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We present the Mathematica package QMeS-Derivation. It derives symbolic functional equations from a given master equation. The latter include functional renormalisation group equations, Dyson-Schwinger equations, Slavnov-Taylor and Ward identities and their modifications in the presence of momentum cutoffs. The modules allow to derive the functional equations, take functional derivatives, trace over field space, apply a given truncation scheme, and do momentum routings while keeping track of prefactors and signs that arise from fermionic commutation relations. The package furthermore contains an installer as well as Mathematica notebooks with showcase examples., 18 pages, 4 figures

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

We reconstruct ghost and gluon spectral functions in 2 þ 1 flavor QCD with Gaussian process regression. This framework allows us to largely suppress spurious oscillations and other common reconstruction artifacts by specifying generic magnitude and length scale parameters in the kernel function. The Euclidean propagator data are taken from lattice simulations with domain wall fermions at the physical point. For the infrared and ultraviolet extensions of the lattice propagators as well as the low-frequency asymptotics of the ghost spectral function, we utilize results from functional computations in Yang-Mills theory and QCD. This further reduces the systematic error significantly. Our numerical results are compared against a direct real-time functional computation of the ghost and an earlier reconstruction of the gluon in Yang-Mills theory. The systematic approach presented in this work offers a promising route toward unveiling real-time properties of QCD, We thank A. Cyrol, F. Gao, L. Kades, J. Y. Lin, J. Papavassiliou, and A. Rothkopf for discussions. We thank P. Boucaud and F. De Soto for an earlier collaboration and for their help in the preparation of the lattice data. We are indebted to the RBC/ UKQCD Collaboration, especially to P. Boyle, N. Christ, Z. Dong, N. Garron, C. Jung, B. Mawhinney, and O. Witzel, for access to the lattices used in this work. We also thank the members of the fQCD Collaboration for discussions and collaborations on related subjects. J. R. Q. acknowledges the support of MICINN Grant No. PID2019-107844GB-C22. Numerical computations have used resources of CINES, GENCI IDRIS (Project No. 52271) and of the IN2P3 computing facility in France. This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy Grant No. EXC 2181/ 1-390900948 (the Heidelberg STRUCTURES Excellence Cluster), under the Collaborative Research Centre SFB 1225 (ISOQUANT), EMMI, and BMBF Grant No. 05P18VHFCA

We discuss the emergence of a low-energy effective theory with quarks, mesons, diquarks and baryons at vanishing and finite baryon density from first principle QCD. The present work also includes an overview on diquarks at vanishing and finite density, and elucidates the physics of transitional changes from baryonic matter to quark matter including diquarks. This set-up is discussed within the functional renormalisation group approach with dynamical hadronisation. In this framework it is detailed how mesons, diquarks, and baryons emerge dynamically from the renormalisation flow of the QCD effective action. Moreover, the fundamental degrees of freedom of QCD, quarks and gluons, decouple from the dynamics of QCD below the respective mass gaps. The resulting global picture unifies the different low energy effective theories used for low and high densities within QCD, and allows for a determination of the respective low energy constants directly from QCD., 30 pages, 17 figures

We investigate the structure of confining and deconfining phases in SU(2) lattice gauge theory via persistent homology, which gives us access to the topology of a hierarchy of combinatorial objects constructed from given data. Specifically, we use filtrations by traced Polyakov loops, topological densities, holonomy Lie algebra fields, as well as electric and magnetic fields. This allows for a comprehensive picture of confinement. In particular, topological densities form spatial lumps which show signatures of the classical probability distribution of instanton-dyons. Signatures of well-separated dyons located at random positions are encoded in holonomy Lie algebra fields, following the semi-classical temperature dependence of the instanton appearance probability. Debye screening discriminating between electric and magnetic fields is visible in persistent homology and pronounced at large gauge coupling. All employed constructions are gauge-invariant without a priori assumptions on the configurations under study. This work showcases the versatility of persistent homology for statistical and quantum physics studies, barely explored to date., Comment: 22 pages, 15 figures, published version

