arXiv subjects

Paul-Hermann Balduf

Publications and source records attributed to Paul-Hermann Balduf.

13 recordsLinked to original sources

arXiv2026

Primitive asymptotics in $ϕ^4$ vector theory

A longstanding conjecture in $ϕ^4_4$ theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with $O(N)$ symmetry. For the 0-dimensional case, we calculate the $N$-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above $\approx 25$ loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in $N$ depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer $N$, while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.

hep-th↗

arXiv2025

Renormalized tropical field theory

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.

math-ph↗

arXiv2025

Statistics and asymptotics of subdivergence-free Feynman integrals in $ϕ^4$ theory

Recent algorithmic improvements have made it possible to evaluate subdivergence-free (=primitive=skeleton) Feynman integrals in $ϕ^4$ theory numerically up to 18 loops. By now, all such integrals up to 13 loops and several hundred thousand at higher loop order have been computed. This data enables a statistical analysis of the typical behaviour of Feynman integrals at large loop order. We find that the average value grows exponentially, but the observed growth rate is accurately described by its leading asymptotics only upwards of 25 loops. This is in contrast with the $N$-dependence of the $ON(N)$-symmetric $ϕ^4$ theory, which is close to its large-order asymptotics already around 10 loops. Secondly, the distribution of integrals has a largely continuous inner part but a few extreme outliers. This makes uniform random sampling inefficient. We find that the value of the integral is correlated with many features of the graph, which can be used for importance sampling. With a naive test implementation we obtained an approximately 1000-fold speedup compared with uniform sampling. This suggests that in future work, Feynman amplitudes at large loop order might be computed numerically with statistical methods, rather than through enumerating and evaluating every individual integral.

hep-th↗

arXiv2025

Ladders and rainbows in Minimal Subtraction

In dimensional regularization with $D=D_0-2ε$, the minimal subtraction (MS) scheme is characterized by counterterms that only consist of singular terms in $ε$. We develop a general method to compute the infinite sums of massless ladder or rainbow Feynman integrals in MS at $D_0$. Our method is based on relating the MS-solution to a kinematic solution at a coupling-dependent renormalization point. If the $ε$-dependent Mellin transform of the kernel diagram of the insertions can be computed in closed form, we typically obtain a closed form expression for the all-order solution in MS. As examples, we consider Yukawa theory and $ϕ^4$ theory in $D_0=4$, and $ϕ^3$ theory in $D_0=6$.

hep-th↗

arXiv2025

The Topological form is the Pfaffian form

For a given graph $G$, Budzik, Gaiotto, Kulp, Wang, Williams, Wu, Yu, and the first author studied a ''topological'' differential form $α_G$, which expresses violations of BRST-closedness of a quantum field theory along a single topological direction. In a seemingly unrelated context, Brown, Panzer, and the second author studied a ''Pfaffian'' differential form $ϕ_G$, which is used to construct cohomology classes of the odd commutative graph complex. We give an explicit combinatorial proof that $α_G$ coincides with $ϕ_G$. We also discuss the equivalence of several properties of these forms, which had been established independently for both contexts in previous work.

math-ph↗

arXiv2024

Combinatorial proof of a Non-Renormalization Theorem

We provide a direct combinatorial proof of a Feynman graph identity which implies a wide generalization of a formality theorem by Kontsevich. For a Feynman graph $Γ$, we associate to each vertex a position $x_v \in \mathbb R$ and to each edge $e$ the combination $s_e = a_e^{-\frac 12} \left( x^+_e - x^-_e \right)$, where $x^\pm_e$ are the positions of the two end vertices of $e$, and $a_e$ is a Schwinger parameter. The "topological propagator" $P_e = e^{-s_e^2}\text d s_e$ includes a part proportional to $\text d x_v$ and a part proportional to $\text d a_e$. Integrating the product of all $P_e$ over positions produces a differential form $α_Γ$ in the variables $a_e$. We derive an explicit combinatorial formula for $α_Γ$, and we prove that $α_Γ\wedge α_Γ=0$.

math-ph↗

arXiv2024

Tubings, chord diagrams, and Dyson--Schwinger equations

We give series solutions to single insertion place propagator-type systems of Dyson--Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson--Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and non-integer values of the insertion parameter $s$. We remark on interesting combinatorial connections and properties.

math.CO↗

arXiv2024

Predicting Feynman periods in $ϕ^4$-theory

We present efficient data-driven approaches to predict Feynman periods in $ϕ^4$-theory from properties of the underlying Feynman graphs. We find that the numbers of cuts and cycles determines the period to approximately 2% accuracy. Hepp bound and Martin invariant allow to predict the period with accuracy much better than 1%. In most cases, the period is a multi-linear function of the parameters in question. Besides classical correlation analysis, we also investigate the usefulness of machine-learning algorithms to predict the period. When sufficiently many properties of the graph are used, the period can be predicted with better than 0.05% relative accuracy. We use one of the constructed prediction models for weighted Monte-Carlo sampling of Feynman graphs, and compute the primitive contribution to the beta function of $ϕ^4$-theory at $L\in \left \lbrace 13, 14, 15, 16 \right \rbrace $ loops. Our results confirm the previously known numerical estimates of the primitive beta function and improve their accuracy. Compared to uniform random sampling of graphs, our new algorithm reaches 35-fold higher accuracy in fixed runtime, or requires 1000-fold less runtime to reach a given accuracy. The data set of all periods computed for this work, combined with a previous data set, is made publicly available. Besides the physical application, it could serve as a benchmark for graph-based machine learning algorithms.

