What are the anti-neutrino flavour oscillation formulae?

All the textbooks and references that I have come across give the neutrino 3-flavor oscillation formulae. However, the formulae for antineutrino oscillations are never given. Is it possible to derive the antineutrino flavour oscillation formulae from those of the neutrinos?

2 Answers

So, now that I've sounded off, T2K has a preliminary finding that contradicts what I said at moderately good confidence.

Figures.

The free-space oscillation formulae depend on the masses of the neutrino flavors (well, on the differences of the squared masses of the mass states), and anti-particles have the same mass as their normal counterparts, so the oscillation of anti-neutrinos in free-space have the same character as those of neutrinos.

There is a small correction on a small correction that is possible do to the differing cross-section for charge scattering of electrons off of nuclei for (anti-)neutrinos traveling through matter than has a proton:neutron ration different from 1:1.

That said, the above considerations take CP symmetry as a postulate. The proposition is testable in atmospheric and beam experiments, and there were hints of evidence for violation in the an early MINOS data-set, though more data saw the significance of the result drop considerably (discussed for instance in DOI:10.1088/0004-637X/758/1/3 AKA arXiv 1012.3245).

The CP and CPT-transformaton property of $P_{\alpha\beta}=P(\nu_\alpha\to\nu_\beta;t)$ are given by $$P_{\alpha\beta}\equiv P(\nu_\alpha\to\nu_\beta;t)\xrightarrow{\text{CP}} P(\bar{\nu}_\alpha\to\bar{\nu}_\beta;t)\equiv P_{\overline{\alpha}\overline{\beta}};~~P_{\alpha\beta}\xrightarrow{\text{CPT}} P_{\overline{\beta}\overline{\alpha}}.$$ Since our theories are CPT invariant, we have, $P_{\alpha\beta}= P_{\overline{\beta}\overline{\alpha}}$ (or $P_{\overline{\alpha}\overline{\beta}}=P_{\beta\alpha}$). We can use it to find the antineutrino oscillation probability by exchaning the labels $\alpha\leftrightarrow\beta$ in the formula for $P_{\alpha\beta}$: $$P_{\overline{\alpha}\overline{\beta}}= \delta_{\alpha\beta}- 4\sum\limits_{i>j} {\text{Re}}(U_{\beta i}^{*} U_{\alpha i} U_{\beta j} U_{\alpha j}^{*}) \sin^2\Delta_{ij}+ 2\sum_{i>j}{\text{Im}}(U_{\beta i}^{*}U_{\alpha i}U_{\beta j} U_{\alpha j}^{*}) \sin(2\Delta_{ij}),$$ which by observation $P_{\overline{\alpha}\overline{\beta}}= P_{\alpha\beta}(U_{\alpha i}\to U_{\alpha i}^*).$

Reference1. Neutrino physics-E. Kh. Akhmedov