g-2 two anomaly of muon's magnetic moment in the TGD framework

g-2 anomaly for muons magnetic moment is a possible indication of new physics. There are two approaches to the estimation of the g-2 in the existing physics.

1. The first one is based on the experimental input from hadron physics and the second one is based on QCD lattice calculations and relies on quarks and gluons (see this. The QCD prediction conforms with the most recent experimental result (2025).
2. The calculation using empirical data from hadronic vacuum polarization, which predicts that the anomalous magnetic moment of muon is larger than the standard model prediction by factor of order 10^-9. TGD supports this approach.

The TGD view of hadrons differs from the QCD view since the TGD view of color is very different. The basic predictions are as follows and follow from Dirac equation for free fermions in H=M⁴×CP₂. The two H-chiralities correspond to quarks and leptons and the different couplings to CP₂ Kähler predict correctly the charge spectrum and electroweak couplings.

1. The Dirac equation in H=M⁴×CP₂ has two variants. First variant assumes just the usual M⁴ geometry without Kähler form whereas the second option assumes that M⁴ has self-dual Kähler form strongly suggested by twistorialization. 8-D masslessness predicts that the mass squared spectra (eigenvalues of the Dirac operator squared) for M⁴ and CP₂ must be same. Mass squared contains scalar d'Alembertian contribution depending on the color partial wave and spin term which does not depend on color partial wave. For properly chosen unit of mass squared, both the d'Alembertian - and spin contribution modulo 3 equals to 1 for quarks and 0 for leptons. It is possible to satisfy M⁴ masslessness condition.
2. Without M⁴ Kähler form 8-D masslessness this gives no additional conditions and one obtains infinite number of color partial waves with mass scale given by CP₂ mass of about 10^-4 Planck masses except for right-handed neutrino which is massless.
3. If Kähler form is assumed, the mass squared spectra for leptons having H chirality +1 as product of M⁴ and CP₂ chiralities is still integer valued for both M⁴ and CP₂ and it is possible to satisfy 8-D masslessness since leptons couple to 3-multiple of both Kähler form of CP₂ and M⁴.
4. The proposed mechanism to produce massless states has been that there must be negative contributions to mass squared identifiable as conformal weight which can be used to reduce the mass squared to zero. p-Adic thermodynamics would give the light masses as thermodynamic mass squared.

How to obtain massless states in M⁴ sense and what theses massless states are? For colored spinor modes they are massless in 8-D sense but massless states in M⁴ sense are required. Several proposals can be considered.

1. The proposal is that the generalized superconformal analogs of Kac-Moody algebras and super symplectic algebras should be used to construct states from the ground states assignable to the modes of the Dirac equation for which states are massless in 8-D sense and have CP₂ mass scale if the mode is colored. A possible exception is covariantly constant right-handed neutrino, which might be tachyonic. Assume that at the fundamental level, the mass squared identifiable as a conformal weight is additive for states of several fundamental fermions.

   For super-conformal generators mass squared correspond to a conformal weight and their action on the fermionic ground states generates a non-negative integer valued mass squared spectrum and mass squared corresponds to a conformal weight with CP₂ mass scale.

   For the 4-D generalization of superconformal algebra hypercomplex and complex degrees of freedom give rise to corresponding conformal weights as longitudinal and transversal conformal weights and the total conformal weight is the difference of these two. This difference could be also negative.

