We use Pade Approximants to obtain improved predictions for the anomalous
magnetic moments of the electron and the muon. These are needed because of the
very precise experimental values presently obtained for the electron, and soon
to be obtained at BNL for the muon. The Pade prediction for the QED
contribution to the anomalous magnetic moment of the muon differs significantly
from the naive perturbative prediction.Comment: 8 pages (LateX); SLAC-PUB-6670, CERN-TH-7451/94, TAUP-2201-94,
  OSU-RN-393/94. Typo correcte

Ellis, John

Karliner, Marek

Samuel, Mark A.

Steinfelds, Eric

English

arXiv

We use Padé Approximants to obtain improved predictions for the anomalous magnetic moments of the electron and the muon. These are needed because of the very precise experimental values presently obtained for the electron, and soon to be obtained at BNL for the muon. The Padé prediction for the QED contribution to the anomalous magnetic moment of the muon differs significantly from the naive perturbative prediction.We use Pade Approximants to obtain improved predictions for the anomalous magnetic moments of the electron and the muon. These are needed because of the very precise experimental values presently obtained for the electron, and soon to be obtained at BNL for the muon. The Pade prediction for the QED contribution to the anomalous magnetic moment of the muon differs significantly from the naive perturbative prediction

Ellis, John R.

CERN Document Server

HEP-PH-9409376SLAC-PUB-6670CERN-TH.7451/94TAUP-2201-94OSU-RN-293/94hep-ph/9409376The Anomalous Magnetic Moments of theElectron and the Muon { Improved QEDPredictions using Pade ApproximantsJohn Ellisy, Marek Karlinerzand Mark A. SamuelxStanford Linear Accelerator CenterStanford University, Stanford, California 94309, USAandEric SteinfeldsDepartment of PhysicsOklahoma State UniversityStillwater, Oklahoma 74078, USAAbstractWe use Pade Approximants to obtain improved predictions for the anomalousmagnetic moments of the electron and the muon. These are needed becauseof the very precise experimental values presently obtained for the electron,and soon to be obtained at BNL for the muon. The Pade prediction for theQED contribution to the anomalous magnetic moment of the muon dierssignicantly from the naive perturbative prediction.Work supported by the Department of Energy, contract DE-AC03-76SF00515.yPermanent address: Theory Division, CERN, CH-1211, Geneva 23, Switzerland;e-mail: johne@cernvm.cern.chzPermanent address: School of Physics and Astronomy, Raymond and Beverly SacklerFaculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel; e-mail:marek@vm.tau.ac.ilxPermanent address: Department of Physics, Oklahoma State University, Stillwater,Oklahoma 74078, USA; e-mail: physmas@mvs.ucc.okstate.edu1Two of the most important tests of quantum electrodynamics (QED) arethe comparisons between theory and experiment of the anomalous magneticmoments of the electron and the muon, aeand arespectively, where a =(g 2)=2: The latest Penning trap measurements of the electron and positronanomalies obtained by the University of Washington group1are:aexpte = 1159652188:4(4:3)  10 12(1)andaexpte+= 1159652187:9(4:3)  10 12(2)The gures in brackets represent the error in the last 2 gures, a conventionwe will follow throughout this paper. Taking the average of eqs (1) and (2),one ndsaexpte= 1159652188:2(3:0)  10 12(3)The most accurate measurement for the muon anomaly comes from theCERN g 2 experiment2in which it was found thataexpt = 1165936(12)  10 9(4)andaexpt+= 1165910(11)  10 9(5)and the combined result isaexpt= 1165923(9)  10 9(6)where correlations are taken into account in combining the errors. A newg 2 muon experiment is being done at Brookhaven National Laboratory(BNL)3, and an improvement in the accuracy by a factor of about 20 isexpected. In order to compare properly theory and experiment, one mustimprove correspondingly the accuracy of the theoretical predictions.In an heroic feat, Kinoshita4has calculated aein eighth order and Ki-noshita, Nizic, Okamoto5and Marciano6have calculated ain eighth or-der. Moreover, there have been some recent improvements in the analyticcalculations7;8of aeand a.There have recently been several papers estimating coecients in Pertur-bative Quantum Field Theory (PQFT) using Pade Approximants9;10. This2procedure is known to give signicant improvements on naive perturbativecalculations