# Testing Bell inequality through $h\rightarrow\tau\tau$ at CEPC

Kai Ma<EMAIL_ADDRESS>Department of Physics, Shaanxi University of
Technology, Hanzhong 723000, Shaanxi, China  Tong Li<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin 300071, China

###### Abstract

The decay of Higgs boson into two spin-1/2 particles provides an ideal system
to reveal quantum entanglement and Bell-nonlocality. Future $e^{+}e^{-}$
colliders can improve the measurement accuracy of the spin correlation of tau
lepton pairs from Higgs boson decay. We show the testability of Bell
inequality through $h\rightarrow\tau\tau$ at Circular Electron Positron
Collider (CEPC). Two realistic methods of testing Bell inequality are
investigated, i.e., Törnqvist’s method and Clauser-Home-Shimony-Holt (CHSH)
inequality. In the simulation, we take into account the detector effects of
CEPC including uncertainties for tracks and jets from $Z$ boson in the
production of $e^{+}e^{-}\rightarrow Zh$. Necessary reconstruction approaches
are described to measure quantum entanglement between $\tau^{+}$ and
$\tau^{-}$. Finally, we show the sensitivity of CEPC to the Bell inequality
violation for the two methods.

###### Contents

  1. I Introduction
  2. II Local Quantum Model and Bell Inequality
    1. II.1 Törnqvist’s method
    2. II.2 Clauser-Home-Shimony-Holt Inequality
  3. III Measurements at Future Lepton Colliders
    1. III.1 Simulation and Detector Effects
    2. III.2 Reconstruction Method
    3. III.3 Reconstruction by using Impact Parameters
  4. IV Sensitivity of CEPC to the Bell Inequality Violation
  5. V Summary

##  I Introduction

It is well known that the most important debate on whether the Quantum
Mechanics (QM) is a complete local theory is the challenge raised by Einstein,
Podolsky, and Rosen (EPR), named EPR paradox Einstein  _et al._ (1935). The
interpretation of the EPR paradox in local hidden variable theory (LHVT) shows
the contradiction of LHVT with QM and presents the non-local nature of QM.
Later on, Bohm et al. proposed a realistic experiment with a system of two
spin-1/2 particles to illustrate the EPR paradox Bohm and Aharonov (1957).
Based on this consideration, Bell established a theorem that the two
particles’ spin correlation satisfies a Bell inequality (BI) in realistic LHVT
Bell (1964). By contrast, the QM predictions may violate this inequality in
some certain parameter space. Clauser, Horne, Shimony and Holt (CHSH) also
generalized the original Bell inequality and established a more practical
inequality Clauser  _et al._ (1969). The test of the Bell inequality delivers
a direct justification if the QM is a complete local theory Bertlmann (2006).

In the past years, the violation of this Bell inequality has been observed in
many low-energy experiments (such as optical experiments) Aspect  _et al._
(1982a, b); Ou and Mandel (1988); Weihs  _et al._ (1998); Bouwmeester  _et
al._ (1999); Pan  _et al._ (2000); White  _et al._ (1999); Yang  _et al._
(2005) as the foundation of the modern quantum information theory. The
predictions of QM are proved to be consistent with the results of these
experiments. However, for testing the completeness of QM beyond the
electromagnetic interaction regime, it is still a challenge of the test of
Bell inequality in high-energy physics (see review Ref. Ding  _et al._ (2007)
and references therein). At $e^{+}e^{-}$ colliders, the testability of BI was
first suggested by using the polarization correlation in the process of
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\rightarrow\pi^{-}p\pi^{+}\bar{p}$
Tornqvist (1981); Hao  _et al._ (2010) or $e^{+}e^{-}\rightarrow
Z\rightarrow\tau^{+}\tau^{-}$ Privitera (1992); Abel  _et al._ (1992); Dreiner
(1992). Based on the CHSH method, dedicated proposals were also raised to test
the BI in the final states of $t\bar{t}$ pair Afik and de Nova (2021);
Fabbrichesi  _et al._ (2021); Severi  _et al._ (2022); Afik and de Nova
(2022); Aguilar-Saavedra and Casas (2022); Afik and de Nova (2023); Aoude  _et
al._ (2022); Varma and Baker (2023) or two weak gauge bosons Barr (2022); Barr
_et al._ (2022); Aguilar-Saavedra (2023); Aguilar-Saavedra  _et al._ (2023);
Fabbrichesi  _et al._ (2023); Aoude  _et al._ (2023); Bernal  _et al._ (2023);
Fabbri  _et al._ (2023); Bi  _et al._ (2023) at the Large Hadron Collider
(LHC). Nevertheless, the spin-0 state formed by a pair of spin-1/2 particles
in the process such as
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\rightarrow\pi^{-}p\pi^{+}\bar{p}$
has the largest entanglement.

The Higgs boson is the only spin-0 elementary particle in the Standard Model
(SM) and can play as a natural spin singlet state to test LHVT through the
Bell inequality at high energies. The properties of SM Higgs boson will be
measured to high precision at future $e^{+}e^{-}$ colliders such as the
Circular Electron Positron Collider (CEPC) Dong  _et al._ (2018). We thus
propose to test the Bell inequality at CEPC through the Higgsstrahlung process
with subsequent decay $h\rightarrow\tau^{+}\tau^{-}$ Altakach  _et al._ (2023)

$\displaystyle e^{+}e^{-}\rightarrow Zh\rightarrow Z\tau^{+}\tau^{-}\;.$ (1)

The tau lepton pair is correlated in the decay process, and Bell inequality
(or quantum entanglement) can be tested by measuring their spin correlation.
However, spin information of the tau leptons can only be partially inferred
from its decay particle. Here we consider only the tau leptons followed by the
1-prong decay mode $\tau^{\pm}\rightarrow\pi^{\pm}\nu_{\tau}$ which is the
best spin analyzer for the tau lepton polarization. In principle, the other
decay modes can also be employed. However, it is more challenging in practice
because of the kinematic reconstruction of the tau lepton as well as limited
spin analyzing power. In order to have higher statistics, the associated $Z$
boson will be reconstructed by both its leptonic and hadronic decay modes.
Furthermore, both Törnqvist’s method Tornqvist (1981) and the CHSH method
Clauser  _et al._ (1969) are explored to evaluate the violation of Bell
inequality. To have a more realistic estimation on the experimental
sensitivities, we investigate the kinematic reconstruction and simulate the
detector effects to reveal the quantum entangled spin correlations in the
decay of tau pairs.

This paper is organized as follows. In Sec. II, we first outline the LHVT and
Bell inequality. Then we show Törnqvist’s method and the CHSH method in terms
of the polarization correlation in decay
$h\rightarrow\tau^{+}\tau^{-}\rightarrow\pi^{+}\bar{\nu}_{\tau}\pi^{-}\nu_{\tau}$.
In Sec. III, we describe the simulation of process $e^{+}e^{-}\rightarrow
Zh\rightarrow Z\tau^{+}\tau^{-}$ and discuss the detector effects as well as
reconstruction methods. The results of projected sensitivity to the Bell
inequality violation are given in Sec. IV. Finally, in Sec. V we summarize our
conclusions.

##  II Local Quantum Model and Bell Inequality

In this section, we describe the original and generalized expressions of Bell
inequality and the realistic methods of testing it in high-energy physics.

