Debye mass of massless ϕ^4-theory to order g^6 at weak coupling

Abstract

We calculate the Debye mass of massless ϕ^4-theory to order g^6 at weak coupling. The contributions to the Debye mass arise from the hard momentum scale of order T and the soft momentum scale of order gT. Effective field theory methods and dimensional reduction are used to separate the contributions from the two momentum scales. The hard contribution can be calculated as a power series in g^2 using naive perturbation theory with bare propagators. The soft contribution is calculated using an effective theory in three dimensions, whose coefficients are power series in g^2. This contribution is a power series in g starting at order g^3. The calculation of the hard part to order g^6. The calculation of the soft part requires calculating the mass parameter in the effective theory to order g^6 and the evaluation of four-loop self-energy diagrams in three dimensions. This gives the Debye mass correct up to order g^6. We discuss the convergence of the perturbative series as well as the loop expansion in three dimensions which implies a selective resummation of higher-order terms.