitudinal response and contributions from non-QE channels have provided some evidence that the scaling of zeroth kind is not fully satisfied by data. In particular, these studies have found fee′ T,exp >fee′ L,exp, a result that has motivated a deep study of the scaling behavior as addressed in the next section. 3.4 Extension of the Superscaling Approach from Relativistic Mean Field Theory: the SuSAv2 Model As defined before, the SuperScaling Approach (SuSA) is based on the scaling properties of the longitudinal response extracted from (e,e′) data to predict charged-current quasielastic (CCQE) neutrino- and antineutrino-nucleus cross sections [85]. Thus, SuSA is based on the hypothesis that the neutrino cross section scales as does the electron scattering cross section. This feature is observed in most of the models based on IA (see, for instance, [85–87]). Accordingly, the SuSA model uses the experimental scaling function fee′ L,exp as a universal scaling function and then builds the different nuclear responses by multiplying it by the corresponding single-nucleon responses. In most IA approaches, one finds that once the single-nucleon cross section is removed in defining the scaling functions, the longitudinal and transverse responses are basically the same. However, within the framework of the RMF theory, one finds that 0th-kind scaling is mildly broken for momentum transfers in the 1 GeV region being fT(ψ′)>fL(ψ′). Moreover, notice that the extraction of fee′ L,exp entails the analysis of the purely-vector longitudinal (e,e′) nuclear response, which combines isoscalar+isovector contributions. In contrast, CC neutrino-nucleus re- 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 53 actions involve only isovector couplings and are mainly dominated by purely transverse responses (TVV +TAA and T′ V A). Thus, one could question the validity of the SuperScaling Approach. This subject was studied in [88] by analyzing the scaling functions evaluated with the RMF model. There, it was found that, contrary to what one might expect, the (e,e′) longitudinal scaling function agrees with the total (νl,l−) one (which is mainly transverse) better than does the transverse scaling function from (e,e′). This result is explained by the different roles played by the isovector and isoscalar nucleon form factors in each process (see [88] for details). These previous studies have motivated a superscaling analysis based on the RMF theory in order to improve the analysis of neutrino reactions. Within the RMF model [89] the bound and scattered nucleon wave functions are solutions of the Dirac-Hartree equation in the presence of energy-independent real scalar (attractive) and vector (repulsive) potentials. Since the same relativistic potential is used to describe the initial and final nucleon states, the model is shown to preserve the continuity equation [90] (this is strictly true for the CC2 current operator [86]); hence the results are almost independent of the particular gauge selected [86, 87]. In the RMF model the nucleons are dynamically and strongly off-shell and, as a consequence, the cross section is not factorized into a spectral function and an elementary lepton-nucleus cross section. The RMF has achieved significant success in describing QE electron scattering data. On the one hand, its validity has been widely proved through comparisons with QE (e,e′) data (see [86] and Sect. 3.5.2). In this connection, an important result is that the model reproduces surprisingly well the magnitude and shape of fee′ L,exp,i.e., it yields an asymmetric longitudinal scaling function, with more strength in the high-ωtail, and with a maximum value (∼0.6) very close to the experimental one. On the other hand, the model predicts fee′ T>fee′ L. For instance, at q=500 MeV/c (1000 MeV/c) the transverse RMF scaling function at the maximum is 13% (20%) larger than the longitudinal one. This violation of zeroth-kind scaling was analyzed in [88], where it was shown that the origin of such an effect lies in the distortion of the lower components of the outgoing nucleon Dirac wave function by the FSI. However, the RMF model also presents some drawbacks. First, it predicts a moderate dependence of the scaling function on the transferred momentum q. For increasing values of qthe RMF model presents: i) a strong shift of the scaling functions to higher ωvalues, ii) too much enhancement of the area under the tail of the functions, and iii) correspondingly too severe a decrease in the maximum of the scaling functions. Despite these shortcomings, the scaling functions obtained within the RMF model reproduce reasonably well the experimental longitudinal scaling function in a wide kinematical region (see Section 3.5). Second, getting results with the RMF model is computationally very expensive, especially when the model is employed to predict neutrino cross sections where one has to fold in the flux distribution of the incident neutrino or to compute totally integrated cross sections. Hence in what follows, after correcting for the too strong q-dependence of the RMF model, we shall implement the main features of the model in a new version of the SuSA approach, called “SuSAv2”, that makes it possible to obtain numerical predictions to compare with data using fast codes, yet retaining the basic physics of the RMF. In summary, in this work we extend the original SuSA model by incorporating in its formalism information from the RMF model. Thus we build the new model in such a way that it reproduces the experimental longitudinal scaling function, produces fee′ T>fee′ L, takes into account the differences in the isoscalar/isovector scaling functions and avoids the problems of the RMF model in the region of high momentum transfer. 54 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 3.4.1 RMF and RPWIA scaling behavior In this section we present a systematic analysis of the scaling functions computed with the Relativistic Mean Field (RMF) and the Relativistic Plane Wave Impulse Approximation (RPWIA). Both models are based on the relativistic impulse approximation (RIA) and provide a fully relativistic description of the scattering process. The bound state Dirac-spinors are the same in both models and correspond to the solutions of the Dirac equation with scalar and vector potentials. The two prescriptions differ in the treatment of the final state: the RPWIA describes the outgoing nucleon as a relativistic plane wave while the RMF model accounts for the FSI between the outgoing nucleon and the residual nucleus using the same mean field as used for the bound nucleon. In what follows, we analyze the scaling functions involved in (e,e′), (ν, µ−) and (ν, µ+) reactions as functions of the momentum transfer q. Due to the large number of existing electron and neutrino experimental data for 12C, most of the calculations presented in this thesis correspond to this nucleus. The extension to other nuclear systems will be addressed in following chapters. We first split all different response functions by isolating the isoscalar (T=0) and isovector (T=1) contributions in electron scattering, and the vector and axial contributions for neutrino and antineutrino induced reactions: VV (vector-vector), AA (axial-axial), VA (vector-axial). This strategy will allow us to extract clear information on how the FSI affect the different sectors of the nuclear current. Furthermore, it will make it easier to explore the relationships between the different responses linked to (e,e′), (ν, µ−) and (ν, µ+) reactions. Assuming charge symmetry, the two channels, Land T, accesible in electron scattering (3.57) can be decomposed as a sum of isoscalar (T=0) and isovector (T=1) contributions. In terms of the scaling functions, the nuclear responses are defined as, Ree′ L,T(q,ω)=1 kFffT=1,ee′ L,T(ψ′)GT=1 L,T(q,ω) +fT=0,ee′ L,T(ψ′)GT=0 L,T(q,ω)g.(3.70) Similarly, the charge-changing muon-neutrino (antineutrino) responses (3.54) can be given as, RVV,ν(ν) L(q,ω)=1 kF fVV,ν(ν) L(ψ′)GVV L(q,ω) (3.71) RAA,ν(ν) CC (q,ω)=1 kF fAA,ν(ν) CC (ψ′)GAA CC (q,ω) (3.72) RAA,ν(ν) CL (q,ω)=1 kF fAA,ν(ν) CL (ψ′)GAA CL (q,ω) (3.73) RAA,ν(ν) LL (q,ω)=1 kF fAA,ν(ν) LL (ψ′)GAA LL (q,ω) (3.74) Rν(ν) T(q,ω)=1 kFffVV,ν(ν) T(ψ′)GVV T(q,ω) +fAA,ν(ν) T(ψ′)GAA T(q,ω)g(3.75) Rν(ν) T′(q,ω)=1 kF fV A,ν(ν) T′(ψ′)GV A T′(q,ω).(3.76) The GK’s terms in (3.70) and (3.71–3.76) are the single-nucleon responses defined in (3.55). Notice that the previous scaling functions fKfor neutrino induced reactions are purely isovector. 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 55 In the following we examine three basic features of the scaling functions in the RPWIA and RMF models: shape, position and height of the peak, and the integrals of the scaling functions over ψ′. 3.4.2 Shape of the scaling functions The goal here is to study the shape of all scaling functions. In Fig. 3.5 (Fig. 3.6), for different values of q, we present the transverse (longitudinal) RMF scaling functions normalized to the maximum value corresponding to a reference function, in this case fVV,ν T, and relocated so that the maximum is at ψ′=0. As already mentioned, the scaling variable ψ′depends on q,ωand Eshi f t. Thus, for each scaling function, Eshi f t is taken so that the maximum is located at ψ′=0. The results within the RPWIA model are presented in Fig. 3.7. We do not present results of fAA CC ,fAA CL ,fAA LL for neutrino and antineutrino scattering, and fT=0 T for electron scattering because they are very sensitive to small effects due to cancellations and/or to the smallness of the denominator (Gfunction) which appears in the definition of the scaling function (3.55). The first three are seen to be insignificant for neutrino reactions as it will be shown in Chapter 7, whereas the fourth does not enter in that case and is known to be a minor correction in the QE regime for electron scattering. The results from the RPWIA model show that all scaling functions have the same shape (see Fig. 3.7). This comment also applies to models based on nonrelativistic and semirelativistic descriptions (see [85,91]). -1 012 3 0 0.2 0.4 0.6 -1 012 3 0 0.2 0.4 0.6 fT vv - ν fT’ VA - ν fT AA - ν fT vv - aν fT’ VA aν fT AA - aν fT (T=1) - e -1 012 3 ψ’ 0 0.2 0.4 0.6 -1 012 3 ψ’ 0 0.2 0.4 0.6 q = 500 MeV/c q = 800 MeV/c q = 1100 MeV/c q = 1400 MeV/c Figure 3.5: Transverse RMF scaling functions normalized to the maximum value corresponding to an arbitrary reference function and relocated at ψ′=0 (see text for details). The convention used to label the different curves is as follows: “e” for electron-induced reactions and “ν” (“aν”) for neutrino- (antineutrino-) induced reactions. 56 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING -1 012 3 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 012 3 0 0.1 0.2 0.3 0.4 0.5 0.6 fL vv - ν fL vv - aν fL T=1 - e fL T=0 - e -1 012 3 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 012 3 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 q = 500 MeV/c q = 800 MeV/c q = 1100 MeV/c q = 1400 MeV/c Figure 3.6: As in Fig. 3.5, but now for the longitudinal RMF scaling functions. -1 012 3 0 0.2 0.4 0.6 0.8 -1 012 3 0 0.2 0.4 0.6 0.8 fL vv - ν fT vv - ν fT’ VA - ν fT AA - ν fL vv - aν fT vv - aν fT’ VA - aν fT AA - aν fL T=1 - e fT T=1 - e fL T=0 - e -1 012 3 ψ’ 0 0.2 0.4 0.6 0.8 -1 012 3 ψ’ 0 0.2 0.4 0.6 0.8 q = 500 MeV/c q = 800 MeV/c q = 1100 MeV/c q = 1400 MeV/c Figure 3.7: As in Fig. 3.5, but in this case the results correspond to RPWIA. Transverse and longitudinal sets are presented together. Within the RMF model, all transverse scaling functions approximately collapse in a single one. On the contrary, the longitudinal responses are grouped in two sets: one corresponding to the pure electron isovector and neutrino (antineutrino) VV-responses, i.e., f T=1,ee′ Land fVV,ν(ν) L, and the other to the isoscalar contribution for electrons, namely, fT=0,ee′ L. This result emerges for all q-values and tends to be rather general. It is also noticeable that the tail is higher and more extended for the transverse responses, whereas for the longitudinal ones it tends to go down faster. 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 57 It is worth observing that in all cases the RMF scaling functions display a much more pronounced asymmetric shape than the RPWIA ones, an effect related to the specific treatment of final state interactions. 3.4.3 Height and position of the peak of the scaling function In the top (bottom) panel in Fig. 3.8 the peak-height of the transverse (longitudinal) set of scaling functions is presented as function of q. The results correspond to RMF and RPWIA predictions. We observe that the peak-heights of the scaling functions within RPWIA are almost q-independent (and very close to RFG value of 3/4), while the RMF ones present a mild q-dependence in the transverse set and a somewhat stronger one for the longitudinal set. It is well known that FSI tend to decrease the peak-height of the responses putting the strength in the tails, especially at high energy loss. This is particularly true for the RMF approach [86, 92] and models based on the Relativistic Green Function (RGF) [93, 94]. Similar effects have also been observed within semirelativistic approaches [85,91]. 400 600 800 1000 1200 1400 1600 q (MeV/c) 0.6 0.65 0.7 0.75 0.8 peak height fT’ VA - ν fT vv - ν fT AA - ν fT’ VA - aν fT vv - aν fT AA - aν fT T=1 - e RPWIA RMF 400 600 800 1000 1200 1400 1600 q (MeV/c) 0.45 0.5 0.55 0.6 0.65 0.7 0.75 peak height fL vv - ν fL vv - aν fL T=1 - e fL T=0 - e RPWIA RMF Figure 3.8: Top panel: Peak height of the transverse set of scaling functions as a function of the transferred momentum q. The upper set of lines corresponds to the prediction within RPWIA (thin lines), while the lower set of lines has been obtained with the RMF model. Bottom panel: As for the top panel, but now for the longitudinal set of scaling functions. 58 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING More specifically, in Fig. 3.8, we see that the discrepancies between the RMF and RPWIA peak-height results average to ∼25% in the transverse set. On the other hand, those discrepancies are more strongly q-dependent in the longitudinal sector, reaching ∼30% (∼70%) in the lower (higher) q-region for the longitudinal isovector responses (blue lines). Finally, the difference between the isoscalar longitudinal (e,e′) scaling function produced by RMF and RPWIA (magenta dashed-dotted lines) is somewhat smaller: ∼20% (∼30%) for lower (higher) q. 400 600 800 1000 1200 1400 1600 q (MeV/c) 10 20 30 40 50 60 70 80 Eshift (MeV) fT’ VA - ν fT vv - ν fT AA - ν fT’ VA - aν fT vv - aν fT AA - aν fT T=1 - e RMF RPWIA 400 600 800 1000 1200 1400 1600 q (MeV/c) 10 20 30 40 50 60 70 80 Eshift (MeV) fL vv - ν fL vv - aν fL T=1 - e fL T=0 - e RMF RPWIA Figure 3.9: Top panel: Shift energy, Eshi f t, needed in order to have the corresponding scaling function peak located at ψ′=0, as function of q. Results for the transverse set of scaling functions. Bottom panel: As for the top panel, but now for the longitudinal set of scaling functions. In Fig. 3.9 we study the position of the peak of the transverse and longitudinal sets. To this scope we display the energy shift, Eshi f t, needed to place the peak of the scaling function at ψ′=0 as a function of q. In the top panel of Fig. 3.9 we see that for the RPWIA transverse scaling function, Eshi f t is almost q-independent, while the corresponding RMF shift increases almost linearly with the momentum transfer. This q-linear dependence of Eshi f t was already observed and discussed within the framework of a semirelativistic model based on the use of the Dirac-equation- based potential [91]. Approximately the same behavior is observed for the longitudinal set (bottom panel in Fig. 3.9), although in this case the RPWIA results are softly linearly dependent on q. It is also worth mentioning that the three transverse scaling functions linked to the same neutrino or antineutrino process, fVV T,fAA Tand fV A T′, collapse in a single line for RMF as for RPWIA. 3.4. THE SUSAV2MODEL:AN EXTENSION FROM THE RMF THEORY 59 From the analysis of Figs. 3.8 and 3.9 one may conclude that fT=1,ee′ Lpresents the same behavior (height and position) as fVV,ν(ν) L(blue lines). The differences between these three curves are approximately constant and arise from the differences in the bound states involved in the reaction: proton+neutron in (e,e′), neutron in (ν, µ−) and proton in (ν, µ+). The Coulomb-FSI, namely, the electromagnetic interaction between the struck nucleon and the residual nucleus, which plays a role when the outgoing nucleon is a proton, could also introduce a difference; however, we find that its effects are negligible and that the differences between, for instance, fVV,ν Land fVV,ν Lin RPWIA (where no Coulomb-FSI are involved) are almost the same as in RMF (see Figs. 3.8 and 3.9). The strong q-dependence of the RMF peak position, which keeps growing with the momentum transfer, is a shortcoming of the model, whose validity is questionable at very high q. Indeed for high qthe outgoing nucleon carries a large kinetic energy so the effects of FSI should be suppressed for such kinematics. In fact, it would be desirable that the RMF results tend to approach the RPWIA ones for increasing momentum transfer, i.e., the scaling functions should become more symmetric, and a saturation of the peak-height reduction and of the energy shift should be observed. That trend is consistent with the scaling arguments [36, 37, 86], i.e., the experimental evidence of a universal scaling function for increasing q. This is one of the motivations to use an alternative model if one aims to reproduce the experimental (e,e′) data at medium-to-high momentum transfers. A possible alternative for the behavior of the peak height, peak position and shape of the scaling functions is to implement the RMF model at low to intermediate-qand the RPWIA one for higher q-values, as it is proposed in following sections. 