It turned also possible to calculate the production amplitude using very reasonable approximation so that the numerics could be restricted to the integral over the phase space of τ-pion. Errors are therefore under analytic control. I cannot of course exclude numerical factors of order unity which are not quite correct since the calculation is really tedious and my calculations skills are not the best ones.

A brief article summarizing the details of the calculation of the τ-pion production cross section can be found from my homepage. Here is the abstract.

* The article summarizes the quantum model for τ-pion production. Various alternatives generalizing the earlier model for electro-pion production are discussed and a general formula for differential cross section is deduced. Three alternatives inspired by eikonal approximation generalizing the earlier model inspired by Born approximation to a perturbation series in the Coulombic interaction potential of the colliding charges. The requirement of manifest relativistic invariance for the formula of differential cross section leaves only two options, call them I and II. The production cross section for τ-pion is estimated and found to be consistent with the reported cross section of about 100 nb for option I under natural assumptions about the physical cutoff parameters (maximal energy of τ-pion center of mass system and the estimate for the maximal value of impact parameter in the collision which however turns out to be unimportant unless its value is very large). For option II the production cross section is by several orders of magnitude too small. Since the model involves only fundamental coupling constants, the result can be regarded as a further success of the τ-pion model of CDF anomaly. Analytic expressions for the production amplitude are deduced in the Appendix as a Fourier transform for the inner product of the non-orthogonal magnetic and electric fields of the colliding charges in various kinematical situations. This allows to reduce numerical integrations to an integral over the phase space of lepto-pion and gives a tight analytic control over the numerics. *

Other aspects of the model are discussed in the chapter Recent Status of Leptohadron Hypothesis of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". The chapter is still under updating since I must recalculate the cross section for the production of electro-pions in heavy ion collisions to test the updated model and also add plots about differential cross sections, which are very strongly concentrated on forward direction and production plane in the case of τ-pion. The singularity concentrated on conical surface is however absent in the relativistic situation. One might hope that the correct estimate for the order of magnitude about 100 nb of the reported cross section under reasonable assumptions about the physical cutoff parameters (maximum energy of leptopion in center of mass system and impact parameter cutoff which is however not significant unless very large impact parameters are allowed) might put bells ringing also in the ears of colleagues.

I might be overoptimistic since even after correct predictions/explanations for all the basic findings, which means

- correct prediction for the lifetime of the lightest new particle in terms of fundamental constants,
- explanation of the three states proposed by CDF group together with correct prediction for their masses differing by powers of two (p-adic length scale hypothesis),
- an explanation for the emergence of jets in terms of the very special kinematics for the decays of leptopions at given p-adic length scale to the low p-adic scale implying that decay produces are almost at rest,
- a unified explanation for the anomalies suggesting the existence of also electropions and muo-pions,
- explanation for the ortopositronium decay rate anomaly and Karmen anomaly,
- possible explanation for the anomaly in anomalous magnetic moment of muon, ...

The text below is not meant to describe the model but serve only as a sample possibly stimulating interest.

The estimate of the cross section involves some delicacies. The model has purely physical cutoffs which must be formulated in a precise manner.

- Since energy conservation is not coded into the model, some assumption about the maximal t-pion energy in cm system expressed as a fraction e of proton's center of mass energy is necessary. Maximal fraction corresponds to the condition m(p
_{t}) £ m(p_{t})g_{1}£ em_{p}g_{cm}in cm system giving [m(p_{t})/(m_{p}g_{cm}) £ e £ 1. g_{cm}can be deduced from the center of mass energy of proton as g_{cm}= Ös2m_{p}, Ös=1.96 TeV. This gives 1.6×10^{-2}< e < 1 in a reasonable approximation. It is convenient to parameterize e ase = (1+d)× m(p _{t})m_{p}× 1 g_{cm}. The coordinate system in which the calculations are carried out is taken to be the rest system of (say) antiproton so that one must perform a Lorentz boost to obtain upper and lower limits for the velocity of t-pion in this system. In this system the range of g

_{1}is fixed by the maximal cm velocity fixed by e and the upper/lower limit of g_{1}corresponds to a direction parallel/opposite to the velocity of proton. - By Lorentz invariance the value of the impact parameter cutoff b
_{max}should be expressible in terms t-pion Compton length and the center of mass energy of the colliding proton and the assumption is that b_{max}=g_{cm}×hbar/m(p_{t}), where it is assumed m(p_{t})=8m(t). The production cross section does not depend much on the precise choice of the impact parameter cutoff b_{max}unless it is un-physically large in which case b_{max}^{2}proportionality is predicted.

The numerical estimate for the production cross section involves some delicacies.

- The power series expansion of the integral of CUT
_{1}using partial fraction representation does not converge since that roots c_{±}are very large in the entire integration region. Instead the approximation A_{1}@ iBcos(y)/D simplifying considerably the calculations can be used. Also the value of b_{1}L is rather small and one can use stationary phase approximation for CUT_{2}. It turns out that the contribution of CUT_{2}is negligible as compared to that of CUT_{1}. - Since the situation is singular for q = 0 and f = 0 and f = p/2 (by symmetry it is enough to calculate the cross section only for this kinematical region), cutoffs
q Î [e _{1}, (1-e_{1})]×p ,f Î [e _{1}, (1-e_{1})]×p/2 ,e _{1}=10^{-3}.The result of the calculation is not very sensitive to the value of the cutoff.

- Since the available numerical environment was rather primitive (MATLAB in personal computer), the requirement of a reasonable calculation time restricted the number of intervals in the discretization for the three kinematical variables g,q,f to be below N
_{max}=80. The result of calculation did not depend appreciably on the number of intervals above N=40 for g_{1}integral and for q and f integrals even N=10 gave a good estimate.

The calculations were carried for the exp(iS) option since in good approximation the estimate for exp(iS)-1 model is obtained by a simple scaling. exp(iS) model produces a correct order of magnitude for the cross section whereas exp(iS)-1 variant predicts a cross section, which is by several orders of magnitude smaller by downwards α_{em}^{2} scaling. As I asked Tommaso Dorigo for an estimate for the production cross section in the discussion inspired by his first blog posting , he mentioned that authors refer to a production cross section is 100 nb, which looks to me suspiciously large (too large by three orders of magnitude), when compared with the production rate of muon pairs from b-bbar. δ=1.5 which corresponds to τ-pion energy 36 GeV gives the estimate &sigma=351 nb. The energy is suspiciously high.

In fact, in the recent posting of Tommaso Dorigo a value of order .1 nb for the production cross section was mentioned. Electro-pions in heavy ion collisions are produced almost at rest and one has Δ v/v∼ .2 giving δ= Δ E/m(π)∼ 2× 10^{-3}. If one believes in fractal scaling, this should be at least the order of magnitude also in the case of τ-pion. This would give the estimate σ∼ 1 nb. For δ= Δ E/m(π)∼ 10^{-3} a cross section σ= .1 nb would result.

The plot for the differential production cross section is here (the scale of the earlier plot was erratic due to the above mentioned error).

For details and background see the updated (and still under updating) chapter Recent Status of Leptohadron Hypothesis.

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