1. QCD Phase Structure

Low energy QCD is governed by two prominent non-perturbative phenomena, dynamical chiral symmetry breaking and confinement. Both are well-incorporated in functional approaches to first principle QCD, see e.g.~the recent works in vacuum QCD with the functional Renormalisation Group (fRG) works [1,2] (fRG) and Dyson-Schwinger equations (DSEs) [3] in the fQCD collaboration, and the reviews [4,5]. These computations have passed systematic error checks, leading to a systematic error estimate on observables of \(\,< 5\% - 10\%\), depending on the specific observable, see the topic Systematics. Moreover, in the vacuum and finite temperature the results agree quantitatively with the available lattice benchmarks. The setup also allows us to study the chiral limit and the relevance of soft modes [6]. All in all this sets the stage for the application of functional approaches to finite density QCD.

Presently, direct computations in first principle QCD at finite density with \(\mu_B/T \gtrsim 4\) are only possible within functional approaches. They (mostly the fRG and DSE approach) are used in the fQCD collaboration to map out the phase structure of QCD: we aim at the location of the crossover line, that of the critical end point or more precisely the onset of new phases such as the moat regime or first order regimes [7,8,9]. We also aim at the computation and prediction of experimental observables such as the fluctuations of conserved charges [10]. We also access realtime and dynamical properties such as spectral functions [9], dynamical critical scaling [11] and transport coefficients that serve, together with full effective potentials, as input for hydrodynamic simulations of heavy-ion collisions [12]. This endeavour goes hand in hand with the systematic improvement of the approximations used in our computational framework, see also the topic Systematics.

References:

[1] Towards quantitative precision in functional QCD I - Ihssen, Pawlowski, Sattler, Wink, arXiv:2408.08413.

[2] Four-quark scatterings in QCD III - Fu, Huang, Pawlowski, Tan, Zhou, arXiv:2502.14388.

[3] Fully coupled functional equations for the quark sector of QCD - Gao, Papavassiliou, Pawlowski, Phys.Rev.D 103 (2021) 9, 094013, arXiv:2102.13053.

[4] The nonperturbative functional renormalization group and its applications - Dupuis, Canet, Eichhorn, Metzner, Pawlowski, Tissier, Wschebor, Phys.Rept. 910 (2021) 1-114, arXiv:2006.04853.

[5] QCD at finite temperature and density within the fRG approach: an overview - Fu, Commun.Theor.Phys. 74 (2022) 9, 097304, arXiv:2205.00468.

[6] Soft modes in hot QCD matter - Braun, Chen, Fu, Gao, Huang, Ihssen, Pawlowski, Rennecke, Sattler, Tan, Wen, Yin, arXiv:2310.19853.

[7] QCD phase structure at finite temperature and density - Fu, Pawlowski, Rennecke, Phys.Rev.D 101 (2020) 5, 054032, arXiv:1909.02991.

[8] Chiral phase structure and critical end point in QCD - Gao, Pawlowski, Phys.Lett.B 820 (2021) 136584, arXiv:2010.13705.

[9] The QCD moat regime and its real-time properties - Fu, Pawlowski, Pisarski, Rennecke, Wen, Yin, arXiv:2412.15949.

[10] Ripples of the QCD Critical Point - Fu, Luo, Pawlowski, Rennecke, Yin, Phys.Rev.D 111 (2025) 3, L031502, arXiv:2308.15508.

[11] Critical dynamics of Model H within the real-time fRG approach - Chen, Tan, Fu, arXiv:2406.00679.

[12] Transport Coefficients in Yang-Mills Theory and QCD - Christiansen, Haas, Pawlowski, Strodthoff, Phys.Rev.Lett. 115 (2015) 11, 112002, arXiv:1411.7986.

