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Hadronic Contribution to (g-2)_{mu}

The recent precise measurement of the muon magnetic anomaly (g-2)_{mu} at BNL opens a window into possible new physics, provided the contribution from hadronic vacuum polarization is well understood. This talk summarizes the development in the evaluation of the leading order hadronic contributions. Significant improvement has been achieved in a series of analyses which is presented historically in three steps: (1), use of tau spectral functions in addition to e+e- cross sections, (2), extended use of perturbative QCD and (3), application of QCD sum rule techniques. The uncertainties, in particular concerning the CVC hypothesis used in step (1), and global quark-hadron duality employed in steps (2) and (3) are discussed. No new analysis results are given in these proceedings.


Introduction
Precision measurements of electroweak observables provide powerful tests of the Standard Model. In the last 10 years significant progress has been achieved in this direction owing to the accurate and complete results from the LEP, SLC and TEVATRON colliders. These measurements yielded for the first time unique information from vacuum polarization effects in weak boson propagators which allowed the mass of the Higgs boson to be significantly bounded. At the other end of the energy scale, the muon magnetic moment can now be measured with a precision such that new physics can be probed, provided all the contributions from the Standard Model can be under control. In this talk, I shall discuss the new precise result on (g − 2) µ from the BNL experiment which deviates from the Standard Model expectation: this exciting situation prompts us to critically examine the status of the theoretical prediction, in particular its most delicate contribution from hadronic vacuum polarization. It turns out that the same physics plays an important role in the analysis of high-energy neutral-current data through the running of the electromagnetic coupling from q 2 = 0 to M 2 Z , relevant for limits on the Higgs mass.
2 The muonic (g − 2) The muon magnetic anomaly a µ receives contributions from all sectors of the Standard Model, the dominant diagrams of which are depicted in Fig. 1. The pure QED contribution, a QED µ = 116584705.7(2.9) × 10 −11 , has been calculated to fourth order which represents a tour de force, only performed by one group 1 (some slight change occurred recently due to computer precision problems in the original calculation 2 ). The fifth order term has been estimated only, but was found to be small 3 . The weak contribution, a weak µ = 152(4) × 10 −11 , is known to two-loops 4 . Large logarithms of ln(M W /m f ) occur, but can be resummed 5 , leading to a robust prediction. The contribution from hadrons stems mainly from vacuum polarization and will be covered in the next section. Its absolute size ≃ 6800(160) (ca. 1995) is such that it must be known to a precision better than 1% if the experiment is to probe the level of the weak part. As it is well known, the first order correction from hadronic vacuum polarization ( Fig. 1) cannot be calculated from "first principles" since most contributions arise from low-mass states, where quark confinement leads to resonances. Fortunately, the result can be expressed as a dispersion integral involving the total cross section for e + e − annihilation into hadrons, or alternatively its ratio R(s) to the point-like cross section, with K(s) ∼ m µ /s, thus giving a large weight to the small s region. An analog integral occurs for the running of α(Q 2 ), where K(s) = (s − Q 2 ) −1 . A small part of the hadronic contribution originates from the so-called light-by-light scattering (see Fig. 1). These diagrams cannot be treated analogously and must be estimated through specific models for the hadron blob. As a consequence the result is less reliable and not known accurately, but not large compared to other contributions. Obviously, the hadronic piece must be known more accurately, by a factor of at least three, before a precise measurement can witness the effect of the weak interaction or a new physics contribution of similar magnitude, such as Supersymmetry. This is the motivation for an increased effort in the last few years to improve the reliability and the accuracy of the hadronic contribution. The experimental progress on a µ is chartered in Fig. 2, together with the levels of the different contributions expected in the Standard Model. While the successive CERN experiments reach enough sensitivity to uncover the expected effect of hadrons, the program underway at BNL (E821) is now at the level of the weak contributions and reaches for a four times smaller sensitivity, thus demanding a corresponding improvement in the accuracy of the hadronic piece.

The precise BNL result
The new value recently announced by E821 7 has a precision three times higher than the previous combined CERN and BNL results 8,9 , a µ + = 11 659 202(16) × 10 −10 .
The quoted uncertainty is dominated by statistics in muon decay counting and the major systematic errors are estimated to 3.5 × 10 −10 for the precession frequency and 4.5 × 10 −10 for the magnetic field (NMR frequency). The E821 experimenters compare their result to the expected SM value with the hadronic contribution from vacuum polarization taken from Ref. 10 , a µ + (SM) = 11 659 159.6(6.7) × 10 −10 .