In the LHVT with the hidden variable being $\lambda$, the Bell inequality can
be phased in terms of the polarization correlation

$P(\vec{a},\vec{b})=\int
d\lambda\,q(\lambda)\cdot\mathcal{P}_{A}(\vec{a},\lambda)\cdot\mathcal{P}_{B}(\vec{b},\lambda)\,,$
(2)

where $\mathcal{P}_{A(B)}(\vec{x},\lambda)$ is the probability of the fermion
$A$ (or $B$) with spin along the direction $\vec{x}=\vec{a}~{}(\vec{b})$ for
given hidden variable $\lambda$, and $q(\lambda)$ is the corresponding
probability distribution of the hidden variable $\lambda$. The original
expression of Bell inequality refers to three independent spatial directions
$\vec{a}$, $\vec{b}$ and $\vec{c}$ as

$\big{|}P(\vec{a},\vec{b})-P(\vec{a},\vec{c})\big{|}\leq
1+P(\vec{b},\vec{c})\;.$ (3)

On the other hand, in QM the quantum average of the correlation operator
$\mathcal{O}(\vec{a},\vec{b})\equiv\big{[}\vec{\sigma}^{A}\cdot\vec{a}\big{]}\;\big{[}\vec{\sigma}^{B}\cdot\vec{b}\,\big{]}$
is given by

$P(\vec{a},\vec{b})=\big{\langle}00\big{|}\big{[}\vec{\sigma}^{A}\cdot\vec{a}\big{]}\;\big{[}\vec{\sigma}^{B}\cdot\vec{b}\,\big{]}\,\big{|}00\big{\rangle}=-\vec{a}\cdot\vec{b}\,,$
(4)

where $\langle 00|$ or $|00\rangle$ refers to a singlet state of the total
spin. After inserting the QM prediction Eq. (4) in Eq. (3), the Bell
inequality Eq. (3) may be violated in some region of phase space. However, in
realistic investigations, the spin correlation of the two fermions $A$ and $B$
can only be transferred to the kinematics of their decay products. In terms of
$h\rightarrow\tau^{+}\tau^{-}$ and tau leptons’ hadronic decay mode
$\tau^{\pm}\rightarrow\pi^{\pm}\nu_{\tau}$, we will describe two existing
methods to perform the test of Bell inequality at high energy colliders.

###  II.1 Törnqvist’s method

In Ref. Tornqvist (1981), Törnqvist suggested to test the BI by using the
polarization correlation in the process

$\displaystyle
e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\rightarrow\pi^{-}p\pi^{+}\bar{p}\;.$
(5)

The parent particle of $\Lambda\bar{\Lambda}$ could be either spin-0
$\eta_{c}$ or spin-1 $J/\psi$. Although there was debate as to whether such
proposal contains controversial assumption Abel  _et al._ (1992); Dreiner
(1992), it is a practical and attractive experiment to test the Bell
inequality. We instead establish the polarization correlation of decay
$h\rightarrow\tau^{+}\tau^{-}\rightarrow\pi^{+}\bar{\nu}_{\tau}\pi^{-}\nu_{\tau}$.
Since the Higgs boson is a scalar and the $\tau$-lepton is a spin-1/2
particle, the decay process $h\rightarrow\tau^{+}\tau^{-}$ provides an ideal
system for testing the Bell inequality. The joint spin density matrix for the
$\tau^{+}\tau^{-}$ system is given by

$\rho_{\tau\bar{\tau}}=\frac{1}{4}\Big{(}1-\vec{\sigma}_{\tau}\cdot\vec{\sigma}_{\bar{\tau}}\Big{)}\,,$
(6)

which means the state with parallel $\vec{\sigma}_{\tau}$ and
$\vec{\sigma}_{\bar{\tau}}$ vanishes because of spin-zero condition. For the
correlation operator $\mathcal{O}(\vec{a},\vec{b})$, one can easily find the
probability is given as

$P(\vec{a},\vec{b})=\big{\langle}00\big{|}\rho_{\tau\bar{\tau}}\mathcal{O}(\vec{a},\vec{b})\big{|}00\big{\rangle}=-\vec{a}\cdot\vec{b}\,.$
(7)

However, spin states of the $\tau$-leptons can not be measured directly at
collider, and can only be accessed by angular distributions of their decay
products. Here we only investigate the 1-prong decay mode
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$, in which the momentum direction of the
charged pion (or equivlently the neutrino) is correlated to the spin direction
of the tau lepton. Thus, this decay mode has the largest spin analyzing power
compared to the cases of the other decay modes. The decay amplitude of the
process $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$ in the rest frame of the mother
particle can be written as

$\mathcal{M}_{\tau}=\frac{1}{\sqrt{4\pi}}\big{(}S+P\vec{\sigma}_{\tau}\cdot\vec{a}\big{)}\,,$
(8)

where $\vec{a}$ is the unit vector along the $\pi^{-}$ momentum direction in
the rest frame of $\tau^{-}$, $S$ and $P$ are the $S$\- and $P$-wave
amplitudes respectively. Similar expression is valid for decay process
$\tau^{+}\rightarrow\pi^{+}\bar{\nu}_{\tau}$ as well. Then, the probability of
having $\pi^{-}$ flying along $\vec{a}$ and $\pi^{+}$ flying along $\vec{b}$
($\vec{b}$ is the unit vector along the $\pi^{+}$ momentum direction in the
rest frame of $\tau^{+}$) becomes

$\widetilde{P}(\vec{a},\vec{b})=\big{\langle}00\big{|}\rho_{\tau\bar{\tau}}\big{[}\mathcal{M}_{\tau}\mathcal{M}_{\bar{\tau}}\big{]}^{\dagger}\;\big{[}\mathcal{M}_{\tau}\mathcal{M}_{\bar{\tau}}\,\big{]}\,\big{|}00\big{\rangle}=\left[\frac{1}{4\pi}\Big{(}\big{|}S\big{|}^{2}+\big{|}P\big{|}^{2}\Big{)}\right]^{2}\Big{(}1+\alpha^{2}\vec{a}\cdot\vec{b}\Big{)}\,,$
(9)

where

$\alpha=-\frac{2\Re SP^{\ast}}{\big{|}S\big{|}^{2}+\big{|}P\big{|}^{2}}\approx
0.573\,.$ (10)

The above value is obtained by fitting in our numerical simulation. One can
see that $\widetilde{P}(\vec{a},\vec{b})$ is a partial measurement of the spin
states of the $\tau$-lepton pair. Its normalized value
$\widetilde{P}^{N}(\vec{a},\vec{b})$ is related to $P(\vec{a},\vec{b})$ by the
following relation

$P(\vec{a},\vec{b})=\frac{1}{\alpha^{2}}\Big{[}1-\widetilde{P}^{N}(\vec{a},\vec{b})\Big{]}\,.$
(11)

The normalized differential cross section is given as

$\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta_{ab}}={1\over
2}\widetilde{P}^{N}(\vec{a},\vec{b})=\frac{1}{2}\left[1-\alpha^{2}\,P(\vec{a},\vec{b})\right]\,,$
(12)

where $\cos\theta_{ab}\equiv\vec{a}\cdot\vec{b}=-P(\vec{a},\vec{b})$. On the
other hand, hidden variable theory predicts Tornqvist (1981)

$\left|P(\vec{a},\vec{b})\right|\leq
1-\frac{2}{\pi}\theta_{ab}\,,~{}~{}\theta_{ab}\in[0,\pi]\;.$ (13)

