A Poincaré-covariant quark+diquark Faddeev equation is used to compute nucleon elastic form factors on $0\leq Q^2\leq 18 \,m_N^2$ ($m_N$ is the nucleon mass) and elucidate their role as probes of emergent hadronic mass in the Standard Model. The calculations expose features of the form factors that can be tested in new generation experiments at existing facilities, e.g. a zero in $G_E^p/G_M^p$; a maximum in $G_E^n/G_M^n$; and a zero in the proton's $d$-quark Dirac form factor, $F_1^d$. Additionally, examination of the associated light-front-transverse number and anomalous magnetisation densities reveals, inter alia: a marked excess of valence $u$-quarks in the neighbourhood of the proton's centre of transverse momentum; and that the valence $d$-quark is markedly more active magnetically than either of the valence $u$-quarks. The calculations and analysis also reveal other aspects of nucleon structure that could be tested with a high-luminosity accelerator capable of delivering higher beam energies than are currently available.
A novel method is employed to compute the pion electromagnetic form factor, F_π(Q^2), on the entire domain of spacelike momentum transfer using the Dyson-Schwinger equation (DSE) framework in quantum chromodynamics (QCD). The DSE architecture unifies this prediction with that of the pion's valence-quark parton distribution amplitude (PDA). Using this PDA, the leading-order, leading-twist perturbative QCD result for Q^2 F_π(Q^2) underestimates the full computation by just 15% on Q^2>~8GeV^2, in stark contrast with the result obtained using the asymptotic PDA. The analysis shows that hard contri…
In contrast to perturbative QCD, the analytic QCD models have running coupling whose analytic properties correctly mirror those of spacelike observables. The discontinuity (spectral) function of such running coupling is expected to agree with the perturbative case at large timelike momenta; however, at low timelike momenta it is not known. In the latter regime, we parametrize the unknown behavior of the spectral function as a sum of (two) delta functions; while the onset of the perturbative behavior of the spectral function is set to be 1.0-1.5 GeV. This is in close analogy with the "minimal h…
Within the Bethe-Salpeter formalism for instantaneous interactions, we describe, along a totally analytic route, the lightest pseudoscalar mesons by quark-antiquark bound states which show at least three indispensable general features, namely, the (almost) masslessness required for pions and kaons to be interpretable as (pseudo) Goldstone bosons, the suitable asymptotic behaviour in the limit of large spacelike relative momenta as determined by the relationship between quark mass function and Bethe-Salpeter amplitudes, and a pointwise behaviour for finite spacelike relative momenta suited for …
We study the pseudoscalar, vector and axial current correlation functions in SU(2)-NJL model with scalar and vector couplings. The correlation functions are evaluated in leading order in number of colors $N_c$. As it is expected in the pseudoscalar channel pions appear as Goldstone bosons, and after fixing the cutoff to reproduce the physical pion decay constant, we obtain well-known current-algebra results. For the vector and axial channels we use essentially that at spacelike momenta the correlation functions can be related to the experimentally known spectral density via dispersion relation…
Newer lattice results indicate that, in the Landau gauge at low spacelike momenta, the gluon propagator and the ghost dressing function are finite nonzero. This leads to a definition of the QCD running coupling, in a specific scheme, that goes to zero at low spacelike momenta. We construct a running coupling which fulfills these conditions, and at the same time reproduces to a high precision the perturbative behavior at high momenta. The coupling is constructed in such a way that it reflects qualitatively correctly the holomorphic (analytic) behavior of spacelike observables in the complex pla…
Some form factors of ground-state hadrons with spin might show a zero crossing at spacelike momenta of electron-hadron scattering. In the timelike region of hadron-antihadron production by electron-positron collisions the form factors become complex. When the hadrons decay, the relative phases between form factors can be measured by the angular distributions of their decay products. Using analyticity and the QCD high-energy limits for constraint-free form factors and for helicity amplitudes, we show how a single zero crossing in the spacelike region is related to the high-energy limit of the p…
