section [1.5em] 1em subsection [3.5em] 1.8em subsubsection [5.5em] 2.5em

#  ACFI-T20-18 CP-violating Higgs Di-tau Decays: Baryogenesis and Higgs
Factories

Shao-Feng<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Shanghai
Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong
University, Shanghai 200240, China  Gang<EMAIL_ADDRESS>Amherst Center
for Fundamental Interactions, Department of Physics, University of
Massachusetts, Amherst, MA 01003, USA.  Pedro<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China  Michael J. Ramsey-
<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University,
Shanghai 200240, China  Amherst Center for Fundamental Interactions,
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA.
Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA
91125, USA.

###### Abstract

We demonstrate how probes of CP-violating observables in Higgs di-tau decays
at prospective future lepton colliders could provide a test of weak scale
baryogenesis with significant discovery potential. Measurements at the
Circular Electron Positron Collider, for example, could exclude a CP phase
larger than $2.9^{\circ}$ ($5.6^{\circ}$) at 68% (95%) C.L. assuming the
Standard Model value for magnitude of the tau lepton Yukawa coupling.
Conversely, this sensitivity would allow for a $5\,\sigma$ discovery for 82%
of the CP phase range $[0,2\pi)$. The reaches of the Future Circular Collider
- ee and International Linear Collider are comparable. As a consequence,
future lepton colliders could establish the presence of CP violation required
by lepton flavored electroweak baryogenesis with at least $3\,\sigma$
sensitivity. Our results illustrate that Higgs factories are not just
precision machines but can also make $\mathcal{O}(1)$ measurement of the new
physics beyond the Standard Model.

##  1 Introduction

The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2] and
subsequent measurements of its properties strongly favor the mechanism of
electroweak symmetry-breaking (EWSB) given by the Standard Model (SM) of
particle physics. In particular, the SM predicts each Higgs boson-fermion
Yukawa coupling to be purely real, with magnitude proportional to the fermon
mass. However, LHC measurements have confirmed this prediction up to only
$\mathcal{O}(10)\%$ precision [3, 4, 5, 6] , leaving considerable room for
physics beyond the Standard Model (BSM) in Higgs boson-fermion interactions.
Precision Higgs boson studies aim to explore these BSM possibilities. Proposed
future lepton colliders, including the Circular Electron-Positron Collider
(CEPC) [7], Future Circular Collider (FCC)-ee [8], and International Linear
Collider (ILC) [9], are designed for this purpose.

The motivations for BSM Higgs interactions are well known, including solutions
to the hierarchy problem, generation of neutrino mass, dark matter, and the
cosmic baryon asymmetry (BAU). In what follows, we focus on the possibility
that the observation of CP-violating effects in Higgs-tau lepton interactions
at a future lepton collider could provide new insight into the BAU problem. As
pointed out by Sakharov[10], a dynamical generation of the BAU requires three
ingredients in the particle physics of the early universe: : (1) non-
conserving baryon number (2) out-of-equilibrium dynamics (assuming CPT
conservation) and (3) C and CP violation. While the SM contains the first
ingredient in the form of electroweak sphalerons, it fails to provide the
needed out-of-equilibrium conditions and requisite CP-violation. The presence
of BSM physics in the dynamics of EWSB could remedy this situation – the
electroweak baryogenesis (EWBG) scenario (for a recent review, see, Ref.
[11]). While flavor-diagonal CP-violating (CPV) interactions relevant to EWBG
are strongly constrained by limits on permanent electric dipole moments (EDMs)
of the electron, neutron, and neutral atoms [12, 13] , the landscape for
flavor non-diagonal CPV is less restricted. Here, we show that searches for
CPV effects in the Higgs di-tau decays at future lepton colliders could
provide an interesting probe of “flavored EWBG” [14, 15, 16, 17, 18] in the
lepton sector.

In addition to being theoretically well-motivated in its own right, EWBG has
the additional attraction of experimental testability. Modifications of the SM
scalar sector necessary for the EWBG out-of-equilibrium conditions provide a
rich array of signatures accessible at the LHC and prospective future
colliders [19]. The signatures of CP violation could appear in either low- or
high-energy experiments. Our present focus is on the possible modification of
the the $\tau$ Yukawa coupling by a nonzero CP phase $\Delta$ as defined in
Eq. (2.1) below and the resulting impact on Higgs decay into a pair of $\tau$
leptons. In this context, it has been known for some time that the $\Delta$
phase can be measured at colliders [20, 21, 32, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31]. An $\mathcal{O}(1)$ modulation in the relevant differential
distribution allows very accurate determination of $\Delta$. The measurements
at LHC can achieve a precision around 10∘[33, 34, 35, 36, 37, 38] at 95%
confidence level (C.L.) by using the full data with 3 ab-1 of integrated
luminosity. Future lepton colliders [39] can further improve the measurements
with higher integrated luminosity, optimized energy for Higgs production, and
cleaner environment. Indeed, it was shown that the projected sensitivity for
the ILC can reach $4.3^{\circ}$ [40] and $2.9^{\circ}$ for the CEPC [41] at
$1\,\sigma$ level. To our knowledge, a detailed study connecting this
sensitivity to the BAU has yet to appear in the literature.

In this work we thus study the capability of the CEPC, FCC-ee and ILC in
probing $\Delta$ and the resulting prospects of testing lepton flavored EWBG
scenario. The projected sensitivities at future lepton colliders are much
better than the current LHC results [42]. In Sec. 2, we first make detailed
comparison of several observables (the neutrino azimuthal angle difference
$\delta\phi_{\nu}$, the polarimeter $\delta\phi_{r}$, the acoplanarity
$\phi^{*}$, and the $\Theta$ variable) to show that the polarimeter [32, 43]
is not just the optimal choice for probing $\Delta$ but can also apply
universally to both the $\tau\rightarrow\pi\nu$ and $\tau\rightarrow\rho\nu$
decay channels. Then we use a simplified smearing scheme to simulate the
detector responses and use $\chi^{2}$ minimization to find the physical
solution of neutrino/tau momentum in Sec. 3. Based on these, we find that the
future lepton colliders can make $5\,\sigma$ discovery of a nonzero CP phase
for 82% of the allowed range. With a combination of these channels, the
$1\,\sigma$ sensitivities can reach $2.9^{\circ},3.2^{\circ}$, and
$3.8^{\circ}$ at the CEPC, FCC-ee, and ILC, respectively. Our result is better
than the previous study for the ILC [40] and the same as Ref. [41] for the
CEPC. Notice that although the leptonic decay mode of $\tau$ is also
considered in addition to the two meson decay modes with more usable events, a
matrix element based observable is adopted [41] instead of the polarimeter
$\delta\phi_{r}$ as we do here, leading to accidentally the same result as
ours. Finally, in Sec. 4, we follow Ref. [14] in analyzing the implications
for lepton flavored EWBG scenario with $3\,\sigma$ sensitivity for the
presence of CP violation at the CEPC, FCC-ee and ILC. We summarize our
findings in Sec. 5.

##  2 CP Phase and Azimuthal Angle Distributions

The SM predicts the Higgs couplings with other SM particles to be proportional
to their masses, including the $\tau$ lepton,
$-y_{\tau}/\sqrt{2}\bar{\tau}_{L}\tau_{R}h+\mathrm{h.c.}=-m_{\tau}/v\bar{\tau}\tau
h$ , where $h$ denotes the SM-like Higgs boson, $v=246$ GeV is the vacuum
expectation value (VEV) and $m_{\tau}$ is the tau lepton mass. Although
$y_{\tau}$ is in general complex, its CP phase can be rotated away without
leaving any physical consequences. However, this is not always true when going
beyond the SM. Any deviation from the SM prediction, for either the $\tau$
Yukawa coupling magnitude or the CP phase, indicates new physics. We first
study the CP phase measurement by explicitly comparing various definitions of
differential distributions in this section and then the detector response
behaviors in Sec. 3. The influence of the Yukawa coupling magnitude deviation
from the SM prediction will be discussed in later parts of this paper.

