In the previous posting I demonstrated that the pion of M

_{89}is rather Higgs like as far as the decays to electro-weak boson pairs are considered since the gauge kinetic term of the action is in both cases responsible for these decays. There is however an anomalous production of gamma pairs due to the anomalous decays of the pion to gamma pairs resulting from axial current anomaly. Earlier I ended up with erraneous hypothesis that the anomaly term alone is enough. The reason for the error was rather banal: I confused number and its inverse with each other! In the following I will demonstrate that the argument works also at the quantitative level.

To see that the model indeed survives also quantitative tests one can consider the decay rate of pion like state to gamma pairs using PCAC. Axial current anomaly tells that the divergence ∂_{μ}A^{μ} of the axial current equals to f_{π}m_{π}^{2}π_{0}, where π_{0} is the neutral pion field. Axial current divergence contains a part proportional to the instanton density for electromagnetic field and this defines the effective action allowing to calculate the production amplitude and rate for gamma pairs.

- From Iztykson-Zuber the decay width of pion to two-gamma would be given as

Γ(π) = α

^{2}m_{π}^{2}/64π^{2}f_{π}^{2}.

f

_{π}is expected to be of order m_{pi}. Let us write f_{π}=Xm_{π}.

- The decay rates of Higgs can be found here. For the decay of Higgs to two photons the rate is

Γ(h)= α

^{2}g_{W}^{2}2^{-10}π^{-3}m_{h}^{3}m_{W}^{-2}.

The prediction is exactly the same in the case of M

_{89}pion. One only replaces scalar with pseudoscalar and Higgs vacuum expectation with that for pseudoscalar and given by PCAC anomaly expressible in terms of instanton density for classical induced em field F_{em}associated with the space-time sheet assignable to colliding quarks and defining the hadronic space-time sheet for M_{89}hadron physics (note that this space-time sheet could be also assicated with colliding protons).

π

_{0}(vac)=-[1/32π^{2}m_{π}^{2}f_{π}]× I , I=ε_{αβγδ}F_{em}^{αβ}F_{em}^{γδ}= 2E•B .

Here F

_{em}is defined by identifying gauge potential as eA_{mu}, which corresponds to the classical gauge potentials in TGD. It is essential that the induced electric and magnetic fields are non-orthogonal: this is true if CP_{2}projection of space-time sheet has dimension larger than d=2: this is actually always the case for preferred extremals so that the generation of the analog of Higgs expectation is basic phenomenon in TGD Universe but does not give rise to massivation. Instanton density I appears as a parameter which is in the first approximation constant.

- The ratio of these rates is for m(π)=m(h)

r == Γ(h)/Γ(π) = X

^{2}[α× sin^{2}(θ_{W})]^{-1}.

Some comments about the result are in order.

- For X= 93/135 holding true for the ordinary neutral pion π
_{0}and m(h)=m(π)=125 GeV this gives r=1.63 and f(π)=1.07m_{W}. Therefore the contribution from the axial anomaly is .61 times the contribution of the gauge kinetic term to the decay rate assuming that the contributions of the amplitudes do not interfere. Interference effects can change the situation. Therefore PCAC anomaly alone is not enough and the prediction for the ratio r== (Γ(h)+Γ(π))/Γ(h), which is 1.61 times higher than predicted by Higgs. Constructive interference can give rise to 3.17 times larger rate and destructive interference to rate which is only .05 of the rate predicted by Higgs alone.

The relative phase of the amplitudes from anomaly and kinetic term is expected to vary and the first guess is that the interference term gives a vanishing contribution average contribution. Local constructive interference in phase space would allow to understand the local values of r above 1.61. The ratio of the observed Higgs to gamma pair signal cross section to the predicted one is certainly consistent with this picture! Note that the anomalous contribution is present also for W and Z since instanton term is non-Abelian and only its vacuum expectation value is Abelian. This means that also the rates to W and Z pairs are enhanced as indeed observed by ATLAS.

- The value of I characterizing the hadronic space-time sheet appears in the kinetic term responsible for the decays and also in the model for the production rate. The expression for the decay rate to gamma pairs involves a relation between Higgs vacuum expectation and Higgs mass provided by standard model. This relationship need not be same for the pion like state.

One cannot predict absolute production rates without a detailed model for the electric and magnetic fields of colliding quarks or protons predicting the instanton density I. This kind of model has been proposed in kenociteallb/leptc.

- Does the production M
_{89}pions provide the only window to M_{89}hadron physics? I have also considered a window which involves transformation of ordinary gluons to those of M_{89}physics and also direct transformation of ordinary hadronic space-time sheet to that of M_{89}physics. If pions are the only window to the new hadron physics, the production of other M_{89}hadrons should take place via the reactions of the pions of M_{89}pion condensate producing other M_{89}hadrons.

- For X= 93/135 holding true for the ordinary neutral pion π

**Addition:**The above picture is attractive but a closer look leads to an objection. If one accepts that gauge theory is a reasonable M

^{4}QFT limit of TGD then- also other aspects of Higgs mechanism related to weak bosons are unavoidable: the pseudo-scalar nature of pion does not matter. In particular, gauge bosons become massive by eating 3 components of the pseudo-scalar: only one neutral state remains in the spectrum! Therefore the pion like state in question cannot be M

_{89}pion but something else - one could call it "Higgsy" pion or "Higgs like state".

The construction of pseudo-scalar like states as axial vectors of imbedding space (pseudo-scalars of M^{4}) carried out earlier in some posting indeed demonstrated that one obtains two kinds of pions - and more generally - meson like states. Pion-like states associated with the long Minkowskian flux tubes connecting wormhole throats assigned with different wormhole contacts and pion-like states associated with short Euclidian flux tubes connecting opposite throats of a given wormhole contact. These two kinds of pion like states would naturally correspond to pions and "Higgsy" pions. There are indications for two pion like states at energies 126 GeV and around 140 GeV and the natural identification would be as Euclidian and Minkowskian M_{89} pions respectively. I will tell about this aspect in detail later.

The conclusion is that TGD based explanation for the new boson is in excellent shape and we can only wait for what experimentalists say. One can also calculate the rates for the decays to fermion pairs and here the main deviations from standard model emerge.

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".

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