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研究了由声子难题(质子半径和μ子g-2的差异)激发的新物理学。 使用轻量级玻色子assuming假设汤川相互作用,则同时解释了这些语音难题。 我们以前的工作将这种标量玻色子的质量mϕ从160 keV限制到60 MeV。 我们通过在计算衰减率时包括耦合到ϕ的所有可能粒子的影响来改善此结果。 这样做涉及到包括夸克的强相互作用物理学,这是计算ηπϕ顶点函数所必需的。 Nambu-Jona-Lasinio模型解决了相关的强相互作用物理问题,该模型考虑了产生组成质量的自发对称性断裂。 我们使用ηπϕ顶点函数重新分析电子束转储实验。 结果是,允许的m 3范围在约160keV和3.5MeV之间。 这个狭窄的范围代表排除或发现这种标量玻色子的诱人目标。 讨论了我们现象学模型的可能的紫外线完成。
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Available online at www.sciencedirect.com
ScienceDirect
Nuclear Physics B 944 (2019) 114638
www.elsevier.com/locate/nuclphysb
Eta decay and muonic puzzles
Yu-Sheng Liu
a,∗
, Ian C. Cloët
b
, Gerald A. Miller
c
a
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
b
Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
c
Department of Physics, University of Washington, Seattle, WA 98195-1560, USA
Received 5
April 2019; received in revised form 30 April 2019; accepted 13 May 2019
Available
online 16 May 2019
Editor: Hong-Jian
He
Abstract
New
physics motivated by muonic puzzles (proton radius and muon g −2 discrepancies) is studied. Using
a light scalar boson φ, assuming Yukawa interactions, accounts for these muonic puzzles simultaneously.
Our previous work limits the existence of such a scalar boson’s mass m
φ
from about 160 keV to 60 MeV. We
improve this result by including the influence of all of the possible particles that couple to the φ in computing
the decay rate. Doing this involves including the strong interaction physics, involving quarks, necessary to
compute the ηπφ vertex function. The Nambu-Jona-Lasinio model, which accounts for the spontaneous
symmetry breaking that yields the constituent mass is employed to represent the relevant strong-interaction
physics. We use the ηπφ vertex function to reanalyze the electron beam dump experiments. The result is that
the allowed range of m
φ
lies between about 160 keV and 3.5 MeV. This narrow range represents an inviting
target for ruling out or discovering this scalar boson. A possible UV completion of our phenomenological
model is discussed.
© 2019 The Authors. 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 proton charge radius measured using the Lamb shift in muonic hydrogen, r
p
=
0.84087(39) fm [1,2], differs from the CODATA average obtained from hydrogen spectroscopy
*
Corresponding author.
E-mail
addresses: mestelqure@gmail.com (Y.-S. Liu), icloet@anl.gov (I.C. Cloët), miller@phys.washington.edu
(G.A. Miller).
https://doi.org/10.1016/j.nuclphysb.2019.114638
0550-3213/© 2019
The Authors. 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
.
2 Y.-S. Liu et al. / Nuclear Physics B 944 (2019) 114638
and e −p scattering, r
p
= 0.8751(61) fm [3], by more than 5σ . Although the discrepancy may
arise from subtle lepton-nucleon non-perturbative effects within the Standard Model (SM), or
experimental uncertainties [4,5], it could also be a signal of new physics involving a violation
of lepton universality [6,7]. The muon anomalous magnetic moment pro
vides another potential
signal of new physics [8]. The BNL measurement [9] differs from the SM prediction by more
than 3σ , a
μ
=a
exp
μ
−a
th
μ
=287(80) ×10
−11
[10,11].
