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Dense QCD matter may exhibit crystalline phases. Their existence is reflected in a moat regime, where mesonic correlations feature spatial modulations. We study the realtime properties of pions at finite temperature and density in QCD in order to elucidate the nature of this regime. We show that the moat regime arises from particle-hole-like fluctuations near the Fermi surface. This gives rise to a characteristic peak in the spectral function of the pion at nonzero \emph{spacelike} momentum. This peak can be interpreted as a new quasi particle, the moaton. In addition, our framework also allows us to directly test the stability of the homogeneous chiral phase against the formation of an inhomogeneous condensate in QCD. We find that the formation of such a phase is highly unlikely for baryon chemical potentials $\mu_B \leq 630$\,MeV., Comment: 20 pages, 16 figures

We study the electron-positron to muon--anti-muon cross-section in the asymptotically safe Standard Model. In particular, we include the graviton contributions to the scattering amplitude, which is computed from momentum-dependent time-like one-particle-irreducible correlation functions. Specifically, we employ reconstruction techniques for the graviton spectral functions. We find that the full asymptotically safe quantum cross section decreases in the ultraviolet with the centre-of-mass energy, and is compatible with unitarity bounds. Importantly, our findings provide non-trivial evidence for the unitarity of the asymptotically safe Standard Model., Comment: 17 pages, 8 figures

We introduce DiFfRG (Discretisation Framework for functional Renormalisation Group flows), a comprehensive computational C++ framework for solving functional Renormalisation Group flows in very general truncation schemes. Its central features are threefold: Firstly, the use of Finite Element Methods (FEM) for efficient, easy to set up and quantitatively reliable computations of field dependences. Secondly, the (simultaneous) setup of large, fully momentum-dependent vertex expansions. Thirdly, efficient time-discretisation methods, incorporating insights from studies of solving theories which exhibit spontaneous symmetry breaking, going hand in hand with shocks and an exponential increase of the information flow velocity in field space. The framework provides a Mathematica package for automatic code generation of flow equations, finite and zero temperature integration routines with support for GPU hardware and extensive parallelisation capabilities. Detailed examples and tutorials are provided and discussed herein which showcase and introduce the framework to the user. We illustrate the capabilities of DiFfRG with four examples, with the complete codes fully included: finite temperature O(N) theory, a Quark-Meson model, SU(3) Yang-Mills theory and four-Fermi flows in the QCD phase diagram.

We study, for the first time, the interplay between colour-confining and chiral symmetry-breaking dynamics in gauge-fermion systems with a general number of flavours and colours. Specifically, we work out the flavour dependence of the confinement and chiral symmetry breaking scales. We connect the QCD-like regime, in quantitative agreement with lattice data, with the perturbative conformal limit, thereby exploring uncharted region of theory space. This analysis is done within the first-principles functional renormalisation group approach to gauge-fermion systems and is facilitated by a novel approximation scheme introduced here. This novel scheme enables a relatively simple access to the confining dynamics. This allows us to investigate the whole landscape of many-flavour theories and to provide a cartography of their phase structure. In particular, we uncover a novel phase with the locking of confining and chiral dynamics at intermediate flavour numbers. We also explore the close-conformal region that displays a walking behaviour. Finally, we provide a quantitative estimate for the lower boundary of the conformal Caswell-Banks-Zaks window, with a $N^{\rm crit}_f(N_c=3)= 9.60^{+0.55}_{-0.53}$. This work offers a self-consistent framework for charting the landscape of strongly interacting gauge-fermion theories necessary to reliably study strongly coupled extensions of the Standard Model of particle physics., Comment: 49 pages, 28 figures

We study the causal structure of the quark propagator with the spectral DSE. The spectral gap equation is solved with the input of the spectral representation of the gluon and a causal STI-construction for the quark-gluon vertex. The latter includes a potential infrared enhancement of the vertex strength of the classical tensor structure that accommodates for the physical strength of chiral symmetry breaking. We find a critical vertex strength, below which the quark has a K\"all\'en-Lehmann representation. While the nature of the first singularity does not change above the critical strength, we find that the quark propagator features at least two additional pairs of complex conjugate poles that are located approximately at the sum of quark pole mass and peak position of the quark-gluon coupling. These additional poles lead to violations of causality, if they persist in $S$-matrix elements. While the vertex strength of the classical tensor structure in full QCD is below the critical one, that of commonly used vertex models, which rely solely on the classical vertex structure, is typically above it. Finally, we discuss how these additional poles could be avoided in full QCD, where part of chiral symmetry breaking is generated by the other tensor structures in the quark-gluon vertex., Comment: 12 pages, 10 figures

