Supersymmetry is very beautiful generalization of the ordinary symmetry concept by generalizing Lie-algebra by allowing grading such that ordinary Lie algebra generators are accompanied by super-generators transforming in some representation of the Lie algebra for which Lie-algebra commutators are replaced with anti-commutators. In the case of Poincare group the super-generators would transform like spinors. Clifford algebras are actually super-algebras. Gamma matrices anti-commute to metric tensor and transform like vectors under the vielbein group (SO(n) in Euclidian signature). In supersymmetric gauge theories one introduced super translations anti-commuting to ordinary translations.
Supersymmetry algebras defined in this manner are characterized by the number of super-generators and in the simplest situation their number is one: one speaks about
N=1 SUSY and minimal super-symmetric extension of standard model (MSSM) in this case. These models are most studied because they are the simplest ones. They have however the strange property that the spinors generating SUSY are Majorana spinors- real in well-defined sense unlike Dirac spinors. This implies that fermion number is conserved only modulo two: this has not been observed experimentally. A second problem is that the proposed mechanisms for the breaking of SUSY do not look feasible.
LHC results suggest MSSM does not become visible at LHC energies. This does not exclude more complex scenarios hiding simplest N=1 to higher energies but the number of real believers is decreasing. Something is definitely wrong and one must be ready to consider more complex options or totally new view abot SUSY.
What is the situation in TGD? Here I must admit that I am still fighting to gain understanding of SUSY in TGD framework. That I can still imagine several scenarios shows that I have not yet completely understood the problem and am working hardly to avoid falling to the sin of sloppying myself. In the following I summarize the situation as it seems just now.
- In TGD framework N=1 SUSY is excluded since B and L and conserved separately and imbedding space spinors are not Majorana spinors. The possible analog of space-time SUSY should be a remnant of a much larger super-conformal symmetry in which the Clifford algebra generated by fermionic oscillator operators giving also rise to the Clifford algebra generated by the gamma matrices of the "world of classical worlds" (WCW) and assignable with string world sheets. This algebra is indeed part of infinite-D super-conformal algebra behind quantum TGD. One can construct explicitly the conserved super conformal charges accompanying ordinary charges and one obtains something analogous to N=∞ super algebra. This SUSY is however badly broken by electroweak interactions.
- The localization of induced spinors to string world sheets emerges from the condition that electromagnetic charge is well-defined for the modes of induced spinor fields. There is however an exception: covariantly constant right handed neutrino spinor νR: it can be de-localized along entire space-time surface. Right-handed neutrino has no couplings to electroweak fields. It couples however to the left handed neutrino by induced gamma matrices except when it is covariantly constant. Note that standard model does not predict νR but its existence is necessary if neutrinos develop Dirac mass. νR is indeed something which must be considered carefully in any generalization of standard model.
Could covariantly constant right handed neutrinos generate SUSY?
Could covariantly constant right-handed spinors generate exact N=2 SUSY? There are two spin directions for them meaning the analog N=2 Poincare SUSY. Could these spin directions correspond to right-handed neutrino and antineutrino. This SUSY would not look like Poincare SUSY for which anticommutator of super generators would be proportional to four-momentum. The problem is that four-momentum vanishes for covariantly constant spinors! Does this mean that the sparticles generated by covariantly constant νR are zero norm states and represent super gauge degrees of freedom? This might well be the case although I have considered also alternative scenarios.
What about non-covariantly constant right-handed neutrinos?
Both imbedding space spinor harmonics and the modified Dirac equation have also right-handed neutrino spinor modes not constant in M4. If these are responsible for SUSY then SUSY is broken.
- Consider first the situation at space-time level. Both induced gamma matrices and their generalizations to modified gamma matrices defined as contractions of imbedding space gamma matrices with the canonical momentum currents for Kähler action are superpositions of M4 and CP2 parts. This gives rise to the mixing of right-handed and left-handed neutrinos. Note that non-covariantly constant right-handed neutrinos must be localized at string world sheets.
This in turn leads neutrino massivation and SUSY breaking. Given particle would be accompanied by sparticles containing varying number of right-handed neutrinos and antineutrinos localized at partonic 2-surfaces.
- One an consider also the SUSY breaking at imbedding space level. The ground states of the representations of extended conformal algebras are constructed in terms of spinor harmonics of the imbedding space and form the addition of right handed neutrino with non-vanishing four-momentum would make sense. But the non-vanishing four-momentum means that the members of the super-multiplet cannot have same masses. This is one manner to state what SUSY breaking is.
What one can say about the masses of sparticles?
The simplest form of massivation would be that all members of the super- multiplet obey the same mass formula but that the p-adic length scales associated with them are different. This could allow very heavy sparticles. What fixes the p-adic mass scales of sparticles? If this scale is CP2 mass scale SUSY would be experimentally unreachable. The estimate below does not support this option.
One can even consider the possibility that SUSY breaking makes sparticles unstable against phase transition to their dark variants with heff =n× h. Sparticles could have same mass but be non-observable as dark matter not appearing in same vertices as ordinary matter! Geometrically the addition of right-handed neutrino to the state would induce many-sheeted covering in this case with right handed neutrino perhaps associated with different space-time sheet of the covering.
This idea need not be so outlandish at it looks first.
The following poor man's estimate supports the idea about dark sfermions and the view that sfermions cannot be very heavy.