A longstanding question in QCD is the origin of the mass gap in the Yang-Mills sector of QCD, i.e., QCD without quarks. In Landau gauge QCD this mass gap, and hence confinement, is encoded in a mass gap of the gluon propagator, which is found both in lattice simulations and with functional approaches. While functional methods are well suited to unravel the mechanism behind the generation of the mass gap, a fully satisfactory answer has not yet been found. In this work we solve the coupled Dyson-Schwinger equations for the ghost propagator, gluon propagator and three-gluon vertex. We corroborate the findings of earlier works, namely that the mass gap generation is tied to the longitudinal projection of the gluon self-energy, which acts as an effective mass term in the equations. Because an explicit mass term is in conflict with gauge invariance, this leaves two possible scenarios: If it is viewed as an artifact, only the scaling solution survives; if it is dynamical, gauge invariance can only be preserved if there are longitudinal massless poles in either of the vertices. We find that there is indeed a massless pole in the ghost-gluon vertex, however in our approximation with the assumption of complete infrared dominance of the ghost this pole is only present for the scaling solution. We also put forward a possible mechanism that may reconcile the scaling solution, with an infrared dominance of the ghost, with the decoupling solutions based on longitudinal poles in the three-gluon vertex as seen in the PT-BFM scheme., 33 pages, 24 figures. Identical to the version published in Phys. Rev. D

In local scalar quantum field theories (QFTs) at finite temperature correlation functions are known to satisfy certain non-perturbative constraints, which for two-point functions in particular implies the existence of a generalisation of the standard K\"{a}ll\'{e}n-Lehmann representation. In this work, we use these constraints in order to derive a spectral representation for the shear viscosity arising from the thermal asymptotic states, $\eta_{0}$. As an example, we calculate $\eta_{0}$ in $\phi^{4}$ theory, establishing its leading behaviour in the small and large coupling regimes., Comment: 21 pages, 3 figures; v3: additional references added as well as some minor text changes, matches published version

We map out the QCD phase structure at finite temperature and chemical potential for 2-flavour and 2 + 1-flavour QCD. This is done within a generalised functional approach to QCD put forward in arXiv:2002.07500 [hep-ph]. Specifically we compute the quark propagator and the finite-temperature and density fluctuations of the gluon propagator and the quark-gluon vertex on the basis of precision data for vacuum correlation functions. The novel ingredient is the direct self-consistent computation of the DSEs for the dressings of the quark-gluon vertex, in contrast to the common use of STI-inspired vertices. For small densities the results for the chiral order parameter agree with the respective lattice and functional renormalisation group results, for large densities the present results are in a quantitative agreement with the latter, including the location of the critical end point., Comment: 8 pages, 5 figures

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

In this contribution, we discuss the asymptotic safety scenario for quantum gravity with a functional renormalisation group approach that disentangles dynamical metric fluctuations from the background metric. We review the state of the art in pure gravity and general gravity-matter systems. This includes the discussion of results on the existence and properties of the asymptotically safe ultraviolet fixed point, full ultraviolet-infrared trajectories with classical gravity in the infrared, and the curvature dependence of couplings also in gravity-matter systems. The results in gravity-matter systems concern the ultraviolet stability of the fixed point and the dominance of gravity fluctuations in minimally coupled gravity-matter systems. Furthermore, we discuss important physics properties such as locality of the theory, diffeomorphism invariance, background independence, unitarity, and access to observables, as well as open challenges., 36 pages, 10 figures. Review of the fluctuation approach to asymptotically safe quantum gravity. v2: refs updated, matches journal version

We present the first direct and non-perturbative computation of the graviton spectral function in quantum gravity. This is achieved with the help of a novel Lorentzian renormalisation group approach, combined with a spectral representation of correlation functions. We find a positive graviton spectral function, showing a massless one-graviton peak and a multi-graviton continuum with an asymptotically safe scaling for large spectral values. We also study the impact of a cosmological constant. Further steps to investigate scattering processes and unitarity in asymptotically safe quantum gravity are indicated., Comment: 12 pages, 7 figures, journal-published version

