  The dispersion relation of an elastic 4-point amplitude in the forward
direction leads to a sum rule that connects the low energy amplitude to the
high energy observables. We perform a classification of these sum rules based
on massless helicity amplitudes. With this classification, we are able to
systematically write down the sum rules for the dimension-6 operators of the
Standard Model Effective Field Theory (SMEFT), some of which are absent in
previous literatures. These sum rules offer distinct insights on the relations
between the operator coefficients in the EFT and the properties of the full
theory that generates them. Their applicability goes beyond tree level, and in
some cases can be used as a practical method of computing the one loop
contributions to low energy observables. They also provide an interesting
perspective for understanding the custodial symmetries of the SM Higgs and
fermion sectors.
