Introductory Chapter

Chapter 1

Particles

1. Overview

of particles.

applications.

3

uniform magnetic field B

Introductory Chapter: Charged

Bringing history back in August 1912, Austrian physicist Victor Hess discovered cosmic rays coming from outer space. These cosmic rays consist of high-energy particles, entering from outer space, such as mainly protons, helium, and heavier nuclei up to uranium. When these cosmic rays come to earth to interact with upper atmosphere, they collide with the nuclei of atoms, creating more high-energy particles such as pions. The charged pions can quickly decay into two particles, a muon and a muon neutrino or antineutrino. Several high-energy particles were also discovered, which is long list. Studies of cosmic rays opened the door to a world class

It is concluded that charged particle is a particle that carries an electric charge. In atomic levels, the atom consists of nucleus around which the electrons turn. The nucleus is formed by proton and neutron and thus carries a positive charge (the proton charge is 1.602 � <sup>10</sup>�<sup>19</sup> Coulombs). The electron carries a negative charge (�1.602 � <sup>10</sup>�<sup>19</sup> Coulombs). An atom is called neutral if the number of protons equals the number of electrons. Thus, an atom can be positive, negative, or neutral. The charged particle is negative when it gains electron from another atom. It is positively charged if it loses electron from it. Applications of charged particles are subjected to control their motion and energy through electric field and magnetic field. Therefore, motion of charged particle in electric and magnetic fields is discussed in order to understand the beam of charged particles and their

> ! , B ! )

is subjected to an elec-

!

! in

(1)

(2)

2. Charged particles motion in an electromagnetic field (E

F ! ¼ qE ! þ qv ! � <sup>B</sup> !

m d v! dt <sup>¼</sup> q E!

! 6¼ 0 !

tromagnetic force called the Lorentz force given by:

The motion of charged particle of mass m and charge q with a velocity v

From Newton second law, the particle's equation of motion is written as:

and uniform electric field E

þ v ! � <sup>B</sup> !

Mahmoud Izerrouken and Ishaq Ahmad

#### Chapter 1

## Introductory Chapter: Charged Particles

Mahmoud Izerrouken and Ishaq Ahmad

#### 1. Overview

Bringing history back in August 1912, Austrian physicist Victor Hess discovered cosmic rays coming from outer space. These cosmic rays consist of high-energy particles, entering from outer space, such as mainly protons, helium, and heavier nuclei up to uranium. When these cosmic rays come to earth to interact with upper atmosphere, they collide with the nuclei of atoms, creating more high-energy particles such as pions. The charged pions can quickly decay into two particles, a muon and a muon neutrino or antineutrino. Several high-energy particles were also discovered, which is long list. Studies of cosmic rays opened the door to a world class of particles.

It is concluded that charged particle is a particle that carries an electric charge. In atomic levels, the atom consists of nucleus around which the electrons turn. The nucleus is formed by proton and neutron and thus carries a positive charge (the proton charge is 1.602 � <sup>10</sup>�<sup>19</sup> Coulombs). The electron carries a negative charge (�1.602 � <sup>10</sup>�<sup>19</sup> Coulombs). An atom is called neutral if the number of protons equals the number of electrons. Thus, an atom can be positive, negative, or neutral. The charged particle is negative when it gains electron from another atom. It is positively charged if it loses electron from it. Applications of charged particles are subjected to control their motion and energy through electric field and magnetic field. Therefore, motion of charged particle in electric and magnetic fields is discussed in order to understand the beam of charged particles and their applications.

#### 2. Charged particles motion in an electromagnetic field (E ! , B ! )

The motion of charged particle of mass m and charge q with a velocity v ! in uniform magnetic field B ! 6¼ 0 ! and uniform electric field E ! is subjected to an electromagnetic force called the Lorentz force given by:

$$
\overrightarrow{F} = q\overrightarrow{E} + q\overrightarrow{v} \times \overrightarrow{B} \tag{1}
$$

From Newton second law, the particle's equation of motion is written as:

$$m\frac{d\vec{v}}{dt} = q\left(\vec{E} + \vec{v} \times \vec{B}\right) \tag{2}$$

#### Charged Particles

Let us assume that the magnetic field is applied in the direction Oz and electric field E ! is applied in the direction Oy.

1. In the case of motion of charged particle through a stationary electric field E ! ¼ E<sup>0</sup> ! and B ! ¼ 0 ! , the equation of motion is:

$$m\frac{d\vec{v}}{dt} = q\vec{E} \quad \Leftrightarrow \, \vec{v} = \frac{q\vec{E}}{m}t + \vec{v\_0} \tag{3}$$

The projection of (3) on the axes gives

$$m\frac{dv\_x}{dt} = 0$$

$$m\frac{dv\_y}{dt} = qE\tag{4}$$

$$m\frac{dv\_x}{dt} = 0$$

We note <sup>Ω</sup> <sup>¼</sup> qB

The solution of (6) is:

Introductory Chapter: Charged Particles DOI: http://dx.doi.org/10.5772/intechopen.82782

> <sup>x</sup> <sup>¼</sup> <sup>E</sup> <sup>B</sup> <sup>t</sup> <sup>þ</sup> 1 Ω

<sup>y</sup> <sup>¼</sup> <sup>1</sup> Ω

z ¼ vz<sup>0</sup> t

charged particle acceleration and beam guidance.

3. Charged particle accelerators

4. Classification of charged particles

4.1 Electrostatic charged particle accelerators

applications:

magnetic.

5

<sup>m</sup> the so-called cyclotron frequency.

vx<sup>0</sup> � <sup>t</sup> B sin ð Þþ <sup>Ω</sup><sup>t</sup>

:<sup>ð</sup> cosð Þ� <sup>Ω</sup><sup>t</sup> <sup>1</sup>Þ þ vy<sup>0</sup>

According to the above equations, we can see that the charged particle can have different trajectories. Depending on the initial conditions, the trajectory could be straight, parabolas, circles, cycloids, spirals, etc. Lorentz force is then a base of

After understanding the concept of controlling and generating the charged particles, different machines were developed to deflect or accelerate the charged particles through electromagnetic fields. The machines that generate and push charged particles to very high speed and energies and contain them in well-defined beams are called charged particle accelerators. A large variety of accelerators were developed since that fabricated by Cockcroft and Walton in 1932. Using such accelerator, the authors achieved the first nuclear reaction using artificially accelerated particle:

p þ Li ! 2He Since then, more and more successful accelerators appeared according to the progress in particle acceleration techniques. Depending on the accelerated particle trajectory, we can distinguish linear accelerators and circular accelerators. The accelerator during the last century can be found chronologically in [1]. Currently available electrostatic accelerators and cyclotrons over the world can produce and accelerate intense and stable charged particle beams with energy varying between few keV and few TeV. Charged particle accelerators are classified as per their

Charged particle accelerators are classified mainly into electrostatic and electro-

In electrostatic accelerators, the static high voltage was generated and then applied across the ion source. The charged particles are accelerated through static electric field generated from static high voltage due to the electrostatic force. These types of accelerators are suitable to accelerate light and heavy ions from keV to few MeV energies. These ion beams, charged particle beams of various energies, are standard research tools in many areas of sciences and engineering having many applications in nuclear physics, atomic physics, medicine, materials science,

vx<sup>0</sup> � <sup>t</sup> B vy<sup>0</sup>

Ω

<sup>Ω</sup> ð Þ <sup>1</sup> � cosð Þ <sup>Ω</sup><sup>t</sup>

sin ð Þ Ωt (7)

The integration of (4) with the initial conditions xð Þ¼ 0 0, yð Þ¼ 0 0, and zð Þ¼ 0 0; vxð Þ¼ 0 vx0, vyð Þ¼ 0 vyo, and vzð Þ¼ 0 vz<sup>0</sup> gives:

$$\begin{aligned} x &= v\_{\ge 0}t \\ y &= \frac{qE}{2m}t^2 + v\_{\ge 0}t \\ z &= v\_{0x}t \end{aligned} \tag{5}$$

If E ! and v ! are collinear, the motion is rectilinear and uniformly accelerated.

2. In the case of motion of charged particle in uniform magnetic field B ! 6¼ 0 ! , the projection of (2) on the axes gives:

$$\begin{aligned} m\frac{dV\_x}{dt} &= qBV\_\mathcal{V} \Leftrightarrow \frac{dV\_x}{dt} = \frac{qB}{m} \ V\_\mathcal{V}\\ m\frac{dV\_\mathcal{V}}{dt} &= -qBV\_\mathcal{X} + qE \Leftrightarrow \frac{dV\_\mathcal{V}}{dt} = -\frac{qB}{m}V\_\mathcal{x} + \frac{q}{m}E\\ m\frac{dV\_x}{dt} &= 0 \end{aligned} \tag{6}$$

Introductory Chapter: Charged Particles DOI: http://dx.doi.org/10.5772/intechopen.82782

We note <sup>Ω</sup> <sup>¼</sup> qB <sup>m</sup> the so-called cyclotron frequency. The solution of (6) is:

$$\begin{aligned} x &= \frac{E}{B}t + \frac{1}{\Omega} \left( v\_{x0} - \frac{t}{B} \right) \sin \left( \Omega t \right) + \frac{v\_{y0}}{\Omega} \left( 1 - \cos \left( \Omega t \right) \right) \\ y &= \frac{1}{\Omega} \left( v\_{x0} - \frac{t}{B} \right) \left( \cos \left( \Omega t \right) - 1 \right) + \frac{v\_{y0}}{\Omega} \sin \left( \Omega t \right) \\ z &= v\_{x0} \ t \end{aligned} \tag{7}$$

According to the above equations, we can see that the charged particle can have different trajectories. Depending on the initial conditions, the trajectory could be straight, parabolas, circles, cycloids, spirals, etc. Lorentz force is then a base of charged particle acceleration and beam guidance.

#### 3. Charged particle accelerators

Let us assume that the magnetic field is applied in the direction Oz and electric

t þ v<sup>0</sup>

dt <sup>¼</sup> qE (4)

<sup>2</sup> <sup>þ</sup> vy0<sup>t</sup> (5)

! 6¼ 0 ! , the

E (6)

! (3)

1. In the case of motion of charged particle through a stationary electric field

, the equation of motion is:

m dvx dt <sup>¼</sup> <sup>0</sup>

m dvy

> m dvz dt <sup>¼</sup> <sup>0</sup>

xð Þ¼ 0 0, yð Þ¼ 0 0, and zð Þ¼ 0 0; vxð Þ¼ 0 vx0, vyð Þ¼ 0 vyo, and vzð Þ¼ 0 vz<sup>0</sup> gives:

x ¼ vx0t <sup>y</sup> <sup>¼</sup> qE 2m t

z ¼ v0zt

2. In the case of motion of charged particle in uniform magnetic field B

dVx dt <sup>¼</sup> qB m Vy

! are collinear, the motion is rectilinear and uniformly accelerated.

dVy

dt ¼ � qB m

Vx <sup>þ</sup> <sup>q</sup> m

field E !

> E ! ¼ E<sup>0</sup> !

Charged Particles

If E !

4

and v

is applied in the direction Oy.

The projection of (3) on the axes gives

m d v! dt <sup>¼</sup> qE ! ⇔ v !¼ qE ! m

The integration of (4) with the initial conditions

projection of (2) on the axes gives:

dt <sup>¼</sup> qBVy <sup>⇔</sup>

dt ¼ �qBVx <sup>þ</sup> qE <sup>⇔</sup>

m dVx

m dVy

m dVz dt <sup>¼</sup> <sup>0</sup>

and B ! ¼ 0 !

After understanding the concept of controlling and generating the charged particles, different machines were developed to deflect or accelerate the charged particles through electromagnetic fields. The machines that generate and push charged particles to very high speed and energies and contain them in well-defined beams are called charged particle accelerators. A large variety of accelerators were developed since that fabricated by Cockcroft and Walton in 1932. Using such accelerator, the authors achieved the first nuclear reaction using artificially accelerated particle:

$$p + Li \rightarrow 2He$$

Since then, more and more successful accelerators appeared according to the progress in particle acceleration techniques. Depending on the accelerated particle trajectory, we can distinguish linear accelerators and circular accelerators. The accelerator during the last century can be found chronologically in [1]. Currently available electrostatic accelerators and cyclotrons over the world can produce and accelerate intense and stable charged particle beams with energy varying between few keV and few TeV. Charged particle accelerators are classified as per their applications:

#### 4. Classification of charged particles

Charged particle accelerators are classified mainly into electrostatic and electromagnetic.

#### 4.1 Electrostatic charged particle accelerators

In electrostatic accelerators, the static high voltage was generated and then applied across the ion source. The charged particles are accelerated through static electric field generated from static high voltage due to the electrostatic force. These types of accelerators are suitable to accelerate light and heavy ions from keV to few MeV energies. These ion beams, charged particle beams of various energies, are standard research tools in many areas of sciences and engineering having many applications in nuclear physics, atomic physics, medicine, materials science,

agriculture, industry, and so on [2–6]. It is an advanced and versatile tool frequently applied across a broad range of discipline and fields.

analysis of ESW is done via fluid modeling. Apart from the high-energy particle physics, charged bodies are also included in this book such as immune effects of negative charged particles dominated by indoor air conditions and many others.

Several techniques of ion beam analysis (IBA) are being used for the study of the chemical composition and structure of surfaces, interface, and thin layers and are

The accelerated charged particle with energy E0 and mass M1 scatters from the analyzed surface containing the particle M2 with energy E1 and scattering angle θ. From the conservation laws of energy and momentum and the known Rutherford cross section, it is possible to deduce the mass M2 and estimate its abundance [7, 8].

