compat=1.1.0

# Light Dilaton in Rare Meson Decays and Extraction of its CP Property

Sudhakantha Girmohanta1,2, Yuichiro Nakai1,2, Yoshihiro Shigekami1,2 and
Kohsaku Tobioka3,4
1Tsung-Dao Lee Institute, Shanghai Jiao Tong University,
520 Shengrong Road, Shanghai 201210, China
2School of Physics and Astronomy, Shanghai Jiao Tong University,
800 Dongchuan Road, Shanghai 200240, China
3Department of Physics, Florida State University, Tallahassee, Florida 32306,
USA
4Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801, Japan

###### Abstract

The dilaton $\phi$ is a pseudo-Nambu-Goldstone boson associated with the
spontaneous breaking of scale invariance in a nearly conformal theory, and
couples to the trace of the stress-energy tensor. We analyze experimental
constraints on a light dilaton with mass in the MeV-GeV range from rare meson
decays. New model-independent inclusive bounds for the $b\to s\phi$ transition
largely exclude the parameter space of a light dilaton that could explain the
muon $g-2$ anomaly. Despite similarities between a dilaton and a Higgs-portal
scalar, the dilaton-photon coupling is enhanced compared to the Higgs-portal
scalar due to contributions from loops of the conformal sector. Consequently,
the shortened lifetime of the dilaton relaxes bounds from $K\to\pi$ \+
invisible searches at the NA62 experiment and constraints from the Big Bang
Nucleosynthesis. We utilize this fact to search for the dilaton signature at a
lepton collider such as the ongoing Belle II experiment. Further, we
demonstrate how to extract the CP property of the dilaton using the variation
of the differential cross-section of $e^{+}e^{-}\to e^{+}e^{-}\phi$ with the
azimuthal angle between the outgoing leptons.

††preprint: KEK–TH–2569

##  I Introduction

The existence of approximate scale invariance with dynamical spontaneous
breaking is an intriguing possibility of physics beyond the Standard Model
(SM) that is able to address the quantum instability and hierarchy of the
electroweak scale with respect to the Planck scale. The AdS/CFT correspondence
Maldacena (1998); Gubser  _et al._ (1998); Witten (1998) (see also refs.
Arkani-Hamed  _et al._ (2001); Rattazzi and Zaffaroni (2001)) tells us that
such a 4D nearly-conformal field theory is holographically dual to the 5D
Randall-Sundrum (RS) model with a warped extra dimension bounded by two
3-branes, called UV and IR branes Randall and Sundrum (1999). Physics on the
IR brane is redshifted due to an exponential warp factor. Here, the existence
of the IR brane in the 5D theory corresponds to the spontaneous breaking of
scale invariance (SBSI). The RS model has a radion degree of freedom to
describe the distance between the two 3-branes. This radion plays the role of
a pseudo-Nambu-Goldstone boson (pNGB) associated with SBSI, called $dilaton$,
in the nearly-conformal theory. The radion (dilaton) mass depends on a
mechanism to stabilize the brane distance, while typically it is somewhat
smaller than the mass scale of the IR brane (the scale of SBSI), which is set
to the TeV scale. However, as pointed out in refs. Bellazzini  _et al._
(2014); Coradeschi  _et al._ (2013), a lighter dilaton can be achieved, and it
has a rich phenomenology Abu-Ajamieh  _et al._ (2018) as the dilaton couples
to SM particles through the trace of the stress-energy tensor.

The radion/dilaton emerges as the lightest degree of freedom from warped
extra-dimensional models, or by duality, from composite dynamics breaking a
nearly conformal theory. Thus, it serves as the key probe into these class of
theories for a low-energy observer. In spite of the efforts in elucidating
dilaton phenomenology at high energy colliders Csaki  _et al._ (2001, 2007), a
detailed phenomenological study was lacking for the MeV-GeV mass range. Ref.
Abu-Ajamieh  _et al._ (2018) partly filled this gap, motivated by the fact
that a dilaton in this mass range can address the muon $g-2$ anomaly Chen  _et
al._ (2016); Marciano  _et al._ (2016); Bennett  _et al._ (2006); Keshavarzi
_et al._ (2018); Abi  _et al._ (2021); Aguillard  _et al._ (2023). Borrowing
the axion search constraints, it was claimed that a direct coupling of the
radion to fermions weakens many bounds and allows a region of the parameter
space that explains the muon $g-2$ anomaly. Nevertheless, a dedicated detailed
study for the dilaton phenomenology is missing for this mass range. In
particular, we note that a dilaton coupling to leptons implies a dilaton
coupling to weak gauge bosons. Specifically, there is a tree-level interaction
between the dilaton and the $W$ boson as the $W$ boson mass breaks the scale
invariance. Besides, one would also expect dilaton couplings to quarks.
Consequently, it is unclear whether the parameter space for the muon $g-2$
anomaly is viable or not when confronted with stringent rare meson decay
constraints111The Kaluza-Klein (KK) mode contribution is typically too small
to address the muon $g-2$ anomaly Beneke  _et al._ (2014). (see ref.
Goudzovski  _et al._ (2023) for a recent review on rare meson decay
phenomenology). Due to reasons we will illustrate later, experimental
constraints on a dilaton resemble those on a Higgs-portal scalar Winkler
(2019). However, there is a key difference, namely that the dilaton has an
enhanced coupling to the photon due to loops of the conformal sector Csaki
_et al._ (2007). We will thoroughly examine the constraints for the
radion/dilaton in the MeV-GeV mass range in detail, which will clarify the
current status.

In the first part of the present paper, we will investigate $B\rightarrow
K\phi$ and $K\to\pi\phi$ transitions for a light dilaton $\phi$ with mass in
the MeV-GeV range. We will calculate the dilaton production from the
$B\rightarrow K\phi$ process at the one-loop level when a dilaton directly
couples to quarks as well as weak gauge bosons. It will be shown that the
current experimental data for the inclusive $b\rightarrow s\phi$ transition
largely exclude the dilaton parameter space to explain the muon $g-2$ anomaly.
Further, in part of the light dilaton parameter space, constraints from
searches for $K\to\pi+$ invisible modes in the NA62 experiment at CERN are
relaxed due to its enhanced coupling with the photon. The constraint from the
Big Bang Nucleosynthesis (BBN) is also comparably weaker due to the shortened
lifetime of the dilaton. On the other hand, searches for the visible signature
modes of dilaton decays such as $K_{L}\to\pi^{0}\gamma\gamma$ and
$K_{L}\to\pi^{0}e^{+}e^{-}$ at the KTeV experiment constrain the dilaton-
photon coupling from a complementary direction in the parameter space. With
mild assumption on the relaxed direct coupling to electrons, a window for the
light dilaton appears that is consistent with the current stringent rare meson
decay constraint and has a large enough coupling to photons that can be
searched for in modes like $e^{+}e^{-}\to\ell^{+}\ell^{-}\phi$ where
$\ell=e,\mu$ at the Belle II experiment.

The results of our analysis for the dilaton parameter space and its heightened
coupling to the photon provide the motivation to proceed one step further and
extract the CP property of the dilaton in a prospective future signal and
analyze its distinction thereof from axions. In the second part of the paper,
we pursue this direction. An imprint of the CP nature of $\phi$ remains on the
distribution of the differential cross-section with a Lorentz-invariant angle
$\Delta\varphi$ to be defined later. We utilize a kinematical method to
illustrate the distinct signature of a CP-odd and CP-even particle, while
thanks to the enhanced dilaton-photon coupling, enough statistics can be
obtained, which is not possible for a Higgs-portal scalar in the focused light
mass range. Our method is not particular to Belle II and can be taken over by
any lepton collider.

The organization of the rest of the paper is given as follows. In section II,
we define the model and summarize the dilaton mass and couplings to the SM.
Then, section III discusses the constraints from rare meson decays and
collider searches for a light dilaton with mass in the MeV-GeV range. Their
implications for the dilaton interpretation of the muon $g-2$ anomaly are
discussed. In section IV, we explore the extraction of the CP nature. Section
V is devoted to conclusions and discussions, while appendix A contains some
relevant details of the kinematical calculations.

##  II The Model

The low-energy effective description of a 4D nearly-conformal field theory is
expressed in terms of the dilaton $\phi$, which is the CP-even pNGB of the
broken scale invariance. Under a scale transformation $x^{\mu}\to
x^{\prime\mu}=e^{-\omega}x^{\mu}$, below the conformal breaking scale
$f_{\phi}$, the dilaton undergoes a shift transformation as
$\phi(x)\to\phi^{\prime}(x^{\prime})=\phi(x)+\omega f_{\phi}$. Here, $\phi$
can be conveniently embedded in a conformal compensator $\chi(x)\equiv
f_{\phi}e^{\phi(x)/f_{\phi}}$, such that under the scale transformation the
compensator transforms linearly
$\chi(x)\to\chi^{\prime}(x^{\prime})=e^{\omega}\chi(x)$. Then, the dilaton
effective Lagrangian can be constructed so that it is symmetric under the
above realization of the scale transformation. As the breaking of the
dilatation current is given by the trace of the energy-momentum tensor
$T^{\mu\nu}$, the coupling of the $\chi$ would be such that it compensates for
the breaking, corrected by explicit conformal breaking effects. Therefore, the
action of the dilaton can be written as Chacko and Mishra (2013); Abu-Ajamieh
_et al._ (2018)

$S_{\phi,{\rm eff}}=\int
d^{4}x\left[\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-V_{\rm
eff}(\phi)+c\frac{\phi}{f_{\phi}}T^{\mu}_{\mu}\right]\ ,$ (1)

where we assume the kinetic term of $\phi$ is canonically normalized, $V_{\rm
eff}$ contains a potential generated for $\phi$ from explicit violation of the
conformal invariance, and $c$ denotes some constant that depends on how the
scale invariance is broken by $T^{\mu}_{\mu}$.

