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We propose an efficacious approach to derive the generalized parton distributions for the pion and proton, based upon prior knowledge of their respective parton distribution functions (PDFs). Our method leverages on integral representations of the electromagnetic form factors derived from the light-front holographic QCD (LFHQCD) formalism, coupled with PDFs computed from continuum Schwinger functional methods at the hadronic scale. Using these techniques, we calculate gravitational form factors and associated mass distributions for each hadron. Remarkably, our calculations yield results that closely match recent lattice QCD simulations conducted near the physical pion mass. This work not only deepens our understanding of hadronic structure but also highlights the efficacy of the LFHQCD approach in modeling fundamental properties of hadrons., Comment: 6 pages, 5 figures

This document presents the initial scientific case for upgrading the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab (JLab) to 22 GeV. It is the result of a community effort, incorporating insights from a series of workshops conducted between March 2022 and April 2023. With a track record of over 25 years in delivering the world's most intense and precise multi-GeV electron beams, CEBAF's potential for a higher energy upgrade presents a unique opportunity for an innovative nuclear physics program, which seamlessly integrates a rich historical background with a promising future. The proposed physics program encompass a diverse range of investigations centered around the nonperturbative dynamics inherent in hadron structure and the exploration of strongly interacting systems. It builds upon the exceptional capabilities of CEBAF in high-luminosity operations, the availability of existing or planned Hall equipment, and recent advancements in accelerator technology. The proposed program cover various scientific topics, including Hadron Spectroscopy, Partonic Structure and Spin, Hadronization and Transverse Momentum, Spatial Structure, Mechanical Properties, Form Factors and Emergent Hadron Mass, Hadron-Quark Transition, and Nuclear Dynamics at Extreme Conditions, as well as QCD Confinement and Fundamental Symmetries. Each topic highlights the key measurements achievable at a 22 GeV CEBAF accelerator. Furthermore, this document outlines the significant physics outcomes and unique aspects of these programs that distinguish them from other existing or planned facilities. In summary, this document provides an exciting rationale for the energy upgrade of CEBAF to 22 GeV, outlining the transformative scientific potential that lies within reach, and the remarkable opportunities it offers for advancing our understanding of hadron physics and related fundamental phenomena., Comment: Updates to the list of authors; Preprint number changed from theory to experiment; Updates to sections 4 and 6, including additional figures

We propose a novel method to construct the Landau thermodynamic potential directly from the fluctuations measured in heavy-ion collisions. The potential is capable of revealing the signal of the critical end-point (CEP) and the first order phase transition (FOPT) of QCD in the system even away from the phase transition region. With the available experimental data, we show that the criterion of the FOPT is negative for most of the collision energies which indicates no signal of FOPT. The data at $\sqrt{s_\mathrm{NN}}=7.7$ GeV with 0-5% centrality shows a different behavior and the mean value of the data satisfies the criterion. However, the uncertainty is still too large to make a certain conclusion. The higher order fluctuations are also required for confirming the signal. We emphasize therefore that new measurements with higher precision for the $C_{1,...,6}$ within 0-5% centrality in the vicinity of $\sqrt{s_\mathrm{NN}}=7.7$ GeV are in demand which may finally reveal the signal of QCD phase transition., Comment: 7 pages, 4 figures

The pion scalar, vector and tensor form factors are calculated within a symmetry-preserving contact interaction model (CI) of quantum chromodynamics (QCD), encompassed within a Dyson-Schwinger and Bethe-Salpeter equations approach. In addition to the traditional rainbow-ladder truncation, a modified interaction kernel for the Bethe-Salpeter equation is adopted. The implemented kernel preserves the vector and axial-vector Ward-Takahashi identities, while also providing additional freedom. Consequently, new tensor structures are generated in the corresponding interaction vertices, shifting the location of the mass poles appearing in the quark-photon and quark tensor vertex and yielding a notorious improvement in the final results. Despite the simplicity of the CI, the computed form factors and radii are compatible with recent lattice QCD simulations., Comment: 11 pages, 8 figures

