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We apply a classical mathematical problem, the moment problem, with its related mathematical achievements, to the study of the parton distribution function (PDF) in hadron physics, and propose a strategy to sieve the moments of the PDF by exploiting its properties such as continuity, unimodality, and symmetry. Through an error-inclusive sifting process, we refine three sets of PDF moments from Lattice QCD. This refinement significantly reduces the errors, particularly for higher order moments, and locates the peak of PDF simultaneously. As our strategy is universally applicable to PDF moments from any method, we strongly advocate its integration into all PDF moment calculations., Comment: 6 pages, 2 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

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

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

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

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 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

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

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

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 $\gamma \gamma^\ast \to \eta_{c,b}$ 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

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

Within contemporary hadron physics there are two common methods for determining the momentum-dependence of the interaction between quarks: the top-down approach, which works toward an ab initio computation of the interaction via direct analysis of the gauge-sector gap equations; and the bottom-up scheme, which aims to infer the interaction by fitting data within a well-defined truncation of those equations in the matter sector that are relevant to bound-state properties. We unite these two approaches by demonstrating that the renormalisation-group-invariant running-interaction predicted by contemporary analyses of QCD's gauge sector coincides with that required in order to describe ground-state hadron observables using a nonperturbative truncation of QCD's Dyson-Schwinger equations in the matter sector. This bridges a gap that had lain between nonperturbative continuum-QCD and the ab initio prediction of bound-state properties., Comment: 6 pages, 2 figures, 1 table

The impulse-approximation expression used hitherto to define the pion's valence-quark distribution function is flawed because it omits contributions from the gluons which bind quarks into the pion. A corrected leading-order expression produces the model-independent result that quarks dressed via the rainbow-ladder truncation, or any practical analogue, carry all the pion's light-front momentum at a characteristic hadronic scale. Corrections to the leading contribution may be divided into two classes, responsible for shifting dressed-quark momentum into glue and sea-quarks. Working with available empirical information, we use an algebraic model to express the principal impact of both classes of corrections. This enables a realistic comparison with experiment that allows us to highlight the basic features of the pion's measurable valence-quark distribution, $q^\pi(x)$; namely, at a characteristic hadronic scale, $q^\pi(x) \sim (1-x)^2$ for $x\gtrsim 0.85$; and the valence-quarks carry approximately two-thirds of the pion's light-front momentum., Comment: 7 pages, 5 figures

We compute the kaon's valence-quark (twist-two parton) distribution amplitude (PDA) by projecting its Poincare'-covariant Bethe-Salpeter wave-function onto the light-front. At a scale \zeta=2GeV, the PDA is a broad, concave and asymmetric function, whose peak is shifted 12-16% away from its position in QCD's conformal limit. These features are a clear expression of SU(3)-flavour-symmetry breaking. They show that the heavier quark in the kaon carries more of the bound-state's momentum than the lighter quark and also that emergent phenomena in QCD modulate the magnitude of flavour-symmetry breaking: it is markedly smaller than one might expect based on the difference between light-quark current masses. Our results add to a body of evidence which indicates that at any energy scale accessible with existing or foreseeable facilities, a reliable guide to the interpretation of experiment requires the use of such nonperturbatively broadened PDAs in leading-order, leading-twist formulae for hard exclusive processes instead of the asymptotic PDA associated with QCD's conformal limit. We illustrate this via the ratio of kaon and pion electromagnetic form factors: using our nonperturbative PDAs in the appropriate formulae, $F_K/F_\pi=1.23$ at spacelike-$Q^2=17\,{\rm GeV}^2$, which compares satisfactorily with the value of $0.92(5)$ inferred in $e^+ e^-$ annihilation at $s=17\,{\rm GeV}^2$., Comment: 7 pages, 2 figures, 3 tables

A rainbow-ladder truncation of QCD's Dyson-Schwinger equations is used to calculate rho- and phi-meson valence-quark (twist-two parton) distribution amplitudes (PDAs) via a light-front projection of their Bethe-Salpeter wave functions, which possess S- and D-wave components of comparable size in the meson rest frame. All computed PDAs are broad concave functions, whose dilation with respect to the asymptotic distribution is an expression of dynamical chiral symmetry breaking. The PDAs can be used to define an ordering of valence-quark light-front spatial-extent within mesons: this size is smallest within the pion and increases through the perp-polarisation to the parallel-polarisation of the vector mesons; effects associated with the breaking of SU(3)-flavour symmetry are significantly smaller than those associated with altering the polarisation of vector mesons. Notably, the predicted pointwise behaviour of the rho-meson PDAs is in quantitative agreement with that inferred recently via an analysis of diffractive vector-meson photoproduction experiments., Comment: 13 pages, 4 figures, 4 tables

