**2. Standard model of particle physics**

The Standard Model (SM) of particle physics (Gottfried and Weisskopf, 1984) was developed throughout the 20th century, although the current formulation was essentially finalized in the mid-1970s following the experimental confirmation of the existence of quarks (Bloom *et al.*, 1969; Breidenbach *et al.*, 1969).

The SM has enjoyed considerable success in describing the interactions of leptons and the multitude of hadrons (baryons and mesons) with each other as well as the decay modes of the unstable leptons and hadrons. However the model is considered to be incomplete in the sense that it provides no understanding of several empirical observations such as: the existence of three families or generations of leptons and quarks, which apart from mass have similar properties; the mass hierarchy of the elementary particles, which form the basis of the SM; the nature of the gravitational interaction and the origin of CP violation.

called red, green and blue quarks. The antiquarks carry anticolors, which for simplicity are called antired, antigreen and antiblue. Each quark or antiquark carries a single unit of color or anticolor charge, respectively. The leptons do not carry a color charge and consequently do not participate in the strong interactions, which occur between particles carrying color charges.

The Generation Model of Particle Physics 3

The SM recognizes four fundamental interactions in nature: strong, electromagnetic, weak and gravity. Since gravity plays no role in particle physics because it is so much weaker than the other three fundamental interactions, the SM does not attempt to explain gravity. In the SM the other three fundamental interactions are assumed to be associated with a local gauge

The strong interactions, mediated by massless neutral spin-1 gluons between quarks carrying a color charge, are described by an *SU*(3) local gauge theory called quantum chromodynamics (QCD) (Halzen and Martin, 1984). There are eight independent kinds of gluons, each of which carries a combination of a color charge and an anticolor charge (e.g. red-antigreen). The strong interactions between color charges are such that in nature the quarks (antiquarks) are grouped into composites of either three quarks (antiquarks), called baryons (antibaryons), each having a different color (anticolor) charge or a quark-antiquark pair, called mesons, of opposite color charges. In the *SU*(3) color gauge theory each baryon, antibaryon or meson is colorless. However, these colorless particles, called hadrons, may interact strongly via residual strong interactions arising from their composition of colored quarks and/or antiquarks. On the other hand the colorless leptons are assumed to be structureless in the SM and consequently do not

The electromagnetic interactions, mediated by massless neutral spin-1 photons between electrically charged particles, are described by a *U*(1) local gauge theory called quantum

The weak interactions, mediated by the massive *W*+, *W*<sup>−</sup> and *Z*<sup>0</sup> vector bosons between all the elementary particles of the SM, fall into two classes: (i) charge-changing (CC) weak interactions involving the *W*<sup>+</sup> and *W*<sup>−</sup> bosons and (ii) neutral weak interactions involving the *Z*<sup>0</sup> boson. The CC weak interactions, acting exclusively on left-handed particles and right-handed antiparticles, are described by an *SU*(2)*<sup>L</sup>* local gauge theory, where the subscript *L* refers to left-handed particles only (Halzen and Martin, 1984). On the other hand, the neutral weak interactions act on both left-handed and right-handed particles, similar to the electromagnetic interactions. In fact the SM assumes (Glashow, 1961) that both the *Z*<sup>0</sup> and the photon (*γ*) arise from a mixing of two bosons, *W*<sup>0</sup> and *B*0, via an electroweak mixing angle

> *γ* = *B*<sup>0</sup> cos *θ<sup>W</sup>* + *W*<sup>0</sup> sin *θ<sup>W</sup>* , (1) *<sup>Z</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>*B*<sup>0</sup> sin *<sup>θ</sup><sup>W</sup>* <sup>+</sup> *<sup>W</sup>*<sup>0</sup> cos *<sup>θ</sup><sup>W</sup>* . (2)

**2.2 Fundamental interactions of the SM**

field.

**2.2.1 Strong interactions**

participate in strong interactions.

**2.2.2 Electromagnetic interactions**

**2.2.3 Weak interactions**

*θW*:

electrodynamics (Halzen and Martin, 1984).

In this section a summary of the current formulation of the SM is presented: the elementary particles and the fundamental interactions of the SM, and then the basic problem inherent in the SM.

#### **2.1 Elementary particles of the SM**

In the SM the elementary particles that are the constituents of matter are assumed to be the six leptons: electron neutrino (*νe*), electron (*e*−), muon neutrino (*νμ*), muon (*μ*−), tau neutrino (*ντ*), tau (*τ*−) and the six quarks: up (*u*), down (*d*), charmed (*c*), strange (*s*), top (*t*) and bottom (*b*), together with their antiparticles. These twelve particles are all spin- <sup>1</sup> <sup>2</sup> particles and fall naturally into three families or generations: (i) *νe*, *e*−, *u*, *d* ; (ii) *νμ*, *μ*−, *c*, *s* ; (iii) *ντ*, *τ*−, *t*, *b* . Each generation consists of two leptons with charges *Q* = 0 and *Q* = −1 and two quarks with charges *Q* = +<sup>2</sup> <sup>3</sup> and *<sup>Q</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>3</sup> . The masses of the particles increase significantly with each generation with the possible exception of the neutrinos, whose very small masses have yet to be determined.