Working with proton and pion valence distribution functions (DFs) determined consistently at the same, unique hadron scale and exploiting the possibility that there is an effective charge which defines an evolution scheme for DFs that is all-orders exact, we obtain a unified body of predictions for all proton and pion DFs - valence, glue, and four-flavour-separated sea. Whilst the hadron light-front momentum fractions carried by identifiable parton classes are the same for the proton and pion at any scale, the pointwise behaviour of the DFs is strongly hadron-dependent. All calculated distributions comply with quantum chromodynamics constraints on low- and high-$x$ scaling behaviour and, owing to emergent hadron mass, pion DFs are the most dilated. These results aid in elucidating the sources of similarities and differences between proton and pion structure., Comment: 8 pages, 4 figures, 2 tables

An algebraic Ansatz for the proton's Poincar\'e-covariant wave function, which includes both scalar and pseudovector diquark correlations, is used to calculate proton valence, sea, and glue distribution functions (DFs). Regarding contemporary data, a material pseudovector diquark component in the proton is necessary for an explanation of the neutron-proton structure function ratio; and a modest Pauli blocking effect in the gluon splitting function is sufficient to explain the proton's light-quark antimatter asymmetry. In comparison with pion DFs, the light-front momentum fractions carried by all identifiable parton classes are the same; on the other hand, the higher moments are different. Understanding these features may provide insights that explain distinctions between Nambu-Goldstone bosons and seemingly less complex hadrons., Comment: 8 pages, 2 figures, 1 table

Based on the recently precise measurements of the electron-positron annihilation reactions into a neutron and anti-neutron pair by BESIII collaboration, the effective form factor of neutron was determined in the time like region, and it was found that the effective form factor of neutron is smaller than the ones of proton. The effective form factors of neutron shows a periodic behaviour, similar to the case of proton. Here, a compared analysis for $\Lambda$, $\Sigma^0$ and $\Xi^0$ is performed. Fits of the available data on the effective form factors of $\Lambda$, $\Sigma^0$ and $\Xi^0$ with charge zero, allow to show a universal phenomenon of the oscillation behavior in their effective form factors. However, this needs to be confirmed by future precise experiments. Both theoretical and experimental investigations of this phenomenon can shed light on the reaction mechanisms of the electron-positron annihilation process., Comment: 4 figures, 5 pages. Experimental data on $\Xi^0$ are included, title is slightly modified

The static properties, masses and decay constants, of pseudoscalar meson triplet in a strongly magnetized medium are studied through the Dyson-Schwinger equation approach treatment of a contact interaction. Complementary to the usual vector-vector form, a symmetry-preserving formulation of couplings has been proposed in this work, without modifying the quark propagator, to control the strength of the $p$-wave component of Bethe-Salpeter amplitude. It is found that, with the help of flexible auxiliary interaction, our simple model is able to reproduce the observation in the lattice QCD simulation, where the spectra of the charged pseudo-scalar meson shows a non-monotonic behavior as the magnetic field grows. The discovery of this work implies the strong magnetic field affects the inner structure of mesons dramatically., Comment: 17 pages, 8 figures

Clear windows onto emergent hadron mass (EHM) and modulations thereof by Higgs boson interactions are provided by observable measures of pion and kaon structure, many of which are accessible via generalised parton distributions (GPDs). Beginning with algebraic GPD Ansaetze, constrained entirely by hadron-scale $\pi$ and $K$ valence-parton distribution functions (DFs), in whose forms both EHM and Higgs boson influences are manifest, numerous illustrations are provided. They include the properties of electromagnetic form factors, impact parameter space GPDs, gravitational form factors and associated pressure profiles, and the character and consequences of all-orders evolution. The analyses predict that mass-squared gravitational form factors are stiffer than electromagnetic form factors; reveal that $K$ pressure profiles are tighter than $\pi$ profiles, with both mesons sustaining near-core pressures at magnitudes similar to that expected at the core of neutron stars; deliver parameter-free predictions for $\pi$ and $K$ valence, glue, and sea GPDs at the resolving scale $\zeta=2$GeV; and predict that at this scale the fraction of meson mass-squared carried by glue and sea combined matches that lodged with the valence degrees-of-freedom, with a similar statement holding for mass-squared radii., Comment: 22 pages, 19 figures, 2 tables