The $\tau$ Yukawa coupling can be generally parametrized as

$\displaystyle\mathcal{L}_{h\tau\tau}=-\kappa_{\tau}\dfrac{m_{\tau}}{v}\bar{\tau}(\cos\Delta+i\gamma_{5}\sin\Delta)\tau
h$ (2.1)

with $\kappa_{\tau}$ being real and positive by definition, and
$\Delta\in[0,2\pi)$ in general. We will consider $\Delta\in[0,\pi]$ since the
CP measurement is insensitive to the multiplication of $\kappa_{\tau}$ by $-1$
[33]. The SM prediction for $\tau$ Yukawa coupling can be recovered with
$\kappa_{\tau}\rightarrow 1$ and $\Delta\rightarrow 0$. A non-zero value of
the CP phase $\Delta$ indicates CP violation in the $\tau$ Yukawa coupling and
can be connected to baryogenesis in the early Universe [14].

Because of the P- and T- violating nature of the second term in Eq. (2.1), the
spin correlation among the two $\tau$ leptons from a Higgs decay is an
especially interesting probe for constraining its value [23]. In practice, one
cannot measure the $\tau$ lepton directly but must rely on its decay products.
The two most promising channels are the $\tau$ decay into $\pi^{\pm}$
($\tau^{\pm}\rightarrow\pi^{\pm}\nu_{\tau^{\pm}}$) and into $\rho^{\pm}$
($\tau^{\pm}\rightarrow\rho^{\pm}\nu_{\tau^{\pm}}\rightarrow\pi^{\pm}\pi^{0}\nu_{\tau^{\pm}}$
) with $\nu_{\tau^{-}}(\nu_{\tau^{+}})$ being the neutrino (anti-neutrino)
from the decay of $\tau^{-}(\tau^{+})$. These two channels contribute 10.82%
and 25.49% of a single $\tau$ decay branching fractions [44], respectively.

###  2.1 Observables

For each $\tau$ decay, one decay plane can be formed by its decay products.
The generic azimuthal angle $\phi$ difference between the two decay planes is
then a good observable for probing $\Delta$, which can be expressed as

$\frac{1}{\Gamma}\frac{d\Gamma}{d\delta\phi}=\frac{1}{2\pi}\left[1+A\cos(2\Delta-\delta\phi)\right]\;,$
(2.2)

where the coefficient $A$ depends on the choice of observable. Note that only
for $\Delta$ differing from integer multiples of $\pi/2$, this distribution
will contain a term odd in $\delta\phi$. There exist a variety of observables
that afford access to $\Delta$, which appears as the azimuthal angle
difference in two decay planes. Here we review several possibilities and
discuss the rationale for our choice of one of them.

Neutrino azimuthal angle difference. For both $\tau$ leptons decaying into a
single charged pion, $\tau^{\pm}\rightarrow\pi^{\pm}\nu_{\tau^{\pm}}$, the
differential distribution of the neutrino momentum azimuthal angle difference
[43] is

$\frac{1}{\Gamma}\frac{d\Gamma(h\rightarrow\pi^{+}\pi^{-}\nu_{\tau}\overline{\nu}_{\tau})}{d\delta\phi_{\nu}}=\frac{1}{2\pi}\left[1-\frac{\pi^{2}}{16}\cos(2\Delta-\delta\phi_{\nu})\right]\
\ \ ,$ (2.3)

where $\delta\phi_{\nu}\equiv\phi_{\nu}-\phi_{\overline{\nu}}$ and
$\phi_{\nu}(\phi_{\overline{\nu}})$ are defined in the $\tau^{-}(\tau^{+})$
rest frame. On the other hand, if both $\tau$’s decay to rho mesons,
$\tau^{\pm}\to\rho^{\pm}(\rightarrow\pi^{\pm}+\pi^{0})\nu_{\tau^{\pm}}$, the
differential distribution of the neutrino azimuthal angle difference
$\delta\phi_{\nu}$ becomes,

$\frac{1}{\Gamma}\frac{d\Gamma(h\rightarrow\rho^{+}\rho^{-}\nu_{\tau}\overline{\nu}_{\tau})}{d\delta\phi_{\nu}}=\frac{1}{2\pi}\left[1-\frac{\pi^{2}}{16}\left(\frac{m_{\tau}^{2}-2m_{\rho}^{2}}{m_{\tau}^{2}+2m_{\rho}^{2}}\right)^{2}\cos(2\Delta-\delta\phi_{\nu})\right]$
(2.4)

with a non-negligible suppression factor,
$(m_{\tau}^{2}-2m_{\rho}^{2})^{2}/(m_{\tau}^{2}+2m_{\rho}^{2})^{2}\sim 0.2$.
As shown in Fig. 1, this significantly reduces the sensitivity to the CP phase
$\Delta$. The neutrino azimuthal angle difference $\delta\phi_{\nu}$ is a good
observable for the $\tau\rightarrow\pi\nu_{\tau}$ decay, but not for the
$\tau\rightarrow\rho\nu_{\tau}$ channel.

Polarimeter. Since the azimuthal angle difference is not necessarily the
optimized choice and multiple definitions of azimuthal angle have been
invented. In Refs. [43, 32, 40], the azimuthal angle difference between the
polarimeter vectors ${\bf r}_{\pm}$ was studied. For the $\tau$ decays, the
polarimeter vectors are defined as

$\displaystyle\tau^{\pm}\rightarrow\pi^{\pm}\nu_{\tau^{\pm}}$ $\displaystyle:$
$\displaystyle{\bf r}_{\pm}\equiv-\hat{{\bf p}}_{\nu_{\tau^{\pm}}},$ (2.5a)
$\displaystyle\tau^{\pm}\rightarrow\rho^{\pm}(\to\pi^{\pm}\pi^{0})\nu_{\tau^{\pm}}$
$\displaystyle:$ $\displaystyle{\bf
r}_{\pm}\equiv-\frac{1}{N_{\pm}}\left[\hat{{\bf
p}}_{\nu_{\tau^{\pm}}}+\frac{2m_{\tau}}{m^{2}_{\rho}-4m^{2}_{\pi}}\frac{E_{\pi^{\pm}}-E_{\pi^{0}_{\pm}}}{E_{\pi^{\pm}}+E_{\pi^{0}_{\pm}}}\left({\bf
p}_{\pi^{\pm}}-{\bf p}_{\pi^{0}_{\pm}}\right)\right],\qquad$ (2.5b)

where ${\bf r}_{\pm}$ is calculated in the corresponding $\tau^{\pm}$ rest
frame, $(E_{\pi^{0}_{\pm}},{\bf p}_{\pi^{0}_{\pm}})$ is the $\pi^{0}$ momentum
in the $\tau^{\pm}$ decay, and $N_{\pm}$ is a normalization factor to ensure
$|{\bf r}_{\pm}|=1$. Then the differential distribution in Eq. (2.2) becomes

$\frac{1}{\Gamma}\frac{d\Gamma}{d\delta\phi_{r}}=\frac{1}{2\pi}\left[1-\frac{\pi^{2}}{16}\cos(2\Delta-\delta\phi_{r})\right],$
(2.6)

for both decay channels including the mixed mode,
$h\rightarrow\rho^{\pm}\pi^{\mp}\nu_{\tau}\overline{\nu}_{\tau}$. From the
neutrino azimuthal angle difference $\delta\phi_{\nu}$ in Eq. (2.4) to the one
of the polarimeter in Eq. (2.6), the amplitude gets amplified by a factor of 5
which is a significant improvement.

