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p-Adic Physics

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Year 2013

New results from PHENIX concerning quark gluon plasma

New results have been published on properties of what is conventionally called quark gluon plasma (QGP) . As a matter fact, this phase does not resemble plasma at all. The decay patterns bring in mind decays of string like objects parallel to the collision axes rather than isotropic blackbody radiation. The initial state looks like a perfect fluid rather than plasma and thus more like a particle like object.

The results of QGP - or color glass condensate (CGC) as it is also called - come from three sources and are very similar. The basic characteristic of the collisions is the cm energy s1/2of nucleon pair. The data sources are Au-Au collisions at RHIC, Brookhaven with s1/2=130 GeV, p-p collisions and p-nucleus collisions at LHC with s1/2=200 GeV and d-Au collisions at RHIC with s1/2=200 GeV studied by PHENIX collaboration.

According to the popular article telling about the findings of PHENIX collaboration the collisions are believed to involve a creation of what is called hot spot. In Au-Au collisions this hot spot has size of order Au nucleus. In d-Au collisions it is reported to be much, much smaller. What does this mean? The size of deuteron nucleus or of nucleon? Or something even much smaller? Hardly so if one believes in QCD picture. If this is however the case, the only reasonable candidate for its size would be the longitudinal size scale of colliding nucleon-nucleon system of order L=hbar/s1/2 if an object with this size is created in the collision. I did my best to find some estimate for the very small size of the hot spot from articles some related to the study but failed (see this, this and this): if I were a paranoid I would see this as a conspiracy to keep this as a state secret;-).

How to understand the findings?

I have already earlier considered the basic characteristics of the collisions. What is called QGP does not behave at all like plasma phase for which one would expect particle distributions mimicking blackbody radiation of quarks and gluons. Strong correlations are found between charged particles created in the collision and the best manner to describe them is in terms of a creation of longitudinal string-like objects parallel to the collision axes.

In TGD framework this observation leads to the proposal that the string like objects could be assigned with M89 hadron physics introduced much earlier to explain strange cosmic ray events like Centauro. The p-adic mass scale assignable to M89 hadron physics is obtained from that of electron (given by p-adic thermodynamics in good approximation by m127= me/51/2) as m89= 2(127-89)/2× me/51/2. This gives m89= 111.8 GeV. This is conveniently below the cm mass of nucleon pair in all the experiments.

In standard approach based on QCD the description is completely different. The basic parameters are now thermodynamical. One assumes that thermalized plasma phase is created and is parametrized by the energy density assignable to gluon fields for which QCD gives the estimate ε ≥ 1 GeV/fm3 and by temperature which is about T=170 GeV and more or less corresponds to QCD Λ. One can think of the collision regions as highly flattened pancake (Lorentz contraction) containing very density gluon phase called color glass condensate, which would be something different from QGP and definitely would not conform with the expectations from perturbative QCD since QGP would be precisely a manifestation of perturbative QGP (see this).

Also a proposal has been made that this phase could be described by AdS/CFT correspondence non-perturbatively - again in conflict with the basic idea that perturbative QCD should work. It has however turned out that this approach does not work even qualitatively as Bee ludicly explains this in her blog article Whatever happened to AdS/CFT and the Quark Gluon Plasma?.

Strangely enough, this failure of QGP and AdS/CFT picture has not created any fuss although one might think that the findings challenging the basic pillars of standard model should be seen as sensational and make happy all those who have publicly told that nothing would be more well-come than the failure of standard model. Maybe particle theorists have enough to do with worrying about the failure of standard SUSY and super string inspired particle phenomenology that they do not want to waste their time to the dirty problems of low energy phenomenology.

A further finding mentioned in the popular article is stronger charm-anticharm suppression in head-on collisions than in peripheral collisions (see this). What is clear that if M89 hadrons are created, they consist of lightest quarks present in the lightest hadrons of M89 hadron physics - that is u and d (and possibly also s) of M89 hadrons, which are scaled variants of ordinary u and d quarks and decay to u and d (and possibly s) quarks of M107 hadron physics. If the probability of creating a hot M89 spot is higher in central than peripheral collisions the charm suppression is stronger. Could a hot M89 spot associated with a nucleon-nucleon pair heat some region around it to M89 hadronic phase so that charm suppression would take place inside larger volume than in periphery?

