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".