Averaging (3) with previous measurements yields where the error is dominated by the statistical experimental error (theoretical systematic errors have been added in quadrature). A discrepancy at the level of 2.6σ is observed and the agreement with the SM can be questioned a .
A 2.6σ deviation is not sufficient to claim for a real discrepancy. It is however large enough to speculate about its origin in order to rightfully question the different ingredients which could be responsible for the effect. Four possibilities can be considered: • A statistical fluctuation? This will be hopefully resolved soon since the E821 Collaboration is presently analyzing their 2000 data with a factor four larger statistics.
• A systematic effect? It is unlikely, since the estimated systematic uncertainty is 2.5 smaller than the statistical error and the (unaccounted) systematics would have to be six times larger than the uncertainty of the systematics accounted for.
• An error in the SM theory value? The QED and electroweak parts are under control at a level of two orders of magnitude smaller than the observed effect. It is then proper to question the hadronic contributions and we turn to this point in detail in the next section.
• New physics? Although it is in my opinion too early to speculate on this chance, it is clear that many possibilities exist, witnessed by the paper flood since the publication of the BNL result (see, e.g., 11,12 ): Supersymmetry, muon substructure, anomalous electroweak couplings, leptoquarks, lepton flavour violation, etc.

Hadronic vacuum polarization for (g − 2) µ -improvements in three steps
Since 1995 several improvements have been applied to the calculations of hadronic vacuum polarization in order to cope with incomplete or unprecise e + e − data. Although QCD predictions were always used at higher energies (> 40 GeV), it became clear that reliable predictions could be made at much lower values. Let me identify the following three steps: (1) Addition of precise τ data using CVC (see, e.g., 13,14 ) (2) QCD predictions at lower energies (see, e.g., 15,16,17,18 ) (3) Constraints from QCD sum rules (see, e.g., 19,10,20 ) a The interpretation of the discrepancy in terms of standard deviations is approximately valid here, since the error is dominated by experimental uncertainties from the aµ measurement and from the hadronic contribution, where Gaussian Bayesian priors have been used to account for the systematic experimental errors. This is common practise, not to be mixed up with a treatment of theoretical parameters which are not statistically distributed quantities, but whose uncertainties are not of dominance here.
(1) Adding precise τ data under CVC The Conserved Vector Current (CVC) hypothesis expresses invariance under SU(2) of the electroweak currents. For the problem at hand it relates the isovector vector electromagnetic and the weak hadronic currents, as occurring in e + e − annihilation and τ decays. From the point of view of strong interactions this corresponds to a factorization of the hadronic physics: hadrons (quark pairs) are created from the QCD vacuum and the probability to produce hadrons with well-defined quantum numbers at a given mass is expressed through spectral functions. At low energy we expect spectral functions to be dominated by resonances, while QCD should provide a good description at sufficiently high energies. The corresponding energy scale must be determined from experiment. The I = 1 vector spectral function v(s) for the two-pion channel is related to the corresponding e + e − cross section and τ branching ratio and invariant mass spectrum: Hadronic τ decays represent a clean environment to study hadron dynamics which is in many ways complementary to e + e − annihilation: • τ data have excellent absolute normalization, because the relevant branching ratios have been measured at LEP with high statistics, large acceptance and very small non-τ background 21 . On the other hand, the shape of the spectral functions is subject to bin-to-bin corrections from resolution effects and handling of fake photons produced by hadron interactions in the electromagnetic calorimeter. Therefore, the determination of the spectral function requires an unfolding procedure which has the effect of strongly correlating the errors of adjacent bins.
• e + e − data have just about the opposite behaviour: the point-to-point normalization is excellent, whereas the systematic uncertainties are highly correlated among the measurements. The overall normalization is a delicate issue, because of radiative corrections and systematic errors from acceptance and luminosity.
The vector and axial-vector spectral functions have been measured at LEP by ALEPH 22,23 and OPAL 24 . Detailed QCD studies have been performed by both collaborations.