Beginning with precise data on the ratio of structure functions in deep inelastic scattering (DIS) from $^3$He and $^3$H, collected on the domain $0.19 \leq x_B \leq 0.83$, where $x_B$ is the Bjorken scaling variable, we employ a robust method for extrapolating such data to arrive at a model-independent result for the $x_B=1$ value of the ratio of neutron and proton structure functions. Combining this with information obtained in analyses of DIS from nuclei, corrected for target-structure dependence, we arrive at a prediction for the proton's valence-quark ratio: $\left. d_v/u_v \right|_{x_B\to 1} = 0.230 (57)$. Requiring consistency with this result presents a challenge to many descriptions of proton structure., Comment: 6 pages, 3 figures

The $\gamma^{(\ast)}+p \to N(1535) \tfrac{1}{2}^-$ transition is studied using a symmetry-preserving regularisation of a vector$\,\otimes\,$vector contact interaction (SCI). The framework employs a Poincar\'e-covariant Faddeev equation to describe the initial and final state baryons as quark+di\-quark composites, wherein the diquark correlations are fully dynamical, interacting with the photon as allowed by their quantum numbers and continually engaging in breakup and recombination as required by the Faddeev kernel. The presence of such correlations owes largely to the mechanisms responsible for the emergence of hadron mass; and whereas the nucleon Faddeev amplitude is dominated by scalar and axial-vector diquark correlations, the amplitude of its parity partner, the $N(1535) \tfrac{1}{2}^-$, also contains sizeable pseudoscalar and vector diquark components. It is found that the $\gamma^{(\ast)}+p \to N(1535) \tfrac{1}{2}^-$ helicity amplitudes and related Dirac and Pauli form factors are keenly sensitive to the relative strengths of these diquark components in the baryon amplitudes, indicating that such resonance electrocouplings possess great sensitivity to baryon structural details. Whilst SCI analyses have their limitations, they also have the virtue of algebraic simplicity and a proven ability to reveal insights that can be used to inform more sophisticated studies in frameworks with closer ties to quantum chromodynamics., Comment: 16 pages, 14 figures, 4 tables

A symmetry-preserving treatment of mesons, within a Dyson-Schwinger and Bethe-Salpeter equations approach, demands an interconnection between the kernels of the quark gap equation and meson Bethe-Salpeter equation. Appealing to those symmetries expressed by the vector and axial-vector Ward-Green-Takahashi identitiges (WGTI), we construct a two-body Bethe-Salpeter kernel and study its implications in the vector channel; particularly, we analyze the structure of the quark-photon vertex, which explicitly develops a vector meson pole in the timelike axis and the quark anomlaous magnetic moment term, as well as a variety of $\rho$ meson properties: mass and decay constants, electromagnetic form factors, and valence-quark distribution amplitudes., Comment: 12 pages, 5 figures

Understanding why the scale of emergent hadron mass is obvious in the proton but hidden in the pion may rest on mapping the distribution functions (DFs) of all partons within the pion and comparing them with those in the proton; and since glue provides binding in quantum chromodynamics, the glue DF could play a special role. Producing reliable predictions for the proton's DFs is difficult because the proton is a three valence-body bound-state problem. As sketched herein, the situation for the pion, a two valence-body problem, is much better, with continuum and lattice predictions for the valence-quark and glue DFs in agreement. This beginning of theory alignment is timely because experimental facilities now either in operation or planning promise to realise the longstanding goal of providing pion targets, thereby enabling precision experimental tests of rigorous theory predictions concerning Nature's most fundamental Nambu-Goldstone bosons., Comment: 7 pages, 5 figures. Accepted for publication in Chinese Physics Letters

The quark mass function is computed both by solving the quark propagator Dyson-Schwinger equation and from lattice simulations implementing overlap and Domain-Wall fermion actions for valence and sea quarks, respectively. The results are confronted and seen to produce a very congruent picture, showing a remarkable agreement for the explored range of current-quark masses. The effective running-interaction is based on a process-independent charge rooted on a particular truncation of the Dyson-Schwinger equations in the gauge sector, establishing thus a link from there to the quark sector and inspiring a correlation between the emergence of gluon and hadron masses., Comment: 9 pages, 4 figures

