**4.5 CP violation in the** *<sup>K</sup>***<sup>0</sup>** <sup>−</sup> *<sup>K</sup>***¯ <sup>0</sup> system**

Gell-Mann and Pais (1955) considered the behavior of neutral particles under the charge-conjugation operator C. In particular they considered the *K*<sup>0</sup> meson and realized that unlike the photon and the neutral pion, which transform into themselves under the C operator so that they are their own antiparticles, the antiparticle of the *K*<sup>0</sup> meson (strangeness *S* = +1), *<sup>K</sup>*¯ 0, was a distinct particle, since it had a different strangeness quantum number (*<sup>S</sup>* <sup>=</sup> <sup>−</sup>1). They concluded that the two neutral mesons, *K*<sup>0</sup> and *K*¯ 0, are degenerate particles that exhibit unusual properties, since they can transform into each other via weak interactions such as

$$K^0 \rightleftharpoons \pi^+ \pi^- \rightleftharpoons \mathbb{R}^0. \tag{42}$$

In order to treat this novel situation, Gell-Mann and Pais suggested that it was more convenient to employ different particle states, rather than *K*<sup>0</sup> and *K*¯ 0, to describe neutral kaon decay. They suggested the following representative states:

$$K\_1^0 = (\mathbf{k}^0 + \bar{\mathbf{k}}^0) / \sqrt{2} \quad K\_2^0 = (\mathbf{k}^0 - \bar{\mathbf{k}}^0) / \sqrt{2} \tag{43}$$

and concluded that these particle states must have different decay modes and lifetimes. In particular they concluded that *K*<sup>0</sup> <sup>1</sup> could decay to two charged pions, while *<sup>K</sup>*<sup>0</sup> <sup>2</sup> would have a longer lifetime and more complex decay modes. This conclusion was based upon the conservation of C in the weak interaction processes: both *K*<sup>0</sup> <sup>1</sup> and the *<sup>π</sup>*+*π*<sup>−</sup> system are even (i.e. *C* = +1) under the C operation.

The particle-mixing theory of Gell-Mann and Pais was confirmed in 1957 by experiment, in spite of the incorrect assumption of C invariance in weak interaction processes. Following the discovery in 1957 of both C and P violation in weak interaction processes, the particle-mixing theory led to a suggestion by Landau (1957) that the weak interactions may be invariant under the combined operation CP.

Landau's suggestion implied that the Gell-Mann–Pais model of neutral kaons would still apply if the states, *K*<sup>0</sup> <sup>1</sup> and *<sup>K</sup>*<sup>0</sup> <sup>2</sup>, were eigenstates of CP with eigenvalues +1 and −1, respectively. Since the charged pions were considered to have intrinsic parity *P<sup>π</sup>* = −1, it was clear that only the *K*<sup>0</sup> <sup>1</sup> state could decay to two charged pions, if CP was conserved.

The suggestion of Landau was accepted for several years since it nicely restored some degree of symmetry in weak interaction processes. However, the surprising discovery (Christenson

together simultaneously, one can calculate that this probability is given by (sin *θ<sup>c</sup>* cos *θc*)<sup>4</sup> = 2.34 <sup>×</sup> <sup>10</sup>−3, using cos *<sup>θ</sup><sup>c</sup>* = 0.9742 (Amsler *et al.*, 2008). Thus, the existence of a small

The Generation Model of Particle Physics 25

decay to the charged 2*π* system without violating CP conservation. Moreover, the estimated

The GM, which contains fewer elementary particles (27 counting both particles and antiparticles and their three different color forms) and only two fundamental interactions (the electromagnetic and strong color interactions), has been presented as a viable simpler

In addition, the GM has provided new paradigms for particle physics, which have led to a new understanding of several phenomena not addressed by the SM. In particular, (i) the mass of a particle is attributed to the energy content of its constituents so that there is no requirement for the Higgs mechanism; (ii) the mass hierarchy of the three generations of leptons and quarks is described by the degree of localization of their constituent rishons and/or antirishons; (iii) gravity is interpreted as a quantum mechanical residual interaction of the strong color interaction, which binds rishons and/or antirishons together to form all kinds of matter and (iv) the decay of the long-lived neutral kaon is understood in terms of mixed-quark states in

