A comment on the impact of CMD-3 $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section measurement on the SM $g_{\mu}-2$ value

V. V. Bryzgalov¹¹1[email protected] and O. V. Zenin²²2[email protected]
NRC “Kurchatov Institute” – IHEP, Protvino, Russia

Abstract

We estimated an impact of the recent CMD-3 measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ total cross section at $0.3<\sqrt{s}<1.2$ GeV on the leading order hadronic contribution $a_{\mu}^{\mathrm{had,LO}}$ to the muon anomalous magnetic moment $a_{\mu}=(g_{\mu}-2)/2$ , in presence of comparably precise ISR measurements of the cross section by BaBar and KLOE experiments being in significant tension with the CMD-3. Assuming that all the experiments are affected by yet unidentified systematic effects, to account for the latter, we scaled the experimental uncertainties following the PDG prescription, thus facilitating a consistent joint fit of the world data on the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ total cross section. The same procedure was applied in all $e^{+}e^{-}\to hadrons$ channels contributing to the dispersive estimate of $a_{\mu}^{\mathrm{had,LO}}$ . Despite an inclusion of the new CMD-3 $\pi^{+}\pi^{-}$ data, our estimate $a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-})=(696.2\pm 2.9)\times 10^{-10}$ is consistent with $a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-})$ values obtained by other authors before publication of the CMD-3 result. Including our $a_{\mu}^{\mathrm{had,LO}}$ value into the SM prediction for $a_{\mu}$ , we obtain $a^{\mathrm{SM}}_{\mu}=(11~{}659~{}184\pm 4_{\mathrm{tot}})\times 10^{-10}$ which is by $4.7\sigma$ smaller than the world average for the experimental value $a^{\mathrm{exp}}_{\mu}=(11~{}659~{}205.9\pm 2.2)\times 10^{-10}$ . We confirm the observation by the CMD-3 authors that their $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ measurement, when taken alone, implies the $a^{\mathrm{SM}}_{\mu}$ prediction consistent with the $a^{\mathrm{exp}}_{\mu}$ at $\sim 1\sigma$ level.

The $e^{+}e^{-}\to\pi^{+}\pi^{-}$ total cross section contributes $\simeq 70\%$ to the dispersive estimate [1] (see [2] for a recent review) of the leading order hadronic term $a_{\mu}^{\mathrm{had,LO}}$ in the SM prediction of the muon anomalous magnetic moment $a^{\mathrm{SM}}_{\mu}=(g_{\mu}-2)/2$ and, on the other hand, introduces a major uncertainty in the $a^{\mathrm{SM}}_{\mu}$ determination.

Following the recent high precision measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section at $0.3<\sqrt{s}<1.2$ GeV by the CMD-3 collaboration [3, 4] being in a significant tension with results previously published by BaBar [5], KLOE [6, 7, 8], CMD-2 [9, 10, 11, 12], SND [13] and SND2k [14] experiments (Fig. 1), we attempted to assess the impact of the CMD-3 result on the joint fit of the available $e^{+}e^{-}\to\pi^{+}\pi^{-}$ total cross section data and its implications for the $a^{\mathrm{SM}}_{\mu}$ . Details of the analysis were reported in [15].³³3The program code [16] was used by the authors for the PDG mini-review “ $\sigma$ and $R$ in $e^{+}e^{-}$ Collisions” [17]. Essentially the same code was used for earlier revisions of the mini-review since [18]. Details on the earliest version of the analysis are given in [19, 20]

