Data-driven results for light-quark connected and strange-plus-disconnected hadronic $g-2$ short- and long-distance windows

We start by defining the window quantities considered in this paper. The master formula for the data-driven (or dispersive) approach to the leading order $a_{\mu}^{\rm HVP}$ is [58, 59, 60]

$$a_{\mu}^{\rm HVP}=\frac{4\alpha^{2}m_{\mu}^{2}}{3}\int_{m_{\pi}^{2}}^{\infty}ds\,\frac{\hat{K}(s)}{s^{2}}\rho_{\rm EM}(s),$$

(1)

where $m_{\pi}$ is the neutral pion mass and $\rho_{\rm EM}(s)$ represents the inclusive EM-current hadronic spectral function, which is related to the $R$ ratio derived from the bare inclusive hadronic electroproduction cross section, $\sigma^{(0)}[e^{+}e^{-}\rightarrow\mbox{hadrons}(+\gamma)]$, by

	$\displaystyle\rho_{\rm EM}(s)$	$\displaystyle=$	$\displaystyle\frac{1}{12\pi^{2}}R(s),$		(2)
	$\displaystyle R(s)$	$\displaystyle=$	$\displaystyle\frac{3s}{4\pi\alpha}\,\sigma^{(0)}[e^{+}e^{-}\rightarrow\mbox{hadrons}(+\gamma)],$

where the kernel $\hat{K}(s)$ is a smoothly varying, monotonically increasing function with $\hat{K}(4m_{\pi}^{2})\approx 0.63$ at the two-pion threshold and $\lim_{s\to\infty}\hat{K}(s)=1$.⁴⁴4For the explicit expressions, we refer to e.g. Ref. [5]. In terms of the so-called time-momentum representation, $a_{\mu}^{\rm HVP}$ can be obtained from the Euclidean-time two-point correlator [61]

$$C(t)=\frac{1}{3}\sum_{i=1}^{3}\int d^{3}x\langle j_{i}^{\rm EM}(\vec{x},t)j_{i}^{\rm EM}(0)\rangle=\frac{1}{2}\int_{m_{\pi}^{2}}^{\infty}ds\,\sqrt{s}\,e^{-\sqrt{s}t}\,\rho_{\rm EM}(s)\quad(t>0),$$

(3)

built from the EM current $j_{i}^{\rm EM}(\vec{x},t)$ as

$$a_{\mu}^{\rm HVP}=2\int_{0}^{\infty}dt\,w(t)C(t),$$

(4)

where the function $w(t)$ is known and can be obtained from $\hat{K}(s)$.⁵⁵5For explicit expressions and a useful approximation see Refs. [61, 62].

The windowed versions (“windows” in what follows) we consider in this work, originally introduced in Ref. [34], are the SD, intermediate (int), and LD windows obtained by inserting the functions

	$\displaystyle W_{\rm SD}(t)$	$\displaystyle=1-\Theta(t,t_{0},\Delta),$
	$\displaystyle W_{\rm int}(t)$	$\displaystyle=\Theta(t,t_{0},\Delta)-\Theta(t,t_{1},\Delta),$
	$\displaystyle W_{\rm LD}(t)$	$\displaystyle=\Theta(t,t_{1},\Delta),$		(5)

respectively, with

$$\Theta(t,t^{\prime},\Delta)=\frac{1}{2}\left(1+\tanh\frac{t-t^{\prime}}{\Delta}\right)$$

(6)

and $t_{0}=0.4\,{\rm fm}$, $t_{1}=1.0\,{\rm fm}$, and $\Delta=0.15\,{\rm fm}$, into the representation of Eq. (4), leading to

	$\displaystyle a_{\mu}^{\rm win}$	$\displaystyle=$	$\displaystyle 2\int_{0}^{\infty}dt\,W_{\rm win}(t)\,w(t)C(t)$
		$\displaystyle=$	$\displaystyle\frac{4\alpha^{2}m_{\mu}^{2}}{3}\int_{m_{\pi}^{2}}^{\infty}ds\,\frac{\hat{K}(s)}{s^{2}}\,\widetilde{W}_{\rm win}(s)\,\rho_{\rm EM}(s),$

where ${\rm win}=\{{\rm SD},{\rm int},{\rm LD}\}$ and the window function in $s$-space is written as

$$\widetilde{W}_{\rm win}(s)=\frac{\int_{0}^{\infty}dt\,W_{\rm win}(t)\,w(t)\,e^{-\sqrt{s}t}}{\int_{0}^{\infty}dt\,w(t)\,e^{-\sqrt{s}t}}.$$

(8)

By construction, $a_{\mu}^{\rm HVP}$ is given by the sum of the three windows defined above.

