Vacuum structure of the Babu-Nandi-Tavartkiladze model of neutrino mass generation

We analyze the vacuum structure of the Babu-Nandi-Tavartkiladze (BNT) model of neutrino mass generation, in which the Standard Model is extended by an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>S</a:mi> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:msub> <a:mrow> <a:mo stretchy="false">)</a:mo> </a:mrow> <a:mrow> <a:mi>L</a:mi> </a:mrow> </a:msub> </a:mrow> </a:math> scalar quadruplet with hypercharge <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>Y</e:mi> <e:mo>=</e:mo> <e:mn>3</e:mn> <e:mo>/</e:mo> <e:mn>2</e:mn> </e:math> and a vectorlike <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>S</g:mi> <g:mi>U</g:mi> <g:mo stretchy="false">(</g:mo> <g:mn>2</g:mn> <g:msub> <g:mo stretchy="false">)</g:mo> <g:mi>L</g:mi> </g:msub> </g:math> triplet fermion with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>Y</k:mi> <k:mo>=</k:mo> <k:mn>1</k:mn> </k:math> , generating neutrino masses via an effective dimension-seven operator. We delineate the theoretical constraints on the model, requiring the scalar potential to be bounded from below in all field directions, ensuring perturbative unitarity of scattering amplitudes, and demanding that the electroweak vacuum corresponds to the global minimum of the potential. We find that the electroweak vacuum is not generically guaranteed to be the global minimum: several charge-breaking stationary points may coexist with—and potentially lie below—it in potential depth. For the electroweak-like vacuum with vanishing quadruplet expectation value, the condition of global stability reduces to two simple mass inequalities involving the doubly and triply charged scalars. In contrast, for the general electroweak vacuum with nonzero doublet and quadruplet expectation values—compatible with neutrino-mass generation—no comparably simple analytic condition emerges, and the stability must in general be assessed for specific choices of scalar couplings. In the special case where the interaction responsible for neutrino-mass generation vanishes, both electroweak configurations coexist, and the bounded-from-below conditions ensure a definite ordering between them. In this limit, the mass inequalities alone are sufficient to guarantee that the general electroweak vacuum is the global minimum. In the physically relevant regime, the results provide practical sufficient criteria and a systematic framework for assessing vacuum stability in the BNT model.