Given that possible origins of a mutual systematic bias were not yet identified for the experiments measuring $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ at $\sim 1\%$ precision, all of them must be treated as is on an equal basis. From statistical viewpoint, this results in a low probability of the joint fit with $\chi^{2}/n_{\mathrm{dof}}=2.18$ (Fig. 2). Before inclusion of the “high” CMD-3 data, the major source of statistical tension was the discrepancy between ISR based measurements of the cross section by BaBar [5] and KLOE [8] experiments leading to the joint fit with $\chi^{2}/n_{\mathrm{dof}}=1.45$ (Fig. 3). For $n_{\mathrm{dof}}\gg 1$ , a consistent fit should yield $\chi^{2}/n_{\mathrm{dof}}\in(1-2\sqrt{2/n_{\mathrm{dof}}},1+2\sqrt{2/n_{\mathrm{dof}}})$ with $\simeq 95\%$ probability. $\chi^{2}/n_{\mathrm{dof}}$ values outside this range indicate either an inadequate parameterization of the cross section or incompatibility between cross section values measured by different experiments. As our model-independent parameterization of the cross section is constructed so that fitting it to mutually compatible measurements should always give $\chi^{2}/n_{\mathrm{dof}}\simeq 1$ [15], the poor fit is mostly due to incompatibility between BaBar, KLOE and CMD-3 experiments discussed in [4].⁴⁴4This situation occurs not only in the $\pi^{+}\pi^{-}$ channel, see $\chi^{2}/{\mathrm{dof}}$ column in the Table 1.

A natural approach to treatment of incompatible measurements is to assume that each experiment is affected by randomly distributed systematic bias that cannot be identified within an isolated experiment and thus unaccounted for by its systematic uncertainty. However, missing systematics can be assessed by observing the distribution of measurements of the same physical quantity made by several independent experiments. The missing systematic uncertainty affecting all the experiments becomes an extra free parameter in the joint fit, with the optimum value maximizing the probability of the fit. A widely used (though debated) PDG prescription [17] is to apply a common scale factor ⁵⁵5Dubbed the Birge factor in literature. to error matrices of all experiments so that the fit yields $\chi^{2}/n_{\mathrm{dof}}=1$ . Being aware of shortcomings of this method, we still use it in our fits for the moment. Effectively, the error matrix of each experiment must be scaled by $(\chi^{2}/n_{\mathrm{dof}})^{orig.}$ obtained in the fit with the unmodified error matrices. At that, the central value of the fitted average cross section remains unchanged, while its uncertainty is scaled by the factor of $\sqrt{(\chi^{2}/n_{\mathrm{dof}})^{orig}}$ .

The result of the joint fit to the $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ data at $0.3<\sqrt{s}<2$ GeV including CMD-3 is shown by the green band in Fig. 2. The fit with unmodified experimental uncertainties gives $\chi^{2}/n_{\mathrm{dof}}=2.18$ due to dramatic tension between BaBar, KLOE and CMD-3 measurements at $0.6<\sqrt{s}<1.0$ GeV.⁶⁶6The fit restricted to the $0.6<\sqrt{s}<1.0$ GeV range would have much larger $\chi^{2}/n_{\mathrm{dof}}$ (see the lower plot in Fig. 2). As explained above, all experimental uncertainties were inflated by $\sqrt{2.18}$ factor to make these measurements compatible. On the plots, the individual measurements are shown with their original uncertainties while the fit result is shown with the uncertainty scaled by $\sqrt{2.18}$ . Scaling up the fit uncertainty accounts for incompatibility between the experiments, as well as for suboptimal parameterization of the cross section.

Substituting the fitted cross section into the dispersion integral for $a_{\mu}^{\mathrm{had,LO}}$ [1] we obtain the contribution of the $\pi^{+}\pi^{-}$ channel in the $0.3<\sqrt{s}<1.937$ GeV range:

a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-}\to\pi^{+}\pi^{-},0.3\div 1.937~{}\mathrm{GeV})=(505.1\pm 1.4_{\mathrm{exp.}e^{+}e^{-}}\pm 1.6_{\mathrm{par.}}\pm 0.6_{\mathrm{rad.}})\times 10^{-10}\,,

(1)

where the first uncertainty is due to experimental uncertainties of the $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ data, the second is the systematic uncertainty due to our cross section parameterization, and the last one is the uncertainty due to radiative corrections applied to the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ data.