A ne
w scalar boson φ (we have concluded in our previous work [12] that other spin-0 and
spin-1 bosons are ruled out), which couples to the muon and proton could explain both the proton
radius and (g − 2)
μ
puzzles [13,14]. Phenomenological motivation for such a scalar boson as a
Higgs portal in the dark sector has been considered theoretically [15–22] and experimentally [23–
27]. We investigate the couplings of this boson to SM fermions, ψ, which appear as Yukawa
terms in the effective Lagrangian, L ⊃ e
f
φ
¯
ψ
f
ψ
f
, where
f
= g
f
/e, e is the electric charge
of the proton, and f is the flavor index. Other authors have pursued this idea, but made further
assumptions relating the couplings to different particle species, mass range, etc. We make no a
priori assumptions regarding signs or magnitudes of the coupling constants. The Lamb shift in
muonic h
ydrogen fixes
μ
and
p
to have the same sign. Without loss of generality, we take both
μ
and
p
to be positive, and
e
and
n
are allowed to have either sign.
Coupling a single scalar to up and do
wn quarks in an effective Lagrangian at the MeV scale
is not consistent with the SU(2)
L
× U(1)
Y
gauge invariance of the Standard Model above the
electroweak symmetry breaking scale. An ultraviolet (UV) completion is needed. As is also well
known [28], while it is difficult to create a viable model of dark scalars with masses in the MeV
range, interesting attempts have been made [20,22].
Electron beam dump e
xperiments have been aimed at searching for new particles [18,29–32].
The typical setup of an electron beam dump experiment involves a beam stopped by a large
amount of material. The ensuing interactions could produce new particles via a bremsstrahlung-
like process. Such particles would pass through a shield re
gion and decay. These new particles
can be detected by their decay products, electron and/or photon pairs, measured by the detector
downstream of the decay region. In our previous work, it was assumed that the new particle only
couples to electrons. In our simple model, considering the φ couplings to other SM particles
could dramatically change the e
xclusion range.
It is w
orthwhile to study the production of a new scalar boson by eta decay. This is be-
cause there are no selection rules preventing φ emission and possible complications involving
strangeness are absent in many channels. We will show that η → π
0
¯
ψ
f
ψ
f
and η → π
0
γγ de-
cay channels are particularly useful. Eta decay to the π
0
¯
ψ
f
ψ
f
final state is forbidden at tree
level in the SM by charge conjugation symmetry, but it is allowed by a virtual φ emission.
Eta decay to π
0
γγ is observed, and the existence of the φ may open up new channels, whose
decay rate should not exceed the observed value. We will use the Nambu–Jona-Lasinio (NJL)
model [33–36], a chiral effective theory of QCD exhibiting dynamical chiral symmetry break-
ing, to provide the strong-interaction input necessary to predict these decay rates. The NJL model
satisfies the soft-pion theorems making it an ideal tool with which to determine the coupling of a
scalar boson to the Goldstone bosons. Therefore, using the current η decay data, we will signifi-
cantly i
mprove the constraints on the new scalar boson.
A recent e
xperiment extracts the proton radius to be r
p
= 0.8335(95) fm [37]by measuring
the 2S − 4P transition frequency in electronic hydrogen. This result agrees with the previ-
ous muonic hydrogen experiments [1,2]but is more than 3 standard deviations away from the
CODATA value [3] that is dominated by many previous hydrogen spectroscopy experiments.
Three possible scenarios can immediately be en
visioned:
Y.-S. Liu et al. / Nuclear Physics B 944 (2019) 114638 3
1. the proton radius puzzle is solved,
2. it is too early to use the ne
w experiment as a replacement for many others,
3. new ph
ysics may coexist with the CODATA value and the new experiment result.