We present a comprehensive study of the quark sector of $2+1$ flavour QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full quark-gluon vertex. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with $2+1$ dynamical quark flavours, obtained from studies with Schwinger-Dyson equations, the functional renormalisation group approach, and large volume lattice simulations. The appropriate renormalisation procedure required in order to self-consistently accommodate external inputs stemming from other functional approaches or the lattice is discussed in detail, and the value of the gauge coupling is accurately determined at two vastly separated renormalisation group scales. Our analysis establishes a clear hierarchy among the vertex form factors. We identify only three dominant ones, in agreement with previous results. The components of the quark propagator obtained from our approach are in excellent agreement with the results from Schwinger-Dyson equations, the functional renormalisation group, and lattice QCD simulation, a simple benchmark observable being the chiral condensate in the chiral limit, which is computed as $(245\,\textrm{MeV})^3$. The present approach has a wide range of applications, including the self-consistent computation of bound-state properties and finite temperature and density physics, which are briefly discussed., Comment: 72 pages, 30 figures. Version including minor changes of version published in PRD. For the benefit of the reader we have also added an Appendix, that contains many more details of the renormalization scheme (MOM${}^2$ scheme) used in this work. In particular we provide a comparison with MOM scheme results in the literature (Appendix A2), as well as the derivation of the RG scheme (Appendix A3)

In covariant gauges, the gluonic mass gap in Yang-Mills theory manifests itself in the basic observation that the massless pole in the perturbative gluon propagator disappears in nonperturbative calculations, but the origin of this behavior is not yet fully understood. We summarize a recent study of the respective dynamics with Dyson-Schwinger equations in Landau-gauge Yang-Mills theory. We identify the parameter that distinguishes the massive Yang-Mills regime from the massless decoupling solutions, whose endpoint is the scaling solution. Similar to the PT-BFM scheme, we find evidence that mass generation in the transverse sector is triggered by longitudinal massless poles., Comment: 10 pages, 6 figures. Proceedings contribution to the XXXIII International (Online) Workshop on High Energy Physics "Hard Problems of Hadron Physics: Non-Perturbative QCD & Related Quests", November 8-12, 2021

We compute the ghost spectral function in Yang-Mills theory by solving the corresponding Dyson-Schwinger equation for a given input gluon spectral function. The results encompass both scaling and decoupling solutions for the gluon propagator input. The resulting ghost spectral function displays a particle peak at vanishing momentum and a negative scattering spectrum, whose infrared and ultraviolet tails are obtained analytically. The ghost dressing function is computed in the entire complex plane, and its salient features are identified and discussed., Comment: 15 pages, 11 figures

We discuss the phase structure of the two-flavour quark-meson model including quantum, thermal, density and critical fluctuations with the functional renormalisation group. This study combines two technical advances in the literature, that are also chiefly important for the quantitative access of the phase boundary of QCD at large density or baryon chemical potential. Specifically we allow for the formation and propagation of shocks as well as a fully self-consistent computation of the order parameter potential for chiral symmetry breaking., Comment: 26 pages, 16 figures

We combine two non-perturbative approaches, one based on the two-particle-irreducible (2PI) action, the other on the functional renormalization group (fRG), in an effort to develop new non-perturbative approximations for the field theoretical description of strongly coupled systems. In particular, we exploit the exact 2PI relations between the two-point and four-point functions in order to truncate the infinite hierarchy of equations of the functional renormalization group. The truncation is "exact" in two ways. First, the solution of the resulting flow equation is independent of the choice of the regulator. Second, this solution coincides with that of the 2PI equations for the two-point and the four-point functions, for any selection of two-skeleton diagrams characterizing a so-called $\Phi$-derivable approximation. The transformation of the equations of the 2PI formalism into flow equations offers new ways to solve these equations in practice, and provides new insight on certain aspects of their renormalization. It also opens the possibility to develop approximation schemes going beyond the strict $\Phi$-derivable ones, as well as new truncation schemes for the fRG hierarchy., Comment: 80 pages, 22 figures

The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson's RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity., v3) Review article, 93 pages + bibliography, 35 figures

