#feynmanintegral — Public Fediverse posts

The #paperOfTheDay is "how to resum Feynman graphs" from 2013. In #quantumFieldTheory , #FeynmanIntegral s are the individual constituents of a perturbation expansion with respect to a small coupling parameter. Since the number of graphs grows factorially with their number of vertices, and (in Bosonic theories) these graphs all have the same sign, this perturbation series is divergent.
One can in some cases improve the situation by a Hubbard-Stratonovich transformation, or loop-vertex expansion. This is a rearrangement of the terms of the perturbation series with respect to a different parameter, such as the parameter 1/N (where N is large) in an O(N) symmetric theory. Morally, this amounts to a "perpendicular" summation: Fix a certain order in 1/N, and sum all orders in a small coupling.
The present article goes even further: The Feynman integral of a graph with E many edges consists of E! many "Hepp sectors", each given by a spanning tree of the graph. The proposal is to use these spanning trees as fundamental object, i.e. first fix a tree and compute the sum of all graphs where that tree contributes, and in a second step sum over all trees.
In the paper, this is envisoned somewhat schematically as an alternative way to compute the exact perturbation series. Today, #tropicalFieldTheory morally realizes this approach as a numerical sampling algorithm, where a Hepp sector is constructed as a primary object, and then any of the corresponding graphs is taken as a numerical sample. #physics
https://link.springer.com/article/10.1007/s00023-013-0299-8

The #paperOfTheDay is a classic in #mathematics from 1909: "Der Eulersche Dilogarithmus Und Seine Verallgemeinerungen" by Niels Nielsen. This article concerns generalizations of logarithms, in particular the dilogarithm. Starting from the fundamental identity
log(x) = integral from 1 to x of 1/t dt,
various authors had thought of "generalized logarithms" by integrating another time. One defines the dilogarithm
Li(x) = - integral from 0 to x of log(1-t)/t dt,
but many other generalizations are conceivable, for example why not allow some other rational function in place of 1/t in the integrand? This was answered by Hill and Kummer, who showed that if F and G are arbitrary rational functions, then any integral of F(t) log(G(t)) dt evaluates to a finite linear combination of Li(x) plus logarithms.
Nielsen gives a comprehensive review of known results about dilogarithms, remarking that notation and definitions in the existing literature are inconsistent and unnecessarily complicated.
In the second part of the article, one class of further generalizations is proposed, today known as "Nielsen polylogarithms", essentially integrals of log^(m-1)(t) log^p(1-t) for arbitrary positive integers m and p. These are truly more general than the dilogarithm.
Funnily, Nielsen starts his work by remarking that dilogarithms appear to be a rather unimportant class of functions, compared to elliptic integrals and hypergeometric functions which have plenty application in #physics . Of course, this changed 30 years later with the advent of #QuantumFieldTheory . Today, generalized polylogarithms are the most common class of special functions encountered in #FeynmanIntegral s.
https://katalog.bibliothek.kit.edu/bib/516451

The #paperOfTheDay is "Landau's Leviathans" from 2026. This paper concerns scattering #amplitudes in #QuantumFieldTheory . Theser are computed through #FeynmanIntegral s, and they express the probability of a scattering event as a function of masses and momenta of all particles involved in the process. This functional dependence is very complicated, as can be seen already in much simpler theories: If one e.g. considers an idealized pendulum under the effect of an external oscillation, the behaviour strongly depends on the frequency of the external oscillation. If it coincides with the pendulum's natural ("eigen") frequency, the pendulum will swing strongly, otherwise rather not. A "response function", as a function of external frequency, therefore has a peak at this particular frequency.
In the same way, the scattering amplitude of quantum fields has all sorts of peaks and singularities when certain combinations of the energies of incoming particles match masses of particles. For practical calculations, it is useful to know when that happens. The present draft introduces a new method, based on algebraic geometry, to determine these points. In principle, this "only" amounts to simplifying or solving systems of polynomial equations with rational coefficients, but in practice this is hardly doable (a polynomial in multiple variables can have extremely many independent coefficients). So they do the actual computations over finite fields: Instead of dealing with huge rational numbers and functions, they compute everything modulo a prime so that all numbers fit into standard integer types, repeat this many times, and finally reconstruct the true rational function. #physics https://arxiv.org/abs/2606.29612