We study the confining and deconfining phases of pure $\mathrm{SU}(3)$ lattice gauge theory with topological data analysis. This provides unique insights into long range correlations of field configurations across the confinement-deconfinement transition. Specifically, we analyze non-trivial structures in electric and magnetic field energy densities as well as Polyakov loop traces and a Polyakov loop-based variant of the topological density. The Betti curves for filtrations based on the electric and magnetic field energy densities reveal signals of electromagnetic dualities. These dualities can be associated with an interchange in the roles of local lumps of electric and magnetic energy densities around the phase transition. Moreover, we show that plaquette susceptibilities can manifest in the geometric features captured by the Betti curves. We also confront the results obtained for $\mathrm{SU}(3)$ with that for $\mathrm{SU}(2)$ and elaborate on the significant differences. Our results demonstrate that topological data analysis can discriminate phase transitions of different order for non-Abelian lattice gauge theories. Moreover, it provides unprecedented insights into the relevant structures in their vicinity., Comment: 13 pages, 8 figures

To deepen our understanding of Quantum Gravity and its connections with black holes and cosmology, building a common language and exchanging ideas across different approaches is crucial. The Nordita Program "Quantum Gravity: from gravitational effective field theories to ultraviolet complete approaches" created a platform for extensive discussions, aimed at pinpointing both common grounds and sources of disagreements, with the hope of generating ideas and driving progress in the field. This contribution summarizes the twelve topical discussions held during the program and collects individual thoughts of speakers and panelists on the future of the field in light of these discussions., Comment: Collection of summaries of twelve topical panel discussions and individual thoughts of speakers and panelists, Nordita Scientific Program "Quantum Gravity: from gravitational EFTs to UV complete approaches". 62 pages + references, no figures

We revise the technique of semiclassical effective dynamics, in particular reexamining the evaluation of Poisson structure of the so-called central moments capturing quantum corrections, providing a systematic, pedagogical, and efficient algorithm for evaluation of said structure. The resulting closed formulae for Poisson brackets involve less summatios than recent results in the literature, thus being more optimal for applications. Found formulae are then applied to a general class of isotropic cosmological models with locally observable configuration variables for the admitted matter fields. In particular, this allowed to formulate a consistent and nontrivial limit or fiducial cell (infrared regulator) removal for models describing spatially noncompact spacetimes., Comment: 15 pages

We study the causal structure for spherically symmetric dust collapse within effective loop quantum gravity. We develop a general strategy (working beyond the dynamical model of our consideration) for constructing double null coordinates, allowing the extraction of conformal diagrams within single coordinate charts. With the methods introduced, we confirm that the homogeneous Oppenheimer-Snyder collapse scenario resembles the Reissner-Nordstr\"om-like picture. For the nonhomogenous collapse scenario, we construct the conformal diagrams, subsequently, we study its relevant properties, in particular, dust particles' trajectories, apparent horizons and shell-crossing singularities. We conclude that a considerable region of spacetime cannot be probed by the model's dynamics because of the formation of the shell-crossings. We also address the problem of the possible (non)-existence of timelike singularity known from the homogeneous collapse scenario., Comment: 18 pages, 6 figures

The loop quantum gravitational collapse of the dust ball in presence of positive cosmological constant is investigated within the Oppenheimer-Snyder collapse scenario. The dust ball interior is described within the framework of loop quantum cosmology, while its exterior geometry is determined by the differentiability of the spacetime metric at the dust ball surface and the assumption of the (vacuum) exterior to be stationary. In order to determine the global causal structure of the investigated spacetime a robust (numerical) method of constructing Penrose-Carter diagrams is built. Unfortunately the presence of cosmological constant does not cure the problems already present in the case of it vanishing -- the exterior geometry resembles that of Reissner-Nordstr\"om-de Sitter black hole, in particular featuring timelike singularties., Comment: 5 figures, 18 pages

