4-fermion twistor amplitudes are basic building bricks of twistor amplitudes in TGD framework. What can one conclude about them on basis of N=4 amplitudes? Instead of 3-vertices as in SYM, one has 4-fermion vertices as fundamental vertices and the challenge is to guess their general form. The basis idea is that N=4 SYM amplitudes could give as special case the n-fermion amplitudes and their supersymmetric generalizations
1. Attempt to understand the physical picture
One must try to identify the physical picture first.
The deformation Δ mk can be expanded in terms of CP2 complex coordinates so that the modes have well defined color hyper-charge and isospin. There are two options to be considered.
- Elementary particles consist of pairs of wormhole contacts connecting two space-time sheets. The throats are connected by magnetic fluxes running in opposite directions so that a closed monopole flux loop is in question. One can assign to the ordinary fermions open string world sheets whose boundary belong to the light-like 3-surfaces assignable to these two wormhole contacts. The question is whether one can restrict the consideration to single wormhole contact or should one describe the situation as dynamics of the open string world sheets so that basic unit would involve two wormhole contacts possibly both carrying fermion number at their throats.
Elementary particles are bound states of massless fermions assignable to wormhole throats. Virtual fermions are massless on mass shell particles with unphysical helicity. Propagator for wormhole contact as bound state - or rather entire elementary particle would be from p-adic thermodynamics expressible in terms of Virasoro scaling generator as 1/L0 in the case of boson. Super-symmetrization suggests that one should replace L0 by G0 in the wormhole contact but this leads to problems if G0 carries fermion number. This might be a good enough motivation for the twistorial description of the dynamics reducing it to fermion propagator along the light-like orbit of wormhole throat. Super Virasoro algebra would emerged only for the bound states of massless fermions.
- Suppose that the construction of four-fermion vertices reduces to the level of single wormhole contact. 4-fermion vertex involves wormhole contact giving rise to something analogous to a boson exchange along wormhole contact. This kind of exchange might allow interpretation in terms of Euclidian correlation function assigned to a deformation of CP2 type vacuum extremal with Euclidian signature.
A good guess for the interaction terms between fermions at opposite wormhole contacts is as current-current interaction jα (x) jα(y), where x and y parametrize points of opposite throats. The current is defined in terms of induced gamma matrices as ‾ΨΓαΨ and one functionally integrates over the deformations of the wormhole contact assumed to correspond in vacuum configuration to CP2 type vacuum extremal metrically equivalent with CP2 itself. One can expand the induced gamma matrix as a sum of CP2 gamma matrix and contribution from M4 deformation Γα = ΓαCP2 + ∂α mkγk. The transversal part of M4 coordinates orthogonal to M2⊂ M4 defines the dynamical part of mk so that one obtains strong analogy with string models and gauge theories.
- One could use CP2 spherical harmonics defined as eigenstates of CP2 scalar Laplacian D2. The scale of eigenvalues would be 1/R2, where R is CP2 radius of order 104 Planck lengths. The spherical harmonics are in general not holomorphic in CP2 complex coordinates ξi, i=1,2. The use of CP2 spherical harmonics is however not necessary since wormhole throats mean that wormhole contact involves only a part of CP2 is involved.
- Conformal invariance suggests the use of holomorphic functions ξ1mξ2n as analogs of zn in the expansion. This would also be the Euclidian analog for the appearance of massless spinors in internal lines. Holomorphic functions are annihilated by the ordinary scalar Laplacian. For conformal Laplacian they correspond to the same eigenvalue given by the constant curvature scalar R of CP2. This might have interpretation as a spontaneous breaking of conformal invariance.
The holomorphic basis zn reduces to phase factors exp(inφ) at unit circle and can be orthogonalized. Holomorphic harmonics reduce to phase factors exp(imφ1)exp(inφ2) and torus defined by putting the moduli of ξi constant and can thus be orthogonalized. Inner product for the harmonics is however defined at partonic 2-surface. Since partonic 2-surfaces represent Kähler magnetic monopoles they have 2-dimensional CP2 projection. The phases exp(imφi) could be functionally independent and a reduction of inner product to integral over circle and reduction of phase factors to powers exp(inφ) could take place and give rise to the analog of ordinary conformal invariance at partonic 2-surface. This does not mean that separate conservation of I3 and Y is broken for propagator.
