Positivity with long-range interactions

A bstract We introduce infrared finite, analytic, crossing symmetric, Regge behaved, and Lorentz invariant amplitudes $$ {\mathcal{M}}_{\mathcal{E}} $$ M E , labeled by the experimental energy resolution $$ \mathcal{E} $$ E for detecting soft photons and gravitons. For $$ \mathcal{E} $$ E exponentially smaller than any hard scale, they also satisfy unitarity and their associated cross sections reproduce the inclusive, infrared-finite cross sections of ordinary amplitudes. These properties make $$ {\mathcal{M}}_{\mathcal{E}} $$ M E suitable for deriving infrared-safe positivity bounds on effective field theories in the presence of long-range forces even in D = 4. As an illustration, we present explicit bounds in the low-energy theory of pions coupled to electromagnetism and gravity.