Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots

TL;DR

The paper develops a framework to extract non-zeta transcendental counterterms in large-$N$ quantum field theories by linking diagram dressings to positive knots. It solves a master two-loop recurrence via Saalschützian ${}_3F_2$ series, enabling all-orders ${ m O}(1/N^3)$ anomalous dimensions and revealing a new zeta-reducible structure. By employing a wreath-product symmetry and knot-theoretic reasoning, the authors enumerate irreducible Euler sums up to level 12, identifying explicit contributions from knots like $8_{19}$, $10_{124}$, and $11_{353}$ and predicting higher-order patterns. They compute high-precision $oldsymbol{\\varepsilon}$-expansions and apply Padé resummation across several models (bosonic sigma-model, phi^4 theory, Gross-Neveu, SUSY sigma-model), achieving accurate 3D estimates and illustrating the deep interplay between renormalization, knot theory, and number theory.

Abstract

Counterterms that are not reducible to $ζ_{n}$ are generated by ${}_3F_2$ hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots $(4,3)=8_{19}$ and $(5,3)=10_{124}$, are found in anomalous dimensions at ${\rm O}(1/N^3)$ in the large-$N$ limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Padé resummations of $\varepsilon$-expansions, which are compared with analytical results in 3 dimensions. The ${\rm O}(1/N^3)$ results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in $1/N$ one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots $10_{139}$ and $10_{152}$ are not generated by such self-energy insertions.

Figures (8)

• Figure 1: A 6-loop graph for the coupling of $\phi^4$ theory, giving a non-zeta counterterm associated with the 8-crossing positive 3-braid knot, $8_{19}$.
• Figure 2: The two-loop two-point function (a) is obtained by cutting the log-divergent tetrahedral vacuum diagram (b) on the line with index $\alpha_6=3d/2-\sum_{n=1}^5\alpha_n$. The index $\alpha_{10}=\alpha_1+\alpha_2+\alpha_3-d/2$ is associated with the vertex where lines 1,2,3 meet.
• Figure 3: Generation of $8_{19}= \sigma_1^{}\sigma_2^3\sigma_1^{}\sigma_2^3$ by self-energy insertions.
• Figure 4: Generation of $10_{124}= \sigma_1^{}\sigma_2^5\sigma_1^{}\sigma_2^3$ by self-energy insertions.
• Figure 5: Generation of the 12-crossing 3-braid $\sigma_1^{}\sigma_2^7\sigma_1^{}\sigma_2^3$ by self-energy insertions.