Hadronic vacuum polarization: $(g-2)_\mu$ versus global electroweak fits

Hadronic vacuum polarization (HVP) is not only a critical part of the Standard Model (SM) prediction for the anomalous magnetic moment of the muon $(g-2)_\mu$, but also a crucial ingredient for global fits to electroweak (EW) precision observables due to its contribution to the running of the fine-structure constant encoded in $\Delta\alpha^{(5)}_\text{had}$. Recent lattice-QCD results for HVP by the Budapest-Marseille-Wuppertal collaboration (BMWc) prompt us to reexamine this interplay. While their result would bring the SM prediction for $(g-2)_\mu$ into agreement with the Brookhaven measurement, this comes at the expense of a $3.2\,\sigma$ deviation with respect to a wide range of $e^+e^-\to\text{hadrons}$ cross-section data. We find that the global EW fit alone provides a competitive, independent determination of $\Delta \alpha^{(5)}_\text{had}$, which lies even below the $e^+e^-$ range, resulting in a tension with the BMWc result whose significance depends on the energy where the bulk of the changes in the cross section occurs. Reconciling experiment and SM prediction for $(g-2)_\mu$ by adjusting HVP, as suggested by BMWc, would thus not necessarily weaken the case for physics beyond the SM (BSM), but merely shift it from $(g-2)_\mu$ to the EW fit. We briefly explore some options of BSM scenarios that could conceivably explain the ensuing tension.

Hadronic vacuum polarization (HVP) is not only a critical part of the Standard Model (SM) prediction for the anomalous magnetic moment of the muon (g − 2)µ, but also a crucial ingredient for global fits to electroweak (EW) precision observables due to its contribution to the running of the fine-structure constant encoded in ∆α (5) had . Recent lattice-QCD results for HVP by the Budapest-Marseille-Wuppertal collaboration (BMWc) prompt us to reexamine this interplay. While their result would bring the SM prediction for (g −2)µ into agreement with the Brookhaven measurement, this comes at the expense of a 3.2 σ deviation with respect to a wide range of e + e − → hadrons cross-section data. We find that the global EW fit alone provides a competitive, independent determination of ∆α (5) had , which lies even below the e + e − range, resulting in a tension with the BMWc result whose significance depends on the energy where the bulk of the changes in the cross section occurs. Reconciling experiment and SM prediction for (g − 2)µ by adjusting HVP, as suggested by BMWc, would thus not necessarily weaken the case for physics beyond the SM (BSM), but merely shift it from (g − 2)µ to the EW fit. We briefly explore some options of BSM scenarios that could conceivably explain the ensuing tension.

I. INTRODUCTION
The SM of particle physics has been established with increasing precision over the last decades. In particular, both the global fits to EW precision data [1][2][3] and to the Cabibbo-Kobayashi-Maskawa (CKM) matrix [4,5] are in general in good agreement with the SM hypothesis and no new particles have been directly observed so far at the large hadron collider (LHC) [6,7].
where in the usual conventions for isospin-breaking effects the integral starts at the threshold s thr = M 2 π 0 due to the e + e − → π 0 γ channel and the kernel func-tionK(s) can be expressed analytically. Global analyses based on a direct integration of cross-section data [30][31][32][33] can now also be combined with analyticity and unitarity constraints for the leading 2π [32,34,35] and 3π [36] channels, covering almost 80% of the HVP contribution, to demonstrate that the experimental data sets are consistent with general properties of QCD. With recent advances in constraining the contribution from hadronic light-by-light scattering (including evaluations [37][38][39][40][41][42][43][44] based on dispersion relations in analogy to Eq. (1), shortdistance constraints [45][46][47], and lattice QCD [48,49]) as well as higher-order hadronic corrections [33,[50][51][52], this data-driven determination of HVP has corroborated the (g − 2) µ tension at the level of 3.5 σ. 2 While the precision of previous results for HVP from lattice QCD [53][54][55][56][57][58][59] was not competitive with the dispersive approach, recently BMWc announced a calculation at almost the same level of precision [60], finding that their result reduces the tension between the SM prediction and the Brookhaven measurement of (g − 2) µ to around 1 σ. From the dispersive perspective the resulting 3.2 σ difference with respect to the integral in Eq. (1) is extremely difficult to understand, as it would imply serious flaws in a host of e + e − cross-section measurements at many different facilities over the last decades. While therefore the results of Ref. [60] certainly need to be scrutinized by other lattice collaborations (see Ref. [61] for a first step to try and resolve the tensions with phe-2 Definite numbers will