*off mass shell*k=107 τ-pion, production of k=107 τ-pion coherent state heated into QCD plasma like state producing colored lepton jets, and production of k=103

*on mass shell*τ-pion with mass scaled up by factor 8 from that of k=107 τ-pion).

The paper proposes that three new particles are involved. The masses for the particles - christened h_{3}, h_{2}, and h_{1} - are assumed to be 3.6 GeV, 7.3 GeV, and 15 GeV. h_{1} is assumed to be pair produced and decay to h_{2} pair decaying to h_{3} pair decaying to a τ pair.

h_{3} mass is assumed to be 3.6 GeV and life-time of 20×10^{-12} seconds. The mass is same as the TGD based prediction for neutral τ-pion mass, whose lifetime however equals to 1.12× 10^{-17} seconds (gamma+gamma decay dominates). The correct prediction for the lifetime provides a strong support for the identification of long-lived state as charged τ-pion with mass near τ mass so that the decay to μ and its antineutrino dominates. Hence the model is not consistent with leptohadronic model.

p-Adic length scale hypothesis predicts that allowed mass scales come as powers of sqrt(2) and these masses indeed come in good approximation as powers of 2. Several p-adic scales appear in low energy hadron physics for quarks and this replaces Gell-Mann formula for low-lying hadron masses. Therefore one can ask whether the proposed masses correspond to neutral tau-pion with p= M_{k}=2^{k}-1, k=107, and its p-adically scaled up variants with p∼ 2^{k}, k= 105, and k=103 (also prime). The prediction for masses would be 3.6 GeV, 7.2 GeV, 14.4 GeV.

This co-incidence cannot of course be taken too seriously since the powers of two in CDF model have rather mundane origin: they follow from the assumed production mechanism producing 8 τ-leptons from h_{1}. One can however spend some time by looking whether it could be realized somehow allowing p-adically scaled up variants of τ-pion.

- The proposed model for the production of muon jets is based on production of k=103 neutral τ-pion (or several of them) having 8 times larger mass than k=107 τ-pion in strong E·B background of the colliding proton and antiproton and decaying via strong interactions to k=105 and k=107 τ-pions.
- The first step would be
π

^{0}_{τ}(103)→ π^{0}_{τ}(105)+π^{+}_{τ}(105)+π^{-}_{τ}(105).This step is not kinematically possible if masses are obtained by exact scaling and if m(π

^{0}_{τ})< m(π^{+/-}_{τ}) holds true as for ordinary pion. p-Adic mass formulas do not however predict exact scaling. In the case that reaction is not kinematically possible, it must be replaced with a reaction in which second charged k=105 pion is virtual and decays weakly. - Second step would consist of a scaled variant of the first step
π

^{0}_{τ}(105)→ π^{0}_{τ}(107)+π^{+}_{τ}(107)+π^{-}_{τ}(107),and the weak decays of the π

^{+/-}_{τ}(105) with mass 2m(τ) to lepton pairs. - The last step would involve the decays of both charged and neutral π
_{τ}(107). The signature of the mechanism would be anomalous gamma pairs with invariant masses 2^{k}×m(τ),k=1,2,3 coming from the decays of neutral τ-pions. - Dimensionless four-pion coupling l determines the decay rates for neutral t-pions appearing in the cascade. Rates are proportional to phase space-volumes, which are rather small by kinetic reasons.

The total cross section for producing single leptopion can be estimated by using the quantum model for leptopion production. Production amplitude is essentially Coulomb scattering amplitude for a given value of the impact parameter b for colliding proton and anti-proton multiplied by the amplitude U(b,p) for producing on mass shell k=103 lepto-pion with given four-momentum in the fields E and B and given essentially by the Fourier transform of E·B. The replacement of the motion with free motion should be a good approximation.

UV and IR cutoffs for the impact parameter appear in the model and are identifiable as appropriate p-adic length scales. UV cutoff could correspond to the Compton size of nucleon (k=107) and IR cutoff to the size of the space-time sheets representing topologically quantized electromagnetic fields of colliding nucleons (perhaps k=113 corresponding to nuclear p-adic length scale and size for color magnetic body of constituent quarks or k=127 for the magnetic body of current quarks with mass scale of order MeV). If one has hbar/hbar_{0} = 2^{7} one could also guess that the IR cutoff corresponds to the size of dark em space-time sheet equal to 2^{7}L(113) = L(127) (or 2^{7}L(127) = L(141)), which corresponds to electron's p-adic length scale. These are of course rough guesses.

Quantitatively the jet-likeness of muons means that the additional muons are contained in the cone q < 36.8 degrees around the initial muon direction. If the decay of p^{0}_{t}(k) can occur to on mass shell p^{0}_{t}(k+2), k=103,105, it is possible to understand jets as a consequence of the decay kinematics forcing the pions resulting as decay products to be almost at rest.