The GM also predicts that the mass of a free neutron is greater than the mass of a free proton so that the free proton is stable. In addition, the model predicts the existence of higher generation quarks in hadrons, which in turn predicts mixed-parity states in hadrons. Further experimentation is required to verify these predictions and thereby strengthen the Generation

Abouzaid, E. *et al.* (2008), Determination of the Parity of the Neutral Pion via its Four-Electron

Aitchison I.J.R. and Hey, A.J.G. (1982), *Gauge Theories in Particle Physics* (Adam Hilger Ltd,

Amsler, C. *et al.* (2008), Summary Tables of Particle Properties, *Physics Letters B*, Vol. 667, Nos.

Aoki, S. *et al.* (2000), Quenched Light Hadron Spectrum, *Physical Review Letters*, Vol. 84, No. 2,

Armstrong, D.S. *et al.* (2005), Strange-Quark Contributions to Parity-Violating Asymmetries in

Arnison, G. *et al.* (1983), Experimental Observation of Isolated Large Transverse Energy

Banner, M. *et al.* (1983), Observation of Single Isolated Electrons of High Transverse

the Forward G0 Electron-Proton Scattering Experiment, *Physical Review Letters*, Vol.

Electrons with Associated Missing Energy, *Physics Letters B*, Vol 122, No. 1, pp.

Momentum in Events with Missing Transverse Energy at the CERN pp Collider,

Decay, *Physical Review Letters*, Vol. 100, No. 18, 182001 (5 pages).

*<sup>L</sup>* meson can

component of the *π*+*π*<sup>−</sup> system with eigenvalue CP = -1 indicates that the *K*<sup>0</sup>

alternative to the SM (61 elementary particles and four fundamental interactions).

decay rate is in good agreement with experimental data (Amsler *et al.*, 2008).

**5. Summary and future prospects**

hadrons and not CP violation.

Model.

**6. References**

Bristol).

1-5, pp. 31-100.

95, No. 9, 092001 (5 pages).

*Physics Letters B*, Vol. 122, Nos. 5-6, pp. 476-485.

pp. 238-241.

103-116.

*et al.*, 1964) of the decay of the long-lived neutral *K*<sup>0</sup> meson to two charged pions led to the conclusion that CP is violated in the weak interaction. The observed violation of CP conservation turned out to be very small (≈ 0.2%) compared with the maximal violations (≈ 100%) of both P and C conservation separately. Indeed the very smallness of the apparent CP violation led to a variety of suggestions explaining it in a CP-conserving way (Kabir, 1968; Franklin, 1986). However, these efforts were unsuccessful and CP violation in weak interactions was accepted.

An immediate consequence of this was that the role of *K*<sup>0</sup> <sup>1</sup> (CP = <sup>+</sup>1) and *<sup>K</sup>*<sup>0</sup> <sup>2</sup> (CP = −1), defined in Eqs. (43), was replaced by two new particle states, corresponding to the short-lived (*K*<sup>0</sup> *<sup>S</sup>*) and long-lived (*K*<sup>0</sup> *<sup>L</sup>*) neutral kaons:

$$K\_S^0 = (K\_1^0 + \epsilon K\_2^0) / (1 + |\epsilon|^2)^{\frac{1}{2}} \text{ , } \ K\_L^0 = (K\_2^0 + \epsilon K\_1^0) / (1 + |\epsilon|^2)^{\frac{1}{2}} \text{ , } \tag{44}$$

where the small complex parameter *�* is a measure of the CP impurity in the eigenstates *K*<sup>0</sup> *S* and *K*<sup>0</sup> *<sup>L</sup>*. This method of describing CP violation in the Standard Model (SM), by introducing mixing of CP eigenstates, is called 'indirect CP violation'. It is essentially a phenomenological approach with the parameter *�* to be determined by experiment.