be included in the upcoming white paper from the Muon g − 2 Theory Initiative. arXiv:2003.04886v1 [hep-ph] 10 Mar 2020 nomenology, but also among different lattice calculations), we explore here its consequences for the global EW fit, a connection that rendered significant modifications of HVP an unlikely explanation of the (g − 2) µ anomaly in the past [62]. HVP enters the global EW fit indirectly via its impact on the running of α. With α most accurately determined as α ≡ α(0), but EW precision data taken around the Z pole, the translation requires, in addition to the leptonic running ∆α lep , a contribution from the top quark ∆α top and, crucially, information on the hadronic running where the dash indicates the principal value of the integral. Apart from a different weight function, this quantity is therefore determined by the same e + e − cross sections, in such a way that a shift in HVP as dramatic as suggested by BMWc will have a profound impact on the EW fit, as we will confirm below. As a reference value from e + e − data we quote [32,33] ∆α (5) had e + e − = 276.  7) are obtained under the hypothesis that the relative change in the cross section occurs only below the indicated scale, but is otherwise energy independent. Such an assumption is supported by a partial ("window") result for HVP given in Ref. [60], which involves a different weight function that removes the contribution from very small and very large energies from the integral, but differs from the phenomenological result by virtually the same relative amount as the entire HVP integral. For definiteness, the projections (5)- (7) have been derived using the integral breakdown from Ref. [31] (Ref. [32] would lead to the same qualitative conclusion, but considers slightly different energy intervals). The significance of the tension with Eq. (4) becomes {4.5, 2.5, 4.5}σ for the three cases, respectively, where in the last case the significance increases again because the dominant uncertainty in the e + e − cross sections arising from the intermediate energy interval drops out (the remaining uncertainty is only 0.3×10 −4 [31]). While the "window" result suggests that  (1 σ). Results that assume the relative change in the cross section to be energy independent (compared to the e + e − data and below the scale indicated in brackets, as explained below Eqs. (5)- (7)) are shown as dashed lines. The colored bands indicate the posteriors within scenario (1), (2), and (3), corresponding to using e + e − data, no input for the prior, and employing the BMWc projection (5), respectively. In addition, we show the 2018 result for the EW fit by the Gfitter group [2], which agrees well with our posterior (2), see Eq. (8), but would slightly reduce the significance of the tension with BMWc. The value derived from aµ = (g − 2)µ/2 is obtained when assuming the absence of BSM physics in aµ and relies on the same scaling assumption as for BMWc, see Eq. (10).
not all the changes can be concentrated at low energies, a definite prediction is not possible without knowledge of the full integrand from Ref. [60]. To illustrate the maximum impact on the EW fit, we will use the projection in Eq. (5) as a reference point. 3 To assess the consequences of a shift in HVP as drastic as suggested by BMWc, we now contrast ∆α (5) had from Eqs. (4) and (5) to a global fit of EW precision data. We find that with modern data and theory calculations the EW fit is sufficiently powerful to provide an independent determination of ∆α (5) had , without assuming any prior input, be it from lattice QCD or e + e − data. We will perform this determination using the Bayesian statistics implemented in the HEPfit package [63].

II. ELECTROWEAK FIT AND HVP
Measurements of the EW observables, as performed at LEP [64,65], are high-precision tests of the SM. The EW sector of the SM can be completely parameterized in terms of the three Lagrangian parameters v, g, and g ; then, other quantities such as the Fermi constant G F and the gauge-boson masses M W , M Z can be expressed in terms of these parameters and their measurements allow for global consistency tests. However, for practical purposes it is more advantageous to choose instead the three quantities with the smallest (relative) experimental error of their direct measurements, i.e., the mass of the Z boson (M Z ), the Fermi constant (G F ), and the fine-structure constant (α). Other EW observables, computed from G F , M Z , and α, include M W , the hadronic Z-pole cross section (σ 0 h ), and the leptonic vector and axial-vector couplings, g V and g A . Assuming the gauge sector to be lepton flavor universal we can thus use the five standard Z observables [65]: M Z , Γ Z , σ 0 had , R 0 , and A 0, FB . Furthermore, the Higgs mass (M H ), the top mass (m t ), and the strong coupling constant (α s ) have to be included as fit parameters as well, since they enter indirectly EW observables via loop effects.
