Lightest pseudoscalar exchange contribution to light-by-light scattering piece of the muon g-2

Lightest pseudoscalar (P= pi^0, eta, eta') exchange contribution to the light-by-light (LbL) scattering piece of the muon anomaly, a_mu=(g_mu-2)/2, has been evaluated using a resonance chiral Lagrangian (R\chi L). Best description of pion transition form-factor (TFF) data is obtained with only tiny violations of one of the relations in the minimal consistent set of short-distance constraints on the anomalous R\chi L couplings. eta^{(')} TFF, predicted in terms of the pi TFF and the eta-eta^' mixing, are in good agreement with measurements. With this input, we obtain a_mu^{P,LbL} = (10.47\pm0.54)x10^-10, consistent with the reference determinations in the literature, albeit with smaller error.

As an electromagnetic property a µ is, first, a stringent test of QED. The one loop contribution computed by Schwinger [3] fixes its size universally to ∼ α/(2π) ∼ 10 −3 for all charged leptons, which differs by six orders of magnitude with the current discrepancy ∼ 3 · 10 −9 . A tremendous effort in computing higher-order terms lead to the complete five-loop QED result [4]. Its error is four orders of magnitude smaller than the previous value and the O(α 5 ) term is only ∼ 5 · 10 −11 . Therefore, this anomaly cannot be attributed to uncalculated or imprecise QED contributions.
Conversely, hadronic effects are important in a µ 1 : hadronic vacuum polarization (LO + NLO) contributes (680.7 ± 4.7) · 10 −10 to a µ [6]. Although with much smaller central value, the Hadronic LbL contribution 1 Electroweak effects are small and well under control at the required precision, a EW µ = (153.6 ± 1.0) · 10 −11 [5]. The total error on the Standard Model prediction for a µ , 6.3·10 −10 , nearly equals the current experimental uncertainty, 6.4·10 −10 . However, forthcoming experiments at FNAL and J-PARC [7] will soon reduce the latter to a fourth. This urges theoreticians to achieve a similar error reduction to benefit fully from the precision of these measurements. This is our main motivation to revisit the dominant contribution to a HLbL µ given by P exchange. We have reconsidered [8] the lightest pseudoscalar (P = π 0 , η, η ) exchange contribution to a HLbL µ , which is one of the possible intermediate states in the fourpoint VVVV Green function with one real and three virtual photons that enters a HLbL µ . This contribution turns out to basically saturate it, due to approximate cancellations between the remaining pieces [2]. Despite a simultaneous chiral and large-N C expansion has been suggested [9] to tackle a HLbL µ , the relative size of the various terms is not fully understood yet. This raises reasonable doubts on the errors obtained studying invividual contributions isolately, which might translate into an increased overall error for a HLbL µ [10].

arXiv:1409.2174v2 [hep-ph] 9 Sep 2014
The main difficulty for computing a HLbL µ has been that, contrary to the hadronic vacuum polarization, there was no way to relate it to measurements using dispersion relations and the optical theorem. However, very recently a formalism has been put forward [11], which would allow to extract the dominant one-and two-pion exchange contributions directly from data.
At the time being, nevertheless, the most precise experimental information that can be obtained on a P,HLbL µ comes from the corresponding T FF, where one photon is real to a very good approximation and the formfactor is measured as a function of the virtuality of the other photon up to roughly 6 GeV. It has been shown [2] that demanding an appropriate short-distance behaviour [12] to the P T FF and to the related VV P Green's function turns out to be crucial for the reliability of the a P,HLbL µ value. Dedicated studies of this question [13] have found that using RχL [14] (rooted in the large-N C limit of QCD [15] and chiral symmetry [16]) there exists a consistent set of short-distance constraints on the VV P Green function and related form factors [17]. For this, the antisymmetric tensor formalism needs to be employed [18] and pseudoscalar resonances be active degrees of freedom [19]. However, it is still an open question whether all short-distance constraints are already imposed working this way, or there are new genuine relations arising from the VVVV Green function which have not been considered yet [20].
In this framework the π T FF can be written [17,19] F π 0 γγ (Q 2 ) = − F 3 and one of the high-energy constraints in the minimal consistent set demands that P 2 cancels the O(Q 0 ) term for Q 2 → ∞. We find that allowing a 4% violation of this condition yields the best fit to current data 2 . Fig. 1 compares our best fit result to all available data. The fully off-shell form factor is also needed to evaluate a P,HLbL µ [2]. It depends only on one additional coupling unrestricted by high-energy behaviour, which can be fixed analysing the π(1300) → γγ and π(1300) → ργ decays [19]. Upon the required integrations [2] one finds a π 0 ,HLbL µ = (6.66 ± 0.21) · 10 −10 , where the corresponding contributions to the error are discussed in detail in our paper [8]. 2 This violation is not due to the difference between BaBar and Belle data points. Excluding BaBar data the violation is only reduced to 3%. , CLEO [22], BaBar [23] and Belle [24] data for the πTFF are confronted to our best fit result using the form-factor in eq. (1).
Chiral dynamics and the η-η mixing (whose uncertainty saturates the error on a η ( ) ,HLbL µ ) allow to relate the π T FF to the η ( ) T FF [8]. Our predictions are compared to data in Figs. 2 and 3. BaBar data CELLO data CLEO data Our upper limit Our lower limit Figure 2: Our predictions for the ηTFF using the π TFF (1) and the η-η mixing are confronted to BaBar [25], CELLO [21] and CLEO [22] data.
The corresponding contributions to a P,HLbL BaBar data CELLO data CLEO data Our upper limit Our lower limit Figure 3: Our predictions for the η TFF using the π TFF (1) and the η-η mixing are confronted to BaBar [25], CELLO [21] and CLEO [22] data.
Our main result is the value a P,HLbL µ = (10.47 ± 0.54) · 10 −10 , for the contribution of the three lightest pseudoscalar mesons (π 0 , η and η ) to the muon anomaly, which is in good agreement with the two reference values: (9.9 ± 1.6) · 10 −10 (Jegerlehner and Nyffeler) and (11.4 ± 1.3) · 10 −10 (Prades, de Rafael and Vainshtein) [2] but has a reduced error, mainly thanks to the new BaBar and Belle data on the P T FF extending to larger energies.
Using the values in the literature for the remaining contributions to a HLbL µ [2] yields a HLbL µ = (11.8 ± 2.0) · 10 −10 , which would translate into a theoretical uncertainty on a µ of ±5.1 · 10 −10 , ∼ 20% smaller than the current estimate. Finally, we also propose [8] the measurement of σ(e + e − → π 0 µ + µ − ) and the corresponding differential distribution as a function of the di-muon invariant mass to better characterize the P T FF and hopefully further reduce the error of a P,HLbL µ .