Muon $g-2$ and $\alpha (M_Z^2)$: A new data-based analysis

This work presents a complete reevaluation of the hadronic vacuum polarization contributions to the anomalous magnetic moment of the muon, $a_{\mu }^{\mathrm{had},\mathrm{VP}}$, and the hadronic contributions to the effective QED coupling at the mass of the $Z$ boson, $\mathrm{\Delta }{\alpha }_{\mathrm{had}}(M_Z^2)$, from the combination of $e^{+}{e}^{-}\to \text{hadrons}$ cross section data. Focus has been placed on the development of a new data combination method, which fully incorporates all correlated statistical and systematic uncertainties in a bias free approach. All available $e^{+}{e}^{-}\to \text{hadrons}$ cross section data have been analyzed and included, where the new data compilation has yielded the full hadronic $R$-ratio and its covariance matrix in the energy range $m_{\pi }\le \sqrt{s}\le 11.2\mathrm{GeV}$. Using these combined data and perturbative QCD above that range results in estimates of the hadronic vacuum polarization contributions to $g-2$ of the muon of $a_{\mu }^{\mathrm{had},\mathrm{LO}\mathrm{VP}}=(693.26\pm 2.46)\times {10}^{-10}$ and $a_{\mu }^{\mathrm{had},\mathrm{NLO}\mathrm{VP}}=(-9.82\pm 0.04)\times {10}^{-10}$. The new estimate for the Standard Model prediction is found to be $a_{\mu }^{\mathrm{SM}}=(11659182.04\pm 3.56)\times {10}^{-10}$, which is $3.7\sigma$ below the current experimental measurement. The prediction for the five-flavor hadronic contribution to the QED coupling at the $Z$ boson mass is $\mathrm{\Delta }{\alpha }_{\mathrm{had}}^{(5)}(M_Z^2)=(276.11\pm 1.11)\times {10}^{-4}$, resulting in ${\alpha }^{-1}(M_Z^2)=128.946\pm 0.015$. Detailed comparisons with results from similar related works are given.

Figure 1

The effect of final state radiation in the $K^{+}{K}^{-}$ channel in the $\varphi$ resonance region. Left panel: the fully inclusive FSR correction $\eta (s)$. Right panel: hard real radiation ${\eta }^{\mathrm{hard},\mathrm{real}}(s)$, estimated with accolinearity cuts used in the two analyses [25, 26]. The $e^{+}{e}^{-}\to K^{+}{K}^{-}$ cross section is also plotted for reference.

Figure 2

The behavior of the inclusive FSR correction, $\eta (s)$, for the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$. The $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ cross section is also plotted for reference.

Figure 3

The differences observed using linear, quadratic, and cubic integration routines in prominent resonance regions in the ${\pi }^{+}{\pi }^{-}$ and $K^{+}{K}^{-}$ channels.

Figure 4

The comparison of the integration of the individual radiative return measurements and the combination of direct scan ${\pi }^{+}{\pi }^{-}$ measurements between $0.6\le \sqrt{s}\le 0.9\mathrm{GeV}$.

Figure 5

Contributing data in the $\rho$ resonance region of the ${\pi }^{+}{\pi }^{-}$ channel plotted against the new fit of all data (left panel), with an enlargement of the $\rho -\omega$ interference region (right panel).

Figure 6

The relative difference of the radiative return and important direct scan data sets contributing to $a_{\mu }^{{\pi }^{+}{\pi }^{-}}$ and the fit of all data. For comparison, the individual sets have been normalized against the fit and have been plotted in the $\rho$ region. The light green band represents the BABAR data and their errors (statistical and systematic, added in quadrature). The yellow band represents the full data combination which incorporates all correlated statistical and systematic uncertainties. However, the width of the yellow band simply displays the square root of the diagonal elements of the total output covariance matrix of the fit.

Figure 7

The effect of the local ${\chi }^2$ inflation and the overall global ${\chi }_{\min }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ in the ${\pi }^{+}{\pi }^{-}$ channel, which is plotted against the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ cross section for reference.

Figure 8

The cross section ${\sigma }^0(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0)$ in the range $0.66\le \sqrt{s}\le 1.937\mathrm{GeV}$, where the prominent $\omega$ and $\varphi$ resonances are clearly visible.

Figure 9

Enlargements of the resonance regions in the ${\pi }^{+}{\pi }^{-}{\pi }^0$ final state.

Figure 10

The cross section ${\sigma }^0(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-})$ in the range $0.6125\le \sqrt{s}\le 1.937\mathrm{GeV}$.

Figure 11

The cross section ${\sigma }^0(e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0)$ in the range $0.850\le \sqrt{s}\le 1.937\mathrm{GeV}$.

Figure 12

The cross section ${\sigma }^0(e^{+}{e}^{-}\to K^{+}{K}^{-})$ in the range $0.9875\le \sqrt{s}\le 1.937\mathrm{GeV}$ and an enlargement of the $\varphi$ resonance. The large influence of the BABAR data (black squares) overwhelms the older data.

Figure 13

The cross section ${\sigma }^0(e^{+}{e}^{-}\to K_S^0{K}_L^0)$ with an enlargement of the $\varphi$ resonance.

Figure 14

The measured cross section ${\sigma }^0(KK\pi )$ compared to the estimate from the previously used isospin relation.