Then we have the following classical region satisfying the Bell inequality

$\begin{cases}\dfrac{1}{2}-\alpha^{2}\Big{(}\dfrac{1}{2}-\dfrac{\theta_{ab}}{\pi}\Big{)}\leq\dfrac{1}{\sigma}\dfrac{d\sigma}{d\cos\theta_{ab}}\leq\dfrac{1}{2}+\alpha^{2}\Big{(}\dfrac{1}{2}-\dfrac{\theta_{ab}}{\pi}\Big{)}\;,&\theta_{ab}\in[0,\pi/2]\\\\[8.53581pt]
\dfrac{1}{2}+\alpha^{2}\Big{(}\dfrac{1}{2}-\dfrac{\theta_{ab}}{\pi}\Big{)}\leq\dfrac{1}{\sigma}\dfrac{d\sigma}{d\cos\theta_{ab}}\leq\dfrac{1}{2}-\alpha^{2}\Big{(}\dfrac{1}{2}-\dfrac{\theta_{ab}}{\pi}\Big{)}\;,&\theta_{ab}\in(\pi/2,\pi]\end{cases}\,.$
(14)

###  II.2 Clauser-Home-Shimony-Holt Inequality

Clauser, Horne, Shimony and Holt (CHSH) generalized the original Bell
inequality Eq. (3) by considering general properties of the quantum density
matrix of a spin-1/2 particles system Clauser  _et al._ (1969). Density matrix
of the quantum state having two spin-1/2 particles can be expressed in general
as

$\rho=\frac{1}{4}\left[\mathbb{I}_{A}\otimes\mathbb{I}_{B}+A_{i}\cdot\big{(}\sigma_{A,i}\otimes\mathbb{I}_{B}\big{)}+B_{j}\cdot\big{(}\mathbb{I}_{A}\otimes\sigma_{B,j}\big{)}+C_{ij}\big{(}\sigma_{A,i}\otimes\sigma_{B,j}\big{)}\right]$
(15)

where $\sigma_{A(B),i}$ and $\mathbb{I}_{A(B)}$ are Pauli matrices and the
unit $2\times 2$ matrix for the particle $A~{}(B)$, respectively. The Bell
operator associated with the quantum CHSH inequality is defined as

$\mathcal{B}_{\rm
CHSH}=\vec{a}\cdot\vec{\sigma}_{A}\otimes\big{(}\vec{b}+\vec{b}^{\prime}\big{)}\cdot\vec{\sigma}_{B}+\vec{a}^{\prime}\cdot\vec{\sigma}_{A}\otimes\big{(}\vec{b}-\vec{b}^{\prime}\big{)}\cdot\vec{\sigma}_{B}\,,$
(16)

where $\vec{a}$, $\vec{a}^{\prime}$, $\vec{b}$, $\vec{b}^{\prime}$ are unit
vectors. Then the CHSH inequality is given by Horodecki  _et al._ (1995)

$\Big{|}{\rm Tr}(\rho\mathcal{B}_{\rm CHSH})\Big{|}\leq 2\;.$ (17)

Again, in practice it is hard to test the above inequality directly because of
the challenge in measuring of the spin directions $\vec{a}$,
$\vec{a}^{\prime}$, $\vec{b}$, $\vec{b}^{\prime}$. Alternatively, the matrix
with following coefficients

$C_{ij}={\rm Tr}\big{[}\rho\sigma_{i}\otimes\sigma_{j}\big{]}\,,$ (18)

can provide an indirect inequality. It was shown that if sum of the two
largest eigenvalues of the matrix $U=C^{T}C$ is larger than 1, then the CHSH
inequality is violated Horodecki  _et al._ (1995).

At colliders, the density matrix can be estimated by angular distributions of
the two spin-1/2 particles’ decay products. The normalized differential cross
section can be generally parameterized as Bernreuther  _et al._ (2015)

$\frac{\sigma^{-1}d\sigma}{d\cos\theta_{A,i}d\cos\theta_{B,j}}=\frac{1}{4}\Big{[}1+A_{i}\cos\theta_{A,i}+B_{j}\cos\theta_{B,j}+C_{ij}\cos\theta_{A,i}\cos\theta_{B,j}\Big{]}\;,$
(19)

where $\theta_{A(B),i(j)}$ are the polar angle of charged particles $A~{}(B)$
from the decays of their mother particles, and measured from the $i(j)$-th
axis. In our case of
$h\rightarrow\tau^{+}\tau^{-}\rightarrow\pi^{+}\bar{\nu}_{\tau}\pi^{-}\nu_{\tau}$,
the cosine quantities of the above polar angles are defined as

$\displaystyle\cos\theta_{\pi^{+},i}=\hat{p}_{\pi^{+}}\cdot\hat{i}\;,~{}~{}~{}\cos\theta_{\pi^{-},j}=\hat{p}_{\pi^{-}}\cdot\hat{j}\;,$
(20)

where the unit vectors $\hat{i}$ and $\hat{j}$ are defined in the rest frames
of the $\tau^{+}$ and $\tau^{-}$, respectively. They belong to a chosen
orthonormal basis $\hat{j}\in\\{\hat{k},\hat{r},\hat{n}\\}$ and satisfy the
relation $\hat{i}=-\hat{j}$. More precisely, we define an unit vector
$\hat{k}$ as the direction of $\tau^{-}$ momentum in the rest frame of the
Higgs boson. In the rest frame of the $\tau^{-}$ lepton, we define an unit
vector $\hat{r}$ in the decay plane of the $\tau^{-}$ lepton and perpendicular
to $\hat{k}$, and an unit vector $\hat{n}=\hat{k}\times\hat{r}$. It was shown
that the matrix $C$ can be estimated as Bernreuther  _et al._ (2015);
Fabbrichesi  _et al._ (2021).

$C_{ij}=-9\int
d\cos\theta_{A,i}d\cos\theta_{B,j}\frac{\sigma^{-1}d\sigma}{d\cos\theta_{A,i}d\cos\theta_{B,j}}\cos\theta_{A,i}\cos\theta_{B,j}\;.$
(21)

Then, one can diagonalize the spin correlation matrix $C^{T}C$ and find the
two largest eigenvalues to test the CHSH inequality. Since the Higgs boson is
considered to be on-shell, and its spin is zero, there is no any invariant
mass and orientation dependencies, compared to the $t\bar{t}$ final states in
Ref. Fabbrichesi  _et al._ (2021).

##  III Measurements at Future Lepton Colliders

At $e^{+}e^{-}$ colliders, the dominant production mode of the Higgs boson is
the so-called Higgsstrahlung channel, $e^{+}e^{-}\rightarrow Zh$. For our
interested mode $h\rightarrow\tau^{+}\tau^{-}$ with subsequent decay channels
$\tau^{\pm}\rightarrow\pi^{\pm}\nu$, two neutrinos appear in the final state.
Hence kinematic reconstruction is necessary in order to measure quantum
entanglement between $\tau^{+}$ and $\tau^{-}$. Next, we describe our
numerical simulation and implementation of the detector effects, and then the
reconstruction approaches in both the leptonic and hadronic decay modes of $Z$
boson.

###  III.1 Simulation and Detector Effects

Our numerical simulations are conducted using the MadGraph5_aMC@NLO Alwall
_et al._ (2014) package, and the quantum entangled spin correlations in the
tau-lepton decay are preserved by the TauDecay Hagiwara  _et al._ (2013)
package. For a realistic simulation, detector resolutions have to be included
for the objects. Charged tracks can be precisely measured by the CEPC detector
for the decay products in $Z\rightarrow\ell^{+}\ell^{-}~{}(\ell=e,\mu)$ or
$\tau^{\pm}\rightarrow\pi^{\pm}\nu$. Table 1 lists typical uncertainties of
the azimuthal angle ($\phi$), rapidity ($\eta$) and magnitude of the
transverse momentum ($|\vec{p}_{T}|$) at CEPC Dong  _et al._ (2018). One can
see that the CEPC uncertainties for tracks are quite small. We smear tracks
(both leptons and pions) by randomly sampling the azimuthal angle, pseudo-
rapidity and transverse momentum according to Gaussian distribution with
standard deviations given in Table 1 de Favereau  _et al._ (2014); Abe  _et
al._ (2010).

