Aspects of the Functional Renormalisation Group

Abstract

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being NPI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.

58 pages, no figures, minor changes

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• Universal behavior of coupled order parameters below three dimensions
• Universal dynamics and non-thermal fixed points in quantum fluids far from equilibrium
• Critical behaviors of the $O(4)$ and $Z(2)$ symmetries in the QCD phase diagram
• Four-quark scatterings in QCD I
• Dynamical generation of low-energy couplings from quark-meson fluctuations
• Sarma phase in relativistic and non-relativistic systems
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• On the self-consistency of off-shell Slavnov-Taylor identities in QCD
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• Asymptotic freedom in $Z_2$-Yukawa-QCD models
• Chiral symmetry restoration at finite temperature within the Hamiltonian approach to QCD in Coulomb gauge
• On the Renormalization of Theories of a Scalar Chiral Superfield
• Application of the functional renormalization group to Bose gases: from linear to hydrodynamic fluctuations
• The zero-dimensional O(N) vector model as a benchmark for perturbation theory, the large-N expansion and the functional renormalization group
• Cooling Stochastic Quantization with colored noise
• Scheme dependence of asymptotically free solutions
• Reparameterisation Invariance and RG equations: Extension of the Local Potential Approximation
• Renormalization flow towards gravitational catalysis in the 3d Gross-Neveu model
• Exact solutions and residual regulator dependence in functional renormalisation group flows
• The Helmholtz Hierarchy: Phase Space Statistics of Cold Dark Matter
• Fermionic excitations at finite temperature and density
• Exact flow equation for bound states
• Twisted bilayer graphene at charge neutrality: competing orders of SU(4) Dirac fermions
• New applications of the renormalization group method in physics -- a brief introduction
• Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems
• The RG flow of Nambu--Jona-Lasinio model at finite temperature and density
• Summing parquet diagrams using the functional renormalization group: X-ray problem revisited
• Regulator scheme dependence of the chiral phase transition at high densities
• Evidence for a novel shift-symmetric universality class from the functional renormalization group
• Phase structure of a three-dimensional Yukawa model
• Critical phenomena at the complex tensor ordering phase transition
• Critical Casimir forces for O(N) models from functional renormalization
• The Numerically Optimized Regulator and the Functional Renormalization Group
• Functional-renormalization-group aided density-functional analysis for the correlation energy of the two-dimensional homogeneous electron gas
• Observable consequences of quantum gravity: Can light fermions exist?
• Ab initio construction of the energy density functional for electron systems with the functional-renormalization-group-aided density functional theory
• Gluon mass scale through the Schwinger mechanism
• Spectral Functions from the Functional Renormalization Group
• Equivalent Fixed-Points in the Effective Average Action Formalism
• Bound states in Functional Renormalization Group
• Functional renormalization group study of neutral and charged pion under magnetic fields in the quark-meson model
• Enhancement of field renormalization in scalar theories via functional renormalization group
• Critical spin dynamics of Heisenberg ferromagnets revisited
• Gradient flow exact renormalization group -- inclusion of fermion fields
• Thermodynamics of the Glueball Resonance Gas
• Discovering and quantifying nontrivial fixed points in multi-field models
• Functional renormalization group study of the Nambu--Jona-Lasinio model at finite temperature and density in an external magnetic field
• Efimov physics from the functional renormalization group
• Spin functional renormalization group for the $J_1{J}_2{J}_3$ quantum Heisenberg model
• Gauge and Poincare' Invariant Regularization and Hopf Symmetries
• New Insight about the Effective Restoration of U_A(1) Symmetry
• Ordinary and exotic mesons in the extended Linear Sigma Model
• Correlations of conserved charges and QCD phase structure
• Conformal invariance and composite operators: A strategy for improving the derivative expansion of the nonperturbative renormalization group
• Four-gluon vertex from the Curci-Ferrari model at one-loop order
• Phenomena at the QCD phase transition in nonequilibrium chiral fluid dynamics (N$\chi$FD)
• Phase diagram of the $J_1$-$J_2$ quantum Heisenberg model for arbitrary spin
• Stability of fixed points and generalized critical behavior in multifield models
• Fermion spectral function in hot strongly interacting matter from the functional renormalization group
• Asymptotic safety in the dark
• Fluctuation-induced higher-derivative couplings and infrared dynamics of the Quark-Meson-Diquark Model
• Dyson-Schwinger approach to strongly coupled theories
• Error estimates and specification parameters for functional renormalization
• Renormalization Flow of Axion Electrodynamics