The physics of strongly correlated systems offers some of the most intriguing physics challenges such as competing orders or the emergence of dynamical composite degrees of freedom. Often, the resolution of these physics challenges is computationally hard, but can be simplified enormously by a formulation in terms of the dynamical degrees of freedom and within an expansion about the physical ground state. Importantly, such a formulation does not only reduce or minimise the computational challenges, it also facilitates the access to the physics mechanisms at play. The tasks of finding the dynamical degrees of freedom and the physical ground state can be systematically addressed within the functional renormalisation group approach with flowing fields which accommodates both, emergent composites as well as the physical ground state. In the present work we use this approach to set up physics-informed renormalisation group flows (PIRG flows): Scale-dependent coordinate transformations in field space induce emergent composites, and the respective flows for the effective action generate a large set of target actions, formulated in these emergent composite fields. This novel perspective on RG flows bears a great potential both for conceptual as well as computational applications: to begin with, PIRG flows allow for a systematic search of the dynamical degrees of freedom and the respective ground state that leads to the most rapid convergence of expansion schemes, thus minimising the computational effort. Secondly, the resolution of the remaining computational tasks within a given expansion scheme can be further reduced by optimising the physics content within a given approximation. Thirdly, the maximal variability of PIRG flows can be used to reduce the analytic and numerical effort of solving the flows within a given approximation., Comment: 35 pages, 16 figures: Correctly compiled figure label

Functional approaches are the only first principle QCD setup that allow for direct computations at finite density. Predictive power and quantitative reliability of the respective results can only be obtained within a systematic expansion scheme with controlled systematic error estimates. Here we set up such a scheme within the functional renormalisation group (fRG) approach to QCD, aiming for full apparent convergence. In the current work we test this setup, using correlation functions and observables in 2+1 flavour vacuum QCD as a natural benchmark case. While the current work includes many evolutionary improvements collected over the past two decades, we also report on three novel important developments: (i) A comprehensive systematic error analysis based on the modular nature of the fRG approach. (ii) The introduction of a fully automated computational framework, allowing for unprecedented access and improvement of the fRG approach to QCD. (iii) The inclusion of the full effective potential of the chiral order parameter. This also gives access to all-order scattering events of pions and to the full momentum dependence of correlation functions, which is a first application of the automated computational framework (ii). The results compare very well to other state-of-the-art results both from functional approaches and lattice simulations, and provide data on general multi-scattering events of pions and the sigma mode for the first time.

The ``Swampland Program'' aims to discriminate consistent-looking effective field theories that do not admit a UV completion in quantum gravity from those that do. While most often developed under the umbrella of string theory, several swampland criteria have been explored also in other contexts, especially asymptotically safe gravity. A comparison between different approaches can help to clarify the dependence of low-energy constraints on UV physics and thereby shed light on the universality of quantum gravity itself. In this short review we summarise what is known about three important swampland conjectures in string theory and in asymptotic safety. We point out future lines of research that can help to understand to what extent swampland conjectures are absolute, i.e. hold in quantum gravity in general, or relative, i.e. belong only to a specific UV framework., Comment: 12 pages, 1 figure, v2: references added

The idea of partial compositeness (PC) in Composite Higgs models offers an attractive means to explain the flavour hierarchies observed in nature. In this talk, predictions of a minimal UV realisation of PC, considering each Standard-Model (SM) fermion to mix linearly with a bound state consisting of a new scalar and a new fermion, are presented, taking into account the dynamical emergence of the composites. Employing the non-perturbative functional renormalisation group, the scaling of the relevant correlation functions is examined and the resulting SM-fermion mass spectrum is analysed., Comment: Contribution to Proceedings of EPS-HEP2023, 21-25 August 2023, Univ. Hamburg and DESY, 10 pages, 3 figures

The chiral crossover of QCD at finite temperature and vanishing baryon density turns into a second order phase transition if lighter than physical quark masses are considered. If this transition occurs sufficiently close to the physical point, its universal critical behaviour would largely control the physics of the QCD phase transition. We quantify the size of this region in QCD using functional approaches, both Dyson-Schwinger equations and the functional renormalisation group. The latter allows us to study both critical and non-critical effects on an equal footing, facilitating a precise determination of the scaling regime. We find that the physical point is far away from the critical region. Importantly, we show that the physics of the chiral crossover is dominated by soft modes even far beyond the critical region. While scaling functions determine all thermodynamic properties of the system in the critical region, the order parameter potential is the relevant quantity away from it. We compute this potential in QCD using the functional renormalisation group and Dyson-Schwinger equations and provide a simple parametrisation for phenomenological applications., Comment: 7+8 pages, 5+4 figures

