Standard Model of Particle Physics Violating Crypto-Nonlocal Realism

It has been well established that quantum mechanics (QM) violates Bell inequalities (BI), which are consequences of local realism (LR). Remarkably QM also violates Leggett inequalities (LI), which are consequences of a class of nonlocal realism called crypto-nonlocal realism (CNR). Both LR and CNR assume that measurement outcomes are determined by preexisting objective properties, as well as hidden variables (HV) not considered in QM. We extend CNR and LI to include the case that the measurement settings are not externally fixed, but determined by hidden variables (HV). We derive a new version of LI, which is then shown to be violated by entangled $B_d$ mesons, if charge-conjugation-parity (CP) symmetry is indirectly violated, as indeed established. The experimental result is quantitatively estimated by using the indirect CP violation parameter, and the maximum of a suitably defined relative violation is about $2.7\%$. Our work implies that standard model (SM) of particle physics violates CNR. Our LI can also be tested in other systems such as photon polarizations.


I. 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 Schrödinger subsequently referred to as entangled [1,2]. 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 [3]. A more experimentally suitable version of BI, called Clauser-Horne-Shimony-Holt inequality [4], was demonstrated to be violated in many experiments, including the ones closing the locality loophole [5,6], the detection loophole [7,8], and both [9][10][11]. To close yet another loophole called measuring setting or freedom of choice loophole, observations of Milky Way stars [12,13] and human choices [14] 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 [15]. This means that even nonlocal realism, at least a subset, cannot save the conflict between local realism and 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 polarizations of entangled photons generated in spontaneous parametric down conversion, first under an additional assumption of rotational invariance [16], then without this assumption [17,18]. LI violation was also demonstrated using polarizations of photons from fibre-based source [19], as well as the orbital angular momenta of photons [20]. Similar phenomena were observed in different degrees of freedom of single particles [21,22]. Various extended discussions have also been made [23][24][25][26].
It is highly interesting to extend the investigations on BI and LI to particle physics, of which 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 photons and nonrelativistic particles governed merely by electromagnetism, and can easily achieve spacelike separation. Besides, 2 one might also wonder whether high energy particles, as excitations of quantum fields, may display nonlocal effects.
Many proposals had been made on BI test in entangled mesons [52,53], and in analogous spin-entangled particles [54,55]. There had been an early experiment using entangled protons to test BI under a few additional assumptions [56]. There was an experiment using entangled B 0 dB 0 d pairs to test BI, in which meson decays act as effective measurement settings [32]. However, it was not regarded as a genuine Bell test, because of the lack of active measurement [34,54,57]. Basically this is a loophole of measurement settings. One can envisage a local HV (LHV) theory in which HV in the source of the particle pairs determine the decay times, modes and even products, and the information is carried by the particles, consequently the two particles are effectively correlated no matter how far away they are separated, rendering the violation of BI. Other approaches to BI 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 the timedependent effective measuring directions due to the decays, as counterparts of the directions of the polarizers measuring the photon polarizations. They lie on a plane and a cone. 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.

