Effective Average Action Research Articles

The unitary conformal field theory behind 2D Asymptotic Safety

Being interested in the compatibility of Asymptotic Safety with Hilbert space positivity (unitarity), we consider a local truncation of the functional RG flow which describes quantum gravity in $d>2$ dimensions and construct its limit of exactly two dimensions. We find that in this limit the flow displays a nontrivial fixed point whose effective average action is a non-local functional of the metric. Its pure gravity sector is shown to correspond to a unitary conformal field theory with positive central charge $c=25$. Representing the fixed point CFT by a Liouville theory in the conformal gauge, we investigate its general properties and their implications for the Asymptotic Safety program. In particular, we discuss its field parametrization dependence and argue that there might exist more than one universality class of metric gravity theories in two dimensions. Furthermore, studying the gravitational dressing in 2D asymptotically safe gravity coupled to conformal matter we uncover a mechanism which leads to a complete quenching of the a priori expected Knizhnik-Polyakov-Zamolodchikov (KPZ) scaling. A possible connection of this prediction to Monte Carlo results obtained in the discrete approach to 2D quantum gravity based upon causal dynamical triangulations is mentioned. Similarities of the fixed point theory to, and differences from, non-critical string theory are also described. On the technical side, we provide a detailed analysis of an intriguing connection between the Einstein-Hilbert action in $d>2$ dimensions and Polyakov's induced gravity action in two dimensions.

Renormalization group equation and scaling solutions for f(R) gravity in exponential parametrization

We employ the exponential parametrization of the metric and a "physical" gauge fixing procedure to write a functional flow equation for the gravitational effective average action in an $f(R)$ truncation. The background metric is a four-sphere and the coarse-graining procedure contains three free parameters. We look for scaling solutions, i.e. non-Gaussian fixed points for the function $f$. For a discrete set of values of the parameters, we find simple global solutions of quadratic polynomial form. For other values, global solutions can be found numerically. Such solutions can be extended in certain regions of parameter space and have two relevant directions. We discuss the merits and the shortcomings of this procedure.

The metric on field space, functional renormalization, and metric–torsion quantum gravity

Searching for new non-perturbatively renormalizable quantum gravity theories, functional renormalization group (RG) flows are studied on a theory space of action functionals depending on the metric and the torsion tensor, the latter parameterized by three irreducible component fields. A detailed comparison with Quantum Einstein–Cartan Gravity (QECG), Quantum Einstein Gravity (QEG), and “tetrad-only” gravity, all based on different theory spaces, is performed. It is demonstrated that, over a generic theory space, the construction of a functional RG equation (FRGE) for the effective average action requires the specification of a metric on the infinite-dimensional field manifold as an additional input. A modified FRGE is obtained if this metric is scale-dependent, as it happens in the metric–torsion system considered.

Solutions to the reconstruction problem in asymptotic safety

Starting from a full renormalised trajectory for the effective average action (a.k.a. infrared cutoff Legendre effective action) $\Gamma_k$, we explicitly reconstruct corresponding bare actions, formulated in one of two ways. The first step is to construct the corresponding Wilsonian effective action $S^k$ through a tree-level expansion in terms of the vertices provided by $\Gamma_k$. It forms a perfect bare action giving the same renormalised trajectory. A bare action with some ultraviolet cutoff scale $\Lambda$ and infrared cutoff $k$ necessarily produces an effective average action $\Gamma^\Lambda_k$ that depends on both cutoffs, but if the already computed $S^\Lambda$ is used, we show how $\Gamma^\Lambda_k$ can also be computed from $\Gamma_k$ by a tree-level expansion, and that $\Gamma^\Lambda_k\to\Gamma_k$ as $\Lambda\to\infty$. Along the way we show that Legendre effective actions with different UV cutoff profiles, but which correspond to the same Wilsonian effective action, are related through tree-level expansions. All these expansions follow from Legendre transform relationships that can be derived from the original one between $\Gamma^\Lambda_k$ and $S^k$.

Loop expansion of the average effective action in the functional renormalization group approach

We formulate a perturbation expansion for the effective action in a new approach to the functional renormalization group method based on the concept of composite fields for regulator functions being their most essential ingredients. We demonstrate explicitly the principal difference between the properties of effective actions in these two approaches existing already on the one-loop level in a simple gauge model.

