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We determine the chiral phase structure of $2+1$-flavour QCD in dependence of temperature and the light flavour quark mass with Dyson-Schwinger equations. Specifically, we compute the renormalised chiral condensate and its susceptibility. The latter is used to determine the (pseudo)critical temperature for general light current quark masses. In the chiral limit we obtain a critical temperature of about 141\,MeV. This result is in quantitative agreement with recent functional renormalisation group results in QCD, and is compatible with the respective lattice results. We also compute the order parameter potential of the light chiral condensate and map out the regime in the phase diagram which exhibits quasi-massless modes, and discuss the respective chiral dynamics., Comment: 20 pages, 13 figures

Fluctuations of conserved charges are sensitive to the QCD phase transition and a possible critical endpoint in the phase diagram at finite density. In this work, we compute the baryon number fluctuations up to tenth order at finite temperature and density. This is done in a QCD-assisted effective theory that accurately captures the quantum- and in-medium effects of QCD at low energies. A direct computation at finite density allows us to assess the applicability of expansions around vanishing density. By using different freeze-out scenarios in heavy-ion collisions, we translate these results into baryon number fluctuations as a function of collision energy. We show that a non-monotonic energy dependence of baryon number fluctuations can arise in the non-critical crossover region of the phase diagram. Our results compare well with recent experimental measurements of the kurtosis and the sixth-order cumulant of the net-proton distribution from the STAR collaboration. They indicate that the experimentally observed non-monotonic energy dependence of fourth-order net-proton fluctuations is highly non-trivial. It could be an experimental signature of an increasingly sharp chiral crossover and may indicate a QCD critical point. The physics implications and necessary upgrades of our analysis are discussed in detail., Comment: 23 pages, 14 figures, 2 tables, published version

We discuss the far-from-equilibrium evolution of $\phi^3$-theory in $1+1$ dimensions with the temporal functional renormalisation group \cite{Gasenzer:2007za, Gasenzer:2010rq}. In particular, we show that this manifestly causal approach leads to novel one-loop exact equations for fully dressed correlation functions. Within this setup, we numerically compute the dynamical propagator. Its behaviour suggests self-similarity far from equilibrium in a restricted momentum regime. We discuss the scaling exponents for our solution, as well as the numerical satisfaction of energy and particle number conservation. We also derive a simple exact representation of the expectation value of the energy-momentum tensor solely in terms of the propagator., Comment: 16 pages, 16 figures. v2: Added comments in Secs. II and III, added refs, corrected typos

We study anomalous chiral symmetry breaking in two-flavour QCD induced by gravitational and QCD-instantons within asymptotically safe gravity within the functional renormalisation group approach. Similarly to QCD-instantons, gravitational ones, associated to a K3-surface connected by a wormhole-like throat in flat spacetime, generate contributions to the 't~Hooft coupling proportional to $\exp(-1/g_N)$ with the dimensionless Newton coupling $g_N$. Hence, in the asymptotically safe gravity scenario with a non-vanishing fixed point coupling $g_N^*$, the induced 't Hooft coupling is finite at the Planck scale, and its size depends on the chosen UV-completion. Within this scenario the gravitational effects on anomalous $U(1)_A$-breaking at the Planck scale may survive at low energy scales. In turn, fermion masses of the order of the Planck scale cannot be present. This constrains the allowed asymptotically safe UV-completion of the Gravity-QCD system. We map-out the parameter regime that is compatible with the existence of light fermions in the low-energy regime., Comment: 14 pages, 7 figures

Physical review / D 102(12), 125016 (2020). doi:10.1103/PhysRevD.102.125016, Published by APS, Melville, NY

