Data-driven approximations to the Hadronic Light-by-Light scattering contribution to the muon (g-2)

The Standard Model (SM) uncertainty on the Muon g-2 ($2a_{\mu}=g_{\mu}-2$) is dominated by the hadronic vacuum polarization (HVP) piece, amounting to $4.0\times 10^{-10}$ (for an overall error of $4.3\times 10^{-10}$) [1] ¹¹1These and the following numbers are quoted -unless otherwise stated- from the Whitepaper of the Muon g-2 Theory Initiative [1] (WP), a collaboration which has been aiming for a community consensus value of the Standard Model prediction of the muon g-2, see https://muon-gm2-theory.illinois.edu/.. This is contributed very mildly by the error of the Hadronic light-by-light (HLbL) scattering part, $1.9\times 10^{-10}$, that we will discuss here ²²2See talks focusing on diverse aspects of the HVP contribution by Matthia Bruno, Christoph Redmer, Francesca de Mori, Álex Miranda, Camilo Rojas and David Díaz Calderón.. Clearly, the most urgent thing is to clarify the discrepancy between the data-driven results [1, 2, 3, 4, 5] and the competitive lattice QCD evaluation, by the BMW collaboration [6], of $a_{\mu}^{\rm HVP}$. To this end, several approaches have been developed, exploiting the so-called windows in Euclidean time [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Ref. [16] (based on the isospin-breaking corrections computed in Ref. [18]), points to nice agreement between data-driven predictions using $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ data [19, 20, 21, 22] (instead of $e^{+}e^{-}\to\pi^{+}\pi^{-}$ measurements) with lattice QCD evaluations. The barely acceptable discrepancy between KLOE [23, 24, 25, 26, 27] and BaBar [28, 29] $e^{+}e^{-}\to\pi^{+}\pi^{-}$ data has been aggravated by the new CMD-3 measurement [30], being this puzzle still not understood (see also e.g. the measurements [31, 32, 33]). Amid this conundrum, halving the error of the SM prediction for $a_{\mu}^{HLbL}$ [1] is still necessary, according to the final precision that the Fermilab experiment will achieve measuring $a_{\mu}$, but not the top priority.

On the contrary, the experimental situation seems crystal-clear: FNAL measurements [34, 35] are extremely consistent with the BNL outcome [36] and their joint picture is fully convincing, yielding

This situation further enhances the pressing need for theoretical progress.

The outsider may wonder why the uncertainty of the $a_{\mu}^{\rm{HLbL}}$ is $\mathcal{O}(20\%)$, while that of the $a_{\mu}^{\mathrm{HVP}}$ is only $\mathcal{O}(0.6)\%$. This much better precision stems from its calculation via a single dispersive integral that is related to the accurately measured $\sigma(e^{+}e^{-}\to\mathrm{hadrons})$ [37] plus a mild contribution from perturbative QCD. On the contrary, a data-driven approach to $a_{\mu}^{\mathrm{HLbL}}$ is very much complicated by the additional loop and multi-scale nature of the problem. Despite enormous advances towards a fully-dispersive computation of $a_{\mu}^{\rm HLbL}$ [38, 39, 40, 41, 42, 43], a completely dispersive evaluation is not feasible yet. This framework provided a rationale for the historical arrangement of the main contributions (starting from the dominance of the pseudoscalar-pole cuts [44]) and could in principle be used up to arbitrary complex multiparticle ones.

Amazingly, the whole $a_{\mu}^{\rm HLbL}$ is basically saturated by the contribution from the lowest-multiplicity cut (even more so because of the approximate cancellations among the other contributions), corresponding to the lightest pseudoscalar ($\pi^{0},\,\eta,\,\eta^{\prime}$) poles, yet it could be related to a combined chiral and large-$N_{C}$ expansion [45]. This can be computed straightforwardly [44] knowing the corresponding pseudoscalar transition form factors (TFFs) as functions of both photons virtuality. See Christoph Redmer’s talk on the precious experimental input to these (and others required for $a_{\mu}^{\rm HLbL}$) TFFs. In addition, there are some theoretical properties constraining these TFFs, like the chiral limit, the singly and doubly virtual asymptotic limits predicted by QCD, analyticity and unitarity … The dispersive evaluation [46, 47] yields a very precise result for the $\pi^{0}$ contribution

confirming the rational approximants’ determination [48]

These results are also supported by e.g. Dyson-Schwinger eqs. evaluations, yielding $a_{\mu}^{\pi^{0},{\rm HLbL}}=\left(62.6\pm 1.3\right)\times 10^{-11}$ [49], and $a_{\mu}^{\pi^{0},{\rm HLbL}}=\left(61.4\pm 2.1\right)\times 10^{-11}$ [78] and by holographic QCD results [51, 52, 53] (see, however, [54]) and chiral Lagrangians including resonances [55, 56]. For the $\eta^{(^{\prime})}$ contributions there is no dispersive computation yet. The rational approximants’ calculation [57, 58, 59, 48] yields

which are the reference values for this contribution. Again, they are supported by the different approaches mentioned before where, in particular, Dyson-Schwinger eqs. results in $a_{\mu}^{\eta,{\rm HLbL}}=\left(15.8\pm 1.2\right)\times 10^{-11}$, $a_{\mu}^{\eta^{\prime},{\rm HLbL}}=\left(14.7\pm 1.9\right)\times 10^{-11}$ [49] and $a_{\mu}^{\eta,{\rm HLbL}}=\left(13.3\pm 0.9\right)\times 10^{-11}$, $a_{\mu}^{\eta^{\prime},{\rm HLbL}}=\left(13.6\pm 0.8\right)\times 10^{-11}$ [78], respectively. From the dispersive and rational approximants calculations, the WP quotes

still to be considered the data-driven SM prediction for this leading contribution to ${\rm HLbL}$, coming from the lightest pseudoscalar poles.