Our data-driven determination of the lqc and the s+lqd contributions to $a_{\mu}^{\rm HVP}$ follows the strategy implemented for the first time in Refs. [54, 55]. We start from the decomposition of the three-light-flavor EM current into its $I=1$ (flavor octet label 3) and $I=0$ (flavor octet label 8) parts, and the related decompositions of $\rho_{\rm EM}(s)$, $C(t)$ and any weighted integrals thereof, into their pure $I=1/0$ (flavor octet labels 33/88) and mixed-isospin (MI, flavor octet labels 38) components. In the isospin limit, the pure $I=1$ contributions are entirely light-quark connected and the corresponding pure $I=0$ lqc contributions exactly 1/9 times their $I=1$ lqc counterparts. Thus, for example, the total lqc contribution to the EM spectral function is

$$\rho^{\rm lqc}_{\rm EM}(s)=\frac{10}{9}\rho_{\rm EM}^{I=1}(s),$$

(9)

and one can obtain the full strange plus light-quark disconnected contribution from the combination

	$\displaystyle\rho^{\rm s+lqd}_{\rm EM}(s)$	$\displaystyle=\rho_{\rm EM}^{I=0}(s)-\frac{1}{9}\rho_{\rm EM}^{I=1}(s)$
		$\displaystyle=\rho_{\rm EM}(s)-\frac{10}{9}\rho_{\rm EM}^{I=1}(s).$		(10)

Our main task is, hence, the identification, with sufficient precision, of the $I=1$ and $I=0$ components of $\rho_{\rm EM}$ and any associated weighted integral quantities. As outlined below, this can be accomplished on a channel-by-channel basis, using isospin symmetry, in the exclusive-mode region of the $R(s)$ data. In this work, we employ the results for the exclusive-mode components of the pre-CMD-3 $R(s)$ data combination of Ref. [13], commonly referred to as ‘KNT19.’ It would be interesting to perform our analysis using other data combinations, such as that of Ref. [12], but this would require access to their exclusive spectra, channel-by-channel and with all correlations, which, to the best of our knowledge, is not publicly available.

Substituting the spectral functions for the lqc and s+lqd components, Eq. (9) and Eq. (10), respectively, into Eq. (2.1), one obtains the respective isospin-limit contributions to each of the $a_{\mu}^{\rm HVP}$ windows, which we denote by $a_{\mu}^{\rm win,lqc}$ and $a_{\mu}^{\rm win,s+lqd}$.

As discussed in detail in Refs. [54, 55, 56, 57], in the isospin limit, modes with positive (negative) $G$-parity have isospin $I=1$ ($I=0$). In Tabs. 1 and 2, we give the contributions of each of the $G$-parity unambiguous modes $X$ present in the KNT19 data combination to the three windows of Eq. (2.1), denoted $[a_{\mu}^{\rm win}]_{X}$, as well as to the total HVP contribution, $[a_{\mu}^{\rm HVP}]_{X}$.⁶⁶6Entries whose mode names include the phrase “low-$s$” are those labelling very-near-threshold contributions for which the underlying $R(s)$ contributions were obtained using Chiral Perturbation Theory in the KNT19 exclusive-mode compilation. Although the central value for $[a_{\mu}^{\rm HVP}]_{X}$ is always the sum of the central values of the contributions to the three windows from the same mode, the uncertainty in the fifth column of Tabs. 1 and  2 includes the effect of correlations among the errors on the three associated window quantities. The results for the intermediate window and for the total HVP were already given in Refs. [56, 54] and are repeated here for completeness.

