Test of Bell Locality Violation in Flavor Entangled Neutral Meson Pair (2024)

Kaiwen Chen1111kwchen@nnu.edu.cn, Zhi-Peng Xing1222zpxing@nnu.edu.cn, Ruilin Zhu1,2,3333rlzhu@njnu.edu.cn1 Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210023, China
2 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China
3 Peng Huanwu Innovation Research Center, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China

Abstract

The quasi-spin entanglement of neutral mesons to test the Bell nonlocality is systematically studied. In the case of CP violation, the concrete expression of the Bell inequality for entangled neutral meson pairs is derived. The violation of Bell inequality in the oscillations of Bd0B¯d0superscriptsubscript𝐵𝑑0superscriptsubscript¯𝐵𝑑0B_{d}^{0}-\bar{B}_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, Bs0B¯s0superscriptsubscript𝐵𝑠0superscriptsubscript¯𝐵𝑠0B_{s}^{0}-\bar{B}_{s}^{0}italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, D0D¯0superscript𝐷0superscript¯𝐷0D^{0}-\bar{D}^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and K0K¯0superscript𝐾0superscript¯𝐾0K^{0}-\bar{K}^{0}italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT meson pair systems are found at certain evolution time includingthe latest experimental data. Our study is helpful for studying CP-violating involved flavor entanglement and testing Bell inequality violation in the current and future high energy experiment facilities.

I Introduction

Quantum entanglement is a unique phenomenon in quantum mechanics (QM), which does not exist in the classical physics. Entanglement in quantum mechanics causes these correlative systems to have instantaneous effects on each other regardless of distance. In 1935, Einstein, Podolsky and Rosen (EPR)Einstein:1935rr proposed a hypothetical experiment to demonstrate that quantum mechanics is incomplete and introduce hidden variable local theory to solve this problem. However, this idea remained untestable until 1965, when Bell proposed the Bell inequalityBell:1964kc , which local realism (LR) would need to satisfy and quantum mechanics would violate under certain circumstances.

The violation of the Bell inequality was initially detected by low-energy photon pairs with spin entanglementAspect:1982fx ; Weihs:1998gy . The two photons are prepared into a singlet state, and the polarizations of the pair of photons are then measured from different directionsHorodecki:2009zz , which proves that quantum mechanics breaks the Bell inequality. Many experiments were further performed using superconducting circuitsStorz:2023jjx , and entangledatomsRosenfeld:2017rka .

Recently, the measurement of quantum entanglement and the test of Bell inequality violation have reattracted a lot of attention in the field of high energy particle physics. In high energy physics, quantum entanglement and the violation of Bell inequality are tested mainly through spin-correlative particle pairs. For a two-qubit system, Bell nonlocality is measured mainly by Clauser-Horne-Shimony-Holt (CHSH) inequalityClauser:1969ny . There have been a lot of works and results in high energy physics for the testing of such systems, e.g., e+eYY¯superscript𝑒superscript𝑒𝑌¯𝑌e^{+}e^{-}\to Y\bar{Y}italic_e start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT → italic_Y over¯ start_ARG italic_Y end_ARG at BESIIIWu:2024asu ; Li:2006fy , e+eτ+τsuperscript𝑒superscript𝑒superscript𝜏superscript𝜏e^{+}e^{-}\to\tau^{+}\tau^{-}italic_e start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_τ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT at BESIIEhataht:2023zzt , through top-quark, tau-lepton and photon pairsFabbrichesi:2022ovb , and the spinning gluons systemGuo:2024jch .For a two-qutrit system, the most commonly used inequality for Bell nonlocality detection is the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequalityCollins:2002sun , which can be generalized to systems of arbitrary dimensions. Great progresses have also been made in quantum entanglement and Bell non-local measurement of such systems, e.g., the B𝐵Bitalic_B meson decays into vector mesonsFabbrichesi:2023idl ; Li:2009rta , the Bcsubscript𝐵𝑐B_{c}italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT meson decays into vector mesonsChen:2024syv ; Geng:2023ffc , and the HZZ𝐻𝑍𝑍H\to ZZitalic_H → italic_Z italic_Z decays with anomalous couplingBernal:2024xhm ; Aguilar-Saavedra:2022wam .

In this work, we are mainly interested in the measurement of Bell nonlocality of entangled neutral meson pairs. We can view the flavor bases of these mixing particles as quasi-spins of the two-body mixing sytem. The flavor eigenstate is generally not the eigenstate of the neutral meson energy, so the polarization direction of the particle in the quasi-spin space will change with time. So we can choose different time points to measure the entangled neutral meson pair just as we can choose different directions to measure the polarization of a particle with spin. This method to test the Bell non-locality has been demonstrated by many previous literaturesBenatti:1997xt ; Foadi:1999sg ; Foadi:2000zz ; Samal:2002mh ; Ancochea:1998nx ; Bertlmann:2001ea ; Bramon:2002yg ; Go:2003tx ; Bramon:2005mg ; Bertlmann:2006hs ; Takubo:2021sdk .Therein the violation of the CHSH inequality is observed in B0B¯0superscript𝐵0superscript¯𝐵0B^{0}-\bar{B}^{0}italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT system using semileptonic B0/B¯0superscript𝐵0superscript¯𝐵0B^{0}/\bar{B}^{0}italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT / over¯ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT decays data at Belle experimentGo:2003tx . However, the CP violation effects in quantum entanglement are not included in these works, which motivatesus to systematically study the quasi-spin entanglement of neutral meson pairs in the most general case. Though the CP violation effects in neutral meson pair mixing are tiny, the precision particle experiments shall help us to capture the nature of quantum flavor entanglement and the nature of CP violation. In addition, the multi-dimensional time evolution and the maximum breaking of Bell inequality is also given.