We explore artificial neural networks as a tool for the reconstruction of spectral functions from imaginary time Green's functions, a classic ill-conditioned inverse problem. Our ansatz is based on a supervised learning framework in which prior knowledge is encoded in the training data and the inverse transformation manifold is explicitly parametrised through a neural network. We systematically investigate this novel reconstruction approach, providing a detailed analysis of its performance on physically motivated mock data, and compare it to established methods of Bayesian inference. The reconstruction accuracy is found to be at least comparable, and potentially superior in particular at larger noise levels. We argue that the use of labelled training data in a supervised setting and the freedom in defining an optimisation objective are inherent advantages of the present approach and may lead to significant improvements over state-of-the-art methods in the future. Potential directions for further research are discussed in detail., Comment: 20 pages, 16 figures

We discuss quantum black holes in asymptotically safe quantum gravity with a scale identification based on the Kretschmann scalar. After comparing this scenario with other scale identifications, we investigate in detail the Kerr-(A)dS and Schwarzschild-(A)dS space-times. The global structure of these geometries is studied as well as the central curvature singularity and test particle trajectories. The existence of a Planck-sized, extremal, zero temperature black hole remnant guarantees a stable endpoint of the evaporation process via Hawking radiation., Comment: 21 pages, 25 figures

We study the asymptotic safety conjecture for quantum gravity in the presence of matter fields. A general line of reasoning is put forward explaining why gravitons dominate the high-energy behavior, largely independently of the matter fields as long as these remain sufficiently weakly coupled. Our considerations are put to work for gravity coupled to Yang-Mills theories with the help of the functional renormalization group. In an expansion about flat backgrounds, explicit results for beta functions, fixed points, universal exponents, and scaling solutions are given in systematic approximations exploiting running propagators, vertices, and background couplings. Invariably, we find that the gauge coupling becomes asymptotically free while the gravitational sector becomes asymptotically safe. The dependence on matter field multiplicities is weak. We also explain how the scheme dependence, which is more pronounced, can be handled without changing the physics. Our findings offer a new interpretation of many earlier results, which is explained in detail. The results generalize to theories with minimally coupled scalar and fermionic matter. Some implications for the ultraviolet closure of the Standard Model or its extensions are given.

We investigate the phase diagram of quantum gravity with a vertex expansion about constantly-curved backgrounds. The graviton two- and three-point function are evaluated with a spectral sum on a sphere. We obtain, for the first time, curvature-dependent UV fixed point functions of the dynamical fluctuation couplings $g^*(R)$, $\mu^*(R)$, and $\lambda_3^*(R)$, and the background $f(R)$-potential. Based on these fixed point functions we compute solutions to the quantum and the background equation of motion with and without Standard Model matter. We have checked that the solutions are robust against changes of the truncation., Comment: 18 pages, 9 figures; v2: typos corrected, refs updated

The skewness and the kurtosis of the baryon number distributions are computed within QCD-improved low energy effective models including quantum thermal and density fluctuations. The results are compared with the Beam Energy Scan experiment at RHIC. The theoretical results agree with the experimental measurements up to errors, for the collision energy $\sqrt{s}\ge 19.6\,\mathrm{GeV}$. For smaller collision energies a discrepancy between theoretical and experimental results develops. This discrepancy partially relates to the lack of precision of the current setup for small collision energies. It is outlined how this deficiency can be overcome., 7 pages, 3 figures, 1 table; updated to match version published in Phy. Rev. D

We investigate the impact of the matter and glue dynamics on baryon number fluctuations and the kurtosis of baryon number distribution. This is done within the framework of QCD-improved low energy effective models. In particular we include the momentum scale dependence of the quark-meson scattering and the non-trivial dispersions of both, quarks and mesons. On the gluonic side we take into account the backreaction of the matter sector on the glue dynamics. It is shown that the above fluctuations lead to a more rapid change of the baryon number fluctuations as well as the kurtosis of with the chiral crossover. We also study the signatures of quark confinement in low energy QCD. It is shown that contrary to the common picture the effective thermal distribution in the presence of confining glue backgrounds does not tend towards the colourless baryonic one. Instead, the dominance of colourless hadronic states is obtained in a subtle interplay of quark and glue contributions to the canonical potential., Comment: 25 pages, 19 figures

