Abstract We consider the energy-dependence of the large-order behavior of forward scattering amplitudes in theories having explicit mass parameters and instanton solutions. We show that in weak coupling theories the exponential part of the Espinosa-Ringwald type cross section can be determined by taking the minimum element of the perturbative series, suggesting that the series is asymptotic. This observation enables us to calculate the nonperturbative cross section by the perturbation of the Borel transform σ (b) about the instanton-anti-instanton singularity. The anomalous cross section in the standard model using the leading energy-dependent σ (b) is presented.