Scaling solutions for the effective action in dilaton quantum gravity are investigated within the functional renormalization group approach. We find numerical solutions that connect ultraviolet and infrared fixed points as the ratio between scalar field and renormalization scale $k$ is varied. In the Einstein frame the quantum effective action corresponding to the scaling solutions becomes independent of $k$. The field equations derived from this effective action can be used directly for cosmology. Scale symmetry is spontaneously broken by a non-vanishing cosmological value of the scalar field. For the cosmology corresponding to our scaling solutions, inflation arises naturally. The effective cosmological constant becomes dynamical and vanishes asymptotically as time goes to infinity., 7 pages, 4 figures

We calculate the gravitational-wave spectra produced by the electroweak phase transition with TeV-scale Beyond-Standard-Model physics in the early universe. Our study captures the effect of quantum and thermal fluctuations within a non-perturbative framework. We discover a universal relation between the mean bubble separation and the strength parameter of the phase transition, which holds for a wide range of new-physics contributions. The ramifications of this result are three-fold: First, they constrain the gravitational-wave spectra resulting from heavy (TeV-scale) new physics. Second, they contribute to distinguishing heavy from light new physics directly from the gravitational-wave signature. Third, they suggest that a concerted effort of gravitational-wave observations together with collider experiments could be required to distinguish between different models of heavy new physics., Comment: 46 pages, 11 figures, 3 tables; JCAP version

We propose a scenario with string theory in the deep ultraviolet, an intermediate asymptotically safe scaling regime for gravity and matter, and the Standard Model in the infrared. This could provide a new perspective to tackle challenges of the two models: For instance, the gravitational Renormalization Group flow could connect a negative microscopic to a positive macroscopic cosmological constant, potentially rendering string theory on an anti-de Sitter background observationally viable. Further, the unitarity of a string-theoretic ultraviolet completion could be inherited by an asymptotically safe fixed point, despite the presence of higher-order interactions. We discuss necessary conditions on the scale of asymptotic safety and the string scale for our scenario to be viable. As a first test, we explore the weak-gravity conjecture in the context of asymptotically safe gravity., 10 pages, 4 figures

We explore asymptotic safety of gravity-matter systems, discovering indications for a near-perturbative nature of these systems in the ultraviolet. Our results are based on the dynamical emergence of effective universality at the asymptotically safe fixed point. Our findings support the conjecture that an asymptotically safe completion of the Standard Model with gravity could be realized in a near-perturbative setting., Comment: 6 pages, 3 figures

Low-energy effective theories have been used very successfully to study the low-energy limit of QCD, providing us with results for a plethora of phenomena, ranging from bound-state formation to phase transitions in QCD. These theories are consistent quantum field theories by themselves and can be embedded in QCD, but typically have a physical ultraviolet cutoff that restricts their range of validity. Here, we provide a discussion of the concept of renormalization group consistency, aiming at an analysis of cutoff effects and regularization-scheme dependences in general studies of low-energy effective theories. For illustration, our findings are applied to low-energy effective models of QCD in different approximations including the mean-field approximation. More specifically, we consider hot and dense as well as finite systems and demonstrate that violations of renormalization group consistency affect significantly the predictive power of the corresponding model calculations., Comment: 18 pages, 5 figures

We present version 3.0 of the Mathematica package DoFun for the derivation of functional equations. In this version, the derivation of equations for correlation functions of composite operators was added. In the update, the general workflow was slightly modified taking into account experience with the previous version. In addition, various tools were included to improve the usage experience and the code was partially restructured for easier maintenance., Comment: 15 pages, 3 figures; code at https://github.com/markusqh/DoFun; v2: added quick start guide, clarified compatibility with Mathematica versions, agrees with published version

We compute correlation functions of three-dimensional Landau-gauge Yang-Mills theory with the Functional Renormalisation Group. Starting from the classical action as only input, we calculate the non-perturbative ghost and gluon propagators as well as the momentum-dependent ghost-gluon, three-gluon, and four-gluon vertices in a comprehensive truncation scheme. Compared to the physical case of four spacetime dimensions, we need more sophisticated truncations due to significant contributions from non-classical tensor structures. In particular, we apply a special technique to compute the tadpole diagrams of the propagator equations, which captures also all perturbative two-loop effects, and compare our correlators with lattice and Dyson-Schwinger results., 10 pages, 11 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

