Burkhardt, Hans

Pietrzyk, B.

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Update of the hadronic contribution to the QEDvacuum polarizationH. Burkhardt, B. PietrzykTo cite this version:H. Burkhardt, B. Pietrzyk. Update of the hadronic contribution to the QED vacuum polariza-tion. Physics Letters B, Elsevier, 2001, 513, pp.46-52. <in2p3-00009886>HAL Id: in2p3-00009886http://hal.in2p3.fr/in2p3-00009886Submitted on 2 Aug 2001HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.LAPP-EXP 2001-03February 2001Update of the hadronic contribution to the QEDvacuum polarizationH. Burkhardt a, B. Pietrzyk baCERN, SL Division, CH-1211 Geneva 23, SwitzerlandbLaboratoire de Physique des Particules LAPP, IN2P3-CNRS, F-74941 Annecy-le-VieuxCedex, FranceAbstractWe have updated the evaluation of the hadronic contribution to the running of the QEDfine structure constant. It is obtained from a dispersion integral over a parametrization ofthe measured cross section of e+e− → hadrons. We find this contribution to be ∆α(5)had(s)= 0.02761 ± 0.00036 at s = m2Z corresponding to 1/α(5)(m2Z) = 128.936 ± 0.046. Theimproved experimental accuracy is mainly due to recent BES measurements. We have alsoupdated and parametrized the hadronic contribution to α as a function of energy.Email addresses: Helmut.Burkhardt@cern.ch (H. Burkhardt),pietrzyk@lapp.in2p3.fr (B. Pietrzyk).URLs: http://home.cern.ch/hbu/aqed/aqed.html (H. Burkhardt),http://alephwww.cern.ch/∼pietrzyk/ (B. Pietrzyk).Preprint submitted to Physics Letters 23 February 20011 IntroductionConsiderable progress has been made in understanding of the electroweak gaugetheory of elementary particle interactions in the last decade. Precision high energydata from LEP, SLC and Tevatron have clearly established the existence of elec-troweak radiative corrections. The dominant correction is provided by the runningof the electromagnetic coupling constant α from its value at vanishing momentumtransfer to high energies.A combined analysis of the high energy experimental data is performed by the LEPelectroweak working group [1]. Recent results from global fits have been reportedin [2]. The results are perfectly consistent with the standard model and allow adetermination of the standard model Higgs boson mass from radiative corrections.The Higgs mass obtained in these fits depends mainly on the value of the top massand the hadronic contribution of the five lighter quarks ∆α(5)had(s) to the running ofthe QED fine structure constant.The running of α can be expressed asα(s) =α(0)1−∆αl(s)−∆α(5)had(s)−∆αtop(s).The leptonic contribution∆αl(s) is known with high accuracy [3], so that its uncer-tainty can be neglected in the present analysis. The contribution from the top quarkloops ∆αtop(s) is treated separately, such that the top quark mass can be treated asparameter in standard model fits.The main subject of this paper is the up-to-date estimate of the contribution fromthe lighter five quark flavours ∆α(5)had(s). It is obtained as dispersion integral [4](Kramers–Kronig rule; the P stands for the principal value of the integral )∆αhad(s) = −α s3πP∞∫4m2πRhad(s′)s′(s′ − s)ds′ . (1)Rhad is the measured QED cross-section of the process e+e− → hadrons, nor-malized to the QED cross-section for lepton-pair production. The integration canbe done numerically or even analytically for energy intervals in which Rhad(s) isparametrized by straight lines. The uncertainty in the result obtained from Eq. (1)is almost entirely due to experimental errors in the determination of Rhad(s), andin particular free of assumptions on light quark masses and nonperturbative QCD.Computational uncertainties, introduced by the particular choice of the parametriza-tion and algorithm used, are negligible compared to the experimental uncertainty in2the data. This has been checked by comparing different programs and