We exploit a method introduced recently to determine parton distribution amplitudes (PDAs) from minimal information in order to obtain light-quark pseudoscalar and vector meson PDAs from the limited number of moments produced by numerical simulations of lattice-regularised QCD. Within errors, the PDAs of pseudoscalar and vector mesons constituted from the same valence quarks are identical; they are concave functions, whose dilation expresses the strength of dynamical chiral symmetry breaking; and SU(3)-flavour symmetry is broken nonperturbatively at the level of 10%. Notably, the appearance of precision in the lattice moments is misleading. The moments also exhibit material dependence on lattice volume, especially for the pion. Improvements need therefore be made before an accurate, unified picture of the light-front structure of light-quark pseudoscalar and vector mesons is revealed., Comment: 7 pages, 4 figures, 2 tables

The pseudoscalar projection of the pion's Poincare'-covariant Bethe-Salpeter amplitude onto the light-front may be understood to provide the probability distribution of the chiral condensate within the pion. Unlike the parton distribution amplitudes usually considered and as befitting a collective effect, this condensate distribution receives contributions from all Fock space components of the pion's light-front wave-function. We compute this condensate distribution using the Dyson-Schwinger equation (DSE) framework and show the result to be a model-independent feature of quantum chromodynamics (QCD). Our analysis establishes that this condensate is concentrated in the neighbourhood of the boundaries of the distribution's domain of support. It thereby confirms the dominant role played by many-particle Fock states within the pion's light-front wave function in generating the chiral condensate on the light-front and verifies that light-front longitudinal zero modes do not play a material role in that process., Comment: 5 pages, 1 figure. To appear in Phys. Lett. B

The gauge principle is fundamental in formulating the Standard Model. Fermion--gauge-boson couplings are the inescapable consequence and the primary determining factor for observable phenomena. Vertices describing such couplings are simple in perturbation theory and yet the existence of strong-interaction bound-states guarantees that many phenomena within the Model are nonperturbative. It is therefore crucial to understand how dynamics dresses the vertices and thereby fundamentally alters the appearance of fermion--gauge-boson interactions. We consider the coupling of a dressed-fermion to an Abelian gauge boson, and describe a unified treatment and solution of the familiar longitudinal Ward-Green-Takahashi identity and its less well known transverse counterparts. Novel consequences for the dressed-fermion--gauge-boson vertex are exposed., Comment: 5 pages, 1 figure

In a strongly-coupled quark-gluon plasma, collective excitations of gluons and quarks should dominate over the excitation of individual quasi-free gluon and quark modes. To explore this possibility, we computed screening masses for ground-state light-quark mesons and baryons at leading-order in a symmetry-preserving truncation scheme for the Dyson-Schwinger equations using a confining formulation of a contact-interaction at nonzero temperature. Meson screening masses are obtained from Bethe-Salpeter equations; and baryon analogues from a novel construction of the Faddeev equation, which employs an improved quark-exchange approximation in the kernel. Our treatment implements a deconfinement transition that is coincident with chiral symmetry restoration in the chiral limit, when both transitions are second order. Despite deconfinement, in all T=0 bound-state channels, strong correlations persist above the critical temperature, T>T_c; and, in the spectrum defined by the associated screening masses, degeneracy between parity-partner correlations is apparent for T >1.3T_c. Notwithstanding these results, there are reasons (including Golberger-Treiman relations) to suppose that the inertial masses of light-quark bound-states, when they may be defined, vanish at the deconfinement temperature; and that this is a signal of bound-state dissolution. Where a sensible comparison is possible, our predictions are consistent with results from contemporary numerical simulations of lattice-regularised QCD., Comment: 21 pages, 12 figures, 1 table

We project onto the light-front the pion's Poincare'-covariant Bethe-Salpeter wave-function, obtained using two different approximations to the kernels of QCD's Dyson-Schwinger equations. At an hadronic scale both computed results are concave and significantly broader than the asymptotic distribution amplitude, \phi_\pi^{asy}(x)=6 x(1-x); e.g., the integral of \phi_\pi(x)/\phi_\pi^{asy}(x) is 1.8 using the simplest kernel and 1.5 with the more sophisticated kernel. Independent of the kernels, the emergent phenomenon of dynamical chiral symmetry breaking is responsible for hardening the amplitude., Comment: 5 pages, 2 figures, 1 table

