Dear Newton,

neutrinos and anti-neutrinos will never really mix differently if the CP phase delta has a trivial value. When you look at the oscillation probabilities, they contain the same set of three mixing angles, no matter if you look at the neutrino or anti-neutrino formulae.

However, in principle the delta can yield a difference. Mathematically, the mixing matrix for anti-neutrinos is the complex conjugate of the mixing matrix for neutrinos, and this is the only place where differences can in principle arise from. However, looking at the reference I had sent you previously, you can see what happens for trivial values of delta:

-If delta=0, then the oscillation formulae for neutrinos and anti-neutrinos are trivially identical.

-If delta=pi, then the oscillation formulae look as if they differed by 2pi, since the probabilities for anti-neutrinos come with -delta instead of +delta for neutrinos. However, since the CP-phase only appears without prefactors within sin or cos terms, the oscillation probabilities are in fact identical for both neutrinos and anti-neutrinos.

Put differently, CP violation really means an observable difference between neutrinos and anti-neutrinos. As long as there is no observable effect, there is no real CP violation.

Best regards,

Alexander

Dear Alexander Merle,

This explanation is true as long as we start with the same set of three mixing angles for neutrinos and anti-neutrinos. But my point is that, can one assume that, there are 3 mixing angles for neutrinos and another 3 mixing angles for anti-neutrinos. In this case Upmns itself are different for particle and anti-particle.

regards,

Newton

Dear Newton,

no, one cannot. The reason is that the masses of neutrinos and anti-neutrinos must be equal, as long as CPT is conserved. And to the very best of our knowledge, we have not observed any CPT violation anywhere. On the contrary, we have extremely stringent limits on such violations: e.g., looking into the PDG, the relative difference between the muon and anti-muon lifetimes is smaller than 10e-5, and the relative difference between the electron and positron masses is even smaller than 10e-9 - and compare this to the few per cent accuracy with which we can determine leptonic mixing angles.

As long as this is the case, we have to assume that the "3D coordinate system" of the anti-neutrino mass eigenstates is identical to that of the neutrino mass eigenstates (which is trivially the case if neutrinos are Majorana fermions). So, as long as CPT is conserved, it does not make sense to consider mixing that is different for neutrinos compared to anti-neutrinos (and even then it would only make sense if neutrinos were Dirac-fermions, which is possible but unlikely, given that even within the Standard Model we do have lepton number violation at the non-perturbative level).

Best regards,

Alexander

Dear Alexander Merle,

Thanks for this explanation. So CPT conservation is  the key point. And if in near future, if we have any signal of CPT violation then one should start talking about mixing that is different for neutrinos compared to anti-neutrinos.

regards,

Newton

Dear Newton,

yes, you're right. The practical problem, though, is that the limites on CPT violation are not only quite stringent, but they also appear in different types of settings. If we e.g. take the limit from the electron/positron mass, we would expect CPT violation to have a relative size of 10e-9 at most. If we hence observe CPT violation in the near future, it will be not more than this small number, and this translates into the difference of "neutrino mixing angles" vs. "anti-neutrino mixing angles", which we would then also expect to be in the realm of 10e-9.

However, our current experiments constrain the values of leptonic mixing angles within about the 10% to 1% range (depending on which mixing angle you're looking at), so we would need an increase in our experimental precision by about 7 order of magnitude at least. This is not realistically achievable with neutrino experiments, certainly not with the current technology, and with very high certainty also not with any future technology, given how elusive and hard to detect neutrinos truly are.

I am thus very skeptical that we could at all find CPT violating effects in the neutrino sector, since these particles are experimentally to difficult to detect. In other sectors, which are easier to control experimentally, it may at some point be possible, though.

Best regards,

Alexander

Consider the interaction term ψ̅Li γμ Oij ψLj WμL + h.c.

Here i,j are generation indices and suffix L denotes left handedness. This two terms contain required information on neutrino mixing valid for neutrinos as well as antineutrinos. There is no other independent term in interaction Lagrangian,  which is valid only for the antineutrino sector. Now the mixing matrix O, for the Dirac case, contains three real angles and one complex phase. Therefore in the O* matrix, mixing angles remain unaltered but the complex phase picks up a negative sign. That's all.