One of the main challenges in simulations on Lefschetz thimbles is the computation of the relative weights of contributing thimbles. In this paper we propose a solution to that problem by means of computing those weights using a reweighting procedure. Besides we present recipes for finding parametrizations of thimbles and anti-thimbles for a given theory. Moreover, we study some approaches to combine the Lefschetz thimble method with the Complex Langevin evolution. Our numerical investigations are carried out by using toy models among which we consider a one-site z^4 model as well as a U(1) one-link model., 11 pages, 9 figures, 2 tables; in revised version added a few references

We investigate the asymptotic safety scenario for a scalar-gravity system. This system contains two avatars of the dynamical Newton coupling, a gravitational self-coupling and a scalar-graviton coupling. We uncover an effective universality for the dynamical Newton coupling on the quantum level: its momentum-dependent avatars are in remarkable quantitative agreement in the scaling regime of the UV fixed point. For the background Newton coupling, this effective universality is not present, but qualitative agreement remains., 24 pages, 10 figures; v2: minor changes, refs updated

For the understanding of fluctuation measurements in heavy-ion collisions it is crucial to develop quantitatively reliable dynamical descriptions which take the non-perturbative nature of QCD near the phase transition into account. We discuss a novel QCD-assisted transport approach based on non-equilibrium chiral fluid dynamics and the effective action of low energy QCD. In this framework, we study the time-evolution of fluctuation measures of the critical mode, notably the kurtosis, for a non-expanding system. From this, we can estimate the equilibration times of critical mode fluctuations in the QCD phase diagram. These allow us to identify both the phase boundary and the critical region near the QCD critical point., Quark Matter 2018 Proceedings

We present non-perturbative correlation functions in Landau-gauge Yang-Mills theory at finite temperature. The results are obtained from the functional renormalisation group within a self-consistent approximation scheme. In particular, we compute the magnetic and electric components of the gluon propagator, and the three- and four-gluon vertices. We also show the ghost propagator and the ghost-gluon vertex at finite temperature. Our results for the propagators are confronted with lattice simulations and our Debye mass is compared to hard thermal loop perturbation theory., Comment: 13 pages + appendix

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 effect of gravitational fluctuations on the quantum effective potential for scalar fields is a key ingredient for predictions of the mass of the Higgs boson, understanding the gauge hierarchy problem and a possible explanation of an---asymptotically---vanishing cosmological constant. We find that the quartic self-interaction of the Higgs scalar field is an irrelevant coupling at the asymptotically safe ultraviolet fixed point of quantum gravity. This renders the ratio between the masses of the Higgs boson and top quark predictable. If the flow of couplings below the Planck scale is approximated by the Standard Model, this prediction is consistent with the observed value. The quadratic term in the Higgs potential is irrelevant if the strength of gravity at short distances exceeds a bound that is determined here as a function of the particle content. In this event, a tiny value of the ratio between the Fermi scale and the Planck scale is predicted., Comment: 18 pages, 5 figures, Version published in PRD

We reconstruct the gluon spectral function in Landau gauge QCD from numerical data for the gluon propagator. The reconstruction relies on two novel ingredients: Firstly we derive analytically the low frequency asymptotics of the spectral function. Secondly we construct a functional basis from a careful consideration of the analytic properties of the gluon propagator in Landau gauge. This allows us to reliably capture the non-perturbative regime of the gluon spectrum. We also compare different reconstruction methods and discuss the respective systematic errors., Comment: 12 pages, 10 figures

We directly calculate spectral functions in the O(N)-model at finite temperature within the framework of the Functional Renormalization group. Special emphasis is put on a fully numerical framework involving four-dimensional regulators preserving Euclidean O(4) and Minkowski Lorentz invariance, an important prerequisite for future applications. Pion and sigma meson spectral functions are calculated for a wide range of temperatures across the phase transition illustrating the applicability of the general framework for finite temperature applications. In addition, various aspects concerning the interplay between the Euclidean and real time two-point function are discussed., 11 pages, 20 figures

