Stochastic Behavior of E ective Field Theories Across Threshold

We explore how the existence of a eld with a heavy mass in uences the low energy dynamics of a quantum eld with a light mass by expounding the stochastic characters of their interactions which take on the form of uctuations in the number of (heavy eld) particles created at the threshold, and dissipation in the dynamics of the light elds, arising from the backreaction of produced heavy particles. We claim that the stochastic nature of e ective eld theories is intrinsic, in that dissipation and uctuations are present both above and below the threshold. Stochasticity builds up exponentially quickly as the heavy threshold is approached from below, becoming dominant once the threshold is crossed. But it also exists below the threshold and is in principle detectable, albeit strongly suppressed at low energies. The results derived here can be used to give a quantitative de nition of the `e ectiveness' of a theory in terms of the relative weight of the deterministic versus the stochastic behavior at di erent energy scales. PACS numbers: 03.70.+k, 11.10.-z, 05.40.+j, -04.62.+v