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Recent Progress on Isospin Breaking Corrections and Their Impact on the Muon G −2 Value *

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Recent Progress on Isospin Breaking Corrections and Their Impact on the Muon G −2 Value * CPC(HEP & NP), 2010, 34(6): 1{6 Chinese Physics C Vol. 34, No. 6, Jun., 2010 Recent progress on isospin breaking corrections and their impact on the muon g −2 value * Gabriel L´opez Castro1) Departamento de F´ısica, Cinvestav, Apartado Postal 14-740, 07000 M´exico, D.F M´exico Abstract We describe some recent results on isospin breaking corrections which are of relevance for predictions had;LO of the leading order hadronic contribution to the muon anomalous magnetic moment aµ when using τ lepton data. When these corrections are applied to the new combined data on the ππ0 spectral function, the had;LO + − prediction for aµ based on τ lepton data gets closer to the one obtained using e e data. Key words isospin breaking, tau decays, muon magnetic moment, radiative corrections PACS 12.40.Bv, 13.35.Dx, 13.40.Em, 13.40.Ks 1 Introduction τ ππν predicted from e+e− data corrected by IB ! effects was underestimated by more than 4σ with re- Currently, the accuracy of the Standard Model spect to the average of direct measurements [4, 6]. prediction of the muon anomalous magnetic moment Given this `e+e− vs τ discrepancy'2), it is believed aµ = (g 2)=2 is limited by the uncertainties of that τ decay data does not provide at present a re- − had;LO hadronic contributions [1, 2]. The dominant term in liable determination of aµ (currently, other use- had;LO the leading order hadronic contribution aµ and ful contributions from τ decay data involve only the an important part of its associated uncertainty is pro- 2π and 4π channels [2]). Even though unidentified vided by the ππ spectral function, which can be mea- errors may be affecting e+e− and/or τ decay data, sured in e+e− annihilations and in τ lepton decays understanding IB corrections becomes crucial to gain (more details about their current status are given confidence about this important 2π contribution and, in the accompanying contribution by Michel Davier when consistency is achieved, to have a more precise had;LO [2]). Owing to the isotopic properties of the elec- prediction of aµ from combined results. tromagnetic and ∆S = 0 weak vector currents, the In this contribution we summarize some recent re- so-called Conserved Vector Current (CVC) hypothe- sults on the isospin breaking corrections that are rel- sis, the spectral functions themselves and their con- evant for understanding such discrepancies. As it is had;LO tributions to aµ must be the same after isospin discussed in [2, 7], their application to the evaluation had;LO breaking (IB) corrections are appropriately applied of aµ from τ data leads to a value [7] that is closer to input data [3]. to e+e−-based calculations. The prediction of the In recent years, a comparison of e+e− and τ based τ ππν branching fraction based on the IB corrected ! measurements of the ππ spectral functions in the e+e− data also shows a reduced discrepancy with re- timelike region, have shown large discrepancies for spect to results of direct measurements [2, 7]. A pre- center of mass energies above the ρ(770) resonance liminary version of this work appeared in Ref. [8]. In peak, beyond the size expected for IB corrections this new version we include a discussion of the com- had;LO [4, 5]. The predictions for aµ based on these plete set of IB corrections and we address some points two sets of data have been in disagreement by more that were unclear in some of our previous reports. than 2σ [5, 6]. Moreover, the branching fraction for Received 25 January 2010 * Supported by Conacyt, M´exico 1) E-mail: glopez@fis.cinvestav.mx + − had;LO 2) Another reason is that e e data is more directly related to aµ through the dispersion integral than τ decay data. ©2009 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 2 Chinese Physics C (HEP & NP) Vol. 34 integral (1). Therefore, it becomes convenient to in- had;LO 2 IB corrections to the aµ disper- troduce the shift in the dispersion integral: 1 sion integral from τ data 2 had;LO − α K(s) (0) ∆a [X ;τ] = ds R − (s) µ 3π2 Z s X × had;LO 2 The leading order hadronic contribution aµ 4mπ can be evaluated by using a combination of experi- RIB(s) mental data and perturbative QCD for the hadronic 1 (3) SEW − vacuum polarization (HVP) function of the photon. produced by the IB corrections. The short-distance At low energies, where QCD does not provide a reli- electroweak radiative effects encoded in S , which able calculation of Green functions, the HVP can be EW includes the re-summation of terms of O(αn lnn(m )) constructed as a sum over exclusive hadronic chan- Z n n − and of O(αα ln (m )), lead to the correction S = nels measured in e+e annihilation. The dispersion s Z EW − 1:0235 0:0003 [4, 13{15]; the quoted uncertainty is integral relating each exclusive e+e X0 channel to ! attributed to neglected corrections of O(αα =π) [15]. ahad;LO is: s µ This term provides the largest of IB effects in 1 − 2 ahad;LO [X ;τ], as it can be seen in Table 1. The had;LO 0 + − α K(s) (0) µ aµ [X ;e e ] = ds R 0 (s) ; (1) 3π2 Z s X remaining IB effects included in RIB(s) are discussed 2 4mπ below. − − 0 (0) 0 Hereafter we focus on the X = π π channel of where RX0 (s) is the ratio of hadronic X to pointlike µ+µ− bare cross sections [1, 2] in e+e− annihilation τ lepton decays. Beyond its rather large contribution had;LO at a center of mass energy ps. The behavior of the to aµ , the precision attained in the measurement QED kernel K(s) 1=s [9], enhances the low-energy of the muon anomalous magnetic moment requires ∼had;LO that the ππ contribution be evaluated below the 1% contributions to aµ in such a way that 91% of it comes from the energy region below 1.8 GeV and accuracy, making crucial the reliable computation of 73% is due to the ππ channel. Further details can be IB corrections [3, 4]. The s-dependent IB correction found in [2]. introduced in (3) is defined as: 3 2 If isospin were an exact symmetry, we would be FSR(s) β0 (s) F0(s) RIB(s) = 3 : (4) able to use in (1) the spectral functions measured in G (s) β−(s) F−(s) EM τ− X−ν decays, where X− is the (I = 1;I = 1) 3 The subscripts i = 0; refer to the electric charge ! 0 − − isotopic partner of the X state. We can define an of the 2π system produced in e+e− annihilation and (0) isotopic analogue of the ratio R 0 (s) as follows (this − X in τ lepton decays, respectively. Each of the fac- quantity is related to the usual spectral function [2, 7] tors in RIB(s) becomes unity in the limit of isospin (0) by R − (s) = 3vX− (s)): X symmetry, thus also RIB(s) = 1 in this limit. (0) 2 In the Standard Model of quarks and leptons in- R − (s) m − 1 dN X = τ BX X 3 6 V 2 N ds × teractions, isospin symmetry is broken by the mass j udj Be X −2 −1 difference of u and d quarks, and by the effects of elec- s 2s tromagnetic interactions. At the hadron level, the IB 1 2 1+ 2 : (2) − mτ mτ effects introduce some model dependence: hadronic In Eq. (2), (1=NX)dNX=ds is the normalized invari- matrix elements that are related by isospin symme- ant mass spectrum of the hadronic final state, and try, get modified in the presence of IB effects by pho- − − denotes the branching fraction of τ X (γ)ν tonic interactions and by the mass and width splitting BX ! τ (throughout this paper, final state photon radiation of hadrons involved. Therefore, the usual procedure is implied for τ branching fractions). For numerical to test isospin symmetry predictions consist in com- purposes [7], we use for the τ lepton mass the value paring `bare' hadronic matrix elements obtained from m = (1776:84 0:17) MeV [10], and for the CKM experimental data by removing the effects of IB cor- τ matrix element V = 0:97418 0:00019 [11], which rections. j udj assumes CKM unitarity. For the electron branching In the following we consider each of the energy- fraction we use e = (17:818 0:032)%, obtained [12] dependent factors that enter in RIB(s) and quantify B had;LO supposing lepton universality. their effects in ∆aµ [ππ;τ]. The correspond- In the presence of IB effects, the spectral func- ing corrections induced in the branching fraction of tion (2) in τ decays must be corrected by the factor τ ππν, which is an independent test of these IB ! RIB(s)=SEW, in order to be used into the dispersion corrections, can be found in Refs. [2, 7]. No. 6 Gabriel L´opez Castro: Recent progress on isospin breaking corrections and their impact on the muon g −2 value 3 2.1 FSR and phase space corrections 1.2 The final state photonic corrections to e+e− ! π+π−, FSR(s), and the ratio of pion velocities 1 β0(s)/β−(s), are the best known corrections to be considered in Eq.
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