- Holomorphic harmonics are very attractive but the problem is that they are annihilated by the ordinary Laplacian. Besides ordinary Laplacian one can however consider
(Conformal Laplacian) defined as
and relating the curvatures of two conformally scaled metrics R denotes now curvature scalar). The overall scale factor and also its sign is just a convention. This Laplacian has the same eigenvalue for all conformal harmonics. The interpretation would be in terms of a breaking of conformal invariance due to CP2 geometry: this could also relate closely to the necessity to assume tachyonic ground state in the p-adic mass calculations.
The breaking of conformal invariance is necessary in order to avoid infrared divergences. The replacement of M4 massless propagators with massive CP2 bosonic propagators in 4-fermion vertices brings in the needed breaking of conformal invariance. Conformal invariance is however retained at the level of M4 fermion propagators and external lines identified as bound states of massless states.
2. How to identify the bosonic correlation functions inside wormhole contacts?
The next challenge is to identify the correlation function for the deformation δ mk inside wormhole contacts.
Conformal invariance suggests the identification of the analog of propagator as a correlation function fixed by conformal invariance for a system defined by the wormhole contact. The correlation function should depend on the differences ξi=ξi,1-ξi,2 of the complex CP2 coordinates at the points ξi,1) and ξi,2 of the opposite throats and transforms in a simple manner under scalings of ξi. The simplest expectation is that the correlation function is power r-n, where r2= [ξ1|2+|ξ2|2 defines U(2) invariant coordinate distance squared. The correlation function can be expanded as products of conformal harmonics or ordinary harmonics of CP2 assignable to ξi,1 and ξi,2 and one expects that the values of Y and I3 vanish for the terms in the expansions: this just states that Y and I3 are conserved in the propagation.
Second approach relies on the idea about propagator as the inverse of some kind of Laplacian. The approach is not in conflict with the general conformal approach since the Laplacian could occur in the action defining the conformal field theory. One should try to identify a Laplacian defining the propagator for δ mk inside Euclidian regions.
- The propagator defined by the ordinary Laplacian D2 has infinite value for all conformal harmonics appearing in the correlation function. This cannot be the case.
- If the propagator is defined by the conformal Laplacian Dc2 of CP2 multiplied by some numerical factor it gives fro a given model besides color quantum numbers conserving delta function a constant factor nR2 playing the same role as weak coupling strength in the four-fermion theory of weak interactions. Propagator in CP2 degrees of freedom would give a constant contribution if the total color quantum numbers for vanish for wormhole throat so that one would have four-fermion vertex.
- One can consider also a third - perhaps artificial option - motivated for Dirac spinors by the need to generalize Dirac operator to contain only I3 and Y. Holomorphic partial waves are also eigenstates of a modified Laplacian D2C defined in terms of Cartan algebra as
D2C== [aY2+bI32]/R2 ,
where a and b suitable numerical constants and R denotes the CP2 radius defined in terms of the length 2π R of CP2 geodesic circle. The value of a/b is fixed from the condition Tr(Y2)=Tr(I32) and spectra of Y and I3 given by (2/3,-1/3,-1/3) and (0,1/2,-1/2) for triplet representation. This gives a/b= 9/20 so that one has
D2C= ((9/20) Y2+ I32]× a/R2 .
In the fermionic case this kind of representation is well motivated since fermionic Dirac operator would be Yk eAkγA+I3k eAkγA, where the vierbein projections YkeAk YkeAk and I3keAk of Killing vectors represent the conserved quantities along geodesic circles and by semiclassical quantization argument should correspond to the quantized values of Y and I3 as vectors in Lie algebra of SU(3) and thus tangent vectors in the tangent space of CP2 at the point of geodesic circle along which these quantities are conserved. In the case of S2 one would have Killing vector field Lz at equator.
Two general remarks are in order.
- That a theory containing only fermions as fundamental elementary particles would have four-fermion vertex with dimensional coupling as a basic vertex at twistor level, would not be surprising. As a matter of fact, Heisenberg suggested for long time ago a unified theory based on use of only spinors and this kind of interaction vertex. A little book about this theory actually inspired me to consider seriously the fascinating challenge of unification.
