# I B, L and Majorana Neutrinos

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1. Jan 25, 2018

### ohwilleke

Background

The Standard Model conserves baryon number, which is equal to (quarks divided by three) minus (anti-quarks divided by three) and lepton number, which is equal to leptons minus antileptons, separately, except in sphaleron processes which conserve B-L, but not B or L separately.

There is no experimental evidence of B non-conservation or L non-conservation (page 22-25 and 27-30) (and also here) under any circumstances. Processes that hypothetically would not conserve B or L, such as proton decay, neutrinoless double beta decay, and flavor changing neutral currents, have not been observed in high precision measurement. These processes are, at a minimum, vanishingly rare under the conditions of the modern universe (including the extreme conditions created in particle physics experiments) and the Standard Model states that they don't happen at all.

As I understand the concept, if neutrinos are Majorana particles, then they are their own anti-particle and indeed Majorana neutrinos are necessary for processes that do not conserve lepton number to exist.

But, neutrinos are absolutely critical in maintaining lepton number conservation in all sorts of relatively common and routinely observed interactions, such as beta decay and other W boson decays, so they must carry lepton number.

In simplified terms, neutrinos are exclusively left-handed in the Standard Model, and antineutrinos are right handed in the Standard Model. It seems that it is possible to distinguish neutrinos from antineutrinos experimentally, even if this may be difficult to do (indeed it is difficult to observe any neutrino full stop).

Question: Do Majorana Neutrinos Have Lepton Number?

But, how can a neutrino have positive or negative lepton number, as they must if all sorts of other lepton number conserving interactions in the Standard Model are to work if they do (indeed, the neutrino was only hypothesized to exist in the first place decades before it was directly observed, in part, because of the need to conserve quantities including lepton number in observed interactions), if a neutrino and an antineutrino are in fact one and the same particle?

It would seem that Majorana neutrinos have to at least preserve some equivalent of lepton number the vast majority of the time, or the entire concept would have been dismissed out of hand as inconsistent with experimental evidence, but it isn't clear to me how it does that.

Parity is not, in and of itself, equivalent to matter-antimatter status in other particles, so why would a property that does not relate to baryon number or lepton number in any other kind of fermion do so with neutrinos?

There seems to be an effort to explain this here, but I don't quite follow the terse explanation given.

Related Question: Does a Majorana Neutrino Model Imply Oscillation Between Neutrino And Antineutrino States?

This question pretty much speaks for itself and also feeds into the next related question. An oscillation between a neutrino and an antineutrino would imply parity violation as well. Neutrinos are known to oscillate between flavors, but I'm not aware of evidence that neutrinos oscillate into antineutrinos and visa versa, even though there are theoretical constructs that would allow this to happen, at least in the Majorana Neutrino case. Does this just happen, the way neutrino flavor oscillation does in these constructs? Or does it take a trigger and energy to oscillate in this way?

Phrased another way, if Majorana neutrinos have lepton number, but don't conserve it, what processes give rise to lepton number violations in Majorana neutrinos?

Feynman diagrams make matter-antimatter annihilation appear equivalent (but rotated in the space-time dimension) to matter getting hit with a high energy photon and turning into antimatter in a very energy intense process that consumes 2mc^2 of energy per interaction.

When matter-antimatter states of hadrons oscillate it involves two parallel interactions of a composite object that has a mix of both matter and antimatter in it from the outset. But, this analogy doesn't work if neutrinos are fundamental particles as they are in the Standard Model. You'd need a preon theory for neutrinos in which they are composite particles made up of matter and antimatter preons, and not just a Majorana neutrino hypothesis for this to happen in an analogous manner. And, particle-antiparticle oscillations in hadrons aren't vanishingly rare and instead are ubiquitous, so you'd have to explain why they are common in hadrons and rare in neutrinos even if a preon theory was correct and the process was analogous.

So, it would be surprising if a neutrino that behaves as matter for lepton number conservation purposes could oscillate into something that behaves as an antineutrino for lepton number conservation purposes without a huge infusion of energy (well, given how light neutrinos are, not so huge, just 2mc^2, which at energy scales on the order of 1meV to 60 meV in a normal mass hierarchy with a near minimal experimentally supported set of neutrino masses, isn't all that much energy, but still isn't nothing either.

Related Question: Does CP Violation In Neutrinos Imply Lepton Number Non-Conservation?

On a related note, suppose that there is CP violation in neutrino oscillation (which the best experimental evidence tends to show even if the exact amount of the CP violation is hard to quantify). And, of course, neutrinos have no charge.

So, does this mean that there has to be parity violation in neutrinos, which would seem to imply lepton number violation in neutrinos (since parity and matter-antimatter status seem to be equivalent in neutrinos)?

Or, does CP violation in neutrinos simply mean that neutrinos and antineutrinos behave in a time-flipped manner in their oscillation equations with respect to each other, since CP violation that honors CPT symmetry collectively is equivalent to time reversal?

Last edited: Jan 25, 2018
2. Jan 26, 2018

### Orodruin

Staff Emeritus
This depends on how you assign lepton number. You could assign positive lepton number to the left-handed state and negative to the right-handed. In that case, the Majorana mass term breaks lepton number. Another option is assigning zero lepton number to the Majorana neutrinos. If you do that, what breaks lepton number is the interaction term with the charged lepton and the W. Where the lepton number is broken depends on how you assign it. Most people would probably assign it in the first manner, such that lepton number is broken by the Majorana mass term.

The most considered process to search for lepton number violation is neutrinoless double beta decay.

No. You can have CP-violation without breaking lepton number. However, if neutrinos are Majorana, there are additional CP-violating phases that come into play. These are not observable in lepton number conserving processes but could be seen in, for example, neutrinoless double beta decay.