On the Leading Order Hadronic Contribution to (g−2)μ

Sum rule constraints dominated by the independent high-scale input, αs(MZ), are shown to be satisfied by I = 1 spectral data from hadronic τ decays, but violated by the pre-2005 electroproduction (EM) cross-section data. Determinations of the Standard Model (SM) hadronic contribution to (g−2)μ incorporating τ decay data are thus favored over those based solely on EM data, implying a SM prediction for (g−2)μ in agreement with current experimental results. After accounting for the accurately known, purely leptonic contributions, the largest of the SM contributions to aμ ≡ (g−2)μ/2 is that due to leading (LO) hadronic vacuum polarization (VP), [ aμ LO had . This contribution can be expressed as a weighted integral, with known kernel, over the electromagnetic (EM) spectral function ρEM(s). The uncertainty on [ aμ LO had which results is at the ∼ 1% level (comparable to the 0.5 ppm experimental uncertainty on aμ [1]) and dominates the uncertainty in the SM prediction for aμ [2]. This determination can, in principle, be improved using CVC, together with data on the I = 1 spectral function, ρI=1(s), obtained from hadronic τ decay. (Isospinbreaking (IB) corrections [3, 4] are needed for accuracy at the 1% level.) Unfortunately, the IB-corrected τ data is in significant disagreement with the corresponding I = 1 EM data, ρEM I=1 lying uniformly below ρ τ I=1 in both the 4π region and that part of the 2π region above the ρ but below 1 GeV2 [5]. The determination of [ aμ LO had employing only EM data leads to a SM aμ prediction ∼ 2.5σ below experiment, while the alternate determination incorporating τ decay data is compatible with experiment at the ∼ 1σ level [5]. We show here that sum rule constraints strongly favor the τ-based determination. For w(s) any function analytic in the region |s| < M, with M > s0, the I = 1 vector (V) current and EM correlators, Π(s), and corresponding spectral functions, ρ(s), satisfy the FESR relations ∫ s0 0 w(s)ρ(s)ds = − 1 2πi ∮ |s|=s0 w(s)Π(s)ds. To suppress duality violation and allow the use of the OPE on the RHS, w(s) should satisfy w(s = s0) = 0 [6]. At scales of ∼ a few GeV2, the OPE representation for V current correlators is essentially entirely dominated by its leading D = 0 component, and hence determined by the single input parameter αs(MZ), whose value is known from independent high-scale studies [7]. If one works with weights w(y), y = s/s0, which satisfy w(1) = 0, and are non-negative and monotonically decreasing for 0 < y < 1, the fact that the EM version of ρI=1 lies uniformly below the corresponding τ version implies that • the normalization and slope with respect to s0 of the EM-based spectral integrals, for a given w(y), should be too low relative to OPE predictions if it is the τ-based TABLE 1. αs(MZ) from fits to the s0 = m2 τ experimental EM and τ spectral integrals, with central D = 2,4 OPE input values Weight [αs(MZ)]EM [αs(MZ)]τ w1 0.1138+0.0030 −0.0035 0.1212 +0.0027 −0.0032 w6 0.1150 +0.0022 −0.0026 0.1195 +0.0020 −0.0022 data which is correct and • the normalization and slope with respect to s0 of the τ-based spectral integrals for a given w(y) should be similarly too high if it is the EM-based data which is correct. Sum rule tests of the two data sets have been performed for a number of different w(y). See Ref. [8] for details on the OPE and spectral integral inputs (including shortand long-distance IB corrections for the τ data and input relevant to the small D > 0 OPE contributions). It is found that both the normalization and slope with respect to s0 of the τ based spectral integrals are in excellent agreement with OPE expectations, while both the normalization and slope of the EM-based spectral integrals are low, particularly if one uses the pre-2005 EM ππ spectral data. Table 1 quantifies the EM normalization problem, giving the values of αs(MZ) needed to bring the OPE and spectral integrals into agreement for the τ and EM cases, at the maximum common available scale, s0 = m2 τ . Results are shown for w(y) = w1(y) = 1−y and w6(y) = 1− 6y/5 + y6/5 (which have zeros of order 1, 2, respectively, at s = s0). The entries are to be compared, e.g., to the high-scale average, αs(MZ) = 0.1195± 0.0016, obtained by excluding (i) τ decay input and (ii) the erroneous heavy quarkonium input [9] from the PDG04 average. Table 2 similarly quantifies the EM slope problem. In the OPE column, “indep” and “fit” label results obtained using (i) the independent high-scale αs(MZ) input and (ii) the fitted αs(MZ) values from Table 1, respectively. We see that lowering the input αs(MZ) value to accommodate the EM spectral integral normalizations at s0 ∼ m2 τ does not resolve the slope problem for the EM spectral integrals. For illustration purposes, the results for the w6 EM case are also displayed in Fig. 1. The dotted and solid lines give the central OPE and OPE error bounds, respectively, the solid dots (with error bars) the EM spectral integrals, with pre-2005 ππ input. The open circles show the shifted “EM” spectral integrals obtained by replacing the EM 2π and 4π data with the equivalent τ data. We see that both the EM slope and normalization problems are resolved if, where the I = 1 V part of the EM and τ data disagree, the τ data are taken to be correct. The recent 2005 SND EM ππ results [10] are compatible with the corresponding τ results, further strengthening the case for the reliability of the τ data, and the agreement between the SM prediction and experimental result for aμ .