There is also the question whether the underlying mechanism relies on specks of hot QGP or some inherent property of nuclei themselves. At the first sight, the latter option could not be farther from the TGD inspired vision. However, in nuclear string model inspired by TGD nuclei consists of nucleons connected by color bonds having quark and antiquark at their ends. These bonds are characterized by rather large p-adic prime characterizing current quark mass scale of order 5-20 GeV for u and d quarks (the first rough estimate for the p-adic scales involved is p≈ 2^k, k=121 for 5 MeV and k= 119 for 20 MeV). These color bonds Lorentz contract in the longitudinal direction so that nearly longitudinal color bonds would shorten to M89 scale whereas transversal color bonds would get only thinner. Could they be able to transform to color bonds characterized by M89 and in this manner give rise to M89 mesons decaying to ordinary hadrons?

Flowers to the grave of particle phenomenology

The recent situation in theoretical particle physics and science in general does not raise optimism. Super string gurus are receiving gigantic prizes from a theory that was a failure. SUSY has failed in several fronts and cannot be anymore regarded as a manner to stabilize the mass of Higgs. Although the existence of Higgs is established, the status of Higgs mechanism is challenged by its un-naturality: the assumption that massivation is due to some other mechanism and Higgs has gradient coupling provides a natural explanation for Higgs couplings. The high priests are however talking about "challenges" instead of failures. Even evidence for the failure of even basic QCD is accumulating as explained above. Peter Higgs, a Nobel winner of this year, commented the situation ironically by saying that he would have not got a job in the recent day particle physics community since he is too slow.

The situation is not much better in the other fields of science. Randy Scheckman, also this year's Nobel prize winner in physiology and medicine has declared boycott of top science journals Nature, Cell and Science. Schekman said that the pressure to publish in "luxury" journals encourages researchers to cut corners and pursue trendy fields of science instead of doing more important work. The problem is exacerbated, he said, by editors who were not active scientists but professionals who favoured studies that were likely to make a splash.

Theoretical and experimental particle physics is a marvellous creation of humankind. Perhaps we should bring flowers to the grave of the particle physics phenomenology and have a five minutes' respectful silence. It had to leave us far too early.

For background see the chapter "New particle physics predicted by TGD" of "p-Adic Physics".

For background see the chapter "New particle physics predicted by TGD".

Has IceCube detected neutrinos coming from decays of p-adically scaled up copies of weak bosons?

This note was inspired by very interesting posting "Storm in IceCube" by Jester. IceCube is a neutrino detector located at South Pole. Most of the neutrinos detected are atmospheric neutrinos originating from Sun but what one is interested in are neutrinos from astrophysical sources.

