Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order

We compute the next-to-next-to-leading order hadronic contribution to the muon anomalous magnetic moment originating from the photon vacuum polarization. The corresponding three-loop kernel functions are calculated using asymptotic expansion techniques which lead to analytic expressions. Our final result, $a_\mu^{\rm had,NNLO} = 1.24 \pm 0.01 \times 10^{-10}$, has the same order of magnitude as the current uncertainty of the leading order hadronic contribution and should thus be included in future analyses.


Introduction
The anomalous magnetic moment of the electron and the muon are measured with high precision and at the same time also accurately predicted including high-order quantum corrections (see, e.g., Refs.[1][2][3][4] for reviews on this topic).Notable recent achievements in this context are the five-loop QED corrections obtained in Refs.[5,6].
In the case of the muon the largest input to the uncertainty comes from hadronic contributions which to a large extend rely on experimental measurements of the cross section σ(e + e − → hadrons).Several groups have performed the leading order (LO) [7][8][9][10] and next-to-leading order (NLO) [8,[11][12][13] analysis.In this paper we compute the next-tonext-to-leading order (NNLO) hadronic corrections to the anomalous magnetic moment of the electron and the muon.We evaluate the three-loop kernels in the limit M µ ≪ m π and show that four expansion terms are sufficient to obtain a precision far below the per cent level.Note that we do not consider the light-by-light contribution where the external photon couples to the hadronic loop (see, e.g., Ref. [14]) but only the contributions involving the hadronic vacuum polarizations.
In the next Section we briefly mention some technical details of our calculation and discuss the NLO contribution.Section 3 contains the results of the various NNLO contributions for the muon anomalous magnetic moment and in Section 4 we apply our results to the anomalous magnetic moment of the electron.We conclude in Section 5.

Technicalities and NLO result
The LO hadronic contribution to the anomalous magnetic moment of the muon (see Fig. 1) can be computed via where α is the fine structure constant and R(s) is given by the properly normalized total hadronic cross section in electron positron collisions with σ pt = 4πα 2 /(3s).A convenient integral representation for K (1) (s) is given by analytic results can be found in Refs.[15,16].A crucial input for the evaluation of a had µ is a compilation of the experimental data for R(s) as obtained by various experiments.In our analysis we use a FORTRAN code which is provided to us by the authors of Ref. [8].This gives us access to both the central values and the upper and lower limit of R(s).However, the use of the latter leads to a vast overestimation of the final uncertainty since we have no information about the correlations of the individual data points.Thus, we use a heuristic method and consider the difference between a had µ as obtained from the central and upper or lower limit of R(s) and divide it by three which leads to realistic (and still conservative) error estimates at LO and NLO.In fact, for the energy region [0.32 GeV, 1.43 GeV] we obtain the LO contribution 608.19 ± 3.97 × 10 −10 which is in good agreement with 606.50 ± 3.35 × 10 −10 from table 5 of Ref. [8].Note that in this paper we do not aim for an improved prediction of the LO or NLO contribution.Rather we present for the first time NNLO hadronic predictions.Obviously, for that purpose, the described prescription for the determination of the uncertainty is sufficient.
At NLO three different contributions are distinguished as shown in Fig. 1(b), (c) and (d).We have computed the kernels K (2a) and K (2b) using the methods of asymptotic expansion [17] and in that way confirmed the results provided in Ref. [11].Ref. [11] also contains analytic expressions for K (2c) (s, s ′ ).It is, however, convenient to work with the one-dimensional integral representation which reads [11] The contributions a are obtained from Eq. (1) after replacing K (1) by either K (2a) or K (2b) and (α/π) 2 by (α/π) 3 .a (2c) µ requires an integration over both s and s ′ and is obtained from Our results for the three contributions read which leads to in good agreement with Refs.[8,13].

