New $\tau$-based evaluation of the hadronic contribution to the vacuum polarization piece of the muon anomalous magnetic moment

We revisit the isospin-breaking and electromagnetic corrections to the decay ${\tau }^{-}\to {\pi }^{-}{\pi }^0{\nu }_{\tau }$, which allow its use as input in the two-pion contribution to the (leading-order) hadronic vacuum polarization part of the muon anomalous magnetic moment. We extend a previous resonance chiral Lagrangian analysis, which included those operators saturating the next-to-leading-order chiral low-energy constants, by including the contributions appearing at the next order. As a result, we improve agreement between the two-pion tau decay and $e^{+}{e}^{-}$ data and reduce the discrepancy between experiment and the Standard Model prediction of $a_{\mu }$ (using $\tau$ input) to the approximately $2\sigma$ level.

Figure 1

One-resonance exchange contributions from the $R\chi T$ to the vector form factors of the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays.

Figure 2

Two-resonance exchange contributions from the $R\chi T$ to the vector form factors of the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays.

Figure 3

Three-resonance exchange contributions from the $R\chi T$ to the vector form factors of the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays.

Figure 4

Anomalous diagrams contributing to the axial tensor amplitude $A^{\mu \nu }$ at $\mathcal{O}(p^4)$.

Figure 5

One-resonance exchange contributions from the $R\chi T$ to the axial-vector form factors of the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays.

Figure 6

Two-resonance exchange contributions from the $R\chi T$ to the axial-vector form factors of the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays.

Figure 7

Three-resonance exchange contributions from the $R\chi T$ to the axial-vector form factors of the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays.

Figure 8

Correction function $G_{\mathrm{EM}}^{(0)}(s)$ in Eq. (51a) (blue dashed line). The solid line shows the $G_{\mathrm{EM}}(s)$ function neglecting the structure-dependent part (SI), i.e., by taking $v_1=v_2=v_3=v_4=a_1=a_2=a_3=a_4=0$; the dashed and dotted lines are the $\mathcal{O}(p^4)$ $G_{\mathrm{EM}}(s)$ function (with either $F_V=\sqrt{2}F$ or $F_V=\sqrt{3}F$ constraints). The blue shaded region is the full $\mathcal{O}(p^6)$ contribution, including (overestimated) uncertainties. The left-hand side plot corresponds to the dispersive parametrization [124], while the right-hand side corresponds to the Guerrero-Pich parametrization [143] of the form factor (the latter was used in Ref. [62]).

Figure 9

Modulus and phase of the pion vector form factor, $f_{+}(s)$. The solid line corresponds to the dispersive representation used in Ref. [124], while the dashed line corresponds to the Guerrero-Pich parametrization [143] employed by Ref. [62].

Figure 10

The ${\pi }^{-}{\pi }^0$ hadronic invariant mass distribution for the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays for $E_{\gamma }^{\mathrm{cut}}=100\mathrm{MeV}$ (black), $E_{\gamma }^{\mathrm{cut}}=200\mathrm{MeV}$ (brown), $E_{\gamma }^{\mathrm{cut}}=300\mathrm{MeV}$ (blue), $E_{\gamma }^{\mathrm{cut}}=400\mathrm{MeV}$ (gray), and $E_{\gamma }^{\mathrm{cut}}=500\mathrm{MeV}$ (red) using only the bremsstrahlung (SI) contribution.

Figure 11

The ${\pi }^{-}{\pi }^0$ hadronic invariant mass distributions for $E_{\gamma }^{\mathrm{cut}}=300\mathrm{MeV}$. The solid and dashed lines represent the $\mathcal{O}(p^4)$ corrections using $F_V=\sqrt{3}F$ and $F_V=\sqrt{2}F$, respectively. The dotted line stands for the bremsstrahlung contribution (SI).

Figure 12

The ${\pi }^{-}{\pi }^0$ hadronic invariant mass distributions for $E_{\gamma }^{\mathrm{cut}}=300\mathrm{MeV}$. The solid and dashed lines represent the $\mathcal{O}(p^4)$ corrections using $F_V=\sqrt{3}F$ and $F_V=\sqrt{2}F$, respectively. The dotted line represents the bremsstrahlung contribution (SI). The dot-dashed red line corresponds to using only SD constraints at $\mathcal{O}(p^6)$, and the blue shaded region overestimates the corresponding uncertainties.

Figure 13

Photon energy distribution for the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays normalized with the nonradiative decay width. The dotted line represents the bremsstrahlung contribution. The solid and dashed lines represent the $\mathcal{O}(p^4)$ corrections using $F_V=\sqrt{3}F$ and $F_V=\sqrt{2}F$, respectively. The dot-dashed red line corresponds to using only SD constraints at $\mathcal{O}(p^6)$ (with overestimated uncertainties in the blue shaded area).

Figure 14

Branching ratio for the ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decays as a function of $E_{\gamma }^{\mathrm{cut}}$. The dotted line represents the bremsstrahlung contribution, and the solid line and dashed lines represent the $\mathcal{O}(p^4)$ corrections using $F_V=\sqrt{3}F$ and $F_V=\sqrt{2}F$, respectively. The dot-dashed red line is the $\mathcal{O}(p^6)$ contribution using only SD constraints and neglecting all other couplings. The blue shaded region overestimates the $\mathcal{O}(p^6)$ uncertainties.

Figure 15

Predictions for the branching ratio at $\mathcal{O}(p^6)$ for a sample of 1000 points, with $E_{\mathrm{cut}}=100$, 300, and 500 MeV from top to bottom.

Figure 16

Ratio of the form factors (FF1) for ${\theta }_{\rho \omega }=(-3.5\pm 0.7)\times {10}^{-3}{\mathrm{GeV}}^2$. The solid line represents the mean value.

Figure 17

Full IB correction factor $R_{\mathrm{IB}}(s)$ for the different orders of approximation in $G_{\mathrm{EM}}(s)$ using the central values given in (FF1). The blue region corresponds to the (overestimated) corrections at $\mathcal{O}(p^6)$ in $G_{\mathrm{EM}}(s)$.

Figure 18

Comparison between the different data sets from BABAR (above) and KLOE (below) with $\mathrm{\Delta }{\mathrm{\Gamma }}_{\pi \pi \gamma }=1.5\mathrm{MeV}$ (left) and $\mathrm{\Delta }{\mathrm{\Gamma }}_{\pi \pi \gamma }=0.45\mathrm{MeV}$ (right) for FF1 and FF2, respectively. The blue region corresponds to the experimental error on ${\sigma }_{\pi \pi (\gamma )}$. The solid and dashed lines represent the contributions with $F_V=\sqrt{3}F$ and $F_V=\sqrt{2}F$ at $\mathcal{O}(p^4)$, respectively. The dotted line is the SI contribution. The red line depicts the envelope of $G_{\mathrm{EM}}(s)$ at $\mathcal{O}(p^6)$, which overestimates the uncertainty at this order. The blue dot-dashed line is the $\mathcal{O}(p^6)$ contribution using only SD constraints.

Figure 19

The ${\tau }^{-}\to {\pi }^{-}{\pi }^0\gamma {\nu }_{\tau }$ decay in the $\tau$-lepton rest frame.

Figure 20

Projection of the kinematic region for the nonradiative decay ${\mathcal{R}}^{III}$ (gray) and the radiative decay ${\mathcal{R}}^{IV}={\mathcal{R}}^{IV/III}\cup {\mathcal{R}}^{III}$ (black and gray) onto the $su$-plane. ${\mathcal{R}}^{IV/III}$ (black) is the kinematic region, which is only accessible to the radiative decay.