text:
\section{\label{sec:level1}Introduction}

In 1935, Einstein, Podolsky and Rosen (EPR) questioned the completeness of QM, by applying  a criterion of LR to a pair of  particles in a quantum state which Schr\"{o}dinger subsequently referred to as entangled~\cite{EPR,sch}. Locality means that two events cannot  have any mutual physical influence if they are spacelike separated, that is, their spatial separation is larger than the distance the fastest physical signal, i.e. the light,  can travel within the time difference between the two events.   In 1964, Bell proposed the first BI satisfied by any local realistic theory while violated by QM~\cite{BI}. A  more experimentally suitable version of BI, called  Clauser-Horne-Shimony-Holt inequality~\cite{CHSHI}, was  demonstrated to be violated in many experiments, including   the ones  closing the locality loophole~\cite{BIAspect,z1}, the detection loophole~\cite{d1,d2},  and both~\cite{BIfree1,free2,free3}. To close yet another loophole called  measuring setting or freedom of choice loophole, observations of Milky Way stars~\cite{as,as2} and human choices~\cite{bigbell} have been employed. Great progress has been made in making use of quantum entanglement in quantum information science.

With  the conflict between LR and QM  well established, it is important to identify which aspects of LR are the sources of the conflict.  For this purpose, Leggett in 2003 proposed the LI, which is satisfied by CNR  and is violated by QM~\cite{LI}. This means that even nonlocal realism, at least a subset,  cannot avoid the conflict with QM, so the source of conflict seems to be more likely realism.  In 2007, a version of LI was  experimentally demonstrated to be  violated by using polarizations of entangled photons generated in spontaneous parametric down conversion, first under an additional assumption of rotational  invariance~\cite{LIExpNature}, then  without this assumption~\cite{Paterek,Branciard}. LI violation was also demonstrated by using polarizations of photons from  fibre-based source~\cite{Eisaman}, as well as  the orbital angular momenta of photons~\cite{Romero}. Similar phenomena were observed in different degrees of freedom of single particles~\cite{Hasegawa,Cardano}. Various extended discussions have also been made~\cite{comment,Colbeck,Branciard2,Oh}.

It is highly interesting to extend the investigations on BI and LI to particle physics, of which standard model (SM) is based on quantum field theory combining QM with special relativity, emphasizing causality  and using  local gauge principle to describe fundamental interactions. Massive and possibly unstable particles governed by strong and weak interactions and flying in relativistic velocities represent a new class beyond both photons and nonrelativistic particles governed merely by electromagnetism, and can still easily achieve spacelike separation. Besides, one might also wonder  whether high energy particles,  as excitations of quantum fields,  may display nonlocal effects.

In particle physics, entanglement, more often called EPR correlation,  has been noted  in pseudoscalar meson  pairs since 1960s~\cite{1960s}. Various discussions were made on its   rigorous verification~\cite{lot}, which was experimentally done  in  $K^0\bar{K}^0$ pairs produced in proton-antiproton annihilation~\cite{cern},  in  $\phi$ resonance~\cite{kloe1,kloe2}, as well as in   $B^0_d\bar{B}^0_d$ pairs produced in $\Upsilon (4S)$ resonance~\cite{go1,go2,bevan}. Entanglement is routinely used to tag mesons  by  identifying their entangled partners~\cite{kloe2,bevan,DT,Betad,Ablikim,detectionEfficience}. Moreover, entangled meson pairs are used in measuring various  parameters~\cite{Ablikim,detectionEfficience,parameter}, and studying  violations of
discrete symmetries~\cite{bernaB,Bd,JDROtherReview}, including CP~\cite{lipkin89,dunietz,ImprovedCPV,bernaB,JDRxzz}, time reversal (T)~\cite{TViolation,TViolationD}, and CPT~\cite{bernabeucpt}. A possible scheme of teleporting mesons was also proposed~\cite{telep}.  There exists similar entanglement in hyperon pairs generated from electron-positron annihilation, which was  used recently to measure the phase between the amplitudes of the  decays to different helicity states~\cite{bes3}.

Many proposals had been made on BI test in entangled mesons~\cite{mesonbi,mesonbi2}, and in the analogous spin-entangled particles~\cite{toernqvist,privitera}. There had been  an early  experiment using entangled protons to test BI under a few additional assumptions~\cite{lamehi}. There was an experiment using entangled $B^0_d\bar{B}^0_d$  pairs to test BI, in which meson decay  acts as effective measurement settings~\cite{go1}. However, it was not regarded as a genuine Bell test,  because of the lack of active measurement~\cite{toernqvist,problemOfMeson,bevan}. Basically this is a manifestation of the  loophole of measurement settings, for the following reason.  One can envisage a local HV (LHV) theory in which HVs in the source of the  particle pairs determine the  times, modes and even products of the decays, and the information is carried  by the particles, consequently  the two particles are secretely correlated no matter how far away they are separated, rendering the violation of BI. Other approaches to BI using entangled high energy particles are difficult to realize, as the alternative bases of measurement are physically  limited.

In this paper, we extend  CNR and LI to include the case that the measurement settings are not externally fixed, but determined by HV, therefore the above situation jeopardizing  BI test in entangled mesons is allowed in CNR, and we propose LI test using entangled neutral $B_d$  mesons. From QM calculation of single particle decays, we identify the time-dependent  effective measuring directions due to the decays, as  counterparts of the directions of the polarizers measuring the photon polarizations.  For different decay times, they all  lie on a plane and a cone, respectively.  For such effective measuring directions, whether it is externally fixed or emerge from averages of measurement outcomes over HV,  we derive a new version of LI, which is violated by QM and entangled $B_d$ mesons. We calculate the measurable quantities  characterizing the relative magnitude of the LI violation, and find their maxima to be  about $2.7\%$. It turns out  that the LI can only be violated when CP symmetry is violated indirectly, i.e. in the mass matrix. Our work establish the true randomness of particle decay, including its time, mode, and product. On the other hand, our new LI can also be tested in other systems such as photon polarizations.

\section{Pseudoscalar Neutral Mesons \label{rev} }

In QM, a neutral pseudoscalar meson $M$ can be regarded as living in a two-dimensional Hilbert space, with basis states $|M^0\rangle$ and $|\bar{M}^0\rangle$, which are flavor eigenstates and  mutual CP conjugates, i.e. $CP|M^0\rangle = |\bar{M}^0\rangle$, $CP |\bar{M}^0\rangle =  |M^0\rangle$.  In this basis, the mass matrix is
 \begin{equation}
H\equiv {\bf M}-\frac{i}{2}{\bf \Gamma }=\left(
\begin{array}{cc}
H_{00} & H_{0\bar{0}} \\
H_{\bar{0}0} & H_{\bar{0}\bar{0}}
\end{array}\right),
\end{equation}
where   $H_{00}\equiv \langle M^0|H|M^0\rangle$, $H_{0\bar{0}}\equiv \langle M^0|H|\bar{M}^0\rangle$, and so on. The eigenstates of $H$ are
\begin{equation}
\begin{split}
&|M_1\rangle=\frac{1}{\sqrt{|p|^2+|q|^2}}[p|M^0\rangle + q|\bar{M}^0\rangle],\\
&|M_2\rangle=\frac{1}{\sqrt{|p|^2+|q|^2}}[p|M^0\rangle -  q|\bar{M}^0\rangle],
\end{split}
\end{equation}
with $p/q \equiv \sqrt{H_{\bar{0}0}/H_{0\bar{0}} }$.  The corresponding eigenvalues are
\begin{equation}
\begin{split}
\lambda _{1}&= m_{1}-\frac{i}{2}\Gamma _{1}=H_{00} +\sqrt{ H_{0\bar{0}}H_{\bar{0}0} },\\
 \lambda _{2}&= m_{2}-\frac{i}{2}\Gamma _{2}
 =H_{00} - \sqrt{ H_{0\bar{0}}H_{\bar{0}0} }.
\end{split}
\end{equation}

$H$ governs the evolution of the meson state
\begin{equation}|\psi(t)\rangle = a(t) |M^0\rangle+ b(t) |\bar{M}^0\rangle,
\end{equation}
with
\begin{equation}
\left(
\begin{array}{c}
a(t) \\
b(t) \\
\end{array}\right)
=U(t)
\left( \begin{array}{c}
 a(0)\\
 b(0) \\
\end{array}
\right),
\end{equation}
where
\begin{equation}
U(t) = \exp(-iHt)=  g_+(t)+g_-(t)
\left( \begin{array}{cc}
0&p/q \\
q/p&0 \\
\end{array}
\right),
\end{equation}
with  $g_{\pm}(t) \equiv \frac{e^{-i\lambda _2 t} \pm e^{-i\lambda _1  t}}{2}$. This leads  to the  mixing phenomena. Especially,  $M^0$ and $\bar{M}^0$ at $t=0$ evolve respectively to
\begin{equation}
\begin{split}
&|M^0(t)\rangle= g_+(t)|M^0\rangle
+\frac{q}{p}g_-(t)|\bar{M}^0\rangle,\\
&|\bar{M}^0(t)\rangle=
\frac{p}{q}g_-(t)|M^0\rangle+g_+(t)|\bar{M}^0\rangle.\\
\end{split}
\end{equation}

For a meson in an arbitrary state, its decay    to some final state $f$ indicates that there  has been a projection or filtering to some basis state $|\phi\rangle$, which decays to $f$,
$|\phi\rangle$ being~\cite{Bd}
\begin{equation}
|\phi\rangle = \frac{1}{\sqrt{|A_f|^2+ |\bar{A}_f|^2} } (\bar{A}_f^*|M^0\rangle+
A_f^*|\bar{M}^0\rangle), \label{phi}
\end{equation}
where $A_f=\langle f |W|M^0\rangle$ and $\bar{A}_f=\langle f |W|\bar{M}^0\rangle$ are,  respectively,  the  amplitudes of the decays from $M^0$ and  $\bar{M}^0$ to the final state $f$.  $ W = \mathcal{U}H_w$, where $H_w$ being the weak decay Hamiltonian while $\mathcal{U}$ being the strong evolution operator and reducing to the identity if  final state interactions are neglected.

A pair of neutral mesons can be produced as  $C=-1$ antisymmetric  entangled state
\begin{equation}
 |\Psi _- \rangle = \frac{1}{\sqrt{2}}\left(|M^0 \rangle |\bar{M}^0 \rangle - |\bar{M}^0 \rangle |M^0 \rangle \right),
 \end{equation}
where in each term, the first and second basis states are those of mesons $a$ and $b$ respectively.

Suppose this two-particle state evolves up to $t_a$, when meson $a$ decays to some final state $f_a$, indicating that there  is a projection or filtering of $a$ to some basis state $|\phi\rangle_a$, which decays to $f_a$. $|\phi\rangle_a$ is $|\phi\rangle$ as given in  (\ref{phi}) for meson a.  The meson $b$ continues to evolve till it decays to some final state $f_b$  at $t_b$,  indicating that there is a projection or filtering of $b$ to some basis  state  $|\phi\rangle_b$, which decays  to $f_b$.  $|\phi\rangle_b$ is $|\phi\rangle$ as given in  (\ref{phi}) for meson b. The time evolution of the entangled state up to the projections can be described as $
P_bU_b(t_b-t_a)P_aU_b(t_a)U_a(t_a)|\Psi _-\rangle
= P_bP_aU(t_b)U(t_a)|\Psi_-\rangle
= P_b P_a|\Psi_-(t_a,t_b) \rangle$,
where $P_a =|\phi\rangle_{aa} \langle \phi|$ and $P_b=|\phi\rangle_{bb} \langle \phi|$ are projection operators, and   the commutativity between operators on $a$ and those on $b$ have been used.  This justifies the  usual use of a state vector with two time variables
\begin{equation}
\begin{split}
&|\Psi _-(t_a,t_b)\rangle \equiv U(t_b)U(t_a)|\Psi_-\rangle \\
&= \frac{1}{\sqrt{2}}\left(|M^0(t_a)\rangle |\bar{M}^0(t_b)\rangle - |\bar{M}^0(t_a)\rangle |M^0(t_b)\rangle \right),
\end{split}
\label{eq.2.3}
\end{equation}
which means that the two mesons decay at $t_a$ and $t_b$, respectively.

Specifically, we use neutral $B_d$ mesons, because of the advantage that  $\Gamma_2 \approx \Gamma _1$, $q/p\approx e^{2i\beta}$, where $2\beta$ is a phase factor, $\beta$ is   given as  $\sin(2\beta)=
0.695$~\cite{PDG}.  Then  $M^0=B^0$, $\bar{M}^0=\bar{B}^0$,  $M_1=B_L$, $M_2=B_H$, and  $U(t)$ is simplified to
\begin{equation}
\begin{split}
&U(t)  =e^{-iMt-\frac{\Gamma}{2} t}\\
&\times \left( \cos\frac{x\Gamma t}{2}+i\sin \frac{x\Gamma t}{2} \left[ \cos(2\beta)\sigma ^x+\sin (2\beta)\sigma ^y\right]\right),
\end{split}
\label{bu}
\end{equation}
where $\sigma ^{i}$, $(i=x,y,z)$, are Pauli operators, $x \equiv (m_H-m_L)/\Gamma$, $M \equiv ( m_H+m_L)/2$ and $\Gamma \equiv (\Gamma_L+\Gamma_H)/2$, the subscripts following  those of $B_H$ and $B_L$.

In Bloch  representation, $|B^0\rangle$, like the horizontally polarized state  of a photon or the spin-up state of an electron, is represented as the vector $(0,0,1)$, while $|\bar{B}^0_d\rangle$, like the vertically   polarized state  of a photon or the spin-down state of an electron, is represented as the vector $(0,0,-1)$.  They can be chosen  as the  ``measuring directions'' or bases of measurement.

