Effective Average Action Research Articles

A universe from a Lagrangian fixed point

In this paper, we investigate the theoretical possibility that a Lagrangian fixed point, when applied to cosmological models, can drive dynamical evolution towards a bouncing universe. We analyze the physics of a Lagrangian fixed point within the context of a gravitational average effective action featuring scale-dependent couplings. To explore this concept, we develop a toy model set in a four-dimensional, spatially flat spacetime, anchored by a Lagrangian fixed point. Solving the cosmological equations of this model analytically, we identify several non-trivial solution branches. These branches are characterized by a modified scale factor and dynamic gravitational couplings, offering new insights into the behavior of cosmological models under these conditions.

Functional renormalization group study of thermodynamic geometry around the phase transition of quantum chromodynamics

We investigate the thermodynamic geometry of the quark-meson model at finite temperature, T, and quark number chemical potential, μ. We extend previous works by the inclusion of fluctuations exploiting the functional renormalization group approach. We use recent developments to recast the flow equation into the form of an advection-diffusion equation. We adopt the local potential approximation for the effective average action. We focus on the thermodynamic curvature, R, in the (μ,T) plane, in proximity of the chiral crossover, up to the critical point of the phase diagram. We find that the inclusion of fluctuations results in a smoother behavior of R near the chiral crossover. Moreover, for small μ, R remains negative, signaling the fact that bosonic fluctuations reduce the capability of the system to completely overcome the fermionic statistical repulsion of the quarks. We investigate in more detail the small μ region by analyzing a system in which we artificially lower the pion mass, thus approaching the chiral limit in which the crossover is actually a second order phase transition. On the other hand, as μ is increased and the critical point is approached, we find that R is enhanced and a sign change occurs, in agreement with mean field studies. Hence, we completely support the picture that R is sensitive to a crossover and a phase transition, and provides information about the effective behavior of the system at the phase transition. Published by the American Physical Society 2024

Approach to the lower critical dimension of the φ^{4} theory in the derivative expansion of the functional renormalization group.

We revisit the approach to the lower critical dimension d_{lc} in the Ising-like φ^{4} theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d≥2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near d_{lc}. We show that the convergence of the fixed-point effective potential is nonuniform in the field when d→d_{lc} with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension d_{lc} and for the behavior of the critical temperature as d→d_{lc}, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.

On the possibility of a novel (A)dS/CFT relationship emerging in Asymptotic Safety

Quantum Einstein Gravity (QEG), nonperturbatively renormalized by means of a certain asymptotically safe renormalization group (RG) trajectory, is explored by solving its scale dependent effective field equations and embedding the family of emerging 4-dimensional spacetimes into a single 5-dimensional manifold, which thus encodes the complete information about all scales. By construction the latter manifold is furnished with a natural foliation. Heuristically, its leaves are interpreted as physical spacetime observed on different scales of the experimental resolution. Generalizing earlier work on the embedding of d-dimensional Euclidean QEG spacetimes in (d + 1)-dimensional flat or Ricci flat manifolds, we admit Lorentzian signature in this paper and we consider embeddings in arbitrary (d + 1)-dimensional Einstein spaces. Special attention is paid to the sector of maximally symmetric metrics, and the fundamental definition of QEG in d = 4 that employs the cross-over trajectory connecting the non-Gaussian to the Gaussian RG fixed point. Concerning the embedding of the resulting family of 4D de Sitter solutions with a running Hubble parameter, we find that there are only two possible 5D spacetimes, namely the anti-de Sitter manifold AdS5 and the de Sitter manifold dS5. To arrive at this result essential use is made of the monotone scale dependence of the running cosmological constant featured by the gravitational effective average action. We show that if the scale invariance of the QEG fixed points extends to full conformal invariance, the 5D picture of the resulting geometric and field theoretic structure displays a novel kind of “AdS/CFT correspondence”. While strongly reminiscent of the usual string theory-based AdS/CFT correspondence, also clear differences are found.