The paper is arranged as follows. In Sec.II, we derive the evolution of the flavor wave function with time in the most general case of neutral entangled mesons, and consider the effect of CP violation on the evolution of the wave function. In Sec.III, we describe how to measure the Bell nonlocality of entangled neutral meson pairs and give the time-dependent CHSH inequality firstly. Then we calculate the specific expression of the CHSH inequality after the probability normalization at a specific point in time. Next we give out the time evolution of CHSH inequality value for different kinds of neutral meson pair system Bd0B¯d0superscriptsubscript𝐵𝑑0superscriptsubscript¯𝐵𝑑0B_{d}^{0}-\bar{B}_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, Bs0B¯s0superscriptsubscript𝐵𝑠0superscriptsubscript¯𝐵𝑠0B_{s}^{0}-\bar{B}_{s}^{0}italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, D0D¯0superscript𝐷0superscript¯𝐷0D^{0}-\bar{D}^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and K0K¯0superscript𝐾0superscript¯𝐾0K^{0}-\bar{K}^{0}italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT. A brief summary is given in the end.

II Evolution of neutral mesons over time

Considering the instantaneous production of a neutral meson pair in which two neutral mesons form a matter-antimatter pair, the neutral meson pair usually make up a flavor singlet through electromagnetic and strong interactions. Take the production of neutral B𝐵Bitalic_B meson pair for example, Υ(4S)Υ4𝑆\Upsilon(4S)roman_Υ ( 4 italic_S ) are largely produced at Belle-II experiments by the electro-positron annihilation and then decay into a Bd0B¯d0superscriptsubscript𝐵𝑑0superscriptsubscript¯𝐵𝑑0B_{d}^{0}\bar{B}_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT pair. In the rest frame of Υ(4S)Υ4𝑆\Upsilon(4S)roman_Υ ( 4 italic_S ), a Bd0B¯d0superscriptsubscript𝐵𝑑0superscriptsubscript¯𝐵𝑑0B_{d}^{0}\bar{B}_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT pair will move in opposite directions with the same magnitude of momentum. Since the JPCsuperscript𝐽𝑃𝐶J^{PC}italic_J start_POSTSUPERSCRIPT italic_P italic_C end_POSTSUPERSCRIPT of the Υ(4S)Υ4𝑆\Upsilon(4S)roman_Υ ( 4 italic_S ) equals 1superscript1absent1^{--}1 start_POSTSUPERSCRIPT - - end_POSTSUPERSCRIPT and the parity is conserved in the decay process, the flavor space wave function of the Bd0B¯d0superscriptsubscript𝐵𝑑0superscriptsubscript¯𝐵𝑑0B_{d}^{0}\bar{B}_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT pair should be antisymmetricBanerjee:2014vga . The same situation can be extended to the D𝐷Ditalic_D and K𝐾Kitalic_K meson pair systems. So the flavor wave function of this kind of neutral meson pair at the initial time t=0𝑡0t=0italic_t = 0 can be written as

|ψ(0)=12(|MM¯|M¯M),ket𝜓012ket𝑀¯𝑀ket¯𝑀𝑀\displaystyle\ket{\psi(0)}=\frac{1}{\sqrt{2}}(\ket{M\bar{M}}-\ket{\bar{M}M}),| start_ARG italic_ψ ( 0 ) end_ARG ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | start_ARG italic_M over¯ start_ARG italic_M end_ARG end_ARG ⟩ - | start_ARG over¯ start_ARG italic_M end_ARG italic_M end_ARG ⟩ ) ,(1)

where M𝑀Mitalic_M represents one kind of Bd0superscriptsubscript𝐵𝑑0B_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, Bs0superscriptsubscript𝐵𝑠0B_{s}^{0}italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, K0superscript𝐾0K^{0}italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, and D0superscript𝐷0D^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT particles while M¯¯𝑀\bar{M}over¯ start_ARG italic_M end_ARG represents the anti-particle. |Mket𝑀\ket{M}| start_ARG italic_M end_ARG ⟩ and |M¯ket¯𝑀\ket{\bar{M}}| start_ARG over¯ start_ARG italic_M end_ARG end_ARG ⟩ are flavor eigenstates of neutral mesons. |MM¯ket𝑀¯𝑀\ket{M\bar{M}}| start_ARG italic_M over¯ start_ARG italic_M end_ARG end_ARG ⟩ denotes the case that one particle M𝑀Mitalic_M flies to the right side and the other particle M¯¯𝑀\bar{M}over¯ start_ARG italic_M end_ARG flies to the left side. It is clear that the initial wave function is maximally flavor entangled state.

However, flavor eigenstates are generally not energy eigenstates of neutral mesons. In order to study the evolution of the wave function over time, we need to express the wave function as a superposition of energy eigenstates. In general, the energy eigenstate of a neutral meson can be written as

|M1=p|M+q|M¯,ketsubscript𝑀1𝑝ket𝑀𝑞ket¯𝑀\displaystyle\ket{M_{1}}=p\ket{M}+q\ket{\bar{M}},| start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩ = italic_p | start_ARG italic_M end_ARG ⟩ + italic_q | start_ARG over¯ start_ARG italic_M end_ARG end_ARG ⟩ ,(2)
|M2=p|Mq|M¯.ketsubscript𝑀2𝑝ket𝑀𝑞ket¯𝑀\displaystyle\ket{M_{2}}=p\ket{M}-q\ket{\bar{M}}.| start_ARG italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩ = italic_p | start_ARG italic_M end_ARG ⟩ - italic_q | start_ARG over¯ start_ARG italic_M end_ARG end_ARG ⟩ .(3)

For example, it is known as |M1=|BLketsubscript𝑀1ketsubscript𝐵𝐿\ket{M_{1}}=\ket{B_{L}}| start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩ = | start_ARG italic_B start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG ⟩ for light B𝐵Bitalic_B meson and |M2=|BHketsubscript𝑀2ketsubscript𝐵𝐻\ket{M_{2}}=\ket{B_{H}}| start_ARG italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩ = | start_ARG italic_B start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_ARG ⟩ for heavy B𝐵Bitalic_B meson. In the above formulation, in order to normalize the wave function it requires |p|2+|q|2=1superscript𝑝2superscript𝑞21|p|^{2}+|q|^{2}=1| italic_p | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_q | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1. We have p=q=12𝑝𝑞12p=q=\frac{1}{\sqrt{2}}italic_p = italic_q = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG in the case of CP conservation while pq𝑝𝑞p\neq qitalic_p ≠ italic_q in the case of CP violation.