However, for a measurement following time evolution, it is more convenient to define an effective time-dependent basis or ``measuring direction''. A state  of a two-state system can be parameterized as
\begin{equation}
 |{\bf u} \rangle =e^{i\zeta} \left( \cos \frac{\theta_{\bf u}}{2}|0\rangle + e^{i\rho_{\bf u}} \sin \frac{\theta_{\bf u}}{2} |1\rangle\right),
\label{eq.2.1}
\end{equation}
where $|0\rangle$ and $|1\rangle$ represent the basis states. We consider its  time evolution that can be  parameterized   as
\begin{equation}
U(\theta _{\bf a},\rho _{\bf a})=\left(\cos\frac{\theta _{\bf a}}{2}-i\sin \frac{\theta _{\bf a}}{2}\left(\cos(\rho _{\bf a})\sigma ^x+\sin (\rho _{\bf a})\sigma ^y\right)\right),
\label{eq.2.2}
\end{equation}
of which (\ref{bu}) is an example, multiplied by  an additional decay factor  $e^{-iMt-\frac{\Gamma}{2} t}$.

Suppose that following the evolution $U(\theta _{\bf a},\rho _{\bf a})$, a signal   is recorded  as $\mathcal{A}=+1$ if $|0\rangle$ is detected, while $\mathcal{A}=-1$ if $|1\rangle$ is detected.  The QM expectation value of $\mathcal{A}$ is
\begin{equation}
\begin{split}
&\bar{\mathcal{A}}({\bf u})=\frac{|\langle  0|U|{\bf u}\rangle|^2-|\langle 1|U|{\bf u}\rangle|^2}{|\langle 0|U|{\bf u}\rangle|^2+|\langle 1|U|{\bf u}\rangle|^2}={\bf u}\cdot {\bf a},
\end{split}
\label{eq.3.qm}
\end{equation}
where
\begin{equation}{\bf u} =\left( \sin \theta_{\bf u}\cos \rho_{\bf u},  \sin \theta_{\bf u} \sin \rho_{\bf u}, \cos \theta_{\bf u}\right)\end{equation}
is the Bloch vector of  $|{\bf u} \rangle$, while
\begin{equation}{\bf a} = \left( -\sin \theta_{a}\sin \rho_{a},  \sin \theta_{a} \cos \rho_{a}, \cos \theta_{a}\right)\end{equation}
is the Bloch vector of  $U^\dagger(\theta _{\bf a},\rho _{\bf a})  |0\rangle$.
This can be easily understood by   regarding $\bar{\mathcal{A}}({\bf u})$   as expectation value of the signal obtained by measuring the initial state $|{\bf u}\rangle$ in the rotated basis $\{U^\dagger|0\rangle, U^\dagger|1\rangle\}$. The rotation $U^\dagger$ of the basis is   realized by evolution.

For a $B_d$ meson, the measurement in the flavor basis \{$|B^0\rangle$, $|\bar{B}^0\rangle$\}, corresponding to $\mathcal{A}_l=\pm1$, can be made through the semileptonic decay channel,  as the direct CP violation or wrong sign decay is negligible~\cite{PDG}. Thus $|\phi\rangle$ in (\ref{phi}) reduces to  $|B^0\rangle$ or $|\bar{B}^0\rangle$, and  one can define
\begin{equation}
\begin{split}
&\bar{\mathcal{A}}_l({\bf u})=\frac{|\langle B^0|U(t)|{\bf u} \rangle|^2-|\langle \bar{B}^0|U(t)|{\bf u}\rangle|^2}{|\langle B^0|U(t)|{\bf u}\rangle|^2+|\langle \bar{B}^0|U(t)|{\bf u}\rangle|^2}={\bf u} \cdot {\bf a}^l(t),
\end{split}
\label{eq.3.3l}
\end{equation}
where
\begin{equation*}
{\bf a}^l (t)=\left(\sin(2 \beta) \sin( x\Gamma t), -\cos(2 \beta) \sin(x\Gamma t), \cos(x\Gamma t)\right).
\end{equation*}

Likewise,   as the direct CP violation is negligible~\cite{PDG}, if the decay products are CP eigenstates $S_{\pm}$,  they signals the projection of  the  meson to CP basis states  $|B_{\pm}\rangle \equiv  \left(|B^0\rangle \pm |\bar{B}^0\rangle\right)/\sqrt{2}$. In this case, $|\phi\rangle$ in (\ref{phi}) reduces to  $|B_\pm\rangle$.

With $B_{\pm}$ corresponding to  $\mathcal{A}_s=\pm 1$, one can define
\begin{equation}
\bar{\mathcal{A}}_s({\bf u})=\frac{|\langle B_+|U(t)|{\bf u}\rangle|^2-|\langle B_-|U(t)|{\bf u}\rangle|^2}{|\langle B_+|U(t)|{\bf u}\rangle|^2+|\langle B_-|U(t)|{\bf u}\rangle|^2}={\bf u}\cdot {\bf a}^s(t),
\label{eq.3.3s}
\end{equation}
where
\begin{equation*}
\begin{split}
&{\bf a}^s (t) =\left(\sin ^2(2 \beta) \cos (x\Gamma t)+\cos ^2(2 \beta),\right.\\
&\left.\sin (4 \beta) \sin ^2\left(x\Gamma t/2\right),-\sin (2 \beta) \sin (x\Gamma t)\right).
\end{split}
\end{equation*}
Eqs.~(\ref{eq.3.3l}) and (\ref{eq.3.3s}) are of the same form as the standard QM result (\ref{eq.3.qm}) because the factor $e^{-\Gamma t}$ exists in all terms  in both denominator and the numerator, and thus cancels.

Note that $|\langle B^0|U(t)|{\bf u} \rangle|^2$, $|\langle \bar{B}^0|U(t)|{\bf u}\rangle|^2$ and $|\langle B_\pm|U(t)|{\bf u}\rangle|^2$ do not depend on the specific decay channels signalling the projection, hence  is not directly observed. In contrast CP asymmetries usually defined
depend on which channels are observed.

However, under the assumption of no wrong sign decays and no direct CP violation, $|\langle B^0|U(t)|{\bf u} \rangle|^2$, $|\langle \bar{B}^0|U(t)|{\bf u}\rangle|^2$ and $|\langle B_\pm|U(t)|{\bf u}\rangle|^2$ and thus the asymmetries we define  above are  related to the rates of decays  in specific channels. For  flavor eigenstates  $l^{\pm}$, as the final states of semileptonic decays,   $$\langle l^+|WU(t)|{\bf u} \rangle= \langle l^+|W |B^0\rangle  \langle B^0|U(t)|{\bf u} \rangle=A_{l^+}\langle B^0|U(t)|{\bf u} \rangle,$$ $$\langle  l^-|WU(t)|{\bf u} \rangle=\langle l^-|W | \bar{B}^0\rangle \langle \bar{B}^0|U(t)|{\bf u}\rangle=\bar{A}_{l^-}\langle \bar{B}^0|U(t)|{\bf u}\rangle.$$
For $|l^-\rangle=CP|l^+\rangle$, we have $A_{l^+}=\bar{A}_{l^-}=A_l$.
For CP eigenstates $S_\pm$ as the final states,  $$ \langle S_+|WU(t)|{\bf u}\rangle = \langle S_+|W|B_+\rangle \langle B_+|U(t)|{\bf u}\rangle,$$  $$ \langle S_-|WU(t)|{\bf u}\rangle = \langle S_-|W|B_-\rangle\langle B_-|U(t)|{\bf u}\rangle.$$
Note that here there is no special relation between  $l^+$ and $l^-$, or  between $S_+$ and $S_-$, as different decay channels may signal a same projection  in the meson Hilbert space. The examples of flavor eigenstates $l^{\pm}$  as the final states include $M^-e^+\nu$, $M^+ e^-\bar{\nu}$, $M^-\mu^+\nu$, $M^+ \mu^-\bar{\nu}$,  etc. The examples of CP eigenstates $S_\pm$ as the final states include $J/\psi K_S$, $J/\psi K_L$, $KK$, $KKK$,  $\pi\pi$, $\pi\pi\pi$, $DD$, etc.

Therefore, the asymmetries $\bar{A}_{l}$ and $\bar{A}_{s}$,  defined above in the meson Hilbert space, can be obtained  from experimentally measurable quantities,
\begin{equation*}
\begin{split}
&\bar{\mathcal{A}}_l({\bf u})=\frac{\frac{|\langle l^+|WU(t)|{\bf u} \rangle|^2}{|\langle l^+|W|B^0\rangle |^2}-\frac{|\langle  l^-|WU(t)|{\bf u}\rangle|^2}{|\langle l^-|W|\bar{B}^0|^2}}{\frac{|\langle l^+|WU(t)|{\bf u} \rangle|^2}{|\langle l^+|W|B^0\rangle |^2}+\frac{|\langle l^-|WU(t)|{\bf u}\rangle|^2}{|\langle l^-|W|\bar{B}^0|^2}},\\
&\bar{\mathcal{A}}_s({\bf u})=\frac{\frac{|\langle S_+|WU(t)|{\bf u}\rangle|^2}{|\langle S_+|W|B_+\rangle|^2}-\frac{|\langle S_-|WU(t)|{\bf u}\rangle|^2}{|\langle S_-|W|B_-\rangle|^2}}{\frac{|\langle S_+|WU(t)|{\bf u}\rangle|^2}{|\langle S_+|W|B_+\rangle|^2}+\frac{|\langle S_-|WU(t)|{\bf u}\rangle|^2}{|\langle S_-|W|B_-\rangle|^2}}.
\end{split}
\end{equation*}

As shown in Fig.~\ref{fig:Polarizer}, with the time passing,  ${\bf a}^l(t)$ rotates on a plane, while ${\bf a}^s(t)$ rotates on a cone whose axis is perpendicular to ${\bf a}^l$ plane. For convenience, we adopt a new coordinate system in which ${\bf a}^l$ plane is the $xy$ plane, then
\begin{equation}
\begin{split}
&{\bf a}^l(\phi_l)=\left(\cos \phi_l,\sin \phi_l,0\right),\\
&{\bf a}^s(\theta _s,\phi_s)=\left(\sin\theta _s \cos \phi_s,\sin \theta _s\sin\phi_s,\cos \theta _s \right),
\end{split}
\label{eq.3.8}
\end{equation}
where $\phi_l= x\Gamma t$ and $\phi_s=x\Gamma t+\pi/2$
are the azimuthal  angles of  ${\bf a}^l$  and ${\bf a}^s$, respectively, $\theta_s=2\beta$   is the polar  angle of   ${\bf a}^s$, and it suffices to consider $0< \theta _s\leq \pi/2$.

${\bf a}^l(\phi_l)$ and ${\bf a}^s(\theta _s,\phi_s)$ are two effective measurement settings or  ``measuring directions''. For $B_d$ mesons, they are time-dependent.  The rotation of   basis or measuring direction  realized by evolution  explains the similarity between decay time and polarizer angle.  But ${\bf a}^l(\phi_l)$ and ${\bf a}^s(\theta _s,\phi_s)$  can also be used, say, for photon polarization, by directly adjusting   $ \phi_l $  and $ (\theta _s,\phi_s)$ in experiments.

\begin{figure}
\resizebox{1.0\hsize}{!}{\includegraphics*{Polarizer.eps}}
\caption{\label{fig:Polarizer}  The effective measuring  directions  ${\bf a}^l$  and ${\bf a}^s$. In a certain  coordinate system, ${\bf a}^l(\phi_l)$  is  on $xy$ plane,  ${\bf a}^s(\theta_s,\phi_s)$ is on a cone.  For $B_d$ mesons,  $\phi_l = x\Gamma t$, $(\phi_s, \theta_s)=(x\Gamma t+\pi/2,2\beta)$,  corresponding to flavor and CP measurements following evolution of time $t$, respectively.  For photon polarizations, ${\bf a}^l$ and  ${\bf a}^s$ are polarizer directions in  Bloch representation, and can be adjusted directly. }
\end{figure}

\section{CNHV  theories \label{sec:level3} }

$|{\bf u}\rangle$ is an eigenstate of  the Pauli operator $\bm{\sigma}\cdot \mathbf{u}$   in the direction of $\mathbf{u}$. A particle in this state has a definite    $\mathbf{u}$.  For a single particle, QM result  can be reproduced by a  realistic or  HV theory, in which the  measurement outcomes are determined by preexisting  properties independent of the measurement, or ``elements of reality'' in the words of EPR.  Thus  $\mathbf{u}$ is  identified  as such an element of reality.

Consider  a HV theory. Suppose a particle with property ${\bf u}$ is measured along direction ${\bf a}$, then the dichotomic measurement  outcome $A=\pm 1$ is determined by the  hidden variables $\lambda$ in addition to the property  ${\bf u}$ and the local parameter ${\bf a}$.  This is also called a local realistic theory, in which the parameter ${\bf a}$ is local.  In  a nonlocal realistic theory,   $A$ also depends on a  non-local parameters, collectively denoted as  $\eta$.   In a crypto-nonlocal HV (CNHV)  theory, the individual properties  of each particle, after averaging  over distribution $\rho_{\mathbf{u} }(\lambda)$ of the hidden variables  $\lambda$,   become local, as indicated in  countless phenomena,
\begin{equation}
 \int d \lambda \rho_{\mathbf{u} }   (\lambda) A(\mathbf{u},\mathbf{a},\eta,\lambda)
 =\bar{A}(\mathbf{u}, \mathbf{a}). \label{ave}
\end{equation}
A concrete example of $\bar{A}(\mathbf{u}, \mathbf{a})$ is  the Malus' law~\cite{LIExpNature}
\begin{equation}
\bar{A}(\mathbf{u}, \mathbf{a})=\mathbf{u} \cdot\mathbf{a}, \label{avem}
\end{equation}
which is consistent with  QM results of photon polarizations. In the last section, we have just shown that it is also consistent with QM result of  the  meson  decay following its evolution.

