Mixed leptonic and hadronic corrections to the anomalous magnetic moment of the muon

Higher-order hadronic corrections to the anomalous magnetic moment of the muon have been evaluated including next-to-next-to-leading-order insertions of hadronic vacuum polarization and next-to-leading-order corrections to hadronic light-by-light scattering. This leaves a set of mixed leptonic and hadronic corrections in the form of double-bubble topologies as the only remaining hadronic effect at ${\mathcal O}(\alpha^4)$. Here, we estimate these contributions by analyzing the respective cuts of the diagrams, suggesting that the impact is limited to $\lesssim 1\times 10^{-11}$ and thus negligible at the level of the final precision of the Fermilab $g-2$ experiment.

The comparison to Eq. (2) shows that while the uncertainties are well under control, even O(α 4 ) contributions do need to be included, given that the HVP contributions at NNLO are at the same level as ∆a exp µ [E989]. This surprising finding in Ref. [13] can be understood from enhancements that trace back to both large logarithms log mµ me from electron loops and large numerical prefactors, as, e.g., expected from leptonic light-by-light topologies. The former also arises for NLO corrections to HLbL scattering, but in this case the corresponding enhancement does not counteract the suppression in α to the same extent [28].
In this Letter we address the remaining class of O(α 4 ) hadronic corrections, so-called double-bubble topologies shown in Fig. 1. These contributions are subtle, in that care is required to avoid double counting with contributions that already enter LO HVP, given that purely hadronic cuts (and, possibly, to some extent mixed hadronic and leptonic cuts) are included in the measured e + e − → hadrons cross section. 2 Numerically, potentially relevant effects are again expected from electron loops, and the subtleties in the definition of the LO HVP contribution further motivate a careful study of the different cuts to ensure that no sizable effects are overlooked. To this end, we first analyze the virtual (two-particle cut) and real (four-particle cut) contributions in QED, following the methods developed in the context of higherorder corrections to heavy-quark production [58][59][60][61][62][63][64][65][66], and then generalize the results to scalar QED to estimate the corrections originating from the leading hadronic channel e + e − → π + π − . This strategy allows us to calculate complicated four-loop contributions in an efficient and transparent manner, since the spectral functions that emerge at intermediate steps directly correspond to physical cross sections.
whereΠ(s) is the renormalized scalar VP function in the sign convention that the fine-structure constant runs as α(s) = α(0)/(1 −Π(s)). We will evaluate Eq. (4) via the dispersion relation since Im Π(s) can be directly related to the cuts of the diagrams. To be explicit, one has the relation where the R-ratio is defined as Of course, the same formula also works for leptonic final states, so that the left diagram in Fig. 1 can be reconstructed from the + 1 − 1 cut, starting at s thr = 4m 2 1 , and As a first step, we work out the results for the cases { 1 2 } = {ee, µe, eµ}, since the separation into the two cuts, referred to as virtual and real contributions, respectively, will allow us to draw first conclusions on the hadronic case.
To this end, we introduce the notation for the two-loop contribution to R(s) and separate the scaling in α/π to obtain the spectral functions ρ V,R 1 2 for the virtual and real parts.   4 2.9 × 10 −11 . The upper panel refers to the QED diagrams (left in Fig. 1), the last line to an estimate of the π + π − contribution in scalar QED (right in Fig. 1). The virtual and real parts correspond to the two-and four-particle cuts, respectively.
The virtual spectral function can be calculated by yet another dispersion relation, where is the spectral function for the inner lepton 2 and we wrote m i = m i as well as  [67]. The resulting integral representation (9) works well for 1 = µ, leading to the result shown in Table I. For 1 = 2 = e one can use the analytic result [58,59], reproduced in Eq. (A2), while for 1 = e, 2 = µ in most of the integration range in Eq. (5) the approximation m 1 = 0 is sufficient. The exception is the region very close to threshold, where a double expansion (A4) in β 1 and m 2 1 /λ 2 should be used instead. Finally, the known real spectral function from the four-particle cut is given in Eq. (A5).
The sum of the real and virtual contributions reproduces the total results from Refs. [2,68], see Table I. 3 We see that by far the dominant contribution arises from with hadronic degrees of freedom. First, we can ignore the case of on outer electron and inner hadronic loop, since the muon example shows that this configuration contributes only < ∼ 10 −12 to a µ . Second, virtual corrections will be included in the e + e − → hadrons data in LO HVP, unless removed by hand through the application of higher-order radiative corrections or in Monte Carlo simulations. This implies that the only potentially missing effect concerns the real radiation of an e + e − pair together with hadronic states. In the muon case, this effect amounts to ∆a µ 0.8 × 10 −11 , which would be negligible for the time being. To corroborate this estimate, we consider a π + π − loop in scalar QED, as a realistic example of the hadronic realization of a quark loop, see right diagram in Fig. 1.
Scalar QED.-To estimate the potentially missing hadronic contributions more quantitatively, we consider the e + e − → π + π − cross section parameterized via the pion vector form factor F V π (s), with β π = 1 − 4M 2 π /s, and evaluate the virtual and real corrections in scalar QED. This strategy is analogous to the calculation of ππγ radiative corrections [75][76][77][78] and captures the infrared enhanced effects, for which the pion can be approximated as a point-like particle.
The calculation of the virtual contribution proceeds in analogy to Eq. (9), where the form factor F πλ (s, λ 2 ) fulfills the dispersion relation From the explicit scalar QED calculation we obtain the compact analytic expression where β = β π , p = 1−β 1+β , l = λ 2 /M 2 π , and Similarly, real radiation from the scalar QED subprocess leads to the spectral function where λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + xz + yz) and Both F πλ (s, λ 2 ) and ρ R π (s) are new results. The numerical evaluation gives the result in the last line of Table I, supporting the conclusions already indicated by the muon example: again, the contribution from the real radiation of e + e − pairs together with finalstate π + π − corresponds to an effect in a µ of less than 1 × 10 −11 . We stress that, while the muon loop is expected to produce results similar to hadronic degrees of freedom given the scales involved, the extent to which the effects in a µ agree is largely coincidental. As shown in Fig. 2, the fermionic spectral functions tend to be larger near threshold (due to the P -wave suppression of the π + π − channel) and for large s (where F V π 1/s leads to a suppression), which is then compensated by the ρ(770) peak at intermediate energies.
Conclusions.-In this Letter we addressed a missing class of hadronic corrections to the anomalous magnetic moment of the muon at O(α 4 ), represented by the double-bubble topologies shown in Fig. 1. As a first step we reproduced the QED configurations involving electron loops by means of the two-and four-particle cuts of these diagrams, and showed that indeed the known results are recovered when adding the virtual and real contributions. In particular, configurations in which the inner loop does not correspond to an electron prove negligible.
Since virtual corrections should be included in the measured e + e − → hadrons cross section, we concluded that the only potentially missed contribution could originate from hadronic final states accompanied by e + e − pair emission. To estimate this effect, we argued that the case with an outer muon should provide a first indication, and then calculated the analog in scalar QED for a charged-pion loop. Both results are remarkably close, albeit to a large extent by coincidence, see Fig. 2, and translate to an effect in a µ of < ∼ 1 × 10 −11 . One would thus need a significant enhancement due to experimental cuts or misidentification of e + e − pairs to produce a relevant effect. It might be interesting to verify in experimental analyses that indeed no enhanced effects from e + e − radiation can occur, but absent such a scenario we conclude that this class of mixed leptonic and hadronic corrections is negligible at the level required for the final precision of the Fermilab g − 2 experiment.