###  II.1 Dilaton interactions

Let us now summarize dilaton interactions with the SM particles. The coupling
to a massive vector boson $V^{\mu}$ can be obtained by noting that the scale
invariance is broken by the mass of the vector boson $m_{V}$. Then, the
dilaton effective coupling in the unitary gauge must be

${\cal L}_{{\rm eff},\chi
V}=\left(\frac{\chi}{f_{\phi}}\right)^{2}m_{V}^{2}V_{\mu}V^{\mu}\ .$ (2)

Expanding $\chi=f_{\phi}e^{\phi/f_{\phi}}$ in terms of $\phi$, one obtains, to
the leading order,

${\cal L}_{{\rm eff},\phi
V}=2\left(\frac{\phi}{f_{\phi}}+\frac{\phi^{2}}{f_{\phi}^{2}}\right)m_{V}^{2}V_{\mu}V^{\mu}\
.$ (3)

Similarly, the scale invariance is broken by the mass of a fermion $f$, and
hence, the dilaton couples to the fermion as

${\cal L}_{{\rm eff},\chi f}=-\left(\frac{\chi}{f_{\phi}}\right)m_{f}\bar{f}f\
,$ (4)

with the fermion mass $m_{f}$. To the linear order, we obtain

${\cal L}_{{\rm eff},\phi f}=-c_{\phi ff}\frac{\phi}{f_{\phi}}m_{f}\bar{f}f\
,$ (5)

where we have introduced $c_{\phi ff}$, a dimensionless parameter that depends
on the mixing between elementary and composite sectors in the UV theory, which
is related to the anomalous dimension of the fermion and used to obtain the
flavor structure of the SM. In the dual 5D picture, it amounts to the
specification of the wavefunction profile of the fermion in the warped extra
dimension. We are interested in a general analysis, and simply parameterize
the model dependence in term of the coefficient $c_{\phi ff}$. The explicit
violation of the conformal invariance by a nearly marginal operator generating
a dilaton mass $m_{\phi}$ corrects the above interactions by ${\cal
O}(m_{\phi}^{2}/f_{\phi}^{2})$, which we neglect in the following discussion.

Although a massless gauge field possesses the traceless energy-momentum tensor
and no scale dependence at the classical level, the running of the gauge
coupling generates a scale dependence. Such a running receives contributions
from elementary and strongly interacting states above the breaking scale of
the scale invariance, and elementary and other light emergent states from the
strong sector below the breaking scale. With the $\beta$-function coefficient
above and below the symmetry breaking scale denoted as $b_{+}$ and $b_{-}$
respectively, the coupling of the dilaton to a massless gauge boson can be
written as Chacko and Mishra (2013)

${\cal L}_{{\rm eff},\phi
A}=\frac{g^{2}}{32\pi^{2}}(b_{-}-b_{+})\frac{\phi}{f_{\phi}}F^{\mu\nu}F_{\mu\nu}\
,$ (6)

where $F$ is the field strength tensor of the massless gauge field $A^{\mu}$,
and $g$ denotes the corresponding gauge coupling. Explicitly, the CFT degrees
of freedom contribute to the 2-point function of the massless fundamental
gauge field Csaki  _et al._ (2007), such that

$b_{+}=-\frac{8\pi^{2}}{g^{2}\ln{\Lambda_{\rm UV}/f_{\phi}}}\ ,$ (7)

where $\Lambda_{\rm UV}$ is the UV cut-off of the theory. Therefore, the
dilaton coupling to a massless gauge boson can be summed up as follows

${\cal L}_{{\rm eff},\phi A}=\left[\frac{1}{4\ln{\Lambda_{\rm
UV}/f_{\phi}}}+\frac{g^{2}}{32\pi^{2}}b_{-}\right]\frac{\phi}{f_{\phi}}F^{\mu\nu}F_{\mu\nu}\
.$ (8)

For concreteness, we will take the cut-off at an intermediate scale, namely
$\Lambda_{\rm UV}\sim 10^{10}\ {\rm GeV}$, while it is straightforward to do
an analysis with other choices.

Finally, the mixing between the SM Higgs field and the dilaton can only arise
from explicit conformal breaking effects and hence is quite suppressed. This
is a non-trivial consequence of the scale invariance and understood as
follows. The Higgs-dilaton potential has a restrictive form due to the scale
invariance, namely

$V(\chi,h)=\left(\frac{\chi}{f_{\phi}}\right)^{4}V_{0}(h)\ ,$ (9)

where $h$ is the SM Higgs field, and $V_{0}$ only contains terms involving
$h$. Therefore, upon expanding $V_{0}$ around the minimum, no linear term in
$h$, that could be responsible for a Higgs-dilaton mixing, appears. Only such
effect can arise from explicit conformal breaking terms, and hence, the
corresponding mixing angle will be suppressed by Chacko and Mishra (2013)

$\theta_{\phi
h}\sim\left(\frac{m_{\phi}^{2}}{16\pi^{2}f_{\phi}^{2}}\right)\frac{v}{f_{\phi}}\ll
1\ ,$ (10)

where $v$ is the Higgs vacuum expectation value (VEV).

Finally, let us compare and contrast the dilaton interactions with those of a
Higgs-portal scalar. A common way to extend the SM with a real singlet scalar
($S$) is to assume that it mixes with the SM Higgs doublet $H$ through the
following term Winkler (2019); Kachanovich  _et al._ (2020)

${\cal L}_{Sh}=\left(g_{1S}S+g_{2S}S^{2}\right)\left(H^{\dagger}H\right)\ .$
(11)

After mass diagonalization, $S$ inherits the SM Higgs interactions suppressed
by the sine of the Higgs-scalar mixing angle ($\theta_{Sh}$). For example, the
singlet coupling to a SM fermion $f$ is

${\cal L}_{Sf}=-\left(\frac{\sin\theta_{Sh}}{v}\right)m_{f}\bar{f}fS\ .$ (12)

Although the dilaton-fermion coupling arises from a different origin, namely
SBSI instead of mixing with the physical Higgs, the effective Lagrangian (12)
looks similar to Eq. (5). However, there is an important difference in the
factor $c_{\phi ff}$ that depends on the anomalous dimension of $f$. Another
key difference is in the coupling with the photon. While $S$ obtains the
photon coupling through loops of SM particles, especially $t,W$ Marciano  _et
al._ (2012), the dilaton also gets additional contribution from loops of the
conformal sector as shown in Eq. (8). These facts will play crucial roles in
our analysis of the dilaton phenomenology.

###  II.2 Dilaton mass

In this section, we will illustrate how the dilaton receives its mass, and
outline how a GeV-scale dilaton can emerge from the underlying theory. The
dilaton receives its mass as a result of an explicit violation of the
conformal symmetry. A general analysis of the dilaton mass can be carried out
by assuming that the initial conformal theory is perturbed by a nearly
marginal operator of mass dimension $4-\epsilon$,

${\cal L}_{\rm CFT}\to{\cal L}_{\rm CFT}+c_{{\cal O}}{\cal O}\ .$ (13)

The scaling of this operator depends on the coupling $c_{{\cal O}}$ and how it
evolves with the renormalization group (RG) flow. At the breaking scale of the
conformal theory, the scaling dimension of the operator will dictate the
dilaton mass.

Alternatively, we can consider a dual 5D picture of the RS model. Here, the
${\mathbb{R}}_{4}\otimes S_{1}/{\mathbb{Z}}_{2}$ metric with the graviton
$g_{\mu\nu}$ and the modulus field $T(x)$ to describe the distance between two
3-branes is

$ds^{2}=G_{AB}dx^{A}dx^{B}=e^{-2k|\theta|T(x)}g_{\mu\nu}(x)dx^{\mu}dx^{\nu}-T(x)^{2}d\theta^{2}\
,$ (14)

where $x^{\mu}$ denotes the 4D coordinate, $k$ is the curvature scale, and
$\theta\in[-\pi,\pi]$ parameterizes the extra dimension with the UV (IR) brane
placed at the fixed point $\theta=0$ ($\theta=\pi$). Integrating out the extra
dimension, the 4D effective action is given by

$\displaystyle S_{\phi,\rm eff}\supset$
$\displaystyle\,\frac{2M_{*}^{3}}{k}\int
d^{4}x\sqrt{-g}\left(1-\left({\phi}/{Z}\right)^{2}\right)R_{4}$
$\displaystyle+\frac{1}{2}\int
d^{4}x\sqrt{-g}\,\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)\ ,$ (15)

where $M_{*}$ is the 5D Planck constant, $R_{4}$ is the Ricci scalar
constructed from $g_{\mu\nu}$, and $\phi(x)\equiv Ze^{-\pi kT(x)}$ with
$Z\equiv\sqrt{24M_{*}^{3}/k}$ denotes the canonically normalized radion. Note
that $N\equiv 4\pi(M_{*}/k)^{3/2}$ is the order of number of colors in the
dual large-$N$ CFT. In Eq. (15), the radion is massless.

To generate a radion mass, we can utilize the classic Goldberger-Wise
mechanism Goldberger and Wise (1999) where a 5D scalar field $\Phi$ with bulk
mass $m_{\Phi}$ is introduced.222 For another mechanism of radion
stabilization, see ref. Fujikura  _et al._ (2020).  It feels brane localized
potentials,

$\displaystyle S_{\Phi}=$ $\displaystyle\int
d^{4}xd\theta\bigg{[}\frac{\sqrt{G}}{2}\left(G^{MN}\partial_{M}\Phi\partial_{N}\Phi-
m_{\Phi}^{2}\Phi^{2}\right)$
$\displaystyle-\sqrt{-g_{h}}\lambda_{h}(\Phi^{2}-v_{h}^{2})^{2}-\sqrt{-g_{v}}\lambda_{v}(\Phi^{2}-v_{v}^{2})^{2}\bigg{]}\
.$ (16)

Here, $g_{h}$ ($g_{v}$) is the induced metric at the UV (IR) brane, and the
mass dimensions for $\Phi,v_{v},v_{h}$ are $3/2$, whereas $\lambda_{h,v}$
carry mass dimension $-2$. When $\lambda_{v,h}$ are sufficiently large, it is
energetically favorable for $\Phi$ to have a fixed VEV $v_{h}$ ($v_{v}$) at
the UV (IR) brane. The solution for the classical equation of motion for
$\Phi$ is then given by