For a pair of particles  from a common source,  with   respective properties $\mathbf{u}$ and $\mathbf{v}$, the measuring direction  of the other particle can serve  as the nonlocal parameter, and one can also assume nonlocal parameters $\eta_a$ and $\eta_a$, which are nonlocal with respect to a and b, respectively.   The  measurement outcomes  along  respective directions ${\bf a}$ and ${\bf b}$  are $  A(\mathbf{u},\mathbf{v},\mathbf{a},\mathbf{b},
\eta_a,\eta_b,\lambda)=\pm 1$ and $  B(\mathbf{v},\mathbf{u},\mathbf{b},\mathbf{a},
\eta_b,\eta_a,\lambda)=\pm 1$.  The local measurement of each particle cannot detect its correlation with the other particle, hence the nonlocal dependence disappears after averaging over the hidden variables,
\begin{equation}
\int d \lambda \rho_{\mathbf{u},\mathbf{v}}   (\lambda) A(\mathbf{u},\mathbf{v},\mathbf{a},\mathbf{b},
\eta_a,\eta_b,\lambda)
=\bar{A}(\mathbf{u}, \mathbf{a}), \label{abar}
\end{equation}
\begin{equation}
\int d \lambda \rho_{\mathbf{u},\mathbf{v}}   (\lambda) B(\mathbf{v},\mathbf{u},\mathbf{b},\mathbf{a},
\eta_b,\eta_a,\lambda)
=\bar{B}(\mathbf{v}, \mathbf{b}). \label{bbar}
\end{equation}
A general physical state is a statistical mixture of subensembles with definite    $ {\bf u}$ and ${\bf v}$.   Hence the final expectation values,  which is experimentally measured, are~\cite{LI,LIExpNature}
\begin{equation}
\begin{split}
&\langle A \rangle = \int d \mathbf{u} F(\mathbf{u}) \bar{A}(\mathbf{u}),\\
&\langle B \rangle = \int d \mathbf{v} F(\mathbf{v}) \bar{B}(\mathbf{u}),
\end{split}
\end{equation}
where $F(\mathbf{u})$ and $F(\mathbf{v})$ are probability distribution of polarizations $\mathbf{u}$ and $\mathbf{v}$, respectively. In case of correlated particles,  they are the reduced ones
\begin{equation}
\begin{split}
F(\mathbf{u})=&\int d\bf{v} F({\bf u},{\bf v}),   \\ F(\mathbf{v})=&\int d\bf{u} F({\bf u},{\bf v}).
\end{split}
\end{equation}

The two-body quantities may indicate correlations.  For definite  $ {\bf u}$ and ${\bf v}$,
\begin{equation}
\begin{split}
&\overline{AB}({\bf u},{\bf v},{\bf a},{\bf b})\\
&=\int d\lambda \rho _{{\bf u},{\bf v}}(\lambda) A({\bf u},{\bf v},{\bf a},{\bf b},\lambda) B({\bf v},{\bf u},{\bf b},{\bf a},\lambda).
\end{split}
\end{equation}
For a general state,
\begin{equation}
E({\bf a},{\bf b})=\int d{\bf u}d{\bf v}F({\bf u}, {\bf v})   \overline{AB}({\bf u},{\bf v},{\bf a},{\bf b}),
\end{equation}
which is  the main quantity to be investigated, as it may differ with the corresponding QM result  when entanglement is present, in which case a probability distribution over subensembles with definite polarizations leads to inequalities violated by the entangled state in QM.

Here we extend CNHV theories to include the case that ${\bf a}$ and ${\bf b}$ are not externally fixed. In each measurement, the  measurement settings $\tilde{\bf a}(\lambda)$ and  $\tilde{\bf b}(\lambda)$ are determined by HV $\lambda$, thus the measurement outcomes are  like $A(\mathbf{u}, \tilde{\bf a}(\lambda), \eta, \lambda)$.   Nevertheless, for  those measurements   with $\tilde{\bf a}(\lambda)= {\bf a} $, we can obtain the average of the outcomes. In the case of a single particle, the average is
\begin{equation}
\begin{split}
&\int d \lambda \rho'_{\mathbf{u},\mathbf{a} }(\lambda) A(\mathbf{u},\tilde{\bf a}(\lambda),\eta,\lambda)=\bar{A}(\mathbf{u}, \mathbf{a})=\mathbf{u} \cdot\mathbf{a},
\end{split}
\label{ave2}
\end{equation}
where $\rho'_{\mathbf{u},\mathbf{a} }(\lambda) \equiv \rho_{\mathbf{u} }(\lambda)\delta(\tilde{\bf a}(\lambda)-\mathbf{a})$ is a shorthand.

Likewise, for two correlated particles,   the outcomes of those measurements     with $\tilde{\bf a}(\lambda)= {\bf a} $ and   $\tilde{\bf b}(\lambda)= {\bf b}$ give rise to
\begin{equation}
\begin{split}
&\int d \lambda \rho_{\mathbf{u},\mathbf{v}} (\lambda)  \delta(\tilde{\bf a}(\lambda)-\mathbf{a})\delta(\tilde{\bf b}(\lambda)-\mathbf{b}) \\    & \times A(\mathbf{u},\mathbf{v},\tilde{\bf a}(\lambda),\tilde{\bf b}(\lambda),\eta_a,\eta_b,\lambda)\\
&=\bar{A}(\mathbf{u}, \mathbf{a})=\mathbf{u}\cdot \mathbf{a},
\end{split}
\label{abar1}
\end{equation}
\begin{equation}
\begin{split}
&\int d \lambda \rho_{\mathbf{u},\mathbf{v}} (\lambda)  \delta(\tilde{\bf a}(\lambda)-\mathbf{a})\delta(\tilde{\bf b}(\lambda)-\mathbf{b})  \\   & \times    B(\mathbf{v},\mathbf{u},\tilde{\bf b}(\lambda),\tilde{\bf a}(\lambda),\eta_b,\eta_a,\lambda)\\
&=\bar{B}(\mathbf{v}, \mathbf{b})=\mathbf{v}\cdot \mathbf{b}.
\end{split}
\label{bbar1}
\end{equation}
\begin{equation}
\begin{split}
&\overline{AB}({\bf u},{\bf v},{\bf a},{\bf b})=\int d\lambda \rho_{\mathbf{u},\mathbf{v}} (\lambda)  \delta(\tilde{\bf a}(\lambda)-\mathbf{a})\delta(\tilde{\bf b}(\lambda)-\mathbf{b})   \\
&\times A({\bf u},{\bf v},\tilde{\bf a}(\lambda),\tilde{\bf b}(\lambda),\eta_a,\eta_b,\lambda)
B({\bf v},{\bf u},\tilde{\bf b}(\lambda),\tilde{\bf a}(\lambda),\eta_b,\eta_a,\lambda).
\end{split}
\label{corab}
\end{equation}

Clearly the original formalism is  a special case of this extension, by externally fixing $\tilde{\bf a}(\lambda)$ to be always $\mathbf{a}$ and $\tilde{\bf b}(\lambda)$ to be always $\mathbf{b}$, independent of $\lambda$.

\section{\label{sec:level4}
LI for measuring directions on a plane and a cone}

We now  consider a pair of particles $a$ and $b$, with the measurement outcomes $A=\pm 1$ and $B=\pm 1$, respectively. The average  of those outcomes $A$  with a same measurement setting ${\bf a}$   satisfy the Malus' Law (\ref{abar1}).  The average  of those outcomes $B$  with a same measurement setting  ${\bf b}$ satisfy the Malus' Law (\ref{bbar1}).  The correlation function is defined  in the way of (\ref{corab}).   $\mathbf{a}$ and  $\mathbf{b}$ are each given as  ${\bf a}^l(\phi_i)$  or ${\bf a}^s(\theta_s,\phi_i)$, $(i=a,b)$, as in (\ref{eq.3.8}).

We first consider   correlation  functions of various combinations  of  ${\bf a}^l $  and ${\bf a}^s $ . Define $\hat{E}^\pm ({\bf a},{\bf b})\equiv E({\bf a},{\bf b} )+ E({\bf b},\pm{\bf b})$,  and rewrite $\hat{E}^{\pm}({\bf a}^s(\theta_s,\phi_a),{\bf a}^l(\phi_b))$ as $  \hat{E}^{\pm}_{sl}(\theta _s,\xi,\varphi)$, where $\xi \equiv (\phi _a+\phi _b)/2$,  $\varphi \equiv  \phi _a-\phi _b$. $\hat{E}^{\pm}_{ll}(\theta _s,\xi,\varphi)$ and $\hat{E}^{+}_{ss}(\theta _s,\xi,\varphi)$ are similarly  defined. Furthermore,  we consider the averages over $\xi$,
$\hat{E}^{-}_{sl} (\theta _s,\varphi)\equiv \int \frac{d\xi }{2\pi}  \hat{E}^{-}_{sl}  (\theta _s,\xi,\varphi)$ and so on.

In the Appendix, we prove the following  LI. The upper bound is given by
\begin{equation}
\begin{split}
& \hat{E}^{-}_{sl} (\theta _s,\varphi _1)+\frac{\pi \cos (\theta _{1}(\theta _s, \varphi _1)) L_1(\theta_s,\varphi _1) }{4\cos(\frac{\varphi _2}{2})} \hat{E}^{-}_{ll}(\theta _s,\varphi _2)  \\
&\leq 2  \left(1+\frac{\pi \cos (\theta _{1}(\theta _s, \varphi _1)) L_1(\theta_s,\varphi _1)  }{4\cos(\frac{\varphi _2}{2})}\right)\\
&-\cos (\theta _{1}(\theta _s, \varphi _1)) L_1(\theta_s,\varphi _1),
\end{split} \label{eq.3.25a}
\end{equation}
where
\begin{equation*}
\begin{split}
&L_1(\theta_s,\varphi)\equiv |{\bf a}^s+{\bf  a}^l|=\sqrt{2+2\cos(\varphi)\sin (\theta _s)},\\
&\theta _{1}(\theta_s,\varphi)=\cos ^{-1}\frac{\cos (\theta _s)}{\sqrt{2+2\cos (\varphi)\sin (\theta _s)}}.
\end{split}
\end{equation*}
With $0<\theta _s<\pi /2$, we have  $\sin(\theta _{1})> 0$, $\cos(\theta _{1})> 0$.

We find two lower bounds. The first is  given as
\begin{equation}
\begin{split}
&\hat{E}^+_{sl}(\theta _s,\varphi _1)+\frac{\pi \cos (\theta _{2}(\theta_s,\varphi _1))L_2(\theta_s,\varphi _1)}{4\left|\sin(\frac{\varphi _2}{2})\right|}\hat{E}^+_{ll}(\theta_s,\varphi _2) \\
&\geq -2\left(1+\frac{\pi \cos (\theta _{2}(\theta_s,\varphi _1)) L_2(\theta_s,\varphi _1)}{4\left|\sin(\frac{\varphi _2}{2})\right|}\right)\\
&+ \cos (\theta _{2}(\theta_s,\varphi _1)) L_2(\theta_s,\varphi _1).
\end{split}
\label{eq.3.29a}
\end{equation}
where
\begin{equation*}
\begin{split}
&L_2(\theta_s,\varphi)\equiv |{\bf a}^s-{\bf  a}^l|=\sqrt{2-2\cos(\varphi)\sin (\theta _s)},\\
&\theta _{2}(\theta_s,\varphi)=\cos ^{-1}\frac{\cos (\theta _s)}{\sqrt{2-2\cos (\varphi)\sin (\theta _s)}}.
\end{split}
\end{equation*}

The second lower bound is given as
\begin{equation}
\begin{split}
& \hat{E}^+_{sl}(\theta_s,\varphi _1)+\frac{\pi \cos (\theta _2(\theta_s,\varphi _1)) L_2(\theta_s,\varphi_1)}{4\sin(\theta _s)\left|\sin(\frac{\varphi _2}{2})\right|}\hat{E}^+_{ss}(\varphi _2) \\
&\geq -2\left(1+\frac{\pi \cos (\theta _2(\theta_s,\varphi _1)) L_2(\theta_s,\varphi_1)}{4\sin(\theta _s)\left|\sin(\frac{\varphi _2}{2})\right|}\right)\\
&+\cos (\theta _2)L_2(\theta_s,\varphi_1).
\end{split}
\label{eq.3.31a}
\end{equation}

Eqs.~(\ref{eq.3.25a}), (\ref{eq.3.29a}) and (\ref{eq.3.31a}) comprise  our LI.

The  correlation functions  averaged over $\xi$ are not directly observable, therefore rotational invariance   or  fair sampling of the averages needs to be assumed for measurements, in order that LI in terms of  these average correlation functions can be experimentally examined~\cite{Branciard,Paterek}. In the case of meson decays, the rotational invariance in Bloch representation  is actually time translational invariance.

To drop this additional assumption, we can redefine each  average in a discrete way,
\begin{equation}
\begin{split}
&\hat{E}^{\pm}_{sl}(\theta_s,\varphi)\equiv \frac{1}{N}\sum _{n=1}^N \hat{E}^{\pm}_{sl}(\theta_s,\xi_n =\frac{2n\pi}{N},\varphi),
\end{split}
\label{eq.3.43}
\end{equation}
and $\hat{E}^{\pm}_{ll}(\theta_s,\varphi)$ and  $\hat{E}^{+}_{ss}(\theta_s,\varphi)$ similarly.  As derived in the Appendix, for  these  discrete average correlation functions, our  LI can be obtained from Eqs.~(\ref{eq.3.25a}), (\ref{eq.3.29a}) and (\ref{eq.3.31a}) by simply replacing $\pi/4$ as $1/2u_N$, where  $u_N\equiv \frac{1}{N}\cot \left(\frac{\pi}{2N}\right)$. $N\geq 2$ is required. As  $N\to \infty$, $u_N\to 2/\pi$, then the discrete version  approaches the continuous version.