The NRA technique is very useful as a tool for the detection and profiling of light

16O nuclear reaction is

elements. The fast charged particle (few MeV) initiates a nuclear reaction with target atom. The reaction products are characteristic for this reaction and can be used to identify the target atom and its concentration. For example, determination

used. This reaction produces alpha particle and excited 12C isotope. The disintegration of the excited 12C to ground state emits a gamma photon with a well-defined

ERDA technique is a unique method to measure the H and D content in thin films. When He ion (alpha particle) interacts with material containing hydrogen (H) and deuterium (D), the H and D will be scattered in the forward direction. From the detection of the forwarded H and D, one can measure the quantitative depth profiling of these elements in the material. Similar experiments can be

Ion beam of energy typically 1–2 MeV induces ionization of the target atom. If the ejected electron belongs to K-shell, an X-ray characteristic of the irradiated element is emitted. Using this technique, qualitative and quantitative analysis can

energy of Eγ = 4.43 MeV, which identifies hydrogen content in material.

performed using heavy ion beam to study light element profiling.

be used where the trace element of about 1 ppm can be achieved [9].

of hydrogen content in material: for this purpose, H(15N, αγ)

6. Charged particle beams for materials analysis

6.1 Rutherford backscattering spectroscopy

6.2 Nuclear reaction analysis (NRA)

6.3 Elastic recoil detection analysis (ERDA)

6.4 Particle-induced X-ray emission (PIXE)

7

explained as below.

Introductory Chapter: Charged Particles DOI: http://dx.doi.org/10.5772/intechopen.82782

#### 4.2 Electromagnetic charged particle accelerators

Electrostatic charged particle accelerators have limitations on its beam energy due to high electrical voltage discharge. To avoid electrical discharge and increase charged particle energies, techniques involved electromagnetic fields instead of electrostatic fields. Electromagnetic acceleration is possible from two mechanisms either nonresonant magnetic induction or resonant circuits or cavities excited by oscillating radio frequency. High-energy charged particle colliders are installed around the world for forefront of scientific discoveries. These colliders are based on electrostatic charged particle accelerators. Through high-energy colliders, standard models are verified experimentally. Moreover, the cutting-edge and important research topic of flavor (particle physics) to search for new physics via charged particles that appears in the different extension of standard model is presented in this book. The latest research on analysis of ultrahigh-energy muon using pairmeter technique is also presented in Geant4 simulation study for iron plates. In this study, the feasibility for detection of high-energy muons at the underground iron calorimeter detector is demonstrated. The basic aim of this study is to detect highenergy muons (1–1000 TeV). The idea of the Eloisatron to Pevatron is also included in this book.

#### 5. Charged particles applications

Charged particles interact with electrons and atom nuclei via Coulomb force, also called electrostatic force. When two charges are placed near to each other, they will be repulsed if they have the same charge or attract each other if they are of opposite charges. Each particle exerts a force on each other given by Coulomb's law expressed as:

$$F = k \frac{Q1Q2}{r^2}$$

where Q1, Q2, and r are the charge of the two particles (1 and 2) in Coulomb and the distance between the charges and k is the proportionality constant:

$$k = \frac{1}{4\pi c}$$

Thus, when accelerated charged particle moves in materials, it interacts with orbital electron and nuclei via Coulomb interaction depending on its energy. At low energy (<0.01 MeV/u), the interaction is with nuclei, known as elastic interaction. This interaction leads to atomic displacement. At high energy (>0.01 MeV/u), the interaction is mainly with orbital electron known as inelastic interaction and leads to ionization and excitation. At very high energy, nuclear reactions can be produced and give rise to new particles (neutron, proton alpha, gamma rays). The basic interaction process of charged particle with matter is well known, and much performed detectors are now available. So ion beam is actually used in several applications. Electrostatic waves in magnetized electron-positron plasmas are covered in this book where the behavior of arbitrary amplitude of electrostatic wave propagation in electron-positron plasma is discussed. The well-known fluid and kinetic approaches have been used to describe linear waves, whereas the nonlinear

agriculture, industry, and so on [2–6]. It is an advanced and versatile tool frequently

Electrostatic charged particle accelerators have limitations on its beam energy due to high electrical voltage discharge. To avoid electrical discharge and increase charged particle energies, techniques involved electromagnetic fields instead of electrostatic fields. Electromagnetic acceleration is possible from two mechanisms either nonresonant magnetic induction or resonant circuits or cavities excited by oscillating radio frequency. High-energy charged particle colliders are installed around the world for forefront of scientific discoveries. These colliders are based on electrostatic charged particle accelerators. Through high-energy colliders, standard models are verified experimentally. Moreover, the cutting-edge and important research topic of flavor (particle physics) to search for new physics via charged particles that appears in the different extension of standard model is presented in this book. The latest research on analysis of ultrahigh-energy muon using pairmeter technique is also presented in Geant4 simulation study for iron plates. In this study, the feasibility for detection of high-energy muons at the underground iron calorimeter detector is demonstrated. The basic aim of this study is to detect highenergy muons (1–1000 TeV). The idea of the Eloisatron to Pevatron is also included

Charged particles interact with electrons and atom nuclei via Coulomb force, also called electrostatic force. When two charges are placed near to each other, they will be repulsed if they have the same charge or attract each other if they are of opposite charges. Each particle exerts a force on each other given by Coulomb's law

Q1Q2 r2 where Q1, Q2, and r are the charge of the two particles (1 and 2) in Coulomb and

F ¼ k

<sup>k</sup> <sup>¼</sup> <sup>1</sup> 4πε

Thus, when accelerated charged particle moves in materials, it interacts with orbital electron and nuclei via Coulomb interaction depending on its energy. At low energy (<0.01 MeV/u), the interaction is with nuclei, known as elastic interaction. This interaction leads to atomic displacement. At high energy (>0.01 MeV/u), the interaction is mainly with orbital electron known as inelastic interaction and leads to ionization and excitation. At very high energy, nuclear reactions can be produced and give rise to new particles (neutron, proton alpha, gamma rays). The basic interaction process of charged particle with matter is well known, and much performed detectors are now available. So ion beam is actually used in several applications. Electrostatic waves in magnetized electron-positron plasmas are covered in this book where the behavior of arbitrary amplitude of electrostatic wave propagation in electron-positron plasma is discussed. The well-known fluid and kinetic approaches have been used to describe linear waves, whereas the nonlinear

the distance between the charges and k is the proportionality constant:

applied across a broad range of discipline and fields.

4.2 Electromagnetic charged particle accelerators

in this book.

Charged Particles

expressed as:

6

5. Charged particles applications

analysis of ESW is done via fluid modeling. Apart from the high-energy particle physics, charged bodies are also included in this book such as immune effects of negative charged particles dominated by indoor air conditions and many others.

### 6. Charged particle beams for materials analysis

Several techniques of ion beam analysis (IBA) are being used for the study of the chemical composition and structure of surfaces, interface, and thin layers and are explained as below.

#### 6.1 Rutherford backscattering spectroscopy

The accelerated charged particle with energy E0 and mass M1 scatters from the analyzed surface containing the particle M2 with energy E1 and scattering angle θ. From the conservation laws of energy and momentum and the known Rutherford cross section, it is possible to deduce the mass M2 and estimate its abundance [7, 8].

#### 6.2 Nuclear reaction analysis (NRA)

The NRA technique is very useful as a tool for the detection and profiling of light elements. The fast charged particle (few MeV) initiates a nuclear reaction with target atom. The reaction products are characteristic for this reaction and can be used to identify the target atom and its concentration. For example, determination of hydrogen content in material: for this purpose, H(15N, αγ) 16O nuclear reaction is used. This reaction produces alpha particle and excited 12C isotope. The disintegration of the excited 12C to ground state emits a gamma photon with a well-defined energy of Eγ = 4.43 MeV, which identifies hydrogen content in material.

#### 6.3 Elastic recoil detection analysis (ERDA)

ERDA technique is a unique method to measure the H and D content in thin films. When He ion (alpha particle) interacts with material containing hydrogen (H) and deuterium (D), the H and D will be scattered in the forward direction. From the detection of the forwarded H and D, one can measure the quantitative depth profiling of these elements in the material. Similar experiments can be performed using heavy ion beam to study light element profiling.

#### 6.4 Particle-induced X-ray emission (PIXE)

Ion beam of energy typically 1–2 MeV induces ionization of the target atom. If the ejected electron belongs to K-shell, an X-ray characteristic of the irradiated element is emitted. Using this technique, qualitative and quantitative analysis can be used where the trace element of about 1 ppm can be achieved [9].

Charged Particles

#### Author details

Mahmoud Izerrouken<sup>1</sup> \* and Ishaq Ahmad<sup>2</sup>


© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

cybergeo/14173

2017;64:95-100

[1] Brissaud I, Baron E. The Race of Particle Accelerators to High Energy and Log Periodicity, Cybergeo: European Journal of Geography

Introductory Chapter: Charged Particles DOI: http://dx.doi.org/10.5772/intechopen.82782

> Physics A: Materials Science & Processing. 2014;117:2275-2279

[8] Husnain G, Ahmad I, Rafique HM, Akrajas AU, Dee CF. Depth-dependent tetragonal distortion study of AlGaN epilayer thin film using RBS and channeling technique. Modern Physics Letters B. 2012;26(14):1250086

[9] Akram W, Madhuku M, Shehzad K, Awais A, Ahmad I, Ahmad I, et al. Roadside dust contamination with toxic metals along Islamabad industrial area. Nuclear Science and Techniques. 2014;

25(3):030201

[Online], Debates, Theory of Relativity Scale and Log Periodicity. Online: December 11, 2007. [Accessed Dec 14, 2018]. http://journals.openedition.org/

[2] Awais A, Hussain J, Usman M, Akram W, Shahzad K, Ali T, et al. The charge state distribution of B, C, Si, Ni, Cu and Au ions on 5 MV Pelletron accelerator. Nuclear Science and Techniques. 2017;28(5):64

[3] Ahmad I, Madhuku M, Sadaf A, Khan S, Hussain J, Ali A, et al. Tailoring the structural and optical characteristics of InGaN/GaN multilayer thin films by 12 MeV Si ion irradiation. Materials Science in Semiconductor Processing.

[4] Ni Z, Ishaq A, Yan L, Gong J, Zhu D. Enhanced electron field emission of carbon nanotubes by Si ion beam irradiation. Journal of Physics D: Applied Physics. 2009;42:075408

[5] Ahmad I, Long Y, Suixia H, Dezhang Z, Xingtai Z. Optical absorption of ion

irradiated multi-walled carbon nanotube sheets in the visible to terahertz ranges. Nuclear Science and

Techniques. 2009;20:197-201

nanotubes under proton beam

1505-1507

9

[6] Ishaq A, Yan L, Gong J, Zhu D. Graphite-to-amorphous structural transformation of multiwalled carbon

irradiation. Materials Letters. 2009;63:

[7] Husnain G, Shu-De Y, Ahmad I, Lin L. Role of substrate temperature on structure and magnetization of Crimplanted GaN thin film. Applied

Introductory Chapter: Charged Particles DOI: http://dx.doi.org/10.5772/intechopen.82782

#### References

[1] Brissaud I, Baron E. The Race of Particle Accelerators to High Energy and Log Periodicity, Cybergeo: European Journal of Geography [Online], Debates, Theory of Relativity Scale and Log Periodicity. Online: December 11, 2007. [Accessed Dec 14, 2018]. http://journals.openedition.org/ cybergeo/14173

[2] Awais A, Hussain J, Usman M, Akram W, Shahzad K, Ali T, et al. The charge state distribution of B, C, Si, Ni, Cu and Au ions on 5 MV Pelletron accelerator. Nuclear Science and Techniques. 2017;28(5):64

[3] Ahmad I, Madhuku M, Sadaf A, Khan S, Hussain J, Ali A, et al. Tailoring the structural and optical characteristics of InGaN/GaN multilayer thin films by 12 MeV Si ion irradiation. Materials Science in Semiconductor Processing. 2017;64:95-100

[4] Ni Z, Ishaq A, Yan L, Gong J, Zhu D. Enhanced electron field emission of carbon nanotubes by Si ion beam irradiation. Journal of Physics D: Applied Physics. 2009;42:075408

[5] Ahmad I, Long Y, Suixia H, Dezhang Z, Xingtai Z. Optical absorption of ion irradiated multi-walled carbon nanotube sheets in the visible to terahertz ranges. Nuclear Science and Techniques. 2009;20:197-201

[6] Ishaq A, Yan L, Gong J, Zhu D. Graphite-to-amorphous structural transformation of multiwalled carbon nanotubes under proton beam irradiation. Materials Letters. 2009;63: 1505-1507

[7] Husnain G, Shu-De Y, Ahmad I, Lin L. Role of substrate temperature on structure and magnetization of Crimplanted GaN thin film. Applied

Physics A: Materials Science & Processing. 2014;117:2275-2279

[8] Husnain G, Ahmad I, Rafique HM, Akrajas AU, Dee CF. Depth-dependent tetragonal distortion study of AlGaN epilayer thin film using RBS and channeling technique. Modern Physics Letters B. 2012;26(14):1250086

[9] Akram W, Madhuku M, Shehzad K, Awais A, Ahmad I, Ahmad I, et al. Roadside dust contamination with toxic metals along Islamabad industrial area. Nuclear Science and Techniques. 2014; 25(3):030201

Author details

Charged Particles

8

Mahmoud Izerrouken<sup>1</sup>

\* and Ishaq Ahmad<sup>2</sup>

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

1 Nuclear Research Center Draria, Algiers, Algeria

2 National Centre for Physics, Islamabad, Pakistan

provided the original work is properly cited.