Functional truncations for the solution of the nonperturbative RG equations

We consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of , where , φ is the averaged order parameter, and corresponds to the minimum of the dimensionless potential , depending on the infrared cut-off scale k. We propose a new approach of functional truncations, using the expansion , where , is an optimization parameter and µ is the exponent, describing the asymptotic. The newly developed method provides accurate estimates of the critical exponents η, ν and also ω. In the case of the local potential approximation (LPA), the discrepancy with the purely numerical solution is for ν and 0.0012 for ω at the s 13 order of truncation. We show that this method is advantageous for estimations beyond the LPA, especially, for the critical exponent ω. In particular, we have obtained , , and for the equation originally derived in 2020 J. Phys. A: Math. Theor. 53 415002 within a new truncation scheme for the effective action. The approach can also be used for the solution of other nonperturbative RG models that have a broad range of applications.

Essential renormalisation group

We propose a novel scheme for the exact renormalisation group motivated by the desire of reducing the complexity of practical computations. The key idea is to specify renormalisation conditions for all inessential couplings, leaving us with the task of computing only the flow of the essential ones. To achieve this aim, we utilise a renormalisation group equation for the effective average action which incorporates general non-linear field reparameterisations. A prominent feature of the scheme is that, apart from the renormalisation of the mass, the propagator evaluated at any constant value of the field maintains its unrenormalised form. Conceptually, the simplifications can be understood as providing a description based only on quantities that enter expressions for physical observables since the redundant, non-physical content is automatically disregarded. To exemplify the scheme’s utility, we investigate the Wilson-Fisher fixed point in three dimensions at order two in the derivative expansion. In this case, the scheme removes all order \partial^2∂2 operators apart from the canonical term. Further simplifications occur at higher orders in the derivative expansion. Although we concentrate on a minimal scheme that reduces the complexity of computations, we propose more general schemes where inessential couplings can be tuned to optimise a given approximation. We further discuss the applicability of the scheme to a broad range of physical theories.

Statefinder analysis of scale-dependent cosmology

We study the statefinder parameters of a cosmological model based on scale-dependent gravity. The effective Einstein field equations come from an average effective action. From the dynamical system, we derive analytical expressions that improve the convergence of the numerical solutions. We determine the statefinder parameters for moderate redshift and compare them with well-known alternatives to ΛCDM.

Scalar-tensor theories within Asymptotic Safety

Asymptotic Safety provides an elegant mechanism for obtaining a consistent high-energy completion of gravity and gravity-matter systems. Following the initial idea by Steven Weinberg, the construction builds on an interacting fixed point of the theories renormalization group (RG) flow. In this work we use the Wetterich equation for the effective average action to investigate the RG flow of gravity supplemented by a real scalar field. We give a non-perturbative proof that the subspace of interactions respecting the global shift-symmetry of the scalar kinetic term is closed under RG transformations. Subsequently, we compute the beta functions in an approximation comprising the Einstein-Hilbert action supplemented by the shift-symmetric quartic scalar self-interaction and the two lowest order shift-symmetric interactions coupling scalar-bilinears to the spacetime curvature. The computation utilizes the background field method with an arbitrary background, demonstrating that the results are manifestly background independent. Our beta functions exhibit an interacting fixed point suitable for Asymptotic Safety, where all matter interactions are non-vanishing. The presence of this fixed point is rooted in the interplay of the matter couplings which our work tracks for the first time. The relation of our findings with previous results in the literature is discussed in detail and we conclude with a brief outlook on potential phenomenological applications.

Gauge Dependence of Effective Average Action

The gauge dependence of effective average action in the functional renormalization group is studied. The effective average action is considered as non-perturbative solution to the flow equation which is the basic equation of the method. It is proven that at any scale of IR cutoff the effective average action depends on gauges making impossible physical interpretation of all obtained results in this method.