2. Thermodynamics and applications

The equation of state (EoS) of strong-interaction matter as a function of the temperature and density and further thermodynamic observables, such as the energy and entropy densities as well as the trace anomaly are of great relevance for phenomenological applications, ranging from heavy-ion collision experiments to neutron stars. In all these applications, an accurate computation of the EoS is key to access the dynamics of these systems. The determination of these observables requires the access of the full momentum and frequency dependence of all the propagators of QCD, and specifically that of the quarks. For other observables or low energy correlation this dynamical information has a sub-leading impact. A prominent example is the effective potential of the chiral order parameter that encodes all order scattering processes of pions. The independence of the latter is at the root of the unreasonable effectiveness of low energy effective theories in QCD [1,2]. The requirement of computing the full momentum and frequency dependence of the QCD propagators for the temperatures and densities under investigations for computing the Eos and further thermodynamic observables makes it indispensable to approach these observables within QCD itself. This allows for an accurate description of hot and dense matter, from perturbatively accessible regimes to non-perturbative domains governed by dynamical symmetry breaking and the formation of condensates. Amongst other demands and the general quest for smaller systematic errors, it is the requirements above that guide our systematic development of more sophistic truncations, In [3], we investigate the impact of strangeness neutrality on the phase structure and thermodynamics of QCD at finite baryon and strangeness chemical potential in temperature and density regimes relevant for heavy-ion collision experiments. In [4], we present a calculation of the zero-temperature equation of state over a wide range of densities. Our results indicate that the speed of sound exceeds the value of the non-interacting quark gas at high densities and even increases as the density is decreased, across a wide range, suggesting the existence of a maximum at supranuclear densities, in accordance with information extracted from neutron-star observations. In [5] we investigate the fluctuations of conserved baryon charge that are related to the chemical potential derivatives of thermodynamic observables.

References:

[1] The nonperturbative functional renormalization group and its applications - Dupuis, Canet, Eichhorn, Metzner, Pawlowski, Tissier, Wschebor, Phys.Rept. 910 (2021) 1-114, arXiv:2006.04853.

[2] QCD at finite temperature and density within the fRG approach: an overview - Fu, Commun.Theor.Phys. 74 (2022) 9, 097304, arXiv:2205.00468.

[3] Strangeness Neutrality and QCD Thermodynamics - Fu, Pawlowski, and Rennecke, SciPost Phys.Core 2, 002 (2020).

[4] From quarks and gluons to color superconductivity at supranuclear densities - Braun, Schallmo, Phys. Rev. D 105, 036003 (2022).

[5] Ripples of the QCD Critical Point - Fu, Luo, Pawlowski, Rennecke, Yin, Phys.Rev.D 111 (2025) 3, L031502, arXiv:2308.15508.

3. Hadronic properties

In functional approaches resonant scattering processes and their properties such as pole masses of the respective hadronic excitations are typically resolved within a combination of DSEs and bound state equations such as Bethe-Salpeter equations (BSE) and Faddeev equations. The fRG approach allows for a very efficient book keeping of resonant scattering channels as well as producing precision results for QCD correlation functions.

This project revolves about two aspects: Firstly, we have initiated the systematic development of a functional Renormalisation Group approach to QCD which is well-adapted to studies of hadron properties (fRG-hadron). Secondly, after an initial pure development phase we use fRG-hadron for the study of non-perturbative properties of hadronic resonances. Amongst the applications are the computation of parton distribution amplitude (PDA) or quasi-PDA of hadrons, transverse-Momentum-Dependent Wave Functions (TMDWF) or quasi-TMDWF, etc.

The development of fRG-hadron has been initiated in a series of work [1-3], cumulating in the setup in first principle QCD in [3]. Specifically, the rôle of the coupled set of the quark gap equation (DSE) and the BSE for the four-quark scattering kernel in the DSE-BSE formulation is taken over by the fRG flow of the four-quark scattering kernel and the quark propagator. This set of coupled flow equations is complemented by the flow of the quark-gluon vertex and the glue sector. The setup in [3] hosts no phenomenological parameters or external input. The only ultraviolet input parameters are the physical ones in QCD: the light and strange quark masses. They are adjusted to the physical ratios of the pion and kaon masses, divided by the pion decay constant. The results for other observables of current first-principles computations are in quantitative agreement with the physical ones. The approach is continuously developed further, drawing also from technical developments in the other project lines in the fRG collaboration.

These developments are accompanied by further ones such as an approach for computing the valence-quark quasi parton distribution amplitude (quasi-PDA) and quasi parton distribution function (quasi-PDF) for the pion with a large longitudinal momentum [4].

References:

[1] Four-quark scatterings in QCD I - Fu, Huang, Pawlowski, Tan, SciPost Phys. 14, 069 (2023), arXiv:2209.13120.

