Test of Bell’s Inequality and Entanglement Measurement in
Collider Experiments
Shion Chen 1
Department of Physics, Graduate School of Science, The University of Tokyo
7-3-1, Hongo, Bunkyo-ku, 113-0033, Tokyo, Japan
26 February 2014

1

chen@icepp.s.u-tokyo.ac.jp

Contents
1 Introduction
1.1

5

Spin, Entanglement and EPR’s argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.1

Description of spin measurement in quantum mechanics . . . . . . . . . . . . . . . . .

6

1.1.2

Entanglement and non-local action . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.3

EPR’s argument and local hidden variable theories . . . . . . . . . . . . . . . . . . . .

11

1.2

Bell’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3

Review of Previous Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.3.1

Photon pair experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.3.2

Loopholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.3.3

Proton pair experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.3.4

K 0 K 0 , B0 B 0 ﬂavor oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

What is still Interesting with testing Bell’s inequality? . . . . . . . . . . . . . . . . . . . . . .

25

1.4

2 Design of the Experiment
2.1

2.2

2.3

2.4

29

Fundamental Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.1.1

Source of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.1.2

Weak decay as polarizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.1.3

Candidate channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Weak Decay as Polarization Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.2.1

Parity violation and ability as a polarizer - in a nut shell . . . . . . . . . . . . . . . . .

33

2.2.2

Parity violation and ability as a polarizer - a closer look . . . . . . . . . . . . . . . . .

35

2.2.3

Comparison with the SG type of spin measurement . . . . . . . . . . . . . . . . . . . .

37

Phenomenology in terms of Testing LRTs - Possible Loopholes . . . . . . . . . . . . . . . . .

40

2.3.1

Interpretation of the polarizer decay in LHVTs . . . . . . . . . . . . . . . . . . . . . .

40

2.3.2

Passiveness of particle decay and the free will loophole . . . . . . . . . . . . . . . . . .

41

2.3.3

Eﬃciency loophole in the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.3.4

Space-like separation of decays and the locality loophole . . . . . . . . . . . . . . . . .

42

2.3.5

Summary of expected loopholes and comparison with other experiments . . . . . . . .

44

Description of the measurement and the overview of detector . . . . . . . . . . . . . . . . . .

44

3 Formulation of Bell Inequalities for Our System

49

3.1

Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.2

Axes optimization using the method of C matrix . . . . . . . . . . . . . . . . . . . . . . . . .

53

i

CONTENTS

1

4 QM Analyses and the Violation of the Bell Inequality

57

4.1

χc0 → ΛΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.2

ηc → ΛΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.3

J/ψ → ΛΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.4

+ −

e e → γ → ττ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.5

Z → ττ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.6

H → ττ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77∗

5 Discussion on Experimental Feasibility
5.1

Distribution of C11 , C33 , Qmax and Sensitivity

79. . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.1.1

Shape of ni n′j distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.1.2

Shape of C11 , C33 distribution vs number of events . . . . . . . . . . . . . . . . . . . .

82

5.1.3

Estimator for Qmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.1.4

Signiﬁcance and the master formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.2

Eﬀect from Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.3
5.4

Eﬀect from contamination of the time-like events . . . . . . . . . . . . . . . . . . . . . . . . .
Eﬀect from detector resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96
96

5.5

Available sample size with current and future experiment and the feasibility . . . . . . . . . .

98

5.5.1

χc0 , ηc → ΛΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.5.2

Z → ττ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.5.3

+ −

e e → γ → τ τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.5.4

H → τ τ → πνπν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100∗

6 Conclusion

105

Acknowledgement

108

2

CONTENTS

Abstract
Is the universe can be only described fully by quantum mechanics? Or are we still allowed to have some hopefor the regaining of extended classical theories? Bell inequality is known as the experimental distinguisher
between them. Though its violation has already been conﬁrmed through a number of photon experiments,from view of the universality and the further understanding toward the mysterious quantum properties, suchas the quantum no-locality and entanglement, other type of experiments have been anticipated for a longtime, which have still only few examples due to technical complications.
In this sense, the particle decays in high energy collider experiments have been supposed to be the newchannels of the test with potential breakthrough, in that unstable particles produced in may serve as thesource of entanglement states, the particle spin can be measured without any special apparatus but using
its spontaneous decay, and that a variety of systems and interactions can be accessed. From 1970’s, it has
been approached from both theory and experiment sides however there has not been a successive result yet.
In addition to these precious studies in consideration, this paper presents (i) a comprehensive reviewof the theoretical background, (ii) a formal establishment of the method, (iii) a formulation of a new Bellinequality, (iv) and a discussion on the feasibility of testing the Bell inequality by this method, with today’shigh speciﬁcation collider machines, as a new proposition of the experiment.
In particular, the six decays below are examined:• ηc → ΛΛ → pπ − p̄π +• χc0 → ΛΛ → pπ − p̄π +• J/ψ → ΛΛ → pπ − p̄π +• e+ e− → γ ∗ → τ + τ − → π − ντ π + ν̄τ• Z → τ + τ − → π − ντ π + ν̄τ• H → τ + τ − → π − ντ π + ν̄τWith analyses, it is demonstrated that all the channels except for the J/ψ → ΛΛ → pπ − p̄π + channel
leads to a violation of the Bell inequality, in which the classical theories and the quantum non-locality canbe tested. Especially, for the two channels ηc , χc0 → ΛΛ → pπ − p̄π + , current experiments already oﬀer
suﬃcient statistics, comparable to 2σ and 3σ of signiﬁcance for the violation respectively, therefore feasible.

CONTENTS

3

The eye sees only what the mind is prepared to comprehend.
Henri Bergson

Chapter 1

Introduction
From the establishment, with no doubt, the quantum mechanics (QM) has experienced a great success; it
explains most of microscopic phenomena that are hardly explained by classical theories, and to an amaze,without having a single counterexample. On the other hand, it roots at a very seemingly artiﬁcial andunnatural foundations such as ”non-realism” that physics quantities generally do not have deﬁnite values
until measured but in superposition, or the issues about measurement and so on; these have been provoking
wide suspicion even up to today. In addition to it, Einstein-Podolsky-Rosen (EPR) pointed out that QM
exhibits even some strange non-locality nature, which decisively illustrated that QM is totally against ourintuition.
Something is terribly wrong. It was quite natural that people started to seek the ways to keep theglorious results of QM, but interpret within the picture of intuitive classical physics during the post-EPR
years. Though the explicit formalism had not been very successful, people were quite optimistic to it, that”some day, an extended classical theory will overtake QM, and we will retain the peaceful empirical view ofphysics. QM is probably no more than an eﬀective theory for the scale of atoms. We should take it easy, to
use them as an ad hoc tools”. It is a quite ironical history that it is such indiﬀerent but decisive attitude
towards QM that encouraged people to apply QM in every ﬁeld of physics as a powerful tool, bringing about
a great progress in entire physics, as a result.
The argument on the foundation had been boring during the period. It was kind of theory-must-bebeautiful type of metaphysical discussion. However, J .S. Bell provided a break through that pulls it toactive science out of philosophy, with his famous Bell’s inequality. It expresses the upper bound that two
particle correlation (especially spin correlation) should satisﬁes in general, providing the realism and the
locality which are both basic principles of classical physics. QM is known to predict larger correlation thanit in some cases, in particular ”entangled states”, that it is to say, it can be experimentally testiﬁed whether
QM or the classical theories are correct through the measurement of the correlation. If the violation isobserved, then ﬁnally we have to prepare to say good-bye to our intuitive and empirical view of physics.
Many experiments using photon pairs were performed from 1970’s and most of them rebutted the classicaltheories, but upheld QM. Still, the classical theories has managed to survive for a long time, by pointing
out the imperfection of experiments, however they are also gradually overcome thus the issue of QM vs theclassical seems to be coming to an end with the victory of QM . . .
5

6

CHAPTER 1. INTRODUCTION

This is the story of the issue regarding to the interpretation of QM. An immediate question should thenbe casted: is there anything else to do? Yes it is, of course, as you guess by the rest of the 100-page thickness.
For example, the vast majority of the tests of Bell’s inequality have been done through optical experiments
however, there are only few examples of tests in non-optical systems. One interest in the test in non-opticalsystems is the universality. In particular, since massive particles have less coherence as read in QM, it isattractive to see in such systems if the classical theories are excluded or the quantum non-locality exhibits.
The quantum non-locality and entanglement itself arouse more profound interest, in that many of the aspects
are not fully understood and still under research today, including the application such as quantum informatics
or quantum computation.
To summarize, the proposition of this paper is approaching these themes by means of testing Bell’sinequality in a new environment, in collider experiments, with a new method and formalism.

Outline of this thesis
The rest of chapter 1 is spent in more speciﬁc introductions, from the deﬁnitions of terms, to the review
of hidden variable theories, Bell’s inequality, overview about the signiﬁcance of the problems or what isstill interesting and so on. In chapter 2, the design of the whole experiments is explained. Starting withintroducing the mechanism of spin measurement with particle decays, review of the candidate decay processes
available for testing Bell’s inequality, and the experimental perfection is described. In chapter 3, a new Bell’sinequality is formulated for the experiment, with introducing new mathematical techniques. The chapter 4shows the result that QM predicts, in which the structure of entanglement are dedicatedly studied and thechannels exhibiting the violation of the inequality is speciﬁed. In the last chapter 5, the candidate channels
are further scrutinized from view of experimental aspects. The expected sensitivity and the feasibility areargued.

1.1

Spin, Entanglement and EPR’s argument

Spin is the carter component throughout the paper. In this section, the characteristics are reviewed mainly
in terms of the measurement. Then, it is extended to two particle spin states with providing the deﬁnitionof entangled states and observing the emergence of the quantum non-locality (EPR’s argument). Lastly, as
solution for the non-locality problem, hidden variable theories are introduced, in comparison with QM.

1.1.1

Description of spin measurement in quantum mechanics

Spin is the intrinsic angular momentum that a particle itself has. In QM, as with angular momentum, it
is quantized and hence a discrete quantity. For simplicity, considering the case with spin one-half particles.
Also as spin has too wide non-trivial nature, here the argument is conﬁned within the description of spinmeasurement in QM.
The most striking feature of spin (or angular momentum) in QM is that it has orientation which isseems to be a continuous property despite it is discretely valued. Specifying an arbitrary direction n, the1.1. SPIN, ENTANGLEMENT AND EPR’S ARGUMENT

7state of a spin can always be expressed as the superposition of the eigenstates toward the direction; |↑⟩, |↓⟩|ψ⟩ = c1 |↑⟩ + c2 |↓⟩ .

(1.1)

Here |↑⟩ and |↓⟩ mean the state that the spin orients parallel and anti-parallel to n respectively.
This expression (1.1) keeps in form even if a diﬀerent direction n is chosen. Providing |c1 |2 + |c2 |2 = 1,
the coeﬃcients c1 and c2 are uniquely determined with respect to the reference direction n, which is deﬁned
as ”quantization axis” (QA).
For single particle spin, a special direction can be deﬁned. There exists such a QA n = s that gives|ψ⟩ = |↑⟩ .
This is eigenstate where the spin points parallel to s. This s is called polarization vector, presenting thedirection of spin. Note however that, since the spin operator σ does not commuting in each component[σi , σj ] = iϵijk σk(i, j, k = 1, 2, 3),

it does not follows that the polarization vector has determined as vector, but fact that it is characterized
by a vector is amusing. From now on, the phrase ”spin points to the direction of s” means that thepolarization vector is s. Phenomenologically, this polarization vector corresponds to the classical spin which
is proportional to magnetic moment.
Besides the polarization vector n is related to the spin operator with(n · σ) |ψ⟩ = |ψ⟩σ is Pauli matrices. Solving this, one ﬁnds the state in which a spin pointing to direction (θ, ϕ) is expressedasθθ|↑⟩z + eiϕ sin |↓⟩z2
2
in the representation where the z-axis is chosen as the QA, as indicated by the subscripts.
|ψ⟩ = cos

(1.2)

Next, the description of measurement in QM is introduced. This usually belongs to a framework named”projection measurement”, which is roughly as below.
Consider a physical quantity to be measured (”observable”) A, and the respective linear operator Â. The
measured system is described generally with a superposition of eigenstates in terms of A valueX|ψ⟩ =ci |ai ⟩ .

(1.3)

i

ai reads the measurable values of A (eigenvalues of Â), which can be discrete or continuous either. |ai ⟩ isthe eigenstate with respective to a = ai . Assume that an ideal measurement is performed onto the state(1.3) and the outcome turns to a = a0 . Then, according to the principle of ”projection measurement”, thestate is projected as
|ψ⟩ =

X

ci |ai ⟩→c0 |a0 ⟩(1.4)

ithrough the measurement. This means that only the components that is consistent to the outcome in thestate remains, the others are projected out. In another form,|ψ⟩ → ⟨a0 |ψ⟩ |a0 ⟩ .

(1.5)

8

CHAPTER 1. INTRODUCTION

About the mechanism of the projection, QM does not tell anything. This scheme is called the ”von Neumann’s
projection postulate” [1], which is incorporated as an axiom in the usual formalism of QM 1 .
Another interest is that how the outcome a = a0 is determined. Unfortunately, as for this QM provides
no hints. Instead, there is an empirical rule called ”Born’s rule”
P (ai ) = |ci |2 ,giving the probability of having A = a0 , a1 , . . . as the outcome [2]. Together with the projection of state
above, this consist of the framework of ”projection measurement”, and the standard QM interpretation
which includes this framework is usually referred as ”Copenhagen interpretation”. There is another ”manyworld interpretation” [3] in contrast, however it is believed that those are equivalent.
Lastly, the two notions are combined and one arrives at the spin and measurement. For later convenience,
let me introduce an explicit device; the Stern-Gerlach instrument (SG) [4]. The overview is shown as Fig.
1.1. This is an instrument that generates an non-uniform magnetic ﬁeld, bending particle trajectories by
the interaction between the ﬁeld applied and the magnetic moment from the particle spins. Speciﬁcally, as
the interaction potential of the ﬁeld B and the magnetic moment µ is
µ · B,

(1.6)

∂B
∂B
∝s·
.
∂z
∂z

(1.7)

the force the particle experiences Fz is
Fz = µ ·s is the polarization vector which is parallel to µ. Thus, it can bend the particles with spin if the gradientof the ﬁeld toward z direction is non-zero. 2 Since the diﬀerent spin state with respective to B results in
opposite bending, the spin can be measured by the position measured by a screen and a detector ahead.
Historically, it is this measurement which discovered that spin is quantized, by observing a two split lines
rather than a continuous pattern.
In the language of projection measurement, it process as below.
For each incident particle, the spin state can be presented as|ψ⟩ = c1 |↑⟩B + c2 |↓⟩B ,(1.8)

with B being the QA. Through the interaction with the B-ﬁeld, this state |ψ⟩ is projected onto either of|↑⟩B|↓⟩B .

(1.9)

One notable thing is that the spin always points parallel or anti-parallel with B after the measurement.
In other words, spins can be ﬂipped to the direction which the observer intends (with two-way
ambiguity though). This nature of spin measurement causes a big problem when it comes to two particle
1 The projection has many other names such as ”reduction of wave packet” or ””reduction of state” and so on. Here it is
conﬁrmed that the name ”projection of state” is consistently used throughout the paper.
2 The concept of ”trajectory of particles” sounds classical, however it is allowed to treat a particle as such a deterministic
object, if the incident particle beam is well collimated that the dispersion of the position is negligible compared with the sizeof instrument [5].

1.1. SPIN, ENTANGLEMENT AND EPR’S ARGUMENT

9systems, as observed in the next section.
Note that, from now on, this particular direction is called ”measurement axis” and for a while this SG
type of measurement is assumed.
z-axisbeam direction
sourcedetectormagnetic field: B

collimatorshaped magnet

Figure 1.1: The overview of the Stern-Gerlach instrument [5]. The shapely peaked magnet creates gradientin the B-ﬁeld. The incident particles tell their spins by exposing the which-path information.

1.1.2

Entanglement and non-local action

In QM, two particle composite states are generally described in a form from a direct product of kets12|ψ⟩ |ϕ⟩ ,(1.10)

where each ket represents its own space of particle state. Let the upper subscripts label particles, which isabbreviated from now on, as long as it is not confusing.
Now consider a case in that a pair of particles with spin one-half forms a spin zero state:|Ψ⟩ =|↑⟩ |↓⟩ − |↓⟩ |↑⟩√,2(1.11)

referred as spin singlet state. This is interesting in many aspects; it has rotational invariance. The representation (1.11) is held no matter how the QA is chosen; it cannot be factorized in a direct product of single
kets. Those unfactorizable states are referred as ”entangled state”. On the other hand, factorable states, forexample, (1.10) are called product states in general.
Then, what does the unfactorability implies? Here let me consider a gendankenexperiment as below.
Consider a pair of particles 1 and 2 of which spin state in the singlet (1.11) emerges at a point in space, thenrun back-to-back each other in a free space. Let the two observers Alice and Bob stand ahead of the particle
1 and 2 respectively, and let them measure the spin by the SG instrument. The setup is shown in Fig. 1.2).
Suppose that Alice and Bob are spatially distant away. Let Alice ﬁrst measure the spin of particle 1, withsetting the measurement axis (B-ﬁeld) of the SG instrument to n, and the outcome is ↑. Accordingly, thespin state of the system becomes projected as|Ψ⟩ =|↑⟩ |↓⟩ − |↓⟩ |↑⟩√2→|↑⟩n |↓⟩n√.
2(1.12)

10

CHAPTER 1. INTRODUCTION

It is fairly reasonable that the spin state of particle 1 is eﬀected due to the measurement, or the interaction
with the B-ﬁeld applied by the SG instrument of Alice. However, it is terribly strange that the spin state ofparticle 2 which does not interacted with anything is projected as well. The undetermined spin of particle
2 determined its polarization vector to −n, without being measured. Alice can ﬂip the orientation of thespin 2 to any direction she wants, with no timing delay after measuring her own spin 1 by tuning
the direction of the B-ﬁeld n, just like working a puppet. In addition, this unbelievable action can easily
be conﬁrmed by the subsequent measurement by Bob. If setting the measurement axis to n as well, theoutcome is, with 100% probability, opposite to that of Alice; ↓.

particle 2
Bobparticle 1

n

Alice

Figure 1.2: The gendankenexperiment of measuring spins of two particles. Observers Alice and Bob respectively have a set of SG instrument literally. Alice measure the spin of particle 1, with setting the B-ﬁeld
to n. (The pictures of SG instrument are referred from [6].)This ”instantaneous” action is devastatingly problematic that the spin state projection on particle 2 takes
place with no delay after the Alice’s measurement, no matter how they are isolated in distance. How canthis ”distant” interaction work?
Usually and historically, the description of interaction is done within the framework of ”local” interaction
with the underlying principle of locality below.
Locality principle

1.1. SPIN, ENTANGLEMENT AND EPR’S ARGUMENT

11

A component in a space-time point is never aﬀected by a change
which occurs in a space-likely isolated point in the space-time coordinates
What it is meant to be is that interaction must take place fairly with the neighbor, and the eﬀect tothe distant points in space must be ”propagated” continuously but never ”jumps”. The propagation must
forwards at a speed less than the light speed, which means, with ﬁnite time. Interestingly, this formalism ofinteraction is strictly and carefully followed even by QM. The reasons why distant interactions are disfavored
are many-fold and profound that here we do not peruse further (please refer [7] [8] [9]). However, the caseof the gendankenexperiment above clearly shows violation of the locality principle above. This nature of
QM is referred as the quantum non-locality.
One remarkable feature about this non-local eﬀect is that it is straightforwardly derived from only theframework of projection measurement and an entangled state. There is no other dynamics can be the causeof it, for example, it takes place all the same even if no interaction is assumed between particle 1 and 2. It
solely arises from the formalism of QM using Hilbert space and its interpretation, which is intrinsic to QM.
Unfortunately, as often in literature, the concepts of entanglement and non-locality are confused in manycases. They are actually independent concepts; the entanglement indicates the generic algebraic structureof unfactorability, while non-locality is referred to such physical insight that the the projection takes placeon the two particles simultaneously and that the coherently superposed entangled state such as (1.11) lasts
forever as long as they are not measured. In the standard QM, the relationship can be summarized as thestatement ”The entangled states can lead to results that can only be interpreted as non-local eﬀect”.
Another interesting point is that the coherency in spin in the entangled state has nothing to do withthe superposition of their wave functions. The non-local link (”spooky link”) is maintained even if the twoparticles are localized in space where the wave functions are completely isolated.

1.1.3

EPR’s argument and local hidden variable theories

This is the so called EPR argument, proposed by Einstein-Podolski-Rosen (1935) [10], pointing out that QM
does not go along with the locality principle. Strictly speaking, they argue over the entanglement of position
and momentum instead of spins 3 , and demonstrated that QM is incomplete if assuming QM and locality
together, but the essence of the same as the gendankenexperiment in 1.1.2.
Motivated by the result of EPR, many attempts are made to recover the straightforward interpretation ofphysics instead of the non-realistic and non-local QM, as well as maintaining the result that QM eloquently
predicts. The major approach was to extending the conventional classical theories into what perfectly
reproduces the prediction of QM.
One of the most poplar approaches is the method the hidden variables in that, physics quantities havedeterministic value in any case, and the ﬂuctuation of measurement outcome which is characterized as thesuperposition of state is regarded as the representation of our ignorance about the state. This is exactly
3 Historically, this is the original argument. The gendankenexperiment of spins are introduced later by D. Bohm(1952) [11],illustrating the essence of the problem.

12

CHAPTER 1. INTRODUCTION

same relationship between Newtonian mechanics and statistical mechanics.
Here the measure of the ignorance is introduced with a set of hidden variables λ, and the ﬂuctuating
measurement outcome A is considered as the consequence of random distribution of λ, thus (A = A(λ)).
Assuming an ideal measurement, the value of A does not change between before and after the measurement.
Therefore, the expectation value of A is denoted as
Z⟨A⟩ = dλ ρ(λ)A(λ)(1.13)

where ρ(λ) is the probability density distribution of λ, characterizing the state of the system.
Reconsidering the gendankenexperiment with from the viewpoint of hid dens variables, one ﬁnds thatit can explain the phenomenon without relying on non-locality. That is to say, it does not follow that theparticle 2 ﬂips its spin instantaneously after the measurement by Alice on the particle 1 with the measurement
axis being n, but the two spins are pointing to n and −n from the beginning, as|ψ⟩ = |↑⟩n |↓⟩nor|ψ⟩ = |↓⟩n |↑⟩n .

(1.14)

There is no non-locality but just pre-determined; this is the attitude of hidden variable theories towards the
EPR problem. In addition, considering a statistical mixture of (1.14) as the initial state, it can perfectly
reproduce the outcomes predicted by QM.
Accordingly as demonstrated above, hidden variable theories do have potentiality of recover the classicaland empirical view of physics, namely, the realism, and if assuming, the locality, without contradicting toexperiment outcome. Those are referred as local hidden variable theories (LHVTs) which may no more thana potentiality, but in principle possible theories.

1.2

Bell’s Inequality

Thorough the studies on the EPR gendankenexperiment, it is found that there are two possible attitudes as
the minimal instrumentalist interpretation of QM: the LHVTs, or the conventional interpretation of QM as
represented by the Copenhagen interpretation. The strong point of LHVTs is after all that it completely
regains the locality as well as the realism which goes along with our intuition, however they are no morethan a potential solution, in that it is even not sure if there exists a speciﬁc and explicit LHVT model thatcan explain all experimental facts as QM does. In contrast, QM goes straightforward only if admitting theweird interpretation of the foundation.
This sort of comparison has long been put in exercise, as a part of philosophical argument, however J.
S. Bell discovered that they have a diﬀerence in the phenomenology rather than in interpretation, and that
there is certainly what can not be reproduced by LHVTs but QM does: the Bell’s inequality. In this section,a brief review of the formulation and a quick phenomenological analysis are proposed. Note that for theconvenience sake for the later analysis, here the CHSH (Clauser-Horne-Shimony-Holt) type inequality [13]is introduced instead of the original argument by Bell (1964) [12]. The argument in『量子力学の基礎』(清水明 著) [15] is extendedly referred.

1.2. BELL’S INEQUALITY

13

Back to the gendankenexperiment, consider how the correlation of the measured spin is expressed in
LHVTs and QM. First of all, let Alice set the measurement axis (B-ﬁeld of the SG instrument) to a, and Bob
to b. Then let A(a), B(b) = ±1 denotes the result of the measurement outcome (±1 corresponds respectively
to ↑, ↓ in the argument above). Correlation is the expectation value of the product
E(a, b) = ⟨ψ|A(a)B(b)|ψ⟩ = ⟨ψ|(σ · a)1 ⊗ (σ · b)2 |ψ⟩ .

(1.15)

When the composite spin state |ψ⟩ in the singlet (1.11)


 √
1
2
1
2
(σ · a) ⊗ (σ · b) |ψ⟩ = (σ · a)1 ⊗ (σ · b)2 |↑⟩b |↓⟩b − |↓⟩b |↑⟩b / 2

 √
1
2
1
2
= (σ · a)1 − |↑⟩b |↓⟩b − |↓⟩b |↑⟩b / 2

The upper subscripts label the particles, while the lower indicates the QA direction which ↑, ↓ represents.
††√
Multiplying |ψ⟩ = (|↑⟩b |↓⟩b − |↓⟩b |↑⟩b ) 2 from the LHS, one has1⟨ψ|(σ · a) ⊗ (σ · b)|ψ⟩ = − (⟨↑ |σ · a| ↑⟩b + ⟨↓ |σ · a| ↓⟩b )2
Deﬁning the opening angle between a and b as θ, according to (1.1),|↑⟩b = cosθθ|↑⟩ + sin |↓⟩a2 a2|↓⟩b = − sinθθ|↑⟩ + cos |↓⟩a ,2 a2(1.16)

therefore,⟨ψ|(σ · a) ⊗ (σ · b)|ψ⟩ = − cos θ = a · b.

(1.17)On the other hand, E(a, b) in LHVTs is expressed by hidden variables λ and the probability distribution
ρ(λ) as follows

Z
E(a, b) =dλ ρ(λ)A(a, b, λ)B(a, b, λ).

(1.18)

This is the most general form. λ can be both continuous or discrete, singular or multiple, contained by bothor either of A, B. Here, it is assumed that Alice and Bob are isolated far apart as described as Fig. 1.3,
and that the measurement on the particle 1 and 2 are performed in such a way that those do not eﬀect eachother (the locality principle), which said,
A = A(a, λ),B = B(b, λ).

(1.19)

In fact, the QM prediction on the result of E(a, b) measurement can be managed to reproduced by
LHVTs, if with a ﬁxed combination of (a, b), since in the case, the state ρ(λ) can be prepared beforehand as
illustrated above. However, in the meantime, if Alice change the direction of the axis from a to c, as well as
Bob changes from b to d randomly during the measurement, the correlations E(a, d), E(c, b), E(c, d) can bemeasured as well in the meantime, and the statement from Bell is that, No LHVTs can reproduce theprediction of QM on the measurement result of E(a, b), E(a, d), E(c, b) and E(c, d) measured atonce (Bell’s theorem).
This is mathematically proved by evaluating the range that a quantity
S = |E(a, b) + E(a, d) + E(c, b) − E(c, d)|.
takes in LHVTs.
Starting with considering an algebraic relation followed by x, y, x′ , y ′ such that −1 ≤ x, y, x′ , y ′ ≤ 1− 2 ≤ x(x′ + y ′ ) + y(x′ − y ′ ) ≤ 2,

(1.20)

14

CHAPTER 1. INTRODUCTION

which is derived easily as|x(x′ + y ′ ) + y(x′ − y ′ )| = ≤ |x(x′ + y ′ )| + |y(x′ − y ′ )|= |x||x′ + y ′ | + |y||x′ − y ′ |≤ |x′ + y ′ | + |x′ − y ′ |= ±2x′ or± 2y ′ .