Results of application of the same procedure to other hadronic final states are shown in the Table 1. The total leading order hadronic contribution to $a^{\mathrm{SM}}_{\mu}$ is then

a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-})=(696.2\pm 1.9_{\mathrm{exp.}e^{+}e^{-}}\pm 2.0_{\mathrm{par.}}\pm 0.8_{\mathrm{rad.}})\times 10^{-10}\,.

(2)

Despite an inclusion of the “high” CMD-3 $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ measurement, our estimate is still consistent with results obtained by the dispersive method by other authors using only pre-2021 $e^{+}e^{-}$ data [2]. Ref. [2] quotes an average value $a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-})=(693.1\pm 4.0_{\mathrm{tot}})\times 10^{-10}$ obtained by merging results from Refs. [21, 22, 23, 24, 25, 26].⁷⁷7We also have a good per final state agreement with [23]. The SM prediction for $a_{\mu}$ including QED, electroweak, hadronic light-by-light, and NLO hadronic vacuum polarization contributions [2] and our $a_{\mu}^{\mathrm{had,LO}}$ value (Eq. 2) is

a^{\mathrm{SM}}_{\mu}=(11~{}659~{}184\pm 4_{\mathrm{tot}})\times 10^{-10}\,,

(3)

which is smaller than the experimental value $a^{\mathrm{exp}}_{\mu}=(11~{}659~{}205.9\pm 2.2)\times 10^{-10}$ [27] by $4.7\sigma$ .

An exclusion of the CMD-3 $\pi^{+}\pi^{-}$ data (Fig. 3) leads to a lower $a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-})=(694.0\pm 2.0_{\mathrm{exp.}e^{+}e^{-}}\pm 1.4_{\mathrm{par.}}\pm 0.4_{\mathrm{rad}})\times 10^{-10}$ which translates to $a^{\mathrm{SM}}_{\mu}=(11659182\pm 4_{\mathrm{tot}})\times 10^{-10}$ , lower than $a^{\mathrm{exp}}_{\mu}$ by $5\sigma$ .

An extreme exercise would be to estimate an impact of the CMD-3 $\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-})$ measurement taken as the dominant experimental input to $a_{\mu}^{\mathrm{had,LO}}(\pi^{+}\pi^{-})$ at $0.3<\sqrt{s}<1.2$ GeV. For this purpose, we completely excluded from the fit all the experiments measuring the cross section with $\sim 1\%$ precision, except CMD-3 (Fig. 4).⁸⁸8In a “CMD-3 only” fit, retaining BaBar data in the $\sqrt{s}>1.2$ GeV range uncovered by CMD-3 would be methodically incorrect due to strong correlations between BaBar data points at $\sqrt{s}<1.2$ GeV (where we ignore BaBar) and $\sqrt{s}>1.2$ GeV. Note that in this case the cross section parameterization uncertainty for the total $a_{\mu}^{\mathrm{had,LO}}$ is slightly lower than the one in the $\pi^{+}\pi^{-}$ channel due to anti-correlation between systematic variations in different channels. This gives $a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-}\to\pi^{+}\pi^{-},0.3\div 1.937~{}\mathrm{GeV})=(529.6\pm 2.8_{\mathrm{exp.}e^{+}e^{-}}\pm 3.3_{\mathrm{par.}}\pm 3.3_{\mathrm{rad.}})\times 10^{-10}$ which corresponds to the total $a_{\mu}^{\mathrm{had,LO}}=(723.4\pm 3.1_{\mathrm{exp.}}\pm 3.1_{\mathrm{par.}}\pm 3.5_{\mathrm{rad.}})\times 10^{-10}$ and, hence, to $a^{\mathrm{SM}}_{\mu}=(11~{}659~{}211\pm 6_{\mathrm{tot}})\times 10^{-10}$ consistent with the experimental value $a^{\mathrm{exp}}_{\mu}=(11~{}659~{}205.9\pm 2.2)\times 10^{-10}$ , as pointed out in Ref. [4].

