  The gradient property of the renormalisation group (RG) flow of multiscalar
theories is examined perturbatively in $d=4$ and $d=4-\varepsilon$ 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.
