# (Non)Existence of electroweak symmetry breaking in AQFT

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G. Scharf in Gauge Field Theories: Spin One and Spin Two (formerly called Quantum Gauge Theories: A True Ghost Story) shows that there is no symmetry breaking needed to produce massive bosons: One can start with massive gauge fields, then show that in third order of perturbation theory the (broken) Higgs potential is derived, and shifting the field produces the "Higgs mechanism" (pg 167). This does not make sense seeing that on the other hand, other expositions on the topic one starts with massless fields then shows how the Higgs mechanism (by definition) breaks some symmetry to give rise to massive fields.

Considering how electroweak symmetry breaking is energy-dependent, I would think statements concerning existence or properties of the effect need to be supported by renormalization group methods:

In AQFT/causal perturbation theory, what is the Higgs mechanism? Is it just the nonzero VEV of the Higgs field or does it include some sort of "breaking"?
If the classical Lagrangian for the electroweak theory is renormalized (to n-th order) and SSB exists (or does not exist), how would one show it? Does it mean at some scale the finitely renormalized mass $m\left(\rho \ge \Lambda_{EW}\right) =0$ ($\neq 0$)?

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In causal perturbation theory there is no Higgs mechanism, unless one counts as it the need to introduce a scalar field to produce a causal description for a vector boson.

In traditional perturbation theory if you start with an action that has the Higgs field replaced by one shifted by a constant, you get directly the broken theory.

Thus perturbatively, the label ''broken symmetry'' is just a verbal convention.

Nonperturbatively, the situation may well be different since which symmetries are broken may depend on the boundary conditions considered (energy scale and chemical potential). Different boundary conditions lead to different asymptotic quasiparticle states. Causal perturbation theory almost exclusively considers only the vacuum sector, where this difference does not show up.

answered Oct 16, 2020 by (15,488 points)

It makes sense that the "Higgs mechanism" and relevant concepts given in Scharf's book could be verbal convention, but does the derivation suffice as rigorous "proof" independent of scale? I heard from another that since causal perturbation theory derives the broken theory, the renormalization group should be used to probe higher energies, but I found that I did not know how to do this or show otherwise (though they may very well be wrong).

The renormalization group only affects the approximation properties of the renormalized series approximation, not the ideal (unknown) nonperturbative solution. The RG is briefly discussed in the second edition of Scharf's book on QED; see also https://www.physicsforums.com/insights/causal-perturbation-theory/

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