UV stability of one-loop radiative corrections in higher-derivative scalar field theory

We consider the theory of a higher-derivative (HD) real scalar field <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>ϕ</a:mi> </a:math> coupled to a complex scalar <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>σ</c:mi> </c:math> , the coupling of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>ϕ</e:mi> </e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>σ</g:mi> </g:math> being given by two types, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>λ</i:mi> <i:mrow> <i:mi>σ</i:mi> <i:mi>ϕ</i:mi> </i:mrow> </i:msub> <i:msup> <i:mi>σ</i:mi> <i:mi>†</i:mi> </i:msup> <i:mi>σ</i:mi> <i:msup> <i:mi>ϕ</i:mi> <i:mn>2</i:mn> </i:msup> </i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>ξ</k:mi> <k:mrow> <k:mi>σ</k:mi> <k:mi>ϕ</k:mi> </k:mrow> </k:msub> <k:msup> <k:mi>σ</k:mi> <k:mi>†</k:mi> </k:msup> <k:mi>σ</k:mi> <k:msup> <k:mrow> <k:mo stretchy="false">(</k:mo> <k:msub> <k:mo>∂</k:mo> <k:mi>μ</k:mi> </k:msub> <k:mi>ϕ</k:mi> <k:mo stretchy="false">)</k:mo> </k:mrow> <k:mn>2</k:mn> </k:msup> </k:math> . We evaluate <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>ϕ</o:mi> </o:math> one-loop corrections <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>δ</q:mi> <q:mi>V</q:mi> <q:mo stretchy="false">(</q:mo> <q:mi>σ</q:mi> <q:mo stretchy="false">)</q:mo> </q:math> to the effective potential of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>σ</u:mi> </u:math> , both the contribution from the positive norm part of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>ϕ</w:mi> </w:math> and that from the (ghost). We show that <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>δ</y:mi> <y:mi>V</y:mi> <y:mo stretchy="false">(</y:mo> <y:msub> <y:mi>σ</y:mi> <y:mrow> <y:mi>cl</y:mi> </y:mrow> </y:msub> <y:mo stretchy="false">)</y:mo> </y:math> at <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:msub> <cb:mi>σ</cb:mi> <cb:mrow> <cb:mi>cl</cb:mi> </cb:mrow> </cb:msub> <cb:mo stretchy="false">→</cb:mo> <cb:mi>∞</cb:mi> </cb:math> , where <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:msub> <fb:mi>σ</fb:mi> <fb:mrow> <fb:mi>cl</fb:mi> </fb:mrow> </fb:msub> </fb:math> is a classical value of <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi>σ</hb:mi> </hb:math> , is positive, implying the stability of <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>δ</jb:mi> <jb:mi>V</jb:mi> <jb:mo stretchy="false">(</jb:mo> <jb:msub> <jb:mi>σ</jb:mi> <jb:mrow> <jb:mi>cl</jb:mi> </jb:mrow> </jb:msub> <jb:mo stretchy="false">)</jb:mo> </jb:math> by the HD one-loop radiative corrections at high energy.