Constraints on the I=1 hadronic tau decay and e^+e^- -->hadrons data sets and implications for (g-2)_mu

Sum rule tests are performed on the spectral data for (i) flavor 'ud' vector- current-induced hadronic tau decays and (ii) e^+ e^- hadroproduction, in the region below s~3-4 GeV^2, where discrepancies exist between the isospin- breaking-corrected charged and neutral current I=1 spectral functions. The tau data is found to be compatible with expectations based on high-scale alpha_s(M_Z) determinations, while the electroproduction data displays two problems. The results favor determinations of the leading order hadronic contribution to (g-2)_mu which incorporate hadronic tau decay data over those employing electroproduction data only, and hence a reduced discrepancy between experiment and the Standard Model prediction for (g-2)_mu.


I. INTRODUCTION
In the Standard Model (SM), the largest of the non-purely-leptonic contributions to the anomalous magnetic moment of the muon, a µ ≡ (g − 2) µ /2, is that due to the leading order (LO) hadronic vacuum polarization (VP), [a µ ] LO had . a µ is currently known to 0.5 ppm [1], with a proposal to reduce this to 0.2 ppm in the near future [2]. The 0.5 ppm uncertainty represents < 1% of [a µ ] LO had , making an accurate determination of [a µ ] LO had crucial to the study of possible non-SM contributions to a µ .
[a µ ] LO had is related to σ [e + e − → hadrons] by the dispersion representation [3] [a µ ] LO had = where the form of K(s) is well-known, and R(s) is the "bare" e + e − → hadrons to e + e − → µ + µ − cross-section ratio. With recent electroproduction data, the uncertainty on [a µ ] LO had from Eq. (1) is comparable to the experimental error on a µ , and dominates the uncertainty in the SM prediction for a µ [4,5,6,7,8]. Since CVC relates the I = 1 electromagnetic (EM) spectral function to the charged current vector spectral function measured in τ − → ν τ + non-strange hadrons, hadronic τ decay data [9,10,11] can, in principle, be used to improve the determination of [a µ ] LO had [12,13]. At the < 1% level necessitated by the current experimental error, isospin-breaking (IB) corrections must be taken into account.
In view of the unsettled experimental situation, we study sum rule constraints on weighted integrals of the I = 1 vector τ decay and EM spectral functions. The weights, w(s), and upper integration limits, s 0 , are chosen such that (i) each spectral integral has a reliable and well-converged OPE representation, and (ii) all relevant OPE input can be obtained from sources independent of the low scale EM and τ data we seek to test. OPE uncertainties are minimized by working with s 0 and w(s) for which the relevant OPE representation is dominated, essentially entirely, by its D = 0 component, and hence determined, essentially entirely, by the single input parameter, α s (M Z ), which can be taken from independent high-scale studies. The τ decay based spectral integrals will be shown to be well reproduced by the corresponding OPE representations. Those based on EM data, in contrast, will be shown to lie consistently below the corresponding OPE values, for positive w(s), and to differ from them in their s 0 dependence. Both features are as expected if the EM spectral data is too low in the disputed region.

II. THE SUM RULE CONSTRAINTS
We study sum rule constraints on the EM spectral function, ρ EM (s), and the sum of the spin J = 0 and 1 components of the charged I = 1 vector current spectral function, ρ V ;ud (s). The former is related to R(s) by ρ EM (s) = R(s)/12π 2 , and to the bare e + e − → hadrons cross-sections, σ bare (s), by Defining where V ud is the flavor ud CKM matrix element, and S EW is an electroweak correction [21]. Contributions to ρ V ;ud (s) to be determined from the experimental decay distribution.

A. Finite Energy Sum Rules
For any correlator, Π(s), with no kinematic singularities, and any w(s) analytic in |s| < M with M > s 0 , analyticity implies the finite energy sum rule (FESR) relation where ρ(s) is the spectral function of Π(s). In QCD, for very large s 0 the OPE representation can be employed on the RHS of Eq. (4). As s 0 is decreased, this representation is expected to break down first near the timelike real s-axis [22]. A range of "intermediate" s 0 will thus exist for which the OPE, though unreliable for general w(s), will remain valid for those w(s) satisfying w(s = s 0 ) = 0. The corresponding FESR's are called "pinched" or pFESR's. For vector (V) and axial vector (A) correlators, and w(s) = s N , OPE breakdown (duality violation) is significant at s 0 ∼ a few GeV 2 [23]. Polynomials w(y) (with y = s/s 0 ) having even a single zero at s = s 0 (y = 1), however, remove such violations for s 0 greater than ∼ 2 GeV 2 [23], even for the flavor ud V-A correlator [24]. In interpreting FESR results, one should bear in mind that very strong correlations exist between spectral integrals corresponding to the same w(y), but different s 0 . Such correlations are particularly strong when w(y) ≥ 0 over the relevant interval, 0 < y ≤ 1, and even more so when w(y) is monotonically decreasing. Similar strong correlations exist between spectral integrals corresponding to different w(y), but fixed s 0 . Correlations among the corresponding OPE integrals are also very strong, especially when the OPE is dominated, as below, by a single (in this case, D = 0) contribution.