The azimuthal angle difference $\delta\phi_{r}\equiv\phi_{{\bf
r}_{+}}-\phi_{{\bf r}_{-}}$ is defined with respect to the $z$ direction,
${\bf z}\equiv{\hat{\bf p}}_{\tau^{-}}$. In the Higgs rest frame,

$\tan\delta\phi_{r}=\frac{{\hat{\bf p}}_{\tau^{-}}\cdot({\bf r}_{+}\times{\bf
r}_{-})}{{\bf r}_{-}\cdot{\bf r}_{+}-({\bf r}_{+}\cdot{\hat{\bf
p}}_{\tau^{-}})({\bf r}_{-}\cdot{\hat{\bf p}}_{\tau^{-}})}.$ (2.7)

For the $\tau^{\pm}\rightarrow\pi^{\pm}\nu_{\tau^{\pm}}$ decay channel, the
polarimeter is along the neutrino momentum direction, namely, ${\bf
r}_{\pm}=-\hat{\bf p}_{\nu_{\tau^{\pm}}}$ as shown in Eq. (2.5a). When di-tau
decay into pions it is the azimuthal angle difference $\phi_{\nu}$. In
contrast, the polarimeter for the
$\tau^{\pm}\rightarrow\rho^{\pm}\nu_{\tau^{\pm}}$ decay channel does not
coincide with any momentum of the final-state particles. For illustration, the
distribution of $\delta\phi_{r}$ for
$h\to\tau^{+}(\rightarrow\rho^{+}\bar{\nu}_{\tau})\tau^{-}(\rightarrow\rho^{-}\nu_{\tau})$
is shown in Fig. 1.

The 4-vector $r_{\pm}=(0,{\bf r_{\pm}})$ serves as the effective spin of the
corresponding $\tau^{\pm}$ leptons. This becomes evident in the total matrix
element of the Higgs decay chain,

$\displaystyle|\mathcal{M}^{\rm total}|^{2}$ $\displaystyle\propto$
$\displaystyle{\rm
Tr}\left[\left(\not{p}_{\tau^{-}}+m_{\tau}\right)\left(1+\gamma_{5}\not{r}_{-}\right)\mathcal{O}\left(\not{p}_{\tau^{+}}-m_{\tau}\right)\left(1-\gamma_{5}\not{r}_{+}\right)\overline{\mathcal{O}}\right],$
(2.8)

with $\mathcal{O}\equiv\cos\Delta+i\gamma_{5}\sin\Delta$,
$\overline{\mathcal{O}}\equiv\gamma^{0}\mathcal{O}^{\dagger}\gamma^{0}$ and
$p_{\tau^{\pm}}\equiv(E_{\tau^{\pm}},{\bf p}_{\tau^{\pm}})$ being the momentum
of $\tau^{\pm}$. If the Higgs boson decays to polarized $\tau$ leptons, it
should be the $\tau$ spin vector $s_{\pm}=(|{\bf
p}_{\tau^{\pm}}|/m_{\tau},E_{\tau^{\pm}}/m_{\tau}{\hat{\bf p}_{\tau^{\pm}}})$
that appears in place of the polarimeter $r_{\pm}$. But since the Higgs decay
chain also contains contribution from the $\tau$ decays, $s_{\pm}$ is replaced
by $r_{\pm}$ to take the extra effects into consideration.

Fig. 1: The differential distributions of
$h\to\tau^{+}(\rightarrow\rho^{+}\bar{\nu}_{\tau})\tau^{-}(\rightarrow\rho^{-}\nu_{\tau})$
for neutrino momentum $\delta\phi_{\nu}$ (red), polarimeter $\delta\phi_{r}$
(black), acoplanarity $\phi^{*}$ (blue), and the $\Theta$ variable (green) at
the truth level for $\Delta=0^{\circ}$.

Acoplanarity. This observable was introduced in Ref. [22, 23] for the
$\tau\rightarrow\rho\nu_{\tau}$ decay mode. In the rest frame of the
$\rho^{+}\rho^{-}$ system, the $\rho$ momenta are back to back. The decay
products of $\rho^{\pm}$ form two decay planes and the angle difference
between them is defined as acoplanarity $\phi^{*}$,

$\tan\phi^{*}\equiv\frac{{\hat{\bf p}}_{\rho^{-}}\cdot[({\bf
p}_{\pi^{+}}\times{\bf p}_{\pi^{0}_{+}})\times({\bf p}_{\pi^{-}}\times{\bf
p}_{\pi^{0}_{-}})]}{({\bf p}_{\pi^{+}}\times{\bf p}_{\pi^{0}_{+}})\cdot({\bf
p}_{\pi^{-}}\times{\bf p}_{\pi^{0}_{-}})}\,.$ (2.9)

This interesting variable requires only the knowledge of the directly
observable momenta of $\pi^{\pm}$ and $\pi^{0}$. However, the oscillation
amplitude of the distribution is suppressed by around 30% in comparison with
the polarimeter as shown in Fig. 1.

The $\Theta$ Variable. For $\tau\rightarrow\rho\nu_{\tau}$ decay, a fourth
observable similar to the usual acoplanarity angle, can be defined as,

$\tan\Theta\equiv\frac{{\hat{\bf p}}_{\tau^{+}}\cdot({\bf E}_{+}\times{\bf
E}_{-})}{{\bf E}_{-}\cdot{\bf E}_{+}-({\bf E}_{+}\cdot{\hat{\bf
p}}_{\tau^{+}})({\bf E}_{-}\cdot{\hat{\bf p}}_{\tau^{+}})},$ (2.10)

with ${\bf E}_{\pm}$ taking analogy to the electromagnetic fields. In the
$\tau^{\pm}$ rest frames, the ${\bf E}_{\pm}$ vector can be expressed as [33],

$\displaystyle{\bf E}_{\pm}$
$\displaystyle\equiv\frac{m^{2}_{\rho}-4m^{2}_{\pi}}{2m_{\tau}}\left[\frac{m^{2}_{\tau}-m^{2}_{\rho}}{m^{2}_{\tau}+m^{2}_{\rho}}\hat{{\bf
p}}_{\nu_{\tau^{\pm}}}+\frac{2m_{\tau}}{m^{2}_{\rho}-4m^{2}_{\pi}}\frac{(E_{\pi^{\pm}}-E_{\pi^{0}})}{(E_{\pi^{\pm}}+E_{\pi^{0}})}\left({\bf
p}_{\pi^{\pm}}-{\bf p}_{\pi^{0}_{\pm}}\right)\right].$ (2.11)

Note that Eq. (2.10) is slightly more general than the one presented in Ref.
[33], where they take the approximation $({\bf E}_{\pm}\cdot{\hat{\bf
p}}_{\tau^{\pm}})\approx 0$. It is very interesting to see that Eq. (2.10) has
very similar form as Eq. (2.7) with the only difference of a proportional
factor $(m^{2}_{\rho}-4m^{2}_{\pi})/2m_{\tau}$. Due to these similarities, the
$\Theta$ variable has roughly the same sensitivity as polarimeter, see Fig. 1.

The comparison in Fig. 1 shows that the polarimeter $\delta\phi_{r}$ and the
$\Theta$ variable are the optimal ones. However, the $\Theta$ variable needs
both momenta of $\pi^{\pm}$ and $\pi^{0}$, limiting its scope to only the
$\tau\rightarrow\rho\nu_{\tau}$ decay mode. In contrast, the polarimeter
method applies for both channels by matching ${\bf r}_{\pm}$ with different
combination of final-state particle momenta as shown in Eq. (2.5). So we adopt
the polarimeter scheme in the following part of this paper.