A proper fixed functional for four-dimensional Quantum Einstein Gravity

Realizing a quantum theory for gravity based on Asymptotic Safety hinges on the existence of a non-Gaussian fixed point of the theory’s renormalization group flow. In this work, we use the functional renormalization group equation for the effective average action to study the fixed point underlying Quantum Einstein Gravity at the functional level including an infinite number of scale-dependent coupling constants. We formulate a list of guiding principles underlying the construction of a partial differential equation encoding the scale-dependence of f(R)-gravity. We show that this equation admits a unique, globally well-defined fixed functional describing the non-Gaussian fixed point at the level of functions of the scalar curvature. This solution is constructed explicitly via a numerical double-shooting method. In the UV, this solution is in good agreement with results from polynomial expansions including a finite number of coupling constants, while it scales proportional to R2, dressed up with non-analytic terms, in the IR. We demonstrate that its structure is mainly governed by the conformal sector of the flow equation. The relation of our work to previous, partial constructions of similar scaling solutions is discussed.

Scaling relations and multicritical phenomena from functional renormalization.

We investigate multicritical phenomena in O(N)+O(M) models by means of nonperturbative renormalization group equations. This constitutes an elementary building block for the study of competing orders in a variety of physical systems. To identify possible multicritical points in phase diagrams with two ordered phases, we compute the stability of isotropic and decoupled fixed point solutions from scaling potentials of single-field models. We verify the validity of Aharony's scaling relation within the scale-dependent derivative expansion of the effective average action. We discuss implications for the analysis of multicritical phenomena with truncated flow equations. These findings are an important step towards studies of competing orders and multicritical quantum phase transitions within the framework of functional renormalization.

Renormalization group flow equations for the proper vertices of the background effective average action

We derive a system of coupled flow equations for the proper vertices of the background effective average action and we give an explicit representation of these by means of diagrammatic and momentum space techniques. This explicit representation can be used as a new computational technique that enables the projection of the flow of a large new class of truncations of the background effective average action. In particular, these can be single- or bifield truncations of local or nonlocal character. As an application we study non-Abelian gauge theories. We show how to use this new technique to calculate the beta function of the gauge coupling (without employing the heat kernel expansion) under various approximations. In particular, one of these approximations leads to a derivation of beta functions similar to those proposed as candidates for an ``all-orders'' beta function. Finally, we discuss some possible phenomenology related to these flows.

Group field theory in dimension4−ϵ

Building on an analogy with ordinary scalar field theories, an $ϵ$-expansion for rank-3 tensorial group field theories with gauge invariance condition is introduced. This allows us to continuously interpolate between the dimension four group $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ and the dimension three SU(2). In the first situation, there is a unique marginal ${\ensuremath{\varphi}}^{4}$ coupling constant, but in contrast to ordinary scalar field theory this model is asymptotically free. In the SU(2) case, the presence of two marginally relevant ${\ensuremath{\varphi}}^{6}$ coupling constants and one ${\ensuremath{\varphi}}^{4}$ super-renormalizable interaction spoils this interesting property. However, the existence of a nontrivial fixed point is established in dimension $4\ensuremath{-}ϵ$, hence suggesting that the SU(2) theory might be asymptotically safe. To pave the way to future nonperturbative calculations, the present perturbative results are discussed in the framework of the effective average action.

Towards a C-function in 4D quantum gravity

We develop a generally applicable method for constructing functions, $C$, which have properties similar to Zamolodchikov's $C$-function, and are geometrically natural objects related to the theory space explored by non-perturbative functional renormalization group (RG) equations. Employing the Euclidean framework of the Effective Average Action (EAA), we propose a $C$-function which can be defined for arbitrary systems of gravitational, Yang-Mills, ghost, and bosonic matter fields, and in any number of spacetime dimensions. It becomes stationary both at critical points and in classical regimes, and decreases monotonically along RG trajectories provided the breaking of the split-symmetry which relates background and quantum fields is sufficiently weak. Within the Asymptotic Safety approach we test the proposal for Quantum Einstein Gravity in $d>2$ dimensions, performing detailed numerical investigations in $d=4$. We find that the bi-metric Einstein-Hilbert truncation of theory space introduced recently is general enough to yield perfect monotonicity along the RG trajectories, while its more familiar single-metric analog fails to achieve this behavior which we expect on general grounds. Investigating generalized crossover trajectories connecting a fixed point in the ultraviolet to a classical regime with positive cosmological constant in the infrared, the $C$-function is shown to depend on the choice of the gravitational instanton which constitutes the background spacetime. For de Sitter space in 4 dimensions, the Bekenstein-Hawking entropy is found to play a role analogous to the central charge in conformal field theory. We also comment on the idea of a `$\Lambda$-$N$ connection' and the `$N$-bound' discussed earlier.

