The interpretation of covariantly constant right-handed neutrinos (briefly νR in what follows) in M4× CP2 has been a continual head-ache. Should they be included to the spectrum or not. If not, then one has no fear/hope about space-time SUSY of any kind and has only conformal SUSY. First some general observations.
- In TGD framework right-handed neutrinos differ from other electroweak charge states of fermions in that the solutions of the modified Dirac equation for them are delocalized at entire 4-D space-time sheets whereas for other electroweak charge states the spinors are localized at string world sheets (see this).
- Since right-handed neutrinos are in question so that right-handed neutrino are in 1-1 correspondence with complex 2-component Weyl spinors, which are eigenstates of γ5 with eigenvalue say +1 (I never remember whether +1 corresponds to right or left handed spinors in standard conventions).
- The basic question is whether the fermion number associated with covariantly constant right-handed neutrinos is conserved or conserved only modulo 2. The fact that the right-handed neutrino spinors and their conjugates belong to unitarily equivalent pseudoreal representations of SO(1,3) (by definition unitarily equivalent with its complex conjugate) suggests that generalized Majorana property is true in the sense that the fermion number is conserved only modulo 2. Since νR decouples from other fermion states, it seems that lepton number is conserved.
- The conservation of the number of right-handed neutrinos in vertices could cause some rather obvious mathematical troubles if the right-handed neutrino oscillator algebras assignable to different incoming fermions are identified at the vertex. This is also suggested by the fact that right-handed neutrinos are delocalized.
- Since the νR:s are covariantly constant complex conjugation should not affect physics. Therefore the corresponding oscillator operators would not be only hermitian conjugates but hermitian apart from unitary transformation (pseudo-reality). This would imply generalized Majorana property.
- A further problem would be to understand how these SUSY candidates are broken. Different p-adic mass scale for particles and super-partners is the obvious and rather elegant solution to the problem but why the addition of right-handed neutrino should increase the p-adic mass scale beyond TeV range?
If the νR:s are included, the pseudoreal analog of N=1 SUSY assumed in the minimal extensions of standard model or the analog of N=2 or even N=4 SUSY is expected so that SUSY type theory might describe the situation. The following is an attempt to understand what might happen. For an earlier attempt see this.
1. Covariantly constant right-handed neutrinos as limiting cases of massless modes
For the first option covariantly constant right-handed neutrinos are obtained as limiting case for the solutions of massless Dirac equation. One obtains 2 complex spinors satisfying Dirac equation nkγku=0 for some momentum direction nk defining quantization axis for spin. Second helicity is unphysical: one has therefore one helicity for neutrino and one for antineutrino.
- If the oscillator operators for νR and its conjugate are hermitian conjugates, which anticommute to zero (limit of anticommutations for massless modes) one obtains the analog of N=2 SUSY.
- If the oscillator operators are hermitian or pseudohermitian, one has pseudoreal analog of N=1 SUSY. Since νR decouples from other fermion states, lepton number and baryon number are conserved.
Note that in TGD based twistor approach four-fermion vertex is the fundamental vertex and fermions propagate as massless fermions with non-physical helicity in internal lines. This would suggest that if right-handed neutrinos are zero momentum limits, they propagate but give in the residue integral over energy twistor line contribution proportional to pkγk, which is non-vanishing for non-physical helicity in general but vanishes at the limit pk→ 0. Covariantly constant right-handed neutrinos would therefore decouple from the dynamics (natural in continuum approach since they would represent just single point in momentum space). This option is not too attractive.
2. Covariantly constant right-handed neutrinos as limiting cases of massless modes
For the second option covariantly constant neutrinos have vanishing four-momentum and both helicities are allowed so that the number of helicities is 2 for both neutrino and antineutrino.
- The analog of N=4 SUSY is obtained if oscillator operators are not hermitian apart from unitary transformation (pseudo reality) since there are 2+2 oscillator operators.
- If hermiticity is assumed as pseudoreality suggests, N=2 SUSY with right-handed neutrino conserved only modulo two in vertices obtained.
- In this case covariantly constant right-handed neutrinos would not propagate and would naturally generate SUSY multiplets.
3. Could twistor approach provide additional insights?
Concerning the quantization of νR:s, it seems that the situation reduces to the oscillator algebra for complex M4 spinors since CP2 part of the H-spinor is spinor is fixed. Could twistor approach provide additional insights?
As discussed, M4 and CP2 parts of H-twistors can be treated separately and only M4 part is now interesting. Usually one assigns to massless four-momentum a twistor pair (λa, ξa') such that one has paa'= λaξa' ( ξ denotes for "\hat(\lambda)" which html does not allow to express). Dirac equation gives λa= +/- (ξa')*, where +/- corresponds to positive and negative frequency spinors.
- The first - presumably non-physical - option would correspond to limiting case and the twistors λ and ξ would both approach zero at the pk→ 0 limit, which again would suggest that covariantly constant right-handed neutrinos decouple completely from dynamics.
- For the second option one can assume that either λ or ξa' vanishes. In this manner one obtains 2 spinors λi, i=1,2 and their complex conjugates ξa'i as representatives for the super-generators and could assign the oscillator algebra to these. Obviously twistors would give something genuinely new in this case. The maximal option would give 4 anti-commuting creation operators and their hermitian conjugates and the non-vanishing anti-commutators would be proportional to δa,bλai(λb)j* and δa,bξa'i(ξa'j)*.
If the oscillator operators are hermitian conjugates of each other and (pseudo-)hermitian, the anticommutators vanish.
An interesting challenge is to deduce the generalization of conformally invariant part of four-fermion vertices in terms of twistors associated with the four-fermions and also the SUSY extension of this vertex.
For details see the new chapter Some fresh ideas about twistorialization of TGD or the article with the same title.