Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 56 (2023)
RIKEN
Accel. Prog. Rep. 56 (2023)

Twelfth-order QED contributions to the muon g −2
R. Yamazaki∗1,∗2 and M. Nio∗2,∗1
The anomalous magnetic moment of the muon, aµ ≡(gµ − 2)/2, has occupied a central role in testing thevalidity of the standard model of elementary particles (SM). The new measurement of aµ at Fermilab1)
is consistent with that at Brookhaven,2) and the longstanding tension between experiment and theory remains unresolved. The average of the two experimentsand the SM prediction3) are given asaµ (Exp) = 116 592 061 (41) × 10−11 ,−11aµ (SM) = 116 591 810 (43) × 10,(1)(2)

respectively, and the diﬀerence is (251 ± 59) × 10−11
corresponding to 4.2 σ.
The theoretical prediction of aµ has been calculated
by considering all three forces of SM. The contribution
from the quantum electrodynamics (QED) is dominant
and has been determined up to the tenth order of theperturbation theory. It has been considered suﬃciently
well known as its uncertainty is 1.0 × 10−12 . The assigned uncertainty was derived from the rough estimateof the leading-order contribution of the twelfth-orderterm. Recently, we have calculated, not guessed, thetwo types of Feynman diagrams shown in Fig. 1 thatare supposed to give the leading contributions in thetwelfth order.

Fig. 1. Some Feynman vertex diagrams of the twelfth-order
QED containing light-by-light scattering subdiagrams.

Because the muon is about two hundred times heavier than the electron, enhancement factors arise in some
QED contributions to aµ . One famous origin of theenhancement is a light-by-light scattering vertex diagram (LL6), in which an external magnetic photon is
attached to an electron loop. Another is a vacuumpolarization subdiagram (VP) consisting of an electronloop. The left diagram of Fig. 1 is a combination of
two enhancement mechanisms of the twelfth order ofthe QED perturbation theory. We have calculated its
numerical contribution taking into account all possible
insertions of three second-order VPs (P2s) to LL6. In
addition, diagrams with insertions of one fourth-order
VP (P4) and two P2s to LL6 were calculated. The coeﬃcients of (α/π)6 , where α is the ﬁne-structure constant, are obtained as follows:∗1∗2

Department of Physics, Saitama University
RIKEN Nishina Center- 62 -(e)a(12)P2(e) P2(e) P2(e) ] = 2415.256 (53), (3)µ [LL6(e)a(12)P2(µ) P2(e) P2(e) ] = 367.974 (15),µ [LL6(4)(e)a(12)P4(e) P2(e) ] = 1451.106 (91),µ [LL6(5)where a superscript (e) or (µ) of LL6, P2, or P4 indicates a species of its fermion loop. The sum of thethree gives the leading contribution of the twelfth-order
QED: α 6× 4230 = 0.665 × 10−12 .
(6)π

It is within the uncertainty assigned to the QED contribution to aµ .
A new enhancement mechanism appears at thetwelfth order. The right diagram of Fig. 1 (LL6 LL6)
contains a “child” LL6 as a subdiagram. When themass of an external fermion is much heavier than thatof a loop fermion, the slope of F1 form factor of LL6
has a large enhancement factor4)
22 mµ2 dF1 (q ) mµ∝∼ 40000 .
(7)
dq 2 q2 =0me

It is uncertain that such a critical enhancement factor exists in the LL6 LL6 contribution to aµ . Therefore,
we calculated LL6 LL6 without any approximation except for numerical integration. The 360 Feynman vertex diagrams of the gauge-invariant set LL6 LL6 canbe reduced to sixteen independent integrals. Each ofthe integrands consists of about 50 MByte text ﬁles. In
principle, no renormalization is required for LL6 LL6.
However, to conduct numerical integration, an integrand needs ultra-violet (UV) counter terms that cancel
the UV divergences of a child LL6 vertex subdiagram
and a light-by-light scattering subdiagram (LL). The
UV counter terms we constructed are the form factor
F1 (0) for a child LL6 and the light-by-light scattering
tensor Πµνρσ (0, 0, 0, 0) for a LL. The sum of the UV
counter terms exactly vanishes when all gauge-invariant
diagrams are summed up. The numerical calculation
has been performed, and the result will be reported
elsewhere.
Numerical calculations were conducted on RIKEN’ssupercomputer HOKUSAI BigWaterfall. This work
was partly supported by KAKENHI 16K05338,20H05646, and 22K03646.
References
1) B. Abi et al. (Muon g−2 Collaboration), Phys. Rev. Lett.
126, 141801 (2021).
2) G. W. Bennett et al. (Muon g −2 Collaboration), Phys.
Rev. D 73, 072003 (2006).
3) T. Aoyama et al., Phys. Rep. 887, 1 (2020).
4) E. H. Wichmann et al., Phys. Rev. 101, 843 (1956).