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Getting a handle on correlation functions

Gernot Eichmann

Abstract

The central objects in a quantum field theory are its n-point correlation functions and matrix elements. Their structure is determined by Lorentz invariance and leads to tensor decompositions whose Lorentz-invariant coefficient functions encode the physics of the process. For growing n, the complexity of these objects may increase considerably and make it challenging to deal with them. Here we give a pedagogical introduction to the topic and provide some tools to manage this complexity, and we will show how symmetries can be used as organizing principles.

Getting a handle on correlation functions

Abstract

The central objects in a quantum field theory are its n-point correlation functions and matrix elements. Their structure is determined by Lorentz invariance and leads to tensor decompositions whose Lorentz-invariant coefficient functions encode the physics of the process. For growing n, the complexity of these objects may increase considerably and make it challenging to deal with them. Here we give a pedagogical introduction to the topic and provide some tools to manage this complexity, and we will show how symmetries can be used as organizing principles.
Paper Structure (2 sections, 7 equations, 2 figures)

This paper contains 2 sections, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Euclidean conventions

Figures (2)

  • Figure S1: (a) A generic $n$-point function has $n$ legs with $n$ momenta, of which only $n-1$ are independent due to momentum conservation. Examples for $n$-point functions and matrix elements: (b) $\pi\pi$ scattering, (c) neutron $\beta$ decay, (d) Higgs-$Z$-boson coupling.
  • Figure S2: (a) Loop integral with a simple pole. (b) Loop integral with multiple poles displaced from the real axis. The orange tracks are the branch cuts arising from the $d^3p$ integration.