Tropical field theory

Preprint (Balduf & Panzer)	2512.21091
Related preprint (Borinsky)	2508.14263
Talk notes 2026	What is tropical field theory?
Poster AvHnet 2026	Tropical field theory
Slides Seminar Leipzig, 5.2026	What is tropical field theory?
Slides at Non-perturbative renormalization 2026	Tropical renormalization at 400 loops
Slides at MathemAmplitudes 2025	The beta function in tropical phi^4 theory
Slides CERN-TH QCD seminar 2.2026	Feynman integrals at very many loops

Perspectives on Motivation

There are different perspectives on what tropical field theory is, among them:

It is a combinatorial construction that extends zero-dimensional field theory to include divergences of Feynman integrals,
It is a graph-by-graph bound on Feynman integrals,
It is a structure-preserving numerical approximation for Feynman integrals by rational numbers,
It is a limiting case of long-range field theory where the momentum space propagator and the spacetime dimension jointly go to zero.
Its static version is a Feynman-diagram expansion of the local potential approximation of the functional renormalization group.

More details on this can be found on a separate page.

Large-order behaviour of perturbation series

One of the most prominent long-term open problems in physics is to find non-perturbative solutions to quantum field theories. In typical cases, such solutions are hard because the (renormalized) perturbation series itself does not converge. Hence, one can not “simply” compute more Feynman integrals to obtain an arbitrarily accurate solution. The divergence of the perturbation series is not a mathematical contradiction; there are many reasonable mathematical functions which do not have a convergent power series expansion. The way to understand this is that these functions have a singularity at the origin when viewed as functions of a complex argument, but they might be perfectly fine as a function of a real parameter $g$ (which in physics typically is the coupling), maybe they are even smooth in the limit $g\rightarrow 0^+$ . Think of the functions $\sqrt{g}$ or $e^{-\frac 1 g}$ . The theory of resurgence asserts that such non-polynomial functional forms can be recovered (in not too pathological cases) from studying the large-order growth rate of the perturbation series. Now unfortunately, such data is usually not available for meaningful quantum field theories because one would need to solve Feynman integrals at large loop order.

In tropical field theory, we can compute the exact coefficients of the loop expansion of the quantum effective potential (i.e. the sum of all renormalized 1PI Feynman integrals at zero external momentum) from a partial differential equation. From this quantity, one can in particular read off the renormalization group functions of the theory. We have computed the beta function in the minimal subtraction scheme to 400 loops, the last coefficient is a rational number of over 17,000 decimal digits, which represents the sum over more than $10^{800}$ vertex Feynman diagrams. The power series $\beta(g)$ is factorially divergent. To visualize this, it is useful to consider the ratio of successive terms,

$r_n := \frac{\beta_{n+1}}{n\cdot \beta_n}.$

A short calculation shows that when $\beta_n$ grows like $\beta_n \sim s\cdot a^{-n-b} \cdot \Gamma(n+b)$ , then $r_n$ grows like $r_n \sim \frac 1 a + \frac b a \cdot \frac 1 n$ . Therefore, one can plot $r_n$ as a function of $\frac 1 n$ and it should have a finite limit at $\frac 1 n \rightarrow 0$ , and approach it linearly with slope $\frac b a$ .

The plot below shows $r_n$ for the tropical beta function in the MS scheme up to 400 loops. We observe that indeed the data approaches a finite limit, which can be shown with other methods to be $\frac 1 a=-3$ .

The limiting slope is $b=\frac 5 2$ , however, this slope only becomes visible at very large loop orders, upwards of 50 loops. Such loop orders are entirely beyond reach in ordinary quantum field theories. Conversely, if one only has access to data below 20 loops, those data points suggest a linear slope, too, but with the wrong growth parameters $a$ and $b$ , indicated in red. A very similar behaviour exists in the zero-dimensional theory. If one assumes that a general quantum field theory is rather more complicated, and more ill-behaved, than the simplified tropical field theory, then the conclusion must be that one can not meaningfully determine asymptotic growth rates even if one can compute all 20-loop Feynman integrals.

For factorially divergent series, one is often interested in their Borel transform. Conceptually, the singularities of the Borel transform (as a function of a complex variable) encode the large-order growth rates, and non-perturbative features. However, it is usually impossible to actually “measure” the Borel transform because this requires numerical data at high perturbative order: As we saw above, one should expect to need more than 25 loops, which is way beyond what can be done in any realistic quantum field theory. The tropical theory is an exception: Here we can use the 400-loop data to actually plot the Borel plane. The animations below show the resulting Borel plane plot when the number of perturbative terms is increased. We clearly see how more and more features can be resolved the more terms are included. First, the beta function in the minimal subtraction scheme:

Relatively early, we see a singularity at the location $-\frac 13$ . This is called the (leading) instanton singularity. The location determines the growth rate $(-3)^n$ in the large-order asymptotics, and the exponent of the singularity corresponds to the parameter $b=\frac 52$ of the large-order growth. Including several hundred terms, we can resolve further singularities, located at (approximate) multiples of the leading instanton. This repetition is a typical feature of Borel transforms of functions in non-linear systems. Notice, in particular, that there are no singularities on the positive real axis. This indicates that a unique non-perturbative resummation can be achieved by integrating along the positive real axis.

This is different if we consider instead a kinematic renormalization scheme as shown below:

Here, we see an additional singularity at the location $+\frac 13$ . This is in physics called a renormalon. Discovering the presence or absence of renormalon singularities is a major finding of our work. So far, these effects had been studied in various limits, e.g. by including only a small subset of Feynman diagrams. Our data includes all Feynman diagrams, without any assumption, and we unequivocally see that renormalons are present in the kinematic scheme, and absent in minimal subtraction.

Research

Perspectives on Motivation

Large-order behaviour of perturbation series