Our LI can be tested using various systems, in which measurement directions  ${\bf a}^l(\phi_l)$ and  ${\bf a}^s(\theta_s,\phi_s)$ can be directly adjusted.

For  meson decays,  $\theta_s=2\beta$ is fixed, while $\phi_l=\phi_l(t) = x\Gamma t$, $\phi_s=\phi _s(t) = x\Gamma t+\frac{\pi}{2}$ are  given by the decay time $t$. We mention that  for  the two particles $a$ and $b$  to be separated in spacelike distance, there is a constraint  on  the decay times $t_a$ and $t_b$. Suppose the particle pairs are generated from a particle at rest and each flies in velocity $v$ to opposite directions. Then spacelike separation means $(1+w)t_a > (1-w) t_b$, where $w=(v/c)/\sqrt{1-v^2/c^2}$.  Consequently there is a constraint on  possible values of $\xi$, but it does not affect the averages over $\xi$, which is an angle mathematically, hence its functions are periodic.

\section{\label{sec:level5}  Testing LI in entangled  \texorpdfstring{$B_d$}{B} mesons}

We now come back to the $C=-1$  $B^0\bar{B}^0$ entangled meson pairs, and we can  write the  correlation functions as
$$E({\bf a}^X(t_a),{\bf a}^Y(t_b)), (X,Y=l,s).$$
By definition,
\begin{equation}
\begin{split}
&E({\bf a}^X(t_a),{\bf a}^Y(t_b))= \\ & [(P(X+,Y+,t_a,t_b)+P(X-,Y-,t_a,t_b)\\ &-
P(X+,Y-,t_a,t_b)-P(X-,Y+,t_a,t_b)]/\\
&[P(X+,Y+,t_a,t_b)+P(X-,Y-,t_a,t_b)\\ &+
P(X+,Y-,t_a,t_b)+P(X-,Y+,t_a,t_b)],
\end{split}
\label{cor}
\end{equation}
where  for convenience, we have invented the shorthand
\begin{equation}
l+ \equiv B^0, \,\, l-\equiv \bar{B}^0, \, \, s\pm \equiv B_\pm,
\end{equation}
which should not be confused with $l^{\pm}$ and $S_{\pm}$, with $\pm$ as the superscript or subscript,  denoting  the final states of decays.

$P(X\pm,Y\pm,t_a,t_b)$ is the probability that the measurement results of a and b are $X\pm$ and $Y\pm$ at $t_a$ and $t_b$, respectively, indicated by the final states of their respective decays. The measurement or filtering or projection in the flavor basis $\{ B^0, \bar{B}^0\}$ is made through  a semileptonic decay to a flavor eigenstate $|l^\pm\rangle$.  The measurement or filtering  or projection  in CP basis $\{ B_+, B_-\}$ is made through a decay into CP eigenstate $|S_\pm\rangle$.  With direct CP violation negligible, we have $A_{l^-}=\bar{A}_{l^+}=0$,  $A_{S_{\pm}}=\pm \bar{A}_{S_{\pm}}$.  Only if $|l^-\rangle =CP |l^+\rangle$, we have $A_{l^+}=\bar{A}_{l^-}$. Note that the decay products of a and b may be different even though  their flavor  or CP eigenvalues are the same, and   may not be CP conjugates even though their flavor eigenvalues are opposites.

The basis of measurement, namely flavor or CP basis, is not actively  chosen  by experiments, but is signalled by the decay products.  Our extension   of CNHV and LI   addresses this issue, as the main achievement of this paper.

A key quantity is the joint decay rate for  particle   a  decaying  to    $f _a$ at $t_a$ while  particle  b  decaying to $f_b$ at $t_b$, $R(f _a,f _b, t_a, t_b) \propto \left|\langle f _a, f _b| W_a W_b | \Psi (t_a,t_b)\rangle \right|^2$. The following joint decay amplitudes  will be needed.

There are four amplitudes in the form of
\begin{equation}
\langle l^\pm_a, l^\pm_b| W_a W_b | \Psi (t_a,t_b)\rangle =
A_{l^\pm_a} A_{l^\pm_b} \langle  l \pm,  l\pm|   \Psi (t_a,t_b)\rangle, \label{ll}
\end{equation}
where,  $\pm$ in $l^\pm_a$ corresponds to the first  $ l \pm$ on RHS,  $\pm$ in   $l^\pm_b$ corresponds to the second $ l \pm$ on RHS.

There are four amplitudes of the form of
\begin{equation}
\begin{split}
&\langle S_\pm^a, S_\pm^b| W_a W_b | \Psi (t_a,t_b)\rangle =\\
&2 A_{S_\pm^a} A_{S_\pm^b} \langle s\pm, s\pm| \Psi (t_a,t_b)\rangle,
\end{split}
\label{ss}
\end{equation}
where we have used $\langle S_\pm|W|B_\pm\rangle =(A_{S_\pm}\pm \bar{A}_{S_\pm})/\sqrt{2}=\sqrt{2} A_{S_\pm }$.

There are four other amplitudes of the form of
\begin{equation}
\begin{split}
&\langle S_\pm^a, l^\pm_b| W_a W_b | \Psi (t_a,t_b)\rangle =\\
&\sqrt{2} A_{S_\pm^a} A_{l^\pm_b} \langle s\pm, l\pm| W_a W_b |  \Psi (t_a,t_b)\rangle.
\end{split}
\label{sl}
\end{equation}

The   experimentally measured quantity  is the  number of the joint events $N(f_a,f_b,t_a,t_b) \propto \epsilon_{f_a,f_b}R(f_a,f_b,t_a,t_b)$,
where $\epsilon_{f_a,f_b}$ is the detection efficiency for that channel~\cite{detectionEfficience}, $R(f_a,f_b,t_a,t_b)$ is proportional to  the modulo square of the joint decay amplitude, as given in   Eqs. (\ref{ll}-\ref{sl}).

Therefore the correlation function (\ref{cor}) can be obtained from event numbers as

\begin{equation}
\begin{split}
&E\left({\bf a}^l(t_a),{\bf a}^l(t_b)\right)\\
&=
\left(\frac{N(l_a^+,l_b^+,t_a,t_b)}{\epsilon_{l_a^+,l_b^+}|A_{l_a^+}|^2|A_{l_b^+}|^2}
+\frac{N(l_a^-,l_b^-,t_a,t_b)}{\epsilon _{l_a^-,l_b^-}|\bar{A}_{l_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.-\frac{N(l_a^+,l_b^-,t_a,t_b)}{\epsilon _{l_a^+,l_b^-}|A_{l_a^+}|^2|\bar{A}_{l_b^-}|^2}
-\frac{N(l_a^-,l_b^+,t_a,t_b)}{\epsilon _{l_a^-,l_b^+}|\bar{A}_{l_a^-}|^2|A_{l_b^+}|^2} \right)\\
&/\left(\frac{N(l_a^+,l_b^+,t_a,t_b)}{\epsilon _{l_a^+,l_b^+}|A_{l_a^+}|^2|A_{l_b^+}|^2}
+\frac{N(l_a^-,l_b^-,t_a,t_b)}{\epsilon _{l_a^-,l_b^-}|\bar{A}_{l_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.+\frac{N(l_a^+,l_b^-,t_a,t_b)}{\epsilon _{l_a^+,l_b^-}|A_{l_a^+}|^2|\bar{A}_{l_b^-}|^2}
+\frac{N(l_a^-,l_b^+,t_a,t_b)}{\epsilon _{l_a^-,l_b^+}|\bar{A}_{l_a^-}|^2|A_{l_b^+}|^2} \right),
\end{split}
\label{eq.4.8.1}
\end{equation}
\begin{equation}
\begin{split}
&E\left({\bf a}^s(t_a),{\bf a}^l(t_b)\right)\\
&=
\left(\frac{N(S_a^+,l_b^+,t_a,t_b)}{\epsilon _{S_a^+,l_b^+}|A_{S_a^+}|^2 |A_{l_b^+}|^2 }
+\frac{N(S_a^-,l_b^-,t_a,t_b)}{\epsilon _{S_a^-,l_b^-}|A_{S_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.-\frac{N(S_a^+,l_b^-,t_a,t_b)}{\epsilon _{S_a^+,l_b^-}|A_{S_a^+}|^2 |\bar{A}_{l_b^-}|^2 }
-\frac{N(S_a^-,l_b^+,t_a,t_b)}{\epsilon _{S_a^-,l_b^+}|A_{S_a^-}|^2  |A_{l_b^+}|^2 }\right)\\
&/\left(\frac{N(S_a^+,l_b^+,t_a,t_b)}{\epsilon _{S_a^+,l_b^+}|A_{S_a^+}|^2 |A_{l_b^+}|^2 }
+\frac{N(S_a^-,l_b^-,t_a,t_b)}{\epsilon _{S_a^-,l_b^-}|A_{S_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.+\frac{N(S_a^+,l_b^-,t_a,t_b)}{\epsilon _{S_a^+,l_b^-}|A_{S_a^+}|^2 |\bar{A}_{l_b^-}|^2 }
+\frac{N(S_a^-,l_b^+,t_a,t_b)}{\epsilon _{S_a^-,l_b^+}|A_{S_a^-}|^2  |A_{l_b^+}|^2 }  \right),
\end{split}
\label{eq.4.8.2}
\end{equation}
\begin{equation}
\begin{split}
&E\left({\bf a}^s(t_a),{\bf a}^s(t_b)\right)\\
&=
\left(
\frac{N(S_a^+,S_b^+,t_a,t_b)}{\epsilon _{S_a^+,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}
+\frac{N(S_a^-,S_b^-,t_a,t_b)}{\epsilon _{S_a^-,S_b^-}|A_{S_a^-}|^2 |A_{S_b^-}|^2 }\right.\\
&\left.-\frac{N(S_a^+,S_b^-,t_a,t_b)}{\epsilon _{S_a^+,S_b^-}|A_{S_a^+}|^2|A_{S_b^-}|^2}
-\frac{N(S_a^-,S_b^+,t_a,t_b)}{\epsilon _{S_a^-,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}\right)\\
&/\left(\frac{N(S_a^+,S_b^+,t_a,t_b)}{\epsilon _{S_a^+,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}
+\frac{N(S_a^-,S_b^-,t_a,t_b)}{\epsilon _{S_a^-,S_b^-}|A_{S_a^-}|^2 |A_{S_b^-}|^2 }\right.\\
&\left.+\frac{N(S_a^+,S_b^-,t_a,t_b)}{\epsilon _{S_a^+,S_b^-}|A_{S_a^+}|^2|A_{S_b^-}|^2}
+\frac{N(S_a^-,S_b^+,t_a,t_b)}{\epsilon _{S_a^-,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}\right),
\end{split}
\label{eq.4.8.3}
\end{equation}
where $\epsilon$'s are the detection efficiencies. In experiments, $|A_{l^{\pm}_{i}}|^2$ and  $|A_{S_{i}^{\pm}}|^2$, $(i=a,b)$,  can be absorbed to the redefinitions of detection efficiencies.

Furthermore, one obtains
\begin{equation}
\begin{split}
&\hat{E}^{ll\pm}(\phi_a,\phi_b) \equiv \\
&E\left({\bf a}^l(x\Gamma t_a),{\bf a}^l(x\Gamma t_b)\right)+E\left({\bf a}^l(x\Gamma t_b),\pm{\bf a}^l(x\Gamma t_b)\right),\\
&\hat{E}^{sl\pm}(\phi_a,\phi_b) \equiv \\
&E\left({\bf a}^s(x\Gamma t_a+\frac{\pi}{2}),{\bf a}^l(x\Gamma t_b)\right)+E\left({\bf a}^l(x\Gamma t_b),\pm{\bf a}^l(x\Gamma t_b)\right),\\
&\hat{E}^+_{ss}(\phi_a,\phi_b)  \equiv
E\left({\bf a}^s(x\Gamma t_a+\frac{\pi}{2}),{\bf a}^l(x\Gamma t_b)\right)\\&+E\left({\bf a}^s(x\Gamma t_b+\frac{\pi}{2}),{\bf a}^s(x\Gamma t_b+\frac{\pi}{2})\right),\\
\end{split}
\label{eq.4.8.new}
\end{equation}
from which the averages  $\hat{E}^{ll\pm}(\varphi)$, $\hat{E}^{sl\pm}(\varphi)$, $\hat{E}^{ss+}(\varphi)$  can be obtained.
Note that we did not define $\hat{E}^{ss-}$, which would not have physical meaning, as   $-{\bf a}^s$ is not on the cone, where all possible ${\bf a}^s$'s lie.
In SM, with $\Delta \Gamma = 0$, $R(f_b,f_b,t_a,t_b)$ can be obtained as
\begin{equation}
\begin{split}
&R(f _a,f _b,t_a, t_b)=\frac{e^{-\Gamma (t_a+t_b)}}{4}\times \left\{(|\xi_-|^2+|\zeta_-|^2)\right.\\
&\left.-(|\xi_-|^2-|\zeta_-|^2)\cos (x\Gamma (t_a-t_b))\right.\\
&\left.-2Im(\zeta_-^*\xi_-)\sin (x \Gamma (t_a-t_b))\right\},
\end{split}
\label{eq.4.3}
\end{equation}
where $\xi_- \equiv -\left(\frac{p}{q}A_{f_a}A_{f_b}-
\frac{q}{p}\bar{A}_{f_a}
\bar{A}_{f_b}\right)$, $\zeta_- \equiv \left(A_{f_a}\bar{A}_{f_b}-
\bar{A}_{f_a}A_{f_b}\right)$.

