View Poll Results: Multiple poll: Check all you agree. Logarithms of lepton mass quotients should be pursued. 24 27.91% Alpha calculation from serial expansion should be pursued 22 25.58% We should look for more empirical relationships 26 30.23% Pythagorean triples approach should be pursued. 21 24.42% Quotients from distance radiuses should be investigated 16 18.60% The estimate of magnetic anomalous moment should be investigated. 26 30.23% The estimate of Weinberg angle should be investigated. 21 24.42% Jay R. Yabon theory should be investigate. 16 18.60% I support the efforts in this thread. 47 54.65% I think the effort in this thread is not worthwhile. 30 34.88% Multiple Choice Poll. Voters: 86. You may not vote on this poll

All the lepton masses from G, pi, e

by arivero
Tags: koide formula, lepton masses
PF Gold
P: 2,921
 Quote by Hans de Vries 0.000 006 263 813 : Difference between electron and muon anomaly 0.000 000 084 639 : First vacuum polarization term of the electron ---------------------------------------------------------------------- 0.000 006 348 452 : Sum of the vacuum polarization terms (the above) 0.000 006 353 732 : (+/-3 000) mass ratio of electron and the W boson. 0.000 000 002 998 : uncertainty from the Z mass
Hmm, I had tried the same with the data truncated at two loops for electron and three for the muon and I got a somehow weaker result. It seems one needs to add the three loop data for the electron, but it is a mess because then we have three new diagrams to exclude.
 Sci Advisor P: 1,135 The other interesting observation is that the missing term becomes equal in both cases to within experimental value. 0.00115877693 : mu/mZ + VP2 0.00115965218 : electron anomaly 0.00000087525 : missing 0.00116504602 : mu/mZ + me/mw (+ VP2 - VP2) 0.0011659208_ : muon anomaly 0.00000087478 : missing 0.00000000047 : missing1 - missing2 0.00000002668 : uncertainty due to Z (cancels if missings are subtracted) 0.00000000299 : uncertainty due to W mass So there may be a single missing term. Regards, Hans PS: VP2 = 0.00000008464 : First vacuum polarization term of the electron which is a second order term. PPS: I'm not entirely sure if the term 0.00116504602 : mu/mZ + me/mw (+ VP2 - VP2) should indeed not include VP2. me/mW - VP2 = difference between muon and electron anomaly. mu/mZ + VP2 = all self energy terms + first vacuum polarization term of the electron anomaly
 PF Gold P: 2,921 Making it in a reverse way: as the missing term is already in the electron anomalous moment, assume it is sort of square of the first term. So it is mu^2/X^2. Solve for X: X=sqrt(mu^2/.000000875015)=112.95 GeV The quantity is interesting in two ways. We suspect of a neutral scalar H0 at 115 GeV, and it could have this role. But also mw*sqrt(2) is 113.87 GeV, so we can use the W particle, which was my first attempt for the missing term. Lacking of more theory, both are equally suitable: values up to 114.5 GeV are covered by the Z indeterminacy. The first has the advantage of not using an arbitrary 1/2 coefficient and it is neutral as the Z, but it has not been discovered (yet?), the second is an already discovered particle but we have used it for the "vacuum polarised graphs", and it is surprising to have it here too, even if squared.
 PF Gold P: 2,921 For the sake of completeness, references. The industry of calculation of the anomalous moment seems to be based in Cornell, around a veteran named T. Kinoshi-ta. Other group does exist in North America around A. Czarnecki http://arxiv.org/abs/hep-ph/9810512, from Czarnecki and Marciano, is the main entry point for the calculation up to order alpha^4. It is regretly a short preprint and it does not separate loop by loop, so one is referred to more detailed bibliography, which is not in the arxiv The five diagrams for order alpha^2 appear well separated in Levine and Wright Phys. Rev. D 8, 3171-3179 (1973) http://prola.aps.org/abstract/PRD/v8/i9/p3171_1. I got from here the specific value we were using above. Also some sums for 40 diagrams of order alpha^3 are presented there. Note that of these, 12 diagrams are vacuum polarisation loops, amounting perhaps to a contribution of 0.37 (alpha/pi)^3 The alpha^2 "polarisation loop", depending of the mass quotient of the external and internal lepton, is studied both analytic and numerically by Li Mendel and Samuel, Phys. Rev. D 47, 1723-1725 (1993) http://prola.aps.org/abstract/PRD/v47/i4/p1723_1 Also Samuel Li and Mendel provide a calculation of tau lepton, Phys. Rev. Lett. 67, 668-670 (1991) http://prola.aps.org/abstract/PRL/v67/i6/p668_1 up to order alpha^3. For this lepton, the contributions for quarks are already noticeable at this scale.
P: 13
 Quote by arivero Also Samuel Li and Mendel provide a calculation of tau lepton, Phys. Rev. Lett. 67, 668-670 (1991) http://prola.aps.org/abstract/PRL/v67/i6/p668_1 up to order alpha^3. For this lepton, the contributions for quarks are already noticeable at this scale.
Alejandro"

Do you and Hans have any close fits relating the tau magnetic moment with any lepton-to-electroweak-boson ratio?