$\Phi(\theta)=Ae^{(4+\epsilon)kT|\theta|}+Be^{-\epsilon kT|\theta|}\ ,$ (17)

where $\epsilon\equiv\sqrt{4+m_{\Phi}^{2}/k^{2}}-2\ll 1$ is the parameter that
controls the explicit breaking of the conformal invariance due to the bulk
scalar mass, and the coefficients $A,B$ are determined from the VEVs for
$\Phi$ at the branes. Plugging Eq. (17) back into the action (II.2) and
integrating out the extra dimension, one gets the effective radion potential,
namely

$V_{\rm eff}(\phi)\simeq
k\left(\frac{\phi}{Z}\right)^{4}v_{v}^{2}\Big{[}(4+2\epsilon)\bigg{\\{}1-\frac{v_{h}}{v_{v}}\left(\frac{\phi}{Z}\right)^{\epsilon}\bigg{\\}}^{2}-\epsilon\Big{]}\
.$ (18)

The potential has a global minimum at

$\frac{\langle\phi\rangle}{Z}=\bigg{[}\frac{v_{h}}{v_{v}}X_{\rm
min}\bigg{]}^{1/\epsilon}\ ,$ (19)

where

$\displaystyle X_{\rm min}$
$\displaystyle=\frac{1}{4+2\epsilon}\Big{[}4+{\epsilon}+\sqrt{\epsilon\left(4+\epsilon\right)}\Big{]}$
$\displaystyle=1+\frac{\sqrt{\epsilon}}{2}-\frac{\epsilon}{4}+{\cal
O}(\epsilon^{3/2})\ .$ (20)

Therefore, the radion mass is given by

$\displaystyle m_{\phi}^{2}$ $\displaystyle=\frac{\partial^{2}V_{\rm
eff}(\phi)}{\partial\phi^{2}}\bigg{|}_{\langle\phi\rangle}$
$\displaystyle=\frac{2v_{v}^{2}}{3M_{*}^{3}}M_{\rm
IR}^{2}\epsilon^{3/2}\left(1+\frac{\sqrt{\epsilon}}{2}\right)+{\cal
O}(\epsilon^{5/2})\ .$ (21)

Here, we have defined the IR mass scale $M_{\rm IR}\equiv ke^{-\pi k\langle
T\rangle}$ with the modulus VEV $\langle T\rangle$ obtained from Eq. (19).
Here, $f_{\phi}\simeq\sqrt{6}M_{\rm IR}$ Csaki  _et al._ (2001).

For a more rigorous treatment, ref. Csaki  _et al._ (2001) solved the coupled
equations of the metric fluctuations and bulk scalar stabilizing the system
for a general class of potentials, and concluded that the radion mass has a
similar form as the naive approximation (21) except that it scales as
$\epsilon^{2}$. Then, the radion mass is suppressed by the small parameter
$\epsilon v_{v}/M_{*}^{3/2}$ compared to the IR mass scale. Concretely, for
$k\simeq M_{\rm Pl}$, with the correct scaling for $\epsilon$, one obtains

$m_{\phi}\simeq 1\ {\rm
GeV}\left(\frac{\epsilon}{1/40}\right)\left(\frac{v_{v}/M_{*}^{3/2}}{0.07}\right)\left(\frac{M_{\rm
IR}}{1\ {\rm TeV}}\right)\ ,$ (22)

where to keep $M_{\rm IR}$ fixed, $v_{h}/v_{v}$ is chosen to be $2.4$ 333See
refs. Coradeschi  _et al._ (2013); Bellazzini  _et al._ (2014); Abu-Ajamieh
_et al._ (2018) for mechanisms that can generate an even smaller dilaton mass
naturally..

###  II.3 Dilaton contribution to muon $g-2$

The anomalous magnetic moment of the muon ($a_{\mu}\equiv{(g_{\mu}-2)}/{2}$)
has been measured precisely by BNL E821 experiment Bennett  _et al._ (2006),
and improved by Fermilab Aguillard  _et al._ (2023) with a precision of $0.20$
parts per million. Therefore, it serves as a precision probe beyond the SM.
Comparison of the SM theory inferred value is in tension with the experimental
value obtained, although precise determination of SM hadronic vacuum
polarization contribution from lattice calculation remains an important
pursuit. Evidently, as the dilaton has a coupling to the muon, the one-loop
diagram has a contribution to the anomalous magnetic moment of the muon
$a_{\mu}$. To the leading order (LO),

$\displaystyle\Delta
a_{\mu}={\displaystyle\frac{c_{\phi\mu\mu}^{2}m_{\mu}^{2}}{8\pi^{2}f_{\phi}^{2}}r_{\phi}^{2}\int_{0}^{1}\\!dx\frac{x^{2}\left(2-x\right)}{r_{\phi}x^{2}-x+1}}\,,$
(23)

where $r_{\phi}\equiv m_{\mu}^{2}/m_{\phi}^{2}$. The current discrepancy of
$\Delta a_{\mu}\equiv a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}$ is Bennett  _et al._
(2006); Keshavarzi  _et al._ (2018); Aoyama  _et al._ (2020); Abi  _et al._
(2021); Aguillard  _et al._ (2023)

$\displaystyle\Delta a_{\mu}=\left(2.49\pm 0.48\right)\times 10^{-9}\ .$ (24)

Here we have used the theoretical calculation of $a_{\mu}$ from the 2020 white
paper Aoyama  _et al._ (2020), and for the experimental value, we take the
world average including the recently released FNAL Run-2 and Run-3 data
Aguillard  _et al._ (2023). Note that the additional contribution due to a
light dilaton, thanks to its CP-property, has the desirable sign, whereas the
LO contribution of the axion has the opposite sign (see ref. Marciano  _et
al._ (2016) for NLO contributions). Motivated by this, it has been argued that
a light radion coupling to muons can alleviate the muon $g-2$ anomaly Abu-
Ajamieh  _et al._ (2018). However, this has to be confronted with stringent
laboratory constraints, especially coming from rare meson decays, which will
be elucidated in the next section.

##  III The dilaton parameter space

In this section, we consider the experimental and astrophysical constraints on
the parameter space of the dilaton. Most stringent constraints come from
searches for decays of $K$ and $B$ mesons, while for a larger dilaton mass
collider constraints become relevant. We will comment on bounds from BBN and
supernovae 1987a. As alluded to earlier, although the dilaton has a similar
coupling to the SM as a Higgs-portal scalar, there are a few important
differences that affect its phenomenology which we elaborate on here. Our
results are shown in Fig. 2.

Figure 1: 1PI diagrams mediating $b\to s\phi$ in the unitary gauge.

###  III.1 Constraints from rare meson decays

Figure 2: Constraints from rare meson decays, collider, cosmology, and
astrophysics on the parameter space of a light dilaton. The cyan region
represents the KTeV bound on $K_{\rm L}\to\pi^{0}\phi(\to\gamma\gamma)$
Abouzaid  _et al._ (2008), while $K_{\rm L}\to\pi^{0}\phi(\to e^{+}e^{-})$
Alavi-Harati  _et al._ (2004) constraint is depicted in orange for $c_{\phi
ee}=0.1,1$ respectively. The darker yellow region corresponds to NA62 bound on
the mode $K^{+}\to\pi^{+}+{\rm inv}$ Cortina Gil  _et al._ (2021) (where
$\phi$ escapes the detector with an effective invisible signature). The
dilaton-photon coupling is calculated from Eq. (8) with $\Lambda_{\rm
UV}=10^{10}$ GeV. The black-shaded region denotes inclusive bounds from $b\to
s\phi$ decay. The purple and brown dashed regions stand for bounds from BBN
and supernova SN1987a respectively. The red region shows the constraint from
semileptonic $B\to K\phi(\to\mu^{+}\mu^{-})$ decay from LHCb Aaij  _et al._
(2017), while the blue region corresponds to exclusion from LEP searches for
$e^{+}e^{-}\to Z^{*}\phi$ Acciarri  _et al._ (1996). Smaller $c_{\phi\mu\mu}$
can reduce $\phi\to\mu^{+}\mu^{-}$ branching ratio as it does not affect
hadronic decay channels, easing the semileptonic $B$ meson decay constraint.
In the green band, the dilaton contribution to the muon $g-2$ can bring
$\Delta a_{\mu}$ within $1\sigma$ deviation from the experiment. Note that the
enhanced coupling to photons for the light dilaton plays a crucial role in
opening up a window of parameters below the muon threshold region.

If the mass of $\phi$ is less than the threshold where the $B\to K\phi$ decay
with on-shell $\phi$ is allowed, we need to take into account constraints from
the $B$ decay processes. In Fig. 1, we show the relevant one-loop diagrams to
the $b\to s$ transition with the dilaton $\phi$. These contributions can be
summarized in the following effective couplings:

$\displaystyle\mathcal{L}_{{\rm eff},\phi sb}\supset$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}V_{u_{i}b}V_{u_{i}s}^{*}\Bigl{(}g_{R}^{B}(q^{2})\bar{s}_{L}b_{R}\phi+g_{L}^{B}(q^{2})\bar{s}_{R}b_{L}\phi\Bigr{)}\,$
$\displaystyle+{\rm h.c.},$ (25)

where $G_{F}$ is the Fermi constant, $V_{ij}$ is an element of the Cabibbo-
Kobayashi-Maskawa (CKM) matrix, $q$ is the four-momentum of $\phi$, and hence,
$q^{2}=m_{\phi}^{2}$ for the on-shell $\phi$. Note that due to the proper
chirality flipping on the fermion line, $g_{R}^{B}(q^{2})$ is expected to be
larger than $g_{L}^{B}(q^{2})$ by a factor of $m_{b}/m_{s}\approx 50$.
Therefore, hereafter, we focus on the coupling $g_{R}^{B}(q^{2})$ only.