The corresponding exclusive-mode contributions to the full lqc window totals are obtained by multiplying the entries in Tab. 1 by 10/9, as per Eq. (9). The resulting sums of lqc contributions from all unambiguous modes are listed in Tab. 5. The contributions of the unambiguous modes to the s+lqd components are given by the combination shown in Eq. (10); the totals of these contributions for each window appear in Tab. 6.

We turn now to the modes with no definite isospin. These are of two distinct types, those for which external information can be used to help separate the different isospin components and those for which this is not possible. Fortunately such external information is available for the channels, $K\bar{K}$, $K\bar{K}\pi$ and $\pi^{0}/\eta+\gamma$, which give the largest of the ambiguous-mode contributions. Contributions from other ambiguous modes all turn out to be small. The strategy used to perform the ambiguous-mode isospin separations is described in Ref. [57] (see also Ref. [54]) and reviewed briefly below.

We start from those ambiguous modes having only $I=0$ and $I=1$ contributions in the isospin limit and for which no external information is available. The small contributions from these modes are treated using a “maximally conservative” prescription, based on the observation that, because of spectral positivity, the $I=0$ and $I=1$ parts of the contribution from the given mode $X$ must both lie between 0 and the full experimental $I=1+0$ total obtained from the KNT19 $R(s)$ data. The $I=0$ and $I=1$ components are then guaranteed to lie, respectively, in the ranges ($50\pm 50$)% and ($50\mp 50$)% times the $I=0+1$ total, with the two errors 100% anticorrelated. The lqc and the s+lqd parts of the mode-$X$ contribution to the EM spectral function then lie in the following ranges [54]

	$\displaystyle[\rho_{\rm EM}^{\rm lqc}]_{X}$	$\displaystyle=\left(\frac{5}{9}\pm\frac{5}{9}\right)[\rho_{\rm EM}]_{X},$
	$\displaystyle[\rho_{\rm EM}^{\rm s+lqd}]_{X}$	$\displaystyle=\left(\frac{4}{9}\pm\frac{5}{9}\right)[\rho_{\rm EM}]_{X}.$		(11)

This maximally conservative separation of $[\rho_{\rm EM}]_{X}$ produces related results for the contributions to the window quantities $[a_{\mu}^{\rm win,lqc}]_{X}$ and $[a_{\mu}^{\rm win,s+lqd}]_{X}$ when used in Eq. (2.1).

It is crucial, however, to have better control over the dominant contributions from ambiguous modes than would be provided by the maximally conservative treatment, especially those arising from the $K\bar{K}$ and, to a lesser extent, the $K\bar{K}\pi$ channels. For these channels, external experimental information can be used to assess the $I=1$ component of the total $I=1+0$ sum. For the $K\bar{K}$ modes ($K^{+}K^{-}$ and $K^{0}\bar{K}^{0}$), this external information comes in the form of BaBar’s measurement of the differential decay distribution of $\tau\to K^{-}K^{0}\nu_{\tau}$ [63], which provides a determination of the $K\bar{K}$ contribution to the charged-current $I=1$ vector spectral function, which in turn, via the conserved vector current (CVC) relation, provides a determination of the $I=1$ part of the $K\bar{K}$ contribution to $\rho_{EM}(s)$, and hence⁷⁷7For our purposes, isospin-breaking corrections to the CVC relation can safely be neglected. In the case of Eq. (12), e.g., a 1% isospin-breaking correction would amount to $0.002\times 10^{-10}$, which can very safely be neglected when compared with other uncertainties entering this determination. an estimate of the $I=1$ part of the $K\bar{K}$ contribution to the various window integrals in the region up to the endpoint, $s=2.7556\leavevmode\nobreak\ {\rm GeV}^{2}$, of the BaBar $\tau$ data. Above this point, we integrate the $I=1+0$ tail of the KNT19 $K\bar{K}$ spectrum, applying the maximally conservative separation to this small remainder to obtain our final $I=1$ and $I=0$ totals for the full exclusive-region $K\bar{K}$ contributions. As an example, for the lqc contribution to the LD window, this procedure gives, using Eq. (9),

$$[a_{\mu}^{\rm LD}]^{\rm lqc}_{K\bar{K}}=\frac{10}{9}(0.1743(84)+0.0065(65))\times 10^{-10}=0.201(12)\times 10^{-10},$$

