Positivity bounds at one-loop level: the Higgs sector

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.