≤2(1.21)

Note that −1 ≤ x = A(a, λ), A(c, λ), B(b, λ), B(d, λ) ≤ 1, substituting x = A(a, λ), y = A(c, λ),R
x = A(b, λ) and y ′ = B(d, λ) into (1.20) and integrating over dλ ρ(λ) immediately leads to′−2 ≤ E(a, b) + E(a, d) + E(c, b) − E(c, d) ≤ 2,

which is say,
S = |E(a, b) + E(a, d) + E(c, b) − E(c, d)| ≤ 2.

(1.22)

This is one of the most popular form of Bell’s inequality, the CHSH inequality. While S is restricted under
2 in LHVTs, which is usually quoted as the classical limit, QM generally breaks the limit. For example, if
in the singlet state, plugging (1.17) into respective terms, one has

S = | − a · b − a · d − c · b + c · d|.

(1.23)√√
Here, with specifying b, c and d with a · c = 0, b = (a + c)/ 2 and d = (a − c)/ 2 (examples are shownin Fig. 1.4), the S value becomes√
S=2 2(1.24)

beyond the limit as described by the inequality (1.22). It is easily shown that (1.24) is the maximum valuethat QM can reach, which is referred as the quantum limit.

One of the biggest reason why the Bell’s inequality is appreciated is that it only requires two factors:
the realism, and the locality, thus greatly general. In other words, a single observation of the violation canmean exclusion of all possible LHVTs. This is amazing that such a powerful inequality can be derived in
the brief discussion.

1.3. REVIEW OF PREVIOUS EXPERIMENTSb

15

particle 2
particle 1

Bob

a
Alice

Figure 1.3: An example of setup of the gendankenexperiment supposed in Bell’s theorem. In order toguarantee the locality condition, the both observers are placed far apart. In the ﬁgure, Alice stands at
Greenland, Bob settles around Afghanistan, waiting for the spins ﬂying from the space. The geographicalcondition and choice of places can be arbitrary, as long as the two spin measurements take place in a spacelikely separated timing, however it is better to have the Internet otherwise they would be bored during theexperiment (the picture of earth and SG instruments are referred from [6] and [14] respectively).

!#

!

$
!"#

$

$"#
!!"#$%&'"#$%!"$%
Figure 1.4: Examples of axes conﬁgurations maximizing S in eq. (1.22).

1.3

Review of Previous Experiments

The ﬁnal judgement is left to experiment. In the section, some noble precedent experiments are introduced
in short, such as one measuring the polarization of photon pairs, one observing spin one-half proton pairs,
as an example of test using particles, and one using the meson oscillation as a sole example of test in collider

16

CHAPTER 1. INTRODUCTION

experiment. A more dedicated review and summary is given by [16] [17].

1.3.1

Photon pair experiments

The most convenient source of spin is the polarization of photon which can be easily measured by very
simple equipment for example birefringent plates which is a type of crystal giving diﬀerent reﬂection indicesrespective to the polarization of photon passing through. The experiment design was ﬁrst established byCrauser, Horne, Shimony and Holt (1969) [13], from then on, there have been bunch of experiments done
up to today, in which some of the milestone experiments are Freedman and Clauser (1972) [19], A. Aspect
et. al. (1981) [20] (1982) [21][22], Y. H. Shih and C. O. Alley (1988) [23]. Here A. Aspect et. al. (1982) [21]is illustrated as an example. The review paper [24] is widely referred.
The two photon singlet state is prepare here again. In Aspect’s experiment, those photon pairs areproduced from the cascade decay of Calcium atoms as Fig. 1.5. This is a good source of singlet states where
the emitted two photons are entangled with a high purity. Fig. 1.6 exhibits the setup of the experiment. A
source is placed at the center and two birefringent plates are settled at the both sides, of which orientations
are respectively a, c for ones at the left side (polarizer 1 and 1’) and b, d for ones at right side of the source(polarizer 2 and 2’). Before the plates, there are optical switches (C1 , C2 ) that determine way the photon
to be guided. Photo-multiplier tubes (PMTs) are set at the both ends, detecting the transmitting photons
through the polarizers.
As the transmission (A, B = +1) and reﬂection (A, B = −1) at the polarizers in orientation of acorresponds to the outcome of ↑, ↓ in the SG type spin measurement with the measurement axes pointing toa, the coincidence rate of the PMT signals N (a, b) is, considering in analogy, proportional to the correlation
amplitude E(a, b) deﬁned above (1.18), therefore the inequality (1.22)
S = |E(a, b) + E(a, d) + E(c, b) − E(c, d)| ≤ 2
can be tested experimentally. Technically, as this inequality has the upper limit of 2, the proportionalcoeﬃcient between N (a, b) and E(a, b) needs to be speciﬁed in precision, in this type of experiments using
coincidence rate, it is preferred to using another extend form (CH74 type inequality [18]):
S := E(a, b) − E(a, d) + E(c, b) + E(c, d) − E(c) − E(b)−1≤S ≤0(1.25)

where E(c) and E(b) are transmission rate of polarizer 1′ and 2 irrespective to the other hand. Now (1.25)
has 0 for the upper limit, hence written in observable N (a, b),S :=N (a, b) − N (a, d) + N (c, b) + N (c, d) − N (c, ∞) − N (∞, b).
N (∞, ∞)

(1.26)∞ labels the case that the corresponding polarizers are absent, whose coincidence count rate are given in
the auxiliary runs.
The measurement gave
S = 0.101 ± 0.020,
violating the inequality (1.26) by 5σ standard deviation.

1.3. REVIEW OF PREVIOUS EXPERIMENTS

17

As seen above, optical experiments generally boast of their huge statistics, leading to an extremely highsigniﬁcance. On the other hand, the detection eﬃciency for these kind of soft photons from atomic cascades
is substantially low. For example the eﬃciency for the PMTs are less than 20%, thus for the coincidence
events far below 4%. Beside the limited solid angle acceptance of detectors, the detected photons count only
on very tiny fraction of the sample. This is the major imperfection of the experiment, so called ”eﬃciency
loophole”, as discussed below.
!!"##$%&#

ν1#)2$ ,-ν$#))$*+#,-#!'!! $($#

ν/!&0#,-#

ν"#!""*.#,-#!""##$%&#

Figure 1.5: Diagram for the calcium cascade decays. Two entangled photons ν1 and ν2 are emitted throughthe atomic de-excitation.

Figure 1.6: A typical setup for the photon pair experiments [24]. The two photons ν1 and ν2 emitted fromthe source (S. center) are guided to the polarizers (1 and 1’ for ν1 , 2 and 2’ for ν2 ) by the optical switches(C1 and C2 respectively). PMTs are set further behind, of which coincidence signals are collected.

1.3.2

Loopholes

Up to today, there have been astonishingly huge statistics of announcements reporting the violation of
Bell inequalities. However, they are never ever perfect experiment which rigorously satisﬁes the conditions
required in the Bell’s theorem. The experimental incompletenesses are often called ”loopholes”. Under sucha setup, the possibility of a LHVT model which reproduce everything QM predicts about measurement
results can not be excluded. Therefore, the experiments so far only negate a subgroup of LHVT models.
Largely, there are two types of loopholes (i) the eﬃciency loophole, and (ii) the locality loophole, which areindividually explained in detail in the following.

The eﬃciency loophole
Since in experiments, it is in eﬀect impossible to detect all the photons punching through the polarizers,
usually it is forced to make an assumption for the detected sample that the sampled sub-ensemble statistically
represents the nature of the entire ensemble, which is to say, the sample is not biased. This is referred to

18

CHAPTER 1. INTRODUCTION

the ”homogeneous assumption” or ”no enhancement assumption [18]).
For most of the case it is a very plausible and reasonable assumption, and the correction to bias can benevertheless applied, however it can never be justiﬁed in a mathematically rigorous sense. There is nodiﬃculty in constructing a LHVT model as rebuttal to the assumption, where the detected events exhibits
large correlation than the undetected events, and the S value calculated from the detected sample surpasses
the classical limit of the inequality (1.22) but never if using the sample of both events [25] [26].
Admittedly, those model are bastard as physics since they assume, for example, unknown and untestableinteractions that are by far even more strange than quantum non-locality, what they meant are no morethan demonstration of the counterexamples.
Without the homogeneous assumption, by considering the worst scenario that the detected sample are
maximally biased, the CHSH inequality (1.22) is modiﬁed with the detection eﬃciency ϵ as:
S = |E(a, b) + E(a, d) + E(c, b) − E(c, d)| ≤

4
− 2.
ϵ

(1.27)√√As the quantum limit is 2 2, ϵ > 2/( 2 + 1) = 0.83 is at least required to observe the violation of (1.27).
By further technical extension of the inequality, the lower bound for the decent necessary eﬃciency canbe dragged down. For example, in the case of the Eberhard inequality (1993) [27], it is ϵ > 0.67, it thebackground level is suﬃciently low.
Since the photon energy is low due to the production mechanism of entangled states, the eﬃciency
loophole has been the weak point for the photon pair experiments for a long time. However, according to
the long-term development of devices and experimental techniques, it has been gradually overcome thatnowadays there is even an experiment that conﬁrms the violation of the Eberhard inequality, with 75%
eﬃciency [28], which closes the eﬃciency loophole.
On the other hand, the experiments using charged particles tends to less serious on the loophole. For
example, the proton pair experiments which is described below or one using ions pair (Rowe (2001) [29])reaches almost 100% eﬃciency.

The locality loophole
In proving the Bell’s theorem, the most important fact used is the locality condition (1.19):
A = A(a, λ)B = B(b, λ)There are mainly two types of situations in experiments that fail to maintain the locality condition (forsimplicity, here we go back to the example of the EPR gendankenexperiment).
(i) The space-like separation of the two measurements (on spins, ) are not strictly guaranteed.
In such a case, it is allowed to explain the correlation in a way that, the measurement result of
Alice propagates to the opposite particle, and then the particle changes its spin according to it, before
measured by Bob. This is nothing but the description of ”local physics” and it does not conﬂict against
the reality as well. Thus the result can be totally explained by LHVTs, even though the violation ofBell’s inequality is measured.

1.3. REVIEW OF PREVIOUS EXPERIMENTS

19(ii) The particles (or photons) can in principle know direction of measurement axes a and b.
In other words, the ideal setup is that those spin or polarization are measured before the informationthat the axes are set to certain directions arrives at them (”delayed choice”). This is essentially related
to the concept of LHVTs during the EPR argument that if the state can be ”prepared” beforehand, it
does not have to ascribe the result of measurement to non-locality.

Recall that LHVTs can make an identical prediction on the correlation amplitude E(a, b), with that
QM predicts on for the singlet state, by considering the case where the spin 2 points along or against
a in the initial state, if Alice measures ﬁrst. But this initial state can not give the concise prediction
on E(c, b) with QM. Hence, the only possible way for LHVTs to make ”correct” prediction on bothE(a, b) and E(c, b) is to prepare diﬀerent initial states according to the axes setting, such that the
spin 2 points along or against a if the measurement axis of Alice’s machine is a, and vice versa. The
reason why the decayed choice is needed and (ii) should be prohibited, is to exclude this last possibility,
otherwise there will be still a slight hope for LHVTs to reproduce the prediction of QM on E(a, b),
E(a, d), E(c, b) and E(c, d), and to violates Bell inequalities.

Associated with it, the free will of observers’ choice have to be assured as well. If the choices of a andb are not independent but in a function relation through a parameter ϕa = a(ϕ), b = b(ϕ)
a = a(ϕ(b)),(1.28)

it is easily shown that Bell inequalities is not satisﬁed even by LHVTs [15]. This is in essence because
the information of axes is eventually sheared with the particles.
If categorizing further, the loophole arising from the situation (i) is usually referred as ”locality loophole”
and (ii) as ”free will” loophole since it is common to the experiments without free will of observers to choose
the axes.
As for the photon pair experiments, those loopholes have been also the challenges. However, the locality
loophole (i) is closed by some experiments. Weihs et al. (1998) [30] and Tittel et al. (1998) [31] are some ofthe examples, where the two detection points are isolated respectively 400m and more than 10km distant,with extended optical ﬁber cables. In recent years, there is even an similar experiments with 144km long
base-line (Uesin et al. (2007) [32]).
To the contrary, particle experiments suﬀer from the locality issue more than eﬃciency loopholes ingeneral, since it is not easy to extend the whole setup. In addition, in the experiments measuring spinsthrough scattering or particles decays, the lack of active and controllable measurement axes will cause thefree will loophole as seen below.
The issues regarding to the free will loophole are basically two-fold; practical possibility of achieving an
ideal decayed choice. It is mostly considered to be solved by randomly reorienting the polarizer during themeasurement, just as done in Aspect’s third experiment (1982) [22] or Weihs et al. (1998) [30] and so on.
The other, more serious problem is about the free will itself. Though in most cases in physics argument,

20

CHAPTER 1. INTRODUCTION

observer’s free will is taken for granted, it is highly non-trivial when it comes to the criteria of who/which
has it. Frankly, there is no good reason to allow only human to be ”free” while the other ”objects” areconsidered to be ruled by physical principles.
An interesting observation is given by J. Conway and S. Kochen with the ”The Free Will Theorem” [33].
It is derived that if the free will of observer is guaranteed, it has to be extended to ”objects”, and evenultimately to elementary particles. Therefore, randomly distributed outcome of measurements, such as spinmeasurements, may have to be interpreted as the representation of those particles’ free will. What this
implies to our Bell’s inequality experiments is thus signiﬁcant, since it can never be ensure that the two
measurements are independent in a true sense. However, since the issue of intrinsic free will is a too muchgeneric problem and needs by far more general insight outside the scope of the thesis, here as is often thecase, we would not go any further, and only the former issue on the delayed choice is considered as the free
will loophole argument.

1.3.3

Proton pair experiment

The experiments of massive particles basically belong to two families: ones using spin entangled proton pairs
0reviewed here, and the other is ones using the ﬂavor oscillation of neutral mesons such as K 0 K or B 0 B

0

as referred in the next section. For the proton experiments so far, there are M. M. Lamehi-Rachti and W.
Mitting (1972) [34], and H. Sakai (2006) [35] etc. but here we conﬁne the discussion in reviewing H. Sakai(2006) for simplicity.
A proton pair from the decay of short-lived 2 He spin-singlet state, populated in the nuclear reaction d+H →2
He + n is in a purity spin singlet state. An shown in Fig. 1.7, these pairs are guided through the event-ready
instruments ﬁrst, and scatters in the graphite target. Since there is anisotropy in the scattering distribution
according to the incident proton spin, the scattered angle has information of the proton spin before scatted,
which measured by the multi-layered hodoscopes settle at the down-stream.
The inequality tested in the experiment is the CHSH type (1.22)
S = |E(a, b) + E(a, d) + E(c, b) − E(c, d)| ≤ 2.

(1.29)a, b, c and d are the measurement axes however here the comparable components do not exist. Instead, a setof virtual axes is considered. Suppose that there are two virtual axes a and b standing in front of the path of
proton 1 and 2 respectively (see Fig. 1.8). Then, think of a set of quantities A(a), B(b) which is measured
as +1 if the proton passes the left-side of the axis, and −1 if it passes the right-side. Considering that the
proton direction is related to the spin before scattered, A(a), B(b) can be straightforwardly regarded as theoutcomes of spin measurement, just as in the EPR gendankenexperiment, and E(a, b) can be deﬁned asabove (1.18).
Note that these axes are virtual thus they can even be chosen after the data taking. This is because theproton trajectory can be acquired in three dimensions, and it is thus uniquely determined if the proton passes
left or right, for any arbitrary setting of the axis. Therefore, with a series of data taking, the dependencyof E(a, b) on a and b is completely reconstructed (tomography), where all events can be applied to eachsetting of (a, b).
Experimentally, E(a, b) is obtained byE(a, b) =1 N++ (a, b) − N+− (a, b) − N−+ (a, b) + N−− (a, b).
P 2 N++ (a, b) + N+− (a, b) + N−+ (a, b) + N−− (a, b)(1.30)

1.3. REVIEW OF PREVIOUS EXPERIMENTS

21

Figure 1.7: The overview of the setup in H. Sakai (2006) [35]. A helium 2 He atom is generated through the(d,2 He) reaction, from which a pair of protons emerge. The proton momentum is selected by the spectrometer(SMART) in the event ready in order to extract pure entangled pair sample. The protons eventually scatterwith the graphite target, and the hodoscope in the rear measure the trajectories 3 dimensionally.

aabA, B=+1

A, B= −1

Figure 1.8: Deﬁnition of the quantity A and B respective to the virtual axes a and b, which are measured
as +1 if the proton passes the left-side of the axes, and −1 if it passes the right-side, Φ is deﬁned as theangle between the planes.

22

CHAPTER 1. INTRODUCTION

Nij (a, b) is the frequency of having events where A(a) = i, B(b) = j (i, j = ±1).
P is the polarization eﬃciency of scattered single proton, deﬁned as
P+ (a) = 1 + P s · a(1.31)

with P± (a) being the probability of having the outcome of A(a) = +1, thus P represents how muchinformation of the spin is translated into the asymmetry of the distribution. The form (1.31) can be derived
from QM calculation, however it is used as the measured result since here the aim is to testing LHVTs.
With the calibration runs, P = 0.16 ∼ 0.24 is obtained. Using (1.31), E(a, b) is derived for the spin singlet
proton pairs, assuming the locality condition,
E(a, b) = −P 2 a · b.

(1.32)

Renormalizing this by dividing into P 2 so that the upper limit becomes to unity, the inequality (1.29) canbe derived in the same manner as described in section 1.2.
The result is S = 2.83 ± 0.24(stat.) ± 0.07(syst.) violating the inequality (1.29) by 99.93% conﬁdence
level, and the tomopraphed function shape of S(Φ) is also in perfect agreement with the QM (Fig. 1.9) (Φis an angle deﬁned in Fig. 1.7).
On the virtual axes, it has to be pointed out that, these can not realize the delayed choice condition,
though they can be chosen well after the data taking because it has nothing to do with the physical resultof protons scattering, while the magnetic ﬁeld of the SG instrument plays an more active role that it canchange the direction of spins if in QM, which makes the diﬀerence from all kinds of LHVTs. In this setup
here, a LHVT model in which thedirection of a scattered proton is determined before it is scattered canreproduces the prediction of QM, which creates therefore the free will loophole.
Another weak point of the massive particle experiments are the statistics. In contrast to photons, it isdiﬃcult to generate huge number of particles in a deﬁnite state. On the other hand, it tends to be easier
to detect particularly for charged particles thus robust against the eﬃciency loophole. For example, theexperiment by Sakai (2006) attains more than 96% eﬃciency.

1.3.4

K 0 K 0 , B0 B 0 ﬂavor oscillation

The experiment proposed in this paper is not actually the ﬁrst attempt of testing Bell’s inequality in colliders.
00

The notable meson oscillation of K 0 K and B 0 B observed in colliders has been put in application for testing00
LHVTs and QM. The analysis by CPLEAR (1999) [36] for K 0 K and Belle (2004, 2007) [37] [38] for B 0 B
are famous. Here Belle (2007) is reviewed.
The ﬂavor number has the identical SU (2) structure called quasi-spin, which represents quite analogous
00properties as spin. A B 0 B pair from decay Υ(4S) → B 0 B is in the ﬂavor singlet as follows:|ψ⟩ =|B 0 B̄ 0 ⟩ − |B̄ 0 B 0 ⟩√2(1.33)The ”measurement” on ﬂavor number is performed when either of the B 0 decays, for example, throughB 0 → D∗ lν. As it follows
B 0 → D∗+ l− ν̄B̄ 0 → D∗− l+ ν,(1.34)

1.3. REVIEW OF PREVIOUS EXPERIMENTS

23

Figure 1.9: The result of H. Sakai (2006) [35]. c and d are ﬁxed so that S is maximized, and Φ is deﬁned as
the angle between a and b. The red line is theoretical prediction by QM, not a ﬁt to the experiment.

by identifying the charge of the lepton, one can ﬁnd the ﬂavor of the decayed b-meson; B 0 or B̄ 0 , as wellas that of the living b-meson as the projection on (1.34) occurs|B 0 B̄ 0 ⟩ − |B̄ 0 B 0 ⟩√2|B 0 B̄ 0 ⟩
√
2

→

or

−

|B̄ 0 B 0 ⟩√,2(1.35)

in which the dynamics of state projection is the identical to the case with spin measurements. Thus, deﬁning
the quantities A, B = ±1 as the charge of the leptons in the ﬁnal state (1.34), the rest of the argument in
the Bell’s theorem seems to work in the same way.
However, one problem peculiar to this case is that, the measurement outcome (A, B) is always (+, −),(−, +), which corresponds to a spin measurement with setting the both measurement axes in the sameorientation. This is apparently boring and of course no diﬀerence is manifested between QM and LHVTs.
Yet, here comes the ﬂavor oscillation saving the situation.
The ﬂavor oscillation is a phenomenon that a neutral meson, here B 0 , exchanges the ﬂavor due to theintermediate box diagrams as Fig. 1.10. If the prompt meson decay at t = t1 reveals it is B 0 , the ﬂavor of
the rest meson is determined as B̄ 0 at the moment. However the time evolution is written as:|ψ(t′ )⟩ = g+ (t) |B 0 ⟩ + g− (t) |B̄ 0 ⟩′′g+ (t) = (−e−λS t + e−λL t )/2′′g− (t) = (e−λS t + e−λL t )/2(1.36)

where the contribution from CP violation is ignored. t′ is deﬁned the time lapse t′ := t − t1 . It is analogous
to the precession of spins, except the damping factor λ which represents the decay rate into the channels
other than (1.34) in interest.

24

CHAPTER 1. INTRODUCTION

t

W

b

d

W

b
t

W

d

b

b

t

d

t

d
W

Figure 1.10: Processes responsible for ﬂavor oscillation. The valence quarks inside B 0 and B̄ 0 exchange theirﬂavor spontaneously driven by this inner dynamics of weak interaction.
Therefore, though the situation that only the lepton charge ± can be measured, the state itself evolves
instead, analogous to that the measurement axes are still ﬁxed but the spin itself rotates. In fact in QM, themeasurements are equivalent between rotated spin with ﬁxed axes and ﬁxed spin with the axes rotated, if
the relative conﬁguration is the same. Thus, selecting a measurement time in the cases of meson oscillation,corresponds to setting a measurement axis to a direction.
In this sense, the correlation amplitude E(t1 , t2 ) is analogously deﬁned as
E(t1 , t2 ) =N++ (t1 , t2 ) − N+− (t1 , t2 ) − N−+ (t1 , t2 ) + N−− (t1 , t2 ),N++ (t1 , t2 ) + N+− (t1 , t2 ) + N−+ (t1 , t2 ) + N−− (t1 , t2 )(1.37)

where Nij (t1 , t2 ) represents the rate of observed events where the two meson decays at t = t1 , t2 yieldingA(t1 , t2 ) = i, B(t1 , t2 ) = j (i, j = ±1).
With this E(t1 , t2 ) and the locality assumption, one can construct an inequality like (1.22) and test it
in experiment. However, as the test of LHVTs, there are bunches of problems, for example the free willloophole since the decay time can not controlled by observers; The assumed locality are also suspicious since
the decays occur one after the other; the eﬃciency is low enough to induce the eﬃciency loophole; ﬁnally,
there is also the ”unitarity problem” typical to this meson experiments, which is in short: here events to the
decay branches other than (1.34) are ignored, and eﬀectively the ﬂavor correlation (1.37) is reconstructedwith the sampled small fraction of events. Thus, the observed violation can be attributed to the potential
biasing due to the sampling. Comprehensive discussion is provided in R. A. Bertlmann et al. (2004) [39]and T. Ichikawa (2008) [40].
For those complications, this kind of experiments usually aims for observing the quantum coherence of(1.33) by comparing the modulating correlation E(t1 , t2 ) (1.37) between data and the prediction of QM
or some other models, rather than excluding the entire class of LHVTs by testing Bell’s inequality. For
example, the standard QM, which interprets (1.33) a coherent superposition, predicts
A(t1 , t2 ) = cos (∆md ∆t),(1.38)

showing a modulation against ∆t but not respective t1 and t2 , representing the inseparability of the state(1.33). On the other hand, if the two mesons are not coherently superposed but in a statistical mixtureρ = |B 0 B̄ 0 ⟩ ⟨B 0 B̄ 0 | + |B̄ 0 B 0 ⟩ ⟨B̄ 0 B 0 |,A(t1 , t2 ) = cos (∆md t1 ) cos (∆md t2 )(1.39)

25

Asymmetry

1.4. WHAT IS STILL INTERESTING WITH TESTING BELL’S INEQUALITY?

1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
0

2

4

6

8

10

12
∆t [ps]010∆t [ps]∆/σtot

6
3
0
-3
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
20 0

Asymmetry∆/σtot

6
3
0
-3
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1

Asymmetry

Asymmetry∆/σtotFigure 1.11: Theoretical prediction for of the function E(t1 , t2 ) against ∆t between the models; the standard
QM (1.38); SD (simultaneous decoherence; (1.39)) and PS (Pompili-Slleri) which is a class of LHVT model
describing meson oscillation [41]. Since the PS provides a range of models with diﬀerent prediction on
E(t1 , t2 ), it is expressed with a band sandwiched by the dotted lines.

10

6
3
0
-3
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
20 0

∆t [ps]10∆t [ps]Figure 1.12: The result of Belle (2007) analysis [38]. The respective plots indicate the comparison between
measurement and the models. The boxes show theoretical value with uncertainty on ∆md . While QM showsa nice consistency with χ2 = 5.2, SD and PS give large deviations respectively χ2 = 174, 31.3.
(see Fig. 1.11). The result of Belle (2007) is presented in Fig. 1.12. The modulation quite agrees with thestandard QM curve.
0

The case of K 0 K is essentially similar, though there are some phenomenological diﬀerences (larger mass
discrepancy between KS0 and KL0 etc.) which is important to other physics.

1.4

What is still Interesting with testing Bell’s inequality?

As mentioned above, the photon experiments evolve day-by-day, and it seems matter of times that theeﬃciency and the locality loophole at once. The triumph of QM is close. Then what should we do in the”post-war” days. Apart from the main theme of thesis, in the section, a review of recent activities relevant
to Bell inequalities is provided from a general viewpoint ﬁrst, and then the motivation of the study done in
the paper is presented.

26

CHAPTER 1. INTRODUCTION

Test of non-local realistic theories
It is shown by the violation of Bell’s inequality that the realism and the locality do not coexist in the nature,then which is wrong after all, or they are both wrong as in QM is next interest. As for this, it has been
great works from both theory and experiment sides: the Leggett inequality.
It is formulated by A, Legett (2003) [43] by extending a Bell’s inequality that expresses the bound of
correlation where any realistic theories, including non-local ones, have to follow. As done in Bell inequalities,
it has been also tested through experiments. With some auxiliary assumption though, the violation isdemonstrated [44]. Thus, the concept of reality is also endangered. It is utterly astonishing that theseemingly trivial statement that physical quantities have determined value provokes this much rebuttal, andthat it is demonstrated through the method of positive science rather than philosophy.

Test of local QM models
On the other hand, what if a combination of the local non-realistic theories? As there has been almost nonecesarity to consider such group of theories so far, there is no great motivation for perform a general testof them. However, testing a speciﬁc sub-group, local-QM is interested. For example, the Bohm-Aharanov
postulate [45], a simple model of QM where the many-body coherence is described to dissolve when they
are far apart. This idea itself is truly reasonable since naively thinking, it is weird that the coherence isconserved forever.
This local-QM model, or inversely quantum non-locality, can be tested by ordinary Bell’s inequalities
without any extension. It is even easy to test, since the it is free from the loopholes aﬀair thus most of theexperiments which are regarded as imperfect tests of LHVT could be good tests of quantum non-locality.

Test of universality of the violation
Aside from non-locality, another interest is casted on if the Bell’s inequality violation is a general natureof the world or only the case in a speciﬁc system. To answer the question, similar tests in various typeof systems are desirable, especially in massive particles system since the violation is less trivial. This isbecause of shorter coherence length of massive particles, leading to a more localized wave function (morelike a particle) compared with massless particles with broaden wave-packet in position.
In addition, tests in massive particles system can lay more emphasis on testing non-locality since theparticles are more localized. In contrast, the violation in photon systems may be able to be explained by
the overlapped wave function rather than the non-locality.