(12)

where the first number in parenthesis is the $I=1$ contribution obtained using BaBar $\tau\to K^{-}K^{0}\nu_{\tau}$ data and the second is the contribution from the tail of the KNT19 spectrum, evaluated using the maximally conservative separation of Eq. (11). Results for the other windows are given in the second row of Tab. 3. The s+lqd LD window $K\bar{K}$ contribution is, similarly, using Eq. (10),

$$[a_{\mu}^{\rm LD}]^{\rm s+lqd}_{K\bar{K}}=\left[13.37(11)-\frac{1}{9}\times 0.181(11)\right]\times 10^{-10}=13.35(11)\times 10^{-10},$$

where the first number in square brackets is the $I=0$ total and the second number (one-ninth of) the $I=1$ part. The dominance of the $I=0$ component is a result of the enhancement produced by the $\phi$ resonance. The analogous results for the other windows are given in the first row of Tab. 4.

For the $K\bar{K}\pi$ contributions, the needed external information is provided by BaBar’s Dalitz plot separation of the $I=1$ and $I=0$ parts of the $K\bar{K}\pi$ cross sections [64]. The BaBar $I=1$ cross sections provide a determination of the $I=1$ part of the $K\bar{K}\pi$ contribution to $\rho_{EM}(s)$, and hence of the $K\bar{K}\pi$ mode contributions to the lqc window quantities. The corresponding $I=0$ contributions are obtained by subtracting the BaBar-based $I=1$ results from the KNT19 $I=1+0$ totals. As an example, we find, for the $K\bar{K}\pi$ contribution to the lqc component of the LD window, the result

$$[a_{\mu}^{\rm LD}]^{\rm lqc}_{K\bar{K}\pi}=\frac{10}{9}[a_{\mu}^{{\rm LD}}]^{I=1}_{K\bar{K}\pi}=0.083(14)\times 10^{-10}.$$

(13)

The corresponding results for the other windows are given in Tab. 3. Similarly, for the $K\bar{K}\pi$ contribution to the s+lqd component of the LD window, we find, using the second line of Eq. (10),

	$\displaystyle[a_{\mu}^{\rm LD}]^{\rm s+lqd}_{K\bar{K}\pi}$	$\displaystyle=[a_{\mu}^{\rm LD}]_{K\bar{K}\pi}-\frac{10}{9}[a_{\mu}^{{\rm LD}}]^{I=1}_{K\bar{K}\pi}$
		$\displaystyle=[0.263(11)-\frac{10}{9}\times 0.075(13)]\times 10^{-10}=0.180(18)\times 10^{-10},$		(14)

with the results for the other windows given in Tab. 4.

For the $K\bar{K}2\pi$ channel, a modest in-principle improvement can be achieved over the purely maximally conservative separation treatment by first using BaBar’s measurement of the $e^{+}e^{-}\to\phi\pi\pi$ cross sections [65] and the PDG $\phi\rightarrow K\bar{K}$ branching fraction to quantify the purely $I=0$ $e^{+}e^{-}\rightarrow\phi(\rightarrow K\bar{K})\pi\pi$ contribution and then applying the maximally conservative separation treatment to the rest of the $K\bar{K}2\pi$ contributions. In practice, the improvement is too small to make a significant impact and the final uncertainties on the $K\bar{K}2\pi$ contributions are still of the order of 100%, as can be seen in Tabs. 3 and 4.

