Reevaluation of the Hadronic Contributions to the Muon g-2 and to \alpha(M_{Z}^{2})

# Reevaluation of the Hadronic Contributions to the Muon g−2 and to α(M2Z)

## Abstract

We reevaluate the hadronic contributions to the muon magnetic anomaly, and to the running of the electromagnetic coupling constant at the -boson mass. We include new cross-section data from KLOE, all available multi-hadron data from BABAR, a reestimation of missing low-energy contributions using results on cross sections and process dynamics from BABAR, a reevaluation of all experimental contributions using the software package HVPTools together with a reanalysis of inter-experiment and inter-channel correlations, and a reevaluation of the continuum contributions from perturbative QCD at four loops. These improvements lead to a decrease in the hadronic contributions with respect to earlier evaluations. For the muon we find lowest-order hadronic contributions of and for the -based and -based analyses, respectively, and full Standard Model predictions that differ by and from the experimental value. For the -based five-quark hadronic contribution to we find . The reduced electromagnetic coupling strength at leads to an increase by in the central value of the Higgs boson mass obtained by the standard Gfitter fit to electroweak precision data.

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## 1 Introduction

The Standard Model (SM) predictions of the anomalous magnetic moment of the muon, , and of the running electromagnetic coupling constant, , are limited in precision by contributions from virtual hadronic vacuum polarisation. The dominant hadronic terms can be calculated with a combination of experimental cross section data, involving annihilation to hadrons, and perturbative QCD. These are used to evaluate an energy-squared dispersion integral, ranging from the threshold to infinity. The integration kernels occurring in the dispersion relations emphasise low photon virtualities, owing to the descend of the cross section, and, in case of , to an additional suppression. In the latter case, about 73% of the lowest order hadronic contribution is provided by the final state,3 while this channel amounts to only 13% of the hadronic contribution to at .

In this paper, we reevaluate the lowest-order hadronic contribution, , to the muon magnetic anomaly, and the hadronic contribution, , to the running at the -boson mass. We include new cross-section data from KLOE [[1]] and all the available multi-hadron data from BABAR [[2], [3], [4], [5], [6], [7], [8], [9]]. We also perform a reestimation of missing low-energy contributions using results on cross sections and process dynamics from BABAR. We reevaluate all the experimental contributions using the software package HVPTools [[10]], including a comprehensive reanalysis of inter-experiment and inter-channel correlations. Furthermore, we recompute the continuum contributions using perturbative QCD at four loops [[11]]. These improvements taken together lead to a decrease of the hadronic contributions with respect to our earlier evaluation [[10]], and thus to an accentuation of the discrepancy between the SM prediction of and the experimental result [[12]]. The reduced electromagnetic coupling strength at leads to an increase in the most probable value for the Higgs boson mass returned by the electroweak fit, thus relaxing the tension with the exclusion results from the direct Higgs searches.

## 2 New Input Data

The KLOE Collaboration has published new cross section data with invariant mass-squared between 0.1 and  [[1]]. The radiative photon in this analysis is required to be detected in the electromagnetic calorimeter, which reduces the selected data sample to events with large photon scattering angle (polar angle between and ), and photon energies above . The new data are found to be in agreement with, but less precise than, previously published data using small angle photon scattering [[13]] (superseding earlier KLOE data [[14]]). They hence exhibit the known discrepancy, on the resonance peak and above, with other data, in particular those from BABAR, obtained using the same ISR technique [[2]], and with data from decays [[15]].

