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My article together with Kimia Shaban has appeared in JHEP today. We have examined the #statistics of #Feynmangraph s in #QFT , and how they can be exploited to efficiently compute #amplitudes at high loop order.
The article is open access, and the dataset is freely available from my website if you want to explore statistics and correlations yourself. Predicting the values of these Feynman integrals could also be interesting as a test case for #machinelearning
link.springer.com/article/10.1

SpringerLinkPredicting Feynman periods in ϕ4-theory - Journal of High Energy PhysicsWe present efficient data-driven approaches to predict the value of subdivergence-free Feynman integrals (Feynman periods) in ϕ4-theory from properties of the underlying Feynman graphs, based on a statistical examination of almost 2 million graphs. We find that the numbers of cuts and cycles determines the period to better than 2% relative accuracy. Hepp bound and Martin invariant allow for even more accurate predictions. In most cases, the period is a multi-linear function of the properties in question. Furthermore, we 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 ϕ4-theory at L ∈ {13, … , 17} 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 is 1000-times faster to reach a desired accuracy, or reaches 32-fold higher accuracy in fixed runtime.The dataset of all periods computed for this work, combined with a previous dataset, is made publicly available. Besides the physical application, it could serve as a benchmark for graph-based machine learning algorithms.

Any physicists around who can help me understand something?

How should I picture neutrinos in a QFT context? If particles are ripples in their field, and we can detect only a teeny-tiny fraction of neutrinos, what does that say about their field? Is it somehow quiet, and only ripples when we make a chance detection of a neutrino? Is it constantly rippling, but so noisily we can hardly ever detect a signal in it?

Or have I got this all wrong?

(Boosts OK, of course.)

Our #workshop on #combinatorics in fundamental #physics has three topic days, November 26-28.

Day 1: Random Geometry for #Quantum #Gravity

Random geometry is a powerful mathematical framework for studying quantum gravity by modeling it as a statistical physics system, where each Boltzmann configuration corresponds to a spacetime geometry selected from an ensemble with a well-defined probability measure. When considering quantum gravity from a lattice perspective, where spacetime is discrete, the challenge of defining a suitable probability measure becomes a combinatorial problem.

Day 2: #Causal Set Theory

Causal Set Theory is an approach to Quantum Gravity in which spacetime is fundamentally discrete and takes the form of a locally finite partial order, or causal set. The twin questions leading much of the research in this field are: How does the continuum physics of General Relativity arise from an underlying discreteness? And what is the quantum nature of a discrete and dynamical spacetime? #causalset

Day 3: Combinatorics in Perturbative #QFT

A unique feature of quantum field theory is the central role that combinatorics plays: from generating Feynman graphs of scalar models to combinatorial maps and higher graph-like objects of sophisticated frameworks such as matrix-/tensor- and group field theories to renormalization Hopf algebras, and the theory of resurgence and asymptotic power series, to name but a few. The focus of this workshop will be on matrix-/tensor-/group-field theories and graph complexes.

Registration is open for everyone, and of course, free of charge.
indico.mitp.uni-mainz.de/event

MITP (Indico)YOUNGST@RS - Combinatorics in Fundamental PhysicsCombinatorics, a branch of mathematics with powerful applications in modern physics, plays a significant role in fundamental areas such as enumerating Feynman diagrams and addressing discreteness in various approaches to Quantum Gravity. This workshop will highlight connections across different fields, encourage collaboration, and contribute to setting new scientific targets for the community. The workshop is designed to build new bridges across adjacent disciplines, encouraging the exchange...

Question for the #QFT people here: What's your favourite way to explain the effective action Γ[Φ]?

The most frequent explanation I see is "quantum analogue of the classical action", but that doesn't really seem to cut it - after all, classical field theories work completely differently from quantum field theories. Most notably, in classical field theories, you don't have correlation functions - you just have a unique field that describes the current state of your system.

Two years ago, I began writing my #doctoralThesis in theoretical #physics. Most effort went into giving a very detailed pedagogical account of what the #renormalization #HopfAlgebra in #QuantumFieldTheory does, and why it is natural and transparent from a physical perspective.
One year ago, my referees recommended in their reports to publish the thesis as a book, and today I received the printed copies!
It was exciting to go through all the steps of actually publishing a book, and I hope that it will be of use to convince physicists that the Hopf algebra structure in #QFT is not a weird mathematical conundrum, but it actually encodes the very way physicists have been thinking of renormalization since the 1950s: Parametrize a theory by quantities one can actually measure, instead of fictional expansion parameters.
link.springer.com/book/10.1007

I recently discovered an excellent #math article by Krajewski and Martinetti that I had overlooked so far: arxiv.org/abs/0806.4309
Basically, renormalization of #Feynmangraph s in #qft is organized in terms of rooted trees, and so are solutions to differential equations, concatenation of differential operators, numerical integration schemes, and the Hopf algebra of power series. It is intuitively clear that all these things must be closely related, but I wasn't aware that there is this article where the relations are actually spelled out in detail. They also include a derivation of Wigner's semicircle law for Gaussian random matrices in the rooted-tree formalism, which is something I didn't think about at all. Learned another unexpected connection 😀 .

arXiv.orgWilsonian renormalization, differential equations and Hopf algebrasIn this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Then, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally, we work out an analogous construction for the Schwinger-Dyson equations, which yields a bijection between planar $ϕ^{3}$ diagrams and a certain class of decorated rooted trees.

New #physics preprint arxiv.org/abs/2403.16217 together with Kimia Shaban! Computing #Feynmangraphs in #qft is hard, but at least for periods at large loop order, their value can be estimated from properties of the underlying graph with a few percent accuracy. This allows us to identify which (out of millions) of graphs contribute most to physical predictions. We wrote a proof-of-concept Monte Carlo program which allows us to compute the primitive beta function of phi4-theory 1000-times faster than the current best approach (which would be using uniform random samples of graphs). This shows that although at high loop order the number of graphs is enormously large, they are "sufficiently regular" that one can get good estimates of physical observables with methods of #statistics instead of solving each and every Feynman integral.

arXiv.orgPredicting Feynman periods in $ϕ^4$-theoryWe 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.
Continued thread

I have been discussing with German Sierra who has worked on rainbow states theoretically. It's interesting that a quench changes local correlations into a global correlation pattern with extensive mutual information.
arxiv.org/abs/2103.08929

It would be interesting to understand whether the experimental non-equilibrium #QFT dynamics gives rise to also other states that the inhomogeneous models that originally yielded rainbow correlation patterns, for example at finite temperatures the phase diagram of subblock entropy includes a crossover in the scaling type.

For now the experiment yielded only 1 type of a rainbow state.