Functional Flow Equations Research Articles

Renormalization flow of nonlinear electrodynamics

We study the renormalization flow of generic actions that depend on the invariants of the field strength tensor of an Abelian gauge field. While the Maxwell action defines a Gaussian fixed point, we search for further non-Gaussian fixed points or rather fixed functions, i.e., globally existing Lagrangians of the invariants. Using standard small-field expansion techniques for the resulting functional flow equation, a large number of fixed points is obtained, which—in analogy to recent findings for a shift-symmetric scalar field—we consider as approximation artifacts. For the construction of a globally existing fixed function, we pay attention to the use of proper initial conditions. Parametrizing the latter by the photon anomalous dimension, both the coefficients of the weak-field expansion are fully determined and those of the large-field expansion can be matched such that a global fixed function can be constructed for magnetic fields. The anomalous dimension also governs the strong-field limit. Our results provide evidence for the existence of a continuum of non-Gaussian fixed points parametrized by a small positive anomalous dimension below a critical value. We discuss the implications of this result within various scenarios with and without additional matter. For the strong-field limit of the 1PI QED effective action, where the anomalous dimension is determined by electronic fluctuations, our result suggests the existence of a singularity free strong-field limit, circumventing the standard conclusions connected to the perturbative Landau pole. Published by the American Physical Society 2024

The conformal sector of Quantum Einstein Gravity beyond the local potential approximation

The anomalous scaling of Newton's constant around the Reuter fixed point is dynamically computed using the functional flow equation approach. Specifically, we thoroughly analyze the flow of the most general conformally reduced Einstein-Hilbert action. Our findings reveal that, due to the distinctive nature of gravity, the anomalous dimension η of the Newton's constant cannot be constrained to have one single value: the ultraviolet critical manifold is characterized by a line of fixed points (g⁎(η),λ⁎(η)), with a discrete (infinite) set of eigenoperators associated to each fixed point. More specifically, we find three ranges of η corresponding to different properties of both fixed points and eigenoperators and, in particular, for the range η<ηc≈0.96 the ultraviolet critical manifolds has finite dimensionality.

UV completion of extradimensional Yang-Mills theory for Gauge-Higgs unification

The SU(N)SU(N) Yang-Mills theory in \mathbb R^4\times S^1ℝ4×S1 spacetime is studied as a simple toy model of Gauge-Higgs unification. The theory is perturbatively nonrenormalizable but could be formulated as an asymptotically safe theory, namely a nonperturbatively renormalizable theory. We study the fixed point structure of the Yang-Mills theory in \mathbb R^4\times S^1ℝ4×S1 by using the functional renormalization group in the background field approximation. We derive the functional flow equations for the gauge coupling and the background gauge-field potential. There exists a nontrivial fixed point for both couplings at finite compactification radii. At the fixed point, gauge coupling and vacuum energy are both relevant. The renormalization group flow of the gauge coupling describes the smooth transition between the ultraviolet asymptotically safe regime and the strong interacting infrared limit.

Renormalised spectral flows

We derive renormalised finite functional flow equations for quantum field theories in real and imaginary time that incorporate scale transformations of the renormalisation conditions, hence implementing a flowing renormalisation. The flows are manifestly finite in general non-perturbative truncation schemes also for regularisation schemes that do not implement an infrared suppression of the loops in the flow. Specifically, this formulation includes finite functional flows for the effective action with a spectral Callan-Symanzik cutoff, and therefore gives access to Lorentz invariant spectral flows. The functional setup is fully non-perturbative and allows for the spectral treatment of general theories. In particular, this includes theories that do not admit a perturbative renormalisation such as asymptotically safe theories. Finally, the application of the Lorentz invariant spectral functional renormalisation group is briefly discussed for theories ranging from real scalar and Yukawa theories to gauge theories and quantum gravity.

Scaling solutions for asymptotically free quantum gravity

We compute scaling solutions of functional flow equations for quantum grav- ity in a general truncation with up to four derivatives of the metric. They connect the asymptotically free ultraviolet fixed point, which is accessible to perturbation theory, to the non-perturbative infrared region. The existence of such scaling solutions is necessary for a renormalizable quantum field theory of gravity. If the proposed scaling solution is con- firmed beyond our approximations asymptotic freedom is a viable alternative to asymptotic safety for quantum gravity.

