Introduction to the functional RG and applications to gauge theories

Abstract

These lectures contain an introduction to modern renormalization group (RG) methods as well as functional RG approaches to gauge theories. In the first lecture, the functional renormalization group is introduced with a focus on the flow equation for the effective average action. The second lecture is devoted to a discussion of flow equations and symmetries in general, and flow equations and gauge symmetries in particular. The third lecture deals with the flow equation in the background formalism which is particularly convenient for analytical computations of truncated flows. The fourth lecture concentrates on the transition from microscopic to macroscopic degrees of freedom; even though this is discussed here in the language and the context of QCD, the developed formalism is much more general and will be useful also for other systems.

60 pages, 14 figures, Lectures held at the 2006 ECT* School "Renormalization Group and Effective Field Theory Approaches to Many-Body Systems", Trento, Italy

References in corpus (7)

• Exact evolution equation for the effective potential
• Infrared Properties of QCD from Dyson-Schwinger equations
• Chiral phase boundary of QCD at finite temperature
• Towards the infrared limit in SU(3) Landau gauge lattice gluodynamics
• Renormalization flow of Yang-Mills propagators
• Renormalization group flows into phases with broken symmetry
• Collective fields in the functional renormalization group for fermions, Ward identities, and the exact solution of the Tomonaga-Luttinger model

Cited by in corpus (217)

