A proton that is not bound into an atomic nucleus is stable (the mean lifetime of a proton which is at least 1.29*10^34 years, which implies that it would happen in not more than 1 in 10^24 protons over the entire 1.38*10^10 year lifetime of the universe), a result which follows naturally in the Standard Model from conservation of baryon number, quark confinement, and the fact that the proton is the baryon with the lowest rest mass.
When a neutron is not bound into an atomic nucleus and is at rest, it has a mean lifetime of about 14 minutes and 42 seconds +/- 1.5 seconds (which is equivalent to a half-life of 10 minutes and 11 seconds +/- 1.0 seconds), and naturally decays via weak force into a proton, an electron, an electron anti-neutrino, each of which has kinetic energy (and in about one in 1000 cases, also into electromagnetic energy in the form of a photon). The kinetic energy and the photon energy combined are equal to the difference in rest mass between the neutron and combined rest mass of the proton, electron and electron anti-neutrino times the speed of light squared, which turns out to be 0.782343 MeV/c2.
This mean lifetime is incredibly long compared to all other unstable particles in physics (except oscillating neutrinos). No other baryon (except the proton) has a mean lifetime of more than 10-10 seconds. No other meson has a mean lifetime of more than 10-8 seconds (the charge pion). The mean lifetimes of the muon (10-6 seconds), the tau lepton (10-13 seconds), and the even shorter mean lifetimes of unhadronized top quarks, the W boson, the Z boson and the Higgs boson, all of which are much less than 10-20 seconds. Gluons don't have a fixed lifetime, per se, but they are typically exchanged between other confined color charged particles in similar time frames, because they travel at speeds approximating the speed of light (a bit less than 3*10^8 meters) over distances on the order of the proton and neutron charge radius (i.e. about 0.8*10-15 meters), which is shorter than the mean muon life time.
The process by which free neutrons normally decay, called beta decay, is what causes nuclear radiation, although the rates of beta decay are lower when neutrons are bound in an atomic nucleus with protons, but the number of neutrons is high in relation to the number of protons.
In contrast, when protons are bound with a number of neutrons that produce a stable atomic isotype, beta decay does not occur and neutrons are stable. In part, this is because the decay of neutrons in a stable nucleus is offset in part by "inverse beta decay" in which an energetic proton emits a neutron, a positron (i.e. anti-electron) and an electron neutrino, and in part by electron capture in which an energetic proton and electron (perhaps one created in the ordinary decay of neutrons in the same atomic nucleus) merge to form a neutron and an electron neutrino (which is the anti-matter equivalent of the electron anti-neutrino produced in ordinary beta decay). There are also lower order (i.e. much less frequent) possible paths by which energetic neutrons and protons can form other kinds of hadrons with or without leptonic decay products.
Empirically, the neutron to proton ratio needed to make an atomic nucleus stable is between 1 and 1.537 (gradually increasing with larger atomic numbers), except in the degenerate cases of hydrogen (a bare proton without a neutron), and helium-3 (two protons and a neutron). As Wikipedia explains:
Neutron-proton ratio (N/Z ratio or nuclear ratio) is the ratio of the number of neutrons to protons in an atomic nucleus. The ratio generally increases with increasing atomic numbers due to increasing nuclear charge due to repulsive forces of protons. Light elements, up to calcium (Z = 20), have stable isotopes with N/Z ratio of one except for beryllium (N/Z ratio=1.25), and every element with odd proton numbers from fluorine to potassium. Hydrogen-1 (N/Z ratio=0) and helium-3 (N/Z ratio=0.5) are the only stable isotopes with neutron–proton ratio under one. Uranium-238 and plutonium-244 have the highest N/Z ratios of any primordial nuclide at 1.587 and 1.596, respectively, while lead-208 has the highest N/Z ratio of any known stable isotope at 1.537.
One can imagine an atomic nucleus which has zero protons and two neutrons, which is called a "bound dineutron". An atomic nucleus with no protons and an arbitrary number of neutrons in known, in general, as neutronium. A bound dineutron particle, like a neutron, would have no electrons and would therefore be chemically inert. It would have a rest mass of about 1.879 GeV before adjusting for mass due to the nuclear binding energy of the two neutrons. Since the strong nuclear force and weak nuclear force only act at short ranges, it would be collisionless and only influenced by gravity at ranges of less than the radius of a typical helium atom nucleus. Thus, if it were stable, it would be an excellent cold dark matter candidate.
Dineutron states were observed in 2012, but were transitory states that were not quite bound, rather than constituting a particle made up of bound neutrons which could in principle be stable. (Incidentally, unlike some neutral mesons, neutrons do not oscillate between neutrons and their anti-matter counterparts, which follows quite naturally from the conservation of baryon number in the Standard Model.) There appears to be a modest shortfall of nuclear binding force between a bound atomic nucleus and the bineutron state, but one could imagine that there could be some special factor that is ignored in other circumstances but is material in a nearly trivial two neutron system. For example, at the high energy scales present in Big Bang nucleosyntheis, the running of the Standard Model parameters with energy scale might permit the existence of bound bineutron states. Similarly lattice QCD studies with higher than physical pion masses (also here) find that bound bineutrons can exist. And other studies can't definitively rule out their existence and they might even play a role in determining alpha decay rates (although experimental evidence pretty clearly rules out truly stable dineutrons).
