About gauge bosons and their decay vertices in TGD framework
The attempt to understand how unitarity of scattering amplitudes emerges led as side track to a more detailed view about gauge bosons as flux tubes carrying monopole flux and consisting of two long portions with Minkowskian signature and two short portions represented by wormhole contacts. Also a more detailed view about decay vertices emerged.
There question is how elementary particles and their basic interaction vertices could be realized in this framework.
This mesonlike picture about weak bosons as bound states of fermions sounds complex as compared with standard model picture. On the other hand only the spinor fields assignable to single fermion family are present.
A couple of comments concerning this picture are in order.
- In TGD framework particle would correspond to pair of wormhole contact associated with closed magnetic flux tube carrying monopole flux. Strongly flattened rectangle with Minkowskian flux tubes as long edges with length given by weak scale and Euclidian wormhole contacts as short edges with CP2 radius as lengths scale is a good visualization. 3-particle vertex corresponding to the replication of this kind of flux tube rectangle to two rectangles would replace 3-vertex of Feynman graph. There is analogy with DNA replication. Similar replication is expected to be possible also for the associated closed fermionic strings.
- Denote the wormhole contacts by A and B and their opposite throats by Ai and Bi, i=1,2. For fermions A1 can be assumed to carry the electroweak quantum numbers of fermion. For electroweak bosons A1 and A2 (for instance) could carry fermion and anti-fermion, whose quantum numbers sum up to those of ew gauge boson. These "corner fermions" can be called active.
Also other distributions of quantum numbers must be considered. Fermion and anti-fermion could in principle reside at the same throat - say A1. One can however assume that second wormhole contact, say A has quantum numbers of fermion or weak boson (or gluon) and second contact carries quantum numbers screening weak isospin.
- The model assumes that the weak isospin is neutralized in length scales longer than the size of the flux tube structure given by electro-weak scale. The screening fermions can be called passive. If the weak isospin of W+/- boson is neutralized in the scale of flux tube, 2 νLνbarR pairs are needed (lepton number for these pairs must vanish) for W-. For Z νbarLνR and νLνbarR are needed. The pairs of passive fermions could reside in the interior of flux tube, at string world sheet or at its corners just like active fermions. The first extreme is that the neutralizing neutrino-antineutrino pairs reside in interior at the opposite long edges of the rectangular flux tube. Second extreme is that they are at the corners of rectangular closed string.
- Rectangular closed string containing active fermion at wormhole A (say) and with members of isospin neutralizing neutrino-antineutrino pair at the throats of B serves as basic units. In scales shorter than string length the end A would behave like fermion with weak isospin. At longer scales physical fermion would be hadron like entity with vanishing isospin and one could speak of confinement of weak isospin.
From these physical fermions one can build gauge bosons as bound states. Weak bosons and also gluons would be pairs of this kind of fermionic closed strings connecting wormhole contacts A and B. Gauge bosons (and also gravitons) could be seen as composites of string like physical fermions with vanishing net isospin rather than those of point like fundamental fermions.
- The decay of weak boson to fermion-antifermion pair would be flux tube replication in which closed strings representing physical fermion and anti-fermion continue along different copies of flux tube structure. The decay of boson to two bosons - say W→ WZ - by replication of flux tube would require creation of a pair of physical fermionic closed strings representing Z. This would correspond to a V-shaped vertex with the edge of V representing closed fermionic closed string turning backwards in time. In decays like Z→ W+W- two closed fermion strings would be created in the replication of flux tube. Rectangular fermionic string would turns backwards in time in the replication vertex and the rectangular strings of Z would be shared between W+ and W-.
See the chapter The Recent View about Twistorialization in TGD Framework or the article More about the construction of scattering amplitudes in TGD framework .
- M8 duality provides a different perspective. In M8 picture these vertices could correspond to analogs of local 3 particle vertices for octonionic superfield, which become nonlocal in the map taking M8=M4× CP2 surfaces to surfaces in H=M4× CP2. The reason is that M4 point is mapped to M4 point but the tangent space at E4 point is mapped to a point of CP2. If the point in M8 corresponds to a self-intersection point the tangent space at the point is not unique and point is mapped to two distinct points. There local vertex in M8 would correspond to non-local vertex in H and fermion lines could just begin. This would mean that at H-level fermion line at moment of replication and V-shaped fermion line pair beginning at different point of throat could correspond to 3-vertex at M8 level.
- The 3-vertex representing replication could have interpretation in terms of quantum criticality: in reversed direction of time two branches of solution of classical field equations would co-incide.