First indications for flavor changing neutral currents?

Tommaso Dorigo talks in his blog posting titled "Hera's intgriguing top candidates" about indications for single top quark production by neutral currents. The eprint by H1 collaboration can be found in the archive.

This kind of processes would be mediated by flavor changing neutral currents forbidden in the standard model. TGD predicts exotic gauge bosons inducing this kind of processes. In TGD framework flavor is due to the topology of the wormhole throat at which the fermionic quantum numbers reside (see this and this). Fermions correspond to CP₂ type vacuum extremals topologically condensed to the background space-time sheet and there is only single wormhole throat. At the throat the induced metric changes its signature from Euclidian to Minkowskian so that the orbit of wormhole throat defines a light-like 3-surface. The topology of the wormhole throat in orientable case is characterized by genus (handle number) so that one obtains sphere, torus, sphere with two handles, etc... There is a nice argument explaining why just the three lowest topologies correspond to stable and light fermions.

Gauge bosons correspond to wormhole contacts-pieces of CP₂ type vacuum extremals with Euclidian signature of metric connecting two Minkowskian space-time sheets. There are two wormhole throats so that gauge bosons are classified by pairs (g₁,g₂) of genera for throats. It is natural to arrange fermions and bosons to representations of dynamical SU(3) group which could be called flavour SU(3) but having not much to do with the ancient flavor SU(3) of Gell-Mann. For fermions/antifermions one obtains triplets/antitriplets and for gauge bosons octet and singlet as the tensor product of triplet and anti-triplet. Singlet is identified as ordinary gauge bosons and octet gives rise to exotic gauge bosons inducing flavor changing neutral currents. If the p-adic mass scale of the exotics is higher than that for standard gauge bosons, one can understand their experimental absence.

CKM mixing in TGD framework is induced by different mixings of topologies of wormhole throat in the case of U and D type quarks. In absence of this mixing the neutral flavor changing currents would change the flavor of both interacting fermions: in the recent case electron and u or c quark transforming to top quark. Already these reactions would be something completely new and could be mediated by both gluons, Z⁰ bosons and also by W bosons in which new kind of charged flavor changing current would be in question. CKM mixing makes possible also reactions in which only other fermion changes its flavor. In the recent case it would be quark whereas electron would transform to electron rather than muon or τ.

If the top in the wrong place is not just dirt- that is statistical fluctuation - what it very probably is - it could be interpreted as a first evidence for the existence of the predicted octet of gauge bosons inducing flavor changing transitions. Also octets of W bosons and gluons are predicted whereas the possibly detected transition would correspond to the exchange of octet Z⁰. Also the possibly higher rate for transitions with correlated change of flavor for electron and top (say e to τ and and u to top) could kill the proposal.

Pieces of something bigger?

John Baez has a very interesting posting about representations of 2-groups. I wish I time to look in more detail what he is saying. I can only I hope that the posting would find readers. John Baez is a mathematical physicists who has the rare gift of representing new mathematical ideas in an extremely inspiring and transparent manner.

My impression was that John and others regard as a problem that the representations for 2-counterparts of Lie groups seem to reduce to representations of permutation groups for a discrete set of objects. The reason is basically that at the level of abstraction they are working the points of n-dimensional space are replaced with n-tuples of linear spaces of varying dimensions. Vector space replaces the point of the vector space. For ordinary vector spaces one has a continuum of different choices of basis vectors transformed to each other by matrices representing group elements. One cannot however superpose linearly vector spaces and representation matrices can just permute the vector spaces forming the components of the vector. For instance, for Poincare group the representations would be induced from the representations of discrete subgroups of Lorentz group known as Fuschian groups and realized in terms of discrete Möbius transformations of complex plane (Fuschian groups).

I am not sure whether I would like to see this as a problem. Categories, quantum groups, n-groups, hyper-finite factors and their inclusions, etc.. arise in TGD in a close association with the notion of finite measurement resolution which among other things led to a stringy formulation of quantum TGD and to a precise formulation of QFT limit of TGD allowing to predict the values of gauge coupling strengths.

At space-time and imbedding space level finite measurement resolution has discreteness as a space-time correlate. Discrete groups are obviously a natural correlate for the finite measurement resolution when one speaks about symmetries. If there is a connection, this discretization - something very concrete- could have also interpretation in terms of an abstraction process in which one replaces points with vector spaces. Could it really be that these mathematicians are becoming conscious of the mathematics needed to realize elegantly the basic physical picture of quantum TGD? Maybe we indeed are pieces of a Very Great Mind and our feeling about working independently is only an illusion. In Great Experiences, which we may have once or twice during lifetime, we can experience directly a contact with this Great Mind and may have the mysterious and paradoxical Brahman=Atman experience of actually being the Great Mind ourselves. Just asking;-).

Sigh of relief

As I have already told, TGD leads naturally to the idea that mere Dirac action with coupling to gauge potentials defines QFT limit of TGD: the reason is that only fermions appear as fundamental particles in TGD framework. This action generates YM action radiatively and predicts all coupling strengths. Somewhat amusingly, this kind of possibility -something extremely beautiful and incredibly simple for me at least- has not been noticed previously (to my best knowledge).

The realization of this idea requires the TGD based notions of zero energy ontology and geometric realization of finite measurement resolution. These concepts indeed lead to a highly unique physical realization of cutoffs in momentum and hyperbolic angle. The reason for two cutoffs is that the momentum integral is Minkowskian: Wick rotation would lead to a mathematical catastrophe in TGD framework and would not be in spirit with Lorentz invariance.

The UV cutoff for hyperbolic angle measured in the rest system of a particle emitting virtual particle is the basic ad hoc element of the model. It is not clear whether it can be deduced within the framework defined by QFT limit or whether only basic quantum TGD predict it. Quantum criticality suggests that the cutoff must be such that a large number of p-adic length scales contributes to the loop integrals so that one obtains interesting coupling constant evolution. In standard QFT the absence of this kind of cutoff would mean that too many length scales involved so that one ends up with divergences. In TGD framework this would mean vanishing gauge boson propagators and a trivial theory. If cutoff is too strong, gauge boson couplings are predicted to be very weak and this is also unsatisfactory situation.

