Finite Electroweak Theory


The Standard Model (SM) of particle physics is an amazing theory. It accounts for all the known families of particles that make up matter, and all their interactions. It even provides a mechanism that allows particles to gain mass. Yet the theory has some shortcomings.

For starters, it has a large number of free parameters. It is one thing to provide a mechanism to generate mass, it is another thing to predict the magnitude of those masses. This, the theory does not do. Individual particle masses, along with other parameters such as coupling constants, are not predicted, but determined through observation.

But these are just minor problems, relatively speaking. There are two much bigger ones.

First, the theory is based on the assumption that neutrinos are massless. But they aren’t. Or, at the very least, there is some mechanism beyond the SM that converts neutrinos of different flavors into one another, because we observed them do just that. What’s wrong with giving neutrinos masses, just like we give masses to charged leptons and quarks? Trouble is, neutrinos are ultra light. The corresponding dimensionless quantity in the model would have to be a very small number. People are suspicious of dimensionless quantities in a physical theory that are very far from unity. But we need either such a quantity or a new feature in the SM to account for neutrino oscillations.

Second, the theory cannot account for quantum gravity, because spin-2 gravity with its dimensioned coupling constant is notoriously nonrenormalizable.

One particular possibility to address these problems is a theory, developed by Moffat and others in 1991 and onwards, which addresses the problem in a somewhat unusual way. Its main feature is a non-local regularization scheme that guarantees that the theory remains finite.

The cornerstone of the SM is the Higgs particle, which really does two things. First, it offers a mechanism through spontaneous symmetry breaking that allows particles to gain mass. Second, it produces exactly the right terms needed to ensure that when we calculate the probability amplitude of some physical processes, we get finite answers. But it is an oddball particle. It is the only spin-0 scalar particle in the theory. The self-interactions of the Higgs boson are a source of additional trouble.

One of the original motivations for Moffat’s theory was the desire to do away with the Higgs boson. The existence of the Higgs has since been confirmed (representing another spectacular triumph for the Standard Model) but other problems remain, so Moffat’s theory remains relevant.

The theory reproduces the SM at low energies, but predicts deviations at energy levels that may be reachable by present-day instruments. This is important… too many theories nowadays rely on postulates that can never, ever, be confronted with experiment!

The basic steps of building the finite electroweak theory (FEW) are as follows.

  • First, we start with a massless SU(2)L×U(1)Y gauge invariant theory that incorporates all the known fermion and boson fields.
  • Next, we change the theory by introducing a non-local operator that has the effect of “smearing” vertices and eliminating all unpleasant infinities. This smearing operator has an energy scale that is fairly low (around 500 GeV). Normally, new physics with an associated energy scale that is this low leads to higher order operators that produce nonsensical results. Not so in this case: the nonlocal operator “eats up” an infinite series of such terms.
  • We then calculate the self-interaction of the fermion fields and find that in addition to the zero-mass, trivial solution, the resulting gap equation also admits a non-trivial solution. This is sufficient to generate fermion masses as an alternative to the Yukawa-couplings to the Higgs field v.e.v., which is the mechanism of the Standard Model.
  • Next, we move on to the vector bosons. Simply doing the same that we do for fermions is not enough to get the right masses. To get the right masses, we introduce into the path integral a nontrivial integration measure that is intimately connected to the nonlocal operator. This measure is constructed such that it breaks the SU(2)L symmetry while leaving the U(1)Y alone, guaranteeing that the photon remains massless. This, too, represents an alternative to the Standard Model’s symmetry breaking mechanism that relies on the Higgs field’s unusual (and in some ways, problematic) quartic “Mexican hat” potential.

The result is a theory in which both fermions and bosons have the right masses, and the rules are almost those of the SM, with two key differences: vector boson masses are running in the vector boson propagators, and the coupling constants are also running with energy.

Some of the more important papers on the FEW are:

  • Nonlocal regularization of gauge theories.
    D. Evens and G. Kleppe and J. W. Moffat and R. P. Woodard. Phys. Rev. D, Vol 43, No 2, 499-519 (1991).
    This paper demonstrates in great detail the nonlocal regularization scheme.
  • Finite electroweak theory without a Higgs particle.
    J. W. Moffat. Mod. Phys. Lett. A, Vol 6, No 11, 1011-1021 (1991).
    The first attempt to use the nonlocal regularization scheme to create a Higgsless finite theory.
  • Prediction of the top quark mass in a finite electroweak theory.
    M. Clayton and J. W. Moffat. Mod. Phys. Lett. A, Vol 6, No 29, 2697-2703 (1991).
    The Higgsless theory can be used to establish an approximate relationship between the heaviest quark, the vector boson masses, and the theory’s own energy scale. If the latter two are known, the former can be predicted. In 1991, the top quark mass was not known; while the prediction is off, it nevertheless demonstrates the power of the nonlocal model.
  • Results from a Finite Electroweak Theory.
    M. Clayton. Master’s Thesis, University of Toronto, 1991.
    In his thesis, Clayton calculates in detail the vector boson masses and the top quark mass.
  • Quantum nonlocal field theory: Physics without infinities.
    N. J. Cornish. Int. J. Mod. Phys. A, Vol 7, No 24, 6121-6157 (1992).
    In this important paper, Cornish explores features of the nonlocal theory, providing details that are not found elsewhere.
  • Electroweak Model Without a Higgs Particle.
    J. W. Moffat. eprint arXiv:0709.4269.
    In this new paper, Moffat revives the nonlocal Higgsless theory, updating it using new observational data.
  • A finite electroweak model without a Higgs particle.
    J. W. Moffat and V. T. Toth. arXiv:0812.1991.
    This paper provides the most complete description yet of the nonlocal electroweak theory. It also includes a pedagogical section that demonstrates how a finite energy scale leads to specific features of the theory without running into the usual problems with higher dimensional operators.
  • The running of coupling constants and unitarity in a finite electroweak theory.
    J. W. Moffat and V. T. Toth. arXiv:0812.1994.
    This companion paper provides explicit calculations for the vector boson masses, the running of coupling constants, and demonstrates how the theory avoids unitarity violations in scattering experiments.

