Nuclear modifications

Although the alternative nuclear models discussed in Section2.4.1are more sophisticated than the RFG model, they generally do not significantly change the total cross section as a function of neutrino energy, although they do significantly change the shape of differential cross sections in various kinematic variables [215]. Therefore, they fail to describe the much larger cross sections measured by experiments with heavy nuclear targets. It has been noted that theorists and experimentalists are working with different definitions of what CCQE is, and that in some sense this may be the cause of the discrepancy [215]. Experimentalists typically define CCQE interactions as interactions with no mesons in the final state (sometimes referred to as CC-0π interactions), which is more inclusive than the CCQE interaction defined in Equation2.2and used by theorists.

(a) 1p–1h (b) ∆ resonance (c) 2p–2h or π production (d) 1p–1h–1π Figure 2.8: W -boson self-energy diagrams used to produce the Nieves model predic-tions for different interaction channels. Solid lines represent particles or holes; double lines represent ∆ resonances; dashed lines represent mesons; and wavy lines repre-sent the incoming and outgoing W -boson. The dotted reprerepre-sents a line for applying a Cutkosky cut: intersected lines are put on mass shell, and represent a possible final state (calculating W self-energy is therefore a convenient way to sum many possible diagrams). The grey circles can be any possible vertex, the possibilities for which are shown in Figure2.9. This figure has been reproduced from Figure 5 of Reference [52].

The Nieves [223] and Martini [224] models look at a large number of possible W -boson self-energy diagrams in nuclear matter and consider diagrams where the interaction is with more than one nucleon to produce a CCQE-like cross section. Tree level ν_l+ n → l⁻ + p interactions are referred to as one-particle, one-hole (1p–1h), or sometimes as true CCQE; higher order two-particle, two-hole (2p–2h) corrections are included in the Nieves and Martini models, and the class of models are often referred to as are referred to as n-particle, n-hole (npnh) models. Note that they both use the LFG as the underlying nuclear model. First and second order diagrams are shown in Figure2.8to illustrate the processes considered in the Nieves model¹⁰. At each of the vertices marked with a grey

10Both Nieves and Martini models include some third order (3p3h) diagrams following the π-less

∆-decay contribution discussed in [225], which forms part of the npnh contribution.

W

N N’

(a) Direct ∆ pole

W

N N’

(b) Crossed ∆ pole

W

N N’’ N’

(c) Direct nucleon pole

W

N N’’ N’

(d) Crossed nucleon pole

W

N N’

(e) Contact

W

N N’

(f) Pion pole

W

N N’

(g) Pion in flight

Figure 2.9: Possible vertices considered in the W self-energy diagrams shown in Figure2.8. This figure has been reproduced from Figure 6 of Reference [52].

circle, each of the seven diagrams shown in Figure 2.9 can be included, and the total cross section prediction involves the summation of all possible diagrams.

Additionally, the Random Phase Approximation (RPA) is a nuclear screening effect that modifies the propagator for interactions in nuclear matter [223,224], and needs to be included in the Martini and Nieves model calculations to find good agreement with data. RPA calculations consider effective interaction terms between particle–hole excitations within the nucleus which change the electroweak coupling in nuclear matter due to strongly interacting nucleons [226]. RPA is illustrated in Figure 2.10, where V indicates the effective interaction; the sum is substituted into the 1p–1h response shown in Figure2.8a, which modifies the cross section for CCQE in an LFG (it is also included in other diagrams shown in Figure 2.8). RPA has a small effect on the overall cross section as a function of neutrino energy, and has a significant effect on the differential cross section as a function of Q² for CCQE interactions.

In the language of the Martini and Nieves models, the 1p–1h interaction is the CCQE interaction considered by theorists, whereas the CC-0π interactions measured by experi-mentalists actually include 1p–1h (with RPA corrections applied) and 2p–2h interactions,

Figure 2.10: RPA modification to CCQE scattering. The effective interactions be-tween particle–hole and ∆–hole excitations are denoted V (∆ is denoted as a double

green line). This figure has been reproduced from Figure 6 of Reference [53].

plus higher order terms which are mostly neglected in the calculations. By including the 2p–2h component, the Martini and Nieves models add additional strength to the CCQE-like cross section which produces good agreement with MiniBooNE neutrino [227,228]

and antineutrino [229,230] CCQE data without requiring an ad hoc inflation of the axial mass. Note that such agreement seems to be impossible for models which do not have an np-nh component [53].