Field-dependent BRST–anti-BRST Lagrangian transformations

We continue our study of finite BRST–anti-BRST transformations for general gauge theories in Lagrangian formalism, initiated in [arXiv:1405.0790 [hep-th] and arXiv:1406.0179 [hep-th]], with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters, and prove the correctness of the explicit Jacobian in the partition function announced in [arXiv:1406.0179 [hep-th]], which corresponds to a change of variables with functionally dependent parameters λa = UaΛ induced by a finite Bosonic functional Λ(ϕ, π, λ) and by the anticommuting generators Ua of BRST–anti-BRST transformations in the space of fields ϕ and auxiliary variables πa, λ. We obtain a Ward identity depending on the field-dependent parameters λa and study the problem of gauge dependence, including the case of Yang–Mills theories. We examine a formulation with BRST–anti-BRST symmetry breaking terms, additively introduced into the quantum action constructed by the Sp(2)-covariant Lagrangian rules, obtain the Ward identity and investigate the gauge independence of the corresponding generating functional of Green's functions. A formulation with BRST symmetry breaking terms is developed. It is argued that the gauge independence of the above generating functionals is fulfilled in the BRST and BRST–anti-BRST settings. These concepts are applied to the average effective action in Yang–Mills theories within the functional renormalization group approach.

A new functional flow equation for Einstein–Cartan quantum gravity

We construct a special-purpose functional flow equation which facilitates non-perturbative renormalization group (RG) studies on theory spaces involving a large number of independent field components that are prohibitively complicated using standard methods. Its main motivation are quantum gravity theories in which the gravitational degrees of freedom are carried by a complex system of tensor fields, a prime example being Einstein–Cartan theory, possibly coupled to matter. We describe a sequence of approximation steps leading from the functional RG equation of the Effective Average Action to the new flow equation which, as a consequence, is no longer fully exact on the untruncated theory space. However, it is by far more “user friendly” when it comes to projecting the abstract equation on a concrete (truncated) theory space and computing explicit beta-functions. The necessary amount of (tensor) algebra reduces drastically, and the usually very hard problem of diagonalizing the pertinent Hessian operator is sidestepped completely. In this paper we demonstrate the reliability of the simplified equation by applying it to a truncation of the Einstein–Cartan theory space. It is parametrized by a scale dependent Holst action, depending on a O(4) spin-connection and the tetrad as the independent field variables. We compute the resulting RG flow, focusing in particular on the running of the Immirzi parameter, and compare it to the results of an earlier computation where the exact equation had been applied to the same truncation. We find consistency between the two approaches and provide further evidence for the conjectured non-perturbative renormalizability (asymptotic safety) of quantum Einstein–Cartan gravity. We also investigate a duality symmetry relating small and large values of the Immirzi parameter (γ→1/γ) which is displayed by the beta-functions in the absence of a cosmological constant.

New method of the functional renormalization group approach for Yang-Mills fields

We propose a new formulation of the functional renormalization group (FRG) approach, based on the use of regulator functions as composite operators. In this case one can provide (in contrast with standard approach) on-shell gauge-invariance for the effective average action.

En route to Background Independence: Broken split-symmetry, and how to restore it with bi-metric average actions

The most momentous requirement a quantum theory of gravity must satisfy is Background Independence, necessitating in particular an ab initio derivation of the arena all non-gravitational physics takes place in, namely spacetime. Using the background field technique, this requirement translates into the condition of an unbroken split-symmetry connecting the (quantized) metric fluctuations to the (classical) background metric. If the regularization scheme used violates split-symmetry during the quantization process it is mandatory to restore it in the end at the level of observable physics. In this paper we present a detailed investigation of split-symmetry breaking and restoration within the Effective Average Action (EAA) approach to Quantum Einstein Gravity (QEG) with a special emphasis on the Asymptotic Safety conjecture. In particular we demonstrate for the first time in a non-trivial setting that the two key requirements of Background Independence and Asymptotic Safety can be satisfied simultaneously. Carefully disentangling fluctuation and background fields, we employ a ‘bi-metric’ ansatz for the EAA and project the flow generated by its functional renormalization group equation on a truncated theory space spanned by two separate Einstein–Hilbert actions for the dynamical and the background metric, respectively. A new powerful method is used to derive the corresponding renormalization group (RG) equations for the Newton- and cosmological constant, both in the dynamical and the background sector. We classify and analyze their solutions in detail, determine their fixed point structure, and identify an attractor mechanism which turns out instrumental in the split-symmetry restoration. We show that there exists a subset of RG trajectories which are both asymptotically safe and split-symmetry restoring: In the ultraviolet they emanate from a non-Gaussian fixed point, and in the infrared they loose all symmetry violating contributions inflicted on them by the non-invariant functional RG equation. As an application, we compute the scale dependent spectral dimension which governs the fractal properties of the effective QEG spacetimes at the bi-metric level. Earlier tests of the Asymptotic Safety conjecture almost exclusively employed ‘single-metric truncations’ which are blind towards the difference between quantum and background fields. We explore in detail under which conditions they can be reliable, and we discuss how the single-metric based picture of Asymptotic Safety needs to be revised in the light of the new results. We shall conclude that the next generation of truncations for quantitatively precise predictions (of critical exponents, for instance) is bound to be of the bi-metric type.

