Gradient properties of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>φ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <

The renormalization group flow of the multiscalar interacting <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>φ</a:mi> <a:mn>3</a:mn> </a:msup> </a:math> theory in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>d</c:mi> <c:mo>=</c:mo> <c:mn>6</c:mn> </c:math> dimensions is known to have a gradient structure, in which suitable generalizations of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>β</e:mi> </e:math> functions <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msup> <g:mi>B</g:mi> <g:mi>I</g:mi> </g:msup> </g:math> emerge as the gradient of a scalar function <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>A</i:mi> </i:math> , <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mo>∂</k:mo> <k:mi>I</k:mi> </k:msub> <k:mi>A</k:mi> <k:mo>=</k:mo> <k:msub> <k:mi>T</k:mi> <k:mrow> <k:mi>I</k:mi> <k:mi>J</k:mi> </k:mrow> </k:msub> <k:msup> <k:mi>B</k:mi> <k:mi>J</k:mi> </k:msup> </k:math> , with a nontrivial tensor <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>T</m:mi> <m:mrow> <m:mi>I</m:mi> <m:mi>J</m:mi> </m:mrow> </m:msub> </m:math> in the space of couplings. This has been shown directly to three loops in schemes such as the modified minimal subtraction scheme ( <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mover accent="true"> <o:mi>MS</o:mi> <o:mo stretchy="true">¯</o:mo> </o:mover> </o:math> ) and can be argued in general by identifying <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>A</s:mi> </s:math> with the coefficient of the topological term of the trace anomaly in <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>d</u:mi> <u:mo>=</u:mo> <u:mn>6</u:mn> </u:math> up to a normalization. In this paper we show that the same renormalization group has a gradient structure in <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>d</w:mi> <w:mo>=</w:mo> <w:mn>6</w:mn> <w:mo>−</w:mo> <w:mi>ϵ</w:mi> </w:math> . The requirement of a gradient structure is translated to linear constraints that the coefficients of the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mover accent="true"> <y:mi>MS</y:mi> <y:mo stretchy="true">¯</y:mo> </y:mover> </y:math> <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>β</cb:mi> </cb:math> functions must obey, one of which is new and pertinent only to the extension to <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>d</eb:mi> <eb:mo>≠</eb:mo> <eb:mn>6</eb:mn> </eb:math> .