  1. Last year the collaboration reported the detection for neutrino cascade events, with with energy around 1 PeV=106 GeV. The atmospheric background decreases rapidly with energy and at these energies the detection of a pair of events at these energies corresponds to about 3 sigma. The recent report tells about a broad excess of events (28 events) above 30 TeV: only about 10 are expected from atmospheric neutrinos alone. The flavor composition is consistent with 1:1:1 ratio of the 3 neutrino species as expected for distant sources for which the oscillations during the travel should cause complete mixing. The distribution of the observed events is consistent with isotropy.
  2. There is a dip ranging from .4 PeV to about 1 PeV and the spectrum has probably a sharp cutoff somewhat above 1 TeV. This suggests a monochromatic neutrino line resulting from the decays of some particle decaying to neutrino and some other particle - possibly also neutrino (see this). Astrophysical phenomena with standard model physics are expected to produce smooth power-law spectrum - typically 1/E2 - rather than peak. The proposal is that the events around 1 PeV could come from the decay of dark matter particles with energy scale of 2 TeV. The observation of two events gives a bound for the life-time of dark matter particle in question: about 1021 years much longer than the age of the Universe. The bound of course depends on what density is assumed for the dark matter.
  3. There is also a continuum excess in the range [.1, .4] PeV. This could result from many-particle decay channels containing more than 2 particles.
What says TGD?
  1. TGD almost-predicts a fractal hierarchy of hadron physics and weak physics labelled by Mersenne primes Mn=2n-1. Also Gaussian primes MG,n= (1+i)n-1 are possible. M107 would correspond to the ordinary hadron physics. M89 would correspond to weak bosons and a scaled up copy of hadron physics, for which there are many indications: in particular, the breaking of perturbative QCD at rather high energies assignable at LHC to proton heavy nucleus collisions. The explanation in terms of AdS/CFT correspondence has not been successful and is not even well-motivated since it assumes strong coupling regime.
  2. The next Mersenne prime is M61 and the first guess is that the observed TeV neutrinos result from the decay of W and Z bosons of scale up copy of weak physics having mass near 1 TeV. The naivest estimate for the masses of these weak bosons is obtained by the naive scaling the masses of ordinary weak bosons by factor 2(89-61)/2=214. For mW=80 GeV and mZ=90 GeV one obtains mW(61)= 1.31 PeV and mZ(61)= 1.47 PeV. The energy of the mono-chromatic neutrino would be about about .65 PeV and .74 PeV in the two cases. This is in the almost empty range between .4 PeV and 1 PeV and too small roughly by a factor of kenosqrt2.

    An improved estimate for upper bound of Z mass is based on the p-adic mass scale m(M89) related to the p-adic mass scale M127 of electron by scaling factor 2(127-89)/2= 219 giving m(89)≈ 120 GeV for me= (5+X)1/2m(127) =.51 MeV and X=0 (X≤ 1 holds true for the second order contribution to electron mass). The scaling by the factor 2(89-61)/2= 214 gives m(61)= 1.96 TeV consistent with the needed 2 TeV. The exact value of weak boson mass depends on the value of Weinberg angle sin2W) and the value of the second order contribution to the mass: m(61) gives upper bound for the mass of Z(61). The model predicts two peaks with distance depending on the value of Weinberg angle of M61 weak physics.

  3. What about the interpretation of the continuum part of anomaly? The proposed interpretation for many-particle decays looks rather reasonable. The simplest possibility is the decay to a pair of light quarks of M61 hadron physics, followed by a decay of quark or antiquark via emission of W boson decaying to lepton-neutrino pair.
To sum up, the existence of scaled up copy of weak physics would mean the last nail to the coffin of GUTs predicting huge desert between weak mass scale and the mass scale of leptoquarks having mass scale of order 10-4 Planck masses. It would also provide a further support for the p-adic length scales hypothesis.

For background see the chapter "New particle physics predicted by TGD" of "p-Adic length scale hypothesis and dark matter hierarchy".

AMS results about dark matter

The results of AMS-02 experiment are published. There is paper, live blog from CERN, and article in Economist. There is also press release from CERN. Also Lubos has written a summary from the point of view of SUSY fan who wants to see the findings as support for the discovery of SUSY neutralino. More balanced and somewhat skeptic representations paying attention to the hypeish features of the announcement come from Jester and Matt Strassler.

The abstract of the article is here. A precision measurement by the Alpha Magnetic Spectrometer on the International Space Station of the positron fraction in primary cosmic rays in the energy range from 0.5 to 350 GeV based on 6.8 × 106 positron and electron events is presented. The very accurate data show that the positron fraction is steadily increasing from 10 to 250 GeV, but, from 20 to 250 GeV, the slope decreases by an order of magnitude. The positron fraction spectrum shows no fine structure, and the positron to electron ratio shows no observable anisotropy. Together, these features show the existence of new physical phenomena.