parametriza-tion methods [5].As in our previous analysis published in 1995 [6], we prefer to represent Rhad bysimple parametrizations, like broad averages and straight lines in the continuumand rely, whenever available, on published world averages. The uncertainty in ourprevious analysis was mainly due to the poor knowledge of the cross sections forc.m.s. energies (√s) from 1 to 5 GeV (see Fig. 2 in [6] ). Recently, the BES Collab-oration [7,8] measured Rhad(s) between 2 and 5 GeV with substantially improvedprecision compared to previous measurements.2 Updated analysisWe updated Rhad(s) and recalculated ∆α(s) using all currently available data. Ourparametrization of Rhad including resonances for centre-of-mass energies up to 10GeV is shown as a solid line in Fig. 1. The relative error of the continuum contribu-tion of Rhad is shown as a band and is given in numbers at the bottom of the figure.Measurements of various experiments are shown with their statistical errors.The contributions from the ρ, narrow resonances and different energy regions to∆α(5)had(m2Z) are summarized in Table 1 and illustrated in Fig. 2. Even with the re-cent significant improvement in measurements at lower energies, it is still the 1 to5 GeV energy range which dominates the uncertainty in ∆α(m2Z).Table 1Contributions to ∆α(5)had(m2Z)Range√s, ∆α Relative CommentsGeV errorρ 0.00350 2.3 % including CMD-2 at VEPP-2MNarrow resonances 0.00184 3.1 % PDG, undressed width, without ψ”..1.05 – 2.0 0.00156 1.5 % as in 1995 paper2.0 – 5.0 0.00381 5.9 % including BES, with ψ”..5 – 7 0.00183 6 % as in 1995 paper7 – 12 0.00304 1.4 % including CLEO> 12 0.01203 0.2 % third-order QCD0.02761 1.3 %The changes from the 1995 analysis are given in Table 2 and discussed below:- In the c.m.s. energy region between 0.61 and 0.96 GeV the results of the pion3Table 2Changes in ∆α(5)had(m2Z) from 1995 analysis∆α(5)had(m2Z) = 0.0280 ± 0.0007 1995 result−0.00030 2 – 5 GeV (BES)−0.00008 above 12 GeV (QCD)−0.00001 other changes (CMD-2, CLEO, PDG)∆α(5)had(m2Z) = 0.02761 ± 0.00036 this analysisform factor measurements with the CMD-2 detector at the VEPP-2M collider areknown to 1.4% systematic uncertainty. We have used the “hidden local parametriza-tion” of [9], with a small extra bump at 1.2 GeV, see Fig. 3. Only statistical errorsare shown in the figures. The overall uncertainty of the ρ region integral, includ-ing the statistical uncertainty, is 2.3% (that of Γee in [9]).- In the c.m.s. energy region between 2 and 5 GeV, results of Rhad measurementsat 85 different energies have been published recently by the BES Collaboration[7], in addition to the 6 points published earlier [8]. These measurements includethe contributions of the charmonium resonances above the ψ’. The number ofmeasured energy points is sufficiently large to allow us to connect and integratethe results directly and to distinguish between correlated systematic uncertaintiesand uncorrelated statistical errors. For the few overlapping energy points from[7] and [8], we used weighted means. The BES data now allows to obtain thedispersion integral in the 2 to 5 GeV energy range to 5.9%.This is a substantial improvement, and in fact the most significant single changecompared to our 1995 analysis, where Rhad in this energy range was only knownto 15%. As always in our analysis, the information from the older measurementsis kept. Here, this was achieved by a slight rescaling of the BES contribution.- In the c.m.s. energy region between 7 and 12 GeV, a new value for Rhad of3.56±0.01±0.07 was measured by the CLEO collaboration [10] at√s = 10.52.We use here the mean value of Rhad = 3.57 with the uncertainty of 1.4% obtainedfrom the combination of the CLEO result with the result of the compilation ofother measurements from our 1995 analysis.