115. Talk in Munich. Leading logs

Last time I have claimed that the leading IR divergences in the loop expansion for the inflaton pair correlation function in theory contribute in the form of expansion

.

Let me now show how these divergences appear. First of all, let us forget about the interaction term and show that the correlation function for the free field on dS background diverges in the IR.

We can write

, (1)

where

(2)

are the eigenmodes of the free scalar field on de Sitter background.

Keeping the leading term in the expansion (2) of the Hankel function, we find that the integral (1) diverges logarithmically on both upper and lower limits. The UV cutoff for the integral is given by the scale , where the Hankel function starts to oscillate rapidly. The IR cutoff for (1) is given by the value of scale factor at the very beginning of inflation, when all modes were still subhorizon: 

.

We therefore find

(3)

If you want to see a more accurate derivation, then I can recommend to take a look at the famous book by Andrei Linde. What I am going to talk about next is not written in any book yet

How to estimate behavior of this correlator in the interacting theory in de Sitter background?

Let us take renormalizations into account and write the effective Lagrangian as

,

where and . The equation of motion for the Heisenberg operator is

. (4)

We just considered the r.h.s. of (4) as a source term and want to write the solution of this equation in the symbolic integral form:

, (5)

where is the solution of the EOMs for , i.e., for the case of free field.

The Eq. (5) is called Yang-Feldman equation and we can easily construct its solution by iterations. It is easy to see that the corresponding expansion for the two point correlator has the following general (and again symbolic ) form:

(6)

If we note that the most divergent contribution comes from in the integral in (5) and take (3) into account, we conclude that the small parameter in the expansion (6) is actually instead of .

This expansion is called the leading log expansion. It was first introduced by Woodard as far as I know (see for example this paper and references therein).