Observables | Uncertainties
---|---
$\phi$ | 0.0002$|\eta|$ \+ 0.000022
$\eta$ | 0.000016$|\eta|$ \+ 0.00000022
$|\vec{p}_{T}|$ | 0.036$|\vec{p}_{T}|$
Table 1: CEPC uncertainties for tracks

However, uncertainties of jet from $Z$ boson’s hadronic decay are relatively
large. Measurement of jet is not only smeared by fragmentation of partons and
the corresponding jet clustering processes, but also by the jet clustering of
reconstructed objects after detector response and its matching to jet at the
generator level Ruan  _et al._ (2018). The results in Ref. Lai ; Lai  _et al._
(2021) indicate that the uncertainty induced by the jet clustering and
matching can be as significant as those from the detector response, and
becomes the dominant uncertainty especially for final state with more than two
jets. Hence, sophisticated jet clustering algorithm has to be used Ruan  _et
al._ (2018). The energy resolution of the jet from light quarks can be
described as Dong  _et al._ (2018)

$\sigma_{\rm jet}(E)=\frac{25.7\%}{\sqrt{E}}\,\oplus\,2.4\%\,.$ (22)

Jets from charm and bottom quarks have slightly larger uncertainties because
of neutrinos in their decays Dong  _et al._ (2018). In consideration of this,
in this paper we also use a smearing algorithm to account for detector
resolutions. Detector responses to the partons (for the channel $Z\rightarrow
q\bar{q}$) are included by smearing energy of the partons according to
Gaussian distribution with standard deviations given in Eq. (22).

To see the impact of the above uncertainties, in Fig. 1, we show the
distributions of the difference between the real value and the smeared value
of transverse momentum $p_{T}$, azimuthal angle $\phi$ and rapidity $\eta$
(defined as $\Delta p_{T}$, $\Delta\phi$ and $\Delta\eta$) for the objects in
different decay modes of $Z$ boson. One can see that, due to the jet energy
smearing, the $p_{T}$ uncertainties of jets in $Z$ boson’s hadronic decay are
quite large compared to those in the leptonic mode. As a result, the $Z$ boson
decaying to dijet is not well reconstructed as shown in Fig. 2.

Furthermore, detection efficiency is also affected by particle identification.
For CEPC, the detector is designed to identify prompt leptons with high
efficiency and high purity Dong  _et al._ (2018). For leptons with energies
above 5 GeV, the identification efficiency is higher than 99% and
misidentification rate is smaller than 2%. For the $\tau$-jet with visible
energy between 20 and 80 GeV, the identification efficiency is above 80% with
a purity closing to 90% Dong  _et al._ (2018), further improvement can be
expected by optimizations. In our simulation, we ignore the momentum
dependence and use an universal identification efficiency 80% estimate
experimental significance for $\tau$-jet. For jets from hadronic decay of the
$Z$ boson, b-jets can be tagged with an efficiency of 80% and a purity of 90%.
Similarly, an efficiency of 60% and a purity of 60% can be achieved for the
c-jet tagging Dong  _et al._ (2018). In our case, since the $Z$ boson is
treated inclusively, jet-tagging is irrelevant to our analysis. Therefore, we
will use an factor of $0.8$ to account for possible efficiency loss in
reconstruction at the detector level.

Figure 1:  Normalized number of events as a function of $\Delta p_{T}$ (left),
$\Delta\phi$ (middle) and $\Delta\eta$ (right) for the objects (blue:
$e^{\pm}$ or jet $j$, green: $\pi^{\pm}$) in leptonic (top) and hadronic
(bottom) decay modes of $Z$ boson.

Figure 2:  Normalized number of events as a function of $\Delta p_{T}$ (left),
$\Delta\phi$ (middle) and $\Delta\eta$ (right) for the $Z$ boson in leptonic
(top) and hadronic (bottom) decay modes.

###  III.2 Reconstruction Method

For decay of the Higgs boson, the degree of freedom of the corresponding phase
space is 8, and 6 of them can be measured thanks to the two charged pions. As
a result, only 2 of them are undetermined. Considering that the decay width of
$\tau$-lepton is very small compared to its mass, it is an excellent
approximation to assume that both $\tau$s are on-shell. With help of the on-
shell conditions, the 8 kinematic degree of freedom can be determined. In the
following studies, we always assume that the $Z$-boson is reconstructed by its
visible decay products, and momentum of the Higgs boson is obtained by energy-
momentum conservation condition, i.e., $p_{h}=p_{e^{+}}+p_{e^{-}}-p_{Z}$. In
the approximation of that $P^{0}=\sqrt{s}$, and $\vec{P}=0$ with
$P=p_{e^{+}}+p_{e^{-}}$, invariant mass of the Higgs momentum is given by
$p_{h}^{2}=s+p_{Z}^{2}-2E_{Z}\sqrt{s}$. Since the decay width of Higgs boson
is also expected to be very small, $p_{h}^{2}\approx m_{h}^{2}$ is again an
excellent approximation. In practice, $p_{h}^{2}\equiv\hat{m}_{h}^{2}$ may
deviate from $m_{h}^{2}$ significantly due to experimental uncertainties in
measurement of the $Z$ boson momentum.

Since the Higgs boson decay isotropically, the reconstruction is done in the
rest frame of $h$. Assuming that both $\tau$-leptons are on-shell, then energy
and magnitude of the $\tau$-lepton momentum in this reference frame can be
obtained directly,

$E_{\tau}^{\star}=\frac{1}{2}\hat{m}_{h}\,,\;\;\;p_{\tau}^{\star}=\frac{1}{2}\hat{m}_{h}\sqrt{1-\frac{4m_{\tau}^{2}}{\hat{m}_{h}^{2}}}\,.$
(23)

Intersection angles between momentum of the $\tau^{\pm}$ and $\pi^{\pm}$ in
this frame are given as,

$\cos\theta_{\pm}^{\star}=\frac{2E_{\tau}^{\star}E_{\pi^{\pm}}^{\star}-m_{\tau}^{2}-m_{\pi^{\pm}}^{2}}{2p^{\star}_{\tau}p^{\star}_{\pi^{\pm}}}$
(24)

Without loss of generality, we define the $z$-axis as the momentum direction
of the negatively charged decay product, and the positively charged decay
product lies in the $x-z$ plane. In this reference frame, momenta of the of
the charged decay products can be written as,

$\displaystyle p_{\pi^{-}}^{\star\mu}$ $\displaystyle=$ $\displaystyle
E_{\pi^{-}}^{\star}\big{(}1,\,0,\,0,\,\beta_{\pi^{-}}^{\star}\big{)}\,,$ (25)
$\displaystyle p_{\pi^{+}}^{\star\mu}$ $\displaystyle=$ $\displaystyle
E_{\pi^{+}}^{\star}\big{(}1,\,\beta_{\pi^{+}}^{\star}\sin\theta_{\pi^{+}}^{\star},\,0,\,\beta_{\pi^{+}}^{\star}\cos\theta_{\pi^{+}}^{\star}\big{)}\,.$
(26)

Furthermore, assuming azimuthal angle of the $\tau^{-}$-lepton is
$\phi_{-}^{\star}$, then it’s momentum can be parameterized as,

$p_{\tau^{-}}^{\star\mu}=E_{\tau}^{\star}\big{(}1,\,\beta_{\tau}^{\star}\sin\theta_{-}^{\star}\cos\phi_{-}^{\star},\,\beta_{\tau}^{\star}\sin\theta_{-}^{\star}\sin\phi_{-}^{\star},\,\beta_{\tau}^{\star}\cos\theta_{-}^{\star}\big{)}\,,$
(27)

where $\beta_{\tau}^{\star}=\sqrt{1-4m_{\tau}^{2}/\hat{m}_{h}^{2}}$. It turns
out that momentum of the $\tau^{+}$-lepton is given as