NNLO hadronic contributions to a µ
We classify the NNLO contributions in analogy to NLO according to the number of hadronic insertions and closed electron loops.This leads to five different kernels which contain the following contributions (see Fig. 2 for sample Feynman diagrams): • K (3a) : two additional photons to the LO Feynman diagram.
• K (3b) : closed electron loop and additional photonic correction; the external photon couples to the muon.
• K (3b,lbl) : light-by-light-type contribution with closed electron loop; the external photon couples to the electron.
• K (3c) : two hadronic insertions; includes also contributions with closed electron and muon loops.
Note that we do not consider contributions with closed tau lepton loops since they are suppressed by an additional factor M 2 µ /M 2 τ .Actually, at NLO these contributions amount to 0.01 × 10 −10 and thus we anticipate that the corresponding NNLO terms are even smaller.
The calculation of K (3a) (s) proceeds in analogy to the corresponding one-and two-loop cases: we apply an asymptotic expansion for √ s ≫ M µ and compute terms up to order (M 2 µ /s) 4 .The minimal value of √ s is given by m π and thus the largest value of the expansion parameter is M 2 µ /m 2 π ≈ 0.6.Note, however, that the contribution from the energy interval [m π , 2m π ] is very small such that in practice the expansion parameter is M 2 µ /(4m 2 π ) ≈ 0.15 or smaller for higher energies.We observe a good convergence of the series as can be seen by considering the difference for a ) by including and neglecting the highest available term which is at the per mil level.For K (3b) and K (3b,lbl) we consider in addition the limit M µ ≫ M e and compute terms up to quartic order in M e .Corrections of order M e /M µ or higher turn out to be negligibly small.In the case of K (3b) the leading term for M e → 0 can be obtained using renormalization group techniques (see, e.g., Ref. [18] where four-loop correction to a µ with closed electron loops have been considered).However, a non-zero electron mass is crucial for the light-by-light-type contribution K (3b,lbl) since the Feynman integrals are divergent in case M e = 0 is chosen.Thus, a non-trivial asymptotic expansion has to be applied.The latter is realized with the help of the program asy [19,20].
For the computation of K (3c) (s, s ′ ) we use asymptotic expansions in the limits s ≫ s ′ ≫ M 2 µ , s ≈ s ′ ≫ M 2 µ and s ′ ≫ s ≫ M 2 µ and construct an interpolating function by combining the results from the individual limits.This procedure can be tested in the case of K (2c) (s, s ′ ) where a comparison to the exact result is possible.In Fig. 3(a) we show K (2c) (s, s ′ ) for √ s = 1 GeV as a function of √ s ′ .1 (For larger values of √ s the convergence properties are even better.)One observes that for each value of √ s ′ there is perfect agreement between the exact result (solid line) and at least one of the approximations (dotted and dashed lines).Furthermore, the final results for a (2c) µ computed from the exact and approximated kernels differ by less than 1%.Fig. 3(b) shows the corresponding results for K (3c) (s, s ′ ).For each value of s ′ we have at least two approximations which agree with each other.Thus it is evident that a function can be defined which agrees piecewise with one of the approximations.Figure 3: (a) Comparison of exact result (solid, black) for K (2c) (s, s ′ ) and the various approximations for s ≫ s ′ (blue, dotted), s ≈ s ′ (orange and red, short and medium dashed) and s ≪ s ′ (green long dashed) for For the kernel of the triple-hadronic insertion, K (3d) (s, s ′ , s ′′ ), we derive a one-dimensional integral representation which is given by We refrain from listing explicit results for the NNLO kernels but provide the results in computer-readable form on the web page [23].
For the computation of a (3b,lbl) µ one inserts the corresponding kernel in Eq. ( 1) and replaces (α/π) 2 by (α/π) 4 .Furthermore, a (3c) µ is obtained from Eq. ( 5) with K (2c) replaced by K (3c) and (α/π) 3 by (α/π) 4 and the three-fold hadronic insertion is calculated from For the individual NNLO contributions we obtain the results Our result is of the same order of magnitude as the uncertainty of the LO hadronic contribution.For example, in Ref. [8] an uncertainty of 3.72 × 10 −10 is quoted due to the statistical and systematic errors of the experimental data.Thus, the NNLO hadronic corrections should be included in the comparison with the experimental result for a µ .

NNLO hadronic contributions to a e
In this Section we apply our results to the electron anomalous magnetic moment, a e .At LO and for K (2a) this means that the lepton mass has to be interpreted as M e .K (2b) is absent and we have checked that K (2c) gives a negligible contribution (see also Ref. [11]).
The situation is analogous at NNLO where we only remain with K (3a) .
At LO and NLO our results for a e read a had,LO e = a which is almost three times larger than the uncertainty of a had,LO e quoted in Ref. [24].It is furthermore of the same order of magnitude as the hadronic light-by-light contribution which amounts to a had,lbl e = 0.035 ± 0.010 × 10 −12 [5].Note that currently both the uncertainty in the theory prediction for a e and the difference between theory and experiment is of order 1 × 10 −12 [5] which is about a factor 40 larger than the result given in Eq. ( 12).

Conclusions
We have computed the NNLO hadronic vacuum polarization corrections to the anomalous magnetic moment of the muon.Five different contributions can be distinguished which are discussed individually.The numerically largest contribution comes from the lightby-light-type diagram with a closed electron loop followed by the photonic corrections and the contribution containing a closed electron two-point function.Multiple hadronic insertions only lead to numerical results which are much smaller.The main result of this paper is given in Eq. (11).
In Ref. [6] the theory prediction a th µ = 116 591 840(59)10 −11 has been compared to the experimental result [25,26] a exp µ = 116 592 089(63)10 −11 which leads to a deviation of 2.9σ.After adding our result in Eq. ( 11) to a th µ this reduces to 2.7σ.As a by-product we have also evaluated the NNLO hadronic corrections to a e .Our result is larger than the uncertainty at LO and of the same order as the hadronic light-by-light contribution.However, it is significantly smaller than both the uncertainty from the fine structure constant and the experimental uncertainty for a e , see the discussion in Ref. [5].

Figure 1 :
Figure 1: LO and sample NLO Feynman diagrams contributing to a had µ .

Figure 2 :
Figure 2: Sample NNLO Feynman diagrams contributing to a had µ .