Permutations, braidings, and amplitudes

Nima Arkani-Hamed et al demonstrate that various twistorially represented on-mass-shell amplitudes (allowing light-like complex momenta) constructible by taking products of the 3-particle on mass-shell amplitude and its conjugate can be assigned with unique permutations of the incoming lines. The article describes the graphical representation of the amplitudes and its generalization. For 3-particle amplitudes, which correspond to ++- and +-- twistor amplitudes, the corresponding permutations are cyclic permutations, which are inverses of each other. One actually introduces double cover for the labels of the particles and speaks of decorated permutations meaning that permutation is always a right shift in the integer and in the range [1,2× n].

Amplitudes as representation of permutations

It is shown that for on mass shell twistor amplitudes the definition using on-mass-shell 3-vertices as building bricks is highly reducible: there are two moves for squares defining 4-particle sub-amplitudes allowing to reduce the graph to a simpler one. The first move is topologically like the s-t duality of the old-fashioned string models and second one corresponds to the transformation black ↔ white for a square sub-diagram with lines of same color at the ends of the two diagonals and built from 3-vertices.

One can define the permutation characterizing the general on mass shell amplitude by a simple rule. Start from an external particle a and go through the graph turning in in white (black) vertex to left (right). Eventually this leads to a vertex containing an external particle and identified as the image P(a) of the a in the permutation. If permutations are taken as right shifts, one ends up with double covering of permutation group with 2× n! elements - decorated permutations. In this manner one can assign to any any line of the diagram two lines. This brings in mind 2-D integrable theories where scattering reduces to braiding and also topological QFTs where braiding defines the unitary S-matrix. In TGD parton lines involve braidings of the fermion lines so that an assignment of permutation also to vertex would be rather nice.

BCFW bridge has an interpretation as a transposition of two neighboring vertices affecting the lines of the permutation defining the diagram. One can construct all permutations as products of transpositions and therefore by building BCFW bridges. BCFW bridge can be constructed also between disjoint diagrams as done in the BCFW recursion formula.

Can one generalize this picture in TGD framework? There are several questions to be answered.

1. What should one assume about the states at the light-like boundaries of string world sheets? What is the precise meaning of the supersymmetry: is it dynamical or gauge symmetry or both?
   
   
2. What does one mean with particle: partonic 2-surface or boundary line of string world sheet? How the fundamental vertices are identified: 4 incoming boundaries of string world sheets or 3 incoming partonic orbits or are both aspects involved?
   
   
3. How the 8-D generalization of twistors bringing in second helicity and doubling the M⁴ helicity states assignable to fermions does affect the situation?
   
   
4. Does the crucial right-left rule relying heavily on the possibility of only 2 3-particle vertices generalize? Does M⁴ massivation imply more than 2 3-particle vertices implying many-to-one correspondence between on-mass-shell diagrams and permutations? Or should one generalize the right-left rule in TGD framework?
   
   

Fermion lines for fermions massless in 8-D sense

What does one mean with particle line at the level of fermions?

1. How the addition of CP₂ helicity and complete correlation between M⁴ and CP₂ chiralities does affect the rules of N=4 SUSY? Chiral invariance in 8-D sense guarantees fermion number conservation for quarks and leptons separately and means conservation of the product of M⁴ and CP₂ chiralities for 2-fermion vertices. Hence only M⁴ chirality need to be considered. M⁴ massivation allows more 4-fermion vertices than N=4 SUSY.
   
   
2. One can assign to a given partonic orbit several lines as boundaries of string world sheets connecting the orbit to other partonic orbits. Supersymmetry could be understoond in two manners.
   
   
   1. The fermions generating the state of super-multiplet correspond to boundaries of different string world sheets which need not connect the string world sheet to same partonic orbit. This SUSY is dynamical and broken. The breaking is mildest breaking for line groups connected by string world sheets to same partonic orbit. Right handed neutrinos generated the least broken N=2 SUSY.
      