$p_{\tau^{+}}^{\star\mu}=E_{\tau}^{\star}\big{(}1,\,-\beta_{\tau}^{\star}\sin\theta_{-}^{\star}\cos\phi_{-}^{\star},\,-\beta_{\tau}^{\star}\sin\theta_{-}^{\star}\sin\phi_{-}^{\star},\,-\beta_{\tau}^{\star}\cos\theta_{-}^{\star}\big{)}\,.$
(28)

Using the equation,
$\vec{p}_{\tau^{+}}^{\;\star}\cdot\vec{p}_{\pi^{+}}^{\;\star}=\cos\theta_{+}^{\star}\big{|}\vec{p}_{\tau^{+}}^{\;\star}\big{|}\big{|}\vec{p}_{\pi^{+}}^{\;\star}\big{|}\,,$
one can immediately have,

$-\sin\theta_{\pi^{+}}^{\star}\sin\theta_{-}^{\star}\cos\big{(}\phi_{-}^{\star}\big{)}-\cos\theta_{\pi^{+}}^{\star}\cos\theta_{-}^{\star}=\cos\theta_{+}^{\star}\,.$
(29)

Then we get following solutions,

$\phi_{-}^{\star}=\pm\arccos\left[-\frac{\cos\theta_{\pi^{+}}^{\star}\cos\theta_{-}^{\star}+\cos\theta_{+}^{\star}}{\sin\theta_{\pi^{+}}^{\star}\sin\theta_{-}^{\star}}\right]\,.$
(30)

Both solutions satisfy all the kinematic constraints, hence there is a two-
fold ambiguity.

We then test the above analytical reconstruction method at parton level. Fig.
3 shows the longitudinal and transverse correlations as a result of the above
analytical solutions for the leptonic decay mode of $Z$ boson. In the left
panel of Fig. 3, the colored densities indicate the true values of
$\cos\theta_{\pi^{+}}$ versus $\cos\theta_{\pi^{-}}$ and the black contours
show the reconstructed values. The above two-fold ambiguity induces a
reduction of the transverse spin correction which is described by the
azimuthal angle difference of the two decay planes $\delta\phi$, as shown in
the right panel of Fig. 3. In the leptonic decay mode of $Z$, one can see that
the above analytical reconstruction method works well.

However, there are some drawbacks in this analytical reconstruction method.
First of all, the two-fold ambiguity of kinematic solutions exists as
mentioned above. One cannot determine the complete $\phi_{-}^{\star}$
distribution at a time. More importantly, due to the energy uncertainty of
jets, the $Z$ boson cannot be well reconstructed in its hadronic decay mode.
As a result, the uncertainties of the Higgs momentum given by analytical
reconstruction are very large in the hadronic decay mode of the $Z$ boson.
Quantum correlation effects are completely washed out, and hence it is nearly
impossible to observe violation of the Bell identity in the hadronic mode.
Therefore, we adopt the other reconstruction method by using impact parameters
in next section.

Figure 3:  Longitudinal (left) and transverse (right) correlations as a result
of analytical reconstruction method for leptonic decay mode of $Z$ boson at
parton level.

###  III.3 Reconstruction by using Impact Parameters

It was shown that impact parameters of the charged $\pi$s are very useful to
reconstruct the full decay kinematics Hagiwara  _et al._ (2017); Jeans and
Wilson (2018); Chen and Wu (2019). The $\tau$-leptons emerging from Higgs
decay are strongly boosted. As a result, its typical decay length $\sim
3000~{}\mu$m is long enough to induce sizable impact parameter for the charged
decay product. The CMS group has used the impact parameter to study CP
property of the interaction between Higgs and the tau pair Tumasyan  _et al._
(2022). Here we adopt a similar method proposed in Ref. Hagiwara  _et al._
(2017). Furthermore, excellent impact parameter resolution can be achieved by
the CEPC vertex detector. The main performance goals for spatial resolution
near the IP can be better than $3~{}\mu$m Ahmad  _et al._ (2015). Here, the
real impact parameters of the pions are smeared according to Gaussian
distribution with standard deviations $\sigma_{\rm IP}=3~{}\mu$m.

We use magnitudes of the $\tau^{\pm}$ momenta, $|\vec{p}_{\tau^{\pm}}|$, as
the free parameters for finding the best fit. For single tau decay, for
instance $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$, the opening angle between
$\tau^{-}$ and $\pi^{-}$ is

$\cos\theta_{\tau^{-}\pi^{-}}=\frac{2E_{\tau^{-}}E_{\pi^{-}}-m_{\tau}^{2}-m_{\pi^{-}}^{2}}{2|\vec{p}_{\tau^{-}}||\vec{p}_{\pi^{-}}|}\,,$
(31)

where $E_{\tau^{-}}=\sqrt{m_{\tau}^{2}+|\vec{p}_{\tau^{-}}|^{2}}$ given by the
on-shell condition. Then, the momentum of $\tau^{-}$ is given as

$\vec{p}_{\tau^{-}}=|\vec{p}_{\tau^{-}}|\cdot\frac{\vec{b}_{\pi^{-}}+\frac{|\vec{b}_{\pi^{-}}|}{\tan\theta_{\tau^{-}\pi^{-}}}\frac{\vec{p}_{\pi^{-}}}{|\vec{p}_{\pi^{-}}|}}{\bigg{|}\vec{b}_{\pi^{-}}+\frac{|\vec{b}_{\pi^{-}}|}{\tan\theta_{\tau^{-}\pi^{-}}}\frac{\vec{p}_{\pi^{-}}}{|\vec{p}_{\pi^{-}}|}\bigg{|}}\,,$
(32)

where $\vec{b}_{\pi^{-}}$ is the impact parameter of the $\pi^{-}$. Momentum
of the neutrino can be obtained by the momentum conservation condition,
$p_{\nu_{\tau}}^{\mu}=p_{\tau^{-}}^{\mu}-p_{\pi^{-}}^{\mu}$. Similarly, one
can obtain momenta of the $\tau^{+}$ and anti-neutrino as functions of the
parameter $|\vec{p}_{\tau^{+}}|$ and the impact parameter $\vec{b}_{\pi^{+}}$.
The best values of the parameters $|\vec{p}_{\tau^{\pm}}|$ are obtained by
minimizing following likelihood function

$\rho=\rho_{BW}(\hat{m}_{\tau\tau},m_{Z},\varGamma_{Z})\cdot\rho_{G}(\hat{m}_{\tau\tau}-m_{Z},\varGamma_{Z})\cdot\prod_{\mu=0,1,2,3}\rho_{G}(\hat{p}_{Z}^{\mu}-p_{Z}^{\mu},\sigma_{Z}^{\mu})\,,$
(33)

where $\rho_{BW}$ is the usual Breit-Wigner distribution for the resonant
production of the $Z$ boson, and $\hat{m}_{\tau\tau}$ is the reconstructed
invariant mass of the tau-lepton pair; $\hat{p}_{Z}^{\mu}$ is the
reconstructed momentum of the $Z$ boson, and $p_{Z}^{\mu}$ is the momentum
obtained by summing momenta of its decay product; $\rho_{G}(x,y)$ is the
Gaussian function with mean value $x$ and variance $y$. Here
$\sigma_{Z}^{\mu}$ is estimated by our numerical simulation.