Jay.
PF Gold
P: 2,921
 Quote by Jay R. Yablon Alejandro" Do you and Hans have any close fits relating the tau magnetic moment with any lepton-to-electroweak-boson ratio?
Hmm the answer past yesterday was yes, the answer today is more towards no. On one side the quantity $$m_e m_\tau/m_W^2$$ in of the right order of magnitude to do further corrections in our calculations, but we do not need it anymore, giving the uncertainty in the mass of Z. On other hand, and more concretely answering your question, the difference between anomalous moment of mu and tau can only be covered with a new quotient $$m_e/m_{X^+}$$, and the mass of the new X+ particle should be around 68-70 GeV. At these energy range, the LEP2 presented an slight "statistical" deviation, but no particle
PF Gold
P: 2,921
 Quote by arivero mw=80525 (+-38)
80425

So I was getting a discrepance between calculations at home and calculations at work.
 P: 13 Dear Alejandro and Hans: I just posted to my web site http://home.nycap.rr.com/jry/FermionMass.htm, a Gordon-like decomposition of the Fermion mass into a term due to electromagnetic charge, and a term due to magnetic moment. This is still an early working draft. I hope this can help you in your efforts by providing a covariant field theory context for your efforts to characterize the magnetic moments. I know that your efforts have helped me recognize that consideration of magnetic moments is likely to be a critical aspect of what I am attempting to do. Best, Jay.
PF Gold
P: 2,921
 Quote by Jay R. Yablon I just posted to my web site http://home.nycap.rr.com/jry/FermionMass.htm, a Gordon-like decomposition of the Fermion mass into a term due to electromagnetic charge, and a term due to magnetic moment. This is still an early working draft.
Probably Hans is also exploring this way, but I am not so optimistic about a direct connection; perhaps a semiclassical effect, could be. But even that is strange to manage. To me, it seems more as if the symmetry breaking mechanism of the electroweak group (and its vacuum value) were needing of the lepton masses is some misterious way.

From our quadratic formulae we can get rather intriguing equations. For instance this one:

$${m_\tau\over m_Z} + {m_\mu\over m_W}= {m_\tau\over m_\mu} a_\mu^{s.e.} + {m_\mu\over m_e} a_\mu^{v.p.}$$

Where $$a_\mu^{s.e.},a_\mu^{v.p.}$$ are the self-energy and vacuum polarisation parts of the muon anomalous magnetic moment; note that the v.p. part depends internally of lepton mass quotients, while the s.e. is mass independent, in QED (in the full electroweak theory new dependences appear).
 PF Gold P: 2,921 Some of the development of the thread uploaded at http://arxiv.org/abs/hep-ph/0503104
 Sci Advisor P: 1,135 Hmm, I was doing something else and ran just incidently into this one: $$\sqrt{ \ 2 \ \frac{m_V}{m_Z} \ \frac{m_{\tau}}{m_e}} \ = \ 137.038 (12)$$ mV is the vacuum expectation value of 246.22046 GeV (according to Jay) The biggest uncertainty is from the tau mass. Regards, Hans $\ \ \alpha \ \ \$ = 1/137.03599911 mV = 246220.46 mZ = 91187.6 (+-2.1) mτ = 1776.99(+0.29-0.26) me = 0.51099892 (+-0.00000004)
PF Gold
P: 2,921
 Quote by Hans de Vries vacuum expectation value of 246.22046 GeV (according to Jay) The biggest uncertainty is from the tau mass.
Also this vacuum should have a high uncertainness. I wonder where did Jay got so many digits from.
P: 1,135
 Quote by arivero Also this vacuum should have a high uncertainness. I wonder where did Jay got so many digits from.
It's basically "one over the square of" : The Fermi Coupling constant 1.16637 (1) times sqrt(2)

So it should be 246.2206 (11)

Regards, Hans.
PF Gold
P: 2,921
 Quote by Hans de Vries That's just the value you want it to have! Do you have you any more data like that.
Funny, I have found a paper which, while aiming toward other goals, also uses a variant of QED discarding vacuum polarisation terms in order to get more explicit formulas. It happens in section 3 of
Phys. Rev. 95, 1300-1312 (1954)
which is titled "3. EXAMPLE: QUANTUM ELECTRODYNAMICS WITHOUT PHOTON SELF ENERGY PARTS".

