120. Talk in Munich. Regularizing inflaton correlation functions

Let me get again back from confinement to eternal inflation , or more precisely, to the infrared behavior of correlation functions of a self-interacting massless scalar field on de Sitter background. In what follows, I will consider the case (a QFT in fixed dS spacetime).

As I have explained in the post about leading logs, naive perturbation theory (expanding correlators at small couplings) seems to break down at scales

, (1)

where  is the self-interaction coupling for the scalar field. The scale (1) is actually quite a bit longer than the scale of eternal inflation, and not surprisingly serious doubts appear that eternal inflation may lead to the regularization of the naively diverging perturbation series. Nevertheless, as I will explain below, this is indeed the case, and to understand how this regularization happens, one has to learn how stochastic formalism for eternal inflation works.

1.  Langevin and Fokker-Planck equations in stochastic formalism

The EoM for the Heisenberg operator for the scalar field (neglecting renormalizations) is

. (2)

Also this equation is simple (at least, it is differential), the Schwinger-Dyson equations for the correlation functions are extremely complicated ? they are integro-differential. It would be stupid to act by brute force and construct their solution in terms of expansion in powers of ? this would just give us seeming IR singularity in the correlation functions and breakdown of the perturbation theory at the scale (1). Instead, to understand IR behavior of the correlation functions of the scalar field, let us actually exploit the following peculiar property of the scalar QFT in de Sitter space: as we know, the modes of the scalar field freeze as long as they cross the de Sitter horizon scale.

We will decompose the Heisenberg operator in (2) into IR (superhorizon) and UV (subhorizon) parts as

(3)

where the second term in the r.h.s. satisfies EoM for a free scalar field, i.e.,

and is a small number.

Substituting the decomposition (3) into (2), we find that

(4)

where

(5)

Although (5) is a very complicated operator equation, all the terms in (4) actually commute with each other, since creation and annihilation operators for a single mode with momentum enter (5) in a single combination. We can therefore consider (4) as a classical equation of motion for the classical quantity . On the other hand, we cannot prescribe any particular numeric value to the quantity (5). Calculating its pair correlation function in the Bunch-Davies vacuum, we find

(6)

i.e., the equation (4) is actually the Fokker-Planck equation describing the random Gaussian walk of the quantity in time.

By Stratonovich prescription, we can immediately derive the corresponding Fokker-Planck equation. It has the form:

. (7)

Note that does not enter the Eq. (7), so it is perfectly applicable for a QFT in fixed de Sitter background.

Equations (6) and (7) are the essence of stochastic formalism for eternal inflation developed by A. Starobinsky back in 1980s.

2. Finiteness of inflaton correlation functions in the infrared. Genesis of leading logs

How to calculate inflaton correlation functions using this formalism? Well, having found the distribution function from the Fokker-Planck eq. (7), we can simply write

. (8)

If the distribution function is finite and normalizable everywhere, then the correlation functions (7) remain finite (integral (8) converges).

The solution for is given by

, (9)

where and are eigenfunctions and the eigenvalues of the operator

and .

The operator (9) is supersymmetric, and all its eigenvalues are positive definite (this shows us that  can be indeed interpreted as probability associated with random walk of ).

Therefore, it is almost guaranteed  that the correlation function (8) is finite.

To understand how leading logs appear in stochastic formalism, we need to

a) recall that and

b) expand the exponents in (9) into power series.

3. Why does stochastic formalism actually keep track of IR divergences?

Or even of IR physics? Indeed, we consider correlation functions of the form (8) where all the points where the operators are taken at are the same. Does not it correspond to UV limit instead?

The answer is negative. When we derive the Langevin equation (4), (5) in the stochastic formalism, we actually perform coarse graining with the scale . The physical reason for that is simple: once inflation has ended, we have a causal contact with our  Hubble patch only. All superhorizon modes are frozen and we are unable to distinguish them from the quasiclassical condensate.

On the other handm, studying inflationary perturbations, we are naturally interested in correlation functions

(10)

with x and x’ taken within our Hubble volume. Divergences in perturbative expansion (i.e., in loop corrections to (10)) come from the superhorizon region.

On the other hand, all correlation functions (10) are the same for the stochastic formalism due to coarse graining that we have used to derive it. Thus, stochastic formalism should correctly capture the most interesting IR physics encoded in the correlation functions of the scalar field .

Some relevant papers

1. While it is rather hard to find the original paper by Starobinsky (the one where stochastic formalism is derived) nowadays, a good technical review of the stochastic formalism is given in the paper by Starobinsky and Yokoyama. It is also explained there why stochastic formalism should reproduce leading logs in the perturbative expansion for a self-interacting scalar field on dS background.
2. Relation between leading logs expansion and stochastic formalism is carefully studied in Prokopec, Tsamis and Woodard, see also references therein to earlier Woodard’s papers.
3. I could certainly recommend you to read my own paper about IR divergences
4. If you want a book where stochastic formalism is presented in understandable way, you will find none. The closest to the understandable one though is the book “Particle physics and inflationary cosmology” by A. Linde. At least, you will learn from there how inflationary random walk works physically.

Update: I have actually found the volume where Starobinsky’s paper is published at, it is even rather cheap. I would seriously recommend reading the very first, original paper, it is actually so beautiful and full of ideas that even an experienced physicist will be able to reread it many times without a slightest chance of getting bored.