We work, in what follows, with pinched polynomial weights, w(y) = m c m y m . The pinching condition, w(1) = 0 implies m c m = 0. For reasons explained below, w(y) is further restricted to be non-negative and monotonically decreasing on 0 ≤ y ≤ 1. Since s 0 is the only scale entering the RHS of Eq. (4), it is obvious, on dimensional grounds, that integrated OPE contributions of dimension D = 2k + 2 scale as 1/s k 0 , up to logarithms. For D ≥ 2, such contributions are absent (up to corrections of O(α s )) when c (D−2)/2 = 0. The structure of the logarithmic integrals, |s|=s 0 ds y k ℓn(Q 2 /µ 2 )/Q D , responsible for the O(α s ) corrections, is such that cancellations inherent in the pinching condition m c m = 0 lead to strong numerical suppressions of these corrections. D ≥ 8 contributions are typically assumed to be negligible, since the relevant condensate values are not known phenomenologically. The much stronger s 0 dependence of such high D contributions allows this assumption to be tested explicitly.
The reason for working with non-negative, monotonically decreasing w(y) is that the EM spectral data for the π + π − and π + π − π 0 π 0 states, which dominate the EM-τ discrepancy, lie uniformly below the IB-corrected τ data. Non-negativity of w(y) then ensures that, for all s 0 , the normalization of the τ -based spectral integrals will be too high if it is the EM data which is correct, while the normalization of the EM spectral integrals will be too low if it is the τ data which is correct. Since the y value for a given experimental bin decreases with increasing s 0 , a monotonically decreasing w(y) similarly ensures that the slope with respect to s 0 of the τ spectral integrals will be too high if the EM data is correct, while the slope with respect to s 0 of the EM spectral integrals will be too low if the τ data is correct. The slope constraint is particularly useful because the slope of the corresponding OPE integrals is very tightly constrained, and only very weakly dependent on the dominant OPE input parameter α s (M Z ).

B. OPE Input
Re-writing the weighted pFESR OPE integrals of the relevant correlator, Π, in terms of the Adler function D(Q 2 ) ≡ −Q 2 dΠ(Q 2 )/dQ 2 , allows potentially large logs in the D = 0 contribution to be summed point-by-point along the integration contour. This "contour-improved" (CIPT) prescription is known to improve the convergence of the integrated D = 0 series [25]. The Adler function is given by whereā = a(Q 2 ) = α s (Q 2 )/π is the running coupling in the MS scheme, and C = 1, 2/3 for the τ , EM cases, respectively. For [27] also exists, based on methods which (i) work well for the coefficients of the D = 0 series [28] and (ii) produced, in advance of the actual calculation, an accurate prediction for the recently computed O(a 3 ) D = 2 coefficient of the (J) = (0 + 1) V+A correlator sum [29].
The leading D = 2 contributions for the τ correlator are O(m 2 u,d ) and numerically negligible. For the EM correlator, up to tiny O(m 2 u,d /m 2 s ) corrections, the D = 2 contributions are determined byā and the running MS strange massm s . At the scales employed here the integrated D = 2 contribution is small. The full expression for [Π EM (Q 2 )] D=2 may be found in Ref. [30]. The D = 4 terms in the OPE of the EM and τ correlators are determined by the RG invariant light quark, strange quark and gluon condensates, l ℓ RGI , ss RGI and aG 2 RGI , up to numerically tiny O(m 4 s ) corrections. The expressions may be found in Refs. [31,32]. The integrated D = 4 contributions are again small at the scales employed. To reduce OPE uncertainties, we concentrate here on weights for which the integrated leading D = 6 contributions are absent. More extensive studies will be reported elsewhere [33]. We assume throughout that contributions with D ≥ 8 may be neglected, but check this assumption for self-consistency, as discussed above.