II. PSEUDOSCALAR NEUTRAL MESONS
In QM, a neutral pseudoscalar meson M can be regarded as living in a two-dimensional Hilbert space, with basis states |M 0 and |M 0 , which are flavor eigenstates and mutual CP conjugates, i.e. CP |M 0 = |M 0 , CP |M 0 = |M 0 . In this basis, the mass matrix is where H 00 ≡ M 0 |H|M 0 , H 00 ≡ M 0 |H|M 0 , and so on. The eigenstates of H are with p/q ≡ H0 0 /H 00 . The corresponding eigenvalues are H governs the evolution of the meson state |ψ(t) = a(t) where . This leads to the mixing phenomena. Especially, M 0 andM 0 at t = 0 evolve respectively to 4 A pair of neutral mesons can be produced as C = −1 antisymmetric entangled state 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 has been a projection of a to some basis state |φ a , which transits to f a . The meson b continues to evolve till it decays to some final state indicating that there is a projection of b to some basis state |φ b , which transits to f b . The time evolution of the entangled state up to the projections can be de- 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 Γ 2 ≈ Γ 1 , q/p ≈ e 2iβ , where 2β is a phase factor, β is given as sin(2β) = 0.695 [58]. Then M 0 = B 0 ,M 0 =B 0 , In Bloch representation, |B 0 , like the horizontally polarized state of a photon or the spinup state of an electron, is represented as the vector (0, 0, 1), while |B 0 d , 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 where |0 and |1 represent the basis states, while the time evolution can be parameterized as U(θ a , ρ a ) = cos θ a 2 − i sin θ a 2 (cos(ρ a )σ x + sin(ρ a )σ y ) .
Suppose that following the evolution U(θ a , ρ a ), a signal A is recorded as where u = (sin θ u cos ρ u , sin θ u sin ρ u , cos θ u ) is the Bloch vector of |u , a = (− sin θ a sin ρ a , sin θ a cos ρ a , cos θ a ) is the Bloch vector of U † (θ a , ρ a )|0 . This can be easily understood by regardingĀ(u) as expectation value of measuring the initial state |ψ in the basis {U † |0 , U † |1 }. This is basis rotation realized by evolution.
Eqs. (12) and (13) are of the same form as the standard QM result (7) because the factor e −Γt exists in all terms in both denominator and the numerator, and thus cancels.
As shown in Fig. 1, with the time passing, a l (t) rotates on a plane, while a s (t) rotates on a cone whose axis is perpendicular to a l plane. For convenience, we adopt a new coordinate system in which a l plane is the xy plane, then a l (φ l ) = (cos φ l , sin φ l , 0), a s (θ s , φ s ) = (sin θ s cos φ s , sin θ s sin φ s , cos θ s ) , where φ l = xΓt and φ s = xΓt + π/2 are the azimuthal angles of a l and a s , respectively, θ s = 2β is the polar angle of a s , and it suffices to consider 0 < θ s ≤ π/2.
The effective measuring directions a l and a s . In a certain coordinate system, (φ s , θ s ) = (xΓt + π/2, 2β), corresponding to flavor and CP measurements following evolution of time t, respectively. For photon polarizations, a l and a s are polarizer directions in Bloch representation, and can be adjusted directly.
a l (φ l ) and a s (θ s , φ 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 a l (φ l ) and a s (θ s , φ s ) can also be used, say, for photon polarization, by directly adjusting φ l and (θ s , φ s ).

III. CNHV THEORIES
|u is an eigenstate of the Pauli operator σ · u in the direction of u. A particle in this state has a definite 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 7 independent of the measurement, or elements of reality. Thus u is identified as such an element of reality.
Consider a HV theory. Suppose a particle with property u is measured along direction a, then the dichotomic measurement outcome A = ±1 is determined by u, a, and hidden variables λ. So much for a local realistic theory. In a nonlocal realistic theory, A also depends on non-local parameters η. In a crypto-nonlocal HV (CNHV) theory, the individual properties of each particle, after averaging over distribution ρ u (λ) of the hidden variables λ, become local, as indicated in countless phenomena, A concrete example ofĀ(u, a) is the Malus' law [16] A which is consistent with QM results of photon polarizations, and of a meson's decay following evolution, as shown in the last section.
For a pair of particles from a common source, with respective properties u and v, the measuring direction of the other particle can serve as the nonlocal parameter, the measurement outcomes along respective directions a and b are A(u, v, a, b, λ) = ±1 and The local measurement of each particle cannot detect its correlation with other particles, hence the nonlocal dependence disappears after averaging over the hidden variables, A general physical state is a statistical mixture of subensembles with definite u and v.
Hence the final expectation values, which is experimentally measured, are [15,16] where F (u) and F (v) are probability distribution of polarizations u and v, respectively.
In case of correlated photons, they are the reduced ones The two-body quantities may indicate correlations. For definite u and v, For a general state, 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 a and b are not externally fixed.
In each measurement, the measurement settingsã(λ) andb(λ) are determined by HV λ, thus the measurement outcomes are like A(u,ã(λ), λ). For a single particle, the average of the outcomes of those measurements withã(λ) = a is is a shorthand. For two correlated particles, the outcomes of those measurements withã(λ) = a and b(λ) = b give rise to where is a shorthand. Clearly the original formalism is a special case of this extension, by externally fixingã(λ) to be always a andb(λ) to be always b, independent of λ. With our extension, all the previous and present derivations of LI remain valid.

IV. 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 = ±1 and B = ±1, respectively. The average of those outcomes A with a same measurement setting a satisfy the Malus' Law (23). The average of those outcomes B with a same measurement setting b satisfy the Malus' Law (24). The correlation function is defined in the way of (25).
a and b given as a l (φ i ) or a s (θ s , φ i ), (i = a, b), as in (14).
The correlation functions averaged over ξ are not directly observable, therefore rotational invariance or fair sampling of averages needs to be assumed for measurements, in order that LI in terms of these average correlation functions can be experimentally examined [17,18].
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, andÊ ± ll (θ s , ϕ) andÊ + ss (θ s , ϕ) similarly. As derived in the Appendix, for these discrete average correlation functions, our LI can be obtained from Eqs. (26), (27) and (28) by simply repla- . N ≥ 2 is required. As N → ∞, u N → 2/π, then the discrete version approaches the continuous version.
Our LI can be tested using various systems, in which measurement directions a l (φ l ) and a s (θ s , φ s ) can be directly adjusted.
For meson decays, θ s = 2β is fixed, while φ l = φ l (t) = xΓt, φ s = φ s (t) = xΓt + π 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.
Consequently there is a constraint on possible values of ξ, but it does not affect the averages over ξ, which is an angle mathematically, hence its functions are periodic.