\*Address all correspondence to: m-izerrouken@crnd.dz

**11**

Section 2

High Energy Particle

Physics

### Section 2

## High Energy Particle Physics

**13**

**Chapter 2**

**Abstract**

**1. Introduction**

Meter Technique

Analysis of Ultra-High Energy

Muons at INO-ICAL Using Pair

The proposed magnetized Iron CALorimeter (ICAL) detector at India-based Neutrino Observatory (INO) is a large-sized underground detector. ICAL is designed to reconstruct muon momentum using magnetic spectrometers as detectors. Muon energy measurements using magnets fail for high energy muons (TeV range), since the angular deflection of the muon in the magnetic field is negligible and the muon tracks become nearly straight. A new technique for measuring the energy of muons in the TeV range, used by the CCFR neutrino detector is known as the pair meter technique. This technique estimates muon energy by measuring the energy

production. In this work we have performed Geant4-based preliminary analysis for iron plates and have demonstrated the feasibility to detect very high energy muons (1–1000 TeV) at the underground ICAL detector operating as a pair meter. This wide range of energy spectrum will not only be helpful for studying the cosmic rays in the Knee region which is around 5 PeV in the cosmic ray spectra but also useful for understanding the atmospheric neutrino flux for the running and upcoming ultra-

ICAL at INO is a 52 ktons detector [1] proposed to be built at Theni district of Tamil Nadu in Southern India. It is designed to study the flavor oscillations of atmospheric neutrinos. The main goal of the ICAL detector is to precisely measure the neutrino oscillation parameters and to determine the neutrino mass hierarchy [1]. At a depth of around 1.2 km underground, the INO-ICAL detector will be the world's biggest magnetized detector to measure cosmic ray muon flux with the capability to distinguish *μ*+ from *μ*−. The existing direct and indirect methods of muon spectrometry at accelerator-based and in cosmic rays (magnetic spectrometers and transition radiation detectors) experiments involve certain technical problems and limitations in the energy region ≥1013 eV. These disadvantages vanquish in this alternate method where the muon energy is estimated by measuring the energy of secondary cascades formed by muons losing their energy in thick layers

this technique, muon energy can be estimated from INO-ICAL detector operating

and e<sup>−</sup> pair

and e<sup>−</sup>. By using

deposited by the muon in several layers of an iron calorimeter through e+

high energy atmospheric neutrino experiments.

**Keywords:** pair meter techniques, cosmic rays, iron calorimeter

of matter, mainly due to the process of direct pair production of e+

*Jaydip Singh, Srishti Nagu and Jyotsna Singh*

#### **Chapter 2**

## Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique

*Jaydip Singh, Srishti Nagu and Jyotsna Singh*

#### **Abstract**

The proposed magnetized Iron CALorimeter (ICAL) detector at India-based Neutrino Observatory (INO) is a large-sized underground detector. ICAL is designed to reconstruct muon momentum using magnetic spectrometers as detectors. Muon energy measurements using magnets fail for high energy muons (TeV range), since the angular deflection of the muon in the magnetic field is negligible and the muon tracks become nearly straight. A new technique for measuring the energy of muons in the TeV range, used by the CCFR neutrino detector is known as the pair meter technique. This technique estimates muon energy by measuring the energy deposited by the muon in several layers of an iron calorimeter through e+ and e<sup>−</sup> pair production. In this work we have performed Geant4-based preliminary analysis for iron plates and have demonstrated the feasibility to detect very high energy muons (1–1000 TeV) at the underground ICAL detector operating as a pair meter. This wide range of energy spectrum will not only be helpful for studying the cosmic rays in the Knee region which is around 5 PeV in the cosmic ray spectra but also useful for understanding the atmospheric neutrino flux for the running and upcoming ultrahigh energy atmospheric neutrino experiments.

**Keywords:** pair meter techniques, cosmic rays, iron calorimeter

#### **1. Introduction**

ICAL at INO is a 52 ktons detector [1] proposed to be built at Theni district of Tamil Nadu in Southern India. It is designed to study the flavor oscillations of atmospheric neutrinos. The main goal of the ICAL detector is to precisely measure the neutrino oscillation parameters and to determine the neutrino mass hierarchy [1]. At a depth of around 1.2 km underground, the INO-ICAL detector will be the world's biggest magnetized detector to measure cosmic ray muon flux with the capability to distinguish *μ*+ from *μ*−. The existing direct and indirect methods of muon spectrometry at accelerator-based and in cosmic rays (magnetic spectrometers and transition radiation detectors) experiments involve certain technical problems and limitations in the energy region ≥1013 eV. These disadvantages vanquish in this alternate method where the muon energy is estimated by measuring the energy of secondary cascades formed by muons losing their energy in thick layers of matter, mainly due to the process of direct pair production of e+ and e<sup>−</sup>. By using this technique, muon energy can be estimated from INO-ICAL detector operating

**Figure 1.** *Schematic view of three modules for the proposed INO-ICAL detector.*

as a pair meter. The primary cosmic rays which are approximately in 50 TeV–50 PeV energy range correspond to this energy range [5].

This work presents a simulation based on the latest version of Geant4 [3] INO-ICAL code, developed by the INO collaboration for momentum reconstruction of muons in GeV energy range using INO-ICAL magnet. We have developed a separate Geant4 code for counting the muon bursts in iron plates for ultra-high energy muon analysis using the pair meter technique. The proposed detector will have a modular structure of total lateral size 48 m × 16 m, subdivided into three modules of size 16 m × 16 m. The height of the detector will be 14.5 m. It will consist of a stack of 151 horizontal layers of ~5.6 cm thick magnetized iron plates interleaved with 4 cm gaps to house the active detector layers. Detector geometry of ICAL magnet is presented in **Figure 1** and details of the components and dimensions are discussed in Ref. [1].

#### **2. Momentum reconstruction analysis with ICAL magnet**

In this section we have discussed the simulation for momentum reconstruction of muons in magnetic field. Details of the detector simulation for muons with energy of few 10's of GeV with older version of INO-ICAL code are already published in Ref. [4]. For simulating the response of high energy (100's of GeV) muons in the ICAL detector, 10,000 muons were propagated uniformly from a vertex randomly located inside 8 m × 8 m × 10 m volume. This is the central region of the central module where the magnetic field is uniform of 1.5 T. In our analysis we have considered only those events whose z coordinate of the input vertex lie within z*in* ≤ 400 cm which comprises the vertex to the central region. The input momentum and zenith angle are kept fixed in each case while the azimuthal angle is uniformly averaged over the entire range –*π* ≤ *ϕ* ≤ *π*. In each case, we have studied the number of reconstructed tracks, the position resolution, including up/down discrimination and the zenith angle resolution. In this chapter, we have followed the same approach as in Ref. [4] for muon response analysis upto energy 500 GeV inside the detector [8]. Momentum reconstruction efficiency in the energy range of 1–400 GeV is shown in **Figure 2** and this energy range at the detector corresponds to the surface muon lying in the energy range 1600–2000 GeV from the top of the

**15**

**Figure 2.**

*(1–20 GeV) and high energy (20–500 GeV).*

*Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique*

surface. Muon will lose around 1600 GeV in the rock overburden [8] to reach at the detector from the top surface. Energetic muons from other directions will also hit the detector since the rock cover in other directions of detector is very huge, so we

The momentum reconstruction efficiency (*εrecon*) is defined as the ratio of the number of reconstructed events, n rec, to the total number of generated events,

**Figure 2** shows the muon momentum reconstruction efficiency as a function of input momentum for different cos*θ* bins, here left and right panels demonstrate detector response for low and high energy muon momentum respectively. One can see that the momentum reconstruction efficiency depends on the incident particle momentum, the angle of propagation and the strength of the magnetic field. As the input momentum increases, the reconstruction efficiency increases for all angles because with increase in energy the particle crosses more number of layers producing more hits in the detector. But at sufficiently high energies, the reconstruction efficiency starts decreasing, since the muons travel nearly straight without being deflected in the magnetic field of the detector. Track reconstruction is done using Kalman Filter techniques [4], tracks for few typical energies are plotted in **Figure 3**, which shows the deflected and undeflected muon tracks depending on the energy

1.High energy muons produce secondary cascades mainly due to electron pair

2.It is one of the most important processes for muon interaction at TeV energies, pair creation cross section exceeds those of other muon interaction processes

*Reconstructed momentum efficiency as a function of the input momentum for different cos θ values at low* 

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (ε*recon*(1 <sup>−</sup> <sup>ε</sup>*recon*)/*Ntotal*) **.**

*Ntotal*

(1)

*DOI: http://dx.doi.org/10.5772/intechopen.81368*

have not incorporated it in our discussion.

of muons in a fixed magnetic field (1.5 T).

in a wide range of energy transfer:

100 MeV ≤ E0 ≤ 0.1*Eμ*, E0 is threshold energy.

Pair meter techniques:

production process.

<sup>ε</sup>*recon* <sup>=</sup> \_\_\_\_ *nrec*

N*total*. We have

with error, *recon* = √

*Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique DOI: http://dx.doi.org/10.5772/intechopen.81368*

surface. Muon will lose around 1600 GeV in the rock overburden [8] to reach at the detector from the top surface. Energetic muons from other directions will also hit the detector since the rock cover in other directions of detector is very huge, so we have not incorporated it in our discussion.

The momentum reconstruction efficiency (*εrecon*) is defined as the ratio of the number of reconstructed events, n rec, to the total number of generated events, N*total*. We have

$$\mathbf{e}\_{recom} = \frac{n\_{\rm{net}}}{N\_{\rm{total}}} \tag{1}$$

with error, *recon* = √ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (ε*recon*(1 <sup>−</sup> <sup>ε</sup>*recon*)/*Ntotal*) **.**

**Figure 2** shows the muon momentum reconstruction efficiency as a function of input momentum for different cos*θ* bins, here left and right panels demonstrate detector response for low and high energy muon momentum respectively. One can see that the momentum reconstruction efficiency depends on the incident particle momentum, the angle of propagation and the strength of the magnetic field. As the input momentum increases, the reconstruction efficiency increases for all angles because with increase in energy the particle crosses more number of layers producing more hits in the detector. But at sufficiently high energies, the reconstruction efficiency starts decreasing, since the muons travel nearly straight without being deflected in the magnetic field of the detector. Track reconstruction is done using Kalman Filter techniques [4], tracks for few typical energies are plotted in **Figure 3**, which shows the deflected and undeflected muon tracks depending on the energy of muons in a fixed magnetic field (1.5 T).

Pair meter techniques:

*Charged Particles*

**Figure 1.**

as a pair meter. The primary cosmic rays which are approximately in 50 TeV–50 PeV

This work presents a simulation based on the latest version of Geant4 [3] INO-ICAL code, developed by the INO collaboration for momentum reconstruction of muons in GeV energy range using INO-ICAL magnet. We have developed a separate Geant4 code for counting the muon bursts in iron plates for ultra-high energy muon analysis using the pair meter technique. The proposed detector will have a modular structure of total lateral size 48 m × 16 m, subdivided into three modules of size 16 m × 16 m. The height of the detector will be 14.5 m. It will consist of a stack of 151 horizontal layers of ~5.6 cm thick magnetized iron plates interleaved with 4 cm gaps to house the active detector layers. Detector geometry of ICAL magnet is presented in **Figure 1** and details of the components and dimensions are discussed in Ref. [1].

In this section we have discussed the simulation for momentum reconstruction of muons in magnetic field. Details of the detector simulation for muons with energy of few 10's of GeV with older version of INO-ICAL code are already published in Ref. [4]. For simulating the response of high energy (100's of GeV) muons in the ICAL detector, 10,000 muons were propagated uniformly from a vertex randomly located inside 8 m × 8 m × 10 m volume. This is the central region of the central module where the magnetic field is uniform of 1.5 T. In our analysis we have considered only those events whose z coordinate of the input vertex lie within z*in* ≤ 400 cm which comprises the vertex to the central region. The input momentum and zenith angle are kept fixed in each case while the azimuthal angle is uniformly averaged over the entire range –*π* ≤ *ϕ* ≤ *π*. In each case, we have studied the number of reconstructed tracks, the position resolution, including up/down discrimination and the zenith angle resolution. In this chapter, we have followed the same approach as in Ref. [4] for muon response analysis upto energy 500 GeV inside the detector [8]. Momentum reconstruction efficiency in the energy range of 1–400 GeV is shown in **Figure 2** and this energy range at the detector corresponds to the surface muon lying in the energy range 1600–2000 GeV from the top of the

**2. Momentum reconstruction analysis with ICAL magnet**

energy range correspond to this energy range [5].

*Schematic view of three modules for the proposed INO-ICAL detector.*

**14**


100 MeV ≤ E0 ≤ 0.1*Eμ*, E0 is threshold energy.

#### **Figure 2.**

*Reconstructed momentum efficiency as a function of the input momentum for different cos θ values at low (1–20 GeV) and high energy (20–500 GeV).*

**Figure 3.** *Fixed energy muons track stored in X-Z plane of the detector going in the downward direction.*


#### **2.1 Average number of burst calculation**

The e<sup>+</sup> and e<sup>−</sup> pair production cross section by a muon of energy E*μ* with energy transfer above a threshold E0 increases as ln2 (2m*e* E*μ*/m*μ*E0) where m*μ* and m*e* are the masses of the muon and the electron respectively [1]. Defining v = E0/E*μ*, above v<sup>−</sup><sup>1</sup> = 10, this cross section dominates over other processes by which the muon loses its energy when it passes through dense matter, generating observable cascades.

**17**

*Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique*

M in the detector with energies above a threshold E0.

*d dv* <sup>≃</sup> \_\_\_\_ <sup>14</sup><sup>α</sup> 9*πt*<sup>0</sup> *ln*( *kmeEμ* \_\_\_\_\_\_

• The pair production cross section depends upon E*μ*/E0 which allows one to estimate the energy of muon by counting the number of interaction cascades

• Differential cross section for pair production is estimated in Ref. [1] and that

where *α* = 1/137, k ≃ 1.8 and t0 is the radiation length(r.l.) of the material, for

• The average number of interaction cascades M above a threshold E0 is given by:

*M*(*E*0,*Eμ*) = *Tt*<sup>0</sup> σ(*E*0,*Eμ*) (3)

where T is target thickness and σ (E0,E*μ*) is the integrated cross section(in unit of

( *kmeEμ* \_\_\_\_\_\_

A muon traversing vertically from the top will cover 151 × 5.6 ≃ 845 cm in iron

calculated using Eq. (3). The number of cascades produced as a function of muon energy is shown in **Figure 4**, which increase with energy in the energy range

The estimation of muon burst energy in iron plates is done by evaluating the

of the detector, that is, the RPC which is a type of spark chamber with resistive electrodes placed parallel to each other. Measurement of penetration depth of the electron in iron gives us a handle to determine the energy loss of the electrons in the iron plates. The energy loss of electron in iron is given by: E = E0 e<sup>−</sup>*x*/*x*<sup>0</sup>

where *x* is distance traveled in the iron plate and *x*0 is the radiation length. It is thus important to determine the range of electron in iron plates which is shown in **Figure 5**.