We present a quantitative analysis of chiral symmetry breaking in two-flavour continuum QCD in the quenched limit. The theory is set-up at perturbative momenta, where asymptotic freedom leads to precise results. The evolution of QCD towards the hadronic phase is achieved by means of dynamical hadronisation in the non-perturbative functional renormalisation group approach. We use a vertex expansion scheme based on gauge-invariant operators and discuss its convergence properties and the remaining systematic errors. In particular we present results for the quark propagator, the full tensor structure and momentum dependence of the quark-gluon vertex, and the four-fermi scatterings., Comment: 18 pages, 7 figures, additional figure, additional references

We investigate the nature of the deconfinement and Roberge-Weiss transition in the heavy quark regime for finite real and imaginary chemical potential within the functional approach to continuum QCD. We extract the critical phase boundary between the first order and cross-over regions, and also explore tricritical scaling. Our results confirm previous ones from finite volume lattice studies., Comment: 6 pages, 11 figures

We compute the Polyakov loop potential in Yang--Mills theory from the fully dressed primitively divergent correlation functions only. This is done in a variety of functional approaches ranging from functional renormalisation group equations over Dyson--Schwinger equations to two-particle irreducible functionals. We present a confinement criterion that links the infrared behaviour of propagators and vertices to the Polyakov loop expectation value. The present work extends the works of [1-3] to general functional methods and sharpens the confinement criterion presented there. The computations are based on the thermal correlation functions in the Landau gauge calculated in [4-6]., 23 pages, 28 figures

We consider a generalized Thirring model in 0+1 dimensions at finite density. In order to deal with the resulting sign problem we employ stochastic quantization, i.e., a complex Langevin evolution. We investigate the convergence properties of this approach and check in which parameter regions complex Langevin evolutions are applicable in this setting. To this end we derive numerous analytical results and compare directly with numerical results. In addition we employ indirect indicators to check for correctness. Finally, we interpret and discuss our findings.

We investigate the quark backreaction on the Polyakov loop and its impact on the thermodynamics of quantum chromodynamics. The dynamics of the gluons generating the Polyakov-loop potential is altered by the presence of dynamical quarks. However, this backreaction of the quarks has not yet been taken into account in Polyakov-loop extended model studies. In the present work, we show within a 2+1 flavour Polyakov-quark-meson model that a quark-improved Polyakov-loop potential leads to a smoother transition between the low-temperature hadronic phase and the high-temperature quark-gluon plasma phase. In particular, we discuss the dependence of our results on the remaining uncertainties that are the critical temperature and the parametrisation of the Polyakov-loop potential as well as the mass of the sigma-meson., 19 pages, 25 figures; version published in Phys. Rev. D

We investigate the infrared (strong-coupling) regime of SU(N)-Yang-Mills theory on a self-dual background. We present an evaluation of the full effective potential for the field strength invariant F_{\mu {\nu}}F^{\mu {\nu}} from non-perturbative gauge correlation functions and find a non-trivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the beta function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators. Consistency between both gauges in the infrared imposes a new upper bound on the infrared exponents of the propagators. For the scaling solution, this bound reads kappa_c < 23/38 which, together with Zwanziger's horizon condition kappa_c> 1/2, defines a rather narrow window for this critical exponent. Current estimates from functional methods indeed satisfy these bounds., Comment: v2 matches published version, refs added, improved estimate of the gluon condensate

The large density of gluons, which is present shortly after a nuclear collision at very high energies, can lead to the formation of a condensate. We identify a gauge-invariant order parameter for condensation based on elementary non-perturbative excitations of the plasma, which are described by spatial Wilson loops. Using real-time lattice simulations, we demonstrate that a self-similar transport process towards low momenta builds up a macroscopic zero mode. Our findings reveal intriguing similarities to recent discoveries of condensation phenomena out of equilibrium in table-top experiments with ultracold Bose gases., 5 pages, 3 figures; v2: journal version, add. section comparing to thermal case and extended discussion