#paperOfTheDay "Integrating out Gluons in Flow equations" from 1996 is another early article about the functional renormalization group #frg , but this time applied to #QCD. The article is relatively long and contains many technicalities, but the main idea is the following: Like every #quantumFieldTheory , QCD contains "quantum fluctuations" on every energy scale, which can be integrated out from high to low energy with the help of a renormalization group flow equation. Unlike the scalar field theories that are often studied as toy models, QCD contains two fundamentally different types of fields: The fermions (quarks), which represent matter, and the bosons (gluons), which are particles of the strong force. Now it turns out that one can arrange the flow equations in such a way that only one type of field is (at first) integrated out, and serves as an "input" for the flow of the other. In principle, this would be exact and yield a full solution of QCD (which still today would be a breakthrough in #physics ), but in practice of course one has to use truncations and approximations. In fact, the computations presented in the paper are rather "coarse" and don't really produce new results; the point is rather to establish the method.
What is interesting is that here, the gluons are integrated out, and one obtains an effective theory for the interaction of matter. This sounds reasonable, but it is the opposite of how lattice simulations (another well-developed approach at non-perturbative QCD) work: There, the gluon field is being simulated, and the fermions are merely a correction term.
https://arxiv.org/abs/hep-ph/9604227

#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 "Relations between short-range and long-range Ising models" from 2014.
The basic version of the Ising model in #statistical #physics is a lattice where every site contains a binary variable, a "spin" that can point up or down. There is a nearest-neighbour interaction which energetically prefers neighbouring spins to point in the same direction. Then, there are "long range" versions, where the interaction also takes into account spins at larger distance, with a weighting factor that decays with some power law with exponent sigma (where sigma=2 reproduces the conventional short-range model). In this class of models, there are thus two parameters: The dimension d, and the parameter sigma.
The present paper is mostly a numerical Monte Carlo study of various such systems at different d and sigma. The guiding question is whether instead of two, there is actually only one parameter. Phrased differently: Given some d and sigma, can I find some other D such that the short-range (sigma=2) model at this D is equivalent to the long-range one at d? It is intuitively plausible that this works close to the interface between long-range and short-range models, making it slightly long-range is mostly the same as slightly changing dimension. However, the authors demonstrate that such relations are in fact only an approximation, and fail for generic values of sigma and d.
A second finding concerns the structure of the correlation functions of certain long-range models, which decay according to a power law (as expected from the #renormalization group), but where the leading correction is another power law (and not exponentially small).
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.062120
https://arxiv.org/abs/1401.6805

#paperOfTheDay "Über die Eigenkräfte der Elementarteilchen I" from 1933.
This is another paper from the very early days of #quantumFieldTheory , concerned with the question of the seemingly infinite self-energy of the electron in its own electromagnetic field, namely: If the electron is point-like, then its classical electromagnetic field should be infinite at its location, which is clearly nonsense.
The present paper presents a more refined relativistic analysis, starting from the assumption that the locations where the electron "generates" the field and where it "feels" it are distinct by a small vector r. If r is space like (i.e. the two locations differ by a distance that is farther than the distance that light could travel in the same time interval), one recovers the familiar divergence. On the other hand, if r is inside the light cone (i.e. the electron "feels" its own field in its causal future or past), the divergence is absent even in the limit r->0. However, this computation only works for a classical electron in a classical electromagnetic field. Using the Dirac equation for the electron, new obstacles appear.
The present article is typical for the time when #quantum theory was being developed, but it was not at all clear how to interpret it, or whether it was even correct. Schrödinger coined the term "Zitterbewegung" for the intuition of the electron making infinitely fine random jumps at light speed; the present paper mentions this Zitterbewegung as an obvious reason for difficulties in the self-energy. Today, I would say that Zitterbewegung can be an intuitive picture, but the laws of classical #physics are simply not valid at so small scales.
https://link.springer.com/article/10.1007/BF01341363

