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5.2.1 Relativistic Fermi Gas

The RFG model is a simple model which is commonly applied to Fermionic physical systems. The assumption is that all particles are in a potential, and form plane-wave states, leading to all states being filled up to a Fermi-level, above which no states are filled. The Fermi-gas model has a flat distribution of states in momentum space (see figure 5.3), and a constant binding energy.

In neutrino-nucleus interactions, the RFG model allows for numerical inte- gration over nucleon states, and as such makes analytical theoretical calculations possible. This model has been used in neutrino interaction generators for many years due to its simplicity.

5.2.2 Local Fermi Gas

The RFG could be described as a global Fermi gas model, meaning that the Fermi momentum and binding energy are considered a constant of the nucleus, and the momentum distribution is not location-dependent. In a Local Fermi Gas (LFG)

model, the same Fermi gas idea is applied, but variations in nuclear density are taken into account. This means the Fermi momentum depends on the location in the nucleus, leading to a smoothed momentum spectrum. The LFG model is not currently implemented in either of the generators used by T2K (NEUT and GENIE), however it has been implemented in NuWro. It will not be discussed further here. 5.2.3 Spectral Function

The discontinuity at the Fermi momentum in the RFG model is unlikely to be a realistic description of nature, so a more physically motivated model would be preferable. “Spectral function” is a generic term for a function that describes the momentum and energy distributions of initial nucleons in a nucleus (the RFG can be described by a spectral function very easily, as it is a simple step-function - see figure 5.3). Spectral functions have been calculated analytically for light nuclei (A 4) [95, 96, 97, 98], and for infinite nuclear matter [99, 100]. For medium-size nuclei, such as carbon and oxygen, various approximations need to be made, but spectral functions can still be built by combining information from electron scattering data with the theoretical calculations from uniform nuclear matter of di↵erent densities. The spectral functions used in NEUT were provided by O. Benhar [101].

The Spectral Function is made up of two di↵erent terms: a mean-field term for single particles, and a term from correlated pairs of nucleons. The correlation term leads to a very long tail in both momentum and binding energy, and accounts for roughly 20% of the total Spectral Function. These initial-state correlations lead to the ejection of a second nucleon (see section 5.7), however the interaction is only with one nucleon, and the kinematics are pure one-particle CCQE.

As the data used to tune the Spectral Functions are frome +pfinal states, the Spectral Functions describe the proton initial state. We approximate the neutron Spectral Function to be the same, and uncertainties discussed in section 5.8 are expected to cover any potential di↵erences between them.

Figure 5.2 shows the oxygen Spectral Function, in which the nuclear shell model energy levels can be clearly seen along the energy axis (labelled as “removal energy”, also referred to as binding energy). These shell orbitals are part of the mean field term. The correlation term extends out to very high momenta and removal energies. Figure 5.3 shows the oxygen Spectral Function projected onto the momentum axis, with the equivalent distribution shown for the RFG model. The correlation term can be clearly seen extending out to very high momenta.

Figure 5.4 demonstrates how the Spectral Function better reproduces the QE peak in electron scattering, while also helping to fill in the “dip” region between the

Figure 5.2: Benhars 2D Spectral Function for oxygen. The shell model orbitals are clearly seen as lobes along the removal energy axis.

momentum/MeV 0 100 200 300 400 500 600 700 800 n(p) -6 10 -5 10 -4 10 -3 10 -2 10 oxygen SF RFG

Figure 5.3: Spectral Function for oxygen projected onto the momentum axis. The black line corresponds to RFG with a Fermi momentum of 220 MeV, green is the spectral function calculated by Benhar [101].

QE peak and the -resonance. This dip is filled in primarily from the highly bound nucleons in the correlation term of the Spectral Function, with contributions from resonant pion production, as well as other non-resonant, and DIS backgrounds.

Figure 5.4: Example of electron scattering data on oxygen compared to both SF and RFG models [102]. “SP” in the legend corresponds to what this thesis refers to as“SF” and “FG” is a global relativistic Fermi gas (RFG) model.

5.2.4 Pauli blocking

In both the RFG and LFG model, all states up to the Fermi level are filled, so particles can’t be ejected in momentum states lower than this level. This naturally leads to a phenomenon known as Pauli blocking which reduces the cross section by reducing the available phase space for the outgoing nucleon. This threshold momentum is commonly referred to as the Fermi momentum, orpF.

In the SF model, Pauli blocking is still expected to occur, but it does not arise from the model as naturally. It is therefore common to use approximations when implementing these models in calculations or simulations. Often these approxima- tions are as crude as applying a hard cut o↵at the value of the Fermi momentum that would be used if using the RFG model. More sophisticated options include applying Pauli blocking to a nucleon according to the probability that the state is filled, which is taken from the spectral function distribution itself. The di↵erences to the total cross section and muon kinematics are minimal, so when the SF model was incorporated into NEUT a hard cut o↵approximation was used. The value of this cut o↵will be known as the Fermi momentum and denotedpFSF to distinguish