Situation is very much analogous to thermodynamical criticality or criticality for spin glass phase. Above critical temperature there is chaos and below it complete order. This vision leads to a model for the cutoff which gives excellent hopes of coupling constant evolution consistent with standard model. The condition that the values of the fine structure constant at electron and intermediate gauge boson mass scales are reproduced correctly fixes the two parameters of the model and the value of the second parameter is consistent with p-adicization and number theoretical vision. Also the behavior in UV is reasonable and the predictions for other bare couplings follow and are sensible. Gauge boson loops allow to understand the non-abelian aspects of coupling constant evolution and asymptotic freedom.

I have been fighting with the precise calculation of the normalization factor of the bosonic propagator determined by fermionic loop integral. I had to work really hardly to end up with formulas consistent with the calculation at the limit of vanishing momentum squared- one factor of two was missing and one exponent n=1 had changed to n=2 and it was really maddening exercise to identify the mistakes! For the original model for which it was assumed that the measurement resolution for the time scale of CDs is maximal and thus of order CP₂ time scale, the range of the allowed hyperbolic angles was predicted to be extremely narrow in longer length scales so that loops effectively disappeared. As a consequence, the coupling constant evolution became essentially trivial except immediately above UV scale. The attempts to save this model were futile. The model for which time scale resolution corresponds to a fraction of p-adic time scale characterizing the sub-CD in question rather than smallest possible sub-CD, loop integrals receive contributions from all scales and a realistic coupling constant evolution is obtained. But as already mentioned, the quantitative expression for cutoff should be deduced from quantum TGD proper or perhaps from the consistency with the results produced by subtraction procedures in standard QFT.

I have not yet corrected the errors in the formulas in previous postings and encourage the interested reader to can consult the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".

Emergent boson propagators, fine structure constant, and hierarchy of Planck constants

I have already discussed the bootstrap approach to S-matrix assuming that boson propagators emerge from fermionic self-energy loops (see this, this, and also the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix").

There are several interesting questions. Are there any hopes that this approach can predict correctly the evolution of gauge coupling constants - in particular that of fine structure constant? The emergence of bosonic propagator from a fermionic loop means that it is inversely proportional to gauge coupling strength and thus to hbar. What does this mean from the point of view of the hierarchy of Planck constants?

Is it possible to understand the value of fine structure constant in bootstrap approach to S-matrix?

The basic test for the theory is whether it can predict correctly the value of fine structure constant for reasonable choice of the UV and IR cutoffs. In the first approximation one can assume that photons has only U(1) couplings to fermions so that the fermion-fermion scattering amplitude at electron's p-adic length scale is determined by the photon propagator alone.

1. One can start from the normalization factor of the inverse of the bare gauge boson propagator

   G_B= [p²g^μν- p^μp^ν]× ∑_Qi² X(η_max),

   where the function X(η_max) can be calculated once the UV cutoff mass squared and in hyperbolic angle is known (Wick rotation would eliminate the hyperbolic cutoff but does not make sense in TGD framework). The sum is over the fermions with charges Q_i. For three lepton and quark generations one would have ∑_Qi²=16. The normalization factor equals to 1/g², where g is the bare gauge boson coupling constant so that one can pose physical constraints in order to deduced information about hyperbolic cutoff.

2. The realization of the cutoff for the mass of the virtual particle in terms of p-adic mass scale m ≤ m(CP₂)/p^1/2 is on a strong basis. The ad hoc assumption is the form for the cutoff in the hyperbolic angle. The cutoff means that the allowed range of 3-momenta for time-like momenta and of energies for time-like momenta of off mass shell particle is rather narrow for a given mass. What is clear is that any extension of the allowed phase space increases the value of X and requires larger p_min for this form of cutoff.

3. The narrow cutoff in the fermionic loop momenta could be interpreted physically in terms of the fermion-anti-fermion bound state character of bosons restricting the range of the virtual momenta of the fermion and anti-fermion to a very narrow range in the rest system of the boson. This is natural if fermion and antifermion reside at the opposite throats of the wormhole contact. In the case of virtual bosons radiated by leptons this restriction would not apply.

4. There is also second interpretation for the narrow cutoff. The rest system of sub-CD in which the fermionic loop is calculated is assumed to be the rest system of the virtual particle. Otherwise one would obtain a breaking of Lorentz invariance. This requirement could provide an alternative justification for the cutoff in cosh(η) since for too large values of η identified as the hyperbolic angle assignable to the lower tip of sub-CD the Lorentz transform of the time coordinate T(p) = pT(CP₂) of the upper tip of sub-CD is T=cosh(η)×pT(CP₂), and could be so large that the upper tip belongs outside CD.

5. The original belief based on an erratic formula for the d⁴k in terms of mass squared and hyperbolic angle was that no gauge boson mass is generated radiatively since space-like and time-like contributions to the loop integral would compensate each other. This belief turned out to be wrong and the requirement that mass term is vanishing fixes uniquely the relationship between hyperbolic cutoffs for time-like and space-like momenta. Hence only the cutoff in time-like region must be fixed.

I have done numerical experimentation with several kinds of cutoffs and done impressive amount of number of numerical errors during this experimentation.

1. The basic constraint on the cutoff is that it predicts reasonably well the values of the fine structure constant at electron and intermediate gauge boson length scales. Also its value in ultraviolet should be reasonable. This suggests that the the cutoff depends on the logarithm of p-adic length scale - that is k. Hence the most plausible cutoff for time-like loop momenta is of the form

   sinh(η)≤ 1+a × k^-b .

   a and b are parameters fixed by the basic constraints. The cutoff for space-like momenta is completely determined by the condition that gauge bosons are massless.
2. Geometrically the cutoff means that the maximal variation of the maximal temporal distance between the tips of the Lorentz transformed CD corresponds to the measurement resolution ΔT=a²T(k)k^-2b. The optimal choice for b is b=1/3 and predicts that the contribution from k^th p-adic length scale to the propagator is inversely proportional to the p-adic length scale. The resulting value of a is a= 0.22050469512552 and predicts correctly the value of fine structure constant both in electron and intermediate gauge boson length scale.