For D = 4 input we use (i) aG 2 = (0.009 ± 0.007) GeV 4 (from the recent reanalysis of charmonium sum rules [34]) and (ii) 2m ℓl ℓ = −m 2 π f 2 π (the GMOR relation). m ss s RGI then follows from conventional ChPT quark mass ratios and the standard estimate ss RGI / l ℓ RGI = 0.8±0.2. For the D = 0, 2 input,ā andm s , we employ exact solutions based on the 4-loop-truncated β and γ functions [35], with initial conditions Eq. (6) reflects the range of results obtained in recent sum rule [29,37] and N f = 2 and 2 + 1 unquenched lattice studies [38]. The N f = 5 value in Eq. (7) is run down to the N f = 3 low-scale region using standard 4-loop running and matching [36], with the same matching scales as used in the recent EM sum rule studies of Refs. [6,20] (HMNT). The input α s (M Z ) in Eq. (7) differs from the PDG 2004 average for the following reasons. First, the PDG average includes hadronic τ decay input, which must be excluded if we wish to test the τ decay data. Second, the PDG average is strongly affected by the quoted low, small-error determination from heavy quarkonium decay [39]. The Quarkonium Working Group, however, has (i) strongly criticized the input to the low central value, (ii) argued that the quoted error is underestimated by a factor of 3−5 [41], and (iii) concluded the method is not competitive with extractions based on perturbative treatments of high scale processes [42]. Eq. (7) is obtained by removing lower scale determinations, including those based on heavy quarkonium and τ decay, from the PDG average [43].

C. Spectral Input
The ALEPH and CLEO τ decay distributions are in good agreement. For definiteness, we use the ALEPH results, for which the covariance matrix is publicly available. A small global rescaling accounts for minor changes in B e , B µ , and the strange branching fraction, B s , since the original ALEPH publication [9]. PDG04 [40] values are used for B e and B µ , while the updated B s value incorporates (i) the new (2004) world averages for [44], (ii) the new (2005) CLEO results for the branching fractions of four-particle modes with kaons [45], and (iii) the higher precision K µ2 value for the K pole contribution [40]. The long-distance EM corrections determined in Ref. [15] are also applied to the dominant τ − → π − π 0 contribution [15] [46].
Detailed discussions and assessments of the EM hadroproduction data base, as of 2002-2003, can be found in Refs. [4,5,6,47]. The exhaustive compilation of Ref. [47] provides useful information on the treatment of radiative and VP corrections for older experiments where such details are absent from the original publications. Detailed assessments of the needed residual VP corrections are also contained in Refs. [4,6]. These corrections are computed using the most recent version of F. Jegerlehner's code [48], generously provided by its author. We also employ the following new, high-precision results, published subsequent to the analyses of Refs. [5,6], and the compilation of Ref. [47]: (i) the final published version of the SND 4π cross-sections [49] (with systematic errors significantly reduced over those of the earlier preprint version); (ii) the updated CMD-2 π + π − π + π − [50] and π 0 γ, ηγ [51] cross-sections; and (iii) the BABAR 3π [52] and π + π − π + π − [53] cross-sections. Since the ππ component of the EM-τ discrepancy is driven by the CMD-2 data, with its very small 0.6% systematic error [16], we employ only CMD-2 data in the CMD-2 region. Where the existence of newer data permits, we exclude older data for which systematic errors are incompletely stated, or absent, in the original publications, and/or the residual radiative/VP corrections to be applied are unknown. Fortunately, data with missing systematic errors for which no such replacement is possible play only a small numerical role. We assign a guess of 20% for these errors in such cases. For the small number of remaining older experiments where the situation with regard to residual VP corrections is unclear, we apply no VP correction. In all such cases, however, (i) the corresponding contributions to the spectral integrals are small, and (ii) the neglected VP corrections are, in any case, much less than the quoted systematic errors. The treatment of "missing mode" contributions, and the use of isospin relations for a number of small contributions where direct experimental determinations are absent, or have large errors, follow the treatments discussed in detail in Refs. [4,6]. More details on the treatment of the EM data will be provided elsewhere [33].

III. ANALYSIS AND RESULTS
For reasons discussed above, we concentrate on pFESR's involving non-negative, monotonically decreasing w(y). The only such degree 1 weight is w 1 (y) = 1 − y. Weights with a double zero at y = 1, which more strongly suppress OPE contributions from the vicinity of the timelike axis, should be even safer from the point of view of potential duality violation. A convenient set of such "doubly-pinched" weights is the family, w N (y) = 1 − N N −1 y + 1 N −1 y N , N ≥ 2. For a given w N , the only non-α s -suppressed ("unsuppressed" in what follows) D > 4 OPE contribution is that with D = 2N + 2. This contribution scales as 1/s N +1 0 relative to the leading integrated D = 0 term. The strong s 0 -dependence allows the neglect of D ≥ 8 contributions (for w N ≥3 ) to be tested for selfconsistency. We have also studied pFESR's based on a number of other weights. Since the results in all cases point to the same conclusion, and OPE uncertainties are reduced for weights having no unsuppressed D = 6 contribution, we focus on the pFESR's for two of the weights defined above, w 1 and w 6 . Other results will be presented elsewhere [33].
