Still about the topology of elementary particles and hadrons

In its recent form TGD allows several options for the model of elementary particles (see this). I wrote this piece of text because I got worried about details of the definition of wormhole contact appearing as basic building brick of elementary particle.

1. Wormhole contacts in 4-D sense (having Euclidian signature of induced metric) modellable as deformed pieces of CP₂ type extremals connecting Minkowskian space-time sheets (representable as graphs of a map M⁴→ CP₂) are identified basic building bricks of elementary particles. 3-D light-like orbits of 2-D wormhole throats- partonic 2-surfaces - at which the signature of induced metric changes from Euclidian to Minkowskian - partonic orbits - are assumed to be carriers of elementary particle quantum numbers.
   
   
2. One can identify simplest wormhole contact as topological sum: two surfaces touch each other. Remove 3-D regions from both space-time sheets and connecting the topologically identical boundaries with a cylinder X²× D¹, where X² has the topology of the boundary characterized by genus. The assumption that X² is boundary requires that its projection to CP₂ is homologically trivial. This is not consistent with the assumption that the flux tube carries monopole flux. These wormhole contacts are unstable and must be distinguished from wormhole contacts mediating monopole flux. I have not however defined the notion precisely enough.
   
   
3. One can consider several situations in which homologically non-trivial wormhole contact appears.
   

   Option I: Assume that the 3-D time=constant sections of 2 Minkowskian space-time sheets are glued together along their boundaries to form a closed 2-sheeted surface and the throats of wormhole contact - partonic 2-surfaces - serve as magnetic charges creating opposite fluxes. One can say that the two throats have opposite homology charges and therefore form a homologically trivial 2-surface to which one can glue the wormhole contact along its boundaries. The flux at sheet B could be seen as return flux from sheet A and the throat could be seen as very short monopole flux tube.
   

   Option II: Assume no gluing along boundaries for the 3-D time=constant sections of 2 Minkowskian space-time sheets. In this case one must assume at least two wormhole contacts to get vanishing homology charges at both sheets. At both space-time sheets the throats of the contacts with opposite homology charges would be connected by monopole fluxes flowing through the wormhole contacts identifiable as a very short monopole flux tube. This makes sese also for the Option I and might be required since is not clear whether space-time having boundaries carrying monopole flux can be glued together.
   

   Remark: One can also consider the light-like orbit of partonic 2-surface connecting its ends (the minimal distance between partonic 2-surfaces vanishes). The homology charges of ends are opposite in ZEO.
   
   

The proper identification of the model of elementary particles remains still open (see this and this). What relevance do these two options this picture have to the model of elementary particles?

1. For Option I leptons and gauge bosons could be identified as single wormhole contact carrying non-trivial homology flux. The size scale of the closed space-time sheet would correspond to the Compton wavelength of the particle. This model is the simplest one at the level of scattering diagrams and was re-considered in the article SUSY in TGD framework.
   

   Even Euclidian regions of single space-time sheet with vanishing homology charge can be considered as a model for leptons and gauge bosons. In this case it is however not clear how to understand how the size scale of the particle as Compton length could be understood at space-time level. This model was one of the first models. I have also considered the identification of the particle as boundary component of Minkowskian space-time surface.
   
   

2. Option II was assumed in the model following the original model for leptons and gauge bosons. It was also proposed that electroweak confinement as dual description of massivation takes place in the sense that the weak charges associated with the two wormhole contacts cancel each other. The size scale of flux tube at given sheet would correspond to the Compton length assignable to the particle. In this case scattering amplitudes are more complex topologically.
   
   

What about baryons?

1. The simplest model assumes that quarks do not differ from leptons and gauge bosons in any manner. The contribution of the quarks to masses of hadrons is very small fraction of total mass, which suggests that color flux tubes carrying also homology charge are present and give the dominating contribution.
   

   One can also consider a structure formed by color magnetic monopole flux tubes carrying most of the hadron mass with Minkowskian signature carrying flux of 2 units branching to two flux tubes carrying 1 unit each. The flux tubes would have length given by hadronic p-adic length scale. The ends of flux tubes would be wormhole throats connected by wormhole contacts to the mirror image of this structure. One can say that homology charges 2,-1,-1 assignable to the throats of single space-time sheet sum up to zero. This brings in mind color hypercharge. Could color confinement have vanishing of homology charge as classical space-time correlate?
   
   

2. In the proposal for the TGD based realization of space-time supersymmetry and the notion of super-space I considered two alternative identification of leptons. Leptons and quarks could correspond to the different chiralities of M⁴× CP₂ spinors and lepton and baryon numbers would be separately conserved. For second option leptons would b local 3-quark composites and therefore analogous to spartners of quarks: this option is possible only in TGD framework and the reason is that color is not spin-like quantum number in TGD framework. Baryon and lepton numbers would not be separately conserved.
   

   One can ask what could be the simplest mechanism inducing the decay of baryon as 3-quark composite involving only 3 wormhole contacts and giving lepton as a local 3-quark composite plus something. Wormhole throats of 3 quarks carrying the quark quantum numbers should fuse together to form a leptonic wormhole throat, and the 3 quark lines representing boundaries of string world sheets should fuse to single line. If the sum of quark homology charges is vanishing, lepton must have a vanishing homology charge unless the reaction involves also a step taking care of the conservation of homology charge as a decay of the resulting wormhole contact with vanishing monopole flux to two wormhole contacts with opposite monopole fluxes. Already the first step of the decay process is quite complex, and one can hope that the rate for the reaction is slow enough.
   
   

See the article SUSY in TGD framework or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.