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "On the Local Borel transform of perturbation theory" from 2010. This article is about perturbative #QuantumFieldTheory , where one computes the terms of the perturbation series in terms of #FeynmanIntegral s. It is well known that the number of Feynman diagrams grows factorially with the loop order, and also there are individual classes of diagrams whose integral grows factorially. Since this is a growth rate faster than exponentially, it implies that the perturbation series does not converge.
For scalar quantum field theories in 2 or even 3 dimensions, this situation is relatively well under control because the second of the two effects is not present. In 4 dimensions, it is more tricky. It was established in the 1970s that at least the two effects do not conspire to give factorial-squared growth, but still, bounds were relatively crude. The present paper aims to give numerically more tight bounds of the magnitude of correlation functions depending on loop order, number of legs, or external momenta.
Notice that this statement is different from "Borel resummability": The latter asks whether the Borel transform is finite and not too quickly growing on the entire positive line, whereas the present article asks whether it has non-zero radius of convergence in the complex plane (e.g. bounded by singularities on the negative line).

Curiously, none of the existing articles, not even this present one, establish a factorial *lower* bound, i.e. from what is proven, it would be conceivable that the perturbation series is actually convergent. However, all numerical data we have is suggestive of factorial growth.
#physics
https://link.springer.com/article/10.1007/s00220-009-0979-x

#paperOfTheDay is "Complex poles and spectral functions of Landau gauge #QCD and QCD-like theories" from 2020.
In #QuantumFieldTheory , one generally wants to compute n-point correlation functions of fields. Of particular interest is the 2-point function, which can be interpreted as describing how a "particle" of that theory moves. If one uses perturbation theory, the leading order of the 2-point function is the "propagator", the quantity that represents the edges in a #FeynmanIntegral . A typical expectation is that a general, non-perturbative 2-point function should still be similar in nature, namely admit a Källen-Lehmann representation, which is qualitatively "an integral over propagators with different mass, weighted by some spectral density function". The density is supposed to describe density of states of the theory. Conversely, the 2-point function as a function of complex energy should have a branch cut on the positive real line.
The present article considers the 2-point functions in theories roughly resembling QCD, and studies their form for complex energy. They find that in addition to the branch cut, these propagators can have pairs of poles in the complex plane off the real axis. In addition, the spectral "density" function can be negative for low energies (and hence can not actually be interpreted as a density).
These findings are based on various approximations and probably not precisely correct, but similar effects have been seen in other work, too. The interpretation, basically, is "confinement": In low-energy QCD, individual quarks or gluons are not meaningful "particles", so their 2-point function is different from that of a typical particle. #physics https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.074044

#paperOfTheDay is another classic from the early days of #quantumFieldTheory : "A relativistic equation for bound-state problems" by Bethe and Salpeter in 1951. In this article, they derive the equation that now bears their name, an integral equation describing bound states.
The starting point is to use perturbative relativistic quantum field theory in the form of #FeynmanIntegral s (which was still a novelty in #physics at that time). But a Feynman integral describes essentially an "instantaneous" interaction, whereas a bound state between two particles means that these interact for a long time (e.g. to make an atom, nucleus and electrons need to attract each other permanently, as opposed to a scattering process, where they merely interact for a short moment). In principle, the sum over all (infinitely many) Feynman diagrams should give the full solution of the theory, including bound states, but in practice this is impossible to compute.
The Bethe-Salpeter equation is essentially a rearrangement of Feynman diagrams: One introduces an "interaction kernel" (which can be determined e.g. from solving Feynman integrals), and this kernel is then used "infinitely often". Of course, the exact kernel can not be computed with reasonable effort, but the approximation is still much better than using only a few Feynman diagrams. One can also view the Bethe-Salpeter equation as a rearranged version of a Dyson-Schwinger equation: An integral equation whose self-consistent solution represents an infinite sum of Feynman diagrams.
https://journals.aps.org/pr/abstract/10.1103/PhysRev.84.1232