New physics has been observed. The findings confirm the earlier findings of Fermi and Pamela also showing positron excess. The experimenters do not give data above 350 GeV but say that the flux of electrons does not change. The press release states that the data are consistent with dark matter particles annihilating to positron pairs. For instance, the flux of the particles is same everywhere, which does not favor supernovae in galactic plane as source of electron positron pairs. According to the press release, AMS should be able to tell within forthcoming months whether dark matter or something else is in question- this sounds rather hypeish statement.

About the neutralino interpretation

Lubos trusts on his mirror neurons and deduces from the body language of Samuel Ting that the flux drops abruptly above 350 GeV as neutralino interpretation predicts.

  1. The neutralino interpretation assumes that the positron pairs result in the decays χχ→ e+e- and predicts a sharp cutoff above mass scale of neutralino due to the reduction of the cosmic temperature below critical value determined by the mass of the neutralino leading to the annilation of neutralinos (fermions). Not all neutralinos annhilate and this would give to dark matter as a cosmic relic.
  2. According the press release and according to the figure 5 of the article the positron fraction settles to small but constant fraction before 350 GeV. The dream of Lubos is that abrupt cutoff takes place above 350 GeV: about this region we did not learn anything yet because the measurement uncertainties are too high. From Lubos's dream I would intuit that neutralino mass should be of the order 350 GeV. The electron/positron flux is fitted as a sum of diffuse background proportional to Ce+/-Ee+/- and a contribution resulting from decays and parametrized as Cs Es exp(-E/Es) - same for electron and positron. The cutoff Es of order Es= 700 GeV: error bars are rather large. The factor exp(-E/Es) does not vary too much in the range 1-350 GeV so that the exponential is probably motivated by the possible interpretation as neutralino for which sharp cutoff is expected. The mass of neutralino should be of order Es. The positron fraction represented in figure 5 of the article seems to approach constant near 350 GeV. The weight of the common source is only 1 per cent of the diffuse electron flux.
  3. Lubos notices that in neutralino scenario also a new interaction mediated by a particle with mass of order 1 GeV is needed to explain the decrease of the positron fraction above 1 GeV. It would seem that Lubos is trying to force right leg to the shoe of the left leg. Maybe one could understand the low end of the spectrum solely in terms of particle or particles with mass of order 10 GeV and the upper end of the spectrum in terms of particles of M89 hadron physics.
  4. Jester lists several counter arguments against the interpretation of the observations in terms of dark matter. The needed annihilation cross section must be two orders of magnitude higher than required for the dark matter to be a cosmic thermal relic -this holds true also for the neutralino scenario. Second problem is that the annihilation of neutralinos to quark pairs predicts also antiproton excess, which has not been observed. One must tailor the couplings so that they favor leptons. It has been also argued that pulsars could explain the positron excess: the recent finding is that the flux is same from all directions.

What could TGD interpretation be?

What can one say about the results in TGD framework? The first idea that comes to mind is that electron-positron pairs result from single particle annihilations but it seems that this option is not realistic. Fermion-antifermion annihilations are more natural and brings in strong analogy with neutralinos, which would give rise to dark matter as a remnant remaining after annihilation in cold dark matter scenario. An analogous scenario is obtained in TGD Universe by replacing neutralinos with baryons of some dark and scaled up variant of ordinary hadron physics of leptohadron physics.

  1. The positron fraction increases from 10 to 250 GeV with its slope decreasing between 20 GeV and 250 GeV by an order of magnitude. The observations suggest to my innocent mind a scale of order 10 GeV. The TGD inspired model for already forgotten CDF anomaly discussed in the chapter The recent status of leptohadron hypothesis of "Hyper-finite factors and hierarchy of Planck constants" suggests the existence of τ pions with masses coming as three first octaves of the basic mass which is two times the mass of τ lepton. I have proposed interpretation of the positron excess ob served by Fermi and Pamela now confirmed by AMS in terms τ pions. The predicted mass of the three octaves of τ pion would be 3.6 GeV, 7.2 GeV, and 14.4 GeV. Could the octaves of τ pion explain the increase of the production rate up to 20 GeV and its gradual drop after that?

