Pseudo-scalar Higgs as Euclidian pion?

The observations of previous postings (see this and this) and earlier work suggest that pion field in TGD framework is analogous to Higgs field.

This raises questions. Assuming that QFT in M⁴ is a reasonable approximation, does a modification of standard model Higgs mechanism allow to approximate TGD description? What aspects of Higgs mechanism remain intact when Higgs is replaced with pseudo-scalar? Those assignable to electro-weak bosons? The key idea allowing to answer these questions is that "Higgsy" pion and ordinary M₈₉ pion are not one and the same thing: the first one corresponds to Euclidian flux tube and the latter one to Minkowskian flux tube. Hegel would say that one begins with thesis about Higgs, represents anti-thesis replacing Higgs with pion, and ends up with a synthesis in which Higgs is transformed to pseudo-scalar Higgs, "Higgsy" pion, or Higgs like state if you wish! Higgs certainly loses its key role in the massivation of fermions.

Can one assume that M⁴ QFT limit exists?

The above approach assumes implicitly - as all comparisons of TGD with experiment - that M⁴ QFT limit of TGD exists. The analysis of the assumptions involved with this limit helps also to understand what happens in generation of "Higgsy" pions.

1. QFT limit involves the assumption that quantum fields and also classical fields superpose in linear approximation. This is certainly not true at given space-time sheet since the number of field like is only four by General Coordinate Invariance. The resolution of the problem is simple: only the effects of fields carried by space-time sheets superpose and this takes place in multiple topological condensation of the particle on several space-time sheets simultaneously. Therefore M⁴ QFT limit can make sense only for many-sheeted space-time.
   
   
2. The light-like 3-surfaces representing lines of Feynman graphs effectively reduce to braid strands and are just at the light-like boundary between Minkowskian and Euclidian regions so that the fermions at braid strands can experience the presence of the instanton density also in the more fundamental description. The constancy of the instanton density can hold true in a good approximation at braid strands. Certainly the M⁴ QFT limit treats Euclidian regions as 1-dimensional lines so that instanton density is replaced with its average.
   
   
3. In particular, the instanton density can be non-vanishing for M⁴ limit since E and B at different space-time sheets can superpose at QFT limit although only their effects superpose in the microscopic theory. At given space-time sheet I can be non-vanishing only in Euclidian regions representing lines of generalized Feynman graphs.
   
   
4. The mechanism leading to the creation of pion like states is assumed to be the presence of strong non-orthogonal electric and magnetic fields accompanying colliding charged particles (see this): this of course in M⁴ QFT approximation. Microscopically this corresponds to the presence of separate space-time sheets for the colliding particles. The generation of "Higgsy" pion condensate or pion like states must involve formation of wormhole contacts representing the "Higgsy" pions. These wormhole contacts must connect the space-time sheets containing strong electric and magnetic fields.
   

Higgs like pseudo-scalar as Euclidian pion?

The recent view about the construction of preferred extremals predicts that in Minkowskian space-time regions the CP₂ projection is at most 3-D. In Euclidian regions M⁴ projection satisfies similar condition. As a consequence, the instanton density vanishes in Minkowskian regions and pion can generate vacuum expectation only in Euclidian regions. Long Minkowskian flux tubes connecting wormhole contacts would correspond to pion like states and short Euclidian flux tubes connecting opposite wormhole throats to "Higgsy" pions.

1. If pseudo-scalar pion like state develops a vacuum expectation value the QFT limit, it provides weak gauge bosons with longitudinal components just as in the case of ordinary Higgs mechanism. Pseudo-scalar boson vacuum expectation contributes to the masses of weak bosons and predicts correctly the ratio of W and Z masses. If p-adic thermodynamics gives a contribution to weak boson masses it must be small as observed already earlier. Higgs like pion cannot give dominant contributions to fermion masses but small radiative correction to fermion masses are possible.
   

   Photon would be massless in 4-D sense unlike weak bosons. If ZEO picture is correct, photon would have small longitudinal mass and should have a third polarization. One must of course remain critical concerning the proposal that longitudinal M² momentum replaces momentum in gauge conditions. Certainly only longitudinal momentum can appear in propagators.
   
   

2. If 3 components of Euclidian pion are eaten by weak gauge bosons, only single neutral pion-like state remain. This is not a problem if ordinary pion corresponds to Minkowskian flux tube. Accordingly, the 126 GeV boson would correspond to the remaining component Higgs like Euclidian pion and the boson with mass around 140 GeV for which CDF has provided some evidence to the Minkowskian M₈₉ pion (see this) and which might have also shown itself in dark matter searches (see this and this).
   
   
3. By the previous construction one can consider two candidates for pion like pseudo-scalars as states whose form apart from parallel translation factor is ‾Ψ₁ j^Akγ_kΨ₂. Here j^A is generator of color isometry either in U(2) sub-algebra or its complement. The state in U(2) algebra transforms as 3+1 under U(2) and the state in its complement like 2+2* under U(2).
   

   These states are analogous of CP₂ polarizations, whose number can be at most four. One must select either of these polarization basis. 2+2* is an unique candidate for the Higgs like pion and can be be naturally assigned with the Euclidian regions having Hermitian structure. 3+1 in turn can be assigned naturally to Minkowskian regions having Hamilton-Jacobi structure.
   

   Ordinary pion has however only three components. If one takes seriously the construction of preferred extremals the solution of the problem is simple: CP₂ projection is at most 3-dimensional so that only 3 polarizations in CP₂ direction are possible and only the triplet remains. This corresponds exactly to what happens in sigma model combining describing pion field as field having values at 3-sphere.
   