In the SM the leptons and quarks are allotted several additive quantum numbers: charge *Q*, lepton number *L*, muon lepton number *Lμ*, tau lepton number *Lτ*, baryon number *A*, strangeness *S*, charm *C*, bottomness *B* and topness *T*. These are given in Table 1. For each particle additive quantum number *N*, the corresponding antiparticle has the additive quantum number −*N*.


Table 1. SM additive quantum numbers for leptons and quarks

Table 1 demonstrates that, except for charge, leptons and quarks are allotted different kinds of additive quantum numbers so that this classification of the elementary particles in the SM is *non-unified*.

The additive quantum numbers *Q* and *A* are assumed to be conserved in strong, electromagnetic and weak interactions. The lepton numbers *L*, *L<sup>μ</sup>* and *L<sup>τ</sup>* are not involved in strong interactions but are strictly conserved in both electromagnetic and weak interactions. The remainder, *S*, *C*, *B* and *T* are strictly conserved only in strong and electromagnetic interactions but can undergo a change of one unit in weak interactions.

The quarks have an additional additive quantum number called "color charge", which can take three values so that in effect we have three kinds of each quark, *u*, *d*, etc. These are often 2 Will-be-set-by-IN-TECH

In this section a summary of the current formulation of the SM is presented: the elementary particles and the fundamental interactions of the SM, and then the basic problem inherent in

In the SM the elementary particles that are the constituents of matter are assumed to be the six leptons: electron neutrino (*νe*), electron (*e*−), muon neutrino (*νμ*), muon (*μ*−), tau neutrino (*ντ*), tau (*τ*−) and the six quarks: up (*u*), down (*d*), charmed (*c*), strange (*s*), top (*t*) and bottom

naturally into three families or generations: (i) *νe*, *e*−, *u*, *d* ; (ii) *νμ*, *μ*−, *c*, *s* ; (iii) *ντ*, *τ*−, *t*, *b* . Each generation consists of two leptons with charges *Q* = 0 and *Q* = −1 and two quarks with

generation with the possible exception of the neutrinos, whose very small masses have yet to

In the SM the leptons and quarks are allotted several additive quantum numbers: charge *Q*, lepton number *L*, muon lepton number *Lμ*, tau lepton number *Lτ*, baryon number *A*, strangeness *S*, charm *C*, bottomness *B* and topness *T*. These are given in Table 1. For each particle additive quantum number *N*, the corresponding antiparticle has the additive

> particle *QLL<sup>μ</sup> L<sup>τ</sup> ASCBT ν<sup>e</sup>* 01 0 0 00000 *e*<sup>−</sup> −11 0 0 0 0 0 0 0 *νμ* 01 1 0 00000 *μ*<sup>−</sup> −11 1 0 0 0 0 0 0 *ντ* 01 0 1 00000 *τ*<sup>−</sup> −11 0 1 0 0 0 0 0

<sup>3</sup> 00 0 <sup>1</sup>

<sup>3</sup> 00 0 <sup>1</sup>

<sup>3</sup> 00 0 <sup>1</sup>

<sup>3</sup> 00 0 <sup>1</sup>

<sup>3</sup> 00 0 <sup>1</sup>

<sup>3</sup> 00 0 <sup>1</sup>

Table 1 demonstrates that, except for charge, leptons and quarks are allotted different kinds of additive quantum numbers so that this classification of the elementary particles in the SM

The additive quantum numbers *Q* and *A* are assumed to be conserved in strong, electromagnetic and weak interactions. The lepton numbers *L*, *L<sup>μ</sup>* and *L<sup>τ</sup>* are not involved in strong interactions but are strictly conserved in both electromagnetic and weak interactions. The remainder, *S*, *C*, *B* and *T* are strictly conserved only in strong and electromagnetic

The quarks have an additional additive quantum number called "color charge", which can take three values so that in effect we have three kinds of each quark, *u*, *d*, etc. These are often

<sup>3</sup> . The masses of the particles increase significantly with each

<sup>3</sup> 0000

<sup>3</sup> 0000

<sup>3</sup> 0100

<sup>3</sup> −10 0 0

<sup>3</sup> 0001

<sup>3</sup> 0 0 −1 0

<sup>2</sup> particles and fall

(*b*), together with their antiparticles. These twelve particles are all spin- <sup>1</sup>

the SM.

charges *Q* = +<sup>2</sup>

be determined.

is *non-unified*.

quantum number −*N*.