In  experiments, it is more convenient to use the time-integrated joint decay rate
\begin{equation}
R(f _a,f _b,\Delta t)=\int _0^{\infty}dt_aR_-(f _a,f _b, t_a, t_a+\Delta t),
\label{eq.4.1}
\end{equation}
which is obtained as
\begin{equation}
\begin{split}
&R(f _a,f _b,\Delta t)= \frac{e^{-\Gamma |\Delta t|}}{8\Gamma}\times \left\{(|\xi_-|^2+|\zeta_-|^2)\right.\\
&\left. - (|\xi_-|^2-|\zeta_-|^2)\cos (x\Gamma \Delta t)\right.\\
&\left.+2Im(\zeta_-^*\xi_-)\sin (x\Gamma \Delta t)\right\},
\end{split}
\label{eq.4.3b}
\end{equation}

It is more rigorous to  test LI of  the discrete version of the average correlation functions, rather than that of the  continuous version. However, it is experimentally   much easier to measure   $N(f_a,f_b,\Delta t)\propto   \epsilon _{f_a,f_b}R(f_a,f_b,\Delta t)$  than   $N_{f_a,f_b}(t_a,t_b)$, consequently it is much easier to test LI in terms of  the continuous version of the average correlation functions.

From (\ref{eq.4.3}) and (\ref{eq.4.3b}), it can be seen that  in SM,  $R(f_a,f_b,t_b+\Delta t,t_b)=2\Gamma e^{-\Gamma (t_a+t_b)} e^{\Gamma |\Delta t|}    R(f_a,f_b, \Delta t)$. Consequently, $\hat{E}(\varphi)$, as an average of $E(\phi _a, \phi _b)$ over $\xi\equiv x\Gamma(t_a+t_b)/2+\pi/2$, can be directly related to $N(f_a,f_b,\Delta t)$ as
\begin{equation}
\begin{split}
&\hat{E}^{ll}(\varphi = x\Gamma \Delta t)
=\\
&\left(\frac{N(l_a^+,l_b^+,\Delta t)}{\epsilon_{l_a^+,l_b^+}|A_{l_a^+}|^2|A_{l_b^+}|^2}
+\frac{N(l_a^-,l_b^-,\Delta t)}{\epsilon _{l_a^-,l_b^-}|\bar{A}_{l_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.-\frac{N(l_a^+,l_b^-,\Delta t)}{\epsilon _{l_a^+,l_b^-}|A_{l_a^+}|^2|\bar{A}_{l_b^-}|^2}
-\frac{N(l_a^-,l_b^+,\Delta t)}{\epsilon _{l_a^-,l_b^+}|\bar{A}_{l_a^-}|^2|A_{l_b^+}|^2}\right)\\
&/\left(\frac{N(l_a^+,l_b^+,\Delta t)}{\epsilon _{l_a^+,l_b^+}|A_{l_a^+}|^2|A_{l_b^+}|^2}
+\frac{N(l_a^-,l_b^-,\Delta t)}{\epsilon _{l_a^-,l_b^-}|\bar{A}_{l_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.+\frac{N(l_a^+,l_b^-,\Delta t)}{\epsilon _{l_a^+,l_b^-}|A_{l_a^+}|^2|\bar{A}_{l_b^-}|^2}
+\frac{N(l_a^-,l_b^+,\Delta t)}{\epsilon _{l_a^-,l_b^+}|\bar{A}_{l_a^-}|^2|A_{l_b^+}|^2} \right),\\
\end{split}
\label{eq.4.9.1}
\end{equation}
\begin{equation}
\begin{split}
&\hat{E}^{sl}(\varphi = x\Gamma\Delta t+\frac{\pi}{2})=\\
&\left(\frac{N(S_a^+,l_b^+,\Delta t)}{\epsilon _{S_a^+,l_b^+}|A_{S_a^+}|^2 |A_{l_b^+}|^2 }
-\frac{N(S_a^-,l_b^-,\Delta t)}{\epsilon _{S_a^-,l_b^-}|A_{S_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.+\frac{N(S_a^+,l_b^-,\Delta t)}{\epsilon _{S_a^+,l_b^-}|A_{S_a^+}|^2 |\bar{A}_{l_b^-}|^2 }
-\frac{N(S_a^-,l_b^+,\Delta t)}{\epsilon _{S_a^-,l_b^+}|A_{S_a^-}|^2  |A_{l_b^+}|^2 }  \right)\\
&/\left(\frac{N(S_a^+,l_b^+,\Delta t)}{\epsilon _{S_a^+,l_b^+}|A_{S_a^+}|^2 |A_{l_b^+}|^2 }
+\frac{N(S_a^-,l_b^-,\Delta t)}{\epsilon _{S_a^-,l_b^-}|A_{S_a^-}|^2|\bar{A}_{l_b^-}|^2}\right.\\
&\left.+\frac{N(S_a^+,l_b^-,\Delta t)}{\epsilon _{S_a^+,l_b^-}|A_{S_a^+}|^2 |\bar{A}_{l_b^-}|^2 }
+\frac{N(S_a^-,l_b^+,\Delta t)}{\epsilon _{S_a^-,l_b^+}|A_{S_a^-}|^2  |A_{l_b^+}|^2 }  \right),\\
\end{split}
\label{eq.4.9.2}
\end{equation}
\begin{equation}
\begin{split}
&\hat{E}^{ss}(\varphi = x\Gamma\Delta t)
=\\
&\left(
 \frac{N(S_a^+,S_b^+,\Delta t)}{\epsilon _{S_a^+,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}
+\frac{N(S_a^-,S_b^-,\Delta t)}{\epsilon _{S_a^-,S_b^-}|A_{S_a^-}|^2 |A_{S_b^-}|^2 }\right.\\
&\left.-\frac{N(S_a^+,S_b^-,\Delta t)}{\epsilon _{S_a^+,S_b^-}|A_{S_a^+}|^2|A_{S_b^-}|^2}
-\frac{N(S_a^-,S_b^+,\Delta t)}{\epsilon _{S_a^-,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}\right)\\
&/\left(\frac{N(S_a^+,S_b^+,\Delta t)}{\epsilon _{S_a^+,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}
+\frac{N(S_a^-,S_b^-,\Delta t)}{\epsilon _{S_a^-,S_b^-}|A_{S_a^-}|^2 |A_{S_b^-}|^2 }\right.\\
&\left.+\frac{N(S_a^+,S_b^-,\Delta t)}{\epsilon _{S_a^+,S_b^-}|A_{S_a^+}|^2|A_{S_b^-}|^2}
+\frac{N(S_a^-,S_b^+,\Delta t)}{\epsilon _{S_a^-,S_b^+}|A_{S_a^+}|^2|A_{S_b^+}|^2}\right).\\
\end{split}
\label{eq.4.9.3}
\end{equation}

Moreover,  the integration over $\xi$ of $\hat{E}^{\pm}({\bf a},{\bf b})=E({\bf a},{\bf b})+E({\bf b},\pm {\bf b})$  can be performed independently for the two  terms on RHS, consequently,
\begin{equation}
\begin{split}
&\hat{E}^+_{ll}(\varphi )=\hat{E}^{ll}(\varphi)+\hat{E}^{ll}(0),\;\;\hat{E}^{-}_{ll}(\varphi )=\hat{E}^{ll}(\varphi)+\hat{E}^{ll}(\pi)\\
&\hat{E}^+_{sl}(\varphi )=\hat{E}^{sl}(\varphi)+\hat{E}^{ll}(0),\;\;
\hat{E}^{-}_{sl}(\varphi )=\hat{E}^{sl}(\varphi)+\bar{E}^{ll}(\pi)\\
&\hat{E}^+_{ss}(\varphi )=\hat{E}^{ss}(\varphi)+\hat{E}^{ss}(0).\\
\end{split}
\label{eq.4.10.new}
\end{equation}

Note that Eqs.~(\ref{eq.4.8.1}-\ref{eq.4.8.3}) and (\ref{eq.4.9.1}-\ref{eq.4.9.3}) are mainly for the use in analyzing  experimental data. QM result  can be obtained simply from $\langle {\bf a}^X,{\bf a}^Y|\Psi_-\rangle=- {\bf a}^X\cdot{\bf a}^Y$, therefore
\begin{equation}
\begin{split}
& \hat{E}^{sl} (\varphi )=-\sin (2\beta)\cos (\varphi ),\;\; \hat{E}^{ll} (\varphi )=-\cos (\varphi ),\\
& \hat{E}^{ss }(\varphi )=-\cos^2(2\beta)-\sin^2(2\beta)\cos (\varphi ). \\
\end{split}
\label{eq.4.10}
\end{equation}
It is  interesting to test our LI using   various systems, in which $\varphi_a$ and  $\varphi_b$ are directly adjusted.

For $B_d$ mesons,  QM result (\ref{eq.4.10})  can also be obtained  from the definition (\ref{cor}) of correlation functions,   with \\ $P(X\pm,Y\pm,t_a,t_b)$ $\propto$ \\$\left|\langle X\pm,Y\pm| W_a W_b | \Psi (t_a,t_b)\rangle \right|^2$   obtained by substituting LHS  of $R(f _a,f _b, t_a, t_b) \propto$ $\left|\langle f _a, f _b| W_a W_b | \Psi (t_a,t_b)\rangle \right|^2$ with the result (\ref{eq.4.3}) of $R(f _a,f _b, t_a, t_b)$, and  RHS  with the joint amplitudes (\ref{ll}-\ref{sl}),  and then having
$A_f$ and  $\bar{A}_f$ cancelled.   For $B_d$ mesons,  the values of $\beta$ and $x$ are given by  $\sin (2\beta )=0.695$, $x=0.769$~\cite{PDG}.

The upper bound of a
our LI violation can be quantified as
\begin{equation}
g^u(\varphi_1,\varphi _2) \equiv \frac{h^u_{L}(\varphi_1,\varphi _2)-h^u_{R}(\varphi_1,\varphi _2)}{\left|h^u_{L}(\varphi_1,\varphi _2)\right|},
\end{equation}
where $h^u_{R}$ and $h^u_{L}(\varphi_1,\varphi _2)$ are RHS and LHS of (\ref{eq.3.25a}), respectively. The first lower bound can be quantified as
\begin{equation}
g^{d1}(\varphi_1,\varphi _2)=\frac{h^{d1}_{R}(\varphi_1,\varphi _2)-h^{d1}_{L}(\varphi_1,\varphi _2)}{\left|h^{d1}_{L}(\varphi_1,\varphi _2)\right|},
\end{equation}
where $h^{d1}_R$ and $h^{d1}_L$ are RHS and LHS of (\ref{eq.3.29a}), respectively.   The second  lower bound can be quantified as
\begin{equation}
g^{d2}(\varphi_1,\varphi _2)=\frac{h^{d2}_{R}(\varphi_1,\varphi _2)-h^{d2}_{L}(\varphi_1,\varphi _2)}{\left|h^{d2}_{L}(\varphi_1,\varphi _2)\right|},
\end{equation}
where $h^{d2}_R$ and $g^{h2}_L$ are RHS and LHS of  (\ref{eq.3.31a}), respectively. Each of these three quantities  larger than $0$ represents the violation of the corresponding bound of LI.

From
$\partial _{\varphi _1}g^u(\varphi _1,\varphi _2)= \partial _{\varphi _1}g^{d1}(\varphi _1,\varphi _2)= \partial _{\varphi _1}g^{d2}(\varphi _1,\varphi _2)=0$, it is determined that the maximum of   $g^{u}$ is on  $\varphi_1=\pi$, while the maxima of $g^{d1}$ and  $g^{d1}$ are both on $\varphi _1=0$.  $L_1(\varphi _1=\pi)=L_2(\varphi _1=0)\approx 0.781$,  $\theta _{1}(\varphi _1=\pi)=\theta _{ 2}( \varphi _1=0)\approx 0.401$.

Furthermore, solving $\partial _{\varphi _2}g^u(\pi,\varphi _2)=0$ numerically, we find that $g^u(\pi, \varphi _2)$ reaches its maximum at $\varphi _2\approx \pm 2.81$.  We also find numerically that  $g^u(\pi,\varphi _2)>0$ when    $2.39<|\varphi _2|<\pi$.
Similarly, solving $\partial _{\varphi _2}g^{d1}(0,\varphi _2)=0$ numerically, we find that $g^{d1}(0,\varphi _2)$ reaches the maximum at $\varphi _2\approx \pm 0.336$. We also find numerically  that $g^{d1}(0,\varphi _2)>0$ when    $ 0<|\varphi _2|<0.75$.

