Imaginary scaling invariance of the one-loop effective potential

Recently, a hitherto unknown class of renormalization group stable relations between parameters of bosonic field theories has been identified and dubbed as the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>r</a:mi> <a:mn>0</a:mn> </a:msub> </a:math> or “GOOFy” symmetries. Here, one-loop properties of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>r</c:mi> <c:mn>0</c:mn> </c:msub> </c:math> invariant two-Higgs doublet model and a minimal symmetric model are discussed. It is concluded that the symmetry is present at the one-loop provided the UV cutoff squared transforms nontrivially under <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>r</e:mi> <e:mn>0</e:mn> </e:msub> </e:math> . The minimal model requires the presence of two real fields.