      
   2. Also single line carrying several fermions would provide realization of generalized SUSY since the multi-fermion state would be characterized by single 8-momentum and helicity. One would have N =4 SUSY for quarks and leptons separately and N =8 if both quarks and leptons are allowed. Conserved total for quark and antiquarks and leptons and antileptons characterize the lines as well.

      What would be the propagator associated with many-fermion line? The first guess is that it is just a tensor power of single fermion propagator applied to the tensor power of single fermion states at the end of the line. This gives power of 1/p²ⁿ to the denominator, which suggests that residue integral in momentum space gives zero unless one as just single fermion state unless the vertices give compensating powers of p. The reduction of fermion number to 0 or 1 would simplify the diagrammatics enormously and one would have only 0 or 1 fermions per given string boundary line. Multi-fermion lines would represent gauge degrees of freedom and SUSY would be realized as gauge invariance. This view about SUSY clearly gives the simplest picture, which is also consistent with the earlier one, and will be assumed in the sequel
      
      

3. The multiline containing n fermion oscillator operators can transform by chirality mixing in 2ⁿ manners at 4-fermion vertex so that there is quite a large number of options for incoming lines with n_i fermions.
   
   
   
4. In 4-D Dirac equation light-likeness implies a complete correlation between fermion number and chirality. In 8-D case light-likeness should imply the same: now chirality correspond to fermion number. Does this mean that one must assume just superposition of different M⁴ chiralities at the fermion lines as 8-D Dirac equation requires. Or should one assume that virtual fermions at the end of the line have wrong chirality so that massless Dirac operator does not annihilate them?
   
   

Fundamental vertices

One can consider two candidates for fundamental vertices depending on whether one identifies the lines of Feynman diagram as fermion lines or as light-like orbits of partonic 2-surfaces. The latter vertices reduces microscopically to the fermionic 4-vertices.

1. If many-fermion lines are identified as fundamental lines, 4-fermion vertex is the fundamental vertex assignable to single wormhole contact in the topological vertex defined by common partonic 2-surface at the ends of incoming light-like 3-surfaces. The discontinuity is what makes the vertex non-trivial.
   
   
2. In the vertices generalization of OZI rule applies for many-fermion lines since there are no higher vertices at this level and interactions are mediated by classical induced gauge fields and chirality mixing. Classical induced gauge fields vanish if CP₂ projection is 1-dimensional for string world sheets and even gauge potentials vanish if the projection is to geodesic circle. Hence only the chirality mixing due to the mixing of M⁴ and CP₂ gamma matrices is possible and changes the fermionic M⁴ chiralities. This would dictate what vertices are possible.
   
   
3. The possibility of two helicity states for fermions suggests that the number of amplitudes is considerably larger than in N=4 SUSY. One would have 5 independent fermion amplitudes and at each 4-fermion vertex one should be able to choose between 3 options if the right-left rule generalizes. Hence the number of amplitudes is larger than the number of permutations possibly obtained using a generalization of right-left rule to right-middle-left rule.
   
   
4. Note however that for massless particles in M⁴ sense the reduction of helicity combinations for the fermion and antifermion making virtual gauge boson happens. The fermion and antifermion at the opposite wormhole throats have parallel four-momenta in good approximation. In M⁴ they would have opposite chiralities and opposite helicities so that the boson would be M⁴ scalar. No vector bosons would be obtained in this manner.

   In 8-D context it is possible to have also vector bosons since the M⁴ chiralities can be same for fermion and anti-fermion. The bosons are however massive, and even photon is predicted to have small mass given by p-adic thermodynamics. Massivation brings in also the M⁴ helicity 0 state. Only if zero helicity state is absent, the fundamental four-fermion vertex vanishes for ++++ and ---- combinations and one extend the right-left rule to right-middle-left rule. There is however no good reason for he reduction in the number of 4-fermion amplitudes to take place.
   
