II-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Veriﬁcation of the QED tenth-order electron g −2:
Diagrams without a fermion loop
A. Hirayama∗1,∗2 and M. Nio∗1,∗2
A comparison between the measured value of theelectron anomalous magnetic moment and its theoretical prediction is the most stringent test of the quantum electrodynamics (QED) and, hence, the standard
model (SM) of elementary particles. The measurement
was performed by trapping a single electron in a cylindrical Penning trap. The deviation of the g value ofthe electron from Dirac’s prediction, ae = (g − 2)/2,
was determined as1)

So far, two independent numerical evaluations ofSet V have been executed by two groups.5,6) There is,however, 4.8 σ tension between the results. The difference is still negligible for the current precision of ae
but will certainly become crucial in the near future.
In Ref. 5), numerical integration was performed on
389 self-energy-like diagrams each, which are concatenations of nine vertex diagrams via the WardTakahashi identity. In contrast, 3,213 integrals derived
from vertex diagrams were evaluated in Ref. 6). Because the renormalization schemes used to construct
integrands are diﬀerent, the numerical result of one integral of the former does not coincide with the sumof the corresponding nine integrals of the latter. The
direct comparison of numerical results of integrals isnontrivial.
The gap can be ﬁlled by calculating the diﬀerencein vertex renormalization constants, δL, of two renormalization schemes for every vertex diagram. The difference in numerical results of Set V integrals can be
expressed in terms of these δL’s and lower-order contributions of the anomalous magnetic moment.
We numerically computed all the necessary δL’s thatare derived from 1 second-, 4 fourth-, 28 sixth-, and269 eighth-order vertex diagrams. The computation
took approximately 120,000 hours in total. This is asmall and quick calculation compared with the numerical evaluation of Set V integrals.
Diagram-by-diagram comparison was performed forthe fourth-, sixth-, and eighth-order cases, and our veriﬁcation method worked very well. For the tenth order,
we examined numerical results representing the 2,232
Set V vertex diagrams that have no self-energy subdiagram and correspond to 135 integrals of self-energylike diagrams. No inconsistency was found in any of
the 135 integrals between Refs. 5) and 6). The investigation of the remaining 4,122 diagrams that have atleast one self-energy subdiagram is in progress and will
be completed soon.
We thank Prof. T. Aoyama and Prof. M. Hayakawa
for useful discussions. This work is supported inpart by JSPS KAKENHI 16K95338, 19K21872, and20H05646. Numerical calculations were conducted on
HOKUSAI-BigWaterfall of RIKEN.

ae [expt.] = 0.001 159 652 180 73 (28) [0.24 ppb]. (1)A new experiment aiming at a 20-fold improvement in
the uncertainty is in progress.
To match the precision of measurement, theory musttake into account small but non-negligible contributions from hadronic and electroweak interactions. The
QED contribution is dominant and required up to thetenth order of perturbation theory because (α/π)5 ∼6.8×10−14 , where α is the ﬁne-structure constant. By
now, all terms up to the eighth order have been ﬁrmly
established. The tenth-order term was obtained only
by one group and has not yet been cross-checked.2) To
theoretically predict ae , a value of α is needed. Recently, two very precise values of α became available,both of which were determined using atom interferometry. The Cs3) and Rb4) atom experiments yielded
α−1 (Cs) = 137.035 999 046 (27)[0.20 ppb],

(2)

−1

[81 ppt],(3)

α(Rb) = 137.035 999 206 (11)respectively. The diﬀerence is about 5.5 σ. A new Cs
experiment has already started to reveal one more digitof α. We then have two SM predictions of ae :ae [theory : α(Cs)] = 0.001 159 652 181 62 (23), (4)ae [theory : α(Rb)] = 0.001 159 652 180 265 (94), (5)where the uncertainty is solely governed by that fromα in both cases. Note that the discrepancy between
measurement and theory is positive in sign for α(Cs),while it is negative for α(Rb).
Considering the accuracy required for future experiments, the tenth-order QED term must be carefully
reexamined. A total of 12,672 Feynman vertex diagrams contributes to the tenth-order term. Amongthem, 6,354 diagrams without a fermion loop form agauge-invariant set called Set V. This is the most difﬁcult set to evaluate, and the uncertainty of the QEDcontribution to ae entirely originates from Set V.
∗1∗2References
1) D. Hanneke et al., Phys. Rev. Lett. 100, 120801 (2008).
2) T. Aoyama et al., Phys. Rev. Lett. 109, 111807 (2012).
3) R. H. Parker et al., Science 360, 191–195 (2018).
4) L. Morel et al., Nature 588, 61–65 (2020).
5) T. Aoyama et al., Phys. Rev. D 97, 036001 (2018).
6) S. Volkov, Phys. Rev. D 100, 096004 (2019).

RIKEN Nishina Center
Department of Physics, Saitama University- 66 -1