- Above the c.m.s. energy of 12 GeV, we previously relied on a direct parametriza-tion of PETRA, PEP and TRISTAN data made by D. Haidt [11]. This parametriza-tion was based on second-order QCD with αs = 0.146+0.031−0.026 at 34 GeV andimplied a value of 0.124+0.021−0.019 at mZ . This purely experimental approach wasmotivated in 1995 by large discrepancies between different values of αs(m2Z)measured in different experiments [12] ranging from the value 0.108 measuredin deep inelastic scattering to the value 0.127 measured from Rhad at LEP. A con-sensus on the value of αs(m2Z) and on the validity of perturbative QCD emergedsince that time. In this paper, we use third-order QCD to describe Rhad above 12GeV, with αs = 0.118 ± 0.002 at mZ [13]. This allows a determination of thedispersion integral in this energy range to 0.2% relative precision.- Narrow resonances were included as in our 1995 papers [6]. The contribution4is proportional to the undressed electronic width Γe. The values and errors areobtained by slight rescaling of the dressed widths listed in the year 2000 versionof the review of particle properties [13]. As mentioned above, the charmoniumresonances above the ψ’ are included in this paper as part of the continuum.3 Comparison of different estimates in the literatureFig. 4 shows a comparison of different estimates of∆α(5)had(m2Z) since 1989 [2,5,6,14–29]. Estimates up to 1995 were discussed in our previous paper [6].A number of analyses has been made since then, based on more theoretical descrip-tions of the hadronic cross section. The results are shown as open circles in Fig. 4.Rather compatible results were obtained from 1997 to 1999. It is interesting to seehowever, that the last of these estimates [29] used results from BES on Rhad(s) atlow c.m.s. energies and found a significantly lower central value than obtained inthe various estimates from 1997 to 1999.A preliminary value was released by us for the ICHEP 2000 in Osaka [2], in goodagreement with the more precise result presented in this paper.4 Parametrization as a function of energyA simple parametrization of the hadronic contribution to the vacuum polarization[30,5] is used in many computer programs. It is described by the formula∆αh(s) ∼= A+ B ln(1 + C|s|)with s or t instead of s, both in GeV2. We have updated the parameters A, B, andC for five different c.m.s. energy regions. This parametrization is better than 0.2σ in the whole t-channel and exact at mZ = 91.1876GeV in the s-channel. Thevalues of the parameters are given in Table 3. Computer code for ∆α with thisparametrization is available from the authors.5 ConclusionOur updated evaluation of the hadronic contributions to the running of the QEDfine structure constant α results in ∆α(5)had(m2Z) = 0.02761± 0.00036 at s = m2Z or1/α(5)(m2Z) = (1−∆αl(s)−∆α(5)had(s))/α(0) = 128.936± 0.046.5Table 3Parametrization of the hadronic vacuum polarizationrange√s, GeV A B C0.0 - 0.7 0.0 0.0023092 3.99253700.7 - 2.0 0.0 0.0022333 4.21917792.0 - 4.0 0.0 0.0024402 3.24966844.0 - 10.0 0.0 0.0027340 2.099509210.0 - mZ 0.0010485 0.0029431 1.0mZ - 10 000.0 0.0012234 0.0029237 1.010 000.0 - 100 000.0 0.0016894 0.0028984 1.0The most important energy range for further improvements is still the low energyregion, mainly from 1 to 5 GeV c.m.s., and efforts for more precise measurementsshould be strongly encouraged.AcknowledgementWe would like to thank Zhengguo Zhao and his BES team, S.I. Eidelman fromNovosibirsk and the LEP electroweak working group with Bob Clare, Martin Gru¨newaldand Gu¨nter Quast.References[1] LEP Electroweak Working Group, http://lepewwg.web.cern.ch/LEPEWWG/.[2] B. Pietrzyk, The global Fit to Electroweak Data, preprint LAPP-EXP 2000-06, toappear in the Proc. of ICHEP2000, Osaka, Japan, July 2000.[3] M. Steinhauser, Phys. Lett. B429 (1998)158.[4] N. Cabbibo and R. Gatto, Phys. Rev. 124 (1961) 1577.[5] H. Burkhardt, F. Jegerlehner, G. Penso and C. Verzegnassi, Z. Phys. C43 (1989) 497.