• The nonperturbative functional renormalization group and its applications
• Baryons as relativistic three-quark bound states
• The QCD phase structure at finite temperature and density
• QCD at finite temperature and chemical potential from Dyson-Schwinger equations
• The QCD Running Coupling
• Inhomogeneous chiral condensates
• Landau gauge Yang-Mills correlation functions
• Towards apparent convergence in asymptotically safe quantum gravity
• Dynamical holographic QCD model for glueball and light meson spectra
• The Renormalization Group flow of unimodular f(R) gravity
• Generalized Parametrization Dependence in Quantum Gravity
• Antiferromagnetic critical point on graphene's honeycomb lattice: A functional renormalization group approach
• Resolving Spacetime Singularities within Asymptotic Safety
• An asymptotically safe solution to the U(1) triviality problem
• Lectures on renormalization and asymptotic safety
• Form Factors in Asymptotic Safety: conceptual ideas and computational toolbox
• Thermodynamics of QCD at vanishing density
• Quantum gravity: a fluctuating point of view
• Non-minimal coupling in Higgs-Yukawa model with asymptotically safe gravity
• Upper bound on the Abelian gauge coupling from asymptotic safety
• Symmetric nuclear matter from the strong interaction
• Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model
• Quantum gravity and Standard-Model-like fermions
• How perturbative is quantum gravity?
• Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system
• Zooming in on fermions and quantum gravity
• On the number of relevant operators in asymptotically safe gravity
• Numerical analytic continuation of Euclidean data
• QCD at finite temperature and density within the fRG approach: An overview
• Quark matter nucleation in neutron stars and astrophysical implications
• Fluctuation-induced modifications of the phase structure in (2+1)-flavor QCD
• Status of the asymptotic safety paradigm for quantum gravity and matter
• Non-perturbative finite-temperature Yang-Mills theory
• The Higgs Mass and the Scale of New Physics
• Brout-Englert-Higgs physics: From foundations to phenomenology
• FormTracer - A Mathematica Tracing Package Using FORM
• On the relevance of matter and glue dynamics for baryon number fluctuations
• Renormalization group flow and symmetry restoration in de Sitter space
• Variable Planck mass from the gauge invariant flow equation
• Fixed-Functionals of three-dimensional Quantum Einstein Gravity
• Real-Time Correlation Functions in the O(N) Model from the Functional Renormalization Group
• On the Functional Renormalization Group approach for Yang-Mills fields
• Asymptotic safety and field parametrization dependence in the f(R) truncation
• QCD Running Couplings and Effective Charges
• Background independent exact renormalization group for conformally reduced gravity
• Quarkyonic phase from quenched dynamical holographic QCD model
• Critical behavior in spherical and hyperbolic spaces
• Fermion-induced quantum criticality in two-dimensional Dirac semimetals: Non-perturbative flow equations, fixed points and critical exponents
• $d=4$ as the critical dimensionality of asymptotically safe interactions
• Lorentz invariance violations in the interplay of quantum gravity with matter
• Solutions to the reconstruction problem in asymptotic safety
• Correlation functions of three-dimensional Yang-Mills theory from the FRG
• Polyakov-Quark-Meson-Diquark Model for two-color QCD
• Fierz-complete NJL model study: fixed points and phase structure at finite temperature and density
• Renormalizable Group Field Theory beyond melonic diagrams: an example in rank four
• An asymptotic safety scenario for gauged chiral Higgs-Yukawa models
• Mass sensitivity of the three-flavor chiral phase transition
• Quantum Einstein-Cartan theory with the Holst term
• Light charged fermions in quantum gravity
• Asymptotic safety, string theory and the weak gravity conjecture
• Nonperturbative quark matter equations of state with vector interactions
• Frequency regulators for the nonperturbative renormalization group: A general study and the model A as a benchmark
• The Planck scale, the Higgs mass and scalar dark matter
• Renormalization group flow of Hořava-Lifshitz gravity at low energies
• Global flow of the Higgs potential in a Yukawa model
• A link that matters: Towards phenomenological tests of unimodular asymptotic safety
• Functional renormalization group analysis of the soft mode at the QCD critical point
• On the impact of Majorana masses in gravity-matter systems
• Fierz-complete NJL model study II: towards the fixed-point and phase structure of hot and dense two-flavor QCD
• Modeling Finite-Volume Effects and Chiral Symmetry Breaking in Two-Flavor QCD Thermodynamics
• Scalar-Tensor theories within Asymptotic Safety
• Asymptotic safety and the cosmological constant
• Essential nature of Newton's constant in unimodular gravity
• Bound state properties from the Functional Renormalisation Group
• DoFun 3.0: Functional equations in Mathematica
• Exact RG Flow Equations and Quantum Gravity
• Shocks and quark-meson scatterings at large density
• Splitting Ward identity
• A covariant variational approach to Yang-Mills Theory at finite temperatures
• The dynamical holographic QCD method for hadron physics and QCD matter