We compute the bound state properties of three-dimensional scalar $\phi^4$ theory in the broken phase. To this end, we extend the recently developed technique of spectral Dyson-Schwinger equations to solve the Bethe-Salpeter equation and determine the bound state spectrum. We employ consistent truncations for the two-, three- and four-point functions of the theory that recover the scaling properties in the infinite coupling limit. Our result for the mass of the lowest-lying bound state in this limit agrees very well with lattice determinations., Comment: 15 pages, 16 figures

We derive the quantum effective action and the respective quantum equations of motion from multi-graviton correlation functions in asymptotically safe quantum gravity. The fully momentum-dependent couplings of three- and four-graviton scatterings are computed within the functional renormalisation group approach and the effective action is reconstructed from these vertices. The resulting quantum equations of motion are solved numerically for quantum black hole geometries. Importantly, the black hole solutions show signatures of quantum gravity outside the classical horizon, which manifest in the behaviour of the temporal and radial components of the metric. Three different types of solutions with distinct causal structures are identified and the phase structure of the solution space is investigated., Comment: 11 pages + appendices, 8 figures; v2: matches published version

In this contribution, we discuss the asymptotic safety scenario for quantum gravity by evaluating the correlation functions of dynamical metric fluctuations. This is done with a functional renormalisation group approach that disentangles dynamical metric fluctuations from the background metric. We detail the derivation of the respective flow equations on space-time manifolds with Euclidean and Lorentzian signatures and discuss the diffeomorphism symmetry constraints on the flow as well as the convergence of systematic vertex expansion schemes. We then proceed with a comprehensive review of results of momentum-dependent correlation functions at vanishing cutoff scale, the phase structure of the asymptotically safe Standard Model, and spectral properties of asymptotically safe gravity from direct computations in space-times with Lorentzian signatures such as the graviton spectral function., Comment: 47 pages, 15 figures. Invited chapter for the "Handbook of Quantum Gravity" (Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer Singapore)

QCD at large density reveals a rich phase structure, ranging from a potential critical end point and inhomogeneous phases or moat regimes to color superconducting ones with competing order effects. Resolving this region in the phase diagram of QCD with functional approaches requires a great deal of quantitative reliability, already for a qualitative access. In the present work, we systematically extend the functional renormalisation group approach to low energy QCD by setting up a fully self-consistent approximation scheme in a low energy effective quark-meson theory. In this approximation, all pointlike multi-scattering events of the mesonic pion and the sigma mode are taken into account in terms of an effective potential as well as all higher quark-antiquark-mesonic scattering orders. As a first application we compute the phase structure of QCD including its low temperature - large chemical potential part. The quantitative reliability of the approximation and systematic extensions are also discussed.

Composite Higgs extensions of the Standard Model provide an explanation for the large hierarchies between the Yukawa couplings. We study their realisation in the context of fundamental partial compositeness where the Standard Model fermions mix linearly with bound states of the new sector, consisting of a fermion and a scalar. The properties of this composite are unravelled with the functional renormalisation group approach using dynamically emergent composites. Specifically, we extract the scaling of correlation functions and provide indicative estimates for the minimal incarnation of the theory., Comment: 16 pages, 4 figures

Renormalisation group approaches are tailor made for resolving the scale-dependence of quantum and statistical systems, and hence their phase structure and critical physics. Usually this advantage comes at the price of having to truncate the full theory at hand, which asks for optimal expansion schemes. In the present work we use a functional renormalisation group (fRG) approach for the effective action which includes general scale-dependent reparametrisations of the theory [1]. This approach is used in an O(N)-theory to set up adaptive RG-flows that correspond to an optimal systematic expansion of the theory about the ground state or rather its full covariance or propagator. These parametrisations are induced by flowing fields that encode the differential reparametrisation steps. The approach is put to work for an investigation of the thermal phase transition in the O(4)-theory in view of applications to QCD. The respective results are compared with those obtained in standard fRG computations., Comment: 19 pages, 14 figures

