Gradient Properties of Perturbative Multiscalar RG Flows to Six Loops

Abstract

The gradient property of the renormalisation group (RG) flow of multiscalar theories is examined perturbatively in $d=4$ and $d=4-\epsilon$ dimensions. Such theories undergo RG flows in the space of quartic couplings ${\lambda }^I$. Starting at five loops, the relevant vector field that determines the physical RG flow is not the beta function traditionally computed in a minimal subtraction scheme in dimensional regularisation, but a suitable modification of it, the $B$ function. It is found that up to five loops the $B$ vector field is gradient, i.e. $B^I=G^{IJ}\partial A/\partial {\lambda }^J$ with $A$ a scalar and $G_{IJ}$ a rank-two symmetric tensor of the couplings. Up to five loops the beta function is also gradient, but it fails to be so at six loops. The conditions under which the $B$ function (and hence the RG flow) is gradient at six loops are specified, but their verification rests on a separate six-loop computation that remains to be performed.

Comments: 8 pages (two-column), 2 figures, ancillary files are included containing constraints, beta function and additional conventions. v2 discussion expanded including an additional appendix with an explicit solution through two loops, accepted for publication in Physics Letters B