Orodruin said:
You do realize that this leaves you only with the possibility of having Majorana masses and that essentially means lepton number violation. Weak interactions do not violate lepton number (at the perturbative level).
I don't agree. I agree that this leaves you with mass that can't arise from interaction with a Higgs field like the other Dirac fermions do. But, I don't agree that Dirac mass arising from a Higgs field interaction and Majorana masses are the only theoretically imaginable means by which mass can arise, even though it is often posed as an either/or possibility. There are more than a dozen well developed
theoretically possible mass generation mechanisms. The SM Higgs mechanism is the one that works for all of the fundamental particles in the Standard Model but the neutrinos. But, there is more than one possible alternatives to it that could be applied to the neutrino case.
Mass could also arise by some other unspecified mechanism. This is the kind of thing that theoretical physicists get paid the big bucks to come up with. Until Higgs and company came along, nobody had imagined that fundamental particles were fundamentally massless and acquired rest mass only through interactions with a scalar field. I'm sure that someone could come up with another alternative, although I don't intend the examples that I myself have given to be anything more than place fillers for "some" unspecified alternative. Honestly, there are lots of mysteries in physics surrounding mass in general and neutrino mass in particular. I am skeptical of the proposition that we have considered every possible solution to the problem even though the question is often presented that way.
In any model in which neutrinos do not acquire mass via a Higgs field Yukawa, you have to do something different than the other fundamental Standard Model particles in any case. It is not as if you would lose theoretical consistency in the process.
Including right-handed neutrinos into your model there is nothing preventing them from having a Majorana mass term and that automatically means different masses. However, the L and R would not be the mass eigenstates - it is just that the heavy eigenstates would be mostly R in the usual seesaw.
There is no other instance in the Standard Model when particles that differ in no respect other than parity have different masses. Indeed, I can't even think of any composite particle for which this is the case. And, there is no terribly compelling reason that neutrinos should be different in this respect.
First of all, the Standard Model already has B+L violation built in through perturbative effects. Second, the vanilla leptogenesis models do not depend on the same CP-violating parameters as neutrinoless double beta decay.
B and L Violation Are Theoretically Possible In The SM, But Doesn't Necessarily Actually Happen
The SM has the sphaleron a theoretically possible process that violates B and L number individually (while preserving B-L). But, this phenomena has never been observed even though it is theoretically possible. And, even if it does exist it doesn't generate enough B and L violation to lead to observed matter-antimatter asymmetry from a starting point of B=0 and L=0 at the Big Bang. And, I would not be at all surprised if it was discovered that there was some accidental symmetry or subtle relationship not yet recognized that actually makes this phenomena impossible.
By analogy, General Relativity recognizes the theoretical possibility that a black hole could have a mass of less than three times the mass of the Sun (these are sometimes called "primordial black holes"). But, no one has ever observed such a black hole, and no one has identified any process after the Big Bang era in which one could form. Even hypothetical processes that could generate primordial black holes in the Big Bang era are very sketchy and have not been worked out in convincing detail (which would require renormalizing all of the laws of physics to extremely high energies beyond the ordinary domain of applicability of these physical laws in the UV direction that has been well confirmed by experiment).
It is entirely possible that no such process ever existed in the Big Bang era either and that black holes with a mass of less than three times the mass of the Sun do not exist now, never have existed, and can't exist at any point in the future, even though they would not violate any of the laws of physics if they did exist. The same thing might be true of sphalerons.
B and L Violation Aren't Theoretically Necessary Or Compelled By Empirical Evidence
Also, there is no a priori reason that aggregate baryon number or aggregate lepton number needs to be zero immediately following the Big Bang. This would be neat and beautiful if it was the case, but ultimately, one can have a perfectly consistent set of laws of fundamental physics in which these numbers at t=0 are not equal to zero, in much the same way that the other fundamental constants of the Standard Model are non-zero at that point. Nobody insists that the fine coupling constant be zero at t=0.
This assumption is not a provable law of Nature and Nature is not obligated to tell us "why" it chose to set the laws of physics and physical constants of Nature at the value that it did.
If theoretical physics has had once vice over the last thirty years, it has been its insistence that "naturalness" is a worthwhile principle from which to generate hypotheses about the laws of Nature, a conceit that has wasted an immense amount of time and resources from very smart people who should know better.
I'm not saying that theories that could get you to aggregate B=0, L=0 for the universe at t=0 aren't well motivated theoretically, if by beauty if nothing else. I'd love to see someone come up with a scientifically demonstrable and provable process by which this could happen.
But, if you don't have to make an assumption to be theoretically consistent, and you also don't have to make that assumption to be consistent with the experimental and observational evidence, you shouldn't insist that only theories that include that assumption are considered. Many of the big breakthrough of modern physics have involved abandoning assumptions that seem obvious or beautiful or natural, but aren't necessary to be theoretically consistent and aren't necessary to be consistent with empirical evidence.
Nothing fundamental in physics breaks down if we abandon the assumption that aggregate B=0, L=0 for the universe at t=0, so it is not absolutely necessary to have either B number violation or L number violation in a fundamental theory of everything (TOE) or a grand unified theory (GUT).
Also, the PMNS is the exact lepton analogue of the CKM. It tells you the mismatch of the mass and interaction eigenstates of the W.
No it isn't.
In the CKM matrix you are looking at transitions from a set containing three fermions via the W+ boson to a disjoint set containing three different fermions, and from the second set via the W- boson to the first set. In short, it involves the interactions of seven particles (eight if W+ and W- count as different particles) in a particular way.
In the PMNS matrix you are looking at transitions from a set containing three fermions without an intermediary to a disjoint set of three masses. In short it involve the interaction of three different particles that can each be in one of three different states. I have not seen any description of the PMNS matrix that suggests that neutrino oscillation is mediated by a boson of any kind (e.g. the W boson). And, unlike the CKM matrix which governs flavor transition of all six quarks, the PMNS matrix only applies to three of the six leptons.
And, of course, if the PMNS matrix were coding a W boson mediated transition, it would require pairs of W boson interactions, because neutrino interactions are charge neutral, but W boson interactions change charge.
They are similar, and they are analogous, but they aren't exactly analogous.
You can have W boson interactions causing, for example, a tau to decay to a tau neutrino and a W- boson which in turn decays to a muon and a muon anti-neutrino, but those interactions are not included in the PMNS matrix which applies only to the neutrino sector. You don't need the equivalent of the CKM matrix or PMNS matrix for charged lepton flavor changes via the W boson because the charged leptons are democratic.
Similarly, you can have in the Standard Model, a regular neutrino that emits a W+ boson giving rise to a charged lepton of the same type, a charged anti-lepton of the same or another type, and a neutrino of the same type as the charged anti-lepton.
An exactly analog to the CKM matrix would be a matrix demonstrating the probability of W boson transitions from charged leptons of a particular type to charged leptons of another type together with a neutrino and visa versa. And, indeed, you can create a matrix that carries out this exact analogy. But, every entry in the matrix would be exactly the positive square root of 1/3 (i.e. 0.5773502 . . .), which would be trivial. Who knows? If the hints of charged lepton non-universality in a few outlier LHC results that have been observed lately turn out to be true, it may be necessary to develop a non-trivial exact analog to the CKM matrix.
But, I have seen nothing in the literature to indicate that the PMNS matrix is coding virtual pairs of these W boson interactions. Neutrino oscillation is not mediated by W bosons or in the Standard Model by any kind of particle.
