** 1. Super-conformal invariance and generalized space-time supersymmetry**

Super-conformal symmetry is behind the space-time supersymmetry in TGD framework. It took long time to get convinced that one obtains space-time supersymmetry in some sense.

- The basic new idea is that the fermionic oscillator operators assignable to partonic 2-surfaces defined the SUSY algebra analagous to space-time SUSY algebra. Without length scale cutoff the number of oscillator operators is infinite and one would have N=∞ SUSY. Finite measurement resolution realized automatically by the dynamics of the modified Dirac action however effectively replaces the partonic two-surface with a discrete set of points which is expected to be finite and one obtains reduction finite value of N. Braids become basic objects in finite measurement resolution at light-like 3-D wormhole throats. String world sheets become the basic objects at the level of 4-D space-time surfaces and it is now possible to identify them uniquely and a connection with the theory of 1-knots, their cobordisms and 2-knots emerges.
The supersymmetry involves several algebras for which the fermionic oscillator operator algebra serves as a building block. Also the gamma matrices of the world of classical worlds (WCW) are expressible in terms of the fermionic oscillator operators so that fermionic anticommutations have purely geometric interpretation. The presence of SUSY in the sense of conservation of fermionic supra currents requires a consistency condition. For induced gamma matrices the surface must be minimal surfaces (extremal of volume action). For Kähler action and Chern-Simons action one must replace induced gammas by modified ones defined by the contractions of canonical momentum densities with the imbedding space gamma matrices.

The super-symmetry in TGD framework differs from that in standard approach in that Majorana spinors are not involved. One has 8-D imbedding space spinors with interpretation as quark and lepton spinors. This makes sense because color corresponds to color partial waves and quarks move in triality t=+/- 1 and leptons in t=0 waves. Baryon and lepton number are conserved exactly.

- A highly non-trivial aspect of TGD based SUSY is bosonic emergence meaning that bosons can be constructed from fermions. Zero energy ontology makes this construction extremely elegant since both massive states and virtual states are composites of massless states. General arguments support the existence of pseudoscalar Higgs but it is not quite clear whether its existence is somehow forbidden by symmetries. Scalar and pseudoscalar Higgs transforming according to 3+1 decomposition under weak SU(2) replace two complex doublets of MSSM if this is not the case. This difference is essentially due to the fact that spinors are not M
^{4}spinors but M^{4}× CP_{2}spinors. A proper notation would be B, h_{B}for the gauge bosons and corresponding Higgs particles and one expects that electroweak mixing characterized by Weinberg angle takes place for neutral Higgs particles and also their super-counterparts.The sfermions associated with left and right handed fermions should couple to fermions via P

_{+/-}=1+/-γ_{5}so that one can speak about left- and right-handed scalars. Maximal mixing between them leads to scalar and pseudoscalar. This observation raises a question about Higgs and its pseudoscalar variant. Could one assume that the initial states are right and left handed Higgs and that maximal mixing leads to scalar and pseudoscalar with scalar eaten by gauge bosons. - Also spin one particles regarded usually as massless must have small mass and this means that Higgs scalar is completely eaten by gauge bosons. Also scalar gluons are predicted and would be eaten by gluons to develop a small mass. This resolves the IR difficulties of massless gauge theories and conjecture to make possible exact Yangian symmetry in the twistor approach to TGD. The disappearance of Higgs means that corresponding limits on the parameters of SUSY are lost. For instance the limits in (tan(β),M
_{SUSY}) coming from Higgs mass do not hold anymore. The disappearence of Higgs means also the disappearance of the little hierarchy problem, which is one of the worst headaches of MSSM SUSY: no Higgs- no Higgs mass to be stabilized.

** 2. Induced spinor structure and purely geometric breaking of SUSY**

Particle massivation and the breaking of SUSY and R-symmetry are the basic problems of both QFT and stringy approach to SUSY. Just by looking the arguments related to how MSSM could emerge from string models make clear how hopelessly ad hoc the constructions are. The notion of modified gamma matrix provides a purely geometric approach to these symmetric breakings involving no free parameters. To my view the failure to realize partially explains the recent situation in the forefront of theoretical physics and LHC findings are now making clear that something is badly wrong.