• Higgs inflation and quantum gravity: An exact renormalisation group approach
• Critical temperature and superfluid gap of the Unitary Fermi Gas from Functional Renormalization
• Scalegenesis and fermionic dark matters in the flatland scenario
• From quantum to classical dynamics: The relativistic $O(N)$ model in the framework of the real-time functional renormalization group
• Vector and Axial-Vector Mesons in Nuclear Matter
• Can quantum fluctuations differentiate between standard and unimodular gravity?
• Functional renormalization group and 2PI effective action formalism
• Self-consistent Spectral Functions in the $O(N)$ Model from the FRG
• Strangeness Neutrality and QCD Thermodynamics
• Two-particle irreducible functional renormalization group schemes---a comparative study
• Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows
• Conformal invariance in the nonperturbative renormalization group: a rationale for choosing the regulator
• BRST in the Exact RG
• Asymptotic freedom and safety in quantum gravity
• Dimensional crossover of nonrelativistic bosons
• Phase structure of NJL model with weak renormalization group
• Universal gravitational-wave signatures from heavy new physics in the electroweak sector
• Exploring new corners of asymptotically safe unimodular quantum gravity
• Towards a Renormalization Group Approach to Density Functional Theory - General Formalism and Case Studies -
• Emergent Hadrons and Diquarks
• Dense two-color QCD from Dyson-Schwinger equations
• Vanishing Beta Function curves from the Functional Renormalisation Group
• Towards coarse graining of discrete Lorentzian quantum gravity
• A glassy phase in quenched disordered graphene and crystalline membranes
• Renormalization-group flow of the effective action of cosmological large-scale structures
• Strangeness neutrality and baryon-strangeness correlations
• Lessons from conformally reduced quantum gravity
• Asymptotically Safe QED
• Truncation Effects in the Functional Renormalization Group Study of Spontaneous Symmetry Breaking
• Convergence of Derivative Expansion in Supersymmetric Functional RG Flows
• Quark propagator in Minkowski space
• Real-time dynamics of the $O(4)$ scalar theory within the fRG approach
• The flat phase of quantum polymerized membranes
• Fractionalized quantum criticality in spin-orbital liquids from field theory beyond the leading order
• Functional renormalization-group calculation of the equation of state of one-dimensional nuclear matter inspired by the Hohenberg--Kohn theorem
• Are there ALPs in the asymptotically safe landscape?
• Gravitational instantons and anomalous chiral symmetry breaking
• From quarks and gluons to color superconductivity at supranuclear densities
• Phase structure and propagators at nonvanishing temperature for QCD and QCD-like theories
• Momentum dependence of quantum critical Dirac systems
• Scaling relations and multicritical phenomena from Functional Renormalization
• Non-universality in the erosion of tilted landscapes
• Nonvanishing gravitational contribution to matter beta functions for vanishing dimensionful regulators
• Tan contact and universal high momentum behavior of the fermion propagator in the BCS-BEC crossover
• On the quark spectral function in QCD
• Background independent exact renormalisation
• Renormalized Functional Renormalization Group
• Dependence of the propagators on the sampling of Gribov copies inside the first Gribov region of Landau gauge
• Pion and $\eta$-meson mass splitting at the two-flavour chiral crossover
• Partial bosonisation for the two-dimensional Hubbard model: How well does it work?
• Sarma phase in relativistic and non-relativistic systems
• Universal behavior of coupled order parameters below three dimensions
• Ward-constrained melonic renormalization group flow
• Critical behaviors of the $O(4)$ and $Z(2)$ symmetries in the QCD phase diagram
• Universal dynamics and non-thermal fixed points in quantum fluids far from equilibrium
• Could reggeon field theory be an effective theory for QCD in the Regge limit?
• Covariant variational approach to Yang-Mills Theory: effective potential of the Polyakov loop
• Functional renormalization group study of the critical region of the quark-meson model with vector interactions
• Field-Dependent BRST-antiBRST Lagrangian Transformations
• Higgs stability-bound and fermionic dark matter
• BRST invariant RG flows
• Ab-initio description of excited states of a one-dimensional nuclear matter with the Hohenberg-Kohn-theorem-inspired functional-renormalization-group method
• Baryon number probability distribution at finite temperature
• Self-consistent $O(4)$ model spectral functions from analytically continued FRG flows
• Functional flows in QED and the modified Ward-Takahashi identity
• Higgs scalar potential coupled to gravity in the exponential parametrization in arbitrary gauge
• On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
• Constraints on discrete global symmetries in quantum gravity
• Spectral functions and in-medium properties of hadrons
• Asymptotic Analysis of the Local Potential Approximation to the Wetterich Equation
• Chiral symmetry restoration at finite temperature within the Hamiltonian approach to QCD in Coulomb gauge