How the bulk of the Universe's visible mass emerges and how it is manifest in the existence and properties of hadrons are profound questions that probe into the heart of strongly interacting matter. Paradoxically, the lightest pseudoscalar mesons appear to be the key to the further understanding of the emergent mass and structure mechanisms. These mesons, namely the pion and kaon, are the Nambu-Goldstone boson modes of QCD. Unravelling their partonic structure and the interplay between emergent and Higgs-boson mass mechanisms is a common goal of three interdependent approaches -- continuum QCD phenomenology, lattice-regularised QCD, and the global analysis of parton distributions -- linked to experimental measurements of hadron structure. Experimentally, the foreseen electron-ion collider will enable a revolution in our ability to study pion and kaon structure, accessed by scattering from the "meson cloud" of the proton through the Sullivan process. With the goal of enabling a suite of measurements that can address these questions, we examine key reactions to identify the critical detector system requirements needed to map tagged pion and kaon cross sections over a wide range of kinematics. The excellent prospects for extracting pion structure function and form factor data are shown, and similar prospects for kaon structure are discussed in the context of a worldwide programme. Successful completion of the programme outlined herein will deliver deep, far-reaching insights into the emergence of pions and kaons, their properties, and their role as QCD's Goldstone boson modes., Comment: 73 pages, 27 figures

Lepton scattering is an established ideal tool for studying inner structure of small particles such as nucleons as well as nuclei. As a future high energy nuclear physics project, an Electron-ion collider in China (EicC) has been proposed. It will be constructed based on an upgraded heavy-ion accelerator, High Intensity heavy-ion Accelerator Facility (HIAF) which is currently under construction, together with a new electron ring. The proposed collider will provide highly polarized electrons (with a polarization of $\sim$80%) and protons (with a polarization of $\sim$70%) with variable center of mass energies from 15 to 20 GeV and the luminosity of (2-3) $\times$ 10$^{33}$ cm$^{-2}$ s$^{-1}$. Polarized deuterons and Helium-3, as well as unpolarized ion beams from Carbon to Uranium, will be also available at the EicC. The main foci of the EicC will be precision measurements of the structure of the nucleon in the sea quark region, including 3D tomography of nucleon; the partonic structure of nuclei and the parton interaction with the nuclear environment; the exotic states, especially those with heavy flavor quark contents. In addition, issues fundamental to understanding the origin of mass could be addressed by measurements of heavy quarkonia near-threshold production at the EicC. In order to achieve the above-mentioned physics goals, a hermetical detector system will be constructed with cutting-edge technologies. This document is the result of collective contributions and valuable inputs from experts across the globe. The EicC physics program complements the ongoing scientific programs at the Jefferson Laboratory and the future EIC project in the United States. The success of this project will also advance both nuclear and particle physics as well as accelerator and detector technology in China., Comment: EicC white paper, written by the whole EicC working group

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

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

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

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 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

We construct the quark-antiquark scattering kernels of Bethe-Salpeter equation from the quark self-energy directly under two specific forms of quark-gluon vertices. The quark dressed tensor vertex is then calculated within this consistent framework and rainbow-ladder(RL) approximation. After employing a simplified nucleon model, the nucleon tensor charge can be defined with the tensor vertex. We then compute the tensor charge with the bare tensor vertex and the dressed vertices obtained in this framework and in RL approximation. The obtained results are consistent with the lattice QCD calculations. We also find that typically the gluon dressing effects suppress the nucleon tensor charge compared to the bare tensor vertex, by about $23\%$ for RL approximation, and turn to be about $13\%$ in this framework., Comment: 9 pages, 6 figures, 3 tables

A symmetry-preserving approach to the continuum bound-state problem in quantum field theory is used to calculate the masses, leptonic decay constants and light-front distribution amplitudes of empirically accessible heavy-light mesons. The inverse moment of the $B$-meson distribution is particularly important in treatments of exclusive $B$-decays using effective field theory and the factorisation formalism; and its value is therefore computed: $\lambda_B(\zeta = 2\,{\rm GeV}) = 0.54(3)\,$GeV. As an example and in anticipation of precision measurements at new-generation $B$-factories, the branching fraction for the rare $B\to \gamma(E_\gamma) \ell \nu_\ell$ radiative decay is also calculated, retaining $1/m_B^2$ and $1/E_\gamma^2$ corrections to the differential decay width, with the result $\Gamma_{B\to \gamma \ell \nu_\ell}/\Gamma_B = 0.47(15)$ on $E_\gamma > 1.5\,$GeV., Comment: 6 pages, 6 figure, 5 tables