Note: An earlier version of this article, written before the discovery of the Higgs boson, emphasized that Moffat’s theory can do away with the Higgs boson altogether. That is true; but it can also live with the Higgs boson, and it has other, desirable features.

 Posted by at 2:55 pm

  7 Responses to “Finite Electroweak Theory”

  1. “It is one thing to provide a mechanism to generate mass, it is another to predict the magnitude of those masses.”
    My guess is that Lestone’s heuristic string theory is by far the best attempt to calculate the fine structure constant.
    “Physics based calculation of the fine structure constant” by J. P. Lestone, 2009
    Does anyone have any criticisms or comments related to the preceding publication?
    In the Los Alamos report LA-UR-16-20131 “Semi-classical Electrodynamics” (January 2016), Lestone wrote:
    “Quantum electrodynamics is complex and its associated mathematics can appear overwhelming for those not trained in this field. Here semi-classical approaches are used to obtain a more intuitive physical feel for several QED processes including electrostatics, Compton scattering, pair annihilation, the anomalous magnetic moment, and the Lamb shift. These intuitive arguments lead to a possible answer to the question of the nature of charge. The corresponding calculated elementary charge is q=1.602177×10^−19 C with a corresponding calculated inverse fine structure constant of α^−1=137.036. These calculations suggest elementary particles have properties that resemble quantum micro black holes, and that electromagnetism and general relativity are intimately connected via virtual Hawking radiation. …”
    Has anyone studied Lestone’s report LA-UR-16-20131 and reached a conclusion on its merits or demerits?

  2. I looked at the 2009 Lestone paper just now, and it is pure numerology. The “relatively simple physics concepts” are just misunderstandings of what, e.g., a virtual particle exchange really means; visualizing elementary particles as tiny but extended objects with many mechanical degrees of freedom goes completely contrary to both quantum theory and experimental evidence; and in the end, he still fails at his goal, resorting to a “guess” to get a number that is sufficiently close to the fine structure constant. And none of this is really new. Eddington was already going to considerable lengths trying to come up with similar “derivations”.

    I also looked at the 2016 paper (which curiously appears to have disappeared from the government Web site but is available on ResearchGate.) It continues in the same vein, with a more sophisticated way of guessing a formula for the fine structure constant (but still missing the mark after the 6th significant digit, which, given the accuracy of QED, is missing by a mile.) And he continues with the misconceptions. In particular, what it calls “relatively simple semiclassical arguments” to help achieve “a better intuitive feel” for quantum field theory, I call a gross mischaracterization, a caricature of both theory and experiment. Finally I am coming to appreciate Schwinger’s dislike of Feynman diagrams: While a helpful computational tool, it gives the completely wrong (and experimentally easily discredited!) impression that particles are the fundamental objects of the universe, not fields.

  3. “… visualizing elementary particles as tiny but extended objects with many mechanical degrees of freedom …” My guess is: Lestone’s heuristic string theory is a wrong idea if and only if there does not exist a higher-dimensional string theoretical duality theorem between virtual particles and virtual quantum fields. Is the following idea wrong? 64 dimensions of virtual particles + 3 dimensions of linear momentum + 3 dimensions of angular momentum + 1 dimension of time = 71 dimensions of string vibration for quantum fields. Who are the 2 best critics of string theory? I think that the answer might be Sheldon Glashow and Burton Richter.
    NOVA | The Elegant Universe | Sheldon Glashow | PBS
    Is “Naturalness” Unnatural? Presentation at SUSY ’06 Prof. Burton Richter

  4. Yes, I can tell immediately that “64 dimensions of virtual particles” is wrong, since it produces far, far too many degrees of freedom compared to what we observe, not to mention that it also reveals a fundamental misunderstanding of what the phrase, “virtual particles,” really means. I also don’t know what a “string theoretical duality theorem between virtual particles and virtual quantum fields” is, but whatever it is appears to bear no resemblance to what I actually do know about quantum field theory.

  5. Does the following publication seem promising?
    “Mass empirics of leptons and quarks” by Gerald Rosen, 2005

  6. Recall a supersymmetry bet with deadline June 16, 2016:
    Supersymmetry and the Crisis in Physics | Not Even Wrong
    If supersymmetry ultimately fails, then what might replace it?

  7. Hmmm. Rosen does get some masses right, but I still think it’s just fancy numerology; several of his quark masses are way off, as a matter of fact. As for the bet, I am firmly on the NO side myself. Replace SUSY? In what sense? What does SUSY do that needs to be done even if SUSY doesn’t apply?