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There are two mass generating mechanisms in the standard model of particle physics (SM). One is related to the Higgs boson and fairly well understood. The other is embedded in quantum chromodynamics (QCD), the SM's strong interaction piece; and although responsible for emergence of the roughly 1 GeV mass scale that characterises the proton and hence all observable matter, the source and impacts of this emergent hadronic mass (EHM) remain puzzling. As bound states seeded by a valence-quark and -antiquark, pseudoscalar mesons present a simpler problem in quantum field theory than that associated with the nucleon. Consequently, there is a large array of robust predictions for pion and kaon properties whose empirical validation will provide a clear window onto many effects of both mass generating mechanisms and the constructive interference between them. This has now become significant because new-era experimental facilities, in operation, construction, or planning, are capable of conducting such tests and thereby contributing greatly to resolving the puzzles of EHM. These aspects of experiment, phenomenology, and theory, along with contemporary successes and challenges, are sketched herein, simultaneously highlighting the potential gains that can accrue from a coherent effort aimed at finally reaching an understanding of the character and structure of Nature's Nambu-Goldstone modes., Comment: 110 pages, 41 figures, 5 tables. Accepted for publication in Prog. Part. Nucl. Phys

Pion and kaon structural properties provide insights into the emergence of mass within the Standard Model and attendant modulations by the Higgs boson. Novel expressions of these effects, in impact parameter space and in mass and pressure profiles, are exposed via $\pi$ and $K$ generalised parton distributions, built using the overlap representation from light-front wave functions constrained by one-dimensional valence distribution functions that describe available data. Notably, e.g. $K$ pressure profiles are spatially more compact than $\pi$ profiles and both achieve near-core pressures of similar magnitude to that found in neutron stars., Comment: 9 pages, 6 figures, 2 tables. Accepted for publication in Phys. Lett. B

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 figures

In this work, the mass of charged pions is investigated in the presence of background magnetic fields stronger than the energy scale of QCD. We introduce an anomaly magnetic momentum term in the Dirac equation and obtain the quark propagator as consequence. We find a novel finite Landau level, denoted as $tl$LL, becoming dominant rather than the conventional lowest Landau level. We examine that, due to the shifting of Landau level, it drives a mass decreasing around $eB\sim 0.8~\mathrm{GeV}^2$ for charged pions and their masses drastically limit to the neutral one at ultra-strong magnetic field, $eB\sim 1.6~\mathrm{GeV}^2$, which is consistent with the recent lattice simulation., Comment: The assumption on the magnitude of the anomalous magnetic moment applied in this paper is hard to achieve

We restrict our computation in the mean field approximation which could lead to a clear critical behavior. We analyze the scaling behavior with different shape of interaction kernel by considering different dressed-gluon models. The critical exponent we obtained is consistent with that in the $3D$ $\textrm{O}(4)$ universality class. The size of critical region is up to $m_{0}^{} \le 2\sim 4\;$MeV in this mean field approximation which sets naturally an upper bound of the critical region since the fluctuations beyond mean-field usually diminish the critical region. Besides, we analyze the possible percentage of the maximum chiral susceptibility and pion mass range at which the chiral phase transition temperature is independent of the current quark mass. The results show that the percentage and the pion mass range depend on the details of interaction kernel, which differs in gluon models., Comment: 7 pages, 5 figures

Strangeness was discovered roughly seventy years ago, lodged in a particle now known as the kaon, $K$. Kindred to the pion, $\pi$; both states are massless in the absence of Higgs-boson couplings. Kaons and pions are Nature's most fundamental Nambu-Goldstone modes. Their properties are largely determined by the mechanisms responsible for emergent mass in the standard model, but modulations applied by the Higgs are crucial to Universe evolution. Despite their importance, little is known empirically about $K$ and $\pi$ structure. This study delivers the first parameter-free predictions for all $K$ distribution functions (DFs) and comparisons with the analogous $\pi$ distributions, i.e. the one-dimensional maps that reveal how the light-front momentum of these states is shared amongst the gluons and quarks from which they are formed. The results should stimulate improved analyses of existing data and motivate new experiments sensitive to all $K$ and $\pi$ DFs., Comment: 8 pages, 5 figures, 1 table

