We explain how to tropicalize scalar quantum field theory and show that tropicalized massive scalar quantum field theory is exactly solvable. This exact solution manifests as a non-linear recursion equation fulfilled by the expansion coefficients of the quantum effective action. Geometrically, this recursion computes specific volumes of moduli spaces of metric graphs and is thereby analogous to Mirzakhani's volume recursions on the moduli space of curves. Building on this exact solution, we construct an algorithm that samples points from the moduli space of graphs approximately proportional to their perturbative contribution. Remarkably, this algorithm requires only polynomial time and memory, suggesting that perturbative quantum field theory computations lie in the polynomial-time complexity class, while all known algorithms for evaluating individual Feynman integrals are exponential in time and memory. To demonstrate the capabilities of the algorithm, we evaluate the primitive contribution to the ϕ^4 beta function at 50 loops with a proof-of-concept implementation.

We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for large $n$. That means, if $n \geq g \geq 2$, then there are isomorphisms $H^k(\mathcal{MG}_{g,n};\mathbb{Q})^{{\mathbb S}_n} \rightarrow H^k(\mathcal{MG}_{g,n+1};\mathbb{Q})^{{\mathbb S}_{n+1}}$ for all $k$.

We compute the weight 2 cohomology of the Feynman transforms of the cyclic (co)operads and , and the top -2 weight cohomology of the Feynman transforms of D and . Using a result of Giansiracusa, we compute, in particular, the top -2 weight cohomology of the handlebody group. We compare the result to the top -2 weight cohomology of the moduli space of curves ℳ_g,n , recently computed by Payne and the last-named author. We also provide another proof of a recent result of Hainaut–Petersen identifying the top weight cohomology of the handlebody group with the Kontsevich graph cohomology.

We introduce an algorithm that samples a set of loop momenta distributed as a given Feynman integrand. The algorithm uses the tropical sampling method and can be applied to evaluate phase-space-type integrals efficiently. We provide an implementation, momtrop, and apply it to a series of relevant integrals from the loop-tree duality framework. Compared to naive sampling methods, we observe convergence speedups by factors of more than 10^6.

We find empirically that the value of Feynman integrals follows a log-Γ distribution at large loop order. This result opens up a new avenue towards the large-order behavior in perturbative quantum field theory. Our study of the primitive contribution to the scalar ϕ4 beta function in four dimensions up to 17 loops provides accompanying evidence. Guided by instanton considerations, we discuss the extrapolation of this contribution to all loop orders.

We give a novel combinatorial interpretation for the perturbative series solutions to a class of Dyson-Schwinger equations. We show how binary tubings of rooted trees with labels from an alphabet on the tubes, and where the labels satisfy certain compatibility constraints, can be used to give series solutions to Dyson-Schwinger equations with a single Mellin transform which is the reciprocal of a polynomial with rational roots in a fully combinatorial way. Further, the structure of these tubings leads directly to systems of differential equations for the anomalous dimension that are ideally suited for resurgent analysis. We give a general result in the distinct root case and investigate the effect of repeated roots, which drastically changes the asymptotics and the transseries structure.

We prove an asymptotic formula for the Euler characteristic of Kontsevich's commutative graph complex. This formula implies that the total amount of commutative graph homology grows super-exponentially with the rank and, via a theorem of Chan, Galatius, and Payne, that the dimension of the top weight cohomology of the moduli space of curves, ℳ_g, grows super-exponentially with the genus g.

The software feyntrop for direct numerical evaluation of Feynman integrals is presented. We focus on the underlying combinatorics and polytopal geometries facilitating these methods. Especially matroids, generalized permutohedra and normality are discussed in detail.

We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in Bayesian statistics, such as evaluating marginal likelihood integrals and sampling from posterior distributions. These are based on a tropical sampling method for evaluating Feynman integrals in physics. We here extend that method from projective spaces to arbitrary toric varieties.

The moduli space of rank n graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like −e−1/4(n/e)n/(nlog⁡n)2 as n goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n.

We present a new computer program, $\texttt{feyntrop}$, which uses the tropical geometric approach to evaluate Feynman integrals numerically. In order to apply this approach in the physical regime, we introduce a new parametric representation of Feynman integrals that implements the causal $i\varepsilon$ prescription concretely while retaining projective invariance. $\texttt{feyntrop}$ can efficiently evaluate dimensionally regulated, quasi-finite Feynman integrals, with not too exceptional kinematics in the physical regime, with a relatively large number of propagators and with arbitrarily many kinematic scales. We give a systematic classification of all relevant kinematic regimes, review the necessary mathematical details of the tropical Monte Carlo approach, give fast algorithms to evaluate (deformed) Feynman integrands, describe the usage of $\texttt{feyntrop}$ and discuss many explicit examples of evaluated Feynman integrals.