Motivation on the study
The primary aim of the experiment described in the thesis is testing universal validity of QM, by observing
the violation of Bell’s inequality in various systems. At least in terms of the variety of systems, current
collider experiments provide a preferable environment: wildly ranged energy scales 1GeV ∼ 1000GeV are1.4. WHAT IS STILL INTERESTING WITH TESTING BELL’S INEQUALITY?

27

available, including particles in variations (bosons and fermions, leptons and hadrons, massive and light etc.)with 3 interactions (strong, weak and electro-magnetic) involved.
00

The K 0 K and B 0 B experiments above are the examples, however the method is hardly extendable totests in other systems, since it has to exploit highly speciﬁc properties of the ﬂavor oscillation. On the other
hand, this thesis focuses again on spin entanglement, which is a quite ubiquitous phenomenon in collider
experiments, trying to extend the horizon of testability of Bell’s inequality in general systems.

Chapter 2

Design of the Experiment
The paper presents an experiment in colliders, using the spin entanglement of two particles from a decay
instead of quasi-spin, which is more close to the Bell’s original argument (section 1.2). This section illustrates
the detail.

2.1

Fundamental Idea

To carry out a Bell’s type experiment, there are two components at least to be maintained.
• Production of entangled states with high purity• Accurate measurement on spin polarization
Though these are in fact the bottleneck in the particle experiment in terms of technology and statistics,
in the following, it is shown that these two can be managed in collider experiments. Note that these are
the decent minimum requirement and there are more other conditions to be satisﬁed to perform a ”better
experiment” for example with fewer loopholes. These details are argued in the later section 2.3.

2.1.1

Source of entanglement

Collider experiments are the experiments that studies the physics in high energy scale, by hitting parcels
face-to-face each other creating a high-energy state. Through the collision process, many unstable particles
are created as intermediate states which decay into other particles usually after a certain short while within
the detector. The entanglement often emerges between the secondary decay particles, just as the meson
00pairs K 0 K from ϕ → K 0 K in which their ﬂavor number (strangeness) are entangled (see section 1.3.4).
The spin entanglement is in fact more ubiquitous phenomenon in colliders. For example, when a pseudoscalar charmonium (cc̄, bound state of a c quark and an anti-c quark) ηc decays into spin one-half fermionspair ΛΛ, they realizes the spin singlet state|ψΛΛ̄ ⟩ =|↑⟩ |↓⟩ − |↓⟩ |↑⟩
√
.
2

(2.1)

This is a maximally entangled state in spin, and a faithful realization of Bohm’s type of gedankenexperiment1.1.2. There are enormous example such as π0 → γγ that demonstrates the entanglement of spin or polarization between the daughter particles, therefore particle decays can serve as the source of entangled state.
29

30

CHAPTER 2. DESIGN OF THE EXPERIMENTThe purity is another problem. The key factor is the performance of selecting particular events where thetarget decay process takes place (”signal”), out of the huge mess with other types of events which is notinterested in. (”background”; BG). Fortunately, this is also the important subject for most of the otherphysics programs in collider experiments (such as new physics search and so on), the detector technology or
know-how in analysis etc. have been developed to a substantially high level that, in a low BG environment
such as the e+ e− collision, with appropriate selection procedures, the BG fraction in the acquired ﬁnal
samples can be suppressed under 2 ∼ 3%, for the channels interested in the paper. In contrast, in a highBG environment, proton-proton collider for example, it is very diﬃcult even with current technologies.

2.1.2

Weak decay as polarizer

Direct measurement on spins of unstable particles in colliders are not feasible in any current colliders,
however instead, the nature endows us the parity-violating weak interaction somehow, particles decaying
through the weak interaction emit their daughters to (against) the direction of their spins; the particles telltheir polarization by themselves. For instance, in the process Λ → pπ, the π tends to emerge against the Λ
spin direction; π from τ → πν prefers the direction of the τ spin and so on. Therefore the spin of parent
particles can be inferred by the direction to which the daughter particles orient in the ﬁnal state. This kindof decays are referred to ”polarimeter decays” in the rest of the contents. Generally, the distribution of adaughter particle C coming from a certain 2-body weak decay B → CD is expressed as:dΓ∝1+αs·ndΩ(2.2)

where the s and n are respectively the spin of the parent B and the direction of motion of the daughter C.
The parameter α indicates the strength between their correlation, characterized only by the decay process,valuing over −1 to 1. Larger α2 enables s to be measured more precisely since n and s are more correlated,thus α2 is the measure of the ”performance” of the decay. The sign of α represents the sign of the correlation,that is to say, s and n tends to be parallel when α > 0 and verse visa. The decay Λ → pπ has a large α value
with α = −0.642 ± 0.013 (PDG 2012) [48] so we let it serve as the ”polarimeter” of Λ. The mechanism of this”self-polarimeter” [46] that the spin is estimated from the direction of the ﬁnal state particles is basicallythe same as that of proton-graphite scattering in the proton pair experiment 1.3.3, but this mechanism isamazing and profound in that the nature prepare such a scheme by itself.

As a matter of course, large α2 is preferred. Tab. 2.1 is the list of B → CD type 2-body weak decay withlarge α2 value. τ → πν boasts of the largest α value with almost 1. In general, leptonic decays channel tend
to have large α, while the number is quite limited. The variety of hadronic decay channels is in contrast huge
however, as generally they have the contribution from the parity conserving strong or electro-magnetic (EM)interaction and with some reasons regarding to kinematics, it is diﬃcult to have α unless it is hyper lucky.
The physics determining the α and its quantitative formulation is given in the next section. Considering
practical utility (statistics, easy to measure etc.), the useful channels in Tab.2.1 are Λ → pπ − , τ − → π − νand their charge conjugation Λ → p̄π + , τ + → π + ν̄.

2.1. FUNDAMENTAL IDEA

31

Table 2.1: The list of ”polarimeter decay”. The upper stacks are hadronic, the lower are leptonic decay
channels. For the hadronic channels, α are referred as the measured value [48], while for the leptonic
channels they show theoretical values (Standerd Model) [49]. Ignoring the tiny eﬀect from the CP violation,
the charge conjugate processes exhibit the identical properties except the sign of α is opposite.
B → CDαBranching of B → CD−
Λ → pπ−0.642 ± 0.013(63.9 ± 0.5)%
Λ → nπ0−0.648 ± 0.045(35.8 ± 0.5)%
Σ+ → pπ0−0.98+0.017(51.57 ± 0.3)%−0.015+Λ+−0.91 ± 0.015(1.07 ± 0.28)%
c → Λπ++Λ c → Σ π0−0.45 ± 0.34
(1.00 ± 0.34)%
τ − → e− ντ ν̄e
0.33(17.83 ± 0.04)%
τ − → µντ ν̄µ0.33(17.41 ± 0.04)%
τ − → π − ντ
1.00
(10.83 ± 0.06)%
τ − → ρ− ντ
0.46(25.52 ± 0.09)%
W − → l− ν̄
1.00(32.6 ± 0.03)%

D1
B1
C1
A

D2
B2
C2
Figure 2.1: The cascade decays A → B1 B2 ; B1 → C1 D1 ; B2 → C2 D2 . Find initial states A such that thesubsequent B1 and B2 are entangled in their spin. The spin of B1 and B2 are measured respectively throughthe direction of motion of C1 and C2 .

2.1.3

Candidate channels

To summarize the result of the two section above, here a cascade decays A → B1 B2 ; B1 → C1 D1 ; B2 →C2 D2 is in consideration in which the spin entanglement is given through A → B1 B2 , and the subsequent
decays B1 → C1 D1 , B2 → C2 D2 have be a certain type of weak decays so as to measure the B1 , B2 spins(Fig. 2.1). Practically, (B1 , B2 ) = (Λ, Λ), (τ − , τ + ), (C1 , C2 ) = (π − , π + ) seem to be promising. The lastthing to be discussed is then the ﬁrst decay A → B1 B2 , the source of ΛΛ or τ − τ + 1 .
The main production channels of ΛΛ are the charmonium decays cc̄ → ΛΛ such as ηc , χc0 and J/ψ. Theτ τ pairs are mainly generated in the e+ e− colliders via e+ e− → γ ∗ → τ τ or e+ e− → Z ∗ → τ τ .
From view of the strength of spin entanglement, the spin 0 initial states are desirable since they lead tomaximally entangled states. As for charmoniums, ηc and χc0 are scalar thus the candidates. Those are
1 In the following, τ + τ − is abbreviated as τ τ if no confusion.

32

CHAPTER 2. DESIGN OF THE EXPERIMENT

massively produced in the recent ”charm factory” experiments such as BES3 [51] or CLEO [52] therefore thestatistics is abundant even though the breaching fractions to ΛΛ are small (typically ∼ 1%). Decays fromscalar to τ τ are not easily accessible. Heavy meson decays ηb → τ τ or χb0 → τ τ etc. can be supposed as
the candidate however they have even not conﬁrmed yet [48]. The other candidate is the Higgs boson decay
H → τ τ 2 . Although, it does include problems in every aspect to test LHVTs and so on with the Higgs
decay (statistics, sample purity, the character of Higgs is not trivial as well etc.), it is interesting to think
how much they can solved in the future Higgs factory project in the ILC (International Collider) [53]. Thisis the ﬁrst application of the Higgs boson, as well as the BI test in the most exotic system.
Vector initial states are the second choices from the viewpoint of spin entanglement but still potentiallyavailable. In particular, since the J/ψ has been the primary target of charm physics for a long time, J/ψ
has incredibly high statistics produced in the charm factories, so does the J/ψ → ΛΛ channel accordingly(∼ 109 ). This is the most promising channel in terms of experimental sensitivity if ΛΛ has enough strong
entanglement. As for the τ τ channel, the Z → τ τ event samples are available, collected in the Z factory
experiment LEP [54] with a large sample size about 106 . This channel is important in that the energy scale
is at the weak scale (∼ 100GeV), which is by far higher than that of other BI experiments done ever.
The initial state A is not necessarily to be at a particle resonance. The direct production via beam collisionsuch as e+ e− → γ ∗ → τ τ is also available as long as the τ spins are entangled. This reaction is very common
which massively occurs in many experiments as continuum background. For example, at the Belle experiment[55], the e+ e− beam energy is tuned to create b-mesons such as Υ(4S), however the e+ e− → γ ∗ → τ τ has ahuge cross-section of 7.8nb which is comparable to ∼ 109 events, assuming the integrated luminosity 771f b−1 ,which is the most promising candidate if considering only statistics.
This type of test of BI, using the spin entanglement of unstable particles generated in colliders, was ﬁrst
proposed by N. A. Törnqvist with cc̄ → ΛΛ → pπ p̄π[46][56]. The DM2 experiment responded to it andcarried out the measurement in the J/ψ → ΛΛ channel [57], however observing no inconsistency to the BI
due to the ineﬀective entanglement of ΛΛ pairs, which will be shown in detail in chapter 4. Recently, thesame test is planned, with the samples of the diﬀerent channel ηc → ΛΛ collected in the BES [58]. From
theoretical side as well, J. P. Baranov provided a new formalism regarding to the test [59]. In the paper,
the author deals with these with diﬀerent formulation of the inequality to test and analysis as illustrated inchapter 3 and 4. The τ τ → πνπν channel used to be intensively studied from both theory and experimentside, in the context of electro-weak theory such as the test of the V-A interaction in CLEO (1999) [62].
A proposal has been made by P. Privitera (1992) [63] that the result of the LEP analysis can be used totest LHVT and quantum non-locality, as this paper suggests. However, it has been controversial on theconditions of the formulation [64], [65], and the ﬁnal result has not pushed yet. The argument will be
reviewed in section 2.3. The H → τ τ → πνπν channel is ﬁrst studied in the paper.
To sum up, this paper discusses the channels of two groups: the Λ series channels cc̄(ηc , χc0 , J/ψ) →ΛΛ → pπ − p̄π + ; the τ series channels Z, H → τ τ → πνπν, e+ e− → γ ∗ → τ τ → πνπν. The corresponding
diagrams are shown as Fig. 2.2. The channels of both series have the common event topology, as a result
highly analogous. Therefore for simplicity, here the case of the Λ series is basically described. The diﬀerence
from the τ series is mentioned if it becomes signiﬁcant.

2 The decay H → τ τ has been veriﬁed in LHC. [50]

2.2. WEAK DECAY AS POLARIZATION ANALYZER

33

p

p
Λ

Λ

̟a-

ηc

Λ
̟a-

χc0

̟a+

J/ψ

̟a-

̟a+

̟a+

Λ

Λ

Λ
p

p

ν
Z/γ*

p

τa-

̟a-

p

ν
τaH

̟a+
τa+

̟a̟a+

τa+

ν

ν

Figure 2.2: The candidate channels analyzed in the paper. ηc → ΛΛ → pπ − p̄π + (upper left), χc0 →ΛΛ → pπ − p̄π + (upper center), J/ψ → ΛΛ → pπ − p̄π + (upper right), Z/γ ∗ → τ τ → πνπν (down left),H → τ τ → πνπν (down right)

2.2

Weak Decay as Polarization Analyzer

The polarimeter decay is interesting by itself, aside from the context of testing Bell’s inequality. In section,
the phenomenology is dedicatedly studied, namely, how the parity violation is associated with the asymmetry in the decay distribution in the polarimeter decays, how much precision the spin measurement oﬀers,
comparisons with the measurement by the SG instrument.

2.2.1

Parity violation and ability as a polarizer - in a nut shell

With a brief relation between the intrinsic parity of particles and the orbital angular momentum, the asymmetry in decay distribution can be derived (here chapter 9 of [47] is partially referred).
We start with intrinsic parity, which is the parity number associated to the particle itself. For elementary
particles, +1 is allocated for bosons. Fermions dose not have unique intrinsic parity 3 but the relative intrinsic parity to anti-fermions are always −1 according to a simple inspection on Dirac equation.Composite
particles are therefore have the opposite intrinsic parity to that of their anti-partners, providing that parity
is multiplicative quantum number. Here we choose a notation that lets baryons and anti-baryons have +1,−1 for their intrinsic parity respectively. For mesons, whose anti-partner are themselves, the intrinsic parities are deﬁned without ambiguity of notation, when the intrinsic angular momentum (spin) is taken into
account. They earn another factor (−1)S (S: spin) as a result, thus scalar mesons (spin 0) have −1, vector
mesons have +1 etc.
3 This is caused by the fact that fermion ﬁeld ψ is not observable [66].

34

CHAPTER 2. DESIGN OF THE EXPERIMENTIn the example of Λ → pπ − , intrinsic parity +1 is attached to proton and Λ baryons while −1 is to π. Assuming the process conserves parity, there must be the contribution (−1)l from the relative orbital angularmomentum l in the ﬁnal state, since the intrinsic parity is not conserved by itself. Considering that Λ is at
rest initially (l = 0) and total angular momentum conservation (initial state: J=1/2, the total spin in theﬁnal stateS=1/2), l is uniquely determined to l = 1 (P-wave). On the other hand, if parity is not conserved
during the decay, l = 0 (S-wave) mode is also allowed, which is the case here.
Total transition amplitude is the superposition of both. With S and P being the amplitude for eachmode, the ﬁnal state is written as|ψ⟩ = S |0, ⟩ |1/2, ⟩ + P |1, ⟩ |1/2, ⟩ .

(2.3)

The indices in kets label |l, m⟩ |S, Sz ⟩; the relative angular momentum between p and π, and the composite
spin. Assume Λ spin points to z-axis initially where J = 1/2, M = 1/2. According to the Clebesh-Gordan
table, (2.3) is speciﬁed to be"r|ψ⟩ = S |0, 0⟩ |1/2, 1/2⟩ + P

2|1, 0⟩ |1/2, 1/2⟩ −
3

r

#
1
|1, 1⟩ |1/2, −1/2⟩ .
3

(2.4)

Note that the latter two terms of P-wave mode can be characterized by a common amplitude P . This is ofrepresentative that the physics is independent of the spatial rotation (Wigner-Eckart’s theorem).
The amplitude that a proton is found to the direction (θ∗ , ϕ∗ ) with spin state |S = 1/2, Sz ± 1/2⟩ is thenr
1 1
⟨1/2, 1/2| ⟨θ∗ , ϕ|ψ⟩ = SY00 − P
Y
(2.5)
3 0
r
2 1
∗
Y ,
⟨1/2, −1/2| ⟨θ , ϕ|ψ⟩ = P
(2.6)
3 1
using the spheric harmonics function Ylm = ⟨θ∗ , ϕ∗ |l, m⟩. The angular distribution for the proton is therefore
derived asdΓΛ∝ | ⟨1/2, 1/2| ⟨θ, ϕ|ψ⟩ |2 + | ⟨1/2, −1/2| ⟨θ∗ , ϕ|ψ⟩ |2
dΩp
r
r
r
r
2
2
∗
1
3
2
3
1
∗
P
cos θ +
Psin θ∗ eiϕ= S√ +
3
4π
3
8π
4π
1
|S|2 + |P |2 − 2Re(SP ∗ ) cos θ∗
4π−2Re(SP ∗ )∝1+cos θ∗ .
|S|2 + |P |2

=

(2.7)

Note that the spin sum is taken after squaring. This is because the proton is supposed to experience
decoherence through the interaction with matter in the detector, in other word, the interference with theother spin state |1/2, ±1/2⟩ dissolves.
The asymmetry cos θ arises from the parity violating S-wave, more strictly, the interference between the
S-wave and the P-wave, and it becomes maximized when the contribution from the S-wave and the P-wave
are equal (P/S| = 1). The distribution for π − is similarly asdΓdΩπ2Re(SP ∗ )cos θ∗ .
|S|2 + |P |2=: 1 + α cos θ.
∝

1+

(2.8)
(2.9)

where θ is the helicity angle; the angle between the π − direction and the Λ spin polarization vector in the2.2. WEAK DECAY AS POLARIZATION ANALYZER

35

Λ rest frame.
Since this argument only exploit general relation of parity and orbital angular momentum, it holds for theother decays such as τ → πν as well.

2.2.2

Parity violation and ability as a polarizer - a closer look

To understand how the parameters S, P (and α) are related to physics, the method of S-matrix of QFT
provides a good insight. Using the eﬀective Lagrangian approach in the lowest level, the matrix element forΛ → pπ − is built as
M ∝ up (cV + cA γ 5 )uΛ .

(2.10)

up , uΛ are the 4-spinors of the proton and the Λ respectively. Coeﬃcients cV , cA represent the fractionof Yukawa and pseudo-Yukawa coupling contributing the process, which here we call V-type and A-type
interaction for the sake of convenience here, analogously to the vector / axial current interaction in theelectro-weak theory. As π is pseudo-scalar, if parity is conserved during the decay, it has to couple only tothe pseudo-Yukawa component (A) of the vertex (2.10), and of course this is not the case again since it is
weak.
Moving to the components of matrix for Λ → pπ. Supposing that the Λ is at rest, the corresponding
4-spinor is√uiΛ = mΛξξ(i = ±).

(2.11)

Here mΛ is the mass of Λ, ξ is 2-spinor, which describes the spin state c1 |↑⟩ + c2 |↓⟩ in the two component
expression T (c1 , c2 ). Taking the QA and z-axis to the direction of the Λ spin, one ﬁnds ξ =T (1, 0).
The 4-spinor for a proton with the momentum p is a bit more complicated: √p · σ ξpj√
ujp (p) =p · σ̄ ξpj
1(E + m − p · σ) ξpj=p(E + m + p · σ) ξpj
2(E + m)
1
(E + m ∓ p) ξpj=p(E + m ± p) ξpj
2(E + m)r

E+m
(1 ± η) ξpj=:(η := p/(E + m))−(1 ∓ η) ξpj
2(2.12)

From the top line to the second line, one usesp√
E+m−p·σ√p · σ := E − p · σ =E+mp√
E+m+p·σ√p · σ̄ := E + p · σ =E+mwith p, E and m being respectively the energy, momentum and mass of the proton. j labels the helicity,which is the spin component with respective to the particle’s direction of motion p̂ := p/|p|, and a helicity
eigen-spinor u±p (p) is deﬁned so as to satisfy the eigen-function of helicity operator


σ
Σ :=
σ
thus,
±
(p̂ · Σ) u±
p (p) = ±up (p).

(2.13)

36

CHAPTER 2. DESIGN OF THE EXPERIMENT

If p̂ is set to the QA, as (p̂ · σ) ξp± = ξp± , ξ is represented as


 
1
0
+
−
ξp =
ξp =
,
0 p
1 p
or if setting the QA to the z-axis,∗cos θ2ξp+ =∗∗eiϕ sin θ2
z

ξp− =



∗

−e−iϕ sin θ2∗cos θ2∗.

(2.14)zwith the polar coordinates of p̂ being (θ∗ , ϕ∗ ).
ū is given as ū := u† γ 0 . Here I promise that all gamma matrices are expressed in the Weyl representation
throughout the paper therefore


1
γ0 =
,
1


γi =

The vertex component cV + cA γ 5 is then

σ



−σ




(i = 1, 2, 3),

γ5 =



−1

1

.



cV − cA.

cV + cA

Substituting (2.11), (2.12), (2.15) into (2.10), one arrives at
cV − cAξ
M ∝ ((1 ± η) ξp†j , (1 ∓ η) ξp†j )cV + cA
ξ(2.15)(j = ±)= [(1 ± η)(cV − cA ) + (1 ∓ η)(cV + cA )] ξp†j ξ= 2(cV ∓ ηcA ) ξp†j ξ.
Using ξ = (1, 0) and (2.14), it follows for respective j as(∗(cV − ηcA ) cos θ2M∝∗∗−(cV + ηcA ) sin θ2 eiϕ(j = +)(j = −)(2.16)

The angular distribution of the proton is obtained by squaring and summing over the spin,
X
dΓ
2
∝
|M|
dΩp
j=±
θ∗
θ
+ |cV + ηcA |2 sin222
= (|cV |2 + η 2 |cA |2 ) − 2η Re(cV c∗A ) cos θ∗∝ |cV − ηcA |2 cos2

The π distribution is given, in the polar coordinates representation (θ, ϕ)dΓ∝ 1 + α cos θdΩπ2η Re(cV c∗A )α=|cV |2 + η 2 |cA |2
Comparing (2.7) with (2.17), one has
S
1 cV
=
P
η cA

=

(2.17)

E + m cVp
cA

It is demonstrated that the mixing of the S-wave and P-wave, which is the source of the asymmetry, depends

2.2. WEAK DECAY AS POLARIZATION ANALYZER

37

on the ratio of the V and A coupling strength (cV /cA ) and a kinematical factor (1/η) with respective tothe outgoing particles. In the ultra-relativistic limit (η → 1), |S/P | = 1 or the maximized asymmetry α2 isachieved when |cV /cA | = 1, referred to the V ± A type interaction. Also in this limit, the extent of parity
violation and the phenomenological distribution asymmetry have a direct correspondence. It does not true
however, when the kinematical factor 1/η falls apart from 1. In this case, the point giving |S/P | = 1 deviatesfrom |cV /cA | = 1, reﬂecting the emerging disagreement between chirality and helicity.
In fact, in Λ → pπ − , the ratio of coupling strength is found to be far away from 1; cV /cA = −0.147±0.004
or cV /cA = −0.0194±0.0005 providing the measured value of α, (Fig. 2.3). Since the ﬁnal state particles aresoft (1/η = 18.7), it can still have a relatively large α value in spite of a small mixing of V and A coupling.
On the other hand, τ → πν gives η = 1, S/P = 1 and ατ = 1, which are the straightforward consequenceof that the neutrino appearing in the ﬁnal state is extremely relativistic and that the interaction vertex is!

exactly V − A.

0

-0.5

-1
-0.2

-0.1

0

CV / CA
Figure 2.3: The relation of α and the ratio of coupling strength cV /cA in the Λ → pπ channel, calculated using(2.17). cV and cA correspond to the contribution from the parity consecrating and parity violating interaction
to the decay. The ratio cV /cA is two-valued, in any case cA is dominant, providing the measurement of
α = −0.642 ± 0.013.

2.2.3

Comparison with the SG type of spin measurement

Before moving to the detail formalism of Bell’s inequality, it worths overviewing the diﬀerence between thepolarimeter decays and the SG type measurement in the statistical and dynamical property in term of spinmeasurement.
The most obvious diﬀerence at a glance might be that the measurement outcome from the SG interment
is digital valued (±), while the polarimeter decay returns a consecutive outcome (the pion direction). It is
however illustrated as below that as long as QM is concerned, these are equivalent as a measurement.
Suppose a process in which the spin of an isolated fermion, with the polarization vector s, is measured

38

CHAPTER 2. DESIGN OF THE EXPERIMENTby a certain device, say the SG instrument. In the case, taking QA along to direction of measurement (hereit is magnetic ﬁeld) n, the particle’s spin state is represented asθθ|ψ⟩ = cos |↑⟩n + eiϕ sin |↓⟩n .
2
2
The probabilities to have the respective outcome ± is accordinglyθP+ = | ⟨↑ |ψ⟩ |2 = cos22θP− = | ⟨↓ |ψ⟩ |2 = sin2 ,2
to sum up,
P± =

1±s·n
.
2

Then, what can we say on s, giving the measurement outcome ±? Naively thinking, if the outcome is +,the true spin s is most likely to point along n and vice versa. To be more sophisticated, a measure of the”likelihood” is introduced as
L=

1+s·n
,
2

(2.18)

which is intact the probability that the outcome ± and the estimation s are consistent. Here s = n leads tothe maximum value of L, and it falls down linearly as s points to an indiﬀerent direction. The estimation
power for the likelihood formalism is usually given by the variance of the function L, namely, the region
giving lnL = lnLmax − 1/2 from the maximum L point is equivalent to one standard deviation contour.
Likelihood L can be also deﬁned for the case of the polarimeter decay, as the probability of observing the
decaying pion to the direction n, when the Λ spin points to s. Using the result (2.8),
1+αs·nL=
4π(2.19)

It is therefore found that the likelihood functions are the same with that of the SG type measurement (2.18),
except a numerical factor α (as for the τ → πν channel ,ατ = 1 thus completely the same). Hereby, theirstatistical property, or knowledge from a measurement are equivalent between them, at least at the level offormalism. The phenomenon that a Λ with spin to s emits a pion to n, has then just the same structure
as an measurement on Λ spin by a SG instrument with the magnetic ﬁeld pointing n (−n) and with theoutcome of + (−). The pion direction plays an equivalent role as the magnetic ﬁeld of the SG instrument.
On the other hand, there is another important character of spin measurement phenomenology; the dynamics of projection. Recall that in the SG case, the spin state is always projected through the measurement,
parallel or anti-parallel with respect to the direction of ﬁeld n, and it is the ultimate cause of the non-local
eﬀect; the projection is performed even onto the state of the unmeasured spin. In order to test this non-local
eﬀect using the polarimeter decays, it has to contain the dynamical mechanism of the projection somehow
in itself as well.
As the spin of an already decayed particle is complicated to deﬁne, here a two particle state, for simplicity
a pair of ΛΛ in the singlet state, is in concern. It is then discussed how the hyperon decayed ﬁrst Λ1 has
impact on the spin of the opposite hyperon Λ2 space-likely apart from Λ1 , analogous to the state projectionin the normal active measurements.
First, as already shown many times, the spin state of Λ1 Λ2 before either of them decays is:|↑⟩ |↓⟩ − |↓⟩ |↑⟩
√
|ψ⟩ =
2
where the left and right components of sub-ket represent the space of Λ1 and Λ2 respectively. Through the2.2. WEAK DECAY AS POLARIZATION ANALYZER

39

prompt decay Λ1 → p1 π1 , the π1 direction is determined on n1 and the proton p1 helicity is on sp , as aconsequence, the state of the sub-system Λ1 is projected into |n1 , sp ⟩ as:|ψ⟩ → |ψ ′ ⟩ = ⟨n1 , sp |ψ⟩ |n1 , sp ⟩ .