A functional RG equation for the c-function

After showing how to prove the integrated c-theorem within the functional RG framework based on the effective average action, we derive an exact RG flow equation for Zamolodchikov's c-function in two dimensions by relating it to the flow of the effective average action. In order to obtain a non-trivial flow for the c-function, we will need to understand the general form of the effective average action away from criticality, where nonlocal invariants, with beta functions as coefficients, must be included in the ansatz to be consistent. We then apply our construction to several examples: exact results, local potential approximation and loop expansion. In each case we construct the relative approximate c-function and find it to be consistent with Zamolodchikov's c-theorem. Finally, we present a relation between the c-function and the (matter induced) beta function of Newton's constant, allowing us to use heat kernel techniques to compute the RG running of the c-function.

Critical temperature and superfluid gap of the unitary Fermi gas from functional renormalization

We investigate the superfluid transition of the unitary Fermi gas by means of the functional renormalization group, aiming at quantitative precision. We extract ${T}_{\mathrm{c}}/\ensuremath{\mu}=0.38(2)$ and $\ensuremath{\Delta}/\ensuremath{\mu}=1.04(15)$ for the critical temperature and the superfluid gap at zero temperature, respectively, within a systematic improvement of the truncation for the effective average action. The key ingredient in comparison to previous approaches consists in the use of regulators which cut off both frequencies and momenta. We incorporate renormalization effects on both the bosonic and the fermionic propagators, include higher-order bosonic scattering processes, and investigate the regulator and specification parameter dependence for an error estimate. The ratio $\ensuremath{\Delta}/{T}_{\mathrm{c}}=2.7(3)$ becomes less sensitive to the relative cutoff scale of bosons and fermions when improving the truncation. The techniques developed in this work are easily carried over to the cases of finite scattering length, lower dimensionality, and spin imbalance.

Stability of chromomagnetic condensation and mass generation for confinement in SU(2) Yang-Mills theory

We show that the Nielsen-Olesen instability of the Savvidy vacuum with a homogeneous chromomagnetic condensation disappears in the framework of the functional renormalization group. This result follows from our observations: (i) the vanishing imaginary part of the effective average action is realized for arbitrary infrared cutoff as a novel fixed point solution of the flow equation for the complex-valued effective average action and (ii) an approximate analytical solution for the effective average action is obtained without the pure imaginary part for large infrared cutoff. This result suggests that there exists a physical mechanism for maintaining the stability or staying on the fixed point even for sufficiently small infrared cutoff. We argue that dynamical gluon mass generation (related to two-gluon bound states identified with glueballs) occurs due to the Becchi-Rouet-Stora-Tyutin-invariant vacuum condensate of mass dimension two without causing instability.

Consistent closure of renormalization group flow equations in quantum gravity

We construct a consistent closure for the beta functions of the cosmological and Newton's constants by evaluating the influence of the fluctuating metric and ghost fields anomalous dimensions on their flow. In this generalized framework we confirm the presence of a UV attractive non-Gaussian fixed-point. Our closure method is general and can be applied systematically to more general truncations of the gravitational effective average action.

Black holes within asymptotic safety

Black holes are probably among the most fascinating objects populating our universe. Their characteristic features found within general relativity, encompassing space–time singularities, event horizons, and black hole thermodynamics, provide a rich testing ground for quantum gravity ideas. We review the status of black holes within a particular proposal for quantum gravity, Weinberg's asymptotic safety program. Starting from a brief survey of the effective average action and scale setting procedures, an improved quantum picture of the black hole is developed. The Schwarzschild black hole and its generalizations including angular momenta, higher-derivative corrections and the implications of extra dimensions are discussed in detail. In addition, the quantum singularity emerging for the inclusion of a cosmological constant is elucidated and linked to the phenomenon of a dynamical dimensional reduction of space–time.

Structural aspects of asymptotically safe black holes

We study the quantum modifications of classical, spherically symmetric Schwarzschild (anti-) de Sitter black holes within quantum Einstein gravity. The quantum effects are incorporated through the running coupling constants Gk and Λk, computed within the exact renormalization group approach, and a common scale-setting procedure. We find that, in contrast to common intuition, it is actually the cosmological constant that determines the short-distance structure of the RG-improved black hole: in the asymptotic UV the structure of the quantum solutions is universal and given by the classical Schwarzschild–de Sitter solution, entailing a self-similarity between the classical and quantum regime. As a consequence asymptotically safe black holes evaporate completely and no Planck-size remnants are formed. Moreover, the thermodynamic entropy of the critical Nariai black hole is shown to agree with the microstate count based on the effective average action, suggesting that the entropy originates from quantum fluctuations around the mean-field geometry.