Hadronic vacuum polarization: Comparing lattice QCD and data-driven results in systematically improvable ways

The precision with which hadronic vacuum polarization (HVP) is obtained determines how accurately important observables, such as the muon anomalous magnetic moment $a_{\mu }$ or the low-energy running of the electromagnetic coupling $\alpha$, are predicted. The two most precise approaches for determining HVP are dispersive relations combined with $e^{+}{e}^{-}\to \text{hadrons}$ cross section data and lattice QCD. However, the results obtained in these two approaches display significant tensions, whose origins are not understood. Here we present a framework that sheds light on this issue and—if the two approaches can be reconciled—allows them to be combined. Via this framework, we test the hypothesis that the tensions can be explained by modifying the R-ratio in different intervals of center-of-mass energy $\sqrt{s}$. As ingredients, we consider observables that have been precisely determined in both approaches. These are the leading hadronic contributions to $a_{\mu }$, to the so-called intermediate window observable, and to the running of $\alpha$ between spacelike virtualities 1 and $10{\mathrm{GeV}}^2$ (for which only a preliminary lattice result exists). Our tests take into account all uncertainties and correlations, as well as uncertainties on uncertainties in the lattice results. For instance, using this framework we show that results obtained in the two approaches can be made to agree, for all three observables, by modifying the $\rho$ peak in the experimental spectrum. More specifically, we show that this requires a common $\sim 5\%$ increase in the contributions of the peak to each of the three observables. This result is robust against the presence or absence of the running of $\alpha$ in the comparison. However, such an increase is much larger than the uncertainties on the measured R-ratio. We also discuss a variety of generalizations of the methods used here, as well as the limits in the information that can be extracted from the R-ratio via a finite set of observables.

Figure 1

Kernels for $a_{\mu }^{\mathrm{LO}\text{−}\mathrm{HVP}}$, $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}$, and $\delta (\mathrm{\Delta }{\alpha }_{\mathrm{had}}^{(5)})$ as a function of Euclidean time (left) and of c.m. energy $\sqrt{s}$ (right). In addition, to clarify the arguments laid out later in the section, we have added the kernels of the short-distance window [Eq. (7) with $t_i=0$ and $t_f=0.4\mathrm{fm}$] and the long-distance one as well [Eq. (7) with $t_i=1\mathrm{fm}$ and $t_f=\infty$]. Some of the kernels are rescaled for better visibility. $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}$, the short-distance and long-distance windows are a contribution to $a_{\mu }^{\mathrm{LO}\text{−}\mathrm{HVP}}$ so that their kernels (green, orange, and red curves, respectively) are contributions to $a_{\mu }^{\mathrm{LO}\text{−}\mathrm{HVP}}$’s kernel (blue curve).

Figure 2

Comparison of lattice and data-driven results for the contribution $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}$ to $a_{\mu }^{\mathrm{LO}\text{−}\mathrm{HVP}}$, from the Euclidean time interval [0.4,1.0] fm [see Eqs. (6) and (7) and subsequent text]. Left: Comparison of lattice results for the isospin-symmetric, $u$ and $d$ connected contribution to the intermediate-window observable, noted $[a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}{]}_{\mathrm{iso}}^{ud}$ (green squares). For each group, only the most recent results are shown. BMW’20 [6], LM’20 [77], ABGP’22 [78], and FHM’23 [13] are obtained with different varieties of staggered fermions; Mainz’22 [10] with $O(a)$-improved-Wilson, ETMC’22 [11] with twisted-mass, and RBC/UKQCD’23 [12] with domain-wall fermions; $\chi \mathrm{QCD}$’22 [79] with overlap valence quarks on either HISQ or domain-wall configurations. The filled squares correspond to fully independent results, while the open ones are obtained using subsets of configurations from other calculations. LM’20 relies on a subset of the configurations used in FHM’23 and $\chi \mathrm{QCD}$’22 on subsets of those used in FHM’23 and RBC/UKQCD’23. The green band corresponds to a weighted average of the fully independent (filled squares) lattice results for $[a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}{]}_{\mathrm{iso}}^{ud}$. Following the procedure discussed in Appendix pp3, the mean is performed without correlations in the determination of the weights, while the uncertainty propagation and ${\chi }^2$ evaluation take them into account. These are obtained by assuming 100% correlation between the total systematic errors of the different calculations. In this way we find $[a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}{]}_{\mathrm{iso}}^{ud}=(206.6\pm 0.9)\times {10}^{-10}$ with a correlated ${\chi }^2/d\mathrm{.}o\mathrm{.}f\mathrm{.}=1.7/4$. Note that a phenomenological estimate of this quantity was reported in Refs. [80, 81]. Right panel: Comparison of the world average (WA) of lattice results (green filled circle and band) with a number of R-ratio results for the intermediate-window observable $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}$ (red diamonds), including the one determined in this paper (filled red diamond and band) and by CEKHLST’22 [14]. The WA for $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}$ is obtained from that for $[a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}{]}_{\mathrm{iso}}^{ud}$, by adding all other quark, QED, and strong-isospin-breaking contributions from Ref. [6]. The resulting average is $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}=(236.1\pm 0.9)\times {10}^{-10}$. The difference of this result with the data-driven one of the present work is $4.2\sigma$, shown as a horizontal arrow.

