Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A beginner's guide to functional methods in particle physics

Markus Q. Huber

TL;DR

This article serves as a beginner’s guide to functional methods in particle physics, outlining the Dyson–Schwinger equations, $n$PI effective actions, bound-state equations, and the functional renormalization group as nonperturbative continuum tools that require truncations for practical use. It presents a practical workflow from truncation choice to observable predictions, culminating in a complete glueball-spectrum calculation in pure Yang–Mills theory to illustrate the approach. Key contributions include a structured overview of truncation schemes, numerical techniques, and the derivation of bound-state kernels from $n$PI actions, along with a concrete three-loop truncation that yields results in good agreement with lattice QCD. The work underscores the complementarities between continuum functional methods and lattice simulations, their handling of gauge theories, and the ongoing development toward fully self-contained calculations with broad applicability in hadron physics and beyond.

Abstract

Functional methods like Dyson-Schwinger equations, the nPI effective action formalism, bound state equations and the functional renormalization group are versatile tools to study quantum field theories. They are exact, nonperturbative equations but have to be truncated for practical calculations. After a general introduction, I focus on their use in particle physics and discuss common truncations and solution techniques. The complete process from choosing a truncation to calculating observable quantities is exemplified by means of the glueball spectrum.

A beginner's guide to functional methods in particle physics

TL;DR

This article serves as a beginner’s guide to functional methods in particle physics, outlining the Dyson–Schwinger equations, PI effective actions, bound-state equations, and the functional renormalization group as nonperturbative continuum tools that require truncations for practical use. It presents a practical workflow from truncation choice to observable predictions, culminating in a complete glueball-spectrum calculation in pure Yang–Mills theory to illustrate the approach. Key contributions include a structured overview of truncation schemes, numerical techniques, and the derivation of bound-state kernels from PI actions, along with a concrete three-loop truncation that yields results in good agreement with lattice QCD. The work underscores the complementarities between continuum functional methods and lattice simulations, their handling of gauge theories, and the ongoing development toward fully self-contained calculations with broad applicability in hadron physics and beyond.

Abstract

Functional methods like Dyson-Schwinger equations, the nPI effective action formalism, bound state equations and the functional renormalization group are versatile tools to study quantum field theories. They are exact, nonperturbative equations but have to be truncated for practical calculations. After a general introduction, I focus on their use in particle physics and discuss common truncations and solution techniques. The complete process from choosing a truncation to calculating observable quantities is exemplified by means of the glueball spectrum.
Paper Structure (25 sections, 63 equations, 16 figures, 1 table)

This paper contains 25 sections, 63 equations, 16 figures, 1 table.

Table of Contents

  1. A beginner's guide to functional methods in particle physics
  2. Introduction
  3. Functional equations
  4. Quantum field theory background
  5. Equations of motion
  6. Dyson-Schwinger equations
  7. $n$PI effective actions and their equations of motion
  8. Bound state equations
  9. Functional renormalization group
  10. Further functional equations
  11. Working with functional equations
  12. Truncations
  13. Vertex expansion
  14. Loop expansion
  15. Derivative expansion
  16. ...and 10 more sections

Figures (16)

  • Figure 1: Examples for different diagram types: 1PI (left), connected but one-particle reducible (middle), and disconnected (right).
  • Figure 2: Differentiation rules: derivatives of a field (left), a propagator (middle), and a vertex (right). Disks represent (here field-dependent) vertices or propagators, crosses represent fields.
  • Figure 3: Relations between different generating functionals and related quantities. LT$_A$, $A\in\{\Phi,D,\Gamma^{(3)}\}$ denotes the Legendre transformation with respect to $A$.
  • Figure 4: Graphical representation of Eq. (\ref{['eq:dse_1deriv']}). Internal propagators are fully dressed. Crosses correspond to fields $\Phi$. Dots represent bare and disks dressed vertices.
  • Figure 5: The DSE for the propagator of the scalar theory from Eq. (\ref{['eq:scalarLagrangian']}).
  • ...and 11 more figures