Elastic and semileptonic transition form factors for the kaon and pion are calculated using the leading-order in a global-symmetry-preserving truncation of the Dyson-Schwinger equations and a momentum-independent form for the associated kernels in the gap and Bethe-Salpeter equations. The computed form factors are compared both with those obtained using the same truncation but an interaction that preserves the one-loop renormalisation-group behaviour of QCD and with data. The comparisons show that: in connection with observables revealed by probes with |Q^2|<~ M^2, where M~0.4GeV is an infrared value of the dressed-quark mass, results obtained using a symmetry-preserving regularisation of the contact-interaction are not realistically distinguishable from those produced by more sophisticated kernels; and available data on kaon form factors do not extend into the domain whereupon one could distinguish between the interactions. The situation is different if one includes the domain Q^2>M^2. Thereupon, a fully consistent treatment of the contact interaction produces form factors that are typically harder than those obtained with QCD renormalisation-group-improved kernels. Amongst other things also described are a Ward identity for the inhomogeneous scalar vertex, similarity between the charge distribution of a dressed-u-quark in the K^+ and that of the dressed-u-quark in the pi^+, and reflections upon the point whereat one might begin to see perturbative behaviour in the pion form factor. Interpolations of the form factors are provided, which should assist in working to chart the interaction between light-quarks by explicating the impact on hadron properties of differing assumptions about the behaviour of the Bethe-Salpeter kernel., Comment: 17 pages, 9 figures, 4 tables

We argue by way of examples that, as a nonlinear integral equation, the gap equation can and does possess many physically distinct solutions for the dressed-quark propagator. The examples are drawn from a class that is successful in describing a broad range of hadron physics observables. We apply the homotopy continuation method to each of our four exemplars and thereby find all solutions that exist within the interesting domains of light current-quark masses and interaction strengths; and simultaneously provide an explanation of the nature and number of the solutions, many of which may be associated with dynamical chiral symmetry breaking. Introducing a stability criterion based on the scalar and pseudoscalar susceptibilities we demonstrate, however, that for any nonzero current-quark mass only the regular Nambu solution of the gap equation is stable against perturbations. This guarantees that the existence of multiple solutions to the gap equation cannot complicate the description of phenomena in hadron physics., Comment: 14 pages, 15 figures

The nucleon's axial charge, g_A, expresses features that are both fundamental to the strong interaction and crucial to its connection with weak interaction physics. We show that dynamical chiral symmetry breaking (DCSB) suppresses the axial-charge of a dressed-quark, g_A^q, at infrared momenta. Since this effect disappears as chiral symmetry is restored, one may argue that g_A vanishes with the restoration of chiral symmetry because no nucleon bound-state survives the associated transition. The suppression of g_A^q is shown to be part of an explanation for a 25% reduction of g_A from its nonrelativistic quark-model value. Critical too, however, is the presence of dressed-quark angular momentum within the nucleon. The value of g_A^q depends on the kernels of the gap and Bethe-Salpeter equations. We find that incorporation of essentially nonperturbative effects associated with DCSB into these kernels inflates the value relative to that obtained at leading-order in a widely used truncation of QCD's Dyson-Schwinger equations. Such corrections also affect the nucleon's axial radius. In both cases, however, agreement with experiment will require similar improvements to the Faddeev kernel and associated interaction current., Comment: 10 pages, 3 figures

We describe a calculation of the spectrum of strange and nonstrange hadrons that simultaneously correlates the dressed-quark-core masses of meson and baryon ground- and excited-states within a single framework. The foundation for this analysis is a symmetry-preserving Dyson-Schwinger equation treatment of a vector-vector contact interaction. Our results exemplify and highlight the deep impact of dynamical chiral symmetry breaking on the hadron spectrum: an accurate description of the meson spectrum entails a similarly successful prediction of the spectrum of baryons, including those with strangeness. The analysis also provides numerous insights into baryon structure. For example, that baryon structure is largely flavour-blind, the first radial excitation of ground-state baryons is constituted almost entirely from axial-vector diquark correlations, and DCSB is the foundation for the ordering of low-lying baryon levels; viz., (1/2)^+, (1/2)^+, (1/2)^-., Comment: 32 pages, 5 figures, 7 tables