The evolution of energy eigenstates can be expressed asBramon:2005mg

|M1(t)=eim1te12Γ1t|M1,ketsubscript𝑀1𝑡superscript𝑒𝑖subscript𝑚1𝑡superscript𝑒12subscriptΓ1𝑡ketsubscript𝑀1\displaystyle\ket{M_{1}(t)}=e^{-im_{1}t}e^{-\frac{1}{2}\Gamma_{1}t}\ket{M_{1}},| start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) end_ARG ⟩ = italic_e start_POSTSUPERSCRIPT - italic_i italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT | start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩ ,(4)
|M2(t)=eim2te12Γ2t|M2,ketsubscript𝑀2𝑡superscript𝑒𝑖subscript𝑚2𝑡superscript𝑒12subscriptΓ2𝑡ketsubscript𝑀2\displaystyle\ket{M_{2}(t)}=e^{-im_{2}t}e^{-\frac{1}{2}\Gamma_{2}t}\ket{M_{2}},| start_ARG italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) end_ARG ⟩ = italic_e start_POSTSUPERSCRIPT - italic_i italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT | start_ARG italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩ ,(5)

where we have considered the decay properties of neutral mesons. misubscript𝑚𝑖m_{i}italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and ΓisubscriptΓ𝑖\Gamma_{i}roman_Γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT represents the mass and the decay width of Misubscript𝑀𝑖M_{i}italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT(i=1,2𝑖12i=1,2italic_i = 1 , 2) meson.

If t1subscript𝑡1t_{1}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and t2subscript𝑡2t_{2}italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are used to represent the evolution time of the left and right side particles respectively, the wave function of the neutral meson pair at a particular moment can be written as

|ψ(t1,t2)=ket𝜓subscript𝑡1subscript𝑡2absent\displaystyle\ket{\psi(t_{1},t_{2})}=| start_ARG italic_ψ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG ⟩ =122pqei(m2t1+m1t2)12(Γ2t1+Γ1t2)(|M2M1\displaystyle\frac{1}{2\sqrt{2}pq}e^{-i(m_{2}t_{1}+m_{1}t_{2})-\frac{1}{2}(%\Gamma_{2}t_{1}+\Gamma_{1}t_{2})}\big{(}\ket{M_{2}M_{1}}divide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 2 end_ARG italic_p italic_q end_ARG italic_e start_POSTSUPERSCRIPT - italic_i ( italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( | start_ARG italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩
eiΔm(t1t2)+12ΔΓ(t1t2)|M1M2),\displaystyle-e^{i\Delta m(t_{1}-t_{2})+\frac{1}{2}\Delta\Gamma(t_{1}-t_{2})}%\ket{M_{1}M_{2}}\big{)},- italic_e start_POSTSUPERSCRIPT italic_i roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT | start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩ ) ,(6)

where Δm=m2m1Δ𝑚subscript𝑚2subscript𝑚1\Delta m=m_{2}-m_{1}roman_Δ italic_m = italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ΔΓ=Γ2Γ1ΔΓsubscriptΓ2subscriptΓ1\Delta\Gamma=\Gamma_{2}-\Gamma_{1}roman_Δ roman_Γ = roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. It is well known that the measurement of flavor eigenstates is a good way that can be measured well experimentally through their decay products. So changing the above formula to flavor eigenstates, one can find that

|ψ(t1,t2)=ket𝜓subscript𝑡1subscript𝑡2absent\displaystyle\ket{\psi(t_{1},t_{2})}=| start_ARG italic_ψ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG ⟩ =B22pq[p2(1A)|MM+pq(1+A)|MM¯\displaystyle\frac{B}{2\sqrt{2}pq}\big{[}p^{2}(1-A)\ket{MM}+pq(1+A)\ket{M\bar{%M}}divide start_ARG italic_B end_ARG start_ARG 2 square-root start_ARG 2 end_ARG italic_p italic_q end_ARG [ italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_A ) | start_ARG italic_M italic_M end_ARG ⟩ + italic_p italic_q ( 1 + italic_A ) | start_ARG italic_M over¯ start_ARG italic_M end_ARG end_ARG ⟩
pq(1+A)|M¯Mq2(1A)|M¯M¯.𝑝𝑞1𝐴ket¯𝑀𝑀superscript𝑞21𝐴ket¯𝑀¯𝑀\displaystyle-pq(1+A)\ket{\bar{M}M}-q^{2}(1-A)\ket{\bar{M}\bar{M}}.- italic_p italic_q ( 1 + italic_A ) | start_ARG over¯ start_ARG italic_M end_ARG italic_M end_ARG ⟩ - italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_A ) | start_ARG over¯ start_ARG italic_M end_ARG over¯ start_ARG italic_M end_ARG end_ARG ⟩ .(7)

For simplicity, we have defined that

A=eiΔm(t1t2)+12ΔΓ(t1t2),𝐴superscript𝑒𝑖Δ𝑚subscript𝑡1subscript𝑡212ΔΓsubscript𝑡1subscript𝑡2\displaystyle A=e^{i\Delta m(t_{1}-t_{2})+\frac{1}{2}\Delta\Gamma(t_{1}-t_{2})},italic_A = italic_e start_POSTSUPERSCRIPT italic_i roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ,(8)
B=ei(m2t1+m1t2)12(Γ2t1+Γ1t2).𝐵superscript𝑒𝑖subscript𝑚2subscript𝑡1subscript𝑚1subscript𝑡212subscriptΓ2subscript𝑡1subscriptΓ1subscript𝑡2\displaystyle B=e^{-i(m_{2}t_{1}+m_{1}t_{2})-\frac{1}{2}(\Gamma_{2}t_{1}+%\Gamma_{1}t_{2})}.italic_B = italic_e start_POSTSUPERSCRIPT - italic_i ( italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT .(9)

The wave function of the neutral meson pair at a given time can then be obtained by simply inserting the parameters of the specific neutral meson into Eq.(7).

