Wilsonian Renormalization Group Research Articles

Emergence of inflaton potential from asymptotically safe gravity

The Asymptotic Safety Hypothesis for gravity relies on the existence of an interacting fixed point of the Wilsonian renormalization group flow, which controls the microscopic dynamics, and provides a UV completion of the theory. Connecting such UV completion to observable physics has become an active area of research in the last decades. In this work we show such connection within the framework of scalar-tensor models. More specifically, we found that cosmological inflation naturally emerges from the integration of the RG flow equations, and that the predicted parameters of the emergent effective potentials provide a slow-roll model of inflation compatible with current observations. Furthermore, the RG evolution of the effective action starting at the UV fixed point, provides a prediction for the initial value of the inflaton field.

Harvesting physical predictions from asymptotically safe quantum field theories

Asymptotic safety is a powerful mechanism for obtaining a consistent and predictive quantum field theory beyond the realm of perturbation theory. It hinges on an interacting fixed point of the Wilsonian renormalization group flow which controls the microscopic dynamics. Connecting the fixed point to observations requires constructing the set of effective actions compatible with this microscopic dynamics. Technically, this information is stored in the UV-critical surface of the fixed point. In this work, we describe a novel approach for extracting this information based on analytical and pseudospectral methods. Our construction is illustrated at the level of the three-dimensional Ising model and easily generalizes to any asymptotically safe quantum field theory. It also constitutes an important step toward setting up a well-founded swampland program within the gravitational asymptotic safety program. Published by the American Physical Society 2024

Euclidean effective theory for partons in the spirit of Steven Weinberg

The standard formulation of parton physics involves light-cone correlations of quark and gluon fields in a hadron, which leads to a widespread impression that it can only be studied through real-time Hamiltonian dynamics or light-front quantization, which are challenged by non-perturbative computations with a pertinent regulator for light-cone/rapidity divergences (or zero modes). As such, standard lattice QCD studies have been limited to indirect parton observables such as first few moments and short-distance correlations, which do not provide the x-distributions without solving the model-dependent inverse problem. Here I describe an alternative formulation of partons in terms of equal-time (or Euclidean) correlators, which allows to compute precision-controlled x-distribution through lattice QCD simulations. This approach is in accord with Weinberg's pioneering idea of effective field theory as well as Wilson's renormalization group, in which the large hadron momentum serves as a natural cut-off for light-cone/rapidity divergences and can ultimately be eliminated through a method like the “perfect action” program in lattice QCD.

Laplacian renormalization group: an introduction to heterogeneous coarse-graining

The renormalization group (RG) constitutes a fundamental framework in modern theoretical physics. It allows the study of many systems showing states with large-scale correlations and their classification into a relatively small set of universality classes. The RG is the most powerful tool for investigating organizational scales within dynamic systems. However, the application of RG techniques to complex networks has presented significant challenges, primarily due to the intricate interplay of correlations on multiple scales. Existing approaches have relied on hypotheses involving hidden geometries and based on embedding complex networks into hidden metric spaces. Here, we present a practical overview of the recently introduced Laplacian RG (LRG) for heterogeneous networks. First, we present a brief overview that justifies the use of the Laplacian as a natural extension of well-known field theories to analyze spatial disorder. We then draw an analogy to traditional real-space RG procedures, explaining how the LRG generalizes the concept of ‘Kadanoff supernodes’ as block nodes that span multiple scales. These supernodes help mitigate the effects of cross-scale correlations due to small-world properties. Additionally, we rigorously define the LRG procedure in momentum space in the spirit of the Wilson RG. Finally, we show different analyses for the evolution of network properties along the LRG flow following structural changes when the network is properly reduced.

