Tropical field theory

Preprint (Balduf & Panzer)	2512.21091
Related preprint (Borinsky)	2508.14263
Talk	MathemAmplitudes 2025

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 diagrams,
It is a graph-by-graph bound on Feynman integrals,
It is a surprisingly accurate 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.

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

Large-order behaviour

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 non-trivial 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 perfectly 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 completely 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 $b$ . Curiously, we have observed a very similar behaviour 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 graphs.

Research

Perspectives on Motivation

Large-order behaviour