Proton electromagnetic form factors: Basic notions, present achievements and future perspectives

Hadron tomography can be investigated by three-dimensional structure functions such as generalized parton distributions (GPDs), transverse-momentum-dependent parton distributions, and generalized distribution amplitudes (GDAs). Here, we extract the GDAs, which are $s\text{\ensuremath{-}}t$ crossed quantities of the GPDs, from cross-section measurements of hadron-pair production process ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ at KEKB. This work is the first attempt to obtain the GDAs from the actual experimental data. The GDAs are expressed by a number of parameters and they are determined from the data of ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ by including intermediate scalar- and tensor-meson contributions to the cross section. Our results indicate that the dependence of parton-momentum fraction $z$ in the GDAs is close to the asymptotic one. The timelike gravitational form factors ${\mathrm{\ensuremath{\Theta}}}_{1}$ and ${\mathrm{\ensuremath{\Theta}}}_{2}$ are obtained from our GDAs, and they are converted to the spacelike ones by the dispersion relation. From the spacelike ${\mathrm{\ensuremath{\Theta}}}_{1}$ and ${\mathrm{\ensuremath{\Theta}}}_{2}$, gravitational densities of the pion are calculated. Then, we obtained the mass (energy) radius and the mechanical (pressure and shear force) radius from ${\mathrm{\ensuremath{\Theta}}}_{2}$ and ${\mathrm{\ensuremath{\Theta}}}_{1}$, respectively. They are calculated as $\sqrt{⟨{r}^{2}{⟩}_{\mathrm{mass}}}=0.32--0.39\text{ }\text{ }\mathrm{fm}$, whereas the mechanical radius is larger $\sqrt{⟨{r}^{2}{⟩}_{\mathrm{mech}}}=0.82--0.88\text{ }\text{ }\mathrm{fm}$. This is the first report on the gravitational radius of a hadron from actual experimental measurements. It is interesting to find the possibility that the gravitational mass and mechanical radii could be different from the experimental charge radius $\sqrt{⟨{r}^{2}{⟩}_{\text{charge}}}=0.672\ifmmode\pm\else\textpm\fi{}0.008\text{ }\text{ }\mathrm{fm}$ for the charged pion. For drawing a clear conclusion on the GDAs of hadrons, accurate experimental data are needed, and it should be possible, for example, by future measurements of super-KEKB and international linear collider. Accurate measurements will not only provide important information on hadron tomography but also possibly shed light on gravitational physics in the quark and gluon level.

Hadron tomography by generalized distribution amplitudes in the pion-pair production process γ * γ → π 0 π 0 and gravitational form factors for pion

We show that the nonperturbative light-front dynamics of relativistic hadronic bound states has a dual semiclassical gravity description on a higher dimensional warped AdS space in the limit of zero quark masses. This mapping of AdS gravity theory to the boundary quantum field theory, quantized at fixed light-front time, allows one to establish a precise relation between holographic wave functions in AdS space and the light-front wavefunctions describing the internal structure of hadrons. The resulting AdS/QCD model gives a remarkably good accounting of the spectrum, elastic and transition form factors of the light-quark hadrons in terms of one parameter, the QCD gap scale. The light-front holographic approach described here thus provides a frame-independent first approximation to the light-front Hamiltonian problem for QCD. This article is based on lectures at the Niccolo Cabeo International School of Hadronic Physics, Ferrara, Italy, May 2011.

Hadronic Form Factor Models and Spectroscopy Within the Gauge/Gravity Correspondence

