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.

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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.