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Convergence Theorem for Non-commutative Feynman Graphs and Renormalization

Iouri Chepelev, Radu Roiban

TL;DR

This work resolves the convergence problem for Feynman graphs in massive non-commutative quantum field theories by formulating and proving a robust convergence theorem that generalizes to arbitrary theories on ${\mathbb R}^d$, capturing how non-commutativity and topology regulate divergences.The authors introduce a topology-aware framework using the genus of embedded surfaces, cycle numbers, and non-planarity indices, along with a refined Schwinger-parametric representation that encodes phase factors via the NC parameter $\Theta$ to control subgraph divergences.They classify divergences into planar and non-planar topologies (Omega, Rings, Com) and demonstrate renormalizability in several models (e.g., Wess-Zumino, $\phi^4$ in $d=2$, $\phi^6$ in $d=2$, and $\phi^*\!\star\!\phi\!\star\!\phi^*\!\star\!\phi$ in $d=4$) with theta-dependent corrections to mass renormalization but theta-independent beta functions.The results provide a rigorous foundation for perturbative NCQFTs, establishing when and how divergences can be subtracted and demonstrating the broader feasibility of renormalization in non-derivative NC theories and certain supersymmetric setups, while outlining limitations in massless cases and more intricate gauge theories.

Abstract

We present a rigorous proof of the convergence theorem for the Feynman graphs in arbitrary massive Euclidean quantum field theories on non-commutative R^d (NQFT). We give a detailed classification of divergent graphs in some massive NQFT and demonstrate the renormalizability of some of these theories.

Convergence Theorem for Non-commutative Feynman Graphs and Renormalization

TL;DR

This work resolves the convergence problem for Feynman graphs in massive non-commutative quantum field theories by formulating and proving a robust convergence theorem that generalizes to arbitrary theories on ${\mathbb R}^d$, capturing how non-commutativity and topology regulate divergences.The authors introduce a topology-aware framework using the genus of embedded surfaces, cycle numbers, and non-planarity indices, along with a refined Schwinger-parametric representation that encodes phase factors via the NC parameter $\Theta$ to control subgraph divergences.They classify divergences into planar and non-planar topologies (Omega, Rings, Com) and demonstrate renormalizability in several models (e.g., Wess-Zumino, $\phi^4$ in $d=2$, $\phi^6$ in $d=2$, and $\phi^*\!\star\!\phi\!\star\!\phi^*\!\star\!\phi$ in $d=4$) with theta-dependent corrections to mass renormalization but theta-independent beta functions.The results provide a rigorous foundation for perturbative NCQFTs, establishing when and how divergences can be subtracted and demonstrating the broader feasibility of renormalization in non-derivative NC theories and certain supersymmetric setups, while outlining limitations in massless cases and more intricate gauge theories.

Abstract

We present a rigorous proof of the convergence theorem for the Feynman graphs in arbitrary massive Euclidean quantum field theories on non-commutative R^d (NQFT). We give a detailed classification of divergent graphs in some massive NQFT and demonstrate the renormalizability of some of these theories.

Paper Structure

This paper contains 18 sections, 18 theorems, 248 equations, 45 figures.

Table of Contents

  1. Introduction
  2. A heuristic proof of the convergence theorem for the non-derivative scalar theories
  3. Statement of theorem and definitions
  4. Renormalization
  5. General discussion.
  6. Wess-Zumino model
  7. $\phi^4$ in $d=2$
  8. $\phi^6$ in $d=2$
  9. $\phi^* \star \phi \star \phi^* \star \phi$ in $d=4$
  10. A comment on non-commutative super Yang-Mills theories (NSYM).
  11. Convergence theorem for non-derivative scalar commutative theories
  12. Convergence theorem for non-commutative theories--Non-derivative case
  13. Outline of the proof
  14. Lemmas and theorems
  15. Convergence theorem for non-commutative theories--General case
  16. ...and 3 more sections

Key Result

Lemma 6.1

If $G$ is an $L$ loop graph and $l$ is one of its lines, then where $X(G, \theta|l)=P(G, \theta)|_{\alpha_l=0}$.

Figures (45)

  • Figure 1: Two-surface with a boundary.
  • Figure 2: Sphere with three holes.
  • Figure 3: Measurement of momentum flow.
  • Figure 4: Measurement of momentum flow along $a$-cycle.
  • Figure 5: Index $j$.
  • ...and 40 more figures

Theorems & Definitions (18)

  • Lemma 6.1
  • Lemma 6.2
  • Lemma 6.3
  • Theorem 1
  • Theorem 2
  • Lemma 6.4
  • Theorem 3
  • Lemma 7.1
  • Lemma 7.2
  • Lemma 7.3
  • ...and 8 more