Another method of introducing CP violation into the SM was proposed by Kobayashi and Maskawa (1973). By extending the idea of 'Cabibbo mixing' (see Subsection 2.2.3) to three generations, they demonstrated that this allowed a complex phase to be introduced into the quark-mixing (CKM) matrix, permitting CP violation to be directly incorporated into the weak interaction. This phenomenological method has within the framework of the SM successfully accounted for both the indirect CP violation discovered by Christenson *et al.* in 1964 and the "direct CP violation" related to the decay processes of the neutral kaons (Kleinknecht, 2003). However, to date, the phenomenological approach has not been able to provide an *a priori* reason for CP violation to occur nor to indicate the magnitude of any such violation.

Recently, Morrison and Robson (2009) have demonstrated that the indirect CP violation observed by Christenson *et al.* (1964) can be described in terms of mixed-quark states in hadrons. In addition, the rate of the decay of the *K*<sup>0</sup> *<sup>L</sup>* meson relative to the decay into all charged modes is estimated accurately in terms of the Cabibbo-mixing angle.

In the CGM the *K*<sup>0</sup> and *K*¯ <sup>0</sup> mesons have the weak eigenstate quark structures [*d*� *s*¯ � ] and [*s*� ¯*d*� ], respectively. Neglecting the very small mixing components arising from the third generation, Morrison and Robson show that the long-lived neutral kaon, *K*<sup>0</sup> *<sup>L</sup>*, exists in a CP = -1 eigenstate as in the SM. On the other hand, the charged 2*π* system:

$$\begin{split} \pi^{+}\pi^{-} &= [\imath d\vec{l}][d\'\imath\mathbb{I}] \\ &= [\imath d\vec{l}][d\imath]\cos^{2}\theta\_{\c} + [\imath\kappa][\imath\imath\mathbb{I}]\sin^{2}\theta\_{\c} + [\imath\kappa][d\imath]\sin\theta\_{\c}\cos\theta\_{\c} \\ &+ [\imath\imath d][\imath\imath\mathbb{I}])\sin\theta\_{\c}\cos\theta\_{\c} . \end{split} \tag{45}$$

For the assumed parities (see Subsection 4.4) of the quarks and antiquarks involved in Eq. (45), it is seen that the first two components are eigenstates of CP = +1, while the remaining two components [*us*¯][*du*¯] and [*u* ¯*d*][*su*¯], with amplitude sin *θ<sup>c</sup>* cos *θ<sup>c</sup>* are not individually eigenstates of CP. However, taken together, the state ([*us*¯][*du*¯]+[*u* ¯*d*][*su*¯]) is an eigenstate of CP with eigenvalue CP = -1. Taking the square of the product of the amplitudes of the two components comprising the CP = -1 eigenstate to be the "joint probability" of those two states existing 24 Will-be-set-by-IN-TECH

*et al.*, 1964) of the decay of the long-lived neutral *K*<sup>0</sup> meson to two charged pions led to the conclusion that CP is violated in the weak interaction. The observed violation of CP conservation turned out to be very small (≈ 0.2%) compared with the maximal violations (≈ 100%) of both P and C conservation separately. Indeed the very smallness of the apparent CP violation led to a variety of suggestions explaining it in a CP-conserving way (Kabir, 1968; Franklin, 1986). However, these efforts were unsuccessful and CP violation in weak

in Eqs. (43), was replaced by two new particle states, corresponding to the short-lived (*K*<sup>0</sup>

where the small complex parameter *�* is a measure of the CP impurity in the eigenstates *K*<sup>0</sup>

Another method of introducing CP violation into the SM was proposed by Kobayashi and Maskawa (1973). By extending the idea of 'Cabibbo mixing' (see Subsection 2.2.3) to three generations, they demonstrated that this allowed a complex phase to be introduced into the quark-mixing (CKM) matrix, permitting CP violation to be directly incorporated into the weak interaction. This phenomenological method has within the framework of the SM successfully accounted for both the indirect CP violation discovered by Christenson *et al.* in 1964 and the "direct CP violation" related to the decay processes of the neutral kaons (Kleinknecht, 2003). However, to date, the phenomenological approach has not been able to provide an *a priori*

reason for CP violation to occur nor to indicate the magnitude of any such violation.

