The γπ → ππ anomaly from lattice QCD and dispersion relations

Stoffer

2019

Self Cite

422

286

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 .

Section: Radiative Channelsmentioning

confidence: 99%

Two-pion contribution to hadronic vacuum polarization

Colangelo

Stoffer

2019

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422

286

“…and f 1 denotes the P -wave projection of the γπ → ππ amplitude, see refs. [103][104][105][106][107] for detailed discussions of these amplitudes. To map this imaginary part onto Im ω , we first write the full pion VFF in the approximation…”

Section: Radiative Channelsmentioning

confidence: 99%

Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization

Colangelo

Kubis³

et al. 2022

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Isospin-breaking (IB) effects in the two-pion contribution to hadronic vacuum polarization (HVP) can be resonantly enhanced, if related to the interference of the ρ(770) and ω(782) resonances. This particular IB contribution to the pion vector form factor and thus the line shape in e+e−→ π+π− can be described by the residue at the ω pole — the ρ-ω mixing parameter ϵω. Here, we argue that while in general analyticity requires this parameter to be real, the radiative channels π0γ, ππγ, ηγ can induce a small phase, whose size we estimate as δϵ = 3.5(1.0)° by using a narrow-width approximation for the intermediate-state vector mesons. We then perform fits to the e+e−→ π+π− data base and study the consequences for the two-pion HVP contribution to the anomalous magnetic moment of the muon, its IB part due to ρ-ω mixing, and the mass of the ω resonance. We find that the global fit does prefer a non-vanishing value of δϵ = 4.5(1.2)°, close to the narrow-resonance expectation, but with a large spread among the data sets, indicating systematic differences in the ρ-ω region.

“…a low-energy theorem that could be tested with future lattice-QCD calculations [113][114][115]. The resonant contributions are described by taking the imaginary part from…”

Section: Introductionmentioning

confidence: 99%

Three-pion contribution to hadronic vacuum polarization

Hoid

Kubis

2019

362

220

We address the contribution of the 3π channel to hadronic vacuum polarization (HVP) using a dispersive representation of the e + e − → 3π amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon (g − 2) µ , both to its absolute value and uncertainty. It is largely dominated by the narrow resonances ω and φ, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for (g − 2) µ an analysis of the available data as model independent as possible becomes critical. Here, we provide such an analysis based on a global fit function using analyticity and unitarity of the underlying γ * → 3π amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various e + e − → 3π data sets. Overall, we obtain a 3π µ | ≤1.8 GeV = 46.2(6)(6) × 10 −10 as our best estimate for the total 3π contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the 2π channel below 1 GeV, this covers nearly 80% of the total HVP contribution, leading to a HVP µ = 692.3(3.3) × 10 −10 when the remainder is taken from the literature, and thus reaffirming the (g−2) µ anomaly at the level of at least 3.4σ. As side products, we find for the vacuum-polarization-subtracted masses M ω = 782.63(3)(1) MeV and M φ = 1019.20(2)(1) MeV, confirming the tension to the ω mass as extracted from the 2π channel.