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The seesaw mechanism is a theoretical model of neutrino masses that has the side benefit of (arguably) naturally explaining why neutrinos are so much lighter than the other massive Standard Model particles. It involves adding heavy "sterile neutrinos" with Majorana masses on the order of the GUT scale. The seesaw mechanism predicts that (for one lepton generation) the geometric mean of the light and heavy neutrino masses should "naturally" be on the order of the other Standard Model particle masses (the story is similar for multiple lepton generations). This prediction seems to be experimentally plausible based on the estimates of the (light) neutrino masses that come from neutrino oscillations and other observations.

The seesaw mechanism also predicts the violation of conservation of total lepton number. The clearest experimental signature of this violation is believed to be the process of neutrinoless double beta decay. Many experiments have tried to detect NDBD, but so far none have succeeded (to the physics community's standards of quality and reproducibility).

  1. What is the order of magnitude of the quantitative rate of NDBD predicted by the seesaw mechanism (with the sterile neutrino Majorana masses on the GUT scale)? (I know there are different variations of the seesaw mechanism, but presumably they all agree to within a few orders of magnitude.)
  2. What is the upper limit on the NDBD rate that hasn't been ruled out by experiment?

I'm curious how these two values compare. Which of these statements is the best summary of the current situation?

A. The seesaw mechanism (with "natural" values for the heavy Majorana masses) predicts NDBD rates that are clearly higher than our current experimental limits. The "natural" seesaw mechanism is in trouble.

B. The seesaw mechanism predicts NDBD rates that are much lower than the current and foreseeable experimental capabilities to detect. We won't be able to confirm or reject the "natural" seesaw mechanism for many years, if ever.

C. The seesaw mechanism predicts NDBD rates that are at or only slightly below our current experimental capabilities to detect. We could plausibly detect it in the near future. (It should be noted that research physicists have a personal incentive to exaggerate the likelihood of this scenario.)

D. The "natural" seesaw mechanism has so many unknowns and so much wiggle room that it can't even nail down the NDBD rate to within a few orders of magnitude. The theory therefore doesn't have much explanatory power, and can at best only give retrodictive explanations rather than predictions.

(Note: the active neutrino masses, and therefore the NDBD rate, can be pushed down arbitrarily low by making the sterile neutrinos sufficiently heavy. But this doesn't solve the neutrino mass hierarchy problem; it simply shifts the question of why the active neutrinos are so light to the question of why the sterile neutrinos are so heavy. In order to preserve naturalness, you need the sterile neutrino masses to be on the same scale as the other large energy scale in the Standard Model - the GUT scale.)

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    $\begingroup$ Required reading. End of sec 2.1. $\endgroup$ Mar 15 '20 at 16:22
  • $\begingroup$ @CosmasZachos So it looks like the answer is D? There is no particular reason to believe that the question of whether neutrinos have Majorana mass terms will be settled in the foreseeable future? $\endgroup$
    – tparker
    Mar 15 '20 at 16:57
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    $\begingroup$ Yeah, looks like D with a dash of B... The mechanism, so far is a pedagogical device to organize thinking in the community. Has been, since its inception, as witnessed. $\endgroup$ Mar 15 '20 at 17:41
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The answer is D, but not because of anyone's malice -- it's just the hand nature has dealt us.

The neutrinoless double beta decay rate is proportional to $|m_{\beta\beta}|^2$, where $m_{\beta\beta}$ is the "effective Majorana mass", which in turn depends on the elements of the PMNS matrix. Here's a plot of $|m_{\beta\beta}|$ as a function of the mass of the lightest neutrino, from a recent review.

enter image description here

There are a couple points to make about this plot.

  • The expected parameter space for the vanilla seesaw mechanism has not been probed at all yet. Many previous experiments were prototypes meant to be scaled up. Unfortunately, every time one of these prototypes sees nothing (as is completely compatible with our expectations), a tidal wave of news releases declares "experiment rules out Majorana masses!" This is a general pattern for BSM experiments that makes us look clueless -- if you just read the news, you would easily get the false impression we've ruled out WIMPs, axions, and sterile neutrinos a hundred times over now.
  • There are no epicycles here. The plot just shows the most basic expectations for the simplest possible seesaw mechanism, assuming only that heavy sterile neutrinos with Majorana mass terms exist, and could have been drawn in much the same way 40 years ago. There are just two options, corresponding to whether the third neutrino is heavier ("normal") or lighter ("inverted") than the rest. You can of course add epicycles to make the experiment harder, but nobody has had to do that.
  • The next generation experiments, under construction now, have a clear physics goal of probing Majorana masses for the inverted neutrino hierarchy. The goalposts have not shifted: the same target has been there for decades, and could have been probed back then with more funding. (In fact, the epicycles go the other way: the less sensitive predecessor experiments could have been motivated by more complicated models, such as having all three neutrino masses high and using a trick to get around the resulting cosmological constraints. We're probing simpler models now than a decade ago.)
  • The "normal" hierarchy, which is possibly more likely, unfortunately can yield destructive interference in $|m_{\beta\beta}|$, dramatically lowering it. Again, it's not a model builder's trick -- that's unfortunately just how the simplest possible scenario plays out.
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    $\begingroup$ This may just be a quibble, but I wouldn't actually describe the situation that you've explained as option D, which I meant to indicate that the seesaw mechanism can't explain whether not we'd expect to have already observed NDBD with our current experiments. I would say that your explanation corresponds to option C for the inverted neutrino hierarchy and option B for the normal hierarchy. $\endgroup$
    – tparker
    Sep 13 '20 at 14:01
  • $\begingroup$ Also, if I could editorialize a bit regarding your point 1: I think that the experimentalists who propose, design, and build these experiments share some of the blame with journalists for giving a misleading impression of their failures to detect BSM physics. They usually describe their future experiments as "may be able to detect [some BSM physics phenomenon]", without conveying any (even qualitative) estimate of the likelihood of the experiment's succeeding. As such, it's natural for non-experts to assume that there was a decent theoretical probability that the experiments would have... $\endgroup$
    – tparker
    Sep 13 '20 at 14:11
  • $\begingroup$ ... detected the phenomenon if it actually occurred in nature, and so the experiment's failure does indeed meaningfully shift the Bayesian estimate of the probability of the phenomenon's existence. I understand that, from a grant-receiving perspective, it doesn't look great to write "There's a small chance that our proposed experiment will detect X, but even if X actually exists then we almost certainly won't detect it." But I think that intellectual honesty requires any experimental proposal to attempt to assess and clearly convey the order-of-magnitude probability of the experiment's ... $\endgroup$
    – tparker
    Sep 13 '20 at 14:14
  • $\begingroup$ ... actually being able to detect the phenomenon in question, given the most natural theoretical models. $\endgroup$
    – tparker
    Sep 13 '20 at 14:17

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