**2.1 Elementary particles of the SM**

<sup>3</sup> and *<sup>Q</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup>

*u* +<sup>2</sup>

*<sup>d</sup>* <sup>−</sup><sup>1</sup>

*c* +<sup>2</sup>

*<sup>s</sup>* <sup>−</sup><sup>1</sup>

*t* +<sup>2</sup>

*<sup>b</sup>* <sup>−</sup><sup>1</sup>

Table 1. SM additive quantum numbers for leptons and quarks

interactions but can undergo a change of one unit in weak interactions.

called red, green and blue quarks. The antiquarks carry anticolors, which for simplicity are called antired, antigreen and antiblue. Each quark or antiquark carries a single unit of color or anticolor charge, respectively. The leptons do not carry a color charge and consequently do not participate in the strong interactions, which occur between particles carrying color charges.

### **2.2 Fundamental interactions of the SM**

The SM recognizes four fundamental interactions in nature: strong, electromagnetic, weak and gravity. Since gravity plays no role in particle physics because it is so much weaker than the other three fundamental interactions, the SM does not attempt to explain gravity. In the SM the other three fundamental interactions are assumed to be associated with a local gauge field.

### **2.2.1 Strong interactions**

The strong interactions, mediated by massless neutral spin-1 gluons between quarks carrying a color charge, are described by an *SU*(3) local gauge theory called quantum chromodynamics (QCD) (Halzen and Martin, 1984). There are eight independent kinds of gluons, each of which carries a combination of a color charge and an anticolor charge (e.g. red-antigreen). The strong interactions between color charges are such that in nature the quarks (antiquarks) are grouped into composites of either three quarks (antiquarks), called baryons (antibaryons), each having a different color (anticolor) charge or a quark-antiquark pair, called mesons, of opposite color charges. In the *SU*(3) color gauge theory each baryon, antibaryon or meson is colorless. However, these colorless particles, called hadrons, may interact strongly via residual strong interactions arising from their composition of colored quarks and/or antiquarks. On the other hand the colorless leptons are assumed to be structureless in the SM and consequently do not participate in strong interactions.

#### **2.2.2 Electromagnetic interactions**

The electromagnetic interactions, mediated by massless neutral spin-1 photons between electrically charged particles, are described by a *U*(1) local gauge theory called quantum electrodynamics (Halzen and Martin, 1984).

#### **2.2.3 Weak interactions**

The weak interactions, mediated by the massive *W*+, *W*<sup>−</sup> and *Z*<sup>0</sup> vector bosons between all the elementary particles of the SM, fall into two classes: (i) charge-changing (CC) weak interactions involving the *W*<sup>+</sup> and *W*<sup>−</sup> bosons and (ii) neutral weak interactions involving the *Z*<sup>0</sup> boson. The CC weak interactions, acting exclusively on left-handed particles and right-handed antiparticles, are described by an *SU*(2)*<sup>L</sup>* local gauge theory, where the subscript *L* refers to left-handed particles only (Halzen and Martin, 1984). On the other hand, the neutral weak interactions act on both left-handed and right-handed particles, similar to the electromagnetic interactions. In fact the SM assumes (Glashow, 1961) that both the *Z*<sup>0</sup> and the photon (*γ*) arise from a mixing of two bosons, *W*<sup>0</sup> and *B*0, via an electroweak mixing angle *θW*:

$$\gamma = B^0 \cos \theta\_W + W^0 \sin \theta\_W \,\,\,\,\,\tag{1}$$

$$Z^0 = -B^0 \sin \theta\_W + W^0 \cos \theta\_W \,. \tag{2}$$

amplitude. Lepton number conservation gives

experiment.

In the SM neutron *β*-decay:

Similarly, Λ<sup>0</sup> *β*-decay:

is interpreted as the sequential transition

*a*(*νe*, *μ*−; *W*−) = *a*(*νμ*,*e*

and *T* are not necessarily conserved in CC weak interaction processes.

found to be slightly weaker (≈ 0.95) than that for muon decay:

is interpreted in the SM as the sequential transition

significantly less (≈ 0.05) than that for muon decay.

for the decay processes (7) and (10) relative to (8).