in many condensed-matter applications11, removes a large partof the discrepancy between experiment and QED calculations of the ortho-positronium decay rate10;12and agrees well with other estimates of higher-order perturbative coecients in QCD9;13.In this paper we will use Pade Approximants (PA's) to estimate, not justthe next term in the perturbation series, but the entire sum of the series(as is frequently done in condensed-matter applications), for both aeanda. We obtain in this way a more accurate theoretical prediction of theQED contribution to a, in particular, which lies outside the errors quotedpreviously.The rst step is to obtain an accurate value for the ne-structure constant. The two most precise measurements of  are14 1= 137:0359979(32) (7)and15 1= 137:0359840(50) (8)We note that these two values dier by more than 2 standard deviations, butnevertheless take the average of eqs (7) and (8) to obtain 1exp= 137:0359939(27) (9)The accuracy of this result limits the precision of tests of QED in the caseof ae, where both theory and experiment are extremely precise. The pertur-bation series for aeis4ae=12()  0:328478965()2+ 1:17611(42)()3  1:434(138)()4(10)and the error in the theoretical prediction is dominated by the error in exp.The [N/M] Pade Approximant to a seriesS = S0+ S1x+ S2x2+ :::+ SN+MxN+M(11)is given by[N=M ] =a0+ a1x+ :::+ aNxN1 + b1x+ :::+ bMxM(12)3where one chooses the coecients ai, bjso that[N=M ] = S +O(xN+M+1) (13)One can use such a PA either to predict the next coecient SN+M+1or toevaluate [N=M ] for the relevant value of x (in our case x =), and obtainan estimate for the sum of the series. Here we do the latter. The PA's areknown to accelerate the convergence of many series by including the eectsof higher (unknown) terms, thus providing a more accurate estimate of theseries11. The PA's also provide reliable estimates of many asymptotic series,as is the case in QED10and QCD9.For our application, we rst construct PA's to aeafter removing an overallmultiplicative factor of (). Our result for the [1/2] PA is[1=2] = 1159652169:1(24:0)  10 12(14)and the [2/1] PA agrees very well with the [1/2]:[2=1] = 1159652169:0(24:0)  10 12(15)The errors consist of 22.8 from  and 7.4 from the theoretical uncertainty.To obtain aeone must add the contribution due to muon diagramsae(muon) = 2:8 10 12(16)the contribution due to  diagramsae(tau) = 0:01  10 12(17)the hadronic contributionae(hadron) = 1:6(2) 10 12(18)and the purely weak contributionae(weak) = 0:05  10 12(19)for a total ofae= 4:5(2)  10 12(20)4Thus the theoretical prediction for aeisae= 1159652173:5(24:0)  10 12(21)Comparing eq (21) with eq (3), we see that there is beautiful agreementbetween theory and experiment:aexpte  ae= 14:7(24:0)  10 12(0:61) (22)As noted before, the error in eq (21) is dominated by the error in . Ifone now assumes that QED is correct, and hence that theory and experimentagree, one obtains a new and more accurate value of : th, where the [1/2]PA givesth= 137:03599228(86) (23)and the [2/1] PA leads toth= 137:03599227(86) (24)Comparing eq (23) with eq (9), one sees that there is beautiful agreementwith the less-precise experimental value. 1expt   1th= 16(28)  10 7(0.57) (25)corresponding to the good agreement in eq (22). We note in passing thatthis provides an a posteriori justication for averaging naively the two mostaccurate measurements14;15of , and that the dierence between the valuesof  1thextracted using the perturbative series and the PA's is just 3 10 7.We now turn to the anomalous magnetic moment of the muon, a. As isusual, we rst consider the dierence6;8a  ae= 1:09433583(7)()2+ 22:869265(4)()3+ 127:00(41)()4(26)In constructing a PA to this series, we must rst remove an overall factor()2from the perturbative series. In this way, we obtain the [1/1] PA value(a  ae)[1=1] = 6194839(12)  10 12(27)whereas the value from the series in eq (26) isa  ae= 6194791(12)  10 12(28)5Adding aefrom eq (21), after subtracting aefrom eq (20), we obtainaQED= 1165847008(12)(27)  10 12(29)where the rst error is due to numerical integrations used in evaluating theperturbative series, and the second error is due to . This should be com-pared the value of Kinoshita and Marciano6aQED= 1165846955(44)(27)  10 12(30)We note that the dierence between these two estimates of aQEDis consid-erably larger than the error propagated from . The reason for our smallererror is that we have