Kantowski–Sachs cosmology, Weyl geometry, and asymptotic safety in quantum gravity

A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvement of the Schwarzschild black hole that removes the r = 0 singularity is presented. It is followed with a RG improvement of the Kantowski–Sachs metric associated with a Schwarzschild black hole interior such that there is no singularity at t = 0 due to the running Newtonian coupling G(t) vanishing at t = 0. Two temporal horizons at [Formula: see text] and [Formula: see text] are found. For times below the Planck scale t < t P, and above the Hubble time t > t H, the components of the Kantowski–Sachs metric exhibit a key sign change, so the roles of the spatial z and temporal t coordinates are exchanged, and one recovers a repulsive inflationary de Sitter-like core around z = 0, and a Schwarzschild-like metric in the exterior region z > R H = 2G o M. The inclusion of a running cosmological constant Λ(t) follows. We proceed with the study of a dilaton-gravity (scalar–tensor theory) system within the context of Weyl’s geometry that permits singling out the expression for the classical potential [Formula: see text], instead of being introduced by hand, and find a family of metric solutions that are conformally equivalent to the (anti) de Sitter metric. To conclude, an ansatz for the truncated effective average action of ordinary dilaton gravity in Riemannian geometry is introduced, and a RG-improved cosmology based on the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric is explored where instead of recurring to the cutoff identification k = k(t) = ξH(t), based on the Hubble function H(t), with ξ a positive constant, one now has [Formula: see text], when [Formula: see text] is a positive-definite dilaton scalar field that is monotonically decreasing with time.

Asymptotically safe gravity with fermions

We use the functional renormalization group equation for the effective average action to study the fixed point structure of gravity-fermion systems on a curved background spacetime. We approximate the effective average action by the Einstein-Hilbert action supplemented by a fermion kinetic term and a coupling of the fermion bilinears to the spacetime curvature. The latter interaction is singled out based on a “smart truncation building principle”. The resulting renormalization group flow possesses two families of interacting renormalization group fixed points extending to any number of fermions. The first family exhibits an upper bound on the number of fermions for which the fixed points could provide a phenomenologically interesting high-energy completion via the asymptotic safety mechanism. The second family comes without such a bound. The inclusion of the non-minimal gravity-matter interaction is crucial for discriminating the two families. Our work also clarifies the origin of the strong regulator-dependence of the fixed point structure reported in earlier literature and we comment on the relation of our findings to studies of the same system based on a vertex expansion of the effective average action around a flat background spacetime.

Why the Cosmological Constant Seems to Hardly Care About Quantum Vacuum Fluctuations: Surprises From Background Independent Coarse Graining

Background Independence is a sine qua non for every satisfactory theory of Quantum Gravity. In particular if one tries to establish a corresponding notion of Wilsonian renormalization, or coarse graining, it presents a major conceptual and technical difficulty usually. In this paper we adopt the approach of the gravitational Effective Average Action and demonstrate that generically coarse graining in Quantum Gravity and in standard field theories on a non-dynamical spacetime are profoundly different. By means of a concrete example, which in connection with the cosmological constant problem is also interesting in its own right, we show that the surprising and sometimes counterintuitive implications of Background Independent coarse graining are neither restricted to high energies, nor to strongly nonperturbative regimes. In fact, while our approach has been employed in most studies of Asymptotic Safety, this particular ultraviolet behaviour plays no essential role in the present context.

Gauge dependence of alternative flow equation for the functional renormalization group

The gauge dependence problem of alternative flow equation for the functional renormalization group is studied. It is shown that the effective two-particle irreducible effective action depends on gauges at any value of IR parameter k. The situation with gauge dependence is similar to the standard formulation based on the effective one-particle irreducible average action.

On Characterizing the Quantum Geometry Underlying Asymptotic Safety

The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background d-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a perturbative regime where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) R^2-operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.

The conformal anomaly and a new exact RG

For scalar field theory, a new generalization of the Exact RG to curved space is proposed, in which the conformal anomaly is explicitly present. Vacuum terms require regularization beyond that present in the canonical formulation of the Exact RG, which can be accomplished by adding certain free fields, each at a non-critical fixed-point. Taking the Legendre transform, the sole effect of the regulator fields is to remove a divergent vacuum term and they do not explicitly appear in the effective average action. As an illustration, both the integrated conformal anomaly and Polyakov action are recovered for the Gaussian theory in d=2.

