The gravitational asymptotic safety program strives for a consistent and predictive quantum theory of gravity based on a nontrivial ultraviolet fixed point of the renormalization group (RG) flow. We investigate this scenario by employing a novel functional renormalization group equation which takes the causal structure of space-time into account and connects the RG flows for Euclidean and Lorentzian signature by a Wick rotation. Within the Einstein-Hilbert approximation, the β functions of both signatures exhibit ultraviolet fixed points in agreement with asymptotic safety. Surprisingly, the two fixed points have strikingly similar characteristics, suggesting that Euclidean and Lorentzian quantum gravity belong to the same universality class at high energies.

Within the gravitational asymptotic safety program, the RG flow of the R 2 truncation in three and four spacetime dimensions is analyzed in detail. In particular, we construct RG trajectories which emanate from the non-Gaussian UV fixed point and possess long classical regimes where the effective average action is well approximated by the classical Einstein-Hilbert action. As an application we study the spectral dimension of the effective QEG spacetimes resulting from these trajectories, establishing that the picture of a multi-fractal spacetime is robust under the extension of the truncated theory space. We demonstrate that regimes of constant spectral dimensions can either be attributed to universal features of RG fixed points or singular loci in the β functions.

We investigate chiral Higgs-Yukawa models with a non-abelian gauged left-handed sector reminiscent to a sub-sector of the standard model. We discover a new weak-coupling fixed-point behavior that allows for ultraviolet complete RG trajectories which can be connected with a conventional long-range infrared behavior in the Higgs phase. This non-trivial ultraviolet behavior is characterized by asymptotic freedom in all interaction couplings, but a quasi conformal behavior in all mass-like parameters. The stable microscopic scalar potential asymptotically approaches flatness in the ultraviolet, however, with a non-vanishing minimum increasing inversely proportional to the asymptotically free gauge coupling. This gives rise to nonperturbative -though weak-couplingthreshold effects which induce ultraviolet stability along a line of fixed points. Despite the weakcoupling properties, the system exhibits non-Gaußian features which are distinctly different from its standard perturbative counterpart: e.g., on a branch of the line of fixed points, we find linear instead of quadratically running renormalization constants. Whereas the Fermi constant and the top mass are naturally of the same order of magnitude, our model generically allows for light Higgs boson masses. Realistic mass ratios are related to particular RG trajectories with a "walking" mid-momentum regime.

We search for asymptotic safety in a Yukawa system with a chiral U(NL)L⊗U(1)R symmetry, serving as a toy model for the standard-model Higgs sector. Using the functional RG as a nonperturbative tool, the leading-order derivative expansion exhibits admissible non-Gaußian fixed-points for 1 ≤ NL ≤ 57 which arise from a conformal threshold behavior induced by self-balanced bosonfermion fluctuations. If present in the full theory, the fixed-point would solve the triviality problem. Moreover, as one fixed point has only one relevant direction even with a reduced hierarchy problem, the Higgs mass as well as the top mass are a prediction of the theory in terms of the Higgs vacuum expectation value. In our toy model, the fixed point is destabilized at higher order due to massless Goldstone and fermion fluctuations, which are particular to our model and have no analogue in the standard model.

We derive an exact functional renormalization group equation for the projectable version of Hořava-Lifshitz gravity. The flow equation encodes the gravitational degrees of freedom in terms of the lapse function, shift vector and spatial metric and is manifestly invariant under background foliation-preserving diffeomorphisms. Its relation to similar flow equations for gravity in the metric formalism is discussed in detail, and we argue that the space of action functionals, invariant under the full diffeomorphism group, forms a subspace of the latter invariant under renormalization group transformations. As a first application we study the RG flow of the Newton constant and the cosmological constant in the ADM formalism. In particular we show that the non-Gaussian fixed point found in the metric formulation is qualitatively unaffected by the change of variables and persists also for Lorentzian signature metrics.