The calculation for the $g_{R,L}^{B}(q^{2})$ is well-known Batell  _et al._
(2011); Winkler (2019) (see Kachanovich  _et al._ (2020) for a general
$R_{\xi}$ gauge calculation). With $c_{\phi qq}=1$, and on-shell $\phi$, one
obtains

$\displaystyle g_{R}^{B}=\frac{3m_{b}m_{t}^{2}}{32\pi^{2}f_{\phi}}\,.$ (26)

Using this coupling, we can estimate the branching ratio for $B\to K\phi$. For
this purpose, we need to calculate the hadronic matrix elements $\langle
K|\bar{s}b|B\rangle$, $\langle K|\bar{s}\gamma^{5}b|B\rangle$. From ref. Ball
and Zwicky (2005), we get

$\displaystyle\langle K|\bar{s}b|B\rangle$
$\displaystyle=\frac{M_{B}^{2}-M_{K}^{2}}{m_{b}-m_{s}}f_{0}^{K}(q^{2})\,,$
(27) $\displaystyle\langle K|\bar{s}\gamma^{5}b|B\rangle$ $\displaystyle=0\,,$
(28)

where $f_{0}^{K}(q^{2})=\frac{0.33}{1-q^{2}/(37.5\ {\rm GeV}^{2})}$ is the
form factor for the $B\to K$ transition Ball and Zwicky (2005). Therefore, we
have the following branching ratio for $B\to K\phi$ process:

$\displaystyle{\rm BR}(B\to K\phi)=$
$\displaystyle\frac{\left|g_{R}^{B}\right|^{2}}{64\pi\Gamma_{B}}\left(\frac{4G_{F}}{\sqrt{2}}\right)^{2}\left|V_{u_{i}b}V_{u_{i}s}^{*}\right|^{2}$
$\displaystyle\times\left(\frac{M_{B}^{2}-M_{K}^{2}}{m_{b}-m_{s}}\right)^{2}\frac{\lambda_{B\to
K\phi}^{1/2}}{M_{B}^{3}}\left[f_{0}^{K}(m_{\phi}^{2})\right]^{2}\,,$ (29)

where $\lambda_{X\to
YZ}=M_{X}^{4}+M_{Y}^{4}+m_{Z}^{4}-2M_{X}^{2}M_{Y}^{2}-2M_{Y}^{2}m_{Z}^{2}-2m_{Z}^{2}M_{X}^{2}$,
and $\Gamma_{B}$ is the total decay width of $B$ meson.

In this paper, we consider the inclusive bound on the coupling $g_{R}^{B}$.
For $b\to s\phi$ decays, the dominant decay mode is $B^{+}\to\bar{c}X$ whose
branching ratio is $97\pm 4\%$. Therefore, we can obtain the inclusive upper
bound on any decay mode associated with $b\to s\phi$ to be

$\displaystyle{\rm BR}(b\to s\phi)<1-{\rm BR}(B^{+}\to\bar{c}X)\lesssim
11\%\,.$ (30)

The above bound is independent of the dilaton decay modes. However,
considering branching ratios similar to a Higgs-portal scalar, a more
stringent bound can be obtained from $\phi$ mediated semileptonic decay
channels for the $B$ meson, such as $B\to K+\mu^{+}\mu^{-}$ from LHCb Aaij
_et al._ (2017). Following ref. Winkler (2019), this constraint is depicted as
the red-shaded region in Fig. 2, while the masked regions correspond to
charmonium resonances. We reiterate that this bound is dependent on
$BR(\phi\to\mu^{+}\mu^{-})$, and therefore model dependent. For example, if
$c_{\phi\mu\mu}\ll 1$, then this branching ratio is suppressed as
$c_{\phi\mu\mu}^{2}$ as the hadronic decay channels do not depend on
$c_{\phi\mu\mu}$. Also, if in some specific model, the dilaton primarily
decays to a dark sector, then this bound is also relaxed. Note that, even
these scenarios can not evade the inclusive $b\to s\phi$ bound we provide in
this work.

Similarly, we also have a constraint from the inclusive $s$ decay. The
diagrams relevant to $K$ decays are the same as in Fig. 1 by replacing initial
$b\to s$ and final $s\to d$. Then, the effective coupling can be estimated
from Eq. (25) as

$\displaystyle\mathcal{L}_{\rm
eff}\supset\frac{4G_{F}}{\sqrt{2}}V_{u_{i}s}V_{u_{i}d}^{*}g_{R}^{K}\bar{d}_{L}s_{R}\phi+{\rm
h.c.}\,,$ (31)

where we have omitted the $g_{L}^{K}$ term due to the suppression of
$m_{d}/m_{s}\approx 1/20$, as in the case for $g_{R,L}^{B}$. Using
$g_{R}^{K}$, the branching ratio for $K\to\pi\phi$ decay can be estimated in a
similar manner. For this decay mode, we also have an inclusive bound, and it
can be read as

$\displaystyle{\rm BR}(s\to d\phi)<\mathcal{O}(0.1)\%\,.$ (32)

However, this is a much weaker constraint as compared to the inclusive $b\to
s\phi$ bound, and therefore, we do not show it in Fig. 2.

Below the threshold mass for the dilaton to decay into two muons, depending on
$f_{\phi}$, the dilaton may decay into two photons, electron-positron pair, or
effectively have a decay length long enough such that it appears as an
invisible mode in a particular experiment. In this mass range, for the
invisible decay signature, the most stringent bound comes from the NA62
experiment which looks for $K^{+}\to\pi^{+}\nu\bar{\nu}$. For a vanilla Higgs-
portal scalar, this NA62 bound Cortina Gil  _et al._ (2021) is the strongest
till the threshold $m_{K}-m_{\pi}$. However, the dilaton can have an enhanced
coupling to photons due to CFT loops, and therefore, can have a decay length
much shorter compared to the Higgs-portal scalar. To illustrate, given the
length $L_{\rm exp}$, and typical energy $E_{\rm exp}$ of a particular
experiment, the fraction of $\phi$ surviving after traversing a length $L_{\rm
exp}$ is

$\displaystyle f_{\rm sur,exp}$ $\displaystyle=e^{-\frac{L_{\rm
exp}}{c\tau_{\phi}\beta_{\phi}\gamma_{\phi}}}\ ,$ (33)

where $\tau_{\phi}$ is the lifetime of $\phi$, and the boost factor
$\beta_{\phi}\gamma_{\phi}\sim E_{\rm exp}/m_{\phi}$. Therefore, the NA62
bound on the branching ratio is translated into the parameter space of the
dilaton taking into account the exponential tail. Specifically, we demand that

${\rm BR}(K\to\pi\phi)\ f_{\rm sur,NA62}<{\rm BR}(K\to\pi+{\rm inv.})_{\rm
NA62}\ ,$ (34)

where the left hand side is calculated from the dilaton interaction Lagrangian
and the right hand side is the experimental constraint taken from the NA62
experiment Cortina Gil  _et al._ (2021). We used the detector length $L_{\rm
NA62}=150$ m, and the energy relevant $E_{\rm NA62}=30$ GeV. In Fig. 2 this is
depicted as a dark yellow region. Note that, for a sufficiently low
$f_{\phi}$, the decay signature of $\phi$ is visible and the NA62 bound is
lifted. This is in sharp contrast to the Higgs-portal case.

On the other hand, if $\phi$ decays visibly, then experimental bounds from
$K_{L}\to\pi^{0}\gamma\gamma$ and $K_{L}\to\pi^{0}e^{+}e^{-}$ need to be
satisfied. We take the KTeV bounds for these modes Alavi-Harati  _et al._
(2004); Abouzaid  _et al._ (2008). As the dilaton is CP even, these bounds are
applied without any CP violating suppression factor as opposed to axions. In
particular, although $K_{L}\to\pi^{0}\gamma\gamma$ leaves a viable window
below the muon production threshold, the stringent $K_{L}\to\pi^{0}e^{+}e^{-}$
constraint nearly closes that window for $c_{\phi ee}=1$. For the KTeV
experiment, we used $L_{\rm KTeV}=1$ m, while the energy relevant was set to
$E_{\rm KTeV}=40$ GeV. It is evident that these visible modes are
complementary to the invisible modes, and cover the opposite direction in the
$f_{\phi}$ axis as a larger $f_{\phi}$ would relax the NA62 bound, but the
KTeV bounds will become more severe in that case. For mapping the KTeV
constraint, we convoluted $K\to\pi\phi$ branching ratio from the theory with
BR$(\phi\to\gamma\gamma)(1-f_{\rm sur,KTeV})$, and BR$(\phi\to
e^{+}e^{-})(1-f_{\rm sur,KTeV})$ for the aforementioned modes, respectively.

Note, however, that depending on the composite nature of the electron in the
UV theory, this is expected to be different from 1, as alluded to in section
II. To be concrete, in the warped extra dimension scenario, this depends on
where the electron is localized, and its coupling to the radion/dilaton can be
easily suppressed by localizing it towards the UV brane Csáki  _et al._
(2021). Therefore, we also show a contour for $c_{\phi ee}=0.1$ in Fig. 2,
which revives the low-mass window. We will choose a benchmark point in this
region to illustrate a technique of model-independent CP property extraction
in the next section.

###  III.2 Collider bounds

The primary collider bound on the dilaton comes from LEP searches for light
Higgs-like particles in the channel $e^{+}e^{-}\to Z^{*}\phi$. The bound
assumes that the $\phi$ coupling is strictly proportional to the SM Higgs
coupling, and hence below the threshold to produce two muons, constraints from
invisible mode searches at L3 are applied. Then, if the model dependence of
the exactly Higgs-like coupling to the muon is relaxed, still the bound from
the invisible mode will be applied Acciarri  _et al._ (1996). Similar bounds
from CMS and LHCb searches above the $B$-meson mass are weaker than the LEP
bound.

A possible search for $e^{+}e^{-}\to\gamma(\phi\to\tau^{+}\tau^{-})$ can be
performed above the $\tau$ production threshold at Belle II. However, the $Z$
boson enjoys a tree-level coupling to the dilaton that is proportional to its
mass, and although the increase in luminosity at Belle II somewhat compensates
for that, the inferred bounds are comparable to the existing LEP bound, at
least in our estimated analysis.

###  III.3 Bounds from astrophysics and cosmology

The parameter space of the light dilaton also receives constraints from the
early Universe cosmology. In particular, if the lifetime of the dilaton is
larger than about $0.1$s, it will spoil the success of the predicted light
element abundances from the BBN due to extra energy injection in the thermal
bath. We utilize ref. Fradette and Pospelov (2017) for the constraint on the
lifetime, together with the dilaton couplings (5), (8) to infer bounds on the
parameter space of the dilaton as shown in Fig. 2. The enhanced coupling to
the photon makes the BBN constraint relaxed compared with the Higgs-portal
scalar case Winkler (2019). While the assumption of a comparable branching
ratio to a Higgs-portal scalar is made, it is expected to give only a marginal
effect on the lifetime bound.

