Interaction Channels

sntools supports multiple different interaction channels described in this section. By default, it will generate events across all supported channels, but it can be restricted to a single channel by using the --channel <value> command line argument, where <value> can be one of ibd, es, o16e or o16eb.

In water Cherenkov detectors like Hyper-Kamiokande, the dominant interaction channel for supernova neutrinos is inverse beta decay, which makes up about 90 % of events. Another important interaction channel is elastic scattering on electrons, which makes up only a few per cent of events but provides precise information on the direction of the supernova. The cross sections for both interactions have been calculated to a high level of precision.

88

3.2 sntools: A Supernova Event Generator

Another important subdominant channel are charged-current interactions of νe

and ¯ν_eon¹⁶O nuclei. While this channel suffers from large theoretical uncertainties, it is very sensitive to the high-energy tail of supernova neutrino fluxes, so that the number of events in this channel can vary greatly between models.

Consistent with recent work on the Super-Kamiokande supernova burst mon-itor [172] and with the Hyper-Kamiokande Design Report [169], I have not included additional subdominant interaction channels like neutral-current interactions on

16O nuclei or neutral- and charged-current interactions on heavier oxygen isotopes.

Due to their low event rates, these channels would have a relatively small influence on the observed event spectra, while introducing additional uncertainties since their cross sections are not well known. Furthermore, their contributions would be mainly at low or medium energies, where inverse beta decay dominates, while their contributions at high energies would be much smaller than that of the charged-current¹⁶O channel. As a result, these channels have a very minor dependence on the supernova model and would contribute little to the analysis presented in this thesis.

Neutral-current scattering on free protons, which may be an important detection channel in scintillator detectors, cannot be detected by Hyper-Kamiokande since the outgoing proton is below the Cherenkov threshold in water [288].

3.2.4.1 Inverse Beta Decay

In Hyper-Kamiokande, inverse beta decay (IBD; ¯νe+p→ⁿ+e⁺) is the dominant interaction channel for supernova neutrinos due to its relatively high cross section and low energy threshold of E^thr_ν ≈ 1.8 MeV, as well as the large number of free protons in the detector. The observed energy of IBD events is closely related to the neutrino energy, making this an excellent channel to reconstruct the ¯ν_espectrum.

In sntools, I have implemented IBD using the full tree-level cross section calculated in reference [289] and including radiative corrections based on the approximation from reference [290].¹⁰ The calculation of the cross section is summarized in ap-pendix A. Due to uncertainties in experimental measurements of input parameters, the overall uncertainty is estimated to be 0.4 % at low energies and reaches about 2 % at Eν = 100 MeV, which is the upper end of the energy range of supernova neutrinos.

10That calculation uses the limit me→0. This approximation is accurate to better than 0.1 % above Ee=1 MeV and the effect in Hyper-Kamiokande, whose energy threshold is much higher than 1 MeV, is completely negligible.

Chapter 3 A Software Toolchain for Supernova Neutrino Events in Hyper-Kamiokande

3.2.4.2 Neutrino-Electron Scattering

In Hyper-Kamiokande, elastic neutrino-electron scattering (ν+e⁻ → ^ν+e⁻) is a subdominant interaction channel due to its low cross section, which is only partially compensated by the large number of electrons in the detector which is 5 (10) times bigger than the number of free protons (¹⁶O nuclei). Elastic scattering events make up only a few per cent of all events but their angular distribution is strongly peaked into a forward direction, pointing away from the supernova. They can therefore be used to determine the direction of a supernova at the fiducial distance of 10 kpc with an accuracy of 3° to 5° in Super-Kamiokande [172] or about 1° in Hyper-Kamiokande [169].

Elastic scattering is the only interaction channel considered here which is sensitive to all neutrino flavours. However, the cross section of ν_eand ¯ν_e, which can interact through both neutral and charged currents, is higher than that of ν_xand ¯ν_x, which can interact only through neutral currents.

In sntools, I have implemented elastic scattering using the tree-level cross sections from standard electroweak theory calculated by ’t Hooft [291] and including one-loop electroweak and QCD corrections as well as QED radiative corrections as calculated in reference [292]. The calculation of the cross section is summarized in appendix A. Similar to the calculation of the IBD cross section above, the uncertainty due to experimental measurements of some input parameters is on a per mille level.

3.2.4.3 Charged-Current Interactions on¹⁶O

In Hyper-Kamiokande, charged-current interactions of νeand ¯νeon¹⁶O nuclei,

νe+¹⁶O→ ^e−+X (3.13)

¯ν_e+¹⁶O→ ^e++X, (3.14)

are a subdominant interaction channel.¹¹ Due to the high energy threshold of both interactions of approximately 15 MeV and 11 MeV, respectively, as well as the steep energy-dependence of the cross sections, the number of events in each channel is a very sensitive probe of the high-energy tail of the supernova neutrino flux. It may vary by more than two orders of magnitude depending on the supernova models and oscillation scenario, making up anywhere from<1 % to over 10 % of all events observed in Hyper-Kamiokande.

In sntools, I have implemented a recent approximation of the cross section for both interaction channels [293], which is based on a new shell model calculation [294].

An earlier calculation used the continuum random phase approximation (CRPA) and tabulated the resulting total cross sections, instead of partial cross sections for

11Charged-current interactions of other neutrino species do not occur, since the energy of supernova neutrinos is too small to produce muons or τ leptons.

90

3.2 sntools: A Supernova Event Generator

Table 3.1: Excitation energies and fitting parameters for charged-current interactions on¹⁶O. Values from reference [293].

each excitation energy [295]. A fit based on those results therefore only considered a simplified scenario where all final nuclear states shared the energy of the ground state [296]. This significantly overestimated the energy spectrum of the emitted e^±. The modern calculation selected 42 different nuclear states and calculated their respective partial cross sections. To simplify the evaluation of the cross section, the authors of reference [293] divide these states into four groups such that the total cross section is

σ(Eν) =

∑

σg(Eν), (3.15)

where the partial cross sections for each group, σ_g, are given by the expression log₁₀

The excitation energies E_gand fit parameters a_g, b_gand c_gof each group of states are shown in table 3.1.

This fit matches the cross sections calculated from the full set of nuclear states to within a few per cent at neutrino energies of up to 100 MeV. For a typical supernova neutrino flux, the difference in the resulting event spectra when using the four groups instead of all 42 nuclear states is also very small.

Chapter 3 A Software Toolchain for Supernova Neutrino Events in Hyper-Kamiokande