The ever-increasing proton lifetime

[ohwilleke]

That is not how probability works.

You will note that I prefaced this statement with the statement "If you engage in the rather deplorable Baysean notion that the many worlds folks try to engaged in that there is a distribution of possible B and L values at the beginning of the universe from which the initial conditions of the universe are chosen on a probabilistic basis," specifically disavowing this technique which I agree is dubious because it formulates a prior distribution without any basis for doing so, when the only reason to use Baysean statistics in the first place is that you have some basis for presuming one prior distribution over another. But, because this kind of reasoning is frequently used by practicing cosmologists, I nonetheless raised it.

If you roll a die, are numbers different from 1 "almost infinitely likely" relative to the single number 1?

If your die has an infinite number of sides, then yes, it is true. The only known limit in principle on the possible values of B or L is that they be on the line of integers, which is, for B and for L, independently, an infinite set that can take any positive or negative integer value.

Looking at the actual values, which are really, really large, is instructive of what kind of probability distribution might make sense, if you are inclined to give the admittedly dubious Baysean multi-worlds approach to fundamental constants of the universe including the initial values of B and L any credit. This size of these values suggest very extreme tails.

Astronomers have been able to estimate the baryon number of the universe to one or two significant digits, but while they have a good estimate of the number of Standard Model leptons in the universe (to one significant digit) they have a less accurate estimate of the lepton number of the universe since they don't know the relative number of neutrinos and antineutrinos.

As one science education website explains:

Scientific estimates say that there are about 300 [neutrinos] per cubic centimeter in the universe. . . . Compare that to the density of what makes up normal matter as we know it, protons electrons and neutrons (which together are called “baryons”) – about 10-7 per cubic centimeter. . . . the size of the observable universe is a sphere about 92 million light-years across. So the total number of neutrinos in the observable universe is about 1.2 x 10^89! That’s quite a lot – about a billion times the total number of baryons in the observable universe.

Thus, the baryon number of the universe is roughly 1.2x10^80, to about one significant digit.

To determine B-L for the universe, the number of protons and number of charged leptons cancel, leaving the number of neutrons (about 5*10^78 given the ratio of protons to neutrons in the universe) minus the number of neutrinos plus the number of antineutrinos. The combined number of neutrinos is 1.2 x 10^89, but we don't have nearly as good of an estimate of the relative number of neutrinos and antineutrinos.

The number of neutrinos in the universe outnumber the number of neutrons in the universe by about 2.4*10^10 (i.e. about 24 billion to 1), so the conserved quantity B-L in the universe (considering only Standard Model fermions) is almost exactly equal to the number of antineutrinos in the universe minus the number of neutrinos in the universe.

As of March 2013, the best available observational evidence suggests that antineutrinos overwhelmingly outnumber neutrinos in the universe. As the abstract of a paper published in the March 15, 2013 edition of the New Journal of Physics by Dominik J Schwarz and Maik Stuke, entitled "Does the CMB prefer a leptonic Universe?" (open access preprint here), explains:

Recent observations of the cosmic microwave background at smallest angular scales and updated abundances of primordial elements indicate an increase of the energy density and the helium-4 abundance with respect to standard big bang nucleosynthesis with three neutrino flavour. This calls for a reanalysis of the observational bounds on neutrino chemical potentials, which encode the number asymmetry between cosmic neutrinos and anti-neutrinos and thus measures the lepton asymmetry of the Universe. We compare recent data with a big bang nucleosynthesis code, assuming neutrino flavour equilibration via neutrino oscillations before the onset of big bang nucleosynthesis. We find a preference for negative neutrino chemical potentials, which would imply an excess of anti-neutrinos and thus a negative lepton number of the Universe. This lepton asymmetry could exceed the baryon asymmetry by orders of magnitude.

Specifically, they found that the neutrino-antineutrino asymmetry supported by each of the several kinds of CMB data was in the range of 38 extra-antineutrinos per 100 neutrinos to 2 extra neutrinos per 100 neutrinos, a scenario that prefers an excess of antineutrinos, but is not inconsistent with zero at the one standard deviation level. The mean value of 118 antineutrinos per 100 neutrinos would mean that B-L is hopelessly out of whack relative to zero.

A 2011 paper considering newly measured PMNS matrix mixing angles (especially theta13), and WMAP data had predicted a quite modest relative neutrino-antineutrino asymmetry (if any), but it doesn't take much of an asymmetry at all to make B-L positive and for the neutrino contribution to this conserved quantity to swamp the baryon number contribution.

It is also worth recalling that while the Standard Model does have one process that violates B and L conservation that even that process conserves B-L. So, unless the ratio of antineutrinos to neutrinos in the universe is just right, it is impossible under any Standard Model process to have initial conditions of B=0 and L=0. To the extent that the ratio of antineutrinos and neutrinos is not exactly 1-1 right down to 89 orders of magnitude, the beauty of B=0 and L=0 that you get in a pure energy initial condition is unattainable anyway even considering sphaleron processes at the very dawn of the Big Bang.

