The Higgs Mechanism with Diagrams: a didactic approach

The manuscript derives the electroweak sector of the standard model (or rather, its physical particle spectrum) in a non-standard way: instead of postulating an action governed by various (e.g. gauge) symmetries and extracting Feynman rules from an appropriate quantization in order to compute probability amplitudes, the authors attempt to derive the electroweak sector by taking a purely diagrammatic approach, i.e., by proceeding directly from a proposed set of probability-conserving Feynman rules. This is an interesting result, since it offers prospects for a formulation of the standard model without relying on elements, in particular gauge invariance, that are problematic in the conventional formulation.

It is not exactly clear how these results are to be interpreted physically, however. In the usual formulation, the ground state becomes degenerate as a result of a phase transition in the very early universe, causing certain physical parameters to change. In particular, the parameter $m^2$ in the scalar field potential becomes negative at the electroweak transition, $\tau \simeq 10^{-10} \, \mbox{s}$, leading to mass generation via the Higgs mechanism. In the diagrammatic approach, the corresponding parameter is always negative however, i.e., $m^2 < 0$. This would correspond to the scalar field potential always having the "Mexican hat" shape that is characteristic of the "broken symmetry" phase in the usual formulation. The representation of the model in which tachyons appear therefore does not correspond to a perturbative expansion around a minimum of energy, but rather around a local maximum. This renders tachyonic fluctuations intrinsically unstable.

In the usual formulation, a clear physical scale distinguishes between the two phases of the theory. At energies above the electroweak scale, $\Lambda \simeq 10^{2} \, \mbox{GeV}$ (or times before the electroweak transition, $\tau \simeq 10^{-10} \, \mbox{s}$), the gauge symmetry is "unbroken" and quarks, leptons and gauge bosons are massless. At energies below $\Lambda$, the vacuum is degenerate and a particular choice of ground state "breaks" the symmetry. In both phases however, the spectrum of physical particles is determined by excitations of the ground state. This is possible because each phase comes equipped with its own ground state.

The authors - experts on QFT - might reason that the preceding remarks merely illustrate the virtues of the diagrammatic approach. Indeed, they explicitly subscribe to the view that "Feynman diagrams $\cdots$ are a more fundamental description $\cdots$ thatn are Lagrangians and actions" (p. 3). Yet, they also state on the same page that the "two approaches are methodologically equivalent". If that is indeed the case, I have trouble understanding why tachyon instability is apparently not an issue in the diagrammatic approach.

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