Summary of $a^{\mathrm{SM}}_{\mu}$ estimates including the above results confronted with the experimental $a^{\mathrm{exp}}_{\mu}$ value is shown in Fig. 5.

In conclusion, we estimated an impact of the recent CMD-3 measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ total cross section at $0.3<\sqrt{s}<1.2$ GeV on the leading order hadronic contribution $a_{\mu}^{\mathrm{had,LO}}$ to the muon anomalous magnetic moment $a_{\mu}=(g_{\mu}-2)/2$ , in presence of comparably precise ISR measurements of the cross section by BaBar and KLOE experiments being in significant tension with the CMD-3. Assuming that all the experiments are affected by yet unidentified systematic effects, to account for the latter, we scaled the experimental uncertainties following the PDG prescription, thus facilitating a consistent joint fit of the world data on the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ total cross section. The same procedure was applied in all $e^{+}e^{-}\to hadrons$ channels contributing to the dispersive estimate of $a_{\mu}^{\mathrm{had,LO}}$ . Despite an inclusion of the new CMD-3 data, our estimate $a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-})=(696.2\pm 2.9)\times 10^{-10}$ is consistent with $a_{\mu}^{\mathrm{had,LO}}(e^{+}e^{-})$ values obtained by other authors before publication of the CMD-3 result. Including our $a_{\mu}^{\mathrm{had,LO}}$ value into the SM prediction for $a_{\mu}$ , we obtain $a^{\mathrm{SM}}_{\mu}=(11~{}659~{}184\pm 4_{\mathrm{tot}})\times 10^{-10}$ which is by $4.7\sigma$ smaller than the world average for the experimental value $a^{\mathrm{exp}}_{\mu}=(11~{}659~{}205.9\pm 2.2)\times 10^{-10}$ . We confirm the observation made in the CMD-3 paper [4] that CMD-3 $\pi^{+}\pi^{-}$ data, when taken alone, imply the $a^{\mathrm{SM}}_{\mu}$ prediction consistent with the $a^{\mathrm{exp}}_{\mu}$ at $\sim 1\sigma$ level.

Acknowledgements.

The authors are indebted to V. B. Anikeev and S. I. Bityukov for valuable discussions on statistical matters.

Refer to caption — Figure 1: (reprinted from [4]) Relative differences between previous measurements of the pion formfactor by BaBar [5], KLOE [6, 7] (middle plot), CMD-2 [9, 10, 11, 12], SND [13] and SND2k [14] (lower plot) experiments and the fit (yellow band) to the CMD-3 data [3, 4] (upper plot).