hep-th↗

arXiv2023

Statistics of Feynman amplitudes in $ϕ^4$-theory

The amplitude of subdivergence-free logarithmically divergent Feynman graphs in $ϕ^4$-theory in 4 spacetime dimensions is given by a single number, the Feynman period. We numerically compute the periods of 1.3 million completed graphs, this represents more than 33 million graphs contributing to the beta function. Our data set includes all primitive graphs up to 13 loops, and non-complete samples up to 18 loops, with an accuracy of ca. 4 significant digits. We implement all known symmetries of the period in a new computer program and count them up to 14 loops. Combining the symmetries, we discover relations between periods that had been overlooked earlier. All expected symmetries are respected by the numerical values of periods. We examine the distribution of the numerically computed Feynman periods. We confirm the leading asymptotic growth of the average period with growing loop order, up to a factor of 2. At high loop order, a limiting distribution is reached for the amplitudes near the mean. A small class of graphs, most notably the zigzags, grows significantly faster than the mean and causes the limiting distribution to have divergent moments even when normalized to unit mean. We examine the relation between the period and various properties of the underlying graphs. We confirm the strong correlation with the Hepp bound, the Martin invariant, and the number of 6-edge cuts. We find that, on average, the amplitude of planar graphs is significantly larger than that of non-planar graphs, irrespective of $O(N)$ symmetry. We estimate the primitive contribution to the 18-loop beta function of the $O(N)$-symmetric theory. We find that primitive graphs constitute a large part of the beta function in MS for $L\rightarrow \infty$ loops. The relative contribution of planar graphs increases with growing $N$ and decreases with growing loop order $L$.

hep-th↗

arXiv2022

Dyson-Schwinger Equations in Minimal Subtraction

We compare the solutions of one-scale Dyson-Schwinger equations in the Minimal subtraction (MS) scheme to the solutions in kinematic (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counter terms in perturbation theory. As concrete examples, we examine three different one-scale Dyson-Schwinger equations, one based on the D=4 multiedge graph, one for the D=6 multiedge graph and one mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson-Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact non-perturbative solution to one of the non-linear DSEs of the toy model.

hep-th↗

arXiv2021

Propagator-cancelling scalar fields

We examine a large class of scalar quantum field theories where vertices are able to cancel adjacent propagators. These theories are obtained as diffeomorphisms of the field variable of a free field. Their connected correlations functions can be computed efficiently with an algebraic procedure dubbed connected perspective in earlier work. Specifically, (1) We compare the Feynman rules in momentum-space to their position-space counterparts, they agree. This is a a-posteriori justification for the Feynman rules of the connected perspective. (2) We extend the connected perspective to also incorporate counterterm vertices. (3) We examine a specific choice of diffeomorphism that assumes the vertices at different valence proportional to each other. We find that these relations lead to various simplifications. (4) We perturbatively compute the connected 2-point function to all orders for an arbitrary diffeomorphism of a massless field. We thereby give a systematic perturbative derivation of the exponential superpropagator known from the literature. (5) We compute a number of one- and two-loop counterterms. We find that for the specific diffeomorphism, the one-loop counterterms arise from the bare vertices by a non-linear redefinition of the momentum. (6) We show that every diffeomorphism fulfils a set of infinitely many Slavnov-Taylor-like identities which express diffeomorphism-invariance of the S-matrix at loop-level. Finally, we comment on surprising similarities between the structure of propagator-cancelling scalar theories and gauge theories .

math-ph↗

arXiv2019

Perturbation theory of transformed quantum fields

We consider a scalar quantum field $ϕ$ with arbitrary polynomial self-interaction in perturbation theory. If the field variable $ϕ$ is repaced by a local diffeomorphism $ϕ(x) = ρ(x) + a_1 ρ^2(x) +\ldots$, this field $ρ$ obtains infinitely many additional interaction vertices. We show that the $S$-matrix of $ρ$ coincides with the one of $ϕ$ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. If tadpole diagrams vanish, the diffeomorphism can be tuned to cancel all contributions of an underlying $ϕ^s$-type self interaction at one fixed external offshell momentum, rendering $ρ$ a free theory at this momentum. Finally, we propose one way to extend the diffeomorphism to a non-local transformation involving derivatives without spoiling the combinatoric structure of the local diffeomorphism.

math-ph↗

arXiv2017

Triple-gluon and quark-gluon vertex from lattice QCD in Landau gauge

We report on preliminary results for the triple-gluon and the quark-gluon vertex in Landau gauge. Our results are based on two-flavor and quenched lattice QCD calculations for different quark masses, lattice spacings and volumes. We discuss the momentum dependence of some of the vertices's form factors and the deviations from the tree-level form.

hep-lat↗