Finally, for the radiative modes $\pi^{0}\gamma$ and $\eta\gamma$, a full decomposition into pure $I=1$, pure $I=0$ and MI components turns out to be possible owing to the strong dominance of the observed exclusive-mode-region cross sections by intermediate vector meson contributions. We briefly outline the decomposition procedure below, referring the reader to App. B of Ref. [57] for further details. We note first that the measured $e^{+}e^{-}\rightarrow\pi^{0}\gamma$ and $e^{+}e^{-}\rightarrow\eta\gamma$ cross sections display prominent narrow $\omega$ and $\phi$ resonance peaks. The normalizations of the underlying $V=\omega$ and $\phi$ contributions to the amplitudes are, of course, set by the measured $V\rightarrow e^{+}e^{-}$ and $V\rightarrow P\gamma$ ($P=\pi,\,\eta$) widths. Less immediately evident visually, but necessarily also present, are broad $\rho$ contributions, with normalizations set by the measured $\rho\rightarrow e^{+}e^{-}$ and $\rho\rightarrow P\gamma$ widths. In Ref. [57] it was shown that, using PDG input for the above widths, the VMD representations of the $e^{+}e^{-}\rightarrow P\gamma$ amplitudes produced by summing over the resulting externally determined $V=\rho,\,\omega$ and $\phi$ contributions accurately reproduce the observed cross sections and, not surprisingly therefore, provide accurate representation of the resulting full HVP and intermediate window integrals. Neglecting additional IB effects in the photon-vector-meson couplings (since the $e^{+}e^{-}\rightarrow\pi^{0}\gamma$ and $e^{+}e^{-}\rightarrow\eta\gamma$ cross sections and associated weighted integrals are already $O(\alpha_{EM})$ and hence first order in IB), the $\rho$ contributions to the two amplitudes come solely from the coupling of the $\rho$ to the $I=1$ (flavor $3$) part of the EM current, and the $\omega$ and $\phi$ contributions solely from the couplings of the $\omega$ and $\phi$ to the $I=0$ (flavor $8$) part. Thus, to first order in IB, the pure $I=1$ (flavor $33$) parts of the cross sections come from the squared modulus of the $\rho$ contribution to the amplitude in question, the pure $I=0$ (flavor $88$) part from the squared modulus of the sum of the $\omega$ and $\phi$ contributions, and the MI (flavor $38$) part from the interference between the $\rho$ and $\omega+\phi$ contributions. The $\pi^{0}\gamma$ and $\eta\gamma$ contributions to the lqc and s+lqd components of the SD and LD windows produced by the resulting $I=1$/$I=0$/MI decompositions are listed in Tabs. 3 and 4, where, for completeness, we also list the corresponding HVP and intermediate window results, reported previously in Ref. [57].

The small contributions from the remaining ambiguous exclusive modes of the KNT19 data compilation, are handled using the maximally conservative separation treatment, and the results, again, collected in Tabs. 3 and 4.

In the inclusive region, i.e., for hadronic squared invariant masses $s>(1.937\leavevmode\nobreak\ {\rm GeV})^{2}$, we use massless three-flavor pQCD (we have checked that strange-quark mass corrections can safely be neglected [54]). The Adler function is exactly known up to $\mathscr{O}(\alpha_{s}^{4})$ [66] and we supplement it with an estimate for the $\mathscr{O}(\alpha_{s}^{5})$ coefficient, as described in our previous works [54, 55, 57]. To this perturbative result, we add an estimate of the duality violating (DV) contribution, which, essentially, captures the residual oscillations in the spectral function due the tails of the higher-mass resonances. To parametrize the DVs, we employ results from our previous study of the $I=1$ spectral function in $\tau\to{\rm hadrons}+\nu_{\tau}$ [67] decays as well as knowledge about the $I=0$ contribution from $e^{+}e^{-}\to{\rm hadrons}$ [68].

Perturbative QCD is in good agreement with inclusive $R(s)$ data from BES [69, 70] and KEDR [71] for $s\geq 4\leavevmode\nobreak\ {\rm GeV}^{2}$, but in some tension with recent, more precise results from BES-III [72] below charm threshold. Because of this tension, and since the DV contribution represents an essential limitation of perturbation theory, we have significantly enlarged the error associated with the use of pQCD in the inclusive region — which is typically assessed using estimates of the size of missing higher-orders in the $\alpha_{s}$ expansion. We consider, therefore, as our final error on the inclusive-region contribution the central value of the DV component. This procedure produces a significantly increased error on the inclusive-region contributions to the lqc and s+lqd components. For the lqc case, the error is enlarged by factors that vary between 3 and 11 (depending on the window) while for the s+lqd component the enlargement can be up to factors of order 30. As we show here, because of the small contribution from the inclusive region in essentially all cases (the exception is the SD window, as could be expected) this increase in error in the perturbative contribution has no meaningful impact on the precision of our final results.

