(g-2)_μ at four loops in QED
Peter Marquard, Alexander V. Smirnov, Vladimir A. Smirnov, Matthias Steinhauser, David Wellmann
aa r X i v : . [ h e p - ph ] S e p ( g − µ at four loops in QED Peter Marquard , , AlexanderV. Smirnov , , VladimirA. Smirnov , ,Matthias Steinhauser ,⋆ , and David Wellmann , Deutsches Elektronen-Synchrotron, DESY, 15738 Zeuthen, Germany Research Computing Center, Moscow State University, 119991, Moscow, Russia Skobeltsyn Instituteof Nuclear Physics of Moscow State University, 119991, Moscow, Russia Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT),76128 Karlsruhe, Germany
Abstract.
We review the four-loop QED corrections to the anomalous magnetic momentof the muon. The fermionic contributions with closed electron and tau contributions arediscussed. Furthermore, we report on a new independent calculation of the universalfour-loop contribution and compare with existing results.
The anomalous magnetic moment of the muon, which is usually written as a µ = ( g − µ /
2, measuresthe deviation from Dirac’s prediction g =
2. Experimentally it is known with high precision frommeasurements at BNL [1, 2] a exp µ =
116 592 089(63) × − . (1)It is expected that the uncertainty will be reduced in the coming years. Actually, there are two experi-ments which are currently under construction, one at Fermilab and one at J-PARC [3–5]Also on the theory side an impressive precision has been reached. However, since many yearsthere is a persistent discrepancy of the order of about 3 sigma. The uncertainty of the theory predictionis dominated by the hadronic contributions, both from the vacuum polarization [6–8] (see Refs. [9–11]for a recent compilations) and the so-called light-by-light part [12].The numerically largest contribution to a µ is given by the QED part which is known up to fiveloops. One-, two- and three-loop corrections are known analytically from Refs. [13–16] and four-and five-loop contributions have been computed in Refs. [17–20] using numerical methods. Thefermionic contributions involving closed tau and electron loops have been cross checked in Refs. [21–23]. Very recently, semi-analytic results for the universal contribution, i.e., the purely photonic andmuon-loop contribution, have been obtained in a remarkable calculation by Laporta [24]. It is basedon an evaluation of Feynman integrals with high-precision (several thousand digits) which was enoughto reconstruct rational coe ffi cients of known transcendental constants with the help of the PSLQ al-gorithm [25]. In addition, there were several contributions which were not recognized as knownconstants. The final result for the four-loop contribution to a µ from [24] is known to 1100 digits. ⋆ e-mail: [email protected] n this work we present results of an independent calculation of the universal contribution.In order to fix the notation we provide numerical results for a µ up to five-loop order which aregiven by (numbers are taken from Refs. [19, 20]) a µ = ( g − µ = α π (2) + ( − .
328 478 . . . + .
094 336 . . . | e ,τ ) (cid:18) απ (cid:19) + (1 .
181 241 . . . + .
869 268 . . . | e ,τ ) (cid:18) απ (cid:19) + ( − .
912 98(84) + .
790 3(60) | e ,τ ) (cid:18) απ (cid:19) + (9 . + . | e ,τ ) (cid:18) απ (cid:19) , where the ellipses indicate that the numbers are truncated and actually more digits are known. The uni-versal part (first number in the brackets) has been separated from electron and tau contributions (sec-ond number), which appears for the first time at two loops. Note that the latter is numerically dominantdue to unsuppressed large logarithms of the ratio of the electron and muon mass, log( m µ / m e ) ≈ . ℓ -loop order such logarithms occur up to the ( ℓ − m µ / m τ . They are numericallysmall.Let us note that up to terms suppressed by m e / m µ the first numbers in the coe ffi cients of Eq. (2)coincide with the anomalous magnetic moment of the electron, a e .It is interesting to note that after inserting the fine structure constant the four-loop coe ffi cientevaluates to a (8) µ = ( − .
912 98 + .
790 3 | e ,τ ) (cid:18) απ (cid:19) ≈ × − , (3)which is of the same order of magnitude as the current di ff erence between the Standard Model predic-tion of a µ and the experimental value given in Eq. (1). Furthermore, it is larger than the uncertaintiesof the hadronic vacuum polarization and light-by-light contributions which are both of the order of40 × − . Thus, an independent cross check of the four-loop QCD contributions is indispensable. The techniques used to obtain the results in Refs. [21–23] and for the universal part, which we reportbelow, have largely been developed in the context of the MS-on-shell quark mass relation in QCD.To obtain the mass relation one has to evaluate on-shell integrals up to four loops which are also thebasis for the anomalous magnetic moment. In fact, we use the same integral families as defined inRefs. [26, 27] and express the four-loop expression for a µ as a linear combination of scalar integrals.The latter are reduced to master integrals with the help of FIRE [28] and
Crusher [29]. Let us mentionthat the reduction of the integrals contributing to a µ is more expensive since vertex integrals (insteadof two-point functions) are considered which are expanded around vanishing momentum transfer ofthe photon. Thus, in the corresponding integrals the total power of the propagators is increased byat least two as compared to the integrals needed for the MS-on-shell relation. For the MS-on-shellrelation we have to evaluate 386 master integrals; a subset of 357 master integrals contribute to a µ .For details concerning their evaluation we refer to Ref. [27]. To obtain the precision mentioned belowsome of the master integrals had to be evaluated with higher precision following the methods of [27].Additional work is needed for the fermionic contributions with closed electron or tau loops. Inboth cases it is appealing to perform an asymptotic expansion either for m e ≪ m µ or m µ ≪ m τ . Theatter is a Euclidean-like asymptotic expansion which can be performed with the help of the program exp [30, 31]. The most complicated integrals which have to be evaluated are four-loop vacuumintegrals which are well studied in the literature (see, for example, Ref. [27] and references therein).All other contributions are of lower loop order and also known analytically. Thus, the four-loopcontribution