Figure 1 shows the available cross section measurements in various panels for different centre-of-mass energies (). The light shaded (green) band indicates the HVPTools average within errors. The deviation between the average and the most precise individual measurements is depicted in Fig. 2. Figure 3 shows the weights versus the different experiments obtain in the locally performed average. BABAR and KLOE dominate the average over the entire energy range. Owing to the sharp radiator function, the available statistics for KLOE increases towards the mass, hence outperforming BABAR above . For example, at KLOE’s small photon scattering angle data [[13]] have statistical errors of , which is twice smaller than that of BABAR (renormalising BABAR to the 2.75 times larger KLOE bins at that energy). Conversely, at the comparison reads (KLOE) versus (BABAR, again given in KLOE bins which are about 4.2 times larger than for BABAR at that energy). The discrepancy between the BABAR and KLOE data sets above causes error rescaling in their average, and hence loss of precision. The group of experiments labelled “other exp” in Fig. 3 corresponds to older data with incomplete radiative corrections. Their weights are small throughout the entire energy domain. The computation of the dispersion integral over the full spectrum requires to extend the available data to the region between threshold and , for which we use a fit as described in Ref. [[10]].

We have modified in this work the treatment of the and resonances, using non-resonant data from BABAR [[3]]. While in our earlier analyses, the resonances were fitted, analytically integrated, and the non-resonant contributions added separately, we now determine all the dominant contributions directly from the corresponding measurements. Hence the and contributions are included in the , , , , spectra. Small remaining decay modes are considered separately. As an example for this procedure, the cross section measurements, featuring dominantly the and resonances, are shown in Fig. 4, together with the HVPTools average.

We also include new, preliminary, cross section measurements from BABAR [[5]], which significantly help to constrain a contribution with disparate experimental information. The available four-pion measurements and HVPTools averages are depicted in Fig. 5 in linear (top) and logarithmic (bottom) ordinates.4

The precise BABAR data [[6], [7], [8], [9]] available for several higher multiplicity modes with and without kaons (which greatly benefit from the excellent particle identification capabilities of the BABAR detector) help to discriminate between older, less precise and sometimes contradicting measurements. Figure 6 shows the cross section measurements and HVPTools averages for the channels (upper left), (upper right), (lower left), and (lower right). The BABAR data supercede much less precise measurements from M3N, DM1 and DM2. In several occurrences, these older measurements overestimate the cross sections in comparison with BABAR, which contributes to the reduction in the present evaluation of the hadronic loop effects.

Finally, Fig. 7 shows the charm resonance region above the opening of the channel. Good agreement between the measurements is observed within the given errors. While Crystal Ball [[47]] and BES [[48]] published bare inclusive cross section results, PLUTO applied only radiative corrections [[49]] following the formalism of Ref. [[50]], which does not include hadronic vacuum polarisation. As in previous cases [[30]] for the treatment of missing radiative corrections in older data, we have applied this correction and assigned a 50% systematic error to it.

Several five and six-pion modes involving ’s, as well as final states are still unmeasured. Owing to isospin invariance, their contributions can be related to those of known channels. The new BABAR cross section data and results on process dynamics thereby allow more stringent constraints of the unknown contributions than the ones obtained in our previous analyses [[30], [31]].

### Pais Isospin Classes.

Pais introduced [[52]] a classification of -pion states with total isospin . The basis of isospin wave functions of a given state belongs to irreducible representations of the corresponding symmetry group, which are characterised by three integer quantum numbers (partitions of ) , obeying the relations and . The total isospin is determined uniquely to be if and are both even, and in the other cases. States are composed by isoscalar three-pion subsystems, isovector two-pion subsystems, and isovector single pions.

Simple examples are for (, -like), and for (, -like). For four pions there are two channels and two isospin classes, related at the cross section level by5

 σ(e+e−→2π+2π−) = 45σ310, (1) σ(e+e−→π+π−2π0) = 15σ310+σ211. (2)

The two isospin classes correspond to the resonant final states for and for .

The states produced in are related to the vector part of specific decays by isospin symmetry (CVC).

### Five-Pion Channels.

There are two five-pion final states, and , of which only the first has been measured. There is only one isospin class , corresponding to and obeying the relation .

BABAR has shown [[6]] that the first channel is indeed dominated by , with some contribution from via the isospin-violating decay . These contributions must be subtracted and treated separately as they do not obey the Pais classification rules. At larger masses there is evidence for a component, which should correspond to contributions above the . Isospin symmetry holds for this contribution.