*<sup>n</sup>*<sup>0</sup> <sup>→</sup> *<sup>p</sup>*<sup>+</sup> <sup>+</sup> *<sup>e</sup>*

*d* → *u* + *W*<sup>−</sup> , *W*<sup>−</sup> → *e*

*μ*<sup>−</sup> → *νμ* + *W*<sup>−</sup> , *W*<sup>−</sup> → *e*

<sup>Λ</sup><sup>0</sup> <sup>→</sup> *<sup>p</sup>*<sup>+</sup> <sup>+</sup> *<sup>e</sup>*

*s* → *u* + *W*<sup>−</sup> , *W*<sup>−</sup> → *e*

In this case the overall coupling strength of the CC weak interactions was found to be

In the SM the universality of the CC weak interaction for both leptonic and hadronic processes is restored by adopting the proposal of Cabibbo (1963) that in hadronic processes the CC weak interaction is *shared* between Δ*S* = 0 and Δ*S* = 1 transition amplitudes in the ratio of cos *θ<sup>c</sup>* : sin *<sup>θ</sup>c*. The Cabibbo angle *<sup>θ</sup><sup>c</sup>* has a value <sup>≈</sup> <sup>13</sup>0, which gives good agreement with experiment

This "Cabibbo mixing" is an integral part of the SM. In the quark model it leads to a sharing of the CC weak interaction between quarks with different flavors (different generations) unlike the corresponding case of leptonic processes. Again, in order to simplify matters, the following discussion (and also throughout the chapter) will be restricted to the first two generations of the elementary particles of the SM, involving only the Cabibbo mixing, although the extension to three generations is straightforward (Kobayashi and Maskawa, 1973). In the latter case, the quark mixing parameters correspond to the so-called Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, which indicate that inclusion of the

The overall coupling strength of the CC weak interactions involved in neutron *β*-decay was

so that there are no CC weak interaction transitions between generations in agreement with

The Generation Model of Particle Physics 5

Unlike the pure leptonic decays, which are determined by the conservation of the various lepton numbers, there is no quantum number in the SM which restricts quark (hadronic) CC weak interaction processes between generations. In the SM the quarks do not appear to form weak isospin doublets: the known decay processes of neutron *β*-decay and Λ<sup>0</sup> *β*-decay suggest that quarks mix between generations and that the "flavor" quantum numbers, *S*, *C*, *B*

−; *W*−) = 0 , (5)

− + *ν*¯*<sup>e</sup>* , (6)

− + *ν*¯*<sup>e</sup>* . (7)

− + *ν*¯*<sup>e</sup>* . (8)

− + *ν*¯*<sup>e</sup>* . (10)

− + *ν*¯*<sup>e</sup>* , (9)

These are described by a *U*(1) × *SU*(2)*<sup>L</sup>* local gauge theory, where the *U*(1) symmetry involves both left-handed and right-handed particles.

Experiment requires the masses of the weak gauge bosons, *W* and *Z*, to be heavy so that the weak interactions are very short-ranged. On the other hand, Glashow's proposal, based upon the concept of a non-Abelian *SU*(2) Yang-Mills gauge theory, requires the mediators of the weak interactions to be massless like the photon. This boson mass problem was resolved by Weinberg (1967) and Salam (1968), who independently employed the idea of spontaneous symmetry breaking involving the Higgs mechanism (Englert and Brout, 1964; Higgs, 1964). In this way the *W* and *Z* bosons acquire mass and the photon remains massless.

The above treatment of the electromagnetic and weak interactions in terms of a *U*(1) × *SU*(2)*<sup>L</sup>* local gauge theory has become known as the Glashow, Weinberg and Salam (GWS) model and forms one of the cornerstones of the SM. The model gives the relative masses of the *W* and *Z* bosons in terms of the electroweak mixing angle:

$$M\_W = M\_Z \cos \theta\_W \,. \tag{3}$$

The Higgs mechanism was also able to cure the associated fermion mass problem (Aitchison and Hey, 1982): the finite masses of the leptons and quarks cause the Lagrangian describing the system to violate the *SU*(2)*<sup>L</sup>* gauge invariance. By coupling originally massless fermions to a scalar Higgs field, it is possible to produce the observed physical fermion masses without violating the gauge invariance. However, the GWS model requires the existence of a new massive spin zero boson, the Higgs boson, which to date remains to be detected. In addition, the fermion-Higgs coupling strength is dependent upon the mass of the fermion so that a new parameter is required for each fermion mass in the theory.

In 1971, t'Hooft (1971a,b) showed that the GWS model of the electroweak interactions was renormalizable and this self-consistency of the theory led to its general acceptance. In 1973, events corresponding to the predicted neutral currents mediated by the *Z*<sup>0</sup> boson were observed (Hasert *et al.*, 1973; 1974), while bosons, with approximately the expected masses, were discovered in 1983 (Arnison *et al.*, 1983; Banner *et al.*, 1983), thereby confirming the GWS model.

Another important property of the CC weak interactions is their universality for both leptonic and hadronic processes. In the SM this property is taken into account differently for leptonic and hadronic processes.