    There is a severe objection against this idea. The energy distribution of τ pions dictates the width of the energy interval in which their decays contribute to the electron spectrum and what suggests itself is that decays of τ pions yield almost monochromatic peaks rather than the observed continuum extending to high energies. Any resonance should yield similar distribution and this suggests that the electron positron pairs must be produced in the two particle annihilations of some particles.

    The annihilations of colored τ leptons and their antiparticles could however contribute to the spectrum of electron-positron pairs. Also the leptonic analogs of baryons could annihilate with their antiparticles to lepton pairs. For these two options the dark particles would be fermions as also neutralino is.

  2. Could colored τ leptons and - hadrons and their muonic and electronic counterparts be really dark matter? ‎ The particle might be dark matter in TGD sense - that is particle with a non-standard value of effective Planck constant hbareff coming as integer multiple of hbar. The existence of colored excitations of leptons and pion like states with mass in good approximation twice the mass of lepton leads to difficulties with the decay widths of W and Z unless the colored leptons have non-standard value of effective Planck constant and therefore lack direct couplings to W and Z.

    A more general hypothesis would be that the hadrons of all scaled up variant of QCD like world (leptohadron physics and scaled variants of hadron physics) predicted by TGD correspond to non-standard value of effective Planck constant and dark matter in TGD sense. This would mean that these new scaled up hadron physics would couple only very weakly to the standard physics.

  3. At the high energy end of the spectrum M89 hadron physics would be naturally involved and also now the hadrons could be dark in TGD sense. Es might be interpreted as temperature, which is in the energy range assigned to M89 hadron physics and correspond to a mass of some M89 hadron. Fermions are natural candidates and the annihilations nucleons and anti-nucleons of M89 hadron physics could contribute to the spectrum of leptons at higher energies. The direct scaling of M89 proton mass gives mass of order 500 GeV and this value is consistent with the limits 480 GeV and 1760 GeV for Es.
  4. There could be also a relation to the observations of Fermi suggesting an annihilation of some bosonic states to gamma pairs with gamma energy around 135 GeV could be interpreted in terms of annihilations of a M89 pion with mass of 270 GeV (maybe octave of leptopion with mass 135 Gev in turn octave of pion with mass 67.5 GeV).

How to resolve the objections against dark matter as thermal relic?

The basic objection against dark matter scenarios is that dark matter particles as thermal relics annihilate also to quark pairs so that proton excess should be also observed. TGD based vision could also circumvent this objection.

  1. Cosmic evolution would be a sequence of phase transitions between hadron physics characterized by Mersenne primes. The lowest Mersenne primes are M2=3, M3=7, M5=31, M_7=127, M13, M17, M19, M31, M61, M89, and M107 assignable to the ordinary hadron physics are involved but it might be possible to have also M127(electrohadrons). There are also Gaussian Mersenne primes MG,n= (1+i)n-1. Those labelled by n=151,157,163,167 and spanning p-adic length scales in biologically relevant length scales 10 nm,..., 2.5 μm.
  2. The key point is that at given period characterised by M_n the hadrons characterized by larger Mersenne primes would be absent. In particular, before the period of the ordinary hadrons only M89 hadrons were present and decayed to ordinary hadrons. Therefore no antiproton excess is expected - at least by the mechanism producing it in the standard dark matter scenarios where all dark and ordinary particles are present simultaneously.
  3. The second objection relates to the cross section, which must be two orders of magnitude larger than required by the cold dark matter scenarios. I am unable to say anything definite about this. The fact that both M89 hadrons and colored leptons are strongly interacting would increase corresponding annilation cross section and leptohadrons could later decay to ordinary leptons.