It is tempting to accept the first scenario, ho
wever it defies the results of decades of the electron-
proton scattering experiments. On the other hand, the preliminary nuclei radii from laser spec-
troscopy of μ
4
He
+
and μ
3
He
+
[38] agree with the electron-nucleus experiments [39,40]. The
PRad experiment [41–43]may shed some light on this direction. For the second approach, one
may argue that the measurement of 2S − 4P transition frequency is very difficult because of
quantum interference effects that involve the details of the experimental setup. It is desirable
to ha
ve a second experiment on regular hydrogen for this transition. Moreover, a more recent
electron hydrogen experiment [44]on the
1
S −
3
S transition finds a radius in agreement with
the CODATA value and earlier hydrogen spectroscopy measurements. For the third approach,
the true value of proton radius may lie within 3 standard deviations of the new experiments and
the old CODATA value, and the muonic hydrogen experiments still signal new physics. In other
wo
rds, the existence of a new scalar meson may not conflict with any of the experiments. We
examine the latter two possibilities here.
This paper is or
ganized as follows: Sec. 2 discusses the Lagrangian, introducing φ couplings to
u and d quarks. A possible UV completion is discussed. The ηπ
0
φ vertex is discussed in Sect. 3.
Sec. 4 presents the φ and η decay rates. Sec. 5 revisits the beam dump experiments. Sec. 6 and
Sec. 7 show the new exclusion region obtained by different η decay channels. Sec. 8 discusses
third scenario which the new physics coexists with the new regular hydrogen experiments and
the old COD
ATA value. A conclusion is given in Sec. 9.
2. Lagrangian
In our previous work, the Lagrangian involved interactions between the φ and nucleons. This
is not sufficient to study effects involving mesons. Coupling between the φ and quarks is exam-
ined here.
2.1. φ couplings to u and d quarks
Here we use a simplified Lagrangian including the new boson φ in the mostly plus metric:
L
φ
⊃−
1
2
(∂φ)
2
−
1
2
m
2
φ
φ
2
+e
f
φ
¯
ψ
f
ψ
f
(1)
where f is the flavor index,
f
= g
f
/e, e is the electric charge, and ψ
f
is the fermion field
(quarks and leptons) in the SM. The couplings to the neutron,
n
, and proton
p
are given by
p
=2
u
+
d
,
n
=2
d
+
u
. (2)
The Lamb shift in muonic hydrogen fixes
μ
and
p
to have the same sign, therefore, we choose
μ
and
p
to be positive, and
n
and
e
are allowed to have either sign. From our previous work
[12,45], the allowed values of
p
and
n
are shown in Figs. 1 and 2 between the solid black,
dotted red, and dashed blue lines. Since
n
can take either sign, we present
n
as a ratio to
p
.
The allowed values of
μ
are shown in Fig. 3. We can find the allowed regions of
u
and
d
in
Fig. 4 and 5, using
p
and
n
in Figs. 1 and 2.
4 Y.-S. Liu et al. / Nuclear Physics B 944 (2019) 114638
Fig. 1. Exclusion plot for
p
(shaded region is excluded at 95% CL). The solid black, dotted red, and dashed blue lines
are from our previous work [12,45] corresponding to combining muonic hydrogen [1–3] with muon g − 2 experiments
[9–11], the binding energy difference of
3
He and
3
H[46–53], and the binding energy of nuclear matter per nucleon
[54]. The thick yellow solid curve is from the preliminary muonic
3
He ion laser spectroscopy experiment [38–40,55]
combining
with the
n
constraint in Fig. 2. The vertical line indicates the allowed mass range obtained in Fig. 11. (For
interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 2. Exclusion plot for
n
/
p
(shaded region is excluded at 95% CL). Since
n
can take either sign, we present
n
as a ratio to
p
. The solid black, dotted red, and dashed blue lines are from our previous work [12,45] corresponding
to the low energy scattering of neutron on
208
Pb [56], the preliminary muonic
4
He ion laser spectroscopy experiment
[38,40,57], and the laser spectroscopy experiment of muonic deuterium [58–60]. The vertical line indicates the allowed
mass range obtained in Fig. 11.
2.2. A concrete UV model
Coupling a single scalar to u and d quarks in an effective Lagrangian at the MeV scale is not
consistent with the SU(2)
L
×U(1)
Y
gauge invariance of the SM above the electroweak symme-
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