#paperOfTheDay is "Analytic structure of three-point functions from countour deformations" from 2022. This article concerns #FeynmanIntegral s, which are the coefficients of the perturbation series for e.g. scattering amplitudes in #quantumFieldTheory . These integrals are functions of momenta and masses, and for specific values of these arguments, they show non-trivial analytic properties. The basic example is the 1-loop massive propagator correction, which starts having an imaginary part when p^2 >= (2*m)^2. Physically, this threshold is the point where the two "virtual" particles in the loop can actually be physical, i.e. there is enough energy to make their rest masses.
Today, there is an active research community in mathematical physics dealing with such questions from the perspective of algebraic geometry: One uses a representation of Feynman integrals in terms of Schwinger parameters, where the integrand is a rational function, so the singularities are a zero locus of the denominator polynomial (and these types of objects -- algebraic varieties -- are well studied in #mathematics ).
The present article, instead, uses the momentum representation. They discuss the 1-loop 2-point and 3-point scalar diagrams in great detail, and comment on generalizations. The momentum representation is more complicated than the parametric one because the integrand is less structured (the integration variables are vector-valued, and can be negative). On the other hand, it has the advantage that it makes very clear the physical reason for analytic properties, so that this article is also a nice pedagogic introduction to the basic mechanism.
https://arxiv.org/abs/2212.02515

#paperOfTheDay "Critical Exponents from the Effective Average Action" from 1993 is one of the early works of what is now known as the functional renormalization group #frg , called at that time "exact non-perturbative evolution equation". In #quantumFieldTheory and statistical #physics , the behaviour of a system is different for different energy scales. This change is captured by the renormalization group: Changing the energy scale gives back a similar system, but with different numerical values of couplings or masses.
The functional renormalization group equation is a "flow equation" for the quantum effective action. Basically, it expresses the change of all correlation functions under change of energy scale. One can also view it as a successive solution of the path integral, where one starts with the classical (tree-level) action, and successively integrates out high-energy modes, so that , when one reaches zero energy, the full path integral has been performed and one has found the full quantum effective action.
Of course, the functional renormalization group equation can not be solved in closed form for any meaningful theory, so one is forced to introduce approximations. One can recover the usual coupling/loop expansion ( #FeynmanIntegral s), but also other types of approximation schemes are possible, for example including only 1PI correlation functions up to a certain number of legs.
The present paper is concerned with O(N) symmetric scalar fields in D=3 space dimensions. They demonstrate that with a suitable low-order approximation of the flow equations, one can indeed compute the critical exponents of this theory to a few percent accuracy.
https://arxiv.org/abs/hep-ph/9308214

#paperOfTheDay is "Correlation functions and zeros of a Gaussian power series and Pfaffians" from 2013.
This paper is a generalisation of the study of random polynomials: They consider random power series, i.e. polynomials with infinitely many terms. These have (almost always) a radius of convergence of unity, so that it only makes sense to study them in the domain (-1,1). There is an accumulation of zeros (=roots) close to the boundaries of this interval.
Given that the coefficients of the power series are random, so are the locations of zeros. The positive locations form an infinite sequence of random numbers, a point process. As such, one can ask about the mean, variance, and all other correlation functions. The main result of the article is that these quantities are given by a Pfaffian (which is an algebraic object similar to a determinant) of some explicitly known matrices.
I got interested in this observation because Pfaffians also show up in #quantumFieldTheory . For example, Isserlis theorem (sometimes called Wicks theorem by physicists) says that the expectation of a product of Gaussian variables is the Pfaffian of their covariances. Or, Pfaffians show up as the integrands in #FeynmanIntegral s in topological field theories.
#mathematics #probabilityTheory
https://projecteuclid.org/journals/electronic-journal-of-probability/volume-18/issue-none/Correlation-functions-for-zeros-of-a-Gaussian-power-series-and/10.1214/EJP.v18-2545.full