Figure 3

Plot illustrating Table 2. The first panel displays, in dark gray, the various $\sqrt{s}$ intervals considered here. The first row corresponds to the $\rho$ peak. In the second panel, we plot the $p$-values that indicate the compatibility of the rescaling hypothesis, in the given interval, with the lattice and data-driven results for $a_{\mu }^{\mathrm{LO}\text{−}\mathrm{HVP}}$ and $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}$ (two observables, blue triangles) and with the results for the additional observable $\delta (\mathrm{\Delta }{\alpha }_{\mathrm{had}}^{(5)})$ (three observables, orange upside-down triangles). These $p$-values are obtained from the ${\chi }^2$ defined in Eq. (19), using the appropriate d.o.f.. The error bars are the statistical uncertainties resulting from those on the lattice covariance matrices determined in Appendix pp2 (see also footnote 12). The corresponding uncertainty distributions are plotted in Figs. 4 and 6 (for the third and eleventh row of this plot, respectively). The filled/empty points correspond to the largest/smallest $p$-value obtained from the four systematic variations of the lattice covariance matrix of Eqs. (b31)–(b34). The vertical dashed line corresponds to 5%. We consider solutions with $p$-values above and close to that line to be compatible with the rescaling hypothesis in the corresponding interval and those far below to be incompatible. The rightmost panel displays the rescaling percentages ${\delta }_1$, corresponding to the $p$-values on the same row that have the same plot symbol. Here the uncertainties are the leading ones, given by the nominal values of the lattice and R-ratio covariance matrices, i.e., not including the small, additional uncertainty associated with the statistical uncertainty on the lattice covariance matrices.

Figure 4

Distributions of ${\chi }^2$ (left) and $p$-value (right) obtained by sampling the bootstrap replicas of the lattice covariance matrix. These figures show the effects of different approaches to determining ${\gamma }_1$ and its uncertainty in the low-mass region below 0.96 GeV. Two moment integrals are considered here, with the lattice covariance matrix 0. In the first row, we consider normalization fits using the full covariance matrices for determining the averaging weights—their comparison with the two plots in the following row is discussed in Appendix pp4. In the middle row, we consider normalization fits with the averaging weights proportional to the inverse of the ${\tilde{\gamma }}_j$ uncertainties squared, as considered in the present section. In the final row, we set the ${\gamma }_b$ normalization coefficients to unity, i.e., we perform no rescaling—these are discussed in Sec. 5a. The blue horizontal error bar indicates the asymmetric variance of the distribution, computed on each side of the nominal value (blue point and blue dashed vertical line). The median (dash-dotted black line), mean (black continuous line), 68.3% and 95.4% quantiles (colored bands) of the distribution are also indicated. The number of degrees of freedom in the ${\chi }^2$ calculation is indicated by the dotted pink line. Note that the scales in the last row of plots (corresponding to no rescaling) are very different from those of the two previous rows (in which rescaling is allowed) to account for the fact that the ${\chi }^2$ and $p$-values with no rescaling are very poor.

