Smooth-cutoff scalar propagator ($C_{\Lambda }(p)=1/f_{\Lambda }((p^2+m^2)/{\Lambda }^2)$)

A smooth-cutoff scalar propagator is the inverse of a positive regulated quadratic kernel. It agrees with the unregulated scalar propagator at low momentum and decreases rapidly at high momentum. Smooth suppression is not identical to vanishing support. Its cutoff derivative is the line weight in an exact renormalization-group flow.