An approach for a unified description of the nucleon electromagnetic form factors in spacelike and timelike regions is presented. The main ingredients of our model are: $i)$ a Mandelstam formula for the matrix elements of the nucleon electromagnetic current; $ii)$ a 3-dimensional reduction of the problem on the Light-Front performed within the so-called {\tt Propagator Pole Approximation} ({\bf PPA}), which consists in disregarding the analytical structure of the Bethe-Salpeter amplitudes and of the quark-photon vertex function in the integration over the minus components of the quark momenta;…
We present a detailed investigation of the spacelike pion's electromagnetic form factor using three-point QCD sum rules that exclusively involve nonlocal condensates. Our main methodological tools are a spectral density which includes $O(α_s)$ corrections, a suitably improved Gaussian ansatz to model the distribution of the average momentum of quarks in the QCD vacuum, and a perturbative scheme that avoids Landau singularities. Using this framework, we obtain predictions for the pion form factor together with error estimates originating from intrinsic theoretical uncertainties owing to the per…
The factorization theorem for exclusive processes in perturbative QCD predicts the behavior of the pion electromagnetic form factor $F(t)$ at asymptotic spacelike momenta $t(=-Q^2)<0$. We address the question of the onset energy using a suitable mathematical framework of analytic continuation, which uses as input the phase of the form factor below the first inelastic threshold, known with great precision through the Fermi-Watson theorem from $ππ$ elastic scattering, and the modulus measured from threshold up to 3 GeV by the BaBar Collaboration. The method leads to almost model-independent uppe…
We consider the gluon propagator in the Landau gauge at low spacelike momenta and with the dressing function $Z(Q^2)$ at the two-loop order. We incorporate the nonperturbative effects by making the (noninteger) powers of the QCD coupling in the dressing function $Z(Q^2)$ analytic (holomorphic) via the Fractional Analytic Perturbation Theory (FAPT) model, and simultaneously introducing the gluon dynamical mass in the propagator as motivated by the previous analyses of the Dyson-Schwinger equations. The obtained propagator has behavior compatible with the unquenched lattice data ($N_f=2+1$) at l…
QCD sum rules are used to calculate the $q^2$ dependence of the $πNN$ coupling $g_{πNN} (q^2)$ in the spacelike region $0.5 \ {\mbox{GeV}}^2 \lesssim q^2 \lesssim 1.5\ {\mbox{GeV}}^2$. We study the Borel sum rule for the three point function of one pseudoscalar and two nucleon currents up to order four in the operator product expansion. The Borel transform is performed with respect to the nucleon momenta, whereas the momentum $q^2$ of the pseudoscalar vertex is kept fixed at spacelike values. The results can be well fitted using a monopole form with a cutoff mass of about $Λ_π\approx 800 \ {\m…
The applications of the consistently coupled Schwinger-Dyson and Bethe-Salpeter approach to the eta-eta' complex are extended to the two-photon transition form factors of eta and eta' for spacelike transferred momenta. We compare our predictions with experiment and some other theoretical approaches.
For the spacelike momenta k of the virtual photon gamma^*, the pi^0(p) gamma^*(k) gamma(k') transition form factor is considered in the coupled Schwinger-Dyson and Bethe-Salpeter approach in conjunction with the generalized impulse approximation using the dressed quark-photon-quark vertices of the Ball-Chiu and Curtis-Pennington type. These form factors are compared with the ones predicted by the vector meson dominance, operator product expansion, QCD sum rules, and the perturbative QCD for the large spacelike transferred momenta k. The most important qualitative feature of the asymptotic beha…
A synopsis exemplifying the employment of Dyson-Schwinger equations in the calculation and explanation of hadron electromagnetic form factors and related phenomena. In particular the contribution: presents the pion form factor computed simultaneously at spacelike and timelike momenta; reports aspects of the evolution of the nucleon and Delta masses with current-quark mass and the correlation of their mass difference with that between scalar and axial-vector diquarks; describes an estimate of the s-quark content of a dressed u-quark and its impact on the nucleon's strangeness magnetic moment; a…
