# An SPDE approach to perturbation theory of $\Phi^{4}_{2}$: asymptoticity and
short distance behavior

Hao Shen Department of Mathematics, University of Wisconsin - Madison, USA
<EMAIL_ADDRESS>,  Rongchan Zhu Department of Mathematics, Beijing
Institute of Technology, Beijing 100081, China; Fakultät für Mathematik,
Universität Bielefeld, D-33501 Bielefeld, Germany<EMAIL_ADDRESS>and
Xiangchan Zhu Academy of Mathematics and Systems Science, Chinese Academy of
Sciences, Beijing 100190, China; Fakultät für Mathematik, Universität
Bielefeld, D-33501 Bielefeld, Germany<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the perturbation theory of $\Phi^{4}_{2}$ model on the
whole plane via stochastic quantization. We use integration by parts formula
(i.e. Dyson-Schwinger equations) to generate the perturbative expansion for
the $k$-point correlation functions, and prove bounds on the remainder of the
truncated expansion using PDE estimates; this in particular proves that the
expansion is asymptotic. Furthermore, we derive short distance behaviors of
the $2$-point function and the connected $4$-point function, also via suitable
Dyson-Schwinger equations combined with PDE arguments.

###### Key words and phrases:

###### 2010 Mathematics Subject Classification:

60H15; 35R60

###### Contents

  1. 1 Introduction

##  1\. Introduction

Physicists’ approach to quantum field theory often relies on perturbation
theory, which is a formal series expansion in terms of a certain coupling
constant, see for instance the standard book [zinn, Chapter 7]. The first goal
of this paper is to discuss this perturbative approach and provide some
estimates on the expansion using SPDE methods. We will focus on the
$\lambda\Phi^{4}$ model in two space dimensions, formally given by

$\displaystyle{\mathord{{\rm
d}}}\nu\varpropto\exp(-\frac{\lambda}{4}\int\,\colon\\!\\!\Phi(x)^{4}\\!\colon{\mathord{{\rm
d}}}x){\mathord{{\rm d}}}\mu,$ (1.1)

where ${\mathord{{\rm d}}}\mu$ is the Gaussian free field with mean zero and
variance $\frac{1}{2}(m-\Delta)^{-1}$ for $m>0$. In physics, interesting
quantities such as correlations are calculated by formally Taylor expanding
the exponential density in the coupling constant $\lambda$, and then applying
Wick theorem for Gaussian measures (which yields Feynman diagrams). Jaffe
[jaffe1965] proved that this formal expansion is a divergent series for any
fixed $\lambda>0$.

A toy model to illustrate this divergent perturbation theory is a $\Phi^{4}$
model in “zero dimension”: consider the following function (“partition
function”)

$\mathcal{Z}(\lambda)=\int_{\mathbb{R}}e^{-x^{2}}e^{-\lambda
x^{4}}{\mathord{{\rm d}}}x\;.$

One has $\mathcal{Z}(0)=\sqrt{\pi}$. For $\lambda>0$, one can formally
calculate it by the expansion $e^{-\lambda
x^{4}}=\sum_{n=0}^{\infty}\frac{(-\lambda)^{n}}{n!}x^{4n}$, since there is
exact formula for Gaussian moments, i.e.

$\mathcal{Z}(\lambda)=\sum_{n=0}^{\infty}\int_{\mathbb{R}}\frac{(-\lambda)^{n}}{n!}x^{4n}e^{-x^{2}}{\mathord{{\rm
d}}}x=\sum_{n=0}^{\infty}\frac{(-\lambda)^{n}}{n!}\frac{(4n-1)!!\sqrt{\pi}}{2^{2n}}\;.{\hbox{\rm(1.2)}\kern
5.69046pt\kern-5.69046pt}:toy{=}e1.1{1.2}$