The Negative Running

Positivity bounds constrain effective field theories by demanding that certain Wilson coefficients remain positive — a requirement derived from the analyticity and unitarity of the UV completion. These bounds are among the sharpest theoretical constraints available for gravitational effective theories. But they are stated at a single energy scale.

arXiv:2603.15755 investigates the one-loop renormalization group evolution of the leading operators in effective field theories of shift-symmetric scalars, photons, and gravitons. Certain non-minimal three-point interactions induce a negative running of the corresponding Wilson coefficients, with beta functions suppressed by the Planck scale.

The coefficients decrease toward the infrared. A coefficient that satisfies positivity at the cutoff scale might violate it at low energies. The bound that was supposed to constrain the theory is itself running — and running in the wrong direction.

The resolution: gravitational interactions generate positive infrared contributions that, after smearing over momentum transfer, are argued to dominate over the negative running. But this dominance requires the number of non-minimally coupled particles to be bounded from above according to the species bound. Too many particle species, and the negative running wins.

The result ties together two independent constraints — positivity bounds and the species bound — through renormalization group flow. The species bound is not just about quantum gravity consistency. It is about whether low-energy physics preserves the positivity that the UV completion demands. The number of species in nature is bounded not by gravity alone but by the requirement that gravity’s own quantum corrections do not destroy the consistency conditions they are supposed to satisfy.