III Bell inequality violation in neutral mesons

The most commonly used extension of Bell inequality for measuring a pair of particles with spin 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG, the CHSH inequalityClauser:1969ny , can be expressed as

S=|E(a,b)+E(a,b)+E(a,b)E(a,b)|2.𝑆𝐸𝑎𝑏𝐸superscript𝑎𝑏𝐸𝑎superscript𝑏𝐸superscript𝑎superscript𝑏2\displaystyle S=|E(\vec{a},\vec{b})+E(\vec{a}^{\prime},\vec{b})+E(\vec{a},\vec%{b}^{\prime})-E(\vec{a}^{\prime},\vec{b}^{\prime})|\leq 2.italic_S = | italic_E ( over→ start_ARG italic_a end_ARG , over→ start_ARG italic_b end_ARG ) + italic_E ( over→ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , over→ start_ARG italic_b end_ARG ) + italic_E ( over→ start_ARG italic_a end_ARG , over→ start_ARG italic_b end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) - italic_E ( over→ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , over→ start_ARG italic_b end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | ≤ 2 .(10)

In the above formula, a𝑎\vec{a}over→ start_ARG italic_a end_ARG and asuperscript𝑎\vec{a}^{\prime}over→ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT mean to choose two different directions to measure the polarization of one of the particles similar to b𝑏\vec{b}over→ start_ARG italic_b end_ARG and bsuperscript𝑏\vec{b}^{\prime}over→ start_ARG italic_b end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for the other particle. The detection of Bell inequality for neutral mesons is somewhat different from the detection for polarized particles.

For neutral mesons, their flavor bases can be regarded as their quasi-spin. The direction of the quasi-spin of neutral mesons changes with time. One can choose four different time points to measure the neutral meson pair, similarly to choose four different directions to measure the spinning particle pair. The CHSH inequality is expressed as

S=|E(t1,t2)+E(t1,t2)+E(t1,t2)E(t1,t2)|2,𝑆𝐸subscript𝑡1subscript𝑡2𝐸superscriptsubscript𝑡1subscript𝑡2𝐸subscript𝑡1superscriptsubscript𝑡2𝐸superscriptsubscript𝑡1superscriptsubscript𝑡22\displaystyle S=|E(t_{1},t_{2})+E(t_{1}^{\prime},t_{2})+E(t_{1},t_{2}^{\prime}%)-E(t_{1}^{\prime},t_{2}^{\prime})|\leq 2,italic_S = | italic_E ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_E ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_E ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) - italic_E ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | ≤ 2 ,(11)

where the correlation function E(t1,t2)𝐸subscript𝑡1subscript𝑡2E(t_{1},t_{2})italic_E ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) is defined as

E(t1,t2)=𝐸subscript𝑡1subscript𝑡2absent\displaystyle E(t_{1},t_{2})=italic_E ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) =P(t1,M;t2,M)+P(t1,M¯;t2,M¯)𝑃subscript𝑡1𝑀subscript𝑡2𝑀𝑃subscript𝑡1¯𝑀subscript𝑡2¯𝑀\displaystyle P(t_{1},M;t_{2},M)+P(t_{1},\bar{M};t_{2},\bar{M})italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_M ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_M ) + italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG )
P(t1,M;t2,M¯)P(t1,M¯;t2,M).𝑃subscript𝑡1𝑀subscript𝑡2¯𝑀𝑃subscript𝑡1¯𝑀subscript𝑡2𝑀\displaystyle-P(t_{1},M;t_{2},\bar{M})-P(t_{1},\bar{M};t_{2},M).- italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_M ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG ) - italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_M ) .(12)

The joint probability density P(t1,X;t2,X)𝑃subscript𝑡1𝑋subscript𝑡2superscript𝑋P(t_{1},X;t_{2},X^{\prime})italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_X ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) is the probability that X𝑋Xitalic_X produced in the left side is measured at t1subscript𝑡1t_{1}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT while Xsuperscript𝑋X^{\prime}italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT produced in the right side is measured at t2subscript𝑡2t_{2}italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

Using Eq.(7) and defining C=|p|2|q|2𝐶superscript𝑝2superscript𝑞2C=\frac{|p|^{2}}{|q|^{2}}italic_C = divide start_ARG | italic_p | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG | italic_q | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG, it is easy to calculate the joint probability density at time (t1,t2)subscript𝑡1subscript𝑡2(t_{1},t_{2})( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )

P(t1,M;t2,M)=C8[e(Γ2t1+Γ1t2)+e(Γ1t1+Γ2t2)]𝑃subscript𝑡1𝑀subscript𝑡2𝑀𝐶8delimited-[]superscript𝑒subscriptΓ2subscript𝑡1subscriptΓ1subscript𝑡2superscript𝑒subscriptΓ1subscript𝑡1subscriptΓ2subscript𝑡2\displaystyle P(t_{1},M;t_{2},M)=\frac{C}{8}[e^{-(\Gamma_{2}t_{1}+\Gamma_{1}t_%{2})}+e^{-(\Gamma_{1}t_{1}+\Gamma_{2}t_{2})}]italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_M ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_M ) = divide start_ARG italic_C end_ARG start_ARG 8 end_ARG [ italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ]
[1cosΔm(t1t2)coshΔΓ(t1t2)2],delimited-[]1𝑐𝑜𝑠Δ𝑚subscript𝑡1subscript𝑡2𝑐𝑜𝑠ΔΓsubscript𝑡1subscript𝑡22\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{%}~{}~{}[1-\frac{cos\Delta m(t_{1}-t_{2})}{cosh\frac{\Delta\Gamma(t_{1}-t_{2})}%{2}}],[ 1 - divide start_ARG italic_c italic_o italic_s roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_c italic_o italic_s italic_h divide start_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 end_ARG end_ARG ] ,(13)
P(t1,M¯;t2,M¯)=18C[e(Γ2t1+Γ1t2)+e(Γ1t1+Γ2t2)]𝑃subscript𝑡1¯𝑀subscript𝑡2¯𝑀18𝐶delimited-[]superscript𝑒subscriptΓ2subscript𝑡1subscriptΓ1subscript𝑡2superscript𝑒subscriptΓ1subscript𝑡1subscriptΓ2subscript𝑡2\displaystyle P(t_{1},\bar{M};t_{2},\bar{M})=\frac{1}{8C}[e^{-(\Gamma_{2}t_{1}%+\Gamma_{1}t_{2})}+e^{-(\Gamma_{1}t_{1}+\Gamma_{2}t_{2})}]italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG ) = divide start_ARG 1 end_ARG start_ARG 8 italic_C end_ARG [ italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ]
[1cosΔm(t1t2)coshΔΓ(t1t2)2],delimited-[]1𝑐𝑜𝑠Δ𝑚subscript𝑡1subscript𝑡2𝑐𝑜𝑠ΔΓsubscript𝑡1subscript𝑡22\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{%}~{}~{}[1-\frac{cos\Delta m(t_{1}-t_{2})}{cosh\frac{\Delta\Gamma(t_{1}-t_{2})}%{2}}],[ 1 - divide start_ARG italic_c italic_o italic_s roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_c italic_o italic_s italic_h divide start_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 end_ARG end_ARG ] ,(14)
P(t1,M;t2,M¯)=18[e(Γ2t1+Γ1t2)+e(Γ1t1+Γ2t2)]𝑃subscript𝑡1𝑀subscript𝑡2¯𝑀18delimited-[]superscript𝑒subscriptΓ2subscript𝑡1subscriptΓ1subscript𝑡2superscript𝑒subscriptΓ1subscript𝑡1subscriptΓ2subscript𝑡2\displaystyle P(t_{1},M;t_{2},\bar{M})=\frac{1}{8}[e^{-(\Gamma_{2}t_{1}+\Gamma%_{1}t_{2})}+e^{-(\Gamma_{1}t_{1}+\Gamma_{2}t_{2})}]italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_M ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG ) = divide start_ARG 1 end_ARG start_ARG 8 end_ARG [ italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ]
[1+cosΔm(t1t2)coshΔΓ(t1t2)2],delimited-[]1𝑐𝑜𝑠Δ𝑚subscript𝑡1subscript𝑡2𝑐𝑜𝑠ΔΓsubscript𝑡1subscript𝑡22\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{%}~{}~{}[1+\frac{cos\Delta m(t_{1}-t_{2})}{cosh\frac{\Delta\Gamma(t_{1}-t_{2})}%{2}}],[ 1 + divide start_ARG italic_c italic_o italic_s roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_c italic_o italic_s italic_h divide start_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 end_ARG end_ARG ] ,(15)
P(t1,M¯;t2,M)=18[e(Γ2t1+Γ1t2)+e(Γ1t1+Γ2t2)]𝑃subscript𝑡1¯𝑀subscript𝑡2𝑀18delimited-[]superscript𝑒subscriptΓ2subscript𝑡1subscriptΓ1subscript𝑡2superscript𝑒subscriptΓ1subscript𝑡1subscriptΓ2subscript𝑡2\displaystyle P(t_{1},\bar{M};t_{2},M)=\frac{1}{8}[e^{-(\Gamma_{2}t_{1}+\Gamma%_{1}t_{2})}+e^{-(\Gamma_{1}t_{1}+\Gamma_{2}t_{2})}]italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over¯ start_ARG italic_M end_ARG ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_M ) = divide start_ARG 1 end_ARG start_ARG 8 end_ARG [ italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - ( roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + roman_Γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ]
[1+cosΔm(t1t2)coshΔΓ(t1t2)2].delimited-[]1𝑐𝑜𝑠Δ𝑚subscript𝑡1subscript𝑡2𝑐𝑜𝑠ΔΓsubscript𝑡1subscript𝑡22\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{%}~{}~{}[1+\frac{cos\Delta m(t_{1}-t_{2})}{cosh\frac{\Delta\Gamma(t_{1}-t_{2})}%{2}}].[ 1 + divide start_ARG italic_c italic_o italic_s roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_c italic_o italic_s italic_h divide start_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 end_ARG end_ARG ] .(16)

Then one need to normalize the probability at time (t1,t2)subscript𝑡1subscript𝑡2(t_{1},t_{2})( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ), otherwise one can not observe the violation of Bell inequality due to the particle decays. One can normalize the joint probability density by the formulaTakubo:2021sdk