If the dilaton is produced on-shell in a supernova explosion, it may carry
away significant energy which may shorten the observed duration of the
neutrino pulse emission. Demanding an upper bound on the energy loss rate due
to the dilaton-nucleon scattering, the constraint can be inferred from
SN1987a. We follow ref. Krnjaic (2016); Ishizuka and Yoshimura (1990) for the
calculation of the energy loss rate per unit volume $Q_{\phi}$, and demand
that

$P_{\rm esc,SN}\ Q_{\phi}V_{\rm SN}\lesssim 10^{53}\ {\rm erg/s}\ ,$ (35)

where $V_{\rm SN}=(4\pi/3)R_{\rm SN}^{3}$ is the supernova volume, $R_{\rm
SN}\sim 10$ km being its radius. $P_{\rm esc,SN}$ denotes the escape
probability for the dilaton without decaying or getting reabsorbed inside the
supernova. The information of the dilaton lifetime enters into the escape
probability as

$P_{\rm esc,SN}\simeq e^{-R_{\rm
SN}/\gamma_{\phi}\beta_{\phi}c\tau_{\phi}}e^{-R_{\rm SN}Q_{\phi}/\rho_{\phi}}\
,$ (36)

where $\gamma_{\phi}\beta_{\phi}$ is the boost factor as before and
$\rho_{\phi}$ is the equilibrium energy density for the dilaton at the
relevant temperature $T_{\rm SN}\sim 30$ MeV. Our inferred constraint is shown
as the brown dashed region in Fig. 2 as this constraint is only estimated at
the order of magnitude level. This constraint resembles the Higgs-portal
scalar because, for the relevant parameter space, the small coupling makes
both decay lengths much bigger than the supernova radius.

Finally, let us comment on the light radion/dilaton explanation of the muon
$g-2$ anomaly. The green band in Fig. 2 represents the parameter space where
the dilaton contribution to the muon $g-2$ makes $\Delta a_{\mu}$ within
$1\sigma$ deviation from the experimentally measured value. However, this
region is excluded from the model-independent $b\to s\phi$ constraint shown as
gray shaded region in Fig. 2 together with the LEP bound in the higher mass
window. A possible caveat for the LEP bound is that here the dilaton branching
ratio is considered exactly similar to the SM Higgs with the same mass,
although marginal modification is expected.

##  IV Extraction of CP property

(a)  (b)

Figure 3: (a) Feynman diagrams for the process $e^{+}e^{-}\to f\bar{f}\phi$
mediated by a virtual photon. The $s$-channel diagram (left) is common for all
$f$, while the $t$-channel (right) contributes for $f=e$ only. (b) a schematic
diagram for the momentum of outgoing particles. We choose an $e^{+}e^{-}$ beam
line to be along with the $z$-axis, while the dotted line indicates an off-
shell photon $\gamma^{*}$. The azimuthal angle between the outgoing fermions
is denoted as $\Delta\varphi$.

Motivated by the enhanced photon coupling to the dilaton, let us now proceed
one step further in extracting the CP property of a prospective signal due to
a light dilaton-mediated process at a lepton collider like Belle II. In
particular, we will illustrate how to distinguish its signature from a CP-odd
axion-like particle (ALP). For a comprehensive exploration of ALPs in the
context of rare meson decays see Ref. Lees  _et al._ (2022). Our method has
the merit that it can be applied in a broader scenario even when $\phi$
couples to some dark sector Dolan  _et al._ (2017) or decays effectively
outside the detector. Furthermore, the technique can be carried over to any
lepton collider machine, as our method is merely kinematical. Following in the
footsteps of the earlier works on extracting the CP property of the SM Higgs
before its discovery Plehn  _et al._ (2002), the strategy relies on the
imprints of the CP nature of $\phi$ in the variation of the differential
cross-section with the azimuthal angle between the outgoing fermions
$\Delta\varphi$.

To establish our notation for this part, let us denote the effective
interaction of the dilaton to the photon as

$\displaystyle\mathcal{L_{{\rm
eff},\phi\gamma}}=-\frac{g_{\phi\gamma\gamma}}{4}\phi F^{\mu\nu}F_{\mu\nu}\,,$
(37)

where $F$ is the electromagnetic field strength tensor. On the other hand, the
coupling of an ALP, or merely an axion, ($a$) to photons can be parametrized
as

$\displaystyle\mathcal{L}_{{\rm
eff},a\gamma}=-\frac{g_{a\gamma\gamma}}{4}aF^{\mu\nu}\widetilde{F}_{\mu\nu}\,,$
(38)

with the dual field strength
$\widetilde{F}_{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$.
For the following discussion, $\phi$ will generically denote both the CP-odd
and CP-even (pseudo)scalars.

###  IV.1 Imprints of CP in $e^{+}e^{-}\to f\bar{f}\phi$

We consider the dilaton production
$e^{+}(p_{1})e^{-}(p_{2})\to\phi(p_{3})f(p_{4})\bar{f}(p_{5})$ where $f$
denotes an SM charged lepton and the four-momentum assignments are shown in
appendix A. Our focus is on the outgoing lepton case as it is easier to
measure their momenta precisely as compared with jets. The Feynman diagram for
this process is depicted in Fig. 3a, where we use the coupling with the
photons, namely $g_{\phi\gamma\gamma}$ in Eq. (37), to produce $\phi$ and then
the off-shell photon can generate the outgoing lepton pair. The azimuthal
angle between the outgoing leptons is the angle of interest $\Delta\varphi$,
and is elucidated in the schematic Fig. 3b. Concretely,

${\widehat{{p_{4}}}_{T}\cdot\widehat{{p_{5}}}_{T}}=\cos(\Delta\varphi)\ ,$
(39)

where $\vec{p_{i}}_{T}$ is the transverse component of the momentum vector
with respect to the beam axis, and the overhat denotes a unit vector.
Manifestly, $\Delta\varphi$ is boost invariant along the beam axis. We note
that at a center of mass (COM) energy lower than the electroweak scale, as
applicable to Belle II, the photon-mediated process dominates over the one
mediated by the $Z$ boson. We will look at the variation of the normalized
differential cross section with $\Delta\varphi$.

For $f\neq e$, only the $s$-channel diagram contributes (left subfigure in
Fig. 3a), while for $f=e$ one also has to include the $t$-channel diagram
(right subfigure in Fig. 3a). The total amplitude for $f=e$ is therefore

$\displaystyle\overline{\left|\cal{M}\right|^{2}}=\overline{\left|\mathcal{M}_{s}+\mathcal{M}_{t}\right|^{2}}=\overline{\left|\mathcal{M}_{s}\right|^{2}}+2{\rm
Re}\overline{\mathcal{M}_{s}^{*}\mathcal{M}_{t}}+\overline{\left|\mathcal{M}_{t}\right|^{2}}\,,$
(40)

while for $f\neq e$,

$\displaystyle\overline{\left|\cal{M}\right|^{2}}=\overline{\left|\mathcal{M}_{s}\right|^{2}}\,.$
(41)

Here overline indicates the average and sum over the initial and final fermion
spins, respectively. We summarize the results for each channel and
interference term, while the details of the calculation including the phase
space integration are presented in appendix A.

(a)  (b)  (c)  (d)

Figure 4: The variation of the normalized differential cross section with
respect to azimuthal angle between the outgoing leptons $\Delta\varphi$ for
the electron in the final state with $m_{\phi}=150$ MeV (top left panel), $5$
GeV (top right panel) and $9$ GeV (bottom left panel) and the muon in the
final state with $m_{\phi}=150$ MeV (bottom right panel). The red solid (black
dot-dashed) curve represents the CP-even (CP-odd) case.

The contribution from the $s$-channel in Eq. (40) is

$\displaystyle\overline{\left|\mathcal{M}_{s}\right|^{2}}$
$\displaystyle\simeq\frac{32\pi^{3}\alpha_{\rm
em}^{2}\alpha_{\phi}}{p_{12}^{2}p_{45}^{2}}\Bigl{[}2\left(p_{14}p_{25}-p_{24}p_{15}\right)^{2}$
$\displaystyle+p_{12}p_{45}\left\\{\left(p_{14}-p_{25}\right)^{2}+\left(p_{24}-p_{15}\right)^{2}+2p_{12}p_{45}\right\\}\Bigr{]}\
,$ (42)

for CP-even $\phi$, and

$\displaystyle\overline{\left|\mathcal{M}_{s}\right|^{2}}$
$\displaystyle\simeq\frac{32\pi^{3}\alpha_{\rm
em}^{2}\alpha_{\phi}}{p_{12}^{2}p_{45}^{2}}\Bigl{[}-2\left(p_{14}p_{25}-p_{24}p_{15}\right)^{2}$
$\displaystyle+p_{12}p_{45}\left\\{\left(p_{14}+p_{25}\right)^{2}+\left(p_{24}+p_{15}\right)^{2}-2p_{12}p_{45}\right\\}\Bigr{]}\
,$ (43)

for CP-odd $\phi$. Here, $\alpha_{\rm em}$ denotes the fine-structure
constant, and we have defined $p_{ij}\equiv p_{i}\cdot p_{j}$ and
$\alpha_{\phi}\equiv g_{\phi\gamma\gamma}^{2}/(4\pi)$. Note that these
expressions are given in the limit of $m_{f}/{\sqrt{s}},\,m_{e}/{\sqrt{s}}\to
0$ for conciseness, while we take the full expressions without taking the
limit into account for our numerical calculations. $\sqrt{s}$ denotes the COM
energy of the collider concerned. The inner products in the denominator are
due to the off-shell photon propagators as usual,
$q_{1}^{2}=(p_{1}+p_{2})^{2}\simeq 2p_{1}\cdot p_{2}$ and
$q_{2}^{2}=(p_{4}+p_{5})^{2}\simeq 2p_{4}\cdot p_{5}$. Notice the differences
in signs for some of the terms in the above equations are a consequence of the
different Lorentz structure of the photon-(pseudo)scalar Lagrangian, which in
turn is connected to their CP property. The $t$-channel contribution in Eq.
(40) is

