Isospin-breaking effects in the three-pion contribution to hadronic vacuum polarization
Martin Hoferichter, Bai-Long Hoid, Bastian Kubis, Dominic Schuh
Abstract
A bstract Isospin-breaking (IB) effects are required for an evaluation of hadronic vacuum polarization at subpercent precision. While the dominant contributions arise from the e + e − → π + π − channel, also IB in the subleading channels can become relevant for a detailed understanding, e.g., of the comparison to lattice QCD. Here, we provide such an analysis for e + e − → 3 π by extending our dispersive description of the process, including estimates of final-state radiation (FSR) and ρ – ω mixing. In particular, we develop a formalism to capture the leading infrared-enhanced effects in terms of a correction factor η 3 π that generalizes the analog treatment of virtual and final-state photons in the 2 π case. The global fit to the e + e − → 3 π data base, subject to constraints from analyticity, unitarity, and the chiral anomaly, gives $$ {\left.{a}_{\mu}^{3\pi}\right|}_{\le 1.8\ \textrm{GeV}}=45.91(53)\times {10}^{-10} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mfenced> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:msubsup> </mml:mfenced> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>1.8</mml:mn> <mml:mspace/> <mml:mi>GeV</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>45.91</mml:mn> <mml:mfenced> <mml:mn>53</mml:mn> </mml:mfenced> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:msup> </mml:math> for the total 3 π contribution to the anomalous magnetic moment of the muon, of which $$ {a}_{\mu}^{\textrm{FSR}}\left[3\pi \right]=0.51(1)\times {10}^{-10} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mi>FSR</mml:mi> </mml:msubsup> <mml:mfenced> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mn>0.51</mml:mn> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:msup> </mml:math> and $$ {a}_{\mu}^{\rho -\omega}\left[3\pi \right]=-2.68(70)\times {10}^{-10} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>−</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mn>2.68</mml:mn> <mml:mfenced> <mml:mn>70</mml:mn> </mml:mfenced> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:msup> </mml:math> can be ascribed to IB. We argue that the resulting cancellation with ρ – ω mixing in e + e − → 2 π can be understood from a narrow-resonance picture, and provide updated values for the vacuum-polarization-subtracted vector-meson parameters M ω = 782 . 70(3) MeV, M ϕ = 1019 . 21(2) MeV, Γ ω = 8 . 71(3) MeV, and Γ ϕ = 4 . 27(1) MeV.