Solving $\partial_{\varphi _2}g^{d2}(0,\varphi _2)=0$ numerically, we  find that $g^{d2}(0,\varphi _2)$ reaches its maximum at $\varphi _2\approx \pm 0.486$, and we find numerically  that $g^{d_2}(0,\varphi _2)>0$ when   $ 0<|\varphi _2|<1.11$. The  $\varphi _2$ range   of  $g^{d2}>0$ is larger than that of  $g^{d1}>0$. The maxima of  $g^u$,  $g^{d1}$ and  $g^{d2}$  are all about $2.7\%$. The results are depicted in Fig.~\ref{Fig:gud}.

\begin{figure*}
\centering{
\subfloat[$g^u(\varphi _1, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gu12}}}
\subfloat[$g^{d1}(\varphi _1, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gd1_12}}}
\subfloat[$g^{d2}(\varphi _1, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gd2_12}}} \\
\subfloat[$g^u(\pi, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gu02}}}
\subfloat[$g^{d1}(0, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gd1_02}}}
\subfloat[$g^{d2}(0, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gd2_02}}}\\
\caption{\label{Fig:gud}
LI violation of entangled $B_d$ mesons. $\theta_s$ is fixed to be  $\sin ^{-1}{0.695}$.      Our LI is violated when $g^u(\varphi_1,\varphi_2)>0$  or $g^{d1}(\varphi_1,\varphi_2)>0$ or $g^{d2}(\varphi_1,\varphi_2)>0$.   $g^u>0$  when $\varphi _1=\pi$, $ 2.39<|\varphi _2|<\pi$, and the maximum is at $\varphi _1=\pm \pi, \varphi _2\approx \pm 2.81$.  $g^{d1} >0$ when $\varphi _1=0$,   $0<|\varphi _2|<0.75$, and the maximum is at $\varphi _1=0, \varphi _2\approx \pm 0.336$. $g^{d2} >0$ when $\varphi _1=0$,   $0<|\varphi _2|<1.11$,  and  the maximum is at $\varphi _1=0, \varphi _2\approx \pm 0.486$. The maxima are all about $2.7\%$.}
}
\end{figure*}

A $B_d$ meson is unstable, and the time interval between the two decays is of the order of the lifetime $\tau _B$~\cite{TViolationD}. So it is better to study  the case  in which $\Delta t$  is of the  order of $\tau _B$, so that the number of events is large. Thus it is easier to test $g^{d2}$, because  in its  violation region, $\Delta t$ is closer to $\tau _B$.

We now focus on how to make measurements to confirm the violation of the second lower bound.  $g^{d2}(\varphi _1,\varphi _2)>0$  can be found   in the regime  $(\varphi _1=0$,  $ 0<|\varphi _2|<1.11)$, as calculated above. The function  $g^{d2}(\varphi _1,\varphi _2)$ contains
$\hat{E}^+_{sl}(\varphi_1)=\hat{E}^{sl}(\varphi_1)+
\hat{E}^{ll}(0)$ and
$\hat{E}^+_{ss}(\varphi_2)=\hat{E}^{ss}(\varphi)+
\hat{E}^{ss}(0)$.
Hence one first  measures $\hat{E}^{ll}(\Delta t_1=0)$ and    $\hat{E}^{sl}(\varphi_1=0)=  \hat{E}^{sl}(\Delta t_1 =-\pi / 2x \Gamma \approx -2.04\tau _B )$,  as shown in Fig.~\ref{Fig:measure}. Thus $\hat{E}^+_{sl}(\varphi_1=0)= \hat{E}^{sl}(\varphi_1=0)+\bar{E}^{ll}(\varphi_1=0)$   is obtained. One also needs to measure  $\hat{E}^{ss}(\Delta t_2 =0)$ and $\hat{E}^{ss}(\Delta t_2)$ with $0<|\Delta t_2|<1.11/x\Gamma \approx 1.44\tau _B$, such that   $\hat{E}^+_{ss}(\varphi_2)$ with     $0<|\varphi _2|<1.11$ is  obtained. Thus one obtains    $g^{d2}(\varphi _1,\varphi _2)>0$ in this regime.   The violation is maximal when $\Delta t_2 \approx 0.486/x\Gamma \approx 0.633\tau _B$, then $g^{d2}(\varphi _1,\varphi _2)\approx
2.7\%$,  as shown in Fig.~\ref{Fig:measure}. Typically, the resolution of the signal is proportional to the inverse of square of event numbers~\cite{TViolationD}. Therefore it  can  be estimated that the expected signal of  LI violation  can be observed when the event number is about $10^4 \sim 10^5$, which can be achieved in  current  experiments~\cite{ImprovedCPV}.

\begin{figure*}
\centering{
\resizebox{0.9\textwidth}{!}{\includegraphics{measure}}
\caption{\label{Fig:measure}
The correlation functions to be measured. The left picture represents   $\hat{E}^{sl}(\Delta t = \pi / 2x\Gamma \approx 2.04\tau _B)=\hat{E}^{sl}(\varphi_1=0)$. The right picture represents $\hat{E}^{ss}(\Delta t = 0.486 / x\Gamma \approx 0.633\tau _B)=\hat{E}^{ss}(\varphi _2\approx 0.486)$. They give rise to $g^{d1}( \varphi_1=0,\varphi _2\approx 0.486)$, which maximally violates the second lower bound of our LI.}
}
\end{figure*}

It is also possible to test LI in polarization-entangled baryon-antibaryon pairs produced in, say,  $J/\Psi \to \Lambda \bar{\Lambda}$ decays, where the polarizations of $\Lambda$ and $\bar{\Lambda}$ can be measured through the angular distribution of the momenta of their decay products pions.  However, the effective measuring directions satisfying the Malus' law are yet to be found out.

\begin{figure*}
\centering{
\subfloat[$g^u(\theta_s,\pi, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gu_3d}}}
\subfloat[$g^{d1}(\theta_s,0, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gd1_3d}}}
\subfloat[$g^{d2}(\theta_s,0, \varphi _2)$]{\resizebox{0.33\textwidth}{!}{\includegraphics{gd2_3d}}}
\caption{\label{Fig:g3d}
LI violation in case $\theta_s$ is a variable. The maxima  of $g^u$, $g^{d1}$ and $g^{d2}$ are still at $\varphi _1= \pi,0,0$, respectively.   $g^u(\theta_s,\pi,\varphi _2)$, $g^{d1}(\theta_s,0,\varphi _2)$ and $g^{d2}(\theta_s,0,\varphi _2)$  are functions of $\theta_s$ and $\varphi _2$.
}}
\end{figure*}

If using other systems such as photon polarizations to test our LI, $\theta_s$ may become a variable. The results of $\partial _{\varphi _1}g^u(\varphi _1,\varphi _2)= \partial _{\varphi _1}g^{d1}(\varphi _1,\varphi _2)= \partial _{\varphi _1}g^{d2}(\varphi _1,\varphi _2)=0$ do not depend on $\theta_s$, hence their maxima are still at $\varphi _1= \pi,0,0$, respectively.  $g^u(\theta_s,\pi,\varphi _2)$, $g^{d1}(\theta_s,0,\varphi _2)$ and $g^{d2}(\theta_s,0,\varphi _2)$  as functions of $\theta_s$ and $\varphi _2$  are shown in Fig.~\ref{Fig:g3d}. Interestingly,   in a  certain range of $\varphi _2$, for any value of $\theta _s$ except $0$ and $\pi/2$,  we always have $g^{d2}>0$, that is,  the second lower bound is always violated.

Furthermore, we numerically found the maximal violations  of the three bounds are all $3.87482\%$  when  $\theta _s=1.18208$, i.e.  \begin{equation}\begin{split}
&g^u(1.18208,\pm \pi, \pm 2.7373)=g^{d1}(1.18208,0, \pm 0.404296)\\&=g^{d2}(1.18208,0, \pm 0.437399)=3.87482\%.\end{split}
\end{equation}
It is also found that
\begin{equation}
g^{d1}(1.18208,0,\varphi_2)>0,\,\mbox{when}\,\, 0<|\varphi_2|<1.0734, \end{equation}
\begin{equation} g^{d2}(1.18208,0,\varphi_2)>0,\,\mbox{when}\, \, 0<|\varphi_2|<1.17078.
\end{equation}
The latter is wider, as   shown in Fig~\ref{Fig:gdmax}.

\begin{figure}
\centering
\resizebox{1.0\hsize}{!}{\includegraphics{gdmax}}
\caption{  \label{Fig:gdmax}
The two lower bounds are maximally violated on $\theta_s=1.18208$, $\varphi_1=0$.   Shown here are  the dependence of  $g^{d1}(\theta_s=1.18208,\varphi_1=0,\varphi_2)$ and  $g^{d2}(\theta_s=1.18208,\varphi_1=0,\varphi_2)$  on $\varphi_2$.  $g^{d1} >0$ when   $0<|\varphi_2|<1.0734$, while   $g^{d2}  >0$ when $0<|\varphi_2|<1.17078$. }
\end{figure}

\section{\label{sec:level6} Summary  and   Discussions}

To summarize,  we have extended the CNHV theories to include the case that the measuring settings, together with the measurement outcomes,  are  not externally fixed, but determined by HVs. The outcomes of those measurements with the same settings give averages satisfying Malus' Law and make up correlation functions. We show that such is the case of meson decays,  which  could be determined by HVs at the source of the meson pairs. This extension does not  change  the validity of LI. Therefore, entangled meson pairs can be used to test LI.

We find that for a $B_d$ meson, the effective measuring directions appearing in Malus' Law are  on a cone and a plane, corresponding to semileptonic decays and decays to CP eigenstates, respectively.  For such effective measuring directions, we present a new  LI.  This can be tested in   $C=-1$ entangled  state of $B^0-\bar{B}^0$ pairs, within the present experimental capability. The expected violation is estimated quantitatively, using the indirect CP violation and other parameters.  Our LI is violated  if there is indirect CP violation. There may be  profound reason for this surprising connection. Besides, our new LI can also be tested in other systems such as photon polarizations, where the measuring directions are simply directions externally fixed.

\begin{acknowledgement}
This work is supported by  National Science Foundation of China (Grant No. 11574054 and No. 11947066).
\end{acknowledgement}

redpajama_set_name: RedPajamaArXiv


EggGang! Does this mean I won his love?
Thanks so much for playing! enjoyed the playthrough!!
egg gang! i thought the jokes were really funny!
Nice Games> If you guys like what I do then subscribe and check out my full and support your boii grow thanks!! I recommend everyone to check this game out!!
Kinda disappointing there wasn't a special ending for just skipping all the days.
Either way, extremely good game!
It will not run on mac. You can still play it through flash but downloading it on a mac doesn't work. The mac unzips the package and then when you try to run it the can says that the game has been damaged and can not be played.
Are you the team that made the fnaf dating sim for phones?
I was giggling toward the end. SUPER clever game, hahaha. Nice work!
I got 300 pts! Egg gang!!!
Quick Egg dating simulator game interesting!! Ending Scared me a little it was short story is horror about life of egg!! Those to like dating simulator or choice making game check it out!!


text:
hygiene operative job description
Look up hygiene operative job description in Wiktionary, the free dictionary.
hygiene operative job descriptionOffering exceptional thermal efficiency, build quality, and desirable minimalist aesthetics, Üni_Slide can achieve frame sightlines as slim as 20mm, and a slim central interlock, while being able to support sliding panels of up to 3m high and 2.2m wide. The product can be assembled in a limitless run of units, with each panel capable of holding up to 320kg of glass. British Glass is organising a one-day seminar on funding for decarbonisation and energy efficiency work on Thursday 2 November (north of England, venue TBC) – which will be attended by British Glass members' and non-members. Martin Nettleton adds: "We invested in the campaign to target those larger installation and distribution companies increasingly dissatisfied with the big fabricators who are starting to compete with them by setting up their own networks of trade counters and distribution outlets. Friends and colleagues gathered to applaud worthy winners across seven categories – and recognise the creativity and dedication which underpin the whole sector's contribution to the economy, to society and to customers, staff and stakeholders. This beautiful bespoke bottle was designed by M&E Design. It perfectly demonstrates how glass can be used as a vessel for ambition, style, and sophistication. The bottle was created with none of the usual visual cues of a standard whiskey bottle. Instead, it was crafted to celebrate the liquid inside and the process behind its creation. The design takes its cues from laboratory glassware found in the micro-distillery and the paneling of a whiskey cask. It has been shaped to refract and bounce light. The side panel embossing produces a Kaleidoscope of patterns within the body of the bottle, bringing Method and Madness together. The British Glass entry features its three-year programme to facilitate collaboration between glass manufacturers and government to create a decarbonisation action plan – setting out the sectors' priorities for energy efficiency and decarbonisation in areas such as research and development, technology implementation, energy infrastructure, recycling, skills and funding. In April of this year all ten of the UK's large-scale glass manufacturers signed up to the voluntary action plan. The whiskey was inspired by masters and apprentices working together at the new Midleton micro-distillery, one of the oldest and most iconic distilleries in the world. The brand combines tradition and innovation to convey a sense of fun and excitement. It uses two opposing patterns throughout the packaging. The straight lines symbolise method and order and the stunning marbled patterns reference madness and liquid experimentation.
1 what is hand hygiene
2 hygiene course
3 crest hygiene
This hygiene for kids page lists articles associated with the title Seo.
If an personal hygiene test questions led you here, you may wish to change the link to point directly to the intended article.
poor personal hygiene

redpajama_set_name: RedPajamaCommonCrawl


Following his surprise music video, "DAYDREAMING" for Radiohead's surprise 2016 album, A Moon Shaped Pool, director Paul Thomas Anderson dropped yet another collaboration with the iconic art-rock outfit out of the warm, blue September sky. In delivering his second video for the album, Anderson has proved himself a cinematic soul-mate to the avant-garde musical character of Radiohead's new album.
The piece, for the track "PRESENT TENSE", dramatically scales back any stylistic pyrotechnics in favor of a subdued, intimate performance video. Sitting against a bare-bones backdrop in warm amber light so dim it could almost be candlelight, Thom Yorke and Jonny Greenwood perform the track through its entirety with their voices, their acoustic guitars, and even a Roland CR78 drum machine. Anderson alternates his camera between two separate setups, framing the two bandmates in static medium shots before punching in for a few shallow-focus closeups that further demonstrate Anderson's fascination with the visual language of portraits. Yorke and Greenwood give themselves entirely to their performances, achieving a heightened state of mind that calls to mind the similar expressions of artistic rapture evidenced in Anderson's 2015 music documentary JUNUN.
From "BURN THE WITCH" to Anderson's own "DAYDREAMING", the music videos that have accompanied the release of A Moon Shaped Pool have been nothing less than creatively-arresting works of art. "PRESENT TENSE" is a much more subdued effort in that sense, but it nevertheless resonates with the conceptual clarity and restrained profundity that has become a hallmark of Anderson's mature aesthetic.
RADIOHEAD: "PRESENT TENSE" is currently available via the Youtube embed above.


