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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 protons 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 Natures most fundamental Nambu-Goldstone bosons.
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The pion distribution amplitude (DA) can be related to the fundamental QCD Greens functions as a function of the quark self-energy and the quark-pion vertex, which in turn are associated with the pion wave function through the Bethe-Salpeter equation. Considering the extreme hard asymptotic behavior in momentum space allowed for a pseudoscalar wave function, which is limited by its normalization condition, we compute the pion DA and its second moment. From the resulting amplitude, representing the field theoretical upper limit on the DA behavior, we calculate the photon-pion transition form factor $F_{pigammagamma^{ast}}(Q^{2})$. The resulting upper limit on the pion transition form factor is compared with existing data published by CLEO, BaBar and Belle collaborations.
In the light of recent experimental progress in determining the pressure and shear distributions in the proton, these quantities are calculated in a model with confined quarks supplemented by the pion field required by chiral symmetry. The incorporation of the pion contributions is shown to account for the long-range distributions, in general agreement with the experimentally extracted quark contributions. The results of the model are also compared with lattice QCD results at unphysically large quark mass.
The $pi^0$ pole constitutes the lowest-lying singularity of the hadronic light-by-light (HLbL) tensor, and thus provides the leading contribution in a dispersive approach to HLbL scattering in the anomalous magnetic moment of the muon $(g-2)_mu$. It is unambiguously defined in terms of the doubly-virtual pion transition form factor, which in principle can be accessed in its entirety by experiment. We demonstrate that, in the absence of a direct measurement, the full space-like doubly-virtual form factor can be reconstructed very accurately based on existing data for $e^+e^-to 3pi$, $e^+e^-to e^+e^-pi^0$, and the $pi^0togammagamma$ decay width. We derive a representation that incorporates all the low-lying singularities of the form factor, matches correctly onto the asymptotic behavior expected from perturbative QCD, and is suitable for the evaluation of the $(g-2)_mu$ loop integral. The resulting value, $a_mu^{pi^0text{-pole}}=62.6^{+3.0}_{-2.5}times 10^{-11}$, for the first time, represents a complete data-driven determination of the pion-pole contribution with fully controlled uncertainty estimates. In particular, we show that already improved singly-virtual measurements alone would allow one to further reduce the uncertainty in $a_mu^{pi^0text{-pole}}$.
We reconsider QCD factorization for the leading power contribution to the $gamma^{ast} gamma to pi^0$ form factor $F_{gamma^{ast} gamma to pi^0} (Q^2)$ at one loop using the evanescent operator approach, and demonstrate the equivalence of the resulting factorization formulae derived with distinct prescriptions of $gamma_5$ in dimensional regularization. Applying the light-cone QCD sum rules (LCSRs) with photon distribution amplitudes (DAs) we further compute the subleading power contribution to the pion-photon form factor induced by the hadronic component of the real photon at the next-to-leading-order in ${cal O}(alpha_s)$, with both naive dimensional regularization and t Hooft-Veltman schemes of $gamma_5$. Confronting our theoretical predictions of $F_{gamma^{ast} gamma to pi^0} (Q^2)$ with the experimental measurements from the BaBar and the Belle Collaborations implies that a reasonable agreement can be achieved without introducing an exotic end-point behaviour for the twist-2 pion DA.
With the aim of extracting the pion charge radius, we analyse extant precise pion+electron elastic scattering data on $Q^2 in [0.015,0.144],$GeV$^2$ using a method based on interpolation via continued fractions augmented by statistical sampling. The scheme avoids any assumptions on the form of function used for the representation of data and subsequent extrapolation onto $Q^2simeq 0$. Combining results obtained from the two available data sets, we obtain $r_pi = 0.640(7),$fm, a value $2.4,sigma$ below todays commonly quoted average. The tension may be relieved by collection and similar analysis of new precise data that densely cover a domain which reaches well below $Q^2 = 0.015,$GeV$^2$. Considering available kaon+electron elastic scattering data sets, our analysis reveals that they contain insufficient information to extract an objective result for the charged-kaon radius, $r_K$. New data with much improved precision, low-$Q^2$ reach and coverage are necessary before a sound result for $r_K$ can be recorded.
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