The predictions for other gauge couplings

One can also look for the predictions for color and electro-weak coupling constants.

1. The loop is proportional to N(B_i) = Tr(Q_i²). The charge matrices are Iⁱ_L for W bosons and I³_L- pQ_em, p=sin²(θ_W) for Z⁰. For the coupling of Kähler gauge potential the charge matrix is Q_K=1 for leptons and Q_K=1/3 quarks: it is easy to see that in this case the normalization factor is same as photon. The traces of non-Abelian charge matrices in fundamental representations are Tr(T_a²)=-1/2 in the standard normalization. For photon and gluons both right and left handed chiralities contribute and W bosons only left handed.

2. This gives the following expressions for the normalization factors N(B_i)

   α(B_i)= (N(γ)/N(B_i))× α_em ,

   with

   N(γ)=N(U(1))= 16 , N(g)= 6 , N(W)=6 , N(Z)= 6-12p+13p² .

   The values of the gauge couplings strengths are given by

   α_s= (8/3)α_em , α(W)=(8/3)α_em , α(Z)= (16/(6-12p+13p²)α_em .

   Electro-weak couplings are unified only if one has p= 12/13, from p=3/8 obtained by definition the ratio α_em/α_W, which is also the typical prediction of GUTs.

3. Coupling constant evolution is assigned with the dependence on IR cutoff with UV cutoff defined by 2-adic length scale. The predictions for the bare couplings for k=2 are α_em^-1= 38.2719, α_s^-1=α_W^-1= 14.5255, and α_Z^-1 = 8.0571 by assuming b=1/3 and posing the above described conditions p²=0 limit for virtual photon mass squared.

Cutoff in the general case

The previous calculations were carried by identifying the UV cutoff as 2-adic length scale. The calculations can be generalized to an UV cutoff defined by any p-adic length scale with p_min ≈ 2^k_min. The Lorentz transforms of sub-CDs must belong inside CD within measurement resolution, which means that the condition sinh(η) ≤ =a× k^-b for p≈ 2^k is satisfied. k ≥ k_min holds of course true.

The definition of the UV cutoff for vertex corrections involves non-trivial delicacies.

1. The problem is following. In the vertex correction for FFB vertex the ends of the virtual boson line in general correspond to fermions with different four-momenta and the hyperbolic angle η must be assigned to the rest system of either initial or final state fermion. The choice means a selection of the arrow of geometric time and breaking of T invariance. The requirement of CPT symmetry is expected to fix the choice.

2. Similar situation is encountered also in basic quantum TGD. In the construction of the counterpart of stringy diagrammatics the CP breaking instanton variant of Kähler action contributes to the modified Dirac action a term whose appearance in the vertices makes the theory non-trivial . One must decide, which end of the line carries the CP breaking CP term. CPT invariance is the natural constraint on the choice. The idea about fermions (anti-fermions) as particles propagating to the geometric future (past) suggests that CP breaking term is associated with the negative energy fermion (positive energy anti-fermion) at the future (past) end of the line. CP symmetry is broken since CP takes fermion to anti-fermion but does not permute the end of the lines. CPT is respected.

3. In the recent case the counterpart of CP and T breaking would be the assignment of the cutoff to the past (future) end in the case of fermions (antifermions). If one assigns the cutoff in both cases to (say) future end, CPT breaking results. It is important to notice that the distinction between future and past is always unique in the rest system of the sub-CD.

How the amplitudes depend on hbar?

TGD predicts a hierarchy of Planck constants and the question concerns the dependence of the loop corrections on hbar.

1. Unless the p-adic cutoff for cosh(η) depends on hbar, boson propagator cannot involve hbar, and this is achieved by putting g=hbar^1/2 so that 1/hbar factor associated with the loop cancels g²=hbar. This means that loops give no powers of 1/hbar as in ordinary quantum field theories. By checking a sufficient number of diagrams one can get convinced that the hbar dependence of the diagram depends on the total number of particles involved with the diagram and is given by the proportionality hbar^{(N_in+N_out)/2-1}.

2. This simple dependence of the amplitudes on hbar suggests that it has actually no physical content. The scaling of the incoming and outgoing wave functions by hbar^-1/2 and the division of the amplitude by hbar indeed makes the amplitudes independent of hbar. In unitarity conditions the 1/hbar factors from d³k/2E factors assignable to intermediate states correspond to the hbar^-1/2 factors of the states involved. Therefore QFT limit defined in this manner does not distinguish between different values of hbar and the difference is seen only at the level of kinematics (1/hbar scaling of the frequencies and wave-vectors for a fixed four-momentum). The difference would become dynamically visible through the fact that the space-time surfaces associated with CDs with different values of hbar are not simply scaled up versions of each other.

3. This result is in contrast with the standard QFT expectations about how the amplitudes should behave as functions of hbar. One of the motivations for the hierarchy of Planck constants was that radiative corrections come in powers of 1/hbar so that large values of Planck constant improves the convergence of the perturbation series in powers of coupling constant strengths. If coupling constants emerge in the proposed manner, this motivation for large values of Planck constants is lost.

Note added: I have updated and shortend this posting several times as the mathematical and physical understanding of the model have developed and as I have discovered various numerical errors in calculations. The recent picture is quite satisfactory but numerical errors could still be present. Hyperbolic cutoff is obviously the ad hoc element of the model and the model hoped to predicting the hyperbolic cutoff from quantum criticality is a work in progress.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".

Still about the emergence of bosonic propagators and vertices

In TGD Universe only fermions are fundamental particles and bosons can be identified as their bound states. This suggest that in the possibly existing QFT type description bosonic propagators and vertices must emerge from the fermionic propagators and from the fundamental fermion-boson vertex appearing in Dirac action with a minimal coupling to gauge bosons. In the earlier posting I discussed how the emergence can be understood in terms of path integral approach (see also the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix"). The nice feature of the approach is that there are no free parameters in the theory. In particular, the counterparts of gauge couplings are predictions of the theory. In this posting I represent some further comments about the resulting Feynman diagrammatics.