text:
Last modified 110 days ago (Oct. 3, 2019)
Hillsboro SENIOR CEnter
Thursday — Spaghetti with meat sauce, Italian vegetables, tossed salad, fruit, garlic bread.
Friday — Taco burger, kidney bean salad, fruit.
Monday — Sweet and sour chicken, steamed white rice, oriental vegetables, cottage cheese, pineapple, roll.
Tuesday — Pulled pork enchiladas, Mexican rice, refried beans, fruit.
Oct. 9 — Taco soup, tossed salad, fruit, carrot raisin salad, cinnamon rolls.
Oct. 10 — Grilled pork chops, baked beans, potato salad, fruit, roll.
Last modified Oct. 3, 2019


Songs against the wind / music Jane L. Komarow ; lyric Jeff Sheppard.
I want to be someone new.
Here at the house of hope.
musical arr.: Kevin Stovall, 1954-.
Oxford, Bodleian Library manuscript Laud misc. 752 : the Giffard Bible ... / Jennifer Mary Sheppard.
Price family and related lines.
The Strategy of corporate survival : a resource dependence ... / Jerry Paul Sheppard.
My North Carolina ancestors of Beaufort, Bladen, Columbus, Martin & Washington counties, NC and Isle of Wight County, VA / Jennifer Modlin Sheppard.
Coping with attention deficit hyperactivity disorder : parent survival program : instructor's manual / Jeanette Nu'Man-Sheppard.
entire text: Chesterfield Community Services Board, employer for hire of Jeanette Nu'Man-Sheppard.
Splash, splash / by Jeff Sheppard ; illustrated by Dennis Panek.
New York : Macmillan, c1994.
Price family and related lines of Beaufort, Martin, Pitt, and Washington Counties, NC : vol. 1 / by Jennifer Modlin Sheppard.
New Matter: additional text & photos, revision & updating.
Prev. reg. as Price family & related lines, 7Nov86, TX 1-935-272.
Singing out of tune : cultural perceptions and national park history on the American Great Plains.
Price Family and related lines of Beaufort, Martin, Pitt and Washington Counties, NC : vol. 2 / by Jennifer Modlin Sheppard.
Prev. reg. 1986, TX 1-935-272.
I want to color / by Jeanne Sheppard.
Angel messages by Jeanne Sheppard.
Prev. reg. as I want to color, 1991, TXu 483-603.
The Right number of elephants / story by Jeff Sheppard ; pictures by Felicia Bond.
I know a bridge / by Jeff Sheppard ; illustrated by Henri Sorensen.
Full moon birthday / story by Jeff Sheppard ; pictures by S. D. Schindler.
New York : Atheneum Books for Young Readers, c1995.
Copyrights records by Schindler, S. D.
Business officer : newsmagazine of the National Association of College and University Business Officers (N A C U B O) / editor, Jeffrey M. Sheppard ... [et al.].
Washington, D. C. : NACUBO.
Vol. 11, no. 8, Feb. 1978.
N A C U B O a.a.d. for National Association of College and University Business Officers.
Copyrights records by Sheppard, Jeffrey M.
Business officer : newsmagazine of the National Association of College and University Business Officers / editor, Jeffrey M. Sheppard ... [et al.].
Vol. 16, no. 12, June 1983.
National Association of College & University Business Officers, employer for hire.


Metasurf is a library for implict surface polygonization. You only need to set a callback that returns the scalar field value at any given point in 3-space, and another callback to accept isosurface vertices. Then at any point just call msurf_polygonize, and the library handles everything else for you.
The following snippet is sufficient to draw the surface of an implicit unit sphere, centered around the origin.
See the examples subdirectory for more examples.
No official stable releases just yet, just grab a copy of the mercurial repository: http://nuclear.mutantstargoat.com/hg/metasurf.
Metasurf is free software, you may use, modify, and redistribute it freely under the terms of the GNU Lesser General Public License (LGPL) v3 (or at your option, any later version published by the Free Software Foundation). See COPYING and COPYING.LESSER for more details.
If you'd like to fix the marching tetrahedra implementation or have any other ideas for improving this library drop me an email at: nuclear@member.fsf.org. Also feel free to submit patches for bugfixes.


text:
The surrealistic Godfather
Magical realism meets the drug trade in 'Birds of Passage'
Michael J. Casey
A still from 'Birds of Passage'
Birds of Passage (Pájaros de verano) — from Colombian directors Cristina Gallego and Ciro Guerra — is a masterpiece. Go see it.
Need more? Very well, here's the story: Set in Colombia and spanning the years 1969 to 1980, Birds is divided into five sections, songs really, that recount the tale of ancient rituals, tight-knit tribes and the sudden and violent encroachment of capitalism thanks to the profitability of the drug trade.
Like most stories depicting the rise and fall of an empire, Birds begins with an attraction. Specifically between the young Zaida (Natalia Reyes) and the bachelor Rapayet (José Acosta) at Zaida's coming out ceremony. With a flurry of images that both disorient and enchant, Gallego and Guerra invoke Dante's first encounter with Beatrice in Vita Nuova: "Here is a God stronger than I who come to rule over me."
Zaida and Rapayet are Wayuus — the indigenous tribe of northern Colombia — and Rapayet lacks the dowry necessary to wed. He teams with close friend Moises (Jhon Narváez) to hustle booze, then coffee and, finally, marijuana to visiting Americans so he can purchase the goats, cattle and necklaces required for Zaida's hand.
Once turned on, Rapayet and Moises find that the marijuana trade is like a fire hose out of control. Suddenly the hills and deserts of Colombia are flooded with money not easily turned off or controlled. Riches may accumulate fast, but violence is always faster, and an empire built upon sand threatens to destroy millenniums worth of ritual and culture.
The past always stands to be lost in the face of progress, and the surrounding Wayuu tribes are not going to stand idly by. Historically speaking, the Wayuus held off Spanish conquistadors in the 18th century, and a couple of machine gun maniacs aren't going to scare them off now.
At the center of this storm: Rapayet, a kingpin who is no Michael Corleone. Haunted by past actions, dreams and an ever-present specter of death that takes the form of a bird, Rapayet is a man in over his head. As the action unfolds, Rapayet has a shot-through look as if he wandered off the pages of a Gabriel García Márquez novel and found himself on the set of The Godfather as directed by Luis Buñuel.
To reiterate, Birds of Passage is a masterpiece. Coming off 2015's equally dreamy Embrace of the Serpent, directors Gallego and Guerra worked closely with Wayuu advisors to faithfully capture the details of the culture. They took those details, decorated them with surrealistic imagery and infused them with flavors so familiar no audience should feel estranged.
Sure, you've seen a hundred movies about crime families; you've probably even seen hundreds more about the perils of drug trafficking. But not like this. It's like a song so familiar you swear you've heard it before. But you haven't, and now you can't forget it.
ON THE BILL: Birds of Passage. April 24–27, Dairy Arts Center, The Boedecker Theater, 2590 Walnut St., Boulder, 303-440-7825, thedairy.org
Previous articleCatfishing
Next articleAlaska: reality & mystique


text:
Trump Prepares for El Paso Visit By Bashing Its Former Congressman
Days after calling to "set aside destructive partisanship," the unifier in chief has logged onto Twitter.
Donald Trump speaks to reporters as he leaves the White House in August.Chip Somodevilla/Getty Images
Mere days ago, following a pair of deadly mass shootings, Donald Trump declared that "now is the time to set destructive partisanship aside and find the courage to answer hatred with unity, devotion, and love." Now, he's returned to form. Ahead of his controversial visit to El Paso, where a gunman parroting his immigration rhetoric shot and killed 22 and wounded more than two dozen, the president accused Beto O'Rourke of pandering to Hispanic voters and politicizing the shooting to his advantage.
"Beto (phony name to indicate Hispanic heritage) O'Rourke, who is embarrassed by my last visit to the Great State of Texas, where I trounced him, and is now even more embarrassed by polling at 1% in the Democrat Primary, should respect the victims & law enforcement — & be quiet!" Trump tweeted late Tuesday following criticism from the former Texas representative and 2020 hopeful over the weekend. "He's not tolerating racism, he's promoting racism," O'Rourke said of Trump in the wake of the shooting. "He's not tolerating violence, he's inciting racism and violence in this country."
Trump, of course, kicked off his campaign by declaring Mexican immigrants "rapists" and has continued to paint Latin American migrants as dangerous criminals, including during a rally in El Paso back in February, for which he still reportedly owes the city more than half a million dollars. Trump alluded to that rally in his Tuesday tweet attacking O'Rourke, who held a dueling rally in the city that same night; the president has falsely claimed that O'Rourke's anti-border wall demonstration had just "200 people, 300 people." (In reality, estimates put the number of attendees at between 7,000 and 15,000—far more than the roughly 6,500 who attended the president's MAGA rally.)
Trump's Tuesday-night tweet escalates the tension between him and El Paso officials, many of whom have discouraged his visit. "From my perspective, he is not welcome here," Democratic Rep. Veronica Escobar said Monday on Morning Joe. "He should not come here while we are in mourning." O'Rourke echoed her sentiments this week, saying that the president "helped to create what we saw in El Paso on Saturday" and that "he's helped to produce the suffering that we are experiencing right now. This community needs to heal."
Trump, of course, is constitutionally incapable of letting criticism go—even when it's coming from a grieving community. Taking a shot at O'Rourke, and suggesting that his grief is some kind of performance to boost his 2020 campaign, is not particularly surprising. It does, however, dispel any doubts about the sincerity of his teleprompter-read words of consolation. He may pay lip service to "unity," but his instinct for division is reflexive. "22 people in my hometown are dead after an act of terror inspired by your racism," O'Rourke shot back at Trump early Wednesday. "El Paso will not be quiet and neither will I."
— Why some Republican insiders fear Kamala Harris the most
— No one is safe: how Saudi Arabia makes dissidents disappear
— What the surfer-mom influencers of Byron Bay reveal about our world
— The horrors of Jeffrey Epstein's private island
— Our September cover story: how Kristen Stewart keeps cool
Leave My Dad Alone!
Ivanka Trump: Let's Focus On the Shootings My Dad Isn't Responsible For
The first daughter is suddenly filled with concern about violence in Chicago, which she never mentioned until her dad was blamed for a mass shooting in El Paso.
By Bess Levin
Beto O'Rourke Condemns Trump Response to El Paso Shooting: " You Know The Shit He's Been Saying"
The Democrat's gifts for empathy and plain speaking made him a crucial figure in his hometown's tragedy—and raised a question: Shouldn't he run for Senate?
"It Was a Bad Headline": Times Chief Says Paper Fumbled Trump Speech
The paper published a pair of headlines on the president's response to mass shootings in El Paso and Dayton that led to mass public pushback.


The actor, who had been shooting in Jodhpur, had arrived in Mumbai recently to celebrated his 53rd birthday with his family. His much-talked-about PDA with wife Kiran Rao at the airport was too cute to handle. Now, the actor is heading back to Jodhpur to resume his shooting for the film.
Aamir was seen in all-black attire. He was seen wearing a plain black tee with black pants and matching shoes.
Aamir, who has been busy with the hectic schedule of his next project 'Thugs Of Hindostan', will be sporting an entirely different look in this Vijay Krishna Acharya-directed film.
The actor has been sporting handlebar moustache for quite some time and has even grown his hair. He has also been sporting a nosepin and earrings for some time. To carry this look, Mr Perfectionist had to go through the painful procedure of getting his nose and ears pierced.
'Thugs Of Hindostan' is reportedly an adaptation of the novel, 'Confessions Of A Thug' written by Philip Meadows Taylor. The film also stars Amitabh Bachchan, Katrina Kaif and Fatima Sana Shaikh in lead roles.


Examines ideas of method and imagination in a variety of texts, in literature, philosophy, and science. Particularly concerned with intellectual backgrounds and methods of inquiry that have shaped modern Western literature. Offered: jointly with CHID 205.


