Monday, October 04, 2010

Unitarity in TGD variant of Grassmannian twistor approach

In previous postings (see this, this, this, and this: see also short file and the pdf article at my homepage) I have developed ideas about the generalization of the Grassmannian twistor program relaying on Yangian symmetry to TGD.

The zero energy ontology inspired view about generalized Feynman diagrams replacies partons with massless on mass shell partons allowed to have also negative energies. This approach predicts manifestly UV and IR finite scattering amplitudes with non-trvial particle exchanges. One of the key questions is the relationship of this approach to the momentum twistor approach of Arkani-Hamed and collaborators (see this and this). Second question is whether this approach is consistent with unitarity formulated in terms of Cutkosky rules.

The approach of Arkani-Hamed and collaborators based on Grassmannian integrals for momentum twistor amplitudes means that loop integrals over four-momenta are replaced with residue integrals around a small sphere p2=ε. This is very much reminiscent of my own proposal for a few years ago based on the idea that the condition of twistorialization forces to accept only massless virtual states (see this and this. I of course soon gave up this proposal as too childish.

This idea seems to however make a comeback in a modified form. At this time one would have only massive and quantized pseudo-momenta located at discrete mass shells whereas physical parton momenta would be lightlike. Can this picture be consistent with unitarity? The discretization of loop momenta was proposed in the vision about bosonic emergence. If loop momenta are replaced with quantized pseudo-momenta also this idea finds a natural place in the vision inspired by zero energy ontology. What this should teach us is that new ideas should be treated in a friendly manner: their first formulation involves almost as a rule mis-understandings and mis-interpretations but they might turn out to be beautiful swans after all.

Before trying to answer the questions posed in the beginning, one must make clear what one could assume in TGD framework.

  1. Physical particles are in the general case massive and consist of collinear fermions at wormhole throats. External partons at wormhole throats must be massless to allow twistorial interpretation. Therefore massive states emerge. This applies also to stringy states.

  2. The simplest assumption generalizing the childish idea is that on mass shell massless states for partons appear as both virtual particles and external particles. Space-like virtual momentum exchanges are possible if the virtual particles can consist of pairs of positive and negative energy fermions at opposite wormhole throats. Hence also partons at internal lines should be massless and this raises the question about the identification of propagators.

  3. Generalized eigenvalue equation for Chern-Simons Dirac operator implies that virtual elementary fermions have massive and quantized pseudo-momenta whereas external elementary fermions are massless. The massive pseudo-momentum assigned with the Dirac propagator of a parton line cannot be identified with the massless real momentum assigned with the fermionic propagator line. The region momenta introduced in Grassmannian approach are something analogous.

    This brings strongly in mind the identification of the pseudo momentum as the counterpart of the region momentum of momentum twistor diagrams so that the external massless fermionic momenta would be differences of the pseudo-momenta. Indeed, since region momenta are determined apart from a common shift, they need not correspond to real momenta. Same applies to pseudo-momenta and one could assume that both internal and external fermion lines carry light-like pseudo-momenta and that external pseudo-momenta are equal to real momenta. It is quite remarkable that the notions of region momentum and pseudo-momentum emerging from totally different approaches seem to naturally correspond to each other.

  4. This picture has a natural correspondence with momentum twistor diagrams of Grassmannian integral approach. For instance, the region momentum appearing in BCFW bridge defining effective propagator is in general massive although the underlying Feynman diagram would contain online massless momenta. In TGD framework massless lines of Feynman graphs associated with singularities would correspond to real momenta of massless fermions at wormhole throats. Also other canonical operations for Yangian invariants involve light-like momenta at the level of Feynman diagrams and would in TGD framework have a natural identification in terms of partonic momenta. Hence partonic picture would provide a microscopic description for the lines of twistor diagrams.

Let us assume being virtual particle means only that the discretized pseudo-momentum is on shell but massive whereas all real momenta of partons are light-like, and that negative partonic energies are possible. Can one formulate Cutkosky rules for unitarity in this framework? What could the unitarity condition

iDisc(T-T)= -TT+

mean now? In particular, are the cuts associated with mass shells of physical particles or with mass shells of pseudo-momenta? Could these two assignments be equivalent?

  1. The restriction of the partons to be massless but having both signs of energy means that the spectrum of intermediate states contains more states than the external states identified as bound states of partons with the same sign of energy. Therefore the summation over intermediate states does not reduce to a mere summation over physical states but involves a summation over states formed from massless partons with both signs of energy so that also space-like momentum exchanges become possible.

  2. The understanding of the unitarity conditions in terms of Cutkosky rules would require that the cuts of the loop integrands correspond to mass shells for the virtual states which are also physical states. Therefore real momenta have a definite sign and should be massless. Besides this bound state conditions guaranteeing that the mass spectrum for physical states is discrete must be assumed. With these assumptions the unitary cuts would not be assigned with the partonic light-cones but with the mass shells associated of physical particles.

  3. There is however a problem. The pseudo-momenta of partons associated with the external partons are assumed to be light-like and equal to the physical momenta.

    1. If this holds true also for the intermediate physical states appearing in the unitarity conditions, the pseudo-momenta at the cuts are light-like and cuts must be assigned with pseudo-momentum light-cones. This could bring in IR singularities and spoil Yangian symmetry. The formation of bound states could eliminate them and the size scale of the largest CD involved would bring in a natural IR cutoff as the mass scale of the lightest particle. This assumption would however force to give up the assumption that only massive pseudo-momenta appear at the lines of the generalized Feynman diagrams.

    2. On the other hand, if pseudo-momenta are not regarded as a property of physical state and are thus allowed to be massive for the real intermediate states in Cutkosky rules, the cuts at parton level correspond to on mass shell hyperboloids and IR divergences are absent.

For details see the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

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