Past exam of the particle physics course of the University of Oxford / 2015 / 4

Assume coherent production and detection, relativistic neutrinos, two active neutrino flavours, propagation in vacuum, negligible packet separation and no absorption. Invert the given real lepton mixing matrix to obtain $|{\nu }_{\mu }⟩=c|{\nu }_2⟩-s|{\nu }_3⟩$ and $|{\nu }_{\tau }⟩=s|{\nu }_2⟩+c|{\nu }_3⟩$, where $c=\cos \theta$, $s=\sin \theta$. With phases ${\varphi }_a=E_{a}t-{\mathbf{p}}_a\cdot \mathbf{x}$ in natural units,Using the paper's opposite sign convention conjugates this probability amplitude and leaves the probability unchanged. For equal momentum, $E_a\simeq p+m_a^2/(2p)$ and $t\simeq L$, so $\Delta \varphi =\Delta m^{2}L/(2E)$. The equal-energy convention gives the same leading result from the momentum difference; evaluating each component at a different arrival time would incorrectly double the phase.Since $\hslash c=1.97327\times 10^{-7}\mathrm{eV}\mathrm{m}$,This gives the stated factor $1.27$. Neutrino oscillation in this approximation measures $|\Delta m^2|$, not the absolute neutrino mass or its ordering.

The Fermilab primary proton beam strikes a target, producing secondary pions and kaons. Horn focusing selects positive parents; their decays ${\pi }^{+}\to {\mu }^{+}{\nu }_{\mu }$ and $K^{+}\to {\mu }^{+}{\nu }_{\mu }$ produce a predominantly muon neutrino beam. A decay tunnel permits decay before the remaining charged secondaries encounter an absorber. A near detector constrains the flux and interaction spectrum, while the distant MINOS detector measures its change.

The alternating scintillation counters and magnetized iron give a sampling calorimeter and a magnetic tracking volume. A charged current ${\nu }_{\mu }N\to {\mu }^{-}X$ event has a long negative muon track and a localized hadronic shower. Estimate $E_{\mu }$ by range if contained or magnetic curvature if exiting, and $E_X$ by the calibrated scintillator pulse-height sum. Reconstruct $E_{\nu }\simeq E_{\mu }+E_X$, with corrections for nuclear binding, invisible energy and detector response. The muon charge separates most wrong-sign muon antineutrino events; short showers without the track are primarily neutral current candidates. The source's $730\mathrm{km}$ baseline is used in the calculations below.

TODO: a numerical estimate from the missing MINOS spectrum cannot be supplied. On source page 7 the graph is absent: PDF Form /Im3 invokes undefined image /Im1 with no image resources, and the archived bundle has the same defect. In particular, no measured minimum energy survives. For maximal mixing angle, the first minimum of muon neutrino survival obeysA restored spectrum or fitted data are needed to choose $E_{\min }$. This is a relation for the survival minimum; a minimum in the raw event spectrum also depends on flux, cross section and smearing. For context only, accelerator and atmospheric results put the splitting at a few $10^{-3}{\mathrm{eV}}^2$, versus the solar splitting about $7.5\times 10^{-5}{\mathrm{eV}}^2$, roughly thirty times smaller. This known scale is not a reconstruction of the absent graph. ^[ref]

Finite event statistics, hadronic shower energy resolution, the muon scale, near-to-far flux extrapolation, interaction and nuclear modeling, backgrounds and detector calibration limit the precision; the baseline itself is comparatively well known. In a two-flavour vacuum fit the sign of $\Delta m^2$ is inaccessible. Evidence for dominant ${\nu }_{\mu }\to {\nu }_{\tau }$ includes the zenith-angle-dependent atmospheric deficit in Super-Kamiokande, approximately preserved active neutral current rates, small electron-flavour appearance at this scale, atmospheric tau appearance and direct tau appearance in OPERA. The latter had already reported four candidates and $4.2\sigma$ evidence in 2014; a later five-candidate discovery result is not needed for this June 2015 answer. ^[ref] These tests constrain a dominant sterile conversion as well as conversion into electron neutrinos.