+ and *e*

 GeV (1–1000 TeV). One can interpret from **Figure 4** that the approximate number of interactions for a 1 TeV muon at threshold energy E0 of 1 GeV is 4. Similarly for a 10 TeV muon, the number of interactions for E0 = 10 GeV is 4 and for 1 GeV is 20. For a 100 TeV muon, the number of interactions for E0 = 100 GeV is around 4, for E0 = 10 GeV is close to 20 and for E0 = 1 GeV is

<sup>9</sup>*πt*0(*ln*<sup>2</sup>

\_\_\_\_\_\_\_\_\_\_\_\_ <sup>9</sup>*M*/7*<sup>T</sup>* <sup>−</sup> *<sup>A</sup>*

*<sup>E</sup>*0*mμ* ) (2)

*<sup>E</sup>*0*mμ* ) <sup>+</sup> *<sup>A</sup>*) (5)

) (4)

). The number of cas-

<sup>−</sup> must hit the active elements

,

) can be

*DOI: http://dx.doi.org/10.5772/intechopen.81368*

expression is given by:

iron t0 = 13.75 g/cm<sup>2</sup>

/g) and A ≃ 1.4.,

cm<sup>2</sup>

103 –10<sup>6</sup>

approximately 50.

*v* \_\_\_

.

*Eμ* = (*E*0*mμ*/*kme*) *exp*(√

<sup>σ</sup>(*E*0,*Eμ*) <sup>≃</sup> \_\_\_\_ <sup>7</sup><sup>α</sup>

**3. Counting the burst using pair meter**

plates, this is equivalent to a path-length of ≃480 r.l.(g/cm<sup>2</sup>

**3.1 Penetration depth of electron in the iron plates**

energy of electron and positron pair, for that *e*

cades produced by high energy muon for a path length of 450 r.l.(g/cm<sup>2</sup>

*Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique DOI: http://dx.doi.org/10.5772/intechopen.81368*


$$
v \frac{d\sigma}{dw} \simeq \frac{14\alpha}{9\pi t\_0} \ln\left(\frac{km\_\epsilon E\_\mu}{E\_0 m\_\mu}\right) \tag{2}$$

where *α* = 1/137, k ≃ 1.8 and t0 is the radiation length(r.l.) of the material, for iron t0 = 13.75 g/cm<sup>2</sup> .

• The average number of interaction cascades M above a threshold E0 is given by:

$$M(E\_0, E\_\mu) = T t\_0 \sigma(E\_0, E\_\mu) \tag{3}$$

$$E\_{\mu} = \left(E\_0 m\_{\mu} / k m\_{\epsilon}\right) \exp\left(\sqrt{9\pi M / 7\alpha T - A}\right) \tag{4}$$

where T is target thickness and σ (E0,E*μ*) is the integrated cross section(in unit of cm<sup>2</sup> /g) and A ≃ 1.4.,

$$\sigma\{E\_0, E\_\mu\} \simeq \frac{7\alpha}{9\pi t\_0} \left( \ln^2\left(\frac{km\_\epsilon E\_\mu}{E\_0 m\_\mu}\right) + A\right) \tag{5}$$

#### **3. Counting the burst using pair meter**

A muon traversing vertically from the top will cover 151 × 5.6 ≃ 845 cm in iron plates, this is equivalent to a path-length of ≃480 r.l.(g/cm<sup>2</sup> ). The number of cascades produced by high energy muon for a path length of 450 r.l.(g/cm<sup>2</sup> ) can be calculated using Eq. (3). The number of cascades produced as a function of muon energy is shown in **Figure 4**, which increase with energy in the energy range 103 –10<sup>6</sup> GeV (1–1000 TeV). One can interpret from **Figure 4** that the approximate number of interactions for a 1 TeV muon at threshold energy E0 of 1 GeV is 4. Similarly for a 10 TeV muon, the number of interactions for E0 = 10 GeV is 4 and for 1 GeV is 20. For a 100 TeV muon, the number of interactions for E0 = 100 GeV is around 4, for E0 = 10 GeV is close to 20 and for E0 = 1 GeV is approximately 50.

#### **3.1 Penetration depth of electron in the iron plates**

The estimation of muon burst energy in iron plates is done by evaluating the energy of electron and positron pair, for that *e* + and *e* <sup>−</sup> must hit the active elements of the detector, that is, the RPC which is a type of spark chamber with resistive electrodes placed parallel to each other. Measurement of penetration depth of the electron in iron gives us a handle to determine the energy loss of the electrons in the iron plates. The energy loss of electron in iron is given by: E = E0 e<sup>−</sup>*x*/*x*<sup>0</sup> , where *x* is distance traveled in the iron plate and *x*0 is the radiation length. It is thus important to determine the range of electron in iron plates which is shown in **Figure 5**.

*Charged Particles*

**16**

v<sup>−</sup><sup>1</sup>

The e<sup>+</sup>

**Figure 3.**

3.Average energy loss for pair production increases linearly with the increase in the muon energy and in the TeV region this process contributes more than 50%

*Fixed energy muons track stored in X-Z plane of the detector going in the downward direction.*

4.Pair meter method for energy reconstruction of high energy muons has been

the masses of the muon and the electron respectively [1]. Defining v = E0/E*μ*, above

 = 10, this cross section dominates over other processes by which the muon loses its energy when it passes through dense matter, generating observable cascades.

and e<sup>−</sup> pair production cross section by a muon of energy E*μ* with energy

(2m*e* E*μ*/m*μ*E0) where m*μ* and m*e* are

of the total energy loss rate.

**2.1 Average number of burst calculation**

transfer above a threshold E0 increases as ln2

used by the NuTeV/CCFR collaboration [6].

#### **Figure 4.**

*Average number of bursts above a threshold E0 versus muon energy for E0 = 1, 10 and 100 GeV, with T fixed to 450 r.l.*

**Figure 5.** *Energy of the electron and its corresponding penetrating range in the iron plates.*

#### *3.1.1 Muon burst of a few typical energies in the iron plates*

A Geant4-based code is developed for simulating the muons burst in iron plates, here horizontal axis is the z-axis of INO-ICAL detector, in which 152 layers of iron plates of width 5.6 cm are placed vertically, interleaved with 2.5 cm gaps for placing the RPC. Muons are propagated using Geant4 particle generator class and generated bursts in the iron plates are counted, muon bursts for a 10, 100, 500 and 1000 GeV are shown in **Figure 6(a)–(d)** respectively [7].

**19**

**Figure 7.**

*technique.*

**Figure 6.**

*Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique*

*DOI: http://dx.doi.org/10.5772/intechopen.81368*

**3.2 Operating ICAL using pair meter technique**

*represents the electron-positron cascade in the x-y plane.*

Cosmic rays are composed of highly energetic particles mostly protons, alpha particles and a small fraction of heavier nuclei that reach the Earth from the outer space. Supernova bursts, quasars and other astronomical events are believed to be the sources of cosmic rays ranging over 1020 eV of energy [9]. The cosmic ray (CR) spectrum depicts a power law behavior over the entire spectral range but displaying two transition regions where the slope of the spectrum changes abruptly. The

*Primary cosmic ray flux (*ϕ ≃ *KE −*α*, where* α ≃ *2.7 and for KNEE (3PeV) α: 2.7* → *3) versus energy of primary particle, and limited range for ICAL to cover the spectrum using magnetic field and pair meter* 

*Cascade generation for a few typical energies. Blue line (muon) represents z-axis of the detector and green line* 

*Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique DOI: http://dx.doi.org/10.5772/intechopen.81368*

**Figure 6.**

*Charged Particles*

**Figure 4.**

*450 r.l.*

**18**

**Figure 5.**

*Energy of the electron and its corresponding penetrating range in the iron plates.*

A Geant4-based code is developed for simulating the muons burst in iron plates, here horizontal axis is the z-axis of INO-ICAL detector, in which 152 layers of iron plates of width 5.6 cm are placed vertically, interleaved with 2.5 cm gaps for placing the RPC. Muons are propagated using Geant4 particle generator class and generated bursts in the iron plates are counted, muon bursts for a 10, 100, 500 and 1000 GeV

*Average number of bursts above a threshold E0 versus muon energy for E0 = 1, 10 and 100 GeV, with T fixed to* 

*3.1.1 Muon burst of a few typical energies in the iron plates*

are shown in **Figure 6(a)–(d)** respectively [7].

*Cascade generation for a few typical energies. Blue line (muon) represents z-axis of the detector and green line represents the electron-positron cascade in the x-y plane.*

#### **3.2 Operating ICAL using pair meter technique**

Cosmic rays are composed of highly energetic particles mostly protons, alpha particles and a small fraction of heavier nuclei that reach the Earth from the outer space. Supernova bursts, quasars and other astronomical events are believed to be the sources of cosmic rays ranging over 1020 eV of energy [9]. The cosmic ray (CR) spectrum depicts a power law behavior over the entire spectral range but displaying two transition regions where the slope of the spectrum changes abruptly. The

#### **Figure 7.**

*Primary cosmic ray flux (*ϕ ≃ *KE −*α*, where* α ≃ *2.7 and for KNEE (3PeV) α: 2.7* → *3) versus energy of primary particle, and limited range for ICAL to cover the spectrum using magnetic field and pair meter technique.*

region around E ~ 5 PeV is where the CR spectrum becomes steeper, is known as the 'knee' region. At higher energy around E ~ 5000 PeV, the CR spectrum flattens at the 'ankle' where the sources of such high energies are believed to be of extragalactic origin. The reason for these two appreciable 'breaks' in the CR spectrum is still unknown. Once accelerated at supernova shocks, cosmic rays have to propagate through the interstellar medium before they can be detected. CR muons are created when cosmic rays enter the Earth's atmosphere where they eventually collide with air molecules and decay into pions. The charged pions (*π*<sup>+</sup> and *π*<sup>−</sup>) decay in flight. All these particles together create into muons (*μ*<sup>+</sup> and *μ*<sup>−</sup>) and neutrinos (ν and *<sup>v</sup>* ¯ ). A cascade called an air shower. Measurement of air shower particles can be interpreted in terms of the energy spectrum and primary cosmic ray composition. To determine these measurements we require calculation of fluxes generated via CR nuclei of mass A, charge Z and energy E.

**Figure 7** represents primary cosmic ray flux versus energy of the primary cosmic ray particle. From **Figure 7**, we can see the limit of the energy range upto which magnetic spectrometers work i.e. around E ~ 104 while the pair meter technique works well in a wide energy range from 5 × 104 –5 × 107 GeV.

#### **4. Results and discussion**

In Section 2, we have discussed the limitation of magnetized ICAL detector to be used as a magnetic spectrometer, which limits the efficiency of the detector to discriminate between *μ*+ and *μ*− at higher energies and reconstruct momentum. Variation in efficiency of muon momentum reconstruction as a function of input momentum is shown in **Figure 2**, which shows a clear fall in efficiency for energetic muons. In **Figure 3**, muon track could be seen, which is undeflected for highly energetic muons. Finally, it is concluded that with ICAL detector we can do analysis for muons in the energy range of 1–400GeV. This corresponds to surface muons in the energy range of 1600–2000 GeV, because muons lose around 1600 GeV energy [3] into the rock overburden to reach at the detector from the top of the INO-ICAL surface. ICAL cannot be used as a magnetic spectrometer for highly energetic cosmic ray muons. For energetic (TeV) muons, pair meter technique [2] can be used for momentum reconstruction as discussed in Section 3 [5]. This technique is tested by a few detectors, since INO-ICAL will be large in dimensions so it will be a perfect machine to test the capability of this technique. We have developed a separate Geant4 code for counting the bursts in the iron plates and also a technique to measure the energy of the bursts with the produced electron pairs into the iron plates. In **Figure 6**, we can see the burst of muons in iron plate. The variation of these burst number is shown in **Figure 4**, which is a function of the muon energy.

#### **4.1 Summary and conclusions**


**21**

**Author details**

provided the original work is properly cited.

Jaydip Singh\*, Srishti Nagu and Jyotsna Singh University of Lucknow, Lucknow, India

\*Address all correspondence to: jaydip.singh@gmail.com

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Analysis of Ultra-High Energy Muons at INO-ICAL Using Pair Meter Technique*

This work is partially supported by Department of Physics, Lucknow University, Department of Atomic Energy, Harish-Chandra Research Institute, Allahabad and INO collaboration. Financially it is supported by government of India, DST project no-SR/MF/PS02/2013, Department of Physics, Lucknow University. We thank Prof. Raj Gandhi for useful discussion and providing hospitality to work in HRI, Dr. Jyotsna Singh for her support and guidance in completing

*DOI: http://dx.doi.org/10.5772/intechopen.81368*

this work from Lucknow University.

**Acknowledgements**

### **Acknowledgements**

*Charged Particles*

region around E ~ 5 PeV is where the CR spectrum becomes steeper, is known as the 'knee' region. At higher energy around E ~ 5000 PeV, the CR spectrum flattens at the 'ankle' where the sources of such high energies are believed to be of extragalactic origin. The reason for these two appreciable 'breaks' in the CR spectrum is still unknown. Once accelerated at supernova shocks, cosmic rays have to propagate through the interstellar medium before they can be detected. CR muons are created when cosmic rays enter the Earth's atmosphere where they eventually collide with

cascade called an air shower. Measurement of air shower particles can be interpreted in terms of the energy spectrum and primary cosmic ray composition. To determine these measurements we require calculation of fluxes generated via CR nuclei of

**Figure 7** represents primary cosmic ray flux versus energy of the primary cosmic

–5 × 107

GeV.

ray particle. From **Figure 7**, we can see the limit of the energy range upto which

In Section 2, we have discussed the limitation of magnetized ICAL detector to be used as a magnetic spectrometer, which limits the efficiency of the detector to discriminate between *μ*+ and *μ*− at higher energies and reconstruct momentum. Variation in efficiency of muon momentum reconstruction as a function of input momentum is shown in **Figure 2**, which shows a clear fall in efficiency for energetic muons. In **Figure 3**, muon track could be seen, which is undeflected for highly energetic muons. Finally, it is concluded that with ICAL detector we can do analysis for muons in the energy range of 1–400GeV. This corresponds to surface muons in the energy range of 1600–2000 GeV, because muons lose around 1600 GeV energy [3] into the rock overburden to reach at the detector from the top of the INO-ICAL surface. ICAL cannot be used as a magnetic spectrometer for highly energetic cosmic ray muons. For energetic (TeV) muons, pair meter technique [2] can be used for momentum reconstruction as discussed in Section 3 [5]. This technique is tested by a few detectors, since INO-ICAL will be large in dimensions so it will be a perfect machine to test the capability of this technique. We have developed a separate Geant4 code for counting the bursts in the iron plates and also a technique to measure the energy of the bursts with the produced electron pairs into the iron plates. In **Figure 6**, we can see the burst of muons in iron plate. The variation of these burst number is shown in **Figure 4**, which is a function of

• The pair meter technique can competently measure muon energy in the energy

• One can probe very high energy muon fluxes and primary cosmic rays in the knee region which will aid in accurate background muon and neutrino flux measurement in the forthcoming detectors designed for ultra-high energy

• Our Geant4 analyses for central module of INO-ICAL detector are successfully performed and variation in the cascade number of varying energy is observed

range of 1–1000 TeV at INO-ICAL detector operating as pair meter.

and *π*<sup>−</sup>) decay in flight. All

and *μ*<sup>−</sup>) and neutrinos (ν and *<sup>v</sup>* ¯ ). A

while the pair meter technique

air molecules and decay into pions. The charged pions (*π*<sup>+</sup>

these particles together create into muons (*μ*<sup>+</sup>

magnetic spectrometers work i.e. around E ~ 104

works well in a wide energy range from 5 × 104

mass A, charge Z and energy E.