The functional renormalization group equation for projectable Hořava-Lifshitz gravity is used to derive the non-perturbative beta functions for the Newton's constant, cosmological constant and anisotropy parameter. The resulting coupled differential equations are studied in detail and exemplary RG trajectories are constructed numerically. The beta functions possess a non-Gaussian fixed point and a one-parameter family of Gaussian fixed points. One of the Gaussian fixed points corresponds to the Einstein-Hilbert action with vanishing cosmological constant and constitutes a saddle point with one IR-attractive direction. For RG trajectories dragged into this fixed point at low energies diffeomorphism invariance is restored. The emergence of general relativity from Hořava-Lifshitz gravity can thus be understood as a crossover-phenomenon where the IR behavior of the theory is controlled by this Gaussian fixed point. In particular RG trajectories with a tiny positive cosmological constant also come with an anisotropy parameter which is compatible with experimental constraints, providing a mechanism for the approximate restoration of diffeomorphism invariance in the IR. The non-Gaussian fixed point is UV-attractive in all three coupling constants. Most likely, this fixed point is the imprint of Asymptotic Safety at the level of Hořava-Lifshitz gravity.

We study the effect of an external magnetic field on the chiral phase transition in the theory of the strong interaction by means of a renormalization-group (RG) fixed-point analysis, relying on only one physical input parameter, the strong coupling at a given large momentum scale. To be specific, we consider the interplay of the RG flow of four-quark interactions and the running gauge coupling. Depending on the temperature and the strength of the magnetic field, the gauge coupling can drive the quark sector to criticality, resulting in chiral symmetry breaking. In accordance with lattice Monte-Carlo simulations, we find that the chiral phase transition temperature decreases for small values of the external magnetic field. For large magnetic field strengths, however, our fixed-point study predicts that the phase transition temperature increases monotonically.Introduction.-The dynamics of gauge theories is expected to be strongly affected by external magnetic fields. This is of great phenomenological relevance for a large variety of systems, ranging from condensed-matter theory [1, 2] over off-central heavy-ion collision experiments [3, 4] to neutron stars [5] and cosmological models [6]. In fact, studies of the influence of an external magnetic field on the phase diagram of the theory of the strong interaction (Quantum Chromodynamics, QCD) and its equation of state have attracted a lot of attention in recent years. In particular, the peculiar inversecatalysis effect is here of great interest [7]. It is related to the observation that the chiral critical temperature decreases with increasing strength of the magnetic field, in contradistinction to purely fermionic models also known from condensed-matter theory where magnetic catalysis is observed, i.e. the critical temperature increases with increasing magnetic field strength [1,8].The observation of inverse catalysis in lattice Monte-Carlo (MC) studies of the chiral phase transition in QCD has come unexpected [7]. This is related to the fact that effective low-energy QCD models are commonly believed to describe correctly many features of the QCD phase diagram, at least on a qualitative level. Since these models are predominantly purely fermionic models being close relatives to the Nambu-Jona-Lasinio model, which has been originally constructed based on analogies to condensed-matter theory [9], it also appeared natural to expect that only magnetic catalysis is at work in QCD [10,11], see Ref.[12] for reviews. Various extensions of low-energy QCD models have been studied, ranging from the inclusion of the orderparameter for deconfinement as obtained from lattice MC simulations [13] to extensions beyond the mean-field approximation [14,15]. Moreover, the effect of a magnetic field on the chiral dynamics has been recently studied using Dyson-Schwinger equations (DSE) [16]. In any case, the very observation of magnetic catalysis is found to be generic, even if effects associated with the chiral anomaly are taken into account [10]. On the other hand, it has been found t...

We investigate the critical behavior of three-dimensional relativistic fermion models with a U(NL)L × U(1)R chiral symmetry reminiscent of the Higgs-Yukawa sector of the standard model of particle physics. We classify all possible four-fermion interaction terms and the corresponding discrete symmetries. For sufficiently strong correlations in a scalar parity-conserving channel, the system can undergo a second-order phase transition to a chiral-symmetry broken phase which is a 3d analog of the electroweak phase transition. We determine the critical behavior of this phase transition in terms of the critical exponent ν and the fermion and scalar anomalous dimensions for NL ≥ 1. Our models define new universality classes that can serve as prototypes for studies of strongly correlated chiral fermions.