(2.20)The state of the sub-system Λ2 is therefore|ψ ′ ⟩ = ⟨n1 , sp |ψ⟩ =⟨n1 , sp | ↑⟩n1 |↓⟩n1 − ⟨n1 , sp | ↓⟩n1 |↑⟩n1√2

Here the QA is taken to the n1 direction as noted by the subscripts, and |↑⟩n1 and |↑⟩n1 indicate that thepolarization vector are respectively along n1 and −n1 in the rest frame.
The amplitude ⟨n1 , sp | ↑⟩n1 and ⟨n1 , sp | ↓⟩n1 can be straightforwardly calculated from the matrix (2.16).
As |↑⟩n is the state where n and the Λ spin is parallel, it reads θ = 0, ϕ = 0, and thus θ∗ = π − θ = π,ϕ∗ = 0. |↓⟩n leads similarly to θ∗ = 0, ϕ∗ = 0. Therefore, they end up in⟨n1 , +| ↑⟩n1 = ⟨n1 , −| ↓⟩n1 = 0⟨n1 , −| ↑⟩n1 = −cV − cA η
⟨n1 , +| ↓⟩n1 = cV − cA η

η := p/(E + m).

It can actually be veriﬁed that the spin of Λ2 is projected to n or −n according to the decay of Λ1 . Whenthe helicity of the ﬁnal state proton p1 is + (sp = +),|ψ ′ ⟩ ∝ (cV − cA η) |↓⟩n1 ,and when sp = −,|ψ ′ ⟩ ∝ (cV + cA η) |↑⟩n1 .
Actually, as the proton helicity sp is not measured in colliders, the Λ2 spin can not be concluded to |↑⟩n1or |↓⟩n1 , with only information of n1 , thus the collected sample contains the both case. Note however that,
this is no more than a statistical mixture, but never a coherent superposition. There is no ambiguity at
event-by-event level, in this sense, the Λ2 spin is forced to point to n1 by the decay.
This is a signiﬁcant result in that the polarimeter decay is in fact does have the dynamics of projection
and the pion direction does perfectly play the role the measurement axis that the SG magnetic ﬁeld does.
The correspondence is not only in form, but is equivalent as long as QM is concerned. However in general,
there is a diﬀerence in if the observer can control the direction of the measurement axis at one’s disposal.
For example, in the SG measurement it is easy: the observer has the right to change the B-ﬁeld direction,
while it is never ever possible in the polarimeter decay because the direction of decay products are chosenrandomly during the ”decoherence” process as they interact macroscopically with the environment. The
signiﬁcance of this diﬀerence in testing LHVTs will be argued in section 2.3.2.
As for the case with τ τ pair, and the result is even more powerful. Since there are only left-handed
neutrino (right-handed anti-neutrino) in the ﬁnal state, the spin state of τ2 is singularly determined as|ψ ′ ⟩ ∝ (cV + cA η) |↑⟩n1 .
if the ﬁrst tau decays into π − ν, and|ψ ′ ⟩ ∝ (cV − cA η) |↓⟩n1 ,if the ﬁrst tau decays into π + ν̄, which can be easily distinguished by identifying the charge of the pion.

40

CHAPTER 2. DESIGN OF THE EXPERIMENT

2.3

Phenomenology in terms of Testing LRTs - Possible Loopholes

According to the discussion so far, it is successfully demonstrated that in QM the polarization decays havethe same properties as the direct measurements such as the SG spin measurement or the photon polarization
measurement by polarizers. The author is now highly motivated to use these to test LHVTs as soon as
possible. However, remember that it is not allowed to test LHVTs with the knowledge of QM in general.
Here we want to prove that the formalism of LHVTs contradicts to reality, thus discussion should be conﬁned
within the context of LHVTs.
In this section, the interpretation of the polarimeter decays in LHVTs and if there is possible diﬀerence
from the conventional spin measurements are analyzed. If so, we try to manage by applying additional
assumption and specify the loopholes according to it.

2.3.1

Interpretation of the polarizer decay in LHVTs

The ﬁrst suspicion is if the relation in the polarimeter decays is nevertheless true in LHVTs, that the parent
spin and the daughter direction are correlated as:
P (n|s) = 1 + α n · s(2.21)

P (n′ |s′ ) = 1 − α n′ · s′ .

(2.22)

P (n|s) is the conditional probability density of the daughter particle is found to n when the parent spin iss. These are indispensable otherwise there will be no way to get access to the spin of the parent particle.
Since it is totally unclear how general LHVTs describe the weak decays, here the possibility to interpret (??)
and (2.22) as experimental fact is sought.
As for the experiments measuring (2.21) and (2.22) for the Λ → pπ channel, J. W. Cronin (1963) [68]or O. E. Overseth(1967)[69] are well-known where the αΛ value is obtained as well. In either of those, apolarized ensemble of Λ was prepared, and the decay distribution of Λ → pπ was measured. Then, howis the Λ polarization guaranteed? It was not conﬁrmed by direct observation in fact, instead was derived
indirectly using QM and some conservation lows as below.
Firstly, the hyperon Λs are produced through the strong interaction reaction π − p → ΛK caused byπ beam injection to the carbon target. On the target, magnetic ﬁeld is applied so that the protons inside
become magnetized (polarization of each proton is not identical but prefers the B-ﬁeld direction on average.).
Providing that the strong interaction conserves parity, and that π and K are both pseudo-scalar with the
intrinsic parity of − while proton and Λ has +, it is easily derived that the relative orbital angular momentum
between K and Λ is zero. Together with the total angular momentum conservation law, it is then shownthat the Λ spin is inherited directly from that of proton which is magnetized. Thus, the Λ polarization canbe assured, in a statistical sense.
Can it be reproduced without the concepts of QM but with only fairly general notions? Check the eachstep of the argument. The magnetization of protons in the target seems to be proper as the magnetization

2.3. PHENOMENOLOGY IN TERMS OF TESTING LRTS - POSSIBLE LOOPHOLES

41

of matter under magnetic ﬁeld can at least be available as empirical fact. Next, the parity conservation
through the Λ production π − p → ΛK. It is still reasonable to assume in LHVTs, but less trivial than theformer one since it is not obvious how to understand the parity of particle generally in LHVTs. However,the spin conservation can be derived directly instead, if assuming the total angular momentum conserves
during the process and that the lowest angular momentum component dominantly contributes to the low
energy scattering π − p → ΛK (the S-wave approximation), therefore the polarization of Λ in the experimentcan be justiﬁed.
Since here only the conservation laws and the S-wave approximation in low energy scatterings are exploited, which are possible to empirically veriﬁed and hence universal in LHVTs, it seems still legal to assert
the relations (2.21) and (2.22) in the context of LHVTs. This justiﬁcation is extremely important in thatthe Bell’s inequality used in the experiment highly relies on (2.21) and (2.22), which is shown in chapter 3.
This sort of demonstration is needed for every channel and every experimental setup, however that it isactually not sure if it succeeds in all channels and setup. For example, the case with the τ channels mightneed some more assumption as illustrated in Able (1992) [64], but anyway empirical, which is by far generalthan requiring conclusions from QM.

2.3.2

Passiveness of particle decay and the free will loopholein section 2.2.3, it is proved that the daughter particle direction n in the polarimeter decay possesses thecomparable function that the spin measurement axis in the EPR gedankenexperiment provides. However,recall the free will loophole arises (see section 1.3.2) if the axes are ﬁxed during the test. Unfortunately
this is exactly the case where n can not be chosen by the observer at will. This is so called ”passiveness of
decay” which is the problem almost in every particle experiment using scattering or decays, including ones
introduced in the previous chapter.
The even worse thing is that, not only the Bell’s inequality can be violated by some LHVTs according to
the free will loophole, it is demonstrated by M. Abel et al. (1992) [64], H. Dreiner (1992) [65], A. Afriat andF. Sellen (1999) [70] that an explicit model can be constructed which perfectly accords to the prediction of
QM, due to the fact that particle direction n is not an uncommuting observable so that the three components
can be determined simultaneously. This is the major conceptual complication of this experiment in testing
LHVTs, which eventually discourages the attempt to analyze Z → τ τ → πνπν at LEP.
This free will loophole is impossible to close though, it does not have to abandon the whole idea. An
assumption with respect to locality is additionally made, so that the LHVTs that fulﬁll the condition can betested. There is also the mission of testing quantum non-locality which is free from the loophole arguments.
As a sign of gratitude for being allowed to exploit another assumption, in turn, we exhibit our best faithby using only the event sample where the two decays of ΛΛ or τ τ are separated space-likely, which strongly
justify the use of it. This issue is discussed in detail during chapter 3.

2.3.3

Eﬃciency loophole in the experiment

Aside from the conceptual problem of the free will, there are also technical complications that let anothersorts of loopholes, for example the eﬃciency loophole or the locality loopholes.

42

CHAPTER 2. DESIGN OF THE EXPERIMENT

As for the eﬃciency loophole, the situation is quite diﬀerent from the cases of photon experiments in thatthe detection eﬃciency is usually not the problem in colliders. Apart from some exception such as neutral
hadrons, neutrinos, and very soft particles, it is almost 100% since collider has thick enough material tointeract with the particle. The bottleneck is the event selection as mention at the begging of the chapter
2.1.1. In this experiment, in particular, a relatively strict selection is required since the BG contamination
does has some bad eﬀect on sensitivity (to be shown in section 5.2). In addition, here only the space-likeevents are in use, as a result, a part of, or possibly vast majority of the signal events will be lost. The
collection eﬃciency of signal events are typically 40% for the Λ series channels where the selection is rather
easy, and 15% ∼ 20% for the τ series channels with less clues due to the two neutrinos appearing in the ﬁnal
state.
These are far below the decent minimum eﬃciency in order to violate the Bell type inequalities (forexample, ϵmin = 0.67 (the Eberhard inequality) [27]), without applying the homogenous assumption. This00

is common to the other experiments in colliders such as one using K 0 K , B 0 B (1.3.4).

2.3.4

Space-like separation of decays and the locality loophole

Locality loophole is also the weak point generally in collider experiments since observers can not control thetiming of decay, as well as orientation of decay products. However, to our good luck, this can be overcomein this case.
Actually we do have some fraction of the unlucky events where the two decays are time-like, and of course
it is impossible to artiﬁcially transform them into the space-like events. The fraction is estimated as follows
for the Λ series channels cc̄(ηc , χc0 , J/ψ) → ΛΛ.
Suppose a pair of ΛΛ emerged at t = 0 running badk-to-back at the same speed βΛ c in the cc̄ rest frame(c is the light speed.). If the hyperon Λ1 and Λ2 decay respectively at t = t1 , t2 (t1 < t2 ) and with the
decay length x1 , x2 (see Fig. 2.4)), the condition that they decay in a space-like conﬁguration in space-timeis that, Λ2 decays before the information of Λ1 decay catches up:(t2 − t1 ) c < x1 + x2
= βΛ c(t1 + t2 )(2.23)

1 + βΛ
t1
1 − βΛ

(2.24)thus,
t2 <Assuming the exponential decay distribution with the lifetime being τΛ , the fraction of the space-like events
ω1 is

Z ∞
ω1 =

1+βΛ
Z 1−β
t1
Λ

dt1
0

βΛ
=
.
2

t1





1
t1
1
t2
dt2
exp −
exp −
τΛ
τΛ
τΛ
τΛ
(2.25)

2.3. PHENOMENOLOGY IN TERMS OF TESTING LRTS - POSSIBLE LOOPHOLES

43

Note that it is equally likely that Λ2 decay before Λ1 does (t1 > t2 ), in total, ω1 is doubled
ω = 2ω1 = βΛ .

(2.26)

Note that result will be identical in any reference of frame, due to Lorentz invariance of theory. The spacelike fraction ω turns to be βΛ itself, independent of lifetime τΛ . βΛ , ω etc. for respective charmonium decays
are listed as Tab.2.2. The space-like fraction is just 65% ∼ 75%, however, those sample can be selectively
extracted, by reconstructing the decay point of the hyperons (secondary vertices) with the help of the 4charged tracks made in the tracker as described in Fig. 2.4. Although the ΛΛ are electrically neutral,leaving no charged tracks, the vertices can be respectively found by extrapolating the tracks of pπ − and p̄π +(for the detail on tracker, see the following section). Accordingly, the space-like condition (2.23) reduces intoa relation of the decay lengths x1 , x2 :x1 + x2 >1|x2 − x1 |.
βΛ

(2.27)

Thanks to the fact that Λ and Λ is long-lived (τΛ = 0.26 ns), the average length reaches 7 − 9cm (see Tab.2.2)
so that the reconstruction accuracy is not problematic.
Table 2.2: Mψ is the cc̄ mass, βΛ expresses the Λ velocity in the cc̄ rest frame.
ChannelMψ (GeV/c2 )βΛAverage decay length (cm)ηc → ΛΛ
2.981
0.663
6.91
χc0 → ΛΛ
3.415
0.757
9.04
J/ψ → ΛΛ
3.096
0.693
7.50

Figure 2.4: A schematic of the process cc̄ → ΛΛ → pπ p̄π. x1,2 is decay length of Λ1,2 = (Λ, Λ). The bluelines shown the charged tracks created by passages of charged particles in the tracker. Although the ΛΛ areelectrically neutral, leaving no charged tracks, the vertices can be respectively found by extrapolating the
tracks of pπ − and p̄π + .
On the other hand, the τ series channels have neutrinos in the ﬁnal state, instead of p and p̄ in the Λseries. As these neutrinos do not interact with matters or gases leaving no signature in the detector, thespace-like selection algorithm does not work any more. However, thank to that the taus decaying from Z or
the Higgs are extremely fast (βτ > 0.999), almost all events are space-like. The direction production channele+ e− → γ ∗ → τ τ needs additional consideration. The space-like fraction ω = βτ depends on the center of44

CHAPTER 2. DESIGN OF THE EXPERIMENT√
mass energy s as:βτ = pτ /Eτ
s
=

1−
r

=
Assuming the run with

√

1−

m2τ
Eτ2
4m2τ
.
s

(2.28)

s = 10.58GeV in the Belle experiment, ω = βτ = 0.94. The contamination from thetime-like events are 6% which is actually not negligible. The impact on sensitivity is quantitative evaluatedin section 5.3.

2.3.5

Summary of expected loopholes and comparison with other experiments

The loophole issues discussed above is summarized in the Tab. 2.3, as well as the comparison with the otherrelevant experiments. Here × indicates the presence of loophole.
As mentioned, the photon pair experiments nowadays come to close all the three, from which non-opticalexperiments still suﬀer much, in particular the locality loophole. However, the experiment proposed in thepaper is suppose to close it. The advantage is striking, compared with the other collier experiments such as00

the K 0 K and B 0 B in terms of loopholes.

Table 2.3: The upper stacks are optical, the lower are non-optical experiments. The raw of Chen(2013) isfor the test proposed in the paper. × means that the respective loophole arises.
Experiments
Exp. type Eﬃciency Locality Free-will
Aspect et al. (1982) [22]photon××
Weihs et al. (1998) [30]photon×
Guistina et al. (2009)
photon××
Rowe et al. (2001) [29]ions××
Sakai et al. (2006) [35]proton××
0 0
CPLEAR (1999) [36]
K K
×
×
×
0
Belle (2007) [37][38]
B0B
×
×
×
+ −
Chen (2013)ΛΛ, τ τ××

2.4

Description of the measurement and the overview of detector

In this experiment, the measured quantities are n and n′ , the direction of the π − π + from the decay ΛΛ →pπ − p̄π + or τ τ → π − νπ + ν̄ deﬁned in the rest frame of their parent particles such as ΛΛ or τ τ .
To obtain n and n′ experimentally therefore, the measurement on π + π − direction in the laboratory frame
is not suﬃcient but these the need additional ”boost back” with respect to the movement of the parent
particle, which means the 3 momenta of ΛΛ or τ τ has to be reconstructed. The case of the Λ series channels
are easy; measuring the momenta of the recoiled p and p̄ is satisfactory. As for the τ series channels, it needs
additional technique (kinematical ﬁt) explained in the following.

2.4. DESCRIPTION OF THE MEASUREMENT AND THE OVERVIEW OF DETECTOR

45

Fortunately, the detector system in colliders is highly suitable for, or perhaps even specialized in, the detailmeasurement of particle momenta and direction. Before the detail illustration of reconstruction algorithmof Λ and τ , here an brief overview of a standard detector complex installed in colliders is given.

A typically setup is as follows: the detector is settled around the collision point of the particle beams(Interaction Point: IP) which has a cylindrical shape covering over the beam pipe (see Fig. 2.5). It hasa layered structure with detectors serving diﬀerent functions. Mostly, there is a vertex detector measuring
the trajectories and decay positions of particles stalled in the innermost layer, to the outside, a tracker
to measure particles momenta; some auxiliary detectors for particle identiﬁcation such as the time of ﬂight(TOF) or Čerenkov counters; a solenoid magnet providing strong B-ﬁeld to the inner detectors; a calorimeterfor particle energy measurement and a muon detector in the outermost layer.

Figure 2.5: The overview of a typical detector in collider experiments. The ILD detector for the future
ILC (International Linear Collier) is shown as an example [71]. The left presents the whole module. The
beam pipe penetrates the center of the cylinder. The component of each detector is displayed in the right.
”ECAL” and ”HCAL” are respectively the electro-magnetic and hadronic calorimeter. For the tracker, TPC(time-projection chamber) is design in ILD.

As a matter of course the tracker is the key part here. The tracker is literally a detector measuring
tracks, created by passage of charged particles. The most of the trackers are subject to a category called
drift chambers. It is ﬁlled with gases such as argon or isobutane etc. with electric and magnetic ﬁeld appliedin. When a charged particle crosses through the chamber, it collides and ionizes the gas atoms and thescattered-away electron start to drift along the electric ﬁeld. They are eventually absorbed by the readoutcausing an avalanche which is converted into electric signals. For readout, there are typically potential wires
put throughout the chamber or silicon PIN diode pads, in which a voltage is applied so as to trigger theavalanche. The signal from a deﬁnite wire gives the information of position where the drifting election isabsorbed. Together with information of ﬁeld and the drift time which is measured as the time duration after
the trigger of the prompt charged particle passage, the position of ionizations can be reconstructed.

46

CHAPTER 2. DESIGN OF THE EXPERIMENT

B

・

e-

p

electrode wires+
E
Figure 2.6: Schematic of momentum measurement in a drift chamber. Magnetic ﬁeld B is applied in thechamber bending the trajectory of an incident charged particle. The charged particle ionizes the atoms on its
path (grey dots), of which electrons are driven into electrodes by electric ﬁeld applied. Using the two or three
dimensional position information of the drift electrons and their drifting time, a track of the charged particle
passage (grey dotted line) can be reconstructed. The perpendicular component of the momentum to the
B-ﬁeld PT is obtained by the curvature radius of the track ρ with 2.29, and the whole three components (blueline) are derived by PT and the extrapolation of the track to its vertex (cross mark), usually approximatedby IP.
Then, how the momentum of the charged particle is measured with the track? It is straightforwardly
performed by bending the trajectory with a strong magnetic ﬁeld B applied in the chamber (usually thesolenoid provided an uniform ﬁeld to the direction of the beam axis) so that the curvature of the track cantell the momentum component perpendicular to the B-ﬁeld (PT ), through the relation
PT = eBρ(2.29)

where ρ is the radius curvature of the bent track.
The determination of direction of motion n is a bit more diﬃcult since the direction before being bent
is in interest (the blue arrow in Fig. 2.6). Thus a ﬁtting and an extrapolation to the IP are needed, withthe assist from the information of vertex detector around the IP. Note that sometimes the track does notstart from IP but from a secondary vertex which is away from IP, just as the pions from Λ or τ as interested
here. When it is high momentum, which is the case for the pions from the τ decay, the approximating thesecondary vertex with IP or the impact parameter (see Fig. 2.7) is quite suﬃcient since high momentum
tracks are hardly bent. Low momentum tracks need accurate speciﬁcation of the secondary vertices because
the tracks are heavily bent, which the Λ series channels can be implement as described above.
The absolute value of the momentum P is then derived from PT the direction of motion n as
PT
P =
sin θ

(2.30)

where θ is the angle between n and B. Up to today, it has achieved a considerably high resolution, typicallyfor momentum dp/p = 0.5% − 1% with charged particles with 1GeV − 10GeV, for direction of motionδθ, δϕ ∼ 1mrad, which is by far better than the required resolution as conﬁrmed in section 5.4.
Next, the reconstruction of 3 momenta of the decayed parent particles ΛΛ and τ τ is discussed. The case

2.4. DESCRIPTION OF THE MEASUREMENT AND THE OVERVIEW OF DETECTOR

47

of ΛΛ is simple. It is just the sum of the momenta obtained from the 4 visible tracks pp , pπ− , pp̄ and pπ+ :pΛ = pp + pπ−pΛ = pp̄ + pπ+(2.31)For the τ τ case, to compensate the absence of neutrino tracks, the information of missing energy momentum (missing 4-momentum pmisµ ) is needed. The missing 4-momentum is deﬁned as the energy momentum
imbalance between the initial and the ﬁnal state, from which the sum of the neutrino contribution can beacquired:pµmis = pµν + pµν̄(2.32)

if the energy and momentum of the collision have to be well-deﬁned such as e+ e− colliders.
As for Z, γ ∗ , H → τ τ → πνπν, the mass shell conditions for τ τ can also be usedm2τ = (p− + pν )µ (p− + pν )µm2τ = (p+ + pν̄ )µ (p+ + pν̄ )µ .

(2.33)

Considering now the degree of freedom of unknown values is 6 (3-momenta of two neutrinos), there are infact already suﬃcient number of conditions (6 conditions) by (2.32) and (2.33), thus it can be analyticallysolved ideally, however this will be severely suﬀered from the measurement errors.
In addition to it, there are still other information can be exploited, for example, the decay planes of τ − → π − νand τ + → π + ν̄, by specifying the planes in which the pion tracks and their impact parameter vectors arecontained in (see Fig. 2.7), which earns 2 more constraints.
These superﬂuous conditions are important in practice that it allows to perform a kinematical ﬁt rather
than an analytical computation with the necessary-suﬃcient number of conditions. By relaxing the exact
relations (2.32) and (2.33) the error propagations are alleviated, that enable a stable reconstruction of τ τ
3-momenta.

pπ

+

π
impact parameter

pτ

+

τ

IP

Figure 2.7: Decay plane of τ → πν and the impact parameter, a plane spanned by the momentum vectors of
π (pink dotted line) (or τ , ν) and IP. In experiments, the extrapolated π track (pink solid line) is acquired
instead of the momentum vector. Usually, the decay plane is then approximated by a plane spanned by”impact parameter” (a vector from IP to the nearest point of the track) instead.
So far up to now, our conclusion is that we do not have to worry about the performance of the detector
system in collider experiments, instead however, the last concern is the eﬀect of strong magnetic ﬁeld tothe Λ or τ spins state. It is true that the state itself will evolve as precession, however the coherence is not

48

CHAPTER 2. DESIGN OF THE EXPERIMENT

disturbed since in general the ﬁeld inside the tracker is usually designed to be uniform, where the trajectoryis not bent through the magnetic ﬁeld-magnetic moment interaction as seen in the SG measurement.
To conclude this section, the measurability and accuracy of n and n′ are fully adequate for either the Λand τ series, as long as an experiment in a e+ e− collider is considered.

Chapter 3

Formulation of Bell Inequalities for
Our System
3.1

Formulation

In this chapter, the Bell’s inequality (BI) is derived, specialized for our own experiment. Apart from thetests reviewed in chapter 1 where discontinuous dichotomic variables are observed, here the continuous π + π −direction n, n′ are in consideration. It is possible to artiﬁcially create a dichotomic variable from continuous
observables as seen in the proton pair experiment (section 1.3.3), however here a straightforward way is taken
to formulate an inequality with the n and n′ are directly embedded in, that envisages the clear propagationfrom the spin correlation of ΛΛ or τ τ to the pions direction. For the convenience sake, the formulation
entirely proceeds in the example of cc̄ → ΛΛ → pπ − p̄π + but it is all true for the τ series channels as well,Firstly, the relation that the Λ Λ polarization vector s, s′ are supposed to fulﬁll is discussed. Deﬁning
si and s′j as the projections to arbitrary unit vector (virtual axes) a, b, c, d as
sa := s · a

sc := s · c

s′b := s′ · bs′d := s′ · d,

(3.1)

the fact −1 ≤ sa , s′b , sc , s′d ≤ 1 and (1.20) lead to a CHSH type algebraic inequality| ⟨ sa s′b ⟩ + ⟨ sa s′d ⟩ + ⟨ sc s′b ⟩ − ⟨ sc s′d ⟩ | ≤ 2.

(3.2)

Note that so far the reality of s and s′ is not assumed, in other words, s and s′ do not have to be
deterministically valued as vectors in which all the three components are uniquely determined. Here s ands′ are even allowed to be the polarization vectors in the QM, of which values do not necessarily correspond tothe measurement outcome. The requirement is only that there is a three-dimensioned vector s characterizing
the orientation spin of a spin one half and it can be assigned to any single particle. As far as this requirement
is satisﬁed, one can immediately arrives at (3.2) as it is merely an algebraic relation.
In the language of LHVT, these s and s′ are actually ones of hidden variables with respect to themeasurement outcome, n and n′ , which is to say,n = n(s, s′ , λ),n′ = n′ (s, s′ , λ).

(3.3)

λ represents the other hidden variables, some of which can contained by n and n′ in common. (3.2) is in49

50

CHAPTER 3. FORMULATION OF BELL INEQUALITIES FOR OUR SYSTEM

eﬀect the restriction imposed on the hidden variables of n and n′ .
Next, as discussed in the section 2.3.1, the crucial relation (??) is introduced as an experimental factthat for an isolated system of a Λ with polarization vector s, the π emerging from the subsequent decay
Λ → pπ prefers the orientation as
P (n|s) = 1 + αΛ n · s,

(3.4)

where n is the π direction (Assumption A). P (n|s) is the probability density function normalized withZdΩn
P (n|s) = 1.
(3.5)
4π
For the Λ → p̄π + , it reads similarly
P (n′ |s′ ) = 1 − αΛ n′ · s′ .

(3.6)

αΛ is determined as αΛ = −0.642 ± 0.013 through experiment.
Using the relations (2.21) and (2.22), the correlation amplitude ⟨na n′b ⟩ is written asZdΩn dΩn′′⟨(n · a)(n · b)⟩ =(n · a)(n′ · b) P (n, n′ )4π 4πZdΩn dΩn′ dΩs dΩs′=(n · a)(n′ · b) P (n, n′ |s, s′ ) P (s, s′ ).
4π 4π 4π 4π(3.7)

with P (x) notes the probability density function (PDF) with respective to the variable x. Here it is thenshown that the inequality with respect to s and s′ (3.3) obtained above can be translated into one written
in the language of n and n′ , by introducing the second assumption that if the two decays of Λ and Λ are ina space-like conﬁguration in space-time, the pions orientations n and n′ depends only on the spin of theirrespective parents hyperons, namely,n = n(s, λ)n′ = n′ (s′ , λ)(3.8)(Assumption B). This is a comparable condition to the locality principle. Hereby, the probabilistic behaviorof n and n′ are assured to be independent, and the factorization of the joint-PDF can be performed as
P (n|s)P (n′ |s′ ) = P (n, n′ |s, s′ )(3.9)

Note that this assumption B is fairly reasonable in terms of LHVT, however, here the author would like tolay emphasis again on that there could be the LHVT models that elude the constrains due to the passivenessof decays, as pointed out in section 2.3.2 (the ”free will” loophole). Those would be such models that thedirection of π + π − is determined beforehand, before the interaction between Λ and Λ is switched oﬀ, thus thepion can in principle have the dependence on the opposite hyperon spin, in which (3.8) is not true anymore.
Now forget the loophole for a while. For the LHVTs which follows the relation 3.8, with the help of (3.9),(3.7) becomes⟨(n · a)(n′ · b)⟩ZdΩn dΩn′ dΩs dΩs′(n · a)(n′ · b) P (n|s) P (n′ |s′ )P (s, s′ )=4π 4π 4π 4πPlugging (2.21) and (2.22) in,ZdΩn dΩn′ dΩs dΩs′=(n · a)(n′ · b)(1 + αΛ n · s)(1 − αΛ n′ · s′ )P (s, s′ ).
4π 4π 4π 4π(3.10)

3.1. FORMULATION

51

Representing n and n′ with the polar coordinates (θ, ϕ) and (θ′ , ϕ′ ) with s and s′ being the zeniths respectively, and with integrating over dΩn dΩn′ , it leads to⟨(n · a)(n′ · b)⟩Z
d cos θdϕ d cos θ′ dϕ′ dΩs dΩs′=
(a1 sin θ cos ϕ + a2 sin θ sin ϕ + a3 cos θ)
4π
4π
4π 4π
× (b1 sin θ′ cos ϕ′ + b2 sin θ′ sin ϕ′ + b3 cos θ′ )(1 + αΛ cos θ)(1 − αΛ cos θ′ )P (s, s′ )Here cos ϕ and sin ϕ vanish with ϕ integration. As a3 = a · s = sa , b3 = b · s′ = s′b ,⟨(n · a)(n′ · b)⟩!
Z
d cos θ d cos θ′=(1 + αΛ cos θ)(1 − αΛ cos θ′ ) cos θ cos θ′22!
ZdΩs dΩs′′′×
sa sb P (s, s )
4π 4π
=−

2
αΛ
⟨ sa s′b ⟩ .
9Hence, the π + π − direction correlation ⟨si sj ⟩ is successfully tied to the ΛΛ spin correlation ⟨si sj ⟩ with abeautiful proportional relation⟨ni nj ⟩ = −2αΛ⟨si sj ⟩9(i = a, c; j = b, d).

