Dispersive analysis of the experimental data on the electromagnetic form factor of charged pions at spacelike momenta
Silvano Simula, Ludovico Vittorio
Abstract
The experimental data on the electromagnetic (em) form factor of charged pions available at spacelike momenta are analyzed using the dispersive matrix (DM) approach [M. Di Carlo , Unitarity bounds for semileptonic decays in lattice QCD, Phys. Rev. D 104, 054502 (2021)] which describes the momentum dependence of hadronic form factors without introducing any explicit parametrization and includes properly the constraints coming from unitarity and analyticity. The unitary bound is evaluated nonperturbatively making use of the results of lattice QCD simulations of suitable two-point correlation functions contributing to the hadronic vacuum polarization term of the muon. Thanks to the DM method we determine the pion charge radius from existing spacelike data in a completely model-independent way and consistently with the unitary bound, obtaining $⟨{r}_{\ensuremath{\pi}}{⟩}_{\mathrm{DM}}=0.703\ifmmode\pm\else\textpm\fi{}0.027\text{ }\text{ }\mathrm{fm}$. This finding differs by $\ensuremath{\simeq}1.6$ standard deviations from the latest PDG [R. L. Workman et al., Review of particle physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).] value $⟨{r}_{\ensuremath{\pi}}{⟩}_{\mathrm{PDG}}=0.659\ifmmode\pm\else\textpm\fi{}0.004\text{ }\text{ }\mathrm{fm}$, which is dominated by the very precise results of dispersive analyses of timelike data coming from measurements of the cross section of the ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ process. We have analyzed the spacelike data using also traditional $z$-expansions, like the Boyd-Grinstein-Lebed (BGL) or Bourrely-Caprini-Lellouch (BCL) fitting functions and adopting a simple procedure that incorporates ab initio the nonperturbative unitary bound in the fitting process. We get $⟨{r}_{\ensuremath{\pi}}{⟩}_{\mathrm{BGL}}=0.711\ifmmode\pm\else\textpm\fi{}0.039\text{ }\text{ }\mathrm{fm}$ and $⟨{r}_{\ensuremath{\pi}}{⟩}_{\mathrm{BCL}}=0.709\ifmmode\pm\else\textpm\fi{}0.028\text{ }\text{ }\mathrm{fm}$ in nice agreement with the DM result. A detailed comparison in a wide range of spacelike momenta between the results of the BGL/BCL fitting procedures and those of the DM method indicates that unitarity must be imposed not only on the fitting function but also on the input data. We have addressed also the issue of the onset of perturbative QCD (pQCD) by performing a sensitivity study of the pion form factor at large spacelike momenta, based only on experimental spacelike data and unitarity. Hence, although the leading pQCD behavior is found to set in only at very large momenta, our DM bands may provide information about the preasymptotic effects related to the scale dependence of the pion distribution amplitude.