Lectures on the functional renormalization group method

Abstract

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained in a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scalar theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

47 pages, 11 figures, final version

References in corpus (7)

• Exact evolution equation for the effective potential
• Optimised Renormalisation Group Flows
• Renormalization Flow of Bound States
• Effective action and density functional theory
• Scheme independence as an inherent redundancy in quantum field theory
• Functional Callan-Symanzik Equation for the Coulomb Gas
• Current-density functional for disordered systems

Cited by in corpus (198)

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• Fixed points of quantum gravity in extra dimensions
• Chiral phase boundary of QCD at finite temperature
• Renormalization Group Approach towards the QCD Phase Diagram
• Global Flows in Quantum Gravity
• Confinement from Correlation Functions
• On the Prospect of Studying Nonperturbative QED with Beam-Beam Collisions
• Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory II
• Asymptotic safety: a simple example
• Geometry and field theory in multi-fractional spacetime
• On the phase structure and thermodynamics of QCD
• Uniqueness of infrared asymptotics in Landau gauge Yang-Mills theory
• Nonthermal fixed points and the functional renormalization group
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• An asymptotically safe solution to the U(1) triviality problem
• Lectures on renormalization and asymptotic safety
• Spectral Functions for the Quark-Meson Model Phase Diagram from the Functional Renormalization Group
• Quantum-gravity effects on a Higgs-Yukawa model
• From Big Bang to Asymptotic de Sitter: Complete Cosmologies in a Quantum Gravity Framework
• Quantum gravity: a fluctuating point of view
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• Upper bound on the Abelian gauge coupling from asymptotic safety
• Quantum gravity fluctuations flatten the Planck-scale Higgs potential
• Entropy signature of the running cosmological constant
• Viability of quantum-gravity induced ultraviolet completions for matter
• Quantum gravity and Standard-Model-like fermions
• Higgs scalar potential in asymptotically safe quantum gravity
• The QCD Phase Boundary from Quark-Gluon Dynamics
• Functional renormalisation group approach to far-from-equilibrium quantum field dynamics
• Numerical analytic continuation of Euclidean data
• On the Phase Structure of QCD in a Finite Volume
• Continuum limit in matrix models for quantum gravity from the Functional Renormalization Group
• On the scaling behavior of the chiral phase transition in QCD in finite and infinite volume
• Generalised global symmetries in states with dynamical defects: the case of the transverse sound in field theory and holography
• UV fixed-point structure of the three-dimensional Thirring model
• Non-perturbative thermal flows and resummations
• Variable Planck mass from the gauge invariant flow equation
• Critical behavior of the (2+1)-dimensional Thirring model
• 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
• Algorithmic derivation of functional renormalization group equations and Dyson-Schwinger equations
• Beyond Miransky Scaling
• Introduction to the nonequilibrium functional renormalization group
• Asymptotic Safety, Fractals, and Cosmology
• Universal Aspects of QCD-like Theories
• Analytic continuation of functional renormalization group equations
• High-accuracy scaling exponents in the local potential approximation
• Baryon number fluctuations in chiral effective models and their phenomenological implications
• Universality and the Renormalisation Group
• Functional renormalization group approach to the sine-Gordon model
• Towards an Asymptotic-Safety Scenario for Chiral Yukawa Systems
• Polyakov-Quark-Meson-Diquark Model for two-color QCD
• Finite temperature spectral functions in the O(N)-model
• Multicritical behavior in models with two competing order parameters
• Asymptotic safety of simple Yukawa systems
• Scale-dependent metric and causal structures in Quantum Einstein Gravity
• Gauge invariant renormalizability of quantum gravity
• Fluctuations and isentropes near the chiral critical endpoint
• Towards phase transitions between discrete and continuum quantum spacetime from the Renormalization Group
• Far-from-equilibrium quantum many-body dynamics
• Renormalisation Flow and Universality for Ultracold Fermionic Atoms
• Manifestly Gauge Invariant QCD
• Thermodynamics of QCD low-energy models and the derivative expansion of the effective action
• Efimov effect from functional renormalization
• Faddeev-Popov ghosts in quantum gravity beyond perturbation theory
• The Planck scale, the Higgs mass and scalar dark matter
• Background independent holographic description : From matrix field theory to quantum gravity
• On the impact of Majorana masses in gravity-matter systems
• Subleading critical exponents from the renormalisation group
• Three-body scattering from nonperturbative flow equations
• Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings
• Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation
• Renormalization-Group Analysis of the Generalized sine-Gordon Model and of the Coulomb Gas for d >= 3 Dimensions
• Functional renormalization group approach and gauge dependence in gravity theories
• Asymptotic safety and the cosmological constant
• Exact scheme independence at one loop
• Fixed Points of Quantum Gravity and the Renormalisation Group
• Exact renormalization group and Phi-derivable approximations