3.4.4 Sum rules In Fig. 3.10, the values of the integrals over ψ′of the different scaling functions within RMF model are presented versus q. These are given by Si(q)=Z∞ −∞ fi(ψ, q)dψ . (3.77) The integration limits, denoted by (−∞,+∞), extend in reality to the range allowed by the kinematics. The above integral in the case of the longitudinal (e,e′) scaling function was shown to coincide, apart from some minor discrepancies ascribed to the particular single-nucleon expressions considered and the influence of the nuclear scale introduced, with the results obtained using the standard expression for the Coulomb Sum Rule (see [95] for details)1. Hence in what follows we denote the functions Si(q) simply as sum rules. We see that all integrals of the transverse set are above unity and increase almost linearly with q. On the contrary, the integrals of fVV,ν(ν) Land fT=1,ee′ L(blue lines) are below unity and decrease with qup to q=1100 MeV/c. From q=900 MeV/c they begin to be stable around the value 0.7. Then, from q=1200 MeV/c to higher q-values the integrals start growing again. However, notice that in that q-region the result of the integrals is very sensitive to the behavior of the tail of these particular scaling functions (see Fig. 3.6). Finally, the values of the integral of the longitudinal isoscalar function, fT=0,ee′ L, is approximately constant and close to unity. The behavior of the integrals of the two longitudinal scaling functions for (e,e′) is consistent with the analysis of the Coulomb sum rule for these two models (see [95]). 1Coulomb sum rule (CSR) in inclusive electron scattering states that by integrating the longitudinal strength over the full range of ωat large q, one should get the total charge (number of protons) of the nucleus. 66 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING When displayed versus the scaling variable ψ′(Fig. 3.14), we notice that the non-blocked scaling function is exactly the same for all kinematics and is only limited for the range of ω-values compatible with the fixed qvalue. As also observed, the PB effects only have an impact in the region of low ψ′which is related to the lower values of ω. 00.5 11.5 22.5 3 0 0.1 0.2 0.3 0.4 0.5 f(ψ’) SuSA SuSAwo PB 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 -1 0 1 2 3 4 5 ψ’ 0 0.1 0.2 0.3 0.4 0.5 f(ψ’) -1 0 1 2 3 4 5 ψ’ 0 0.1 0.2 0.3 0.4 0.5 q=50 MeV q=100 MeV q=250 MeV q=500 MeV Figure 3.14: Superscaling function versus ψ′at different q-fixed values and evaluated for the SuSA model with (SuSA) and without (SuSAwoPB) Pauli blocking. . 0 0,1 0,2 0,3 0,4 ω (GeV) 0,1 0,2 0,3 0,4 0,5 fblocked = f(ψ) - f(Φ) f(Φ) = f(-ω,q) f(ψ) = f(ω,q) q = 570 MeV/c 0 0,1 0,2 0,3 0,4 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 q = 400 MeV/c 0 0,1 0,2 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 q = 250 MeV/c -0,05 00,05 0,1 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 q = 150 MeV/c 00,05 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 0,6 q = 100 MeV/c -0,025 00,025 ω (GeV) 0 0,1 0,2 0,3 0,4 0,5 0,6 q = 50 MeV/c Figure 3.15: Superscaling function versus ωat different q-fixed values and evaluated for the SuSA model with (fblocked) and without (f(ψ)) Pauli blocking. The mirror function (f(Φ)) is also shown as reference. 3.5. ANALYSIS OF THE SUSAV2MODEL 67 Next we show the effects of PB in the SuSA and SuSAv2 models when comparing with (e,e′) data on 12C. For this purpose, we display in Fig. 3.16 the SuSA results with and without PB and compare them with a few sets of data at the kinematics in which PB effects are significant, i.e., very low q. In the region of low-ω, the PB effects can be observed on the width and peak height of the cross sections. In general we conclude that the agreement between SuSA and data improves when PB is introduced. SuSA without PB (green-dashed) produces cross sections too wide, while SuSA with PB (brown) provides narrower cross sections in better agreement with data. This is particularly true in panels (1) and (2) in Fig. 3.16. The same comments apply to Fig. 3.17 where SuSAv2 with and without PB is compared with the same set of low-qdata. The lowest energy transfer data, corresponding to the excitation of resonant and collective states, cannot be described by any of the present models. 00.05 0.1 0.15 0 50000 100000 150000 200000 SuSA w PB SuSA w/o PB εi=400, θe=36º, q~238 00.05 0.1 0 20000 40000 60000 εi=280, θe=60º, q~260 0.1 0.2 0.3 0.4 0 500000 1000000 1500000 dσ/dΩ/dω (nb/sr/GeV) εi=1300, θe=11.9º, q~270 0 0.1 0.2 0 50000 100000 εi=480, θe=36º, q~285 00.05 0.1 0.15 ω (GeV/c) 0 10000 20000 30000 εi=320, θe=60º, q~296 0.1 0.2 ω (GeV/c) 0 200000 400000 600000 800000 εi=1300, θe=13.5º, q~305 (1) (2) (3) (4) (5) (6) Figure 3.16: SuSA with and without Pauli Blocking is compared with (e,e′) data. Eshi f t =10 MeV has been employed. Data taken from [100,101]. 68 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 00.05 0.1 0.15 0 50000 100000 150000 200000 SuSAv2 w PB SuSAv2 w/o PB εi=400, θe=36º, q~238 00.05 0.1 0 20000 40000 60000 εi=280, θe=60º, q~260 0.1 0.2 0.3 0.4 0 500000 1000000 1500000 dσ/dΩ/dω (nb/sr/GeV) εi=1300, θe=11.9º, q~270 0 0.1 0.2 0 50000 100000 εi=480, θe=36º, q~285 00.05 0.1 0.15 ω (GeV/c) 0 10000 20000 30000 εi=320, θe=60º, q~296 0.1 0.2 ω (GeV/c) 0 200000 400000 600000 800000 εi=1300, θe=13.5º, q~305 (1) (2) (3) (4) (5) (6) Figure 3.17: SuSAv2 with and without Pauli Blocking is compared with (e,e′) data. Data taken from [100,101]. A clear difference between SuSA and SuSAv2 (Figs. 3.16 and 3.17) is that the latter clearly overestimates the data in the region below and close to the peak whereas the SuSA model underestimates all data for medium and high q-values (Figs. 3.19-3.20). However, in all cases the maximum is placed at ω.50−60 MeV where, as discussed in Sect. 3.5.2, the validity of the models based on IA is questionable and no definitive conclusions can be drawn based on comparison of model and data in this ω-region. Nevertheless, the agreement of the SuSAv2 model with data in the low-energy range can be improved by determining more accurately the q0transition parameter between RMF and RPWIA contributions, as shown in Sect. 6.1, in such a way that the effects of FSI via the RMF model be more significant for lower kinematics, thus reducing the resulting cross section. The effect of Pauli Blocking is also of relevance for the analysis of CCQE neutrino cross sections, particularly at low kinematics, as will be shown in Chapter 7. 3.5. ANALYSIS OF THE SUSAV2MODEL 69 3.5.2 Analysis of the SuSAv2 model for (e,e′) reactions within the QE regime In this section we present a systematic comparison of inclusive 12C(e,e′) experimental cross sections and the predictions for the QE process within RMF, SuSA and SuSAv2 models. As mentioned, data correspond to the total inclusive cross section which includes contributions from several channels, mainly: QE scattering, inelastic scattering, many-nucleon emission, etc. Here we only focus on the QE process, whereas a more detailed analysis of the inclusive cross section including inelastic processes and 2p-2h MEC contributions will be addressed in Chapter 6. Therefore, one expects that the models do not reproduce the total inclusive experimental data corresponding to kinematical situations in which non-QE contributions play some role. Thus, the main interest of the systematic analysis presented in this section is the comparison between SuSAv2 predictions and those from the SuSA and RMF models. Full analyses of the inclusive (e,e′) cross section (including descriptions of QE and non-QE contributions) have been presented with some success in the past [84, 85] within the context of the semiphenomelogical SuSA. Our aim in the following is to complete the description of the inclusive process within the context of SuSAv2 model, via the inclusion of the inelastic spectrum and 2p-2h contributions. This will be detailed in the next chapters. In Figs. 3.18-3.20 we present the comparison of the (e,e′) experimental data and models. Due to the large amount of available data on 12C(e,e′) at different kinematics (see [100,101]) in these three figures we only show some representative examples. A more detailed analysis will be drawn in Chapter 7. Each figure is labeled by the incident electron energy, εi(in MeV), the scattering angle, θe, and the transferred momentum corresponding to the center of the quasielastic peak, q (in MeV/c). Pauli Blocking has been included in the SuSA and SuSAv2 models following the procedure described in [96,97] and detailed in Section 3.5.1. 00.05 0.1 0.15 0 50000 100000 150000 200000 dσ/dΩ/dω (nb/sr/GeV) εi=400, θe=36º, q~239 00.05 0.1 0 20000 40000 60000 εi=280, θe=60º, q~260 0 0.1 0.2 ω (GeV) 0 50000 100000 150000 dσ/dΩ/dω (nb/sr/GeV) εi=480, θe=36º, q~285 0 0.1 0.2 ω (GeV) 0 10000 20000 εi=361, θe=60º, q~333 (a) (b) (c) (d) Figure 3.18: Comparison of inclusive 12C(e,e′) cross sections and predictions of the RMF (red), SuSA (green-dashed) and SuSAv2 (brown) models (see text for details). Set of panels corresponding to low-qvalues. Data are taken from [100,101]. 70 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING 00.1 0.2 0 5000 10000 dσ/dΩ/dω (nb/sr/GeV) εi=440, θe=60º, q~401 0.2 0.4 0.6 0 2000 4000 6000 εi=961, θe=37.5º, q~585 0 0.2 0.4 0.6 0.8 ω (GeV) 0 10000 20000 30000 dσ/dΩ/dω (nb/sr/GeV) εi=2500, θe=15º, q~660 0.2 0.4 0.6 0.8 ω (GeV) 0 500 1000 1500 2000 εi=1299, θe=37.5º, q~792 (e) (f) (g) (h) Figure 3.19: Continuation of Fig. 3.18. Set of panels corresponding to medium-qvalues. Data are taken from [100,101]. 0.2 0.4 0.6 0.8 200 400 600 800 1000 dσ/dΩ/dω (nb/sr/GeV) εi=1501, θe=37.5º, q~917 0.2 0.4 0.6 0.8 0 200 400 600 εi=3595, θe=20º, q~1316 11.5 ω (GeV) 0 2 4 6 8 10 dσ/dΩ/dω (nb/sr/GeV) εi=4045, θe=30º, q~2247 2.2 2.4 2.6 2.8 3 ω (GeV) 0 0.1 0.2 0.3 εi=4045, θe=55º, q~3457 (i) (j) (k) (l) Figure 3.20: Continuation of Fig. 3.18. Set of panels corresponding to high-qvalues. Data are taken from [100,101]. 3.5. ANALYSIS OF THE SUSAV2MODEL 71 The panels in Figs. 3.18-3.20 are organized according to the value of the transferred momentum (at the center of the QE peak) in three sets: low-q(from q=238 to q=333 MeV/c) in Fig. 3.18, medium-q(from q=401 to q=792 MeV/c) in Fig. 3.19 and, high-q(from q=917 to q=3457 MeV/c) in Fig. 3.20. The only phenomenological parameters entering in the calculation are the Fermi momentum kFand the energy shift Eshi f t. For these we use kF=228 MeV/c (see [37]) in both SuSA and SuSAv2 models. A constant energy shift of 20 MeV is employed in SuSA [37] while a q-dependent function, the one described in Sect. 3.5, is used for Eshi f t in the SuSAv2 model. Preliminary values for the SuSAv2 transition parameters of q0=800 MeV/c and ω0=200 MeV have been applied. We begin commenting on the low-qpanels presented in Fig. 3.18. The main contributions to the cross section from non-QE processes such as inelastic processes contributions (∆-resonance) and MEC, are very small, even negligible, in this low-qregion. In spite of that, when the transferred energy is small (ω.50 −60 MeV) other processes such as collective effects contribute to the cross section making questionable the treatment of the scattering process in terms of IA-based models. This could explain, in part, the general disagreement between models and data in that ωregion in (a), (b) and (c) panels. Moreover, we can observe that the SuSA model seems to be shifted to high ωvalues with regard to the experimental data whereas the SuSAv2 model matches with the experimental QE peak position. At the same time, the SuSAv2 results are higher than the SuSA ones, which is mainly due to the natural enhancement on the RMF transverse scaling function. Some clarifications are called for regarding the RMF results in Fig. 3.18, where sharp resonances appear at very low ωvalues. These correspond to 1p-1h excitations with the phase shift of a given partial wave going through 90 degrees. With more complicated many-body descriptions these sharp features are smeared out. In summary, in order to test the goodness of the models in the kinematical situation of Fig. 3.18, one should focus on the study of the tails of the cross sections where large enough ω-values (ω&50 −60 MeV) are involved. There, one observes that SuSA predictions are clearly overshifted to high ω-values while RMF and SuSAv2 models fit the data reasonably well. In addition, as expected, SuSA results are systematically below SuSAv2 and RMF ones at the QEP. We now discuss the results for medium-qvalues presented in Fig. 3.19. First of all, one should mention that for the kinematics of this figure, in addition to the QE process, non-QE contributions are essential to describe the experimental cross sections. For instance, in panels (f), (g) and (h) the ∆-peak appears clearly defined at ωvalues above the QE peak. In panel (e) one sees that in the region around the center of the QE-peak, the RMF prediction is slightly above the SuSAv2 one, being closer to the experimental data. This is consistent with the behavior of the RMF scaling function studied in Sect. 3.4.3 (see Fig. 3.8), namely, the peak-height of the RMF scaling functions increases for decreasing q-values. If the main non-QE contributions are not included in the modeling it is hard to conclude which model is better to reproduce the purely QE cross section. However, it seems reasonable to conclude that SuSAv2 improves the agreement with data compared to SuSA. For instance, in the situation of panel (e), it would be needed that non-QE processes would contribute more than 20% around the QE-peak in order to SuSA fits the height of the data. A 20% fraction of the cross section linked to ∆-resonance and MEC contributions is probably too much for that kinematics. Similar comments and conclusions apply to the results in panel (d) of Fig. 3.18. 72 3. CHARGED-CURRENT QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING For q-values close to 650 MeV/c (panels (f) and (g)) RMF and SuSAv2 produce very similar results because of the way in which SuSAv2 has been defined (see Sect. 3.5). For higher q-values, q&792 MeV/c ((h) panel), SuSAv2 and RMF predictions begin to depart from each other. In particular, RMF results tend to shift the peak to higher ωvalues and to place more strength in the tail while SuSAv2 cross sections tend to be more symmetrical due to the increasing dominance of the RPWIA scaling behavior (see Sect. 3.5). This difference is more evident for higher q-values, as observed in panels (j)-(l) of Fig. 3.20. It is important to point out that for the kinematics presented in Fig. 3.20 the non-QE contributions are not only important but they become dominant in the cross sections. This is the case presented in panels (k) and (l) where the QE-peak is not even visible in the data. Next, we summarize the main conclusions from the present comparison of the SuSA and SuSAv2 models in the QE regime with the inclusive (e,e′) data: •Regarding the enhancement of the transverse response, RT, in SuSAv2 compared with SuSA: prior to the addition of non-QE contributions (which will be addressed in Chapter 6), the most clear indications that support the SuSAv2 assumptions arise from the comparison with data at kinematical situations in which non-QE effects are supposed to be small (panels (e) and (d) in Figs. 3.18 and 3.19, respectively). •Regarding the energy-shift analysis: within the SuSA model we have used a constant energy shift of 20 MeV/c. From the comparison with the low-qset of experimental data, Fig. 3.18, one concludes that 20 MeV is a too large shift. On the contrary, the comparison with the high-qset of data, Fig. 3.20, suggests that 20 MeV is probably too small. Then, one is led to conclude that a constant energy shift is not the best option to reproduce (e,e′) data. These results support the idea of introducing a q-dependent energy shift such as we made in the SuSAv2 model. The theoretical justification of this assumption was already discussed in Sect. 3.5. In Chapter 7 we will show that the SuSAv2 model describes more accurately the neutrino induced reactions than the semiphenomelogical SuSA aproach. This result mainly comes from the q-dependence on the energy shift, the enhancement on the transverse response via RMF prescriptions as well as the possibility of separating scaling functions into isoscalar and isovector contributions. Finally, we should add a remark concerning the RMF/RPWIA transition parameters, q0and ω0, appearing in Eq. 3.91. In the previous results (Figs. 3.18-3.20), we have applied a fixed value of these parameters which have shown a reasonable agreement with some experimental data at the QE peak. However, the transition between the RMF and RPWIA models depends on the particular kinematics involved, namely on the momentum transfer q. Accordingly, the transition parameter, q0, is expected to increase with qin such a way that the RMF contribution will be dominant at low kinematics where FSI effects are of high relevance, whereas the RPWIA description is needed at higher kinematics. Therefore we introduce a dependence on the momentum transfer qthat determines explicitly the relative RMF and RPWIA contributions at different kinematics. This analysis is carried out together with the extension of the SuSAv2 formalism to the inelastic spectrum and applied to the study of the (e,e′) data in Chapter 6. Chapter 4 2p-2h MEC contributions for electroweak reactions In this Chapter, we evaluate and discuss the impact of two-particle two-hole meson-exchange currents (2p-2h MEC) on electron- and neutrino-nucleus cross sections. The 2p-2h MEC responses are calculated within the RFG model in which a fully Lorentz and transitionally invariant calculation can be developed. In order to reduce the computational time, we make use of an accurate parametrization of these responses. 