##  3 Measurements at Future Lepton Colliders

Future lepton colliders [45] are designed to produce millions of Higgs events.
The three prominent candidate colliders are the CEPC [7], FCC-ee [8] and ILC
[46]. The CEPC experiment [7] is expected to have around $1.1\times 10^{6}$
Higgs events. This comes from an integrated luminosity of 5.6 ab-1 with two
interaction points (IP) and 7 years of running at $\sqrt{s}=240$ GeV. The FCC-
ee has a higher luminosity and 4 interaction points, but runs in the Higgs
factory mode for only 3 years resulting in a 5 ab-1 of integrated luminosity
or equivalently $1.0\times 10^{6}$ Higgs events [8]. The ILC, on the other
hand, has a significantly lower integrated luminosity at 2 ab-1, but is able
to produce polarized electrons/positrons which increases the cross section
significantly, effectively raising its number of Higgs production to
$0.64\times 10^{6}$ [9]. The configuration of these three experiments and the
expected numbers of Higgs events at the benchmark luminosities have been
summarized in Table 1 for comparison.

| Integrated luminosity | $\sqrt{s}$ | Number of Higgs bosons
---|---|---|---
CEPC [7] | 5.6 ab-1 | 240 GeV | $1.1\times 10^{6}$
FCC-ee [8] | 5 ab-1 | 240 GeV | $1.0\times 10^{6}$
ILC [9] | 2 ab-1 | 250 GeV | $0.64\times 10^{6}$
Table 1: Configurations (integrated luminosity, energy $\sqrt{s}$, and Higgs
production rate) at the future lepton colliders CEPC, FCC-ee, and ILC.

In this section, we study the detector responses, including the smearing
effects, selection cuts, and momentum reconstruction ambiguities. With around
$650\sim 1100$ events, the uncertainty at the level of $14\%\sim 18\%$ is much
smaller than the expected $60\%$ modulation in the CP measurement. This allows
a $5\,\sigma$ discovery potential for approximately 80% of the allowed range
in $[0,2\pi)$ of the CP phase $\Delta$ and a determination of $\Delta$ with
the accuracy of $2.9^{\circ}\sim 3.8^{\circ}$.

###  3.1 Simulation and Detector Responses

At lepton colliders, the Higgs boson is mainly produced in the so-called
Higgsstrahlung process, $e^{+}e^{-}\rightarrow Zh$, with an associated $Z$
boson. This channel allows a model-independent measurement of the Higgs
properties thanks to the recoil mass reconstruction method [47]. The Higgs
event is first selected by reconstructing the $Z$ boson without assuming any
Higgs coupling with the SM particles. The Higgs boson momentum can be either
derived from the $Z$ boson momentum using energy-momentum conservation or
reconstructed from the Higgs decay products. Since there are always two
neutrinos in the final state of $h\rightarrow\tau\tau$ events, the $Z$ boson
momentum is needed to reconstruct the Higgs momentum as the initial condition
of the Higgs decay kinematics to fully recover the two neutrino momenta.

We use MadGraph [48] and TauDecay [49] packages to simulate the spin
correlation in the Higgs decay chains. For a realistic simulation, both
detector response and statistical fluctuations have to be taken into
consideration. In order to perform fast detector simulation we construct a
simplified smearing algorithm which is validated by comparing with Delphes
[50] output.

Using the recoil mass method, the smearing should in principle be applied to
the $Z$ momentum. Nevertheless, since the Higgs and $Z$ bosons are back to
back in the center of mass frame, we can directly smear the Higgs momentum.
Defining the $z$-axis along the Higgs momentum, only its $P_{z}$ component is
affected by the $Z$ boson decay modes while the other two, $P_{x}$ and
$P_{y}$, have independent smearing behaviors. To select the Higgsstrahlung
events, those with the reconstructed $Z$ invariant mass outside the range
$80~{}{\rm GeV}<\sqrt{p^{2}_{Z}}<100$ GeV are discarded. The momentum
uncertainties of Higgs smearing have been summarized in the left part of Table
2.

Higgs Smearing
---
| Observables | Uncertainty
---|---
$P_{x,y}$ | 1.82 GeV
$P_{z}$ ($Z\rightarrow jj)$ | 2.3 GeV
$P_{z}$ ($Z\rightarrow l\overline{l})$ | 0.57 GeV

Pion Smearing Observables Uncertainty $\phi$ 0.0002$|\eta|$\+ 0.000022 $\eta$
0.000016$|\eta|$ \+ 0.00000022 $|{\bf p}_{T}|$ 0.036$|{\bf p}_{T}|$

Table 2: Left: Uncertainties of the Higgs boson [41] and Right: pion momentum
smearing parameters to be consistent with the Delphes configurations
delphes_card_CircularEE.tcl [51] for the CEPC/FCC-ee and delphes_card_ILD.tcl
[52] for the ILC.

The pion momentum smearing is performed by randomly sampling the azimuthal
angle $\phi$ and the pseudo-rapidity $\eta$ according to Gaussian distribution
[53, 50]. In addition, the transverse momentum $|{\bf p}_{T}|$ is sampled with
a Log-normal like distribution from Ref. [50],

$\displaystyle|{{\bf p}_{T}^{\rm rec}}|=\exp\left(\log|{{\bf
p}_{T}}|-\frac{\epsilon}{2}\sqrt{1+\frac{\sigma^{2}}{|{{\bf
p}_{T}}|^{2}}}\right),$ (3.1)

with $\epsilon$ being a random number following a Gaussian distribution
centered in 0 with error 1 and $N$ a normalization factor. For $\tau$ decay
into $\rho^{\pm}$, the reconstructed $\rho$ invariant mass is required to be
within the range of $0.3~{}{\rm GeV}<\sqrt{p^{2}_{\rho}}<1.2~{}$GeV. The
uncertainties of $(\phi,\eta,|{\bf p}_{T}|)$ for pions are summarized in the
right part of Table 2.

Fig. 2: The pion smearing effects simulated by Delphes (black) and our
simplified algorithm (blue). Notice that in the  panel $(b)$ the numbers in
the horizontal axis are multiplied by a factor $\times 10^{5}$ for better
visualization.

Although our simplified smearing algorithm is admittedly less sophisticated,
our results are broadly compatible with those commonly adopted in the
literature. A complete analysis in momentum reconstruction and detailed cuts
was performed in Refs. [50, 41]. For validation, we compare our smearing
algorithm to the Delphes simulation with the configurations cards
delphes_card_CircularEE.tcl [51] for the CEPC/FCC-ee [54] and
delphes_card_ILD.tcl [52] for the ILC. Fig. 2 shows the smeared distributions
of the pion kinematic variables simulated with Delphes (black) vs our
simplified smearing (blue). We can see that the results of these two
simulations agree with each other quite well. In this work, we take the
simplified smearing algorithm for a fast simulation.

To obtain the total number of expected events, one needs to consider several
branching ratios. First, the $Z$ boson can only be reconstructed if it decays
into either leptons or jets with $80\%$ of branching ratio in total [44].
Also, since the decay branching ratio of Higgs decaying into two $\tau$
leptons is $6.64\%$ [44], only around 5.3% of the actual Higgs events
associated with $Z$ production are available for the CP measurement. Further
suppression comes from the branching fraction of the decay of $\tau$ into
$\pi$ or $\rho$. And we arrive at 7704 events at the CEPC, 7003 events at the
FCC-ee, and 4482 events at the ILC. Taking into account the identification of
$\tau$ jets and tagging of the Higgs boson and other selection cuts [40], we
obtain an overall efficiency, $\epsilon=0.145,0.144,0.142$ for
$(\pi,\pi),(\pi,\rho)$ and $(\rho,\rho)$ decay modes, respectively.