Physics is an experimental science; and a constructive feedback between theory and extant and forthcoming experiments is necessary if an understanding of nonperturbative QCD is to be achieved. The Dyson-Schwinger equations connect confinement with dynamical chiral symmetry breaking, both with the observable properties of hadrons, and hence can plausibly provide a means of elucidating the empirical content of strong QCD. We illustrate these points via comments on: in-hadron condensates; dressed-quark anomalous chromo- and electro-magnetic moments; the self-limiting magnitudes of such moments and pion-loop contributions to the gap equation; deep inelastic scattering; the spectra of mesons and baryons; the critical role played by hadron-hadron interactions in producing these spectra; and nucleon elastic and transition form factors., Comment: 15 pages. Contribution to the proceedings of the "International Workshop on QCD Green's Functions," Confinement and Phenomenology, 5-9 September 2011, Trento, Italy

We survey contemporary studies of hadrons and strongly interacting quarks using QCD's Dyson-Schwinger equations, addressing: aspects of confinement and dynamical chiral symmetry breaking; the hadron spectrum; hadron elastic and transition form factors, from small- to large-Q^2; parton distribution functions; the physics of hadrons containing one or more heavy quarks; and properties of the quark gluon plasma., Comment: 56 pages. Summary of lectures delivered by the authors at the "Workshop on AdS/CFT and Novel Approaches to Hadron and Heavy Ion Physics," 2010-10-11 to 2010-12-03, hosted by the Kavli Institute for Theoretical Physics, China, at the Chinese Academy of Sciences

Using a framework in which all elements are constrained by Dyson-Schwinger equation studies in QCD, and therefore incorporates a consistent, direct and simultaneous description of light- and heavy-quarks and the states they constitute, we analyze the accuracy of SU(4)-flavor symmetry relations between {\pi}{\rho}{\pi}, K{\rho}K and D{\rho}D couplings. Such relations are widely used in phenomenological analyses of the interactions between matter and charmed mesons. We find that whilst SU(3)-flavor symmetry is accurate to 20%, SU(4) relations underestimate the D{\rho}D coupling by a factor of five., Comment: 5 pages, two figures

Ground-state, radially-excited and exotic scalar-, vector- and flavoured-pseudoscalar-mesons are studied in rainbow-ladder truncation using an interaction kernel that is consonant with modern DSE- and lattice-QCD results. The inability of this truncation to provide realistic predictions for the masses of excited- and exotic-states is confirmed and explained. On the other hand, its application does provide information that is potentially useful in proceeding beyond this leading-order truncation, e.g.: assisting with development of projection techniques that ease the computation of excited state properties; placing qualitative constraints on the long-range behaviour of the interaction kernel; and highlighting and illustrating some features of hadron observables that do not depend on details of the dynamics., Comment: 10 pages, 6 figures, 2 tables

The in-pseudoscalar-meson condensate can be represented through the pseudoscalar-meson's scalar form factor at zero momentum transfer. With the aid of a mass formula for scalar mesons, revealed herein, the analogue is shown to be true for in-scalar-meson condensates. The concept is readily extended to all hadrons so that, via the zero momentum transfer value of any hadron's scalar form factor, one can readily extract the value for a quark condensate in that hadron which is a measure of dynamical chiral symmetry breaking., Comment: 5 pages, 3 figures

A Dyson-Schwinger equation calculation of the light hadron spectrum, which correlates the masses of meson and baryon ground- and excited-states within a single framework, produces a description of the Roper resonance that corresponds closely with conclusions drawn recently by EBAC. Namely, the Roper is a particular type of radial excitation of the nucleon's dressed-quark core augmented by a material meson cloud component. There are, in addition, some surprises., Comment: 4 pages, 3 figures. Contribution to the Proceedings of "NSTAR2011 - The 8th International Workshop on the Physics of Excited Nucleons," Thomas Jefferson National Accelerator Facility, Newport News, Virginia USA, 17-20 May 2011

We explain a form for the rainbow-ladder kernel whose momentum-dependence is consonant with modern DSE- and lattice-QCD results, and assess its capability as a tool in hadron physics. In every respect tested, this form produces results for observables that are at least equal to the best otherwise obtained in a comparable approach. Moreover, it enables the natural extraction of a monotonic running-coupling and -gluon-mass., Comment: 5 pages, 1 figure, 2 tables