For leptonic CC weak interaction processes, each of the charged leptons is assumed to form a weak isospin doublet (*i* = <sup>1</sup> <sup>2</sup> ) with its respective neutrino, i.e. (*νe*,*e*−), (*νμ*, *μ*−), (*ντ*, *τ*−), with each doublet having the third component of weak isospin *i*<sup>3</sup> = (+<sup>1</sup> <sup>2</sup> , <sup>−</sup><sup>1</sup> <sup>2</sup> ). In addition each doublet is associated with a different lepton number so that there are no CC weak interaction transitions between generations. Thus for leptonic processes, the concept of a universal CC weak interaction allows one to write (for simplicity we restrict the discussion to the first two generations only):

$$a(\nu\_{\varepsilon}, e^-; \mathcal{W}^-) = a(\nu\_{\mu}, \mu^-; \mathcal{W}^-) = \mathcal{g}\_{w} \,. \tag{4}$$

Here *a*(*α*, *β*; *W*−) represents the CC weak interaction transition amplitude involving the fermions *α*, *β* and the *W*− boson, and *gw* is the universal CC weak interaction transition 4 Will-be-set-by-IN-TECH

These are described by a *U*(1) × *SU*(2)*<sup>L</sup>* local gauge theory, where the *U*(1) symmetry

Experiment requires the masses of the weak gauge bosons, *W* and *Z*, to be heavy so that the weak interactions are very short-ranged. On the other hand, Glashow's proposal, based upon the concept of a non-Abelian *SU*(2) Yang-Mills gauge theory, requires the mediators of the weak interactions to be massless like the photon. This boson mass problem was resolved by Weinberg (1967) and Salam (1968), who independently employed the idea of spontaneous symmetry breaking involving the Higgs mechanism (Englert and Brout, 1964; Higgs, 1964).

The above treatment of the electromagnetic and weak interactions in terms of a *U*(1) × *SU*(2)*<sup>L</sup>* local gauge theory has become known as the Glashow, Weinberg and Salam (GWS) model and forms one of the cornerstones of the SM. The model gives the relative masses of the *W* and *Z*

The Higgs mechanism was also able to cure the associated fermion mass problem (Aitchison and Hey, 1982): the finite masses of the leptons and quarks cause the Lagrangian describing the system to violate the *SU*(2)*<sup>L</sup>* gauge invariance. By coupling originally massless fermions to a scalar Higgs field, it is possible to produce the observed physical fermion masses without violating the gauge invariance. However, the GWS model requires the existence of a new massive spin zero boson, the Higgs boson, which to date remains to be detected. In addition, the fermion-Higgs coupling strength is dependent upon the mass of the fermion so that a new

In 1971, t'Hooft (1971a,b) showed that the GWS model of the electroweak interactions was renormalizable and this self-consistency of the theory led to its general acceptance. In 1973, events corresponding to the predicted neutral currents mediated by the *Z*<sup>0</sup> boson were observed (Hasert *et al.*, 1973; 1974), while bosons, with approximately the expected masses, were discovered in 1983 (Arnison *et al.*, 1983; Banner *et al.*, 1983), thereby confirming the GWS

Another important property of the CC weak interactions is their universality for both leptonic and hadronic processes. In the SM this property is taken into account differently for leptonic

For leptonic CC weak interaction processes, each of the charged leptons is assumed to form a

doublet is associated with a different lepton number so that there are no CC weak interaction transitions between generations. Thus for leptonic processes, the concept of a universal CC weak interaction allows one to write (for simplicity we restrict the discussion to the first two

Here *a*(*α*, *β*; *W*−) represents the CC weak interaction transition amplitude involving the fermions *α*, *β* and the *W*− boson, and *gw* is the universal CC weak interaction transition

each doublet having the third component of weak isospin *i*<sup>3</sup> = (+<sup>1</sup>

*a*(*νe*,*e*

<sup>2</sup> ) with its respective neutrino, i.e. (*νe*,*e*−), (*νμ*, *μ*−), (*ντ*, *τ*−), with

−; *W*−) = *a*(*νμ*, *μ*−; *W*−) = *gw* . (4)

<sup>2</sup> , <sup>−</sup><sup>1</sup>

<sup>2</sup> ). In addition each

*MW* = *MZ* cos *θ<sup>W</sup>* . (3)

In this way the *W* and *Z* bosons acquire mass and the photon remains massless.

involves both left-handed and right-handed particles.

bosons in terms of the electroweak mixing angle:

parameter is required for each fermion mass in the theory.

model.

and hadronic processes.

generations only):

weak isospin doublet (*i* = <sup>1</sup>

amplitude. Lepton number conservation gives

$$a(\upsilon\_{\varepsilon}, \mu^{-}; W^{-}) = a(\upsilon\_{\mu}, e^{-}; W^{-}) = 0 \; \; \; \; \tag{5}$$

so that there are no CC weak interaction transitions between generations in agreement with experiment.