• New applications of the renormalization group method in physics -- a brief introduction
• Evidence for a novel shift-symmetric universality class from the functional renormalization group
• Exact solutions and residual regulator dependence in functional renormalisation group flows
• Ab initio construction of the energy density functional for electron systems with the functional-renormalization-group-aided density functional theory
• Critical phenomena at the complex tensor ordering phase transition
• The RG flow of Nambu--Jona-Lasinio model at finite temperature and density
• Discovering and quantifying nontrivial fixed points in multi-field models
• Spectral Functions from the Functional Renormalization Group
• Functional renormalization group study of the Nambu--Jona-Lasinio model at finite temperature and density in an external magnetic field
• Existence and Construction of Exact FRG Flows of a UV-Interacting Scalar Field Theory
• Static approach to renormalization group analysis of stochastic models with spatially quenched disorder
• Critical Wess-Zumino models with four supercharges from the functional renormalization group
• Asymptotic safety in the dark
• Computing First-Passage Times with the Functional Renormalisation Group
• Correlations of conserved charges and QCD phase structure
• Error estimates and specification parameters for functional renormalization
• Shape Dynamics and Effective Field Theory
• On the complex structure of Yang-Mills theory
• Renormalization Group Flow of the Higgs Potential
• Five-dimensional scale-dependent black holes with constant curvature and Solv horizons
• Covariant variational approach to Yang-Mills Theory: Thermodynamics
• Fixed Point Structure of Gradient Flow Exact Renormalization Group for Scalar Field Theories
• Asymptotic safety, quantum gravity, and the swampland: a conceptual assessment
• Chiral and deconfinement phase transition in the Hamiltonian approach to QCD in Coulomb gauge
• The Effect of Quantum Fluctuations in the High-Energy Cold Nuclear Equation of State and in Compact Star Observables
• Chiral Perturbation Theory Reflections on Effective Theories of the Standard Model
• A Lorentzian renormalisation group equation for gauge theories
• Vainshtein in the UV and a Wilsonian analysis of derivatively coupled scalars
• Dimensional crossover in ultracold Fermi gases from Functional Renormalisation
• One-meson-loop NJL model: Effect of collective and noncollective excitations on the quark condensate at finite temperature
• The Nordic-walking mechanism and its explanation of deconfined pseudocriticality from Wess-Zumino-Witten theory
• Lectures in Quantum Gravity
• A comprehensive approach to the physics of mesons
• Center phase transition from matter propagators in (scalar) QCD
• Spectral functions from the functional renormalization group at finite temperature and density
• UV completion of extradimensional Yang-Mills theory for Gauge-Higgs unification
• Gradient Flow Exact Renormalization Group for Scalar Quantum Electrodynamics
• Optimized regulator for the quantized anharmonic oscillator
• Effect of momentum anisotropy on quark matter in the quark-meson model
• QCD propagators and vertices from lattice QCD (in memory of Michael Müller-Preußker)
• Quantum-classical transition in the Caldeira-Leggett model
• Looking at QED with Dyson-Schwinger equations: basic equations, Ward-Takahashi identities and the two-photon-two-fermion irreducible vertex
• Exact renormalization group for wave functionals
• Phase Structure and Dynamics of QCD-- A Functional Perspective
• Towards a Higgs mass determination in asymptotically safe gravity with a dark portal
• Tunable tachyon mass in the PT-broken massive Thirring model
• The weak-gravity bound and the need for spin in asymptotically safe matter-gravity models
• A Rigorous Derivation of the Functional Renormalisation Group Equation
• Scaling solutions for asymptotically free quantum gravity
• Coarse-graining in time with the Functional Renormalisation Group: Relaxation in Brownian Motion
• Vacuum Energy and Renormalization of the Field-Independent Term
• Comment on "The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization"
• Physics-informed neural networks for solving functional renormalization group on a lattice
• Higher derivative extension of the functional renormalization group
• Addressing energy density functionals in the language of path-integrals II: Comparative study of functional renormalization group techniques applied to the (0+0)-D $O(N)$-symmetric ${\phi }^4$-theory
• Renormalization of the bilocal sine-Gordon model
• Gauge-covariant stochastic neural fields: Stability and finite-width effects
• Relativistic BEC extracted from a complex FRG flow equation
• Functional Renormalization Group Flows and Gauge Consistency in QED
• Gauge dependence of effective average action
• Hamiltonian approach to QCD in Coulomb gauge at zero and finite temperature
• Flat Bundles on Function Manifolds and Evolution Equations in Quantum Field Theories
• Composite operators in functional renormalization group approach
• Reconstruction of observed mechanical motions with Artificial Intelligence tools
• Non-perturbative SQED beta function using functional renormalization group approach and the NSVZ exact beta function
• Landau Theory of Causal Dynamical Triangulations