**4. Results and discussion**

**20**

the muon energy.

**4.1 Summary and conclusions**

neutrino experiments.

in the iron plates.

This work is partially supported by Department of Physics, Lucknow University, Department of Atomic Energy, Harish-Chandra Research Institute, Allahabad and INO collaboration. Financially it is supported by government of India, DST project no-SR/MF/PS02/2013, Department of Physics, Lucknow University. We thank Prof. Raj Gandhi for useful discussion and providing hospitality to work in HRI, Dr. Jyotsna Singh for her support and guidance in completing this work from Lucknow University.

### **Author details**

Jaydip Singh\*, Srishti Nagu and Jyotsna Singh University of Lucknow, Lucknow, India

\*Address all correspondence to: jaydip.singh@gmail.com

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References** Chapter 3

[1] Ahamed S, Sajjad Atahar M, et al. (INO Collab). Physics potential of the ICAL detector at the Indian based neutrino observatory. 9 May 2017; arXiv:1505.07380

From the Eloisatron to the

In the late 1970s the experimental physics community was active in promoting the Large Electron Positron (LEP) collider and its associated experiments to study the Z- and W-bosons, and with the expectation that the tunnel could subsequently house a hadron collider (LHC), providing a center-of-mass energy for discoveries at the frontier of knowledge. At this time, Antonino Zichichi, who had chaired a Working Group in charge of promoting LEP among the community of experimental and accelerator physicists, realized that one should envisage building as large a ring as possible, for which LEP/LHC would be but a scale model, and it was thus the idea of the Eloisatron, or ELN, in a ring of about 300 km in circumference, was born. CERN and IHEP, China, are now engaged in studies for future colliders of 100 km in circumference, aiming to extend center-of-mass energy in hadron collisions to 100 TeV by using very high field magnets. The ELN idea lives on, but it is time to envision an update. A ring of diameter 300 km would make possible the installation of a sequence of increasingly complex accelerators culminating in one eventually capable of providing a center-of-mass energy of 1000 TeV, i.e. a peta-electron-volt

Keywords: particle physics, standard model, CERN, colliders, ISR, LEP, LHC, FCC,

Following the success of the first hadron collider, the Intersecting Storage Rings (ISR) [1] at CERN in the early 1970s, colliders have been the experimentalists' main tool for exploring, in laboratory conditions, particle physics at the frontier of knowledge. Enthusiasm for the large collider that was to become the Large

Electron-Positron collider (LEP), was amplified in 1976 by the workings of the socalled LEP Working Group [2], which produced several visionary reports regarding the future possibilities of an accelerator of about 30 km in circumference [3]. Thanks to technological advances at the ISR, the Z- and W-bosons foreseen in the Standard Model were first observed (1984) in the proton-antiproton collider in the SPS tunnel at CERN [4], before LEP was completed. The lepton collider LEP, an accelerator of 27 km in circumference targeted detailed study of the intermediate bosons that had been previously discovered, providing unprecedented precision that helped to entrench the Standard Model of fundamental physics between 1989 and 2000 as being the best, albeit incomplete, description of the physical world of elementary particles at the present time [5]. In parallel, the feasibility of performing

Pevatron

Thomas Taylor

Abstract

or PeV.

23

1. Introduction

Eloisatron, 100 TeV, 1 PeV, Pevatron

[2] Kokoulin RP, Petrukhin AA. Theory of the pairmeter for high energy muon measurements. Nuclear Instruments and Methods in Physics Research. 1987;**A263**(1988):468-479

[3] Groom DE, Mokhov NV, Striganov S. Muon stopping power and range tables 10 MeV-100 TeV. Atomic Data and Nuclear Data Tables. 2001;**76**(2)

[4] Chatterjee A et al. (INO Collab). JINST 9(2014) PO 7001; July 2014

[5] Gandhi R, Panda S. Probing very high energy prompt muon and neutrino fluxes and the cosmic ray knee via underground muons. 31 August 2006; arXiv:hep-ph/0512179

[6] Chikkatur AP, Bugel L, et al. Test of a calorimetric technique for measuring the energy of cosmic ray muon in TeV energy range. Zeitschrift für Physik. 1997;**74**:279-289

[7] Agostinelli S et al. GEANT4: A simulation toolkit, GEANT4 collaboration. Nuclear Instruments and Methods in Physics Research Section A. 2003;**506**:250 http://geant4.cern.ch/

[8] Singh J, Singh J, et al. Atmospheric muons charge ratio analysis at the INO-ICAL detector. 29 Jun 2018; arXiv:1709.01064

[9] Ramesh N, Hawron M, Martin C, Bachri A. Flux variation of cosmic muons. Journal of the Arkansas Academy of Science. 2011;**65**:67-72

#### **References** Chapter 3

## From the Eloisatron to the Pevatron

Thomas Taylor

#### Abstract

In the late 1970s the experimental physics community was active in promoting the Large Electron Positron (LEP) collider and its associated experiments to study the Z- and W-bosons, and with the expectation that the tunnel could subsequently house a hadron collider (LHC), providing a center-of-mass energy for discoveries at the frontier of knowledge. At this time, Antonino Zichichi, who had chaired a Working Group in charge of promoting LEP among the community of experimental and accelerator physicists, realized that one should envisage building as large a ring as possible, for which LEP/LHC would be but a scale model, and it was thus the idea of the Eloisatron, or ELN, in a ring of about 300 km in circumference, was born. CERN and IHEP, China, are now engaged in studies for future colliders of 100 km in circumference, aiming to extend center-of-mass energy in hadron collisions to 100 TeV by using very high field magnets. The ELN idea lives on, but it is time to envision an update. A ring of diameter 300 km would make possible the installation of a sequence of increasingly complex accelerators culminating in one eventually capable of providing a center-of-mass energy of 1000 TeV, i.e. a peta-electron-volt or PeV.

Keywords: particle physics, standard model, CERN, colliders, ISR, LEP, LHC, FCC, Eloisatron, 100 TeV, 1 PeV, Pevatron

#### 1. Introduction

Following the success of the first hadron collider, the Intersecting Storage Rings (ISR) [1] at CERN in the early 1970s, colliders have been the experimentalists' main tool for exploring, in laboratory conditions, particle physics at the frontier of knowledge. Enthusiasm for the large collider that was to become the Large Electron-Positron collider (LEP), was amplified in 1976 by the workings of the socalled LEP Working Group [2], which produced several visionary reports regarding the future possibilities of an accelerator of about 30 km in circumference [3]. Thanks to technological advances at the ISR, the Z- and W-bosons foreseen in the Standard Model were first observed (1984) in the proton-antiproton collider in the SPS tunnel at CERN [4], before LEP was completed. The lepton collider LEP, an accelerator of 27 km in circumference targeted detailed study of the intermediate bosons that had been previously discovered, providing unprecedented precision that helped to entrench the Standard Model of fundamental physics between 1989 and 2000 as being the best, albeit incomplete, description of the physical world of elementary particles at the present time [5]. In parallel, the feasibility of performing

**22**

*Charged Particles*

arXiv:1505.07380

1987;**A263**(1988):468-479

[1] Ahamed S, Sajjad Atahar M, et al. (INO Collab). Physics potential of the ICAL detector at the Indian based neutrino observatory. 9 May 2017;

[2] Kokoulin RP, Petrukhin AA. Theory of the pairmeter for high energy muon measurements. Nuclear Instruments and Methods in Physics Research.

[3] Groom DE, Mokhov NV, Striganov S. Muon stopping power and range tables 10 MeV-100 TeV. Atomic Data and Nuclear Data Tables. 2001;**76**(2)

[4] Chatterjee A et al. (INO Collab). JINST 9(2014) PO 7001; July 2014

[5] Gandhi R, Panda S. Probing very high energy prompt muon and neutrino fluxes and the cosmic ray knee via underground muons. 31 August 2006;

[6] Chikkatur AP, Bugel L, et al. Test of a calorimetric technique for measuring the energy of cosmic ray muon in TeV energy range. Zeitschrift für Physik.

collaboration. Nuclear Instruments and Methods in Physics Research Section A. 2003;**506**:250 http://geant4.cern.ch/

[8] Singh J, Singh J, et al. Atmospheric muons charge ratio analysis at the INO-ICAL detector. 29 Jun 2018;

[9] Ramesh N, Hawron M, Martin C, Bachri A. Flux variation of cosmic muons. Journal of the Arkansas Academy of Science. 2011;**65**:67-72

[7] Agostinelli S et al. GEANT4: A simulation toolkit, GEANT4

arXiv:hep-ph/0512179

1997;**74**:279-289

arXiv:1709.01064

comparable experiments with leptons with linear colliders was ably demonstrated at SLAC, and a more powerful proton-antiproton collider was put into service at the Fermilab Tevatron [6]. Experimental particle physics had well and truly embraced the advantage of laboratory-based colliders for probing deeper into the unknown. In the quest for still higher center-of-mass energies, the 80 km Superconducting Super Collider (SSC) [7] in the USA was destined to provide conditions for experiments at center-of-mass energies of 25– 30 TeV, but due to funding problems this was not to be. It was therefore at the Large Hadron Collider [8], installed in the LEP tunnel, that events revealing the elusive Higgs boson were observed in 2012 [9]. Now the community is studying the possibility of providing the discovery potential of still higher center-of-mass energies, as well as precision measurements of the Higgs boson in a lepton collider in much the same way as was done at LEP for the Z- and W-bosons.

a minimum cost route to a 100 TeV center-of-mass hadron collider: the idea being to finish the SCC tunnel in Texas, in which an injector synchrotron would be installed, and to bore a second tunnel, of length 270 km to house the main collider, using affordable 5 T superconducting dipoles. The SCC tunnel could also house an

vicinity of the SCC is propitious for tunneling, the per meter cost being less than that near CERN, at least in the sections of limestone [16] (but for a global costing this would have to be offset by the value of the existing laboratory infrastructure at CERN). However, unlike for LHC, where there was an identified goal (to reveal Higgs), indicating an appropriate energy, it is now desirable to foresee being able to access far higher collision energies, so the goal should perhaps evolve. It is thus proposed that one should consider a ring of about 300 km in diameter, enabling the installation of a sequence of accelerators culminating in one capable of achieving a hadronic center-of-mass energy of 1000 TeV, i.e. a PeV, or peta-electron-volt. In this chapter, a vision is presented of what would probably be the world's Ultimate Circular Collider, based on proven and foreseeable accelerator physics and engi-

It is first assumed that a Higgs Factory, either in the form of a linear collider or

e collider, it is supposed to be principally an instru-

LEP/LHC tunnel) will have done its work before the first phase of experimentation starts at the UCC. While the tunnel could (and should, initially) undoubtedly be

ment for performing frontier physics using hadron collisions. The second assumption is that suitable stable sites can be identified for efficiently boring or excavating an approximately circular tunnel of about 300 km in diameter. For the first phase of operation of the hadron collider, the accelerator should be capable of delivering comfortably a center-of-mass energy of 100 TeV in at least two experiments using the simplest (and cheapest) possible guiding magnet system. This accelerator would serve as the injector for subsequent upgrades of the collision energy, culminating in 1000 TeV and (like the FCC) would depend on the development of affordable high performance superconducting material. The first phase would feature extremely simple magnets based on the use of existing low-cost superconductors. This, together with the fact that synchrotron radiation, and other effects associated with the deflection of stiff beams would not be a problem, could mean that the first phase of the proposed machine may even be cost-competitive with the hadron

There are undoubted advantages in using present laboratory infrastructure if at all possible, as has been done to render the sequence of accelerators at CERN both performant and affordable. The longstanding international nature of Geneva also facilitates the hosting of such a center. It is thus evident that one candidate should be CERN. However, the terrain is not ideal, as the tunnel would have to pass through many kilometers of mixed rock including limestone, which would complicate the process and have clear implications on the scale, cost and timescale of civil engineering. Other sites must therefore be considered for a larger circular accelerator than the FCC, and the cost analysis should consider what would be required to

e collider in the

phase 1 of an FCC or CEPC (or possibly "LEP3", an ultimate e<sup>+</sup>

versions of FCC/CEPC. An added incentive to read on….

e collider to be used as a Higgs factory. The terrain in the

efficient circular e<sup>+</sup>

From the Eloisatron to the Pevatron

DOI: http://dx.doi.org/10.5772/intechopen.80579

2. The requirement

used to house an exciting e+

3. The site

25

neering science: i.e. the UCC, or Pevatron.