We compute non-perturbative spectral functions in a scalar $\phi^4$-theory in three spacetime dimensions via the spectral functional renormalisation group. This approach allows for the direct, manifestly Lorentz covariant computation of correlation functions in Minkowski spacetime, including a physical on-shell renormalisation. We present numerical results for the spectral functions of the two- and four-point correlation functions for different values of the coupling parameter. These results agree very well with those obtained from another functional real-time approach, the spectral Dyson-Schwinger equation., Comment: 22 pages, 13 figures

The dust ball collapse is studied in the context of the "rainbow metric" approach (where the matter content is supplemented with a scalar field perturbation) to the Oppenheimer-Snyder collapse scenario within the framework of loop quantum cosmology. The global spacetime structure is determined for this scenario and subsequently used to evaluate the entanglement entropy via a slight adaptation of existing formulas. The resulting model is shown to qualitatively resemble the Reissner-Nordstr\"om black hole spacetime, in particular, it still contains singularities, which allows us to define the entropy only for portions of the null infinity. The consequence of these results for the black hole information loss paradox is discussed. Furthermore, the results in presence of the scalar field are used in discussion regarding the viability of the selected scenario, in particular, the assumption of stationarity used to determine the exterior metric., Comment: BibTex, 18 pages, 5 figures

Phase transitions in a non-perturbative regime can be studied by ab initio Lattice Field Theory methods. The status and future research directions for LFT investigations of Quantum Chromo-Dynamics under extreme conditions are reviewed, including properties of hadrons and of the hypothesized QCD axion as inferred from QCD topology in different phases. We discuss phase transitions in strong interactions in an extended parameter space, and the possibility of model building for Dark Matter and Electro-Weak Symmetry Breaking. Methodological challenges are addressed as well, including new developments in Artificial Intelligence geared towards the identification of different phases and transitions., Comment: 94 pages, 23 figures, GGI workshop "Phase Transitions in Particle Physics" review - v2: minor revisions and typo corrected, matches accepted version on PPNP

One of critical components of Loop Quantum Gravity (LQG) and Cosmology (LQC) -- Thiemann regularization procedure is non-unique. Different choices of particular prescriptions lead to models which differ in both mathematical structure and physical predictions. Here we briefly recall a set of such prescriptions proposed in the literature in context of isotropic LQC on the example of a flat universe with massless scalar matter content. For the one least investigated so far, further called Yang-Ding-Ma prescription, a detailed analysis of its mathematical structure and resulting quantum dynamics is performed, confirming and extending the results obtained so far by phenomenological methods. In order to probe the dynamics, a relatively robust method (working in the approximation of the macroscopic universe) of evaluating quantum trajectories is devised. Said method is a variant of a semiclassical treatment that allows to express the trajectories analytically as function of internal clock and a set of certain central moments -- constants of motion encoding quantum corrections up to arbitrary order in systematic manner. As a test of method's robustness analogous evaluation of the quantum trajectory in volume is performed for those of other prescriptions, for which it is applicable. The limitations of the treatment are further briefly discussed., Comment: RevTex, BibTex, 38 pages, 2 figures, extension of the MSc thesis arXiv:2112.07786

We compute masses of the two lightest glueballs from spectral reconstructions of timelike interaction channels of the four-gluon vertex in Landau gauge Yang-Mills theory. The Euclidean spacelike dressings of the vertex are calculated with the functional renormalisation group. For the spectral reconstruction of these Euclidean data, we employ Gaussian process regression. The glueball resonances can be identified straightforwardly and we obtain $m_{sc} = 1870(75)~$ MeV as well as $m_{ps} = 2700(120)~$ MeV, in accordance with functional bound state and lattice calculations., Comment: 10 pages, 6 figures