Similarly, ∆α (5) had enters indirectly to encode the hadronic information needed to evolve α(µ 2 ) from µ = 0, where its most precise measurements are performed, to the scale µ = M Z , where it is needed for the EW fit. A key new development compared to Ref. [62] is that with modern EW input, especially a definite Higgs mass M H , the EW fit is now sufficiently over-constrained that it is possible to actually determine ∆α (5) had from the fit, without using any prior information neither from e + e − data nor from lattice QCD [2]. Furthermore, using ∆α (5) had from e + e − data or from BMWc as an input, one can compare the goodness of the resulting fit and analyze the tensions (pulls) within the fit. We consider three different scenarios: (1) EW fit using ∆α (5) had e + e − from e + e − data as a prior; (2) EW fit without any experimental or theoretical constraint on ∆α (5) had (using a large flat prior), with the posterior of ∆α (5) had EW solely (albeit indirectly) determined by EW precision data; (3) EW fit with BMWc projection (5) as a prior for ∆α (5) had BMWc . Note that scenario (1) corresponds to the standard approach used previously in the literature.
We perform the global fit within these three scenarios in a Bayesian framework using the publicly available HEPfit package [63], whose Markov Chain Monte Carlo (MCMC) determination of posteriors is powered by the Bayesian Analysis Toolkit (BAT) [66]. The results of the three scenarios are shown in Table I. In scenario (1) we find consistency between the value from e + e − data and the other observables of the global fit, as can be seen from the good agreement between the measurement and the posterior of ∆α (5) had . In scenario (2) we find a posterior of ∆α (5) had EW = 270.2(3.0) × 10 −4 .
Note that this value (see Fig. 1 for the comparison with other determinations) has a larger error than the one obtained in scenario (1) because no additional input (experimental or from lattice QCD) has been used and its posterior is entirely determined (indirectly) from the global EW fit. Our value is compatible with the 2018 Gfitter result of 271.6(3.9) × 10 −4 [2]. In particular, we observe that this independent determination (8) of the hadronic running largely agrees with Eq. (4), but differs from Eq. (5) at the level of 4.2 σ, demonstrating that if the changes to the cross section were equally distributed over all energies, the BMWc result would stand in significant conflict with the EW fit (Eqs. (6) and (7) would imply a tension of 3.1 σ and 2.4 σ, respectively, while the e + e − result (4) lies 1.8 σ above Eq. (8)). The same conclusion also derives from scenario (3), in which posterior and measurement (lattice determination) of ∆α (5) had are no longer in good agreement. Furthermore, the pulls of several measurements are significantly increased compared to scenario (1), signaling significant tensions within the EW fit. These tensions within scenario (3) are also confirmed by its information criterion (IC) value [67,68] of 36, which is significantly higher than the IC values of scenarios (1) and (2) of 20.5 and 17, respectively. In the terms defined in Ref. [68], this constitutes "very strong" evidence for scenarios (1) and (2) compared to scenario (3).

III. BSM PHYSICS IN THE EW FIT
As demonstrated most conclusively in terms of Eq. (8), removing the tension between SM prediction and experiment for (g − 2) µ according to BMWc leads, in general, to tensions within the EW fit. Thus, the hints for BSM physics are difficult to be removed in this way, but always shifted at least to some extent from (g − 2) µ to the EW fit. Therefore, the question arises if there are BSM scenarios that would impact the EW fit in the observed manner, while leaving (g − 2) µ unaffected.
As can be seen from Table I, the main tensions (largest pulls) of the fit in scenario (3) are in the W mass and even more pronounced in where g A (g V ) is the axial-vector (vector) coupling of charged leptons to the Z [69]. Another notable pull in scenario (3) appears in sin 2 θ lept eff(Had.coll.) , while the pull in A 0,b FB , the second-most significant one in the standard fit, is one of the few that becomes mitigated.
In order to get a shift in A , an effect in g V /g A is necessary. In the EFT language [82,83], this shift can be   [69], which are so precisely measured that the posteriors are identical to their direct measurements. Concerning the W mass computation, HEPfit provides both the option of using the precise numerical formula from Ref. [76] as well as the usual determination of MW from GF , MZ , and α [77], with radiative corrections encoded in ∆r (which is known up to 3-loop O(α 3 ) EW [78] and O(αα 2 s , α 2 αs) EW-QCD contributions [78][79][80][81]). We opt for the latter possibility. . At tree level, these operators can be modified by vector-like leptons or a Z boson coupling to right-handed leptons and mixing with the SM Z [84]. Furthermore, these effects are expected to affect the closely related observable A 0, FB as well, where also a tension in scenario (3) arises.