Smoothing of field configurations is highly important for precision calculations of physical quantities on the lattice. We present a cooling method based on Stochastic Quantization with a built-in UV momentum cutoff. The latter is implemented via a UV-regularized, hence colored, noise term. Our method is tested in a two-dimensional scalar field theory. We show, that UV modes can be removed systematically without altering the physics content of the theory. The approach has an interpretation in terms of the non-perturbative (Wilsonian) renormalization group that facilitates the physics interpretation of the cutoff procedure. It also can be used to define the maximal colored cooling applicable without changing the theory., Comment: 21 pages, 25 figures, 1 table, revised version with minor changes

A phenomenologically viable theory of quantum gravity must accommodate all observed matter degrees of freedom and their properties. Here, we explore whether a toy model of the Higgs-Yukawa sector of the Standard Model is compatible with asymptotically safe quantum gravity. We discuss the phenomenological implications of our result in the context of the Standard Model. We analyze the quantum scaling dimension of the system, and find an irrelevant Yukawa coupling at a joint gravity-matter fixed point. Further, we explore the impact of gravity-induced couplings between scalars and fermions, which are non-vanishing in asymptotically safe gravity., 13 pages + appendix, 10 figures, simplified basis for induced couplings

We study the nature of the confinement phase transition in d=3+1 dimensions in various non-abelian gauge theories with the approach put forward in [1]. We compute an order-parameter potential associated with the Polyakov loop from the knowledge of full 2-point correlation functions. For SU(N) with N=3,...,12 and Sp(2) we find a first-order phase transition in agreement with general expectations. Moreover our study suggests that the phase transition in E(7) Yang-Mills theory also is of first order. We find that it is weaker than for SU(N). We show that this can be understood in terms of the eigenvalue distribution of the order parameter potential close to the phase transition., 15 pages

The theory of real-time quantum many-body dynamics as put forward in Ref. [arXiv:0710.4627] is evaluated in detail. The formulation is based on a generating functional of correlation functions where the Keldysh contour is closed at a given time. Extending the Keldysh contour from this time to a later time leads to a dynamic flow of the generating functional. This flow describes the dynamics of the system and has an explicit causal structure. In the present work it is evaluated within a vertex expansion of the effective action leading to time evolution equations for Green functions. These equations are applicable for strongly interacting systems as well as for studying the late-time behaviour of nonequilibrium time evolution. For the specific case of a bosonic N-component phi^4 theory with contact interactions an s-channel truncation is identified to yield equations identical to those derived from the 2PI effective action in next-to-leading order of a 1/N expansion. The presented approach allows to directly obtain non-perturbative dynamic equations beyond the widely used 2PI approximations., 20 pp., 6 figs; submitted version with added references and typos corrected.

We relate quark confinement, as measured by the Polyakov-loop order parameter, to color confinement, as described by the Kugo-Ojima/Gribov-Zwanziger scenario. We identify a simple criterion for quark confinement based on the IR behaviour of ghost and gluon propagators, and compute the order-parameter potential from the knowledge of Landau-gauge correlation functions with the aid of the functional RG. Our approach predicts the deconfinement transition in quenched QCD to be of first order for SU(3) and second order for SU(2) -- in agreement with general expectations. As an estimate for the critical temperature, we obtain T_c=284MeV for SU(3)., Comment: 5 pages, 1 figure

We derive the two-loop effective action for covariantly constant field strength of pure Yang-Mills theory in the presence of an infrared scale. The computation is done in the framework of the worldline formalism, based on a generalization procedure of constructing multiloop effective actions in terms of the bosonic worldline path integral. The two-loop beta-function is correctly reproduced. This is the first derivation in the worldline formulation, and serves as a nontrivial check on the consistency of the multiloop generalization procedure in the worldline formalism., 8 pages, 4 figures

We study the consistency of dynamical fermionic matter with the asymptotic safety scenario of quantum gravity using the functional renormalisation group. Since this scenario suggests strongly coupled quantum gravity in the UV, one expects gravity-induced fermion self-interactions at energies of the Planck-scale. These could lead to chiral symmetry breaking at very high energies and thus to large fermion masses in the IR. The present analysis which is based on the previous works \cite{Christiansen:2015rva, Meibohm:2015twa}, concludes that gravity-induced chiral symmetry breaking at the Planck scale is avoided for a general class of NJL-type models, regardless of the number of fermion flavours. This suggests that the phase diagram for these models is topologically stable under the influence of gravitational interactions., 14 pages, 9 figures