1. Consider first the exponent of the action exp(iS_c) resulting in fermionic path integral. The exponent
   exp[i∫ d⁴xd⁴y ξbar(x)G_F(x-y)ξ(y)]=
   exp[i∫ d⁴kξbar(-k)G_F(k)ξ(k)]
   is combinatorially equivalent with the sum over n-point functions of a theory representing free fermions constructed using Wick's rules that is by connecting n Grassmann spinors and their conjugates in all possible ways by the fermion propagator G_F.

2. The action of
   exp[i∫ d⁴x (δ/δ ξbar(x))γ • A(x) (δ/δ ξ(x))]
   = exp[i∫ d⁴k d⁴k₁ (δ/δ ξbar(k-k_1))γ• A(-k) (δ/δ ξ (k_1))]
   on diagrams consisting of n free fermion lines gives sum over all diagrams obtained by connecting fermion and anti-fermion ends of two fermion lines and inserting to the resulting vertex A(k) such that momentum is conserved. This gives sum over all closed and open fermion lines containing n ≥2 boson insertions. The diagram with single gauge boson insertion gives a term proportional to A_μ(k=0) ∫ d^4k k^μk^-2, which vanishes.

3. S_c as obtained in the fermionic path integral is the generating functional for connected many-fermion diagrams in an external gauge boson field and represented as sum over diagrams in which one has either closed fermion loop or open fermion line with n ≥2 bosons attached to it. The two parts of S_c have interpretation as the counterparts of YM action for gauge bosons and Dirac action for fermions involving arbitrary high gauge invariant n-boson couplings besides the standard coupling. An expansion in powers of γ^μD_μ is suggestive. Arbitrary number of gauge bosons can appear in the bosonic vertices defined by the closed fermion loops and gauge invariance must pose strong constraints on the bosonic part of the action if expressible in terms of bosonic gauge invariants. The closed fermion loop with n=2 gauge boson insertions defines the bosonic kinetic term and bosonic propagator. The sign of the kinetic terms comes out correctly thanks to the minus sign assigned to the fermion loop.

4. Feynman diagrammatics is constructed for S_c using standard Feynman rules. In ordinary YM theory ghosts are needed for gauge fixing and this seems to be the case also now.

5. One can consider also the presence of Higgs bosons. Also the Higgs propagator would be generated radiatively and would be massless for massless fermions as the study of the fermionic self energy diagram shows. Higgs would be necessary CP₂ vector in M⁴×CP₂ picture and E⁴ vector in M⁸=M⁴×E⁴ picture. It is not clear whether one can describe Higgs simply as an M⁴ scalar. Note that TGD allows in principle Higgs boson but - according to the recent view - it does not play a role in particle massivation.

The diagrammatics differs from the Feynman diagrammatics of standard gauge theories in some respects.

1. 1-P irreducible self energy insertions involve always at least one gauge boson line since the simplest fermionic loop has become the inverse of the bosonic propagator. Fermionic self energy loops in gauge theories tends to spoil asymptotic freedom in gauge theories. In the recent case the lowest order self-energy corrections to the propagators of non-abelian gauge bosons correspond to bosonic loops since fermionic loops define propagators. Hence asymptotic freedom is suggestive.

2. The only fundamental vertex is AFFbar vertex. As already found, there seems no point in attaching to the vertex an explicit gauge coupling constant g. If this is however done n-boson vertices defined by loops are proportional to gⁿ. In gauge theories n-boson vertices are proportional to g^n-2 so that a formal consistency with the gauge theory picture is achieved for g=1. In each internal boson line the g² factor coming from the ends of the bosonic propagator line is canceled by the g^-2 factor associated with the bosonic propagator. In S-matrix the division of the bosonic propagator from the external boson lines implies gⁿ proportionality of an n-point function involving n gauge bosons. This means asymmetry between fermions and bosons unless one has g=1. Gauge couplings could be identified by transferring the normalization factor of gauge boson propagators to fermion-boson interaction vertices so that bosonic propagators would have standard normalization. The counterparts of gauge coupling constants could be identified from the amplitudes for 2-fermion scattering by comparison with the predictions of standard gauge theories. The small value of effective g obtained in this manner would correspond to a large deviation of the normalization factor of the radiatively generated boson propagator from its standard value.

3. Furry's theorem holding true for Abelian gauge theories implies that all closed loops with an odd number of Abelian gauge boson insertions vanish. This conforms with the expectation that 3-vertices involving Abelian gauge bosons must vanish by gauge invariance. In the non-abelian case Furry's theorem does not hold true so that non-Abelian 3-boson vertices are obtained.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".

Twistors and TGD: a summary

The encounter between twistors and TGD turned out to be extremely fruitful. I spent some time with the idea about replacing loop momenta in Feynman diagrams with light-like ones in order to achieve twistorialization (or rather spinorialization) of Feynman graphs but - as it sometimes happens - a silly idea stimulated the right question, and after 31 years of hard work I have a proposal for precise rules of Feynman diagrammatics producing UV finite and unitary S-matrix. I glue below the introduction to the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix" in the hope that it give an overall view about the situation.

Twistors - a notion discovered by Penrose - have provided a fresh approach to the construction of perturbative scattering amplitudes in Yang-Mills theories and in N=4 supersymmetric Yang-Mills theory. This approach was pioneered by Witten. The latest step in the progress was the proposal by Nima Arkani-Hamed and collaborators that super Yang Mills and super gravity amplitudes might be formulated in 8-D twistor space possessing real metric signature (4,4). The questions considered below are following.

1. Could twistor space could provide a natural realization of N=4 super-conformal theory requiring critical dimension D=8 and signature metric (4,4)? Could string like objects in TGD sense be understood as strings in twistor space? More concretely, could one in some sense lift quantum TGD from M⁴×CP₂ to 8-D twistor space T so that one would have three equivalent descriptions of quantum TGD.