- Suppose that the decays to three pions can take place as on mass shell decays so that pions are very nearly at rest. The distribution of decay products m[`(n)] in the decays of p
^{±}(105) is spherically symmetric in the rest frame and the energy and momentum of the muon are given by[E,p] = [m(t)+ m ^{2}(m)4m(t),m(t)- m ^{2}(m)4m(t)] . The boost factor g = 1/Ö{1-v

^{2}} to the rest system of muon is g = x+(4x)^{-1}~ 18, x= m(τ)/m(μ). - The momentum distribution for m
^{+}coming from p^{+}_{t}is spherically symmetric in the rest system of p^{+}. In the rest system of m^{-}the momentum distribution is non-vanishing only for when the angle q between the direction of velocity of m^{-}is below a maximum value of given by tan(q_{max}) = 1 corresponding to a situation in which the momentum m^{+}is orthogonal to the momentum of m^{-}(the maximum transverse momentum equals to m(m)vg and longitudinal momentum becomes m(m)vg in the boost). This angle corresponds to 45 degrees and is not too far from 36.8 degrees. - At the next step the energy of muons resulting in the decays of p
^{±}(103)[E,p] = [ m(t) 2+ m ^{2}(m)2m(t), m(t) 2- m ^{2}(m)2m(t)] , and the boost factor is g

_{1}= (x + x^{-1})/2 ~ 9, x= m(τ)/m(μ). q_{max}satisfies the condition tan(q_{max}) = g_{1}v_{1}/gv @ 1/2 giving q_{max}@ 26.6 degrees.

If on mass shell decays are not possible, the situation changes since either of the charged pions is off mass shell. In order to obtain similar result the virtual should occur dominantly via states near to on mass shell pion. Since four-pion coupling is just constant, this option does not seem to be realized.

Additional signatures of the model come from very peculiar kinematics of lepto-pion production. The produced τ-pions are restricted in the scattering plane of the colliding charges and produced very nearly at rest in cm frame. In the rest frame of the target the produced τ-pions are concentrated on the cone with opening angle cos(θ)= β/v_{cm}, where v_{cm}= 2v/(1+v^{2}) and v is the velocity of τ-pion in the rest system of the proton.

For details and background see the chapter Recent Status of Leptohadron Hypothesis of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

## 7 comments:

And of course Woit continues to delete my comments also, even when they are certainly on topic.

Hi Matti,

I have noticed that there seems to be a correspondence between the Zitterbewegung of David Hestenes and the Beltrami flow of thermophysics.

I saw that some of your papers dealt with Beltrami flow.

To Kea:

I noticed that my comment about Peter's censoring activities had dropped away as I added details to the model and Peter does not deserve rewriting the comment;-). I any case, this kind of intellectual inhonesty makes me sad.

To Doug:

The generalizaton of 3-D Beltrami flows to 4-D onesappears in the construction of extremals of Kahler action here.

So called CP_2 type vacuum extremals serving as a model for elementary particles have random 1-D light-like curve as an M^4 projection and the interpretation in terms of random zitterbewung is attractive.

Light-like 3-surfaces which are fundamental objects in quantum TGD could be also interpreted locally as random motion of partonic 2-surfaces with light-velocity and also here one might speak about zitterbewegung.

If one would allow completely random motion (arbitrary velocity), TGD would reduced to topological QFT so that lightlikeness is absolutely essential.

This relates closely to the massivation: random lightlike orbit looks like motion with subluminal velocity in given length scale so that particle behaves like massive particle.

p-Adic thermodynamics provides the mathematical description for this massivation.

Also p-adic length scale hypothesis follows from basic quantum TGD by using analogy with Brownian motion and the fact that time scales T=2^kT_0 appear quantum TGD in fundamental role.

Matti

Hi Matti,

I became interested in mathematical dynamics and found that this was being used by thermophysicists in fluid dynamics, plasma physicists and physiologists for blood flow.

There is a Springer book, from the RAS, searchable on Amazon, that has great diagrams with the text:

SV Alekseenko, PA Kuibin and VL Okulov, "Theory of Concentrated Vortices: An Introduction", 2007, under 500 pages.

Beltrami flows are also used in medical imaging.

Hi, Matti!

How did you do those equations? I would also like to add some equations on my blog, but I do not know how to do them. I know some LaTeX, though...

I use latex to write the text and then translate it to html using tth_exe: you find the program easily from web. For complicated equations some additional editing is required. There are some features. For instance, /font^P:s appear in html file must be transformed to /font:s by find and replace.

Best,

Matti

Ok. Thanks, Matti!

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