Connection with strange cosmic ray events and strange observations at RHIC and LHC

The model could also allow to understand the strange ultrahigh energy cosmic ray events (Centauros,etc) suggesting a formation of a blob ("hot spot" of exotic matter in atmosphere and decaying to ordinary hadrons. In the center of mass system of atmospheric particle and incoming cosmic ray cm energies are indeed of order M89 mass scale. As suggested, these hot spots would be hot in p-adic sense and correspond to p-adic temperature assignable to M89. Also the strange events observed already at RHIC in heavy ion collisions and later at LHC in proton-heavy ion collisions, and in conflict with the perturbative QCD predicting the formation of quark gluon plasma could be understood as a formation of M89 hot spots (see this). The basic finding was that there were strong correlations: two particles tended to move either parallel or antiparallel, as if they had resulted in a decay of string like objects. The AdS/CFT inspired explanation was in terms of higher dimensional blackholes. TGD explanation is more prosaic: string like objects (color magnetic flux tubes) dominating the low energy limit of M89 hadron physics were created.

The question whether M89 hadrons, or their cosmic relics are dark in TGD sense remains open. In the case of colored variants of the ordinary leptons the decay widths of weak bosons force this. It however seems that a coherent story about the physics in TGD Universe is developing as more data emerges. This story is bound to remain to qualitative description: quantitative approach would require a lot of collective theoretical work.

Also CDMS claims dark matter

Also CDMS (Cryogenic Dark Matter Search) reports new indications for dark matter particles: see the Nature blog article Another dark matter sign from a Minnesota mine. Experimenters have observed 3 events with expected background of .7 events and claim that the mass of the dark matter particle is 8.6 GeV. This mass is much lighter than what has been expected: something like 350 GeV was suggested as explanation of the AMS observations. The low mass is however consistent with the identification as first octave of tau-pion with mass about 7.2 GeV for which already forgotten CDF anomaly provided support for years ago (as explained above p-adic length scale hypothesis allows octaves of the basic mass for leptopion which is in good approximation 2 times the mass of the charged lepton, that is 3.6 GeV). The particle must be dark in TGD sense, in other words it must have non-standard value of effective Planck constant. Otherwise it would contribute to the decay widths of W and Z.

For background see the chapter New Particle Physics Predicted by TGD: Part I.

3 sigma evidence for kaons of M89 hadron physics?

The news about Moriond conference (for details see for the posting of Phil Gibbs) did not bring anything really new concerning the situation with Higgs. The two-photon discrepancy is still there although the production rate is now about 1.6 times higher than predicted. The error bars are however getting narrower so that there are excellent reasons to hope/fear that unexpected kind of new physics is trying to tell about itself. Also the masses deduced from gamma pair and Z pair decay widths are slightly different.

The TGD-based explanation would be in terms of M89 hadron physics, a fractal copy of ordinary hadron physics with 512 times higher overall mass scale. If the pion of this new physics has mass not too far from 125 GeV its decays to gamma and Z pairs would affect the observed decay rates of Higgs to gamma and Z pairs if one assumes just standard model. Fermi anomaly suggests mass of about 135 GeV for the pion of M89 hadron physics. The observations of RHIC and those from proton-heavy nucleus collisions - correlated pairs of charged particles moving in same or opposite directions- could be understood in terms of decays of M89 mesons behaving like hadronic strings in low energies in the relevant energy scale.

Lubos tells in his recent posting about 3 sigma excess for new charged and neutral particles with mass around 420 GeV. They would be produced as pairs of charged and neutral particle. M89 physics based explanation would be in terms of kaons of M89 hadron physics. The naive scaling by the ratio r=m(π+107)/m(K+107) of masses of ordinary pion and kaon predicts that the M89 pion should have mass m(π+89)= r× 420 GeV. This would give m(π+89)=119 GeV not too far from 125 GeV to affect the apparent decay rates of Higgs to gamma and Z pairs since its width as strongly interacting particle decaying to ordinary quarks and gluons is expected to be large. This mass however deviates from the 135 GeV mass suggested by Fermi data by 18 per cent.

For background see the chapter New Particle Physics Predicted by TGD: Part I.

Right-handed neutrino as inert neutrino?

There is a very interesting posting by Jester in Resonaances with title How many neutrinos in the sky?. Jester tells about the recent 9 years WMAP data and compares it with earlier 7 years data. In the earlier data the effective number of neutrino types was Neff= 4.34 +/- 0.87 and in the recent data it is Neff= 3.26 +/- 0.35. WMAP alone would give Neff = 3.89 +/- 0.67 also in the recent data but also other data are used to pose constraings on Neff.

