The rainbow modified-ladder approximation and degenerate pion

Correlation functions can be described by the corresponding equations, $viz.$, gap equation for quark propagator and the inhomogeneous Bethe-Salpeter equation for vector dressed-fermion-Abelian-gauge-boson vertex in which specific truncations have to be implemented. The general vector and axial-vector Ward-Green-Takahashi identities require these correlation functions to be interconnected, in consequence of this, truncations made must be controlled consistently. It turns out that if the rainbow approximation is assumed in gap equation, the scattering kernel in Bethe-Salpeter equation can adopt the ladder approximation, which is one of the most basic attempts to truncate the scattering kernel. Additionally, a modified-ladder approximation is also found to be a possible symmetry-preserving truncation scheme. As an illustration of this approximation for application a treatment of pion is included. Pion mass and decay constant are found to be degenerate in ladder and modified-ladder approximations, even though the Bethe-Salpeter amplitude are with apparent distinction. The justification for the modified-ladder approximation is examined with the help of the Gell-Mann-Oakes-Renner (GMOR) relation.Comment: 9 pages, 4 figure

Thermal properties of π\pi and ρ\rho meson

We computed the pole masses and decay constants of $\pi$ and $\rho$ meson at finite temperature in the framework of Dyson-Schwinger equations and Bethe-Salpeter equations approach. Below transition temperature, pion pole mass increases monotonously, while $\rho$ meson seems to be temperature independent. Above transition temperature, pion mass approaches the free field limit, whereas $\rho$ meson is about twice as large as that limit. Pion and the longitudinal projection of $\rho$ meson decay constants have similar behaviour as the order parameter of chiral symmetry, whereas the transverse projection of $\rho$ meson decay constant rises monotonously as temperature increases. The inflection point of decay constant and the chiral susceptibility get the same phase transition temperature. Though there is no access to the thermal width of mesons within this scheme, it is discussed by analyzing the Gell-Mann-Oakes-Renner (GMOR) relation in medium. These thermal properties of hadron observables will help us understand the QCD phases at finite temperature and can be employed to improve the experimental data analysis and heavy ion collision simulations.Comment: 8 pages, 4 figures, matched the published versio

Two Photon Transition Form Factor of cˉc\bar{c}c Quarkonia

The two photon transition of $\bar{c}c$ quarkonia are studied within a covariant approach based on the consistent truncation scheme of the quantum chromodynamics Dyson-Schwinger equation for the quark propagator and the Bethe--Salpeter equation for the mesons. We find the decay widths of $\eta_{c}^{} \to \gamma\gamma$ and $\chi_{c0,2}^{} \to \gamma\gamma$ in good agreement with experimental data. The obtained transition form factor of $\eta_{c}^{} \to \gamma\gamma^{\ast}$ for a wide range of space-like photon momentum transfer squared is also in agreement with the experimental findings of the BABAR experiment. As a by-product, the decay widths of $\eta_{b}^{},\chi_{b0,2}^{} \to \gamma\gamma$ and the transition form factor of $\eta_{b}^{}, \chi_{c0,b0}^{} \to\gamma\gamma^{\ast}$ are predicted, which await for experimental test

Leading-twist parton distribution amplitudes of S-wave heavy-quarkonia

The leading-twist parton distribution amplitudes (PDAs) of ground-state $^1S_0$ and $^3S_1$ $c\bar c$- and $b\bar b$-quarkonia are calculated using a symmetry-preserving continuum treatment of the meson bound-state problem which unifies the properties of these heavy-quark systems with those of light-quark bound-states, including QCD's Goldstone modes. Analysing the evolution of $^1S_0$ and $^3S_1$ PDAs with current-quark mass, $\hat m_q$, increasing away from the chiral limit, it is found that in all cases there is a value of $\hat m_q$ for which the PDA matches the asymptotic form appropriate to QCD's conformal limit and hence is insensitive to changes in renormalisation scale, $\zeta$. This mass lies just above that associated with the $s$-quark. At current-quark masses associated with heavy-quarkonia, on the other hand, the PDAs are piecewise convex-concave-convex. They are much narrower than the asymptotic distribution on a large domain of $\zeta$; but nonetheless deviate noticeably from $\varphi_{Q\bar Q}(x) = \delta(x-1/2)$, which is the result in the static-quark limit. There are also material differences between $^1S_0$ and $^3S_1$ PDAs, and between the PDAs for different vector-meson polarisations, which vanish slowly with increasing $\zeta$. An analysis of moments of the root-mean-square relative-velocity, $\langle v^{2m}\rangle$, in $^1S_0$ and $^3S_1$ systems reveals that $\langle v^4\rangle$-contributions may be needed in order to obtain a reliable estimate of matrix elements using such an expansion, especially for processes involving heavy pseudoscalar quarkonia.Comment: 6 pages, 2 figures, 3 table