Pt1,t2(X,X)=P(t1,X;t2,X)X,XP(t1,X;t2,X),subscript𝑃subscript𝑡1subscript𝑡2𝑋superscript𝑋𝑃subscript𝑡1𝑋subscript𝑡2superscript𝑋subscript𝑋superscript𝑋𝑃subscript𝑡1𝑋subscript𝑡2superscript𝑋\displaystyle P_{t_{1},t_{2}}(X,X^{\prime})=\frac{P(t_{1},X;t_{2},X^{\prime})}%{\sum_{X,X^{\prime}}P(t_{1},X;t_{2},X^{\prime})},italic_P start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_X , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = divide start_ARG italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_X ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_X , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_X ; italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG ,(17)

where X𝑋Xitalic_X and Xsuperscript𝑋X^{\prime}italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT denote M𝑀Mitalic_M or M¯¯𝑀\bar{M}over¯ start_ARG italic_M end_ARG. After normalization, the expression for the joint probability density can be written as

Pt1,t2(M,M)=C2(1F)(C+1)2(C1)2F,subscript𝑃subscript𝑡1subscript𝑡2𝑀𝑀superscript𝐶21𝐹superscript𝐶12superscript𝐶12𝐹\displaystyle P_{t_{1},t_{2}}(M,M)=\frac{C^{2}(1-F)}{(C+1)^{2}-(C-1)^{2}F},italic_P start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_M , italic_M ) = divide start_ARG italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_F ) end_ARG start_ARG ( italic_C + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_C - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F end_ARG ,(18)
Pt1,t2(M¯,M¯)=(1F)(C+1)2(C1)2F,subscript𝑃subscript𝑡1subscript𝑡2¯𝑀¯𝑀1𝐹superscript𝐶12superscript𝐶12𝐹\displaystyle P_{t_{1},t_{2}}(\bar{M},\bar{M})=\frac{(1-F)}{(C+1)^{2}-(C-1)^{2%}F},italic_P start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over¯ start_ARG italic_M end_ARG , over¯ start_ARG italic_M end_ARG ) = divide start_ARG ( 1 - italic_F ) end_ARG start_ARG ( italic_C + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_C - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F end_ARG ,(19)
Pt1,t2(M,M¯)=C(1+F)(C+1)2(C1)2F,subscript𝑃subscript𝑡1subscript𝑡2𝑀¯𝑀𝐶1𝐹superscript𝐶12superscript𝐶12𝐹\displaystyle P_{t_{1},t_{2}}(M,\bar{M})=\frac{C(1+F)}{(C+1)^{2}-(C-1)^{2}F},italic_P start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_M , over¯ start_ARG italic_M end_ARG ) = divide start_ARG italic_C ( 1 + italic_F ) end_ARG start_ARG ( italic_C + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_C - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F end_ARG ,(20)
Pt1,t2(M¯,M)=C(1+F)(C+1)2(C1)2F,subscript𝑃subscript𝑡1subscript𝑡2¯𝑀𝑀𝐶1𝐹superscript𝐶12superscript𝐶12𝐹\displaystyle P_{t_{1},t_{2}}(\bar{M},M)=\frac{C(1+F)}{(C+1)^{2}-(C-1)^{2}F},italic_P start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over¯ start_ARG italic_M end_ARG , italic_M ) = divide start_ARG italic_C ( 1 + italic_F ) end_ARG start_ARG ( italic_C + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_C - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F end_ARG ,(21)

where

F=cosΔm(t1t2)coshΔΓ(t1t2)2.𝐹Δ𝑚subscript𝑡1subscript𝑡2ΔΓsubscript𝑡1subscript𝑡22\displaystyle F=\frac{\cos\Delta m(t_{1}-t_{2})}{\cosh\frac{\Delta\Gamma(t_{1}%-t_{2})}{2}}.italic_F = divide start_ARG roman_cos roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG roman_cosh divide start_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 end_ARG end_ARG .(22)

In combination with formula in Eq.(12), the expression of the correlation function including the CP violation effects can be obtained as

E(t1,t2)=(C1)2(C+1)2F(C+1)2(C1)2F.𝐸subscript𝑡1subscript𝑡2superscript𝐶12superscript𝐶12𝐹superscript𝐶12superscript𝐶12𝐹\displaystyle E(t_{1},t_{2})=\frac{(C-1)^{2}-(C+1)^{2}F}{(C+1)^{2}-(C-1)^{2}F}.italic_E ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = divide start_ARG ( italic_C - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_C + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F end_ARG start_ARG ( italic_C + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_C - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_F end_ARG .(23)

It is obvious that there will be four completely independent time variables in the expression of CHSH inequality S𝑆Sitalic_S at the end, which makes the experimental measurement more complex. So in the following, we will fix the value of t1subscript𝑡1t_{1}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, t2subscript𝑡2t_{2}italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and let S𝑆Sitalic_S evolve as a function of other two time potints t1superscriptsubscript𝑡1t_{1}^{\prime}italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and t2superscriptsubscript𝑡2t_{2}^{\prime}italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. In addition, it is best to measure the point in time as quickly as possible due to the existence of decay of the neutral mesons. So we will focus on the earliest S𝑆Sitalic_S maximum value. Of course, due to relativistic time dilatation effect, the neutral mesons can survive for a while to be detected if they are produced with a higher energy.

ParticlesLifetime(ps)ΔmΔ𝑚\Delta mroman_Δ italic_m(ps-1)ΔΓΔΓ\Delta\Gammaroman_Δ roman_Γ(ps-1)C𝐶Citalic_C
Bd0superscriptsubscript𝐵𝑑0B_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT,B¯d0superscriptsubscript¯𝐵𝑑0\bar{B}_{d}^{0}over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT1.517±0.004plus-or-minus1.5170.0041.517\pm 0.0041.517 ± 0.0040.5069similar-to\sim 00.998
Bs0superscriptsubscript𝐵𝑠0B_{s}^{0}italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT,B¯s0superscriptsubscript¯𝐵𝑠0\bar{B}_{s}^{0}over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT1.520±0.005plus-or-minus1.5200.0051.520\pm 0.0051.520 ± 0.00517.7650.0830.9994
D0superscript𝐷0D^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT,D¯0superscript¯𝐷0\bar{D}^{0}over¯ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT0.410±0.001plus-or-minus0.4100.0010.410\pm 0.0010.410 ± 0.0010.009930.031481.0121
KSsubscript𝐾𝑆K_{S}italic_K start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT,89.54±0.04plus-or-minus89.540.0489.54\pm 0.0489.54 ± 0.04,0.0052920.011151.0064
KLsubscript𝐾𝐿K_{L}italic_K start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT51160±210plus-or-minus5116021051160\pm 21051160 ± 210

