Hadronic vacuum polarization: three-pion channel

. The 3 π -channel contribution to hadronic vacuum polarization (HVP) in the anomalous magnetic moment of the muon ( g − 2) µ is examined based on a dispersive representation of the γ ∗ → 3 π amplitude. This decay amplitude is reconstructed from dispersion relations, fulﬁlling the low-energy theorem of QCD. The global ﬁt function is applied to the data sets of the 3 π channel below 1 . 8GeV, which constitutes the second-largest exclusive contribution to HVP and its uncertainty. The dominant ω - and φ -peak regions in the e + e − → 3 π cross section as well as the non-resonant regions are precisely described to obtain our best estimate. The ﬁnal result, a 3 πµ | ≤ 1 . 8GeV = 46 . 2(6)(6) × 10 − 10 , reduces the model dependence owing to the fundamental principles of analyticity and unitarity and provides a cross check for the compatibility of the di ﬀ erent e + e − → 3 π data sets. A combination of the current analysis and the recent similar treatment of the 2 π channel yields a dispersive determination of almost 80% of the entire HVP contribution. Our analysis rea ﬃ rms the muon anomaly at 3 . 4 σ level, when the rest of the contributions is taken from the literature.

The leading hadronic contribution, HVP, can be evaluated using e + e − → hadrons cross section data as input into a dispersion integral [24,25]. Among the most recent data compilations of HVP [26][27][28][29], [26,27] were performed following direct data-integration methods without additional model assumptions, where tensions between conflicting data sets are typically considered by a local error inflation. On the other hand, for low-multiplicity channels like 2π, a reconstruction of global fit functions from analyticity, unitarity, and low-energy theorems make it possible to confront the data with constraints from QCD. Such an investigation has been recently completed for the 2π channel below 1 GeV [30].

HVP contribution to a µ
The leading-order HVP contribution to a µ reads [24,25] where α is the fine-structure constant in the Thomson limit, and the kernel function is given bŷ R had (s) is expressed in terms of the hadronic cross section as R had (s) = 3s 4πα 2 σ 0 (e + e − → hadrons), when the electron mass is neglected. Conventions need to be specified for the radiative corrections to the cross section so that higher-order insertions of HVP and HLbL scattering can be performed consistently [41][42][43]. In this regard, the "bare" cross section σ 0 (e + e − → hadrons) should be inclusive of final-state radiation (FSR), but exempt from initial state radiation (ISR) and vacuum polarization (VP). As a consequence, s thr is no longer equal to the two-pion threshold, but s thr = M 2 π 0 due to the π 0 γ channel.
Concerning the radiative corrections to the 3π channel, the ISR correction has usually been accounted for in the Monte Carlo generator of each experiment. We use the bare cross section provided by the experiment in which VP is already removed. Otherwise we uniformly apply the VP routine from [27]. We also constructed another VP function based on the 2π and 3π channels [30,31] and conclude that the details of the VP routine prove irrelevant at the current level of precision. In the dispersive approach, the representation of the γ * → 3π amplitude is only valid in pure QCD. Consequently, experimental data inclusive of FSR is incompatible with the dispersive treatment at first sight, because the photon contribution from FSR should be removed before the fit to extract quantities defined in pure QCD and added in a consistent perturbative manner afterwards. This strategy has been carried out in the 2π channel with the help of a scalar-QED approximation [30]. The HVP contribution of the 3π channel is more than one order of magnitude smaller. Hence, the total size of the 3πγ final-state contribution could be naively estimated as 0.3 × 10 −10 , which is only borderline relevant at the present level of accuracy. Therefore, the FSR correction is neglected here and relegated to future work.
The iterative solution of the integral equation (9) is performed for a(q 2 ) → 1, using the recently extracted phase shift δ(s) from [30] and a cutoff parameter Λ 3π = 2.5 GeV. Variations of δ(s) [54,55] and Λ 3π produce irrelevant systematic uncertainties compared to other sources.
The full a(q 2 ) dependence in (9) can be restored as an overall normalization of the iterative solution and contains the information of the isoscalar-photon coupling to 3π. It is determined by the low-energy theorem (5) at q 2 = 0, and is accessible in e + e − → 3π in the decay region q 2 > 9M 2 π . We take the improved parameterization from [20,21]: in which the subtraction constant α A is determined by the chiral anomaly (corrected by quark-mass renormalization) [56,57], The function A is given by the sum of resonance contributions with V = ω, φ, ω (1420), and ω (1650). The energydependent widths Γ ω/φ (q 2 ) of ω/φ include all the main decay channels, see [20,21] for more details. For ω and ω , with masses and widths taken from the PDG [58], we assume a 100% branching ratio to 3π. We allow the masses and widths of ω and φ to vary: noticeable differences between the VP-subtracted parameters and the PDG are expected, since radiative effects are subsumed in the PDG parameters. Non-resonant, inelastic effects are taken into account in (11) by the conformal polynomial The inelastic threshold s inel = 1 GeV 2 is motivated by the nearby KK threshold, and the second parameter s 1 is set to −1 GeV 2 . Further constraints are implemented to eliminate the S -wave cusp in the conformal polynomial (14) and to ensure that the sum rule is fulfilled exactly. Constructed in such a way, a(q 2 ) thus obeys the low-energy theorem from the chiral anomaly, preserves analyticity of F (s, t, u; q 2 ), and is flexible enough to describe the data up to 1.8 GeV.
Finally, the relation between the bare cross section (neglecting the electron mass) and the scalar amplitude F (s, t, u; q 2 ) is given by and F (s, t, u; q 2 ) 2 .
(17) The integration boundaries are given as and In the following we briefly discuss the data sets to which we perform our fit using (16) and (17).