used the new more precise values in ref. [8].If one now adds the hadronic16and the weak6contributionsa(had) = 7011(76)  10 11(31)anda(weak) = 195(10)  10 11(32)one obtains the theoretical valuea= 116591907(77)  10 11(33)The error is dominated by the error in a(had), and new, more preciseexperiments are underway in Novosibirsk and Frascati17to reduce this error.Comparing eq (33) with eq (6), we obtainaexpt  a= 4(9)  10 9(0:4) (34)The error in the dierence between theory and experiment is dominated bythe experimental error in eq (6), which should be reduced by a factor of 20in the forthcoming BNL experiment3.In summary, we have used PA to obtain new more precise values forthe QED values of aeand a. These PA values, in eect, estimate theunknown higher-order contributions, and should be more precise than thenaive perturbative values used previously. It would be interesting to compareour estimates with values obtained in a dierent way, for example usingthe eective charge approach13which agrees very well with PA's in QCDapplications9. Although smaller than some of the other uncertainties, the6shift we nd in a, in particular, is signicantly larger than other theoreticaluncertainties and the error due to .J. E., M. K. and M. S. would like to thank the theory group at SLAC for itsvery kind hospitality. The research of M.K. was supported in part by grantNo. 90-00342 from the United States-Israel Binational Science Foundation(BSF), Jerusalem, Israel, and by the Basic Research Foundation administeredby the Israel Academy of Sciences and Humanities. The research of M.A.S.was supported by the U.S. Department of Energy under Grant No. DE-FG02-94ER40852.References[1] R. S. Van Dyck, Jr. P. B. Schwinberg and H. G. Dehmelt, Phys. Rev.Lett. 59, 26(1987).[2] J. Bailey et al , Phys. Lett. 68B, 191(1977).[3] Proposal for Brookhaven National Laboratory experiment E821, A Pre-cision Measurement of the Muon g 2 Value at the level of 0.35 ppm,E. Benedict et al. (1986); co-spokesmen V.W. Hughes, W.M. Morse andB.L. Roberts.[4] T. Kinoshita, in Quantum Electrodynamics, edited by T. Kinoshita(World Scientic, Singapore, 1990), p.218.[5] T. Kinoshita, B. Nizic and Y. Okamoto, Phys. Rev. D41, 593(1990).[6] T. Kinoshita and W. J. Marciano, in Quantum Electrodynamics [3],p.419.[7] M. A. Samuel and G. Li, Phys. Rev. D44, 3935(1991) and D48,1879(1993) erratum; G. Li, R. Mendel and M. A. Samuel, Phys. Rev.D47, 1723(1993).[8] S. Laporta and E. Remiddi, Phys. Lett. B301, 440(1993); S. Laporta andE. Remiddi, Phys. Lett. B265, 182(1991); S. Laporta, Phys. Lett. B312,495(1993); S. Laporta, Nuovo Cimento 106, 675(1993); S. Laporta, Phys.Lett. B328, 522(1994); S. Laporta, Phys. Rev. D47, 4793(1993).7[9] M. A. Samuel, G. Li and E. Steinfelds, Phys. Rev. D48, 869(1993); Phys.Lett. B323, 188(1994); M. A. Samuel, G. W. Li and E. Steinfelds, OnEstimating Perturbative Coecients in Quantum Field Theory, Con-densed Matter Theory and Statistical Physics, Oklahoma State Univ.Research Note 278, August(1993); M. A. Samuel and G. W. Li, Esti-mating Perturbative Coecients in High Energy Physics and CondensedMatter Theory, Oklahoma State Univ. Research Note 275(1992), to bepublished in Intl. Jrnl. of Theo. Phys.(1994).[10] M. A. Samuel and G. W. Li, Phys. Lett. B331, 114(1994).[11] For a review, see George A. Baker, Jr., Essentials of Pade Approximants,Academic Press (1975).[12] I.B. Khriplovich and A.I. Milstein, Budker Inst. report BINP-94-30, hep-ph/9404213.[13] A. L. Kataev and V. Starshenko, CERN-TH.7198/94, hep-ph/9405294and CERN-TH.7400/94, hep-ph/9409395; see also P.M. Stevenson,Phys. Rev. D23, 2916(1981); G. Grunberg, Phys. Lett. 95B, 70(1980),E { ibid. 110B, 501(1982); Phys. Rev. D29, 2315(1984). For an alterna-tive approach to reducing ambiguities in higher orders, see S.J. Brodsky,G.P. Lepage and P.B. Mackenzie, Phys. Rev. D28, 228(1983), S.J. Brod-sky and H.J. Lu, SLAC-PUB-6481, hep-ph/9405218.[14] M. E. Cage et al, IEEE Trans. on Instrum. Meas. 38, 284(1989).[15] E. R. Williams et al, IEEE Trans on Instrum. Meas. 38, 233(1989).[16] L. Martinovic and S. Dubnicka, Phys. Rev. D42, 884(1990).[17] For a discussion of the feasibility of such an experiment, see V.W. Hughesand T. Kinoshita, Comments Nucl. Part. Phys. 14, 341(1985).8

The Anomalous Magnetic Moments of the Electron and the Muon - Improved QED Predictions Using Pade Approximants

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