2. Could one construct the preferred extremals of Kähler action in terms of twistors -may be by mimicking the construction of hyper-quaternionic resp. co-hyper-quaternionic surfaces in M⁸ as surfaces having hyper-quaternionic tangent space resp. normal space at each point with the additional property that one can assign to each point x a plane M²(x) subset M⁴ as sub-space or as sub-space defined by light-like tangent vector in M⁴. Could one mimic this construction by assigning to each point of X⁴ regarded as a 4-surface in T a 4-D plane of twistor space satisfying some conditions making possible the interpretation as a tangent plane and guaranteing the existence of a map of X⁴ to a surface in M⁴×CP₂. Could twistor formalism help to resolve the integrability conditions involved?

3. Could one modify the notion of Feynman diagram by allowing only massless loop momenta so that twistor formalism could be used in elegant manner to calculate loop integrals and whether the resulting amplitudes are finite in TGD framework where only fermions are elementary particles? Could one modify Feynman diagrams to twistor diagrams by replacing momentum eigenstates with light ray momentum eigenstates completely localized in transversal degrees of freedom?

The arguments of this chapter suggest some these questions might have affirmative answers.

Twistors at space-time level

Consider first the twistorialization at the classical space-time level.

1. One can assign twistors to only 4-D Minkowski space (also to other than Lorentzian signature). One of the challenges of the twistor program is how to define twistors in the case of a general curved space-time. In TGD framework the structure of the imbedding space allows to circumvent this problem.

2. The lifting of classical TGD to twistor space level is a natural idea. Consider space-time surfaces representable as graphs of maps M⁴→ CP₂. At classical level the Hamilton-Jacobi structure required by the number theoretic compactification means dual slicings of the M⁴ projection of the space-time surface X⁴ by stringy word sheets and partonic two-surfaces. Stringy slicing allows to assign to each point of the projection of X⁴ two light-like tangent vectors U and V parallel to light-like Hamilton-Jacobi coordinate curves. These vectors define components tildeμ and λ of a projective twistor, and twistor equation assigns to this pair a point m of M⁴. The conjecture is that for preferred extremals of Kähler action this point corresponds to the M⁴ projection of the point in the natural M⁴ coordinates associated with the upper or lower tip of causal diamond CD. If this conjecture is correct one can lift the M⁴ projection of the space-time surface in CD×CP₂ subset M⁴×CP₂ to a surface in PT×CP₂, where CP₃ is projective twistor space PT=CP₃. Also induced spinor fields and induced gauge fields can be lifted to twistor space.

3. If one can fix the scales of the tangent vectors U and V and fix the phase of spinor λ one can consider also the lifting to 8-D twistor space T rather than 6-D projective twistor space PT. Kind of symmetry breaking would be in question. The proposal for how to achieve this relies on the notion of finite measurement resolution. The scale of V at partonic 2-surface X² subset dCD×X³_l would naturally correlate with the energy of the massless particle assignable to the light-like curve beginning from that point and thus fix the scale of V coordinate. Symplectic triangulation in turn allows to assign a phase factor to each strand of the number theoretic braid as the Kähler magnetic flux associated with the triangle having the point at its center. This allows to lift the stringy world sheets associated with number theoretic braids to their twistor variants but not the entire space-time surface. String model in twistor space is obtained in accordance with the fact that N=4 super-conformal invariance is realized as a string model in a target space with (4,4) signature of metric. Note however that CP₂ defines additional degrees of freedom for the target space so that 12-D space is actually in question.

4. One can consider also a more general problem of identifying the counterparts for the preferred extremals of Kähler action with arbitrary dimensions of M⁴ and CP₂ projections in 10-D space PT×CP2. The key idea is the reduction of field equations to holomorphy as in Penrose's twistor representation of solutions of positive and negative frequency parts of free fields in M⁴. A very helpful observation is that CP₂ as a sub-manifold of PT corresponds to the 2-D space of null rays of the complexified Minkowski space M⁴_c. For the 5-D space N subset PT of null twistors this 2-D space contains 1-dimensional light ray in M⁴ so that N parametrizes the light-rays of M⁴. The idea is to consider holomorphic surfaces in PT_±×CP₂ (± correlates with positive and negative energy parts of zero energy state) having dimensions D=6,8, 10; restrict them to N×CP₂, select a sub-manifold of light-rays from N, and select from each light-ray subset of points which can be discrete or portion of the light-ray in order to get a 4-D space-time surface. If integrability conditions for the resulting distribution of light-like vectors U and V can be satisfied (in other words they are gradients), a good candidate for a preferred extremal of Kähler action is obtained. Note that this construction raises light-rays to a role of fundamental geometric object.

Twistors and Feynman diagrams

The recent successes of twistor concept in the understanding of 4-D gauge theories and N=4 SYM motivate the question of how twistorialization could help to understand construction of M-matrix in terms of Feynman diagrammatics or its generalization.

1. One of the basic problems of twistor program is how to treat massive particles. Massive four-momentum can be described in terms of two twistors but their choice is uniquely only modulo SO(3) rotation. This is ugly and one can consider several cures to the situation.
   1. Number theoretic compactification and hierarchy of Planck constants leading to a generalization of the notion of imbedding space assign to each sector of configuration space defined by a particular CD a unique plane M² subset M⁴ defining quantization axes. The line connecting the tips of the CD selects also unique rest frame (time axis). The representation of a light-like four-momentum as a sum of four-momentum in this plane and second light-like momentum is unique and same is true for the spinors λ apart from the phase factors (the spinor associated with M² corresponds to spin up or spin down eigen state).

   2. The tangent vectors of braid strands define light-like vectors in H and their M⁴ projection is time-like vector allowing a representation as a combination of U and V. Could also massive momenta be represented as unique combinations of U and V?

   3. One can consider also the possibility to represent massive particles as bound states of massless particles.

   It will be found that one can lift ordinary Feynman diagrams to spinor diagrams and integrations over loop momenta correspond to integrations over the spinors characterizing the momentum.

2. One assign to ordinary momentum eigen states spinor λ but it is not clear how to identify the spinor tildeμ needed for a twistor.
   1. Could one assign tildeμ to spin polarization or perhaps to the spinor defined by the light-like M² part of the massive momentum? Or could λ and tildeμ correspond to the vectors proportional to V and U needed to represent massive momentum?