We calculate the spectral function of light quark flavours in 2+1 flavour vacuum QCD in the isospin-symmetric approximation. We employ spectral Dyson-Schwinger equations and compute the non-perturbative quark propagator directly in real-time, using recent spectral reconstruction results from Gaussian process regression of gluon propagator data in 2+1 flavour lattice QCD. Our results feature a pole-like peak structure at time-like momenta larger than the propagator's gapping scale as well as a negative scattering continuum, which we exploit assuming an analytic pole-tail split during the iterative solution. The computation is augmented with a general discussion of the impact of the quark-gluon vertex and the gluon propagator on the analytic structure of the quark propagator. In particular, we investigate under which conditions the quark propagator shows unphysical complex poles. Our results offer a wide range of applications, encompassing the ab-initio calculation of transport as well as resonance properties in QCD., Comment: 17 pages, 7 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

We apply the Local Discontinuous Galerkin discretisation to flow equations of the O(N)-model in the Local Potential Approximation. The improved stability is directly observed by solving the flow equation for various $N$ and space-time dimensions $d$. A particular focus of this work is the numerical discretisation and its implementation. The code is publicly available, and is explained in detail here. It is realised as a module within the high performance PDE framework DUNE., Comment: 18 pages, 10 figures

In the present work we set up a general functional renormalisation group framework for the computation of complex effective actions. For explicit computations we consider both flows of the Wilsonian effective action and the one-particle irreducible (1PI) effective action. The latter is based on an appropriate definition of a Legendre transform for complex actions, and we show its validity by comparison to exact results in zero dimensions, as well as a comparison to results for the Wilsonian effective action. In the present implementations of the general approaches, the flow of the Wilsonian effective action has a wider range of applicability and we obtain results for the effective potential of complex fields in $\phi^4$-theories from zero up to four dimensions. These results are also compared with results from the 1PI effective action within its range of applicability. The complex effective action also allows us to determine the location of the Lee-Yang zeros for general parameter values. We also discuss the extension of the present results to general theories including QCD., Comment: 29 pages, 24 figures

We present a comprehensive non-perturbative study of the phase structure of the asymptotically safe Standard Model. The physics scales included range from the asymptotically safe trans-Planckian regime in the ultraviolet, the intermediate high-energy regime with electroweak symmetry breaking to strongly correlated QCD in the infrared. All flows are computed with a self-consistent functional renormalisation group approach, using a vertex expansion in the fluctuation fields. In particular, this approach takes care of all physical threshold effects and the respective decoupling of ultraviolet degrees of freedom. Standard Model and gravity couplings and masses are fixed by their experimental low energy values. Importantly, we accommodate for the difference between the top pole mass and its Euclidean analogue. Both, the correct mass determination and the threshold effects have a significant impact on the qualitative properties, and in particular on the stability properties of the specific ultraviolet-infrared trajectory with experimental Standard Model physics in the infrared. We show that in the present rather advanced approximation the matter part of the asymptotically safe Standard Model has the same number of relevant parameters as the Standard Model, and is asymptotically free. This result is based on the novel UV fixed point found in the present work: the fixed point Higgs potential is flat but has two relevant directions. These results and their analysis are accompanied by a thorough discussion of the systematic error of the present truncation, also important for systematic improvements., Comment: 41 pages, 19 figures. v3: summary of approximations and suggestions from referees v2: discussion of full eigenfunctions of fixed point solution added in Section V, typos corrected, citations added

We derive renormalised finite functional flow equations for quantum field theories in real and imaginary time that incorporate scale transformations of the renormalisation conditions, hence implementing a flowing renormalisation. The flows are manifestly finite in general non-perturbative truncation schemes also for regularisation schemes that do not implement an infrared suppression of the loops in the flow. Specifically, this formulation includes finite functional flows for the effective action with a spectral Callan-Symanzik cutoff, and therefore gives access to Lorentz invariant spectral flows. The functional setup is fully non-perturbative and allows for the spectral treatment of general theories. In particular, this includes theories that do not admit a perturbative renormalisation such as asymptotically safe theories. Finally, the application of the Lorentz invariant spectral functional renormalisation group is briefly discussed for theories ranging from real scalar and Yukawa theories to gauge theories and quantum gravity., Comment: 31 pages, 1 figure