Combined OPE errors for the various pFESR's are obtained by adding in quadrature uncertainties associated with the OPE input parameters (D = 4 condensates, m s , and α s (M Z )) and the truncation/residual scale dependence of the integrated D = 0, 2 series. The latter are estimated to be twice the magnitude of the last term in the corresponding truncated series. The resulting OPE error estimate is somewhat more conservative than that employed by HMNT.
Results for the EM case are presented in Figures 1, 2, those for the τ case in Figures 3,  4. The dashed lines represent the central OPE results, the solid lines the upper and lower edges of the OPE error bands. Because of the strong correlations, the OPE band is better thought of as a bundle of allowed parallel lines than as a generally allowed region. We see that, in the region 2 GeV 2 < s 0 < m 2 τ , both the magnitude and slope of the integrals over the τ decay distribution are in good agreement with OPE expectations. In contrast, the EM spectral integrals are consistently low relative to OPE expectations (particularly for s 0 greater than ∼ 2.5 GeV 2 ) and have slopes with respect to s 0 significantly lower than those of the OPE curves. Both features of the EM results are as expected if the τ data is correct, and hence the EM data low, in the disputed region.
The implications of the normalizations of the EM and τ spectral integrals can be quantified by working out the α s (M Z ) required to match the OPE and spectral sides of   Table I for both the EM and τ cases. The agreement between the τ -decay and independent high-scale determinations is excellent [54]. In contrast, the EM data corresponds to α s (M Z ) ∼ 2σ lower than the high-scale determination. The y 6 term of w 6 (y), in principle, produces an unsuppressed D = 14 OPE contribution scaling as 1/s 6 0 (1/s 7 0 relative to the leading D = 0 term). Such a contribution, if present, would contaminate the extraction of α s (M Z ). There is, however, no evidence for such a contribution, at a level which would impact our analysis, in the s 0 dependence of either the EM or τ w 6 -weighted spectral integrals [55]. The excellent agreement between the α s (M Z ) extracted using different doubly-pinched weights, with potential unsuppressed D > 6 contributions of different dimension, provides further evidence in support of the absence of such D > 6 contributions [56,57].
To quantify the disagreement between the EM OPE and experimental slope values, we work out the correlated errors for the slopes with respect to s 0 of the OPE and spectral integrals. The correlations are such that the uncertainty on the OPE side is rather small. In particular, the slope is quite insensitive to α s (M Z ). These points are illustrated in Table II, which shows the spectral integral and OPE slope values for the EM w 1 and w 6 pFESR's. The OPE entries labelled "indep" are those obtained using the independent, high-scale fit value for α s (M Z ). Those labelled "fit", in contrast, correspond to the α s (M Z ) values obtained by fitting to the relevant s 0 = m 2 τ spectral integrals, as given in Table I. We see that, even if one were willing to tolerate the lower central α s (M Z ) values implied by the EM spectral integrals, such a lowering of α s (M Z ) would have negligible impact on the OPE vs. spectral integral slope discrepancy problem.

IV. DISCUSSION AND CONCLUSIONS
We have shown that weighted spectral integrals constructed using I = 1 hadronic τ decay data are in good agreement with OPE expectations, while those involving EM data (i) require a value of α s (M Z ) ∼ 2σ below that given by high-scale determinations and (ii) correspond to a slope with respect to s 0 in ∼ 2.5σ disagreement with the OPE prediction. The slope problem, moreover, cannot be cured simply by adopting the lower α s (M Z ) values, shown in Table I, which would bring the normalization of the OPE and spectral integrals into agreement for s 0 ≃ m 2 τ . The insensitivity of the slope to α s (M Z ) also means that the agreement between the OPE expectation and the observed slope for the τ decay spectral integrals represents a non-trivial test of the τ data.
One possibility is that the problems with the EM sum rules might be attributable to the presence of residual duality violation at the intermediate scales studied here; the success of the OPE in predicting both the slope and magnitude of the τ -decay-based spectral integrals over the whole of the region 2 GeV 2 < s 0 < m 2 τ , however, renders such an explanation highly implausible. The results thus point to the reliability of the τ data, and to the likelihood of either (i) a problem with the experimental EM spectral distribution, or (ii) the presence of as-yet-unidentified non-one-photon physics contributions in the experimental EM cross-sections. This in turn suggests that a µ determinations which incorporate τ decay data are to be favored over those employing EM spectral data only.
