Chiral perturbation theory for gradient flow observables

We construct the next-to-leading-order chiral Lagrangian for scalar and pseudoscalar densities defined using the gradient flow, for flow times much smaller than the square of the pion wavelength. We calculate the chiral condensate and the pion decay constant to this order from operators at positive flow time, and confirm results obtained earlier in the chiral limit. We also calculate the quark-mass dependence of the scales t(0) and w(0) defined from the scalar gluon density and find that nonanalytic terms in the quark mass only enter at next-to-next-to-leading order.