There also appear to be a set of phenomena common to not quite bound systems of baryons.
Neutron balls would not make good dark matter particles because they would clump together, and eventually make neutrons stars or other much larger objects. Free neutrons would much rather stick to nuclei. Also, we can rule of neutrons as dark matter particles because more neutrons in the early universe would effect the concentration of helium and deuterium.
In order to be consistent with all data sets, dark matter needs to be a non-electrically-charged fermion with a rest mass of ~2-10 keV.
Even if two neutrons could be stable, it doesn't imply that more in a nucleus would be. More generally, the experimental evidence tends to point pretty strongly towards dineutron states being quite unstable.
I wouldn't be surprised, however, if consideration of short lived dineutron states could help to resolve the lithium problem in Big Bang Nucleosynthesis and perhaps also shed light on hints of low levels of Boron that is observed in very old stars and not predicted by BBN.
WDM does fit the data better than CDM, but the 2-10 keV rest mass is as much a function of the mean velocity of DM particles implied if they are a thermal relic, as they do with the mass itself. If DM is generated in a manner that is not a thermal relic that produces the same mean velocity and mean density of DM particles, then the rest mass limitation is not so great, and one could imagine neutronium particles being produced by means other than thermal relic production.
The biggest problem with WDM at 2-10 keV is that particle physics experiments pretty definitively rule out any weakly interacting particle in that mass range. So, we can't have a 2-10 keV WIMP. We would need to have right handed (i.e. "sterile") 2-10 keV particle without electric charge, and the data favor a singlet dominate dark sector over a multi-generational one (although, if there were three generations of right parity dark matter particles and 2-10 keV was the lightest and the heavier ones decayed into the lighter ones with a mean lifetime of even 10 million years or less, that would work out.
A 2-10 keV DM particle could have a tiny Yukawa coupling to the Higgs boson in addition to its gravitational coupling (which would be experimentaly impossible to measure) even though it couldn't couple to photons or W bosons or Z bosons. Also, if it were a composite particle that was color charged, the gluon field would make them at a minimum not much lighter than a proton (ca. 970 GeV) (the lightest fermionic hadron) since even with zero rest mass quarks, lattice QCD predicts that a proton would weigh about 840 GeV +/-.
DM-genesis would still be a problem, however and some self-interaction term might make sense although it would require a new boson as well.
Also would DM have lepton number, baryon number, or some other DM number? Would it have anti-particles (presumably LH anti-matter and RH matter)?
Would it be generated from high energy gravitons much like photons can create particle-antiparticle pairs? What kind of circumstances would generate 4-20 keV gravitons and cause them to condense into matter?
Would it have spin-1/2 or spin-3/2?
The 3.53 keV line certainly is suggestive of a 7.06 keV or 1.265 keV DM particle. Or perhaps they would be weird and it would take three of them to annihilate since they would not be right on the matter-antimatter axis but would be skewed from it in three ways (a bit like three color quarks v. two direction electric charge) implying 2.58ish keV which would be right on target.
My best guess (based off of what is consistent with known data sets within experimental error) is that dark matter is a 7.1 keV right-handed neutrino of spin 1/2 and an electron lepton number of +/-1. It's anti-particle would have the same rest mass, but with an electron lepton number of opposite sign (-/+1).
This particle would be formed before the temperature of the universe fell below MeV, and would be a non-thermal relic from the big bang with a non-zero(but small) decay rate into its left-handed active electron neutrino.
As you mentioned above, a right-handed neutrino (since it has rest mass) interacts with the Higgs Boson. So, it would be generated during the "electro-weak era" of the Big Bang when the temperatures were on the order of GeV to TeV.
The big questions I have are: are there muon and taon versions of this right-handed neutrino? If so, what are their rest masses?
Why is the mass of the left-handed one less than the mass of the right handed neutrino? (Since no other particle we know has a different mass when it changes from left to right handed polarization.)
Can right handed neutrinos cancel with left handed anti-neutrinos?
Going back to my first comment regarding your post...the reason I wrote that comment is that any type of neutron matter can't be dark matter because neutrons fail a number of important criterion. For example, dark matter can't interact via the E&M force (neutrons can because they are made of quarks.) Dark matter can't clump together (neutrons can in the form of neutron stars.) And finally, the rest mass of dark matter is roughly in the range of 2-10 keV, which means that neutrons are too heavy to be dark matter.
"Why is the mass of the left-handed one less than the mass of the right handed neutrino?"
This is one of the big reasons that I do not believe that non-weakly interacting leptons with no EM charge can truly be "right handed NEUTRINOS" and probably need something to put them in another category entirely. I don't believe that there are right handed counterparts to the Standard Model neutrinos.
Instead, I think it is more likely that DM particles, if they do exist, are either:
In the gravitational sector together with the spin-2 graviton and lacks either baryon number or lepton number or interactions with color charge, weak force or EM, likely as a singlet gravitino with spin-1/2 or spin-3/2 on the gravitational side of a gravito-weak unification that shares a common Higgs bosons but has no other overlap.
For example, DM is made of composite particles that have a right parity lepton/left parity anti-lepton with charge +/- 0.5 that comes in confined pairs bound by something analogous to color charge that is confining and mediated by a new spin-1 vector boson that does not interact with any other Standard Model particles except DM leptons and perhaps themselves. The DM confining force might be weaker than the strong force by a factor of about 10^6, thus keeping the mass lower.
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