#paperOfTheDay is "Perturbative renormalization and ifrared finiteness in the Wilson renormalization group: the massless scalar case" from 1993. The point of this paper is to prove renormalizability of massless scalar #quantumFieldTheory (which had been known for decades at that point) , but from a new perspective. Namely, instead of discussing the properties of #FeynmanIntegral s, the authors set up a set of integral equations for the quantum effective action in presence of UV and IR momentum cutoffs, and then show that the renormalized versions of these equations stay finite as either of the two cutoffs is removed, thus proving UV and IR finiteness of the renormalized theory.
Notice that this paper appeared in the early 1990s, at the same time as many foundational articles of the #functionalRenormalizationGroup , but the present article uses a custom derivation and a version of functional renormalization group that is not obviously equal to e.g. the Wetterich equation (although, as the authors discuss, it is a version of Polchinski's equation, and my impression is that all these functional renormalization group equations are to some extent equivalent up to changes of variables).
Regardless of whether renormalizability had been known, it is of course very important to check if an emergent new formulation of quantum field theory reproduces this result, or perhaps leads to new insights (or difficulties).
https://www.sciencedirect.com/science/article/abs/pii/S0370157321000156

The #paperOfTheDay : "Nonperturbative study of the fermion propagator in quenched QED in covariant gauges using a renormalizable truncation of the Schwinger-Dyson equation" from 1993 does what the title says.
Concretely, the Dyson-Schwinger equations are an infinite set of integral equations between all correlation functions of a #quantumFieldTheory . These equations are believed to contain all information about the theory in question, but they can not be solved exactly. In practice, one has to first truncate the system to a finite number of equations, and secondly also make some assumptions about the solutions of the remaining equation (e.g. expand in power series or solve numerically to finite accuracy).
One issue with this is that the truncations and assumptions can easily be inconsistent, for example break gauge symmetry or be non-renormalizable. This is easily understood from the perspective of #FeynmanIntegral s: The sum of all integrals has the desired properties, but these rely on identities or cancellations. A random subset, in general, will break the symmetries. The point of the present article is to set up a more consistent truncation than what had been used before. They use it to examine whether the fermion in QED dynamically acquires a mass. The result is that indeed, if the coupling is strong enough, a QED-type theory can produce a mass by itself, even if the input Lagrangian was massless. The threshold is alpha~1, which, if I understand the conventions correctly, is much larger than the physical value, but such things are always tricky because the computation still is only a coarse approximation.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.48.4933

The #paperOfTheDay is "Phase Transition in Uniaxial Ferroelectrics" from 1969. This paper considers a problem in condensed matter #physics (at that time rather called solid state physics): Electrical dipoles in e.g. a 3-dimensional lattice have an interaction that does not just affect adjacent dipoles, but decays like a power law over the entire lattice. This gives rise to a phase transition, and the paper computes the properties of that phase transition, such as the behaviour of the specific heat using methods of field theory such as #FeynmanIntegral s.
From today's perspective, this is all pretty standard, but it should be seen in its historic context: The #renormalization group in its modern form was only discovered in the early 1970s, together with the understanding of universality: A system consisting of many interacting "objects" behaves, close to a critical point, in an "universal" way that depends only on few parameters such as symmetries and dimension. By now, it is widely known that the critical behaviour of any such system can be computed with the methods of another, e.g. one can use perturbative field theory for lattices, or lattice simulations for field theory. The present paper already contains much of this insight, in particular the appendix notes that a generalization to an O(N)-symmetric field theory would be straightforward.
https://jetp.ras.ru/cgi-bin/e/index/e/29/6/p1123?a=list

Monday's #paperOfTheDay is "Renormalization Group flows between Gaussian Fixed Points" from 2022. This preprint concerns scalar #quantumFieldTheory with different choices of the propagator. Conventionally, one has (in a massless theory) a propagator of the form 1/p^2, corresponding to a kinetic term of second derivatives. However, there could be (i.e. it is generated by quantum fluctuations) also 2-point interactions proportional to more derivatives, in particular a fourth derivative. This raises the question whether one can equivalently use that term as the propagator, i.e. assign the value 1/p^4 to edges in #FeynmanIntegral s, and use the other term as an interaction vertex. In principle that works, but it leads to a number of technical issues such as having states with negative norm (ghosts).
The present preprint takes a different perspective: At low energies (consider e.g. plane waves with long wavelength), a fourth derivative will be numerically small, while it dominates at high energy. One can therefore view the transition from one choice of propagator to the other as a #renormalization group flow that starts in the UV with a fourth derivative, and arrives at a second derivative in the IR. An analogous argument has long been known for a mass term (i.e. 2-point term with zero derivatives): In the UV, the kinetic term p^2 determines the behaviour of the field (e.g. UV convergence of Feynman integrals), whereas at low energy, every propagator is essentially constant 1/m^2. Notice that all these transitions are taken at fixed spacetime dimension, whereas #tropicalFieldTheory is an analogous limit to zero derivatives in zero dimensions, which gives a different result.
https://arxiv.org/abs/2207.10596v1