Unlike the pure leptonic decays, which are determined by the conservation of the various lepton numbers, there is no quantum number in the SM which restricts quark (hadronic) CC weak interaction processes between generations. In the SM the quarks do not appear to form weak isospin doublets: the known decay processes of neutron *β*-decay and Λ<sup>0</sup> *β*-decay suggest that quarks mix between generations and that the "flavor" quantum numbers, *S*, *C*, *B* and *T* are not necessarily conserved in CC weak interaction processes.

In the SM neutron *β*-decay:

$$n^0 \to p^+ + e^- + \overline{v}\_{\varepsilon'} \tag{6}$$

is interpreted as the sequential transition

$$d \to \mathfrak{u} + \mathcal{W}^- \,, \quad \mathcal{W}^- \to \mathcal{e}^- + \bar{\nu}\_{\mathcal{C}} \,. \tag{7}$$

The overall coupling strength of the CC weak interactions involved in neutron *β*-decay was found to be slightly weaker (≈ 0.95) than that for muon decay:

$$
\mu^- \to \nu\_{\mu} + \mathcal{W}^- \,, \qquad \mathcal{W}^- \to \mathcal{e}^- + \bar{\nu}\_{\mathcal{e}} \,. \tag{8}
$$

Similarly, Λ<sup>0</sup> *β*-decay:

$$
\Lambda^0 \to p^+ + e^- + \vec{v}\_{\ell'} \tag{9}
$$

is interpreted in the SM as the sequential transition

$$s \to \mathfrak{u} + \mathcal{W}^- \,, \quad \mathcal{W}^- \to \mathfrak{e}^- + \mathfrak{v}\_\mathcal{e} \,. \tag{10}$$

In this case the overall coupling strength of the CC weak interactions was found to be significantly less (≈ 0.05) than that for muon decay.

In the SM the universality of the CC weak interaction for both leptonic and hadronic processes is restored by adopting the proposal of Cabibbo (1963) that in hadronic processes the CC weak interaction is *shared* between Δ*S* = 0 and Δ*S* = 1 transition amplitudes in the ratio of cos *θ<sup>c</sup>* : sin *<sup>θ</sup>c*. The Cabibbo angle *<sup>θ</sup><sup>c</sup>* has a value <sup>≈</sup> <sup>13</sup>0, which gives good agreement with experiment for the decay processes (7) and (10) relative to (8).

This "Cabibbo mixing" is an integral part of the SM. In the quark model it leads to a sharing of the CC weak interaction between quarks with different flavors (different generations) unlike the corresponding case of leptonic processes. Again, in order to simplify matters, the following discussion (and also throughout the chapter) will be restricted to the first two generations of the elementary particles of the SM, involving only the Cabibbo mixing, although the extension to three generations is straightforward (Kobayashi and Maskawa, 1973). In the latter case, the quark mixing parameters correspond to the so-called Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, which indicate that inclusion of the

**2.3 Basic problem inherent in SM**

basis for this scheme.

complicated movements.

in Subsection 3.1.

approximately elliptical planetary orbits.

phenomena, which the SM is unable to address.

**3. Generation model of particle physics**

which will be discussed in Subsection 3.2.

The basic problem with the SM is the classification of its elementary particles employing a diverse complicated scheme of additive quantum numbers (Table 1), some of which are not conserved in weak interaction processes; and at the same time failing to provide any physical

The Generation Model of Particle Physics 7

A good analogy of the SM situation is the Ptolemaic model of the universe, based upon a stationary Earth at the center surrounded by a rotating system of crystal spheres refined by the addition of epicycles (small circular orbits) to describe the peculiar movements of the planets around the Earth. While the Ptolemaic model yielded an excellent description, it is a complicated diverse scheme for predicting the movements of the Sun, Moon, planets and the stars around a stationary Earth and unfortunately provides no understanding of these

Progress in understanding the universe was only made when the Ptolemaic model was replaced by the Copernican-Keplerian model, in which the Earth moved like the other planets around the Sun, and Newton discovered his universal law of gravitation to describe the

The next section describes a new model of particle physics, the Generation Model (GM), which addresses the problem within the SM, replacing it with a much simpler and unified classification scheme of leptons and quarks, and providing some understanding of

The Generation Model (GM) of particle physics has been developed over the last decade. In the initial paper (Robson, 2002) a new classification of the elementary particles, the six leptons and the six quarks, of the SM was proposed. This classification was based upon the use of only three additive quantum numbers: charge (*Q*), particle number (*p*) and generation quantum number (*g*), rather than the nine additive quantum numbers (see Table 1) of the SM. Thus the new classification is both simpler and unified in that leptons and quarks are assigned the same kind of additive quantum numbers unlike those of the SM. It will be discussed in more detail

Another feature of the new classification scheme is that all three additive quantum numbers, *Q*, *p* and *g*, are required to be conserved in all leptonic and hadronic processes. In particular the generation quantum number *g* is strictly conserved in weak interactions unlike some of the quantum numbers, e.g. strangeness *S*, of the SM. This latter requirement led to a new treatment of quark mixing in hadronic processes (Robson, 2002; Evans and Robson, 2006),

The development of the GM classification scheme, which provides a unified description of leptons and quarks, indicated that leptons and quarks are intimately related and led to the development of composite versions of the GM, which we refer to as the Composite Generation

Model (CGM) (Robson, 2005; 2011a). The CGM will be discussed in Subsection 3.3.