   2. Or is something more profound needed? The notion of light-ray is central for the proposed construction of preferred extremals. Should momentum eigen states be replaced with light ray momentum eigen states with a complete localization in degrees of freedom transversal to light-like momentum? This concept is favored both by the notion of number theoretic braid and by the massless extremals (MEs) representing "topological light rays" as analogs of laser beams and serving as space-time correlates for photons represented as wormhole contacts connecting two parallel MEs. The transversal position of the light ray would bring in tildeμ. This would require a modification of the perturbation theory and the introduction of the ray analog of Feynman propagator. This generalization would be M⁴ counterpart for the highly successful twistor diagrammatics relying on twistor Fourier transform but making sense only for the (2,2) signature of Minkowski space.

3. In perturbation theory one can also consider the crazy idea of restricting the loop momenta to light-like momenta so that the auxiliary M² twistors would not be needed at all. This idea failed but led to a first precise proposal for how Feynman diagrammatics producing unitarity and UV finite S-matrix could emerge from TGD, where only fermions are elementary particles and all coupling constants are in principle predictions of the theory. Emergence would mean that the fundamental action is just the Dirac action with gauge boson couplings and containing no bosonic kinetic term, that the perturbative functional integral over the fermion fields in the construction of the effective action induces bosonic kinetic term radiatively, and that a further perturbative functional integral over the gauge boson fields gives an effective action in which all bosonic n-point functions have emerged from the fermionic dynamics. Physically this would mean that bosons interact only when the wormhole contact representing boson and carrying fermion and antifermion quantum numbers at the opposite light-like wormhole throats decays to a pair of fermion and anti-fermion represented by CP₂ type extremals with single wormhole throat only. Even fermionic propagators would emerge radiatively from the modified Dirac operator in more fundamental description . What is remarkable is that p-adic length scale hypothesis and the notion of finite measurement resolution lead to a precise proposal how UV divergences are tamed in a description taking into account the finite measurement resolution.

To sum up, perhaps the most important outcome of the interaction of twistor approach with TGD is a proposal for precise Feynman rules allowing to construct unitary and UV finite S-matrix. This realizes a 31 year old dream to a surprisingly high degree. Everything would emerge radiatively from the modified Dirac operator and boson-fermion vertices dictated by the charge matrix of the boson coding boson as a fermion-antifermion bilinear.

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".

Bootstrap approach to obtain a unitary S-matrix

In TGD framework S-matrix must be constructed without the help of path integral. The replacement of the loop momenta with light-like momenta does not eliminate UV divergences and the worst situation is encountered for gauge boson vertex corrections. This suggests a bootstrap program in which one starts from very simple basic structures and generates the remaining n-point functions as radiative corrections. The success of twistorial unitary cut method in massless gauge theories suggests that its basic results such as recursive generation of tree diagrams might be given a status of axioms. The idea that loop momenta are light-like cannot be however be taken too seriously. Also massive particles should be treated in practical approach.

The dream

Let us summarize the first variant of the dream about bootstrap approach.

1. In Construction of Quantum Theory: M-Matrix of "Towards M-Matrix" I have discussed how both field theoretic and stringy variants of the fermion propagator could arise via radiative self energy insertions described by a fundamental 2-vertex giving a contribution proportional to p^kγ_k and leading a propagator containing the counterpart as a mass term expressed in terms of CP₂ gamma matrices so that massive particles can have fixed M⁴×CP₂ chirality.

2. In TGD 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. Purely bosonic gauge boson couplings would be generated radiatively from triangle and box diagrams involving only fermion-boson couplings. Even bosonic propagator would be generated as a self-energy loop: bosons would propagate by decaying to fermion-antifermion pair and then fusing back to the boson. Gauge theory dynamics would be emergent and bosonic couplings would have form factors with IR and UV behaviors allowing finiteness of the loops constructed from them.

As already found this dream about emergence is killed by the general arguments already discussed demonstrating that one encounters UV divergences already in the construction of gauge boson propagator for both light-like and free loop momenta. The physical reason for the emergence of these divergences and also their cure at the level of principle is well-understood in TGD Universe.

1. The description in terms of number theoretic braids based on the notion of finite measurement resolution should resolve these divergences at the expense of locality.

2. Zero energy ontology brings into the picture also the natural breaking of translational and Lorentz symmetries caused by the selection of CD. This breaking is compensated at the level of configuration space since all Poincare transforms of CDs are allowed in the construction of the configuration space geometry.

3. If this approach is accepted then for given CD there are natural IR and UV cutoffs for 3-momentum (perhaps more naturally for these than for mass squared). IR cutoff is quantified by the temporal distance between the tips of CD and UV cutoff by similar temporal distance of smallest CD allowed by length scale resolution. If the hypothesis that the temporal distances come as octaves of fundamental time scale given by CP₂ time scale T₀ and implying p-adic length scale hypothesis, the situation is fixed. A weaker condition is that the distances come as prime multiples pT₀ of T₀.

4. QFT type idealization would make sense in finite measurement resolution and the loop integrals would be both IR and UV finite.

This leads to a modified form of the dream.

1. Concerning propagators there are two options: only fermionic propagators are allowed and bosonic propagators emerge or both fermionic and bosonic propagators appear as fundamental objects. Only boson-fermion coupling characterizing the decay of a wormhole contact to two CP₂ type almost vacuum extremals with single wormhole throat carrying fermion and anti-fermion number would be feeded to the theory as something given and all vertices involving more than one boson would result as radiative corrections. Boson-fermion coupling would be proportional to Kähler coupling strength fixed by quantum criticality and very near or equal to fine structure constant at electron's p-adic length scale for the standard value of Planck constant. If not anything else, this approach would be predictive.

2. This approach could be tried to both free and light-like loop momenta. For free loop momenta the cutoff would be naturally associated with the mass squared of the virtual particle rather than the energy of a massless particle. Despite its Lorentz invariance one could criticize this kind of UV cutoff because it allows arbitrarily small wavelengths not in accordance with the vision about finite measurement resolution.

