Testing the Standard Model at the precision frontier: the anomalous magnetic moment of the muon

The anomalous magnetic moment of the muon (g − 2)μ is one of the most precisely measured quantities in particle physics (0.54 ppm). There is a long-standing discrepancy of 3-4 standard deviations between the direct measurement of (g − 2)μ and its theoretical evaluation. This theoretical prediction is subdivided into three contributions: QED, weak and hadronic. The QED and weak parts can be determined in perturbative approaches with very high precision. Thus, the hadronic uncertainty dominates the total theoretical uncertainty. Within the hadronic uncertainty, the largest contribution stems from the vacuum polarization term, which can be evaluated with the measurement of the inclusive hadronic cross section in e+e− annihilation. The second largest contribution to the hadronic uncertainty stems from the so-called Light-by-Light amplitudes. They have to be evaluated via theoretical models. These models require transition form factor measurements as input. Existing and future measurements of the relevant hadronic cross sections and transition form factors are presented.


Introduction
There are various ways to test the Standard Model (SM) of particle physics. Many approaches include searches for new particles or phenomena at the high energy frontier. Another approach is to measure SM observables with high precision and compare the measurement to the SM prediction. One particular test at this high precision frontier is the measurement of the anomalous magnetic moment of the muon a μ = 0.5 · (g − 2) μ . On the one hand, it is one of the most precisely measured observables in particle physics. On the other hand, there is a discrepancy of 3-4 standard deviations between the direct measurement of (g − 2) μ and its theoretical evaluation [1,2]. This theoretical prediction is subdivided into three contributions: QED, weak and hadronic. The QED and weak parts can be determined in perturbative approaches with very high precision [3]. At low energies perturbation theory cannot be used to calculate the hadronic contribution a had μ . Thus, the hadronic uncertainty dominates the total theoretical uncertainty. Within the hadronic uncertainty, the largest contribution stems from the vacuum polarization (VP) term. It is possible to relate this contribution a had,VP μ via a dispersion relation (Eq. (1)) to hadronic cross sections, which typically are a e-mail: hafner@kph.uni-mainz.de measured in e + e − energy scan experiments at low energies.
The Kernel function K(s) is proportional to s −1 [4]. As an alternative to the energy scan experiments, the study of the Initial State Radiation (ISR) events at flavor-factories allows independent measurements of exclusive hadronic cross sections. This method allows high statistics e + e − experiments running at a fixed center-of-mass (c.m.) energy to access processes at lower effective c.m. energies by studying events with a high energy photon emitted from the initial state. The use of this technique at high luminosity φand B-factories has been discussed in detail in Refs. [5][6][7]. The second largest contribution to the hadronic uncertainty stems from the so-called hadronic Light-by-Light (LbL) amplitudes. They have to be evaluated via theoretical models. These models require transition form factor measurements as input.
Existing and future measurements of (g − 2) μ , the relevant hadronic cross sections, and transition form factors are presented.

Experiments
The most precise measurement of a μ = (11659208.9 ± 6.4) · 10 −10 has been performed at BNL [8]. The production of the polarized muon beam is illustrated in Fig. 1 (left). It is essential that the decay of π + → μ + ν violates parity, which leads to longitudinal polarization of the muons. These muons are guided into a storage ring with constant magnetic field B, and thus perform a rotating motion with cyclotron frequency ω c . The spin of the muons is precessing with the frequency ω s . This leads to a change of the helicity with the Lamor frequency ω a (Eq. 2).
The last equality of Eq. 2 is true for a particular choice of the so-called magical muon energy E magic = 3.098 GeV. Finally the decay of μ + → e +ν ν again violates parity. Thus, the e + direction remembers the μ + polarization and the number of detected e + above an energy threshold is modulated with ω a , see Fig. 1 (right). This allows the extraction of a μ . Future experiments at Fermilab [9] and JPARC [10] intend to further improve the precision by a factor of 4.

Meson-Photon transition form factors
The uncertainty of the hadronic LbL contribution [11], Δa had,LbL μ = 2.6 · 10 −10 , to the theoretical estimate of (g − 2) μ is larger than the expected uncertainties of the upcoming direct (g − 2) μ measurements, ∼ 1.5 · 10 −10 . The LbL estimates can be improved with experimental input, especially from photon-meson transition form factors (TFF). Here, the contribution of the π 0 TFF is dominating.

η/η -photon transition form factors
The η TFF measurement as a function of momentum transfer Q 2 from BABAR [12] and CLEO [15] is shown in Fig. 2 (middle). For BABAR the systematic uncertainty is 2.9%, dominated by model and π 0 reconstruction uncertainties of the ηπ + π − π 0 final state The η → π + π − η, with η → γγ measurements are shown in Fig. 2 (right). The BABAR systematic uncertainty of 3.5% is dominated by model and η reconstruction uncertainties. The η -TFF stays below the asymptotic expectation.
More data is needed in order to solve this puzzling situation for the TFFs at large Q 2 and as input for a had,LbL μ at small Q 2 , especially for the π 0 final state. 01001-p.3

Hadronic cross sections
The dominant contribution to the hadronic part of (g − 2) μ stems from the VP, requiring hadronic cross section measurements as input. The e + e − → π + π − cross section has the largest contribution (∼ 75%).
2.3.1 σ(e + e − → π + π − ) Fig. 3 (left) shows the precise σ measurements that have been performed in the ρ peak region. The ρω interference is clearly visible. The relative difference between the two most precise measurements, from BABAR and KLOE, is shown in Fig. 3 (right). We observe a disagreement of several standard deviations at and above the ρ resonance peak, leading to a large uncertainty of a μ . Additional high precision measurements are expected from BES-III, CMD-III and SND in the near future.  Left: cross section of e + e − → π + π − in the ρ-peak region from OLYA [16], CMD [17], SND [18], DM1 [19], KLOE [20], and BABAR [21]. Right: relative difference between BABAR and KLOE.

σ(e + e − → KK)
Recently, the e + e − → K + K − final state has been published by BABAR [21], see Fig. 4. In the φ peak region, this measurement is in agreement within 2 standard deviations to the CMD2 [22] and SND [23] data. At larger cms energies (right) a different shape in comparison to SND is observed. In Fig. 5 (left), the invariant K 0 S K 0 L -mass distribution by the BABAR [24] experiment in the φ peak region is shown, from which the partial electronic width of φ to K 0 S K 0 L is extracted. This observable is in agreement with the CMD2 [25] result. At larger energies, Fig. 5 (right), a clear structure is visible, which might be due to the φ(1680) resonance.

Multihadronic final states
In Fig. 6, we see the first measurement of the hadronic cross section of the K 0 S K 0 L π + π − (left), K 0 S K 0 S π + π − (middle), and K 0 S K 0 L K + K − (right) final states measured by BABAR. This will allow for the first time the direct extraction of the (g − 2) μ contribution of these final states. Up to date the value has been estimated by means of isospin relations. Figure 6. Hadronic cross section of the K 0 S K 0 L π + π − (left), K 0 S K 0 S π + π − (middle), and K 0 S K 0 L K + K − (right) final states measured by BABAR. 01001-p.5

Conclusion and outlook
Including hadronic cross section and TFF measurements the remaining difference between theoretical prediction and measured value of 3.0-3.5 standard deviations [1,2] for the muon anomaly still poses an open question and requires further studies. The measurement of KK(π + π − /K + K − ) and await to be included in the world averages. Additional direct (g − 2) μ measurements and hadronic cross section and TFF measurements will help to shed more light in the observed difference.
I want to thank the organizers for the warm hospitality and the organization of this beautiful conference in Cracow.