• The two dimensional N=(2,2) Wess-Zumino Model in the Functional Renormalization Group Approach
• Dyson--Schwinger approach to Hamiltonian Quantum Chromodynamics
• Fixed Point Structure of Gradient Flow Exact Renormalization Group for Scalar Field Theories
• ${\mathbb{Z}}_4$-symmetric perturbations to the XY model from functional renormalization
• Gauge Invariant Cutoff QED
• Renormalization Group Flow of the Higgs Potential
• Functional-renormalization-group approach to classical liquids with short-range repulsion: a scheme without repulsive reference system
• Thermodynamics of Bose gases from functional renormalization with a hydrodynamic low-energy effective action
• Covariant variational approach to Yang-Mills Theory: Thermodynamics
• Renormalization of QCD_2
• On the complex structure of Yang-Mills theory
• Renormalization of the 4PI effective action using the functional renormalization group
• Stability of the Fulde-Ferrell-Larkin-Ovchinnikov states in anisotropic systems and critical behavior at thermal $m$-axial Lifshitz points
• Conserved and non-conserved Noether currents from the quantum effective action
• Chiral and deconfinement phase transition in the Hamiltonian approach to QCD in Coulomb gauge
• Stability of chromomagnetic condensation and mass generation for confinement in SU(2) Yang-Mills theory
• FlowPy - a numerical solver for functional renormalization group equations
• Variational and Dyson--Schwinger Equations of Hamiltonian Quantum Chromodynamics
• Functional renormalisation group for few-nucleon systems: SU(4) symmetry and its breaking
• Asymptotic safety in the sine-Gordon model
• Chiral symmetry breaking in accelerating and rotating frames
• Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism
• A Lorentzian renormalisation group equation for gauge theories
• A unified quark-nuclear matter equation of state from the cluster virial expansion within the generalized Beth-Uhlenbeck approach
• Spontaneous mass generation and the small dimensions of the Standard Model gauge groups U(1), SU(2) and SU(3)
• From Euclidean to Minkowski space with the Cauchy-Riemann equations
• Renormalization group flow equations with full momentum dependence
• Super-Strong Coupling NJL model in arbitrary space-time dimensions
• Stochastic dynamics for Group Field Theories
• Spectral functions from the functional renormalization group at finite temperature and density
• Solving the QCD non-perturbative flow equation as a partial differential equation and its application to the dynamical chiral symmetry breaking
• Ghost-gluon vertex in the presence of the Gribov horizon: General kinematics
• 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
• On the Yang-Mills two-loop effective action with wordline methods
• Four-boson bound states from a functional renormalisation group
• The Nordic-walking mechanism and its explanation of deconfined pseudocriticality from Wess-Zumino-Witten theory
• Gross-Neveu-XY quantum criticality in moiré Dirac materials
• Center phase transition from matter propagators in (scalar) QCD
• Lectures in Quantum Gravity
• Flowing bosonization in the nonperturbative functional renormalization-group approach
• Real-time observables from Euclidean thermal correlation functions
• Criticality of the $O(2)$ model with cubic anisotropies from nonperturbative renormalization
• Polyakov loop, gluon mass, gluon condensate and its asymmetry near deconfinement
• A comprehensive approach to the physics of mesons
• Hydrodynamic collective modes for cold trapped gases
• UV completion of extradimensional Yang-Mills theory for Gauge-Higgs unification
• Quantum-classical transition in the Caldeira-Leggett model
• The rainbow modified-ladder approximation and degenerate pion
• On the phase structure of extra-dimensional gauge theories with fermions
• Effect of momentum anisotropy on quark matter in the quark-meson model
• Conformal invariance constraints in the $O(N)$ models: a first study within the nonperturbative renormalization group
• Gradient Flow Exact Renormalization Group for Scalar Quantum Electrodynamics
• Few-body hierarchy in non-relativistic functional renormalization group equations and a decoupling theorem
• Optimized regulator for the quantized anharmonic oscillator
• (Non) Gauge Invariance of Wilsonian Effective Actions in (Supersymmetric) Gauge Theories : A Critical Discussion
• Quark Confinement and the Renormalization Group
• On the low-energy spectrum of an SU(2) gauge theory with dynamical fermions
• Chiral criticality and glue dynamics
• Functional renormalization of QCD in $1+1$ dimensions: four-fermion interactions from quark-gluon dynamics
• Flow equation of functional renormalization group for three-body scattering problems
• Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes
• The weak-gravity bound and the need for spin in asymptotically safe matter-gravity models
• Renormalisation of the two-dimensional border-collision normal form
• Quantum interface unbinding transitions
• Phonon renormalization and Pomeranchuk instability in the Holstein model
• Towards a Higgs mass determination in asymptotically safe gravity with a dark portal
• Phase Structure and Dynamics of QCD-- A Functional Perspective