   

4. Minkowskian and Euclidian signatures correspond naturally to the decompositions 3+1 and 2+2*, which could be assigned to quaternionic and co-quaternionic subspaces of SU(3) Lie algebra or imbedding space with tangent vectors realized in terms of the octonionic representation of gamma matrices.
   

One can proceed further by making objections.

1. What about kaon, which has a natural 2+2* composition but can be also understood as 3+1 state? Is kaon is Euclidian pion which has not suffered Higgs mechanism? Kaons consists of usbar, dsbar and their antiparticles. Could this non-diagonal character of kaon states explain why all four states are possible? Or could kaon corresponds to Minkowskian triplet plus singlet remaining from the Euclidian variant of kaon? If so, then neutral kaons having very nearly the same mass - so called short lived and long lived kaons - would correspond to Minkowskian and Euclidian variants of kaon. Why the masses if these states should be so near each other? Could this relate closely to CP breaking for non-diagonal mesons involving mixing of Euclidian and Minkowskian neutral kaons? Why CP symmetry requires mass degeneracy?
   
   
2. Are also M₁₀₇ electroweak gauge bosons? Could they correspond to dark variant of electro-weak bosons with non-standard value of Planck constant? This would predict the existence of additional - possibly dark - pion-like state lighter than ordinary pion. The Euclidian neutral pion would have mass about (125/140)× 135 ∼ 125 MeV from scaling argument. Interestingly, there is evidence for satellites of pion: they include also state which are lighter than pion. The reported masses of these states would be M = 62, 80, 100, 181, 198, 215, 227.5, and 235 MeV. 125 MeV state is not included. The interpretation of these states is as IR Regge trajectories in TGD framework.
   

How the vacuum expectation of the pseudo-scalar pion is generated?

Euclidian regions have 4-D CP₂ projection so that the instanton density is non-vanishing and Euclidian pion generates vacuum expectation. In the following an attempt to understand details of this process is made using the unique Higgs potential consistent with conformal invariance.

1. One should realize the linear coupling of Higgs like pion to instanton density. The problem is that Tr(F^Fπ) since π does not make sense as such since π is defined in terms of gamma matrices of CP₂ and F in terms of sigma matrices. One can however map gamma matrices to sigma matrices in a natural manner by using the quaternionic structure of CP₂. γ₀ corresponding to e₀ is mapped to unit matrix and γ_i to the corresponding sigma matrix: γ_i→ ε_ijkσ^jk. This map is natural for the quaternionic representation of gamma matrices. What is crucial is the dimension D=4 of CP₂ and the fact that it has U(2) holonomy.
   
   
   
2. Vacuum expectation value derives from the linear coupling of pion to instanton density. If instanton density is purely electromagnetic, one obtains correct pseudo-scalar Higgs vacuum expectation commuting with photon.
   
   

   
   

3. If the action density contains only the mass term m²π^/2 plus instanton term 1/32π²f_ππ I,
   where I is the instanton density, one obtains the standard PCAC relation between the vacuum expectation of the pion field and instanton density.
   

   π₀= (1/32π²f_πm_π²) I .
   

   This relation appears also in the model for leptopion production. In the standard model the mass term must be tachyonic. This leads to so called hierarchy problem hierarchy problem. The source of the problem are the couplings of Higgs to fermions proportional to the mass of fermion. The radiative corrections to Higgs mass squared are positive and proportional to fermion mass so that top quark gives the dominating contribution. This implies that the sign of the mass squared can become positive and the state with vanishing vacuum expectation value of Higgs field becomes the ground state. In the recent situation this is not a problem since fermions couple to the pion like state only radiatively.
   
   

4. When one adds to the fields appearing in the classical instanton term the quantum counter parts of electroweak gauge fields, one obtains an action giving rise to the anomaly term inducing the anomalous decays to gamma pairs and also other weak boson pairs. The relative phase between the instanton term and kinetic term of pion like state is highly relevant to the decay rate. If the relative phase corresponds to imaginary unit then the rate is just the sum of the anomalous and non-anomalous rates since interference is absent.
   

What is the window to M₈₉ hadron physics?

Concerning the experimental testing of the theory one should have a clear answer to the question concerning the window to M₈₉ hadron physics. One can imagine several alternative windows.

1. Two gluon states transforming to M₈₉ gluons could be one possibility proposed earlier. The model contains a dimensional parameter characterizing the amplitude for the transformation of M₁₀₇ gluon to M₈₉ gluon. Dimensional parameters are not however well-come.
   
   
2. Instanton density as the portal to new hadron physics would be second option but works only in the Euclidian signature. One can however argue that M₈₉ Euclidian pions represent just electroweak physics and cannot act as a portal.
   
   
3. Electroweak gauge bosons correspond to closed flux tubes decomposing to long and short parts. Two short flux tubes associated with the two wormhole contacts connecting the opposite throats define the "Higgsy" pions. Two long flux tubes connect two wormhole contacts at distance of order weak length scale and define M₈₉ pions and mesons in the more general case. In the case of weak bosons the second end of long flux tube contains neutrino pair neutralizing the weak isospin so that the range of weak interactions is given the length of the long flux tube. For M₈₉ the weak isospins at the ends need not sum up to zero and also other states that neutrino pair are allowed, in particular single fermion states. This allows an interpretation as electroweak "de-confinement" transition producing M₈₉ mesons and possibly also baryons. This kind of transition would be rather natural and would not requite any specific mechanisms.
   

For a TGD based discussion of the general theoretical background for Higgs and possible TGD inspired interpretation of the new particle as pionlike state of scaled variant of hadron physics see Is it really Higgs?. See also the chapter " New particle physics predicted by TGD: Part I of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".