Spacelike Form Factors Research Articles

Revisiting the time-ordering issue of TMD soft factors: causality, coordinate space analyticity and new equalities

We show that as a result of causality-constrained coordinate space analyticity, the Drell-Yan-shape transverse-momentum dependent soft factor in the exponential regulator allows Euclidean-type parametric representations without cuts, to all orders in perturbation theory. Moreover, it is identical to another soft factor defined with a single time-ordering that has a natural interpretation as a space-like form factor. Furthermore, this relation continues to hold for a larger class of TMD soft factors that interpolate between three different rapidity regulators: the off-light-cone regulator, the finite light-front length regulator, and the exponential regulator.

Electromagnetic form factors of Σ+ and Σ− in the vector-meson dominance model

Based on the recent measurements of the and processes by the Beijing Spectrometer III (BESIII) collaboration, the electromagnetic form factors of the hyperon Σ+ and Σ− in the timelike region are investigated using the vector-meson dominance model, where the contributions of the ρ, ω, and ϕ mesons are taken into account. The model parameters are determined from the BESIII experimental data of the timelike effective form factors ∣G eff∣ of the Σ+ and Σ− baryons for center-of-mass energies from 2.3864 to 3.02 GeV. It is found that we can provide quantitative descriptions of the available data using as few as one adjustable model parameter. We then progress to an analysis of the electromagnetic form factors in the spacelike region and evaluate the spacelike form factors of the hyperons Σ+ and Σ−. The electromagnetic form factors obtained for the Σ+ and Σ− baryons are comparable with those of other model calculations.

Pion light-cone distribution amplitude from the pion electromagnetic form factor

We suggest to probe the pion light-cone distribution amplitude, applying a dispersion relation for the pion electromagnetic form factor. Instead of the standard dispersion relation, we use the equation between the spacelike form factor $F_\pi(Q^2)$ and the integrated modulus of the timelike form factor. For $F_\pi(Q^2)$, the QCD light-cone sum rule with a dominant twist-2 term is used. Adopting for the pion twist-2 distribution amplitude a certain combination of the first few Gegenbauer polynomials, it is possible to fit their coefficients $a_{2,4,6,...}$ (Gegenbauer moments) from this equation, employing the measured pion timelike form factor. For the exploratory fit we use the data of the BaBar collaboration. The results definitely exclude the asymptotic twist-2 distribution amplitude. Also the model with a single $a_2\neq 0$ is disfavoured by the fit. Considering the models with $a_{n>2}\neq 0$, we find that the fitted values of the second and fourth Gegenbauer moments cover the intervals $a_2 (1 \mbox{GeV}) = (0.22 - 0.33) $, $a_4 (1 \mbox{GeV}) = (0.12 - 0.25) $. The higher moments starting from $a_{8}$ are consistent with zero, albeit with large uncertainties. The spacelike pion form factor obtained in two different ways, from the dispersion relation and from the light-cone sum rule, agrees, within uncertainties, with the measurement by the Jefferson Lab $F_\pi$ collaboration.

Electromagnetic form factors of Λ hyperon in the vector meson dominance model

We perform an analysis on the electromagnetic form factors of the $\Lambda$ hyperon in the time-like reaction $e^+e^-\rightarrow \Lambda\bar\Lambda$ by using a modified vector meson dominance model. We consider both the intrinsic structure components and the meson clouds components. For the latter one, we not only include the contributions from the $\phi$ and $\omega$ mesons, but also take into account the contributions from the resonance states $\omega(1420)$, $\omega(1650)$, $\phi(1680)$ and $\phi(2170)$. We extract the model parameters by combined fit to the time-like effective form factor $|G_{\rm{eff}}|$, the electromagnetic form factor ratio $|G_E/G_M|$ and the relative phase $\Delta\Phi$ of the $\Lambda$ hyperon from the BaBar and BESIII Collaborations. We find that the vector meson dominance model can simultaneously describe these observables. Particularly, the inclusion of the resonance states in the model is necessary for explaining the ratio $|G_E/G_M|$ in a wide range of $\sqrt{s}$ as well as the large phase angle. With the fitted parameters, we predict the single and double polarization observables, which could be measured in polarized annihilation reactions. Moreover, we analytically continue the expression of the form factors to space-like region and estimate the space-like form factors of $\Lambda$ hyperon.