Scaling solution for field-dependent gauge couplings in quantum gravity

Quantum gravity can determine the dependence of gauge couplings in a scalar field, which is related to possible fifth forces and time varying fundamental “constants”. This prediction is based on the scaling solution of functional flow equations. For momenta below the field-dependent Planck mass the quantum scale invariant standard model emerges as an effective low energy theory. For a small non-zero value of the infrared cutoff scale the time-variation of couplings or apparent violations of the equivalence principle turn out to be negligibly small for the present cosmological epoch, unless some further quantum scale symmetry violation beyond the standard model comes into play. More sizable effects of quantum scale symmetry violation are expected during nucleosynthesis or before. Scaling solutions relate field dependence and dependence on the renormalization scale. We find asymptotically free gauge couplings for the standard model coupled to gravity, while for grand unified models the gauge couplings are asymptotically safe with non-zero values at the ultraviolet fixed point. The scaling solution of asymptotically safe metric quantum gravity yields restrictions for model building, as limiting the number of scalars in grand unified theories.

Sequence of phase transitions in a model of interacting rods.

In a system of interacting thin rigid rods of equal length 2ℓ on a two-dimensional grid of lattice spacing a, we show that there are multiple phase transitions as the coupling strength κ=ℓ/a and the temperature are varied. There are essentially two classes of transitions. One corresponds to the Ising-type spontaneous symmetry-breaking transition and the second belongs to less-studied phase transitions of geometrical origin. The latter class of transitions appears at fixed values of κ irrespective of the temperature, whereas the critical coupling for the spontaneous symmetry-breaking transition depends on it. By varying the temperature, the phase boundaries may cross each other, leading to a rich phase behavior with infinitely many phases. Our results are based on Monte Carlo simulations on the square lattice and a fixed-point analysis of a functional flow equationon a Bethe lattice.

Constructing CFTs from AdS flows

We study the renormalization group flow equations for correlation functions of weakly coupled quantum field theories in AdS. Taking the limit where the external points approach the conformal boundary, we obtain a flow of conformally invariant correlation functions. We solve the flow for one- and two-point functions and show that the corrections to the conformal dimensions can be obtained as an integral over the Mellin amplitude of the four-point function. We also derive the flow of the Mellin amplitude for higher n-point functions. We then consider the flows at tree level and one loop (in AdS), and show that one obtains exactly the recursion relations for the corresponding Mellin amplitudes derived earlier by Fitzpatrick et al. [1] at tree level and Yuan [2, 3] at one loop. As an application, we furthermore compute one-loop corrections to the conformal dimensions for some operators in the CFT dual to an O(N) scalar model in AdS.

Asymptotic freedom and safety in quantum gravity

We compute non-perturbative flow equations for the couplings of quantum gravity in fourth order of a derivative expansion. The gauge invariant functional flow equation for arbitrary metrics allows us to extract β-functions for all couplings. In our truncation we find two fixed points. One corresponds to asymptotically free higher derivative gravity, the other is an extension of the asymptotically safe fixed point in the Einstein-Hilbert truncation or extensions thereof. The infrared limit of the flow equations entails only unobservably small modifications of Einstein gravity coupled to a scalar field. Quantum gravity can be asymptotically free, based on a flow trajectory from the corresponding ultraviolet fixed point to the infrared region. This flow can also be realized by a scaling solution for varying values of a scalar field. As an alternative possibility, quantum gravity can be realized by asymptotic safety at the other fixed point. There may exist a critical trajectory between the two fixed points, starting in the extreme ultraviolet from asymptotic freedom. We compute critical exponents and determine the number of relevant parameters for the two fixed points. Evaluating the flow equation for constant scalar fields yields the universal gravitational contribution to the effective potential for the scalars.

Cosmological functional renormalization group, extended Galilean invariance, and approximate solutions to the flow equations

The functional renormalisation group is employed to study the non-linear regime of late-time cosmic structure formation. This framework naturally allows for non-perturbative approximation schemes, usually guided by underlying symmetries or a truncation of the theory space. An extended symmetry that is related to Galilean invariance is studied and corresponding Ward identities are derived. These are used to obtain (formally) closed renormalisation group flow equations for two-point correlation functions in the limit of large wave numbers (small scales). The flow equations are analytically solved in an approximation that is connected to the 'sweeping effect' previously described in the context of fluid turbulence.