Data sets and combined fits
We list the data sets which are included in our analysis in Table 1. The Novosibirsk experiments SND [59][60][61][62] and CMD-2 [63][64][65][66] provide the most precise measurements for the low-energy and ω-and φ-resonance regions. These data were obtained from direct energy scans. Only the BaBar experiment [70], which covers the higher-energy range above the φ, used the ISR technique. For completeness, we also take into account older data from DM1 [67], DM2 [68], and ND [69]. The statistical uncertainties of all data sets are given in diagonal form without further correlations. In contrast, the systematic uncertainties are divided into normalization uncertainties usually arising from luminosity measurement or detection efficiency, which are 100% correlated by definition, and other sources of systematic effects (for instance model assumptions and background) restricted to localized energy regions, which should not be considered correlated. We follow the experimental documentation to the best of our knowledge and consider a systematic error of normalization type whenever given as a percentage, and the remaining one as a diagonal error. Additionally, BaBar [70] explicitly states that the systematic errors of the data obtained by ISR for different energy bins are fully correlated.
It is well known that a normalization uncertainty may introduce a potential bias in the fit. More explicity, a χ 2minimization with an empirical covariance matrix V(i, j) including a normalization uncertainty, will converge to a solution that is biased towards a lower value, as firstly observed by D'Agostini [71]. This bias can be severe for a large number of data points or large normalization uncertainties. We follow the iterative fit method proposed by the NNPDF collaboration [72] to eliminate the bias. To this end, the corrected iterative covariance matrix reads where V stat (i, j) is the statistical covariance matrix, and the systematic covariance matrix V syst (i, j) is determined by multiplying the normalization factors with the fit model in each iteration step. Rapid convergence to the final solution is expected after a few iteration steps. Our combined fit results are shown in Table 2, where we apply p conf = 2 . . . 4 free parameters in the conformal polynomial, added together with other dispersive uncertainties to estimate the systematic uncertainty of our analysis. We include the fits with diagonal errors and full ones simultaneously to better monitor the role of correlations, but use the latter as our final estimate. Combinations of different data sets reveal acceptable consistency and statistics except for [66], wherein the last few data points of each scan below 5 nb cause unacceptable χ 2 and the third scan produces a significantly lower φ mass. Accordingly, we have to remove the data with very small cross sections mentioned above and allow for a rescaling in the right ballpark with the energy-calibration uncertainties in the third scan data. In addition, we discard the DM2 data [68] since they disagree with both SND [62] and BaBar [70] high-statistics data. Our central value is fixed by the best statistical fit with p conf = 3, as illustrated in the left plot of Figure 1. In particular, we find the vacuum-polarization-subtracted ω and φ parameters where the fit errors are inflated by the scale factor following the PDG prescription [58].  [62,68,70]. In the last column, normalization-type systematic uncertainties are assumed to be 100% correlated.
of radiative effects, which reveals a perfect agreement between (23) and the PDG average [58] for the φ, with an anticipated downward shift of 0.26 MeV for the mass and negligible correction to the width. In view of a downward shift of 0.13 MeV and an upward shift of 0.06 MeV for the mass and width of the ω, this analysis solidifies the tension with the VP-subtracted ω mass extracted from the 2π channel [30].

Consequences for a µ
We find our final result for the 3π contribution to a µ to be It is in very good agreement with a 3π µ | ≤1.8 GeV = 46.2(1.5) × 10 −10 from [26], moreover, providing a reduced uncertainty. The deviation from [27] is mainly due to the different interpolation methods applied to the integration of the data: a linear interpolation reproduces the central value a 3π µ | ≤1.8 GeV = 47.7(9)×10 −10 from [27], while higher-order interpolations produce smaller values closer to (25). 1 Estimates as low as a 3π µ | ≤2.0 GeV = 44.3(1.5)×10 −10 [74] are not supported by our analysis. We find a larger contribution from the near threshold region, a 3π µ | ≤0.66 GeV = 0.02×10 −10 , twice as large as the value from [75]. The estimate of [75] follows the analyses [76,77], which are based on a model with a contact term determined by the WZW anomaly and vector meson dominance. This model is depicted in the right plot of Figure 1, where it undershoots the lowestenergy data points.

Conclusions and outlook
We have reported on a comprehensive dispersive analysis of the 3π contribution to HVP. We first constructed the γ * → 3π amplitude imposing constraints from analyticity, unitarity, and the low-energy theorem of QCD. We performed the fit to the combined data sets using the aforementioned amplitude, and eliminated the potential bias related to the normalization uncertainties. After these setups, we were able to check the consistencies of different data sets. Identifying the tensions in [66], our analysis completes another independent check for the 3π-channel HVP contribution with the global fit function fulfilling QCD constraints, complementary to strategies based on direct integration of the data.   (9) 10 4 × ξ 1.9(7) 1.8 (7) 1.8 (7 Table 2. Fits to the combined data sets as shown in Table 1, with fit uncertainties given in brackets.  Table 1 (left) and the comparison to the estimate from [76,77] (blue curve) in the near threshold region (right). The black error band represents the fit uncertainty.
We extracted the masses and widths of the ω and φ from our central fit and raised the potential impacts of radiative effects. The tension between the ω mass extracted from the 2π and 3π channels survives. Furthermore, our result of the 3π-channel contribution to a µ scrutinized the discrepancies standing in the literature before. Detailed analysis of FSR is in progress and further improvements are expected in light of new data inputs.