# Fun with the Abelian Higgs model

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## Abstract

In calculations of the elementary scalar spectra of spontaneously broken gauge theories there are a number of subtleties which, though it is often unnecessary to deal with them in the order-of-magnitude type of calculations, have to be taken into account if fully consistent results are sought for. Within the “canonical” effective-potential approach these are, for instance: the need to handle infinite series of nested commutators of derivatives of field-dependent mass matrices, the need to cope with spurious IR divergences emerging in the consistent leading-order approximation and, in particular, the need to account for the fine interplay between the renormalization effects in the one- and two-point Green functions which, indeed, is essential for the proper stable vacuum identification and, thus, for the correct interpretation of the results. In this note we illustrate some of these issues in the realm of the minimal Abelian Higgs model and two of its simplest extensions including extra heavy scalars in the spectrum in attempt to exemplify the key aspects of the usual “hierarchy problem” lore in a very specific and simple setting. We emphasize that, regardless of the omnipresent polynomial cut-off dependence in the one-loop corrections to the scalar two-point function, the physical Higgs boson mass is always governed by the associated symmetry-breaking VEV and, as such, it is generally as UV-robust as all other VEV-driven masses in the theory.

## Keywords

Higgs Boson Higgs Mass Vacuum Expectation Value Abelian Higgs Model Break Gauge Theory## Notes

### Acknowledgements

The work was partially supported by the Marie Curie Intra European Fellowship within the 7th European Community Framework Programme FP7-PEOPLE-2009-IEF, contract number PIEF-GA-2009-253119, by the Marie-Curie Career Integration Grant within the 7th European Community Framework Programme FP7-PEOPLE-2011-CIG, contract number PCIG10-GA-2011-303565, by the EU Network grant UNILHC PITN-GA-2009-237920, by the Spanish MICINN grants FPA2008-00319/FPA and MULTIDARK CAD2009-00064 (Consolider-Ingenio 2010 Programme), by the Generalitat Valenciana grant Prometeo/2009/091 and by the Research proposal MSM0021620859 of the Ministry of Education, Youth and Sports of the Czech Republic. I am indebted to Luca di Luzio for reading through an early version of the manuscript.

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