The idea of the Eloisatron, or ELN, to be installed in a tunnel of about 300 km of circumference, was born in 1979. Zichichi, an experimental physicist who had played an active role in the first g-2 experiment at CERN, and led experiments at the ISR, clearly understood the value of storage rings and colliders. He had been very active in promoting work on LEP and its associated experiments [2]—fully aware of the fact that the tunnel could be used subsequently to house a hadron collider. He realized the importance of equipping as large a ring as possible to enable to perform experiments at the highest possible center-of-mass energy, and for which LEP/LHC would be but a scale model. It was thus that he came up with the idea of the Eloisatron this being the largest that could be accommodated on the island of Sicily. Zichichi argued that such an instrument could be built for roughly the same cost as a bridge that was envisioned to join the island to the mainland, and this and several other sites in Italy were considered (perhaps the most appropriate being in the geologically stable island of Sardinia) [10]. Kjell Johnsen, who had previously led the highly innovative ISR project, was put in charge of the first studies for such an accelerator. However, this was to be overshadowed by work on the ill-fated 83 km Superconducting Synchrotron Collider (SSC) project in the USA, and on the LHC at CERN, only to be revived later in the ephemeral Very Large Hadron Collider (VLHC) studies for a 233 km long collider at Fermilab [11]—itself largely inspired by the Eloisatron concept. Nevertheless, it should be emphasized that it was due to the recognition of the importance of studies on detectors associated with the potential ELN (and a fortiori the LHC) that the LAA project was born. This auxiliary project provided the framework for vital work to be performed on detectors, enabling the adoption of techniques to address problems inherent with equipment required to observe very high energy collisions at the LHC [12].

Today we have the LHC, which will be upgraded to provide increased luminosity from 2026. Studies for a future generation of accelerator/colliders focus on either a linear collider (for leptons) aimed at detailed study of the Higgs (or Higgs-like) events seen at the LHC, or a large (100 km) circular collider. The idea is that this would first provide e+ e collisions (as a simpler alternative to the linear collider for a so-called Higgs factory), both in the European Future Circular Collider (FCC) and Chinese Circular Electron-Positron Collider (CEPC) versions, to be followed by installing a hadron collider with discovery potential [13, 14]. To achieve a soughtafter 100 TeV center-of-mass energy, the new machine would require a large number of yet-to-be-developed, very high field superconducting magnets (16 T dipoles). The experimental physics community hopes that at least one of the large circular machines will be constructed, even if, as likely, it does not quite reach the presently advertised performance, it will allow groundbreaking studies both in particle and in accelerator physics and technology. However, the philosophy behind the idea of the ELN lives on. Closest to the ELN concept, there is a proposal [15] for

#### From the Eloisatron to the Pevatron DOI: http://dx.doi.org/10.5772/intechopen.80579

comparable experiments with leptons with linear colliders was ably demonstrated at SLAC, and a more powerful proton-antiproton collider was put into service at the Fermilab Tevatron [6]. Experimental particle physics had well and truly embraced the advantage of laboratory-based colliders for probing deeper into the unknown. In the quest for still higher center-of-mass energies, the 80 km Superconducting Super Collider (SSC) [7] in the USA was destined to provide conditions for experiments at center-of-mass energies of 25– 30 TeV, but due to funding problems this was not to be. It was therefore at the Large Hadron Collider [8], installed in the LEP tunnel, that events revealing the elusive Higgs boson were observed in 2012 [9]. Now the community is studying the possibility of providing the discovery potential of still higher center-of-mass energies, as well as precision measurements of the Higgs boson in a lepton collider in much the same way as was done at LEP for the

The idea of the Eloisatron, or ELN, to be installed in a tunnel of about 300 km of

Today we have the LHC, which will be upgraded to provide increased luminosity from 2026. Studies for a future generation of accelerator/colliders focus on either a linear collider (for leptons) aimed at detailed study of the Higgs (or Higgs-like) events seen at the LHC, or a large (100 km) circular collider. The idea is that this

a so-called Higgs factory), both in the European Future Circular Collider (FCC) and Chinese Circular Electron-Positron Collider (CEPC) versions, to be followed by installing a hadron collider with discovery potential [13, 14]. To achieve a soughtafter 100 TeV center-of-mass energy, the new machine would require a large number of yet-to-be-developed, very high field superconducting magnets (16 T dipoles). The experimental physics community hopes that at least one of the large circular machines will be constructed, even if, as likely, it does not quite reach the presently advertised performance, it will allow groundbreaking studies both in particle and in accelerator physics and technology. However, the philosophy behind the idea of the ELN lives on. Closest to the ELN concept, there is a proposal [15] for

e collisions (as a simpler alternative to the linear collider for

circumference, was born in 1979. Zichichi, an experimental physicist who had played an active role in the first g-2 experiment at CERN, and led experiments at the ISR, clearly understood the value of storage rings and colliders. He had been very active in promoting work on LEP and its associated experiments [2]—fully aware of the fact that the tunnel could be used subsequently to house a hadron collider. He realized the importance of equipping as large a ring as possible to enable to perform experiments at the highest possible center-of-mass energy, and for which LEP/LHC would be but a scale model. It was thus that he came up with the idea of the Eloisatron this being the largest that could be accommodated on the island of Sicily. Zichichi argued that such an instrument could be built for roughly the same cost as a bridge that was envisioned to join the island to the mainland, and this and several other sites in Italy were considered (perhaps the most appropriate being in the geologically stable island of Sardinia) [10]. Kjell Johnsen, who had previously led the highly innovative ISR project, was put in charge of the first studies for such an accelerator. However, this was to be overshadowed by work on the ill-fated 83 km Superconducting Synchrotron Collider (SSC) project in the USA, and on the LHC at CERN, only to be revived later in the ephemeral Very Large Hadron Collider (VLHC) studies for a 233 km long collider at Fermilab [11]—itself largely inspired by the Eloisatron concept. Nevertheless, it should be emphasized that it was due to the recognition of the importance of studies on detectors associated with the potential ELN (and a fortiori the LHC) that the LAA project was born. This auxiliary project provided the framework for vital work to be performed on detectors, enabling the adoption of techniques to address problems inherent with equipment required to observe very high energy collisions at the LHC [12].

Z- and W-bosons.

Charged Particles

would first provide e+

24

a minimum cost route to a 100 TeV center-of-mass hadron collider: the idea being to finish the SCC tunnel in Texas, in which an injector synchrotron would be installed, and to bore a second tunnel, of length 270 km to house the main collider, using affordable 5 T superconducting dipoles. The SCC tunnel could also house an efficient circular e<sup>+</sup> e collider to be used as a Higgs factory. The terrain in the vicinity of the SCC is propitious for tunneling, the per meter cost being less than that near CERN, at least in the sections of limestone [16] (but for a global costing this would have to be offset by the value of the existing laboratory infrastructure at CERN). However, unlike for LHC, where there was an identified goal (to reveal Higgs), indicating an appropriate energy, it is now desirable to foresee being able to access far higher collision energies, so the goal should perhaps evolve. It is thus proposed that one should consider a ring of about 300 km in diameter, enabling the installation of a sequence of accelerators culminating in one capable of achieving a hadronic center-of-mass energy of 1000 TeV, i.e. a PeV, or peta-electron-volt. In this chapter, a vision is presented of what would probably be the world's Ultimate Circular Collider, based on proven and foreseeable accelerator physics and engineering science: i.e. the UCC, or Pevatron.

#### 2. The requirement

It is first assumed that a Higgs Factory, either in the form of a linear collider or phase 1 of an FCC or CEPC (or possibly "LEP3", an ultimate e<sup>+</sup> e collider in the LEP/LHC tunnel) will have done its work before the first phase of experimentation starts at the UCC. While the tunnel could (and should, initially) undoubtedly be used to house an exciting e+ e collider, it is supposed to be principally an instrument for performing frontier physics using hadron collisions. The second assumption is that suitable stable sites can be identified for efficiently boring or excavating an approximately circular tunnel of about 300 km in diameter. For the first phase of operation of the hadron collider, the accelerator should be capable of delivering comfortably a center-of-mass energy of 100 TeV in at least two experiments using the simplest (and cheapest) possible guiding magnet system. This accelerator would serve as the injector for subsequent upgrades of the collision energy, culminating in 1000 TeV and (like the FCC) would depend on the development of affordable high performance superconducting material. The first phase would feature extremely simple magnets based on the use of existing low-cost superconductors. This, together with the fact that synchrotron radiation, and other effects associated with the deflection of stiff beams would not be a problem, could mean that the first phase of the proposed machine may even be cost-competitive with the hadron versions of FCC/CEPC. An added incentive to read on….

#### 3. The site

There are undoubted advantages in using present laboratory infrastructure if at all possible, as has been done to render the sequence of accelerators at CERN both performant and affordable. The longstanding international nature of Geneva also facilitates the hosting of such a center. It is thus evident that one candidate should be CERN. However, the terrain is not ideal, as the tunnel would have to pass through many kilometers of mixed rock including limestone, which would complicate the process and have clear implications on the scale, cost and timescale of civil engineering. Other sites must therefore be considered for a larger circular accelerator than the FCC, and the cost analysis should consider what would be required to

develop (or establish) suitable infrastructure for another, possibly entirely new, laboratory. As such an accelerator must be a fully international endeavor, and it will certainly take decades to achieve the ultimate goal, mechanisms will have to be enacted that render it attractive and guarantee its perennity. What springs to mind is a development of the successful CERN model, and CERN should most definitely be central to its establishment. Suitable geologically stable sites with good tunneling attributes certainly exist in China, Europe, Russia and the United States, but sociopolitical support will be as important as geographical location.

as science proper. It has been observed that clustering cities can lead to the creation of hotbeds of efficiency, creativity and innovation, and it is on this premise that China, for example, has already identified 19 regions that it intends to endow with the necessary infrastructure [17]. Other large nations are also considering moving in the same direction. It is opined that the fundamental research orientation of the complex suggested here would be more effective in providing the impetus for getting such a city cluster to work, than either bureaucratic edict or simply hoping

It is foreseen that such a collider would be operated in three major phases, each

A large fraction of the experimental particle physics community is presently of the opinion that to provide worthwhile discovery potential the next hadron collider after the LHC should deliver a center-of-mass energy of about 100 TeV. This is the ultimate goal of the FCC as seen today, and it would be the target for the first phase of the UCC. For the 300 km diameter accelerator considered here, this would imply

having a main dipole magnet field of 1.7 T. The transmission line magnet (pipetron), first developed for the VLHC study [11] provides such a field very efficiently. As shown in Figure 2, this device consists of field-shaping iron poles and yoke excited by a single superconducting cable carrying up to 100 kA and is extremely cost effective. Since the VLHC study, new cables based on magnesium diboride (MgB2) material, which work comfortably at 20 K as opposed to 4.2 K for Nb-Ti, have been developed at CERN for interconnecting equipment required for the luminosity upgrade of the LHC [18]. As the specific heat of metals is proportional to the cube of the absolute temperature, such a cable is very stable, and, compared with the previous study for the VLHC, its use would also lead to a simplification of the associated cryogenic envelope and cooling system. To make best use of this technology the main magnet guiding and focusing system could be of the combined function type featured in the ISR. The estimated cost for such a

Schematic cross-section of a transmission line magnet for bending the counter-rotating beams of elementary particles in a collider. The magnet is excited by a large current (about 100 kA) flowing in a superconducting cable. Such a device can deliver a field of up to about 1.7 T—sufficient for delivering a center-of-mass energy of

that lavish infrastructure would somehow engender efficiency.

4. The accelerator/collider complex

From the Eloisatron to the Pevatron

DOI: http://dx.doi.org/10.5772/intechopen.80579

phase taking its performance to new heights.

4.1 Phase 1

Figure 2.

27

100 TeV to protons in a collider of 300 km in diameter.

Considering the evident urbanization of the planet, this new collider would provide the incentive for a farseeing nation or region to combine the excitement of creating a laboratory to explore the very forefront of natural science, with that of establishing a cluster of cities, including some that are radically new. These should feature all the latest developments in sustainability and form a living exhibition of what can be done to enhance the quality of life and quest for perfection. Besides accelerator scientists and experimental physicists, architects, engineers, social scientists, artists and philosophers should all share the excitement of working together to create such a holistic ensemble showing the way for a harmonious future of the region of the world that is farsighted enough to seize the opportunity. We consider the establishment of at least four major agglomerations, or sub-cities, each housing at least 5 million inhabitants clustered around a circle defined by the accelerator, inter-linked by rapid train and highway systems. Ideally, to facilitate the setting up of the complex, at least one or two of the cities would be developments of existing conurbations. Each city would feature its own local subway system. The airport should be located approximately at the center of the circle with rapid local trains connecting to each of the mainline stations at the city nodes. Such an arrangement is shown schematically in Figure 1, but the actual layout would depend on local geography and a consensus based on overall requirements and planning, and responding to the constraints of sustainability, comfort and efficiency. Some of the glue holding the enterprise together would be the pride of hosting a forefront laboratory probing the mysteries of science using a unique instrument. It is confidently expected that such a complex of cities would be a breeding ground for experimentation in urban living as well as in physics research and associated technological developments and would pioneer advances in social well-being as well

#### Figure 1.

Schematic layout of a cluster of cities hosting a very large particle accelerator/collider. The large circle represents the main ring. It is supposed that the injector synchrotron would be located at one of the cities, and the two major experiments at two of the other cities.

From the Eloisatron to the Pevatron DOI: http://dx.doi.org/10.5772/intechopen.80579

as science proper. It has been observed that clustering cities can lead to the creation of hotbeds of efficiency, creativity and innovation, and it is on this premise that China, for example, has already identified 19 regions that it intends to endow with the necessary infrastructure [17]. Other large nations are also considering moving in the same direction. It is opined that the fundamental research orientation of the complex suggested here would be more effective in providing the impetus for getting such a city cluster to work, than either bureaucratic edict or simply hoping that lavish infrastructure would somehow engender efficiency.

#### 4. The accelerator/collider complex

It is foreseen that such a collider would be operated in three major phases, each phase taking its performance to new heights.