- A common problem of all these options seems to be that the 4-fermion coupling strength is of order R2 - about 108 times gravitational coupling strength and quite too weak if one wants to understand gauge interactions. It turns out however that color partial waves for the deformations of space-time surface propagating in loops can increase R2 to the square Lp2= pR2 of p-adic length scale. For D2C assumed to serve as an propagator in an effective action of a conformal field theory one can argue that large renormalization effects from loops increase R2 to something of order pR2.
3. Do color quantum numbers propagate and are they conserved in vertices?
The basic questions are whether one can speak about conservation of color quantum numbers in vertices and their propagation along the internal lines and the closed magnetic flux loops assigned with the elementary particles having size given by p-adic length scale and having wormhole contacts at its ends. p-Adic mass calculations predict that in principle all color partial waves are possible in cm degreees of freedom: this is a description at the level of imbedding space and its natural counterpart at space-time level would be conformal harmonics for induced spinor fields and allowance of all of them in generalized Feynman diagrams.
- The analog of massless propagation in Euclidian degrees of freedom would correspond naturally to the conservation of Y and I3 along propagator line and conservation of Y and I3 at vertices. The sum of fermionic and bosonic color quantum numbers assignable to the color partial waves woul be conserved. For external fermions the color quantum numbers are fixed but fermions in internal lines could move also in color excited states.
- One can argue that the correlation function for the M4 coordinates for points at the ends of fermionic line do not correlate as functions of CP2 coordinates since the distance between partonic 2-surface is much longer than CP2 scale but do so as functions of the string world sheet coordinates as stringy description strongly suggests and that stringy correlation function satisfying conformal invariance gives this correlation. One can however couner argue that for hadrons the color correlations are different in hadronic length scale. This in turn suggests that the correlations are non-trivial for both the wormhole magnetic flux tubes assignable to elementary particles and perhaps also for the internal fermion lines.
- I3 and Y assignable to the exchanged boson should have interpretation as an exchange of quantum numbers between the fermions at upper and lower throat or change of color quantum numbers in the scattering of fermion. The problem is that induced spinors have constant anomalous Y and I3 in given coordinate patch of CP2 so that the exchange of these quantum numbers would vanish if upper and lower coordinate patches are identical. Should one expand also the induced spinor fields in Euclidian regions using the harmonics or their holomorphic variants as suggested by conformal invariance?
The color of the induced spinor fields as analog of orbital angular momentum would realized as color of the holomorphic function basis in Euclidian regions. If the fermions in the internal lines cannot carry anomalous color, the sum over exchanges trivializes to include only a constant conformal harmonic. The allowance of color partial waves would conform with the idea that all color partial waves are allowed for quarks and leptons at imbedding space level but define very massive bound states of massless fermions.
- The 4-fermion vertex would involve a sum over the exchanges defined by spherical harmonics or - more probably - by their holomorphic analogs. For both the spherical and conformal harmonic option the 4-fermion coupling strength would be of order R2, where R is CP2 length. The coupling would be extremely weak - about 108 times the gravitational coupling strength G if the coupling is of order one. This is definitely a severe problem: one would want something like Lp2, where p is p-adic prime assignable to the elementary particle involved.
This problem provides a motivation for why a non-trivial color should propagate in internal lines. This could amplify the coupling strength of order R2 to something of order Lp2=pR2. In terms of Feynman diagrams the simplest color loops are associated with the closed magnetic flux tubes connecting two elementary wormhole contacts of elementary particle and having length scale given by p-adic length scale Lp. Recall that νL (νR)c pair or its conjugate neutralizes the weak isospin of the elementary fermion. The loop diagrams representing exchange of neutrino and the fermion associated with the two wormhole contacts and thus consisting of two fermion lines assignable to "long" strings and two boson lines assignable to "short strings" at wormhole contacts represent the first radiative correction to 4-fermion diagram. They would give sum over color exchanges consistent with the conservation of color quantum numbers at vertices. This sum, which in 4-D QFT gives rise to divergence, could increase the value of four-fermion coupling to something of order Lp2= kpR2 and induce a large scaling factor of $D^2_C$.
- Why known elementary fermions correspond to color singlets and triplets? p-Adic mass calculations provide one explanation for this: colored excitations are simply too massive. There is however evidence that leptons possess color octet excitations giving rise to light mesonlike states. Could the explanation relate to the observation that color singlet and triplet partial waves are special in the sense that they are apart from the factor 1/(1+r2)1/2 , r2=∑ |ξi|2 for color triplet holomorphic functions?