(3.11)

This formula is ﬁrst derived by J. P. Baranov (2008) [59] with arguing the correlation of measured spinvalues of ΛΛ: ⟨A(a)B(b)⟩, instead of ⟨si s′j ⟩. There are some suspicious points in the paper:• Measurements on the ΛΛ spins before they decay are assumed, which is never the case in reality.
• Spins are described to point to the measurement axis when they are measured, which is the notion of
QM.
Then the author and Y. Nakaguchi provided the new version of the proof [72] as above.
With (3.11), the inequality for the ΛΛ spins (3.2) is transformed into
Q := | ⟨ na n′b ⟩ + ⟨ na n′d ⟩ + ⟨ nc n′b ⟩ − ⟨ nc n′d ⟩ | ≤

2
2αΛ
9

(3.12)Q is deﬁned as the LHS of the inequality. This is the BI that is used in the experiment. In contrast to 3.2, itis written in the language of experimentally observable n and n′ . The classical limit is 2α2 /9 (CL), while in
QM, due to the immediate non-local spin ﬂip as described in 2.2.3, the correlation is enhanced such that the√
LHS of (3.12) can reach 2 2α2 /9 when the Λ and Λ spins are in singlet, which is referred as the quantum
limit (QL). This BI is extended to the other channels with reading αΛ to the respective α value. Several
comments are attached to this inequality.
• Thank to the fact that the hyperon spins s, s′ are not measured directly, the inequality can be derived
without requiring them to be realistic. Here the only what they are required to be is to satisfy 3.4
and 3.6 and the rest of the properties belonging to s and s′ are not assumed. Thus inequality (3.12)
could be more general than the ordinary BI formulation that requires the reality of spins as well as the
locality, which is to say, it can serve to test general HVTs rather than LHVTs that are tested with the52

CHAPTER 3. FORMULATION OF BELL INEQUALITIES FOR OUR SYSTEM
conventional BIs. However, as this generality problem is a complicated subject, here for simplicity, therange in interest is conﬁned in testing LHVTs, since it is plausible that (3.12) is at least valid for any
LHVTs that satisfy the assumption A and B.

• To realize the condition described in the assumption B (3.8) (the locality condition), the hyperons
decays should be space-like, otherwise it requires by far strong assumption to assure the factorization(3.9). It is illustrated in the previous chapter that it is experimentally possible to extract only the
space-like events as for the Λ series channels, and the contamination from the time-like events are
negligible as for the τ series channels except for the e+ e− → γ ∗ → τ τ . The time-like contamination infact requires the modiﬁcation on the RHS of (3.12), which is discussed in the later section 5.3.

• BG events generally do not follow the BI (3.12) since the assumption A (2.21) (2.22) are not valid for
those events. As a result, the restriction imposed on them becomes looser than (3.12). The reﬂectionof the BG events to the BI is argued in the section 5.2• One remarkable character of this BI is that, the relative magnitude of the violation does not depend√on the channel of polarimeter decay. The ratio of the QL to the CL is always 2, regardless of thevalue of α2 . This is because, as discussed more in detail in chapter 4, ⟨ni n′j ⟩ has the same dependency(proportional to α2 ) as the CL (2α2 /9). Hence when α2 decreases, ⟨ni n′j ⟩ and the CL decline by thesame extent. This fact implies that the BI (3.12) is essentially an inequality in terms of the hyperons
spins, which is just converted into one with respect to the poins correlation with (3.11).
Therefore, not only Λ → pπ and τ → πν but any other weak decays even with small α2 value canviolate the BI in principle. This is an important conclusion in that potentially there are more channels
available, which expands the opportunity to test LHVTs and the quantum non-locality in various
systems.

• Thus theoretically, the BI (3.12) can violate if only α2 is not zero. But it would be extremely diﬃcult√
to conﬁrm it in experiment since the absolute disparity between the QL and the CL 2( 2 − 1)α2 /9close to zero as α2 → 0. After all, large α2 is preferred.

• From view of experimental feasibility, a noticeable character in the test using BI (3.12) is that theoverall 4 terms can be measured at one run, in contrast to the ones measuring the coincidence rate
such as the photon pair experiments. The axes conﬁguration a, b, c, d can be chosen after the datataking, and the data can be applied to each of the setting. Thus, it is 4 times more economical instatistics than the coincident rate type experiments. Considering that the statistics is limited in thecollider experiments, this 4 times diﬀerence is huge.
Here a, b, c, d are the virtual measurement axes, as seen in the proton pair experiment (section 1.3.3),
which have no impact on the measure net outcome therefore play no physical role, in construct to themagnetic ﬁeld of the SG instrument. The maximum value or the dependence of Q on the axes a, b, c, d aredetermined uniquely if the data of n, n′ are taken. Thus it is expected to be more economical to ﬁx theaxes in such a conﬁguration that the Q is maximized, and eliminate them from the representation. In the3.2. AXES OPTIMIZATION USING THE METHOD OF C MATRIX

53

next section, it is demonstrated that this optimization can be performed totally analytically and the BI is
reduced into an extremely concise representation which allow us to recognize the BI is violated at a glance.

3.2

Axes optimization using the method of C matrix

This section is entirely about linear algebra. The theme is to ﬁnd the conﬁguration of unit vectors a, b, c,d that maximize a value
Q := | ⟨ na n′b ⟩ + ⟨ na n′d ⟩ + ⟨ nc n′b ⟩ − ⟨ nc n′d ⟩ |.
Before computing, let me re-write the correlation amplitude ⟨ na n′b ⟩ in terms of a 3 × 3 matrix Ĉ deﬁned
asĈij := ⟨ ni n′j ⟩where the indices i, j = 1, 2, 3 correspond to x, y, z in the Cartesian coordinates respectively. With thecorrelation matrix C, ⟨ na n′b ⟩ is reduced into the form of a bilinear⟨ na n′b ⟩ =

3 X
3
X

ai bj ⟨ ni n′j ⟩ = aT Ĉ b .

(3.13)

i=1 j=1

Accordingly, Q becomes
Q = | aT Ĉ (b + d) + cT Ĉ (b − d) |.

(3.14)

Here the physical part (Ĉ) and the rest (a, b, c, d) are well separated.
Next, ﬁnd the maximum Q. It is obvious that considering Q as Q = aT Ĉ(b + d) + cT Ĉ(b − d) instead
do not lose any generality. Also, as Q will be absolute-valued eventually, here we do not care if the obtained
value is the maximum or the minimum (A simple symmetry argument proves they are identical.).
The method of Lagrange multipliers (MLM) is utilized. The constraint conditions are:aT a = 1,bT b = 1cT c = 1,

dT d = 1.

(3.15)

With the respective multipliers ξa , ξb , ξc and ξd , one builds a scalar function L as:
L =−aT Ĉ (b + d) + cT Ĉ (b − d)1111ξa (aT a − 1) − ξb (bT b − 1) − ξc (cT c − 1) − ξd (dT d − 1).
22
2
2

Setting all the derivatives of L to zero,
∂L
=0
∂aT
∂L
=0
∂b
∂L
=0
∂cT
∂L
=0
∂d
∂L
=0
∂ξα

(3.16)(3.17)⇐⇒Ĉ(b + d) − ξa a = 0(3.18)⇐⇒(a + c)T Ĉ − ξb bT = 0(3.19)⇐⇒Ĉ(b − d) − ξc c = 0(3.20)⇐⇒(a − c)T Ĉ − ξd dT = 0(3.21)⇐⇒(3.15)(α = a, b, c, d).

54

CHAPTER 3. FORMULATION OF BELL INEQUALITIES FOR OUR SYSTEM

Multipling aT , cT (b, d) from the LHSs of (3.18)∼ (3.21) and using (3.15), the multipliers ξα are shown to beξa = aT Ĉ(b + d)

ξc = cT Ĉ(b − d)ξb = (a + c)T Ĉbξd = (a − c)T Ĉd.

(3.22)

Substituting these back into (3.19) and (3.21) givesb=d=Ĉ T (a + c)
(a + c)T ĈbĈ T (a − c)(a − c)T Ĉd=:Ĉ T (a + c)ρ=:Ĉ T (a − c).
σ(3.23)

b and d are parallel to Ĉ T (a + c) and Ĉ T (a − c)respectively. The normalization factor ρ and σ are taken
to keep |b| and d being 1:qρ = (a + c)Ĉ T Ĉ(a + c)σ=q(a − c)Ĉ T Ĉ(a − c)(3.24)

Here, a 3 × 3 symmetric matrix S := Ĉ T Ĉ = Ĉ Ĉ T is introduced. Eliminating b and d in (3.18) and (3.20),
with (3.22) and (3.23), one ﬁnds
(σ + ρ)Sa + (σ − ρ)Sc= µa

(σ − ρ)Sa + (σ + ρ)Sc= νc,

(3.25)whereµ:= (σ + ρ)aT Sa + (σ − ρ)aT Sc

ν:= (σ + ρ)cT Sa + (σ − ρ)cT Sc .

These end up in the eigen-equations with respect to a and c as−4ρσS 2 + (σ + ρ)(µ + ν)S a4ρσS 2 + (σ + ρ)(µ + ν)S c(3.26)= µνa= µνc.

(3.27)

Seemingly these have a terribly intricate structure, since ρ, σ, µ and ν all contain a and c in a very
complicated way. However, a simple attempt of applying the LHS of (3.27) as T := ±4ρσS 2 +(σ +ρ)(µ+ν)S
to the eigenvectors of S: vi reveals
±4ρσS 2 + (σ + ρ)(µ + ν)S vi = ±4ρσλ2i + (σ + ρ)(µ + ν)λi vi ,(3.28)

with λi being the eigenvalues of S and the relation Sv i = λi vi . This shows that, every eigenvector of S
is the eigenvector of R as well. Though vi are illustrated to be necessary, whether they provide suﬃcientsolutions for (3.27) needs a little more argument. First, when S has three independent eigenvectors, it isobviously the case that vi has 1 to 1 correspondence to the eigenvectors of T thus no other solutions exist.
When S possesses less than two independent eigenvectors is a bit problematic, however in the case, S is the
projection operator into the eigenspace RS . According to the nature of projector, S 2 and thus the linear
combination of S 2 and S are also shown to be the projector into RS , so is T . Therefore, the class of T is
proven to be at least no more than that of S in general, which guarantees that vi gives the full descriptionof solutions for (3.27).

3.2. AXES OPTIMIZATION USING THE METHOD OF C MATRIX55

As the norm of a and c are 1,
a = ±vic = ±vj|vi | = 1

(i, j = 1, 2, 3).

(3.29)

Plugging these into (3.23) and (3.24), one obtains b, d and the other coeﬃcients. Considering the case with(a, c) = (vi , vj ) ﬁrst,q(λi + λj )(1 + viT · vj )q
σ = (λi + λj )(1 − viT · vj )ρ=When (i) a = c (i = j),
ρ=2

p

λi

σ=0

T

Ĉ vi
b= √
λi

d = (arbitrary unit vector)p
Q = 2aT Ĉb = 2 λi .
When (ii) a ̸= c (i ̸= j),ρ=σ=(3.30)p
λi + λjĈ T (vi − vj )d= p
λi + λjĈ T (vi + vj )b= p
λi + λjp
Q = 2 λi + λj .

(3.31)

Here the fact is used that S = Ĉ T Ĉ is a symmetric matrix which has eigenvectors orthogonal to each other(viT · vj = δij ).
The other combinations (a, c) = (vi , −vj ), (−vi , vj ), (−vi , −vj ) lead to the same result. The maximum
Q is derived from (3.31), with letting λi and λj being the two largest eigenvalues of S.
Lastly, a typical concern with respect to MLM is reviewed that generally maxima (minima) derived from
MLM could be local (fake) maxima where the real maximum (minimum) value does not sit at extrema butat the boundary of parameter space. However it is not case here, since the parameter space of a, b, c, andd are consecutive ”spheres” that have no boundary.
To summarize,
Qmax = 2p

λ1 + λ2 ≤2α2.
9(3.32)

where λ1 and λ2 are the largest two eigenvalues of Ĉ T Ĉ. It is not pleasant to contain values with measurement
error in the upper limit, dividing with the LHS of (3.32),
9
Cij =⟨ ni n′j ⟩2α2p
Qmax = 2 λ1 + λ2

(i, j = 1, 2, 3)

Qmax ≤ 1.
This is the target expression. The classical limit is 1, while the QM reaches(3.33)√

2 at maximum.

Chapter 4

QM Analyses and the Violation of the
Bell Inequality
In this chapter, a series of QM calculation is carried out to illustrate the pattern of spin entanglement of ΛΛor τ τ in the candidate channels; χc0 , ηc , J/ψ → ΛΛ; Z, H → τ τ ; e+ e− → γ ∗ → τ τ , as well as to ﬁnd out
if they actually give the large Qmax values that violate the BI derived in the previous chapter (3.33). Since
QM is assumed to be the ”truth” here, these are therefore the expected Qmax values in experiment.
Speciﬁcally, for each channel, the distribution of π + π − in the ﬁnal state is calculated using the methodof S-matrix in QFT, from which a C matrix is build⟨nx n′x ⟩ ⟨nx n′y ⟩ ⟨nx n′z ⟩C = ⟨ny n′x ⟩ ⟨ny n′y ⟩ ⟨ny n′z ⟩⟨nz n′x ⟩ ⟨nz n′y ⟩ ⟨nz n′z ⟩√and the Qmax value is speciﬁed as Qmax = 2 λ1 + λ2 (λ1,2 is the largest two eigenvalues of C T C). The
channel giving Qmax larger than the classical limit 1 is entitled to be an possible experiment channel, which
will be scrutinized further in chapter 5.

The result for the scalar channels ηc , χc0 → ΛΛ and H → τ τ is trivial and boring that these apparently√
gives Qmax = 2, the quantum limit. This can be easily derived with some symmetry arguments, however
it is still pedagogical since the entanglement structure is clear and gives a good insight for the non-trivial
channels such as ones with vector initial state.
The vector channels are interesting since the relativistic eﬀect comes into play where the orbital angularmomentum intervenes into the spin state creating highly non-trivial pattern of entanglement. Experimentally,
these vector initial states have much more magnitude of statistics than the scalar channels. The result is
important in considering the experimental feasibility.

The calculation itself is nothing but nasty, however the structure and the implication arising during theprocess is amusing.

57

58

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY

4.1

χc0 → ΛΛ

Begin with the simplest case; the scalar initial state channel χc0 → ΛΛ → pπ p̄π. The total matrix element
M is the product of sub-matrices corresponding to each decay:
X
jM=
MiΛ Mij
χc0 MΛ

(4.1)

i,ji, j = ± denote the helicity of Λ and Λ respectively, of which sum are taken in the total matrix since they
are unobserved intermediate modes. The sub-matrices are written as below:i jMijχc0 ∝ ūΛ vΛ

(4.2)

MiΛ ∝ ūspp (1 + cΛ γ 5 )uiΛsjMjΛ̄ ∝ v̄Λ̄
(1 + cΛ̄ γ 5 )vp̄p̄

(4.3)
(4.4)

u and v are the 4-spinor of particle or anti-particle, sp and sp̄ the helicity of p and p̄ in the ﬁnal state
respectively. In forming the vertex of χc0 → ΛΛ, the eﬀective Lagrangian approach is adopted since hadrons
are composite particles bounded by the strong force in which the perturbation is not guaranteed to work.
With the fact that χc0 is scalar and an assumption that χc0 → ΛΛ conserve parity, the coupling can be
uniquely determined as Yukawa coupling. Note that the parity conservation should not be taken for granted
in general since the weak interaction could come into play. Fortunately from theoretical and experimental
observation, it is well-known that the electro-magnetic interaction (EM) and the strong interaction contributein the charmonium annihilation cc̄ → ΛΛ dominantly [73], thus this parity conservation assumption is
supposed to safely holds for the case here.
The matrix (4.3) is the same as (2.10) in the chapter 2. cΛ cΛ̄ are related to cV and cA appearing in(2.10) throughcΛ := −cΛ̄ =

cA
.
cV

(4.5)

Start with χc0 → ΛΛ ﬁrst. Suppose that Λ ﬂies with the momentum p and Λ the −p which is back-to-backeach other in the χc0 rest frame (χc0 -RF), and recalling that the notation of helicity of anti-particle andanti-spinor is opposite [74] √
p · σ ξ −j
√
vj =
,
(4.6)
− p · σ̄ ξ −jthe sub-matrix turns to bejMχijc0 = ūiΛ (p)vΛ̄
(−p)



−i
E+m
(1 ± η)ξ−p( (1 ∓ η)ξp†i , (1 ± η)ξp†i )γ 0−i−(1 ∓ η)ξ−p
2


−i
E+m
(1 ± η)ξ−p=( (1 ± η)ξp†i , (1 ∓ η)ξp†i )−i−(1 ∓ η)ξ−p
2
=

(4.7)

where EΛ , pΛ and mΛ are Λ (Λ) energy, momentum and mass respectively. Here the 2-spinor ξp± is deﬁned
the same as (2.14) in chapter 2, so as η := pΛ /(EΛ + mΛ ). Representing p with polar angle coordinates(θ, ϕ), one ﬁnds that −p corresponds to (π − θ, ϕ + π), thus
+
ξ−p
= ξp−

−
ξ−p
= −ξp+ ..

(4.8)

4.1. χC0 → ΛΛ59

Plugging these into the above,
E+m((1 ± η)ξp†i , (1 ∓ η)ξp†i )2= (1 + η)2 − (1 − η)2 δij



Mij
χc0 =

±(1 ± η) ξpi∓(1 ∓ η) ξpi= 2pΛ · δij(4.9)

Letting |ψΛΛ̄ ⟩ note the ΛΛ helicity state, as Mij
χc0 is proportional to the coeﬃcients of the helicity eigenstatei, j = ±,Mijχc0 ∝ ⟨i, j|ψΛΛ̄ ⟩ .
Therefore|ψΛΛ̄ ⟩ =X⟨i, j|ψΛΛ̄ ⟩ |i, j⟩

i,j∝|++⟩ + |−−⟩√.
2This is the helicity triplet state with maximal entanglement, which is equivalent to the spin triplet state|↑↓⟩ − |↓↑⟩√.
(4.10)|ψΛΛ̄ ⟩ →2
if boosting back to their respective Λ rest frame.
It sounds a bit weird that the outcome of composite spin of the 2 Λs is not zero despite they are born froma spin 0 particle. An ad hoc explanation is provided from parity transformation property of Yukawa, similar
to the discussion in section 2.2.1. Because of that the parity is conserved through χc0 → ΛΛ and that the
intrinsic parity of χc0 , Λ and Λ are respectively +, +, −, the relative orbital angular momentum betweenΛΛ has to be l = 1, which immediately leads to the total spin S = 1.
On the other hand, we know the spin conservation rule in non-relativistic system, however, these twoseemingly contradicting facts are barely reconciled, since Mχc0 → 0 when pΛ → 0.
Next we come to the total matrix M and obtain the π + π − angular distribution. The relation between them
is

X
dσ
2
2
∝
|M| =: |M|dΩn dΩn′ s ,s
p p̄
i
X h
′
∗j
j′
ij
∗ij
i
=
M∗i
Λ MΛ Mχc0 MΛ̄ MΛ̄ Mχc0 .

(4.11)

i,j,i′ ,j ′Note that the sum of sp , sp̄ , expressed by the over line, is take after squaring M while the helicity of ΛΛ is2summed before it. Accordingly, there becomes two types of terms in (4.11); terms such as |M+Λ | which canbe interpreted by itself as the transition probability from Λ with the helicity + to pπ (”direct terms”) and−the others such as M∗+
Λ MΛ which do not mean straightforwardly the transition probability. (”interferenceterms”). This is the essential part of the QM formalism that the spin coherence of ΛΛ is considered tokeep during the decay process and the helicity of intermediate ΛΛ state is not deﬁned. As seen later, thisinterference terms play a role in enhancing the correlation amplitude Qmax , giving the major impetus to theBI violation.
′

∗j
j
i
Deﬁne L and L as Lii′ := M∗iΛ MΛ and L := MΛ̄ MΛ̄ and substitute (4.9) into (4.11),
2|M| = 4p2Λ L++ L++ + L−− L−− + L−+ L+− + L+− L−+ .
′(4.12)

60

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY

Here L±± corresponds to the direct terms while L±∓ to the interference terms.

A great advantage in using S-matrix is that every sub-matrices are guaranteed to be invariant under any
proper Lorentz transformations, that is to say, we are allowed to evaluate each individual piece of the matrix
in diﬀerent respective frame. For example, Mχc0 is calculated in the χc0 -RF above, as for MΛ and MΛ̄ we
can jump the other frame, namely the Λ and the Λ-RF respectively.
′iLii′ := M∗i
Λ MΛ
X
′
=
ūiΛ (1 − c∗Λ γ 5 )ūspp uspp (1 + cΛ γ 5 )uiΛ
sp
′

5 i
= ūiΛ (1 − c∗Λ γ 5 )(p/p + mp )(1 + cΛ γ )uΛ




mp
pµp · σµ1 + c∗Λ
1 + cΛ
i= ūΛ
1 + c∗Λpµp · σ̄µmp′mp (1 + c∗Λ )(1 − cΛ )pp · σ|1 + cΛ |2= ūiΛuiΛ .
pp · σ̄|1 − cΛ |2
mp (1 − c∗Λ )(1 + cΛ )As the Λ-RF is considered, u±
Λ =

√


1 + cΛ

′uiΛ
(4.13)± ±mΛ T (ξΛ, ξΛ ). Here the ”direction of boost” is deﬁned as the directionof relative movement of the Λ-RF to the χc0 -RF, and the z-axis is taken to it as well. Denoting ξ ± as the
eigenspinor along or against the z-axis,



+
ξΛ
=

1
0


,

−
ξΛ
=



0
1


.

(4.14)

Using these relations, one arrives′i†iLii′ = 2mΛ ξΛ
((1 − |c|2 )mp + (1 + |c|2 )Ep − (c + c∗ )pp · σ)ξΛ′

= 2mΛ K ξ i† (1 − αΛ n · σ)ξ i(
L±± = 2mΛ K(1 ± αΛ nz )=L±∓ = 2mΛ KαΛ n± .

(n± := nx ± iny )(4.15)

with K, αΛ and n being
K := (1 − |c|2 )mp + (1 + |c|2 )EpαΛ := (c + c∗ )/K
pπ
pp
n :=
=− .
pπ
pp

(4.16)

Same calculation is carried out for L in the Λ-RF:′

Ljj ′ = MjΛ̄ MjΛ̄
 ′
∗
X j
s
s
j
=
v̄Λ̄ (1 − cΛ γ 5 ) vp̄p̄v̄Λ̄
(1 − cΛ γ 5 ) vp̄p̄
sp̄
′

j
∗ 5 j
= v̄Λ̄
(1 − cΛ γ 5 )(p/p̄ − mp )(1 + cΛ γ )vΛ̄
′

−j†
−j
= 2mΛ ξΛ̄K(1 + αΛ n′ · σ)ξΛ̄.

(4.17)

Here the coordinate system is taken the same as one introduced in the calculation in the Λ-RF above, inorder to visualize more clearly the relation of Λ and Λ spin as well as the direction of π + and π − . As the
direction of Λ boost is taken to the z-axis, that for Λ becomes the −z direction. The x, y axes are takensuch that (x, y, z) forms a right-handed coordinate system. Fig. 4.1 is the summary of these relations. For

4.1. χC0 → ΛΛ61+
the sake of consistency, deﬁne ξΛ̄as the state where the spin points to the Λ direction (−z direction):
 


0
1
+
−
ξΛ̄ =
(4.18), ξΛ̄ =.
1
0

With these expressions, one obtains(
L±± = 2mΛ K(1 ± αΛ n′z )L±∓ = −2mΛ KαΛ n′∓(4.19)(n′± := n′x ± in′y ).

The result of L and L (4.15),(4.19) can be summarized as a 2 × 2 matrix as below, with the helicity indicesof Λ and Λ being the legs:
1 + αΛ nz
L=
αΛ n−

αΛ n+
1 − α Λ nz




,

L=

1 + αΛ n′z−αΛ n′+−αΛ n′−
1 − αΛ n′z
(4.20)The diagonals and the oﬀ-diagonals corresponds to the direct and the interference terms respectively.

y
x

̟+

z

Λ

y

x

̟-

Λ rest framezΛΛ rest frame

Figure 4.1: The coordinate system in the Λ and Λ RF. z-axis is deﬁned as the orientation of Λ boost whichis the direction of the relative movement of Λ-RF to the χc0 -RF. The x, y axes are taken such that (x, y, z)
forms a right-handed coordinate system.

Now all preparation has been done. With (4.15) and (4.19) thrown into (4.12), one eventually arrives at
dσ
2
∝ |M|
dΩn dΩn′
2= 16p2Λ m2Λ K 2 (1 + αΛ nz )(1 + αΛ n′z ) + (1 − αΛ nz )(1 − αΛ n′z ) − αΛ(n+ n′− + n− n′+ )22= 1 + αΛnz n′z − αΛ(nx n′x + ny n′y )2∝ 1 + αΛ(−nx n′x − ny n′y + nz n′z )(4.21)The interpretation of this π + π − distribution is fairly clear. With respect to the Λ and Λ boost direction(”longitudinal direction”), π − and π + tend to be in the same orientation, while they prefer to be anti-parallelwith respect to the ”transverse” direction. Recall that π − disfavors the Λ spin direction and that π + favors
Λ spin direction, this is nothing but the demonstration of ΛΛ spin triple state. (Fig. 4.2)
The ﬁrst two terms L++ L++ and L−− L−− in (4.21) originates from the direct terms, while the latter two
L−+ L+− and L+− L−+ are from the interference terms. The direct terms only responsible for longitudinalcorrelation as they describe the mode where ΛΛ helicities are determined respectively to (+, −) or (−, +),which are the spin eigen-states along boost directions thus never create correlation to the transverse direction.

62

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY

̟+

s’

n’

s
Λ

χc0

Λ̅

n

̟Figure 4.2: The relation of ΛΛ spins and π + π − directions in the χc0 → ΛΛ → pπ p̄π channel.

Finally, a C matrix is obtained as
9
Cij = 2 ⟨ ni n′j ⟩
αΛ


Z
9
dσ
= 2
dΩn dΩn′ ni n′j/σtot
αΛdΩΛ dΩn dΩn′
R2
9 dΩn dΩn′ ni n′j |M|,= 2 R
αΛ dΩ dΩ dΩ ′ |M|2
Λ

therefore,

n


C=

(4.22)

n



1/2



1/2

(4.23)

1/2
C11 and C22 correspond to the transverse, C33 to the longitudinal correlation. This unambiguous way ofrepresenting correlation is the biggest advantage of introducing the C matrix.
Qmax is derived by ﬁnding the two largest eigenvalues of C T C. Here it is simply
C T C = diag(1/4, 1/4, 1/4)
1
λ1 = λ2 =√4
Qmax = 2 > 1(4.24)As expected, Qmax overwhelms the classical upper bound that the BI describes, and it is identical to the√quantum limit 2.