In the Standard Model, baryon number (the number of quarks minus the number of antiquarks, divided by three) is conserved as is lepton number (the number of charged leptons and neutrinos minus the number of charged antileptons and antineutrinos), except in sphaleron process. Per Wikipedia:

A sphaleron (Greek: σφαλερός "weak, dangerous") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and it is involved in processes that violate baryon and lepton number. Such processes cannot be represented by Feynman diagrams, and are therefore called non-perturbative. Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy (in the infinite-dimensional field space), much like the saddle point of the surface z(x,y)=x2−y2 in three dimensional analytic geometry.

In the standard model, processes violating baryon number convert three baryons to three antileptons, and related processes. This violates conservation of baryon number and lepton number, but the difference B−L is conserved. In fact, a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, and hence a quark may be converted to 2 anti-quarks and an anti-lepton, and an anti-quark may be converted to 2 quarks and a lepton. A sphaleron is similar to the midpoint (\tau=0) of the instanton, so it is non-perturbative. This means that under normal conditions sphalerons are unobservably rare. However, they would have been more common at the higher temperatures of the early universe.

The trouble is that if you start with B=0 and L=0, as you would expect to in a Big Bang comprised initially of pure energy, it is hard to determine how you end up with the observed values of B and L in the universe which are so far from zero.

The mainstream view among physicists, although there are some theorists who dissent from this analysis (there was a link for this to a paper from Helesinki U. but it went bad and I can't figure out how to fix it), is that Standard Model sphaleron processes in the twenty minutes during which Big Bang Nucleosynthesis is believed to have taken place, or the preceding ten seconds between the Big Bang and the onset of Big Bang Nucleosynthesis, can't account for the massive asymmetry between baryons made of matter and baryonic anti-matter that is observed in the universe (also here) without beyond the Standard Model physics (also here).

For example, even if you have BSM processes that allow neutrinoless double beta decay and proton decay, that isn't good enough. Those processes have to produce the huge B and L numbers seen today for the entire universe fast, in just half an hour or so.

In any case, the line between Big Ban Nucleosynthesis which is figured out using Standard Model physics with some very basic assumptions and compared to real evidence that matches it except for a modest Lithium-7 problem, and the ten seconds before, is really where the science of cosmology gives way to speculation. Physics is incredibly powerful, but our understanding of the first ten seconds out of 13 and change billion years still has mostly questions and few answers. Still, the credibility we get from applying the SM to BBN means that once you get to BBN the BSM physics have to be pretty modest to non-existent at that point. You need to fit more or less almost all of your BSM physics that violations B number and L number and B-L number conservation into the first ten seconds or however long the pre-BBN period actually was, which means you need a phase transition and it has to be an incredibly fast and efficient process working in one direction on the B side and in one direction on the L side, and unless you get a perfect balance of neutrinos and antineutrinos in the universe today, you still can't have B=0, L=0, which is pretty pointless with B=0 but L not equal to 0, in terms of beauty.

Likewise, after that point, sphaleron processes should be so rare that they can't explain the baryon asymmetry of the universe (BAU), which is one of the great unsolved problems in physics, unless you come to terms with the totally arbitrary assumption that the initial conditions of the universe had equal amounts of matter and antimatter, which has no real solid basis except that it seems pretty and unique.

Also, even if you have an alternative to sphaleron processes, most of the popular supersymmetric and other BSM models which have B and L violations, like the sphaleron process, conserve B-L at least, so if the neutrino-antineutrino balance isn't perfect it doesn't work anyway.

(By the way, if your DM has the appropriate matter v. antimatter character to partially balance out a B-L issue with ordinary matter, you still have a problem if your DM is in the keV mass range or more because the number of neutrinos is so huge that any meaningful imbalance there can't be meaningfully mitigated with DM because there are orders of magnitude fewer DM particles than neutrinos. The only kind of dark matter that is numerous enough to address a B-L imbalance due to neutrinos not being evenly balanced between neutrinos and antineutrinos to many significant digits is axion-like dark matter because only it has enough particles to rival the number of neutrinos in the universe. And, if you use axion-like DM to address a B-L imbalance you also need to have all of it created prior to BBN if the processes affecting ordinary matter conserve B-L.)

Another reference that could have been included earlier: As of a 2006 paper discussing the experimental evidence for baryon number non-conservation (and citing S. Eidelman et al. (Particle Data Group), Phys. Lett. B592 (2004) 1): "No baryon number violating processes have yet been observed." Some of the processes that B-L conservation might make possible in some beyond the standard model theories, such as proton decay, are also not observed.