Sieving parton distribution function moments via the moment problem

Reconstructing parton distribution function (PDF) from the corresponding Mellin moments belongs to a classical mathematical problem: the moment problem, which has been overlooked for years in the contemporary hadron community. We propose the strategy to sieve the moments leveraging PDF properties such as continuity, unimodality, and symmetry. Through an error-inclusive sifting process, we refine three sets of lattice QCD PDF moments. This refinement significantly reduces the errors, particularly for higher order moments, and locates the peak of PDF simultaneously. As our method is universally applicable to PDF moments from any methodology, we strongly advocate its integration into all PDF moment calculations.Comment: 6 pages, 2 figure

Symmetry, symmetry breaking, and pion parton distributions

Pion valence, glue and sea distributions are calculated using a continuum approach to the two valence-body bound-state problem. Since the framework is symmetry preserving, physical features of the distributions are properly expressed. The analysis reveals that the emergent phenomenon of dynamical chiral symmetry breaking causes a hardening of the valence-quark distribution function, ${q}^\pi(x)$. Nevertheless, this distribution exhibits the $x\simeq 1$ behaviour predicted by quantum chromodynamics (QCD). At the scale $\zeta_2:=2\,$GeV, the following momentum fractions are predicted: $\langle x_{\rm valence} \rangle = 0.48(3)$, $\langle x_{\rm glue} \rangle = 0.41(2)$, $\langle x_{\rm sea} \rangle = 0.11(2)$. Evolving to $\zeta=5.2\,$GeV, the result for ${q}^\pi(x)$ agrees with that computed using lattice QCD. These outcomes should both spur improved analyses of existing experiments and stimulate efforts to obtain new data on the pion distribution functions using available and envisioned facilities.Comment: 13 pages, 7 figures, 2 table

γ∗γ→η,η′\gamma^\ast \gamma \to \eta, \eta^\prime transition form factors

Using a continuum approach to the hadron bound-state problem, we calculate $\gamma^\ast \gamma \to \eta, \eta^\prime$ transition form factors on the entire domain of spacelike momenta, for comparison with existing experiments and in anticipation of new precision data from next-generation $e^+ e^-$ colliders. One novel feature is a model for the contribution to the Bethe-Salpeter kernel deriving from the non-Abelian anomaly, an element which is crucial for any computation of $\eta, \eta^\prime$ properties. The study also delivers predictions for the amplitudes that describe the light- and strange-quark distributions within the $\eta, \eta^\prime$. Our results compare favourably with available data. Important to this at large-$Q^2$ is a sound understanding of QCD evolution, which has a visible impact on the $\eta^\prime$ in particular. Our analysis also provides some insights into the properties of $\eta, \eta^\prime$ mesons and associated observable manifestations of the non-Abelian anomaly.Comment: 16 pages, 7 figures, 3 table

A fresh look at the generalized parton distributions of light pseudoscalar mesons

We present a symmetry-preserving scheme to derive the pion and kaon generalized parton distributions (GPDs) in Euclidean space. The key to maintaining crucial symmetries under this approach is the treatment of the scattering amplitude, such that it contains both the traditional leading-order contributions and the scalar/vector pole contribution automatically, the latter being necessary to ensure the soft-pion theorem. The GPD is extracted analytically via the uniqueness and definition of the Mellin moments and we find that it naturally matches the double distribution; consequently, the polynomiality condition and sum rules are satisfied. The present scheme thus paves the way for the extraction of the GPD in Euclidean space using the Dyson-Schwinger equation framework or similar continuum approaches.Comment: 5 pages, 2 figures, references adde

Deformation and fracture characteristics of ferrite/bainite dual-phase steels

The deformation and fracture characteristics of a low carbon Si&ndash;Mn steel with ferrite/bainite dual&ndash;phase structure were investigated by thermo&ndash;mechanical controlled process (TMCP). The results showed that the curves of the instantaneous work&ndash;hardening factor n* value versus true strain &epsilon; are made up with three stages during uniform plastic deformation: n* value is relatively higher at stage I, decreases slowly with &epsilon; in stage II, and then decreases quickly with &epsilon; in stage III. Compared tothe equiaxed ferrite/bainite dual&ndash;phase steel, the quasi&ndash;polygonal ferrite/bainite dual&ndash;phase steel shows higher tensile strength and n*value in the low strain region. The voids or micro&ndash;cracks formed not only at ferrite&ndash;bainite interfaces but also within ferrite grains in the necked region, which can improve the property of resistance to crack propagation by reducing local stress concentration of the crack tips.<br /