Here one can learn that interference terms give the important contribution. Without them, which is thecase of the most simple LHVT model,





C=


1/2

Qmax = 1
The longitudinal correlation is maximum though, Qmax can only reach the classical limit.

(4.25)

4.2. ηC → ΛΛ

4.2

63

ηc → ΛΛThe analysis of ηc → ΛΛ is quite similar to the χc0 → ΛΛ channel. In fact the only diﬀerence from χc0 → ΛΛis that ηc is pseudo-scalar. The total matrix readsj
M = MiΛ Mij
ηc MΛ

(4.26)i 5 j
Mij
ηc ∝ ūΛ γ vΛ̄(4.27)

Again, assuming the parity conservation in ηc → ΛΛ, the pseudo-Yukawa type of vertex is obtained. MΛ
and MΛ̄ are same as ones in the previous section (4.3) and (4.4). The sub-matrix Mηc → ΛΛ becomes
i
5 j
Mij
ηc = ūΛ (p)γ vΛ̄ (−p)



−i
E+m
(1 ± η)ξ−p( (1 ∓ η)ξp†i , (1 ± η)ξpi,,† )γ 0 γ 5−i−(1 ∓ η)ξ−p
2


−i
E+m
−(1 ± η)ξ−p=.
( (1 ± η)ξp†i , (1 ∓ η)ξp†i )−i−(1 ∓ η)ξ−p2=(4.28)+−
Using ξ−p
= −ξp+ (4.8),= ξp− , ξ−pE+m∓(1 ± η)ξpi†i†i=((1 ± η)ξp , (1 ∓ η)ξp )∓(1 ∓ η)ξpi
2(−(1 + η)2 − (1 − η)2 = −2EΛ (i = j = +)= δij
(1 − η)2 (1 + η)2 = −2EΛ (i = j = −)(4.29)

Just as in the case of χc0 → ΛΛ, only the terms which have the same helicity for Λ and Λ survive. With acareful observation on the signs, the helicity state of ΛΛ reads|++⟩ − |−−⟩√
|ψΛΛ̄ ⟩ ∝,2

(4.30)

which is the helicity singlet state with the maximal entanglement. Using L and L, the total matrix is written
as



2
|M| = 4EΛ2 L++ L++ + L−− L−− − L−+ L+− − L+− L−+(4.31)The only diﬀerence from the χc0 channel is the opposite sign of the interference terms. Plugging the result(4.15) obtained in the previous section,22|M| = 16EΛ2 m2Λ K 2 (1 + αΛ nz )(1 + αΛ n′z ) − (1 − αΛ nz )(1 − αΛ n′z ) + αΛ(n+ n′− + n− n′+ )22= 32EΛ2 m2Λ K 2 1 + αΛnz n′z + αΛ(nx n′x + ny n′y )22∝ (1 + αΛ
nz n′z ) + αΛ(nx n′x + ny n′y )2= 1 + αΛ(nx n′x + ny n′y + nz n′z )(4.32)The transverse correlation (correlation to the x, y direction, perpendicular to the ΛΛ movement) is opposite
to the χc0 → ΛΛ case, thus the n and n′ prefer to be anti-parallel. This results from the fact that Λ and Λ
are in helicity singlet state (Fig. 4.3).

64

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY

̟+

̟-

s’
ηc

n’

Λ

n

s
Λ̅
Figure 4.3: The relation of ΛΛ spins and π + π − directions in the ηc → ΛΛ → pπ p̄π channel.

The C matrix and Qmax value is thereforeC=
Qmax =



1/2



1/2
√

1/2
2.

(4.33)

It again reaches the quantum limit and the BI is again maximally violated.

4.3

J/ψ → ΛΛThe case with J/ψ is a bit complicated since it depends on its initial polarization. A brilliant analysis valid
for arbitrary J/ψ polarization is delivered by N. A. Tornqvist [56], however here rather the structure ofentanglement is in interest, we still keep our straightforward approach.

We ﬁrst start with the fact that J/ψs has to be in either pure left-handed or pure right-handed transverse
polarization, as far as they are directly produced via e+ e− → γ ∗ → J/ψ Fig. 4.4 from e+ e− collision. This
−
+ −
+ −
is essentially due to the helicity structure of EM coupling that only (e+R , eL ) and (eL , eR ) contribute to e eannihilation (L/R denote the left/right helicity) when they are highly relativistic. Consider J/ψ mass is∼ 3GeV and thus the e+ e− have to be typically 1.5GeV, this assumption is fairy reasonable (me /Ee < 10−6 ).
−+ −The coupling of e+
L eR gives rise of right-handed, and eR eL of the left-handed J/ψ. As long as an unpolarizede+ e− beam is assumed (which is more common than not in practice), they are equally produced. Note that
here ”unpolarized” means a statistical mixture of equal number of left and right-handed e+ e− in the beam,
therefore the polarization of J/ψ is deﬁnitely R or L event by event, rather than in the coherent superposition
state. This is an extremely important diﬀerence so that here no interference should be considered between
the two polarizations L and R for J/ψs.

4.3. J/ψ → ΛΛ

65

e+
γ*

c
c

eFigure 4.4: Diagram of the charmonium production. e+ e− do not directly couple to quarks for the lepton
number conservation. Thus the intermediation of virtual photon γ ∗ is needed.

Coming to the matrix of J/ψ → ΛΛ, generally it has the form of:aψ µ jijiµMψ,s ∝ ϵs µ ūΛ γ +
q vΛ̄
2pΛ

(4.34)

i, j = ± are the indices of ΛΛ helicity and s = L, R notes the J/ψ polarization.

This vertex is less trivial than the case of χc0 , ηc → ΛΛ but the terms can be enormously limited giving
parity conservation (EM and strong interaction is dominant). As J/ψ is vector (parity −, spin 1), it has tobe vector coupling. (without parity conservation, pseudo-vector coupling is also available). To the lowest
order of the eﬀective Lagrangian, the J/ψ ﬁeld Aµ only couples to the current made of Λ the ﬁeld ψψ̄γ µ ψ,ψ̄∂ µ ψ.

(4.35)

Therefore, the interaction terms in the eﬀective Lagrangian are written using coeﬃcients c1 , c2 as
c1 Aµ ψ̄γ µ ψ + c2 Aµ ψ̄∂ µ ψ.

(4.36)The vertex factor in the matrix is identical to those represented in the momentum space:
ϵµ ū(c1 γ µ + c2 q µ ),(4.37)

where q µ is momentum transfer q µ := pµΛ − pµΛ̄ . (4.34) is then derived with re-parametrizing c1 and c2 intoaψ .

j
is calculated. As the ΛΛ go back-toFor simplicity, consider the ﬁrst tern in (4.34) for now; ϵµ s ūiΛ γ µ vΛ̄back in the J/ψ rest frame,E+mjūiΛ (p)γ µ vΛ̄(−p) =( (1 ∓ η)ξp†i , (1 ± η)ξp†i )γ 0 γ µ
2

=

E+m
( (1 ∓ η)ξp†i , (1 ∓ η)ξp†i )2Note that σ µ = (1, −σ), σ µ = (1, σ).



σµ

!
−j
(1 ± η)ξ−p−j−(1 ∓ η)ξ−p
(1 ± η)ξpj.
σµ−(1 ∓ η)ξpj(4.38)

66

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY
For the terms with i = j = ±, one obtains
E+m(1 − η 2 )(ξp†i σ µ ξpi − ξp†i σ µ ξpi )
2


0
=
−2mΛ ξp†i σξpi0
 ∓ sin θ .
= 2mΛ 

0∓ cos θIūiΛ (p)γ µ vΛ̄(−p) =(4.39)

Here θ is the polar angle with e− beam direction deﬁned as z-axis (zenith).
For the terms with i ̸= j (i = −j = ±),E+mjūiΛ (p)γ µ vΛ̄(−p) =(1 ∓ η)ξp†i σ µ ξpj − (1 ± η)ξp†i σ µ ξpj(4.40)
 2∓2pΛ ξp†i ξpj=∓2EΛ ξp†i σξpj
0=∓2EΛ ξp†i σξpj0
 − cos θ = 2EΛ (4.41)
 ±i 
± sin θ√Here the notation ϵµL,R =T (0, 1, ±i, 0)/ 2 is employed (± in the RHS corresponds to L, R in the LHS
jrespectively). ϵs µ ūiΛ γ µ vΛ̄with respective i, j, s turns to as in the Tab. 4.1. Following the presentationjfor each polarization of J/ψ and e+ e−
Table 4.1: ϵs µ ūiΛ γ µ vΛ̄(i, j)√s=R√s=L(+, +)
2msinθΛ√√ 2mΛ sin θ(+, −)
2E
(1
+
cos
θ)
−
Λ
√
√ 2EΛ (1 − cos θ)(−, +) − 2E
(1
−
cos
θ)
2E
√Λ
√Λ (1 + cos θ)(−, −)− 2mΛ sin θ− 2mΛ sin θ

above, The ΛΛ helicity state is√√√√|ψΛΛ̄ ⟩R ∝ 2mΛ sin θ |++⟩ − 2mΛ sin θ |−−⟩ + 2EΛ (1 + cos θ) |+−⟩ − 2EΛ (1 − cos θ) |−+⟩√√√√
|ψΛΛ̄ ⟩L ∝ 2mΛ sin θ |++⟩ − 2mΛ sin θ |−−⟩ − 2EΛ (1 − cos θ) |+−⟩ + 2EΛ (1 + cos θ) |−+⟩ .

(4.42)(4.43)

It is terribly non-trivial if it is entangled, though it is visually entangled in that it has a complicateddependence on kinematical factors (mΛ , EΛ and θ) and is seemingly impossible to factorize into a productof two kets. Here an example is demonstrated that the visual impression has nothing to do with the extend
of entanglement.
Suppose a non-relativistic limit (EΛ → mΛ ). As mentioned above, ΛΛ helicity eigenstates reduce to spin
eigenstates with respect to ΛΛ orientation,|+⟩Λ → |↑⟩Λ ,|−⟩Λ → |↓⟩Λ

(4.44)

4.3. J/ψ → ΛΛ67

The subscripts indicate the choice of the QA. Re-writing into a representation with the beam direction being
the QA, (4.44) becomesθθ|+⟩Λ = |↑⟩Λ = cos |↑⟩e + eiϕ sin |↓⟩e ,22|−⟩Λ = |↓⟩Λ = −e−iϕ sinθθ|↑⟩e + cos |↓⟩e22(4.45)

where (θ, ϕ) is the polar and azimuthal angles with respect to the beam axis (see Fig. 4.5). The spin statefor Λ is obtained by replacing θ → π − θ, ϕ → −ϕ in (4.45).
θθθθ|+⟩Λ̄ = sin |↑⟩e − eiϕ cos |↓⟩e ,|−⟩Λ̄ = e−iϕ cos |↑⟩e + sin |↓⟩e .
2222(4.46)

Applying these to (4.45) and (4.46), one ﬁnds|ψΛΛ̄ ⟩R ∝ |↑↑⟩e ,|ψΛΛ̄ ⟩L ∝ |↓↓⟩e .

(4.47)

These are actually product states. Physically, this is the conclusion from the spin conservation law inthe non-relativistic limit, therefore it is deterministically predicted that ΛΛ spin state is |↑↑⟩e when J/ψpolarization is R (spin 1 pointing to the +z direction), vice versa. This result implicates the vector initialstate is just boring if non-relativistic, at the same, that the ΛΛ helicity may be entangled with the relativistic
eﬀect that mixing spin and orbital angular momentum.
As shown above, it is generally challenging to systematically evaluate the entanglement from the statevector directly though 1 , (4.43) and (4.42) nevertheless allows us to infer the property qualitatively.
For instance, the terms proportional to mΛ forms the helicity singlet regardless of the Λ direction θ. Here
we call this ”scalar mode” as it looks spin singlet.
In contrast, the latter two terms proportional to EΛ are complicated. First of all, the entanglement dependson θ. It is enhanced when the Λ goes to the transverse direction and the state comes asymptotically to√
2 EΛ (|+−⟩ − |−+⟩) as θ ∼ π/2, on the other hand, it is suppressed when Λ is forward and dissolved into
a product state with being θ ∼ 0, π. For the sake of convenience, here the contribution from these terms isdeﬁned ”vector mode”. In a non-relativistic limit the scalar mode and the vector mode contribute equally,
while the vector mode overcomes when the system becomes relativistic.

y
x

e+

̟+

z

Λ

y

x

̟-

Λ rest frame
z

Λ

θ

e-

Λ rest frame

Figure 4.5: The coordinates system in the Λ and Λ RF. x, z axes are taken such that they span the ee →J/ψ → ΛΛ reaction plane. z-axis is referred to the orientation of Λ boost which is the direction of therelative movement of Λ-RF to the χc0 -RF. The y axis are taken such that (x, y, z) forms a right-handed
coordinate system.
1 There does exist a formula that decides if the state is entangled [75], however no well-deﬁned measure of entanglement has
been established yet.

68

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY
Phenomenologically, this power balance can be understood by the angular distribution of ΛΛ.
dσ∝ | ⟨ψΛΛ̄ |ψΛΛ̄ ⟩ |2R + | ⟨ψΛΛ̄ |ψΛΛ̄ ⟩ |2LdΩΛ
= 8m2Λ sin2 θ + 4EΛ2 (1 + cos2 θ)∝ (EΛ2 + m2Λ ) + (EΛ2 − m2Λ ) cos2 θ∝1+EΛ2 − m2Λcos2 θ.
EΛ2 + m2Λ(4.48)The scalar mode contribute to the factor sin2 θ, while the vector mode to the 1 + cos2 θ dependency. Ina non-relativistic limit the both dependency cancels each other leading to an isotropic distribution, where
no angular orbital momentum comes into eﬀect, consistent to the spin conservation rule. The angulardependence give rise as becoming relativistic, and in an ultra-relativistic limit, it comes to a pure P-waveaψ µdistribution 1 + cos2 θ. Now that we almost understand the behavior, let the second term of (4.34)
ϵ qµ2pΛ
join in the game. Since it does not have the legs of spinor space, it is essentially Yukawa coupling, thus theresult of χc0 → ΛΛ (4.9) can be simply referred.
The factor ϵµ qµ is computed as below, with the taking a certain notion of ϵµqµ = pµΛ − pµΛ̄ = T (0, 2pΛ )√√ϵµL qµ = − 2(px − ipy ) = − 2pΛ e−iϕ sin θ√√
ϵµR qµ = − 2(px + ipy ) = − 2pΛ eiϕ sin θ.
Therefore, the matrix ϵµ qµ ūv becomes


√
qµ
j
µ i
ϵs ūΛ aψvΛ̄= ∓ 2aψ pΛ sin θ · δij
2pΛ(s = L, R; i = j = ±)(4.49)(4.50)(4.51)(4.52)

Here the ϕ is set to zero, exploiting the rotational invariance of the system. According to (4.52), it apparently
contributes to the terms of scalar mode only. Eﬀectively, it replaces the Λ mass mΛ in the Tab. 4.1 withm′Λ := mΛ − aψ pΛ .
Similar to L and L, the result of the overall matrix 4.34 can be summarized in a 2 × 2 matrix whose
components corresponds to the ΛΛ helicity (i, j) as
√ ′
√


 
M++
M+−
2mΛ sin θ2E
Λ (1 + cos θ)
ψ,R
ψ,R
√
√
Mψ,R :==M−+
M−−− 2EΛ (1 − cos θ)− 2m′Λ sin θ
ψ,Rψ,RMψ,L := M†ψ,R .

(4.53)

which will be utilized later on.
The modiﬁcation mΛ → m′Λ due to the second term in (4.34) also requires change in J/ψ → ΛΛ angular
distribution, since the parameter m′Λ tunes the ratio of contribution from the scalar mode to the vectormode.
dσ
E 2 − m′2Λ
∝ 1 + Λ2
cos2 θdΩΛEΛ + m′2Λ= 1 + A cos2 θ.

(4.54)

In fact m′2
Λ is determined through the measurement of this distribution. The recent result is A = 0.65 ±+0.22′
0.11(stat.) ± 0.03(syst.) (BES2) [76], giving m′2
Λ = 0.51−0.19 . mΛ is two value at the moment with ambiguity

4.3. J/ψ → ΛΛ69

in the sign, however as pointed later, the sign of m′Λ contributes to the interference term of J/ψ → ΛΛ̄ →pπ − p̄π − hence it explicitly eﬀects to the correlation pattern of π + π − direction, which can easily be observed.

Finally, the total matrix J/ψ → ΛΛ̄ → pπ − p̄π − is


X X 
2
j
∗j ′
∗i′ j ′
∗i′
|M|s =MiΛ Mij
M
M
M
M
Λ
ψ,s
ψ,s
Λ̄
Λ̄
sp ,sp̄ i,j,i′ ,j ′=X

i,j,i′ ,j ′

=

X

′

′ ′

∗i Mi Mij Mj M∗j M∗i j
MΛ
Λ
ψ,s
ψ,s
Λ̄
Λ̄
′

′ ′

∗i j
′
Li′ i Mijψ,s Ljj Mψ,si,j,i′ ,j ′hi
= Tr LMψ,s LM†ψ,s .

(4.55)

Here L and L is as deﬁned in (4.20). Together with (4.53) one arrives athi2|M|s = Tr LMψ LM†ψ

1 + αΛ nz
α Λ n+m′Λ sin θEΛ (1 + cos θ)= 8m2Λ K 2 Tr[αΛ n−
1 − αΛ nz−EΛ (1 − cos θ)−m′Λ sin θ−EΛ (1 − cos θ)m′Λ sin θ
1 + αΛ n′z −αΛ n′−].
−αΛ n′+ 1 − αΛ n′z
EΛ (1 + cos θ)−m′Λ sin θ(4.56)
When s = R, this comes eventually to2

2
2|M|R = 16m2Λ K 2 E 2 [(1 + cos2 θ)(1 − αΛnz n′z ) + αΛsin2 θ(−nx n′x + ny n′y ) + 2αΛ cos θ(nz − n′z )2+ Γ2 1 + αΛ(−nx n′x − ny n′y + nz n′z ) sin2 θ2+ Γ αΛ
sin 2θ(nx n′z + nz n′x ) − 2αΛ sin θ(nx − n′x ) ](4.57)

Note that Γ := m′Λ /EΛ . The ﬁrst line shows the contribution from the vector mode (∝ EΛ2 ), the second line′
is from the scalar mode (∝ m′2
Λ ) and the third is from their interference (∝ EΛ mΛ ).

The case of s = L is the same except the opposite sign of 2αΛ cos θ(nz − n′z ) and −2αΛ sin θ(nx − n′x ) in
(4.57) (which corresponds to the third term in the ﬁrst line, the second term in the third line respectively).
These terms cancel, provided that J/ψ is produced through unpolarized e+ e− beam, and the distributionof π + π − results in

2
dσ
1 X
|M|s
∝
′
dΩΛ dΩn dΩn
2
s=L,R

2
2
∝ (1 + cos2 θ)(1 − αΛnz n′z ) + αΛsin2 θ(−nx n′x + ny n′y )2+ Γ 2 1 + αΛ(−nx n′x − ny n′y + nz n′z ) sin2 θ+ Γ sin 2θ(nx n′z + nz n′x )Integrating over dΩΛ , the π + π − distribution is obtained asdσ2
2∝ 2(1 − αΛnz n′z ) + αΛ(−nx n′x + ny n′y )dΩn dΩn′2+ Γ2 1 + αΛ(−nx n′x − ny n′y + nz n′z )(4.58)(4.59)

Here the interference term of scalar and the vector mode vanishes as sin 2θ is odd with respect to θ. The C

70

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY

matrix is found to be


C=

1

2(2 + Γ2 )−1 − Γ2
1 − Γ2

(4.60)−2 + Γ2

2
2
C11
> C22
can be easily conﬁrmed provided Γ2 > 0. Thus Qmax is√q
5 − 2Γ2 + 2Γ4
2 + C2 =Qmax = 2 C11.
33
2 + Γ2(4.61)+0.0922
With the measured value of Γ2 = m′2
Λ /EΛ = 0.212−0.080 ,Qmax = 0.976 ± 0.048.
The result is subtle to the hell. The Qmax is consistent to either above or below 1. This is highly unfortunate
case: if there is only the contribution from the vector mode (comparable to the case with Γ = 0 here), theC matrix isCv = 



−1/4



1/4

(4.62)−1/2

or if with only scalar mode, (comparable to the case with Γ → ∞ here)


−1/2
.
−1/2
Cs = 
(4.63)
1/2√√
The corresponding Qmax values are Qmax = 5/2 and Qmax = 2 respectively, thus both violate the BI bythemselves. In this case in question, the contributions are mixed weighted with 1 : Γ/2, as a result, the mostpowerful correlation C33 cancel partially, suppressing Qmax . The relation between Qmax and Γ is shown in
Fig.4.6. Since the measurement uncertainty on the parameter m′2
Λ is still huge, the possibility of having
Qmax far excessing 1 has not been excluded. Though J/ψ → ΛΛ has enormous statistics (∼ 106 ), ∼ 2σdeviations is at least needed for realistic experimentally observable violation, which is not promising.

2
2
2
Figure 4.6: Dependence of Qmax on Γ2 . Γ2 is m′2
Λ /EΛ = (mΛ − aψ pΛ ) /EΛ in the J/ψ → ΛΛ channel,222∗
Γ = mτ /Eτ in the Z, γ → τ τ channel. The colored lines are the measured or theoretical values of their Γ,
and the red band means the error for the J/ψ → ΛΛ channel.

4.3. J/ψ → ΛΛ71

Well, can we save the J/ψ by any chance? For example, what if only the events with Λ oriented to a
speciﬁc direction are sampled? According to (4.58), the C matrix without integrating over θ is−(1 + Γ2 ) sin2 θ0Γ sin 2θ
1

C(θ) =
0(1 − Γ2 ) sin2 θ
02(1 + Γ2 ) + 2(1 − Γ2 ) cos2 θ22Γ sin 2θ0
−2 + (1 + Γ ) sin θ(4.64)

in which the explicit dependence of θ is contained. Fig. 4.7 displays the dependency for all components ofC(θ).
As C(θ) has non-zero values in the oﬀ-diagonals C13 , C31 , to obtain Qmax , C T C has to be diagonalized√
and the largest two eigenvalues λ1 , λ2 have to be speciﬁed by deﬁnition. The result of Qmax = λ1 + λ2 isshown in Fig. 4.8 (red solid line). For any θ, Qmax surpasses the classical limit 1. This is of representativethat ΛΛ is actually entangled. The black points and the error bar are the result of one pseudo-experimentwith 106 events using MC simulation. The bin of cos θ = 0 gives the largest violation with 7.5σ standard
deviation.

72

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY
C11 vs cos θΛC12 vs cosθΛ

C13 vs cosθΛ

0.2

0
0.05

0.1
-0.2
0

0
-0.4

-0.1
-0.05
-1

-0.5

0

0.5

1

C21 vs cosθΛ

-1

-0.5

0

0.5

1

C22 vs cosθΛ

-0.2

-1

-0.5

0

0.5

1

-0.5

0

0.5

1

-0.5

0

0.5

1

C23 vs cosθΛ

0.04
0.05

0.3
0.02
0.2
0

0
0.1

-0.02

-0.05

-0.04
-1

-0.5

0

0.5

1

C31 vs cosθΛ

0 -1

-0.5

0

0.5

1

C32 vs cosθΛ

0.2

-1

C33 vs cosθΛ

-0.3
0.05

0.1

-0.35
-0.4

0
0

-0.45

-0.1

-0.5
-0.2

-0.05
-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

Figure 4.7: Dependency of each components of C(θ) against cos θ. The red line the calculated theoreticalline from (4.64), the black points and the error bars are the result of one pseudo-experiment with 106 eventsusing MC simulation. For C13 and C31 , two cases Γ > 0 (red line and black points) and Γ < 0 (blue line
and blue points) are considered.

4.3. J/ψ → ΛΛ

73

Qmax

1.3

1.2

1.1

1

classical limit: 1
-1

-0.5

0

0.5

1

cos"!
Figure 4.8: Qmax calculatedwith events sampled with a speciﬁc range of θ. The red line is theoreticalpprediction Qmax = 2 λ1 (θ) + λ2 (θ) (λ1,2 is the largest two eigenvalues of C T (θ)C(θ)), and the black points
corresponds to the result of one pseudo-experiment with 106 events using MC simulation

This result is perfectly preferable, but only if the procedure of the arbitrary θ sampling is legal in
LHVT. Actually it is substantially doubtful, as θ should be regarded as a hidden variable in principle. Thus
letting samples with a speciﬁc value of θ represent the entire ensemble is just equivalent with setting thehomogeneous assumption as referred in section 1.3.2. It is true that a subclass of LHVT models which ishomogeneous in θ follows the Qmax ≤ 1 in any θ, however it is not the case in general, and practically
impossible to deduce how large the subclass is.
Moreover as J/ψ is vector which is not homogenous in direction, the assumption of having same correlation
value in any θ is much less plausible compared with the case of χc0 or ηc which are isotropic scalar. In fact,
this result of QM is a very good counter-example to it: the correlation amplitude could be Qmax biased byan arbitrary sampling of events.

To conclude, it does have some conceptual failure to test LHVT using this channel. However it is stillinteresting in the texture of testing QM non-locality, since it predicts the non-trivial θ dependence of Qmax
(Fig. 4.8) and the modulation of C(θ) (Fig. 4.7), which is sensitive to the decoherence of ΛΛ. To test thechange in modulation amplitude, one needs large statistics in order to cope with the binning. Fortunately
the J/ψ channel perfectly fulﬁlls the requirement that the expected available sample size 106 is demonstrated
to be enough.

An additional bonus for this channel, though it has nothing to do with neither LHVT nor QM non-lcality,
is that the unknown sign of the parameter Γ or m′Λ can be unambiguously determined by the sign of the
C31 and C13 modulation (Fig. 4.7). The red line is theoretical prediction if Γ is positive, while the blue lineis the case with negative Γ. The black points are the result of a pseudo-experiment with 106 events using
MC simulation. The error bars are safely small enough to distinguish the sign of Γ.

74

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY

4.4

e+ e− → γ ∗ → τ τIn this and next section, the direct production of τ τ pair through the s-channel annihilation of e+ e− is in
discuss. There are two processes with diﬀerent intermediate gauge particles: (i) the e+ e− → γ ∗ → τ τ andthe (ii) e+ e− → Z → τ τ . Generally, there exists the interfere with each other, however, here the argument
is conﬁne to only two scenario:
1. The center of mass energy of e+ e−√

s is far below the weak scale mZ (∼ 90GeV). （s << m2Z ）2. The center of mass energy of e+ e− is around the Z resonance. （s ∼ m2Z ）In fact, the center mass energy for most of the e+ e− collider experiments subject to either of the condition,
since there has been supposed to be little physics motivation in the energy region between the two. In eithercase the contribution from the interference is negligible since one of the two is dominant compared with theother. This section discusses the case 1. and 2. will be analyzed in the next section.
The matrices of each sub-processes of e+ e− → γ ∗ → τ τ are
i µ j
Mij
γ ∝ ϵs µ ūτ γ vτ̄(4.65)Miτ ∝ ūsνν (1 − γ 5 )uiτ(4.66)

Mjτ̄ ∝ v̄τ̄j (1 − γ 5 )vν̄sν̄

(4.67)In (4.65), only the tree-level process is considered. For the same reason with the ee → J/ψ as illustrated−+−
above, when ee are relativistic, only the combinations (e+L , eR ) and (eR , eL ) contribute to the annihilation.