Arbeitsgemeinschaft: QFT and Stochastic PDEs

Quantum field theory (QFT) is a fundamental framework for a wide range of phenomena is physics. The link between QFT and SPDE was first observed by the physicists Parisi and Wu (1981), known as Stochastic Quantisation. The study of solution theories and properties of solutions to these SPDEs derived from the Stochastic Quantisation procedure has stimulated substantial progress of the solution theory of singular SPDE, especially the invention of the theories of regularity structures and paracontrolled distributions in the last decade. Moreover, Stochastic Quantisation allows us to bring in more tools including PDE and stochastic analysis to study QFT.This Arbeitsgemeinschaft starts by covering some background material and then explores some of the advances made in recent years. The focus of this Arbeitsgemeinschaft is QFT models such as the \Phi^{4} , sine-Gordon and Yang–Mills models as examples to discuss stochastic quantisation and SPDE methods and their applications in these models. We introduce the key ideas, results and applications of regularity structure and paracontrolled distributions, construction of solutions of the SPDEs corresponding to these models, and use the PDE method to study some qualitative behaviors of these QFTs, and connections with the corresponding lattice or statistical physical models. We also discuss some other topics of QFT, such as Wilsonian renormalisation group, log-Sobolev inequalities and their implications, and various connections between these topics and SPDEs.

Beyond Wilson? Carroll from current deformations

At extreme energies, both low and high, the spacetime symmetries of relativistic quantum field theories (QFTs) are expected to change with Galilean symmetries emerging in the very low energy domain and, as we will argue, Carrollian symmetries appearing at very high energies. The formulation of Wilsonian renormalisation group seems inadequate for handling these changes of the underlying Poincare symmetry of QFTs and it seems unlikely that these drastic changes can be seen within the realms of relativistic QFT. We show that contrary to this expectation, changes in the spacetime algebra occurs at the very edges of parameter space. In particular, we focus on the very high energy sector and show how bilinears of U(1) currents added to a two dimensional (massless) scalar field theory deform the relativistic spacetime conformal algebra to conformal Carroll as the effective coupling of the deformation is dialed to infinity. We demonstrate this using both a symmetric and an antisymmetric current-current deformation for theories with multiple scalar fields. These two operators generate distinct kinds of quantum flows in the coupling space, the symmetric driven by Bogoliubov transformations and the antisymmetric by spectral flows, both leading to Carrollian CFTs at the end of the flow.

On the running and the UV limit of Wilsonian renormalization group flows

In nonperturbative formulation of quantum field theory, the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation. Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations.

Rigorous Wilsonian renormalization group for impurity models with a spectral gap

Rigorous Wilsonian renormalization group for impurity models with a spectral gap

Stochastic quantization and holographic Wilsonian renormalization group of conformally coupled scalar in AdS$$_{4}$$

Stochastic quantization and holographic Wilsonian renormalization group of conformally coupled scalar in AdS$$_{4}$$

Stochastic quantization and holographic Wilsonian renormalization group of scalar theory with generic mass, self-interaction and multiple trace deformation

We explore the mathematical relationship between holographic Wilsonian renormalization group (HWRG) and stochastic quantization (SQ) of scalar field theory with its generic mass, self-interaction and [Formula: see text]-multiple-trace deformation on the [Formula: see text]-dimensional conformal boundary defined in AdS[Formula: see text] space–time. We understand that once we define our Euclidean action, [Formula: see text] as [Formula: see text], then the stochastic process will reconstruct the HWRG data via solving Langevin equation and computing stochastic correlation functions. The [Formula: see text] is given by [Formula: see text], where [Formula: see text] is the boundary counter term and [Formula: see text] is the boundary deformation which gives a boundary condition. In our study, we choose the boundary condition adding (marginal)[Formula: see text]-multiple-trace deformation to the holographic dual field theory. In this theory, we establish maps between fictitious time, [Formula: see text] evolution of stochastic [Formula: see text]-point, ([Formula: see text])-point correlation functions and the (AdS)radial, [Formula: see text] evolution of [Formula: see text]-multiple-trace and ([Formula: see text])-multiple-trace deformations, respectively, once we take identifications of [Formula: see text] and between some of the constants appearing in both sides.