SU(2) breaking
If the τ data is to be used in the vacuum polarization calculations, that is we identify v π ± π 0 (s) with v π + π − (s), it is mandatory to consider in detail the amount of CVC violation 13,25,26 . Isospin breaking is expected mainly from electromagnetic effects and it has to be corrected for the calculation of the integral (2). The sources considered in the analysis and their quantitative effects on a had µ are given in Table 1. The dominant contribution comes from short distance electroweak radiative corrections to the effective four-fermion coupling τ − → (dū) − ν τ . It can be absorbed into an overall multiplicative electroweak correction S EW = 1.0194 27,28 , while remaining perturbative electroweak corrections are of order α n (m τ ) ln n (M Z /m τ ) 0.3 n which is safe to ignore. The sub-leading non-logarithmic short distance correction, calculated to order O(α) at quark level 28 , 5α(m τ )/12π ≃ 0.0010, is also small. Additional intermediate-distance corrections have been computed only for τ − → π − ν τ and the overall correction was found to be dominated by the leading logarithm from the short distance contribution 29 . The electromagnetic π ± − π 0 mass splitting affects the measured cross section through phase space corrections.
Electromagnetic corrections also affect the pion form factor, in particular the width of the ρ resonance(s): the ρ − ω mixing, not present in τ decays; the π ± − π 0 and ρ ± − ρ 0 mass splitting; electromagnetic decays. The occurrence of second class currents is expected to be proportional to the mass splitting-squared of the light u, d quarks which is negligible. We observe that most of the effects cancel, so that the net correction applied corresponds to approximately the pure short-distance radiative correction S EW . It is important to further investigate isospin-violating contributions to keep the increasing precision of the τ data exploitable for the purpose discussed here.
The use of τ data improves the precision on the evaluation of a had µ by a factor of 1.6 13 . Pion velocity: β − = β 0 −7.1 (due to EM π − -π 0 mass splitting: affects cross section) Form Factor 13, 25 : Total correction −9.9 ± 2.4 (2) Replacing poor data by QCD prediction The data driven analysis 6,13 shows that to improve the precision on the dispersion integral, a more accurate determination of the hadronic cross section between 2 GeV and 10 GeV is needed, where some poorly measured and sparse data points dominate the final error. Indeed, QCD analyses using τ spectral functions 23,24 revealed the excellent applicability of the Operator Product Expansion (OPE) 30,31 at the scale of the τ mass, m τ ≃ 1.8 GeV, and below. The OPE organizes perturbative and nonperturbative contributions to a physical observable through the concept of global quark-hadron duality. Using moments of spectral functions, dimensional nonperturbative operators contributing to the τ hadronic width have been determined experimentally and found to be small. The evolution to lower energy scales proved (to some surprise) the validity of the OPE down to about 1.1 GeV. An analog analysis based on spectral moments of e + e − cross section measurements has been performed in Ref. 16 (and more recently in Ref. 32 -yielding compatible results). The theoretical prediction of these moments and of the total hadronic cross section in e + e − annihilation, R(s 0 ), at a given energy-squared, s 0 , involves the Adler D-function 33 , related to the former via Massless perturbative QCD predictions of D are available 34 to order (α s /π) 3 (see also Ref. 35 for a heroic effort to go beyond this). The OPE of D also includes second order quark mass corrections far from the production threshold 36 and the first order dimension D = 4 nonperturbative term involving the gluon condensate, (α s /π)GG , and the quark condensates, m fqf q f , for the quark flavours f (see, e.g., Ref. 37 ). The complete dimension D = 6 and D = 8 operators are parameterized phenomenologically using the saturated vacuum expectation values O 6 and O 8 , respectively. The nonperturbative operators (with the exception of the quark condensates, which are obtained from PCAC relations) are determined experimentally by means of a combined fit of the theoretical moments to data. It results in a very small contribution from the OPE power terms to the lowest moment at the scale of 1.8 GeV (repeated and confirmed at 2.1 GeV). This is in agreement with the findings from the τ analyses. The calculation accounts for theoretical uncertainties, including the dependence on the choice of the renormalization scheme and scale, the uncertainty on the strong coupling, the missing term (α s /π) 4 and the ambiguity between contour-improved and fixed-order perturbation theory (see Refs. 16,23 ). Additional sources are the uncertainties on the running quark masses and on the nonperturbative contributions. In spite of the implicit assumption of local duality for the theoretical prediction of R, the evaluation of the dispersion integral (2) turns the duality globally, i.e., remaining nonperturbative resonance oscillations are averaged over the integrated energy spectrum b .