Thursday's #paperOfTheDay is "Tropical Mathematics" from 2009.
I'm currently developing a version of #QuantumFieldTheory called #tropicalFieldTheory . The present article is background on what "tropical" means in #mathematics : This term has first appeared in the context of #computerScience in the 1970s, and it was coined in honor of the early work being done in São Paulo, Brasil. The basic idea is to consider a special type of (mathematical) ring: A typical example of a ring would be the real numbers, together with addition and multiplication. Now, the "tropical semiring" is the real numbers and infinity, but "addition" is replaced by "taking minimum", while "multiplication" is replaced by "addition". This strange object behaves well in many ways. For example, in the usual ring of real numbers one would have
7 + 2*3 = 7+6 = 13
in the tropical semiring, the same equation becomes
min{ 7, 2+3 } = min { 7, 5 }= 5.
The tropical semiring is only SEMI because taking minimum does not always have an inverse: There is no x such that min {x,5}=8 .
In the following decades, tropical arithmetics has been developed into a full mathematical theory. In particular, ome has tropical polynomials, where the conventional addition of monomials is replaced by taking minimums. This is exactly what we do in tropical field theory: The #FeynmanIntegral s are integrals over rational functions, and we replace their denominators and numerators by tropical polynomials.
Today's article was written before tropical field theory, but it discusses a nice application from #biology : One can compute phylogenetic trees with the help of tropical algebraic geometry.
https://arxiv.org/abs/math/0408099

#paperOfTheDay for Wednesday is "Dimensional renormalization: The number of dimensions as a regularizing parameter" from 1972. As the title suggests, this is one of the articles that first introduced dimensional regularization.
In perturbative #QuantumFieldTheory (or statistical physics), one encounters #FeynmanIntegral s which are divergent. These divergences are eventually removed through #renormalization , but in order to even get to that point, one first needs to assign some value to these integrals. This is called regularization. Various methods of regularization are known, but the typical problem is that they destroy symmetries of the theory. Dimensional regularization was a breakthrough for practical computation of Feynman integrals because it respects many symmetries.
The basic idea is to define an integral for non-integer dimension of spacetime. This is done, essentially, by analytic continuation: We know what it means to take a first, second, third etc derivative of a function, and to integrate it once, twice, thrice etc. If the function is spherically symmetric (i.e. depends only on the radius of spherical coordinates), then the "count" of the integrals or derivatives appears as an explicit number in intermediate steps. For example, the volume element in 3 dimensional spherical coordinates is r^2*dr*(angular part), where the exponent "2" represents dimension D=2+1=3. Basically, you could insert any number in place of the "2", and declare this to be the D-dimensional integral. Of course, in reality this is more sophisticated, but the basic idea is very much in this spirit.
https://link.springer.com/article/10.1007/BF02895558

#paperOfTheDay : "#Renormalization of a scalar field theory in strong coupling" from 1972.
Recall that phi^6 theory in 3 dimensions is a perturbatively renormalizable scalar #QuantumFieldTheory model, and in perturbation theory (using #FeynmanIntegral s), one expects there to be counterterms for the phi^6, phi^4, and phi^2 interactions to accommodate their anomalous scale dependence. In the present paper, Wilson uses a different approach, and introduces an approximation scheme for the quantum effective action, which is not inherently related to conventional perturbation theory. In this, he finds that the so-approximated model only acquires anomalous flow for phi^2, but not for phi^4 and phi^6. The approximation is relatively coarse, so one should not take this as a "solution" of phi^6 theory, but rather as a concrete example of what could in principle happen in a strongly coupled interacting field theory.
Note that with such results, there is no contradiction with perturbation theory: Low-order perturbation theory describes a behaviour very close to a free theory, but perturbation series are divergent and asymptotic. This means that the true functional form only emerges after resummation, and is in general very different from "inserting a large number for the coupling into the low-order perturbation series".
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.6.419

