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Title: Two-pion contribution to hadronic vacuum polarization

Abstract

We present a detailed analysis of e + e -π + π - data up to \( \sqrt{s}=1 \) 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, $$\langle$$$r^{2}_{π}$$$\rangle$$ = 0 .429(1)(4) fm 2.

Authors:
 [1];  [2];  [3]
  1. Univ. of Bern (Switzerland)
  2. Univ. of Washington, Seattle, WA (United States)
  3. Univ. of California, San Francisco, CA (United States)
Publication Date:
Research Org.:
Univ. of Washington, Seattle, WA (United States); Univ. of California, San Diego, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1609328
Grant/Contract Number:  
FG02-00ER41132; SC0009919
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Volume: 2019; Journal Issue: 2; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics

Citation Formats

Colangelo, Gilberto, Hoferichter, Martin, and Stoffer, Peter. Two-pion contribution to hadronic vacuum polarization. United States: N. p., 2019. Web. doi:10.1007/jhep02(2019)006.
Colangelo, Gilberto, Hoferichter, Martin, & Stoffer, Peter. Two-pion contribution to hadronic vacuum polarization. United States. doi:10.1007/jhep02(2019)006.
Colangelo, Gilberto, Hoferichter, Martin, and Stoffer, Peter. Fri . "Two-pion contribution to hadronic vacuum polarization". United States. doi:10.1007/jhep02(2019)006. https://www.osti.gov/servlets/purl/1609328.
@article{osti_1609328,
title = {Two-pion contribution to hadronic vacuum polarization},
author = {Colangelo, Gilberto and Hoferichter, Martin and Stoffer, Peter},
abstractNote = {We present a detailed analysis of e+e- → π+π- data up to \( \sqrt{s}=1 \) 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 aππμ|≤ 1 GeV = 495.0(1.5)(2.1) × 10-10 and aππμ|≤ 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, $\langle$$r^{2}_{π}$$\rangle$ = 0.429(1)(4) fm2.},
doi = {10.1007/jhep02(2019)006},
journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 2,
volume = 2019,
place = {United States},
year = {2019},
month = {2}
}

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Cited by: 16 works
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Works referenced in this record:

Remarks on higher-order hadronic corrections to the muon <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math>
journal, July 2014


Pion Form Factor
journal, July 1979


Electromagnetic form Factor of the pion
conference, November 2011


Measuring the leading hadronic contribution to the muon g-2 via $$\mu e$$ μ e scattering
journal, March 2017


Electromagnetic Charge Radius of the Pion at High Precision
journal, September 2017


ππ scattering
journal, June 2001


Towards a data-driven analysis of hadronic light-by-light scattering
journal, November 2014


Dispersion relation for hadronic light-by-light scattering: two-pion contributions
journal, April 2017

  • Colangelo, Gilberto; Hoferichter, Martin; Procura, Massimiliano
  • Journal of High Energy Physics, Vol. 2017, Issue 4
  • DOI: 10.1007/JHEP04(2017)161

Hadronic Vacuum Polarization Contribution to the Anomalous Magnetic Moments of Leptons from First Principles
journal, July 2018


Suggested Boson-Lepton Pair Couplings and the Anomalous Magnetic Moment of the Muon
journal, April 1968


Asymptotic behavior of the pion form factor in quantum chromodynamics
journal, February 1980

  • Efremov, A. V.; Radyushkin, A. V.
  • Theoretical and Mathematical Physics, Vol. 42, Issue 2
  • DOI: 10.1007/BF01032111

A measurement of the electromagnetic size of the pion from direct elastic pion scattering data at 50 GeV/c
journal, October 1977


On the pion decay constant
journal, September 2010


Model-independent approach to <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>η</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mi>γ</mml:mi></mml:math> and <mml:math altimg="si2.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mi>γ</mml:mi></mml:math>
journal, January 2012


Dispersive approach to hadronic light-by-light scattering
journal, September 2014

  • Colangelo, G.; Hoferichter, M.; Procura, M.
  • Journal of High Energy Physics, Vol. 2014, Issue 9
  • DOI: 10.1007/JHEP09(2014)091

Chiral perturbation theory: Expansions in the mass of the strange quark
journal, January 1985


Rescattering Effects in the Hadronic-Light-by-Light Contribution to the Anomalous Magnetic Moment of the Muon
journal, June 2017


On the solution of certain singular integral equations of quantum field theory
journal, April 1958


Roy–Steiner equations for γγ→ππ
journal, September 2011


Precision calculation of the <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mi>d</mml:mi></mml:math> scattering length and its impact on threshold πN scattering
journal, January 2011


Review of Particle Physics
journal, August 2018


The decay of a light Higgs boson
journal, October 1990


CODATA recommended values of the fundamental physical constants: 2014
journal, September 2016


A generalization of the Watson theorem
journal, October 1973


Anomalous decay and scattering processes of the $$\eta $$ η meson
journal, June 2015


Charge symmetry, quarks and mesons
journal, October 1990


Exclusive processes in quantum chromodynamics: Evolution equations for hadronic wavefunctions and the form factors of mesons
journal, November 1979


Chiral perturbation theory to one loop
journal, November 1984


The Interpretation of Low Energy Proton-Proton Scattering
journal, January 1950


Pion-Pole Contribution to Hadronic Light-By-Light Scattering in the Anomalous Magnetic Moment of the Muon
journal, September 2018


Form factors of the isovector scalar current and the $$\eta \pi $$ η π scattering phase shifts
journal, October 2015