#### 4.1 Phase 1

develop (or establish) suitable infrastructure for another, possibly entirely new, laboratory. As such an accelerator must be a fully international endeavor, and it will certainly take decades to achieve the ultimate goal, mechanisms will have to be enacted that render it attractive and guarantee its perennity. What springs to mind is a development of the successful CERN model, and CERN should most definitely be central to its establishment. Suitable geologically stable sites with good tunneling attributes certainly exist in China, Europe, Russia and the United States, but socio-

Considering the evident urbanization of the planet, this new collider would provide the incentive for a farseeing nation or region to combine the excitement of creating a laboratory to explore the very forefront of natural science, with that of establishing a cluster of cities, including some that are radically new. These should feature all the latest developments in sustainability and form a living exhibition of what can be done to enhance the quality of life and quest for perfection. Besides accelerator scientists and experimental physicists, architects, engineers, social scientists, artists and philosophers should all share the excitement of working together to create such a holistic ensemble showing the way for a harmonious future of the region of the world that is farsighted enough to seize the opportunity. We consider the establishment of at least four major agglomerations, or sub-cities, each housing at least 5 million inhabitants clustered around a circle defined by the accelerator, inter-linked by rapid train and highway systems. Ideally, to facilitate the setting up of the complex, at least one or two of the cities would be developments of existing conurbations. Each city would feature its own local subway system. The airport should be located approximately at the center of the circle with rapid local trains connecting to each of the mainline stations at the city nodes. Such an arrangement is shown schematically in Figure 1, but the actual layout would depend on local geography and a consensus based on overall requirements and planning, and responding to the constraints of sustainability, comfort and efficiency. Some of the glue holding the enterprise together would be the pride of hosting a forefront laboratory probing the mysteries of science using a unique instrument. It is confidently expected that such a complex of cities would be a breeding ground for experimentation in urban living as well as in physics research and associated technological developments and would pioneer advances in social well-being as well

Schematic layout of a cluster of cities hosting a very large particle accelerator/collider. The large circle represents the main ring. It is supposed that the injector synchrotron would be located at one of the cities, and the two major

political support will be as important as geographical location.

Charged Particles

Figure 1.

26

experiments at two of the other cities.

A large fraction of the experimental particle physics community is presently of the opinion that to provide worthwhile discovery potential the next hadron collider after the LHC should deliver a center-of-mass energy of about 100 TeV. This is the ultimate goal of the FCC as seen today, and it would be the target for the first phase of the UCC. For the 300 km diameter accelerator considered here, this would imply having a main dipole magnet field of 1.7 T. The transmission line magnet (pipetron), first developed for the VLHC study [11] provides such a field very efficiently. As shown in Figure 2, this device consists of field-shaping iron poles and yoke excited by a single superconducting cable carrying up to 100 kA and is extremely cost effective. Since the VLHC study, new cables based on magnesium diboride (MgB2) material, which work comfortably at 20 K as opposed to 4.2 K for Nb-Ti, have been developed at CERN for interconnecting equipment required for the luminosity upgrade of the LHC [18]. As the specific heat of metals is proportional to the cube of the absolute temperature, such a cable is very stable, and, compared with the previous study for the VLHC, its use would also lead to a simplification of the associated cryogenic envelope and cooling system. To make best use of this technology the main magnet guiding and focusing system could be of the combined function type featured in the ISR. The estimated cost for such a

#### Figure 2.

Schematic cross-section of a transmission line magnet for bending the counter-rotating beams of elementary particles in a collider. The magnet is excited by a large current (about 100 kA) flowing in a superconducting cable. Such a device can deliver a field of up to about 1.7 T—sufficient for delivering a center-of-mass energy of 100 TeV to protons in a collider of 300 km in diameter.

magnet system for the UCC is of the order of twice the cost of the magnet system for the 8.3 T system for the (very much smaller) LHC, and probably more affordable (and certainly less risky) than that of the presently envisaged high field magnet system for the FCC. The accelerator would require an injector system to supply 4 TeV protons, consisting of a linear accelerator and two pulsed synchrotrons.

• Accumulation of skills of the workforce

• Location in an internationally-oriented city

each of these when planning how to undertake the work.

• Skill in the use of infrastructure to build increasingly sophisticated accelerators

A new aspiring global laboratory should consider carefully the implication of

The CERN convention is a visionary 32-page document that spells out clearly the purpose of the Organization and the rules for its governance. The governing body is the CERN Council, made up of two delegates from each member state. The Council is assisted by the Finance Committee—which deals with material and personnel budgets, and the Scientific Policy Committee—which advises on the research agenda. The convention is drawn up in such a way as to vest the Council with significant autonomy and authority to negotiate and take decisions in the interest of the Organization, to empower its scientists, and to reduce to a strict minimum the

Council allocates the annual budget, with funds provided by the member states

Scientists, engineers and technicians are encouraged to hone their skills through their work on projects. This enables them to write comprehensive specifications for equipment that is available industrially, to follow up constructively the contracts, and to design and prototype special equipment that is not available on the market. The laboratory maintains well-equipped workshops for this purpose and for that of resolving technical problems which may occur due to accidents or malfunction of

CERN maintains control of projects. The normal way of acquiring equipment is through buying from industry to a tight specification, written by staff competent in the field, and close technical follow-up. Cost is minimized by in-house design and

prototyping, and by limiting the risk to manufacturers by confining the

in proportion to Net Nation Income. Any activity—in particular research and development—suffers from erratic funding. In the case of CERN this is avoided by a

procedure of rolling forecasts: each year the budget for the following year is established, together with firm estimates for the next 2 years and provisional estimates for the following 2 years. This has provided the laboratory with a stable funding profile and enabled planning of both the day-to-day running and that of the medium and long-term scientific program. Purchasing of equipment is subject to strict rules that favor the lowest bid for a supply satisfying carefully drawn-up specifications. While there is not a policy of fair return ("juste retour"), some effort

is put into distributing contracts fairly among member states.

• Tight technical control of projects

DOI: http://dx.doi.org/10.5772/intechopen.80579

From the Eloisatron to the Pevatron

5.1 The convention

bureaucracy.

5.2 Funding

5.3 Skills

equipment.

5.4 Control

29

#### 4.2 Phase 2

By adding a pair of rings, powered by coils wound from classical Nb-Ti conductor cooled to 4.2 T, delivering a dipole field of about 5 T, the second phase of the UCC would provide a center-of-mass energy of up to 330 TeV. By using preaccelerated beams from the transmission line arrangement, the new magnet system would only be required to increase the momentum of circulating particles by a factor of 3.5 to 4, simplifying the process and enabling the inclusion of beam screen within a relatively small magnet aperture. It is reasonable to assume that, by careful design, the cost per meter of such a magnet system would be less than half that of the LHC system, and as the system is entirely classical, the technological risk would be low. The layout of the transmission line magnet in the tunnel will have to be such that the new magnet system can be installed during shutdowns without disrupting the Phase 1 operation.

#### 4.3 Phase 3

It is confidently expected that during the construction and exploitation of phases 1 and 2 of the UCC, a vigorous R&D program on high field superconducting materials, and on their engineering into reliable, cost-effective cables, will have delivered conductors comparable to Nb-Ti today. That being the case, a third set of rings, with dipoles providing fields of up to 18 T, would allow us to envisage a center-of-mass energy of 1000 TeV, or 1 PeV, the ultimate goal of the complex. It is emphasized that such superconductors do not exist at present and that once they have been identified, it will also be necessary to identify a long-term use for the material that generates the need for its engineering development, just as MRI provided the "killer application" for Nb-Ti, and which led to it becoming an affordable commodity. The sole application to particle accelerators, being "one-off" in nature, is not sufficient. The push for increasingly high field magnets for NMR is an immediate application, but essentially small-scale; as seen today, the most likely long-term large-scale application would be for fusion containment—should that develop into a viable energy source. The technological development of such material, identification of applications, and the industrialization of the manufacture are examples of the high-tech activity that one could expect to flourish in the collider city complex.

#### 5. The host laboratory

It is generally recognized that CERN provides a good example of how to organize a large international laboratory [19]. This is not an accident. The success of the laboratory is based on a combination of several important principles:


A new aspiring global laboratory should consider carefully the implication of each of these when planning how to undertake the work.

#### 5.1 The convention

magnet system for the UCC is of the order of twice the cost of the magnet system for the 8.3 T system for the (very much smaller) LHC, and probably more affordable (and certainly less risky) than that of the presently envisaged high field magnet system for the FCC. The accelerator would require an injector system to supply 4 TeV protons, consisting of a linear accelerator and two pulsed synchrotrons.

By adding a pair of rings, powered by coils wound from classical Nb-Ti conductor cooled to 4.2 T, delivering a dipole field of about 5 T, the second phase of the UCC would provide a center-of-mass energy of up to 330 TeV. By using preaccelerated beams from the transmission line arrangement, the new magnet system would only be required to increase the momentum of circulating particles by a factor of 3.5 to 4, simplifying the process and enabling the inclusion of beam screen within a relatively small magnet aperture. It is reasonable to assume that, by careful design, the cost per meter of such a magnet system would be less than half that of the LHC system, and as the system is entirely classical, the technological risk would be low. The layout of the transmission line magnet in the tunnel will have to be such that the new magnet system can be installed during shutdowns without disrupting

It is confidently expected that during the construction and exploitation of phases

It is generally recognized that CERN provides a good example of how to organize a large international laboratory [19]. This is not an accident. The success of the

laboratory is based on a combination of several important principles:

1 and 2 of the UCC, a vigorous R&D program on high field superconducting materials, and on their engineering into reliable, cost-effective cables, will have delivered conductors comparable to Nb-Ti today. That being the case, a third set of rings, with dipoles providing fields of up to 18 T, would allow us to envisage a center-of-mass energy of 1000 TeV, or 1 PeV, the ultimate goal of the complex. It is emphasized that such superconductors do not exist at present and that once they have been identified, it will also be necessary to identify a long-term use for the material that generates the need for its engineering development, just as MRI provided the "killer application" for Nb-Ti, and which led to it becoming an affordable commodity. The sole application to particle accelerators, being "one-off" in nature, is not sufficient. The push for increasingly high field magnets for NMR is an immediate application, but essentially small-scale; as seen today, the most likely long-term large-scale application would be for fusion containment—should that develop into a viable energy source. The technological development of such material, identification of applications, and the industrialization of the manufacture are examples of the high-tech activity that one could expect to flourish in the collider

4.2 Phase 2

Charged Particles

the Phase 1 operation.

4.3 Phase 3

city complex.

28

5. The host laboratory

• A simple, well drawn-up convention

• Consistent rules-based funding and purchasing

The CERN convention is a visionary 32-page document that spells out clearly the purpose of the Organization and the rules for its governance. The governing body is the CERN Council, made up of two delegates from each member state. The Council is assisted by the Finance Committee—which deals with material and personnel budgets, and the Scientific Policy Committee—which advises on the research agenda. The convention is drawn up in such a way as to vest the Council with significant autonomy and authority to negotiate and take decisions in the interest of the Organization, to empower its scientists, and to reduce to a strict minimum the bureaucracy.

#### 5.2 Funding

Council allocates the annual budget, with funds provided by the member states in proportion to Net Nation Income. Any activity—in particular research and development—suffers from erratic funding. In the case of CERN this is avoided by a procedure of rolling forecasts: each year the budget for the following year is established, together with firm estimates for the next 2 years and provisional estimates for the following 2 years. This has provided the laboratory with a stable funding profile and enabled planning of both the day-to-day running and that of the medium and long-term scientific program. Purchasing of equipment is subject to strict rules that favor the lowest bid for a supply satisfying carefully drawn-up specifications. While there is not a policy of fair return ("juste retour"), some effort is put into distributing contracts fairly among member states.

#### 5.3 Skills

Scientists, engineers and technicians are encouraged to hone their skills through their work on projects. This enables them to write comprehensive specifications for equipment that is available industrially, to follow up constructively the contracts, and to design and prototype special equipment that is not available on the market. The laboratory maintains well-equipped workshops for this purpose and for that of resolving technical problems which may occur due to accidents or malfunction of equipment.

#### 5.4 Control

CERN maintains control of projects. The normal way of acquiring equipment is through buying from industry to a tight specification, written by staff competent in the field, and close technical follow-up. Cost is minimized by in-house design and prototyping, and by limiting the risk to manufacturers by confining the

requirement to that of satisfying engineering standards: CERN specifically bears the technical risk for the correct functioning of complex equipment. While the LHC collider was mainly funded via CERN, that of the experiments was mainly financed via the participating institutes and universities, which led to frequent use of "inkind" supplies. It was found to be necessary to ensure compliance of such equipment by tight control from the host laboratory. For this to work it is essential to have dedicated, competent, experienced and respected staff which is empowered with appropriate authority.

the forming of city clusters calls for the injection of capital investment on a huge scale, and the additional percentage cost of hosting a large research facility of the type discussed here would be small, whereas the publicity and enhancement of the attractiveness of the cluster would be considerable. It is anticipated that in the medium term, competition between the clusters will grow, and this will in turn accelerate the performance—and stimulate the desire to host big science projects.

While the FCC and the CEPC are thought of as being the next, and possibly final step in colliders, these do not reach the size dreamt of already 40 years ago—the Eloisatron, or ELN. The proposal for an accelerator in the USA, using the defunct SSC tunnel to house the injector gets close [15], but in the study presented here the possibility of going still further is addressed. It is suggested that the increasing desire of people to live in cities, and the expected increase in efficiency (and wellbeing) provided by setting up of clusters of cities, may provide an opportunity to consider associating such a city cluster with a collider that could ultimately deliver

interactions at a center-of-mass energy of 1 PeV, the Pevatron.

7. Conclusion

From the Eloisatron to the Pevatron

DOI: http://dx.doi.org/10.5772/intechopen.80579

Author details

Thomas Taylor1,2

31

1 AT Scientific LLC, Bernex, Switzerland

provided the original work is properly cited.

\*Address all correspondence to: tom.taylor@cern.ch

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 CERN, Geneva, Switzerland

#### 5.5 Infrastructure

The maintenance and development of infrastructure is of vital importance for a laboratory. As CERN has evolved, it capitalized on existing accelerators and associated equipment to build successive increasing complex and energetic accelerators and colliders at minimum cost. In parallel, there has been a continuing development in the technology of particle detectors required for the evolving experiments, and of course in that of the supporting informatics hardware and software. The maintenance of efficient and well-equipped workshops has also been of vital importance for the Laboratory. It is an understatement to say that the experience of laboratories established harboring the specific intention of excluding integrated workshops (i.e. relying exclusively on purchasing) has not been good.