We report progress on the development of Perturbative Integral Representation Ans\"atze to compute the nucleon Faddeev Wave function, using an explicit quark-diquark picture. Our formalism is able to handle non-pointlike diquark and to mimic the strong, dynamical correlations observed when solving the Faddeev Equation. We then project the wave function in order to compute the leading-twist Parton Distribution Amplitude., Comment: 6 pages, 2 figures, contribution to the proceedings of the XXII International Conference on Few-Body Problems in Physics (FB22) 2018, July 9-13, Caen, France

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 tables

A continuum approach to quark-antiquark bound-states is used to determine the electromagnetic form factors of pion-like mesons with masses $m_{0^-}/{\rm GeV}=0.14$, $0.47$, $0.69$, $0.83$ on a spacelike domain that extends to $Q^2 \lesssim 10\,$GeV$^2$. The results enable direct comparisons with contemporary lattice-QCD calculations of heavy-pion form factors at large values of momentum transfer and aid in understanding them. They also reveal, inter alia, that the form factor of the physical pion provides the best opportunity for verification of the factorised hard-scattering formula relevant to this class of exclusive processes and that this capacity diminishes steadily as the meson mass increases., Comment: 6 pages, 2 figures, 1 table

It is thought that strong interactions within the Standard Model can generate bound-states in which non-Abelian gauge-bosons play a dual role, serving both as force and matter fields. In this context we introduce a novel approach to the hybrid-meson (valence-gluon+quark+antiquark) bound-state problem in relativistic quantum field theory. Exploiting the existence of strong two-body correlations in the gluon-quark, $q_g=[gq]$, and gluon-antiquark, $\bar q_g=[g\bar q]$ channels, we argue that a sound description of hybrid properties can be obtained by solving a coupled-pair of effectively two-body equations; and, consequently, that hybrids may be viewed as highly-correlated $q_g \bar q \leftrightarrow q \bar q_g$ bound-states. Analogies may be drawn between this picture of hybrid structure and that of baryons, in which quark+quark (diquark) correlations play a key role. The potential of this formulation is illustrated by calculating the spectrum of light-quark isovector hybrid mesons., Comment: 5 pages, 2 figures, 1 table

Algebraic Ansaetze for the Poincar\'e-covariant Bethe-Salpeter wave functions of the pion and kaon are used to calculate their light-front wave functions (LFWFs), parton distribution amplitudes (PDAs), quasi-PDAs (qPDAs), valence parton distribution functions (PDFs), and quasi-PDFs (qPDFs). The LFWFs are broad, concave functions; and the scale of flavour-symmetry violation in the kaon is roughly 15%, being set by the ratio of emergent masses in the $s$-and $u$-quark sectors. qPDAs computed with longitudinal momentum $P_z =1.75\,$GeV provide a semiquantitatively accurate representation of the objective PDA; but even with $P_z=3\,$GeV, they cannot provide information about this amplitude's endpoint behaviour. On the valence-quark domain, similar outcomes characterise qPDFs. In this connection, however, the ratio of kaon-to-pion $u$-quark qPDFs is found to provide a good approximation to the true PDF ratio on $0.3\lesssim x \lesssim 0.8$, suggesting that with existing resources computations of ratios of quasi-parton-distributions can yield results that support empirical comparison., Comment: 10 pages, 8 figures

We explain how the emergent phenomenon of dynamical chiral symmetry breaking ensures that Poincar\'e covariant analyses of the three valence-quark scattering problem in continuum quantum field theory yield a picture of the nucleon as a Borromean bound-state, in which binding arises primarily through the sum of two separate contributions. One involves aspects of the non-Abelian character of Quantum Chromodynamics that are expressed in the strong running coupling and generate tight, dynamical color-antitriplet quark-quark correlations in the scalar-isoscalar and pseudovector-isotriplet channels. This attraction is magnified by quark exchange associated with diquark breakup and reformation, which is required in order to ensure that each valence-quark participates in all diquark correlations to the complete extent allowed by its quantum numbers. Combining these effects, we arrive at a properly antisymmetrised Faddeev wave function for the nucleon and calculate, e.g. the flavor-separated versions of the Dirac and Pauli form factors and the proton's leading-twist parton distribution amplitude. We conclude that available data and planned experiments are capable of validating the proposed picture., Comment: Contribution to the proceedings of the Workshop: Critical Stability of Quantum Few-Body Systems (Crit17). Oct. 16-20, 2017. Dresden, Germany