Final state	$a_{\mu}(\mathrm{had},\mathrm{LO})$ $\times 10^{10}$ 000.000 (exp.) (par.) (rad.)	$\sqrt{s}\,[\mathrm{GeV}]$	$\chi^{2}/\mathrm{dof}$	References
$\pi^{+}\pi^{-}(\gamma)$	505.147 (1.367) (1.551) (0.606)	0.3 $\div$ 1.937	2.18	[28, 3, 9, 13, 11, 12, 8, 14, 29, 30, 31, 32, 5, 33, 34, 10]
$\pi^{+}\pi^{-}\pi^{0}$	48.481 (0.967) (0.629) (0.066)	0.66 $\div$ 1.937	1.79	[35, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]
$\pi^{+}\pi^{-}2\pi^{0}$	18.778 (0.431) (0.509) (0.067)	0.85 $\div$ 1.937	1.94	[46, 47, 48, 49, 50, 51, 52, 53, 43, 54]
$2\pi^{+}2\pi^{-}$	15.397 (0.181) (0.060) (0.043)	0.6125 $\div$ 1.937	2.34	[46, 47, 49, 55, 56, 57, 58, 52, 59, 60, 61, 62, 43, 63, 64]
$K^{+}K^{-}$	23.211 (0.188) (0.072) (0.009)	0.985 $\div$ 1.937	1.99	[65, 66, 67, 68, 69, 70, 40, 71, 72]
$K_{S}K_{L}$	13.188 (0.130) (0.000) (0.000)	1.00371 $\div$ 1.937	0.95	[73, 67, 38, 74, 75, 76, 40, 77, 78]
$\pi^{0}\gamma$	4.359 (0.093) (0.049) (0.000)	0.59986 $\div$ 1.38	1.70	[79, 80, 81]
$K_{S}K^{+}\pi^{-}+K_{S}K^{-}\pi^{+}$	1.814 (0.100) (0.000) (0.000)	1.24 $\div$ 1.937	0.99	[82]
$2\pi^{+}2\pi^{-}\pi^{0}$	1.746 (0.043) (0.000) (0.009)	1.0125 $\div$ 1.937	0.00	[49, 83, 63]
$2\pi^{+}2\pi^{0}2\pi^{-}$	1.728 (0.198) (0.034) (0.000)	1.3125 $\div$ 1.937	1.99	[84, 85, 63]
$2\pi^{+}2\pi^{-}3\pi^{0}$	0.099 (0.013) (0.002) (0.001)	1.575 $\div$ 1.937	0.57	[86]
$3\pi^{+}3\pi^{-}$	0.240 (0.014) (0.000) (0.012)	1.3125 $\div$ 1.937	0.00	[84, 87, 88, 85, 63]
$3\pi^{+}3\pi^{-}\pi^{0}$	0.020 (0.004) (0.001) (0.000)	1.6 $\div$ 1.937	0.65	[89]
$\eta\gamma$	0.691 (0.051) (0.000) (0.000)	0.6 $\div$ 1.354	1.36	[79, 38, 90, 91]
$\eta\pi^{+}\pi^{-}$	0.575 (0.019) (0.000) (0.000)	1.15 $\div$ 1.937	1.18	[92, 83, 93, 94, 43]
$K^{+}K^{-}\pi^{0}$	0.202 (0.050) (0.000) (0.001)	1.44 $\div$ 1.937	0.54	[95, 96]
$K^{+}K^{-}\pi^{0}\pi^{0}$	0.100 (0.011) (0.000) (0.000)	1.5 $\div$ 1.937	1.32	[97]
$K^{+}K^{-}\pi^{+}\pi^{-}$	0.799 (0.033) (0.000) (0.000)	1.4 $\div$ 1.937	0.00	[95, 98, 99, 97]
$K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$	0.129 (0.024) (0.000) (0.000)	1.6125 $\div$ 1.937	1.63	[83]
$K_{S}K_{L}\eta$	0.238 (0.059) (0.000) (0.000)	1.575 $\div$ 1.937	1.31	[100]
$K_{S}K_{L}\pi^{0}$	0.839 (0.114) (0.000) (0.000)	1.425 $\div$ 1.937	1.50	[100]
$K_{S}K_{L}\pi^{0}\pi^{0}$	0.137 (0.043) (0.000) (0.000)	1.35 $\div$ 1.937	0.00	[100]
$K_{S}K_{L}\pi^{+}\pi^{-}$	0.166 (0.028) (0.000) (0.000)	1.425 $\div$ 1.937	0.00	[77]
$K_{S}K^{+}\pi^{-}\pi^{0}+K_{S}K^{-}\pi^{+}\pi^{0}$	0.640 (0.044) (0.000) (0.000)	1.51 $\div$ 1.937	1.08	[101]
$K_{S}K_{S}\pi^{+}\pi^{-}$	0.066 (0.007) (0.000) (0.000)	1.63 $\div$ 1.937	1.37	[77]
$\omega(783)\eta$	0.035 (0.002) (0.000) (0.000)	1.34 $\div$ 1.937	0.85	[102, 103]
$\omega(783)<\pi^{0}\gamma>\pi^{0}$	0.894 (0.021) (0.000) (0.000)	0.75 $\div$ 1.937	1.56	[104, 105, 106, 107, 47, 108, 109, 110, 111]
$\omega(783)<\pi^{+}\pi^{-}\pi^{0}>\pi^{+}\pi^{-}$	0.098 (0.005) (0.000) (0.000)	1.15 $\div$ 1.937	1.10	[112, 92, 83, 45]
$\omega\eta\pi^{0}$	0.055 (0.043) (0.000) (0.000)	1.5 $\div$ 1.937	1.16	[113]
$\phi(1020)\eta$	0.067 (0.003) (0.000) (0.000)	1.56 $\div$ 1.937	0.98	[114, 83, 82]
$\pi^{+}\pi^{-}2\pi^{0}\eta$	0.117 (0.019) (0.000) (0.000)	1.625 $\div$ 1.937	0.85	[115]
$\pi^{+}\pi^{-}3\pi^{0}$	1.067 (0.112) (0.000) (0.000)	1.125 $\div$ 1.937	0.68	[115]
$\pi^{+}\pi^{-}\pi^{0}\eta$	0.663 (0.075) (0.000) (0.000)	1.394 $\div$ 1.937	0.82	[102]
$p\bar{p}$	0.030 (0.001) (0.000) (0.000)	1.889 $\div$ 1.937	1.24	[116, 117, 118, 119]
$n\bar{n}$	0.028 (0.006) (0.000) (0.000)	1.89 $\div$ 1.937	1.24	[120, 121]
$2hadron(hadrons)$	43.509 (0.722) (0.661) (0.000)	1.937 $\div$ 11.199	1.35	[122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156]
pQCD	2.065 (0.000) (0.002) (0.000)	$>$ 11.1990
ChPT $\pi\pi,\pi^{0}\gamma$	0.538 (0.000) (0.013) (0.000)	0.2792 $\div$ 0.3000
$\Psi(1S)$	6.495 (0.000) (0.124) (0.000)	3.0969
$\Psi(2S)$	1.631 (0.000) (0.057) (0.000)	3.6861
$\Upsilon(1S)$	0.054 (0.000) (0.002) (0.000)	9.4604
$\Upsilon(2S)$	0.021 (0.000) (0.003) (0.000)	10.0234
$\Upsilon(3S)$	0.014 (0.000) (0.002) (0.000)	10.3551
$\Upsilon(4S)$	0.010 (0.000) (0.001) (0.000)	10.5794
Total	696.181 (1.925) (1.953) (0.813)