Subsection 3.4 discusses the fundamental interactions of the GM.

third generation would have a minimal effect on the overall coupling strength of the CC weak interactions.

Cabibbo mixing was incorporated into the quark model of hadrons by postulating that the so-called weak interaction eigenstate quarks, *d*� and *s*� , form CC weak interaction isospin doublets with the *u* and *c* quarks, respectively: (*u*, *d*� ) and (*c*,*s*� ). These weak eigenstate quarks are linear superpositions of the so-called mass eigenstate quarks (*d* and *s*):

$$d' = d\cos\theta\_{\mathcal{C}} + s\sin\theta\_{\mathcal{C}} \tag{11}$$

and

$$ds' = -d\sin\theta\_{\mathcal{L}} + s\cos\theta\_{\mathcal{L}}\,. \tag{12}$$

The quarks *d* and *s* are the quarks which participate in the electromagnetic and the strong interactions with the full allotted strengths of electric charge and color charge, respectively. The quarks *d*� and *s*� are the quarks which interact with the *u* and *c* quarks, respectively, with the full strength of the CC weak interaction.

In terms of transition amplitudes, Eqs. (11) and (12) can be represented as

$$a(\mathfrak{u}, d'; W^{-}) = a(\mathfrak{u}, d; W^{-}) \cos \theta\_{\mathfrak{c}} + a(\mathfrak{u}, \mathfrak{s}; W^{-}) \sin \theta\_{\mathfrak{c}} = \mathfrak{g}\_{\mathfrak{w}} \tag{13}$$

and

$$a(c, s'; W^{-}) = -a(c, d; W^{-})\sin\theta\_{\mathcal{C}} + a(c, s; W^{-})\cos\theta\_{\mathcal{C}} = \mathcal{g}\_{\mathcal{W}}\,. \tag{14}$$

In addition one has the relations

$$a(u, s'; W^-) = -a(u, d; W^-) \sin \theta\_\mathcal{c} + a(u, s; W^-) \cos \theta\_\mathcal{c} = 0 \tag{15}$$

and

$$a(c, d'; W^-) = a(c, d; W^-) \cos \theta\_\mathcal{c} + a(c, s; W^-) \sin \theta\_\mathcal{c} = 0 \,. \tag{16}$$

Eqs. (13) and (14) indicate that it is the *d*� and *s*� quarks which interact with the *u* and *c* quarks, respectively, with the full strength *gw*. These equations for quarks correspond to Eq. (4) for leptons. Similarly, Eqs. (15) and (16) for quarks correspond to Eq. (5) for leptons. However, there is a fundamental difference between Eqs. (15) and (16) for quarks and Eq. (5) for leptons. The former equations do not yield zero amplitudes because there exists some quantum number (analagous to muon lepton number) which is required to be conserved. This lack of a selection rule indicates that the notion of weak isospin symmetry for the doublets (*u*, *d*� ) and (*c*,*s*� ) is dubious.

Eqs. (13) and (15) give

$$a(\mathfrak{u}, d; \mathcal{W}^-) = \mathcal{g}\_{\mathcal{W}} \cos \theta\_{\mathcal{C}} \; \; \; \; \; a(\mathfrak{u}, \mathfrak{s}; \mathcal{W}^-) = \mathcal{g}\_{\mathcal{W}} \sin \theta\_{\mathcal{C}} \; \; \; \; \; \tag{17}$$

Thus in the two generation approximation of the SM, transitions involving *d* → *u* + *W*<sup>−</sup> proceed with a strength proportional to *g*<sup>2</sup> *<sup>w</sup>* cos2 *<sup>θ</sup><sup>c</sup>* <sup>≈</sup> 0.95*g*<sup>2</sup> *<sup>w</sup>*, while transitions involving *<sup>s</sup>* <sup>→</sup> *<sup>u</sup>* <sup>+</sup> *<sup>W</sup>*<sup>−</sup> proceed with a strength proportional to *<sup>g</sup>*<sup>2</sup> *<sup>w</sup>* sin<sup>2</sup> *<sup>θ</sup><sup>c</sup>* <sup>≈</sup> 0.05*g*<sup>2</sup> *<sup>w</sup>*, as required by experiment.