The Coulomb scattering between three charged pions is shown to be perturbatively calculable using scalar QED for the relative momenta accessible to high-energy experiments of particle collisions. For the triplet relative momenta, $Q_3 \gtrsim 50$ MeV/$c$, the calculated three-pion correlation function at $O(α^2)$ is very similar to the commonly-used triple-product of pair factors -- Riverside approximation. At lower $Q_3$, but above the lowest existing experimental measurement of $\sim15$ MeV/$c$, terms of $O(α^3)$ or higher are needed to obtain a convincing description of three-pion Coulomb i…
The leading order evolution of parton orbital angular momenta is treated in the probabilistic manner. As a result, the splitting probability matrix, while coinciding with the recent explicit calculations, is expressed in the terms of the standard spin-dependent and spin-averaged kernels. The implied relations between spin and orbital angular momenta evolution kernels may be interpreted as resulting from the ambiguity in the separation of the angular momenta to the spin and orbital contributions and are likely to be valid for higher orders and twists. Consequently, the orbital and total angular…
The evaluation of the numerically dominant pseudoscalar-pole contribution to hadronic light-by-light scattering in the muon g-2 involves the pseudoscalar-photon transition form factor F_{P gamma^* gamma^*}(-Q_1^2, -Q_2^2) with P = pi^0, eta, eta^\prime and, in general, two off-shell photons with spacelike momenta Q_{1,2}^2. We determine which regions of photon momenta give the main contribution for hadronic light-by-light scattering. Furthermore, we analyze how the precision of future measurements of the single- and double-virtual form factor impacts the precision of a data-driven estimate of …
The pion transition form factor $G(q^2)$ is computed on the entire domain of spacelike and timelike momenta using a quantum field theory continuum approach. In analytical continuation of the function $G(Q^2)$ we utilized the Gauge Technique with the quark propagator determined from Minkowski space solution of QCD Dyson-Schwinger equations. The scale is set up by the phenomena of dynamical chiral symmetry breaking, which is a striking feature of low energy QCD.
The electromagnetic pion form factor, $F_π(q^2)$, is calculated for spacelike-$q^2$ in impulse approximation using a confining quark propagator, $S$, and a dressed quark-photon vertex, $Γ_μ$, obtained from realistic, nonperturbative Dyson-Schwinger equation studies. Good agreement with the available data is obtained for $F_π(q^2)$ and other pion observables, including the decay $π^0 \rightarrow γ\,γ$. This calculation suggests that soft, nonperturbative contributions dominate $F_π(q^2)$ at presently accessible~$q^2$.
A simple model for the hadronic contribution to the photon vacuum polarization function $Π_{had}(q^2)$, for spacelike momenta, is presented. For small momenta, the two loop contribution from the pseudoscalar meson octet is computed from the chiral Lagrangian. The light quark contribution (which at low momentum gives the ${\cal O}(q^6)$ counterterm in the chiral Lagrangian) is calculated within a relativistic constituent quark model incorporating the momentum dependence of the quark mass. The perturbative gluons of QCD are included in a standard fashion. The total result is close to an estimate…
A simple method is used to extrapolate the form factor for the process gamma* gamma *--->pi^0 when two virtual photons produce a pi ^0 from the region of large spacelike photon virtualities to the experimentally accessible case when one of the photons is real. The method is also applied to the study of the axial anomaly and omega -rho -pi coupling and to the pion charge and charge radius. The results compare favourably with the experimental measurements.
We present a determination of the pion charge radius from high precision data on the pion vector form factor from both timelike and spacelike regions, using a novel formalism based on analyticity and unitarity. At low energies, instead of the poorly known modulus of the form factor, we use its phase, known with high accuracy from Roy equations for $ππ$ elastic scattering via the Fermi-Watson theorem. We use also the values of the modulus at several higher timelike energies, where the data from $e^+e^-$-annihilation and $τ$-decay are mutually consistent, as well as the most recent measurements …
The behavior of the vector Adler function at spacelike momenta is studied in the framework of a covariant chiral quark model with instanton-like quark-quark interaction. The model reproduces the Adler function and $V-A$ correlator extracted from the ALEPH and OPAL data on hadronic $τ$ lepton decays, transformed into the Euclidean domain via dispersion relations. The leading order contribution from hadronic part of the photon vacuum polarization to the anomalous magnetic moment of the muon, $a_μ^{\mathrm{hvp}(1)} $, is estimated.