Table 1: Summary of

a_{\mu}^{\mathrm{had,LO}}

contributions from the exclusive

\sigma(e^{+}e^{-}\to hadrons)

measurements at

\sqrt{s}<1.937

GeV and inclusive

\sigma(e^{+}e^{-}\to 2hadron(hadrons))

measurements at

1.937<\sqrt{s}<11.199

GeV. The first uncertainty is the experimental one scaled by

\sqrt{\chi^{2}/n_{\mathrm{dof}}}

in channels with

\chi^{2}/n_{\mathrm{dof}}>1

. The second uncertainty is due to our parameterization of the experimental cross section used in the

a_{\mu}^{\mathrm{had,LO}}

dispersion integral. The last uncertainty is due to radiative corrections applied to the experimental data. At

\sqrt{s}<0.3

GeV we use ChPT parameterization for

e^{+}e^{-}\to\pi^{0}\gamma

\pi^{+}\pi^{-}

cross sections. The 3-loop pQCD parameterization of the total

e^{+}e^{-}\to hadrons

cross section is used at

\sqrt{s}>11.199

GeV. Contributions of the narrow

\Psi(1,2S)

and

\Upsilon(1-4S)

resonances are accounted for using the Breit–Wigner parameterization. See details in Ref. [15].

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A comment on the impact of CMD-3 e+​e−→π+​π−e^{+}e^{-}\to\pi^{+}\pi^{-} cross section measurement on the SM gμ−2g_{\mu}-2 value

Abstract

Acknowledgements.

References

A comment on the impact of CMD-3 $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section measurement on the SM $g_{\mu}-2$ value