Are light-like loop momenta consistent with unitarity?

The original version of this note was written before it had become clear that the replacement of momenta rotating in loops with massless momenta - suggested by the expressibility of the momenta of internal lines in terms of twistors - very probably implies very bad UV behavior for general Feynman diagrams if one assumes only fermion lines. This unless symmetries lead to miraculous cancellation of divergences so that the idea does not look promising.

One can however consider the possibility of a modification of gauge theory with self energy diagrams excluded. A rough check shows that gauge boson vertices are very probably UV divergent due to the presence of momentum factors in all three three-vertices. One can however wonder whether a bootstrap approach -very natural in TGD framework and strongly encouraged by the success of twistorial unitary cut method in gauge theories - taking fermion-boson couplings as fundamental couplings and purely bosonic couplings as radiatively generated ones could lead to UV and IR finite theory by allowing only light-like loop momenta. In this case gauge theory dynamics would be emergent and resulting bosonic couplings could have form factors with IR and UV behaviors allowing finiteness of loops constructed from them. The TGD inspired physical motivation for this view is that bosons are identified as bound states of fermion and antifermion at opposite wormhole throats so that bosonic n-vertex would correspond to the decay of bosons to fermion pairs in the loop.

The following argument shows that one could achieve unitarity also in this framework by simple further assumption about the proposed rules. As a matter fact, so called generalized cut rules combined with twistor approach allow to construct massless S-matrix using just cut rules and dispersion relation from tree diagrams without any need for the calculation of loop integrals and this approach might apply also in TGD framework where one does not have any Feynman rules in the ordinary sense of the word.

If massless negative energy states are allowed in loops the analog of S-matrix constructed using modified Feynman rules might not be unitary. It is also questionable whether one can exclude negative energy particles from the final states. In TGD framework also physical picture allows to challenge these basic rules.

1. In negative energy ontology S-matrix is replaced with M-matrix representing time-like entanglement coefficients between positive and negative energy parts of the zero energy state. M-matrix need not be unitary. The proposal is however that M-matrix decomposes into a product of square root of a positive definite diagonal density matrix and unitary S-matrix just as Schrödinger amplitude decomposes into a product of modulus and phase. This would mean unification of statistical physics and quantum theory at fundamental level. S-matrix would be still something universal and only the density matrix would be state dependent.

2. One of the long standing issues of TGD has been whether one should allow besides positive energy states also negative states regarded as analogs of phase conjugate laser beams to be distinguished from antiparticles which can be also seen analogs of negative energy particles. The TGD based identification of bosons as pairs of wormhole throats carrying fermion numbers is most elegant if the second wormhole carries phase conjugate fermion with negative energy. The minimal deviation from standard physics picture would allow negative energy light-like momenta in loops interpreted as phase conjugate particles and their appearance in only loops would explain why they are rare. One can however consider phase conjugate states also as incoming and final states. The interpretation would be that they result through time reflection from the lower boundaries of sub-CDs whereas negative energy parts of zero energy state correspond to reflection from the upper boundary of CD.

In standard approach unitarity conditions i(T-T^†) = -TT^† are expressed very elegantly in terms of Cutkosky rules.

1. The difference i(T-T^†) corresponds to the discontinuity of the Feynman diagram in a channel in which one has N parallel lines. For instance, 2-2 scattering by boson exchange corresponds to cut for a box diagram.

2. T^† is obtained by changing the sign of ε in Feynman propagators giving contribution to T so that one has 1/(p²-m²+iε)→ 1/(p²-m²-iε). The subtraction of Feynman propagators with different signs of ε in T-T^† gives just an integral over on mass shell states with positive energy and therefore TT^†.

3. Analyticity in momentum space allows to use dispersion relations deduce also the "real" part of the amplitude so that in principle one could avoid loop calculations altogether. In twistor approach where only on mass shell momenta allow a nice description in terms of twistors the unitary cut method developed by Bern, Dixon, Dunbar and Kosower (see this) approach is very natural and generalized cuts are utilized in twistor approach heavily to deduce information about amplitudes using only tree diagrams as a starting point.

For the proposed modification one obtains standard unitarity if the intermediate lines appearing in cuts contain always only Feynman propagators. The generalized Feynman rules should guarantee this automatically. Since the goal is to construct unitary S-matrix without the constraints coming from standard QFT formalism, and since the lines allowing to cut the diagram are topologically very special, nothing hinders from posing the rule that light-like loop momenta do not appear in the lines allowing a cut. It seems however that the bad UV behavior of proposed diagrammatics based on four-fermion vertices and light-like loop momenta does not allow to construct Feynman diagrams in the proposed manner. Rather, the unitary cut method looks much more natural manner to build up S-matrix in TGD framework provided one can define tree diagrams using 4-fermion vertices. One must also remember that in the general TGD vision about the construction of M-matrix involves no local vertices. The vertices are defined in terms of N-point functions of conformal field theory and each braid strand correspond to a point where fermion emerges. Therefore local four-fermion vertex would mean giving up the non-locality due to the replacement of particles with partonic 2-surfaces containg fermions at braid strands.

Note added: It turned out that p-adic length scale hypothesis and finite measurement resolution lead to a beautiful realization of the bootstrap approach assuming only fermionic propagator and fermion boson couplings as given. The approach reduces to strict Feynman rules allowing to generated bosonic propagators and bosonic vertices radiatively. The notion of light-like loop momenta turned out to be a failure without which I had never realized that in TGD framework the only possible manner to generate bosonic propagators and vertices is by radiative corrections!

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".