$\displaystyle\overline{\left|\mathcal{M}_{t}\right|^{2}}$
$\displaystyle\simeq\frac{32\pi^{3}\alpha_{\rm
em}^{2}\alpha_{\phi}}{p_{14}^{2}p_{25}^{2}}\Bigl{[}2\left(p_{12}p_{45}-p_{24}p_{15}\right)^{2}$
$\displaystyle+p_{14}p_{25}\left\\{\left(p_{12}-p_{45}\right)^{2}+\left(p_{24}-p_{15}\right)^{2}+2p_{14}p_{25}\right\\}\Bigr{]}\
,$ (44)

for CP-even $\phi$, and

$\displaystyle\overline{\left|\mathcal{M}_{t}\right|^{2}}$
$\displaystyle\simeq\frac{32\pi^{3}\alpha_{\rm
em}^{2}\alpha_{\phi}}{p_{14}^{2}p_{25}^{2}}\Bigl{[}-2\left(p_{12}p_{45}-p_{15}p_{24}\right)^{2}$
$\displaystyle+p_{14}p_{25}\left\\{\left(p_{12}+p_{45}\right)^{2}+\left(p_{24}+p_{15}\right)^{2}-2p_{14}p_{25}\right\\}\Bigr{]}\
,$ (45)

for CP-odd $\phi$. For this channel, the (inverse of) off-shell photon
propagators are $(q^{\prime}_{1})^{2}=(p_{1}-p_{4})^{2}\simeq-2p_{1}\cdot
p_{4}$ and $(q^{\prime}_{2})^{2}=(p_{2}-p_{5})^{2}\simeq-2p_{2}\cdot p_{5}$.
Note that these expressions can be obtained from the $s$-channel results by
replacing $p_{2}\leftrightarrow p_{4}$, namely, $p_{12}\leftrightarrow
p_{14}$, $p_{25}\leftrightarrow p_{45}$. The interference term is

$\displaystyle\overline{\mathcal{M}_{s}^{*}\mathcal{M}_{t}}$
$\displaystyle\simeq\frac{16\pi^{3}\alpha_{\rm
em}^{2}\alpha_{\phi}}{p_{12}p_{14}p_{25}p_{45}}\Bigl{[}2\left(p_{12}p_{45}-p_{14}p_{25}\right)^{2}+\left(p_{24}-p_{15}\right)^{2}$
$\displaystyle\times\left(p_{12}p_{45}+p_{14}p_{25}\right)-p_{15}p_{24}\left(p_{15}^{2}+p_{24}^{2}\right)\Bigr{]}\
,$ (46)

for CP-even $\phi$, and

$\displaystyle\overline{\mathcal{M}_{s}^{*}\mathcal{M}_{t}}$
$\displaystyle\simeq\frac{16\pi^{3}\alpha_{\rm
em}^{2}\alpha_{\phi}}{p_{12}p_{14}p_{25}p_{45}}\Bigl{[}-2\left(p_{12}p_{45}-p_{14}p_{25}\right)^{2}$
$\displaystyle+\left(p_{24}+p_{15}\right)^{2}\left(p_{12}p_{45}+p_{14}p_{25}\right)$
$\displaystyle-p_{15}p_{24}\left(p_{15}^{2}+p_{24}^{2}\right)\Bigr{]}\ ,$ (47)

for CP-odd $\phi$. We note that the $t$-channel cross section can be larger
than the $s$-channel one because we can take small $(q^{\prime}_{1,2})^{2}$
for two off-shell photons. On the other hand, for the $s$-channel process, we
cannot take small $q_{1}^{2}$ as it is determined by the COM energy.

For the purpose of illustrating the distinct CP property, let us choose a few
benchmark points from the allowed parameter space in Fig. 2. We take
$m_{\phi}=0.15,5,9$ GeV with $f_{\phi}=4$ TeV. For $\Lambda_{\rm UV}=10^{10}$
GeV, using Eq. (8), the inferred value of $g_{\phi\gamma\gamma}\sim 1.7\times
10^{-5}$ GeV-1, which also satisfies constraints on the (pseudo)scalar-photon
coupling Bauer  _et al._ (2022). This choice allows us to portray the behavior
with the dilaton mass for three distinct regions for a collider like Belle II
with $\sqrt{s}\sim 10$ GeV.

###  IV.2 Angular distribution of differential cross section

In Fig. 4, we show the result of the variation of the normalized differential
cross section with respect to $\Delta\varphi$.

(a)  (b)

Figure 5: Variation of the expected number of signal and principal background
events with respect to the azimuthal angle between the outgoing $e^{+}e^{-}$
pair for a representative benchmark of $m_{\phi}=5$ GeV, where we have assumed
the Belle II integrated luminosity as $50$ab-1. (a) We consider visible decay
mode $\phi\to\mu^{+}\mu^{-}$ and hence show the SM background $e^{+}e^{-}\to
e^{+}e^{-}\mu^{+}\mu^{-}$ (purple dashed), scaled down by $3\times 10^{-3}$. A
lower bound $m_{ee}>0.1$ GeV was employed in order to trim down the collinear
peak towards low $\Delta\varphi$. This background profile is flat with
$\Delta\varphi$, aiding with the discrimination of the CP-even (red solid)
from the CP-odd (black dot-dashed) cases. (b) We consider invisible decay mode
for $\phi$ with the same mass and show the SM background (gray dashed) from
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ where the taus decay to electrons. This is
peaked towards large $\Delta\varphi$ and we have scaled the background by
$3\times 10^{-4}$.

For the case with the electron in the final state, the distinct signatures of
CP-odd and CP-even cases are evident. Apart from this angular difference, the
total cross sections are similar for both cases. Numerically, for
$m_{\phi}\ll\sqrt{s}$, we find

$\displaystyle\sigma_{e^{+}e^{-}\to e^{+}e^{-}\phi}\simeq
50\times\left(\frac{g_{\phi\gamma\gamma}}{10^{-5}}\right)^{2}~{}{\rm ab}\,.$
(48)

Let us comment on the specific case of Belle II as an application. Their final
integrated luminosity goal is $50{\rm ab}^{-1}$, and hence, we can expect
$\sim 2500$ events in total. It should be emphasized that for the Higgs portal
case, with the same given $f_{\phi}$, we get $g_{\phi\gamma\gamma}\sim
10^{-6}$ GeV-1 Marciano  _et al._ (2012), and expect $\sim 25$ events, which
is two orders of magnitude below the number of events for the dilaton case,
and not enough for extracting the CP property.

The qualitative behavior of Fig. 4 can be understood as follows. For the CP-
odd pseudoscalar, the amplitude is proportional to the Levi-Civita tensor, and
therefore, four distinct momenta are required to contract all Lorentz
indices.444 Although the squared amplitude has $g^{\mu\nu}$ term as well as
$p_{i}^{\mu}p_{j}^{\nu}$, its contribution is smaller than the
$p_{i}^{\mu}p_{j}^{\nu}$ terms with $\Delta\varphi\neq 0,\pi$. This is evident
in Fig. 4. Therefore, the amplitude will diminish if the transverse components
of $p_{4}$ and $p_{5}$ are parallel, which is realized with $\Delta\varphi=0$
or $\pi$. For the CP-even scalar, the inner product of $p_{4}\cdot p_{5}$ has
an important role in the behavior. If we focus on the inner product of
transverse components, we have

$\displaystyle\vec{p_{4}}_{T}\cdot\vec{p_{5}}_{T}=\sqrt{E_{4}^{2}-m_{f}^{2}}\sqrt{E_{5}^{2}-m_{f}^{2}}s_{\theta_{f}}s_{\theta_{\bar{f}}}c_{\Delta\varphi}\,,$
(49)

and hence the squared amplitude will decrease around $\Delta\varphi=\pi/2$.
Here $E_{4}$, $E_{5}$, $\theta_{f}$, $\theta_{\bar{f}}$ are the energies and
polar angles of the outgoing fermions, and are illustrated in Fig. 3b. See
appendix A.1 for further details. Our results agree with ref. Plehn  _et al._
(2002) qualitatively.

From Fig. 4 it is clear that the relative difference of the CP-odd vs. CP-even
cases decreases with an increase in $m_{\phi}$ due to the reduction of the
available phase space. Besides, the total cross section also decreases

$\displaystyle\sigma_{e^{+}e^{-}\to
e^{+}e^{-}\phi}\simeq\begin{cases}{\displaystyle
6\times\left(\frac{g_{\phi\gamma\gamma}}{10^{-5}}\right)^{2}~{}{\rm
ab}}&\text{($m_{\phi}=5$ GeV)}\,,\\\\[8.61108pt] {\displaystyle
0.4\times\left(\frac{g_{\phi\gamma\gamma}}{10^{-5}}\right)^{2}~{}{\rm
ab}}&\text{($m_{\phi}=9$ GeV)}\,.\end{cases}$ (50)

Therefore, it is easier to distinguish the CP property for the lighter mass.

For nearly all the range of $\Delta\varphi$, the differential cross section is
dominated by the $t$-channel contribution, due to the fact that both (the
inverse of) propagators of the virtual photon can be small,
$(q^{\prime}_{1,2})^{2}\sim 0$. For $s$-channel, on the other hand, only
$q_{2}^{2}$ can be small for $\Delta\varphi\sim 0$. We can check this directly
as

$\displaystyle q_{2}^{2}$ $\displaystyle=p_{4}^{2}+2p_{4}\cdot
p_{5}+p_{5}^{2}$
$\displaystyle=2\left[m_{e}^{2}+E_{4}E_{5}-|\vec{p_{4}}||\vec{p_{5}}|\left(c_{\theta_{f}}^{2}+s_{\theta_{f}}^{2}c_{\Delta\varphi}\right)\right]$
$\displaystyle\to
2\left(m_{e}^{2}+E_{4}E_{5}-|\vec{p_{4}}||\vec{p_{5}}|\right)\quad(\Delta\varphi\to
0)\,.$ (51)

If $|\vec{p_{4}}||\vec{p_{5}}|\approx E_{4}E_{5}$, we obtain $q_{2}^{2}\approx
2m_{e}^{2}\ll 2E^{2}$, and hence, the $s$-channel contribution is enhanced.