Decay modes | Branching ratio
---|---
$Z\rightarrow$ vis. | 80%
$h\rightarrow\tau^{+}\tau^{-}$ | 6.64%
$\tau\rightarrow\pi\nu_{\tau}$ | 10.82%
$\tau\rightarrow\rho\nu_{\tau}$ | 25.49%

$\tau$ decay products | Number of Higgs decay events
---|---
CEPC | FCC-ee | ILC
| before | after | before | after | before | after
$(\pi,\pi)$ | 684 | 99 | 622 | 90 | 398 | 58
$(\pi,\rho)$ | 3223 | 465 | 2930 | 423 | 1875 | 271
$(\rho,\rho)$ | 3797 | 541 | 3451 | 491 | 2209 | 314

Table 3:  Left:  Branching fractions associated with the entire reaction. The
values were obtained from [44]. Right: Expected event numbers at the CEPC [7]
with the integrated luminosity $\mathcal{L}=5.6~{}$ab-1, FCC-ee [8] with
$\mathcal{L}=5~{}$ab-1 and ILC [46] with $\mathcal{L}=2~{}$ab-1. The expected
numbers of events before and after selection cuts are shown in the columns
“before” and “after”, respectively, with the overall cut efficiencies taken
from Ref. [40].

The expected event numbers before and after applying the selection
efficiencies are shown in Table 3 for comparison. In total, roughly 1105,
1004, and 643 events of the
$h\to\tau^{+}\tau^{-},\tau^{\pm}\to\pi^{\pm}/\rho^{\pm}\nu_{\tau^{\pm}}$ decay
chains can be reconstructed at the CEPC, FCC-ee, and ILC, respectively.

###  3.2 Ambiguities in Momentum Reconstruction

Experimentally, in order to reconstruct the $\tau$ momentum, it is unavoidable
to first obtain the neutrino momentum which is not directly detectable. With
two neutrinos in the final state, we need to constrain two 4-vector momenta.
Since the Higgs momentum can be fully reconstructed from the $Z$ boson
counterpart, only one neutrino momentum is independent due to energy-momentum
conservation. The 4 degrees of freedom can be constrained by the on-shell
conditions of the two neutrinos and the two $\tau$ leptons.

Unfortunately, the solutions have a two-fold ambiguity. Since on-shell
conditions are in quadratic forms, one sign can not be uniquely fixed. For
completeness, we summarize the solution here in terms of the $\tau^{-}$
momentum defined in the Higgs rest frame,

$\displaystyle{\bf p}_{\tau_{-}}$ $\displaystyle=$
$\displaystyle\sqrt{p^{2}_{h}-4m^{2}_{\tau}}\left[\sin\theta_{\tau}\left(\cos\phi_{\tau}{\hat{\bf
n}}_{1}+\sin\phi_{\tau}{\hat{\bf n}}_{2}\right)\pm\cos\theta_{\tau}{\hat{\bf
n}}_{3}\right].$ (3.2)

The unit base vectors $\hat{\bf n}_{i}$ are constructed in terms of the
primary decay mesons, $X_{\pm}\equiv\pi^{\pm},\rho^{\pm}$,

$\displaystyle{\hat{\bf n}}_{1}={\hat{\bf p}}_{X_{+}}\,,\quad{\hat{\bf
n}}_{2}=\frac{{\hat{\bf p}}_{X_{-}}-({\hat{\bf p}}_{X_{+}}\cdot{\hat{\bf
p}}_{X_{-}}){\hat{\bf p}}_{X_{+}}}{\sqrt{1-({\hat{\bf
p}}_{X_{+}}\cdot{\hat{\bf p}}_{X_{-}})^{2}}},\qquad{\hat{\bf
n}}_{3}=\frac{{\hat{\bf p}}_{X_{+}}\times{\hat{\bf
p}}_{X_{-}}}{\sqrt{1-({\hat{\bf p}}_{X_{+}}\cdot{\hat{\bf
p}}_{X_{-}})^{2}}}\,.$ (3.3)

The first base vector $\hat{\bf n}_{1}$ is along the momentum of $\pi^{+}$ or
$\rho^{+}$ while the third one $\hat{\bf n}_{3}$ is perpendicular to the
momentum of both primary mesons. Finally, $\hat{\bf n}_{2}$ is simply the one
perpendicular to both $\hat{\bf n}_{1}$ and $\hat{\bf n}_{3}$. The polar
angles of the $\tau$ momentum can be reconstructed as,

$\displaystyle\sin\theta_{\tau}\cos\phi_{\tau}$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{\tau}+m^{2}_{X}-m_{h}E_{X_{+}}}{|{\bf
p}_{X_{+}}|\sqrt{m^{2}_{h}-4m^{2}_{\tau}}},$ (3.4a)
$\displaystyle\sin\theta_{\tau}\sin\phi_{\tau}$ $\displaystyle=$
$\displaystyle\frac{m_{h}E_{X_{-}}-m^{2}_{\tau}-m^{2}_{X_{-}}}{|{\bf
p}_{X_{-}}||s_{X_{-}X_{+}}|\sqrt{m^{2}_{h}-4m^{2}_{\tau}}}+\frac{m_{h}E_{X_{+}}-m^{2}_{\tau}-m^{2}_{X_{+}}}{|{\bf
p}_{X_{+}}||s_{X_{-}X_{+}}|\sqrt{m^{2}_{h}-4m^{2}_{\tau}}}c_{X_{-}X_{+}}\;,\qquad$
(3.4b)

where
$(s_{X_{-}X_{+}},c_{X_{-}X_{+}})\equiv(\sin\theta_{X_{-}X_{+}},\cos\theta_{X_{-}X_{+}})$
and $\theta_{X_{-}X_{+}}$ is the angle between the momentum of $X_{+}$ and
$X_{-}$.

However, in Eq. (3.2) the $\pm$ sign in front of $\hat{\bf n}_{3}$ reflects
the fact that both solutions obey all the constraints from energy-momentum
conservation and the correct solution cannot be unambiguously obtained. This
sign ambiguity can significantly decrease the CP sensitivity, especially for
the neutrino azimuthal angle distribution. Using momentum conservation, the
result in Eq. (2.7) for $\tan\delta\phi_{\nu}$ can be written in the same form
by substituting ${\bf p}_{\nu}$ by ${\bf p}_{X^{\pm}}$, hence,
$\tan\delta\phi_{\nu}\propto{\hat{\bf p}}_{\tau^{-}}\cdot({\bf
p}_{X^{+}}\times{\bf p}_{X^{-}})=\pm\cos\theta_{\tau}/\sqrt{1-({\hat{\bf
p}}_{X_{+}}\cdot{\hat{\bf p}}_{X_{-}})^{2}}$. In other words,
$\delta\phi_{\nu}$ can have both positive and negative solutions with the same
magnitude. This would not be a big problem for the symmetric distribution of
$\delta\phi_{\nu}$ around its origin, such as those curves in Fig. 1 with
$\Delta=0^{\circ}$. But it causes significant issues for other $\Delta$ values
and effectively flattens the curve for $\Delta=\pm 45^{\circ}$.

This ambiguity can be solved by measuring other decay information. An
especially useful quantity is the impact parameter [55, 56], the minimum
distance of charged meson trajectory to the $\tau$ leptons production point.
The impact parameter measurement essentially removes the two-fold ambiguity
for the $\tau$ Yukawa CP measurement at future lepton colliders [24, 53]. A
more recent study with spatial resolution of 5 $\mu$m can be found in Refs.
[41, 40].