In Figs. 4 and 5, after using the method of impact parameters, we show the
observable uncertainties for the reconstructed $\tau$ lepton and the
reconstructed $Z$ mass in different decay modes of $Z$ boson, respectively. It
turns out that the reconstruction in hadronic mode is as good as that in
leptonic mode for azimuthal angle, rapidity and the reconstructed $Z$ boson
mass. The reconstructed transverse momentum $p_{T}$ in hadronic mode is still
worse than that in leptonic mode. We also display the reconstructed
longitudinal and transverse correlations for hadronic and leptonic decay modes
of $Z$ in Fig. 6. As seen in the bottom panel, the reconstructed transverse
correlation agrees well with the true result at parton level. It turns out
that the method of impact parameters has no the drawback of two-fold
ambiguity.

Figure 4:  Normalized number of events as a function of $\Delta p_{T}$ (left),
$\Delta\phi$ (middle) and $\Delta\eta$ (right) for the reconstructed $\tau$
lepton in leptonic (blue) and hadronic (red) decay modes of $Z$ boson.  Figure
5:  Normalized number of events as a function of reconstructed $Z$ mass
$m_{f\bar{f}}^{\rm Rec}$ in leptonic (red) and hadronic (blue) decay modes of
$Z$ boson.

Figure 6:  Reconstructed longitudinal (top) and transverse (bottom)
correlations for hadronic (top left) and leptonic (top right) decay modes of
$Z$ boson.

Finally, in Fig. 7, we show the distribution of $\cos\theta_{\pi\pi}$ for
hadronic (red) and leptonic (blue) decay modes of $Z$ and compare with the
Bell inequality. The LHVT holds between the two dashed lines (gray fitted
region) given by the inequality in Eq. (14). The black solid line shows the
normalized differential cross section in Eq. (12) and the result at parton
level is shown in green dots. The simulation results after using the method of
impact parameters are represented by the red and blue histograms for hadronic
and leptonic decay modes of $Z$ boson, respectively. Naively seen from the
distributions, they stay outside the region satisfying the Bell inequality and
agree with the QM/QFT prediction in black solid line.

Figure 7: Reconstructed distributions of $\cos\theta_{\pi\pi}$ for Törnqvist’s
test of Bell inequality. The gray-fitted region is the phase space consistent
with classical prediction.

Fig. 8 shows the reconstructed angular distributions of the charged pions. In
general, quantum entanglement disappears when we observe any single angular
observable of the six angles. This is because quantum correlations among
$\pi^{+}$ and $\pi^{-}$ are integrated out, as shown by the flat distributions
in the middle panel for the $\pi^{+}$ and the right panel for both $\pi^{+}$
and $\pi^{-}$. Since the $x-z$ plane is by definition spanned by the momentum
direction of the $\tau^{-}$ and $\pi^{-}$, the observable
$\cos\theta_{\pi^{-},n}$ can only be zero as shown in the middle panel. The
nontrivial distributions shown in the left-panel of the Fig. 8 are purely
kinematic. Since $d\sigma/d\cos\theta_{\pi^{-},k}$ is proportional to a
constant and $\theta_{\pi^{-},r}={\pi\over 2}-\theta_{\pi^{-},k}$, we have

$\frac{\sigma^{-1}d\sigma}{d\cos\theta_{\pi^{-},r}}=\frac{\sigma^{-1}d\sigma}{d\sin\theta_{\pi^{-},k}}\propto{\rm
const.}\times\cot\theta_{\pi^{-},r}\,,$ (34)

which is essentially reconstructed in our approach as shown by the red-solid
and green-dashed lines in the left-panel of the Fig. 8. Similarly, the
asymmetric distribution for the $\pi^{+}$ is due to the fact that we have
defined $\cos\theta_{\pi-,r}>0$, which can give a nontrivial integration on
the differential cross section
$d\sigma/d\cos\theta_{\pi-,r}d\cos\theta_{\pi+,r}\propto
1+C_{rr}\cos\theta_{\pi-,r}\cos\theta_{\pi+,r}$ (the integration region is
limited to the range $\cos\theta_{\pi-,r}\in[0,\,1]$). The CHSH inequality is
buried in the coefficients $C_{ij}$.

Figure 8:  Reconstructed angular distributions of $\pi^{\pm}$ for
$\cos\theta_{\pi,r}$ (left), $\cos\theta_{\pi,n}$ (middle) and
$\cos\theta_{\pi,k}$ (right).

##  IV Sensitivity of CEPC to the Bell Inequality Violation

At the CEPC with $\sqrt{s}=240$ GeV, the total cross section of the
Higgsstrahlung process is

$\sigma_{Zh}=196.2\,{\rm fb}\,.$ (35)

A huge number of the Higgs boson events will be produced with an expected
integrated luminosity of $\mathcal{L}=5.6~{}{\rm ab}^{-1}$ Dong  _et al._
(2018). However, since both the branching ratios
$\mathcal{B}(h\rightarrow\tau\tau)=6.32\%$ and
$\mathcal{B}(\tau\rightarrow\pi\nu_{\tau})=10.82\%$ are small, only hundreds
of the events are available to test the Bell inequality. The following
kinematic cuts are used to select well-reconstructed events, and match to the
real detector configuration

$\displaystyle p_{T}(\ell/j)>10\,{\rm
GeV},~{}|\eta(\ell/j)|<3,~{}\big{|}m^{\rm Rec.}_{ff}-m_{Z}\big{|}<10\,{\rm
GeV}.$ (36)

The efficiency for the above kinematic cuts is 0.645 for the decay mode
$Z\rightarrow\ell\ell$, and 0.648 for the hadronic decay mode $Z\rightarrow
jj$. Furthermore, as we have mentioned, a universal jet reconstruction
efficiency 0.8 will be used in our following estimation. The number of events
can be further reduced by $\tau$-jet identification which is assumed to be 0.8
for a purity closing to 90% at the CEPC Dong  _et al._ (2018). Table 2 gives
the expected number of events at the CEPC.

CEPC (240 GeV, $5.6~{}{\rm ab}^{-1}$) | $Z\rightarrow\ell\ell$ | $Z\rightarrow jj$
---|---|---
No. of Events | 55 | 568
Kin. Cuts and jet reconstruction | 22 | 151
$\tau$-identification | 14 | 97
Table 2: Number of events used to test the Bell inequality at the CEPC.

The experimental sensitivity for the Törnqvist’s approach is studied by
defining the following asymmetric observable

$\mathcal{A}=\frac{N(\cos\theta_{\pi\pi}<0)-N(\cos\theta_{\pi\pi}>0)}{N(\cos\theta_{\pi\pi}<0)+N(\cos\theta_{\pi\pi}>0)}\,.$
(37)

The analytical prediction of the observable $\mathcal{A}$ is
$\mathcal{A}=0.164$ in QM or $\mathcal{A}=0.119$ in LHVT. The experimental
sensitivity at the CEPC is estimated by performing 10000 pseudo-experiments.
We obtain $\mathcal{A}=0.155\pm 0.287$ for $Z\rightarrow\ell\ell$ channel and
$\mathcal{A}=0.140\pm 0.098$ for $Z\rightarrow jj$ channel, respectively, as
listed in Table 3. Both central values are closer to the prediction of QM. The
channel from the hadronic decay mode of $Z$ boson produces more events and
gives more reasonable result with small error. In the CHSH approach, the
violation of Bell inequality refers to the fact that the sum of the two
largest eigenvalues of the matrix $U=C^{T}C$ (denoted by $m_{1}+m_{2}$) is
larger than 1. It turns out that both channels lead to $m_{1}+m_{2}>1$, as
listed in Table 3. As we can see, for both the Törnqvist’s and CHSH approachs,
the Bell inequality can be tested at the CEPC at $1\sigma$ level. It is
expected that the sensitivity can be further improved by using sophisticated
jet reconstruction method and enhanced $\tau$-jet identification efficiency.