The available data points together with the theoretical prediction (crossed hatched band) are shown in Fig. 3. Good agreement is found between theory and the newest BES measurements 38 , while older data are significantly higher. The preceeding discussion justifies the application of QCD predictions for R between 1.8 GeV and the DD production threshold, as well as from 5 GeV up to infinity 16 . This yields a factor of 1.3 improvement on the precision of a had µ and a factor of 2.4 better accuracy on ∆α had (M 2 Z ). Similar precise analyses are performed in Ref. 17 (applying a renormalization of experimental data on the cc resonances using QCD predictions of the continuum and assuming systematics to be correlated) and Ref. 18 (see also Ref. 39 for a recent theory driven analysis). b A systematic uncertainty is introduced through the cut at explicitly 1.8 GeV so that non-vanishing oscillations may give rise to a bias after integration. The associated (small) systematic error is estimated by means of fitting different oscillating curves to the data around the cut region 16 .
(3) Improving data with QCD sum rules It was shown in Refs. 19,10 that the previous determinations can be further improved by using finite-energy QCD sum rule techniques in order to access theoretically energy regions where perturbative QCD fails locally. In principle, the method uses no additional assumptions beyond those applied in the previous section. The idea is to reduce the data contribution to the dispersion integrals by subtracting analytical functions from the singular integration kernel in Eq. (2), and adding the subtracted part subsequently by using theory only. Two approaches have been applied in Ref. 10 : first, a method based on spectral moments is defined by the identity with P n (s) = s 0 dt p n (t). The regular functions p n (s) approximate the kernel K(s)/s in order to reduce the contribution of the non-analytic first integral in Eq. (9), which is evaluated using experimental data. The second integral in Eq. (9) can be calculated theoretically in the framework of the OPE. The functions p n (s) are chosen in order to reduce the uncertainty of the data integral which is not necessarily equivalent to a reduction of its contribution. A second approach 10 involving local quark-hadron duality uses the dispersion relation of the Adler D-function for space-like Q 2 = −q 2 and quark flavours f , to approximate the integration kernel. The theoretical errors of both approaches are evaluated in close analogy to the analysis presented in the previous Section. The improvement in accuracy on the dispersion integrals obtained from these constraints is weak for a had µ but valuable for ∆α had (M 2 Z ). Table 2 shows the experimental and theoretical evaluations of a had µ for the distinguished energy regions. Experimental errors between different lines are assumed to be uncorrelated, whereas theoretical errors, but those from the cc and bb thresholds which are quark mass dominated, are added linearly. The combination of the theoretical and experimental evaluations of the dispersion relation yields the final result (see Table 2 for a note concerning the theoretical error):  14.31 ± 0.50 exp ± 0.21 theo (5 -9.3) udsc 6.87 ± 0.11 theo (9.3 -12) udscb 1.21 ± 0.05 theo (12 -∞) udscb 1.80 ± 0.01 theo (2m t -∞) t ≈ 0 (2m π -∞) udscbt 692.4 ± 5.6 exp ± 2.6 theo * The theoretical error accounts for uncertainties concerning the QCD prediction only. Due to the correlated average procedure applied in Ref. 13 , uncertainties from CVC and radiative corrections are folded into the systematic part of the experimental error.

Conclusions and perspectives
Much effort has been undertaken during the last years to ameliorate the theoretical predictions on a had µ . The currently most precise value obtained for the hadronic contribution is 10 a had µ = (692.4 ± 6.2) × 10 −10 . Figure 4 gives a chronological compilation of published results.  Fairly good agreement is observed among the newest evaluations, so that the discrepancy between the BNL value and the Standard Model varies between 2σ and 2.6σ. To maintain the sensitivity on interesting physics of the experimental improvements to be expected from BNL, more theoretical effort is needed. In particular, a better precision on a had µ requires further studies of the following items.
• Radiative corrections in e + e − annihilation data • SU(2) breaking: let me recall that the τ data not only provide precise and in many ways complementary cross section measurements, but they also constitute a powerful cross check. The current a had µ evaluation being wrong would require not only the e + e − data to have unaccounted systematics, but also that CVC violation is much larger than expected, since the e + e − and τ data are mutually (fairly) compatible.
• More experimental information. In particular, complementary e + e − measurements from, e.g., new precision experiments, or analyses of radiative events using data from existing e + e − factories.
(Also the contribution from hadronic LBLS is of much importance but not the subject of this talk.) It is obvious that using the τ data and CVC represents only an auxiliary remedy to cope with the lack of precision in the e + e − measurements. Certainly, the preferable scenario would be to improve the latter well below the 1% accuracy so that one does not need to include the former. Moreover in such a situation precise tests of CVC could be performed, providing insight into the interesting physics of possible violations.