Could one lift Feynman diagrams to twistor space?

In previous posting I already considered the question how TGD could be lifted from 8-D M⁴×CP₂ to 8-D twistor space with motivation coming from N=4 super-conformal invariance requiring that the target space in which strings live has metric signature (4,4). In the articles of Witten and Nima Arkani-Hamed and collaborators the possibility of twistor diagrammatics is considered. This inspired a crazy morning hour speculation, which I will represent since I find it difficult to imagine what I could still lose in this crazy and cruel world;-).

1. The arguments start from ordinary momentum space perturbation theory. The amplitudes for the scattering of massless particles are expressed in terms of twistors after which one performs twistor Fourier transform obtaining amazingly simple expressions for the amplitudes. For instance, the 4-pt one loop amplitude in N=4 SYM is extremely simple in twistor space having only values '1' and '0' in twistor space and vanishes for generic momenta.

2. Also IR divergences are absent in twistor transform of the scattering amplitude but are generated by the transform to the momentum space. Since plane waves are replaced with light rays, it is not surprising that the IR divergences coming from transversal degrees of freedom are absent. Interestingly, TGD description of massless particles as wormhole throats connecting two massless extremals extends ideal light-ray to massless extremal having finite transversal thickness so that IR cutoff emerges purely dynamically.

3. This approach fails at the level of loops unless one just uses the already calculated loops. The challenge would be a generalization of the ordinary perturbation theory so that loops could be calculated in twistor space formulation.

The vision about lifting TGD from 8-D M⁴×CP₂ to 8-D twistor space suggests that it should be possible to lift also ordinary M⁴ propagators to propagators to twistor space. The first problem is that the momenta of massive virtual particles do not allow any obvious unique representation in terms of twistors. Second problem relates to massive incoming momenta necessarily encountered in stringy picture even if one forgets massivation of light states by p-adic thermodynamics.

1. Could one restrict loop momenta to be light-like?

Could one somehow circumvent the first problem, say by bravely modifying the notion of loop? Could this modification even allow to get rid of UV divergences? The following argument- which might well be one of those arguments which come and go- suggests that this is the case and that also the second problem could be circumvented.

1. At the level of tree diagrams representing 2-particle scattering of massless particles by particle exchange there are no problems. The propagator involves the difference of the two massless momenta and makes sense in twistor space.

2. Twistor picture poses very strong constraint on the notion of loop: loop momenta must be expressible in terms of light-like momenta. This is achieved if only massless momenta rotate in loops so that one can express the momenta appearing in internal lines in terms of the incoming momenta and massless loop momenta and therefore express propagators in terms of incoming twistors and virtual twistor. For instance, for self energy diagram involving two N-vertices the incoming light-like momentum would decompose to N-1 light-like momenta plus one off mass shell momentum expressible in terms of light-like momenta. The odd ball momentum can be assigned to any of the internal lines of the vertex.

3. The four-dimensional loop momentum integral would reduce to 3-dimensional integral over light-cone boundary in momentum space (over both future and past directed light-cones as it seems). The integral over the phase of the loop twistor is not needed unless it appears in the vertices. Since the only mass scale associated with the loop momentum is μ= 0, there are excellent hopes of getting rid of UV divergences. In terms of conformal invariance this kind definition of loop looks extremely natural and in the case of M-matrix unitarity constraint cannot be used to argue that ordinary loops are the only possibility.

The detailed rules possibly realizing this idea are based on the idea of using different propagators. The retarded and advanced Feynman propagators can be assigned naturally to the outgoing and incoming particles and represent classical propagation from the lower or upper light-like boundary of CD. The difference of retarded and advanced propagator -G_- has in its Fourier expansion only light-like momenta, which means also classical propagation. The interpretation is that virtual particle energies to either time direction from sub-CD as on mass shell particle. Feynman propagator in turn is essentially quantal and can be used for the loop momenta, which are not light-like (also advanced or retarded propagator could be considered).

One can check whether these rules give finite results by calculating some simplest loops. For detailed computations see this. It is easy to check what these rules would give in the general case assuming only fermions and couplings involving no gradients. The basic observation is that due to the vanishing of k_i² in the denominators of the ordinary Feynman propagators give at worst a contribution behaving like k⁰ whereas loop lines give a contribution behaving as k. The integral over the light-cone gives k² for each loop. For ordinary loops one has 1/k behavior. Hence the amplitude diverges as

μ^3L ,

where L is the number of light-like loops. For ordinary loops with I the number of internal lines one would have

μ^3L-I .

Therefore UV behavior becomes worse than for ordinary Feynman propagators in the general case.