QED in the exact renormalization group

Abstract The functional flow equation and the quantum master equation are consistently solved in perturbation for chiral symmetric QED with and without four-fermi interactions. Due to the presence of a momentum cutoff, unconventional features related to gauge symmetry are observed even in our perturbative results. In the absence of the four-fermi couplings, a one-loop calculation gives us the standard results of anomalous dimensions and the beta function for the gauge coupling, and therefore the Ward identity, Z1 = Z2. This is a consequence of the regularization-scheme independence in the one-loop computation. We also find a photon mass term. When included, four-fermi couplings contribute to the beta function and the Ward identity is also modified, Z1 ≠ Z2, due to a term proportional to the photon mass multiplied by the four-fermi couplings.

Quantum discontinuity fixed point and renormalization group flow of the Sachdev-Ye-Kitaev model

We determine the global renormalization group (RG) flow of the Sachdev-Ye-Kitaev (SYK) model. From a controlled truncation of the infinite hierarchy of the exact functional RG flow equations, we identify several fixed points. Apart from a stable fixed point, associated with the celebrated non-Fermi liquid state of the model, we find another stable fixed point related to an integer-valence state. These stable fixed points are separated by a discontinuity fixed point with one relevant direction, describing a quantum first-order transition. Most notably, the fermionic spectrum continues to be quantum critical even at the discontinuity fixed point. This rules out a description of the transition in terms of a local effective Ising variable as is established for classical transitions. We propose an entangled quantum state at phase coexistence as a possible physical origin of this critical behavior.

Fundamental scale invariance

We propose fundamental scale invariance as a new theoretical principle beyond renormalizability. Quantum field theories with fundamental scale invariance admit a scale-free formulation of the functional integral and effective action in terms of scale invariant fields. They correspond to exact scaling solutions of functional renormalization flow equations. Such theories are highly predictive since all relevant parameters for deviations from the exact scaling solution vanish. Realistic particle physics and quantum gravity are compatible with this setting. The non-linear restrictions for scaling solutions can explain properties as an asymptotically vanishing cosmological constant or dynamical dark energy that would seem to need fine tuning of parameters from a perturbative viewpoint. As an example we discuss a pregeometry based on a diffeomorphism invariant Yang-Mills theory. It is a candidate for an ultraviolet completion of quantum gravity with a well behaved graviton propagator at short distances.

Fluctuating vector mesons in analytically continued functional RG flow equations

In this work we study contributions due to vector and axial-vector meson fluctuations to their in-medium spectral functions in an effective low-energy theory inspired by the gauged linear sigma model. In particular, we show how to describe these fluctuations in the effective theory by massive (axial-)vector fields in agreement with the known structure of analogous single-particle or resonance contributions to the corresponding conserved currents. The vector and axial-vector meson spectral functions are then computed by numerically solving the analytically continued functional renormalization group flow equations for their retarded two-point functions at finite temperature and density in the effective theory. We identify the new contributions that arise due to the (axial-)vector meson fluctuations, and assess their influence on possible signatures of a QCD critical end point and the restoration of chiral symmetry in thermal dilepton spectra.

Stationary, isotropic and homogeneous two-dimensional turbulence: a first non-perturbative renormalization group approach

We study the statistical properties of stationary, isotropic and homogeneous turbulence in two-dimensional (2D) flows, focusing on the direct cascade, that is on large wave-numbers compared to the integral scale, where both energy and enstrophy are provided to the fluid. Our starting point is the 2D Navier–Stokes equation in the presence of a stochastic forcing, or more precisely the associated field theory. We unveil two extended symmetries of the Navier–Stokes action which were not yet identified, one related to time-dependent (or time-gauged) shifts of the response fields and existing in both 2D and 3D, and the other to time-gauged rotations and specific to 2D flows. We derive the corresponding Ward identities, and exploit them within the non-perturbative renormalization group formalism, and the large wave-number expansion scheme developed in Tarpin et al (2018 Phys. Fluids 30 055102). We consider the flow equation for a generalized n-point correlation function, and calculate its leading order term in the large wave-number expansion. At this order, the resulting flow equation can be closed exactly. We solve the fixed point equation for the 2-point function, which yields its explicit time dependence, for both small and large time delays in the stationary turbulent state. On the other hand, at equal times, the leading order term vanishes, so we compute the next-to-leading order term. We find that the flow equations for equal-time n-point correlation functions are not fully constrained by the set of extended symmetries, and discuss the consequences.