Next, we discuss about the process $e^{+}e^{-}\to\phi\mu^{+}\mu^{-}$. The
result is shown in the bottom right panel of Fig. 4. Here, we set
$m_{\phi}=150$ MeV with a fixed $f_{\phi}$. It is found that for the range of
$\pi/2\leq\Delta\varphi<\pi$, the distributions for the CP-even and CP-odd
cases are slightly different: increasing for the CP-even case, while
decreasing for the CP-odd case. However, this might not be enough to
distinguish the CP property of $\phi$, compared with the electron case. This
is because, for the process with the muon in the final state, we have
$s$-channel only, and the total cross section is much smaller than the
electron case, due to no enhancement from $t$-channel. We obtain

$\displaystyle\sigma_{e^{+}e^{-}\to\mu^{+}\mu^{-}\phi}\simeq
0.02\times\left(\frac{g_{\phi\gamma\gamma}}{10^{-5}}\right)^{2}~{}{\rm ab}\,\
\ (m_{\phi}=0.15\ {\rm GeV})\,,$ (52)

both for the CP-even and CP-odd cases. Therefore, we conclude that
$e^{+}e^{-}\to\phi e^{+}e^{-}$ will be the ideal process for our kinematical
method to distinguish the CP property.

A detailed analysis with the Belle II detector setup is left for a future
study while we comment on the possible background for the $e^{+}e^{-}\phi$
mode briefly. The dilaton heavier than the dimuon threshold can decay to
dimuon, and then we consider the $e^{+}e^{-}\mu^{+}\mu^{-}$ background for
this signal with MadGraph 5 Alwall  _et al._ (2014). After removing the
collinear peak by placing a lower bound on the $e^{+}e^{-}$ invariant mass,
$m_{ee}>0.1$ GeV, we consider the bin size of 80 MeV for the dimuon invariant
mass inferred by the detector resolution of the signal peak. The expected
yield of the background is 2$\times 10^{4}$, which is not significant even
before the detailed selections. Furthermore, the trimmed resulting background
distribution is flat with respect to $\Delta\varphi$, which is distinct from
the signal distribution. This background is shown as the purple dashed line in
Fig. 5a, where we have scaled it by $3\times 10^{-3}$. Therefore, the
difference between CP-odd and CP-even particles is expected to be seen in this
channel.

When the dilaton is effectively invisible, the potential background arises
either from undetected photons or from the production of a tau lepton pair
that decays to $e^{+}e^{-}$, associated with neutrinos. In a benchmark with
the lowest mass $m_{\phi}=150$ MeV, the background estimate studied for the
muonic force mediator can be applied Jho  _et al._ (2019). Assuming the
background structure of $\mu^{+}\mu^{-}+$inv is similar for $e^{+}e^{-}+$inv,
almost all the background can be removed. For the higher mass $m_{\phi}=5,9$
GeV, the dilaton signature becomes invisible if it dominantly decays into some
dark sector particles. We find that the undetected photon background that
peaks at the zero missing mass is negligible, but the tau lepton background is
sizable, particularly in large $\Delta\varphi$, based on MadGraph 5. The
background is selected to have missing mass around the dilaton mass up to the
detector resolution of 560(110) MeV for $m_{\phi}=5(9)$ GeV, and the
background shape is included in Fig. 5b as the gray dashed line, where we have
scaled it by $3\times 10^{-4}$. Dedicated analysis, like reconstructing the
tau lepton pair using the collinear approximation, could reduce this type of
background.

##  V Conclusions

The existence of approximate scale invariance with dynamical spontaneous
breaking is a promising candidate of physics beyond the SM to address the
electroweak naturalness problem, and the dilaton is the key probe into such
scenario for a low-energy observer. In the present paper, we have analyzed the
constraints on a light dilaton with a mass spanning the MeV-GeV range, imposed
by experimental data primarily from rare meson decays. We provided a new
inclusive bound from the $b\to s\phi$ transition. This bound, together with
the collider bounds in the higher mass window, elucidates the status of the
light dilaton explanation of the muon $g-2$ anomaly. It has been shown that
despite the parallels between the dilaton and a Higgs-portal scalar, the
dilaton coupling to the photon is substantially enhanced due to the
involvement of loops from the conformal sector. Consequently, the dilaton’s
reduced lifetime partly mitigates constraints from searches for $K\to\pi$ \+
invisible processes at the NA62 experiment and from considerations related to
BBN. By capitalizing on this distinctive aspect, we have developed a strategy
for extracting the CP property of the dilaton at a lepton collider, such as
the ongoing Belle II experiment, using the variation of the differential
cross-section of $e^{+}e^{-}\to e^{+}e^{-}\phi$ with the azimuthal angle
between the outgoing leptons.

## Acknowledgements

YN is supported by the Natural Science Foundation of China under grant No.
12150610465. KT is supported by in part the US Department of Energy grant DE-
SC0010102 and JSPS Grant-in-Aid for Scientific Research (Grant No. 21H01086).

##  Appendix A Details of the calculations

###  A.1 Kinematics

In the discussion of section IV, we assign the four momenta as follows:

$\displaystyle e^{-}$ $\displaystyle:p_{1}=(E,0,0,p_{z})\,,$ (53)
$\displaystyle e^{+}$ $\displaystyle:p_{2}=(E,0,0,-p_{z})\,,$ (54)
$\displaystyle f$
$\displaystyle:p_{4}=(E_{4},|\vec{p_{4}}|s_{\theta_{f}}c_{\varphi_{f}},|\vec{p_{4}}|s_{\theta_{f}}s_{\varphi_{f}},|\vec{p_{4}}|c_{\theta_{f}})\,,$
(55) $\displaystyle\bar{f}$
$\displaystyle:p_{5}=(E_{5},|\vec{p_{5}}|s_{\theta_{\bar{f}}}c_{\varphi_{f}+\Delta\varphi},|\vec{p_{5}}|s_{\theta_{\bar{f}}}s_{\varphi_{f}+\Delta\varphi},|\vec{p_{5}}|c_{\theta_{\bar{f}}})\,,$
(56)

where we consider the center of mass (COM) frame of $e^{+}e^{-}$. Here, we use
$s_{\alpha}\equiv\sin\alpha$ and $c_{\alpha}\equiv\cos\alpha$. For the azimuth
angle of $p_{5}$, we define $\Delta\varphi$ as a deviation from $\varphi_{f}$.
Note that we set the $z$-axis along with the $e^{+}e^{-}$ beam line, and use
the four-momentum conservation to parameterize the momentum of $\phi$, which
is denoted as $p_{3}$, by the other parameters:

$\displaystyle p_{3}^{\mu}=p_{1}^{\mu}+p_{2}^{\mu}-p_{4}^{\mu}-p_{5}^{\mu}\,.$
(57)

The parameterizations of angles in $p_{4}$ and $p_{5}$ are defined for later
convenience. The other constraints come from the $p^{2}=m^{2}$ relations,
which are

$\displaystyle(p_{1})^{2}=m_{e}^{2}~{}~{}$
$\displaystyle\Rightarrow~{}~{}p_{z}=\sqrt{E^{2}-m_{e}^{2}}\approx E\,,$ (58)
$\displaystyle(p_{4})^{2}=m_{f}^{2}~{}~{}$
$\displaystyle\Rightarrow~{}~{}|\vec{p}_{4}|=\sqrt{E_{4}^{2}-m_{f}^{2}}\,,$
(59) $\displaystyle(p_{5})^{2}=m_{f}^{2}~{}~{}$
$\displaystyle\Rightarrow~{}~{}|\vec{p}_{5}|=\sqrt{E_{5}^{2}-m_{f}^{2}}\,,$
(60)

where we take a limit of $m_{e}^{2}/E^{2}\to 0$. Note that for $m_{f}$, it is
not trivial to take a limit of $m_{f}^{2}/E_{4,5}^{2}\to 0$, because $E_{4,5}$
can be as small as the $m_{f}$ scale. Still, the limit $m_{f}^{2}/E^{2}\to 0$
can be justified for $f=e,\mu$ and other light fermions.

Now, we can calculate all the inner products straightforwardly, and the
results can be read as

$\displaystyle p_{1}\cdot p_{2}$ $\displaystyle=2E^{2}-m_{e}^{2}\approx
2E^{2}\,,$ (61) $\displaystyle p_{1}\cdot p_{3}$ $\displaystyle\approx
E\Big{(}2E-E_{4}-E_{5}+\sqrt{E_{4}^{2}-m_{f}^{2}}c_{\theta_{f}}$
$\displaystyle{\hskip
113.81102pt}+\sqrt{E_{5}^{2}-m_{f}^{2}}c_{\theta_{\bar{f}}}\Big{)}\,,$ (62)
$\displaystyle p_{1}\cdot p_{4}$ $\displaystyle\approx
E\left(E_{4}-\sqrt{E_{4}^{2}-m_{f}^{2}}c_{\theta_{f}}\right)\,,$ (63)
$\displaystyle p_{1}\cdot p_{5}$ $\displaystyle\approx
E\left(E_{5}-\sqrt{E_{5}^{2}-m_{f}^{2}}c_{\theta_{\bar{f}}}\right)\,,$ (64)
$\displaystyle p_{2}\cdot p_{3}$ $\displaystyle\approx
E\Big{(}2E-E_{4}-E_{5}-\sqrt{E_{4}^{2}-m_{f}^{2}}c_{\theta_{f}}$
$\displaystyle{\hskip
113.81102pt}-\sqrt{E_{5}^{2}-m_{f}^{2}}c_{\theta_{\bar{f}}}\Big{)}\,,$ (65)
$\displaystyle p_{2}\cdot p_{4}$ $\displaystyle\approx
E\left(E_{4}+\sqrt{E_{4}^{2}-m_{f}^{2}}c_{\theta_{f}}\right)\,,$ (66)
$\displaystyle p_{2}\cdot p_{5}$ $\displaystyle\approx
E\left(E_{5}+\sqrt{E_{5}^{2}-m_{f}^{2}}c_{\theta_{\bar{f}}}\right)\,,$ (67)
$\displaystyle p_{3}\cdot p_{4}$
$\displaystyle=(2E-E_{5})E_{4}-m_{f}^{2}+\sqrt{E_{4}^{2}-m_{f}^{2}}\sqrt{E_{5}^{2}-m_{f}^{2}}f_{\theta}\,,$
(68) $\displaystyle p_{3}\cdot p_{5}$
$\displaystyle=(2E-E_{4})E_{5}-m_{f}^{2}+\sqrt{E_{4}^{2}-m_{f}^{2}}\sqrt{E_{5}^{2}-m_{f}^{2}}f_{\theta}\,,$
(69) $\displaystyle p_{4}\cdot p_{5}$
$\displaystyle=E_{4}E_{5}-\sqrt{E_{4}^{2}-m_{f}^{2}}\sqrt{E_{5}^{2}-m_{f}^{2}}f_{\theta}\,,$
(70)

where “$\approx$” corresponds to the limit $m_{e}^{2}/E^{2}\to 0$ as in Eq.
(58), and we define