Supersymmetric and Conformal Features of Hadron Physics

The QCD Lagrangian is based on quark and gluonic fields—not squarks nor gluinos. However, one can show that its hadronic eigensolutions conform to a representation of superconformal algebra, reflecting the underlying conformal symmetry of chiral QCD. The eigensolutions of superconformal algebra provide a unified Regge spectroscopy of meson, baryon, and tetraquarks of the same parity and twist as equal-mass members of the same 4-plet representation with a universal Regge slope. The predictions from light-front holography and superconformal algebra can also be extended to mesons, baryons, and tetraquarks with strange, charm and bottom quarks. The pion q q ¯ eigenstate has zero mass for m q = 0 . A key tool is the remarkable observation of de Alfaro, Fubini, and Furlan (dAFF) which shows how a mass scale can appear in the Hamiltonian and the equations of motion while retaining the conformal symmetry of the action. When one applies the dAFF procedure to chiral QCD, a mass scale κ appears which determines universal Regge slopes, hadron masses in the absence of the Higgs coupling. One also predicts the form of the nonperturbative QCD running coupling: α s ( Q 2 ) ∝ e − Q 2 / 4 κ 2 , in agreement with the effective charge determined from measurements of the Bjorken sum rule. One also obtains viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. The combination of conformal symmetry, light-front dynamics, its holographic mapping to AdS 5 space, and the dAFF procedure thus provide new insights, not only into the physics underlying color confinement, but also the nonperturbative QCD coupling and the QCD mass scale.

Hadron Tomography and Its Application to Gravitational Radii of Hadrons

Hadron tomography has been investigated by three-dimensional structure functions, such as generalized parton distributions (GPDs) and generalized distribution amplitudes (GDAs). The GDAs are $s$-$t$ crossed quantities of the GPDs, and both functions probe gravitational form factors for hadrons. We determined the pion GDAs by analyzing Belle data on the differential cross section for the two-photon process $\gamma^* \gamma \to \pi^0 \pi^0$. From the determined GDAs, we calculated timelike gravitational form factors of the pion and they were converted to the spacelike form factors by using the dispersion relation. These gravitational form factors $\Theta_1$ and $\Theta_2$ indicate mechanical (pressure, shear force) and gravitational-mass (or energy) distributions, respectively. Then, gravitational radii are calculated for the pion from the form factors, and they are compared with the pion charge radius. We explain that the new field of gravitational physics can be developed in the microscopic level of quarks and gluons.

Rho-pion transition form factors in the kT -factorization formalism revisited

We revisit the evaluations for the spacelike and timelike $\rho \pi$ transition form factors $F_{\rho\pi}(Q^2)$ and $G_{\rho\pi}(Q^2)$ with the inclusion of the the next-to-leading order (NLO) QCD contributions in the framework of the $k_T$ factorization theorem. The infrared divergence is regularized by the transversal momentum carried by external valence quarks, and ultimately absorbed into the meson wave functions. In the region of $ Q^2 \leqslant 2 \, \textrm{GeV}^2 $, where PQCD factorization apporach applicable, the NLO contribution can bring no larger than $ 35\%$ enhancement to the spacelike form factor $F_{\rho\pi}(Q^2)$. For the timelike form factor derived under the kinematic exchanging symmetry, this contribution is also under control when the momentum transfer squared is large enough. We also prolong our prediction into the small $Q^2$ region by taking the Lattice QCD results into account, and subsequently obtain the coupling $g_{\rho\pi\gamma} = G_{\rho\pi}(0)=0.596$.

Form Factor Measurements at BESIII for an Improved Standard Model Prediction of the Muon g-2

The anomalous part of the magnetic moment of the muon, (g-2)[Formula: see text], allows for one of the most precise tests of the Standard Model of particle physics. We report on recent results by the BESIII Collaboration of exclusive hadronic cross section channels, such as the 2[Formula: see text], 3[Formula: see text], and 4[Formula: see text] final states. These measurements are of utmost importance for an improved calculation of the hadronic vacuum polarization contribution of (g-2)[Formula: see text], which currenty is limiting the overall Standard Model prediction of this quantity. BESIII has furthermore also intiatated a programme of spacelike transition form factor measurements, which can be used for a determination of the hadronic light-by-light contribution of (g-2)[Formula: see text] in a data-driven approach. These results are of relevance in view of the new and direct measurements of (g-2)[Formula: see text] as foreseen at Fermilab/USA and J-PARC/Japan.

