Yukawa scalar self energy at two loop and $\langle \phi^2 \rangle$ in the inflationary de Sitter spacetime

Abstract

We have considered the Yukawa theory at two loop in the inflationary de Sitter spacetime, for a massless minimally coupled scalar and a massless fermion. The one loop computation for the same has been investigated in detail in the earlier literatures. The chief motivation behind this study is the fact that at one loop, the scalar self energy contains only fermions, which are conformally invariant. At two loop, there are two diagrams, each containing one internal scalar line, thereby breaking the conformal invariance. This should result in the appearance of infrared secular logarithms in the scalar self energy. The renormalisation of this two loop self energy has been performed. We next compute the loop corrected coincident two point correlation function, $\langle \phi^2\rangle$, due to the self energies. The expectation value has been taken with respect to the initial Bunch-Davies vacuum. We argue that the late time secular contribution from the local or UV self energy must dominate the non-local or IR ones in the present case, from the point of view of the powers of these large logarithms of the scale factor. This corresponds to the fact that fermion lines do not show any IR secular effect. The leading behaviour of $\langle \phi^2\rangle$ at one and two loop are respectively found to be $\ln^3 a$ and $\ln^4 a$. These are hybrids of UV and IR logarithms, where the latter originate from the massless and minimal two external scalar lines. A resummed expression for $\langle \phi^2\rangle$ has also been computed. The same is found to be bounded and decreasing monotonically with the increasing magnitude of the Yukawa coupling. Accordingly, the dynamically generated scalar mass increases with the increasing coupling.