Infra-red enhanced loops in quadratic gravity

It has been suggested that logarithmically enhanced infra-red loop corrections arising in theories with four derivatives correspond to a physical running of couplings, rendering quadratic gravity asymptotically free. We find that these effects depend on the gauge and on the field parameterisation. We compute physical on-shell amplitudes and find genuine infra-red log-enhanced loop corrections, that are process-dependent and cannot be absorbed into running couplings. As a byproduct, we show that the effective action of quadratic gravity at tree level and without matter fields is just Einstein gravity, so positivity bounds are satisfied despite the presence of ghosts.