Concerning the W mass, this shift can be understood as an effect in the EW T parameter [85][86][87][88] generated by O φD . Here, a possible explanation could be given in terms of the minimal supersymmetric SM (MSSM), where a necessarily constructive effect (increasing the value of M W with respect to the SM) is predicted [89] as confirmed by current fits [90]. Furthermore, composite Higgs models have been known for a long time to be prime candidates to solve the EW hierarchy problem, and can give rise to sizable effects in the EW precision data, in particular in the S and T parameters [91][92][93][94]. Usually, to protect tree-level modifications of the T parameter, custodial symmetry is imposed. Nonetheless, its value can still be substantially modified via fermion resonances, as shown for instance in Refs. [93][94][95].
Since the BMWc result reduces the tension between the SM prediction and experiment for (g − 2) µ to the 1 σ level, one could go even further and determine HVP by demanding agreement (within the uncertainties) between experiment and the remaining part of the SM prediction. This means that (g − 2) µ measurements could be used to determine HVP under the assumption that it is free of BSM effects and, more crucially, assuming a certain energy dependence of the changes in the cross section. A naive scaling up to M Z with respect to Eq. (4) would lead to ∆α (5) by definition even larger than Eq. (5), and with an error that would decrease to about 1.0 for the final E989 precision [96]. The comparison of the different values for ∆α (5) had is shown in Fig. 1, with the ones affected by the scaling assumption indicated by dashed lines. In view of these different scenarios it is worthwhile to assess the impact of future determinations of ∆α (5) had on the global EW fit.
For this purpose, we remove the measurements of two observables with large pulls (M W and A ) from the fit and predict their posterior as a function of ∆α (5) had (without assigning an error to ∆α (5) had for each point sampled). We choose M W and A as representatives here given that these are two of the observables that mainly drive the tensions in scenario (3), while the slight improvement in A 0,b FB is by far not sufficient to balance their effect. We also note that A exhibits the biggest tension already in the standard scenario (1), a tension that is further exacerbated in scenario (3). The corresponding results are depicted in Fig. 2 had together with its preferred ranges from e + e − data and the BMWc projection (5). See main text for details. ranges for ∆α (5) had as well as the measurements for M W and A are included. Therefore, the differences between the posteriors and the measurements, for a given value of ∆α (5) had , would need to be explained by BSM physics to restore the goodness of the global EW fit. Again, we see that HVP derived from e + e − data does not require a BSM component, while for the projection (5) the EW fit is no longer consistent without a significant BSM contribution.

IV. CONCLUSIONS
In this article we reexamined the impact of HVP on (g − 2) µ and the global EW fit in light of the recent lattice-QCD calculation by BMWc. On the one hand, the commonly used result for HVP from e + e − data leads to a consistent global EW fit, but generates the well-known discrepancy with the measurement of (g − 2) µ . On the other hand, the new BMWc result for HVP leads to a SM prediction for (g − 2) µ consistent with the Brookhaven measurement, but is not only in tension with the e + e − data, but also leads to tensions within the EW fit, via the change in the hadronic running of the fine-structure constant ∆α (5) had , whose significance depends on the energy scale where the changes in the cross section occur. Our analysis assumes a naive scaling with respect to the e + e − data below different thresholds, see Eqs. (5)-(7), given that the full result cannot be inferred from Ref. [60]. Further information on the claimed changes in HVP compared to the e + e − cross sections is thus critical to assess their impact on the EW fit.
Either way, a significant shift in HVP, as suggested by the BMWc result, can in principle account for the experimental value of (g − 2) µ , but at the expense of generating tensions within the EW fit. As seen from Fig. 2, we observe that for any of the values of ∆α (5) had assumed in Eqs. (5)-(7), the shifts predicted by the EW fit for M W and A always occur into the direction in which the tension with respect to their measured value increases. These tensions, which, in principle, could end up anywhere between the red and gray bands, would call for an explanation in terms of BSM physics just as the one in (g − 2) µ would. However, the kind of BSM scenarios required here would be notably different from the ones necessary to explain (g − 2) µ . E.g., a tension in the prediction for M W with respect to the measured value could be explained in models that generate a sizable effect in the T parameter. Here, composite models (or in the dual picture models with extra dimensions) come to mind. On the other hand, the tension in g A could be resolved in models with vector-like leptons. Furthermore, since extra-dimensional or composite models not only lead to sizable effects in the S and T parameters, but also possess vector-like fermions, these models are prime candidates for reconciling the EW fit in case the BMWc result were confirmed. We stress that such an outcome would imply severe deficiencies in e + e − cross sections affecting in the same way different channels measured at different experiments and facilities over decades. However, our analysis reaffirms that even if that were the case and the need for BSM physics eliminated in (g − 2) µ , other tensions in the SM would likely arise elsewhere.