On the scaling of composite operators in asymptotic safety

The Asymptotic Safety hypothesis states that the high-energy completion of gravity is provided by an interacting renormalization group fixed point. This implies non-trivial quantum corrections to the scaling dimensions of operators and correlation functions which are characteristic for the corresponding universality class. We use the composite operator formalism for the effective average action to derive an analytic expression for the scaling dimension of an infinite family of geometric operators int {d}^dxsqrt{g}{R}^n . We demonstrate that the anomalous dimensions interpolate continuously between their fixed point value and zero when evaluated along renormalization group trajectories approximating classical general relativity at low energy. Thus classical geometry emerges when quantum fluctuations are integrated out. We also compare our results to the stability properties of the interacting renormalization group fixed point projected to f (R)-gravity, showing that the composite operator formalism in the single-operator approximation cannot be used to reliably determine the number of relevant parameters of the theory.

Functional renormalization group approach and gauge dependence in gravity theories

We investigate the gauge symmetry and gauge fixing dependence properties of the effective average action for quantum gravity models of general form. Using the background field formalism and the standard BRST-based arguments, one can establish the special class of regulator functions that preserves the background field symmetry of the effective average action. Unfortunately, regardless the gauge symmetry is preserved at the quantum level, the non-invariance of the regulator action under the global BRST transformations leads to the gauge fixing dependence even under the use of the on-shell conditions.

Cosmology of Bianchi type-I metric using renormalization group approach for quantum gravity

We study the anisotropic Bianchi type-I cosmological model at late times, taking into account quantum gravitational corrections in the formalism of the exact renormalization group flow of the effective average action for gravity. The cosmological evolution equations are derived by including the scale dependence of Newton’s constant G and cosmological constant . We have considered the solutions of the flow equations for G and at next to leading order in the infrared cutoff scale. Using these scale dependent G and in Einstein equations for the Bianchi-I model, we obtain the scale factors in different directions. It is shown that the scale factors eventually evolve into FLRW universe for known matter like radiation. However, for dust and stiff matter we find that the universe need not evolve to the FLRW cosmology in general, but can also show Kasner type behaviour.

BRST in the exact renormalization group

AbstractWe show, explicitly within perturbation theory, that the quantum master equation and the Wilsonian renormalization group flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naïvely break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov–Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov–Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to the choice of the renormalization scheme.

Background independent quantum field theory and gravitating vacuum fluctuations

The scale dependent effective average action for quantum gravity complies with the fundamental principle of Background Independence. Ultimately the background metric it formally depends on is selected self-consistently by means of a tadpole condition, a generalization of Einstein’s equation. Self-consistent background spacetimes are scale dependent, and therefore “going on-shell” at the points along a given renormalization group (RG) trajectory requires understanding two types of scale dependencies: the (familiar) direct one carried by the off-shell action functional, and an equally important indirect one related to the continual re-adjustment of the background geometry. This paper is devoted to a careful delineation and analysis of certain general questions concerning the indirect scale dependence, as well as a detailed explicit investigation of the case where the self-consistent metrics are determined predominantly by the RG running of the cosmological constant. Mathematically, the object of interest is the spectral flow induced by the background Laplacian which, on-shell, acquires an explicit scale dependence. Among other things, it encodes the complete information about the specific set of field modes which, at any given scale, are the degrees of freedom constituting the respective effective field theory. For a large class of RG trajectories (Type IIIa) we discover a seemingly paradoxical behavior that differs substantially from the off-shell based expectations: A Background Independent theory of (matter coupled) quantum gravity looses rather than gains degrees of freedom at increasing energies. As an application, we investigate to what extent it is possible to reformulate the exact theory in terms of matter and gravity fluctuations on a rigid flat space. It turns out that, in vacuo, this “rigid picture” breaks down after a very short RG time already (less than one decade of scales) because of a “scale horizon” that forms. Furthermore, we critically reanalyze and refute the frequent claim that the huge energy densities one obtains in standard quantum field theory by summing up to zero-point energies imply a naturalness problem for the observed small value of the cosmological constant.