Complete Tenth-Order QED Contribution to the Muon g-2

Complete Tenth-Order QED Contribution to the Muon


We report the result of our calculation of the complete tenth-order QED terms of the muon . Our result is in units of , which is about 4.5 s.d. larger than the leading-logarithmic estimate . We also improved the precision of the eighth-order QED term of , obtaining in units of . The new QED contribution is , which does not resolve the existing discrepancy between the standard-model prediction and measurement of .


The anomalous magnetic moment of the muon has been studied extensively both experimentally and theoretically since it provides one of the promising paths in exploring possible new physics beyond the standard model. For this purpose it is crucial to know the prediction of the standard model as precisely as possible.

On the experimental side the current world average of the measured is Bennett et al. (2004); Roberts (2010):


New experiments designed to improve the precision further are being prepared at Fermilab Lee Roberts (2011) and J-PARC Iinuma (2011).

In the standard model, can be divided into electromagnetic, hadronic, and electroweak contributions


At present (hadronic) is the largest source of theoretical uncertainty. The uncertainty comes mostly from the hadronic vacuum-polarization (v.p.) term, being the fine-structure constant. The lattice QCD simulations have attempted to evaluate this contribution Blum (2003); Gockeler et al. (2004); Aubin and Blum (2007); Feng et al. (2011); Boyle et al. (2012); Della Morte et al. (2012). At present, most accurate evaluations must rely on the experimental information. Three types of measurements are available for this purpose: (1) , (2) , (3) . These processes have been investigated intensely by many groups Davier et al. (2011); Jegerlehner and Szafron (2011); Hagiwara et al. (2011).We list here one of them Hagiwara et al. (2011):


which overlaps other values based on the data Davier et al. (2011); Jegerlehner and Szafron (2011) and makes the standard-model prediction closest to the experiment (1). The next-to-leading-order (NLO) hadronic vacuum-polarization contribution is also known Hagiwara et al. (2011):


The hadronic light-by-light scattering contribution (l-l) is of similar size as , but has a much larger theoretical uncertainty Melnikov and Vainshtein (2004); Bijnens and Prades (2007); Prades et al. (2009); Nyffeler (2009)


where the uncertainty covers almost all values obtained in different publications.

The electroweak contribution has been calculated up to 2-loop order Fujikawa et al. (1972); Czarnecki et al. (1996); Knecht et al. (2002); Czarnecki et al. (2003):


Since this uncertainty is 30 times smaller than the experimental precision of (1), it can be regarded as known precisely.

The primary purpose of this letter is to report the complete numerical evaluation of all tenth-order QED contribution to . It leads to a sizable reduction of the uncertainty of the previous estimate by the leading-log approximations Kinoshita and Nio (2006a); Kataev (2006). We have also improved the numerical precision of the eighth-order QED contribution including the newly evaluated tau-lepton contribution. Together they represent a significant reduction in the theoretical uncertainty of the QED part of .

The QED contribution to can be evaluated by the perturbative expansion in :


where is finite thanks to the renormalizability of QED and can be written as


is independent of mass and universal for all leptons. , and are known exactly Schwinger (1948); Petermann (1957); Sommerfield (1958); Laporta and Remiddi (1996). Mass dependence is known analytically for and for Samuel and Li (1991); Li et al. (1993); Laporta (1993a); Laporta and Remiddi (1993); Czarnecki and Skrzypek (1999). We reevaluated them using the latest values of the muon-electron mass ratio and/or the muon-tau mass ratio Mohr et al. (2012). In the same order of terms as shown on the right-hand-side of (8), the results are summarized as follows:

Figure 1: Vertex diagrams representing 13 gauge-invariant subsets contributing to the lepton at the eighth-order. Solid and wavy lines represent lepton and photon lines, respectively.

The value of has been obtained mostly by numerical integration Laporta (1993b); Kinoshita and Nio (2006b); Aoyama et al. (2007); ?. They arise from 13 gauge-invariant sets whose representative diagrams are shown in Fig. 1. We have reevaluated some of them for further check and improvement of numerical precision. The results for the mass-dependent terms are summarized in Table 1.

Table 1: The eighth-order mass-dependent QED contribution from 12 gauge-invariant groups to muon , whose representatives are shown in Fig. 1. The mass-dependence of is .

From the data listed in Table 1 and the value of from Refs. Kinoshita and Nio (2006b); Aoyama et al. (2007); ?; Aoyama et al. (2012a), we obtain the following value for the eighth-order QED contribution :

Figure 2: Self-energy-like diagrams representing 32 gauge-invariant subsets contributing to the lepton at the tenth order. Solid lines represent lepton lines propagating in a weak magnetic field.