Using a continuum approach to the hadron bound-state problem, we calculate $γ^\ast γ\to η, η^\prime$ 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 $η, η^\prime$ properties. The study also delivers predictions for the amplitudes that describe the light- and strange-quark distributions …
The behavior of the vector Adler function at spacelike momenta is studied in the framework of a covariant chiral quark model with instanton-like quark-quark interaction. This function describes the transition between the high energy asymptotically free region of almost massless current quarks to the low energy hadronized regime with massive constituent quarks. The model reproduces the Adler function and V-A correlator extracted from the ALEPH and OPAL data on hadronic τlepton decays, transformed into the Euclidean domain via dispersion relations. The leading order contribution from hadronic pa…
The $γ^\ast γ\to π^0$ transition form factor, $G(Q^2)$, is computed on the entire domain of spacelike momenta using a continuum approach to the two valence-body bound-state problem in relativistic quantum field theory: the result agrees with data obtained by the CELLO, CLEO and Belle Collaborations. The analysis unifies this prediction with that of the pion's valence-quark parton distribution amplitude (PDA) and elastic electromagnetic form factor, and demonstrates, too, that a fully self-consistent treatment can readily connect a pion PDA that is a broad, concave function at the hadronic scal…
Generalized parton distributions (GPDs) are three-dimensional structure functions of hadrons, and they can reveal the orbital-angular-momentum contributions to the nucleon spin. Therefore, GPDs are important for solving the proton spin puzzle. The generalized distribution amplitudes (GDAs) are the $s$-$t$ crossed quantities of the GPDs, and the GDAs can be investigated in two-photon process ($γ^* γ\to h\bar h$) which is accessible at KEKB. The pion GDAs were obtained by analyzing the Belle measurements for $π^0 π^0$ production in the $e^+ e^-$ collision. From the obtained GDAs, the form factor…
The evaluation of the numerically dominant pseudoscalar-pole contribution to hadronic light-by-light scattering in the muon g-2 involves the pseudoscalar-photon transition form factor F_{P gamma^* gamma^*}(-Q_1^2, -Q_2^2) with P = pi^0, eta, eta^\prime and, in general, two off-shell photons with spacelike momenta Q_{1,2}^2. We show, in a largely model-independent way, that for pi^0 (eta, eta^\prime) the region of photon momenta below about 1 (1.5) GeV gives the main contribution to hadronic light-by-light scattering. We then discuss how the precision of current and future measurements of the s…
We extend recently developed methods used for determining the electromagnetic charge radius and $a_μ^{ππ}$ to obtain a determination of the electromagnetic form factor of the pion, $F_π^V(t)$, in several significant kinematical regions, using a parametrization-free formalism based on analyticity and unitarity, and with the inclusion of precise inputs from both timelike and spacelike regions. On the unitarity cut, below the first inelastic threshold we use the precisely known phase of the form factor, known from $ππ$ elastic scattering via the Fermi-Watson theorem, and above the inelastic thres…
We present a construction of the pion electromagnetic form factor where the transition from large-Nc Regge vector meson dominance models with infinitely many resonances to perturbative QCD is built in explicitly. The construction is based on an appropriate assignment of residues to the Regge poles, which fulfills the constraints of the parton-hadron duality and perturbative QCD. The model contains a slowly falling off non-perturbative contribution which dominates over the perturbative QCD radiative corrections for the experimentally accessible momenta. The leading order and next-to-leading ord…
We discuss two-photon and hadronic production of $B_c$ mesons in nonrelativistic bound state approximation and to lowest order in the coupling constants $α$ and $α_s$. It is shown that in photon-photon collisions, heavy quark fragmentation is dominated by recombination of $\bar b$ and $c$ quarks up to the highest accessible transverse momenta. In contrast, in hadroproduction, which at high energies mainly involves gluon--gluon collisions, the fragmentation mechanism dominates at transverse momenta $p_T > m_{B_c}$, providing a simple and satisfactory approximation of the complete $O(α_s^4)$ res…
Perturbative QCD in mass independent schemes leads in general to running coupling $a(Q^2)$ which is nonanalytic (nonholomorphic) in the regime of low spacelike momenta $|Q^2| \lesssim 1 \ {\rm GeV}^2$. Such (Landau) singularities are inconvenient in the following sense: evaluations of spacelike physical quantities ${\cal D}(Q^2)$ with such a running coupling $a(κQ^2)$ ($κ\sim 1$) give us expressions with the same kind of singularities, while the general principles of local quantum field theory require that the mentioned physical quantities have no such singularities. In a previous work, certai…