Channels | Obs. | QM | Clas. | Exp. @ CEPC
---|---|---|---|---
$Z\rightarrow\ell\ell$ | $\mathcal{A}$ | 0.164 | 0.119 | $0.155\pm 0.287$
$m_{1}+m_{2}$ | $>1$ | $\leq 1$ | $2.12\pm 1.11$
$Z\rightarrow jj$ | $\mathcal{A}$ | 0.164 | 0.119 | $0.140\pm 0.098$
$m_{1}+m_{2}$ | $>1$ | $\leq 1$ | $1.20\pm 0.37$
Table 3: The results of observables testing the Bell inequality in Törnqvist’s
method and the CHSH approach. The experimental predictions are given for the
CEPC with colliding energy $\sqrt{s}=240\,{\rm GeV}$ and a total luminosity of
$5.6~{}\,{\rm ab}^{-1}$.

##  V Summary

Since spin state can not be directly measured at collider, it is a challenge
of the test of quantum entanglement and Bell-nonlocality in high-energy
collider physics. However, testing Bell-nonlocality in high energy scattering
process is essentially important because it provides a unique way to address
the quantum entanglement at high energy scale. We investigate the testability
of Bell inequality through $h\rightarrow\tau^{+}\tau^{-}$, which is an ideal
system to observe the LHVT violation, at future $e^{+}e^{-}$ collider CEPC. We
demonstrated how to use angular distributions of decay products of the spin-
correlated $\tau$-pair to address the Bell-nonlocality. Future $e^{+}e^{-}$
colliders can improve the measurement accuracy of the spin correlation of tau
lepton pairs from Higgs boson decay. Two realistic methods of testing Bell
inequality, i.e., Törnqvist’s method and the CHSH inequality are studied in
terms of the polarization correlation in decay
$h\rightarrow\tau^{+}\tau^{-}\rightarrow\pi^{+}\bar{\nu}_{\tau}\pi^{-}\nu_{\tau}$.

We simulate the production of $e^{+}e^{-}\rightarrow Zh\rightarrow
Z\tau^{+}\tau^{-}$ as well as the $Z$ boson’s leptonic and hadronic decay
modes. The detector effects of CEPC including uncertainties for tracks and
jets from $Z$ boson are taken into account. We also describe necessary
reconstruction approaches to measure quantum entanglement between $\tau^{+}$
and $\tau^{-}$. Finally, we find that for both the Törnqvist’s and CHSH
approachs, the Bell inequality can be tested at the CEPC at $1\sigma$ level.
Further improvements are expected by employing sophisticated jet
reconstruction method and enhanced $\tau$-jet identification efficiency.

## Acknowledgements

T.L. would like to thank Xue-Qian Li for helpful discussion. T.L. is supported
by the National Natural Science Foundation of China (Grants No. 11975129,
12035008) and “the Fundamental Research Funds for the Central Universities”,
Nankai University (Grant No. 63196013). K.M. was supported by the Innovation
Capability Support Program of Shaanxi (Program No. 2021KJXX-47).