Unfortunately (or fortunately?) I made direct checks for scalar couplings of fermions first and in this case the simplest fermionic self-energy loop vanishes for on shell particle. Also box and polygon diagrams are finite for scalars if external momenta are on mass shell but not for off mass shell momenta. The optimistic hope was that k_i→ -k_i symmetry could cancel the integral for large loop momenta more generally with positive and negative energies giving compensating contributions for large loop energies.

One can however consider the possibility of a modification of gauge theory by taking fermionic propagator and fermion boson couplings as fundamental ones (only fermions appear as fundamental particles in TGD framework). Both gauge boson propagators and couplings would be generated radiatively. The TGD inspired physical motivation for this view is that bosons are identified as bound states of fermion and antifermion at opposite wormhole throats. The idea is that bosons propagate only by decaying to a pair of fermion and antifermion and then fusing back. Bosonic n-vertex would correspond to the decay of bosons to fermion-antifermion pairs in the loop. In this manner one would completely avoid the introduction of bosonic propagators as primary structures.

p-Adic length scale hypothesis and zero energy ontology suggests a fractal structure for the fermionic propagator implying that it contains a scale factor depending on the p-adic length scale approaching to zero fast enough so that loop integrals reducing to a sum over octaves of momentum scale are finite. This approach could work for both Feynman diagrams (it does!) and their variants with light-like loop momenta (it doesn't!). This kind of description would be applied above p-adic momentum length scale defining the momentum resolution (determined by the size of the smallest CD included) so that no dramatic deviations from standard QFT picture would be predicted for ordinary Feynman propagators in the loops.

A possible manner to get over the problems would be by treating bosons as elementary particles and by allowing no self energy loops. For gauge theory couplings one would have only the modifications of standard type of loops with physical polarizations for on mass shell bosons and simple counting arguments support the view that UV divergences are absent for massless states. The replacement of loop momenta with massless ones improves also IR behavior.

2. Purely twistorial formulation of Feynman graphs with light-like loop momenta

One can also lift the 3-dimensional integral d³k/2E to an integral over twistor variables, which means that complete twistorialization of Feynman diagrams is possible if the loop integrals involve only light-like momenta (recall that this idea failed). This formulation generalizes to the case when loop momenta are massive but requires the introduction of an auxiliary twistor corresponding to momenta restricted to the preferred plane M² subset M⁴ predicted by the number theoretical compactification and hierarchy of Planck constants.

1. It is convenient to introduce double cylindrical coordinates λ_i= ρ_i exp(i(φ+/- ψ)) in twistor space. The integration over overall phase φ gives only a 2π factor since ordinary Feynman amplitude has no dependence on this variable so that the non-redundant variables are ρ₁,ρ₂,ψ.

2. The condition is that the integral measure d⁴uX of the spinor space with a suitable weight function X is equivalent with the measure d³k/2E in cylindrical coordinates. This gives

   d⁴u X= dφd³k/2E

   when the integrand does not depend on φ.

3. In cylindrical coordinates this gives

   2ρ₁ρ₂dρ₁dρ₂dψ Xδ(U-k_z)δ(V-k_x)δ(W-k_z)= 1 ,

   U= (ρ₁²-ρ₂²)/2 ,

   V= ρ₁ρ₂cos(ψ)/2 ,

   W= ρ₁ρ₂sin(ψ)/2 .

   Here the functions U, V, and W are obtained form the representations of k_z,k_x,k_y in terms of spinor and its conjugate.

4. Taking U,V,W as integration variables one has

   2ρ₁ρ₂[∂(ρ₁,ρ₂,ψ)/∂ (U,V,W)]×X = 1 .

5. The calculation of the Jacobian gives X= (ρ₁²+ρ₂²)/4= E/2 so that one has the equivalence

   (1/4π)d⁴u ↔ d³k/2E .

6. Similar lifting can be carried out for the integration measure defined at light-cone boundary in M⁴. If the integrations in generalized Feynman diagrams are over amplitudes depending on light-like momenta and coordinates of the light-like boundaries of CDs in given length scales coming as T_n= 2ⁿT₀ or T_p=pT₀ the integrals of momentum space and light-one can be transformed to integrals over twistor space in given length scale. Twistorialization requirement obviously gives a justification for the basic assumption of zero ontology that all transition amplitudes can be formulated in terms of data at the intersections of light-like 3-surfaces with the boundaries of CDs.

7. It should be emphasized that there is no need to keep the phase angle φ as a redundant variable is the interpretation as Kähler magnetic flux is accepted. In fact, Kähler magnetic fluxes are expected to appear as zero modes define external parameters in the amplitudes.

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".