  In this paper, we promote the convex cone method of positive bounds from tree
level to loop level. This method is general and can be applied to obtain
leading $s^2$ positivity bounds on the forward scattering process in the
standard model effective field theory. To obtain the loop level bounds, the
original tree level bounds are modified by loop corrections, which involve low
dimensional coefficients. New positivity bounds being valid at one loop level
on the four-Higgs scattering have been provided. We study some specific
ultraviolet models to check the validity of the new bounds. In addition, the
renormalisation group effect on positivity is explored. We point out that as
long as the new bounds are satisfied at the cutoff scale $\Lambda$, they will
also be satisfied at all scales.