My #paperOfTheDay was "Calculation of dimensionally regularized box graphs in the zero mass case" from 1979. Perturbative #QuantumFieldTheory is based on power series whose coefficients are sums of #FeynmanIntegral s. Each of these integrals is graphically depicted by a graph. The present paper is concerned with a 1-loop "box" integral, that is, a cycle graph with 4 external edges. Its solution is a dilogarithm, which is obtained here by using a version of Schwinger parameters. By now, this result is widely known and well understood, and it is often used as an example, or as a test case for novel methods.
https://link.springer.com/article/10.1007/BF01577398

#paperOfTheDay for my #dailyPaperChallenge on Wednesday: "Graphical functions and single-valued multiple polylogarithms" from 2013. This is one of the foundational articles for the theory of graphical functions, a framework to compute a certain class of #FeynmanIntegral s in #physics They work for massless integrals that depend on two kinematical parameters (i.e. 3-point functions or confomral 4-point functions), and the key is to interpret these two parameters as a single complex number, and then use methods of complex analysis. Graphical functions are by far the most powerful method for computing such Feynman integrals, recently for example they are being used for the beta function of phi^4 theory at 8 loops. The paper is rather long, the section about single-valued multiple polylogarithms is actually a separate thing that isn't too relevant for graphical functions as such. https://arxiv.org/abs/1302.6445

For my #dailyPaperChallenge , today I read "Rationalizability of square roots". This paper is about a problem one often faces in #FeynmanIntegral s in theoretical #physics: The integrand might be mostly rational, but contain a square root of some rational function R in several variables. Integrating square roots is not nice, so the idea is: can I find a rational change of variables such that R=P^2, where P is a rational function? If that is possible, the square root of R is simply replaced by P and the integral becomes much easier. If the rational functions are in only one variable, the answer is relatively simple, it works when the degree of the square-free factor is at most 2. For functions in several variables, this is increasingly more complicated and can be tackled with methods of algebraic geometry. https://doi.org/10.1016/j.jsc.2020.12.002

On 10-12 November, there is the online workshop "loop-the-loop-2" about #FeynmanIntegral calculus and its applications in gravity and particle physics. This could be a good way to catch up with latest developments in the area if you're a PhD student in #physics or #maths. Registration is still open.
https://indico.dfa.unipd.it/event/1569/overview

Currently, I'm working on a problem in #quantum field theory where we use #FeynmanIntegral s. These integrals are depicted by graphs, and they can be divergent when a graph has too many edges for a given number of vertices. The task is to identify all subgraphs that are divergent. This is a coproduct: It produces multiple terms, and each term is a list of 2 elements. The first element is one or multiple divergent subgraphs, and the second element is the remainder. It is surprising how many terms the coproduct has even for small graphs. For my example, even if the red graph is rather small, there are already 15 combinations of divergent subgraphs. To compute a physically sensible result, one needs to sum over all original graphs, and subtract all these combinations of subgraphs. #physics #research

At the #CAP #Physics Congress and the Theory Canada meeting, I gave two talks about the statistical distribution of #FeynmanIntegral s and how their correlations can be used for efficient sampling at high loop order. The slides are now available from my website!
https://paulbalduf.com/research/statistics-periods/

In #QuantumFieldTheory, scattering amplitudes can be computed as sums of (very many) #FeynmanIntegral s. They contribute differently much, with most integrals contributing near the average (scaled to 1.0 in the plots), but a "long tail" of integrals that are larger by a significant factor.
We looked at patterns in these distributions, and one particularly striking one is that if instead of the Feynman integral P itself, you consider 1 divided by root of P, the distribution is almost Gaussian! To my knowledge, this is the first time anything like this has been observed. We only looked at one quantum field theory, the "phi^4 theory in 4 dimensions". It would be interesting to see if this is coincidence for this particular theory and class of Feynman integrals, or if it persists universally.
More background and relevant papers at https://paulbalduf.com/research/statistics-periods/
#quantum #physics #statistics