A concept of SUb-atmospheric Radio-frequency Engine (SURE) designed for near space environment is reported. The antenna wrapping quartz tube consists of two solenoid coils with variable separation distance, and is driven by radio-frequency power supply (13.56~MHz-1~kW). The discharge involves inductive coupling under each solenoid coil and capacitive coupling between them. This novel scheme can ionize the filling air efficiently for the entire pressure range of 32\sim 5332 Pa in near space. The formed plasma density and temperature are up to 2.23\times 10^{18}~m^{-3} and 2.79 eV, respectively. The influences of separation distance, input power, filling pressure and the number of solenoid turns on discharge are presented in detail. This air-breathing electric propulsion system has no plasma-facing electrode and does not require external magnetic field, and is thereby durable and structurally compact and light., Comment: 7 pages, 5 figures

We propose a practical procedure to extrapolate the space-like quark propagator onto the complex plane, which follows the Schlessinger Point Method and the spectral representation of the propagator. As a feasible example, we employ quark propagators for different flavors, obtained from the solutions of the corresponding Dyson-Schwinger equation (DSE). Two different truncations are employed. Thus, the analytical structure of the quark propagator is studied, capitalizing on the current-quark mass dependence of the observed features., Comment: 8 pages, 6 figures. It includes a new example with a beyond Rainbow-Ladder truncation

The valence-quark distribution function of the pion has been of interest for decades; particularly, the profile it should adopt when $x\to1$ (the large-$x$ behavior) is subject of a long-standing debate. In the light-front holographic QCD (LFHQCD) approach, this behavior is controlled by the so called reparametrization function, $w_\tau(x)$, which is not fully determined from first principles. We show that, owing to the flexibility of $w_\tau(x)$, the large-$x$ profile $u^{\pi}(x)\sim (1-x)^{2}$ can be contained within the LFHQCD formalism. This is in contrast with a previous LFHQCD study~(Guy F. de Teramond et al., Phys. Rev. Lett., 120(18), 2018) in which $u^{\pi}(x)\sim (1-x)^{1}$ is found instead. Given our observations, augmented by perturbative QCD and recent lattice QCD results, we state that the large-$x$ exponent of "2" cannot be excluded., Comment: 5 pages, 4 figures

We predict the masses of the lowlying $B_c$ mesons with $J^P = 0^-,\,1^-,\,0^+,\,1^+,\,2^+$, using a flavor dependent interaction pattern which gives an unified successful description of the light, heavy-light and heavy mesons and is also appliable to the radial excited heavy mesons. The errors are controlled carefully. With the errors from the RL approximation subduced, our predictions are consistent with the lQCD and quark model results, which supports strongly that the flavor dependent interaction pattern is reasonable. Our predictions provide significant guides to the experiment search of the $B_c$ mesons.

We give out a simple way to connect the parton distribution functions defined in Minkowskian space and the nonperturbative QCD methods grounded in Euclidean space (e.g., lattice QCD(LQCD), Dyson-Schwinger (DS) equations, functional renormalization group (FRG) approach) in this work. We combine the MIT bag model with the DS equation approach to calculate the longitudinal distribution function, transverse distribution function and scalar distribution function in a proton at renormalization point $\mu = 2\,\text{GeV}$. We look then insight into the dressed effects on the axial, tensor and scalar charges in a nucleon to some extent. This method can be regard as a new bridge between the Euclidean non-perturbative approaches and the Minkowskian space physics., Comment: 16 pages, 7 figures, and 6 tables. Typos corrected

The mass relation ${M_{0^{+}}+3M_{1^{+\prime}}+5M_{2^{+}}= 9M_{1^{+}}}$ miraculously holds for the $P$-wave charmonium $(c\bar{c})$ and bottomonium $(b\bar{b})$ systems with soaring precision. The origin of such relation can be addressed from Quark Models, and have been confirmed experimentally in a limited number of cases. In this connection, we propose $M_{0^{+}}+5M_{2^{+}}=3(M_{1^{+\prime}}+M_{1^{+}})$ as an extension to the $P$-wave $B_{c}$ case. In order to test its applicability, we employ a variety of Quark Model predictions for the $B_c$ mass spectrum. Our numerical analysis confirms such formula is accurate up to very small deviations., Comment: 3 pages, no figures

A symmetry-preserving continuum approach to the two valence-body bound-state problem is used to calculate the valence, glue and sea distributions within the pion; unifying them with, inter alia, electromagnetic pion elastic and transition form factors. The analysis reveals the following momentum fractions at the scale $\zeta_2:=2\,$GeV: $\langle x_{\rm valence} \rangle = 0.48(3)$, $\langle x_{\rm glue} \rangle = 0.41(2)$, $\langle x_{\rm sea} \rangle = 0.11(2)$; and despite hardening induced by the emergent phenomenon of dynamical chiral symmetry breaking, the valence-quark distribution function, ${q}^\pi(x)$, exhibits the $x\simeq 1$ behaviour predicted by quantum chromodynamics (QCD). After evolution to $\zeta=5.2\,$GeV, the prediction for ${q}^\pi(x)$ matches that obtained using lattice-regularised QCD. This confluence should both stimulate improved analyses of existing data and aid in planning efforts to obtain new data on the pion distribution functions., Comment: 6 pages, 1 figure. To be published as a Letter in Chinese Physics C. arXiv admin note: substantial text overlap with arXiv:1905.05208