• Scaling solutions for asymptotically free quantum gravity
• Exact renormalization group for wave functionals
• The Charm and Beauty of Strong Interactions
• Spectral properties and observables in ultracold Fermi gases
• Quantum field theories of relativistic Luttinger fermions
• Dynamical symmetry breaking in Georgi-Glashow chiral-gauge theories
• Preferred foliation effects in Quantum General Relativity
• Background Effective Action with Nonlinear Massive Gauge Fixing
• Examination of background effects on light-nuclei yield ratio in relativistic heavy-ion collisions
• The Equal-Time Quark Propagator in Coulomb Gauge
• Vacuum Energy and Renormalization of the Field-Independent Term
• An Extension of Pohlmeyer's Theorem
• Nonperturbative Renormalization Group for the Landau-de Gennes Model
• Functional renormalization group study of a four-fermion model with $CP$ violation
• Renormalization of the bilocal sine-Gordon model
• Interacting two-particle states in the symmetric phase of the chiral Nambu--Jona-Lasinio model
• Pion observables calculated in Minkowski and Euclidean spaces with Ansätze for quark propagators
• Higher derivative extension of the functional renormalization group
• Partial wave analysis for the in-hadron condensate
• On the Lorentz symmetry in conformally reduced Quantum Gravity
• 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
• Patterns of gauge symmetry in the background field method
• Implementation of Cluster expansion for hot QCD matter
• Functional renormalization group without functional integrals: implementing Hilbert space projections for strongly correlated electrons via Hubbard X-operators
• Renormalization Flow of Nonlinear Electrodynamics
• Continuous order-to-order quantum phase transitions from fixed-point annihilation
• Tunneling with physics-informed RG flows in the anharmonic oscillator
• Renormalizing the Quark-Meson-Diquark Model
• A basis of the gradient flow exact renormalization group for gauge theory
• From negative to positive cosmological constant through decreasing temperature of the Universe: connection with string theory and spacetime foliation results
• Towards a unified viewpoint of Gribov--Zwanziger and Serreau--Tissier gauge fixing
• Functional renormalization group for extremely correlated electrons
• The gap equations of background field invariant Refined Gribov-Zwanziger action proposals and the deconfinement transition
• An all order renormalizable Refined-Gribov-Zwanziger model with BRST invariant fermionic horizon function in linear covariant gauges
• A compared analysis of the susceptibility in the O(N) theory
• Physics-informed operator flows and observables
• Explicit construction of the energy-momentum tensor in the large N limit
• Study of the Emergence of a Gluon Mass Scale from Center Vortices Using a Wave-Functional Formalism
• Bad Metal Behavior and Lifshitz Transition of a Nagaoka Ferromagnet
• QCD chiral phase diagram from weak functional renormalization group
• Subtleties in the calculation of correlation functions for hot and dense systems
• Functional Renormalization Group Analysis of $O(3)$ Nonlinear Sigma Model and Non-Abelian Bosonization Duality
• Sensitivity of Nonrenormalizable Trajectories to the Bare Scale
• The effective potential of the Polyakov loop in the Hamiltonian approach to QCD
• Superfluidity in multicomponent fermions via the functional renormalization group
• Vertex functions and their flow equations from the 2PI effective action
• Gauge dependence of effective average action
• Indications against dynamical CPT symmetry restoration in quantum gravity
• Functional Renormalization Group Flows and Gauge Consistency in QED
• Spin functional renormalization group for dimerized quantum spin systems
• Operator product expansion coefficients from the nonperturbative functional renormalization group
• Functional renormalization group approach to phonon modified criticality: anomalous dimension of strain and non-analytic corrections to Hooke's law
• Multi-Regulator Functional Renormalization Group for Many-Fermion Systems
• Semiclassical partition function for the double-well potential
• Benchmarking regulator-sourced 2PI and average 1PI flow equations in zero dimensions
• Non-Perturbative $S$-matrix Renormalization
• Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes
• BRST, Ward identities, gauge dependence, and a functional renormalization group
• The $\theta$-vacuum from functional renormalisation
• Hamiltonian approach to QCD in Coulomb gauge at zero and finite temperature
• Fluctuation-induced first-order superfluid transition in unitary $\mathrm{SU}(N)$ Fermi gases
• Functional renormalization with interaction flows: A single-boson exchange perspective and application to electron-phonon systems
• Renormalization group approach to the elastic properties of graphene bilayers
• Emergence of dynamical tensor fields in composite models of gravity
• Phase diagram of Model C in the parametric space of order parameter and space dimensions
• Critical behavior of isotropic systems with strong dipole-dipole interaction from the functional renormalization group
• Perturbative Renormalisation Group Improved Black Hole Solution and its Quasinormal Modes