Abstract
Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results.
Highlights
1.1 The problemDealing with scalar potentials is one of the ubiquitous tasks one faces when building models beyond the Standard Model (SM)
We present a novel approach to algorithmically establish the bounded from below (BFB) conditions for any polynomial scalar potential
A somewhat similar systematic method of deriving the exact BFB conditions exists for models, in which the Higgs potential can be written in terms of independent positivedefinite field bilinears ri
Summary
Dealing with scalar potentials is one of the ubiquitous tasks one faces when building models beyond the Standard Model (SM). A potential can be bounded from below even if there exist some flat directions of the quartic potential, that is, subspaces of Rn in which the quartic term in (2) is exactly zero In this case, one needs to require that, within these subspaces, the lower-degree terms in the scalar potential grow and not decrease at large values of the fields. One needs to require that, within these subspaces, the lower-degree terms in the scalar potential grow and not decrease at large values of the fields This situation was called in [4] stability in the weak sense. In many cases, the main effect of quantum corrections can be absorbed into running parameters of the renormalization-group-improved potential { a} without changing the polynomial structure of the potential In these cases, the mathematical task of establishing the BFB conditions remains unchanged
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