Another ambiguity comes from the detector resolutions. The $\tau$ momentum is
reconstructed from the smeared Higgs and meson momentum. This reconstruction
is realized with energy and momentum conservation, assuming narrow width
approximation for the $\tau$ momentum, $p_{\tau^{\pm}}^{2}=m_{\tau}^{2}$. Both
smearing and finite width could lead to nonphysical solutions in Eq. (3.4),
for example $\sin\theta_{\tau}\sin\phi_{\tau}>1$. For those events, we follow
a similar procedure introduced in Ref. [41]. We try to find the solution for
the (anti-)neutrino momenta optimally consistent with all the information we
have on each event (including four-momentum conservation) by minimizing the
function

$\displaystyle\chi^{2}_{\rm rec}=\sum_{i=0}^{3}\left(\frac{(p^{\rm
rec}_{\tau^{+}})_{i}+(p^{\rm rec}_{\tau^{-}})_{i}-(p^{\rm
rec}_{h})_{i}}{\sigma_{h}}\right)^{2}+\left(\frac{(p^{\rm
rec}_{\tau^{+}})^{2}-m_{\tau}^{2}}{\sigma_{\tau}}\right)^{2}+\left(\frac{(p^{\rm
rec}_{\tau^{-}})^{2}-m_{\tau}^{2}}{\sigma_{\tau}}\right)^{2},$ (3.5)

where $i=0,\cdots,3$ runs over the 4-momentum components of each particle
momentum. We adopt the uncertainties as $\sigma_{h}=4.0$ GeV and
$\sigma_{\tau}=0.2$ GeV [41]. The $\chi^{2}_{\rm rec}$ function is minimized
over the 6 kinematic parameters of the unmeasured neutrino momentum: the
pseudo-rapidity, azimuthal angle, and absolute value of the momentum for both
neutrino/anti-neutrino. Then the $\tau$ momentum is then obtained with energy
momentum conservation, $p^{\rm rec}_{\tau^{\pm}}=p_{\nu^{\pm}}+p^{\rm
rec}_{X^{\pm}}$. The best fit at the minimum of $\chi^{2}_{\rm rec}$
approximates the physical solution. We keep the event if the minimum solution
is consistent with the mass cuts. Otherwise, the event is discarded.

Fig. 3: Smeared differential distributions of $\delta\phi_{r}$ in the decay
mode
$h\rightarrow\tau^{+}(\rightarrow\rho^{+}\overline{\nu}_{\tau})\tau^{-}(\rightarrow\rho^{-}\nu_{\tau})$.
Left: The distribution for CP phases $\Delta=0^{\circ}$ (red),
$\Delta=45^{\circ}$ (blue), $\Delta=90^{\circ}$ (green) at the CEPC; Right: A
comparison of the distributions at the CEPC (red), FCC-ee (blue), and ILC
(green) for $\Delta=0^{\circ}$. In both panels, the error bars indicate the
statistical uncertainties.

The final result of the differential distribution for the
$h\rightarrow\tau^{+}(\rightarrow\rho^{+}\overline{\nu}_{\tau})\tau^{-}(\rightarrow\rho^{-}\nu_{\tau})$
process is plotted in Fig. 3. The left panel shows the differential
distributions for $\Delta=0^{\circ}$ (red), $\Delta=45^{\circ}$ (blue), and
$\Delta=90^{\circ}$ (green), respectively. Being divided into 20 bins [33,
40], there are $25\sim 35$ events in each bin on average. The corresponding
statistical uncertainty at the level of $17\%\sim 20\%$ is much smaller than
the oscillation amplitude, $\pi^{2}/16\approx 62\%$. The event rate at the
CEPC are large enough to constrain the modulation pattern as elaborated in
Sec. 3.3. The right panel shows the spectrum at the three future candidate
lepton colliders, CEPC (red), FCC-ee (blue), and ILC (green), respectively,
for comparison. While CEPC and FCC-ee have comparable spectrum, ILC has much
lower event rate and hence larger fluctuations.

It is interesting to see that for $\Delta=90^{\circ}$, the differential
distribution of $\delta\phi_{r}$ has only $\cos\delta\phi_{r}$ but no
$\sin\delta\phi_{r}$ in Eq. (2.6). In other words, the observable that we
measure has only CP conserving contribution that does not change under CP
transformation. However, the distributions in the left panel of Fig. 3 show
that the difference between $\Delta=0^{\circ}$ and $\Delta=90^{\circ}$ is
maximal. This is because $\cos 2\Delta=\pm 1$ take the two extreme values with
opposite signs.

###  3.3 Discovery Potential and Sensitivity of the CP Phase

To evaluate the CP measurement sensitivities, we adopt a $\chi^{2}$ function
defined according to the Poisson distribution,

$\displaystyle\chi^{2}\equiv\sum_{i}2(N_{i}^{\rm test}-N_{i}^{\rm
true})+2N_{i}^{\rm true}\log(N_{i}^{\rm true}/N_{i}^{\rm test}),$ (3.6)

where $i=1,\cdots,20$ runs over all the 20 bins of the $\delta\phi_{r}$
differential distribution. Since we are studying the projected sensitivity at
future lepton colliders, there is no real data available yet. Instead, we
simulate the measurement with some assumed true values of the CP phase
$\Delta$ to produce a set of pseudo-data $N^{\rm true}_{i}$ and then fit these
pseudo-data with some test values $N^{\rm test}_{i}$. The event numbers
$N^{\rm true}_{i}$ and $N^{\rm test}_{i}$ are functions of the true value
$\Delta^{\rm true}$ and $\Delta^{\rm test}$, respectively.

Fig. 4:  Left:  The CP phase discovery potential at the CEPC for
$\kappa_{\tau}=1$. The green shaded regions represent the results from various
decay modes: only $(\rho,\rho)$ (light green), $(\rho,\rho)+(\pi,\rho)$
(green) and the fully combined one $(\rho,\rho)+(\pi,\rho)+(\pi,\pi)$ (dark
green), with the boundaries describing the $\sqrt{\chi^{2}_{\rm CPV}}$ values
according to Eq. (3.7) given $\Delta^{\rm{true}}$. The black dashed lines with
$\sqrt{\chi_{\rm{CPV}}^{2}}=1.96,5$ mark the sensitivities at $95\%$ C.L. and
$5\sigma$, respectively. Right:  Sensitivity of all the channels at the CEPC
(green), FCC-ee (blue), and ILC (red). In both panels, the region outside of
the gray bands are excluded at 95% C.L. by the current CP measurement at the
LHC [42].

The discovery ability of a nonzero CP phase can be parametrized as the smaller
one of the two $\chi^{2}$ values between the given $\Delta^{\rm true}$ and the
CP conserving cases $\Delta^{\rm test}=0^{\circ}$ or $\Delta^{\rm
test}=180^{\circ}$,

$\displaystyle\chi^{2}_{\rm CPV}(\Delta^{\rm true})\equiv{\rm
min}[\chi^{2}(\Delta^{\rm true},\Delta^{\rm
test}=0^{\circ}),\chi^{2}(\Delta^{\rm true},\Delta^{\rm test}=180^{\circ})].$
(3.7)

Fig. 4 shows the $\sqrt{\chi^{2}_{\rm CPV}}$ distribution as a function of
$\Delta$ assuming $\kappa_{\tau}=1$. The sensitivities for the di-$\tau$ decay
into $(\rho,\rho)$, $(\rho,\rho)+(\pi,\rho)$, and the full combination
$(\rho,\rho)+(\pi,\rho)+(\pi,\pi)$ are depicted in light green, green, and
dark green regions, respectively. For different decay channels, the
differential distributions have the same amplitude $\pi^{2}/16$ as indicated
in Eq. (2.6). So the main difference in the sensitivities is due to the event
rates: the branching ratio of the $\tau\rightarrow\pi\nu_{\tau}$ is only
$10.8\%$, in comparison with the $25\%$ for the
$\tau\rightarrow\rho\nu_{\tau}$ channel. As indicated by the black dashed
lines, 95% of the values of $\Delta$ can be tested above $95\%$ C.L. and 82%
of the parameter space can be tested at even more than $5\sigma$. The
sensitivity peaks at $\Delta=\pm 90^{\circ}$ where Eq. (2.6) takes the most
different value from that of $\Delta=0^{\circ}$ or $180^{\circ}$ with more
than $10\,\sigma$ significance. In the right panel, we also show the
comparison of the sensitivities at the CEPC (green), FCC-ee (blue), and ILC
(red). As expected, the CEPC has the highest sensitivity due to the higher
number of events.

| 68% C.L. for $m=1$ | 95% C.L. for $m=1$ | 95% C.L. for $m=2$
---|---|---|---
CEPC | 2.9∘ | 5.6∘ | 7.0∘
FCC-ee | 3.2∘ | 6.3∘ | 7.8∘
ILC | 3.8∘ | 7.4∘ | 9.3∘

Table 4: The CP phase precision at the CEPC, FCC-ee, and ILC for $m$
parameter(s)

.

