If neutrinos are Majorana particles it is possible that neutrinoless double decay happens, with a diagram like this one:


However it seems to me that this diagram requires the Majorana neutrino to flip its chirality: for instance if it is produced by the 'lower' W its chirality is right-handed but in order to be absorbed by the 'upper' W it needs to be a left-handed state.

If chirality switches are possibile, why and how? It seems to me that if this is the case it must be a very important matter but I don't see a lot of people talking about it, so my guess is that there is something that I'm misunderstanding. Is chirality a well defined property for majorana particles?

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    $\begingroup$ The usual Dirac mass flips the fermion chirality. So what is the big deal? $\endgroup$
    – MadMax
    Jan 18 at 23:25

1 Answer 1


This review might help. It is impossible to have a chiral (Weyl) Majorana neutrino. An easy mnemonic for the only part of your diagram that is relevant, the virtual $W^-W^-\to e^- e^-$ part, is the following, that eschews Majoranery for L-SU(2) quantum numbers (so, charge) and lepton number violation, achievable through a huge Majorana mass term violating lepton number for the sterile/inert SU(2) singlet neutrino component.

It is a schematic fantasy trail-map mnemonic which you may well have been exposed to in the simplest see-saw mechanism; I use it to summarize the basic fact to students who don't want to get caught up in charge conjugation and Lorentz considerations, amply discussed on this site here. Consider only one generation for simplicity.

Think of a $W^-$ zipping along (you chose the "lower" one) and resolving to a (L) $e^-$ and an active antineutrino, $\overline{\nu_L}$, so a R part of an SU(2) antidoublet. This antineutrino will have an "adventure" as it propagates, and ultimately turn to an active (L) neutrino, $\nu_L$, part of an SU(2) doublet, which is hit by the second $W^-$ and turns into a (L) $e^-$.

The "adventure" is a long spin 1/2 line which starts as a R antifermion and ends up as a L fermion, active under SU(2) in different ways. It has three steps.

  1. As you may have learned from the see-saw paradigm, the propagating active neutrino states are not pure doublets or antidoublets. They have an infinitesimal component of an SU(2) inactive, singlet field $\eta$, i.e. the propagating field at the first stage of the "adventure" is really $\overline{\nu_L}+\theta \overline{\eta_L} $ for a minuscule parameter $\theta$. (This parameter is loosely a normal charged lepton -like mass over a huge gut mass for the inactive component.) So the first step is to convert the active antineutrino to an inactive one with the same chirality, with strength $\theta$.

  2. The second step of the "adventure" is to have the R antineutrino flip its chirality and lepton number through the huge Majorana mass M term for η, and turn to $\eta_L$, out of sight, at the GUT scale; and with no SU(2) numbers to flip. Mass terms flip chirality. Steady, now, this "conversion" has a huge likelihood, given the huge M. Postpone judgement for a second.

  3. The third and final step is for the $\eta_L$ to overlap back to the mass-matrix-eigenstate neutrino, whith which it overlaps by θ as seen above. Now it is mostly $\nu_L$ active, part of the SU(2) doublet, ready to absorb the other $W^-$ and turn to an electron.

So the three factors for the entire "adventure" multiply to ~ $\theta^2 M$, which, wonder of wonders!, is actually the see-saw value for the neutrino mass, less than an eV. The huge M of the chirality flipping, lepton number violating mass term was overwhelmed by the square of θ .

As you might confirm in the review article cited, with real formulas, this basic pattern is robust: it obtains on all see-saws, i.e. the simplistic story outlined can be twisted and generalized without crucial dependence on details.

  • Neutrinoless double β decay amps basically go like the real lab neutrino masses.
  • $\begingroup$ Thank you very much, this is very helpful but there are a few things I'm not quite sure about yet. Could give a reference on the review or somewhere else about the first point? the fact that active states have an infinitesimal inactive field component. And also, while it kind of makes sense, how exactly does the majorana mass term allow the chirality flip? I mean it is not quite clear to me why the mere fact that there is a mass term would allow this to happen. Thank you again for your helpful answer $\endgroup$
    – Crucio
    Jan 18 at 9:29
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    $\begingroup$ Seesaw. $\endgroup$ Jan 18 at 9:50
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    $\begingroup$ I added the link in the answer. I've deliberately resisted writing down the Majorana mass term, as it confuses some students, with misleading RR or LLs. It does flip chirality, as you may confirm from the Weinberg operator in a world without Dirac masses, and without extra species, such as R neutrinos. In a fantasy world with just L neutrinos and R antineutrinos, the Weinberg term , a Majorana mass, converts a propagating L neutrino to a R antineutrino. $\endgroup$ Jan 18 at 17:00

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