- The space-time supersymmetry is broken. The reason is that the modified gamma matrices are superpositions of M
^{4}and CP_{2}gamma matrices. This implies mixing of M^{4}chiralities, which is direct symptom of massivation and is responsible for Higgs like aspects of massivation: p-adic thermodynamics is second and completely new aspect. There is hierarchy of supersymmetries according to the strength of breaking. Right handed neutrino generates supersymmetry which is broken only by the mixing of right handed neutrino with left handed one and induced by the mixing of gamma matrices. This corresponds to the supersymmetry analogous to that of MSSM. The supersymmetries generated by other fermionic oscillator operators with electroweak quantum numbers break the super-symmetry in much more dramatic manner but the basic algebra remains and should allow an elegant formulation of TGD in terms of generalized super fields (see this). - One important implication is R-parity breaking due to the transformation of right handed neutrino of superpartner to left-handed neutrino. If this takes place fast enough, the process sP→ P+ν becomes possible. The universal decay signature would be lonely neutrino representing missing energy without accompanying charged lepton. This means that the experimental limits on sparticle masses deduced assuming R-symmetry do not hold anymore. For instance, the masses of charginos and neutralinos can be considerably lower than weak mass scale as suggested by strange 1995 event (see the earlier posting). The recent very high lower bounds of squark masses putting them above 800 GeV assume also R-symmetry and therefore need not hold true if the decays sh→ q+ν and sg→ g+ν take place fast enough. A further implication is that the scale of SUSY can be weak mass scale- say the p-adic mass scale of 105 GeV corresponding to Mersenne prime M
_{89}.

** 3. p-Adic length scale hypothesis and breaking of SUSY by a selection of p-adic length scale**

p-Adic length scale symmetry leads to a completely new view about SUSY breaking which leads to extremely strong prediction. The basic conjecture is that if the p-adic length scales associated with particles and sparticle are same, their masses are identical. The basic aspect of SUSY breaking is therefore different p-adic mass scales of particle and sparticle. By p-adic length scale hypothesis the masses of particle and superpartner therefore differ by a power of 2^{1/2}. This is extremely powerful prediction and given only minimal kinematical constraints on the event suggesting super-symmetry allows to deduce the mass of super-partner. Some examples give an idea about
what is involved.

- The idea about M
_{89}as the prime characterizing both electroweak scale and SUSY mass scale leads to the proposal that all superpartners correspond to the same p-adic mass scale characterized by k=89. For instance, the masses of sfermions would be given in first order p-adic thermodynamics calculation bym

_{sL}/GeV = (262, 439, 945) ,m

_{sν}/GeV = (235, 423, 938) ,m

_{sU}/GeV = (262, 287, 893) ,m

_{sD}/GeV = (235, 287, 900) . - A good example is the already mentioned 1995 event suggesting the decay cascade involving selectron, chargino and neutralino. One can deduce the mass estimate for all these sparticles just from loose constraints on the mass intervals and obtains for the mass of selectron the estimate 131 GeV which corresponds to M
_{91}instead of M_{89}. This event allowed also to estimate the masses of Zino and corresponding Higgsino. The results are summarized by the following table:m(se)=131 GeV , m(sZ

^{0})=91.2 GeV , m(sh)=45.6 GeV . - In case of the mixing of gaugino and corresponding Higgsino the hypothesis means that the mass matrix is of such form that its eigenvalues have same magnitude but opposite sign. For instance, for the mixing of wino and h
_{W}one would haveM

_{11}= M_{2}=-μM

_{12}=M_{21}= 2^{1/2}M_{W}cos(β) .The masses of the resulting two states would be same but they could correspond to different p-adic primes so that mass scales would differ by a power of 2