The Nieves model [223] was limited to E_ν . 1.2 GeV because only the lowest delta resonance was included in the calculation (see Figure2.9). In a later paper [54], the ob-servation was made that as the neutrino energy increases the cross section is relatively stable as a function of energy and three-momentum (|~q|) transfer. The cross section for the Nieves multi-nucleon–neutrino model for 3 GeV neutrinos and antineutrino in-teractions on a carbon target is shown as a function of energy and momentum transfer in Figure 2.11. The top peak comes from the ∆ component, the bottom non-∆ peak fills in the dip region [54]. By imposing a cut on the three momentum transfer, the model can be extended up to Eν ≤ 10 GeV for low momentum transfer events, which is acceptable for many experiments where mostly forward going (low four-momentum transfer) events are measured. The nominal three-momentum cutoff is |~q| ≤ 1.2 GeV, but the authors of the model note that variations of ± 0.1 GeV can change the model cross section by up to 10% due to the large amount of phase space included or omitted in the calculation. With this high energy extension, the Nieves model should in principle be valid for experiments at higher energies (see Figure 2.3).

Finally, an effective np–nh model called the Transverse Enhancement Model (TEM) is available and is motivated by electron scattering data. This is described in detail in Chapter6, so is not covered here. It is unclear how reliable an electron scattering based multi-nucleon–neutrino interaction can be because it is unclear what the multi-nucleon enhancement to the axial response should be. For the TEM the axial response is not enhanced, and the enhancement to the axial response cannot be extracted from electron scattering data. There is an interesting discussion in Reference [215] on the difference between the enhancement of the axial response in the Martini and Nieves models. In

(a) Neutrinos

(b) Antineutrinos

Figure 2.11: Figures reproduced from Figure 2 of Reference [54], which show the Nieves multi-nucleon–neutrino cross section as a function of momentum and energy

transfer. This example is for 3 GeV (anti)neutrinos on a carbon target.

the Nieves model, the axial response is not enhanced, whereas in the Martini model, the enhancement to the magnetic and axial response is assumed to be identical. This difference may account for the large difference in total cross section predicted by the two models.

Experimental Setup

Usually, T2K theses have a chapter describing the T2K experiment. This thesis is no different in this regard: an overview of the T2K experiment is given in Section 3.1.

Additionally, published data from two other experiments, MINERνA and MiniBooNE, are an integral part of the analysis work presented in Chapters 4, 5 and 6, so brief descriptions of both of these experiments are given in Sections 3.2(MINERνA) and3.3 (MiniBooNE).

3.1 The T2K experiment

The T2K (Tokai to Kamioka) experiment is a long-baseline neutrino oscillation experi-ment designed to make high precision measureexperi-ments of various neutrino mixing param-eters using a high intensity off-axis muon neutrino beam.

The experimental setup is shown in Figure 3.1. A high purity ν_µ beam is produced at the Japan Proton Accelerator Research Complex (J-PARC) on the east coast of Japan. There is a near detector complex located 280 m downstream of the target, which is designed to measure the unoscillated beam intensity, purity and direction to high precision. The flavour composition of the beam is then measured 295 km downstream of the production point at the far detector, Super-Kamiokande (SK), which measures oscillations in the flux. Both SK and the off-axis near detector are designed to be at an angle of 2.5^◦ with respect to a direct line between each detector and the target.

This technique produces a narrow-band beam, which allows greater precision oscillation measurements to be made. The peak neutrino energy at the off-axis angle is 0.6 GeV, which was selected such that the far detector is at the first oscillation maximum.

40

This section gives an overview of the design of the detectors that make up the T2K experiment. A more thorough description of the T2K experiment can be found in Reference [55]. A detailed description of the T2K oscillation analysis strategy is given in Chapter 7.

Figure 3.1: Location of T2K near and far detectors relative to neutrino production at J-PARC. This figure has been reproduced from Figure 1 of Reference [55].