We present the first quantum field theory calculation of the pointwise behaviour of the leading-twist parton distribution amplitudes (PDAs) of the proton and its lightest radial excitation. The proton's PDA is a broad, concave function, whose maximum is shifted relative to the peak in QCD's conformal limit expression for this PDA; an effect which signals the presence of both scalar and pseudovector diquark correlations in the nucleon, with the scalar generating around 60% of the proton's normalisation. The radial-excitation is constituted similarly, and the pointwise form of its PDA, which is negative on a material domain, is the result of marked interference between the contributions from both types of diquark; particularly, the locus of zeros that highlights its character as a radial excitation. These features originate with the emergent phenomenon of dynamical chiral-symmetry breaking in the Standard Model., Comment: 5 pages, 2 figures

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 formulae are accurate to within 25% at momentum transfers $Q^2 \approx 8\,$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 $Q^2 \approx 6\,$GeV$^2$. Its subsequent $Q^2$-evolution is accurately described by the hard scattering formulae. Projections for 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., Comment: 7 pages, 4 figures

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.

We propose a new numerical method to compute parton distribution amplitude(PDA) from the Euclidean Bethe-Salpeter wave function. The essential step is to extract the weight function in the Nakanishi representation of the Bethe-Salpeter wave function in Euclidean space, which is an ill-posed inversion problem, via the maximum entropy method(MEM). The Nakanishi weight function as well as the corresponding light-front PDA can be well determined. We confirm the previous works on PDA computation therein the different method has been performed.

The $\gamma \gamma^\ast \to \eta_{c,b}$ transition form factors are computed using a continuum approach to the two valence-body bound-state problem in relativistic quantum field theory, and thereby unified with equivalent calculations of electromagnetic pion elastic and transition form factors. The resulting $\gamma \gamma^\ast \to \eta_c$ form factor, $G_{\eta_c}(Q^2)$, is consistent with available data: significantly, at accessible momentum transfers, $Q^2 G_{\eta_c}(Q^2)$ lies well below its conformal limit. These observations confirm that the leading-twist parton distribution amplitudes (PDAs) of heavy-heavy bound-states are compressed relative to the conformal limit. A clear understanding of the distribution of valence-quarks within mesons thus emerges; a picture which connects Goldstone modes, built from the lightest-quarks in Nature, with systems containing the heaviest valence-quarks that can now be studied experimentally, and highlights basic facts about manifestations of mass within the Standard Model., Comment: 4 pages, 2 figures, 1 table

In principle, the strong-interaction sector of the Standard Model is characterised by a unique renormalisation-group-invariant (RGI) running interaction and a unique form for the dressed--gluon-quark vertex, $\Gamma_\mu$; but, whilst much has been learnt about the former, the latter is still obscure. In order to improve this situation, we use a RGI running-interaction that reconciles both top-down and bottom-up analyses of the gauge sector in quantum chromodynamics (QCD) to compute dressed-quark gap equation solutions with 1,660,000 distinct Ansaetze for $\Gamma_\mu$. Each one of the solutions is then tested for compatibility with three physical criteria and, remarkably, we find that merely 0.55% of the solutions survive the test. Plainly, therefore, even a small selection of observables places extremely tight bounds on the domain of realistic vertex Ansaetze. This analysis and its results should prove useful in constraining insightful contemporary studies of QCD and hadronic phenomena., Comment: 6 pages, 7 figures

