The renormalization group flow of the multiscalar interacting ${\phi }^3$ theory in $d=6$ dimensions is known to have a gradient structure, in which suitable generalizations of the $\beta$ functions $B^I$ emerge as the gradient of a scalar function $A$, ${\partial }_{I}A=T_{IJ}{B}^J$, with a nontrivial tensor $T_{IJ}$ in the space of couplings. This has been shown directly to three loops in schemes such as the modified minimal subtraction scheme ($\overline{\mathrm{MS}}$) and can be argued in general by identifying $A$ with the coefficient of the topological term of the trace anomaly in $d=6$ up to a normalization. In this paper we show that the same renormalization group has a gradient structure in $d=6-ϵ$. The requirement of a gradient structure is translated to linear constraints that the coefficients of the $\overline{\mathrm{MS}}$ $\beta$ functions must obey, one of which is new and pertinent only to the extension to $d\ne 6$.

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• Anomalies
• Conformal field theory
• Renormalization group