charged modes is estimated accurately in terms of the Cabibbo-mixing angle. In the CGM the *K*<sup>0</sup> and *K*¯ <sup>0</sup> mesons have the weak eigenstate quark structures [*d*�

Recently, Morrison and Robson (2009) have demonstrated that the indirect CP violation observed by Christenson *et al.* (1964) can be described in terms of mixed-quark states in

respectively. Neglecting the very small mixing components arising from the third generation,

For the assumed parities (see Subsection 4.4) of the quarks and antiquarks involved in Eq. (45), it is seen that the first two components are eigenstates of CP = +1, while the remaining two components [*us*¯][*du*¯] and [*u* ¯*d*][*su*¯], with amplitude sin *θ<sup>c</sup>* cos *θ<sup>c</sup>* are not individually eigenstates of CP. However, taken together, the state ([*us*¯][*du*¯]+[*u* ¯*d*][*su*¯]) is an eigenstate of CP with eigenvalue CP = -1. Taking the square of the product of the amplitudes of the two components comprising the CP = -1 eigenstate to be the "joint probability" of those two states existing

= [*u* ¯*d*][*du*¯] cos<sup>2</sup> *θ<sup>c</sup>* + [*us*¯][*su*¯] sin2 *θ<sup>c</sup>* + [*us*¯][*du*¯] sin *θ<sup>c</sup>* cos *θ<sup>c</sup>*

+[*u* ¯*d*][*su*¯]) sin *θ<sup>c</sup>* cos *θ<sup>c</sup>* . (45)

*<sup>L</sup>*. This method of describing CP violation in the Standard Model (SM), by introducing mixing of CP eigenstates, is called 'indirect CP violation'. It is essentially a phenomenological

*<sup>L</sup>* = (*K*<sup>0</sup>

<sup>2</sup> <sup>+</sup> *�K*<sup>0</sup>

2) 1 <sup>2</sup> , *K*<sup>0</sup> <sup>1</sup> (CP = <sup>+</sup>1) and *<sup>K</sup>*<sup>0</sup>

<sup>1</sup>)/(1 + |*�*|

2) 1

*<sup>L</sup>* meson relative to the decay into all

*s*¯ �

*<sup>L</sup>*, exists in a CP = -1 eigenstate

] and [*s*� ¯*d*�

],

<sup>2</sup> (CP = −1), defined

<sup>2</sup> , (44)

*<sup>S</sup>*) and

*S*

interactions was accepted.

*K*0 *<sup>S</sup>* = (*K*<sup>0</sup>

long-lived (*K*<sup>0</sup>

and *K*<sup>0</sup>

An immediate consequence of this was that the role of *K*<sup>0</sup>

<sup>1</sup> <sup>+</sup> *�K*<sup>0</sup>

hadrons. In addition, the rate of the decay of the *K*<sup>0</sup>

as in the SM. On the other hand, the charged 2*π* system:

*π*+*π*<sup>−</sup> = [*u* ¯*d*�

Morrison and Robson show that the long-lived neutral kaon, *K*<sup>0</sup>

][*d*� *u*¯]

<sup>2</sup>)/(1 + |*�*|

approach with the parameter *�* to be determined by experiment.

*<sup>L</sup>*) neutral kaons:

together simultaneously, one can calculate that this probability is given by (sin *θ<sup>c</sup>* cos *θc*)<sup>4</sup> = 2.34 <sup>×</sup> <sup>10</sup>−3, using cos *<sup>θ</sup><sup>c</sup>* = 0.9742 (Amsler *et al.*, 2008). Thus, the existence of a small component of the *π*+*π*<sup>−</sup> system with eigenvalue CP = -1 indicates that the *K*<sup>0</sup> *<sup>L</sup>* meson can decay to the charged 2*π* system without violating CP conservation. Moreover, the estimated decay rate is in good agreement with experimental data (Amsler *et al.*, 2008).