A valuable approach to the analysis of hadron physics observables is provided by QCD's equations-of-motion; namely, the Dyson-Schwinger equations. Drawing from a diverse collection of predictions, we revisit: $\gamma \gamma^* \to$ neutral pseudoscalar transition form factors, their corresponding valence-quark distribution amplitudes and a recent result on the pion distribution functions., Comment: 5 pages, 3 figures. Contribution to the proceedings of the XVIII International Conference on Hadron Spectroscopy and Structure, HADRON-2019. August 16-21, Guilin, China

Starting from the axial vector Ward-Green-Takahashi identity, and the necessity of preserving this identity when attempting to calculate the properties describing pseudoscalar mesons, we describe a Bethe-Salpeter kernel which takes into account the non-Abelian anomaly and therefore is suitable to describe the flavor mixing states $\eta,\eta^\prime$. We then report results for the two photon transition form factors of $\eta,\eta^\prime$, and compare them with existing experimental data., Comment: 5 pages, 1 figure; proceeding of the 18th International Conference on Hadron Spectroscopy and Structure (HADRON2019), Guilin, China, 16-21 Aug. 2019

We sketch the calculation of the pion structure functions within the DSE framework, following two alternative albeit consistent approaches, and discuss then their QCD evolution, the running driven by an effective charge, from a hadronic scale up to any larger one accessible to experiment., Comment: INPC 2019 proceedings

Understanding the origin and dynamics of hadron structure and in turn that of atomic nuclei is a central goal of nuclear physics. This challenge entails the questions of how does the roughly 1 GeV mass-scale that characterizes atomic nuclei appear; why does it have the observed value; and, enigmatically, why are the composite Nambu-Goldstone (NG) bosons in quantum chromodynamics (QCD) abnormally light in comparison? In this perspective, we provide an analysis of the mass budget of the pion and proton in QCD; discuss the special role of the kaon, which lies near the boundary between dominance of strong and Higgs mass-generation mechanisms; and explain the need for a coherent effort in QCD phenomenology and continuum calculations, in exa-scale computing as provided by lattice QCD, and in experiments to make progress in understanding the origins of hadron masses and the distribution of that mass within them. We compare the unique capabilities foreseen at the electron-ion collider (EIC) with those at the hadron-electron ring accelerator (HERA), the only previous electron-proton collider; and describe five key experimental measurements, enabled by the EIC and aimed at delivering fundamental insights that will generate concrete answers to the questions of how mass and structure arise in the pion and kaon, the Standard Model's NG modes, whose surprisingly low mass is critical to the evolution of our Universe., Comment: 16 pages, 12 figures, to appear in the European Physical Journal A - "Hadrons and Nuclei"

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 tables

We study the most recently observed excited $B_{c}$ states with the Dyson-Schwinger equation and the Bethe-Salpeter equation approach of continuum QCD. The obtained $M_{B^+_{c}(2S)}=6.813(16)\text{GeV}$, $M_{B^{*+}_{c}(2S)}=6.841(18)\text{GeV}$ and the mass splitting $M_{B_c^+(2S)}-M^{\text{rec}}_{B_c^{*+}(2S)} \approx 0.039 \text{GeV}$ agree with the observations very well. Moreover we predict the leptonic decay constant $f_{B^+_{c}(2S)}=-0.165(10)\text{GeV}$, $f_{B^{*+}_{c}(2S)}=-0.161(7)\text{GeV}$ respectively., Comment: 5 pages, 2 figures

A flavor dependent kernel is constructed based on the rainbow-ladder truncation of the Dyson-Schwinger and Bethe-Salpeter equation approach of Quantum Chromodynamics. The quark-antiquark interaction is composed of a flavor dependent infrared part and a flavor independent ultraviolet part. Our model gives a successful and unified description of the light, heavy and heavy-light ground pseudoscalar and vector mesons. For the first time, our model shows that the infrared enhanced quark-antiquark interaction is stronger and wider for the lighter quark., Comment: 6 pages, 3 figures