4.1 Introduction As shown in Fig. 4.1, the 2p-2h MEC process takes place when a weak or electromagnetic boson from the leptonic current is exchanged by a pair of nucleons (2-body current) leading to the emission of two nucleons from the primary vertex. These states, where two nucleons are promoted above the Fermi level leaving two holes inside the Fermi sea, are known to give a large contribution in the so-called “dip region”, corresponding to excitation energies lying between the quasielastic (QE) and ∆(1232) excitation peaks. These contributions are essential for a correct interpretation of Figure 4.1: Schematic view of the 2p-2h MEC process for lepton-nucleus interactions. current and forthcoming neutrino oscillation experiments, which strongly relies on our understanding of neutrino-nucleus scattering at intermediate energies (from 0.5 to 10 GeV) and in particular 73 74 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS of the nuclear-structure effects involved. At these kinematics, it has been proved that processes beyond the IA, in which MEC play a major role, give a significant positive contribution to the cross section which helps to account for the discrepancy observed in (e,e′) processes between theory and experiment in the “dip” region as well as for the discrepancies between some recent neutrino CCQE measurements (e.g., MiniBooNE, NOMAD, MINERνA, T2K) [20, 21,24,46,47,56]. Indeed, the inclusion of 2p-2h MEC contributions has allowed to explain these data without modifying any effective parameter (such as the axial mass MA) [30,31,39,102,103]. All this supports the need to consider mechanisms such as final-state interactions (included in our QE description via RMF theory), nuclear correlations or MEC, in particular through their contribution to multinucleon knock-out around and beyond the QE peak as suggested by explicit modeling [30,31,104,105]. As commented in Chapter 3, the 2p-2h MEC processes together with inelastic contributions and, in particular, meson production via baryon resonances such as the ∆, are responsible of scaling violations in (e,e′) reactions, which are more prominent in the transverse [84,85]. However, even below the meson production threshold there are scaling violations in the transverse response [36], one source of which is clearly the MEC contributions, again predominantly transverse. These two-body currents can excite both one-particle one-hole (1p-1h) and two-particle two-hole (2p- 2h) states. Most studies of electromagnetic (e,e′) processes performed for low-to-intermediate momentum transfers with MEC in the 1p-1h sector (see, e.g., [106–109]) have shown a small reduction of the total response at the QE peak, mainly due to diagrams involving the electroexcitation of the ∆resonance. Nevertheless, they are roughly compensated by the positive contributions of correlation diagrams, where the virtual boson couples to a correlated pair of nucleons. In this work we shall therefore neglect them and restrict our attention to 2p-2h final states, computed in a fully relativistic way. As discussed in previous works [105, 110–113], relativity is an essential ingredient in the analysis of 2p-2h processes at momentum transfers above 400-500 MeV/c. At these kinematics, the non-relativistic reduction can lower the resulting 2p-2h MEC cross section around a∼40% or more [105]. This statement is in connection with the kinematical regions of interest for neutrino oscillation experiments which extend to relativistic domains. At these q-values, the static approximation used for the ∆propagator in the non-relativistic calculations of 2p-2h transverse response function [114] fails to explain the “dip” region. Moreover, the presence of nucleon-nucleon correlation interactions involving the one-nucleon current may lead to the excitation of 2p-2h final states, and interference between these processes and those involving MEC should be analyzed to assess their relevance. These effects, taken into account in the RFG-based descriptions of 2p-2h provided by Nieves et al. [31] and Martini [30], are not included explicitly in our RFG MEC model, that relies on a hybrid description where the oneparticle emission already contains contributions of nuclear ejections due to nuclear correlations — via scaling functions from the SuSAv2 model. Explicit calculations of the correlation-MEC interference terms are still in progress and their contributions will be presented in further works. The contributions considered in this thesis englobe the 2p-2h states excited by the action of meson-exchange currents within a fully relativistic framework (see [110,112,113,115] for details), involving virtual ∆resonances as well as the seagull (contact) and pion-in-flight currents obtained in previous works [110, 111]. Deviations from the Fermi gas model 2p-2h responses produced by ingredients such as final-state interactions, finite nuclear effects or nuclear correlations are expected to be moderate, which would result in small corrections in the impulsive cross section as the MEC contributions are also moderate. The previous assumption is based on previous works at low-to-intermediate momentum transfers for 12C and 40Ca within the framework of the continuum shell model [114,116] as well as in other analyses [117–119]. 4.2. GENERAL FORMALISM 75 4.2 General formalism The evaluation of the 2p-2h MEC contributions is performed within an exact microscopic calculation, where the two-body current is the sum of seagull, pion-in-flight, pion-pole and ∆-pole operators and the basis wave functions are non-interacting Dirac spinors. The features of the RFG model allows for a fully Lorentz covariant calculation of the MEC. In Refs. [110, 113, 115] it has been calculated for the first time the fully relativistic weak (with vector and axial components) charged meson-exchange currents for neutrino-nucleus interaction in both longitudinal and transverse channels as well as a complete analysis for electromagnetic reactions. The numerical integration method is described in [105,113,120] where the 2p-2h MEC hadronic tensor is calculated considering two particles p′ 1and p′ 2above the Fermi momentum in the final state, p′ i>kF, and two holes h1and h2below the Fermi momentum, hi<kF, Wµν 2p−2h=V (2π)9Zd3p′ 1d3h1d3h2M4 E1E2E′ 1E′ 2 Θ(p′ 1,p′ 2,h1,h2) rµν(p′ 1,p′ 2,h1,h2)δ(E′ 1+E′ 2−E1−E2−ω),(4.1) where by momentum conservation, p′ 2=h1+h2+q−p′ 1and Eiand E′ iare the on-shell energies of the holes and particles. The only two parameters in the description are the Fermi momentum kF associated to the nuclear species and the separation energy, i.e.,Eshi f t. Pauli-blocking effects are also considered through the step function Θ(p′ 1,p′ 2,h1,h2)=θ(p′ 2−kF)θ(p′ 1−kF)θ(kF−h1)θ(kF−h2).(4.2) The elementary 2p-2h hadronic tensor rµν is defined in terms of the two-body MEC antisymmetrized matrix element jµ(1′,2′,1,2)A, rµν (p′ 1,p′ 2,h1,h2)=1 4X s1s2s′ 1s′ 2X t1t2t′ 1t′ 2 jµ(1′,2′,1,2)∗ Ajν(1′,2′,1,2)A.(4.3) The MEC operator is written as the sum of four contributions, seagull (a,b), pion-in-flight (c), pion-pole (d,e), and ∆pole (f–i), as shown in Fig. 4.2, jµ MEC =jµ sea +jµ π+jµ pole +jµ ∆.(4.4) The different contributions are characterized in terms of how the virtual boson is attached to the hadronic vertex. The seagull or contact terms are associated to the attachment of the virtual boson to the N Nπvertex whereas the pion-in-fligh operator is referred to the direct interaction of the boson with the virtual pion. At variance with the pion-in-flight current, the pion-pole terms has only the axial component and therefore it is absent in the electromagnetic case. This contribution could be considered as the “axial counterpart” of the pion-in-flight term, in the sense that it contains two pion propagators. MEC contributions involving the virtual ∆resonance (∆-pole terms) are also of relevance for the description. For completeness, it is worth pointing out that those diagrams that correspond to the excitation of a 2p2h+πstate, hence to pion production [121], are implicitly included in the phenomenological inelastic scaling function described in Chapter ??. The exact evaluation of the 2p-2h hadronic tensor (4.1) in a fully relativistic way involves numerical seven-dimensional integrations of a huge number of terms. This makes the computation, exactly performed in [105,110,113,120], highly non-trivial. In order to reduce the computational time as well as to ease the implementation of the results in Monte Carlo generators used in the 82 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS 050 100 150 200 0 0.01 0.02 0.03 0.04 R2p-2h MEC (MeV-1) CC CL LL T T’ LEM TEM TAA TVV 0 100 200 300 400 500 600 ω (MeV) -0.01 0 0.01 0.02 0.03 R2p-2h MEC (MeV-1) 0 200 400 600 800 1000 ω (MeV) -0.01 -0.005 0 0.005 0.01 R2p-2h MEC (MeV-1) Figure 4.9: Comparison between 2p-2h MEC transverse (T=TVV +TAA and T′=T′ V A) response functions and the longitudinal ones (CC,CL and LL) at q=200 MeV/c (top panel), q=600 MeV/c (mid panel) and q=1000 MeV/c (bottom panel). 4.4. 2P-2HMEC RESPONSES FOR (νl,l)REACTIONS 83 We can also notice a larger relevance of the weak longitudinal contributions with respect to the electromagnetic case. Nevertheless, when computing the total MEC neutrino cross section, the contribution of the CC and LL channels is roughly compensated by that of the negative CL response, so that for neutrino energies below ∼1 GeV the net longitudinal contribution plays a minor role in the total MEC response. This is illustrated in Fig. 4.10, where the L,Tand T′contributions to the 2p-2h MEC cross section are displayed versus the neutrino energy. At higher energies the Land T′contributions become comparable, both being much smaller that the dominant Tone. The balance between the longitudinal and transverse 2p-2h channel discussed above is somehow different from the one emerging in the electromagnetic case. As described in Section 4.3, the longitudinal electromagnetic MEC response is indeed negligible with regard to the transverse one. However, as illustrated in Fig. 4.10, we notice that when computing the total 2p-2h MEC weak cross section the longitudinal contribution is dominated by the axial channel and thus it plays a more relevant role compared with the purely-vector electromagnetic case. Concerning the transverse responses, it is noticeable that the magnitude of the pure axial and vector channels to the cross section is very similar. Moreover, the vector-axial interference contribution reaches its maximum around Eν∼1 GeV and decreases at higher energies as a consequence of the behavior of the leptonic factor VT′and the axial form factor GA(Q2), which vanish as |Q2|increases (see Chapter 2 for details). 00.5 11.5 22.5 Eν (GeV) 0 1 2 3 4 σMEC (10-39cm2) L T T’ L+T+T’ 00.5 11.5 22.5 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 σMEC (10-39cm2) LVV LAA TVV TAA T’VA Figure 4.10: Separation into components of the total 2p-2h MEC νµcross section displayed versus neutrino energy Eν. The total longitudinal (L), transverse (T) and transverse interference (T′) contributions are shown (left panel) as well as the net cotnribution (L+T+T′). Longitudinal and transverse channels are decomposed into vector and axial contributions (right panel). 0500 1000 1500 2000 ω (MeV) 0 0.01 0.02 0.03 RT (MeV-1) q: 200-2000 MeV/c (steps: 200 MeV/c) TAA TVV 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 RT’ VA (MeV-1) q: 200-2000 MeV/c (steps: 200 MeV/c) Figure 4.11: Comparison between 2p-2h MEC axial transverse (TAA) response functions and the vector ones (TVV ) versus ω(left panel). The transverse vector-axial interference term (T′ V A) is also shown (right panel). The curves are displayed from left to right in steps of q=200 MeV/c. 84 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS The analysis of the evolution with qof the individual transverse components (see Fig. 4.11) shows that the axial term is larger than the vector one at low-intermediate kinematics (q<800 MeV/c) whereas the opposite occurs at higher kinematics. The interference vector-axial response functions yield an intermediate result between the axial and vector responses. The present evaluation of the 2p-2h MEC responses and their corresponding fits also have the merit of covering a very wide qrange, including the tail of the responses at high ψ′and ωvalues. In general, (e,e′) data are rarely available when ω→qand hence the high-ωregion was ignored in some prescriptions. In contrast, for CCQE reactions one must integrate over a broad neutrino spectrum and hence, potentially, the high-ωregion may be relevant. This has motivated us to make an accurate description of the MEC contributions also for these kinematics. In Fig. 4.5, the analysis of the RM EC T,VV results versus ωshows a small contribution below q<300 MeV/c as well as the relevance of the tail in the response at q>800 MeV/c. On the other hand, the tail of the MEC responses at high q(q>1000 MeV/c) which appears at ω&1000 MeV does not contribute significantly to the cross section at lower kinematics, as can be deduced from Fig. 4.12. In fact, if we neglect the tail of the MEC response in our parametrization, no significant differences emerge except at neutrino energies above 1 GeV where the complete approach yields somewhat larger contributions, as seen in Fig. 4.13. It can also be deduced from Fig. 4.12 that no significant MEC contributions appear for q>2000 MeV/c, and the same is true for large ω > 1000 MeV, showing that the kinematics where the MEC give the largest contribution to the cross section corresponds to q∼500 −1500 MeV/c and ω∼250 −1000 MeV, as also stated in [115]. A more detailed analysis of the relevant kinematic regions for the entire 2p-2h MEC contribution will be shown in Chapter 7. Moreover, the comparison between the electromagnetic 2p-2h MEC calculations and the (e,e′) data will be addressed in Chapter 6 together with the QE and inelastic contributions. 0.1 110 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 1.2 σν (10-39cm2) MEC MEC, q>100 MeV/c MEC, q>250 MeV/c MEC, q>500 MeV/c MEC, q>1000 MeV/c MEC, q>2000MeV/c 0.1 110 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 1.2 σν (10-39cm2) MEC MEC, ω>50 MeV MEC, ω>100 MeV MEC, ω>250 MeV MEC, ω>500 MeV MEC, ω>1000 MeV MEC, ω>2000MeV Figure 4.12: Total 2p-2h MEC νµcross section for the TVV channel per target nucleon evaluated excluding all contributions coming from transferred momentum (upper panel) and energy (lower panel) below some selected values, as indicated in the figure. The main merit of the parametrization provided here for 2p-2h MEC electromagnetic and weak reactions on 12C is that it translates a sophisticated and computationally demanding microscopic calculation into a smooth parametrization which depends on the values of the transfer variables of the process, easing its implementation into Monte Carlo neutrino event simulations used in the analysis of experiments. In the following section, we study the extension of this parametrization to other nuclei. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 85 0.1 110 Eν (GeV) 0 0.2 0.4 0.6 0.8 1 1.2 σν (10-39cm2) MEC (TVV) no tail MEC (TVV) Figure 4.13: Comparison between the total 2p-2h MEC νµcross section per target nucleon for the TVV channel with and without considering the tail contribution in the responses. 