Using lattice configurations for quantum ​​​​​chromodynamics (QCD) generated with three domain-wall fermions at a physical pion mass, we obtain a parameter-free prediction of QCD’s renormalisation-group-invariant process-independent effective charge, . Owing to the dynamical breaking of scale invariance, evident in the emergence of a gluon mass-scale, GeV, this coupling saturates at infrared momenta: . Amongst other things: is almost identical to the process-dependent (PD) effective charge defined via the Bjorken sum rule; and also that PD charge which, employed in the one-loop evolution equations, delivers agreement between pion parton distribution functions computed at the hadronic scale and experiment. The diversity of unifying roles played by suggests that it is a strong candidate for that object which represents the interaction strength in QCD at any given momentum scale; and its properties support a conclusion that QCD is a mathematically well-defined quantum field theory in four dimensions.

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.

With a modified chemical potential dependent effective model for the gluon propagator, we try to locate the critical end point (CEP) of strongly interacting matter in the framework of Dyson-Schwinger equations (DSE). Beyond the chiral limit, we find that Nambu solution and Wigner solution could coexist in some area. Using the CornwallJackiw-Tomboulis (CJT) effective action, we show that these two phases are connected by a first order phase transition. We then locate CEP as the end point of the first order phase transition line. Meanwhile, based on CJT effective action, we give a direct calculation for the chiral susceptibility and thereby study the crossover.

Within the framework of Dyson-Schwinger equations , we discuss the chiral phase transition of QCD with a chiral chemical potential ${\mu }_5$ as an additional scale. We focus especially on the issues related to the widely accepted as well as interesting critical end point (CEP). With the help of a scalar susceptibility, we find that there might be no ${\mathrm{CEP}}_5$ in the $T-{\mu }_5$ plane, and the phase transition in the $T-{\mu }_5$ plane might be totally crossover when $\mu <50\mathrm{MeV}$, which has apparent consistency with the lattice QCD calculation. Our study may also provide some useful hints to some other studies related to ${\mu }_5$.

Analyses of the pion valence-quark distribution function (DF), "Equation missing", which explicitly incorporate the behaviour of the pion wave function prescribed by quantum chromodynamics (QCD), predict "Equation missing", $$\beta (\zeta \gtrsim m_p)>2$$β ( ζ ≳ m p ) > 2 , where $$m_p$$m p is the proton mass. Nevertheless, more than forty years after the first experiment to collect data suitable for extracting the $$x\simeq 1$$x ≃ 1 behaviour of "Equation missing", the empirical status remains uncertain because some methods used to fit existing data return a result for "Equation missing" that violates this constraint. Such disagreement entails one of the following conclusions: the analysis concerned is incomplete; not all data being considered are a true expression of qualities intrinsic to the pion; or QCD, as it is currently understood, is not the theory of strong interactions. New, precise data are necessary before a final conclusion is possible. In developing these positions, we exploit a single proposition, viz. there is an effective charge which defines an evolution scheme for parton DFs that is all-orders exact. This proposition has numerous corollaries, which can be used to test the character of any DF, whether fitted or calculated.

We consider the fidelity of the vector meson dominance (VMD) assumption as an instrument for relating the electromagnetic vector-meson production reaction $$e + p \rightarrow e^\prime + V + p$$e + p → e ′ + V + p to the purely hadronic process $$V + p \rightarrow V+p$$V + p → V + p . Analyses of the photon vacuum polarisation and the photon-quark vertex reveal that such a VMD Ansatz might be reasonable for light vector-mesons. However, when the vector-mesons are described by momentum-dependent bound-state amplitudes, VMD fails for heavy vector-mesons: it cannot be used reliably to estimate either a photon-to-vector-meson transition strength or the momentum dependence of those integrands that would arise in calculations of the different reaction amplitudes. Consequently, for processes involving heavy mesons, the veracity of both cross-section estimates and conclusions based on the VMD assumption should be reviewed, e.g., those relating to hidden-charm pentaquark production and the origin of the proton mass.