Dispersion theoretical analysis of the nucleon form factors

We review the achievements of the project C.2 , that deals with a dispersion theoretical analysis of the nucleon form factors. We have analyzed the world data for the space-like and time-like form factors to extract nucleon radii and vector meson coupling constants. Our proton charge radius r P E ≃ 0.85 fm has gained prominence in the so-called proton radius puzzle, see Ref. [1]. We also have analyzed the strong energy dependence of the e + e − → pp amplitude at the nucleon-antinucleon threshold, two-photon exchange effects and isospin violation in the isovector nucleon form factors.

ANALYTIC CONTINUATION OF NUCLEONS FORM FACTORS IN TIME-LIKE REGION WITH SKYRME MODEL

Using the Skyrme model for nucleons we test the possibility to obtain time-like form factors by analytic continuation of space-like ones. We present a method based on a direct computation of the space-like form factors in the Breit frame, where they are defined as Fourier transform of electromagnetic current. A Padé approximation of the current is implemented in order to solve the integral. The results are compared to data both in space-like and time-like region.

Electromagnetic Structure of the Proton, Pion, and Kaon by High-Precision Form Factor Measurements at Large Timelike Momentum Transfers

The electromagnetic structure of the lightest hadrons, proton, pion, and kaon is studied by high-precision measurements of their form factors for the highest timelike momentum transfers of |Q2|=s=14.2 and 17.4 GeV2. Data taken with the CLEO-c detector at sqrt[s]=3.772 and 4.170 GeV, with integrated luminosities of 805 and 586 pb(-1), respectively, have been used to study e+ e- annihilations into π+ π-, K+ K-, and pp. The dimensional counting rule prediction that at large Q2 the quantity Q2F(Q2) for pseudoscalar mesons is nearly constant, and should vary only weakly as the strong coupling constant αS(Q2) is confirmed for both pions and kaons. However, the measurements are in strong quantitative disagreement with the predictions of the existing quantum chromodynamics-based models. For protons, it is found that the timelike form factors continue to remain nearly twice as large as the corresponding spacelike form factors measured in electron elastic scattering, in significant violation of the expectation of their equality at large Q2. Further, in contrast to pions and kaons, a significant difference is observed between the values of the corresponding quantity |Q4|G(M)(|Q2|)/μ(p) for protons at |Q2|=14.2 and 17.4 GeV2. The results suggest the constancy of |Q2|G(M)(|Q2|)/μ(p), instead, at these large |Q2|.

Spacelike pion form factor from analytic continuation and the onset of perturbative QCD

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 $\pi\pi$ elastic scattering, and the modulus measured from threshold up to 3 GeV by the BaBar Collaboration. The method leads to almost model-independent upper and lower bounds on the spacelike form factor. Further inclusion of the value of the charge radius and the experimental value at $-2.45 \gev^2$ measured at JLab considerably increases the strength of the bounds in the region $ Q^2 \lesssim 10 \gev^2$, excluding the onset of the asymptotic perturbative QCD regime for $Q^2< 7\gev^2$. We also compare the bounds with available experimental data and with several theoretical models proposed for the low and intermediate spacelike region.

Realizing vector meson dominance with transverse charge densities

The transverse charge density in a fast-moving nucleon is represented as a dispersion integral of the imaginary part of the Dirac form factor in the timelike region (spectral function). At a given transverse distance b the integration effectively extends over energies in a range sqrt{t} ~< 1/b, with exponential suppression of larger values. The transverse charge density at peripheral distances thus acts as a low-pass filter for the spectral function and allows one to select energy regions dominated by specific t-channel states, corresponding to definite exchange mechanisms in the spacelike form factor. We show that distances b ~ 0.5 - 1.5 fm in the isovector density are maximally sensitive to the rho meson region, with only a ~10% contribution from higher-mass states. Soft-pion exchange governed by chiral dynamics becomes relevant only at larger distances. In the isoscalar density higher-mass states beyond the omega are comparatively more important. The dispersion approach suggests that the positive transverse charge density in the neutron at b ~ 1 fm, found previously in a Fourier analysis of spacelike form factor data, could serve as a sensitive test of the the isoscalar strength in the ~1 GeV mass region. In terms of partonic structure, the transverse densities in the vector meson region b ~ 1 fm support an approximate mean-field picture of the motion of valence quarks in the nucleon.