2. Massless Dirac equation requires 8-D masslessness for the fundamental fermions defining the ground states of generalized super-conformal representations. Could it be possible to have negative integer values of M⁴ mass squared for covariantly constant right-handed neutrino. If so, these states are off mass shell states in 8-D sense. The addition of a tachyonic right-handed covariantly constant neutrinos is analogous to a supersymmetry. Could these tachyonic contributions to the positive mass squared coming from the superconformal degrees of freedom make possible M⁴ masslessness? Or could also two superconformal contributions sum up to negative contribution. Also this allow to build states with vanishing M⁴ mass squared. If the addition of neutrino-pairs correspond to the action of a tachyonic super-conformal generator reducing the mass squared, these intuitive views would be equivalent.
3. There is also a second way to see the problem. M⁸-H duality (see this and this) predicts that M⁴ ⊂ H corresponds to some quaternionic normal space M⁴ of a four-surface Y⁴ ⊂ M⁸ with Euclidian signature of number theoretically induced metric. Fermions correspond to discrete points of Y⁴ and the deformation of Y⁴ rotate the quaternionic normal space M⁴ locally so that 8-D massless states are massless in M⁴ sense. In other words, the 8-momentum is in the direction of the normal space. 8-D masslessness corresponds to M⁴ masslessness. The coordinate system of M⁸ could be regarded as analogous to a system in which the transverse part of light-like momentum is vanishing.
4. What does this kind of deformation of Y⁴ mean? Could it correspond to a local G₂ automorphism respecting quaternionicity. This automorphism modifies color representations. At the level this would suggest that the color representations of leptons and quarks combinbe to irreducible representations of G₂ with different values of M⁴ mass squared and that one can obtain also vanishing mass squared value. One can however argue that local G₂ transformation can have no physical effect.

   However, TGD predicts that conformal algebras have non-negative conformal weights and define a hierarchy of isomorphic sub-algebras with conformal weights coming as integer multiples of the weights of the full algebra. The gauge action of the sub-space of A spanned by generators with conformal weights h≤ n would transform to dynamical symmetries and only the generators A_nk and their commutators with A would annihilate the physical states. Could these local G₂ deformations transform in this way to dynamical symmetries. This transmutation from gauge - to dynamical symmetries might correspond physically to the addition of pairs of tachyonic right handed neutrinos and left handed neutrinos? This would also give electroweak screening and the reduction of color so that massless color singlets for leptons are obtained.

To sum up, the basic prediction following from mere Dirac equations in H with M⁴ Kähler form is that baryons and mesons rather than quarks and gluons are the basic dynamical units at low energies and colored fundamental fermions appear as dynamical units only in the CP₂ mass scales. This has a direct relevance for the understanding of muon's anomalous g-2 anomaly. This means that the g-2 must be calculated by using baryons as basic units and therefore hadronic data. p-Adic length scale hypothesis and also the infinite number of color partial waves predicts a hierarchy of scaled up hadron physics and electroweak physics. If this picture is correct, the g-2 anomaly can be seen as a support for new physics. The scaled up hadron physics and possible scaled variants of weak bosons could give rise to the g-2 anomaly but TGD does not have the machinery to estimate it precisely enough. In any case, the situation is extremely interesting from the TGD point of view.

There is however an objection against this view. The successes of QCD suggests that also the description in terms of massless quarks should make sense and should correspond to a phase different from the hadronic phase.

1. The induction of the spinor structure to the space-time surface is a fundamental piece of TGD. This gives a induced/modified Dirac equation and the generalized holomorphic solutions of this equation are massless in the sense that the square of the modified Dirac operator annihilates them. The conjugates of the holomorphic gamma matrices annihilate these modes and implies that the spin term involving the induced K\"ahler form vanishes and does not give rise to mass squared term.
2. The induction of spinor structure (see this) by restricting the H modes to the space-time surface however requires a generalized holomorphic solution basis for H, which makes sense only in finite regions of M⁴ and CP₂ inside which the holomorphic modes remain finite. It is not clear whether this basis is locally orthogonal to the solution basis of ordinary Dirac equation in H. These modes must remain finite in X⁴. Since space-time surfaces are enclosed inside CDs with a finite size scale and CP₂ type Euclidean regions connecting two Minkowskian space-time sheets (see this) have holes as 3-D light-like partonic orbits, these modes can remain finite.
3. If the notion of the induced/modified spinor structure really makes sense, one can speak of two phases: free quarks and gluons and hadrons. The hadronic phase corresponds to the modes of the Dirac operator of H massless in 8-D sense but extremely massive in M⁴ sense. The holomorphic modes of the X⁴ Dirac operator correspond to massless quarks and gluons and also leptons. These descriptions should be dual to each other. The challenge is to understand this phase transition, which means breaking of conformal invariance and can be seen as a generalization of the phase transition from Higgs=0 phase to a phase with non-vanishing Higgs expectation.

See the article About Dirac equation in H=M⁴×CP₂ assuming Kähler structure for M⁴ or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.