For completeness, Table 4 summarizes the expected precision of the $\Delta$
measurement at future lepton colliders at 68% C.L. and 95% C.L. for $m=1$
parameter ($\Delta$), or $m=2$ parameters ($\Delta$ and $\kappa_{\tau}$).
Notice that our estimation at $1\,\sigma$ level is slightly better than the
$4.4^{\circ}$ with $1\,\mbox{ab}^{-1}$ in only the
$\tau\rightarrow\rho\nu_{\tau}$ decay channel [33] or $4.3^{\circ}$ with
$2\,\mbox{ab}^{-1}$ in both $\tau$ decay channels [40] at the ILC. For the
CEPC, our result is the same as the $2.9^{\circ}$ in Ref. [41]. Notice that in
addition to the two mesonic decay channels, the leptonic decay channel
$\tau\rightarrow\ell\nu\bar{\nu}$ is also considered in Ref. [41] with the
matrix element based observable that is different from our polarimeter
$\delta\phi_{r}$. We can clearly see from Table 4 that the future lepton
colliders can differentiate the CPV scenario from the CP-conserving one very
well.

##  4 Prospects of Constraining New Physics

As the aforementioned analysis shows, there remains significant potential for
discovering CP violation in the $h\to\tau^{+}\tau^{-}$ decay at prospective
future lepton colliders. We now draw the connection with the lepton flavored
EWBG scenario, following the treatment given in Ref. [14] for concrete
illustration (see Refs.[15, 16, 17, 18]).This discussion exemplifies future
lepton colliders are not only precision machines but can also make an
$\mathcal{O}(1)$ measurement of BSM physics effects.

###  4.1 Two Higgs Doublet Model

The set up in Ref. [14] relies on the type III Two Higgs Doublet Model (THDM)
[57, 58], wherein the two scalar doublet fields before EWSB are denoted as
$\Phi_{1,2}$. Both neutral scalars inside $\Phi_{1,2}$ acquire nonzero VEVs,
$v_{1}$ and $v_{2}$, respectively, with
$v\equiv\sqrt{v_{1}^{2}+v_{2}^{2}}=246$ GeV. The neutral components can mix
with each other to form three neutral massive scalar fields after one neutral
Goldstone boson is eaten by the $Z$ boson. We assume a CP-invariant scalar
potential, namely, only the real parts of the two neutral scalars can mix with
each other but not with the imaginary parts,

$\displaystyle H\equiv c_{\alpha}{\rm Re}[\Phi^{0}_{1}]+s_{\alpha}{\rm
Re}[\Phi^{0}_{2}],\quad h\equiv-s_{\alpha}{\rm
Re}[\Phi^{0}_{1}]+c_{\alpha}{\rm Re}[\Phi^{0}_{2}],\quad A\equiv-s_{\beta}{\rm
Im}[\phi_{1}^{0}]+c_{\beta}{\rm Im}[\phi_{2}^{0}],$ (4.1)

where $s_{\alpha}\equiv\sin\alpha$, $c_{\alpha}\equiv\cos\alpha$,
$\tan\beta\equiv v_{2}/v_{1}$, and Re and Im denote the real and imaginary
parts, respectively. Note that $\alpha$ is the mixing angle from the neutral
scalar mass matrix diagonalization. The neutral particle masses are ordered as
$m_{H},m_{A}>m_{h}\approx 125$ GeV, so that $h$ is the SM-like Higgs boson.

In the Type-III THDM, the Yukawa interaction for each doublet field has the
same structure as the SM Yukawa interaction;

$\displaystyle\mathcal{L}_{Y}=-\overline{L}Y_{1}\ell_{R}\Phi_{1}-\overline{L}Y_{2}\ell_{R}\Phi_{2}+{\rm
h.c.}$ (4.2)

In this way, both Higgs doublets can contribute its neutral components to
couple with the $\tau$ lepton [29],

$\displaystyle-\frac{m_{\tau}}{v}\overline{\tau}_{L}\tau_{R}\left[\left(s_{\beta-\alpha}+\frac{N_{\tau\tau}}{m_{\tau}}c_{\beta-\alpha}\right)h+\left(c_{\beta-\alpha}-\frac{N_{\tau\tau}}{m_{\tau}}s_{\beta-\alpha}\right)H+iAN_{\tau\tau}\right],$
(4.3)

where $N_{\tau\tau}$ is a complex parameter related to the matrix elements of
$Y_{1,2}$. Following the parametrization of Eq. (2.1), the $\tau$ Yukawa
coupling becomes

$\displaystyle\kappa_{\tau}(\cos\Delta+i\sin\Delta)=s_{\beta-\alpha}+\frac{N_{\tau\tau}}{m_{\tau}}c_{\beta-\alpha}.$
(4.4)

Notice that CP violation arises due to the imaginary part of $N_{\tau\tau}$.
Moreover, for the particular texture, $Y_{j,22}=Y_{j,23}=0$,
$Y_{1,33}=Y_{2,33}=Y_{33}$ and $Y_{1,32}=r_{32}Y_{2,32}$, one can write the
imaginary part of the Jarskog invariant $J_{A}$ of the Yukawa interaction in
Eq. (4.2) as,

$\displaystyle\mathrm{Im}[J_{A}]=-\mathrm{Im}[r_{32}]|Y_{2,32}|^{2}=\frac{2m^{2}_{\tau}}{v^{2}c_{\beta-\alpha}}\kappa_{\tau}\sin\Delta\,.$
(4.5)

It is the imaginary part of the Jarlskog invariant that controls the size of
the BAU in early universe through lepton flavored baryogenesis [14]. Rewriting
Eq. (4.5) gives

$\sin\Delta=\frac{v^{2}c_{\beta-\alpha}}{2m^{2}_{\tau}\kappa_{\tau}}\mathrm{Im}[J_{A}].$
(4.6)

Thus, one may connect the $\tau$ Yukawa CP phase $\Delta$, which can be
measured at future lepton colliders, with CPV source for baryogenesis during
the era of EWSB in the early universe.

###  4.2 Sensitivity to the Baryogenesis Scenario

To make this connection concrete, we plot in Fig. 5 the 95% C.L. constraints
on $\kappa_{\tau}\cos\Delta$ and $\kappa_{\tau}\sin\Delta$ from present and
future collider probes and from lepton flavored EWBG. For generality, we also
set $\kappa_{\tau}^{\rm test}$ to be free to obtain a full picture on a two-
dimensional plot. The CP sensitivity is then depicted as the contours around
the true value $\Delta^{\rm true}=0^{\circ}$ and $\kappa^{\rm true}_{\tau}=1$
in the left pannel, with the green, blue (dashed), and red (dot dashed)
contours indicating the 95% C.L. sensitivities. The green dotted lines from
the origin $\kappa_{\tau}=0$ are added to show that the contour size
corresponds to roughly $7^{\circ}$ at 95% C.L. Consistent with the previous
observation, the CEPC and FCC-ee have comparable precision while that of the
ILC is slightly weaker due to different luminosities. For all three cases, the
pink region allowing for successful explanation of BAU is outside the 95% C.L.
contour. In other words, the lepton flavored BAU mechanism as given in Ref.
[14] could be excluded at better than than 95% C.L. . For comparison, we also
show the projected $\tau$ Yukawa CP measurement at the High Luminosity
(HL-)LHC [38] with the integrated luminosity of $3\,\mbox{ab}^{-1}$ as the
black contour, which will be further elaborated below. It is clear that even
with the HL-LHC, the THDM BAU mechanism can only be tested with barely 95%
C.L. The CP measurements at future lepton colliders can significantly improve
the situation.