Renormalization and ultraviolet sensitivity of gauge vertices in universal extra dimensions

When computing radiative corrections in models with compactified extra dimensions, one has to sum over the entire tower of Kaluza-Klein excitations inside the loops. The loop corrections generate a difference between the coupling strength of a zero-mode gauge boson and the coupling strength of its Kaluza-Klein excitation, although both originate from the same higher-dimensional gauge interaction. Furthermore, this discrepancy will in general depend on the cutoff scale and assumptions about the UV completion of the extra-dimensional theory. In this article, these effects are studied in detail within the context of the minimal universal extra dimension model (MUED). The broad features of the cutoff scale dependence can be captured through the solution of the functional flow equation in five-dimensional space. However, an explicit diagrammatic calculation reveals some modifications due to the compactification of the extra dimension. Nevertheless, when imposing a physical renormalization condition, one finds that the UV sensitivity of the effective Kaluza-Klein gauge-boson vertex is relatively small and not very important for most phenomenological purposes. Similar conclusions should hold in a larger class of extra-dimensional models besides MUED.

Gauge-invariant fields and flow equations for Yang–Mills theories

We discuss the concept of gauge-invariant fields for non-abelian gauge theories. Infinitesimal fluctuations around a given gauge field can be split into physical and gauge fluctuations. Starting from some reference field the gauge-invariant fields are constructed by consecutively adding physical fluctuations. An arbitrary gauge field can be mapped to an associated gauge invariant field. An effective action that depends on gauge-invariant fields becomes a gauge-invariant functional of arbitrary gauge fields by associating to every gauge field the corresponding gauge-invariant field. The gauge-invariant effective action can be obtained from an implicit functional integral with a suitable “physical gauge fixing”. We generalize this concept to the gauge-invariant effective average action or flowing action, which involves an infrared cutoff. It obeys a gauge-invariant functional flow equation. We demonstrate the use of this flow equation by a simple computation of the running gauge coupling and propagator in pure SU(N)-Yang–Mills theory.

Gauge-invariant flow equation

We propose a closed gauge-invariant functional flow equation for Yang–Mills theories and quantum gravity that only involves one macroscopic gauge field or metric. It is based on a projection on physical and gauge fluctuations. Deriving this equation from a functional integral we employ the freedom in the precise choice of the macroscopic field and the effective average action in order to realize a closed and simple form of the flow equation.

Nonseparable frequency dependence of the two-particle vertex in interacting fermion systems

We derive functional flow equations for the two-particle vertex and the self-energy in interacting fermion systems which capture the full frequency dependence of both quantities. The equations are applied to the hole-doped two-dimensional Hubbard model as a prototype system with entangled magnetic, charge and pairing fluctuations. Each fluctuation channel acquires substantial dependencies on all three Matsubara frequencies, such that the frequency dependence of the vertex cannot be accurately represented by a channel sum with only one frequency variable in each term. At the temperatures we are able to access, the leading instabilities are mostly antiferromagnetic, with an incommensurate wave vector. However, at large doping, a divergence in the charge channel occurs at a finite frequency transfer, if the vertex flow is computed without self-energy feedback. This enigmatic instability was already observed in a calculation by Husemann et al. [Phys. Rev. B 85, 075121 (2012)], who used an approximate separable ansatz for the frequency dependence of the vertex. We identify a simple mechanism for this instability in terms of a random phase approximation for the charge channel with a frequency dependent effective magnetic interaction as input. In spite of the strong momentum and frequency dependence of the vertex, the self-energy has a Fermi liquid form. At the moderate interaction strength where our approach is applicable, we obtain a moderate reduction of the quasi-particle weight and a sizable decay rate with a pronounced momentum dependence. Nevertheless, the self-energy feedback into the vertex flow turns out to be crucial, as it suppresses the unphysical finite frequency charge instability.

Competing many-body instabilities in two-dimensional dipolar Fermi gases

Experiments on quantum degenerate Fermi gases of magnetic atoms and dipolar molecules begin to probe their broken symmetry phases dominated by the long-range, anisotropic dipole-dipole interaction. Several candidate phases including the p-wave superfluid, the stripe density wave, and a supersolid have been proposed theoretically for two-dimensional spinless dipolar Fermi gases. Yet the phase boundaries predicted by different approximations vary greatly, and a definitive phase diagram is still lacking. Here we present a theory that treats all competing many-body instabilities in the particle-particle and particle-hole channel on equal footing. We obtain the low temperature phase diagram by numerically solving the functional renormalization-group flow equations and find a nontrivial density wave phase at small dipolar tilting angles and strong interactions, but no evidence of the supersolid phase. We also estimate the critical temperatures of the ordered phases.