Figure 5

Distributions of the normalization rescaling factor ${\gamma }_1$ (left) and its uncertainty propagated from the covariance matrices of the lattice QCD and dispersive results (right), obtained by sampling the bootstrap replicas of the lattice covariance matrix. These figures show the effects of different approaches to determining ${\gamma }_1$ and its uncertainty in the low-mass region below 0.96 GeV. Two moment integrals are considered here, with the lattice covariance matrix 0. In the first row, we consider normalization fits using the full covariance matrices of the ${\tilde{\gamma }}_j$ for determining the averaging weights. In the second row, we consider normalization fits with the averaging weights proportional to the inverse of the ${\tilde{\gamma }}_j$ uncertainties squared. The blue horizontal line indicates the asymmetric variance of the distribution, computed on each side of the nominal value (dashed blue vertical line). The median (dash-dotted black line), mean (black continuous line), 68.3% and 95.4% quantiles (colored bands) of the distribution are also indicated.

Figure 6

Distributions of ${\chi }^2$ (left) and $p$-value (right), obtained by sampling the bootstrap replicas of the lattice covariance matrix. These figures show the effects of different approaches to determining ${\gamma }_1$ and its uncertainty in the high-mass region above 3.0 GeV. Two moment integrals are considered here, with the lattice covariance matrix 3. In the first row, we consider normalization fits using the full covariance matrices for determining the averaging weights—their comparison with the two plots in the following row is discussed in Appendix pp4. In the middle row, we consider normalization fits with the averaging weights proportional to the inverse of the ${\tilde{\gamma }}_j$ uncertainties squared, as considered in the present section. In the final row, we set the ${\gamma }_b$ normalization coefficients to unity, i.e., we perform no rescaling—these are discussed in Sec. 5a. The blue horizontal error bar indicates the asymmetric variance of the distribution, computed on each side of the nominal value (blue point and blue dashed vertical line). The median (dash-dotted black line), mean (black continuous line), 68.3% and 95.4% quantiles (colored bands) of the distribution are also indicated. The number of degrees of freedom in the ${\chi }^2$ calculation is indicated by the dotted pink line. Note that the scales in the last row of plots (corresponding to no rescaling) are very different from those of the two previous rows (in which rescaling is allowed) to account for the fact that the ${\chi }^2$ and $p$-values with no rescaling are very poor.

Figure 7

Distributions of the normalization rescaling factor ${\gamma }_1$ (left) and its uncertainty propagated from the covariance matrices of the lattice QCD and dispersive results (right), obtained by sampling the bootstrap replicas of the lattice covariance matrix. These figures show the effects of different approaches to determining ${\gamma }_1$ and its uncertainty in the high-mass region above 3.0 GeV. Two moment integrals are considered here, with the lattice covariance matrix 3. In the first row, we consider normalization fits using the full covariance matrices of the ${\tilde{\gamma }}_j$ for determining the averaging weights. In the second row, we consider normalization fits with the averaging weights proportional to the inverse of the ${\tilde{\gamma }}_j$ uncertainties squared. The blue horizontal line indicates the asymmetric variance of the distribution, computed on each side of the nominal value (dashed blue vertical line). The median (dash-dotted black line), mean (black continuous line), 68.3% and 95.4% quantiles (colored bands) of the distribution are also indicated.

Figure 8

Histograms for the estimation of the total, statistical, and systematic lattice covariance matrices for the connected $ud$ (a), the connected $s$ (b), and disconnected (c) contributions to $a_{\mu }^{\mathrm{LO}\text{−}\mathrm{HVP}}$ and $a_{\mu ,\mathrm{win}}^{\mathrm{LO}\text{−}\mathrm{HVP}}$. These are obtained via the marginalization of Eq. (b25) to these individual contributions, as described in the text. Each of the three plots has a heat map in its center, illustrating the two-dimensional covariance distribution of the total combined uncertainty. As usual, colors from white to blue represent higher to lower probability densities. These densities are calculated using the random sampling discussed in the text. The solid and dashed ellipses in these maps indicate the $1\sigma$ and $2\sigma$ errors determined from the corresponding quantiles. In the upper and right panels around the heat maps, histograms of the marginalized distributions are shown. These are calculated exactly without random sampling. The two green bands show the $1\sigma$ and $2\sigma$ uncertainties determined by the corresponding quantiles of the distribution evaluated symmetrically around the median.