The very well-known pseudoscalar electromagnetic form factors are the key objects to determine their box contributions to $a_{\mu}^{{\rm HLbL}}$. The dispersive result for the $\pi$ case

was later on confirmed by Schwinger-Dyson evaluations $a_{\mu}^{\pi-box,{\rm HLbL}}=-(15.7\pm 0.4)\times 10^{-11}$ [49], and $a_{\mu}^{\pi-box,{\rm HLbL}}=-(15.6\pm 0.2)\times 10^{-11}$ [60]. For the Kaon case, the early evaluation of ref.  [61], $a_{\mu}^{K-box,{\rm HLbL}}=-(0.46\pm 0.02)\times 10^{-11}$ was slightly revised within Dyson-Schwinger and then also using a dispersive framework [62], both agreeing on

The SM prediction comes from eqs. (6) and (7), still coinciding with the WP number [1]

Now we turn to another contribution coming from two-particle cuts, that associated to pseudoscalars rescattering. For the pions case, the dispersive evaluation [42, 43] is quite precise for the contribution associated to the $\pi$-pole left-hand cut (LHC):

where contributions from $D-$ and higher orders partial waves were covered by the uncertainty. This agrees with other evaluations [63, 64, 65, 66] that include additional scalar contributions, converging to [66]

again in accord with the WP [1]. Similarly, the tensors contribution [67]

is unchanged with respect to Ref. [1].

The part which has been evolving less trivially since 2020 corresponds to the axial-vector contributions, which should be regarded together with the remaining perturbative QCD constraints.

Melnikov and Vainshtein [68] put forward that pseudoscalar poles alone cannot satisfy short-distance QCD restrictions and emphasized the importance of axial-vectors to fulfil this requirement. Modern studies coincide in smaller values for these contributions than initially advocated.

Ref. [69] clarified ambiguities about bases arising because of axials off-shellness and, together with ref. [77], emphasized the relationship between short-distance, axial anomaly constraints, and the axial contributions (with possible relevant role of pseudoscalar resonances, see also [78]), a hot topic since then. Refs. [40, 70, 69] gave rise to the WP number [1]

This was accompanied by the estimation of the contribution from light-quark loops and remaining QCD short-distance constraints ($SDCs$) [72, 73, 74]

Given their correlation, these two contributions were combined with errors added linearly (uncertainties are combined quadratically, unless otherwise stated) to [1]

Finally, the $c$-quark contribution (with uncertainty to be added linearly to the Eq. (14)) is [78, 71, 72, 73, 74]

The leading-order $a_{\mu}^{\rm HLbL}$ contributions is obtained from Eqs. (5), (8), (10), (11), Eqs. (14), and (15), yielding

Progress since the WP on axials and/or SDCs has improved the understanding of the regime where all photon virtualities are large, and when one of them is much smaller than the other two [75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89]. However, different model calculations considering axial-vector mesons and SDCs [90, 51, 77, 84, 52, 53] suggest a shift in the central value around

larger than previously estimated, (14), but compatible within errors. Using Eq. (17), the overall contribution would then be

which is closer to the latest lattice QCD evaluations by the Mainz [91] ($(109.6\pm 15.9)\times 10^{-11}$) and RBC/UKQCD [92] ($(124.7\pm 14.9)\times 10^{-11}$) collaborations (to be compared to $(78.7\pm 35.4)\times 10^{-11}$ [93] by RBC, used in the WP). At $NLO$ [94] the central value and its uncertainty are increased by only $(2\pm 1)\times 10^{-11}$.

These observations evince that a better understanding of the role of axial vector mesons and the intermediate energy region is an important step towards a more precise and reliable estimate for the HLbL contribution. Progress in this direction continues [75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89].

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  The WP number, $a_{\mu}^{{\rm HLbL},LO}=(92\pm 19)\times 10^{-11}$ [1], still stands as the data-driven SM prediction for $a_{\mu}^{{\rm HLbL},LO}$.

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  The dominant uncertainty comes from short-distance + axial contributions (correlated uncertainties), with improved understanding since the WP, where still work needs to be done. This may shift the SM prediction slightly, to $a_{\mu}^{{\rm HLbL},LO}=(102\pm 17)\times 10^{-11}$.

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  Measurement of di-photon resonance couplings (particularly for axials) would be very helpful.

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  Lattice QCD has just reached a comparable uncertainty to the data-driven determinations of this piece, thereby reducing the uncertainty through their combination to $\leq 10\times 10^{-11}$, in agreement with the sought accuracy by the time of the final publication of the $a_{\mu}$ measurement by the FNAL experiment. So the ball is on ${\rm HVP}$’s court.