Wilsonian renormalization group and thermal field theory in the Schwinger-Keldysh closed-time-path formalism

We study perturbative Wilsonian renormalization group (RG) for the scalar <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>ϕ</a:mi><a:mn>4</a:mn></a:msup></a:math> theory at finite temperature to one loop order in the Schwinger-Keldysh closed-time-path (CTP) formalism. By explicitly integrating out the UV modes, we show how effective interactions coupling the two branches of the CTP contour arise. This provides a microscopic derivation of the type of effective CTP actions used in the literature. While the full effective theory has no critical points, we instead proceed to study critical properties of the RG flow in a reduced space of complex couplings that characterize certain time-ordered correlators and show the existence of two novel fixed points in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi><c:mo>=</c:mo><c:mn>4</c:mn></c:math> spacetime dimensions. We relate one to the Wilson-Fisher fixed point known to exist only in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>4</e:mn><e:mo>−</e:mo><e:mi>ε</e:mi></e:math> dimensions, and the other to the standard Gaussian fixed point.