The authors are a M.Gell-Mann and a F.E.Low, from Illinois.
P: 1,135
 Quote by arivero Funny, I have found a paper which, while aiming toward other goals, also uses a variant of QED discarding vacuum polarisation terms in order to get more explicit formulas. It happens in section 3 of Phys. Rev. 95, 1300-1312 (1954) which is titled "3. EXAMPLE: QUANTUM ELECTRODYNAMICS WITHOUT PHOTON SELF ENERGY PARTS". The authors are a M.Gell-Mann and a F.E.Low, from Illinois.
There are related follow-ups:

Quantum Electrodynamics at Small Distances,
Baker, Johnson, 1969.
http://prola.aps.org/abstract/PR/v183/i5/p1292_1

Quantum Electrodynamics Without Photon Self-Energy Parts
S. L. Adler and W. A. Bardeen 1971
http://prola.aps.org/abstract/PRD/v4/i10/p3045_1

Regards, Hans
 PF Gold P: 2,921 Baker and Johnson have actually a whole forest of papers. Following this review of classic bibliography, I also come across to the formula $${M_0^2 \over M_V^2}= {3 \over 2 \pi} \alpha$$ which is world-famous, but I was unaware. Regretly it is about a single scalar charged particle, not a fermion, and the quotient against the vector boson gets this square dependence. The formula was found in Radiative Corrections as the Origin of Spontaneous Symmetry Breaking by Sidney Coleman and Erick Weinberg, Phys. Rev. D 7, 1888-1910 (1973) They even have a generalisation to SU(3)xU(1). Incidentaly, one of these authors was contacted about our preprint 0503104, here is his statement: Given the current state of knowledge in the field, speculations concerning approximate numerical coincidences such as the ones you discuss do not constitute the degree of substantial new physics that is required for publication
 Sci Advisor HW Helper P: 1,204 Hans, I am impressed. It's too many digits for the accuracy. If the standard model is an effective field theory from a deeper level, then the fine structure constant should be calculated from a series in that deeper (unified) level. One supposes that such a unified field theory would be extremely strongly coupled (otherwise it'd be visible), and that our usual perturbation methods would fail, and therefore that calculations would be impossible. However, this is a way out of this. Our usual experience with bound states is that when two particles are bound together, we expect the bound state to have a higher mass than either of the particles contributing to it. Of course the total mass is a little less than the sum of the masses, $$E=mc^2$$ and all that. But if the particles are extremely strongly bound, then the mass of the bound state could be negligible compared to the mass of either free particle. For example, the mass of a free up quark is unknown, but all indications are that it would require a lot of energy to make one, so its mass should be extremely large. Present experimental limits say it should be much larger than the mass of a proton. Now doing quantum mechanics in such a nonperturbational region might seem impossible, but this is not necessarily the case. In fact, infinite potential wells make for simple quantum mechanics problems. Perturbation theory may not be needed or appropriate. Looking at QFT from the position eigenstate representation point of view, the creation operators for elementary particles have to work in infinitesimal regions of space. Suppose we want to do physics in that tiny region. The natural thing we'll do is to use a Gaussian centered at the position. One way of representing the potential energy between two objects bound by extreme energies is to suppose that they each stress space-time (in the general relativistic stress-energy manner), but in ways that are complementary. Thus the sum makes for less stress to space-time than either of the separate particles. In that case, if we represent the stress of each particle with a Gaussian, we end up deriving a potential energy that, for very low energies, works out as proportional to the square of distance. This is the classic harmonic oscillator problem, and the solution in QM is well known without any need for perturbation theory. Now your series for the fine structure constant used a Gaussian form. Coincidence? I doubt it. My suspicion is that this is a clue. My guess is that there is a unified field theory with equal coupling constants for everything, and that the strong force is strong because it has fewer coupling constants multiplied together in it. That would have to do with the factor in the exponential. This all has to do with my bizarre belief that even the leptons are composite particles. Carl Brannen As an aside, I once decided to see if the sum of inverses of cubes $$\zeta(3) = \sum 1/n^3$$ could be summed similarly to how the sum of inverses of squares or fourth powers could be summed. I wrote a C++ program that computed the sum out to 5000 decimal places (which requires a lot of elementary mathematics as the series converges very very slowly), and then did various manipulations on it to search for a pattern. The most useful thing to know, when trying to determine if a high precision number is rational, is the series obtained by taking the fractional part of an approximation and inverting it. It's been over a decade, but I seem to recall that the name for this is the "partial fraction expansion".
 PF Gold P: 2,921 Attached (!) I have drawn the whole elementary particle spectrum at logarithmic scale. Honoring Yablon, I have put a 1/137 line also between the tau and the electroweak vacuum. There are four clearly distinguished zones, usually called the electromagnetic breaking, the chiral breaking, the hadronic scale (or SU(3) gap) and the electroweak breaking scale. SO I have encircled them with green rectangles. (EDITED: If you are using the Microsoft Explorer, you will need to expand the jpg to full screen or almost) Attached Thumbnails

 Related Discussions General Physics 3