Figure 15

The cross section ${\sigma }^0(KK\pi \pi )$ compared to the previous estimate using isospin relations.

Figure 16

The fit of inclusive $R$ data in the region of the newly included BABAR $R_b$ data, where the BABAR $R_b$ is shown as light blue markers. The resonance structures of the $\mathrm{\Upsilon }(5S)$ and $\mathrm{\Upsilon }(6S)$ states are clearly visible.

Figure 17

The combination of inclusive $R$ data in the region $1.937\le \sqrt{s}\le 4.50\mathrm{GeV}$. For comparison, the fit and the contributing data are plotted against the estimate of pQCD, represented by the dashed line and grey band.

Figure 18

Compilation of inclusive data in range $1.937\le \sqrt{s}\le 11.2\mathrm{GeV}$. The dashed line and surrounding grey band shows the estimate from pQCD for comparison. The yellow band represents the total uncertainty of the inclusive data combination.

Figure 19

The energy region between $1.8\le \sqrt{s}\le 2.2\mathrm{GeV}$ where the fit of inclusive $R$ ratio measurements (light blue band) replaces the sum of exclusive hadronic final states (red band) from 1.937 GeV to 2 GeV. The patterned light blue band and patterned red band show the continuation of the inclusive data combination below 1.937 GeV and the continuation of the exclusive sum above 1.937 GeV, respectively. The recent KEDR data are individually marked and included in the inclusive data fit. The light green band shows the data combination of old inclusive hadronic cross section data that exist between $1.43\le \sqrt{s}\le 2.00\mathrm{GeV}$, which were previously discussed in [9] and are not used due to their lack of precision. The estimate from pQCD is included for comparison as a dashed line with an error band which is dominated by the variation of the renormalization scale $\mu$ in the range $\frac{1}{2}\sqrt{s}<\mu <2\sqrt{s}$.

Figure 20

Pie charts showing the fractional contributions to the total mean value (left pie chart) and $(\mathrm{error}{)}^2$ (right pie chart) of both $a_{\mu }^{\mathrm{had},\mathrm{LO}\mathrm{VP}}$ (upper panel) and $\mathrm{\Delta }{\alpha }_{\mathrm{had}}^{(5)}(M_Z^2)$ (lower panel) from various energy intervals. The energy intervals for $a_{\mu }^{\mathrm{had},\mathrm{LO}\mathrm{VP}}$ are defined by the boundaries $m_{\pi }$, 0.6, 0.9, 1.43, 2.0, and $\infty \mathrm{GeV}$. For $\mathrm{\Delta }{\alpha }_{\mathrm{had}}^{(5)}(M_Z^2)$, the intervals are defined by the energy boundaries $m_{\pi }$, 0.6, 0.9, 1.43, 2.0, 4.0, 11.2, and $\infty \mathrm{GeV}$. In both cases, the $(\mathrm{error}{)}^2$ includes all experimental uncertainties (including all available correlations) and local ${\chi }_{\min }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ inflation. The fractional contribution to the $(\mathrm{error}{)}^2$ from the radiative correction uncertainties are shown in black and indicated by “rad.”

Figure 21

Contributions to the total hadronic $R$-ratio from the different final states (upper panel) and their uncertainties (lower panel) below 1.937 GeV. The full $R$-ratio and its uncertainty is shown in light blue in each plot, respectively. Each final state is included as a new layer on top in decreasing order of the size of its contribution to $a_{\mu }^{\mathrm{had},\mathrm{LO}\mathrm{VP}}$.

Figure 22

The resulting hadronic $R$-ratio shown in the range $m_{\pi }\le \sqrt{s}\le 11.1985\mathrm{GeV}$, where the prominent resonances are labeled.

Figure 23

The normalized difference of the clusters of the ${\pi }^{+}{\pi }^{-}$ data fit from the KNT18 analysis with respect to those from the HLMNT11 analysis in the range $0.6\le \sqrt{s}\le 0.9\mathrm{GeV}$. The width of the yellow band represents the total uncertainties of the clusters of the HLMNT11 analysis. The ${\pi }^{+}{\pi }^{-}$ cross section is displayed for reference.

Figure 24

Comparison of recent and previous evaluations of $a_{\mu }^{\mathrm{had},\mathrm{LO}\mathrm{VP}}$ determined from $e^{+}{e}^{-}\to \text{hadrons}$ cross section data. The analyses listed in chronological order are DEHZ03 [81], HMNT03 [7], DEHZ06 [82], HMNT06 [8], FJ06 [83], DHMZ10 [84], JS11 [85], HLMNT11 [9], FJ17 [79], and DHMZ17 [78]. The prediction from this work is listed as KNT18, which defines the uncertainty band that the other analyses are compared to.

Figure 25

A comparison of recent and previous evaluations of $a_{\mu }^{\mathrm{SM}}$. The analyses listed in chronological order are DHMZ10 [84], JS11 [85], HLMNT11 [9], FJ17 [79], and DHMZ17 [78]. The prediction from this work is listed as KNT18, which defines the uncertainty band that other analyses are compared to. The current uncertainty on the experimental measurement [1, 2, 3, 4] is given by the light blue band. The light grey band represents the hypothetical situation of the new experimental measurement at Fermilab yielding the same mean value for $a_{\mu }^{\exp }$ as the BNL measurement, but achieving the projected fourfold improvement in its uncertainty [5].