Since the details of our perturbative description in the inclusive region, as well as of the parametrization of the DVs that we employ, have already been extensively discussed in Ref. [54, 57] we, here, simply quote the final numbers for the pQCD+DV contributions in Tabs. 5 and 6.

The final step is to estimate electromagnetic (EM) and strong-isospin-breaking (SIB) contributions to the results discussed above. These contributions must be subtracted before comparing our results with those of isospin-symmetric lattice QCD. We follow the treatment discussed extensively in our previous works [54, 55, 57] and here simply summarize the application of this framework to the SD and LD windows. Our isospin-symmetric results, as is the case for many lattice groups, correspond to a definition of the isospin limit of QCD in which all pions have the neutral pion mass.

We work to first order in IB and start from the observation that, to this precision, SIB appears only in the MI component of $\rho_{\rm EM}(s)$ while EM contributions appear in all of the pure $I=1$, pure $I=0$ and MI components. MI contributions, in general, produce small IB “contaminations” of the nominally pure $I=1$ $G$-parity positive and nominally pure $I=0$ $G$-parity negative exclusive-mode contributions discussed above. These must be subtracted, mode-by-mode, to arrive at data-driven isospin-limit lqc and s+lqd results suitable for comparison to the corresponding lattice results. It is expected, however, that the dominant such contaminations will be those in the $2\pi$ and $3\pi$ channels, which are strongly enhanced by the effects of $\rho-\omega$ mixing through the processes $e^{+}e^{-}\to\omega\to\rho\to 2\pi$ and $e^{+}e^{-}\to\rho\to\omega\to 3\pi$. To estimate the resulting dominant MI contaminations we use the results of Ref. [73, 74, 75] where the $2\pi$ and $3\pi$ electroproduction data were fitted with dispersive representations incorporating the effects of $\rho-\omega$ mixing. Since these estimates are obtained using experimental input, they, of course, include both the EM and SIB components of the MI contributions.

The resulting estimates for the MI components of the nominally $I=1$ $2\pi$-mode contributions to HVP integral and the three windows discussed in this paper can be found in Tab. I of Ref. [73] and read

	$\displaystyle[a_{\mu}^{\rm SD}]_{\pi\pi}^{\rm MI}\times 10^{10}=0.06(1),$
	$\displaystyle[a_{\mu}^{\rm int}]_{\pi\pi}^{\rm MI}\times 10^{10}=0.86(6),$
	$\displaystyle[a_{\mu}^{\rm LD}]_{\pi\pi}^{\rm MI}\times 10^{10}=2.87(12),$
	$\displaystyle[a_{\mu}^{\rm HVP}]_{\pi\pi}^{\rm MI}\times 10^{10}=3.79(19).$		(15)

In the $2\pi$ channel, this MI contribution, in spite of the enhancement due to the $\rho-\omega$ mixing, never exceeds 0.85% of the $2\pi$ total given in Tab. 1. With no analogous narrow, nearby resonance enhancements of this type expected for other nominally $I=1$ modes, we consider it safe to assume that the total MI contamination present in the contributions from these other modes will not exceed $1\%$ of the sum of their contributions. In this spirit, we add to the results of Eq. (15) an additional uncertainty equal to 1% of the sum of all non-2$\pi$ nominally $I=1$ contributions, from both unambiguous and ambiguous modes (the latter with the exception of the radiative modes, $\pi^{0}\gamma$ and $\eta\gamma$, where the VMD representation provides a full $I=1$/$I=0$/MI separation and there is, thus, no MI contamination of either the $I=1$ or $I=0$ contribution). The non-$2\pi$, $I=1$ totals for each window, obtained by adding to the unambiguous-mode $I=1$ results of Tab. 1 the $I=1$ components of the ambiguous-mode contributions that can be inferred from Tab. 3 (excluding the radiative channels $\pi^{0}\gamma$ and $\eta\gamma$), are the following