We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized integration algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.

Flow Oriented Perturbation Theory (FOPT) is a novel approach to Feynman diagrams based on the coordinate (position) space description of Quantum Field Theories (QFT). FOPT offers interesting features regarding the computation of higher-loop Feynman amplitudes such as combinatorial and canonical Feynman rules, explicit infrared singularity factorization on a per-diagram level and the potential to have manifest cancellation of real and virtual singularities. In these proceedings we briefly summarize the derivation of FOPT and present its Feynman rules for covariant diagrams, S-matrix elements and cut diagrams in massless scalar QFT, supported by examples. We then discuss the extension of FOPT to massless fermion fields and indicate steps towards the treatment of massive lines in arbitrary dimensions.

Abstract Feynman periods are Feynman integrals that do not depend on external kinematics. Their computation, which is necessary for many applications of quantum field theory, is greatly facilitated by graphical functions or the equivalent conformal four-point integrals. We describe a set of transformation rules that act on such functions and allow their recursive computation in arbitrary even dimensions. As a concrete example we compute all subdivergence-free Feynman periods in ϕ 3 theory up to six loops and 561 of 607 Feynman periods at seven loops analytically. Our results support the conjectured existence of a coaction structure in quantum field theory and suggest that ϕ 3 and ϕ 4 theory share the same number content.

Graphical functions are special position space Feynman integrals, which can be used to calculate Feynman periods and one- or two-scale processes at high loop orders. With graphical functions, renormalization constants have been calculated to loop orders seven and eight in four-dimensional $\phi^4$ theory and to order five in six-dimensional $\phi^3$ theory. In this article we present the theory of graphical functions in even dimensions $\geq4$ with detailed reviews of known properties and full proofs whenever possible.

This paper explains how to use quantum field theory techniques to find formal power series that encode the virtual Euler characteristics of Out(F_n) and related graph complexes. Finding such power series was a necessary step in the asymptotic analysis of χ(Out(F_n)) carried out in the authors' previous paper.

We perform an all-order resurgence analysis of a quantum field theory renormalon that contributes to an anomalous dimension in six-dimensional scalar ϕ3 theory and is governed by a third-order nonlinear differential equation. We augment the factorially divergent perturbative expansion associated to the renormalon by asymptotic expansions to all instanton orders, in a conjectured and well-tested formula. A distinctive feature of this renormalon singularity is the appearance of logarithmic terms, starting at second-instanton order in the trans-series. To highlight this and to illustrate our methods, we also analyze the trans-series for a closely related second-order nonlinear differential equation that exhibits a similarly resonant structure but lacks logarithmic contributions.

Perturbative expansions in quantum field theory diverge for at least two reasons: the number of Feynman diagrams increases dramatically with the loop number and the process of renormalization may make the contribution of some diagrams large. We give an example of the second problem, from an ultra-violent renormalon of $\phi^3$ theory in 6 dimensions, where we can compute to very high loop-order. Taming this renormalon involves recent work on resurgence. This challenge is much more demanding than the corresponding problem for Yukawa theory in 4 dimensions.

We apply the method of graphical functions that was recently extended to six dimensions for scalar theories, to $\phi^3$ theory and compute the $\beta$ function, the wave function anomalous dimension as well as the mass anomalous dimension in the $\overline{\mbox{MS}}$ scheme to five loops. From the results we derive the corresponding renormalization group functions for the Lee-Yang edge singularity problem and percolation theory. After determining the $\varepsilon$ expansions of the respective critical exponents to $\mathcal{O}(\varepsilon^5)$ we apply recent resummation technology to obtain improved exponent estimates in 3, 4 and 5 dimensions. These compare favourably with estimates from fixed dimension numerical techniques and refine the four loop results. To assist with this comparison we collated a substantial amount of data from numerical techniques which are included in tables for each exponent.

We analyze the asymptotically free massless scalar $\phi^3$ quantum field theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-series solutions which yield the non-perturbative completion of the divergent perturbative solutions to the Kreimer-Connes Hopf-algebraic Dyson-Schwinger equations for the anomalous dimension. This scalar conformal field theory is asymptotically free and has a real Lipatov instanton. In the Hopf-algebraic approach we find a trans-series having an intricate Borel singularity structure, with three distinct but resonant non-perturbative terms, each repeated in an infinite series. These expansions are in terms of the renormalized coupling. The resonant structure leads to powers of logarithmic terms at higher levels of the trans-series, analogous to logarithmic terms arising from interactions between instantons and anti-instantons, but arising from a purely perturbative formalism rather than from a semi-classical analysis.