Fig. 5:  Left: The 95% C.L. constraints on the $\tau$ Yukawa coupling at the
CEPC (green), FCC-ee (blue), and ILC (red) assuming the true value
$\Delta^{\rm true}=0^{\circ}$ and $\kappa^{\rm true}_{\tau}=1$. The gray band
gives 95% C.L. constraints from the current LHC signal strength measurements
[5, 6] while the black contour denotes the expected 95% C.L. constraint from
the combined measurements of $\mu_{\tau\tau}$ [59, 60] and $\Delta$ [38] at
the HL-LHC. Right:  The 3$\sigma$ contours for each collider assuming central
values $\Delta^{\rm true}=\pm 13.3^{\circ}$ and $\kappa^{\rm true}_{\tau}=1$
corresponding to the minimum $|\kappa_{\tau}\sin\Delta|$ compatible with the
BAU.

We also include the constraints from the measurement of the $h\to\tau\tau$
decay signal strength $\mu_{\tau\tau}$, which is proportional to
$\kappa_{\tau}^{2}$. The current data at the LHC indicate
$\mu_{\tau\tau}=1.09^{+0.35}_{-0.30}$ at ATLAS [5] and
$\mu_{\tau\tau}=0.85^{+0.12}_{-0.11}$ at CMS [6], which are depicted as the
gray region. In other words, the current measurement at LHC is still quite
crude with at least 10% uncertainty. At the HL-LHC, the $1\,\sigma$
uncertainty of $\mu_{\tau\tau}$ can be further improved to $5\%$ [59, 60],
which is further combined with the CP measurement [38] that is shown as the
black contour. The future lepton colliders can significantly improve the
sensitivities to 0.8% at the CEPC [61], 0.9% at the FCC-ee [8], and 1.09% at
the ILC [61], which are shown as the rings in the left panel of Fig. 5. Note
that these rings with inclusive $\tau$ decays are much narrower than the width
of the contours or equivalently the marginalized sensitivity on
$\kappa_{\tau}$ after integrating out the CP phase $\Delta$ from the original
two-dimensional distributions. The discrepancy comes from the fact that the
$\tau\rightarrow\pi\nu_{\tau}$ and $\tau\rightarrow\rho\nu_{\tau}$ channels
contribute only a very small fraction ($\sim 13\%$) of the inclusive decay
events. The strength measurement can provide very important complementary info
and reduce the parameter space to be explored.

Fig. 6: The CP discovery capability of the CEPC as a function of the $\Delta$
and $\kappa_{\tau}$ true values. The black dashed lines represents several
typical values of the significance, $\sqrt{\chi^{2}_{\rm CPV}}=5,10,15,20$.
The green region represents the space parameter where the sensitivity is below
95% C.L. The pink region represents the parameter space that can explain the
BAU in the lepton flavored EWBG scenario [14].

Instead of assuming the SM values $\kappa^{\rm true}_{\tau}=1$ and
$\Delta^{\rm true}=0^{\circ}$, it is interesting to ask the whether the lepton
flavored EWBG scenario can explain the BAU and at the same time produce a
signal that is distinguishable from the SM. To address this question, we show
in the right panel of Fig. 5 the similar contours around $\Delta^{\rm
true}=\pm 13.3^{\circ}$ and $\kappa^{\rm true}_{\tau}=1$ that is at the
boundary of the BAU region. Under this assumption, the CEPC and FCC could
establish the presence of CPV in the $\tau$ Yukawa interaction with $3\sigma$
significance, while for the ILC the significance would be somewhat weaker.

It is also interesting to investigate the behavior of the CP violation
sensitivity when one varies the assumed true values of $\kappa_{\tau}$. This
can be observed from Fig. 6 where we show the sensitivity as a function of the
CP phase $\Delta$ and the coupling strength $\kappa_{\tau}$. The dashed gray
lines give several typical sensitivities $\sqrt{\chi^{2}_{\rm
CPV}}=5,10,15,20$. Note that the dashed gray lines expand with larger $\tau$
Yukawa coupling due to event number enhancement. This is especially
significant for small $\kappa_{\tau}$ while for large values of
$\kappa_{\tau}$ the CP sensitivity does not change substantially. The BAU-
compatible region has a lower limit at $\kappa_{\tau}\approx 0.25$ due to the
lower limit on $\kappa_{\tau}\sin\Delta$ according to Fig. 5 and most of the
BAU-compatible region falls inside the $\sqrt{\chi^{2}_{\rm CPV}}=5$ curve,
corresponding to $5\,\sigma$ discovery.

##  5 Conclusions

Explaining the origin of the baryon asymmetry of the Universe is a key open
problem at the interface of particle and nuclear physics with cosmology. An
essential ingredient in the explanation is the presence of BSM CP violation.
In the electroweak baryogenesis scenario, the relevant CPV interactions would
have generated the BAU during the era of EWSB. The corresponding mass scale
makes these interactions in principle experimentally accessible. While null
results for permanent EDM searches place strong constraints on new flavor
diagonal, electroweak scale CPV interactions, flavor changing CPV effects are
significantly less restricted. Lepton flavored EWBG draws on this possibility,
with interesting implications for CPV in the tau-lepton Yukawa sector.

In this work, we have shown how measurements of CPV observable in Higgs di-tau
decays at prospective future lepton colliders could test this possibility,
with significant discovery potential if it is realized in nature. After making
a detailed comparison of the four differential distributions of the neutrino
azimuth angle $\delta\phi_{\nu}$, polarimeter $\delta\phi_{r}$, acoplanarity
$\phi^{*}$, and the $\Theta$ variable for the first time as well as various
detector responses, we explore the prospects of CP measurement in the $\tau$
Yukawa coupling at future lepton colliders. With $(5.6,5,2)\,\mbox{ab}^{-1}$
luminosity, the $1\,\sigma$ uncertainty can reach
$2.9^{\circ},3.2^{\circ},3.8^{\circ}$ at the CEPC, FCC-ee, and ILC,
respectively. This allows the possibility of distinguishing the attainable
EWBG from the CP conserving case with $3\,\sigma$ sensitivity. The future
lepton colliders are not just precision machines for detailing our
understanding of the Higgs boson, but can also make $\mathcal{O}(1)$
measurement of the possible new physics beyond the SM.

## Acknowledgements

SFG is sponsored by the Double First Class start-up fund (WF220442604)
provided by Tsung-Dao Lee Institute, Shanghai Jiao Tong University and the
Shanghai Pujiang Program (20PJ1407800). SFG is also grateful to Kai Ma for
sharing his PhD thesis with derivations on the differential distribution of
the $\tau\rightarrow\pi\nu_{\tau}$ decay as well as Manqi Ruan, Xin Chen, and
Dan Yu for useful discussions. GL would like to thank Shou-hua Zhu for helpful
discussions. MJRM was supported in part under National Natural Science
Foundation of China grant number 19Z103010239. GL and MJRM were supported in
part under U.S. Department of Energy contract number DE-SC0011095.

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