Abstract
We perform an old school, one-loop renormalization of the Abelian–Higgs model in the Unitary and Rξ gauges, focused on the scalar potential and the gauge boson mass. Our goal is to demonstrate in this simple context the validity of the Unitary gauge at the quantum level, which could open the way for an until now (mostly) avoided framework for loop computations. We indeed find that the Unitary gauge is consistent and equivalent to the Rξ gauge at the level of β-functions. Then we compare the renormalized, finite, one-loop Higgs potential in the two gauges and we again find equivalence. This equivalence needs not only a complete cancellation of the gauge fixing parameter ξ from the Rξ gauge potential but also requires its ξ-independent part to be equal to the Unitary gauge result. We follow the quantum behavior of the system by plotting Renormalization Group trajectories and Lines of Constant Physics, with the former the well known curves and with the latter, determined by the finite parts of the counter-terms, particularly well suited for a comparison with non-perturbative studies.
Highlights
Broken gauge theories are of great physical interest, notably because of the Brout-Englert-Higgs mechanism [1]
Diagrams with four external Higgs fields contribute through their divergent parts to the running of the Higgs quartic self coupling λ and through their finite parts they contribute to the one-loop scalar potential
Since we are interested here only in the common subsector that consists of the Z-mass term and the scalar potential, we can carry out the renormalization program on the Unitary gauge Lagrangean and only when we arrive at the stage where we analyse the finite, renormalized scalar potential where finite corrections become relevant, we may have to distinguish between the two gauges if necessary
Summary
Broken gauge theories are of great physical interest, notably because of the Brout-Englert-Higgs (or Higgs) mechanism [1]. The scalar effective potential computed with the background field method on the other hand is notoriously known to be ξ-dependent already at one-loop, which renders its physicality (and the relevance of the precision triviality and instability bounds derived from it) a delicate matter [5, 6] It is an open issue how to define an unambiguous, physical, quantum potential in spontaneously broken gauge theories. The Unitary gauge is one where only physical fields propagate and it is commonly used in textbooks in order to demonstrate the physical spectrum of spontaneously broken gauge theories at the classical level It is rarely used though for loop calculations, we are not aware of a complete renormalization work in this gauge. We have a number of Appendices where auxiliary material can be found
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