Two-pion contribution to hadronic vacuum polarization

Abstract

We present a detailed analysis of e + e − → π + π − data up to √ s = 1 GeV in the framework of dispersion relations. Starting from a family of ππ P -wave phase shifts, as derived from a previous Roy-equation analysis of ππ scattering, we write down an extended Omnès representation of the pion vector form factor in terms of a few free parameters and study to which extent the modern high-statistics data sets can be described by the resulting fit function that follows from general principles of QCD. We find that statistically acceptable fits do become possible as soon as potential uncertainties in the energy calibration are taken into account, providing a strong cross check on the internal consistency of the data sets, but preferring a mass of the ω meson significantly lower than the current PDG average. In addition to a complete treatment of statistical and systematic errors propagated from the data, we perform a comprehensive analysis of the systematic errors in the dispersive representation and derive the consequences for the two-pion contribution to hadronic vacuum polarization. In a global fit to both time-and space-like data sets we find a ππ µ | ≤1 GeV = 495.0(1.5)(2.1) × 10 −10 and a ππ µ | ≤0.63 GeV = 132.8(0.4)(1.0) × 10 −10 . While the constraints are thus most stringent for low energies, we obtain uncertainty estimates throughout the whole energy range that should prove valuable in corroborating the corresponding contribution to the anomalous magnetic moment of the muon. As side products, we obtain improved constraints on the ππ P -wave, valuable input for future global analyses of low-energy ππ scattering, as well as a determination of the pion charge radius, r 2 π = 0.429(1)(4) fm 2 .

$${\hbox {BHLS}}_2$$, a new breaking of the HLS model and its phenomenology

“…As for < r 2 π ± >, Table 5 shows an agreement of our result with the experimental data from NA7 and Fermilab at the 1σ level or better. The result of G. Colangelo et al [114] (CHS) is derived from a global fit of the timelike and spacelike pion form factor data based on a parametrization through the Omnès representation of F π (s) taking into account isospin breaking effects (ω) and prescriptions for the asymptotics of the ππ P -wave phase shift; this estimate is in perfect agreement with our own fit result. Using an elaborate method mixing the low energy ππ phase information and some form factor modulus information, B. Ananthanarayan et al [115] recently derived the value flagged by ACD in Table 5 which is very close to ours.…”

“…are different, but more importantly for this energy range, the dipion BaBar data included in the study [114] and discarded in ours, certainly contribute to the difference. One may also compare the BHLS 2 results with others for the contribution to a HVP−LO µ (π + π − ) from the energy region (0.6 ≤ √ s ≤ 0.9) GeV.…”

“…One may also compare the BHLS 2 results with others for the contribution to a HVP−LO µ (π + π − ) from the energy region (0.6 ≤ √ s ≤ 0.9) GeV. KNT18 [71] gets (369.41 ± 1.32) × 10 −10 and CHS [114] (369.8 ± 1.3 f it ± 1.3 syst ) × 10 −10 using BaBar data at low energies. BHLS2 running with KLOE10/12, CMD-2, SND, BESIII and Cleo-c data in the π + π − channel yields (367.81 ± 0.77) × 10 −10 ; replacing the KLOE10/12 data by the KLOE85 combination [70], one gets (367.61 ± 0.64) × 10 −10 .…”

Rho resonance, timelike pion form factor, and implications for lattice studies of the hadronic vacuum polarization

“…Our results compare well to an independent calculation of the charge radius on the same ensembles, obtained from the slope of the pion form factor in terms of the spacelike momentum transfer Q 2 [63]. When lowering the pion mass, our results for r 2 π and the curvature approach the phenomenological values [67,68]. The resulting mass dependence of the squared radius is compatible with the results in Ref.…”

Contribution of neutral pseudoscalar mesons to aμHLbL within a Schwinger-Dyson equations approach to QCD

“…The most recent evaluations of the hadronic vacuum polarization contribution to a µ [27][28][29][30][31] have reduced its error, reaching the same level of uncertainty as the HLbL contribution. The latter must be diminished to fully benefit from the very precise forthcoming measurements at FNAL and J-PARC 2 .…”