• Splitting Ward identity
• 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
• Effect of short-range interactions on the quantum critical behavior of spinless fermions on the honeycomb lattice
• Gluon condensation and scaling exponents for the propagators in Yang-Mills theory
• Black hole entanglement entropy and the renormalization group
• Scaling functions for the O(4)-model in d=3 dimensions
• Phase transition and critical behavior of d=3 chiral fermion models with left/right asymmetry
• Anomalous scaling at non-thermal fixed points of Burgers' and Gross-Pitaevskii turbulence
• Meson Spectral Functions at Finite Temperature and Isospin Density with Functional Renormalization Group
• Renormalizable parameters of the sine-Gordon model
• Comparison of renormalization group schemes for sine-Gordon type models
• BRST in the Exact RG
• Asymptotic freedom and safety in quantum gravity
• Functional renormalization group at finite density and Bose condensation
• Fermionic spectral functions with the Functional Renormalization Group
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• Exact Renormalisation Group Equations and Loop Equations for Tensor Models
• N=1 Wess Zumino Model in d=3 at zero and finite temperature
• Renormalization-group flow of the effective action of cosmological large-scale structures
• Truncation Effects in the Functional Renormalization Group Study of Spontaneous Symmetry Breaking
• Onset of symmetry breaking by the functional RG method
• Dynamical Locking of the Chiral and the Deconfinement Phase Transition in QCD
• Spontaneous symmetry breaking and optimization of functional renormalization group
• A Manifestly Gauge Invariant and Universal Calculus for SU(N) Yang-Mills
• Renormalization-Group Analysis of Layered Sine-Gordon Type Models
• Can we see quantum gravity? Photons in the asymptotic-safety scenario
• Quantum censorship in two dimensions
• KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables
• Real-time methods for spectral functions
• Gravitational instantons and anomalous chiral symmetry breaking
• The fate of non-polynomial interactions in scalar field theory
• Generalized universality in the massive sine-Gordon model
• Collective magnetic fluctuations in Hubbard plaquettes captured by fluctuating local field method
• Wilsonian renormalisation and the exact cut-off scale from holographic duality
• Renormalization group in the internal space
• Effective field equations and scale-dependent couplings in gravity
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• TASI Lectures on Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems
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• Critical exponents in quantum Einstein gravity
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• Self-consistent $O(4)$ model spectral functions from analytically continued FRG flows
• Universality From Very General Nonperturbative Flow Equations in QCD
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• Towards Functional Flows for Hierarchical Models
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• Interplay of fixed points in scalar models
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• Cooling Stochastic Quantization with colored noise
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• Fermionic excitations at finite temperature and density
• Efimov physics from the functional renormalization group
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• Spectral Functions from the Functional Renormalization Group
• Unifying renormalization group and the continuous wavelet transform
• Renormalization Flow of Axion Electrodynamics
• Invariant formulation of the Functional Renormalisation Group method for $U(n)\times U(n)$ symmetric matrix models
• Stability of fixed points and generalized critical behavior in multifield models
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• Renormalization of QCD_2
• FlowPy - a numerical solver for functional renormalization group equations
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• Fine-tuning and vacuum stability in Wilsonian effective action
• Effective scalar potential in asymptotically safe quantum gravity
• Asymptotic safety in the sine-Gordon model
• A low-rank approach to the computation of path integrals
• Real-time observables from Euclidean thermal correlation functions
• Vainshtein in the UV and a Wilsonian analysis of derivatively coupled scalars
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• Quantum-classical transition in the Caldeira-Leggett model
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• Scaling violation and the magnetic equation of state in chiral models
• Renormalization-Group Running Induced Cosmic Inflation
• Vacuum Energy and Renormalization of the Field-Independent Term
• Functional renormalization group study of a four-fermion model with $CP$ violation
• Inflationary Potentials from the Exact Renormalisation Group
• Current-density functional for disordered systems
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• On the running and the UV limit of Wilsonian renormalization group flows
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• Wilsonian renormalisation group and thermal field theory in the Schwinger-Keldysh closed-time-path formalism
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• Quantum fields in the non-perturbative regime - Yang-Mills theory and gravity
• Scaling behavior in interacting systems: joint effect of anisotropy and compressibility
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• Existence theorem on the UV limit of Wilsonian RG flows of Feynman measures