4.5 Density dependence of 2p-2h meson-exchange currents In this section, we analyze the density dependence of the 2p-2h MEC contributions in leptonnucleus interactions in order to extend the above parametrization on 12C to other nuclei. This is connected to the growing interest in the extension to heavier nuclei, such as 16O, 40Ar, 56Fe and 208Pb, used in ongoing and future neutrino experiments. Therefore, an estimation of the density dependence of the 2p-2h MEC responses would be extremely useful to extrapolate the results from the current parametrization on 12C to other nuclei. In Chapter 3, the “superscaling” behavior of the inclusive electron scattering data was analyzed for various nuclei (see Figs. 3.2 and 3.3), showing that, below the quasielastic peak, these data were independent on the momentum transfer (scaling of first kind) and on the mass number (scaling of second kind). On the contrary, some scaling violation emerged for energy transfer around and above the QE peak, mainly in the transverse channel and ascribed to reaction mechanisms different from one-nucleon knockout, such as 2p-2h MEC contributions. This superscaling behavior was also reproduced through the SuSAv2 scaling functions arising from the RMF theory. More specifically, it is worth mentioning that the reduced QE cross section, i.e. the QE cross section divided by the appropriate single-nucleon one, scales as ∼A/kF,kFbeing the Fermi momentum. This can be deduced by simple inspection of the equations described in Sections 3.2-3.4. The Fermi momentum of most nuclei belongs to the range 200-300 MeV/c (see Table 3.1 for a detailed comparison). In what follows we explore the density dependence of the 2p-2h nuclear responses. Since the behavior with density of the nuclear response is not expected to depend very much on the specific channel or on the nature of the probe, for simplicity we focus on the electromagnetic 2p-2h transverse response, which largely dominates over the longitudinal one. Our starting point is therefore the electromagnetic transverse response for Z=Nnuclei, RMEC T, associated with meson-exchange currents carried by the pion and by the ∆-resonance, evaluated within the model of [110]. In a previous work [117] based on the non-relativistic Fermi gas, it has been suggested that the nuclear dependence of the 2p-2h MEC responses scales as Ak2 F. Thus, in order to test this scaling 86 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS rule within our relativistic approach, we remove the single-nucleon physics from the problem by defining the following reduced response (per nucleon) H FMEC T(q,ω)≡1 η2 F RMEC T(q,ω) Gee′ T(τ),(4.9) where the single-nucleon term Gee′ Twas previously defined in Section 3.2.3 and depends on the proton (GMp) and neutron (GMn) magnetic form factors. For simplicity here we neglect in the single-nucleon dividing factor small contributions coming from the motion of the nucleons (Wee′ 2∆ term), where the electric form factor contributes, which depend on the Fermi momentum (see Section 3.2.3). Note that the reduced response H FMEC Tis divided by η2 F=(kF/mN)2, together with the single-nucleon factor Gee′ Tin the search for the Ak2 Fscaling. We can also introduce a 2p-2h MEC scaling function, fMEC T, defined analogously to the transverse scaling function coming from the one-body response, fMEC T(q,ω)= RMEC T(q,ω) Gee′ T(τ)=η2 FkFH FMEC T.(4.10) The particular case of asymmetric nuclei, Z,Nrequires more involved formalism and will be addressed in future work, although preliminary studies indicate that the qualitative behavior with kFdoes not change significantly unless N−Zis very large. 4.5.1 Analysis of results In Fig. 4.14 we show the dependence on qof the above described 2p-2h MEC scaling function, fMEC T, for 12C. Unlike the QE case, where the scaling functions for the RFG and RMF models “collapse” into a single one for all qvalues, the MEC scaling functions depend largely on qand spread far beyond the QE region (−1≤ψ′≤+1 within the RFG model). This behavior is more pronounced for q: 400−600 MeV/c. As qincreases, the MEC scaling functions are more similar and their maxima tend to lower ψ′values, getting closer to the QE peak (ψ′=0). Furthermore, we can also notice that at very high momentum transfers the 2p-2h MEC contributions are very significant in the deep scaling region (large negative ψ′values), to the extent that they may even provide a contribution similar to the QE one at very low ω. Nevertheless, the QE and 2p-2h MEC contributions for low ωand high qare very small. In general, the observed trend with qof the 2p-2h MEC responses is consistent with the violation of first-kind scaling exhibited by the MEC in [111]. In Fig. 4.15 we display RMEC Tas a function of the energy transfer ωfor momentum transfers qranging from 200 to 2000 MeV/c and three values of the Fermi momentum kFfrom 200 to 300 MeV/c. Unlike the 1-body quasielastic case, it clearly appears that the 2p-2h response functions increase with kF,i.e. with the mass number of the nuclear species. This motivates the search for a second-kind scaling behavior in the 2p-2h MEC regime. In order to study the kF-dependence of the responses, we fix the momentum transfer to a specific value in Fig.4.16, where we show the response RMEC Tfor q=800 MeV/c (upper panels) and the same three values of kFused above. In the lower panels of Fig. 4.16 we display the scaled 2p-2h MEC response, H FMEC T, as a function of the MEC scaling variable ψ′ MEC (q,ω, kF) and of the quasielatic one ψ′≡ψ′ QE (q,ω, kF). The MEC scaling variable is defined in the Appendix C, in analogy with the usual QE scaling variable, to adjust the maximum of the 2p-2h results at ψ′ MEC ≈0. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 87 The results show that the reduced 2p-2h response roughly scales as k2 Fwhen represented as a function of ψ′ MEC,i.e., the scaled 2p-2h MEC responses shown blend at the peak into a universal result. This scaling law is very accurate at the peak of the 2p-2h response, while it is violated to some extent at large negative values of the scaling variable. These effects in the “deep scaling” region are reduced when considering the usual scaling variable ψ′ QE devised for quasielastic scattering. -1 012 3 4 5678 9 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 fT MEC (ψ’) 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 Figure 4.14: Transverse 2p-2h MEC scaling functions fMEC Tfor 12C versus the usual QE scaling variable ψ′ QE from q=400 MeV/c to 1500 MeV/c. 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 ω (GeV) 0 2 4 6 8 10 12 14 RT MEC (GeV-1) kF=200 MeV/c kF=250 MeV/c kF=300 MeV/c q: 200-2000 MeV/c Figure 4.15: The 2p-2h MEC response plotted versus ωfor three values of the Fermi momentum kFand for different values of the momentum transfer. The curves are displayed from left to right in steps of q=200 MeV/c. 88 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ω (GeV) 0 1 2 3 4 5 6 7 RT MEC (GeV-1) kF=200 MeV/c kF=250 MeV/c kF=300 MeV/c -2 0 2 4 6 8 10 ψ’QE 0 1 2 3 4 5 6 7 RT MEC (GeV-1) -2 -1 012 3 4 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ -2 0 2 4 6 8 10 ψ’QE 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ Figure 4.16: Upper panels: the 2p-2h MEC response plotted versus ω(left panel) and ψ′ QE (right panel) for q=800 MeV/c and Fermi momentum kFvarying between 200 and 300 MeV/c. Lower panels: the corresponding scaled 2p-2h MEC response H FMEC Tplotted versus the scaling variables ψ′ MEC (left panel) and ψ′ QE (right panel). In Fig. 4.17 the scaled 2p-2h MEC response is now plotted versus ψ′ MEC for four values of q and kF. In particular, we have considered the cases of 12C (kF=228 MeV/c) and 40Ca (kF=241 MeV/c). The first is clearly relevant for ongoing neutrino oscillation studies whereas the second is a symmetric nucleus lying close to the important case of 40Ar for upcoming neutrino experiments. Here we see that the same kF-dependence is valid for different values of qas long as Pauli blocking is not active, namely q>2kF. At lower qand in the deep scaling region this type of scaling is mildly broken. In particular, this corresponds to q.500 MeV/c and the lowest ωvalues. Nevertheless, this is not a relevant kinematic region for the analysis of 2p-2h MEC cross sections on neutrino reactions as shown in Fig. 4.12 ensuring an accurate extension of the 12C-parametrization to other nuclei. For completeness, we compare in Fig. 4.18 the scaled responses H FMEC Tfor different powers of ηFfor q=1000 MeV/c, which reinforces the idea of the Ak2 Fscaling behavior observed in Fig. 4.17 also for the same value of q. In particular, no scaling behavior is observed in terms of AkF(left panel) even though some agreement between different kFvalues appears in the deep scaling region. On the contrary, the analysis of the Ak3 Fdependence (right panel) only produces similar results for high ψ′ MEC values which corresponds to high energy transfers. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 89 -2 -1.5 -1 -0.5 00.5 11.5 ψ’MEC 0 1 2 3 4 FT MEC (GeV-1) ∫ q=500 MeV/c -2 -1 012 3 4 5 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ q=1000 MeV/c -2 -1 012 3 4 5678 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) ∫ q=1500 MeV/c -2 -1 012 3 4 5678 ψ’MEC 0 1 2 3 4 5 6 7 FT MEC (GeV-1) kF=200 MeV/c 12C (kF=228 MeV/c) 40Ca (kF=241 MeV/c) kF=300 MeV/c ∫ q=2000 MeV/c Figure 4.17: The scaled 2p-2h MEC response H FMEC Tplotted versus the scaling variable ψ′ MEC for different values of qvarying from 500 to 2000 MeV/c and Fermi momentum kFbetween 200 and 300 MeV/c. The particular cases of 12C and 40Ca are shown. -2 -1 012 3 4 5 ψ’MEC 0 0.5 1 1.5 2 2.5 RT MEC/GT/ηF (GeV-1) kF=200 MeV/c 12C (kF=228 MeV/c) 40Ca (kF=241 MeV/c) kF=300 MeV/c -2 -1 012 3 4 5 ψ’MEC 0 5 10 15 20 25 30 35 RT MEC/GT/ηF 3 (GeV-1) Figure 4.18: The scaled 2p-2h MEC response H FMEC Tfor different powers of ηn F≡(kF/mN)n,n=1 (left panel) and n=3 (right panel) plotted versus the scaling variable ψ′ MEC for q=1000 MeV/c and Fermi momentum kFvarying between 200 and 300 MeV/c. 90 4. 2P-2HMEC CONTRIBUTIONS FOR ELECTROWEAK REACTIONS After analyzing the k2 Fdependence of the MEC responses as well as the 1/kFdependence for the QE regime in terms of the superscaling behavior, we can draw out some conclusions about the relative importance of both contributions. Regarding the kFdependence for different nuclei shown in Table 3.1, we can deduce that for lighter nuclei, where kFis changing more rapidly with increasing A, the size of the MEC relative to the QE peak changes noticeably as Abecomes larger. As Aincreases toward heavier nuclei, the nuclear density saturates, causing kFto slowly approach the nuclear matter value of kF∼250 −260 MeV/c. This implies that for heavier nuclei all contributions will scale approximately as A. Therefore, while the relative MEC contribution will be largest for heavy nuclei, it changes most rapidly when comparing cross sections for light nuclei. This is also observed in the analysis of (e,e′) inclusive data shown in Chapter 6 for different nuclei. 4.5.2 Neutrino-Oxygen 2p-2h MEC responses Finally, we focus on the 2p-2h MEC responses for 16O, which is relevant for recent neutrino experiments [122]. In particular, in the T2K experiment the far detector may have different nuclear targets, mineral oil and water, and it is then crucial to understand how to extrapolate the results from one target to another. In Fig. 4.19 we analyze the 2p-2h MEC responses for neutrino interactions on 16O compared with 12C for two of the most relevant contributions (TVV and T′ V A), based on microscopic calculations by Amaro et al. [105]. As previously mentioned, the 2p-2h MEC responses are evaluated within the RFG model in a fully relativistic framework, for which the values of the Fermi momentum and energy shift used are, kF=228 MeV/c, Eshi f t =20 MeV for 12C and kF=230 MeV/c, Eshi f t =16 MeV for 16O. The kFand Eshi f t values corresponding to 16O are consistent with the analysis of electron scattering data, as detailed in Chapter 6. We observe in Fig. 4.19 (upper panels) that the results for 16O are larger than the 12C ones, which is mainly due to the different Anumbers as the kFvalues are very similar. At the same time, the 16O responses are slightly shifted to lower ωvalues with respect to the 12C ones as a consequence of the smaller Eshi f t for oxygen. Using now the previous result that the 2p-2h MEC contributions scale as Ak2 Fwe can multiply the 12C results (which are parametrized) by the ratio between the Anumbers and the k2 Fvalues for both nuclei, A(16O)k2 F(16O) A(12C)k2 F(12C)≈1.35 ,(4.11) in order to reproduce the 16O ones. This is presented in the lower panels of Fig. 4.19 for the TVV and T′ V A channels, showing the high degree of accuracy of the scaling law for extrapolating the 2p- 2h responses from carbon to oxygen for all values of qand leading to the conclusion that the Ak2 F dependence is also widely fulfilled by the 2p-2h weak reactions. Note also that the different energy shifts have been corrected by shifting the carbon results 8 MeV to the left.3. Similar comments also apply to all the remaining weak 2p-2h MEC responses. The comparison of our theoretical prescription with 16O (e,e′) and (νµ, µ−) data will be addresed in Chapters 6 and 7. 3Although the shift difference between 12C and 16O is, ∆Eshi f t =Eshi f t (12C) −Eshi f t (16O) =4 MeV, we have to double this value when considering two-nucleon emission. 4.5. DENSITY DEPENDENCE OF 2P-2H MESON-EXCHANGE CURRENTS 91 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 RTVV (MeV-1) 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 0.03 RT’ (MeV-1) 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 RTVV (MeV-1) 0500 1000 1500 2000 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 0.03 RT’ (MeV-1) Figure 4.19: 2p-2h MEC vector-vector transverse (TVV ) response and the axial-vector interference (T′ V A) one. Upper panels: Comparison between the results for 12C (dots) and 16O (solid). Bottom panels: comparison by re-scaling the 12C results with a factor 1.35 (see text). The curves are displayed from left to right in steps of q=200 MeV/c from q=200 MeV/c up to 2000 MeV/c. Summarizing, we have shown that the 2p-2h electromagnetic and weak MEC response functions roughly grow as Ak2 Ffor Fermi momentum varying from 200 to 300 MeV/c. This scaling law is excellent around the MEC peak for intermediate and high values of qwhereas it starts to break down around the Pauli-blocking region (q=2kF), where 2p-2h MEC do not contribute significanly to the neutrino cross sections. Compared to the behavior of the 1-body response, which scales as A/kF, the relative importance of the 2p-2h contribution grows as k3 F. This result allows one to get an estimate of the relevance of these contributions for a variety of nuclei, of interest in ongoing and future neutrino scattering experiments, and should facilitate the implementation of 2p-2h effects in Monte Carlo generators. 98 5. DEEP INELASTIC SCATTERING FORMALISM where, similarly to the QE regime, we can define: UL=UCC =U00 =κ2 τf1+τρ2W2(τ, ρ)−W1(τ, ρ)+W2(τ, ρ)D(κ,τ, ρ)g(5.37) ULL =U33 =λ2 κ2U00 (5.38) UCL =−1 2U03 +U30=−λκ κ2U00 (5.39) UT=U11 +U22 =2W1(τ, ρ)+W2(τ, ρ)D(κ, τ, ρ) (5.40) UT′=−i 2U12 −U21=±τ κ"1 2(ǫF+ǫ0)+λ ρ#W3(τ, ρ),(5.41) where Dis defined as: D(κ, τ, ρ)=1 ǫF−ǫ0(ρ)ZǫF ǫ0(ρ) dǫZ2π 0 dΦ 2πη×D κ2 =τ κ2(1 3fǫ2 F+ǫFǫ0(ρ)+ǫ0(ρ)2g+λǫF+ǫ0(ρ)+λ2)−(1 +τ) +(ρ−1) τ κ2λǫF+ǫ0(ρ)−τ(ρ+1) =ξF1−ψ2 X"1+ξFψ2 X−λ κψXqξF2+ξFψ2 X+τ 3κ2ξF1−ψ2 X#. (5.42) As previously discussed, and for a fixed value of the invariant mass µX, the RFG yields a scaling function fRFG(ψ′ X)=fRFG L(ψ′ X)=fRFG T(ψ′ X)=3 4(1 −ψ′2 X)θ(1 −ψ′2 X).(5.43) where ψ′ Xalso contains the corresponding energy shift, Eshi f t, as for the QE regime, ψ′ X≡1 √ξF λ′−τ′ρ′ q(1 +λ′ρ′)τ′+κpτ′(τ′ρ′2+1) ,(5.44) with ρ′defined as ρ′≡2H·Q Q2=1+1 4τ′µ′ X 2−1;µ′ X= W′ X MN =1 MNω′+Eh2−p2 X.(5.45) Then, we can indentify in Eq. (5.36), the term which englobes all the nuclear dependence of the interaction, fRFG, and replace it by the one arising from the SuSA or SuSAv2 models (fmodel), in a similar way as done for the QE regime in Chapter 3. Therefore, we calculate the inelastic nuclear responses for each specific model (RFG, SuSA or SuSAv2) as RK inel (κ, τ)=N η3 FκξFZµmax X µmin X dµXµXfmodel (ψ′ X)UK.(5.46) The previous formalism allows for a complete description of the inelastic cross sections on lepton-nucleus interactions where the nuclear effects are included by means of the SuSAv2 model. 5.3. INELASTIC STRUCTURE FUNCTIONS 99 The procedure is similar to that applied for the QE regime (see Chapter 3), keeping the same functional form for the RMF and RPWIA scaling functions (3.89) but using a different scaling variable (ψ′ X) and a different q0transition parameter for the blending function (3.91). The specific details about the application of the SuSAv2 model for the inelastic spectrum as well as the estimation of the q0transition parameter between the RMF and RPWIA prescriptions for both quasielastic and inelastic regimes will be presented in Chapter 6, where a detailed analysis of the existing (e,e′) data and the 2p-2h MEC contributions is also provided. 5.3 Inelastic structure functions In this Section, the properties of single-nucleon inelastic structure functions for protons and neutrons are analyzed. The specific details about the existing parametrizations and empirical models are detailed in Section 5.4. As defined in Section 5.2, the inelastic structure functions F1,F2and F3depend on two variables Q2and x, or, equivalently, τand ρ. To determine these inelastic functions for given Q2and x, measurements of the differential cross section at different scattering angles and incoming electron beam energies are needed. This will help in obtaining phenomenological fits of the single-nucleon inelastic structure functions for protons and neutrons. Specific information on the proton inelastic form factors can be obtained from the analysis of electron-proton (ep) and electron-deuteron (eD) scattering processes. On the contrary, the neutron case is much harder as no spedific electronneutron (en) processes are available. Thus, information on neutron inelastic form factors relies directly on the joint analysis of the two scattering processes mentioned above, namely, ep and eD. Experimentally it is observed that both F1and F2are almost independent of Q2in the limit of high Q2and ν. This is known as Bjorken x-scaling [123] F1,2(x,Q2)→F1,2(x).