- The generation of many-sheeted covering has interpretation in terms of breaking of conformal invariance. The sub-algebra for which conformal weights are n-tuples of integers becomes the algebra of conformal transformations and the remaining conformal generators do note represent gauge degrees of freedom anymore. They could however represent conserved conformal charges still.
- This generalization of conformal symmetry breaking gives rise to infinite number of fractal hierarchies formed by sub-algebras of conformal algebra and is also something new and a fruit of an attempt to avoid sloppy thinking. The breaking of conformal symmetry is indeed expected in massivation related to the SUSY breaking.
A rough model for the neutrino mixing in TGD framework
- Neutrino mixing rate should correspond to the mass scale of neutrinos known to be in eV range for ordinary value of Planck constant. For heff/h=n it is reduced by factor 1/n, when mass kept constant. Hence sfermions could be stabilized by making them dark.
- A very rough order of magnitude estimate for sfermion mass scale is obtained from Uncertainty Principle: particle mass should be higher than its decay rate. Therefore an estimate for the decay rate of sfermion could give a lower bound for its mass scale.
- Assume the transformation νR→ νL makes sfermion unstable against the decay to fermion and ordinary neutrino. If so, the decay rate would be dictated by the mixing rate and therefore to neutrino mass scale for the ordinary value of Planck constant. Particles and sparticles would have the same p-adic mass scale. Large heff could however make sfermion dark, stable, and non-observable.
The mixing of right- and left handed neutrinos would be the basic mechanism in the decays of sfermions. The mixing mechanism is mystery in standard model framework but in TGD it is implied by both induced and modified gamma matrices. The following argument tries to capture what is essential in this process.
The following provides as more detailed view.
- Conformal invariance requires that the string ends at which fermions are localized at wormhole throats are light-like curves. In fact, light-likeness gives rise to Virasosoro conditions.
- Mixing is described by a vertex residing at partonic surface at which two partonic orbits join. Localization of fermions to string boundaries reduces the problem to a problem completely analogous to the coupling of point particle coupled to external gauge field. What is new that orbit of the particle has corner at partonic 2-surface. Corner breaks conformal invariance since one cannot say that curve is light-like at the corner. At the corner neutrino transforms from right-handed to left handed one.
- In complete analogy with Ψbar;γtAtΨ vertex for the point-like particle with spin in external field, the amplitude describing nuR-νL transition involves matrix elements of form νbarRΓt(CP2)ZtνL at the vertex of the CP2 part of the modified gamma matrix and classical Z0 field.
How Γt is identified? The modified gamma matrices associated with the interior need not be well-defined at the light-like surface and light-like curve. One basis of weak form of electric magnetic duality the modified gamma matrix corresponds to the canonical momentum density associated with the Chern-Simons term for Kähler action. This gamma matrix contains only the CP2 part.
To get some idea about orders of magnitude assume that the CP2 projection of string boundary is geodesic circle thus describable as Φ= ω t, where Φ is angle coordinate for the circle and t is Minkowski time coordinate. The contribution of CP2 to the induced metric gtt is Δ gtt =-R2ω2.
- Let us denote by ΓtCP2(in/out) the CP2 part of the modified gamma matrix at string at at partonic 2-surface and by Z0t the value of Z0 gauge potential along boundary of string world sheet. The direction of string line in imbedding space changes at the partonic 2-surface. The question is what happens to the modified Dirac action at the vertex.
- For incoming and outgoing lines the equation
D(in/out)Ψ(in/out)= pk(in,out)γk Ψ(in/out) ,
where the modified Dirac operator is D(in/out)=Γt(in/out)Dt, is assumed. νR corresponds to "in" and νR to "out". It implies that lines corresponds to massless M4 Dirac propagator and one obtains something resembling ordinary perturbation theory.
It also implies that the residue integration over fermionic internal momenta gives as a residue massless fermion lines with non-physical helicities as one can expect in twistor approach. For physical particles the four-momenta are massless but in complex sense and the imaginary part comes classical from four-momenta assignable to the lines of
generalized Feynman diagram possessing Euclidian signature of induced metric so that the square root of the metric determinant differs by imaginary unit from that in Minkowskian regions.
- In the vertex D(in/out) could act in Ψ(out/in) and the natural idea is that νR-νL mixing is due to this so that it would be described the classical weak current couplings νbarR ΓtCP2(out)Z0t(in)νL and νbarR ΓtCP2(out)Z0t(in)νL.
- In the first approximation string end is a light-like curve in Minkowski space meaning that CP2 contribution to the induced metric vanishes. Neutrino mixing vanishes at this limit.
- For a non-vanishing value of ω R the mixing and the order of magnitude for mixing rate and neutrino mass is expected to be R∼ ω and m∼ ω/h. p-Adic length scale hypothesis and the experimental value of neutrino mass allows to estimate m to correspond to p-adic mass to be of order eV so that the corresponding
p-adic prime p could be p≈ 2167. Note that k=127 defines largest of the four Gaussian Mersennes MG,k= (1+i)k-1 appearing in the length scale range 10 nm - 2.5 μm. Hence the decay rate for ordinary Planck constant would be of order R∼ 1014/s but large value of Planck constant could reduced it dramatically. In living matter reductions by a factor 10-12 can be considered.
See the chapter Does the QFT Limit of TGD Have Space-Time Super-Symmetry? or the article What went wrong with symmetries?.