[6] H. Burkhardt and B. Pietrzyk, Phys. Lett. B356 (1995) 398.[7] J.Z. Bai et al., BES Collab., preprint hep-ex/0102003.[8] J.Z. Bai et al., BES Collab., Phys. Rev. Lett. 84 (2000) 594.[9] R.R. Akhmetshin et al., CMD-2 Collab., hep-ex/9904027.6[10] R. Ammar et al., CLEO Collab., Phys. Rev. D57 (1998) 1350.[11] D. Haidt, in ‘Directions in High Energy Physics’, Vol. 14, Precision Tests of theStandard Electroweak Model, Ed. P. Langacker, World Sc. Comp., 1995.[12] S. Bethke - page 213, J. Casaus - page 219, D. A. Harris - page 247, Proc. of theXXXth Rencontres de Moriond, Les Arcs, March 18-25, 1995, Ed. J. Traˆn ThanhVan, Editions Frontieres.[13] Review of Particle Properties, Eur. Phys. J. C15 (2000) 1.[14] F. Jegerlehner in ‘Testing the Standard Model’, eds. Cveticˇ, Langacker, WorldScientific, Singapore, 1991, p. 476.[15] R.B. Nevzorov, A.V. Novikov and M.I.Vysotsky, JETP Lett. 60 (1994) 399.[16] B.V. Geshkenbein and V.L. Morgunov, Phys. Lett. B340 (1994) 185, B352 (1995) 456.[17] A.D. Martin and D. Zeppenfeld, Phys. Lett. B345 (1994) 558.[18] M.L. Swartz, SLAC–PUB–6710, Nov. 1994, hep-ph/9411353.[19] M.L. Swartz, Phys. Rev. D53 (1996) 5268.[20] S. Eidelman and F. Jegerlehner, Z. Phys. C67 (1995) 585.[21] K. Adel and F. Yndurain, FTUAM-95-32, Sept. 1995, hep-ph/9509378.[22] R. Alemany et al., Eur. Phys. J. C2 (1998) 123.[23] M. Davier and Ho¨cker, Phys. Lett. B419 (1998) 419.[24] J. Ku¨hn and M. Steinhauser, Phys. Lett. B437 (1998) 425.[25] S. Groote, J. G. Ko¨rner, K. Schilcher and N. F. Nasrallah, Phys. Lett. B 440 (1998)375.[26] M. Davier and Ho¨cker, Phys. Lett. B435 (1998) 427.[27] F. Jegerlehner, Hadronic Effects in (g-2) and αQED(MZ ) : Status and Perspectives,Proc. of Int. Symp. on Radiative Corrections, Barcelona, Sept. 1998, Ed. Joan Sola`,World Scientific, Singapore, 1999, page 75.[28] J. Erler, Phys. Rev. D59 (1999) 054008.[29] A.D. Martin, J. Outhwaite and M.G. Ryskin, Phys. Lett. B492 (2000) 69.[30] H. Burkhardt, TASSO Note 192 (1981) and DESY F35–82–03 (thesis).7012345670 1 2 3 4 5 6 7 8 9 10Bacci et al.Cosme et al.Mark IPlutoCornell,DORISCrystal BallMD-1 VEPP-4VEPP-2M NDDM2BES 1999BES 2001Burkhardt, Pietrzyk 200115 % 5.9 % 6 % 1.4 %rel. err. cont.Rhadρ,ω,φ Ψ's Υ's√s in GeVFig. 1. Rhad including resonances. Measurements are shown with statistical errors. In ad-dition there are overall systematic errors (up to 20% in case of Mark I). The relative un-certainty assigned to our parametrization is shown as band and given with numbers at thebottom.8> 12 GeV7 - 12 GeV5 - 7 GeV2.0 - 5 GeV1.05 - 2.0 GeVnarrow resonancesρ> 12 GeV7 - 12 GeV5 - 7 GeV2.0 - 5 GeV 1.05 - 2.0 GeVnarrow resonancesρcontributions at mZin magnitudein uncertaintyBurkhardt, Pietrzyk 2001Fig. 2. Relative contributions to ∆α(5)had(m2Z) in magnitude and uncertainty.91100.4 0.6 0.8 1 1.2 1.4NA7VEPP-2M TOFVEPP-2M OLYAVEPP-2M CMDACO DM1 Quenzer et al.VEPP-2M Koop et al.VEPP-2M CMD-2√s in GeV| Fπ(s) |2Fig. 3. The pion form factor.100.026 0.027 0.028 0.029 0.03Burkhardt, Jegerlehner, Verzegnassi, Penso 1989Jegerlehner 1992Nevzorov 1994Geshkenbein, Morgunov 1994Geshkenbein, Morgunov 1995Eidelman, Jegerlehner 1995Burkhardt, Pietrzyk 1995Adel, Yndurain 1995Alemany, Davier, Höcker 1997Davier, Höcker 1997Kühn, Steinhauser 1998Groote, et al. 1998Davier, Höcker 1998Jegerlehner 1999Erler 1999Osaka 2000Martin, Outhwaite, Ryskin 2000Burkhardt, Pietrzyk 2001Martin, Zeppenfeld 1994Swartz 1994Swartz 1995update of Burkhardt, Pietrzyk 1995this paper∆αhad(mz)2Fig. 4. Comparison of recent estimates of ∆α(5)had(m2Z). Estimates based on dispersion in-tegration of the experimental data are shown with solid dots and estimates relying on addi-tional theoretical assumptions shown as open circles.11

Update of the hadronic contribution to the QED vacuum polarization

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