Soft thermal contributions to 3-loop gauge coupling

We analyze 3-loop contributions to the gauge coupling felt by ultrasoft (“mag-
netostatic”) modes in hot Yang-Mills theory. So-called soft/hard terms, originating from dimension-six operators within the soft effective theory, are shown to cancel 1097/1098 of the IR divergence found in a recent determination of the hard 3-loop contribution to the soft gauge coupling. The remaining 1/1098 originates from ultrasoft/hard contributions, induced by dimension-six operators in the ultrasoft effective theory. Soft 3-loop contributions are likewise computed, and are found to be IR divergent, rendering the ultrasoft gauge coupling non-perturbative at relative order O(α3/2). We elaborate on the implications of these findings for effective theory studies of physical observables in thermal QCD.