Some Applications of Renormalization Group to Few-Body Problems in EFT

General aspects of applying renormalization group in perturbative calculations are briefly discussed. Next, an application of the Wilsonian renormalization group with multitude of cutoff parameters to halo EFT for P-wave shallow states is considered.

Renormalization group flow to effective quantum mechanics in the IR in an emergent dual holographic description for spontaneous chiral symmetry breaking

Implementing the Wilsonian renormalization group (RG) transformation in a nonperturbative way, we construct an effective holographic dual description with an emergent extradimension identified with an RG scale. Taking the large$-N$ limit, we obtain an equation of motion of an order-parameter field, here the chiral condensate for our explicit demonstration. In particular, an intertwined structure manifests between the first-order RG flow equations of renormalized coupling functions and the second-order differential equation of the order-parameter field, thus referred to as a nonperturbative RG-improved mean-field theory. Assuming translational symmetry as a vacuum state, we solve these nonlinear coupled mean-field equations based on a matching method between UV- and IR-regional solutions. As a result, we find an RG flow from a weakly-coupled chiral-symmetric UV fixed point to a strongly-correlated chiral-symmetry broken IR fixed point, where the renormalized velocity of Dirac fermions vanishes most rapidly and effective quantum mechanics appears at IR. Furthermore, we translate these RG flows of coupling functions into those of emergent metric tensors and extract out geometrical properties of the emergent holographic spacetime constructed from the UV- and IR-regional solutions. Surprisingly, we obtain the volume law of entanglement entropy in the Ryu-Takayanagi formula, which implies appearance of a black hole type solution in the limit of infinite cutoff even at zero temperature. We critically discuss our field theoretic interpretation for this solution in terms of potentially gapless multi-particle excitation spectra.

Determining wave equations in holographic QCD from Wilsonian renormalization group

We show a possible way to build the AdS/CFT correspondence starting from the quantum field theory side based on renormalization group approach. An extra dimension is naturally introduced in our scheme as the renomalization scale. The holographic wave equations are derived, with the potential term being determined by the QFT properties. We discover that only around the fixed point, i.e. the conformal limit, the potential in the bulk equations can be fully constrained, and upon this foundation, the correspondence is build. We demonstrate this fact using a 3D scalar theory in which, besides the trivial fixed point, there exists the Wilson-Fisher fixed point. From the energy scalings around those fixed points, we determine the behavior of the potential in the bulk equations.

Incompleteness of the large-N analysis of the O(N) models: Nonperturbative cuspy fixed points and their nontrivial homotopy at finite N.

We summarize the usual implementations of the large-N limit of O(N) models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit N→∞. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as N increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of N. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all N>0 and show that they play an important role for the tricritical physics of O(N) models. Finally, we show that some of these fixed points are bivalued when they are considered as functions of d and N thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at N=∞ and d=3 is shown to play a crucial role for the internal consistency of all our results.

Wilsonian renormalization group for a multitrace matrix model

The Wilsonian renormalization group approach to matrix models is outlined and applied to multitrace matrix models with an emphasis on the computation of the fixed points which could describe the phase structure of noncommutative scalar phi-four theory.

On the reconstruction problem in quantum gravity

Path integrals and the Wilsonian renormalization group provide two complementary computational tools for investigating continuum approaches to quantum gravity. The starting points of these constructions utilize a bare action and a fixed point of the renormalization group flow, respectively. While it is clear that there should be a connection between these ingredients, their relation is far from trivial. This results in the so-called reconstruction problem. In this work, we demonstrate that the map between these two formulations does not generate non-localities at quadratic order in the background curvature. At this level, the bare action in the path integral and the fixed-point action obtained from the Wilsonian renormalization group differ by local terms only. This conclusion does not apply to theories coming with a physical ultraviolet cutoff or a fundamental non-locality scale.