The hadronic vacuum polarization contribution to the muon g − 2 from lattice QCD
journal, October 2017

  • Morte, M. Della; Francis, A.; Gülpers, V.
  • Journal of High Energy Physics, Vol. 2017, Issue 10
  • DOI: 10.1007/JHEP10(2017)020

Dispersive analysis of the scalar form factor of the nucleon
journal, June 2012

  • Hoferichter, M.; Ditsche, C.; Kubis, B.
  • Journal of High Energy Physics, Vol. 2012, Issue 6
  • DOI: 10.1007/JHEP06(2012)063

Dispersive analysis of the pion transition form factor
journal, November 2014


Some General Relations between the Photoproduction and Scattering of π Mesons
journal, July 1954


Dispersive analysis of $$\eta \rightarrow 3 \pi $$ η → 3 π
journal, November 2018


ρ 0–γ mixing in the neutral channel pion form factor $F^{(e)}_{\pi}(s)$ and its role in comparing e + e − with τ spectral functions
journal, May 2011


Dispersion relation for hadronic light-by-light scattering: theoretical foundations
journal, September 2015

  • Colangelo, Gilberto; Hoferichter, Martin; Procura, Massimiliano
  • Journal of High Energy Physics, Vol. 2015, Issue 9
  • DOI: 10.1007/JHEP09(2015)074

A dispersive treatment of $$K_{\ell 4}$$ K ℓ 4 decays
journal, April 2015


Dispersion relation for hadronic light-by-light scattering: pion pole
journal, October 2018

  • Hoferichter, Martin; Hoid, Bai-Long; Kubis, Bastian
  • Journal of High Energy Physics, Vol. 2018, Issue 10
  • DOI: 10.1007/JHEP10(2018)141

Dispersive analysis of ω→3π and ϕ→3π decays
journal, May 2012


Regge analysis of the ππ scattering amplitude
journal, February 2012


Muon $$g-2$$ g - 2 estimates: can one trust effective Lagrangians and global fits?
journal, December 2015


Finite-Width Corrections to the Vector-Meson-Dominance Prediction for ρ e + e
journal, July 1968


Roy–Steiner-equation analysis of pion–nucleon scattering
journal, April 2016


Unified dispersive approach to real and virtual photon-photon scattering at low energy
journal, September 2013


Extraction of resonance parameter
conference, January 2012

  • Sato, T.
  • THE 8TH INTERNATIONAL WORKSHOP ON THE PHYSICS OF EXCITED NUCLEONS: NSTAR 2011, AIP Conference Proceedings
  • DOI: 10.1063/1.3701187

Precision muon physics
journal, September 2015


Dispersive analysis of the decay η → 3π
journal, May 1996


Radiative Corrections to π e Scattering
journal, August 1964


La résonance dans la diffusion méson π— méson π et le moment magnétique anormal du méson μ
journal, January 1961


A new parametrization for the scalar pion form factors
journal, December 2018


Pion Scattering Lengths
journal, September 1966


Dispersion relations for $$\eta '\rightarrow \eta \pi \pi $$ η ′ → η π π
journal, July 2017


Hadronic contributions to αμ below one GeV
journal, April 2004


Muon g – 2 theory: The hadronic part
journal, January 2018


Elastic-Scattering Measurement of the Negative-Pion Radius
journal, February 1982


Precision calculation of threshold scattering, πN scattering lengths, and the GMO sum rule
journal, December 2011


Virtual Photon–Photon Scattering
journal, January 2014

  • Hoferichter, Martin; Colangelo, Gilberto; Procura, Massimiliano
  • International Journal of Modern Physics: Conference Series, Vol. 35
  • DOI: 10.1142/S2010194514604001

The muon g−2
journal, June 2009


Higher order hadronic corrections to the anomalous magnetic moment of the muon
journal, March 1976


Charge symmetry breaking in pion-deuteron scattering
conference, July 2013

  • Hoferichter, Martin; Baru, Vadim; Hanhart, Christoph
  • Proceedings of The 7th International Workshop on Chiral Dynamics — PoS(CD12)
  • DOI: 10.22323/1.172.0093

On the $\pi\pi$ π π continuum in the nucleon form factors and the proton radius puzzle
journal, November 2016

  • Hoferichter, M.; Kubis, B.; Ruiz de Elvira, J.
  • The European Physical Journal A, Vol. 52, Issue 11
  • DOI: 10.1140/epja/i2016-16331-7

Factorization and asymptotic behaviour of pion form factor in QCD
journal, July 1980


Phenomenological Lagrangians
journal, April 1979


On the use of the covariance matrix to fit correlated data
journal, July 1994

  • D'Agostini, G.
  • Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 346, Issue 1-2
  • DOI: 10.1016/0168-9002(94)90719-6

Roy-Steiner equations for pion-nucleon scattering
journal, June 2012

  • Ditsche, C.; Hoferichter, M.; Kubis, B.
  • Journal of High Energy Physics, Vol. 2012, Issue 6
  • DOI: 10.1007/JHEP06(2012)043

Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order
journal, June 2014


A new parameterization for the pion vector form factor
journal, August 2012


    Works referencing / citing this record:

    b → sℓ+ℓ− transitions in two-Higgs-doublet models
    journal, June 2019

    • Crivellin, Andreas; Müller, Dario; Wiegand, Christoph
    • Journal of High Energy Physics, Vol. 2019, Issue 6
    • DOI: 10.1007/jhep06(2019)119

    Rho resonance, timelike pion form factor, and implications for lattice studies of the hadronic vacuum polarization
    journal, March 2020