While the disagreement between the EM and high-scale determinations of α s (M Z ) is only ∼ 2σ, even with significantly lower high-scale input, e.g., the 2002 PDG average α s (M Z ) = 0.1172 ± 0.0020 used by HMNT, the EM normalization, and even more so the EM slope, would still require, on average, upward fluctuations in ρ EM (s). Since K(s)/s > 0, such fluctuations would typically also increase [a µ ] LO had . Thus, even ignoring the slope problem and assessing the EM data as moderately consistent, within errors, with the OPE constraints, the fact that the spectral integrals lie consistently below the corresponding OPE constraint values, for any sensible input α s (M Z ), points to the likelihood of a [a µ ] LO had value higher than the current central EM-data-based value. Two further points are of relevance to assessing the implications of our results for the value of [a µ ] LO had . First, we find that, replacing the EM π + π − , π + π − π 0 π 0 and π + π − π + π − data with the equivalent τ data resolves completely both the normalization and slope problems for the resulting modified "EM" spectral integrals [58]. Second, it is readily demonstrated that the pFESR's employed are sensitive to, not just the discrepancies in the 4π region, but also those in the 2π region. (This is relevant since the [a µ ] LO had integral is dominated by the 2π spectral contribution.) In fact, for the w 1 pFESR, the shift in the EM spectral integral associated with the modification of the 2π part of the EM spectral function represents 82% of the full shift at s 0 = 2 GeV 2 and 32% at s 0 = m 2 τ . The corresponding figures for the w 6 pFESR are 87% at s 0 = 2 GeV 2 and 45% at s 0 = m 2 τ . Thus, even though the pFESR's employed are relatively more sensitive to the 4π spectral contributions than is the [a µ ] LO had integral, a clear sensitivity to the 2π component remains, making the constraints associated with these pFESR's highly relevant to the [a µ ] LO had problem.
In conclusion, all the sum rule tests performed favor the reliability of the τ decay data, and point to problems with the EM data. We conclude that, at present, determinations of [a µ ] LO had employing IB-corrected τ decay data are more reliable than those based on EM data alone, and hence that there is no clear sign of a discrepancy between the current experimental value for a µ and the SM prediction.
NOTE ADDED: Subsequent to the submission of this paper, new results for the e + e − → π + π − cross-sections were released by the SND Collaboration [59]. As would be expected from the sum rule results above, the SND cross-sections are compatible with the τ → ν τ ππ data, but in significant disagreement with the KLOE ππ data, in the disputed region.
in excellent agreement with those obtained using the corresponding w 6 sum rule.
[57] The ALEPH [9] and OPAL [10] collaborations have, in fact, performed extractions of the effective D = 6, 8 condensate combinations for the ud V correlator by fitting to the s 0 = m 2 τ spectral integrals for a number of different "spectral weights". The reader is cautioned that these extractions are performed assuming D > 8 contributions are absent, despite the fact that, for all but one of the weights employed, such contributions are, in principle, present in unsuppressed form. The analogous assumption for the ud V-A correlator has been tested explicitly and found to be incompatible with data [24], making it also suspect for the V case. One can nonetheless take the extracted values as representative of what might be expected for the size of such effects and investigate the impact on the sum rules used in this analysis. By construction, no unsuppressed D = 6 contributions are present in the sum rules considered. While the D = 8 term does not contribute to the w 1 or w 6 pFESR's, it does contribute to the w 3 pFESR. Using the larger of the ALEPH and OPAL D = 8 fitted values, it is straightforward to demonstrate that, at s 0 = m 2 τ , the resulting D = 8 contribution is at the level of 0.2% of the dominant D = 0 term, producing a shift in the value of α s (M Z ) extracted using the w 3 sum rule of < 0.0003, which, as claimed, is much smaller than the ∼ .0020 error associated with data uncertainties. This observation simply confirms the conclusion obtained by studying the s 0 dependence of the various sum rules and the consistency of the α s (M Z ) extractions from sum rules where non-α s -suppressed contributions with different D > 6 would be present [56].
[58] The normalizations of the modified EM integrals are now in full agreement with the highscale-input OPE constraint bands, for all s 0 > 2 GeV 2 . The modified spectral integral slopes are also in good agreement with OPE expectations (the modified central values are 0.00947 for the w 1 and 0.00807 for the w 6 pFESR, to be compared with the OPE predictions 0.00943 ± 0.0008 and 0.00811 ± 0.00009 already reported in Table II.