We consider the pion-photon transition form factor at low to intermediate spacelike momenta within the theoretical framework of light-cone sum rules. We derive predictions which take into account all currently known contributions stemming from QCD perturbation theory up to the next-to-next-to-leading order (NNLO) and by including all twist terms up to order six. In order to enable a more detailed comparison with forthcoming high-precision data, we also estimate the main systematic theoretical uncertainties, stemming from various sources, and discuss their influence on the calculations --- in p…
We calculate the $π^0\to γ^*γ^*$ transition form factor ${\cal F}_{π^0γ^*γ^*}(q_1^2,q_2^2)$ in lattice QCD with two flavors of quarks. Our main motivation is to provide the input to calculate the $π^0$-pole contribution to hadronic light-by-light scattering in the muon $(g-2)$, $a_μ^{\rm HLbL;π^0}$. We therefore focus on the region where both photons are spacelike up to virtualities of about $1.5~$GeV$^2$, which has so far not been experimentally accessible. Results are obtained in the continuum at the physical pion mass by a combined extrapolation. We reproduce the prediction of the chiral an…
New, accurate measurements of the pion and kaon electromagnetic form factors are expected in the near future from experiments at electron-positron colliders,using the radiative return method. We construct a model for the timelike pion electromagnetic form factor, that is valid also at momentum transfers far above the $ρ$ resonance. The ansatz is based on vector dominance and includes a pattern of radial excitations expected from dual resonance models.The form factor is fitted to the existing data in the timelike region, continued to the spacelike region and compared with the measurements there…
Recent investigations of the Dyson-Schwinger equations and Monte-Carlo lattice calculations resulted in a coherent description of the fully dressed gluon, ghost and quark propagators in Landau gauge QCD. In the Dyson-Schwinger approach the infrared behaviour of these propagators is determined analytically. For finite spacelike momenta the gluon, ghost and quark propagators are compared to available corresponding results of lattice Monte-Carlo calculations. For all three propagators an almost quantitative agreement is found. These results for the non-perturbative propagators allow an analytical…
A method of calculating Feynman diagrams from their small momentum expansion [1] is extended to diagrams with zero mass thresholds. We start from the asymptotic expansion in large masses [2] (applied to the case when all $M_i^2$ are large compared to all momenta squared). Using dimensional regularization, a finite result is obtained in terms of powers of logarithms (describing the zero-threshold singularity) times power series in the momentum squared. Surprisingly, these latter ones represent functions, which not only have the expected physical `second threshold' but have a branchcut singulari…
We calculate upper and lower bounds on the modulus of the pion electro magnetic form factor on the unitarity cut below the $ωπ$ inelastic threshold, using as input the phase in the elastic region known via the Fermi-Watson theorem from the $ππ$ $P$-wave phase shift, and a suitably weighted integral of the modulus squared above the inelastic threshold. The normalization at $t=0$, the pion charge radius and experimental values at spacelike momenta are used as additional input information. The bounds are model independent, in the sense that they do not rely on specific parametrizations and do not…
We investigate the gluon, ghost and quark propagators in the Landau gauge with dynamic quarks. We perform a one-loop calculation in a model where the standard Faddeev-Popov Lagrangian is complemented by a mass term for the gluons which is seen as a minimal way of taking into account the effect of the Gribov copies. The analytic results are compared with lattice data obtained in four dimension and for two, three and four quark flavors. The gluon and ghost propagators are reproduced with a few percent accuracy in the whole range of accessible momenta. The scalar part of the quark propagator is f…
We set up a framework for the study of the power-suppressed three-parton contribution to the pion electromagnetic form factor in the $k_T$ factorization theorem. It is first shown that the gauge dependence proportional to parton transverse momenta from the two-parton Fock state and the gauge dependence associated with the three-parton Fock state cancel each other. After verifying the gauge invariance, we derive the three-parton-to-three-parton $k_T$-dependent hard kernel at leading order of the coupling constant, and find that it leads to about 5% correction to the pion electromagnetic form fa…