Valence and nonvalence vertex functions for timelike and spacelike nucleon form factors within light-front dynamics

A unified model for spacelike and timelike hadron form factors within light-front dynamics successfully applied to the pion is extended to the nucleon to study the role of valence and nonvalence vertex functions. Preliminary results for −10(GeV/c)2 ≤ q2 ≤ 10(GeV/c)2 are presented.

Conformal symmetry and pion form factor: Space- and timelike region

We extend a recent analysis of the pion electromagnetic form factor constrained by the conformal symmetry to explore the timelike region. We show explicitly that the timelike form factor obtained by the analytic continuation of the spacelike form factor correctly satisfies the dispersion relation. Our results indicate that the quark spin and dynamical mass effects are crucial to yield the realistic features of the vector meson dominance phenomena.

Spacelike and timelike form factors forω→πγ*andK*→Kγ*in the light-front quark model

We investigate space- and timelike form factors for $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\pi}{\ensuremath{\gamma}}^{*}$ and ${K}^{*}\ensuremath{\rightarrow}K{\ensuremath{\gamma}}^{*}$ decays using the light-front quark model constrained by the variational principle for the QCD-motivated effective Hamiltonian. The momentum dependent spacelike form factors are obtained in the ${q}^{+}=0$ frame and then analytically continued to the timelike region. Our prediction for the timelike form factor ${F}_{\ensuremath{\omega}\ensuremath{\pi}}({q}^{2})$ is in good agreement with the experimental data. We also find that the spacelike form factor ${F}_{{K}^{*\ifmmode\pm\else\textpm\fi{}}{K}^{\ifmmode\pm\else\textpm\fi{}}}({Q}^{2})$ for charged kaons encounters a zero because of the negative interference between the two currents to the quark and the antiquark.

Timelike and spacelike hadron form factors, Fock state components and light-front dynamics

A unified description of spacelike and timelike hadron form factors within a light-front model was successfully applied to the pion. The model is extended to the nucleon to study the role of q q ¯ pair production and of nonvalence components in the nucleon form factors. Preliminary results in the spacelike range 0 ≤ Q 2 ≤ 10 ( GeV / c ) 2 are presented.

The quark form factor at higher orders

We study the electromagnetic on-shell form factor of quarks in massless perturbative QCD. We derive the complete pole part in dimensional regularization at three loops, and extend the resummation of the form factor to the next-to-next-to-leading contributions. These results are employed to evaluate the infrared finite absolute ratio of the time-like and space-like form factors up to the fourth order in the strong coupling constant. Besides for the pole structure of higher-loop QCD amplitudes, our new contributions to the form factor are also relevant for the high-energy limit of massive gauge theories like QED. The highest-transcendentality component of our results confirms a result recently obtained in N=4 Super-Yang-Mills theory.

Structure of the nucleon from electromagnetic timelike form factors

Recent experimental data on space-like and time-like form factors of the nucleon are reviewed in light of a model of the nucleon with an intrinsic (quark-like) structure and a meson cloud. The analysis points to the astonishing result that the proton electric space-like form factor vanishes at Q^2~8 (GeV/c)^2 and becomes negative beyond that point. The intrinsic structure is estimated to have a r.m.s. radius of ~0.34 fm, much smaller than the proton r.m.s. radius ~0.82 fm. The calculations are in perfect agreement with space-like proton data, but deviate drastically from space-like neutron data at Q^2>1 (GeV/c)^2. Analysis of the time-like data appears to indicate excellent agreement with both proton and neutron data in the entire range of measured q^2=-Q^2 values.

π,K+,andK0electromagnetic form factors

The rainbow truncation of the quark Dyson-Schwinger equation is combined with the ladder Bethe-Salpeter equation for the meson amplitudes and the dressed quark-photon vertex in a self-consistent Poincar\'e-invariant study of the pion and kaon electromagnetic form factors in impulse approximation. We demonstrate explicitly that the current is conserved in this approach and that the obtained results are independent of the momentum partitioning in the Bethe-Salpeter amplitudes. With model gluon parameters previously fixed by the condensate, the pion mass and decay constant, and the kaon mass, the charge radii and spacelike form factors are found to be in good agreement with the experimental data.