Text: The Origin of the Magnetic Moments of the Electron and the Muon To summarize the revolving electron and muon models, a pion captures a negative mass tachyon and becomes, overall, a less massive muon. The muon captures still another negative mass tachyon, and becomes an even lighter electron. The orbital velocities of the revolving charged particles are constant at the speed of light, with only the orbital dimensions and the overall energy of the system changing during the transition from one particle system to another. Because these transition from pion to muon and from muon to 3 electron are monopole and not dipole transitions, no radiation would be expected of them. This is observed to be the case experimentally. Further, based on this model, it is mandatory that the byproducts of high energy electron-positron collisions include muons and pions and gamma rays. This is observed to be the case experimentally. NOTE: (With apologies) At this time there is a some difficulty getting some of the Greek characters to print on the html page, so we are forced to use pi and mu for pion and muon for the moment, except in graphics or those equations that derived from the equation editor and imported from another document. This also holds for h bar, or Planck's constant divided by 2 pi. The masses of the muon's and electron's tachyons are MTmu = Mmu - Mpi = -33.9091 MeV, (1) MTe = me - Mmu = -105.147388 MeV. (2) Next, we will need to utilize half of these masses as binding energies. I.e., we have ETmu = - 16.9546 MeV. (3) ETe = - 52.573694. (4) The sum of these energies is ETmu + ETe = - 16.9546 - 52.573694 = - 69.5283 MeV. (5) Next, examine Fig. 2. It is a composite of two particle conversion curves. The mu -> e curve on the left is well known and is contained in most particle physics books. The right most curve, the direct pi -> e conversion curve, is less well known, the direct conversion of a pion into an electron is relatively rare, about one in 104 pion conversions. It should be noted that accurate fits to the mu -> e curve have been produced by the V-A theory. The generally accepted assumption is that two neutrinos are produced by the decay of an electron into a muon, and the shape of the curve is determined by the relative angles of emission of the two neutrinos. That is to say, the curves are normally considered to be decay spectra. Neutrinos have been observed. And, the residual energies of this model are 20 eV for the electron model, and 123 MeV for the muon model, more than enough to account for the estimated masses of the neutrinos. The interpretation used here is that the reaction during the capture of a tachyon by a muon has a residual energy whose distribution is described by the mu-> e curve. However, if the reaction energy is greater than that of the binding energy of the electron's tachyon to the charged particle, there will be no capture and hence, no electrons will be produced. The point at which this happens, 52.6 MeV, is the cutoff energy of the mu -> e curve. This compares nicely with the energy of Eq. 4. The pi -> mu capture, on the other hand, produces monoenergetic muons at an energy 4.119 MeV, so that there is no cutoff energy. Therefore, another approach must be taken. So compare Eq. 5 with the 69.5 MeV cutoff energy of the -> e curve. The double tachyon capture implies that the total binding energy of the muon and electron's tachyons is half of sum of their masses, and hence, the binding energy of the muon's tachyon is also half of its mass energy. Note, incidentally, that the difference in the two cutoff energies is 16.9 MeV, which is half the muon's tachyon's mass energy as given in Eq. 3. Because of its negative mass, a revolving tachyon will have an inwardly directed force, not an outwardly directed force. This inwardly directed force of the tachyon balances the outwardly directed force of the orbiting charged particle, thus maintaining the particle systems in tightly bound orbits. The balance conditions are similar to that of a helium balloon (a negative mass analog) on one end of a massless rod balanced by a less massive weight placed between the balloon and a pivot on the other end of the rod. Because of the negative mass, the center of mass of the system is at the pivot, and is thus external to the line connecting the charged particle's orbit and the tachyon. This is shown in Fig. 2. Based on the above, in general, the magnitude of the binding energy, which is the same as the ground state energy, is given by Considering the above, the de Broglie wavelength for the tachyon is given simply by where h is Planck's constant, MT is the mass of the tachyon in grams, and ET is the energy of the tachyon. (NOTE, again that we will be forced to sometimes use the word lambda for the wavelength.) Using Eq. 6 for the energy in Eq.7, we have It could be argued that it is naive to apply this simple equation to tachyons and ignore relativity. But there is no experimental evidence one way or the other as to how they behave. Certainly it is no more naive than extending the Lorenz transformation to hyperluminal regions and concluding that tachyons have an imaginary mass as has been the accepted practice. Therefore, we will work with what we have and see how the model develops. If we assume a single de Broglie wavelength, lambda, for the circumference of the tachyon's orbit around the charged particle, we may divide equation 8 by 2 pi. This gives us the tachyon's orbital radius, r lambda T, as it orbits the charged particle in the charged particle's frame of reference. That is, Here, the