[2] Four-quark scatterings in QCD II - Fu, Huang, Pawlowski, Tan, SciPost Phys. 17, 148 (2024), arXiv:2401.07638.

[3] Four-quark scatterings in QCD III - Fu, Huang, Pawlowski, Tan, Zhou, arXiv:2502.14388.

[4] Quasi parton distributions of pions at large longitudinal momentum - Zhang, Huang, Fu, arXiv:2502.15384.

4. Systematics

Functional QCD approaches build upon diagrammatic master equations for the QCD effective action. These master equations are one loop (fRG) or two loop (DSE) exact (figures). By taking derivatives with respect to the gluon, ghost, as well as quark and potential composite hadronic fields these master equations lead to towers of one- or two-loop exact loop relations between QCD correlation functions. These relations are fully non-perturbative and its constituents are the full n-point vertices and propagators of QCD. The closed form of these relations (one loop or two-loop) allows for a comprehensive access to systematic error estimates, see [1,2,3,4]:

Vertex expansion: Instead of an infinite number of diagrams left out as in perturbation theory we typically expand this tower of relations in the number n of fields in a scattering vertex. Accordingly, within a given order n of this vertex expansion we are left with a finite number of n+1 point functions we assumed to be irrelevant and that are left out in the diagrammatic relations. Their impact is readily evaluated by 'dialing' their strengths within physically allowed ranges. Moreover, we can compute their strength within the approximation itself and test their irrelevance. The latter tests the self-consistency of the approximation and is only comprehensive in this sense if it is based on closed relations. In general, the vertex expansion converges very rapidly due to the phase space suppression of higher order scatterings. The only exception are resonant channels in higher order vertices, and the prime example are resonant channels in the four-quark scattering vertices. These resonant channels are accommodated with the fRG approach with emergent composites or a combination of bound state equations (BSE, Faddeev and four-body equations) and the DSE.

Emergent composites and systematic error control: Resonant channels in scattering vertices undo the otherwise generic phase space suppression of higher order vertices and lead to a slow convergence of the vertex expansion. These resonant channels can be captured completely by composite exchange fields and this is formulated in terms of an exact flow equation that relates the dynamics of the emergent composites and that of the resonant channels. This allows us to efficiently also capture the higher order scatterings of the resonant channels and, most importantly, restores the rapid convergence of the vertex expansion for the full tower of loop relations.

Derivative expansion: For asymptotically small momenta, the infrared regime of QCD is governed by the pion dynamics, and admits a well-controlled expansion in pion scatterings, chiral perturbation theory. Moreover, scattering vertices can be expanded in terms of the small momentum, measured in the infrared mass scales of QCD. This expansion is called the derivative expansion, and its versatility, rapid convergence and small systematic error has been shown in many applications. These applications include low energy effective models of QCD, but also general condensed matter and statistical systems. With the LEGO principle these checks can be straightforwardly used in QCD.

LEGO principle [4]: These checks of the systematic error of a given order of the vertex expansion elucidates a general principle behind the systematic error analysis in functional approaches, based on their modularity. The constituents in the loop relation for a given vertex can be ordered in the different sectors of QCD: the pure glue sector, the matter sector, and the interface sector connecting those two (the LEGO blocks). These sectors are only connected by a few constituents (the LEGO studs), and a separate systematic error analysis of the different sectors can be done by varying the strength of the studs. The systematic error analysis of the whole system is then constructed by combining the systematic error estimates of the different sectors.

References:

[1] The nonperturbative functional renormalization group and its applications - Dupuis, Canet, Eichhorn, Metzner, Pawlowski, Tissier, Wschebor, Phys.Rept. 910 (2021) 1-114.

[2] QCD at finite temperature and density within the fRG approach: an overview - Fu, Commun.Theor.Phys. 74 (2022) 9, 097304.

[3] QCD equation of state and thermodynamic observables from computationally minimal Dyson-Schwinger Equations - Lu, Gao, Liu, Pawlowski, Phys.Rev.D 110 (2024) 1, 014036, arXiv:2310.18383.

[4] Towards quantitative precision in functional QCD I - Ihssen, Pawlowski, Sattler, Wink, arXiv:2408.08413.