The following considerations lead to the conclusion that bosonic propagators could emerge from fermionic ones in the quantum field theory type description and that this description is also favored by the basic structure of quantum TGD. This kind of formulation would simplify enormously the definition of the theory.

Quantitative realization of UV finiteness in terms of p-adic length scale hypothesis and finite measurement resolution

p-Adic fractality suggests an elegant realization of the notion of finite measurement resolution implying the finiteness of the ordinary Feynman integrals automatically but predicting divergences for light-like loop momenta.

1. For the four-momenta above cutoff-momentum scale defined by the measurement resolution characterized by p-adic mass scale one cannot detect any details of the wave function of the particle inside sub-...-sub-CDs in question. Only the position of sub-...-sub-CD inside CD can be measured with a resolution defined by the cutoff scale. Therefore the number of detectable momentum eigen states does not anymore increase as the momentum scale is doubled but remains unchanged.

2. Unitarity realized in terms of the Cutkosky rules and in consistency with the finite measurement resolution requires that the density of states factor d³k/2E receives a reduction factor 2^-2 as the momentum scale is doubled above the resolution scale in the Feynman integral. This gives an effective reduction factor μ^-2L to the Feynman integral.

3. The cutoffs will be posed on both mass squared and hyperbolic angle. This conforms with the p-adic length scale hypothesis emerging from p-adic mass calculations and with the geometry of CDs. p-Adic length scales come as L_p propto p^1/2, p≈ 2^k rather than L_p propto p as the proportionality T(p)= pT(CP₂) of the temporal distance between tips of the CD combined with Uncertainty Principle would suggest. The reason is that light-like randomness of partonic 3-surfaces means Brownian motion so that L_p propto T(p)^1/2 and M_p propto T(p)^-1/2 follows. To avoid confusions note that for the conventions that I have used T(p) corresponds to the secondary p-adic length scale T_p,2= p^1/2T_p. For electron T(p) corresponds to .1 seconds.

Definition of loop integration

Consider now definition of the integration measure for loop momenta.

1. It is far from obvious whether the usual definition based on Wick rotation of the Euclidian variant of the integral makes sense in the recent case. The definition based on Wick rotation would eliminate the divergence in the hyperbolic angle leave only a cutoff in k² > 0 and give quadratic resp. logarithmic divergences for n=1 resp. n=2. This prescription is not favored by the picture suggested by the geometry CDs.

2. The most natural integration measure is just the standard M⁴ volume element d⁴k, which can be written as

   d⁴k=k³dk× sinh²(η)dη dΩ , k=(k^μk_μ)^1/2

   for time-like momenta and

   ⁴k=k³dk×× cosh²(η)dη dΩ, k=(-k^μk_μ)^1/2

   for space-like momenta. The original calculations contained a silly error due to the naive generalization of the Euclidian integration measure by replacing sin³(θ) with sinh³(η).

3. The geometry of CDs requires IR and UV cutoffs in both mass squared and hyperbolic angle giving
   p_max^-1/2 ≤ (m/m(CP₂)≤ p_min^-1/2 ,
   sinh(η)≤ sinh(η_max).
4. The primes p_max and p_min correspond to IR and UV cutoffs and p_min≥ 2 holds true naturally in QFT limit since stringy excitations having mass scale given by CP₂ mass are not included. This means that all loop integrals are finite if also hyperbolic cutoff is present.
5. The justification for the cutoff in |sinh(η)| comes either from the requirement that the Lorentz transformed sub-CDs to which the fermion loop can be associated remain inside CD within the measurement resolution for temporal distance in the scale corresponding to T(k) or from the condition that the decomposition of the gauge boson to a pair of fermion and antifermion at opposite wormhole throats restricts the range of the virtual momenta to momenta almost at rest in the rest system of boson. The condition that coupling constant evolution is realistic fixes the form of hyperbolic cutoff for time-like momenta in high precision and it remains to be seen whether quantum criticality can be used to predict the hyperbolic cutoff from first principles.
6. Hyperbolic cutoffs can and must be different for time-like and space-like momenta and the cancellation of the mass term from the bosonic propagator fixes the relationship between these cutoffs uniquely. Hyperbolic cutoffs can and must depend on p-adic length scale so that the integral over loop momenta decomposes to integral over momenta corresponding to p-adic half octaves with a fixed hyperbolic cutoff in each half octave.

The conclusion is that the definition of loop integrals as Euclidian integrals would lead to a catastrophe via the generation of gauge boson mass proportional to the cutoff mass whereas the Minkowskian definition with the notion of cutoff motivated by p-adic length scale hypothesis and hierarchy of causal diamonds keeps gauge bosons massless provided the cutoffs for hyperbolic angle for time-like and space-like loop momenta are related in a unique manner and the only contribution to the boson mass comes from mass terms in the fermionic propagators.

Could bosonic propagators emerge?

My views about whether bosonic propagators can emerge or not have been fluctuating wildly during last weeks. The following argument however suggests that emergent bosonic propagation is a mathematically consistent notion and conforms with the special features of quantum TGD.

1. In basic quantum TGD modified Dirac equation containing induced spinor connection as induced gauge boson field defines the theory and the exponent of Kähler action emerges as Dirac determinant. The natural guess is that this structure is preserved in the sense that Feynman diagrammatics is defined by Dirac action coupled to gauge potentials but containing no kinetic term for gauge potentials with kinetic terms emerging from the fermionic loops and the values of gauge couplings following as predictions of the formalism.

2. One can try to formulate this idea in terms of path integral formalism. Couple gauge bosonic field A resp. Grassmann valued fermion fields Ψ to external currents j resp. Grassmann valued external currents ξ and perform Legendre transform giving the exponent of the effective action as a functional integral over Ψ and A. The functional derivatives of the effective action with respect to the ξ and j allow to deduce N-point functions.