## References

  * Einstein  _et al._ (1935) Albert Einstein, Boris Podolsky, and Nathan Rosen, “Can quantum mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
  * Bohm and Aharonov (1957) D. Bohm and Y. Aharonov, “Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky,” Phys. Rev. 108, 1070–1076 (1957).
  * Bell (1964) J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics Physique Fizika 1, 195–200 (1964).
  * Clauser  _et al._ (1969) John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt, “Proposed experiment to test local hidden variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
  * Bertlmann (2006) R. A. Bertlmann, “Entanglement, Bell inequalities and decoherence in particle physics,” Lect. Notes Phys. 689, 1–45 (2006), arXiv:quant-ph/0410028 .
  * Aspect  _et al._ (1982a) Alain Aspect, Philippe Grangier, and Gerard Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91–97 (1982a).
  * Aspect  _et al._ (1982b) Alain Aspect, Jean Dalibard, and Gerard Roger, “Experimental test of Bell’s inequalities using time varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982b).
  * Ou and Mandel (1988) Z. Y. Ou and L. Mandel, “Violation of Bell’s Inequality and Classical Probability in a Two-Photon Correlation Experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
  * Weihs  _et al._ (1998) Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, and Anton Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998), arXiv:quant-ph/9810080 .
  * Bouwmeester  _et al._ (1999) Dik Bouwmeester, Jian-Wei Pan, Matthew Daniell, Harald Weinfurter, and Anton Zeilinger, “Observation of three photon Greenberger-Horne-Zeilinger entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999), arXiv:quant-ph/9810035 .
  * Pan  _et al._ (2000) J. W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement,” Nature 403, 515–519 (2000).
  * White  _et al._ (1999) Andrew G. White, Daniel F. V. James, Philippe H. Eberhard, and Paul G. Kwiat, “Nonmaximally entangled states: Production, characterization, and utilization,” Phys. Rev. Lett. 83, 3103–3107 (1999).
  * Yang  _et al._ (2005) Tao Yang, Qiang Zhang, Jun Zhang, Juan Yin, Zhi Zhao, Marek Żukowski, Zeng-Bing Chen, and Jian-Wei Pan, “All-versus-nothing violation of local realism by two-photon, four-dimensional entanglement,” Phys. Rev. Lett. 95, 240406 (2005).
  * Ding  _et al._ (2007) Yi-Bing Ding, Jun-li Li, and Cong-Feng Qiao, “Bell Inequalities in High Energy Physics,” (2007), arXiv:hep-ph/0702271 .
  * Tornqvist (1981) Nils A. Tornqvist, “Suggestion for Einstein-podolsky-rosen Experiments Using Reactions Like $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\rightarrow\pi^{-}p\pi^{+}\bar{p}$,” Found. Phys. 11, 171–177 (1981).
  * Hao  _et al._ (2010) Xi-Qing Hao, Hong-Wei Ke, Yi-Bing Ding, Peng-Nian Shen, and Xue-Qian Li, “Testing the Bell Inequality at Experiments of High Energy Physics,” Chin. Phys. C 34, 311–318 (2010), arXiv:0904.1000 [hep-ph] .
  * Privitera (1992) Paolo Privitera, “Decay correlations in e+ e- —$>$ tau+ tau- as a test of quantum mechanics,” Phys. Lett. B 275, 172–180 (1992).
  * Abel  _et al._ (1992) S. A. Abel, M. Dittmar, and Herbert K. Dreiner, “Testing locality at colliders via Bell’s inequality?” Phys. Lett. B 280, 304–312 (1992).
  * Dreiner (1992) Herbert K. Dreiner, “Bell’s inequality and tau physics at LEP,” in  _2nd Workshop on Tau Lepton Physics_ (1992) arXiv:hep-ph/9211203 .
  * Afik and de Nova (2021) Yoav Afik and Juan Ramón Muñoz de Nova, “Entanglement and quantum tomography with top quarks at the LHC,” Eur. Phys. J. Plus 136, 907 (2021), arXiv:2003.02280 [quant-ph] .
  * Fabbrichesi  _et al._ (2021) M. Fabbrichesi, R. Floreanini, and G. Panizzo, “Testing Bell Inequalities at the LHC with Top-Quark Pairs,” Phys. Rev. Lett. 127, 161801 (2021), arXiv:2102.11883 [hep-ph] .
  * Severi  _et al._ (2022) Claudio Severi, Cristian Degli Esposti Boschi, Fabio Maltoni, and Maximiliano Sioli, “Quantum tops at the LHC: from entanglement to Bell inequalities,” Eur. Phys. J. C 82, 285 (2022), arXiv:2110.10112 [hep-ph] .
  * Afik and de Nova (2022) Yoav Afik and Juan Ramón Muñoz de Nova, “Quantum information with top quarks in QCD,” Quantum 6, 820 (2022), arXiv:2203.05582 [quant-ph] .
  * Aguilar-Saavedra and Casas (2022) J. A. Aguilar-Saavedra and J. A. Casas, “Improved tests of entanglement and Bell inequalities with LHC tops,” Eur. Phys. J. C 82, 666 (2022), arXiv:2205.00542 [hep-ph] .
  * Afik and de Nova (2023) Yoav Afik and Juan Ramón Muñoz de Nova, “Quantum Discord and Steering in Top Quarks at the LHC,” Phys. Rev. Lett. 130, 221801 (2023), arXiv:2209.03969 [quant-ph] .
  * Aoude  _et al._ (2022) Rafael Aoude, Eric Madge, Fabio Maltoni, and Luca Mantani, “Quantum SMEFT tomography: Top quark pair production at the LHC,” Phys. Rev. D 106, 055007 (2022), arXiv:2203.05619 [hep-ph] .
  * Varma and Baker (2023) Mira Varma and O. K. Baker, “Quantum Entanglement in Top Quark Pair Production,” (2023), arXiv:2306.07788 [hep-ph] .
  * Barr (2022) Alan J. Barr, “Testing Bell inequalities in Higgs boson decays,” Phys. Lett. B 825, 136866 (2022), arXiv:2106.01377 [hep-ph] .
  * Barr  _et al._ (2022) Alan J. Barr, Pawel Caban, and Jakub Rembieliński, “Bell-type inequalities for systems of relativistic vector bosons,” (2022), arXiv:2204.11063 [quant-ph] .
  * Aguilar-Saavedra (2023) J. A. Aguilar-Saavedra, “Laboratory-frame tests of quantum entanglement in $H\rightarrow WW$,” Phys. Rev. D 107, 076016 (2023), arXiv:2209.14033 [hep-ph] .
  * Aguilar-Saavedra  _et al._ (2023) J. A. Aguilar-Saavedra, A. Bernal, J. A. Casas, and J. M. Moreno, “Testing entanglement and Bell inequalities in $H\rightarrow ZZ$,” Phys. Rev. D 107, 016012 (2023), arXiv:2209.13441 [hep-ph] .
  * Fabbrichesi  _et al._ (2023) M. Fabbrichesi, R. Floreanini, E. Gabrielli, and L. Marzola, “Bell inequalities and quantum entanglement in weak gauge bosons production at the LHC and future colliders,” (2023), arXiv:2302.00683 [hep-ph] .
  * Aoude  _et al._ (2023) Rafael Aoude, Eric Madge, Fabio Maltoni, and Luca Mantani, “Probing new physics through entanglement in diboson production,” (2023), arXiv:2307.09675 [hep-ph] .
  * Bernal  _et al._ (2023) Alexander Bernal, Paweł Caban, and Jakub Rembieliński, “Entanglement and Bell inequalities violation in $H\rightarrow ZZ$ with anomalous coupling,” (2023), arXiv:2307.13496 [hep-ph] .
  * Fabbri  _et al._ (2023) Federica Fabbri, James Howarth, and Theo Maurin, “Isolating semi-leptonic $H\rightarrow WW^{*}$ decays for Bell inequality tests,” (2023), arXiv:2307.13783 [hep-ph] .
  * Bi  _et al._ (2023) Qi Bi, Qing-Hong Cao, Kun Cheng, and Hao Zhang, “New observables for testing Bell inequalities in $W$ boson pair production,” (2023), arXiv:2307.14895 [hep-ph] .
  * Dong  _et al._ (2018) Mingyi Dong _et al._ (CEPC Study Group), “CEPC Conceptual Design Report: Volume 2 - Physics & Detector,” (2018), arXiv:1811.10545 [hep-ex] .
  * Altakach  _et al._ (2023) Mohammad Mahdi Altakach, Priyanka Lamba, Fabio Maltoni, Kentarou Mawatari, and Kazuki Sakurai, “Quantum information and CP measurement in $H\rightarrow\tau^{+}\tau^{-}$ at future lepton colliders,” Phys. Rev. D 107, 093002 (2023), arXiv:2211.10513 [hep-ph] .
  * Horodecki  _et al._ (1995) R. Horodecki, P. Horodecki, and M. Horodecki, “Violating bell inequality by mixed spin-12 states: necessary and sufficient condition,” Physics Letters A 200, 340–344 (1995).
  * Bernreuther  _et al._ (2015) Werner Bernreuther, Dennis Heisler, and Zong-Guo Si, “A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions,” JHEP 12, 026 (2015), arXiv:1508.05271 [hep-ph] .
  * Alwall  _et al._ (2014) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,” JHEP 07, 079 (2014), arXiv:1405.0301 [hep-ph] .
  * Hagiwara  _et al._ (2013) Kaoru Hagiwara, Tong Li, Kentarou Mawatari, and Junya Nakamura, “TauDecay: a library to simulate polarized tau decays via FeynRules and MadGraph5,” Eur. Phys. J. C 73, 2489 (2013), arXiv:1212.6247 [hep-ph] .
  * de Favereau  _et al._ (2014) J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi (DELPHES 3), “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” JHEP 02, 057 (2014), arXiv:1307.6346 [hep-ex] .
  * Abe  _et al._ (2010) Toshinori Abe _et al._ (Linear Collider ILD Concept Group -), “The International Large Detector: Letter of Intent,”  (2010), 10.2172/975166, arXiv:1006.3396 [hep-ex] .
  * Ruan  _et al._ (2018) Manqi Ruan _et al._ , “Reconstruction of physics objects at the Circular Electron Positron Collider with Arbor,” Eur. Phys. J. C 78, 426 (2018), arXiv:1806.04879 [hep-ex] .
  * (46) Pei-Zhu Lai, “Jet Reconstruction at the CEPC (2017),” CEPC-REC-2017-002.
  * Lai  _et al._ (2021) Pei-Zhu Lai, Manqi Ruan, and Chia-Ming Kuo, “Jet performance at the circular electron-positron collider,” JINST 16, P07037 (2021), arXiv:2104.05029 [hep-ex] .
  * Hagiwara  _et al._ (2017) Kaoru Hagiwara, Kai Ma, and Shingo Mori, “Probing CP violation in $h\rightarrow\tau^{-}\tau^{+}$ at the LHC,” Phys. Rev. Lett. 118, 171802 (2017), arXiv:1609.00943 [hep-ph] .
  * Jeans and Wilson (2018) D. Jeans and G. W. Wilson, “Measuring the CP state of tau lepton pairs from Higgs decay at the ILC,” Phys. Rev. D 98, 013007 (2018), arXiv:1804.01241 [hep-ex] .
  * Chen and Wu (2019) Xin Chen and Yongcheng Wu, “Probing the CP-Violation effects in the $h\tau\tau$ coupling at the LHC,” Phys. Lett. B 790, 332–338 (2019), arXiv:1708.02882 [hep-ph] .
  * Tumasyan  _et al._ (2022) Armen Tumasyan _et al._ (CMS), “Analysis of the $CP$ structure of the Yukawa coupling between the Higgs boson and $\tau$ leptons in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” JHEP 06, 012 (2022), arXiv:2110.04836 [hep-ex] .
  * Ahmad  _et al._ (2015) Muhammd Ahmad _et al._ , “CEPC-SPPC Preliminary Conceptual Design Report. 1. Physics and Detector,” (2015).