To be precise, Neff could include instead of fourth neutrino species also some other weakly interacting particle. The only criterion for contributing to Neff is that the particle is in thermal equilibrium with other massless particles and thus contributes to the density of matter considerably during the radiation dominated epoch.

Jester also refers to the constraints on Neff from nucleosynthesis, which show that Neff∼ 4 us slightly favored although the entire range [3,5] is consistent with data.

It seems that the effective number of neutrinos could be 4 instead of 3 although latest WMAP data combined with some other measurements favor 3.
Addition:Later however a corrected version of the eprint appeared telling that the original estimate of Neff contained a mistake and the correct estimate is Neff=3.84+/- 0.40.

What could Neff=4 mean in TGD framework?

  1. One poses to the modes of the modified Dirac equation the following condition: electric charge is conserved in the sense that the time evolution by modified Dirac equation does not mix a mode with a well-defined em charge with those with different em charge. The implication is that all modes except pure right handed neutrino are restricted at string world sheets. The first guess is that string world sheets are minimal surfaces of space-time surface (rather than those of imbedding space). One can also consider minimal surfaces of imbedding space but with effective metric defined by the anti-commutators of the modified gamma matrices. This would give direct physical meaning for this somewhat mysterious effective metric.

    For the neutrino modes localized at string world sheets mixing of left and right handed modes takes place and they become massive. If only 3 lowest genera for partonic 2-surfaces are light, one has 3 neutrinos of this kind. The same applies to all other fermion species. The argument for why this could be the case relies on simple observation: the genera g=0,1,2 have the property that they allow for all values of conformal moduli Z2 as a conformal symmetry (hyper-ellipticity). For g>2 this is not the case. The guess is that this additional conformal symmetry is the reason for lightness of the three lowest genera.

  2. Only purely right-handed neutrino is completely delocalized in 4-volume so that one cannot assign to it genus of the partonic 2-surfaces as a topological quantum number and it effectively gives rise to a fourth neutrino very much analogous to what is called sterile neutrino. Delocalized right-handed neutrinos couple only to gravitation and in case of massless extremals this forces them to have four-momentum parallel to that of ME: only massless modes are possible. Very probably this holds true for all preferred extremals to which one can assign massless longitudinal momentum direction which can vary with spatial position.
  3. The coupling of νR is to gravitation alone and all electroweak and color couplings are absent. According to standard wisdom delocalized right-handed neutrinos cannot be in thermal equilibrium with other particles. This according to standard wisdom. But what about TGD?

    One should be very careful here: delocalized right-handed neutrinos is proposed to give rise to SUSY (not N=1 requiring Majorana fermions) and their dynamics is that of passive spectator who follows the leader. The simplest guess is that the dynamics of right handed neutrinos at the level of amplitudes is completely trivial and thus trivially supersymmetric. There are however correlations between four-momenta.

    1. The four-momentum of νR is parallel to the light-like momentum direction assignable to the massless extremal (or more general preferred extremal). This direct coupling to the geometry is a special feature of the modified Dirac operator and thus of sub-manifold gravity.
    2. On the other hand, the sum of massless four-momenta of two parallel pieces of preferred extremals is the - in general massive - four-momentum of the elementary particle defined by the wormhole contact structure connecting the space-time sheets (which are glued along their boundaries together since this is seems to be the only manner to get rid of boundary conditions requiring vacuum extremal property near the boundary). Could this direct coupling of the four-momentum direction of right-handed neutrino to geometry and four-momentum directions of other fermions be enough for the right handed neutrinos to be counted as a fourth neutrino species in thermal equilibrium? This might be the case!

    One cannot of course exclude the coupling of 2-D neutrino at string world sheets to 4-D purely right handed neutrinos analogous to the coupling inducing a mixing of sterile neutrino with ordinary neutrinos. Also this could help to achieve the thermal equilibrium with 2-D neutrino species.
For background see the chapter SUSY in TGD Universe .

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