Thus eﬀectively, the intermediate photon γ ∗ only has transverse polarization R or L, represented by ϵs µ in
(4.65).
We do not have to be obsessed with the calculation since it can be settled almost entirely in analogy ofJ/ψ → ΛΛ: Firstly, the result of (4.65) is the same as the ﬁrst term of (4.34). (4.66) and (4.67) corresponds
to (4.3) and (4.4) with substituting cΛ = 1 and cΛ̄ = −1 respectively. clm and cΛ̄ only accounts for thepolarimeter performance α of which the C matrix, Qmax and the BI are all independent. Therefore, fromnow on, the only thing we have to do is interpreting the entire result of the J/ψ → ΛΛ channel with replacingΓ = m′Λ /EΛ to Γ = mτ /Eτ . Accordingly the C matrix and Qmax end up in−1 − Γ21
1 − Γ2C=
2(2 + Γ2 )2
−2 + Γ
√2
4
5 − 2Γ + 2ΓQmax =
2 + Γ2(4.68)The dependency of Qmax against Γ is illustrated in Fig. 4.6. Here Γ, the ratio of contribution from the scalar
and the vector mode can be tuned by the beam energy. By (4.68), one ﬁnds Qmax > 1 is fulﬁlled when√√
Γ2 > 3 + 2 2orΓ2 < 3 − 2 2.
√

s = 2Eτ , it turns out that the center mass energy√
2mτ
= 8.58(GeV)(4.69)s> √2−1√√
is at least needed to violate the BI and greater s is preferred. From view of s and statistics, the most
Using 0 < Γ = mτ /Eτ < 1,

4.5. Z → τ τ

75promising candidate is the Belle experiment with√s = 10.58(GeV), which gives

Qmax = 1.037.
With the massive statistics in Belle, this much of violation may enough for experimental sensitivity, however
here only the tree-level diagram is considered thus the minor change from loop correction can potentiallychange the conclusion, for example, the anomalous magnetic moment of electron or tau give rise a term justas the second term in (4.34) which can slightly tune Γ. Since the violation is tiny if any, the result is highly
sensitive to the minor uncertainty. Further study such as including loop corrections is needed.

4.5

Z → ττ

In the previous section 4.4, an important fact is derived that faster τ suppresses the Γ value, in other words,
enhances the contribution from the vector mode and Qmax . From the viewpoint, Z → τ τ is a premium
channel in that the two τ s are so fast that the approximation of ultra-relativistic limit can be adopted.
(β ≃ 0.999, Γ := m2τ /Eτ2 ≃ 0.0015).

The matrix for Z → τ τ → πνπν is all the same to ee → γ ∗ → τ τ → πνπν, except the sub-matrix of
Z → τ τ is
i µ
5 i
Mij
Z,s ∝ ϵs,µ ūτ γ (cV − cA γ )vτ

cV = T3 − 2Q sin2 θw = 0.0378
1
cV = T3 = − .
2(4.70)s denotes the polarization of the Z (s = L, R) and sin2 θw is the Weinberg angle, with valued as sin2 θw =0.2397 ± 0.0013 [48]. The ﬁrst term, the vector current term, is proportional to the ﬁrst term in (4.34) which
results in Tab. 4.1. The latter term, the axial current term is thenE+mūiτ (p)γ µ γ 5 vτj (−p) =( (1 ∓ η)ξp†i , (1 ± η)ξp†i )γ 0 γ 5 γ µ
2

=

E+m
( (1 ∓ η)ξp†i , (1 ± η)ξp†i )
2



−σ µ

!
−j
(1 ± η)ξ−p−j−(1 ∓ η)ξ−p
(1 ± η)ξpi.
σµ−(1 ∓ η)ξpi

For the case with i = j = ±,E+m
(1 − η 2 )(−ξp†i σ µ ξpi − ξp†i σ µ ξpi )
2


−2mτ


0

= δij 00ūiτ (p)γ µ γ 5 vτi (−p) =(4.71)

76

CHAPTER 4. QM ANALYSES AND THE VIOLATION OF THE BELL INEQUALITY

For the case with i ̸= j, (i = −j = ±)E+m−(1 ∓ η)ξp†i σ µ ξpj − (1 ± η)ξp†i σ µ ξpj
 2

0
=
∓2pΛ ξp†i σξpj0
 − cos θ = 2pτ  ±i  .
± sin θūiτ (p)γ µ γ 5 vτi (−p) =(4.72)

Summing up the two terms, the value of the matrix MZ is found to be as Tab. 4.2 for respective s, i, j.

(i, j)(+, +)(+, −)(−, +)(−, −)

Table 4.2: The result of ϵs,µ ūiτ γ µ (cV − cA γ 5 )vτj
√s=R
√s=L
2mτ sin θ
2mτ sin θ
√
√
2(c
E
+
c
p
)(1
+
cos
θ)
−
2(c
E
+ cA pτ )(1 − cos θ)
V
τ
A
τ
V
τ
√
√
− 2(cV Eτ√+ cA pτ )(1 − cos θ)2(cV Eτ √+ cA pτ )(1 + cos θ)− 2mτ sin θ− 2mτ sin θ

Under the ultra-relativistic limit mΛ → 0, terms of (i, j) = (+, +), (−, −) vanish. Comparing the twotables Tab. 4.1 and Tab. 4.2 in the limit, they are proportional to each other, thus the angular distributionin π + π − in the Z → τ τ → πνπν channel corresponds to the result of the J/ψ → ΛΛ → pπpπ channel withreplacing the αΛ in (4.57) with ατ = 1 and with Γ being 0:2|M|R = 16m2Λ K 2 E 2 (1 + cos2 θ)(1 − ατ2 nz n′z ) + ατ2 sin2 θ(−nx n′x + ny n′y ) + 2αΛ cos θ(nz − n′z ) ]2|M|L = 16m2Λ K 2 E 2 (1 + cos2 θ)(1 − ατ2 nz n′z ) + ατ2 sin2 θ(−nx n′x + ny n′y ) − 2αΛ cos θ(nz − n′z ) ](4.73)(4.74)

Assuming it is produced via unpolarized e+ e− annihilation. However, unlike the example of ee → J/ψ
+− +above, since the coupling strength of Z to e−
L eR and to eR eL are diﬀerent, the number of outgoing left-

handed and right-handed Z bosons are not equal (note that helicity of Z is still deterministic by each event).
However this asymmetry has no impact eventually, since the diﬀerences between L (4.74) and R (4.73) areR
only in terms linear to ni or n′j which vanishes with integral dΩn dΩn′ when computing the C matrix. 2
Therefore, it simply reduces to the case of J/ψ → ΛΛ (4.60) with Γ → 0:


−1/4

1/4
C=
−1/2
√
5
Qmax =≃ 1.118 > 1.
2

(4.75)

Although the correlation is weaker than the channels with scalar initial state such as χc0 and ηc , it is enoughto overcome the classical limit.

2

2 The contribution from longitudinal polarization mode of Z (∼ O( mτ )) is negligible.
m2
Z

4.6. H → τ τ

4.6

77

H → ττ

As long as the CP-even SM Higgs is concerned, the result is identical to the case of the χc0 → ΛΛ channel(4.23),


C=



−1/2

Qmax =√

−1/2(4.76)

1/2
2,

resulting in maximal violation of the BI.
Though now it seems less likely giving the result from LHC [77], in principle Higgs is possible to containthe CP-odd component as well. For pure pseudo-scalar Higgs (purely CP-odd), the result will be identicalto the case of ηc → ΛΛ (4.2):





1/2

C=



1/2

Qmax =√(4.77)

1/2
2.

The case of the mixture is a bit bothering as it is a coherent superposition of CP-even and CP-odd that is
not equal to a simple averaging of (4.76) and (4.77) thus we trace back to the matrix for H → τ τ :M ∝ ūτ (a + bγ 2 )vτ .

(4.78)

Matrices corresponding to the τ − → π − ν and τ + → π + ν̄ are (4.66) and (4.67) respectively. Fortunately the
result of the χc0 channel (4.12) and the ηc channel (4.31) can be referred and the total matrix is derived to2|M| ∝ L++ L++ + L−− L−− + r(L−+ L+− + L+− L−+ )a E 2 − b2 p2τ
r := 2 τ2
a Eτ + b2 p2τ
2≃(4.79)a −b.
a2 + b22

2

It is worth pointing that the interference terms (L−+ L+− + L+− L−+ ) are weaken due to the mixing of CP
phase. As the direct term, (L++ L++ + L−− L−− ) contributes to C33 , while the interference terms to C11
and C22 , it is shown that



 1
C=
Qmax =

2r

1
2r

p
1 + r2 ≥ 1

1
2(4.80)

where r = 1 with the SM Higgs. The violation of the BI is substantially suppressed if the Higgs CP is highly
mixed.

Chapter 5

Discussion on Experimental
Feasibility
Theoretical analysis in the previous section reveals that all candidates except J/ψ → ΛΛ have Qmax > 1. In
addition to it, this section tries to include numerous experimental eﬀects and discusses the realistic sensitivity,
to be speciﬁc, it proceeds as:• Estimation on required and available statistics to conﬁrm Qmax > 1 with a certain level of signiﬁcance.
• Evaluation of eﬀect from the contamination of background events.
• Evaluation of eﬀect from ﬁnite detector resolution.

5.1

Distribution of C11 , C33 , Qmax and Sensitivity

Our top-most interest is Qmax , which is calculated from the C matrix. Here we start with the basic elementsof C matrix ⟨ni n′j ⟩ , (i, j = 1, 2, 3), observing the statistical behavior in detail. In practical situations, these
correlation amplitudes can only be estimated from simple algebraic averaging
1 X(ni n′j )k .
NN⟨ni n′j ⟩ =k=1which will inevitably suﬀer from limited precision due to the statistical ﬂuctuation when the available sample
size is ﬁnite.

5.1.1

Shape of ni n′j distribution

Before heading to correlation ⟨ni n′j ⟩, the event-by-event distribution of ni n′j is worth noting, in that it has
a terribly non-trivial shape and statistical features.
According to the result of chapter 4, the distribution of n and n′ can be generally reduced toXdσ∝ 1 + α2wij ni n′j .
dΩn dΩn′
i,j

(5.1)

Here i, j = 1, 2, 3 correspond to the Cartesian coordinates x, y, z in their parent rest frame as deﬁned before.
The projection to the 2 variable space of ni , n′j is made by integrating over the rest of all degree of freedom79

80

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY(DOF). For instance, the angular distribution for ηc → ΛΛ channeldσ2∝ 1 + αΛ(nx n′x + ny n′y + nz n′z )dΩn dΩn′has 4 DOFs, corresponding to the 2 πs direction (θ, ϕ), (θ′ , ϕ′ ). The 2D joint probability density function(PDF) for nz =: cos θ and n′z =: cos θ′ is straightforwardly obtained by integrating with the other DOFs (ϕ,ϕ′ ), thus
P (nz , n′z ) =

Z

dϕdϕ′

dσ
2
∝ 1 + αΛnz n′zdΩn dΩn′with a smooth slope as shown in Fig. 5.1. The 1D PDF with respect to x := nz n′z is then derived via anormal procedure of variable transformation in which a factor responsible for metric variation (see Fig. 5.2)


Z 1
dx
x
1
x + dxM (x) =−
= 2 lndx=
2dnz
dnz dn′z
n
n
|x|
z
z
x
is applied, therefore,
P (x) = P (nz , n′z )M (x)1∝ (1 + α2 x) ln.
|x|
It is illustrated that the singularity at x = 0 in the distribution (Fig.5.3) purely arises from the metric factornz

M (x) but physical consequences.

1
800
0.5
700
0

600
500

-0.5

400
-1
-1

-0.5

0

0.5

1

n’z
Figure 5.1: 2D PDF of nz , n′z in the ηc → ΛΛ → pπ − p̄π + channel: P (nz , n′z ) = 1 + α2 nz n′z

5.1. DISTRIBUTION OF C11 , C33 , QMAX AND SENSITIVITY

81

n’z
1

M(x)

x+dx = nz n’z
x = nz n’z

x
-x

-1

x

nz

1

-x

-1

arb.unit

Figure 5.2: The metric factor M (x) = dx/dnz dn′z corresponding to the transformation x := nz n′zhEntries 1000000
Mean
0.04603RMS
0.3299

10000

5000

0
-1

-0.5

0

0.5

1

nz*n’zFigure 5.3: Distribution of x := nz n′z in the ηc → ΛΛ → pπ − p̄π + channel. The black histogram shows MC
1simulation, while the red line is the ﬁt by function P (x) ∝ (1 + α2 x) ln |x|with perfect accordance.
It has long tails in the both side and diverges at x = 0. The asymmetry represents the correlation of ππin their direction. With a closer analysis, one easily ﬁnds
• In spite of the divergence at x = 0, the integration over x ∈ [−1, 1] is ﬁnite, with a constant value

82

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY
independent of α.
′1(1 + α2 x)2
dx
ln
2
2α
|x|
−1

1
Z 1
2 2
(1 + α x)
1
1
−1
=
ln
−
dx(1 + α2 x)22α2|x| −1 2α2 −1
x
Z 1
1
= 2
dx2α2
α 0
=2

Z 1

1
dx(1 + α x) ln
=
|x|
−1
2

Z 1



(5.2)

• P (x) is positive throughout x ∈ [−1, 1], when α2 ≤ 1.
This is the case here that α ≤ 1 is guaranteed by physics (see 2.2) and |x| ≤ 1 is a trivial fact as|nz |, |n′z | ≤ 1.
Accordingly, P (x) is likely to have good reason to be treated as a PDF, though strange to some extent. It
seems to be something like delta-function. Normalizing with (5.2)
1
1
P (x) = (1 + α2 x) ln
.
2
|x|

(5.3)

The expectation value µ and RMS σ are
Z 1
α2
µ=
dx xP (x) =
9
−1
Z 1

1
σ =
dx (x − µ) P (x) =
9
−1
r
1
α4
σ=
1−
3
9
2

2

(5.4)


α4
1−
9



(5.5)

This µ is the true value of ⟨nz n′z ⟩ (estimated value with inﬁnitely large sample size). It greatly depends
on the performance of polarimeter α2 because good polarimeter results in larger asymmetry of distribution
hence larger µ value. The notable fact is in the RMS. The distribution has a substantial extended width
and the dependence on α2 is almost nothing. α2 = 1 gives the minimum width of σ = 0.3305 while themaximum is σ = 0.3333 with α2 = 0. This extended intrinsic width is the nature of statistics rather thanphysics.
The other combinations of ni n′j i, j = 1, 2, 3 follow the same argument and their distributions are described as

1
1(1 + wij α2 x) ln
(5.6)
2
|x|
where wij are the same as in (5.1). As later discussed, the large intrinsic width in the distributions deduces
P (x) =

a signiﬁcant conclusion: Since there is already a heavy ﬂuctuation even with ideal measurement in inﬁnite
accuracy, the smearing eﬀect from measurement error of n, n′ to ni n′j is almost negligible.

5.1.2

Shape of C11 , C33 distribution vs number of events

Following the result above, uncertainty of the estimation for ⟨ni n′j ⟩ in an experiment with sample size N is
PN
studied, which is namely the statistical ﬂuctuation of y = N1 k=1 (ni n′j )k over many pseudo-experiments.

5.1. DISTRIBUTION OF C11 , C33 , QMAX AND SENSITIVITY

83

According to the central limit theorem (CLT), with large enough sample size N , the distribution of y (deﬁned
as Q(y)) converge into a Gaussian with the mean value µ′ and RMS σ ′ ofµ′ = µ =

α2
9

√

1
σ = σ/ N = √
3 N
′

r
1−

α4
9

(5.7)

where µ, σ is the mean and RMS of ni n′j distribution, equal to (5.4) and (5.5) respectively.
How much N is then needed for this conversion and that the approximation (5.7) holds? Here a MC
simulation is carried out and Q(y) over pseudo-experiments are given in Fig. 5.4. The convergence isamazingly fast so that Q(y) is already nearly Gaussian with N ∼ 2, 3 and Fig. 5.5 shows that it perfectly
agrees with the approximation of CLT when N ≥ 2. P (x) is revealed to have an extremely convergenceproperty.
Since C33 is just proportional to ⟨nz n′z ⟩:C33 =9⟨nz n′z ⟩ ,2α

the distribution is a one with Q(x) scaled by the factor 2α9 2 . The mean and RMS are
1
µ33 =
2
r
1
9
σ33 = √− 1.
2 N α4(5.8)

The mean µ33 is equal to the true value thus it is unbiased regardless of N . The other components of C
matrix follow the same distribution, namely Gaussians centered around the true values ﬂuctuating with the
1 q9magnitude of σ33 = √4 − 1.
2 N α

84

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY
C33 with n=1

C33 with n=2

300

200

150

200

100
100
50

0
-1

-0.5

0

0.5

1

0
-1

-0.5

0

C33 = < nz n’z >C33 with n=3

0.5

1

C33 = < nz n’z >C33 with n=4

200
200
150
100

100

50
0
-1

-0.5

0

0.5

1

0
-1

-0.5

C33 = < nz n’z >

0

0.5

1

C33 = < nz n’z >RMS of C33

Figure 5.4: Distribution Q(y) in the ηc → ΛΛ → pπ − p̄π + channel. y is an algebraic average of nz n′z which
follows according to (5.3). Each histogram show the case with N = 1, 2, 3, 4, the sample size used to calculate
y. N = 1 gives the same distribution as (5.3). Center limit is already conﬁrmed with N ≥ 2.

0.2

0.1

0
0

100

200

300

Number of Events
Figure 5.5: RMS of Q(y)q
distribution with respect to sample size N . The red line is the approximated value

from the CLT: σ = 3√1N

1 − α9 .
4

5.1. DISTRIBUTION OF C11 , C33 , QMAX AND SENSITIVITY

5.1.3

85

Estimator for Qmax

All preparation for analyzing statistical behavior of Qmax is now done, except one thing: the choice of
estimator for Qmax is not trivial and actually problematic. There are two candidate methods to deﬁne:• Following the deﬁnition of Qmax literally. That is to say, estimating all components of C matrix,
building S = C t C then take the two biggest eigenvalue of S matrix λ1 , λ2 to form the estimator of
Qmax
QM
max = 2p
λ1 + λ2 .

• On the other hand, in either 6 channels studied in chapter 4, QM predicts the zero values of oﬀ-diagonals
in C matrix, leading Qmax to
QD
max =

q
2 + C2 .
C1133

The other possible way to estimating Qmax is use this value, in other word ignoring the oﬀ-diagonalcomponents of C, even if the measured values are non-zero.
D
Naively thinking, there is no reason to take QD
max for solution because Qmax = Qmax is not valid in theory

other than QM in general. It sounds logically failed to use a measure only valid in QM to test the othertheories. However as demonstrated in 3.2, the conﬁguration maximizing Q value for a given C is case whereDM
S = C T C is strictly diagonalized. So it is always QD
max ≤ Qmax . Qmax always gives conservative estimation

but never ever cheating.
This is as well the reason why QM
max should not use. Since the contribution of non-zero oﬀ-diagonal comM
ponent always enhance the QM
max value, statistical ﬂuctuation of oﬀ-diagonals only to biases Qmax positively.
M
To summarize, QD
max is at least no less conservative than Qmax and as long as QM prediction, where theDoﬀ-diagonals are all zero, is supposed to be true, QM
max will suﬀer from heavier bias than Qmax .
D
Fig. 5.6 and Fig. 5.7 are the comparison of QM
max (blue) and Qmax (red) distribution over pseudo-experiments

and their mean/RMS respectively for ηc → ΛΛ channel. The dotted line is the true value QM predicts, fromwhich deviation means the bias. Though QM
max experience less statistical ﬂuctuation, it has apparently moreDsevere bias and slower center limit than QD
max . This is the reason Qmax is more suitable as the estimator of

Qmax . Fig. 5.8 and Fig. 5.9 are same plots for the Z → τ τ channel, which shows the same tendency.

86

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY
Qmax with n=500

Qmax with n=1000

60
40
40
20
20

0
0

1

2

0
0

3

1

2

Qmax
Qmax with n=1500

3

Qmax
Qmax with n=2000

100
80
60
50

40
20
0
0

1

2

0
0

3

1

2

Qmax

3

Qmax

RMS of Qmax

Mean of Qmax

D
Figure 5.6: Distribution of QM
max (blue) and Qmax (red) for the ηc → ΛΛ channel. The dotted line means
the true (theoretical) value of Qmax in QM.

2

1.8

0.3

1.6
0.2
1.4
1000

2000

3000

Number of Events

1000

2000

3000

Number of EventsD
Figure 5.7: Mean and RMS of QM
max (blue) and Qmax (red) distribution for the ηc → ΛΛ channel. The
dotted line means the true (theoretical) value of Qmax in QM. The case with χc0 → ΛΛ is exactly the same.

5.1. DISTRIBUTION OF C11 , C33 , QMAX AND SENSITIVITY
Qmax with n=200

87

Qmax with n=500

60
80
40

60
40

20
20
0

0.5

1

0

1.5

0.5

1

1.5

Qmax
Qmax with n=1000

Qmax
Qmax with n=1500

150
100
100
50
50

0

0.5

1

0

1.5

0.5

1

Qmax

1.5

Qmax

RMS of Qmax

Mean of Qmax

D
Figure 5.8: Distribution of QM
max (blue) and Qmax (red) for the Z → τ τ channel. The dotted line means thetrue (theoretical) value of Qmax in QM.

1.3

0.2

0.15

1.2
0.1

1.1
0.05
500

1000 1500 2000

500

Number of Events

1000 1500 2000

Number of EventsD
Figure 5.9: Mean and RMS of QM
max (blue) and Qmax (red) distribution for the Z → τ τ channel. The dottedline means the true (theoretical) value of Qmax in QM. The case with ee → γ ∗ → τ τ and H → τ τ are exactly
the same.

Several comments on statistical properties of QD
max are attached.

88

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY• As QD
max depends on C11 and C33 in a non-linear way,q
2 + C2
QD
=
2
C11
max
33
it does suﬀer from some bias and a positive non-Gaussian in the distribution when N is small, despite
C11 and C33 always follow a bias-free Gaussian. In most the case, this non-Gaussianity vanishes
because the expected available sample is suﬃciently abundant. The only exception is H → τ τ where
the sample size could be poor as N ∼ 20).

• The non-Gaussianity vanishes almost with
(
150
N∼
20

(Λ series)(τ series).

(5.9)

See also Fig. 5.10 and Fig. 5.11 where the explicit convergence of the distribution is shown. The
non-Gaussianity arise from the ﬂuctuation of C11 and C33 . Since the extent only depends on α (5.8),
the performance of polarimeter, it is common with the channels in the same series.

• The RMS of QD
max distribution is easily found when it converges into Gaussian, considering error
propagation from C11 and C33 . Assuming no statistical correlation between C11 and C33 ,

2

2
∂ QD
∂ QD
max
max
2
2
2
σ(QD
)
=
σ
+
σ33
max
11
∂C11
∂C33
!2
!2
C33
C11
2
2
σ11 + 2 p 2
σ33= 2p 22
2
C11 + C33
C11 + C33
4
2 2
2 2
= 2
2 (C11 σ11 + C33 σ33 ).
C11 + C33
σ11 and σ33 are referred from (5.8) then1
σ11 = σ33 = √2 N=: σC .

r

9
−1
α4

Therefore22σ(QDmax ) = 4σCσ(QDmax ) = 2σC
1
=√
N

r

9
−1
α4

(5.10)

D
To conclude, one deﬁnes the experimental value of Qmax as QD
max . From now on, Qmax designates Qmax

if no special remark accompanies.

5.1. DISTRIBUTION OF C11 , C33 , QMAX AND SENSITIVITY
Qmax with n=20

89

Qmax with n=50

150

100

100

50

50

0
0

2

4

0
0

6

Qmax = 2*sqrt(C +C233)
11

2

11

Qmax with n=100

Qmax with n=150

150

150

100

100

50

50

0
0

1

2

3

4

4

Qmax = 2*sqrt(C2 +C233)

2

0
0

1

2

3

Qmax = 2*sqrt(C2 +C233)

Qmax = 2*sqrt(C +C233)
11
2

11

Figure 5.10: QD
max distribution with sample size of N = 20, 50, 100, 150 in case of channels of the Λ series.
It comes to Gaussian around N ∼ 150.
Qmax with n=2

Qmax with n=5

2000
1500

1000

1000
500
500
0
0

2

4

6

8

0
0

Qmax with n=10

2

4

6

Qmax = 2*sqrt(C2 +C233)

Qmax = 2*sqrt(C +C233)
11
2

11

Qmax with n=20

200
2000

150
100

1000
50
0
0

1

2

3

4

Qmax = 2*sqrt(C +C233)
11
2

0
0

1

2

3

Qmax = 2*sqrt(C2 +C233)
11

Figure 5.11: QD
max distribution with sample size of N = 2, 5, 10, 20 in channels of the τ series. It comes toGaussian around N ∼ 20.

90

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY

5.1.4

Signiﬁcance and the master formula

Here we ﬁnally derive the expected signiﬁcance of observed violation of BI, with respect to sample size N .
The signiﬁcance is deﬁned as:∆ =:µ(Qmax ) − CLσ(Qmax )(5.11)

When N is suﬃciently large so that Qmax distributes according to Gaussian, this is just the standard
deviation, for example, ∆ = 1 means 1σ deviation. When N is not enough large then it is not the case,(5.11) gives a more conservative evaluation of signiﬁcance since the non-Gaussianic tail enhance the value ofσ(Qmax ).

In an ideal setup, µ(qmax) is identical with the theoretical Qmax value derived in chapter 4:√2(ηc , χc0 → ΛΛ; H → τ τ )

√


 5
(Z → τ τ )µ(Qmax ) =
√2


2

5 − 2Γ + 2Γ4
4m2τ

2

;
Γ
:=
(e+ e− → γ ∗ → τ τ )2 + Γ2s√√s is the e+ e− center of mass energy. Assuming s = 10.58(GeV) of Belle experiment, µ(Qmax ) = 1.037.
CL is the classical limit of BI: 1, in an ideal case.
When N is larger than the threshold (5.9) and Qmax distributes according asymptotically to Gaussian,substituting (5.10) gives∆ =: (µ(Qmax ) − CL) √α2 √N.
9 − α4(5.12)

This is the master formula of signiﬁcance.

One ﬁnds the dependency of (5.12) can be factorized in the 3 factors; the ﬁrst part µ(Qmax ) - CLrepresents the extent of spin entanglement of ΛΛ or τ τ pair, the indispensable component for violating√
BI; the second factor α2 / 9 − α4 indicates the dependence on the polarimeter power of decay Λ → pπ orτ → πν, in which larger α2 reﬂects the spin entanglement more to the observable Qmax , enhancing the√signiﬁcance; the last factor is statistics, with the N dependency. The function relation to N with respect
to each channel is illustrated in Fig. 5.12. The sample size giving 1σ, 2σ, 3σ signiﬁcance is as in Tab. 5.1.

Table 5.1: Sample size corresponds to 1σ, 2σ, 3σ signiﬁcance with respect to each channel
ChannelN(1σ) N(2σ) N(3σ)ηc , χc0 → ΛΛ
600
2200
5000
Z → ττ
300
1200
1700
ee → γ ∗ → τ τ
6000
26500 60000
H → ττ
45
180
400

Significance (!)