(5.47) Under the previous kinematical conditions, a relationship between F1and F2is given by the Callan- Gross relation [124] F2(x)=2xF1(x),(5.48) originally based on the parton model. Some violation effects on the Callan-Gross relation are related to second order QCD corrections [125,126], which appear at xapproaching zero, i.e. in the highly deep-inelastic region, as well as to mesonic contributions in the nuclear medium. The violation due to mesonic and other nuclear effects are shown to be noticeable only in the region of low xand Q2[127] where the inelastic structure functions are very reduced. Violation of Callan-Gross relation in nuclei is of current interest in ongoing experiments at JLab [128,129], where the measurements of F1and F2on nuclear targets will provide important information on this subject [130]. Both the Bjorken Scaling and the Callan-Gross relationship can be explained assuming that DIS is dominated by the scattering of a single virtual photon (boson) from a point-like quark within the proton (see Fig. 5.2). At very high kinematics where nucleon resonance structures are not substantially relevant, the parton distribution functions arising from pQCD (perturbative QCD) provide a proper representation of the inelastic structure functions. The range of validity of this approach will be discussed in Section 5.4.4. 100 5. DEEP INELASTIC SCATTERING FORMALISM Figure 5.2: DIS process for electromagnetic e-p reactions described in terms of inelastic structure functions (left panel) and of the quark-parton model (right panel). 5.3.1 Extension to the weak sector The description of the deep-inelastic regime for weak interactions implies the analysis of an additional structure function, F3(W3), related to the parity violating contribution associated to the V−Ainterference. An accurate determination of this weak function is hard to achieve from neutrino experiments as well as from parity-violating electron scattering [131, 132] due to the large uncertainties associated to the cross section measurements. Nevertheless, within the quark-parton model, we can establish a relationship among the electromagnetic and weak structure functions and between F2and F3[74, 133, 134]. This is based on the assumption that the corresponding structure functions Wican be written in terms of quark Qand antiquark Qdistributions [135,136] F2=νW2=Q+Q(5.49) F3=xνW3=Q − Q (5.50) and, hence, xνW3=νW2−2Q.(5.51) For electron scattering, the isoscalar F2structure function of the nucleon, defined as the average of the proton and neutron structure functions, is given (at leading order in αsand for three flavors) by FeN 2=1 2Fep 2+Fen 2=5x 18 u+u+d+d+x 9(s+s),(5.52) where u(u),d(d) and s(s) are the distributions for the up, down and strange quarks (antiquarks), respectively. The quark distributions are defined to be those in the proton and the factors 5/18 and 1/9 arise from the squares of the quark charges. For neutrino scattering, the corresponding F2 structure function is given by FνN 2=x(u+u+d+d+s+s),(5.53) where quark charges are not considered. In the moderate and large-xregion, where strange quarks are suppressed, the weak and electromagnetic F2structure functions approximately satisfy, FeN 2≈5x 18 u+u+d+d≈5 18 FνN 2.(5.54) Under this assumption, which has been analyzed in connection with experimental results [135, 137–139], one can readily obtain the weak structure functions from the existing parametrization of the electromagnetic structure functions and the antiquark distribution.1 1In this work, the inelastic cross sections are only calculated and compared with data for electromagnetic reactions. Their extension to the weak sector and the construction of the appropriate isoscalar and isovector contributions needed for CC and NC neutrino reactions will be accounted for in further works. 5.4. PARAMETRIZATION OF THE INELASTIC STRUCTURE FUNCTIONS 101 5.4 Parametrization of the inelastic structure functions In this section, we analyze two different parametrizations of the inelastic structure functions and a Parton Distribution Function (PDF) model, showing their capability to reproduce (e,e′) data at intermediate kinematics for the inelastic regime. 5.4.1 Bodek-Ritchie parametrization This approach is based on phenomenological fits of the single-nucleon inelastic structure functions W1and W2for ep and en, being the latter extracted from deuteron data. The Bodek-Ritchie parametrization is a general fit of Refs. [133, 134, 140, 141], which describes both the deep inelastic and resonance regions, covering the entire inelastic spectrum. The parametrization fits the SLAC data published in [140], covering a |Q2|range from 0.1 to 30 GeV2, and including scaling violations in terms of a modified scaling variable ωw. The structure function W2is described by νWep(en) 2(ν,Q2)=B(WX,Q2)g(ωw)ωw/ω , (5.55) g(ωw)= 7 X n=3 Cn(1 −1/ωw)n,(5.56) ωw=2MNν+a2 Q2+b2.(5.57) The modulating function B(WX,Q) contains 12 parameters representing the masses, widths, and amplitudes of the cross section for electroproduction of the four most prominent nucleon resonances, and eight parameters representing the WXdependence of the low-WXnonresonant contribution and single-pion production threshold. The modulating function is close to unity in the deep-inelastic region (WX>2 GeV). The rest of parameters are defined in [133] where aand b are the same for proton (Wep 2) and neutron (Wen 2) and the Cncoefficients are different for protons and neutrons. The Wep 1and Wen 1structure functions can be deduced from the Callan-Gross relation (5.48). The W3functions, related to V A interference, are obtained using the relation (5.51) with the antiquark distribution, Q, given by Q(x,Q2)=1 2 1−1 ωw!7 B(WX,Q2)g(0)ωw ω.(5.58) The different Wifunctions for protons and neutrons within the Bodek-Ritchie parametrization will be shown in 5.4.4 in comparison with other parametrizations. 5.4.2 Bosted-Christy parametrization Although the Bodek-Ritchie [133] parametrization has been widely used for the analysis of the highly-inelastic scattering region [142], in recent years new studies, both theoretical and experimental, of the nucleon structure functions in the resonance region have been performed, indicating the need for more sophisticated parametrizations. The kinematical region where nucleon resonances contribute is essential for the analysis of (e,e′) data and pion production results from inclusive neutrino measurements. Therefore, it is more convenient to employ more actual expressions for Wep(en) 1,2. In this sense, the Bosted-Christy (B-C) parametrization for the proton [143] and neutron [144] structure functions seems to reproduce better the electromagnetic behavior in the resonance region. 102 5. DEEP INELASTIC SCATTERING FORMALISM This approach is based on an empirical fit to describe the measurements of inclusive inelastic electron-proton and electron-deuteron cross sections in the kinematic range of four-momentum transfer 0 ≤ |Q2|<8 GeV2and final state invariant mass 1.1<WX<3.1 GeV, thus starting roughly from the pion production region to the highly-inelastic region. The fit is constrained by the high precision longitudinal and transverse (L/T) separated cross section measurements from JLab Hall C [145]. Compared to previous fits, it covers a wider kinematic range, fits both transverse and longitudinal cross sections, and features smooth transitions to the photoproduction data at Q2=0 and DIS data at high |Q2|and WX. At the same time, it provides an excellent description of the resonant structures seen in inclusive (e,e′) cross sections. As it will be shown in 5.4.4, its agreement with data in the region of the ∆-peak within the SuSAv2 model is better than using the previous Bodek-Ritchie parametrization. 5.4.3 PDF model: GRV98 The Glück-Reya-Vogt GRV98 model [146] employs effective leading order (LO) Parton Distribution Functions for quarks and antiquarks to get the inelastic structure functions F1,F2and F3. In this thesis, we use a recent update [147] designed for both inelastic neutrino- and electron-nucleon scattering cross sections. The model describes existing inelastic neutrino-nucleon scattering measurements and has been developed to analyze neutrino oscillation experiments in the few GeV region. The PDFs are extracted from global fits to various sets of deep inelastic scattering data at high energies and high |Q2|, where non-perturbative QCD effects are negligible. These effects are relevant for lower kinematic regions. In this sense, an scaling variable (ξw(x,Q2)) is employed to construct effective LO PDFs that account for the contributions from target mass corrections, non-perturbative QCD effects, and higher order QCD terms. The non-perturbative effects from spectator quarks are required to be considered for |Q2|<1 GeV2. These corrections together with effects from low energy scattering data are parametrized and included in the description. This model also accounts for nucleonic resonances and includes photoproducion data above the ∆(1232). In general, the model gives a reasonable average cross section in the resonance region and beyond as shown in [147] but it seems to be inadequate to describe inelastic (e,e′) data at lowintermediate kinematics, below |Q2|<1 GeV2. Regarding this, we show in Section 5.4.4 that the GRV98 prescription only matches the empirical Bosted-Christy and Bodek-Ritchie structure functions, based on fits of electron scattering data, at |Q2|&5 GeV2. 5.4.4 Comparison of the different parametrizations Next, we present a comparison of the different parametrizations for the single-nucleon inelastic structure functions described above, also confronting them with (e,e′) experimental data in the inelastic regime. The electromagnetic inelastic structure functions for protons and neutrons Wep(en) 1,2are displayed versus the Bjorken scaling variable xin Figure 5.3. The comparison is carried out for the Bodek-Ritchie and Bosted-Christy parametrizations as well as for the GRV98 PDFs prescription from low to high-Q2values. As observed, the GRV98 PDFs produces an average of the inelastic structure functions through the parton distribution functions whereas the Bodek-Ritchie and 5.4. PARAMETRIZATION OF THE INELASTIC STRUCTURE FUNCTIONS 103 Bosted-Christy parametrizations, based on phenomenological fits of the structure functions, reproduce in a more realistic way the resonance structures observed in ep and ed reactions. The three models appear to be similar at |Q2|above 5 GeV2for low xvalues where DIS processes dominate whereas some minor differences appear as xgets closer to 1, i.e., where resonance effects are more significant. On the contrary, the differences among these parametrizations increase when moving to lower |Q2|values where resonant contributions are more relevant in the entire x range, even at low xvalues. Indeed, at |Q2|=1 GeV2, the GRV98 is roughly an average of the empirical Bodek-Ritchie and Bosted-Christy fits but does not reproduce the resonant structures at all. The comparison at the lowest |Q2|values produces the most diverging picture between the empirical fits and the GRV98 PDFs model. Therefore, its applicability at energies of relevance for current neutrino experiments is rather questionable. Focusing on the empirical fits, we can observe small differences between the proton structure functions Wep 1,2for the Bodek-Ritchie and Bosted-Christy parametrizations (see Fig. 5.3), being the latter a bit larger at low |Q2|and high x. Concerning the neutron structure functions, Wen 1,2, they are significantly greater at lower |Q2|and large xfor the Bosted-Christy fit. Although, in general, the neutron structure functions remain below the proton ones, the opposite occurs for the Bosted- Christy parametrization at the lowest |Q2|values. All these differences between both parametrizations may be due to the more accurate analysis of the resonances included in the Bosted-Christy fit as well as to the different procedures to disentangle the neutron structure functions from the proton and deuteron ones. Similar comments also apply to Fig. 5.4, where the sum of proton and neutron contributions are shown for FeN 2and FeN 3within the Bosted-Christy and Bodek-Ritchie fits. The antiquark distribution (Q) obtained through the procedure described in Section 5.4.1 (see also Ref. [133]), is also displayed. As can be noticed, the antiquark distribution is more relevant at |Q2|<1 GeV2and low xvalues, thus implying that the main differences between the F3and F2functions appear at these kinematics. In Fig. 5.4, the ωdependence with xis also displayed, showing how ωincreases as xtends to lower values. Regarding the Bosted-Christy approach, some divergences appear at very low xfor higher |Q2|values. Nevertheless, they correspond to the highly-inelastic regime and extremely high ω, which, in general, corresponds to kinematically forbidden regions. The comparison of the three parametrizations with (e,e′) data is shown in Fig. 5.5 for two representative cases at intermediate kinematics where the QE and ∆peaks are easily recognizable. At the beginning of the highly-inelastic region, i.e. large ωvalues, where inelastic structure functions are more similar (see Fig. 5.3), the three models tend to be closer. It can be also noticed that the GRV98 PDFs overstimate the experimental data in the ∆-resonance region. At the same time, its contribution in the “dip” region between the QE and the ∆peaks is too large in such a way that the total result, after considering QE and 2p-2h MEC contributions, would clearly overestimate the data. On the contrary, the Bodek-Ritchie parametrization underestimates the ∆region as well as the beginning of the highly-inelastic regime which may be in connection with its poorer description of the resonance structures with respect to the Bosted-Christy one. As commented for Figs. 5.3 and 5.4, the larger Bosted-Christy structure functions result here in an increased inelastic cross section. At the same time, the accurate inclusion of resonances in this parametrization allows for a better description of the (e,e′) data in the ∆-peak region without contributing remarkably to the “dip” region. As can be deduced from Fig. 5.4, the differences between the empirical fits decrease as going deeper into the inelastic regime, i.e., larger ωvalues. 104 5. DEEP INELASTIC SCATTERING FORMALISM Therefore, on the basis of the previous analysis, the Bosted-Christy parametrization emerges as the most appropriate approach for the analysis of the inelastic regime. In this sense, the reference inelastic structure functions Fito be employed in our theoretical description are shown in Fig. 5.6. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x 0 0.1 0.2 0.3 0.4 0.5 0.6 Q2=1.0 GeV2/c2 0.2 0.4 0.6 0.8 1 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q2=5.0 GeV2/c2 Figure 5.3: Inelastic structure functions Wep 1(black lines), Wen 1(red lines), Wep 2(green lines), Wen 2(blue lines) in terms of xfor the Bodek-Ritchie parametrization (solid lines), Bosted-Christy parametrization (dot-dashed lines) and GRV98 PDFs (dashed lines) at Q2=0.14,1.0 and 5.0 GeV2. 5.4. PARAMETRIZATION OF THE INELASTIC STRUCTURE FUNCTIONS 105 0 0,1 0,2 0 0,1 0,2 0,3 0,4 0,5 0,6 νW2, xνW3 or Q xνW3 (B-R) νW2 (B-R) Q νW2 (B-C) xνW3 (B-C) 0 0,2 x 0 1 2 3 4 5 6 7 8 9 10 ω (GeV) ω vs x Q2=0.1 GeV2/c2, Ei= 2.0 GeV, q=0.6 GeV/c, ω=0.51 GeV 0 0,1 0,2 0,3 0,4 0,5 0,6 0 0,2 0,4 0,6 0,8 νW2, xνW3 or Q 0 0,2 0,4 0,6 x 0 1 2 3 4 5 6 7 8 9 10 ω (GeV) Q2=0.5 GeV2/c2, Ei= 2.0 GeV, q=1.0 GeV/c, ω=0.71 GeV 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0 0,1 0,2 0,3 0,4 0,5 0,6 νW2, xνW3 or Q 0 0,2 0,4 0,6 0,8 x 0 1 2 3 4 5 6 7 8 9 10 ω (GeV) Q2=1.0 GeV2/c2, Ei= 2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 νW2, xνW3 or Q 0 0,2 0,4 0,6 0,8 1 x 0 5 10 15 20 ω (GeV) Q2=5.0 GeV2/c2, Ei=10.0 GeV, q=5.0 GeV/c, ω=4.47 GeV 0 0,2 0,4 0,6 0,8 1 0 0,2 0,4 0,6 0,8 νW2, xνW3 or Q 0 0,2 0,4 0,6 0,8 1 x 0 20 40 60 80 100 120 140 ω (GeV) Q2=10.0 GeV2/c2, Ei=30.0 GeV, q=9.0 GeV/c, ω=8.43 GeV Figure 5.4: Comparison of the inelastic structure functions FeN 2=νWeN 2=νWen 2+νWep 2 and FeN 3=xνWeN 3=xνWen 3+xνWep 3for the Bodek-Ritchie (B-R) and Bosted-Chisty (B-C) parametrizations. The antiquark distribution, QeN =Qen +Qep, is also shown. The ωdependence with xis also displayed. 106 5. DEEP INELASTIC SCATTERING FORMALISM 0 0,1 0,2 0,3 0,4 0,5 ω (GeV) 0 500 1000 1500 2000 2500 Bosted-Christy Bodek GRV98 E=680 MeV, θ=60o, qQE=610 MeV/c 0,2 0,4 0,6 ω (GeV) 0 1000 2000 3000 4000 E=1108 MeV, θ=37.5o, qQE=674.6 MeV/c Figure 5.5: Comparison of inclusive 12C(e,e′) double differential cross sections and predictions for the inelastic regime of the Bodek-Ritchie parametrization (solid lines), Bosted-Christy parametrization (dot-dashed lines) and GRV98 PDFs (dashed lines) at different kinematics (incident electron beam and scattering angle) in terms of the energy transferred to the nucleus (ω). Experimental data taken from [100, 101]. The y-axis represents d2σ/dΩ/dωin nb/GeV/sr. The value of qat the QE peak (qQE) is shown as reference. 