to a µ containing tau leptons is known analytically as a series in m µ / m τ which is rapidlyconverging [21].To obtain an expansion of the electron-loop contribution in m e / m µ an asymptotic expansion aroundthe on-shell limit has to be performed. The complicated integrals one has to compute are either of on-shell type (as for the universal contribution) or integrals which contain linear propagators of the form1 / p · q where q = m µ is the external momentum and p is a loop momentum. Some integrals of this typecan be computed analytically, others are computed numerically using FIESTA [32]. In Refs. [22, 23]expansion terms up to order m e / m µ have been computed which show a good convergence behaviour.Let us remark that the numerically dominant contribution arises from the light-by-light-type contribu-tions which have been computed in [22].In Refs. [17–19] a completely di ff erent technique has been used to compute the four-loop correc-tions to a µ . In a first step a finite expression is constructed by generating the proper countertermstogether with four-loop diagrams which is afterwards integrated numerically.In Ref. [24], similar to our approach, all occurring integrals are reduced to a small set of masterintegrals. However, di ff erent software is used and most probably also a di ff erent basis of masterintegrals is chosen. Furthermore, Ref. [24] manages to obtain high-precision numerical expressionsfor all master integrals whereas we have chosen a more automated approach and stopped manipulatingthe integrals once the desired precision has been reached. Let us start with discussing the universal part to a µ which consists of the pure photon contributionand the contribution with closed muon loops. It can be subdivided into six gauge invariant subsets; arepresentative diagram for each one is shown in the first column of Tab. 1. The second column in Tab. 1contains the corresponding results from Ref. [33], this work, and Ref. [24], respectively (from top tobottom). The results from Ref. [33] are taken from Table I of that reference and the uncertaintiesare added in quadrature in case several contributions had to be combined. The uncertainty of theresults obtained in this work are the quadratically combined results from the individual ǫ coe ffi cientsof the master integrals. We refrain from introducing a “security factor” (as, e.g., in Ref. [27]) for theuniversal contribution since the four-loop result for a µ has also been computed by two other groups.There is no uncertainty in the result provided in Ref. [24].Within the given uncertainties the results from [33] and this work agree with the semi-analyticexpressions of [24]. In most cases our uncertainty is at the per cent level or below, except for thecontribution in the second row where a 40% uncertainty is observed. Note, that the absolute size ofthe uncertainty is of the same order as the one in the first and third row. However, due to cancellationsfrom individual contributions, the central value is significantly smaller.In the following we summarize the four-loop QED contributions and compare the results from thedi ff erent groups. Denoting the coe ffi cient of ( α/π ) by a (8) µ we haveuniversal e − τ e − + τ a (8) µ = − . + . + . + . a (8) µ = − .
912 98(84) + . + . + . a (8) µ = − . . . . [24]epresentative Contribution of a µ Feynman diagram − . ± . − . ± . − . . ± . . ± . . − . ± . − . ± . − . − . ± . − . ± . − . . ± . . ± . . . . . Table 1.
The three numbers given in each row (from top to bottom) are taken from [33], this work, and [24],respectively.
Note that the uncertainties in the first line in the parts involving a tau lepton are due to the leptonmasses only. After multiplication with ( α/π ) we obtain for the three equations( − . + . . + . + . × − this work and [21–23]( − . + . + . + . × − [19]( − . . . . + . . . ) × − [24]The uncertainty of our result is about two orders of magnitudes larger. It is nevertheless much smallerthan the current and foreseen uncertainties from both experiment and the hadronic contributions. Thiscan be seen by considering the di ff erence between the experimental result and the Standard Modelprediction which is given by (see, e.g., Ref. [19]) a µ (exp) − a µ (SM) ≈ × − . The uncertainty is about two orders of magnitude larger than our numerical uncertainty cited above.This remains even true after applying the improvements by a factor 4. Thus, it can be claimed thatthe four-loop contribution for a µ is cross-checked: There are three independent calculations for theuniversal part and the electron and tau contributions have been computed by two independent groups.Let us finally remark on a e . The Standard Model prediction given in Ref. [24] reads a e (SM) =
115 965 218 . × − , (4)here the three uncertainties have their origin in the numerical accuracy of the five-loop calculation,the hadronic and electroweak corrections and the fine structure constant. Due to the result of Ref. [24]an additional uncertainty of “(60)”, which is still present in [33], has been removed. Note that ourresult for the universal part of a µ can also be applied to a e . However, since it has an uncertainty whichis two orders of magnitude larger than the one cited in [33] it is not competitive to [33] and [24]. We summarize all four-loop QED contributions to the anomalous magnetic moment of the muon.They have been computed for the first time in Refs. [17–19]. An independent cross check of the tau-loop contributions can be found in Ref. [21] where analytic results are provided for the expansion in m µ / m τ . The electron-loop contributions have been cross checked in Refs. [22, 23] where an asymp-totic expansion in m e / m µ has been used. An independent semi-analytic calculation of the universal(purely photonic and muon-loop) contribution has been obtained in Ref. [24]. In this work we provideyet another independent cross check. In summary, all four-loop QED contributions to a µ have beencomputed by at least two groups independently using completely di ff erent methods. Acknowledgments
We thank the High Performance Computing Center Stuttgart (HLRS) and the Supercomputing Cen-ter of Lomonosov Moscow State University for providing computing time used for the numericalcomputations with
FIESTA . P.M. was supported in part by the EU Network HIGGSTOOLS PITN-GA-2012-316704.
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