### **2.3 Basic problem inherent in SM**

6 Will-be-set-by-IN-TECH

third generation would have a minimal effect on the overall coupling strength of the CC weak

Cabibbo mixing was incorporated into the quark model of hadrons by postulating that the

The quarks *d* and *s* are the quarks which participate in the electromagnetic and the strong interactions with the full allotted strengths of electric charge and color charge, respectively. The quarks *d*� and *s*� are the quarks which interact with the *u* and *c* quarks, respectively, with

Eqs. (13) and (14) indicate that it is the *d*� and *s*� quarks which interact with the *u* and *c* quarks, respectively, with the full strength *gw*. These equations for quarks correspond to Eq. (4) for leptons. Similarly, Eqs. (15) and (16) for quarks correspond to Eq. (5) for leptons. However, there is a fundamental difference between Eqs. (15) and (16) for quarks and Eq. (5) for leptons. The former equations do not yield zero amplitudes because there exists some quantum number (analagous to muon lepton number) which is required to be conserved. This lack of a selection rule indicates that the notion of weak isospin symmetry for the doublets

Thus in the two generation approximation of the SM, transitions involving *d* → *u* + *W*<sup>−</sup>

) and (*c*,*s*�

; *W*−) = *a*(*u*, *d*; *W*−) cos *θ<sup>c</sup>* + *a*(*u*,*s*; *W*−) sin *θ<sup>c</sup>* = *gw* (13)

; *W*−) = −*a*(*c*, *d*; *W*−) sin *θ<sup>c</sup>* + *a*(*c*,*s*; *W*−) cos *θ<sup>c</sup>* = *gw* . (14)

; *W*−) = −*a*(*u*, *d*; *W*−) sin *θ<sup>c</sup>* + *a*(*u*,*s*; *W*−) cos *θ<sup>c</sup>* = 0 (15)

; *W*−) = *a*(*c*, *d*; *W*−) cos *θ<sup>c</sup>* + *a*(*c*,*s*; *W*−) sin *θ<sup>c</sup>* = 0 . (16)

*a*(*u*, *d*; *W*−) = *gw* cos *θ<sup>c</sup>* , *a*(*u*,*s*; *W*−) = *gw* sin *θ<sup>c</sup>* . (17)

*<sup>w</sup>*, while transitions involving

*<sup>w</sup>*, as required by

*<sup>w</sup>* sin<sup>2</sup> *<sup>θ</sup><sup>c</sup>* <sup>≈</sup> 0.05*g*<sup>2</sup>

*<sup>w</sup>* cos2 *<sup>θ</sup><sup>c</sup>* <sup>≈</sup> 0.95*g*<sup>2</sup>

*d*� = *d* cos *θ<sup>c</sup>* + *s* sin *θ<sup>c</sup>* (11)

� = −*d* sin *θ<sup>c</sup>* + *s* cos *θ<sup>c</sup>* . (12)

, form CC weak interaction isospin

). These weak eigenstate quarks

so-called weak interaction eigenstate quarks, *d*� and *s*�

are linear superpositions of the so-called mass eigenstate quarks (*d* and *s*):

*s*

In terms of transition amplitudes, Eqs. (11) and (12) can be represented as

doublets with the *u* and *c* quarks, respectively: (*u*, *d*�

the full strength of the CC weak interaction.

*a*(*u*, *d*�

*a*(*c*,*s* �

> *a*(*u*,*s* �

> > *a*(*c*, *d*�

) is dubious.

proceed with a strength proportional to *g*<sup>2</sup>

*<sup>s</sup>* <sup>→</sup> *<sup>u</sup>* <sup>+</sup> *<sup>W</sup>*<sup>−</sup> proceed with a strength proportional to *<sup>g</sup>*<sup>2</sup>

In addition one has the relations

interactions.

and

and

and

(*u*, *d*�

experiment.

) and (*c*,*s*�

Eqs. (13) and (15) give

The basic problem with the SM is the classification of its elementary particles employing a diverse complicated scheme of additive quantum numbers (Table 1), some of which are not conserved in weak interaction processes; and at the same time failing to provide any physical basis for this scheme.

A good analogy of the SM situation is the Ptolemaic model of the universe, based upon a stationary Earth at the center surrounded by a rotating system of crystal spheres refined by the addition of epicycles (small circular orbits) to describe the peculiar movements of the planets around the Earth. While the Ptolemaic model yielded an excellent description, it is a complicated diverse scheme for predicting the movements of the Sun, Moon, planets and the stars around a stationary Earth and unfortunately provides no understanding of these complicated movements.

Progress in understanding the universe was only made when the Ptolemaic model was replaced by the Copernican-Keplerian model, in which the Earth moved like the other planets around the Sun, and Newton discovered his universal law of gravitation to describe the approximately elliptical planetary orbits.

The next section describes a new model of particle physics, the Generation Model (GM), which addresses the problem within the SM, replacing it with a much simpler and unified classification scheme of leptons and quarks, and providing some understanding of phenomena, which the SM is unable to address.
