Renormalized tropical field theory

TL;DR

This work develops tropical field theory as a renormalizable model for quantum field theory by tropicalizing Feynman integrals, focusing on $O(N)$-symmetric $\phi^4_4$ theory. It derives a tropical loop equation and recurrence relations that enable exact, high-order computation of renormalized amplitudes in both MS and MOM schemes, and analyzes their large-order behavior via Borel-plane singularities. The results reveal a clean instanton-driven structure with negative-axis poles and potential Borel summability in MS, while MOM exhibits renormalons on the positive axis and richer scheme-dependent singularities; these differences illuminate the role of renormalization schemes in nonperturbative features. The study also demonstrates strong correlations between tropical and non-tropical amplitudes, providing a robust testing ground for understanding renormalization, asymptotics, and resummation in quantum field theory, with explicit high-loop data for beta functions and critical exponents across $N$.

Abstract

We introduce tropical scalar field theory as a model for renormalizable quantum field theory, and examine in detail the case of quartic self-interaction and internal $O(N)$ symmetry. This model arises in a formally zero-dimensional limit of critical long-range models, but nevertheless its Feynman integrals exhibit strong numerical correlations with the ordinary 4-dimensional theory. The tropical theory retains the full complexity of renormalization with nested and overlapping vertex subdivergences and infinitely many primitive graphs. We compute the perturbation series of the tropical renormalization group functions exactly to 400 loops and study their asymptotic growth. In the minimal subtraction scheme, we find only an arithmetic sequence of singularities on the negative real axis in the Borel plane. These singularities are confluent and imply that the large-order perturbative asymptotics contain logarithmic and fractional power corrections. The absence of any further singularities suggests these series are Borel summable. In contrast, in a kinematic subtraction scheme, the singularity structure on the negative axis changes, and we find additional singularities on the positive real axis.

Figures (10)

• Figure 1: Poles (dots) and zeros (crosses) of the order-200 Padé approximant of the 400-loop tropical beta function in minimal subtraction. The actual singularities are accumulation points of poles, they lie on the negative real Borel axis, at multiples of $u=-\frac{1}{3}$. Scattered individual poles are artefacts. The interpretation of such plots is discussed in e.g. costin_conformal_2021balduf_asymptotic_2026.
• Figure 2: Poles of the order-100 Padé approximant of the 200-loop tropical beta function in a kinematic renormalization scheme . There are still singularities at multiples of $u=-\frac{1}{3}$ on the negative real axis, but in addition there is a singularity on the positive real axis at $u=+\frac{1}{3}$, which is the expected location for the leading renormalon.
• Figure 3: Schematic phase diagram of long-range $\phi^4$ theory and emergence of tropical $\phi^4$ theory as a blow-up of the origin. The blue curve indicates the crossover $\xi_\star = 1-\eta$ (data from onsager_crystal_1944guillou_accurate_1985guillou_accurate_1987kompaniets_minimally_2017). The red dot and arrow indicates the perturbative expansion in $D=D_0-2\epsilon$ dimensions for a fixed $D_0$ and $\xi$. Tropical field theory amounts to a blow-up of the origin, as indicated as a green arc in the left panel. The parameter $\epsilon$ of the perturbative expansion of tropical theory corresponds to the angle of approaching the origin in the non-tropical theory, indicated in orange. $\epsilon=2$ is a vertical approach and reproduces ordinary zero-dimensional $\phi^4$ theory, compare \ref{['sec:PDE_other']}.
• Figure 4: Hepp bound and period of primitive graphs up to 18 loops, double logarithmic plot. Within each loop order, the correlation is very close. Figure taken from balduf_statistics_2023.
• Figure 5: A graph with 1PI subgraphs $\gamma_j$. (a): Removing the edge $e_1$ creates a graph with two bridges $f_1,f_2$. The edge $e_2$ plays no special role and becomes part of a 1PI component $\gamma_1'=\gamma_1\cup e_2$. This choice of edge $e$ induces a cycle with three 1PI components $\left \lbrace \gamma_1,\gamma_2,\gamma_3 \right \rbrace$. (b) For the same graph, choosing $e_2$ as the first edge to remove means that the remaining graph $\gamma=\gamma_1\cup \gamma_2\cup \gamma_3\cup e_1\cup e_2\cup f_1\cup f_2$ is still 1PI. Hence, this choice produces a cycle with only a single 1PI component.

Theorems & Definitions (22)

• Remark 1
• Lemma 2
• Proposition 1
• Proposition 2
• proof
• Theorem 1
• Remark 3
• Remark 4
• Lemma 5: e.g. balduf_primitive_2024
• Proposition 3