The estimation procedure for the unknown five-pion contribution is as follows: , with  [[51]], , and is considered separately. There is no contribution from , and the contribution of with non purely pionic decays is taken from with  [[51]].

### Six-Pion Channels.

There are three channels and four isospin classes with the relations ():

 σ(3π+3π−) = 2435σ510+35σ330, (3) σ(2π+2π−2π0) = 835σ510+25σ411+25σ330+σ321, (4) σ(π+π−4π0) = 335σ510+35σ411, (5)

where the lowest-mass resonant states are for , for , for , and for .

BABAR has measured [[7]] the cross sections (3) and (4), and observed only one state per event in the fully charged mode, thus favouring over in (3). The process is dominated by up to . A small contribution is also found, but only the cross section for is given.

To estimate the cross section (5) the relative contributions of and need to be known, which can be constrained from data. The corresponding isospin relations for the spectral functions are

 v(τ−→2π+3π−π0ντ) = 1635v510+45v411 (6) +15v330+12v321, v(τ−→π+2π−3π0ντ) = 1035v510+15v411 (7) +45v330+12v321, v(τ−→π−5π0ντ) = 935v510. (8)

The branching fractions of the first two modes have been measured by CLEO. As for the final states, they are dominated by production, and , with branching fractions and , respectively, to be compared to total branching fractions of and ( subtracted). This yields the bound . The limit for is looser than that quoted in Ref. [[30]], where the partition was assumed to be negligible.

The estimation procedure for the missing six-pion mode is as follows: , with  [[51]], and ; is treated separately, and the contribution from non-pionic decays is given by .

### K¯¯¯¯Kπ Channels.

The measured final states are and , with missing (). Except for a very small contribution, these processes are governed by (dominant) and transitions below . Both amplitudes () contribute. The fit of the Dalitz plot in the first channel yields the moduli of the two amplitudes and their relative phase as a function of mass. Hence everything is determined, as seen from the following relations (labels written in the order with the given decay modes):

 σ(K+K−π0+K−K+π0) = 16|A0−A1|2, (9) σ(K0SK0Lπ0+K0LK0Sπ0) = 16|A0+A1|2, (10) σ(K0K−π++¯¯¯¯K0K+π−) = 13|A0+A1|2, (11) σ(K+¯¯¯¯K0π−+K−K0π+) = 13|A0−A1|2. (12)

The measured cross section (no ordering here) is therefore equal to , and for the dominant part. Note that, unlike it was assumed in Ref. [[30], [31]], in general is not equal to .

The complete contribution is obtained from , with , where contributions from non-hadronic decays are neglected, whereas decays to are already counted in the multi-pion channels.

### K¯¯¯¯K2π Channels.

The channels measured by BABAR are and  [[9]]. They are dominated by , with not in a , and smaller contributions from and .

In the dominant mode one can have and amplitudes. The different charge configurations can be obtained via and amplitudes, where, however, is not favoured because it would have predicted , whereas a ratio of roughly 4:1 has been measured [[9]]. In the following we assume a pure state, so that the relevant cross sections read (labels in the order , appropriately summing over )

 σ(K±π0K∓π0) = 118|A0−A1|2, (13) σ(K0π±K∓π0) = 19|A0−A1|2, (14) σ(K0π0K0π0) = 118|A0+A1|2, (15) σ(K±π∓K0π0) = 19|A0+A1|2, (16) σ(K0π0K±π∓) = 19|A0+A1|2, (17) σ(K±π∓K±π∓) = 29|A0+A1|2, (18) σ(K±π0K0π∓) = 19|A0−A1|2, (19) σ(K0π±K0π∓) = 29|A0−A1|2. (20)

The inclusive cross section is thus obtained as follows: get and (note that the published BABAR cross section table for already includes the branching fraction for ). In lack of more information, we assume , with a 100% error, and obtain .

### K¯¯¯¯K3π Channels.

BABAR has only measured the final state  [[6]], which is dominated by up to . The channel has been measured, and the remaining amplitude is negligible. The dominance does not apply to the missing channels and , but their dynamics (for instance ) should be seen in the measured mode, so it may be small, at least below .

