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

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