V. TESTING LI IN ENTANGLED B d MESONS
We now come back to the C = −1 B 0B0 entangled meson pairs, and we can write the correlation functions as E(a X (t a ), a Y (t b )), (X, Y = l, s). By definition, where for convenience, we have invented the shorthand Note that the decay products of a and b may be different even though their flavors or CP eigenvalues are the same, and may not be CP conjugates even though their flavor eigenvalues are opposites.
A key quantity is the joint decay rate for particle a decaying to f a at t a while particle Hamiltonians governing the decays of meson a and b, respectively. The following joint decay amplitudes will be needed. There are four amplitudes written as where, ± in l ± a corresponds to the first l± on RHS, ± in l ± b corresponds to the second l± on RHS. There are four amplitudes written as where we have used There are four other amplitudes written as The experimentally measured quantity is the joint event number where ǫ fa,f b is the detection efficiency for that channel [38], is proportional to the modulo square of the joint decay amplitude, as given in Eqs. (31,32,33).
Therefore the correlation function (30) can be obtained from event numbers as In experiments, |A l ± a/b | 2 and |A S ± a/b | 2 can be absorbed to the redefinitions of detection efficiencies.
Furthermore, one obtainŝ from which the averagesÊ ll± (ϕ),Ê sl± (ϕ),Ê ss+ (ϕ) can be obtained. Note that we did not defineÊ ss− , which would not have physical meaning, as −a s is not on the cone, where all possible a s 's lie. In SM, with ∆Γ = 0, where In experiments, it is more convenient to use the time-integrated joint decay rate 13 which is obtained as It is 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 , consequently it is much easier to test LI in terms of the continuous version of the average correlation functions.
From (36) and (38), it can be seen that in SM, Moreover, the integration over ξ ofÊ ± (a, b) = E(a, b) + E(b, ±b) can be performed independently for the two terms on RHS, consequently, Note that Eqs. (34) and (39) are mainly for the use in analyzing experimental data. QM result can be obtained simply from a X , a Y |Ψ − = −a X · a Y , thereforê E sl (ϕ) = − sin(2β) cos(ϕ),Ê ll (ϕ) = − cos(ϕ), It is interesting to test our LI using various systems, in which ϕ a and ϕ b are directly adjusted.
For B d mesons, QM result (41) can also be obtained from the definition (30) of correlation , f b , t a , t b ), and RHS with the joint amplitudes (31), (32), (33), and then having A f andĀ f cancelled. For B d mesons, the values of β and x are given by sin(2β) = 0.695, x = 0.769 [58].
The upper bound of a our LI violation can be quantified as where h u R and h u L (ϕ 1 , ϕ 2 ) are RHS and LHS of (26), respectively. The first lower bound can be quantified as where h d1 R and h d1 L are RHS and LHS of (27), respectively. The second lower bound can be quantified as where h d2 R and g h2 L are RHS and LHS of (28), respectively. Each of these three quantities larger than 0 represents the violation of the corresponding bound of LI.
It is also possible to test LI in polarization-entangled baryon-antibaryon pairs produced in, say, J/Ψ → ΛΛ decays, where the polarizations of Λ andΛ can be measured through the (a) g u (θ s , π, ϕ 2 ) FIG. 4: LI violation in case θ s is a variable. The maxima of g u , g d1 and g d2 are still at ϕ 1 = π, 0, 0, respectively. g u (θ s , π, ϕ 2 ), g d1 (θ s , 0, ϕ 2 ) and g d2 (θ s , 0, ϕ 2 ) are functions of θ s and ϕ 2 . 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.

VI. 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 HV. The outcomes of those measurements with the same settings give averages satisfying Malus' Law and make up correlation functions. This is the case of meson decays, which could be determined by HV 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 derive a new LI, for a l (φ l ) on a plane and a s (θ s , φ s ) on a cone.
All the results remain valid in the special case thatã(λ) andb(λ) are externally set to be always a and b) respectively.