Over the period of more than nine years we have numerically evaluated all 32 gauge-invariant sets of diagrams that contribute to Kinoshita and Nio (2006a); Aoyama et al. (2008b, 2010a, c); ?; ?; ?; ?; ?; ?; Aoyama et al. (2012a), whose representative diagrams are shown in Fig. 2. The results for mass-dependent terms are summarized in Table 2. Some simple diagrams were evaluated analytically or in the asymptotic expansion in Kataev (1992); ?; Broadhurst et al. (1993); Laporta (1994); Aguilar et al. (2008); Baikov et al. (2008). The results are consistent with our numerical ones.

From the data listed in this Table and the value of from Ref. Aoyama et al. (2012a), we obtain the complete tenth-order result:


The uncertainty is attributed entirely to the statistical fluctuation in the Monte-Carlo integration of Feynman amplitudes by VEGAS Lepage (1978). This is 20 times more precise than the previous estimate, , obtained in the leading-logarithmic approximation Kinoshita and Nio (2006a). This is mainly because we had underestimated the magnitude of the contribution of the Set III(a). Note also that (11) is about s.d. larger than the leading-log estimate. The numerical values of for are summarized in Table 3.

Table 2: Tenth-order mass-dependent contribution to the muon from 31 gauge-invariant subsets shown in Fig. 2. The mass-dependence of is .
order with with
Table 3: Contributions to muon from QED perturbation term . They are evaluated with two values of the fine-structure constant determined by the Rb experiment and by the electron ().

In order to evaluate (QED) using (7), a precise value of is needed. At present, the best non-QED is the one obtained from the measurement of Bouchendira et al. (2011), combined with the very precisely known Rydberg constant and Mohr et al. (2012):


Actually, we have a more precise value of which is derived from the measurement Hanneke et al. (2008, 2011) and theory of the electron Aoyama et al. (2012a):


where the first three uncertainties are due to the eighth-order term, tenth-order term, and the hadronic and electroweak terms, involved in the evaluation of . The fourth uncertainty comes from the measurement of . At present the difference between (12) and (13) is much smaller than the current uncertainty in the measurement of so that one may use either one of these two. However, some caution must be exercised to employ to calculate , when more accurate experiment of becomes available, because theoretical calculation of is strongly correlated with that of .

Substituting (9), (10), and (11) in Eq. (7) and using (12), we obtain


where the uncertainties are from the lepton mass ratios, the eighth-order term, the tenth-order term, and the value of in (12), respectively. If we use the value of in (13) instead, we get


Note that the uncertainties of the lepton mass ratios, the eighth-order term, the tenth-order terms, and are improved by factors 1.7,  1.3,  20,  and 1.5, respectively, compared with given in Eq. (99) of Ref. Jegerlehner and Nyffeler (2009).

The difference between (14) and (15) is less than so that we may use either one as far as comparison with the current experimental data is concerned.

In view of the rather large value of one might wonder how large might be. As a matter of fact it is not difficult to estimate its size. For this purpose note that the dominant contribution to comes from the Group IV(a) and the dominant contribution to comes from the Set VI(a). Both are integrals obtained by inserting several second-order vacuum-polarization loops into the virtual photon lines of the sixth-order diagram which contains a light-by-light scattering electron loop. Analogously the leading contribution to the twelfth-order term will come from insertion of three ’s in , namely,


noting that and the factor 10 accounts for the possible ways of insertion of . Including the contribution of other diagrams, the size of the 12th-order term might be as large as . This is larger than the uncertainty of the 10th-order term in (14) so that it would be desirable to obtain at least a crude evaluation of this term.

Adding (3), (4), (5), (6), and (14), and using from (12), the theoretical value of in the standard model is given by


We have therefore


The size of discrepancy between theory and experiment has not changed much, since the tenth-order QED contribution is not a significant source of theoretical uncertainties. Let us emphasize, however, that the complete calculation of enables us to concentrate on improving the precision of the hadronic contributions.

We thank J. Rosner for a helpful comment. This work is supported in part by the JSPS Grant-in-Aid for Scientific Research (C)20540261 and (C)23540331. T. K.’s work is supported in part by the U. S. National Science Foundation under Grant No. NSF-PHY-0757868. T. K. thanks RIKEN for the hospitality extended to him while a part of this work was carried out. Numerical calculations were conducted on RSCC and RICC supercomputer systems at RIKEN.


  1. preprint: RIKEN-QHP-26


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