In the calculation, the relevant parameter inputs for the neutral mesons are listed in Table.1. Fixing two time points and varying two other time points, the time evolution of CHSH inequality S for different neutral meson pair system over time is shown inFigs.1, 2, 3 and4 respectively. The yellow plane represents the value of S𝑆Sitalic_S over time. The blue translucent plane represents the maximum value predicted by local realism (LR). There are two kinds of time dependence for CHSH inequality S. The first is a periodic change in the value of S𝑆Sitalic_S caused by factor cosΔm(t1t2)𝑐𝑜𝑠Δ𝑚subscript𝑡1subscript𝑡2cos\Delta m(t_{1}-t_{2})italic_c italic_o italic_s roman_Δ italic_m ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). The second is the depression of the maximum value of the adjacent period S𝑆Sitalic_S caused by factor coshΔΓ(t1t2)2𝑐𝑜𝑠ΔΓsubscript𝑡1subscript𝑡22cosh\frac{\Delta\Gamma(t_{1}-t_{2})}{2}italic_c italic_o italic_s italic_h divide start_ARG roman_Δ roman_Γ ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG 2 end_ARG.

For simplicity, we can fix the initial time t1=t2=0subscript𝑡1subscript𝑡20t_{1}=t_{2}=0italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 ps, naming we first observe the neutral meson pair produced instantaneously.In the case of neutral Bd0B¯d0subscriptsuperscript𝐵0𝑑subscriptsuperscript¯𝐵0𝑑B^{0}_{d}-\bar{B}^{0}_{d}italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT system, the value of S𝑆Sitalic_S reaches its maximum value 2.5 for the first time with t1=10.3superscriptsubscript𝑡110.3t_{1}^{\prime}=10.3italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 10.3 ps and t2=2.1superscriptsubscript𝑡22.1t_{2}^{\prime}=2.1italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 2.1 ps, or swap the two time values because of the symmetry of the two times. In addition, the maximum value of S𝑆Sitalic_S in each oscillation period does not decay with time because of ΔΓBd0=0ΔsubscriptΓsubscriptsuperscript𝐵0𝑑0\Delta\Gamma_{B^{0}_{d}}=0roman_Δ roman_Γ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0. For Bs0B¯s0subscriptsuperscript𝐵0𝑠subscriptsuperscript¯𝐵0𝑠B^{0}_{s}-\bar{B}^{0}_{s}italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT system, the value of S𝑆Sitalic_S reaches its maximum 2.5 for the first time at t1=0.29superscriptsubscript𝑡10.29t_{1}^{\prime}=0.29italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 0.29 ps, t2=0.06superscriptsubscript𝑡20.06t_{2}^{\prime}=0.06italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 0.06 ps. One can see that the oscillation period of the S𝑆Sitalic_S value of Bs0B¯s0subscriptsuperscript𝐵0𝑠subscriptsuperscript¯𝐵0𝑠B^{0}_{s}-\bar{B}^{0}_{s}italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT system is very short because the value of ΔmBs0Δsubscript𝑚superscriptsubscript𝐵𝑠0\Delta m_{B_{s}^{0}}roman_Δ italic_m start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is very large. At the same time, the depression of the maximum value of S𝑆Sitalic_S between adjacent periods can hardly been reflected due to ΔmBs0ΔΓBs0much-greater-thanΔsubscript𝑚superscriptsubscript𝐵𝑠0ΔsubscriptΓsuperscriptsubscript𝐵𝑠0\Delta m_{B_{s}^{0}}\gg\Delta\Gamma_{B_{s}^{0}}roman_Δ italic_m start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≫ roman_Δ roman_Γ start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT unless the evolution of S𝑆Sitalic_S is plotted on a larger time scale. For D0D¯0superscript𝐷0superscript¯𝐷0D^{0}-\bar{D}^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT system, one can find that the first time that S𝑆Sitalic_S reaches its maximum 2.3 is at t1=496superscriptsubscript𝑡1496t_{1}^{\prime}=496italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 496 ps and t2=80.3superscriptsubscript𝑡280.3t_{2}^{\prime}=80.3italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 80.3 ps. For K0K¯0superscript𝐾0superscript¯𝐾0K^{0}-\bar{K}^{0}italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT sytem, the situation will be quite different. Because the value of ΔmKΔsubscript𝑚𝐾\Delta m_{K}roman_Δ italic_m start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT is very small, the oscillation period of K0superscript𝐾0K^{0}italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT is very long. Meanwhile, ΔΓK02ΔmK0ΔsubscriptΓsuperscript𝐾02Δsubscript𝑚superscript𝐾0\Delta\Gamma_{K^{0}}\approx 2\Delta m_{K^{0}}roman_Δ roman_Γ start_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≈ 2 roman_Δ italic_m start_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT causes a significant decrease in the maximum value of S𝑆Sitalic_S between adjacent periods. In this case, only a very small number of peaks are sufficient to violate CHSH inequality. After searching, we fix the time t1=300subscript𝑡1300t_{1}=300italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 300 ps and t2=372subscript𝑡2372t_{2}=372italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 372 ps. The only peak of S𝑆Sitalic_S that violates Bell inequality is 2.34 at the time t1=444.9superscriptsubscript𝑡1444.9t_{1}^{\prime}=444.9italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 444.9 ps, t2=227.1superscriptsubscript𝑡2227.1t_{2}^{\prime}=227.1italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 227.1 ps.

