Standard Model Gauge Couplings from Gauge-Dilatation Symmetry Breaking

Abstract

It is well known that the self-energy of the gauge bosons is quadratically divergent in the Standard Model when a simple cutoff is imposed. We demonstrate phenomenologically that the quadratic divergences in fact unify. The unification occurs at a surprisingly low scale, $\Lambda _\mathrm {u}\approx 4\times 10^7$ GeV. Suppose now that there is a spontaneously broken rotational symmetry between the space-time coordinates and gauge theoretical phases. The symmetry-breaking pattern is such that the gauge bosons arise as the massless Goldstone bosons, whereas the dilatonic mode acts as the massive (Higgs) boson, whose vacuum expectation value determines the gauge couplings. In this case, the quadratic divergences or the tadpoles of the gauge boson self-energy should indeed unify because these divergences need to be cancelled by a universal dilatonic contribution, assuming dynamical symmetry breaking. If there is dynamical symmetry breaking, we are in principle able to calculate the value of the gauge couplings as well as the scale hierarchy $\Lambda _\mathrm {cut}/\Lambda _\mathrm {u}$. We perform this calculation by adopting a naive quartic symmetry-breaking potential which unfortunately violates local gauge invariance. Using tadpole-cancellation and dilatonic self-energy conditions, the value of $\Lambda _\mathrm {cut}$ is then found to be approximately $4\times 10^{18}$ GeV in the Feynman gauge and $5\times 10^{15}$ GeV in the Landau gauge. The cancellation of an anomaly in the dilaton self-energy requires that the number of fermionic generations equals three. The symmetry-breaking needs to be driven by some other mass-generating mechanism such as electroweak symmetry breaking. Our estimation for $\Lambda _\mathrm {u}$ is of the correct order if $\Lambda _\mathrm {cut}\approx 5\times 10^{15}$ GeV.