Abstract
We observe that biquadratic potentials admit non-trivial flat directions when the determinant of the quartic coupling matrix of the scalar fields vanishes. This consideration suggests a new approach to the problem of finding flat directions in scale-invariant theories, noticeably simplifying the study of scalar potentials involving many fields. The method generalizes to arbitrary quartic potentials by requiring that the hyperdeterminant of the tensor of scalar couplings be zero. We demonstrate our approach with detailed examples pertaining to common scalar extensions of the Standard Model.
Highlights
We offer a new implementation of the Gildener-Weinberg method based on the following observation: a biquadratic scalar potential admits a flat direction if the determinant of its quartic coupling matrix vanishes
Our technique noticeably simplifies the study of biquadratic potentials involving multiple scalar fields and allows us to identify the orientation of the flat direction in a straightforward manner
We present an alternative implementation of the Gildener-Weinberg approach, demonstrating that a flat direction appears in a biquadratic scalar potential if the determinant of the quartic coupling matrix vanishes
Summary
Scale-invariant models have taken the spotlight as possible solutions to fundamental issues such as the hierarchy problem [1], inflation [2,3,4,5,6], and cosmological gravitational wave background [7,8]. We investigate the appearance of flat directions in scale-invariant scalar potentials, which, in the Gildener-Weinberg approach [9], ensures a successful application of the Coleman-Weinberg mechanism [10] for the radiative generation of mass scales.. We offer a new implementation of the Gildener-Weinberg method based on the following observation: a biquadratic scalar potential admits a flat direction if the determinant of its quartic coupling matrix vanishes. Our technique noticeably simplifies the study of biquadratic potentials involving multiple scalar fields and allows us to identify the orientation of the flat direction in a straightforward manner. We exemplify the method for a biquadratic two-field potential [22] We exemplify the method for a biquadratic two-field potential [22] (see Ref. [23] and references therein), a biquadratic three-field potential [24,25,26], and a general twofield potential [24]
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