	$\displaystyle[a_{\mu}^{\rm SD}]_{{\rm non}-2\pi}^{I=1}\times 10^{10}=9.12(34),$
	$\displaystyle[a_{\mu}^{\rm int}]_{{\rm non}-2\pi}^{I=1}\times 10^{10}=25.66(76),$
	$\displaystyle[a_{\mu}^{\rm LD}]_{{\rm non}-2\pi}^{I=1}\times 10^{10}=6.68(17),$
	$\displaystyle[a_{\mu}^{\rm tot}]_{{\rm non}-2\pi}^{I=1}\times 10^{10}=41.5(1.2).$		(16)

Our final estimates for the nominally $I=1$ MI contaminations consist of the numbers given in Eq. (15) with 1% of the central values of Eq. (16) added in quadrature as an additional uncertainty. These results are to multiplied by $10/9$ to convert them to the corresponding lqc MI contaminations and subtracted from the sum of the uncorrected lqc results obtained above, shown in the first three lines of Tab. 5. The resulting MI corrections, to be added to the other entries, are listed in line four of this table.

The procedure employed to estimate the $I=0$ MI IB contribution is very similar: we use the results of Ref. [73] as estimates of the MI contaminations, $[a_{\mu}^{\rm win}]_{3\pi}^{\rm MI}$, present in the nominally $I=0$ $3\pi$ window contributions and add to those results an additional uncertainty of 1% of the total of all non-3$\pi$, nominally $I=0$ contributions, both unambiguous and ambiguous — again with the exception of those from the radiative modes $\pi^{0}\gamma$ and $\eta\gamma$, where the MI contributions has been explicitly determined and the $I=0$ contributions determined using the VMD representation contain no MI contamination. The $3\pi$ MI IB contributions from Tab. I of Ref. [73] are

	$\displaystyle[a_{\mu}^{\rm SD}]_{3\pi}^{\rm MI}\times 10^{10}=-0.13(3),$
	$\displaystyle[a_{\mu}^{\rm int}]_{3\pi}^{\rm MI}\times 10^{10}=-1.03(27),$
	$\displaystyle[a_{\mu}^{\rm LD}]_{3\pi}^{\rm MI}\times 10^{10}=-1.52(40),$
	$\displaystyle[a_{\mu}^{\rm HVP}]_{3\pi}^{\rm MI}\times 10^{10}=-2.68(70).$		(17)

The corresponding $I=0$ non-$3\pi$ totals, obtained by adding to the results of Tab. 2 the $I=0$ ambiguous-mode contributions inferable from Tabs. 4 and 3 (again excluding the $\pi^{0}\gamma$ and $\eta\gamma$ mode contributions) are found to be

	$\displaystyle[a_{\mu}^{\rm SD}]_{{\rm non}-3\pi}^{I=0}\times 10^{10}=5.04(30),$
	$\displaystyle[a_{\mu}^{\rm int}]_{{\rm non}-3\pi}^{I=0}\times 10^{10}=22.53(59),$
	$\displaystyle[a_{\mu}^{\rm LD}]_{{\rm non}-3\pi}^{I=0}\times 10^{10}=14.00(14),$
	$\displaystyle[a_{\mu}^{\rm HVP}]_{{\rm non}-3\pi}^{I=0}\times 10^{10}=41.56(97).$		(18)

Adding 1% of these non-3$\pi$, $I=0$ totals as an additional uncertainty to the results of Eq. (17) and combining these results with the $I=1$ MI IB contributions as per Eq. (10), we find the s+lqd MI corrections shown in the 5-th row of Tab. 6.

With the full EM+SIB MI corrections in hand, the remaining IB effects to be dealt with are the EM corrections to the pure $I=1$ and $I=0$ window contributions. At this point, although several EM contributions have been estimated from experimental data [75, 74, 73], other potentially non-negligible EM effects have not (see the discussion in the appendix of Ref. [55]). We have decided, therefore, to rely on lattice EM data for our estimates of the $I=1$ and $I=0$ EM corrections. As discussed below, these corrections are very small in the lqc case and completely negligible for the s+lqd components, which means that our final results are still (almost) purely data-driven.