The description of the VV'P form factors (V,V' stands for vector particles and P for a pseudoscalar meson) for different particles virtualities remains a challenge for the theory of strong interactions While their chiral limit is well understood, recent measurements of the gamma^* omega pi^0 and gamma^* gamma pi^0 form factors at high photon virtualities seem to depart from the simplest scaling behavior suggested by QCD Here we attempt to describe them in their whole measured energy regimes within the Resonance Chiral Theory, a framework which naturally incorporates the chiral limit constraints and extends to higher energies by including the resonances as active fields 
Specifically, we obtained an accurate description of the data up to 2 GeV on the former form factor by including three multiplets of vector resonances Good agreement with measurements of the latter was possible even in the single resonance approximation, although we propose to measure the e^+e^- to mu^+mu^-pi^0 cross-section and di-muon invariant mass distribution to better characterize this form factor We have then evaluated the pion exchange contribution to the muon g-2 obtaining (666\pm021)x10^{-10} with an accurate determination of the errors We have also recalled that approximating the whole pion exchange by the pion pole contribution underestimates the corresponding result for the anomaly (by (15,20)%) Based on these results, we have predicted the eta^(\prime) transition form factors obtaining good agreement with data and obtained their respective contributions to the muon anomaly In this way, the contribution of the three lightest pseudoscalars to it yields (1047\pm054)x10^{-10}, in agreement with previous evaluations but with smaller error

$VV^\prime P$ form factors in resonance chiral theory and the $\pi-\eta-\eta^\prime$ light-by-light contribution to the muon $g-2$

The form factors for the transitions πγ, ηγ, η′γ and η c γ are analyzed within the modified perturbative approach in which quark transverse degrees of freedom are retained. The results for the form factors are compared to experiment in detail. As compared to previous calculations within the same approach only little modifications of the meson distribution amplitudes are required in general in order to achieve reasonable agreement with experiment. Only for the πγ form factor a strong contribution from the second Gegenbauer term is found. It also commented on the case of two virtual photons and on the transition form factors in the time-like region.

The form factors for the photon to pseudoscalar meson transitions—an update

Axial anomaly as a collective effect of meson spectrum

Transition form factors of ${\ensuremath{\pi}}^{0}$, $\ensuremath{\eta}$, ${\ensuremath{\eta}}^{\ensuremath{'}}$ mesons are investigated by means of the anomaly sum rule---an exact nonperturbative relation which follows from the dispersive representation of the axial anomaly. Considering the problem of contributions of operators originated from a non-(local) operator product expansion, we found that they are required by the available set of experimental data, including the most recent data from the Belle Collaboration (which, if taken alone, can be described without such contributions, although are compatible with them). In this approach, we analyzed the experimental data on $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ meson transition form factors and obtained the constraints on the decay constants and mixing parameters.

Transition form factors and mixing of pseudoscalar mesons from anomaly sum rule

The relation for transition form factors of $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ is obtained by combining the exact nonperturbative QCD sum rule, following from the dispersive representation of axial anomaly, and quark-hadron duality. It is valid at all virtual photon momenta and allows one to express the transition form factors entirely in terms of meson decay constants. This relation is in a good agreement with experimental data.

Axial anomaly and mixing: From real to highly virtual photons

A connection between the axial anomaly and the vector meson dominance model is revealed for the processes of pseudoscalar meson-photon transitions. The analytic continuation of the anomaly sum rules, which are exact QCD relations following follow from the dispersive representation of the axial anomaly, to the time-like region is performed. Using these sum rules, the transition form factors of π0, η, and η′ mesons in this region are calculated. A good agreement with the available experimental data is found.

Axial Anomaly and Vector Meson Dominance Model

The pion-photon transition form factor is studied by employing two types of sum rules: light cone sum rules (LCSR) and anomaly sum rules (ASR). By comparing the predictions for the pion-photon transition form factor, obtained from these two approaches, the applicability limit of the LCSRs at low momenta is determined. Reciprocally, the ASR threshold dependence on the momentum was extracted using our LCSR-based method in combination with two different types of pion distribution amplitudes and found that at higher ${Q}^{2}$ it approaches a constant.

Armen Oganesian

Papers

Axial anomaly as a collective effect of meson spectrum

Transition form factors and mixing of pseudoscalar mesons from anomaly sum rule

Axial anomaly and mixing: From real to highly virtual photons

Axial Anomaly and Vector Meson Dominance Model

Matching lightcone- and anomaly-sum-rule predictions for the pion-photon transition form factor