# Eligio Lisi: The Electron Neutrino as a Superposition of States

The electron neutrino plays a prominent role in neutrino physics: it is the partner of the lightest (=stable) charged lepton. So it can be easily produced and absorbed in a variety of processes, can be used to probe different kinematical regimes, and dynamically feel the background electrons with a number density N_e, giving rise to the MSW effect.

In the last decade we have learned that the electron neutrino is a superposition of at least three massive states. The coefficients of this superposition (to the best of our current knowledge) are 0.82, 0.55, -0.16. The energy level splittings are proportional to 7.5 E-5 eV^2 (the states 1 and 2) and 2.4 E-3 eV^2 (state 3 with respect to the others).

If we had started with a long-baseline reactor experiment in KamLand, we would have determined an accurate value of the dm_12^2, and would have been faced with the hierarchy of two possible situations. The only way to solve it would be with solar neutrinos, because we need to beat the oscillation of the 1- and 2- state with some Q-driven oscillation with known sign of Q and a similar scale. Then the next question we would have asked in this parallel universe is the one of unitarity: electron neutrinos could leak into a state nu_3, with a splitting dm^2 large enough to be unresolved.

We then would have moved to short baseline experiments from reactor neutrinos, to access the high dm^2 scale. We now know that there is a component of electron neutrinos due to a nu_3. Then we stumble in a new hierarchy problem: the state with the smallest nu_e component may be the lightest (inverted hierarchy) or the heaviest (normal hierarchy). We might solve this with two Q-driven electron neutrino oscillations.

This whole pattern may repeat itself with more sterile neutrinos. One could have additional states nu_4, nu_5, and many more patterns of hierarchy.

We can learn a lot from electron neutrinos, but we must ask ourselves what else we may learn with muon neutrinos too. With only electron-neutrino oscillations, one cannot probe complex coefficients, nor the distribution of muon and tau neutrino flavours, such as the angle theta_23. So one must do a global neutrino data analysis, where one sees the interplay between the different oscillation channels. It is useful to show a progression of constraints (see arxiv:1205.5254) by incorporating first the long-baseline plus solar and KamLand results, then adding short-baseline reactor experiments, and finally adding the atmospheric neutrino information.

Current data in the plane sin^2 theta_23 versus sin^2 theta_13, once all other parameters are marginalized away, show to slightly favor non-maximal mixing, with values of sin^2 theta_23 at 0.4 and 0.6. This both for the normal and inverted hierarchy. Solar and KamLand data prefer a value of sin^2 theta_13 of 0.02, so this is unable to lift the degeneracy. Adding short-baseline reactor constraints, the lower value of sin^2 theta_23 becomes slightly favoured: the first octant solution. This especially in the scenario of a normal hierarchy.

Finally if we add the atmospheric neutrino data, we end up with a marked preference for the first octant. As for the CP violating phase, delta, it is basically unconstrained. If one includes short-baseline results still delta remains free to vary. With atmospheric nu data, we get a marked preference for delta of about pi, since this helps fitting sub-GeV electron-like excess seen in SuperKamiokande.

So if we make an overall synopsis of the oscillation parameters, one sees that the less-well-constrained parameter is of course delta; as for sim^2_23, the first octant is preferred but both are allowed at three-sigma. There are however no hints for a preference in the hierarchy. As for the fractional 1-sigma accuracy in the parameters, the dm_12^2 parameter has 2.6% accuracy, while sin^2 theta_23 has a 14% accuracy. The electron neutrino can thus be described as a superposition of three states with the mentioned coefficients.

The graph below, showing the measurement of the relevant parameters as a function of the number of standard deviations, is the best possible summary of the situation.

A final message from Egidio was that it is important that the MINOS and T2K and SK collaborations perform full-fledged three-neutrino analyses of combined appearance and disappearance datasets, because in 2-nu analyses the errors are comparable to the value of dm^2_12.

An exercise, but I would call it a scenario, was then proposed. Assuming that m_2b is non-zero (Klapdor et al.), and the recent claim from the south-pole collaboration that the sum of neutrino masses is in the 0.1-0.54 eV plane, then there is a narrow region of compatibility with present data. Not excluded yet by KamLand-Zen and EXO. If we believe it, then the mass of each neutrino is expected to be slightly below 0.2 eV. This is in the reach of Planck, and of double-beta decay experiments ! So this scenario, while not totally crazy, could be proven this year. Very interestingly, since the allowed region is very small, one can really tell if the Majorana phase is positive in this scenario.

In conclusion, we still have a lot to learn about the electron neutrino as a superposition of states, in many different and maybe surprising ways.

“As for the CP violating phase, delta, it is basically unconstrained. If one includes short-baseline results still delta remains free to vary. With atmospheric nu data, we get a marked preference for delta of about pi, since this helps fitting sub-GeV electron-like excess seen in SuperKamiokande.”

This is the first time that I can recall seeing a published result indicating an experimental suggestion of a non-zero CP violating phase in the PMNS matrix. Is this a real effect made possible by a longer data set, or could it reflect early undiscovered experimental issues with SuperK that only got smoothed other later. Both inferrences seem plausible given that this is the oldest of the experiments if I recall correctly by quite a bit.

You are right ohwilleke, delta_CP is still unconstrained. I think the (quite mild) constrain I reported about above is only there if you entertain yourself with combinations that may be subject to unwanted assumptions… The time to measure delta_CP has not yet come, but see other talks for interesting prospects.

Cheers,

T.