The missing channels are estimated as follows: . We assume, within a systematic error of 50%, , treat separately, and compute the non-pionic contribution by . Contributions from and below are neglected.

### η4π Channels.

BABAR has measured  [[6]], where the state has , . Because , we assume the same ratio for the process with the same quantum numbers. We thus estimate , and assign a systematic error of 25% to it.

## 4 Data averaging and integration

In this work, we have extended the use of HVPTools6 to all experimental cross section data used in the compilation.7 The main difference of HVPTools with respect to our earlier software is that it replaces linear interpolation between adjacent data points (“trapezoidal rule”) by quadratic interpolation, which is found from toy-model analyses, with known truth integrals, to be more accurate. The interpolation functions are locally averaged between experiments, whereby correlations between measurement points of the same experiment and among different experiments due to common systematic errors are fully taken into account. Incompatible measurements lead to error rescaling in the local averages, using the PDG prescription [[51]].

The errors in the average and in the integration for each channel are obtained from large samples of pseudo Monte Carlo experiments, by fluctuating all data points within errors and along their correlations. The integrals of the exclusive channels are then summed up, and the error of the sum is obtained by adding quadratically (linearly) all uncorrelated (correlated) errors.

Common sources of systematic errors also occur between measurements of different final state channels and must be taken into account when summing up the exclusive contributions. Such correlations mostly arise from luminosity uncertainties, if the data stem from the same experimental facility, and from radiative corrections. In total eight categories of correlated systematic uncertainties are distinguished. Among those the most significant belong to radiative corrections, which are the same for CMD2 and SND, as well as to luminosity determinations by BABAR, CMD2 and SND (correlated per experiment for different channels, but independent between different experiments).

## 5 Results

A compilation of all contributions to and to , as well as the total results, are given in Table 2. The experimental errors are separated into statistical, channel-specific systematic, and common systematic contributions that are correlated with at least one other channel.

Table 1 quotes the specific contributions of the various cross section measurements to . Also given are the corresponding CVC-based branching fraction predictions. The largest (smallest) discrepancy of () between prediction and direct measurement is exhibited by KLOE (BABAR). It is interesting to note that the four determinations in Table 1 agree within errors (the overall of their average amounts to 3.2 for 3 degrees of freedom), whereas significant discrepancies are observed in the corresponding spectral functions [[10]]. Since we cannot think of good reasons why systematic effects affecting the spectral functions should necessarily cancel in the integrals, we refrain from averaging the four values with a resulting smaller error. The combined contribution is instead computed from local averages of the spectral function data that are subjected to local error rescaling in case of incompatibilities.

The contributions of the and resonances in Table 2 are obtained by numerically integrating the corresponding undressed8 Breit-Wigner lineshapes. Using instead the narrow-width approximation, , gives compatible results. The errors in the integrals are dominated by the knowledge of the corresponding bare electronic width .

Sufficiently far from the quark thresholds we use four-loop [[11]] perturbative QCD, including quark mass corrections [[53]], to compute the inclusive hadronic cross section. Non-perturbative contributions at were determined from data [[54]] and found to be small. The errors of the contributions given in Table 2 account for the uncertainty in (we use from the fit to the hadronic width [[55]]), the truncation of the perturbative series (we use the full four-loop contribution as systematic error), the full difference between fixed-order perturbation theory (FOPT) and, so-called, contour-improved perturbation theory (CIPT) [[56]], as well as quark mass uncertainties (we use the values and errors from Ref. [[51]]). The former three errors are taken to be fully correlated between the various energy regions (see Table 2), whereas the (smaller) quark-mass uncertainties are taken to be uncorrelated. Figure 8 shows the comparison between BES data [[48]] and the QCD prediction below the threshold between 2 and . Agreement within errors is found.9

### Muon magnetic anomaly.

Adding all lowest-order hadronic contributions together yields the estimate (this and all following numbers in this and the next paragraph are in units of )