We start with a discussion of the lqc EM contributions. For the total HVP, the EM contribution was published in 2021 by BMW [30] and corrected in their more recent paper [33]. The corrected, 2024 result,

$$\Delta_{\rm EM}a_{\mu}^{\rm HVP,lqc}\times 10^{10}=-1.57(42)(35)=-1.57(55),$$

(19)

has to be subtracted from our HVP result before comparison with the isospin-symmetric lattice HVP value. BMW also provided the EM correction to the intermediate window lqc component [30],

$$\Delta_{\rm EM}a_{\mu}^{\rm int,lqc}\times 10^{10}=-0.035(59),$$

(20)

which constitutes a very small correction with essentially no impact on the final results of our previous works [56, 57].

For the SD and LD windows no number was provided in the BMW papers. This leaves a contribution of about $1.535\times 10^{-10}$ to be split between these two windows. Here, we make the assumption that the EM contribution to the lqc component of the SD window is negligible, which means that the entirety of the remaining $1.535\times 10^{-10}$ is attributed to the LD window. Two arguments support this assumption. First, a significant portion of the EM contribution to the SD window is amenable to a perturbative calculation. This gives only a tiny correction, which the Mainz collaboration estimated to be 0.03$\times 10^{-10}$ [47]. Second, the same Mainz paper quotes an initial direct lattice simulation result for the SD EM contribution equal to $0.15(15)\%$ of the sum of the corresponding light and strange connected contributions, corresponding to a central value of $0.085\times 10^{-10}$ [47]. This is compatible (though smaller in magnitude) with what we expect to be a conservative estimated bound on the magnitude of the total EM contribution, equal to $\alpha_{\rm EM}$ times the total of the unambiguous and ambiguous exclusive-mode SD lqc contributions plus $\alpha_{\rm EM}/\pi$ times the corresponding pQCD contribution. This yields the bound

	$\displaystyle|\Delta_{\rm EM}a_{\mu}^{\rm SD,lqc}|$	$\displaystyle\leq\alpha_{EM}\left([a_{\mu}^{\rm SD}]_{\rm unamb}^{\rm lqc}+[a_{\mu}^{\rm SD}]_{\rm amb}^{\rm lqc}\right)+\frac{\alpha_{\rm EM}}{\pi}[a_{\mu}^{\rm SD}]_{\rm pQCD}^{\rm lqc}$
		$\displaystyle=\alpha_{\rm EM}(25.99+0.75)\times 10^{-10}+\frac{\alpha_{\rm EM}}{\pi}20.28\times 10^{-10}$
		$\displaystyle=0.24\times 10^{-10},$		(21)

where $[a_{\mu}^{\rm SD,lqc}]_{\rm unamb}$, $[a_{\mu}^{\rm SD,lqc}]_{\rm amb}$, and $[a_{\mu}^{\rm SD}]_{\rm pQCD}^{\rm lqc}$ are, respectively, the unambiguous-mode, ambiguous-mode and the pQCD contributions to $a_{\mu}^{\rm SD,lqc}$, given in Tab. 5. This result is a factor of nearly $3$ times larger than the uncertainty on the Mainz result. To be conservative, we assign this larger estimate as the uncertainty on our EM SD assumption. Our final estimates for the EM IB contribution to the lqc SD and LD windows are then

	$\displaystyle\Delta_{\rm EM}a_{\mu}^{\rm SD,lqc}\times 10^{10}=0.00(24),$
	$\displaystyle\Delta_{\rm EM}a_{\mu}^{\rm LD,lqc}\times 10^{10}=1.54(55).$		(22)

For the s+lqd component, the EM corrections to the intermediate window and to the total HVP can be obtained from the published BMW results via the diagrammatic approach explained in detail in Ref. [54]. These corrections turn out to be tiny due, in part, to strong cancellations in the numerically dominant contributions arising from light-quark EM valence-valence connected and disconnected diagrams. For the intermediate window, for example, we obtain the following EM correction to the s+lqd component [56]

$\displaystyle\Delta_{\rm EM}a_{\mu}^{\rm int,s+lqd}\times 10^{10}=0.012(11),$

(23)

more than 60 times smaller than our final error. Since the final relative uncertainties in the s+lqd SD and LD windows are larger than that of the intermediate window, and since the EM contributions to s+lqd intermediate window and total HVP results are so small, we believe it to be safe to neglect the EM corrections for the s+lqd component in the case of the SD and LD windows as well.