Test of analyticity and unitarity for the pion form-factor data around the $ρ$ resonance

Abstract

High-statistics data on the e + e − → π + π − cross section and the pion vector form factor have been obtained recently by several collaborations. Unfortunately, there are some tensions between different datasets, especially the most precise ones, which have not been resolved so far. Additional independent constraints on the data are therefore of interest. We consider a parametrization-free method of analytic extrapolation proposed recently, which is based on a mixed phase and modulus extremal problem and combines rigorous upper and lower bounds with numerical simulations to account for the statistical distributions of the input and output values. Spacelike data on the form factor and measurements of the modulus in the region (0.65 − 0.71) GeV are used as input. In previous works, the formalism was applied for extrapolating the form factor to low energies. In the present work, we use it as a stringent and model-independent test of consistency with analyticity and unitarity for the high-statistics data around the ρ resonance. The study reveals some inconsistencies, in particular below the ρ peak the BABAR data are slightly higher than the band of extrapolated values, while above the ρ peak all the data are situated at the lower edge of the band. The implications of the results on the two-pion vacuuum polarization contribution to the anomalous magnetic moment of the muon are briefly discussed.

Constraints on the two-pion contribution to hadronic vacuum polarization

“…This phase shift, in turn, is strongly constrained by ππ Roy equations [50][51][52][53], which further limits the permissible changes in F V π (s), see Refs. [57][58][59][60][61][62][63][64] for representations that exploit this intimate connection between the VFF and ππ scattering. Below 1 GeV inelastic effects are small, but at the level of precision necessary here, have to be taken into account.…”

Study of pion vector form factor and its contribution to the muon $(g-2)$

“…Using a parametrization-free formalism based on analyticity and unitarity, the PVFF and its contribution to a HVP µ were investigated in Refs. [68][69][70] for the energy range around the ρ(770) resonance. As we have already known that, about 73% of the LO hadronic contribution and about 60% of the total uncertainty are given by e + e − annihilated to the π + π − (γ) final states, which are dominated by the ρ(770) resonance.…”