#paperOfTheDay : "How soon after a zero-temperature quench is the fate of the Ising model sealed?" from 2013.
As is well known, several methods of #quantumFieldTheory and statistical #physics can be used to study the behaviour of systems in equilibrium, and in particular at the critical point. For example, the #Ising model describes a lattice of spin variables, and one can compute critical exponents for the correlation length, assuming that the model has reached a steady state for a fixed temperature.
The present article studies the Ising model, but in a different situation: A (somewhat low) temperature is given, but the model is initialized in a fully random state (which would be the equilibrium state at very high temperature). As the simulation starts, the model moves towards its steady state: Neighbouring spins start to align, and clusters of a certain size are formed. Qualitatively, the size of the clusters in equilibrium is known, but their precise shape and orientation depends on the particular (random) simulation. This information must therefore emerge at some point after the initialization of the simulation. The present paper asks: When? The outcome is that this happens very early, in particular, long before the equilibrium is reached. But also, it's not the first cluster that survives. Instead, several clusters emerge after very few time steps, some disappear, some rearrange, but then one configureation "wins", and for a long time all that happens is that this clustering grows into its final equilibrium shape.
This paper is a nice (full of pictures!) example for properties of the Ising model beyond the usual equilibrium critical exponents story.
https://arxiv.org/abs/1312.1712

#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

The #paperOfTheDay is "The Universe Fan" from 2026. It concerns a very special type of #quantumFieldTheory , describing the cosmic microwave background, that goes by the name "wave function of the universe". The precise #physics application is maybe not so much the point, but rather, that this theory has a curious mathematical feature. Namely, the tree level amplitudes can be described in a nice way.
Recall that tree level amplitudes (i.e. correlation functions of classical field theory) have their name because their Feynman diagrams are trees without closed loops. In position-space, these trees represent integrals over internal vertices, in momentum space they are just products of propagators without any integral. The difficulty is therefore not the individual "Feynman integral", but rather the fact that with many external legs, there can be many trees with different topologies and orientations, which all depend on certain linear combinations of momenta. It is a recurrent theme in quantum field theory that the sum of all Feynman integrals is usually much simpler than one would guess, but still, one usually needs to enumerate all diagrams and compute all their integrals in order to find this sum.
In the present case, there is a shortcut: The authors consider Laplace transforms of the tree amplitudes, so that they are actually integrals. It then turns out that the individual trees correspond to different domains of integration, a so-called fan (i.e. a sum of wedges in coordinate space). Crucially, this fan can be described with a rather simple formula, thus defining the full (tree-level) amplitude without ever generating all tree graphs.
#mathematics
https://arxiv.org/abs/2602.21194

The #paperOfTheDay is a classic of #mathematics : "On the average number of real roots of a random algebraic equation" from 1943. This article examines random polynomials of degree n, which have the form p(x)= c_0 + c_1*x + ... + c_n*x^n, where the coefficients c_j are independent random, following a Gaussian distribution with standard deviation unity and mean zero. Every polynomial of degree n has exactly n complex roots, that is, there are exactly n (not necessarily distinct) complex numbers x_j such that p(x_j)=0. On the other hand, the number of real roots might be smaller or even zero: The polynomial p(x)=x^2+1 does not have any real root.
The question studied in the paper is: For a random polynomial with large degree n, how many roots are real? The outcome is that the mean of this number, for large n, grows like 2/Pi*log(n). Since the total number of roots is n, this implies that for a large polynomial, almost all roots are not real.
This paper is from before the time of computers, so it was impossible at that time to actually generate large random polynomials and determine their roots numerically. Today we can do this. Below, I have added a plot of the mean number of roots for n<400, together with the n->oo asymptotics (orange curve). At n<400, the actual number of roots appears to be slightly larger than the asymptotic, this is not concerning. What is notable, however, is the huge standard deviation: A "typical" random polynomial at n=100 will have anything between 2 and 5 real roots, and at n=400 maybe between 2 and 6 (out of the total 400 complex roots it has!).
https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-49/issue-4/On-the-average-number-of-real-roots-of-a-random/bams/1183505112.full
#statistics