^{1/2}. This formula applies also zino and sh_{Z}and photino and sh_{γ}. One possibility is that heavier weak gaugino corresponds to intermediate gauge boson mass scale and light gaugino to 1/2 of this scale: lighter mass scales are forbidden by the decay widths of weak gauge bosons. The exotic event of 1995 suggest that heavier zino has very nearly the same mass as Z^{0}and the lighter one mass which equal to one half of Z^{0}mass. This would mean M_{2}<< M_{Z}. - If one accepts the MSSM formula (see the reference)
m

_{μ}^{2}= m_{sL}^{2}+M_{Z}^{2}cos(2β)/2 ,relating neutrino and sneutrino masses, one can conclude that cos(β)=1/2

^{1/2}is the only possible option so that tan(β)=1 is obtained. This value is excluded by R-parity conserving MSSM but could be consistent with the explanation of g-2 anomaly of muon in terms of loops involving weak gauginos and corresponding higgsinos. - One must distinguish between right- and left handed sfermions sF
_{R}and sF_{L}. These states couple via 1+/-γ_{5}to fermions and are not therefore either pure scalars or pure pseudoscalars. One expects that a maximal mixing of left and right handed sfermions occurs and leads to scalar and pseudoscalar. Mass formula is naturally same for the states for the same value of p-adic prime and also same value of p-adic prime is suggestive. It might however happen that a p-adic mass scales are different for scalars and pseudoscalars. This would allow to have light and heavy variants of squarks and sleptons with scalars probably being the lighter ones.Around 1996 there was a lot of talk about R

_{B}anomaly and Aleph anomaly and some SUSY models of R_{B}anomaly and Aleph anomaly based on R-parity breaking were proposed. No one talks about these anomalies today which suggest that they were statistical fluctuations. One could however spend a few minutes by pretending to take them seriously. If R_{B}anomaly were real, the decay rate of Z^{0}to bbbar pair would be slightly higher than predicted. This might be understood if sb_{R}and st_{R}are light and have masses about 55 GeV not too far above m_{Z}/2. This would make possible decays via loops involving decay to virtual sb_{R}or st_{R}pair decaying to b-pair by an exchance of light chargino or neutralino.Similar mechanism could apply to the claimed Aleph anomaly in which a pair of dijets is produced in e

^{+}e^{-}annihilation. What would happen that virtual photon would produce first to st_{S}st_{Sc}or sb_{S}sb_{Sc}pair. Here subscript "S" refers to scalar associated with fermion and "PS" would refer to the pseudoscalar. Both final states would in turn decay to b quark and chargino or neutralino. If st_{S}or sb_{S}has mass 55 GeV it could explain the anomaly. k=97 would give m_{st}= 55.8 GeV and m_{sb}= 56.25 GeV so that quantitatively the idea seems to work. Chargino and neutralino would have masses m≥ 45 GeV from Z^{0}decay width and satisfying m< 55 GeV to make the decay of squark to chargino and neutralino.

- Since there is no Higgs, there are no bounds to parameters from Higgs mass and little hierarchy problem is avoided.
- The basic element is R-parity breaking reflected in the possibility of decays of sparticles to particle and lonely neutrino not balanced by charge lepton. This together with the absence of Higgs would allow to circumvent various mass limits deduced from LHC and its predecessors and only the limits from the decay width of intermediate gauge bosons on charginos and neutralinos would remain.
- The assumption that the masses of particles and sparticles are same for same p-adic length scale and that the choice of p-adic length scale breaks SUSY means that sparticle masses can be deduce from those of particles apart from a scaling by a power of 2
^{1/2}. This is a powerful and directly testable prediction and predicts the 2× 2 mixing matrices for charginos and neutralinos completely apart from the parameter M=-μ. An attractive idea is that the m_{SUSY}corresponds to the p-adic mass scale associated with Mersenne prime M_{89}characterizing electro-weak length scale except perhaps light chargino and neutralino. For M_{89}sfermions would have for this option masses in the range 200-1000 GeV. - Muon g-2 anomaly could be understood from the predictions tan(β)=1, m
_{SUSY}≈ 105 GeV. Scalar and pseudoscalar sfermions could have different mass scales but there seems to be no recently accepted anomalies requiring this.

For more details see the chapter p-Adic Particle Massivation: New Physics of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".