At the level of quarks, the scattering would be completely trivial, which looks strange. One would obtain a product of quark propagators connecting the points at mass shells with opposite energies plus entanglement coefficients arranging them at positive and negative energy light-cones to groups which are Galois singlets.
This is completely analogous to the OZI role. In QCD it is of course violated by generation of gluons decaying to quark pairs. In TGD, gauge bosons are also quark pairs so that there is no problem of principle.
There is an objection against this picture.
- If particle reactions are mere recombinations of Galois singlets with Galois singlets, the quark and antiquark numbers Nq and Nqbar of quark and antiquark numbers are separately conserved (as also their difference Nq-Nqbar). This forbids many reactions, for instance those in which a gauge boson is emitted unless one assumes that many quark states are superpositions of states with a varying total quark number N. This would mean that the extremely simple re-combinatorics picture is lost.
- Crossing symmetry, which is a symmetry of standard QFTs, suggests a solution to the problem. Crossing symmetry would mean that one can transfer quarks between initial and final states by changing the sign of the quark four-momentum so that momentum conservation is not violated. Crossing means analytic continuation of the scattering amplitude by replacing incoming (outgoing) momentum p with outgoing (incoming) momentum -p. The scattering amplitudes for reactions for which the quark number is conserved can be constructed using mere recombinatorics, and the remaining amplitudes would be obtained by crossing.
- Crossing must respect the Galois singlet property. For instance, the crossing of a single quark destroys Galois singlet. Unless one allows destruction and recombination of Galois singlets, the crossing can apply only to Galois singlets. These rules bring to mind the vanishing of twistor amplitudes when one gluon has negative helicity and the remaining gluons have positive helicity.
For a summary of earlier postings see Latest progress in TGD.