Test of Bell Locality Violation in Flavor Entangled Neutral Meson Pair (1)
Test of Bell Locality Violation in Flavor Entangled Neutral Meson Pair (2)
Test of Bell Locality Violation in Flavor Entangled Neutral Meson Pair (3)
Test of Bell Locality Violation in Flavor Entangled Neutral Meson Pair (4)

If we vary all the four measurement point for quasi-spin, the value of S𝑆Sitalic_S is 2.18 with t1=1subscript𝑡11t_{1}=1italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 ps, t2=0.5subscript𝑡20.5t_{2}=0.5italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.5 ps, t1=0subscriptsuperscript𝑡10t^{\prime}_{1}=0italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 ps and t2=1.5subscriptsuperscript𝑡21.5t^{\prime}_{2}=1.5italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1.5 ps, while the maximum value of S𝑆Sitalic_S becomes 2.83 with t1=1.55subscript𝑡11.55t_{1}=1.55italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1.55 ps, t2=3.10subscript𝑡23.10t_{2}=3.10italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 3.10 ps, t1=0subscriptsuperscript𝑡10t^{\prime}_{1}=0italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 ps and t2=4.65subscriptsuperscript𝑡24.65t^{\prime}_{2}=4.65italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 4.65 ps for Bd0B¯d0subscriptsuperscript𝐵0𝑑subscriptsuperscript¯𝐵0𝑑B^{0}_{d}-\bar{B}^{0}_{d}italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT system. For Bs0B¯s0subscriptsuperscript𝐵0𝑠subscriptsuperscript¯𝐵0𝑠B^{0}_{s}-\bar{B}^{0}_{s}italic_B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT system, the maximum value of S𝑆Sitalic_S becomes 2.83 with t1=0.50subscript𝑡10.50t_{1}=0.50italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.50 ps, t2=0.54subscript𝑡20.54t_{2}=0.54italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.54 ps, t1=0.59subscriptsuperscript𝑡10.59t^{\prime}_{1}=0.59italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.59 ps and t2=0.45subscriptsuperscript𝑡20.45t^{\prime}_{2}=0.45italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.45 ps. For D0D¯0superscript𝐷0superscript¯𝐷0D^{0}-\bar{D}^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT system, the maximum value of S𝑆Sitalic_S becomes 2.77 with t1=80.6subscript𝑡180.6t_{1}=80.6italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 80.6 ps, t2=156.4subscript𝑡2156.4t_{2}=156.4italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 156.4 ps, t1=232.2subscriptsuperscript𝑡1232.2t^{\prime}_{1}=232.2italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 232.2 ps and t2=4.71subscriptsuperscript𝑡24.71t^{\prime}_{2}=4.71italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 4.71 ps. For K0K¯0superscript𝐾0superscript¯𝐾0K^{0}-\bar{K}^{0}italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT system, the value of S𝑆Sitalic_S is 2.13 with t1=29.7subscript𝑡129.7t_{1}=29.7italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 29.7 ps, t2=59.3subscript𝑡259.3t_{2}=59.3italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 59.3 ps, t1=89subscriptsuperscript𝑡189t^{\prime}_{1}=89italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 89 ps and t2=0subscriptsuperscript𝑡20t^{\prime}_{2}=0italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 ps, while the maximum value of S𝑆Sitalic_S becomes 2.35 with t1=205subscript𝑡1205t_{1}=205italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 205 ps, t2=278subscript𝑡2278t_{2}=278italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 278 ps, t1=351subscriptsuperscript𝑡1351t^{\prime}_{1}=351italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 351 ps and t2=133subscriptsuperscript𝑡2133t^{\prime}_{2}=133italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 133 ps. Note that the choices of time measurement points are not unique when the CHSH inequality is max broken. In general, the violation of the Bell inequality generally exists in the above processes as long as the appropriate time observation point is chosen. By choosing these appropriate time observation point, the experiment collaboration such as Belle-II, BESIII and future Super tau-charm factories(STCF) and Circular Electron Positron Collider(CEPC) can test the Bell inequality and explore what extent the Bell locality is brokenAchasov:2023gey ; CEPCStudyGroup:2023quu . Among these neutral meson pair, the test of Bell locality violation in D0D¯0superscript𝐷0superscript¯𝐷0D^{0}-\bar{D}^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - over¯ start_ARG italic_D end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT system is relatively difficult compared with other neutral meson entanglement systems in experiments due to its short lifetime and narrow mass splitting.

IV conclusion

In this paper, the time evolution of the entangled wave function of neutral meson pair is derived theoretically where minor CP violation parameters are also taken into account. In order to investigate the violation of Bell inequality during the oscillation of neutral mesons, the concrete expression of CHSH inequality including CP violation effect is calculated. By studying the oscillations of mesons Bd0superscriptsubscript𝐵𝑑0B_{d}^{0}italic_B start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, Bs0superscriptsubscript𝐵𝑠0B_{s}^{0}italic_B start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, D0superscript𝐷0D^{0}italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and K0superscript𝐾0K^{0}italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, we find that there are violations of Bell inequality in these processes as long as the appropriate time node is selected to measure. The maximum breaking for CHSH inequality is obtained. It shows that these oscillations of neutral mesons accord with the prediction of quantum mechanics (QM) and cannot be explained by local realism (LR). This makes a contribution to test the violation of Bell inequality in high-energy physics. It also helps us better understand flavor entanglement and Bell non-locality in particle physics.

Acknowledgments

The work is supported by NSFC under grant Nos. 12322503, 12075124, 12375088 and 12335003 and by Natural Science Foundation of Jiangsu under Grant No. BK20211267.

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Test of Bell Locality Violation in Flavor Entangled Neutral Meson Pair (2024)
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