Mastodawn

#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

Critical Exponents from the Effective Average Action

We compute the critical behaviour of three-dimensional scalar theories using a new exact non-perturbative evolution equation. Our values for the critical exponents agree well with previous precision estimates.

arXiv.org

Paul Balduf May 7

#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

Paul Balduf May 2

#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

Paul Balduf Apr 29

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

Paul Balduf Apr 19

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

Journal of Experimental and Theoretical Physics

Paul Balduf Apr 14

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

Renormalization Group flows between Gaussian Fixed Points

A scalar theory can have many Gaussian (free) fixed points, corresponding to Lagrangians of the form $ϕ\,\Box^kϕ$. We use the non-perturbative RG to study examples of flows between such fixed points. We show that the anomalous dimension changes continuously in such a way that at the endpoints the fields have the correct dimensions of the respective free theories. These models exhibit various pathologies, but are nonetheless interesting as examples of theories that are asymptotically free both in the infrared and in the ultraviolet. Furthermore, they illustrate the fact that a diverging coupling can actually correspond to a free theory.

arXiv.org

Paul Balduf Mar 28

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

Tropical Mathematics

These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical approach'' in mathematics, which has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields. It offers an an elementary introduction to this subject, touching upon Arithmetic, Polynomials, Curves, Phylogenetics and Linear Spaces. Each section ends with a suggestion for further research. The bibliography contains numerousreferences for further reading in this field.

arXiv.org

Paul Balduf Mar 26

#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

Dimensional renorinalization : The number of dimensions as a regularizing parameter - Il Nuovo Cimento B (1971-1996)

We perform an analytic extension of quantum electrodynamics matrix elements as (analytic) functions of the number of dimensions of space(ν). The usual divergences appear as poles forν integer. The renormalization of those matrix elements (forν arbitrary) leads to expressions which are free of ultraviolet divergences forν equal to 4. This shows thatν can be used as an analytic regularizing parameter with, advantages over the usual analytic regularization method. In particular, gauge invariance is mantained for anyν.

SpringerLink

Paul Balduf Mar 10

#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

Paul Balduf Mar 7

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

Calculation of dimensionally regularized box graphs in the zero mass case - Zeitschrift für Physik C Particles and Fields

We evaluate the non trivial one loop graphs for massless particles which are relevant for the QCD description ofe + e − → three jets. Especially we present a method which allows a quick calculation of the imaginary part of box graphs.

SpringerLink