3. The exponent Z=exp(G_c(j,ξ,ξbar)) defined as the functional Fourier transform of the action exponential is the key quantity since its functional derivatives at origin define the connected Green's functions. Z can be calculated in two steps. At the first step one functionally integrates over Ψ and its conjugate. This can be done perturbatively and gives the generating functional for connected fermionic Green's functions for ξ and its conjugate as a functional of gauge fields A. This functional is also analogous to action and contains bosonic kinetic term which is of correct form by the preceding observations. Also interaction terms for A are included and since the original system is gauge invariant also the effective action must be gauge invariant and should reduce to Yang-Mills action in the lowest orders. Perturbation theory is therefore possible and one can calculate effective action by performing the functional integral over A using the induced propagators and vertices. At this step fields ξ are in the role of non-dynamical external fields just as A was at the first step and all propagators are bosonic. From the resulting exponential one can generate connected Green's functions as functional derivatives with respect to the sources.

4. It seems that the proposed description avoids the most obvious divergences. In particular, the tadpole term from A_μΨbar(x)γ^μΨ(x) proportional to the fermion propagator D_F(x,x) proportional to an integral of form ∫d⁴k k^μ/k² and thus vanishing.

5. The bosonic kinetic term would be proportional to the over all gauge coupling g² if one expresses gauge potential in the form gA. This decomposition is however not natural in TGD since the induced spinor connection corresponds to gA with no explicit value of g being specified. In the case of simplest tree diagram describing 2→ 2 fermion scattering that the g² coming from the ends of the boson line is canceled by the 1/g² coming from the bosonic propagator so that the predictions of the theory do not depend on the value of g in the lowest order. This looks strange but would conform with the absence of bosonic kinetic term in the primary action making it impossible to identify the value of g in standard manner. One can however say that the numerical coefficient given by the fermionic loop integrals defining the bosonic propagator predicts the values of gauge couplings g through the comparison of their values with the prediction of standard gauge theory for say 2→ 2 scattering. The sign of the kinetic terms comes out correctly thanks to the minus sign assigned to the fermion loop. This picture would conform with the vision that TGD predicts all gauge couplings. Maybe the emergence of gauge boson propagators and vertices could be seen as one aspect of quantum criticality.

These arguments suggest that the notion of emergent gauge boson propagation makes sense mathematically and is favored also by the general structure of quantum TGD. On the other hand, the preceding arguments allow the presence of the bosonic propagators as fundamental objects and do not force to take seriously the idea about emergent gauge boson propagation. This motivates the attempt to debunk the notion once and for all. Consistency with p-adic mass calculations might provide the needed killer argument.

1. The resulting bosonic mass squared would be in the lowest order sum over products of masses of fermion pairs coupling to the boson. It is far from clear whether this prediction is quantitatively consistent with the predictions of the p-adic mass calculations. This possibility is not of course excluded: boson mass squared is quadratic in fermion masses coupling to the boson and the p-adic primes associated with the fermions are naturally those associated with the boson rather than free fermions so that at least the mass scale comes out correctly. This picture conforms also qualitatively with the fact that mass squared is identified as conformal weight and the eigenvalue of modified Dirac operator related closely to the ground state contribution to the mass can be regarded as complex squares root of conformal weight.

2. Note that even photon is predicted to be massive unless the fermion and antifermion associated with photon and other massless particles are massless or in so low p-adic temperature that the thermal mass is negligible. Also the p-adic prime associated with massless bosons could be so large that the mass is small.

3. Boson masses are of course emergent in the sense that they are determined by the masses of the fermion and anti-fermion, which they consist of. The question is whether the emergence of masses takes place via loops rather than p-adic mass calculations in the proposed sense and whether these pictures are equivalent. That loops could provide the fundamental description for boson masses is suggested by the asymmetry between bosons and fermions in the recent form of p-adic mass calculations. The p-adic temperature for bosons must be T_p ≤ 1/2 whereas T_p=1 holds true for fermions, and for fermions the analog of Higgs contribution is negligible whereas for gauge bosons it dominates.

4. It could be also possible to code p-adic thermodynamics into the Feynman diagrammatics in a more refined manner so that loops would give only corrections to the masses obtained from p-adic mass calculations. Instead of simply feeding in the results of p-adic mass calculations as mass parameters of the fermionic propagators, one could replace S-matrix with M-matrix involving the square root of density matrix describing the real counterpart of the partition function characterizing p-adic thermodynamics. Zero energy state would represent a square root of thermodynamical ensemble involving massless ground states and their conformal excitations rather than only ground states with thermal masses.

The emergence of fermionic Feynman propagator

The emergence of the fermionic propagators from the fundamental propagator 1/D defined by the modified Dirac equation is an attractive starting point for the improved variant of the dream.

1. The fundamental two-vertex would basically reflect the non-determinism of Kähler action implying the breaking of the effective 3-dimensionality (holography) of the dynamics, and would generate the fermion propagator from the propagator 1/D associated with the modified Dirac action behaving as Minkowski scalar and expressible in terms of CP₂ gamma matrices. The vertex would be characterized as p^kγ_k. This would give

   G_F= i/[p^kγ_k-D] .

   This expression is consistent with cut unitarity.

2. The propagator G_- is usually identifiable in terms of classical propagators as G_-=G_ret-G_adv and it seems that one an assume that this propagator is just i×(γ^kp_k-D)δ(p²)sign(p₀). It is perhaps needless to restate that light-like loop momenta do not lead to a finite theory under the assumptions motivated by p-adic length scale hypothesis.

From this Feynman propagator and its bosonic counterpart one can build all diagrams and get finite results for a finite momentum cutoff forced by the finite measurement resolution. One could of course worry whether the introduction of the p-adic length scale hierarchy might lead to problems with analyticity and unitarity. It is now clear that the idea about massless loop momenta fails. The idea did not however live for vain since it led to the first concrete quantitatively precise conjecture about how gauge theory could emerge as an approximation of quantum TGD from the basic physical picture behind TGD. I am of course the first admit that the proposed scenario looks horribly ugly against the extreme elegance of gauge theories like N=4 SYM. The tough challenge is to find an elegant mathematical realization of the proposed physical picture and twistor approach might be of considerable help here.

Note added: I have updated and shortened this posting several times as the vision about bosonic emergence has evolved. The recent picture seems rather stable. I have left out detailed formulas and encourage the reader to consult the summary of the recent situation concerning bosonic emergence in quantum TGD framework given in the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".