### Estimate for electro-pion production cross section in heavy ion collisions

I have described in earlier postings the model explaining the CDF anomaly as evidence for colored excitations of leptons - one of the basic predictions of TGD distinguishing it from standard model- forming bound states identified as leptopions: τ-pion in this case. For 15 years ago I ended up with this kind of model as an explanation of anomalous electron-positron pair production in heavy ion collisions near Coulomb wall: electron-positron pairs would have resulted from electro-pions. The difficulty of this model was that the total production cross section was roughly by an order of magnitude smaller than the reported one using the maximal value of the impact parameter, which looked reasonable at that time.

The work with CDF anomaly led to a generalization and modification of the original leptopion model and it is important to check that the modified model can reproduce also the cross section for the production of electro-pions. The maximal value of the impact parameter allowing this turns out to be essentially 1 Angstrom corresponding to photon energy of 8.1 keV: this X-ray energy has same scale as the rest energy difference between exotic and ordinary variants of nuclei predicted by TGD and discussed in the previous posting, which suggests a connection. Note that atomic radius would in TGD framework represent a fundamental length scale of also nuclear physics realized as the size scale of "field bodies" associated with nuclei and implied by the topological quantization of classical fields in TGD Universe. The following piece of text summarizes the result of calculation. For details the interested reader can consult the links at the end.

The numerical estimate for lepto-pion production cross section (giving estimate cross section for the production of electron-positron pairs) is carried out for thorium with (Z=90,A=232). The value of the collision velocity of the incoming nucleus in the rest frame of the second nucleus is taken as b = .1. From the width dv/v=.2 of velocity distribution in the same frame the upper bound g £ 1+d, d @ 2×10^{-3} for the Lorentz boost factor of electro-pion in cm system is deduced. The cutoff is necessary because energy conservation is not coded to the structure of the model.

As expected, the singular contribution from the cone v_{cm}cos(q) = b, v_{cm} = 2v/(1+v^{2}) gives the dominating contribution to the cross section. This contribution is proportional to the value of b_{max}^{2} at the limit f = 0. Cutoff radius is taken to be b_{max}=150 ×g_{cm}hbar/m(p_{e})=1.04 A. The numerical estimate for the cross section using the parameter values listed comes out as s = 5.5 mb to be compared with the rough experimental estimate of about 5 mb. The interpretation would be that the space-time sheet associated with colliding nuclei during the collision has this transversal size in cm system. At this space-time sheet the electric and magnetic fields of the nuclei interfere.

From this one can cautiously conclude that lepto-pion model is consistent with both electro-pion production and t-pion production in proton antiproton collisions. One can of course criticize the large value of impact parameter and a good justification for 1 Angstrom should be found. One could also worry about the singular character of the amplitude making the integration of total cross section somewhat risky business using the rather meager numerical facilities available. The rigorous method to calculate the contribution near the singularity relies on stepwise halving of the increment Δθ as one approaches the singularity. The calculation gives twenty smaller result as that with constant value of Δθ. Hence it seems that one can trust on the result of calculation at least at the order of magnitude level.

This figure gives the differential production cross section for g_{1} = 1.0319. Obviously the differential cross section is strongly concentrated at the cone due to singularity of the production amplitude for fixed b.

The important conclusion is that the same model can reproduce the value of production cross section for both electro-pions explaining the old electron-positron anomaly of heavy ion collisions and τ-pions explaining the CDF anomaly of proton-antiproton collisions at cm energy sqrt(s)= 1.96 TeV with essentially same and rather reasonable assumptions (do not however forget the large maximal value of the impact parameter!).

In the case of electro-pions one must notice that depending on situation the final states are gamma pairs for electro-pion with mass very nearly equal to twice the electron mass. In the case of neutral τ-pion the strong decay to three p-adically scaled down versions of τ-pion proceeds faster or at least rate comparable to that for the decay to gamma pair. For higher mass variants of electro-pion for which there is evidence (for instance, one with mass 1.6 MeV) the final states are dominated by electron-positron pairs. This is true if the primary decay products are electro-baryons of form (say) e_{ex}= e_{8}ν_{8}ν_{c,8} resulting via electro-strong decays instead of electrons and having slightly larger mass than electron. Otherwise the decay to gamma pair would dominate also the decays of higher mass states. A small magnetic moment type coupling between e, e_{ex} and electro-gluon field made possible by the color octet character of colored leptons induces the mixing of e and e_{ex} so that e_{ex} transforms to e by emission of photon. The anomalous magnetic moment of electron poses restrictions on the color magnetic coupling.

For details and background see the updated (and still under updating) chapter Recent Status of Leptohadron Hypothesis of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

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