A symmetry-preserving truncation of the two-body light-quark bound-state problem in relativistic quantum field theory is used to calculate the leading-twist parton distribution amplitudes (PDAs) of scalar systems, both ground-state and radial excitations, and the radial excitations of vector mesons. Owing to the fact that the scale-independent leptonic decay constant of a scalar meson constituted from equal-mass valence-constituents vanishes, it is found that the PDA of a given scalar system possesses one more zero than that of an analogous vector meson. Consequently, whereas the mean light-front relative momentum of the valence-constituents within a vector meson is zero, that within a scalar meson is large, an outcome which hints at a greater role for light-front angular momentum in systems classified as $P$-wave in quantum mechanical models. Values for the scale-dependent decay constants of ground-state scalar and vector systems are a by-product of this analysis, and they turn out to be roughly equal, viz. $\simeq 0.2\,$GeV at an hadronic scale. In addition, it is confirmed that the dilation characterising ground-state PDAs is manifest in the PDAs of radial excitations too. The impact of $SU(3)$-flavour symmetry breaking is also considered. When compared with pseudoscalar states, it is a little stronger in scalar systems, but the size is nevertheless determined by the flavour-dependence of dynamical chiral symmetry breaking and the PDAs are still skewed toward the heavier valence-quark in asymmetric systems., Comment: 10 pages, 3 tables, 5 figures

A symmetry-preserving truncation of the two-body bound-state problem in relativistic quantum field theory is used to compute the leading-twist parton distribution amplitudes (PDAs) for the first radial excitations of the $\pi$- and $K$-mesons. In common with ground states in these channels, the PDAs are found to be dilated with respect to the relevant conformal-limit form and skewed toward the heavier valence-quark in asymmetric systems. In addition, the PDAs of radially-excited pseudoscalar mesons are not positive definite, owing to the fact that dynamical chiral symmetry breaking (DCSB) forces the leptonic decay constant of such states to vanish in the chiral limit. These results highlight that DCSB is expressed visibly in every pseudoscalar meson constituted from light-quarks. Hence, so long as its impact is empirically evident in the pseudoscalar members of a given spectrum level, it is unlikely that chiral symmetry is restored in any of the hadrons that populate this level., Comment: 9 pages, 3 figures

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 a meson's momentum at a characteristic hadronic scale and vanishing 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)-flavour symmetry breaking in meson parton distribution functions is modulated by the flavour dependence of dynamical chiral symmetry breaking. Corrections to these leading-order formulae 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 predict 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., Comment: 10 pages, 3 figures. arXiv admin note: text overlap with arXiv:1406.5450

Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing stringent relationships between the kernels of the one- and two-body problems, which must be preserved in any veracious treatment of mesons as bound-states. In this connection, one may view the dressed gluon-quark vertex, $\Gamma_\mu^a$, as fundamental. We use a novel representation of $\Gamma_\mu^a$, in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quark-antiquark Bethe-Salpeter kernel, $K$, that is symmetry-consistent with a given quark gap equation. A strength of the scheme is its ability to expose and capitalise on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of $H$-diagrams in $K$, which are two-particle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of $\Gamma_\mu^a$ in a Bethe-Salpeter kernel nor expressed as a member of the class of crossed-box diagrams. Thus, there are no general circumstances under which the WGT identities essential for a valid description of mesons can be preserved by a Bethe-Salpeter kernel obtained simply by dressing both gluon-quark vertices in a ladder-like truncation; and, moreover, adding any number of similarly-dressed crossed-box diagrams cannot improve the situation., Comment: 6 pages, 8 figures

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 tables

We compute all kaon and pion parton distribution amplitudes (PDAs) to twist-three and find that only the pseudotensor PDA can reasonably be approximated by its conformal limit expression. At terrestrially accessible energy scales, the twist-two and pseudoscalar twist-three PDAs differ significantly from those functions commonly associated with their forms in QCD's conformal limit. In all amplitudes studied, SU(3) flavour-symmetry breaking is typically a 13% effect. This scale is determined by nonperturbative dynamics; namely, the current-quark-mass dependence of dynamical chiral symmetry breaking. The heavier-quark is favoured by this distortion, for example, support is shifted to the s-quark in the negative kaon. It appears, therefore, that at energy scales accessible with existing and foreseeable facilities, one may obtain reliable expectations for experimental outcomes by using these "strongly dressed" PDAs in formulae for hard exclusive processes. Following this procedure, any discrepancies between experiment and theory will be significantly smaller than those produced by using the conformal-limit PDAs. Moreover, the magnitude of any disagreement will either be a better estimate of higher-order, higher-twist effects or provide more realistic constraints on the Standard Model., Comment: 14 pages, 4 figures, 2 tables. arXiv admin note: text overlap with arXiv:1406.3353