4. Why twistorialization in CP2 degrees of freedom?
A couple of comments about twistorialization in CP2 degrees of freedom are in order.
5. Reduction of scattering amplitudes to subset of N =4 scattering amplitudes
- Both M4 and CP2 twistors could be present for the holomorphic option. M4 twistors would characterize fermionic momenta and CP2 twistors to the quantum numbers assignable to deformations of CP2 type vacuum extremals. CP2 twistors would be discretized since I3 and Y have discrete spectrum and it is not at all clear whether twistorialization is useful now. There is excellent motivation for the integration over the flag-manifold defining the choices of color quantization axes. The point is that the choice of conformal basis with well-defined Y and I3 breaks overall color symmetry SU(3) to U(2) and an integration over all possible choices restores it.
- Four-fermion vertex has a singularity corresponding to the situation in which p1, p2 and p1+p2 assignable to emitted virtual wormhole throat are collinear and thus all light-like. The amplitude must develop a pole as p3+p3= p1+p2 becomes massless. These wormhole contacts would behave like virtual boson consisting of almost collinear pair of fermion and anti-fermion at wormhole throats.
N=4 SUSY provides quantitative guidelines concerning the actual construction of the amplitudes.
Some words of critism is in order.
- For single wormhole contact carrying one fermion, one obtains two N=2 SUSY multiplets from fermions by adding to ordinary one-fermion state right-handed neutrino, its conjugate with opposite spin, or their pair. The net spin projections would be 0, 1/2 ,1 with degeneracies (1,2,1) for fermion helicity 1/2 and (0,-1/2, -1) with same degeneracies for fermion helicity -1/2. These N=2 multiplets can be imbedded to the N=4 multiplet containing 24 states with spins (1,1/2,0,-1/2,-1) and degeneracies given by (1, 4, 6, 4, 1). The amplitudes in N=2 case could be special cases of N=4 amplitudes in the same manner as they amplitudes of gauge theories are special cases of those of super-gauge theories. The only difference would be that propagator factors 1/p2 appearing in twistorial construction would be replaced by propagators in CP2 degrees of freedom.
- In twistor Grassmannian approach to planar SYM one obtains general formulas for n-particle scattering amplitudes with k positive (or negative helicities) in terms of residue integrals in Grassmann manifold G(n,k). 4-particle scattering amplitudes of TGD, that is 4-fermion scattering amplitudes and their super counterparts would be obtained by restricting to N=2 sub-multiplets of full N=4 SYM. The only non-vanishing amplitudes correspond for n=4 to k=2=n-2 so that they can be regarded as either holomorphic or anti-holomorphic in twistor variables, an apparent paradox understandable in terms of additional symmetry as explained and noticed by Witten. Four-particle scattering amplitude would be obtained by replacing in Feynman graph description the four-momentum in propagator with CP2 momentum defined by I3 and Y for the particle like entity exchanged between fermions at opposite wormhole throats. Analogous replacement should work for twistorial diagrams.
- In fact, single fermion per wormhole throat implying 4-fermion amplitudes as building blocks of more general amplitudes is only a special case although it is expected to provide excellent approximation in the case of ordinary elementary particles. Twistorial approach could allow the treatment of also n>4-fermion case using subset of twistorial n-particle amplitudes with Euclidian propagator. One cannot assign right-handed neutrino to each fermion separately but only to the elementary particle 3-surface so that the degeneration of states due to SUSY is reduced dramatically. This means strong restrictions on allowed combinations of vertices.
For details see the new chapter Some fresh ideas about twistorialization of TGD or the article with the same title.
- Should one use CP2 twistors everywhere in the 3-vertices so that only fermionic propagators would remain as remnants of M4? This does not look plausible. Should one use include to 3-vertices both M4 and CP2 type twistorial terms? Do CP2 twistorial terms trivialize as a consequence of quantization of Y and I3?
- Nothing has been said about modified Dirac operator. The assumption has been that it disappears in the functional integration and the outcome is twistor formalism. The above argument however implies functional integration over the deformations of CP2 type vacuum extremals.