## Final Report Summary - STRONG QCD (Strong-Interaction Physics and the Phase Diagram of Quantum Chromodynamics)

Strong-interaction matter fuels the stars and makes up almost the entire mass of the luminous universe. The underlying theory of quarks and gluons, Quantum Chromodynamics (QCD), completely specifies the interactions. However, these are so complex and non-linear that they have yet to be fully understood. Indeed, it is these strong interactions that under normal conditions confine quarks and gluons into the interior of hadrons. Understanding the generation of their masses, the confinement of quarks and gluons, the different phases of QCD at extreme temperatures or densities and the transitions between them are some of the great challenges in physics.

This project addresses these outstanding problems in theoretical particle and strong interaction physics, which receive significant attention also in various major experimental programs around the world, such as PHENIX and STAR at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven, ALICE at the Large Hadron Collider (LHC) at CERN, or the Compressed Baryonic Matter (CBM) experiment planned at the Facility for Antiproton and Ion Research (FAIR) at GSI. The results advance our knowledge of how the fundamental laws of physics describe the subatomic structure of nature and its cosmological origins.

The central goal of this research project was to further unveil the nature of hadrons and their interactions through combining state-of-the-art functional methods of non-perturbative local quantum field theory in the continuum and the ab-initio and first principles approach of lattice gauge theory and simulations. To achieve this goal the project addressed three work packages: (a) the understanding of QCD Green's functions in the non-perturbative domain; (b) the study of the dynamics of topological objects and their role in the deconfinement transition; (c) the development of suitable tools to investigate the phase diagrams of QCD and QCD-like theories, especially at finite baryon density.

(a) QCD Green's functions

The correlation functions of QCD, or QCD Green’s functions, are the fundamental building blocks for hadron phenomenology and strong-interaction matter studies based on functional continuum methods. Built on a solid understanding of the pure gauge theory’s vacuum correlations, there has recently been considerable progress in extensions to finite temperature, to including dynamical quarks, and to finite baryon density.

In this project, the infrared behavior of the QCD propagators has first been studied with lattice implementations of the Landau in the strong coupling limit in various space-time dimensions. This research highlighted ambiguities of current lattice implementations of the Landau gauge and emphasized the differences between lattice and continuum methods based on the Becchi-Rouet-Stora-Tyutin (BRST) symmetry. Detailed studies of the behavior of the propagators of Landau gauge QCD over the deconfinement phase transitions at finite temperature for two and three colors with Monte-Carlo simulations on the lattice were performed. These studies demonstrated the characteristic differences between first and second-order phase transitions as reflected in these propagators for the first time.

Our precision determinations of the strong coupling αS from the perturbative behavior of these propagators as extracted from lattice simulations with 2 and 2+1 flavors of dynamical quarks progressed very well, and our results were updated regularly.

We have also continued our studies of the QCD Green's functions from their equations of motion, the Dyson-Schwinger equations (DSEs) of QCD, where we included for the first time the 3-point gluon-ghost vertex function dynamically in the iterative solutions in various spacetime dimensions. This demonstrated that DSE studies and lattice simulations are in very good quantitative agreement nowadays. We have furthermore recently solved a coupled system of DSEs for the pure gauge theory that dynamically includes the full set of three-point vertex functions for the first time. This brought DSE studies to a new level and provided good evidence for the convergence of suitable vertex expansions in order to add more systematics to these widely used non-perturbative continuum methods for strong-interaction physics.

The BRST symmetry underlying the covariantly gauge-fixed continuum theory was formulated with the methods of supersymmetric quantum mechanics. This explicitly demonstrated that the Neuberger 0/0 problem of perfect cancellation of all Gribov copies on the lattice can be avoided.

(b) Topological objects in the deconfinement transition

Topological defects such as monopoles, vortices and domain walls are of considerable interest in the Standard Model (SM) and its extensions. For example, they are often identified as the objects responsible for driving phase transitions that correspond to the breaking of some symmetry. This research component is therefore directed towards these defects and the phase structure of the SM, and is motivated in particular by the deconfinement transition in QCD.

There is compelling evidence that static charges, or infinitely heavy quarks, are confined in pure SU(N) gauge theory by center vortices, which are the analogues of `knots' in liquid crystals or flux tubes in superconductors. At low temperatures, the proliferation of certain center vortices disorders Wilson loops and leads to confinement. Above a critical deconfinement temperature, these center vortices are suppressed and static charges are approximately free.

