Third-order perturbation expansion of the two-point correlation function of the dissipative quantum φ4 theory

Abstract

We analytically calculate the perturbation expansion of the two-point correlation function (propagator) of the scalar, dissipative quantum ${\ensuremath{\varphi}}^{4}$ theory up to third order in the interaction parameter $\ensuremath{\lambda}$. The calculations are carried out with two different methods. The first is based on a brute force computation of all interactions appearing in the perturbation expansion term by term at the given order of the perturbation, supported by the use of a symbolic mathematical code. The second uses an efficient perturbation scheme which allows one to hierarchically organize terms within a given order in the perturbation expansion based on the number of occurrences of the free evolution operator ${\mathcal{R}}_{s}^{\mathrm{free}}$; in addition, it allows one to utilize information from already computed lower-order terms in the calculation of higher-order terms. The two methods yield the same result. In the limit of zero dissipation (zero friction coefficient $\ensuremath{\gamma}$), our result coincides with that obtained from the Lagrangian approach in $D=d+1$ dimensions ($d$ denotes the number of space dimensions, while $D$ includes the time dimension) with the use of $D$-dimensional vectors, re-expressed in terms of only $d$-dimensional space integrals by integrating out the zero component. We have also verified that the value of the critical coupling constant ${\ensuremath{\lambda}}^{*}$ as a function of space dimensionality $d$, which can be used to evaluate perturbation expansions at the physical length scale ${\ensuremath{\ell}}_{m}$, comes out correctly from our third-order calculation.