Tap East is a brewery pub offering more than 100 ales, stouts and beers to choose from as well as 17 draught lagers.
Do you work at Tap East?
What did you think of Tap East?
This small bar in the Westfield shopping centre, just a few seconds from Stratford station, is certainly in a handy location whether you live nearby, are escaping the shops or are commuting into town. However, for anyone other than beer lovers it may not be worth going out of your way to visit Tap East.
Tap East calls itself a specialist beer bar and microbrewery; and micro is the word. This bar, rather hidden away at the back of Westfield in Stratford, is tiny, with only enough seating for around six groups. The location is clever – what more could bored husbands or weary, thirsty shoppers want than a comfortable pub full of over 100 beers, right inside the shopping centre? Overall, the décor is modern and quirky, with stripes of multi-coloured wood propping up the bar, beer mats covering the walls and beer barrels for tables. It's a nice, bright space but it could do with being a lot bigger if hoping to cater for such a large shopping centre.
Just like Westfield, it's almost always busy at Tap East so getting a seat isn't easy, but it is definitely a great spot for setting down your shopping bags and refueling, or dropping off bored friends while you finish your search for that last Christmas present. Although Tap East have done a great job making it feel like a pub in its own right, it's still difficult to forget you're in a shopping centre, which doesn't make for the most relaxed drinking session. It certainly doesn't seem like a destination bar; you probably wouldn't make plans to meet friends here for a night out unless you're all big beer fans.
There's a limited bar menu, or 'Tap Eats', which changes regularly. These snacks suit the beer offering fairly well, though. Try the Margherita pizza at a reasonable £5.70 or the bratwurst at £3.50 if you're hoping to soak up those ales.
As a microbrewery, they've certainly got a lot of choice when it comes to beer. Draught beers change daily, but you can expect to find the brewery's Scarred Belgian, Coffee In The Morning and Six Pence Stout on tap. As well as in house brews, there's often Grolsch Blond, Fullers Frontier and Liefmann's Fruitesse on one of the 16 fonts, plus Big Leaf Maple and Viven Imperial IPA. Tap East also stocks over 100 bottled beers.
For a bar in a shopping centre, Tap East has a lot more charm than you might expect. This is an obvious spot for shoppers or commuters to meet or take a load off, and it's definitely attractive to beer drinkers since there's such a huge selection, but its size and small food menu are unlikely to pull in punters far and wide.


text:
Home > SAE Glasgow: Live Lounge
SAE Glasgow: Live Lounge
SAE Glasgow invites you to an exclusive Live Lounge session on 15 February. Three acts will each perform a trio of their best tracks in front of an audience.
The session will be filmed, taking place between 5.00-7.00 pm.
It's a great chance to explore the wealth of musical talent that the flourishing Glaswegian music scene has to offer.
The acts that will be performing are:
Flew The Arrow
Flew The Arrow is Lee McGilvray, a fast-rising contemporary folk artist from the small town of Ayr on the West Coast of Scotland. He melds fleeting finger-style melodies with percussive beats through an array of pleasant open tunings, that are accompanied by his poetic lyrics and powerful voice.
SUPA & Da Kryptonites
SUPA & Da Kryptonites are a 9 piece Funk, Ska, Reggae, Soul, Hip-Hop, Swing and Grime Band. They were Winners of The Best Unsigned Band Comp 2015 by The Sunday in Association with XFM & the Wickerman Festival, and were also selected as STV's Ones To Watch For 2016.
Megan Black
Megan Black is a young musician, vocalist, songwriter and recording artist from Livingston, Scotland. Megan's music mainly features the influence of early jazz and blues music, however, Megan always aims to develop her music in many different genres to grow as an artist and work with many different musicians and in diverse music scenes around the country.
Gathering an array of inspiring sounds from Ella Fitzgerald to Stevie Nicks, Megan's influences have helped her pen some originals of her own and is conjuring a sound for timeless music that keeps the generations to come ever connected to what galvanised them. Lining up, alongside instrumental three-piece band Leonard the Rabbit, at defining venues such as King Tut's, La Belle Angele and The Hug and Pint, Megan has made a considerable impression at venues across the country.
There will also be an opportunity for guests to network with other artists and see the amazing audio production studios SAE Glasgow has to offer.
Check out and share our Facebook Event Page.


Welcome to Alamosa, where you'll be inspired by the beautiful landscape, charmed by the laid-back friendliness, and amazed by the rich culture and history. Here you'll find what you've been looking for: blue skies, purple sunsets, emerald forests, red rocks, white fluffy snow, and a population proud of its diversity in culture, lifestyle, and heritage. It is no wonder we named our state after its colorful qualities. Now, it's time for you to experience how vivid life can be in this land of limitless possibilities. Read stories, get acquainted with all of the hidden corners of Alamosa, and start planning a trip you'll never forget.
Take the time to explore all of the corners of Colorado, and us at Colorado Hotel Service will do our best to provide you with hotels, motels, inns, and Bed and Breakfast locations in over 180 cities across Colorado. All of our hotels have been approved by the Colorado Hotel and Motel Association. All of our hotels offer instant online hotel reservations.


text:
From its founding in 1955, in a former TV repair shop converted to a chapel in Burlington's New North End, Christ Church Presbyterian (CCP) has been on a different path.
Called by founding pastor Bill Hollister to take the gospel of Jesus seriously, CCP turned its focus outward to the city throughout the '50's and '60's, becoming involved in work in the correctional system and creating a coffee house and bookstore on Church Street. Health professionals, businesspeople and educators met regularly to consider what it meant to live out the gospel in their professions.
The church's outreach in the late '60's was marked by the founding of the Burlington Ecumenical Action Ministry (BEAM), a coalition of Burlington churches and businesses that initiated a variety of community projects including drug counseling and neighborhood revitalization.
In 1978, the church moved to a chapel on the UVM campus, and refugee assistance became a focus. In 1983, the church sponsored a family from Cambodia. This was followed by the declaration of CCP as a Sanctuary Church and in 1985 a family of five, fleeing violence in Guatemala, lived in our family room for 16 months.
Meanwhile, in 1984, the church had taken a stand with gay, lesbian, bisexual, transgender and questioning (GLBTQ) persons, declaring itself to be a "More Light" church offering full acceptance into the life and leadership of the church to all. This same year, a three-year effort by a group from CCP led to the opening of the Ronald McDonald House in Burlington.
As "More Light"-related issues continued, CCP took a leadership role in adopting a "Resolution of Dissent" in 1997 in response to the denomination's ban on ordination of GLBTQ persons. This active dissent continued until 2003, when the PC(USA)'S General Assembly resolved the dispute in support of ordination.
Following Founding Pastor Bill Hollister's departure in 1986 and a time of interim leadership, Rev. Mike Brown and Rev. Becky Strader were called to be our co-pastors. Shortly after their arrival, we welcomed New Alpha Baptist Church into shared use of our building for several years.
In 1990, CCP joined the federation of Burlington churches known as "JUMP" (the Joint Urban Ministry Project) serving the pressing needs of people who are homeless and impoverished in our city. This work continues, and CCP continues to support this with JUMP volunteers, financial commitment and leadership positions.
In 2004, Rev. Mike Brown became solo pastor. In 2005, Vermont Interfaith Action (VIA) was established with an opening convention at St. Joseph's Church in Burlington; Mike Brown was instrumental in the formation of VIA. In 2007, CCP established its affiliation with the Center for Progressive Christianity, and has continued its study of progressive theology and liturgy.
Recognizing the limitations of our lease on the UVM campus, a Strategic Planning Task Force was convened in 2007 to develop a long-range strategic plan and vision. This group completed its work in 2009, recommending that CCP affiliate with another church. There was extensive research and conversation, but ultimately, these efforts did not lead to an affiliation and CCP needed to regroup.
In 2012, UVM let us know that they would not be renewing our lease on campus. After a period of research and outreach, the Cathedral Church of St. Paul (Episcopal) offered us space in their community room and two years later, a space of our own in a former bookstore space in the building. This move from our chapel on campus was a difficult one for the congregation, but church members remained active and highly engaged throughout the transition. This new space has served us well and we have made it our own.
In 2015, Pastor Mike Brown retired as planned, and following a search, Rev. Deadra Ashton became our pastor until 2018 when she left to pursue work as a hospital chaplain. Following her resignation, the Worship Committee provided leadership for Sunday services, using a network of local pastors, both active and retired, who were willing to lead our services. Rev. Marjorie MacNeill was among these pastors and provided pulpit supply and pastoral care. In 2019, we signed a contract with Marjorie as our Transitional Pastor, with a focus on helping us live into and implement our new vision and support our individual missions. Our work with VIA and JUMP continues, as do other CCP sponsored missions including support for TeacHaiti , a residence for newly-released ex-offenders and meditation groups.
Marjorie MacNeill continues as our Transitional Pastor now, in 2020, as we launch our search for a new pastor to move with us into the next chapter of our journey as a community of faith.


Showing results for tags 'diagnosis'.
Had a tiny fight with my psychiatrist, he seems to be dobuting my diagnosis. In the hospital they tried to say i was bpd with bipolar, but my psychologist says i cant be bpd and even my psychiatrist used to think im not. I dont have any bpd traits other than emotional inestability. Anyways im mad bc people dobut my psychotic symptoms he thinks is just something from my imagination or something "typical" from me. Im mad bc the new medication is not working and now i have panic attacks i didnt had before. The hospital kinda helped kinda made things worse. I feel like my doctor has left me behind, he doesnt care anymore, if he had answered or seen me 15days before when i asked, before i had my big crisis i wouldnt had ended in hospital. Im scheduling an appointment with a new doctor. But i feel so lost, so sad, like no one can help me, not even doctors. Its not fair. This illness is destroying my life. Has anyone ever had this kind of problems with doctors? Have you had a moment where medication made things worse?
snakes posted a topic in I've *Still* Got Issues!
I am having a lot of problems. I am enrolled in a mental health program, and I (finally) see the psychiatrist in 2 weeks, but I am basically holding on my a thread until then, because it feels like my only hope. I'm 34 and I still have no real diagnosis, because I've never been able to access medical care for long enough to get one. The current tentative one is Bipolar II with Eating Disorder in remission, but I've been given ADHD, GAD, OCD, OCPD, MDD in the past, plus Gender Dysphoria and some autoimmune activity as well. Also pretty deep poverty. I am cripplingly depressed and dysfunctional. I've had repeated "bad periods" of my life, where I'll be previously be "neurotic but stable", but then start slipping downwards into complete non-function. When this happens I usually lose my job and my partner, and more than once my home and most of my personal possessions -- this has happened over and over. This current "bad period" started last summer, where I just started to feel a little more tired, but now I've had to drop out of everything in life since can barely leave the house, am increasingly suicidal. The only reason I'm not homeless this time is a family member stepped in to help my bills (which no one has ever done before). And the ED is starting to resurface after many years in remission because I literally can't afford to buy food and don't qualify for state assistance. Some days I have overwhelming depression and hopelessness and can barely get out of bed. Some days I feel constant looming terror like something bad will happen and I'm afraid to leave my room. Some days I seem to be emotionally all over place. Some days I feel completely groggy, like my head is in a fog, and have difficulty making sentences or even playing solitaire. As far as I can see all of this is totally random and happens for no reason. However -- and this is big -- from the outside most people see nothing wrong. I'm never "sick enough" to have an "excuse" for my problems. I've had several different brief periods with talk-therapy, usually 6 months max, and I hate them because they've never seemed to do anything beneficial. I talk about my feelings and do whatever little "tips" they tell me and then go home and nothing is changed. Currently I'm supposed to be doing "affirmations" (yet again) but I simply cannot -- they say that negative self-talk is "a self-fulfilling prophesy", but the fact is I didn't have negative self-talk when I was younger and first having problems. Every evidence of my adult life shows that I fail out of everything and can't function as an adult. I have zero confidence in myself because I have zero evidence of capability. I feel defective. My mom and some other people are pushing me to get disability, but I've tried to tell them I don't even have a diagnosis -- being a dysfunctional loser isn't a legal disability. They are also pinning a lot of hopes on the Psychiatrist, as if taking a pill is the solution to all problems instantaneously. I know it doesn't work that way, because even if they put me on the "right" meds the first time around, they don't start working right away anyway. I'm enrolled in college but I am barely hanging on, and I don't know if my problems are going to get better fast enough to avoid getting kicked out. And if I lose that, I have absolutely nothing left in life. I just wish I had an "explanation", and some "validation", but it feels like most people think that even if I do have genuine problems, they aren't actually bad enough to explain my dysfunctional life. So I don't know what I'm asking or what kind of support I need right now.


Canon point and shoot 2017 Canon EOS 5D Mark II Full Frame dslr Camera (Body Only) (OLD model) : Camera & Photo. AdvertentieBij foto- en videospecialist vind je snel lle producten van Canon.
Camera Lens Adapters, Mounts & Tubes for sale AdvertentieHet grootste aanbod Camera's.


text:
Netflix Dominates Video Streaming: There Will Be Blood
By Jeff Bertolucci
PCWorld |
No wonder everybody's gunning for Netflix. The video-streaming service is more popular than many of us imagined. A new study by market research firm The NPD Group shows that Netflix's share of streamed or downloaded digital movies was a competition-crushing 61 percent between January and February 2011.
Yes, Netflix streams 6 out of every 10 digital movies. And its competitors? Comcast is a distant second with a tiny 8 percent of the market. The rest of the also-rans include Apple, DirecTV, and Time Warner, all tied for third at 4 percent.
It's no secret that home Internet providers don't care for Netflix's popular all-you-can-stream video service. AT&T, Comcast, Time Warner, and the rest of the ISP posse feel that Netflix is getting a free ride on their data networks. An October 2010 report by Canadian broadband equipment supplier Sandvine didn't help matters much when it stated that Netflix streaming accounts for more than 20 percent of downstream traffic during peak hours, and is most active between 8 to 10 p.m.
Adding insult to injury is the fact that Netflix' 20 million or so subscribers are less likely to rent movies and TV shows from the cable and satellite guys' on-demand services. Ouch.
Revenge of the ISP
ISPs are fighting back with data caps, which will no doubt impact even moderate Netflix users. Beginning in May, AT&T's DSL and U-Verse customers will have monthly data caps of 150GB and 250GB, respectively. Time Warner has launched trial programs with data caps and overage fees in some markets, but public outcry has caused it to put usage-based pricing on hold, at least for now. Comcast caps monthly data usage at 250GB.
AT&T says its average DSL subscriber uses only 18 GB per month, and that only 2 percent of its customers will hit the data cap. But as I wrote nearly a year ago, soon we'll all be data hogs. That 2-percent figure will almost certainly rise video streaming becomes even more mainstream.
A high-definition Netflix video stream uses about 2GB of data per hour, according to company spokesman Steve Swasey. If you're an AT&T DSL customer who watches one 2-hour HD movie on Netflix every night, you're already at 120GB a month, which leaves 30GB for the rest of your online activities.
Is AT&T's data-cap argument disingenuous? A convenient way to strike back at Netflix? I believe it is, and that the fight has just begun.
Contact Jeff Bertolucci via Twitter (@jbertolucci) or at jbertolucci.blogspot.com.