We investigate BRST invariance in Landau gauge Yang-Mills theory with functional methods. To that end, we solve the coupled system of functional renormalisation group equations for the momentum-dependent ghost and gluon propagator, ghost-gluon, and three- and four-gluon vertex dressings. The equations for both, transverse and longitudinal correlation functions are solved self-consistently: all correlation functions are fed back into the loops. Additionally, we also use the Slavnov-Taylor identities for computing the longitudinal correlation functions on the basis of the above results. Then, the gauge consistency of the solutions is checked by comparing the respective longitudinal correlation functions. We find good agreement of these results, hinting at the gauge consistency of our setup., Comment: 27 pages, 19 figures

We consider the coupled set of spectral Dyson-Schwinger equations in Yang-Mills theory for ghost and gluon propagators, which gives us access to the ghost and gluon spectral functions. The set-up is used for a systematic analytic evaluation of the constraints on generalised spectral representations in Yang-Mills theory that are most relevant for informed spectral reconstructions. We also provide numerical results for the coupled set of spectral functions for a large range of potential mass gaps of the gluon, and discuss the limitations and extensions of the present work., Comment: 31 pages, 16 figures

Lattice simulations along with studies in continuum QCD indicate that non-perturbative quantum fluctuations lead to an infrared regularisation of the gluon propagator in covariant gauges in the form of an effective mass-like behaviour. In the present work we propose an analytic understanding of this phenomenon in terms of gluon condensation through a dynamical version of the Higgs mechanism, leading to the emergence of color condensates. Within the functional renormalisation group approach we compute the effective potential of covariantly constant field strengths, whose non-trivial minimum is related to the color condensates. In the physical case of an SU(3) gauge group this is an octet condensate. The value of the gluon mass obtained through this procedure compares very well to lattice results and the mass gap arising from alternative dynamical scenarios., Comment: 22 pages, 9 figures

We derive the temporal renormalisation group flow of the energy-momentum tensor at the example of a general scalar theory. The local causal structure of the temporal renormalisation group flow allows to monitor and control causality, unitarity and general conservation laws at each infinitesimal renormalisation group step. We explore energy-conserving truncations in a comparison of generic flows and the causal temporal flow of the energy-momentum tensor. We also observe that the temporal regulator preserves scale invariance, which is violated for generic momentum regulators. Specifically we discuss the relation of these terms to the trace anomaly of the energy-momentum tensor. Moreover, we show that the causal temporal flow of the energy-momentum tensor can be integrated analytically, and demonstrate, that the result is consistent with energy conservation., Comment: 9 pages, 4 figures

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 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 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 reconstruct ghost and gluon spectral functions in 2+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 towards unveiling real-time properties of QCD., Comment: 12 pages, 5 figures

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., Comment: 33 pages, 24 figures. Identical to the version published in Phys. Rev. D

Path integrals with complex actions are encountered for many physical systems ranging from spin- or mass-imbalanced atomic gases and graphene to quantum chromo-dynamics at finite density to the non-equilibrium evolution of quantum systems. Many computational approaches have been developed for tackling the sign problem emerging for complex actions. Among these, complex Langevin dynamics has the appeal of general applicability. One of its key challenges is the potential convergence of the dynamics to unphysical fixed points. The statistical sampling process at such a fixed point is not based on the physical action and hence leads to wrong predictions. Moreover, its unphysical nature is hard to detect due to the implicit nature of the process. In the present work we set up a general approach based on a Markov chain Monte Carlo scheme in an extended state space. In this approach we derive an explicit real sampling process for generalized complex Langevin dynamics. Subject to a set of constraints, this sampling process is the physical one. These constraints originate from the detailed-balance equations satisfied by the Monte Carlo scheme. This allows us to re-derive complex Langevin dynamics from a new perspective and establishes a framework for the explicit construction of new sampling schemes for complex actions., Comment: 29 pages, 9 figures

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 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 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., Comment: 30 pages, 17 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

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)

We reconstruct the Lorentzian graviton propagator in asymptotically safe quantum gravity from Euclidean data. The reconstruction is applied to both the dynamical fluctuation graviton and the background graviton propagator. We prove that the spectral function of the latter necessarily has negative parts similar to, and for the same reasons, as the gluon spectral function. In turn, the spectral function of the dynamical graviton is positive. We argue that the latter enters cross sections and other observables in asymptotically safe quantum gravity. Hence, its positivity may hint at the unitarity of asymptotically safe quantum gravity., Comment: 22 pages, 8 figures, 2 tables

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 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., Comment: 18 pages, 4 figures