In the functional renormalisation group approach to gauge theory, the Ward-Takahashi identity is modified due to the presence of an infrared cutoff term. It take the most accessible form for the Wilsonian effective action. In the present work we solve these identities, partially, for the Wilson effective action of QED. In particular, we compute the longitudinal part of the photon two point vertex function as a momentum- dependent function in the presence of the cutoff k. The resultant Wilsonian effective action carries form factors that originate from the modified Ward-Takahashi identity. We show how this result carries over to the one-particle-irreducible effective action., Comment: 15 pages

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being NPI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation., Comment: 58 pages, no figures, minor changes

The Renormalisation Group is a versatile tool for the study of many systems where scale-dependent behaviour is important. Its functional formulation can be cast into the form of an exact flow equation for the scale-dependent effective action in the presence of an infrared regularisation. The functional RG flow for the scale-dependent effective action depends explicitly on the choice of regulator, while the physics does not. In this work, we systematically investigate three key aspects of how the regulator choice affects RG flows: (i) We study flow trajectories along closed loops in the space of action functionals varying both, the regulator scale and shape function. Such a flow does not vanish in the presence of truncations. Based on a definition of the length of an RG trajectory, we suggest a practical procedure for devising optimised regularisation schemes within a truncation. (ii) In systems with various field variables, a choice of relative cutoff scales is required. At the example of relativistic bosonic two-field models, we study the impact of this choice as well as its truncation dependence. We show that a crossover between different universality classes can be induced and conclude that the relative cutoff scale has to be chosen carefully for a reliable description of a physical system. (iii) Non-relativistic continuum models of coupled fermionic and bosonic fields exhibit also dependencies on relative cutoff scales and regulator shapes. At the example of the Fermi polaron problem in three spatial dimensions, we illustrate such dependencies and show how they can be interpreted in physical terms., 22 pages, 16 figures

We study a $\phi^4$-theory at finite temperature in a finite volume. Quantum, thermal and volume fluctuations are treated with the functional renormalisation group. Specifically, we focus on the interplay of temperature and length scales driving the system. We find that thermodynamical observables at finite volume such as the pressure approach the infinite volume limit similarly to that of the vanishing temperature limit. We also advance the functional renormalisation group method at finite volume. In particular, we identify requirements for suitable regulators that admit the exponential thermal and finite volume decay properties., Comment: 20 pages, 20 figures

We put forward a functional renormalisation group approach for the direct computation of real time correlation functions, also applicable at finite temperature and density. We construct a general class of regulators that preserve the space-time symmetries, and allows the computation of correlation functions at complex frequencies. This includes both imaginary time and real time, and allows in particular the use of the plethora of imaginary time results for the computation of real time correlation functions. We also discuss real time computation on the Keldysh contour with general spatial momentum regulators. Both set-ups give access to the general momentum and frequency dependence of correlation functions., 21 pages, 13 figures; v2: extended discussion of the analytical continuation, references added; v3: published version; v4: corrected typos

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 calculate the shear viscosity over entropy density ratio $\eta/s$ in Yang--Mills theory from the Kubo formula using an exact diagrammatic representation in terms of full propagators and vertices using gluon spectral functions as external input. We provide an analytic fit formula for the temperature dependence of $\eta/s$ over the whole temperature range from a glueball resonance gas at low temperatures, to a high-temperature regime consistent with perturbative results. Subsequently we provide a first estimate for $\eta/s$ in QCD., Comment: 5 pages, 4 figures

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 investigate the ultraviolet behaviour of quantum gravity within a functional renormalisation group approach. The present setup includes the full ghost and graviton propagators and, for the first time, the dynamical graviton three-point function. The latter gives access to the coupling of dynamical gravitons and makes the system minimally self-consistent. The resulting phase diagram confirms the asymptotic safety scenario in quantum gravity with a non-trivial UV fixed point. A well-defined Wilsonian block spinning requires locality of the flow in momentum space. This property is discussed in the context of functional renormalisation group flows. We show that momentum locality of graviton correlation functions is non-trivially linked to diffeomorphism invariance, and is realised in the present setup., Comment: 4 pages, 3 figures