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?


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


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 $P_{\alpha\beta}$: \begin{equation} 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}), \end{equation} which by observation $P_{\overline{\alpha}\overline{\beta}}= P_{\alpha\beta}(U_{\alpha i}\to U_{\alpha i}^*).$

Reference1. Neutrino physics-E. Kh. Akhmedov

  1. Neutrino oscillation

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