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Graphical functions with spin

Oliver Schnetz

TL;DR

This work generalizes graphical functions to theories with spin, introducing spinful propagators, associated partitioning and duality frameworks, and an effective Laplace operator that governs edge appending. It demonstrates how spinful Feynman integrals, especially in Yukawa-$\phi^4$ theory, can be reduced to combinations of scalar integrals via partitioning and algebraic dualities, enabling IBP-free computations up to high loop orders. Key advances include a detailed inversion machinery for the spinful effective Laplace operator, a robust set of graphical-function identities, and the concept of constructible graphs that organizes reductions to scalar primitives. The results establish consistency of the framework, reveal comprehensive identities (including star-triangle variants), and culminate in an eight-loop completed-graph example with explicit multi-zeta values, underscoring the method’s power and potential for exploring the motivic structure of quantum field theory.

Abstract

The theory of graphical functions is generalized from scalar theories to theories with spin, leading to a numerator structure in Feynman integrals. The main part of this article treats the case of positive integer spin, which is obtained from spin $1/2$ theories after the evaluation of $γ$ traces. As an application (in this article used mainly to prove consistency and efficiency of the method), we calculate primitive Feynman integrals in Yukawa-$φ^4$ (Gross-Neveu-Yukawa) theory up to loop order eight.

Graphical functions with spin

TL;DR

This work generalizes graphical functions to theories with spin, introducing spinful propagators, associated partitioning and duality frameworks, and an effective Laplace operator that governs edge appending. It demonstrates how spinful Feynman integrals, especially in Yukawa- theory, can be reduced to combinations of scalar integrals via partitioning and algebraic dualities, enabling IBP-free computations up to high loop orders. Key advances include a detailed inversion machinery for the spinful effective Laplace operator, a robust set of graphical-function identities, and the concept of constructible graphs that organizes reductions to scalar primitives. The results establish consistency of the framework, reveal comprehensive identities (including star-triangle variants), and culminate in an eight-loop completed-graph example with explicit multi-zeta values, underscoring the method’s power and potential for exploring the motivic structure of quantum field theory.

Abstract

The theory of graphical functions is generalized from scalar theories to theories with spin, leading to a numerator structure in Feynman integrals. The main part of this article treats the case of positive integer spin, which is obtained from spin theories after the evaluation of traces. As an application (in this article used mainly to prove consistency and efficiency of the method), we calculate primitive Feynman integrals in Yukawa- (Gross-Neveu-Yukawa) theory up to loop order eight.

Paper Structure

This paper contains 25 sections, 4 theorems, 147 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Propagators
  3. Feynman integrals
  4. One-point integrals
  5. Two-point functions
  6. Three-point functions and the definition of graphical functions
  7. Identities for graphical functions
  8. Edges between external vertices
  9. Permutation of external vertices
  10. Product factorization
  11. Appending an edge
  12. The effective Laplace operator $\space\qed_\lambda^\alpha$
  13. Inverting $\space\qed_\lambda^\alpha$ in the regular case
  14. The log-divergent case
  15. Appending an edge with spin or with weight $\neq1$
  16. ...and 10 more sections

Key Result

Lemma 3

Let $G$ be a graph with an external label $0$ and $N_G=0$. Then, the integral (Pdef) does not depend on the choices of $0$ and $z_1$. That is, $P_G^\alpha$ is well defined and for every $G'$ which is $G$ with a different external label $0$ ($0$ in $G$ is internal in $G'$ and vice versa) we have $N_{ Moreover, $P_G^\alpha=0$ if the total spin $|\alpha|$ of $G$ is odd.

Figures (12)

  • Figure 1: Appending an edge to the vertex $z$ in $G$ gives $G_1$.
  • Figure 2: The asymptotic expansion of graphical functions at $z=0$. Bold lines stand for sets of edges.
  • Figure 3: Constructing the three-star (right) in $D=2\lambda+2$ dimensions by appending an edge to the two-star (left). The weights are as indicated.
  • Figure 4: Integration over $z$ by appending an edge of weight $1$.
  • Figure 5: Example of the graphical reduction of a two-point function in $\phi^4$ theory. The two-point function can be calculated by following the arrows starting from the trivial graphical function which is equal to $1$. The symbol $\Delta^{-1}$ means appending an edge and $\int\mathrm{d} z\mathrm{d}{\overline{z}}$ is integration over $z$.
  • ...and 7 more figures

Theorems & Definitions (26)

  • Example 1
  • Example 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Example 5
  • Example 6
  • Example 7
  • Lemma 8
  • ...and 16 more