Resummation of local and non-local scalar self energies via the Schwinger–Dyson equation in de Sitter spacetime

We consider a massless and minimally coupled self interacting quantum scalar field in the inflationary de Sitter spacetime. The scalar potential is taken to be a hybrid of cubic and quartic self interactions, \(V(\phi )= \lambda \phi ^4/4!+\beta \phi ^3/3!\) (\(\lambda >0\)). Compared to the earlier well studied \(\beta =0\) case, the present potential has a rolling down effect due to the \(\phi ^3\) term, along with the usual bounding effect due to the \(\phi ^4\) term. We begin by constructing the Schwinger–Dyson equation for the scalar Feynman propagator up to two loop, at \({\mathcal {O}}(\lambda )\), \({{\mathcal {O}}}(\beta ^2)\), \({{\mathcal {O}}}(\lambda ^2)\) and \({\mathcal {O}}(\lambda \beta ^2)\). Using this equation, we consider first the local part of the scalar self energy and compute the rest mass squared of the scalar field, dynamically generated via the late time non-perturbative secular logarithms, by resumming the daisy-like graphs. The logarithms associated here are sub-leading, compared to those associated with the non-local, leading terms. We also argue that unlike the quartic case, considering merely the one loop results for the purpose of resummation does not give us any sensible result here. We next construct the non-perturbative two particle irreducible effective action up to three loop and derive from it the Schwinger–Dyson equation once again. This equation is satisfied by the non-perturbative Feynman propagator. By series expanding this propagator, the resummed local part of the self energy is shown to yield the same dynamical mass as that of the above. We next use this equation to resum the effect of the non-local part of the scalar self energy in the Feynman propagator, and show that even though the perturbatively corrected propagator shows secular growth at late times, there exists one resummed solution which is vanishing for large spacelike separations, in qualitative agreement with the well known result found via the stochastic formalism.