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x 0 0,2 0,4 0,6 0,8 2Mx(W1 p+W1 n) 2MxW1 n 2MxW1 p Q2=1.0 GeV2/c2, Ei=2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x 0 0,1 0,2 0,3 0,4 0,5 ν(W2 p+W2 n) νW2 n νW2 p Q2=1.0 GeV2/c2, Ei=2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x 0 0,1 0,2 0,3 0,4 0,5 xν(W3 p+W3 n) xνW3 n xνW3 p Q2=1.0 GeV2/c2, Ei=2.0 GeV, q=1.41 GeV/c, ω=1.0 GeV Figure 5.6: Reference inelastic structure functions F1=2M xW1(top panel), F2=νW2(mid panel) and F3=xνW3(bottom panel) for protons and neutrons and the sum of both contributions at Q2=1 GeV2. The calculations are performed within the Bosted-Christy parametrization. Chapter 6 Analysis of inclusive electron scattering within the SuSAv2-MEC model Although the focus in this PhD thesis concerns the study of CC neutrno-nucleus scattering processes, it is essential first to test our theoretical predictions against the available inclusive electron scattering data. This was already shown in Chapter 3 but restricting ourselves to the pure QE regime. In this chapter the SuSAv2 model is extended to include the complete inelastic spectrum — resonant, nonresonant and deep inelastic scattering — described in Chapter 5. We also consider the impact of 2p-2h meson-exchange currents following the procedure detailed in Chapter 4. The predictions of the full SuSAv2-MEC model are first compared with the existing inclusive 12C(e,e′) data and later also applied to other nuclei. The capability of the model to describe electron scattering data with accuracy gives us confidence in its subsequent extension, and validity, when applied to recent neutrino oscillation experiments. This subject will be addressed in Chapter ??. 6.1 SuSAv2 model for quasielastic and inelastic regimes In Chapter 3 a very detailed description of the SuSAv2 model was given, that, as known, incorporates the predictions from the RMF theory and a transition to the RPWIA model at high values of the momentum transfer. This transition between both RMF and RPWIA regimes is governed by a blending function whose explicit expression is also given in Chapter 3 (see Eqs. 3.89—3.91 for details). Here our interest is to extend the SuSAv2 model to the inelastic regime so its predictions can be compared with data covering the entire energy spectrum. This requires to have a good control of the transition parameters (q0,ω0) that determine the relative strength of the RMF and RPWIA responses, and how the transition between them evolves as the transfer momentum varies. Accordingly, the transition parameter, q0, is expected to increase with qin such a way that the RMF contribution will be dominant at low kinematics whereas the RPWIA one starts to be relevant at higher energies. Therefore we introduce a dependence of the parameter q0on the momentum transfer qthat determines the relative RMF and RPWIA contributions at different kinematics. Moreover, this transition occurs in a region of width ω0, which is fixed at 200 MeV. The particular procedure to determine the q0-behavior with qis in accordance to the best fit to a large amount of (e,e′) experimental data on 12C in a wide kinematical region, covering from low to high q-values (q: 239 −3432 MeV/c). For this analysis, 12C is employed as target reference due to the ample variety of existing data for electron scattering as well as its relevance for neutrino oscillation experiments. The method applied to determine the RMF/RPWIA transition in the SuSAv2 model in both QE and inelastic regimes is based on a reduced-χ2analysis of the data sets. 107 1146. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL Some comments concerning the “dip” region between the QE and the ∆peaks are also in order. This is the region where the QE and the inelastic contributions overlap the most and where FSI effects that modify in a significant way the tail of the QE curve at large ω-values can introduce an important impact. Moreover, the role of the 2p-2h MEC effects is essential because its maximum contribution occurs in this region. Thus, only a realistic calculation of these ingredients beyond the IA can describe successfully the behavior of the cross section. To conclude, the accordance between theory and data in the inelastic regime, where a wide variety of effects are taken into account, also gives us a great confidence in the reliability of our calculations. The inelastic part of the cross section is dominated by the ∆-peak that mainly contributes to the transverse response function. At low electron scattering angles the longitudinal QE response function dominates the cross section and the inelastic contribution is smaller (as will be shown in Section 6.2.4). The opposite holds at large scattering angles, where the ∆-peak contribution is important. On the other hand, for increasing values of the transferred momentum the peaks corresponding to the ∆and QE domains become closer, and their overlap increases significantly. This general behaviour is clearly shown by our predictions compared with data. In those kinematical situations where inelastic processes are expected to be important, our results for the QE peak are clearly below the data which is compensated by the larger inelastic contribution. On the contrary, when the inelastic contributions are expected to be small, the QE theoretical predictions get closer to data. Note also the excellent agreement in some situations (bottom panels on Fig. 6.4) even being aware of the limitations and particular difficulties in order to obtain phenomenological fits of the inelastic structure functions, and the reduced cross sections at these kinematics. 6.2.2 Sensitivity of the model It is important to point out the novelties introduced in this PhD thesis compared with some previous studies. With regards to the original “SuSA” results shown in [84], that were based only on the superscaling function extracted from the analysis of the longitudinal (e,e′) data and assuming the transverse function to be equal (scaling of zeroth kind), in the SuSAv2 approach the enhancement in the transverse channel introduced by the RMF model is incorporated. Moreover, the role of FSI is carefully examined by making use of the evolution of the scaling function from the RMF responses to the RPWIA ones as the momentum transfer goes up. This explains why the present analysis provides a much more accurate description of the data. Notice that the SuSAv2 model makes both QE and inelastic results higher than the SuSA ones. A similar outcome can be also observed in Sections 3.5.2 and 3.5.1 (see also [153]) where the study was restricted to the QE region and a fixed value of q0that can be appropriate for the specific kinematics considered was used. On the contrary, here the aim is to provide a model capable of reproducing (e,e′) cross sections for a very wide selection of kinematics and including in each case the whole energy spectrum. This is consistent with the q-dependence shown by q0in both regimes, QE and inelastic. We have also tested the sensitivity of our results to different choices in the values of ω0,q0and Eshi f t for two representative kinematical situations (see Fig. 6.5). Concerning the ω0parameter, a variation of ±100 MeV leads to negligible effects, hence the value of χ2is basically the same (upper panels in Fig. 6.5). In the case of q0and Eshi f t, variations of the order of ±100 MeV/c (in q0) and ±5 MeV (Eshi f t) lead to differences within ∼20% on χ2, but still providing a very good representation of the data (see results presented in the middle and bottom panels of Fig. 6.5). Note however that q0is a dynamical parameter running with q, whereas the value of Eshi f t is determined by the right location of the maxima in the scaling functions. Hence a significant variation of these three values does not imply a worsening in the agreement with data. 6.2. ANALYSIS OF (e,e′)EXPERIMENTAL DATA 115 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 3000 ω0+100 MeV, χ2 red=4.15 ω0-100 MeV, χ2 red=4.33 χ2 red=4.15 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 10000 20000 30000 40000 50000 60000 70000 ω0+100 MeV, χ2 red=12.22 ω0-100 MeV, χ2 red=12.23 χ2 red=12.22 E=1930 MeV, θ=16o, qQE=536.3 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 3000 q0+100 MeV/c, χ2 red=4.59 q0-100 MeV/c, χ2 red=4.25 χ2 red=4.15 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 10000 20000 30000 40000 50000 60000 70000 q0+100 MeV/c, χ2 red=10.98 q0-100 MeV/c, χ2 red=14.30 χ2 red=12.22 E=1930 MeV, θ=16o, qQE=536.3 MeV/c 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 500 1000 1500 2000 2500 3000 Eshift-5 MeV, χ2 red=7.37 Eshift+5 MeV, χ2 red=3.44 χ2 red=4.15 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 10000 20000 30000 40000 50000 60000 70000 Eshift-5 MeV, χ2 red=16.89 Eshift+5 MeV, χ2 red=8.60 χ2 red=12.22 E=1930 MeV, θ=16o, qQE=536.3 MeV/c Figure 6.5: Comparison of inclusive 12C(e,e′) cross sections and predictions of the QE-SuSAv2 model (long-dashed red line), 2p-2h MEC model (dot-dashed brown line) and inelastic-SuSAv2 model (long dot-dashed orange line). The sum of the three contributions is represented with a solid blue line. It is also shown the total contribution by shifting ω0(top panels), q0(middle panels) and Eshi f t (bottom panels). The y-axis represents d2σ/dΩ/dωin nb/GeV/sr, whereas the x-axis represents ωin GeV. 1166. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL 6.2.3 Relevance of the RMF/RPWIA effects Next, we discuss the relevance of the RMF and RPWIA approaches in the SuSAv2 model. Whereas the RMF provides an excellent description of the experimental longitudinal scaling function extracted from data taken at intermediate q-values, producing the required asymmetry and the enhancement of the transverse response, the RPWIA approach yields much more suitable results at higher values of the momentum transfer where FSI effects are significantly reduced. In Fig. 6.6 we present the cross sections for a set of kinematical situations showing the isolated contributions emerging from the two models in the case of the QE regime. Notice that we consider the effects introduced by the blending function and the q0parameters in the RMF and RPWIA results. The percentage of the two contributions is given in each panel. As shown, for those kinematics that correspond to the lower values of qQE (top panels) the RMF response contributes the most. As qQE increases, the RPWIA contribution becomes relatively more important, approaching the RMF one (see panels in the middle). Finally, for the higher qQE-values (bottom panels) the behavior reverses with the RPWIA result being the main one responsible for the QE response. 00.05 0.1 0 10000 20000 30000 40000 50000 RMF (86.3 %) RPWIA (13.7 %) 00.05 0.1 E=280 MeV, θ=60o, qQE=263.739 MeV/c 0 0.1 0.2 0.3 0.4 0 10000 20000 30000 40000 50000 RMF (76.7 %) RPWIA (23.3 %) 0 0.1 0.2 0.3 0.4 E=620 MeV, θ=36o, qQE=367 MeV/c 0 0.1 0.2 0.3 0 2000 4000 6000 8000 RMF (68.8 %) RPWIA (32.2 %) 0 0.1 0.2 0.3 E=500 MeV, θ=60o, qQE=456.6 MeV/c 00.05 0.1 0.15 0.2 0.25 0 500 1000 1500 2000 2500 3000 RMF (66.3 %) RPWIA (33.7 %) 00.05 0.1 0.15 0.2 0.25 E=400 MeV, θ=90o, qQE=489 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 RMF (52.6 %) RPWIA (47.4 %) 0 0.1 0.2 0.3 0.4 0.5 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.2 0.4 0.6 0.8 0 5000 10000 15000 20000 25000 30000 RMF (45.7 %) RPWIA (54.3 %) 0 0.2 0.4 0.6 0.8 E=2500 MeV, θ=15o, qQE=658.6 MeV/c 0.2 0.3 0.4 0.5 ω (GeV) 0 100 200 300 400 500 RMF (44.2 %) RPWIA (55.8 %) E=560 MeV, θ=145o, qQE=795 MeV/c 0.2 0.4 0.6 0.8 ω (GeV) 0 100 200 300 400 500 600 RMF (34.8 %) RPWIA (65.2 %) 0.2 0.4 0.6 0.8 E=3595 MeV, θ=20o, qQE=1316 MeV/c 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω (GeV) 0 10 20 30 40 50 RMF (30.7 %) RPWIA (69.3 %) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E=4045 MeV, θ=30o, qQE=2247 MeV/c Figure 6.6: Comparison of RMF and RPWIA contributions in the QE regime. Also shown for reference the predictions of the total QE-SuSAv2 model (long-dashed red line) and the total inclusive contribution (solid blue line). The y-axis represents d2σ/dΩ/dωin nb/GeV/sr. To make clearer how both RMF and RPWIA approaches contribute within the SuSAv2 model, in Fig. 6.7 we present the specific percentages ascribed to the two contributions and how they vary with qQE. The main variation in the two cases is produced in the region of intermediate 6.2. ANALYSIS OF (e,e′)EXPERIMENTAL DATA 117 qQE-values, namely, 250 .qQE .700 MeV/c. Here, the relative RMF contribution quickly diminishes as qQE increases whereas the opposite occurs for the RPWIA. Note that at qQE ∼700 MeV/c both models produce basically the same answer (∼50%) crossing each other, whereas for qQE .500 MeV/c RPWIA gives a very minor contribution, that is, FSI are essential to describe data at these kinematics. Finally, at higher qQE the RPWIA increases slowly, whereas the RMF decreases, although in both cases some kind of saturation seems to emerge approaching the RPWIA percentage to ∼60 −70% (∼30 −40% for the RMF). Although not presented here for simplicity, similar conclusions arise for the RMF and RPWIA contributions in the inelastic regime. 0500 1000 1500 2000 2500 3000 qQE (MeV/c) 0 20 40 60 80 100 % RPWIA % RMF Figure 6.7: Comparison of percentages corresponding to the RMF and RPWIA contributions in the QE regime as a function of qQE. 6.2.4 Separate L/T analysis The separate analysis of the longitudinal and transverse response functions of 12C is presented in Fig. 6.8. We compare our predictions with data taken from Jourdan [82] based on a Rosenbluth separation of the (e,e′) world data. In each case we isolate the contributions corresponding to the QE, inelastic and 2p-2h MEC sectors. Three kinematical situations corresponding to fixed values of the momentum transfer have been considered in Fig. 6.8: q=300 MeV/c (top panel), 380 MeV/c (middle) and 570 MeV/c (bottom). As observed, the longitudinal channel is totally dominated by the QE contribution. Only at very large values of ωdoes the inelastic process enter giving rise to a minor response, whereas the effects due to 2p-2h MEC are negligible. This result is in accordance with previous work [82,93,151,154], and it clearly shows that the longitudinal response is basically due to the IA. On the contrary, the transverse sector shows an important sensitivity to MEC and inelastic processes. Note that the inelastic transverse response gives rise to the high tail shown by data at large ω-values, whereas the 2p-2h MEC can modify significantly the transverse response in the dip region as well as in the maximum of the QE peak. It is also worth mentioning that the natural enhancement in the transverse response arising from the RMF model is necessary in order to reproduce the separate L/T data, rejecting the idea of 0-th kind scaling. 1186. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL 050 100 150 200 250 300 0 0.01 0.02 0.03 0.04 R (MeV-1) L T 050 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 R (MeV-1) 050 100 150 200 250 300 350 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 R (MeV-1) Figure 6.8: Analysis of the longitudinal (solid lines) and transverse responses (dashed lines) in (e,e′) scaterring at q=300 MeV/c (top panel), q=380 MeV/c (middle panel) and q=570 MeV/c (bottom panel). QE, MEC and inelastic contributions are shown, respectively, as green, blue and orange lines. The total response is shown by the black lines. Data taken from [82]. 6.3. EXTENSION OF THE SUSAV2-MEC MODEL TO OTHER NUCLEI 119 From results in Fig. 6.8 we observe that the model leads to a reasonable agreement with data in both channels, although some discrepancies also emerge. Notice that the longitudinal prediction at q=300 MeV/c (q=380 MeV/c) overstimates data by ∼12% (∼15%). It is important to point out that SuSAv2 is based on the existence of the scaling phenomenon for (e,e′) data, and this is completely fulfilled when the value of qis large enough (q≥400 MeV/c). Therefore, the extension of the superscaling approach to low q-values is not well established even though a good agreement at low kinematics has been achieved in the previous section. Furthermore, the minor discrepancies observed may also be due to the specific Rosenbluth separation method used in [82], which introduces some level of model dependence through y-scaling assumptions and the treatment of radiative corrections. 6.3 Extension of the SuSAv2-MEC model to other nuclei Once analyzed the capability of the SuSAv2 model to reproduce the 12C(e,e′) data, we extend the previous formalism to the analysis of electron scattering data on other nuclei. For this purpose, no differences in the scaling functions are assumed for the different nuclei except for the values used for the Fermi momentum and energy shift (see Table 3.1 for details). The use of the same scaling functions for different nuclear systems is consistent with the property of scaling of second type, i.e, independence of the scaling function with the nucleus, and it also follows from the theoretical predictions provided by the RMF and RPWIA models on which SuSAv2 relies. This has been studied in detail in previous works (see [32, 86–88, 155]) where the electromagnetic and weak scaling functions evaluated with the RMF and RPWIA approaches have been compared for 12C, 16O and 40Ca. In Figure 6.9 we compare the general RMF scaling functions for these nuclei, which exhibit no remarkable differences in terms of the nuclear species. In this sense, we apply the reference scaling functions for 12C to the analysis of QE and inelastic regimes in other nuclei. The extension of the 2p-2h MEC contributions to other targets was previously described in Section 4.5. -2 -1 012 3 4 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 f(ψ’) 40Ca 16O 12C Figure 6.9: Analysis of second kind scaling within the RMF model for 12C, 16O and 40Ca. In the case of 16O, the kFand Eshi f t-values selected (kF=230 MeV/c, Eshi f t =16 MeV/c) are also consistent with the general trend observed in [37], i.e., an increase of the Fermi momentum 1206. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL with the nuclear density. This is at variance with some previous works [85–87,91] where 16O was described by using kF=216 MeV/c and Eshi f t =25 MeV. Although both sets of values lead to small differences in the cross sections, the present choice does provide a more consistent analysis of the superscaling behavior in the deep scaling region, and more importantly, it also improves the comparison with electron scattering data. In particular, the analysis of the scaling behavior of the 16O(e,e′) data in the deep scaling region below the QE peak (ψ′=0), where no scaling violations are expected, leads to the conclusion that the present choice of kFand Eshi f t-values works better when comparing with other nuclei (see Figure 6.10). -2 -1.5 -1 -0.5 00.5 11.5 2 ψ’ 0.01 0.1 1 f(ψ’) 4He, Ei=3595 MeV, θe=16o 12C, Ei=3595 MeV, θe=16o 27Al, Ei=3595 MeV, θe=16o 56Fe, Ei=3595 MeV, θe=16o 4He, Ei=730 MeV, θe=37.1o 12C, Ei=730 MeV, θe=37.1o 16O (kF=216 MeV/c), Ei=730 MeV, θe=37.1o 16O (kF=230 MeV/c), Ei=730 MeV, θe=37.1o Figure 6.10: Experimental scaling data for various nuclei and for different values of the incident energy (Ei) and scattering angle (θe). The 16O data are show for two different values of kF. The kF-values for other nuclei are described in Table 3.1. In accordance with the previous analysis, we show in Fig. 6.11 the predictions of the SuSAv2- MEC model for six different kinematical situations, corresponding to the available (e,e′) data on 16O. In all the cases we present the separate contributions for the QE, 2p-2h MEC and inelastic regimes. The 2p-2h MEC responses are extrapolated from the exact calculation performed for 12C assuming the scaling law R2p2h∼Ak2 Fdeduced in Ref. [156] as well as in Section 4.5. The inclusive cross sections are given versus the transferred energy (ω), and each panel corresponds to fixed values of the incident electron energy (Ei) and the scattering angle (θ). Whereas the latter is fixed to 320[157] except for one case (center panel on the top, i.e, θ=37.10) [158], the electron energy values run from 700 MeV (left-top panel), where the QE peak dominates, to 1500 MeV (right-bottom) with the inelastic channel giving a very significant contribution. This is due to the values of the transferred momentum qinvolved in each situation. Although qis not fixed in each of the panels, i.e., it varies as ωalso varies, the range of q-values allowed by the kinematics increases very significantly as the electron energy grows up (for fixed scattering angles). Thus, for higher Ei the two regimes, QE and inelastic, overlap strongly, the inelastic processes being responsible for the large cross sections at increasing values of ω. This different range of q-values spanned in each panel also explains the relative role played by the RMF versus the RPWIA approaches. 6.3. EXTENSION OF THE SUSAV2-MEC MODEL TO OTHER NUCLEI 121 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 10000 20000 30000 40000 50000 60000 70000 80000 QE 2p-2h MEC Inelastic Total 16O, Ei=700MeV, θ=32o 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 5000 10000 15000 20000 25000 30000 16O, Ei=737MeV, θ=37.1o 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 10000 20000 30000 40000 16O, Ei=880MeV, θ=32o 0 0.2 0.4 0.6 0.8 ω (GeV) 0 5000 10000 15000 16O, Ei=1080MeV, θ=32o 0 0.2 0.4 0.6 0.8 ω (GeV) 0 2000 4000 6000 8000 10000 16O, Ei=1200MeV, θ=32o 0 0.2 0.4 0.6 0.8 1 ω (GeV) 0 1000 2000 3000 4000 16O, Ei=1500MeV, θ=32o Figure 6.11: Comparison of inclusive 16O(e,e′) cross sections and predictions of the SuSAv2- MEC model. The separate contributions of the pure QE response (dashed line), the 2p-2h MEC (dot-dashed), inelastic (double-dot dashed) are displayed. The sum of the three contributions is represented with a solid blue line. The yaxis represents d2σ/dΩ/dωin nb/GeV/sr. Data from Refs. [157] and [158]. Although not shown in the figure for simplicity, whereas the RMF response dominates at lower Ei-values (panels from left to right on the top), the reverse occurs, that is, the scaling function is essentially given by the RPWIA prediction, as Eiincreases together with q(panels on the bottom). As observed, the SuSAv2-MEC predictions are in very good accordance with data for all kinematical situations. Although the relative role of the 2p2h-MEC effects is rather modest compared with the QE and inelastic contributions, at the peak of the 2p2h response the three contributions are comparable in size, as also observed for 12C. For completeness, we also present in Figure 6.12 the calculations for the heavier target 40Ca (kF=241 MeV/c, Eshi f t =28 MeV) where the comparison with data is again very precise from forward to very backward angles. The analysis of these results is relevant because of the similarity with 40Ar, a target of interest for recent and forthcoming neutrino oscillation experiments. Note also that the 2p-2h MEC contributions are more prominent for 40Ca than for 12C and 16O with respect to the QE regime due to the different kFdependence of the QE and 2p-2h MEC contributions, A/kFand Ak2 F, respectively [156]. To conclude, the comparison with a very light nucleus as 4He (kF=200 MeV/c, Eshi f t =15 MeV) is displayed in Fig. 6.13. In this situation, the SuSAv2 model underestimates the QE peak which may be a consequence of the strong vector and scalar potentials arising from the RMF prescription. Therefore, the applicability of the SuSAv2 model and, specifically, the 12C RMF scaling functions and their associated FSI effects may be questionable for very light nuclei, i.e. for very low Fermi momentum. We can also observe, in accordance with the density dependence shown in Section 4.5, that the 2p-2h MEC relative contribution is significantly smaller than the one observed for heavier nuclei. 1226. ANALYSIS OF INCLUSIVE ELECTRON SCATTERING WITHIN THE SUSAV2-MEC MODEL 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 5000 10000 15000 20000 25000 30000 40Ca, Ei=681MeV, θ=45.5o 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 5000 10000 15000 40Ca, Ei=841MeV, θ=45.5o 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 5000 10000 15000 20000 QE Inelastic 2p-2h Total 40Ca, Ei=560MeV, θ=60o 0 0.1 0.2 0.3 ω (GeV) 0 1000 2000 3000 40Ca, Ei=400MeV, θ=140o Figure 6.12: Comparison of inclusive 40Ca(e,e′) cross sections and predictions of the SuSAv2- MEC model. The separate contributions of the pure QE response (dashed line), the 2p-2h MEC (dot-dashed), inelastic (double-dot dashed) are displayed. The sum of the three contributions is represented with a solid blue line. The yaxis represents d2σ/dΩ/dωin nb/GeV/sr. Data from Refs. [159] and [160]. 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 2000 4000 6000 8000 10000 12000 Total QE Inelastic 2p-2h MEC 4He, Ei=737MeV, θ=37.1o 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 ω (GeV) 0 400 800 1200 4He, Ei=3595MeV, θ=16o Figure 6.13: Comparison of inclusive 4He(e,e′) cross sections and predictions of the SuSAv2- MEC model (solid blue line). The separate contributions of the pure QE response (dashed line), the 2p-2h MEC (dot-dashed), inelastic (double-dot dashed) are displayed. The yaxis represents d2σ/dΩ/dωin nb/GeV/sr. Data from Refs. [158] and [161]. 6.4. CONCLUSIONS 123 6.4 Conclusions In this Chapter the SuSAv2-MEC model applied to electron scattering is extended to the whole energy spectrum, incorporating the contributions coming from the QE, inelastic and two-body meson exchange currents. Within this framework a general “blending” function is introduced to make the transition between the RMF and RPWIA responses. This function is constructed in terms of a parametrization of the optimized blending region given by a transition parameter, q0, and it has been applied consistently to the QE as well as to the inelastic regimes, using the same scaling functions with an analogous RMF-RPWIA transition. Although the use of more free parameters, as ω0 and/or the shift energy, leads to an even better agreement with data in some particular cases, the specific parametrization assumed is not critical, and indeed, the present model is capable of reproducing very successfully the whole energy spectrum of 12C(e,e′) data at very different kinematics. In particular, the SuSAv2 model reproduces accurately the position, width and maximum of the QE peak for all kinematics, whereas the use of the single-nucleon inelastic structure functions together with the SuSAv2 scaling functions has led to a precise description of the ∆-resonance region and the entire inelastic regime. Accordingly, the application of the phenomenological SuSA approach to the inclusive (e,e′) reactions would clearly result in an underprediction of the data, as it was previously suggested in Section 3.5.2. Remarkably, we have also shown that the SuSAv2-MEC model gives a reasonably good description not only of the cross sections but also of the separate longitudinal and transverse response functions. This is a very important test for models used in neutrino scattering studies, since in this case the balance between the L and T channels is different from the (e,e′) case. All this gives us a great confidence in the reliability of the model, providing a solid benchmark to assess its validity when extended to the description of neutrinonucleus scattering. In this case, not only new responses contribute, but also the wide neutrino energy band implied by the typical accelerator-based neutrino fluxes makes it difficult to reconstruct the neutrino energy. Thus, ingredients beyond the ones usually assumed within the IA can have a significant impact on the analysis of data. A basic feature of our present study, apart from the appliance of the SuSAv2 model to the QE and inelastic regions, concerns the evaluation of the two-body meson exchange currents in both longitudinal and transverse channels. This fully relativistic calculation has allowed for a consistent evaluation of high energies/momenta experimental data. Furthermore, the parametrization of the exact results for the 2p-2h MEC responses has allowed us to avoid the computationally demanding microscopic calculation for the entire set of kinematics required for the experimental data presented here. Additionally, this analysis has been extended to other nuclear species leading to an accurate description of the experimental data for nuclei of relevance in forthcoming neutrino experiments. Despite the smaller amount of available (e,e′) experimental data for some nuclei with respect to 12C, the growing interest of experimental collaborations on (e,e′) measurements, together with the extension of neutrino experiments to other nuclear targets, is expected to shed light on their nuclear effects and the associated parameters that play a role in most theoretical description for both electron and neutrino reactions. To conclude, we emphasize the importance of scaling arguments that provide a proper description of electron scattering data before the analysis be extended to neutrino reactions. In this sense, the fully relativistic analysis presented in this chapter for the entire energy spectrum and the different nuclear responses is of crucial importance for the analysis of neutrino reactions. 130 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS antineutrino energy fluxes, the quality of the agreement with data is rather similar in the two cases. 0 0.4 0.8 1.2 1.6 2 0 5 10 15 20 d2σ/dTµ/dcosθµ 0.9 < cosθµ < 1.0 0 0.4 0.8 1.2 1.6 0 5 10 15 20 25 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 0.7 < cosθµ < 0.8 0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 20 MiniBooNE QE+MEC QE MEC 0.6 < cosθµ < 0.7 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 d2σ/dTµ/dcosθµ 0.5 < cosθµ < 0.6 0 0.2 0.4 0.6 0.8 1 0 5 10 15 0.4 < cosθµ < 0.5 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 0.3 < cosθµ < 0.4 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 0.2 < cosθµ < 0.3 0 0.2 0.4 0.6 0.8 Tµ (GeV) 0 2 4 6 8 10 12 d2σ/dTµ/dcosθµ 0.1 < cosθµ < 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 0.0 < cosθµ < 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -0.1 < cosθµ < 0.0 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 2 4 6 8 10 -0.2 < cosθµ < -0.1 Figure 7.3: MiniBoone flux-folded double differential cross section per target nucleon for the νµ CCQE process on 12C. Results are given in 10−39 cm2/GeV and displayed versus the µ−kinetic energy Tµfor various bins of cos θµobtained within the SuSAv2-MEC approach. QE and 2p-2h MEC results are also shown separately. Data are from [20]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 d2σ/dTµ/dcosθµ -0.3 < cosθµ < -0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 -0.4 < cosθµ < -0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 -0.5 < cosθµ < -0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 -0.6 < cosθµ < -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 d2σ/dTµ/dcosθµ -0.7 < cosθµ < -0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -0.8 < cosθµ < -0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -0.9 < cosθµ < -0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 2 4 6 8 10 -1.0 < cosθµ < -0.9 Figure 7.4: As for Fig. 7.3 but considering more backward kinematics. Data are from [20]. 7.1. CCQE νµAND νµREACTIONS AT MINIBOONE AND NOMAD KINEMATICS 131 0 0.4 0.8 1.2 1.6 0 2 4 6 8 10 12 d2σ/dTµ/dcosθµ 0.9 < cosθµ < 1.0 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 0.7 < cosθµ < 0.8 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 0.6 < cosθµ < 0.7 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 d2σ/dTµ/dcosθµ 0.5 < cosθµ < 0.6 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 30.4 < cosθµ < 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1 1.5 2 2.5 0.3 < cosθµ < 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 20.2 < cosθµ < 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 d2σ/dTµ/dcosθµ 0.1 < cosθµ < 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 0.0 < cosθµ < 0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 -0.1 < cosθµ < 0.0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.2 < cosθµ < -0.1 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 d2σ/dTµ/dcosθµ -0.3 < cosθµ < -0.2 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.4 < cosθµ < -0.3 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.5 < cosθµ < -0.4 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.6 < cosθµ < -0.5 Figure 7.5: As for Fig. 7.3 but considering now the νµCCQE process on 12C. Data are from [21]. A further general comment on the previous results from Figs. 7.3—7.5 is in order: the RMF predictions (shown in [163]) produce almost identical results to the ones from the SuSAv2 model at MiniBooNE kinematics. A similar comparison was also shown in Chapter 3 for the analysis of (e,e′) data at intermediate q-values. To complete the previous discussion on the double differential cross sections, we present in Figs. 7.6 and 7.7 the results averaged over the muon kinetic energy bins as functions of the muon scattering angle for neutrinos and antineutrinos, respectively. These graphs complement the previous ones, and prove the capability of the model to reproduce the data for a large variety of kinematic situations. The 2p-2h MEC contributions increase the pure QE response by ∼25 −35% (depending on the particular region explored) and are shown to be essential in order to describe the data. As observed, the total model tends to overpredict the data measured at angles close to zero and Tµin the vicinity of ∼0.8−1 GeV. This is consistent with results in previous figures and the inability of the model to describe properly data at very small angles. However, the largest discrepancy between theory and data occurs at the smallest muon kinetic energy bins considered, i.e., 0.2<Tµ<0.4, in particular, for neutrinos (Fig. 7.6) and angles bigger than 900(cos θµ<0). As seen, the data are higher by ∼25−30% than theoretical predictions. This outcome is consistent with the results shown in the panels on the bottom in Figs. 7.3-7.5. 132 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 1.0 < Tµ < 1.1 0.4 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 0.9 < Tµ < 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 0.8 < Tµ < 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.7 < Tµ < 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 4 8 12 16 20 24 0.6 < Tµ < 0.7 -0.2 0 0.2 0.4 0.6 0.8 1 0 4 8 12 16 20 0.5 < Tµ < 0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 2 4 6 8 10 12 14 16 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.4 < Tµ < 0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 2 4 6 8 10 12 14 0.3 < Tµ < 0.4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 2 4 6 8 10 12 0.2 < Tµ < 0.3 Figure 7.6: MiniBoone flux-folded double differential cross section per target nucleon for the νµ-12C CCQE process displayed versus cos θµfor various bins of Tµobtained within the SuSAv2- MEC approach. QE and 2p-2h MEC results are also shown separately. Data are from [20]. 