Physics Letters B 797 (2019) 134855
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Physics Letters B
www.elsevier.com/locate/physletb
Single pseudoscalar meson pole and pion box contributions to the
anomalous magnetic moment of the muon
Gernot Eichmann
a
, Christian S. Fischer
b,∗
, Esther Weil
b
, Richard Williams
b
a
CFTP, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
b
Institut für Theoretische Physik, Justus-Liebig Universität Gießen, 35392 Gießen, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received
24 April 2019
Received
in revised form 22 July 2019
Accepted
9 August 2019
Available
online 20 August 2019
Editor:
B. Grinstein
We present results for single pseudoscalar meson pole contributions and pion box contributions to
the hadronic light-by-light (LBL) correction of the muon’s anomalous magnetic moment. We follow the
recently developed dispersive approach to LBL, where these contributions are evaluated with intermediate
mesons on-shell. However, the space-like electromagnetic and transition form factors are not determined
from analytic continuation of time-like data, but directly calculated within the functional approach
to QCD using Dyson-Schwinger and Bethe-Salpeter equations. This strategy allows for a systematic
comparison with a strictly dispersive treatment and also with recent results from lattice QCD. Within
error bars, we obtain excellent agreement for the pion electromagnetic and transition form factor and
the resulting contributions to LBL. In addition, we present results for the η and η
pole contributions
and discuss the dynamical effects in the η − η
mixing due to the strange quarks. Our result for the
total pseudoscalar pole contributions is a
PS-pole
μ
= 91.6 (1.9) × 10
−11
and for the pion-box contribution
we obtain a
π−box
μ
=−16.3 (2)(4) × 10
−11
.
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The anomalous magnetic moment a
μ
=
1
2
(g − 2)
μ
of the muon
is currently under intense scrutiny from both theory and exper-
iment.
With a persistent discrepancy of about 3–4 standard de-
viations
between the theoretical Standard Model (SM) predictions
and experimental determinations [1], a
μ
is considered a potential
candidate for the observation of physics beyond the SM. In order
to identify such contributions, both theory and experiment nee d
to improve their precision beyond the 0.54 parts per million level
that has been achieved by E821 at Brookhaven [2,3]. Two new ex-
periments
at Fermilab [4] and J-PARC [5]are under way, aiming to
reduce the experimental error by a factor of four.
However, the error budget of the theoretical SM prediction is
dominated by hadronic contributions that probe non-perturbative
QCD and at present mask any potential signals of new physics.
The most relevant of these are hadronic vacuum polarisation (HVP)
and light-by-light (LBL) scattering effects; the latter of which are
*
Corresponding author.
E-mail
addresses: Gernot.Eichmann@tecnico.ulisboa.pt (G. Eichmann),
Christian.Fischer@physik.uni-giessen.de (C.S. Fischer),
Esther.D.Weil@physik.uni-giessen.de (E. Weil),
Richard.Williams@theo.physik.uni-giessen.de (R. Williams).
the focus of this work and are shown diagrammatically in Fig. 1.
While the currently accepted estimate on hadronic LBL stems from
a combination of calculations based on low-energy effective mod-
els [6],
see [7]for a recent overview, there are great efforts both
from lattice QCD [8–18]as well as dispersion theory [19–28]to
improve this estimate.
Within
the functional approach via Dyson-Schwinger and Bethe-
Salpeter
equations (DSEs and BSEs), meson exchange contribu-
tions
to LBL as well as an (incomplete) determination of quark-
loop
effects (see Fig. 2) have been presented and discussed in
Refs. [29–31]. In the same framework, the dispersive results for
hadronic vacuum polarisation have been reproduced on the level
of 2-3 percent [32].
A
principal challenge for the functional approach is to provide
a reliable error estimate. In all practical calculations the tower of
DSEs must be truncated, and it is extremely hard to quantify the
systematic error of neglected contributions. Within the class of
rainbow-ladder truncations employed thus far for a
μ
, insight can
be gained only through comparison with both experimental results
and other approaches whose error estimates are well-defined.
Subsequently,
when a given truncation scheme is known to
perform well for certain observables, it can be expected to per-
form
equally well for related ones. Fortunately, the rainbow-ladder
scheme used in the context of a
μ
passes this test. As summarised
https://doi.org/10.1016/j.physletb.2019.134855
0370-2693/
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.