5.2. EFFECT FROM BACKGROUND

91

5

H→ττ
4

Z/γ*→ττ3

ηc, χco →ΛΛ̅
2

ee→ττ

1

(√s=10.58GeV)
0
10

103

102

104

Number of events
Figure 5.12: The signiﬁcance ∆ in the function of sample size N with respect to each channel

5.2

Eﬀect from Background

There are two independent eﬀects brought by BG.
(i) Correlation amplitude ⟨ni n′j ⟩ will be smeared, reducing the Qmax value. BG events tend to be accidental
where the two pions candidate are less like to correlated. Therefore the observed correlation amplitude⟨ni n′j ⟩ is supposed to be diluted. Here a BG model is assumed in which the two pions candidate
have no correlation: ⟨ni n′j ⟩ = 0, giving the most conservative estimation of the smearing eﬀect. The
distribution of x = ni n′j for BG events comes to
P (x) =

1
1
ln
.
2 |x|

which is same as (5.3) with α = 0. The distribution of x = ni n′j for signal events is (5.3). Assuming asimple mixture of signal and BG events, the observed distribution of x = ni n′j will be11
1
1
Pb (x) = (1 − b) (1 + α2 x) ln+ b ln
2
|x|
2 |x|

1
1
2 2
∝
1 + (1 − b) α x ln
2|x|(5.13)

where b represent the fraction of BG events. So, it eﬀectively shifts the α value in the distribution for
signal events as
α2 → (1 − b)α2 .
The mean value and the RMS are
α2µ = (1 − b)9s
 2 2α12
1 − (1 − b)σ=3
9(5.14)

92

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY

Hence the Cij =9⟨ni n′j ⟩ has the mean and RMS of
2α2µ = (1 − b)C̃ij
r
1
9
√
σ=
− (1 − b)24
α
2 N(5.15)

respectively. The tilde mark represent theoretical value. The statistical behavior of Qmax is found,
repeating the argument in 5.1.3µ(Qmax ) = (1 − b)Q̃maxr
1
9σ(Qmax ) = √− (1 − b)2 .
N α4(5.16)(5.17)
As smeared by BG, the mean value µ(Qmax ) scale down from theoretical value Q̃max by the factor
1 − b. On the other hand RMS is almost independent of b, remains the same magnitude as the casewithout BG. The signiﬁcance curves with diﬀerent BG levels b are illustrated as Fig. 5.13∼5.17 foreach channel individually.
The expected BG level is, for the ηc , χc0 → ΛΛ channel, less than 0.5% in current analysis. (though
the signal selection criteria might be too strict for our analysis).
The situation in the τ series channel are more severe as only π + π − appear in the ﬁnal state. Nevertheless, the BG level can be managed to keep below 2% in the Z → τ τ at LEP and the e+ e− → γ ∗ → τ τ
at Belle, with appropriate procedures.
H → τ τ is the most challenging channel in suppressing BG because it has ∼ 10 times more cross-section
than that of the signal. Current analysis achieves 45% in BG level with 25% collection eﬃciency [78],however, which still leaves much to be desired. For example in reconstructing the mass mτ τ , thecollinear approximation is applied which blurs much the peak of mτ τ as shown in Fig. 5.18, as a result,the cut on mτ τ is not eﬀective that allows much residual BG such as ee → ZZ → τ τ q q̄ (pink dottedline) to sit under the peak. Most of them are ”irreducible” BG channels that have the same ﬁnal sate
topology as the signal channel, which can only be distinguished by mτ τ . However, this is supposed tobe drastically improved by implementing the kinematical ﬁt as described in the bottom of section 2.4,
instead of the collinear approximation. The peak of mτ τ will be sharpen with the width being about
3GeV, then the BG level will be accordingly reduced to around 15%, while the corresponding width is
now about 10GeV with the collinear approximation.

5.2. EFFECT FROM BACKGROUND

4

BG 0%

＠

BG 1%

χc0→ΛΛ ( BES3)

BG 2%

3

Z→ττ channel

Significance (!)

Significance (!)ηc, χc0 →ΛΛ̅ channel

93

2

Z→ττ (@ILC, gigaZ)

BG 0%

12
Z→ττ (@LEP, 1exp)

BG 2%

10

BG 4%

8
Z→ττ (@LEP, 4exp)

6
4＠

ηc→ΛΛ ( BES3)

1

2
01000 2000 3000 4000 5000

Number of Events N

103

105

104

Number of Events N

Figure 5.13: Expected signiﬁcance with each BGlevel in the ηc , χc0 → ΛΛ channel.

Figure 5.14: Expected signiﬁcance with each BGlevel in the Z → τ τ channel.

1

BG 0%
Z→ττ (@LEP, 1exp)

BG 2%

0.8

BG 4%

0.6

Significance (!)

ee→γ*→ττ channel (√s=10.58 GeV)

Z→ττ channel

Significance (!)

102

Q=Q0, BG 0%@Belle, 771fb-1 (eff.: 5%)4
Q=1.02*Q0, BG 2%@Belle, 771fb-1 (eff.: 10%)0.4

2
Z→ττ (@LEP, 4exp)

Q=Q0, BG 0.5%

0.2
100 200 300 400 500

Number of Events N
Figure 5.15: Expected signiﬁcance with each BGlevel in the Z → τ τ channel in a wider scale of
N

200

400

600

800

!103
1000

Number of Events N
Figure 5.16: Expected signiﬁcance with each BGlevel in the ee → γ ∗ → τ τ channel. The eﬀectof the contamination from the time-like events(evaluated in section 5.3) is included. Q0 is theoretical value Qmax = 1.037 with tree-level calculation. The blue line indicates the case where thehigher order corrections enhance Qmax by 2%,
with considering the BG level of 2%. The sensitivity highly depends on the uncertainty.

94

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY

Significance (!)

H→ττ channel
BG 0%

2

@ILC, 250fb-1@ILC, 1000fb-1@ILC, 2000fb-1

1.5

BG 5%
BG 10%BG 15%

1
0.5
50

100

150

200

Number of Events N
Figure 5.17: Expected signiﬁcance with each BGlevel in the H → τ τ channel.
allAll
Entries18309
qqH, H->τ τ
MeanqqH, H->others 122.9
llH
RMS
15.26
ττH

all

80

qqqq
γ γ ->qqqq
qqll
qqτ τ
γ γ ->qqll
γ γ ->qqτ τ
qqlν
qqτ ν
γ γ ->qqlν
γ γ ->qqτ νother SM bkg

70
60
50
40
30
20
10
0
0

20

40

60

80

100

120

140

160

180

200

Figure 5.18: Simulation of the H → τ τ ; Z → q q̄ channel with ILD full simulation (with 120GeV Higgs mass).
The horizontal axis indicates the reconstructed mass omτ τ . Signal (blue line) peaks around the Higgs mass
120GeV, while the other BG is continually distributed. The collinear approximation is a rough method thatbroadens much the signal peak, allowing ee → ZZ → τ τ q q̄ (pink dotted line) to overlap with.

(ii) The classical limit of the inequality will be enforced to rise. Due to the contamination of BG, theachievable correlation Qmax in LHVT is enhanced, which sounds contradictory to the statement abovethat BG generally smears out the correlation. The latter statement is true in reality, while the former
is a theoretically introduced fact.

5.2. EFFECT FROM BACKGROUND

95

Recall that our Bell’s inequality (3.12) relies upon the equation (3.11), however it results from therelation between the Λ or τ spin and the π direction (2.21) which is required as experimental results.
This (3.11) holds true for the signal processes ΛΛ → pπpπ and τ τ → πνπν but in principle it has no
reason to hold for the other channels.
Without (3.11), the restriction which can be generally imposed on the direction of two particles will
be dramatically loosen as| ⟨ na n′b ⟩ + ⟨ na n′d ⟩ + ⟨ nc n′b ⟩ − ⟨ nc n′d ⟩ | ≤ 2.

(5.18)

This is no more than an algebraic inequality that 3 vectors generally follow. On the other hand, theinequality for the signals is| ⟨ na n′b ⟩ + ⟨ na n′d ⟩ + ⟨ nc n′b ⟩ − ⟨ nc n′d ⟩ | ≤

2α2
,
9

(5.19)

2α2+ b × 2.
9(5.20)

therefore, the sample of admixture of signal and BG follows| ⟨ na n′b ⟩ + ⟨ na n′d ⟩ + ⟨ nc n′b ⟩ − ⟨ nc n′d ⟩ | ≤ (1 − b) ×

where b is the BG level deﬁned above. Since the upper limit for the BG inequality (5.18) is 9/2α2 =9 ∼ 20 times greater than that of signal one (5.19), the impact of BG is huge.
In language of Qmax ,Qmax ≤ CL
9
CL = (1 − b) × 1 + b × 2
α


9
=1+b
−1
α2
Considering the quantum limit of Qmax is√(5.21)(5.22)(5.23)

2 at the most, the inequality (5.20) becomes extremely diﬃcult

to violate even with a tiny fraction of BG. Plugging αΛ = 0.642 ± 0.013 and ατ = 0.998 ± 0.005 in (5.23), b
has to be suppressed below 1.7% for the Λ series channels and 5.2% for the τ series channels otherwise Qmax
never overcomes the upper limit of (5.20).
By the way, how can the BG level b be estimated? MC simulation is the most common way, in which aparticular model is assumed for example SM. We know that this is not ideal way here as long as the test of
LHVT is considered because it is based on QM. However practically it is almost impossible to estimate BG
without assuming any single model even though some experimental fact can be used (more precisely, thereis even diﬃcult to know if it is really impossible). This is another loophole arising in the phase of analysis.
In practice, a certain model has to be assumed for BG in an empirical way, such as the ”uncorrelated BG”
discussed in (i) above. In fact this not an absurd assumption at all, considering it almost never happens thatthe BG sample realize the maximum correlation 2 described as the BG inequality (5.18) such as when all
π − ﬂies to the direction of a and all π + to the b. (5.18) is a theoretical rigorous inequality but compromise
too much.
Thus eventually, with this ”no correlation assumption” for BG, the inequality (3.12) is allowed to usedagain, and the impact of BG ends up in only the smearing eﬀect illustrated in (i).

96

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY

5.3

Eﬀect from contamination of the time-like events

As pointed out in section 2.3, the τ series channels will experience the contamination of the time-likeevents in which the decays of τ + and τ − are in the time-like conﬁguration in space time, especially thee+ e− → τ τ → πνπν which ends up in having 6% of the events time-like.
Similar to the rhetoric adopted above, the inequality that the time-like sample does not follow (3.11), however
the relation (2.21) and (2.22) should hold for both time-like and space-like events.
P (n|s) = 1 + α n · s
P (n′ |s′ ) = 1 − α n′ · s′ .
Thus the eventual restriction imposed on the time-like events is√| ⟨ na n′b ⟩ + ⟨ na n′d ⟩ + ⟨ nc n′b ⟩ − ⟨ nc n′d ⟩ | ≤ 2 2.

(5.24)

Therefore, the classical limit for the inequality that the sample with fraction t of the time-like events
contaminated in becomes
1 · (1 − t) +√2 · t,(5.25)

which is 1.025 for t = 0.06 in the e+ e− → τ τ → πνπν channel.
This eﬀect is included in calculating the sensitivity as in (5.16). Though the absolute eﬀect is small, thisconsequence is severe in the case.

5.4

Eﬀect from detector resolution

In this section, the eﬀect due to the ﬁnite detector resolution is studied. Qualitatively it will deﬁnitely smear
the correlation ⟨ni n′j ⟩, however, the impact is almost negligible as mentioned in section 5.1.1.
This is because the ni n′j distribution has a very large width which overwhelms the measurement error onn and n′ unless the resolution is terrible. Fig. 5.19 display the nz n′z distribution in the ηc → ΛΛ → pπpπchannel. The red and blue lines show the case without and with measurement error respectively. δ stands
for the resolution of nz and n′z . They are in perfect agreement up to δnz ≤ 0.1. However, the resolution
for a typical measurement on charged particle momentum and the direction are σpT /pT ∼ 1% and σθ , σϕ ∼10−3 (rad) respectively, which is presumed safely below δnz ≤ 0.1.
Fig. 5.20 shows the Qmax value dependence on δ, assuming a ﬁxed number of sample size N . The points
are the mean value and the error bars are the RMS (not the error of the mean). The eﬀect of measurement
error is tiny compared with the statistical ﬂuctuation.

δ=0
Smearing by Det. resolution 0%δ=0.1
Smearing by Det. resolution 10%
arb. unit

97

arb. unit

5.4. EFFECT FROM DETECTOR RESOLUTION

1

0.5

0.5

0
-1

1

-0.5

0

0.5

0
-1

1

-0.5

0

0.5

δ=0.2
Smearing by Det. resolution 20%

δ=0.5
Smearing by Det. resolution 50%

arb. unit

arb. unit

nz n’z

1

0.5

0
-1

1
nz n’z

1

0.5

-0.5

0

0.5

1

0
-1

-0.5

0

0.5

nz n’z1
nz n’zMean of Qmax

Figure 5.19: Simulated nz n′z distribution in the ηc → ΛΛ channel with (blue) and without (red) measurement
error. The measurement error is implemented by randomly multiplying a coeﬃcient to nz and n′z which
follows according to a Gaussian distribution with the RMS of δ for each event (”smearing”). Cases with
δ = 0, 0, 1, 0.2, 0, 5 are displayed, which shows almost no impact until δ = 0.1. The result is common to theother 4 channels as well.

1.8

bar: RMS of Qmax
1.6
1.4
1.2
&c / %c0 $ # # (2500evt)

1*

Z/( $ '+ '- (1600evt)
H $ '+ '- (100evt)

0.8
00

5
10
15
20
0.15
0.2
0.05
0.1
cos"! Resolution (%)δ

Figure 5.20: The dependence of Qmax on the smearing δ with a ﬁxed N . The points indicate the mean value
and the error bars are the RMS. The realistic value of δ here is expected well below 0.1, thus the eﬀect ofmean drop is negligible compared with the statistical error, as far as thess order of statistics is concerned.

98

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY

5.5

Available sample size with current and future experiment andthe feasibility

Last but not least, the realistic number of available sample size and the corresponding signiﬁcance is estimatedfor each channel. The conclusion plots are Fig. 5.13∼5.17. The colored band indicates the expected available
sample size with respect to each scenario.

5.5.1

χc0 , ηc → ΛΛExperimentally, ηc and χc0 are produced through the radiative deexcitation of J/ψ and ψ ′ respectively:
J/ψ → ηc → γ

ψ ′ → χc0 → γ

The branching of each decay in the later cascade ηc , χc0 → ΛΛ → pπpπ is displayed in Tab. 5.2.
Table 5.2: Branching ratio of each decay in cc̄ → ηc , χc0 → ΛΛ → pπpπChannelBranching fraction
J/ψ → ηc + γ(1.7 ± 0.4) × 10−2′
ψ → χc0 + γ(9.7 ± 0.3) × 10−2ηc → Λ + Λ(1.41 ± 0.17) × 10−3
χc0 → Λ + Λ(3.3 ± 0.4) × 10−4−Λ→p+π
(6.39 ± 0.05) × 10−1+
Λ→p+π
(6.39 ± 0.05) × 10−1
J/ψ → γηc ; ηc → ΛΛ → pπpπ(9.8 ± 2.6) × 10−6′
ψ → γχc0 ; χc0 → ΛΛ → pπpπ (1.31 ± 0.16) × 10−5

Both the ηc and χc0 mode maintain very tiny breaching fraction with the order of 10−5 . Hence substantially large number of ηc or χc0 sample is required. Only the charm factory experiments, for example
BES3 or CLEO, can potentially meet the demand. Either of them is e+ e− collision experiment and has beencollecting enormous statistics at J/ψ and ψ ′ resonance.
As for BES3, about 1.2 × 109 of J/ψ and 3.0 × 108 of ψ ′ are produced by the end of 2013. The number of
events of J/ψ → ηc γ; ηc → ΛΛ → pπpπ, ψ ′ → χc0 γ; χc0 → ΛΛ → pπpπ are therefore 1.2 × 104 and 4 × 103
respectively.
Considering the eﬃciency for detection, selection and the fraction of events where the decays Λ → pπ − andΛ → p̄π + is separated in space-like, the total eﬃciency is expected to 20 ∼ 30%, assuming the detector of
BES3. The eﬃciencies for each step throughout the event acquisition is listed in Tab. 5.3. The cos θΛ cut
is due to the limited angular acceptance of tracker. Tracking reconstruction eﬃciency is calculated as theprobability to have more than 3 tracks out of pπ p̄π, based on the oﬃcial number of tracking eﬃciency of
single charged particle from BES3 publication [79]. The cut of reconstructed mass is for vetoing backgrounds
since for signal events, the reconstructed mass mpπ , mp̄π and mpπp̄π should be around the mass shell of Λ,Λ and ηc (or χc0 ) respectively. Eﬃciency for the cuts is deducted from BES3 analysis of χc0 → ΛΛ [80].
Fraction of the space-like events is calculated in 2.3.4 and the result is exhibited in Tab. 2.2.
Finally, the available sample size of ηc and χc0 mode is found to be around 4000 and 1500 respectively,
which corresponds to roughly 3.5σ and 2σ of signiﬁcance.

5.5. AVAILABLE SAMPLE SIZE WITH CURRENT AND FUTURE EXPERIMENT AND THE FEASIBILITY99
Table 5.3: The eﬃciency ﬂow of ηc , χc0 → ΛΛ → pπ − p̄π + events assuming BES3. The bracket number isassumed.
| cos θΛ | < 0.93
Tracking reconstruction (3 & 4 prong)
Reconstructed mass cut
Space-like decays
Total

5.5.2

ηc → ΛΛ
0.86
0.95
(0.6)
0.66
0.32

χc0 → ΛΛ
0.86
0.98
(0.6)
0.76
0.38

Z → ττ

From the viewpoint of statistics of Z, LHC sounds like an ideal experiment, however it is proton-proton
collider thus the center of mass energy is unknown. Since one desires to measure n, n′ here which is theπ + π − direction in τ + τ − rest frame respectively, as pointed out in section 2.4, the 4 momentum of each tau
has to be reconstructed where the missing 4 momentum is required to be precisely grasped. Therefore forthe channels of τ series, e+ e− collider experiment is the only possible option. LEP is the most promising
candidate in its rich statistics of Z. It has 4 detectors along the ring (DELPHI; ALEPHE; L3; OPAL)
and each collected about 106 of e+ e− → Z events [81]. The list of branchings for respective decays in
Z → τ τ → πνπν is shown in Tab. 5.4. Thus about 400 Z → τ τ → πνπν events are generated per
experiment. The collection eﬃciency is typical ∼ 20% [82], the number of available event sample is expected
to be around 100 for one experiment, 400 for 4 experiments, which will give a low signiﬁcance ∆ = 0.4, 1.0,
corresponding to 75%CL, 85%CL respectively.

Table 5.4: Branching ratio of each decay in Z → τ − τ + → π − νπ + ν̄τChannelBranching fraction
Z → ττ
(3.37 ± 0.0008)%
τ − → π − ντ
(10.83 ± 0.0006)%
τ + → π + ν̄τ(10.83 ± 0.0006)%
Z → τ − τ + → π − νπ + ν̄τ (3.95 ± 0.04) × 10−4

Other than LEP, ILC could be the potential candidate of experiment. Though of course the Higgs
factory is the physics program of paramount importance in ILC, there is also a project called ”gigaZ” isbeing planned [53]. This is a optional run depending on the result of the Higgs factory, aiming at ultra-ﬁne
measurement of electro-weak physics. It is plan to generate 109 of e+ e− → Z events in ILC, which counts
103 times larger statistics than that in LEP. The corresponding signiﬁcance reaches 4σ − 5σ, realizing anobservation with high sensitivity.

100

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY

5.5.3

e+ e− → γ ∗ → τ τ√
According to the tree-level calculation, in order to have µ(Qmax ) > 1 in this channel, s > 8.6GeV is at least√needed and s = 10.58(GeV ) run of Belle experiment gives µ(Qmax ) = 1.03 which barely violates the BI.
Instead, this channel has by far abundant statistics. with the integrated luminosity L = 771fb−1 collectedby 2010 run, 7 × 108 of e+ e− → τ τ , 8 × 106 of e+ e− → τ τ → πνπν are produced.
The drawback of this channel is that it is terribly vulnerable against theoretical uncertainty or experimental eﬀects, (e.g. contamination of BG, detector resolution etc.) since the true value µ(Qmax ) is justslightly above the CL, it will easily be close to or underneath CL when the correlation are smeared. The
quantitative estimation of the sensitivity does not make any sense at the moment though, for reference, here
the result of the case where the higher order corrections do not contribute to Qmax is provided as Fig. 5.17.

The eﬀect of the contamination from the time-like events (evaluated in section 5.3) is included. Considering the realistic BG level is ∼ 2%, there are almost no sensitivity. Fig. 5.17 also shows that it is sensitive
to the deviation of theoretical Qmax value (the blue line represent the case where the higher order corrections
enhance Qmax by 2%).

5.5.4

H → τ τ → πνπνAs τ τ needs full reconstruction, ILC is the only candidate. For Higgs production, there is 2 main production√
mode; ZH associated production mode (Fig.5.21) and WW fusion mode (Fig. 5.22). In s = 250, 350GeV√
run, ZH channel is dominant while WW fusion are the main production process in high-energy runs ( s >500GeV) (Fig. 5.23). Here only the ZH channel is available since the WW fusion process generates additional
neutrinos which makes τ τ reconstruction impossible.

e+

Z

e+

Z*

ν
W
H

e-

H

Figure 5.21: The ZH associated production. The
dominant Higgs√ production channel at ILC low
energy runs ( s = 250, 350GeV).

W

e-

ν

Figure 5.22: The WW fusion. The dominant
Higgs √production channel at ILC high-energy
runs ( s = 500, 1000GeV).

5.5. AVAILABLE SAMPLE SIZE WITH CURRENT AND FUTURE EXPERIMENT AND THE FEASIBILITY101-

P(e , e+)=(-0.8, 0.3), M =125 GeV

400

and

SM all f f h
Zh

Cross section (fb)center of mass energyhWW fusion

300

ZZ fusion

200
100
0
200 250 300 350 400 450 500

s (GeV)

Figure 5.23: Cross-sections for each Higgs production channels in ILC.
Analysis of the ZH channel is still challenging in many aspects. First, reconstruction of τ τ requires
precise reconstruction of the missing 4 momentum. In the ZH case, this means that energy and momentumof the recoiled Z has also to be reconstructed in precision. Though it is easy when Z decays into lepton
pairs as Z → ee, µµ but those accounts for only 6% of the branchings. From statistical point of view, eventsof Z → q q̄ (braching: 70%) are inevitably counted on.
Fortunately, this is also the requirement from the original ILC physics program, dedicated detector
systems [83] and reconstruction algorithms (”Particle Flow Algorithm” [84] etc.) have been developed,
specialize for precise measurement of q q̄ jets from Z or W bosons. The up-to-date ILD full simulation
indicates typically σ/E ∼ 4% of single jet energy resolution can be achieved. (Fig. 5.24). As for momentum,
an approximate study is carried out using the sample of Z → q q̄ where the Z is at rest with respect tolaboratory frame, to test if the 3 momentum can be reconstructed to 0. The result is shown in Fig. 5.25.
Each component of the reconstructed 3 momentum peaks at the true value 0 but does have residual by 2GeV
on average. Though the Z boson appearing in the ZH process is not at rest, from this result the momentum
resolution of the recoiled Z can be inferred to the order of a few GeV, which is the precision of missing
momentum as well. Rigid numerical estimation of the impact due to the imperfect precision of the missing
4 momentum on τ τ reconstruction or Qmax is still under study. However as discussed later in section 5.4,
this order of measurement precision is suﬃcient that the eﬀect is supposed to be negligible.

CHAPTER 5. DISCUSSION ON EXPERIMENTAL FEASIBILITY
rms90/Ejet [%]

102

Total
Resolution
Confusion

4

Other
Leakage

3

2

1

0

0

50

100

150

200

250

EJET/GeV

Figure 5.24: Distribution of reconstructed energy of 2 jets from W → q q̄ and Z → q q̄ events (left) and thesingle jet energy resolution in the function of jet energy (right).[84].

hyhz

Entries5472
Mean
0.07563
RMS
2.099
Underflow
9
Overflow
7

Entries5472
Mean-0.01134
RMS
1.807
Underflow
4
Overflow
3

hx

300

Entries

5472

Mean

0.02436

RMS

2.126

Underflow

10

Overflow

10

200

300

200

300

200
100

100

0
-10

-5

0

5

100

0
-10

10

-5

0

2

0
-10

10

4
2

0

-2

-2

-2

-4
-2

0

2

4

Px (GeV)

5

10

2

0

-4

0

4

0

-4

-5

Pz (GeV)

Pz (GeV)

Pz (GeV)Py (GeV)45

Py (GeV)Px (GeV)

-4
-4

-2

0

2

4

Py (GeV)

-4

-2

0

2

4

Px (GeV)Figure 5.25: Reconstructed momentum of Z by summing the momenta of the particles in the ﬁnal state of
Z → q q̄ for x, y, z components (upper raw). The distributions are settled at zero, with sharp Gaussianic
peaks (red line is the ﬁt) and long tails. The RMS is typically 2GeV for each component. The lower raws
show absence of the correlation between them.
Finally, as done in the other channels above, the number of aﬀordable event sample are estimated. The
branching of each decay of H → τ τ → πνπν is listed as Tab. 5.5). The total number of Higgs depends
on run projections. Here 3 scenarios are assumed where the integrated luminosity is 250fb−1 ; 1000fb−1 ;2000fb−1 respectively and the center of mass energies are all 250GeV. The number of produced Higgs is
8 × 104 , 32 × 104 and 64 × 104 and the expected number of events for H → τ τ → πνπν is around 45, 180,
and 360 respectively. The collection eﬃciency in current analysis is 25% [78], however it still leaves much

5.5. AVAILABLE SAMPLE SIZE WITH CURRENT AND FUTURE EXPERIMENT AND THE FEASIBILITY103
Table 5.5: Branching ratio for each decay in H → τ τ → πνπν. The number for H → τ τ is calculated fromSM with Higgs mass of 125 GeV assumed.
channelbranching fraction
H → ττ
(6.32 ± 0.36)%
τ − → π − ντ
(10.83 ± 0.0006)%
τ + → π + ν̄τ(10.83 ± 0.0006)%
Z → q q̄, e+ e− , µ+ µ−(76.6 ± 0.06)%
e+ e− → Z ∗ → ZH; H → τ − τ + → π − νπ + ν̄τ (5.70 ± 0.33) × 10−4

to be improved and this number can be supposed to rise, at maximally optimistic expectation (including
some anticipation), up to around 50% in future. With the assumption, the available sample size will be 20,
90, and 180, which lead to only poor sensitivity that the violation of the BI can be conﬁrmed with 80%CLwith 1000fb−1 , 95%CL with 2000fb−1 , if ignoring the contamination of BG. Here the impact of BG therefore
a realistic number of signiﬁcance should be something like 68%CL with 2000fb−1 , assuming 15% of BGlevel. Nevertheless the author thinks that this is a very interesting result in that this is the ﬁrst attempt at
application of Higgs boson.

Chapter 6

Conclusion
Bell inequality is an inequality that lets us learn whether the nature prefers classical physics or quantum
ones, as well as signatures of quantum non-locality. While the violation has been extensively veriﬁed inphoton experiments, tests with massive particles are still impotent which however have only few examples
yet.
The particle decays in high energy collider experiments have been supposed to be new channels of thetesting Bell inequality, using the spin entanglement from decays of particle, and the magical spin measurement
scheme with weak decays provided by the nature, which enables the test with various types of systems andinteractions as well.
On the basis of previous studies, this paper provided (i) a comprehensively review of the theoreticalbackground, (ii) a formal establishment of the method which is free from the locality loophole for the ﬁrst
case in non-optical experiments, (iii) a formulation of a new Bell inequality, (iv) and a discussion on thefeasibility of testing the Bell inequality by this method, with today’s high speciﬁcation collider machines, asa new proposition of the experiment.
In particular, the six decays below are examined:• ηc → ΛΛ → pπ − p̄π +• χc0 → ΛΛ → pπ − p̄π +• J/ψ → ΛΛ → pπ − p̄π +• e+ e− → γ ∗ → τ + τ − → π − ντ π + ν̄τ• Z → τ + τ − → π − ντ π + ν̄τ• H → τ + τ − → π − ντ π + ν̄τto violate the Bell inequality
Qmax =p

λ1 + λ2

Qmax ≤ 1
in the quantum mechanics.

105

106

CHAPTER 6. CONCLUSION
The result of the analyses are √2(ηc , χc0 → ΛΛ → pπ − p̄π + , H → τ + τ − → π − ντ π + ν̄τ )










0.976 ± 0.048(J/ψ → ΛΛ → pπ − p̄π + )Qmax =√

5/2(Z → τ + τ − → π − ντ π + ν̄τ )






√



4m2τ
5 − 2Γ2 + 2Γ4; Γ2 :=(e+ e− → γ ∗ → τ + τ − → π − ντ π + ν̄τ ).
22+Γs

The conclusions are that it is in principle possible to conﬁrm the violation of the inequality
for the all channels except J/ψ → ΛΛ → pπ − p̄π + and, with studies on experimental aspects, thatcurrent experiments can already aﬀord the statistics to observe the violation in the channels
ηc , χc0 → ΛΛ → pπ − p̄π + , with the signiﬁcance of 2σ and 3σ respectively.

107
Maybe knowledge is as fundamental, or even more fundamental than reality.
Anton Zeillinger

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