   

Partonic surfaces as 3-vertices

At space-time level one could identify vertices as partonic 2-surfaces.

1. At space-time level the fundamental vertices are 3-particle vertices with particle identified as wormhole contact carrying many-fermion states at both wormhole throats. Each line of BCFW diagram would be doubled. This brings in mind the representation of permutations and leads to ask whether this representation could be re-interpreted in TGD framework. For this option the generalization of the decomposition of diagram to 3-particle vertices is very natural. If the states at throats consist of bound states of fermions as SUSY suggests, one could characterize them by total 8-momentum and helicity in good approximation. Both helicities would be however possible also for fermions by chirality mixing.
   
   
2. A genuine decomposition to 3-vertices and lines connecting them takes place if two of the fermions reside at opposite throats of wormhole contact identified as fundamental gauge boson (physical elementary particles involve two wormhole contacts).

   The 3-vertex can be seen as fundamental and 4-fermion vertex becomes its microscopic representation. Since the 3-vertices are at fermion level 4-vertices their number is greater than two and there is no hope about the generalization of right-left rule.
   
   

OZI rule implies correspondence between permutations and amplitudes

The realization of the permutation in the same manner as for N=4 amplitudes does not work in TGD. OZI rule following from the absence of 4-fermion vertices however implies much simpler and physically quite a concrete manner to define the permutation for external fermion lines and also generalizes it to include braidings along partonic orbits.

1. Already N=4 approach assumes decorated permutations meaning that each external fermion has effectively two states corresponding to labels k and k+n (permutations are shifts to the right). For decorated permutations the number of external states is effectively 2ⁿ and the number of decorated permutations is 2× n!. The number of different helicity configurations in TGD framework is 2ⁿ for incoming fermions at the vertex defined by the partonic 2-surface. By looking the values of these numbers for lowest integers one finds 2n≥ 2ⁿ: for n=2 the equation is saturated. The inequality log(n!) >nlog(n)/e)+1 (see Wikipedia) gives

   log(2n!)/log(2ⁿ)geq; log(2)+ 1+nlog(n/e)/nlog(2)= log(n/e)/log(2) +O(1/n)

   so that the desired inequality holds for all interesting values of n.
   
   

2. If OZI rule holds true, the permutation has very natural physical definition. One just follows the fermion line which must eventually end up to some external fermion since the only fermion vertex is 2-fermion vertex. The helicity flip would map k→ k+n or vice versa.
   
   
3. The labelling of diagrams by permutations generalizes to the case of diagrams involving partonic surfaces at the boundaries of causal diamond containing the external fermions and the partonic 2-surfaces in the interior of CD identified as vertices. Permutations generalize to braidings since also the braidings along the light-like partonic 2-surfaces are allowed. A quite concrete generalization of the analogs of braid diagrams in integrable 2-D theories emerges.
   
   
4. BCFW bridge would be completely analogous to the fundamental braiding operation permuting two neighboring braid strands. The almost reduction to braid theory - apart from the presence of vertices conforms with the vision about reduction of TGD to almost topological QFT.
   
   

To sum up, the simplest option assumes SUSY as both gauge symmetry and broken dynamical symmetry. The gauge symmetry relates string boundaries with different fermion numbers and only fermion number 0 or 1 gives rise to a non-vanishing outcome in the residue integration and one obtains the picture used hitherto. If OZI rule applies, the decorated permutation symmetry generalizes to include braidings at the parton orbits and k→ k +/- n corresponds to a helicity flip for a fermion going through the 4-vertex. OZI rule follows from the absence of non-linearities in Dirac action and means that 4-fermion vertices in the usual sense making theory non-renormalizable are absent. Theory is essentially free field theory in fermionic degrees of freedom and interactions in the sense of QFT are transformed to non-trivial topology of space-time surfaces.

See the chapter Classical part of the twistor story of "Towards M-matrix" or the article Classical part of the twistor story.