We describe expressions for pion and kaon dressed-quark distribution functions that incorporate contributions from gluons which bind quarks into these mesons and hence overcome a flaw of the commonly used handbag approximation. The distributions therewith obtained are purely valence in character, ensuring that dressed quarks carry all the meson’s momentum at a characteristic hadronic scale and vanish as $(1-x{)}^2$ when Bjorken-$x\to 1$. Comparing such distributions within the pion and kaon, it is apparent that the size of $SU(3)$-flavor symmetry breaking in meson parton distribution functions is modulated by the flavor dependence of dynamical chiral symmetry breaking. Corrections to these leading-order formulas may be divided into two classes, responsible for shifting dressed-quark momentum into glue and sea quarks. Working with available empirical information, we build an algebraic framework that is capable of expressing the principal impact of both classes of corrections. This enables a realistic comparison with experiment which allows us to identify and highlight basic features of measurable pion and kaon valence-quark distributions. We find that whereas roughly two thirds of the pion’s light-front momentum is carried by valence dressed quarks at a characteristic hadronic scale; this fraction rises to 95% in the kaon; evolving distributions with these features to a scale typical of available Drell-Yan data produces a kaon-to-pion ratio of $u$-quark distributions that is in agreement with the single existing data set, and predicts a $u$-quark distribution within the pion that agrees with a modern reappraisal of $\pi N$ Drell-Yan data. Precise new data are essential in order to validate this reappraisal and because a single modest-quality measurement of the kaon-to-pion ratio cannot be considered definitive.

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 ${\mu }_5$.

A symmetry-preserving approach to the two valence-body continuum bound-state problem is used to calculate the elastic electromagnetic form factors of the $\rho$-meson and subsequently to study the evolution of vector-meson form factors with current-quark mass. To facilitate a range of additional comparisons, $K^{*}$ form factors are also computed. The analysis reveals that vector mesons are larger than pseudoscalar mesons; composite vector mesons are nonspherical, with magnetic and quadrupole moments that deviate $\sim 30\%$ from point-particle values; in many ways, vector-meson properties are as much influenced by emergent mass as those of pseudoscalars; and vector-meson electric form factors possess a zero at spacelike momentum transfer. Qualitative similarities between the electric form factors of the $\rho$ and the proton, $G_E^p$, are used to argue that the character of emergent mass in the Standard Model can force a zero in $G_E^p$. Moreover, the existence of a zero in vector-meson electric form factors entails that a single-pole vector-meson dominance model can only be of limited use in estimating properties of off-shell vector mesons, providing poor guidance for systems in which the Higgs mechanism of mass generation is dominant.

In this paper, we use the recent updated source properties of GW170817 to constrain the hybrid equation of state (EOS) constructed by a three-window modeling between the hadronic EOS and quark EOS. Specifically, the hadronic EOS is described by the $\mathrm{NL}3\omega \rho$ model whose corresponding pure neutron star (NS) is already excluded by the constraint of tidal deformability (TD) from GW170817, and the quark EOS is calculated with the $2+1$ flavors Nambu-Jona-Lasinio (NJL) model. We also consider four other constraints on the hybrid EOS. As a result, we find the parameter set ($B^{\frac{1}{4}},\tilde{\mu },\mathrm{\Gamma }$) can be well constrained, indicating the possible existence of the hybrid star (HS) with a crossover inside. The type of the two stars in the binary system for nine representative hybrid EOSs is shown in this paper too. Furthermore, the HSs restricted by five constraints do not suggest a pure quark core but a mixed phase in the center.