#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 "Derivative expansion of the exact renormalization group" from 1994 is a follow-up on yesterday's paper, by the same author. Here, he uses the (then) new functional #renormalization group #FRG equations and introduces a certain expansion in momenta, which he then studies for the scalar flip-symmetric model (i.e. the avatar of phi^4 theory). This gives rise to two non-linear differential equations, which can be solved numerically, and produce numerical values of the critical exponents rather close to the correct ones.
It seems to me that this method is surprisingly simple -- solving few differential equations instead of coupled integral equations -- and the author claims repeatedly that it can be systematically improved at will. It didn't entirely become clear to me why he does not do that. After all, computing critical exponents for phi^4 theory is one of the globally accepted benchmarks for methods in field theory and statistics, which would give much credibility to this new method. Perhaps the concrete technical challenges were too big, even if a systematic expansion is possible conceptually.
I would be interested to know if now, 30 years later, this systematic momentum expansion has been continued to higher order, or if it has been replaced by another method.
doi.org/10.1016/0370-2693(94)90767-6