Regulator dependence in the functional renormalization group: A quantitative explanation

The search for controlled approximations to study strongly coupled systems remains a very general open problem. Wilson's renormalization group has shown to be an ideal framework to implement approximations going beyond perturbation theory. In particular, the most employed approximation scheme in this context, the derivative expansion, was recently shown to converge and yield accurate and very precise results. However, this convergence strongly depends on the shape of the employed regulator. In this paper we clarify the reason for this dependence and justify, simultaneously, the most commonly employed procedure to fix this dependence, the principle of minimal sensitivity.

Holographic models of composite Higgs in the Veneziano limit. Part II. Fermionic sector

We continue our study of strongly-coupled, approximately scale-invariant gauge theories with a large number of flavours, which provide a suitable ultraviolet completion of the composite-Higgs scenario. We identify the requisite operators to realise partial compositeness of the Standard-Model fermions. In order to compute the spectrum of composite fermionic states, we extend the bottom-up holographic models, which we previously introduced to capture the main features of the non-perturbative dynamics in the Veneziano limit, by adding fermion fields in the bulk. We identify regions in parameter space where some fermionic bound states become light, depending in particular on the number of flavours, the operator scaling dimensions, and the bulk Yukawa couplings. We also observe a dense spectrum of states, when multi-scale dynamics is induced by a large backreaction of bulk scalars on the geometry. Adapting the formalism of the holographic Wilsonian renormalisation group, we study the linear coupling between the composite and elementary fermions, as a function of energy scale. We find that, in some circumstances, the associated operators are dangerously irrelevant: the renormalisation-group flow gives rise to a large linear coupling in the infrared, even when it is irrelevant from the point of view of the ultraviolet fixed point. We finally compute the partially composite spectrum, correlate it with the analysis of the flow, and assess the potential phenomenological implications, e.g. for the top-quark partners.

Emergent dual holographic description as a nonperturbative generalization of the Wilsonian renormalization group

In holographic duality, dynamics along the emergent extra-dimensional space describes a renormalization group (RG) flow of the corresponding quantum field theory (QFT). Following this idea, we develop an emergent holographic description of a QFT, where not only the information of the RG flow is introduced into an IR holographic dual effective field theory (HDEFT), but also the UV information of the QFT is encoded in the HDEFT through the IR boundary condition. In particular, we argue that this dual holographic construction is self-consistent within the assumption of bulk locality, showing the following two aspects: The solution of the Hamilton-Jacobi equation is given by the IR boundary effective action, and the Ward identity involving the QFT energy-momentum tensor current is satisfied naturally. We discuss the role of the RG $\beta$-function in the bulk effective dynamics of the metric tensor near a conformally invariant fixed point. We claim that this emergent dual gravity theory generalizes the perturbative Wilsonian RG framework into a non-perturbative way.

On holographic Wilsonian renormalization group of massive scalar theory with its self-interactions in AdS

Holographic model of massive scalar field with its self-interaction $\lambda\phi^n$ in AdS space is able to give a logarithmic scale dependence to marginal multi trace deformation couplings on its dual conformal field theory, where $\lambda$ is the self-interaction coupling of the scalar field, $\phi$ and $n$ is an integral number. In arXiv:1501.06664, the authors realize this feature by looking at bulk scalar solutions near AdS boundary imposing a specific boundary condition between the coefficients of non-normalizable and normalizable modes of the scalar field excitations. We study the same holographic model to see scale dependence of marginal deformations on the dual conformal field theory by employing completely different method: {\it holographic Wilsonian renormalization group}. We solve Hamilton-Jacobi equation derived from the holographic model of massive scalar with $\lambda\phi^n$ interaction and obtain the solution of marginal multi trace deformations up to the leading order in $\lambda$. It turns out that the solution of marginal multi trace deformation also presents logarithmic behavior in energy scale near UV region.