$\displaystyle f_{\theta}\equiv
s_{\theta_{f}}s_{\theta_{\bar{f}}}c_{\Delta\varphi}+c_{\theta_{f}}c_{\theta_{\bar{f}}}\,.$
(71)

Note that Eq. (70) can be rewritten by using the four-momentum conservation of
Eq. (57) and Eqs. (59), (60):

$\displaystyle(p_{4}+p_{5})^{2}=(p_{1}+p_{2}-p_{3})^{2}=4E^{2}-4EE_{3}+m_{\phi}^{2}\,,$
and $\displaystyle(p_{4}+p_{5})^{2}=p_{4}^{2}+2p_{4}\cdot
p_{5}+p_{5}^{2}=2m_{f}^{2}+2p_{4}\cdot p_{5}\,,$

where we use $(p_{3})^{2}=m_{\phi}^{2}$ and
$E_{3}=\sqrt{|\vec{p_{3}}|^{2}+m_{\phi}^{2}}$. Comparing these two relations,
we obtain

$\displaystyle p_{4}\cdot p_{5}$
$\displaystyle=\frac{1}{2}\left(-4E^{2}+4EE_{4}+4EE_{5}+m_{\phi}^{2}\right)-m_{f}^{2}\,.$
(72)

Eq. (70) and Eq. (72) should be the same, and hence we can obtain the solution
for one of $(E_{4},E_{5})$ (or one of $(|\vec{p_{4}}|,|\vec{p_{5}}|)$) in
terms of the other parameters. For example, the solution of $E_{5}$ can be
expressed as

$\displaystyle E_{5,\pm}^{0}=$
$\displaystyle\frac{1}{2\left[(2E-E_{4})^{2}-(E_{4}^{2}-m_{f}^{2})f_{\theta}^{2}\right]}$
$\displaystyle\hskip
5.0pt\times\biggl{[}(2E-E_{4})(4E^{2}-4EE_{4}+2m_{f}^{2}-m_{\phi}^{2})$
$\displaystyle\hskip
23.99997pt\pm\sqrt{(E_{4}^{2}-m_{f}^{2})F_{5}(E,E_{4},f_{\theta}^{2})}\biggl{]}\,,$
(73)

where we have defined the following function:

$\displaystyle F_{5}(E,E_{4},X)\equiv$
$\displaystyle\Bigl{[}(4E^{2}-4EE_{4}+2m_{f}^{2}-m_{\phi}^{2})^{2}$
$\displaystyle\hskip 10.00002pt-4(2E-E_{4})^{2}m_{f}^{2}\Bigr{]}X$
$\displaystyle\hskip 10.00002pt+4m_{f}^{2}(E_{4}^{2}-m_{f}^{2})X^{2}\,.$ (74)

Note that the relation obtained from Eqs. (70), (72) is symmetric under
$E_{4}\leftrightarrow E_{5}$ exchange. We emphasize that Eq. (73) is
originally from the $p_{3}^{2}=m_{\phi}^{2}$ relation, and hence, we can use
this solution for the integral of $E_{5}$ to obtain the cross section.

###  A.2 Phase space integration

To calculate the cross section for the process, we need to perform the phase
space integration. In general, there are 9 parameters before applying the
four-momentum conservation, and hence, we need to perform 9 integration for
the three-body decay process:

$\displaystyle d\Phi_{3}(p_{1}+p_{2};p_{3},p_{4},p_{5})=$
$\displaystyle\left(\prod_{i=3}^{5}\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}\right)$
$\displaystyle\times\delta^{4}\left(p_{1}+p_{2}-p_{3}-p_{4}-p_{5}\right)\,.$
(75)

The delta functions reduce the number of integrals to 5, and in our
calculation, we choose independent parameters as

$\displaystyle
E_{4}\,,~{}\theta_{f}\,,~{}\theta_{\bar{f}}\,,~{}\varphi_{f}\,,~{}\Delta\varphi\,.$
(76)

Note that $E_{5}$ is parameterized by Eq. (73), and $p_{1}$ and $p_{2}$ are
written by input parameters, the beam energy $E$ and electron mass $m_{e}$. To
deal with the delta functions appropriately, we can use the following
relation:

$\displaystyle\int\\!\frac{d^{3}\vec{p_{i}}}{2E_{i}}=\int\\!d^{4}p_{i}\theta(E_{i})\delta\left(p_{i}^{2}-m_{i}^{2}\right)\,.$
(77)

Then, integration over $p_{3}$ can be done by the delta functions in Eq. (75),
which gives

$\displaystyle d\Phi_{3}(p_{1}+p_{2};p_{3},p_{4},p_{5})$
$\displaystyle=\frac{d^{3}p_{4}d^{3}p_{5}}{(2\pi)^{9}4E_{4}E_{5}}\theta(2E-E_{4}-E_{5})$
$\displaystyle\times\delta\left((p_{1}+p_{2}-p_{4}-p_{5})^{2}-m_{\phi}^{2}\right)\,.$
(78)

$d^{3}p_{4}d^{3}p_{5}$ can be replaced by

$\displaystyle d^{3}p_{4}d^{3}p_{5}$ $\displaystyle=\big{|}\det{\cal
J}\big{|}|\vec{p_{4}}|d\theta_{f}d\varphi_{f}d|\vec{p_{5}}|d\theta_{\bar{f}}d\Delta\varphi$
$\displaystyle=|\vec{p_{4}}|^{2}|\vec{p_{5}}|^{2}s_{\theta_{f}}s_{\theta_{\bar{f}}}d|\vec{p_{4}}|d\theta_{f}d\varphi_{f}d|\vec{p_{5}}|d\theta_{\bar{f}}d\Delta\varphi\
,$ (79)

with the Jacobian matrix ${\cal
J}\equiv\frac{\partial(p_{4x},p_{4y},p_{4z},p_{5x},p_{5y},p_{5z})}{\partial(|\vec{p_{4}}|,|\vec{p_{5}}|,\theta_{f},\varphi_{f},\theta_{\bar{f}},\Delta\varphi)}$.

The integration over $\varphi_{f}$ gives $2\pi$, because all inner dots and
$E_{5}$ solution in Eq. (73) are independent of $\varphi_{f}$.
$d|\vec{p_{i}}|$ can be replaced by $dE_{i}$ as

$\displaystyle
d|\vec{p_{i}}|=\frac{E_{i}dE_{i}}{\sqrt{E_{i}^{2}-m_{i}^{2}}}=\frac{E_{i}dE_{i}}{|\vec{p_{i}}|}\,.$
(80)

The delta function is

$\displaystyle 0$ $\displaystyle=(p_{1}+p_{2}-p_{4}-p_{5})^{2}-m_{\phi}^{2}$
$\displaystyle=4E^{2}-4E(E_{4}+E_{5})+2E_{4}E_{5}+2m_{f}^{2}$
$\displaystyle\hskip
11.99998pt-2\sqrt{E_{4}^{2}-m_{f}^{2}}\sqrt{E_{5}^{2}-m_{f}^{2}}f_{\theta}-m_{\phi}^{2}$
$\displaystyle\equiv f_{\delta}(E_{5})\,,$ (81)

and hence, it is easy to perform the $E_{5}$ integration by using the solution
in Eq. (73). For this calculation, we need to take into account the Jacobian
appropriately, and in this case,

$\displaystyle\delta\left((p_{1}+p_{2}-p_{4}-p_{5})^{2}-m_{\phi}^{2}\right)=\frac{\delta(E_{5}-E_{5}^{0})}{\left|f_{\delta}^{\prime}(E_{5}^{0})\right|}$
$\displaystyle=\left|2E_{4}-4E-\frac{2E_{5}^{0}\sqrt{E_{4}^{2}-m_{f}^{2}}f_{\theta}}{\sqrt{E_{5}^{0}-m_{f}^{2}}}\right|^{-1}\delta(E_{5}-E_{5}^{0})\,,$
(82)

with the solution $E_{5}^{0}$. Finally, we have

$\displaystyle d\Phi_{3}(p_{1}+p_{2};p_{3},p_{4},p_{5})$
$\displaystyle=\frac{\sqrt{E_{4}^{2}-m_{f}^{2}}\sqrt{(E_{5}^{0})^{2}-m_{f}^{2}}}{(2\pi)^{8}4\left|f_{\delta}^{\prime}(E_{5}^{0})\right|}$
$\displaystyle\times
dE_{4}dc_{\theta_{f}}dc_{\theta_{\bar{f}}}d\Delta\varphi\,.$ (83)

Hence, the result of the differential cross section for the process is

$\displaystyle d\sigma$
$\displaystyle=\frac{(2\pi)^{4}}{4E^{2}|\vec{v_{1}}-\vec{v_{2}}|}\overline{\left|\cal{M}\right|^{2}}d\Phi_{3}(p_{1}+p_{2};p_{3},p_{4},p_{5})$
$\displaystyle=\frac{\sqrt{E_{4}^{2}-m_{f}^{2}}\sqrt{(E_{5}^{0})^{2}-m_{f}^{2}}}{32E^{2}(2\pi)^{4}\left|f_{\delta}^{\prime}(E_{5}^{0})\right|}\overline{\left|\cal{M}\right|^{2}}dE_{4}dc_{\theta_{f}}dc_{\theta_{\bar{f}}}d\Delta\varphi\,,$
(84)

where $\overline{\left|\cal{M}\right|^{2}}$ is corresponding averaged squared
amplitude (with appropriate replacement for $E_{5}\to E_{5}^{0}$),
$|\vec{v_{1}}-\vec{v_{2}}|$ is the relative velocity of the beam, which is $2$
for COM frame.

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