We discuss the QCD phase structure at finite temperature and chemical potential for 2-flavor and $2+1$-flavor QCD. The results are achieved by computing QCD correlation functions within a generalized functional approach that combines Dyson-Schwinger equations (DSE) and the functional renormalization group (fRG). In this setup fRG precision data from [A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D 97, 054006 (2018).] for the vacuum quark-gluon vertex and gluon propagator of 2-flavor QCD used as input, and the respective DSEs are expanded about this input. While the vacuum results for other correlation functions serve as a self-consistency check for functional approaches, the results at finite temperature and density are computed, for the first time, without the need of phenomenological infrared parameters.

Beginning with results for the leading-twist two-particle distribution amplitudes of $$\pi $$π - and K-mesons, each of which exhibits dilation driven by the mechanism responsible for the emergence of hadronic mass, we develop parameter-free predictions for the pointwise behaviour of all $$\pi $$π and K distribution functions (DFs), including glue and sea. The large-x behaviour of each DF meets expectations based on quantum chromodynamics; the valence-quark distributions match extractions from available data, including the pion case when threshold resummation effects are included; and at $$\zeta _5=5.2\,$$ζ 5 = 5.2 GeV, the scale of existing measurements, the light-front momentum of these hadrons is shared as follows: $$\langle x_{\mathrm{valence}} \rangle ^\pi = 0.41(4)$$⟨ x valence ⟩ π = 0.41 ( 4 ) , $$\langle x_{\mathrm{glue}} \rangle ^\pi = 0.45(2)$$⟨ x glue ⟩ π = 0.45 ( 2 ) , $$\langle x_{\mathrm{sea}} \rangle ^\pi = 0.14(2)$$⟨ x sea ⟩ π = 0.14 ( 2 ) ; and $$\langle x_{\mathrm{valence}} \rangle ^K = 0.42(3)$$⟨ x valence ⟩ K = 0.42 ( 3 ) , $$\langle x_{\mathrm{glue}} \rangle ^K = 0.44(2)$$⟨ x glue ⟩ K = 0.44 ( 2 ) , $$\langle x_{\mathrm{sea}} \rangle ^K = 0.14(2)$$⟨ x sea ⟩ K = 0.14 ( 2 ) . The kaon’s glue and sea distributions are similar to those in the pion, although the inclusion of mass-dependent splitting functions introduces some differences on the valence-quark domain. This study should stimulate improved analyses of existing data and motivate new experiments sensitive to all $$\pi $$π and K DFs. With little known empirically about the structure of the Standard Model’s (pseudo-) Nambu-Goldstone modes and analyses of existing, limited data being controversial, it is likely that new generation experiments at upgraded and anticipated facilities will provide the information needed to resolve the puzzles and complete the picture of these complex bound states.

We present a comprehensive study of the quark sector of $2+1$ flavor 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 in the one-loop dressed approximation. 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 flavors, obtained from studies with Schwinger-Dyson equations, the functional renormalization group approach, and large volume lattice simulations. The appropriate renormalization 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 renormalization 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 renormalization group, and lattice QCD simulation, a simple benchmark observable being the chiral condensate in the chiral limit, which is computed as $(245\mathrm{MeV}{)}^3$. 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.

A continuum approach to the kaon and pion bound-state problems is used to reveal their electromagnetic structure. For both systems, when used with parton distribution amplitudes appropriate to the scale of the experiment, Standard Model hard-scattering formulas are accurate to within 25% at momentum transfers $Q^2\approx 8{\mathrm{GeV}}^2$. There are measurable differences between the distribution of strange and normal matter within the kaons, e.g. the ratio of their separate contributions reaches a peak value of 1.5 at $Q^2\approx 6{\mathrm{GeV}}^2$. Its subsequent $Q^2$ evolution is accurately described by the hard scattering formulas. Projections for the ratio of kaon and pion form factors at timelike momenta beyond the resonance region are also presented. These results and projections should prove useful in planning next-generation experiments.