subscript lambdaT refers to the de Broglie wavelength, and h_bar = h/2 pi. While the original model used this concept, another way of looking at it is to consider that both the tachyon and charged particle revolve around the common, external center of mass. The tachyon has some 207 de Broglie wavelengths in its orbit, which is, in this case, larger than that of the charged particles orbit. We will now explore the balance conditions for a negative mass particle that is coupled to a positive mass. This is illustrated in Fig. 2. For the electron, we define For the muon, The equation describing the balance of this system for the electron model is Mmurce + MTerTe = 0, (12) Mmurce + MTe(rce + rTe) = 0, where we used the fact that rTe = rce + rlambda Te. Using Eq. 2 (for MTe) in Eq. 12, we have that Mmurce + (me - Mmu)(rce + rTe) = 0, (13) Mmurce + merce + merlambda Te - Mmurce - Mmurlambda Te = 0. The Mmurce terms cancel, so that Eq. 13 becomes, after a little rearrangement, rceme = (Mmu - me)rlambda Te . (14) Dividing both sides of 14 by me, and then using Eq.10, we obtain rce = ( Re - 1 )rlambda Te . (15) Also, rewrite Eq. 2 using Eq. 10 to obtain MTe = me - Mmu = (1 - Re)me . (16) Using Eq. 9 for rlambda Te, Eq. 15 becomes Using MTe as defined by Eq. 16, we eliminate (Re - 1) and MTe from Eq. 17 so that for the electron. Using an identical approach for the muon model, the orbital radius of the muon's pion is The magnetic moment of a current loop is, in general, mu = IA, (20) where I is the current in the loop, and A is its area. (Note that mu is not to be confused with the subscript mu representing the muon.) Current is, in general, given by the number of charges passing a point multiplied by the charge per particle. Also, recall that in the gaussian system of units, the charge in statcoulombs divided by the speed of light is the unit of charge used to calculate the magnetic field. Hence, the current at a point caused by a single charged particle revolving about a center point is where f is the frequency of the particle's rotation, and for a light speed particle is given by where c is the velocity of the charged particle and rc is its orbital radius. Hence, the magnetic moment of a single, revolving charged particle is obtained from Eqs. 20, 21, and 22, as where pi*rc2 is used for the area, A, of the current loop of Eq. 20. Eq. 23 then becomes Using equation 18 in Eq. 24, the magnetic moment of the electron is Using Eq. 20 in Eq.24, the magnetic moment for the muon is These are the Bohr magnetons for the electron and muon respectively. These values for the magnetic moments agree with experiment to within 0.17 % for the electron and 0.12 % for the muon. No particular significance is attached to the plus and minus versions of the magnetic moments at this time. But to take it a step further, by requiring that the electron's charged particle have an integral number of wavelengths, the accuracy of the electron's magnetic moment is improved to within 39 parts per million. That is, the gyromagnetic ratio is g/2 = 1.0011208. (QED does better than this, but with many years and hundreds of workers, it should!) It should be noted, for contrast, that the self-energy calculation for the electron provides the well known classical electron radius of 2.8179 fm, which is far smaller than that of the electron as given above. However, it is less than twice that of the muon. No particular significance is attached to this, however. But it is interesting to note that if we divide the electron's charged particle's radius (the reduced Compton wavelength) by the classical electron radius, the result is the fine structure constant. Again, the significance of this with respect to this model, if any, is not clear at this time. One objection that may be raised is that the electron is much larger than the high energy scattering data indicates it is rather small. The electron's charged particle's orbit has a radius of 386.15933 fm, and the muon's charged particle's orbital radius is 1.8675947 fm. In spite of these large orbital radii, the actual scattering cross section of muons and electrons would be expected to be much smaller at high energies because the actual charged particle itself is no larger than the pion. That is, the upper most limit of its radius is 0.185 fm (2.15 Mb). This does not contradict the much lower experimental value of 5 - 30 Nb. (No lower limit is available from the model.) Insofar as the question of a revolving charge radiating its energy away, we could, after the manner of Bohr with his hydrogen atom, merely hypothesize that its orbit is a bound, ground state orbit, and leave it with no further explanation. After all , this model uses a revolving charged particle to produce extensive agreement with experiment in spite of the radiation problem, so that obviously, conventional electrodynamics would appear not to hold here. But to suggest two paths of investigation to address this radiation problem, first consider the significance of Eq. 27. The possibility that an electron or muon is a pseudo photon looping back on itself is one possible path of investigation. The second is to provide a detailed analysis of a light speed particle's radiation characteristics. There is no a priori reason to assume that a light speed particle should radiate in the same manner as a subluminal particle, nor is there a definite a prior reason to assume that it does not. But to reiterate, this model does produce extensive agreement with experiment in spite of this radiation question.

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