In this work package the gauge invariant studies of center vortices and their dynamics over the deconfinement phase transitions of SU(N) gauge theories were extended with various numbers of colors N and in three and four dimensions. In particular, this allowed further high-precision studies of the universal aspects of the second-order transitions of SU(2) and SU(3) in three dimensions. These studies revealed for the first time a self-duality between center vortices and the confining electric fluxes. As a byproduct we obtained an explicit analytic formula generalising the well-known duality transformations of the two-dimensional q-state Potts models to finite volumes with different boundary conditions. The new finite-volume duality transformation holds for all q-state Potts models on arbitrary graphs with torus topology in two dimensions.

Another important result from this work package was our implementation of the SU(2) x U(1) toy model with dynamical Wilson quarks of two colors and half-integer electric charge with respect to the Abelian U(1) factor in lattice simulations. This provided a first demonstration of the non-perturbative effect of the fractionality of the quarks' charges due to additional disorder that brings back the phase transition of the pure SU(2) gauge theory despite the presence of fractionally charged dynamical quarks. This attracted considerable attention as it might change or view on confinement and disorder in the QCD vacuum when QCD is embedded in the Standard Model. This research naturally connects to ongoing studies of the same effect in corresponding Higgs models with fractionally charged scalars where an analogous effect can be simulated at much reduced costs.

(c) QCD-like theories at finite density

The main focus of the 2nd funding period was thereby the application of functional methods to the phase diagram of strong-interaction matter, in particular at finite baryon density. It was intended for example to compare functional renormalization group results, which have predicted a critical end point in this phase diagram where the first-order chiral transition boundary ends, with lattice simulations. Since this region is not directly accessible in Monte-Carlo simulations based on lattice QCD due to the fermion sign problem, it became more and more evident during the 1st period that it was worthwhile to enlarge the original focus by including QCD-like theories without this problem in the studies.

Particular highlights of the 2nd project period include the first ab-inito simulations within the framework of lattice field theory of a gauge theory with fermionic baryons at finite density. Replacing the SU(3) gauge group of QCD by the exceptional group G2 which contains SU(3) as a subgroup one obtains G2-QCD with many common features but without fermion sign problem. This allowed us to study the phase diagram of this QCD-like theory in large-scale Monte-Carlo simulations. We have determined the spectrum of this theory and we obtained evidence of a low-temperature liquid-gas phase transition to nuclear matter as expected in QCD.

Another replacement theory without fermion sign problem is two-color QCD, with only two instead of the usual three colors of QCD. Since the baryons are bosons composed of two quarks in this case, which undergo Bose-Einstein condensation, one can exploit analogies with ultracold fermionic quantum gases. Based on available QCD methods we have developed a new effective description for two-color QCD, including the effects of fluctuations due to collective mesonic and baryonic excitations. We applied the same techniques in parallel to QCD at finite isospin density and successfully tested this description for both cases against lattice simulations. Combinations of isospin and baryon chemical potential in the effective description furthermore allowed us to identify the universal behavior of polarized fermionic quantum gases at unitarity in this relativistic model of strong-interaction matter.

An important new development over the last year was that we now have the capacity to calculate real-time quantities such as spectral functions and transport coefficients directly from the functional renormalization group. This became possible with a new method to solve analytically continued flow equations for two-point correlation functions, which explicitly realizes Baym-Mermin boundary conditions for the uniqueness of this continuation. This method was successfully tested in the O(N) model and later extended to a study of the mesonic spectral functions in the quark-meson model at finite temperature and density. This allowed us to study their behaviour over the chiral crossover where chiral symmetry is gradually restored, for example, or the critical fluctuations in the sigma correlator in the vicinity of the critical endpoint.

In relation to other strongly interacting systems, all our non-perturbative methods for QCD, lattice Monte-Carlo simulations, functional renormalization group studies and Dyson-Schwinger equations, have also been applied and tested for fermions on a two-dimensional honeycomb lattice in order to describe the electronic properties and phases of graphene. The electronic Lifshitz transition of the corresponding tight-binding model, whose density of states could be measured in microwave photonic crystals with a Dirac spectrum, was identified as an Excited State Phase Transition as known from nuclear and atomic physics. The measurement of the density of states by an experimental group within the Collaborative Research Center, SFB 634, of the Deutsche Forschungsgemeinschaft at the TU Darmstadt is but one example of the synergy effects that were gained by embedding this project into the research environment at the host institution.