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 1.0 < Tµ < 1.1 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 0.9 < Tµ < 1.0 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 0.8 < Tµ < 0.9 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.7 < Tµ < 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 0.6 < Tµ < 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 0.5 < Tµ < 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cosθµ 0 1 2 3 4 5 6 7 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0.4 < Tµ < 0.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 1 2 3 4 5 0.3 < Tµ < 0.4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cosθµ 0 0.5 1 1.5 2 2.5 3 3.5 40.2 < Tµ < 0.3 Figure 7.7: As for Fig. 7.6, but now for the νµCCQE process on 12C. Data are from [21]. 7.1. CCQE νµAND νµREACTIONS AT MINIBOONE AND NOMAD KINEMATICS 133 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cosθµ 0 5 10 15 20 25 dσ/dcosθµ (10-39cm2) MiniBooNE QE+MEC MEC QE -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cosθµ 0 2 4 6 8 10 12 dσ/dcosθµ (10-39cm2) 00.5 11.5 2 Tµ (GeV) 0 2 4 6 8 10 12 14 16 dσ/dTµ (10-39cm2/GeV) 00.5 11.5 2 Tµ (GeV) 0 0.5 1 1.5 2 2.5 3 dσ/dTµ (10-39cm2/GeV) Figure 7.8: MiniBooNE flux-averaged CCQE νµ-12C (νµ-12C) differential cross section per nucleon as a function of the muon scattering angle (top panels) and of the muon kinetic energy (bottom panels). The left panels correspond to neutrino cross sections and the right ones to antineutrino reactions. Data are from [20,21]. In Fig. 7.8 results are presented for the MiniBooNE flux averaged CCQE νµ(νµ)−12C differential cross section per nucleon as a function of the muon scattering angle (top panels) and the muon kinetic energy (bottom panels). The integration over the muon kinetic energy has been performed in the range 0.2 GeV <Tµ<2.0 GeV. Panels on the left (right) correspond to neutrinos (antineutrinos). As shown, and in consistency with previous results, the SuSAv2-MEC model is capable of reproducing the magnitude as well as the shape of the experimental cross section in all of the cases. 7.1.2 Analysis of longitudinal/transverse channels for neutrino reactions In this section we study in detail the relevance of the different longitudinal and transverse channels that contribute to the QE and 2p-2h MEC MiniBooNE cross sections, also accounting for the corresponding axial and vector contributions which arise from the hadronic currents. An analysis on the different channels for the 2p-2h MEC nuclear responses and the total cross section was addressed in Chapter 4 (see Figs. 4.9, 4.10 and 4.11), showing a predominance of the transverse over the longitudinal ones, whereas for the latter the contributions arising from vector currents were negligible in comparison with the axial ones. Moreover, the separate transverse channels, TVV ,TAA and T′ V A, whereas showing some remarkable differences for different qvalues, contribute in a similar way to the total cross section. This is due to the relevant kinematic regions (see Section 7.3 for details). Here we explore the relevance of the different channels in the specific kinematics of the MiniBooNE experiment. In Fig. 7.9 the separate 2p-2h MEC contributions to the different channels (L,TVV ,TAA and T′ V A) corresponding to the MiniBooNE double differential cross section at different bins of the muon scattering angle. 134 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS 00.5 11.5 2 0 1 2 3 4 5 d2σ/dTµ/dcosθµ (10-39cm2/GeV) 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 TOTAL L TVV TAA T’VA 0.8 < cosθµ < 0.9 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 d2σ/dTµ/dcosθµ (10-39cm2/GeV) 0.3 < cosθµ < 0.4 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 10.3 < cosθµ < 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 d2σ/dTµ/dcosθµ (10-39cm2/GeV) -0.2 < cosθµ < -0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.2 < cosθµ < -0.1 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.2 0.4 0.6 0.8 1 d2σ/dTµ/dcosθµ (10-39cm2/GeV) -0.6 < cosθµ < -0.5 0 0.1 0.2 0.3 0.4 0.5 Tµ (GeV) 0 0.1 0.2 0.3 0.4 0.5 -0.6 < cosθµ < -0.5 Figure 7.9: Comparison of the different 2p-2h MEC channels for the νµ(left panels) and νµ(right panels) MiniBooNE double differential cross section. Results in Fig. 7.9 show the differences between the TAA and TVV contributions, the latter being shifted to higher Tµvalues by about 50 MeV for all angular bins. At very forward angles, i.e., lower q-values, the global magnitude of the AA channel is greater than the VV one, in accordance with the results observed in Fig. 4.11. This figure also shows clearly that at qof the order of 400 MeV/c the TVV and TAA responses differ roughly by a factor 2 at the maximum. This difference decreases for higher q-values. Concerning the interference T′ V A component, its magnitude is not so different from the VV and AA ones at very forward angles, being on the contrary the most relevant contribution at larger angles. Finally, although the longitudinal channel gives the smallest global contribution, its role is essential in order to interpret antineutrino scattering at backward angles. This is a consequence of the negative T′ V A term for antineutrino reactions that almost cancels out the TVV +TAA contribution. 7.1. CCQE νµAND νµREACTIONS AT MINIBOONE AND NOMAD KINEMATICS 135 The conclusions extracted from the previous analysis on the 2p-2h MEC cross section also apply for the separate QE contributions to neutrino and antineutrino cross sections. The different QE channels are analyzed for the MiniBooNE double differential cross sections in Fig. 7.10, where the transverse contribution predominate at all kinematics whilst the net longitudinal channel, even being a very small contribution, is essential to describe antineutrino data at backward kinematics together with the 2p-2h longitudinal one. 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 22 24 d2σ/dcosθµ/dTµ (10-39cm2/GeV) MiniBooNE SuSAv2 SuSAv2, L SuSAv2, T SuSAv2, TVV SuSAv2, TAA SuSAv2, T’VA 0.8 < cosθµ < 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 0.1 < cosθµ < 0.2 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 Tµ (GeV) 0 1 2 3 4 5 6 7 8 9 d2σ/dcosθµ/dTµ (10-39cm2/GeV) 0 0.1 0.2 0.3 0.4 0.5 0.6 Tµ (GeV) 0 0.5 1 1.5 2 2.5 3 3.5 Figure 7.10: Separation into components of the MiniBooNE CCQE νµ(top panel) and νµ(botoom panel) double-differential cross section per nucleon displayed versus Tµfor various bins of cos θµ within the SuSAv2 approach. The MiniBooNE data [20,21] are also shown for reference. In Fig. 7.11 we show the breakdown of the total neutrino cross sections into individual L(= LVV +LAA), T(=TVV +TAA), TVV ,TAA and T′ V A contributions, with the last occurring as a positive (constructive) term in the neutrino cross section and a negative (destructive) term in the antineutrino one. The sign of the T′ V A channel represents the main difference between the total neutrino and antineutrino cross sections (see also Fig. 7.1). Apart from the opposite sign in the V A response, some minor differences between neutrino and antineutrino cross sections arise from the different Coulomb distortions of the emitted lepton (see Section 7.3.2 for details) and the final nuclei involved in the CC neutrino (nitrogen) and antineutrino (boron) scattering processes on carbon. We also notice that below 1 GeV the T′ V A response is higher than the TVV one and of the same order as the TAA one. Note that the maximum of the V A contribution is around the peak of the MiniBooNE neutrino flux. On the contrary, the effects of V A contributions to the cross sections are negligible at energies above 10 GeV as a consequence of the small axial form factor GAand leptonic factor VT′ at high Eνand Q2values (see Chapter 2 for details). This is also in agreement with some previous QE results [19]. As a consequence, for very high νµ(νµ) energies (above ∼10 GeV) the total cross section for neutrinos and antineutrinos is very similar. Only the Land Tchannels contribute for the higher values explored by NOMAD experiment. On the contrary, in the region explored by the MiniBooNE collaboration, the main contributions come from the two transverse T,T′channels. 136 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS 0.1 110 100 Eν (GeV) 0 2 4 6 8 10 12 14 16 σν (10-39cm2) SuSAv2 SuSAv2, TVV SuSAv2, TAA 0.1 110 100 0 2 4 6 8 10 12 14 16 SuSAv2, T’VA SuSAv2, T SuSAv2, L Figure 7.11: Separation into components of the CCQE νµcross section per nucleon on 12C displayed versus neutrino energy Eνwithin the SuSAv2 approach. The MiniBooNE [20] and NOMAD [24] data are also shown for reference. The asymmetry between neutrino and antineutrino cross sections introduced by the V A interference is analyzed in Fig. 7.12 where we show the experimental difference (σνµ−σνµ)exp from MiniBooNE, together with the corresponding theoretical prediction from the SuSAv2-MEC model. This difference is approximately equal to 2 (σνµ)SuSAv2 T′ V A +2 (σνµ)M EC T′ V A , apart from the mi- nor differences between neutrino and antineutrino reactions described above. The result from the purely QE responses is also shown, reinforcing the importance of the 2p-2h MEC contributions for the analysis of the MiniBooNE experiment and their relevance in the interference channel. 0.1 110 100 Eν (GeV) 0 2 4 6 8 10 σν-σν (10-39cm2) MiniBooNE (νµ-νµ) QE+MEC (νµ-νµ) QE (νµ-νµ) Figure 7.12: Experimental difference between neutrino and antineutrino cross sections (σνµ−σνµ) from MiniBooNE, together with the corresponding theoretical prediction from SuSAv2+MEC. The purely QE response is also shown. MiniBooNE data are from [20,21]. 7.2. CCQE-LIKE SCATTERING IN THE MINERνAEXPERIMENT 137 The asymmetry of the nuclear effects for neutrino and antineutrino is important for CP violation studies and it has been analyzed in previous works [3, 115, 164]. However, the inherent difficulties related to the different neutrino and antineutrino fluxes result in a potential obstacle for the interpretation of experiments aimed at the measurement of the CP violation angle. In this context, some preliminary studies on the CP asymmetry from the MINERνA Collaboration have been recently published [165]. Moreover, CP violation effects in the lepton sector related to the νµ and νecross sections will be addressed in following sections. 7.2 CCQE-like scattering in the MINERνA experiment In this section we apply the SuSAv2-MEC model to the analysis of the MINERνA experiment. The MINERνA Collaboration has recently measured differential cross sections for muonic and electron neutrino and antineutrino charged-current quasielastic scattering on a hydrocarbon target [46, 47]. The “quasielastic” events are defined, in this case, as containing no mesons in the final state (CCQE-like) thus also including contributions from multi-nucleon excitations. The energy flux extends up to 10 GeV and is peaked at Eν∼3 GeV for both neutrinos and antineutrinos, i.e., in between MiniBooNE and NOMAD energy ranges. Therefore, its analysis can provide valuable information on the role played by 2p-2h meson-exchange currents in the nuclear dynamics [110,166,167]. 7.2.1 Analysis of charged-current muonic neutrino results In previous studies of the MINERνA Collaboration [46, 47], the RMF, SuSA and SuSAv2 models [97,153] as well as other theoretical approaches [168–170] were able to reproduce the MINERνA data without the inclusion of np-nh excitations, unlike the MiniBooNE results. It is important to point out that the MINERνA kinematics, larger than the MiniBooNE one, extends into the pion production region and hence the experimental subtraction of these effects to isolate CCQE-like events is more critical. Note that in these experiments only the information related to muon variables is considered. Moreover, the recently improved analysis of the MINERνA flux results [171] leads to an increase of the MINERνA CCQE-like experimental cross sections. Considering the reevaluation of the MINERνA flux [171,172], we present in Fig. 7.13 the flux averaged CCQE νµ(νµ) differential cross section per nucleon as a function of the reconstructed four-momentum Q2 QE. This reconstructed magnitude is obtained following the procedure introduced in Section 7.1 and also detailed in [46,47]. The top panel in Fig. 7.13 refers to νµ−12C whereas the bottom panel contains predictions and data for νµ−CH. The mean energy of the MINERνA muonic flux is much higher than the Mini- BooNE one, about 3.5 GeV for both νµand νµ. As observed, significant contributions of the 2p-2h MEC, of the order of ∼35 −40% (∼25%) at the maxima for νµ(νµ), are needed in order to reproduce the experimental data that correspond to a new analysis performed by the MINERνA collaboration [171,172]. These data exceed by ∼20% the ones already presented in previous publications [46,47] (using the uncorrected MINERνA flux) that, on the other hand, were consistent with calculations based exclusively on the impulse approximation (see [97]). Thus, the new MINERνA analysis shows major consistency with the MiniBooNE data and invalidates the previous conclusion [97] that the MINERνA results can be reproduced without the 138 7. ANALYSIS OF CHARGED-CURRENT NEUTRINO INDUCED REACTIONS inclusion of 2p-2h contributions. In spite of the very different muon neutrino (antineutrino) energy fluxes between MiniBooNE and MINERνA, 2p-2h MEC effects remain very significant (on average, 25 −35%) being their contribution essential in order to reproduce the data. For completeness, it is worth mentioning that in the specific conditions of MINERνA, it clearly appears that, even if the neutrino energy is as large as 3 GeV, the process is largely dominated by relatively small energy and momentum transfer, namely, ω < 500 MeV, q<1000 MeV, whereas contributions below ω < 50 MeV, q<200 MeV govern the lowest Q2 QE region. More specific details can be found in Section 7.3 and, additionally, in Ref. [97]. 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 Q2 QE (GeV2) 0 5 10 15 20 dσ/dQ2 QE (10-39cm2/GeV2/neutron) Minerva QE 2p-2h MEC QE + 2p-2h MEC νµ - 12C 0 0.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2 Q2 QE (GeV2) 0 2 4 6 8 10 12 14 16 dσ/dQ2 QE (10-39cm2/GeV2/proton) νµ - CH Figure 7.13: Flux-folded νµ−12C CCQE (upper panel) and νµ−CH (lower panel) scattering cross section per target nucleon as a function of Q2 QE and evaluated in the SuSAv2 and SuSAv2-MEC models. MINERνA data are from [172]. 7.2. CCQE-LIKE SCATTERING IN THE MINERνAEXPERIMENT 139 7.2.2 Analysis of charged-current electron neutrino results The previous conclusion holds also for the νeCCQE-like differential cross sections on hydrocarbon published by MINERνA in [53]. These results are presented in Fig. 7.14 as a function of the electron energy (top-left panel), electron angle (top-right) and reconstructed four-momentum (bottom-left). Compared to the νµ(νµ) fluxes, the νeand νeones have roughly the same shape in the region of the peak but the tail region at large energies is significantly higher in the electronic case which softly increases the average νeenergy up to 3.6 GeV. A detailed comparison of the νe versus νµcross sections is addressed in Section 7.4. 12 3 4 5678 9 10 Ee (GeV) 0 0.5 1 1.5 2 2.5 dσ/dEe (10-39cm2/GeV/nucleon) 0510 15 20 25 30 35 θe (deg) 0 0.1 0.2 0.3 0.4 0.5 dσ/dθe (10-39cm2/degree/nucleon) MINERνA MEC QE QE+MEC 00.5 11.5 2 Q2 QE (GeV2) 0 2 4 6 8 10 12 dσ/dQ2 QE (10-39cm2/GeV2/nucleon) 00.5 11.5 2 Q2 QE (GeV2) 0.5 1 1.5 (dσνe+νe/dQ2 QE)/(dσνµ/dQ2 QE) Figure 7.14: MINERνA flux-integrated differential νe-12C CCQE-like cross section per nucleon vs. electron energy (top left) and electron angle (top right). The bottom panels show the νedifferential cross section vs. Q2 QE (bottom left) and the ratio between the flux averaged CCQE νe+νeand νµcross sections versus Q2 QE (bottom right) compared with the SuSAv2-MEC prediction (solid red line). Data are from [53]. In Fig. 7.14, results are shown for the pure QE response based on the IA, the 2p-2h MEC contribution and the total response. In all the cases the contribution at the maximum coming from the 2p-2h MEC is roughly 30 −35% compared with the pure QE response. These results are similar to the ones already presented for muon neutrinos (antineutrinos), and they show the importance of 2p-2h effects in order to explain the behavior of data. As observed, the model is capable of reproducing successfully the data. For completeness, we present in the right-bottom panel the results corresponding to the ratio between the flux averaged CCQE νe+νeand νµcross sections versus the reconstructed four-momentum. We compare the predictions of the model (red curve) with the data, leading to a similar result as when compared with GENIE predictions [53], i.e., the SuSAv2-model predicts a rather flat ratio while the experimental data seem to grow up to Q2 QE ≈1 GeV2. Taking into account that the νeflux is negligible in comparison with the νe