Yet another lesson on the stability conditions in multi-Higgs potentials
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Springer
Received: June 10, Revised: October 1, Accepted: October 10, Published: November 20,
2020 2020 2020 2020
Igor P. Ivanov and Francisco Vazão Centro de Física Teórica de Particulas, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
E-mail: [email protected], [email protected] Abstract: We discuss a rather common but often unnoticed pitfall which arises when deriving the bounded-from-below (BFB) conditions in multi-Higgs models with softly broken global symmetries. Namely, necessary and sufficient BFB conditions derived for the case with an exact symmetry can be ruined by introducing soft symmetry breaking terms. Using S4 and A4 -symmetric three-Higgs-doublet models as an example, we argue that all published necessary and sufficient BFB conditions, even those which are correct for the exactly symmetric case, are no longer sufficient if soft symmetry breaking is added. Using the geometric formalism, we derive the exact necessary and sufficient BFB conditions for the 3HDM with the symmetry group S4 , either exact or softly broken, and review the situation for the A4 -symmetric case. Keywords: Beyond Standard Model, Higgs Physics, Global Symmetries ArXiv ePrint: 2006.00036
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)104
JHEP11(2020)104
Yet another lesson on the stability conditions in multi-Higgs potentials
Contents 1 Introduction A4 3HDM scalar potential Two parametrizations Linking the BFB conditions with minimization The literature on the BFB conditions in A4 3HDM A quick check for the rephasing invariant model
3 BFB conditions from geometry 3.1 Geometric treatment of the rephasing invariant model 3.2 Is checking neutral minima enough? 3.3 Adding soft symmetry breaking
2 2 4 5 6 7 7 8 9
4 The S4 -symmetric case 4.1 The case with an exact S4 symmetry 4.2 Softly broken S4
10 10 12
5 The A4 -symmetric case
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6 Conclusions
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A The bilinear formalism in 3HDM
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B The BFB conditions for the rephasing invariant model: direct calculations
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orbit space of the S4 3HDM Flat directions Circumscribed polyhedron The arc C.3.1 Formulation of the problem C.3.2 The main solution C.3.3 The “trivial” solution C.4 The net result: description of the (x, y) plane
C The C.1 C.2 C.3
–i–
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2 The 2.1 2.2 2.3 2.4
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1
Introduction
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That is, if a minimum happened to lie along such a direction, it would be a charge-breaking vacuum.
–1–
JHEP11(2020)104
Theoretical search for New Physics beyond the Standard Model (SM) is driven, in the absence of direct experimental indications, by “educated guesses”. Theorists introduce a set of new fields, construct their interaction lagrangian based on a desired set of symmetries or anticipated phenomenological features, and then calculate observable consequences. New free parameters entering the lagrangian are either fixed by fitting the data or scanned over in search for viable and interesting versions of
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