#### 5.6 Location

Geneva is a city with a long tradition of hosting international organizations. This activity is an important source of income for the city, and it makes a corresponding effort to simplify the bureaucratic problems that can occur with international staffing. Permanent staff is not subject to local income tax, and goods are not subject to value-added tax, which helps to keeps costs under control. The city also has a conveniently located international airport and an efficient public transport system that provides excellent access to the laboratory. The laboratory itself lies astride the frontier between Switzerland and France, so that both countries are in fact host states and provide facilities over and above their reglementary contributions to compensate for the advantages incurred. The country or region wishing to host a laboratory providing facilities for international big science projects is strongly advised to set up a framework of a similar nature.

#### 6. City clusters and project funding

One of the reasons for proposing to associate the new collider with a cluster of cities is that it could facilitate the funding. If it is recognized that to host the collider has some value—be it for education, innovation, regional pride or some other factor —then investing in the success of the project could be rendered acceptable, compared with less well-focused requests to central government. Viewed alone, the cost of a very large particle accelerator/collider, just as an array of telescopes for astronomy, or a nuclear fusion device, is perceived to loom large in a national budget, even though much of the cost is simply pumping money around the economy. And to put the figures into perspective, the cost of running CERN, presently the largest collider facility in the world, is equivalent to that of a proverbial annual cup of coffee of the population of the member states—a small price to pay for motivational news generated directly by the laboratory, without considering the economic fallout, and innovation derived from the activity [20, 21]. The complexity inherent in

#### From the Eloisatron to the Pevatron DOI: http://dx.doi.org/10.5772/intechopen.80579

the forming of city clusters calls for the injection of capital investment on a huge scale, and the additional percentage cost of hosting a large research facility of the type discussed here would be small, whereas the publicity and enhancement of the attractiveness of the cluster would be considerable. It is anticipated that in the medium term, competition between the clusters will grow, and this will in turn accelerate the performance—and stimulate the desire to host big science projects.

### 7. Conclusion

requirement to that of satisfying engineering standards: CERN specifically bears the technical risk for the correct functioning of complex equipment. While the LHC collider was mainly funded via CERN, that of the experiments was mainly financed via the participating institutes and universities, which led to frequent use of "inkind" supplies. It was found to be necessary to ensure compliance of such equipment by tight control from the host laboratory. For this to work it is essential to have dedicated, competent, experienced and respected staff which is empowered

The maintenance and development of infrastructure is of vital importance for a laboratory. As CERN has evolved, it capitalized on existing accelerators and associated equipment to build successive increasing complex and energetic accelerators and colliders at minimum cost. In parallel, there has been a continuing development in the technology of particle detectors required for the evolving experiments, and of course in that of the supporting informatics hardware and software. The maintenance of efficient and well-equipped workshops has also been of vital importance for the Laboratory. It is an understatement to say that the experience of laboratories established harboring the specific intention of excluding integrated workshops (i.e.

Geneva is a city with a long tradition of hosting international organizations. This activity is an important source of income for the city, and it makes a corresponding effort to simplify the bureaucratic problems that can occur with international staffing. Permanent staff is not subject to local income tax, and goods are not subject to value-added tax, which helps to keeps costs under control. The city also has a conveniently located international airport and an efficient public transport system that provides excellent access to the laboratory. The laboratory itself lies astride the frontier between Switzerland and France, so that both countries are in fact host states and provide facilities over and above their reglementary contributions to compensate for the advantages incurred. The country or region wishing to host a laboratory providing facilities for international big science projects is strongly

One of the reasons for proposing to associate the new collider with a cluster of cities is that it could facilitate the funding. If it is recognized that to host the collider has some value—be it for education, innovation, regional pride or some other factor —then investing in the success of the project could be rendered acceptable, compared with less well-focused requests to central government. Viewed alone, the cost of a very large particle accelerator/collider, just as an array of telescopes for astronomy, or a nuclear fusion device, is perceived to loom large in a national budget, even though much of the cost is simply pumping money around the economy. And to put the figures into perspective, the cost of running CERN, presently the largest collider facility in the world, is equivalent to that of a proverbial annual cup of coffee of the population of the member states—a small price to pay for motivational news generated directly by the laboratory, without considering the economic fallout, and innovation derived from the activity [20, 21]. The complexity inherent in

with appropriate authority.

relying exclusively on purchasing) has not been good.

advised to set up a framework of a similar nature.

6. City clusters and project funding

5.5 Infrastructure

Charged Particles

5.6 Location

30

While the FCC and the CEPC are thought of as being the next, and possibly final step in colliders, these do not reach the size dreamt of already 40 years ago—the Eloisatron, or ELN. The proposal for an accelerator in the USA, using the defunct SSC tunnel to house the injector gets close [15], but in the study presented here the possibility of going still further is addressed. It is suggested that the increasing desire of people to live in cities, and the expected increase in efficiency (and wellbeing) provided by setting up of clusters of cities, may provide an opportunity to consider associating such a city cluster with a collider that could ultimately deliver interactions at a center-of-mass energy of 1 PeV, the Pevatron.

#### Author details

Thomas Taylor1,2

1 AT Scientific LLC, Bernex, Switzerland

2 CERN, Geneva, Switzerland

\*Address all correspondence to: tom.taylor@cern.ch

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Johnsen K. The ISR and accelerator physics. Particle Accelerators. 1986;18: 167-182

[2] Zichichi A, editor. ECFA-LEP Working Group 1979 Status Report, ECFA/79/39. 1980

[3] Camilleri L et al. Physics with very high energy e+ e colliding beams. CERN 76-18. 1976

[4] Gareyte J. The SPS p-pbar collider. CERN-SPS-84-03-DI-(MST). 1984

[5] Schopper H. LEP—The Lord of the Collider Rings at CERN 1980–2000. Berlin: Springer Verlag; 2009

[6] Wilson RR. The Tevatron. FERMILAB-TM-0753. 1978

[7] Jackson JD, editor. Conceptual design of the superconducting super collider. SSC-SR-2020. 1986

[8] Evans L, editor. The Large Hadron Collider. Lausanne: EPFL Press; 2009

[9] Randall L. Higgs discovery: the power of empty space. London: The Bodley Head; 2012

[10] Barletta WA, Leutz H, editors. Supercolliders and Super-Detectors. Singapore: World Scientific Publishing; 1993

[11] VLHC Design Study Group. Design study for a phased very large hadron collider. Fermilab-TM-2149. 2001

[12] Zichichi A et al. The LAA Project— Report No. 1. CERN/LAA. 1986

[13] Bogomyagkov AV et al. Projects for ultra-high-energy circular colliders at CERN. Physics of Particles and Nuclei Letters. 2016;13(7):870-875

[14] CEPC-SppC Preliminary conceptual design report, volume II: accelerator. Beijing: Institute of High Energy Physics, CAS; October 2015

Chapter 4

Abstract

B-meson physics.

1. Introduction

predictions [3–6],

and quark sectors.

33

<sup>Δ</sup>a<sup>μ</sup> � <sup>Δ</sup>aexp

<sup>μ</sup> � <sup>Δ</sup>ath

anomalies in <sup>B</sup>-meson decay like <sup>B</sup> ! <sup>K</sup>ð Þ <sup>∗</sup> <sup>μ</sup>þμ� and <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> τν [7–16].

physics. After listing some examples of them, the applications to some flavor physics will be discussed focusing on some specific cases. We find it interesting to consider new charged particles which are related to flavor physics in both lepton

Particle

Takaaki Nomura

Flavor Physics and Charged

We have new charged particles in many scenarios of physics beyond the Standard Model where these particles are sometimes motivated to explain experimental anomalies. Furthermore, such new charged particles are important target at the collider experiments such as the Large Hadron Collider in searching for a signature of new physics. If these new particles interact with known particles in the Standard Model, they would induce interesting phenomenology of flavor physics in both lepton and quark sectors. Then, we review some candidate of new charged particles and its applications to flavor physics. In particular, vector-like lepton and leptoquarks are discussed for lepton flavor physics and

Keywords: flavor physics, charged particle from beyond the standard model, B-meson decay, vector-like lepton/quark, leptoquarks, charged scalar boson

Charged particles are often considered in the physics beyond the Standard Model (BSM) of particle physics as new heavy particles which are not observed at the experiments. Such charged particles can have rich phenomenology since it would interact with particles in the Standard Model (SM). Furthermore they are motivated to explain some experimental anomalies indicating deviation from predictions in the SM. For example, some charged particle interaction can accommodate with the anomalous magnetic moment of the muon, ð Þ g � 2 <sup>μ</sup>, which shows a long-standing discrepancy between experimental observations [1, 2] and theoretical

where a<sup>μ</sup> ¼ ð Þ g � 2 <sup>μ</sup>=2. This difference reaches to 3:6σ deviation from the prediction. In addition, new charged particles are introduced when we try to explain

In this chapter, we review some candidates of new charged particles from BSM

<sup>μ</sup> <sup>¼</sup> ð Þ� <sup>28</sup>:<sup>8</sup> � <sup>8</sup>:<sup>0</sup> <sup>10</sup>�<sup>10</sup>, (1)

[15] Assadi S, McIntyre P, et al. Higgs factory and 100 TeV hadron collider: Opportunity for a new world laboratory within a decade. 2014. Available from: https://arxiv.org/pdf/1402.5973

[16] Ballantyne B. Future proof. London: World Market Intelligence Ltd., Tunnels and Tunnelling; June 2016

[17] A tale of 19 megacities. London: The Economist magazine; June 23, 2018

[18] Ballarino A. Design of an MgB2 feeder system to connect groups of superconducting magnets to remote power converters. Journal of Physics: Conference Series. 2010;234:032003

[19] Fabjan CF et al., editors. Technology Meets Research. Singapore: World Scientific Publishing; 2016

[20] Schmied H. A study of the economic utility resulting from CERN contracts. CERN-1975-005. 1975

[21] Barbalat O. Technology transfer from particle physics: The CERN experience 1974–1997. CERN-BLIT-97-1. 1997

#### Chapter 4

References

Charged Particles

ECFA/79/39. 1980

high energy e+

76-18. 1976

167-182

[1] Johnsen K. The ISR and accelerator physics. Particle Accelerators. 1986;18:

[3] Camilleri L et al. Physics with very

[4] Gareyte J. The SPS p-pbar collider. CERN-SPS-84-03-DI-(MST). 1984

[5] Schopper H. LEP—The Lord of the Collider Rings at CERN 1980–2000. Berlin: Springer Verlag; 2009

[7] Jackson JD, editor. Conceptual design of the superconducting super collider.

[8] Evans L, editor. The Large Hadron Collider. Lausanne: EPFL Press; 2009

[9] Randall L. Higgs discovery: the power of empty space. London: The

[10] Barletta WA, Leutz H, editors. Supercolliders and Super-Detectors. Singapore: World Scientific Publishing;

[11] VLHC Design Study Group. Design study for a phased very large hadron collider. Fermilab-TM-2149. 2001

[12] Zichichi A et al. The LAA Project—

[13] Bogomyagkov AV et al. Projects for ultra-high-energy circular colliders at CERN. Physics of Particles and Nuclei

Report No. 1. CERN/LAA. 1986

Letters. 2016;13(7):870-875

[6] Wilson RR. The Tevatron. FERMILAB-TM-0753. 1978

SSC-SR-2020. 1986

Bodley Head; 2012

1993

32

e colliding beams. CERN

[14] CEPC-SppC Preliminary conceptual design report, volume II: accelerator. Beijing: Institute of High Energy Physics, CAS; October 2015

[15] Assadi S, McIntyre P, et al. Higgs factory and 100 TeV hadron collider: Opportunity for a new world laboratory within a decade. 2014. Available from: https://arxiv.org/pdf/1402.5973

[16] Ballantyne B. Future proof. London: World Market Intelligence Ltd., Tunnels

[17] A tale of 19 megacities. London: The Economist magazine; June 23, 2018

[19] Fabjan CF et al., editors. Technology Meets Research. Singapore: World

[20] Schmied H. A study of the economic utility resulting from CERN contracts.

[21] Barbalat O. Technology transfer from particle physics: The CERN experience 1974–1997. CERN-BLIT-

[18] Ballarino A. Design of an MgB2 feeder system to connect groups of superconducting magnets to remote power converters. Journal of Physics: Conference Series. 2010;234:032003

and Tunnelling; June 2016

Scientific Publishing; 2016

CERN-1975-005. 1975

97-1. 1997

[2] Zichichi A, editor. ECFA-LEP Working Group 1979 Status Report,

## Flavor Physics and Charged Particle

Takaaki Nomura

#### Abstract

We have new charged particles in many scenarios of physics beyond the Standard Model where these particles are sometimes motivated to explain experimental anomalies. Furthermore, such new charged particles are important target at the collider experiments such as the Large Hadron Collider in searching for a signature of new physics. If these new particles interact with known particles in the Standard Model, they would induce interesting phenomenology of flavor physics in both lepton and quark sectors. Then, we review some candidate of new charged particles and its applications to flavor physics. In particular, vector-like lepton and leptoquarks are discussed for lepton flavor physics and B-meson physics.

Keywords: flavor physics, charged particle from beyond the standard model, B-meson decay, vector-like lepton/quark, leptoquarks, charged scalar boson

#### 1. Introduction

Charged particles are often considered in the physics beyond the Standard Model (BSM) of particle physics as new heavy particles which are not observed at the experiments. Such charged particles can have rich phenomenology since it would interact with particles in the Standard Model (SM). Furthermore they are motivated to explain some experimental anomalies indicating deviation from predictions in the SM. For example, some charged particle interaction can accommodate with the anomalous magnetic moment of the muon, ð Þ g � 2 <sup>μ</sup>, which shows a long-standing discrepancy between experimental observations [1, 2] and theoretical predictions [3–6],

$$
\Delta a\_{\mu} \equiv \Delta a\_{\mu}^{\text{exp}} - \Delta a\_{\mu}^{\text{th}} = (28.8 \pm 8.0) \times 10^{-10},\tag{1}
$$

where a<sup>μ</sup> ¼ ð Þ g � 2 <sup>μ</sup>=2. This difference reaches to 3:6σ deviation from the prediction. In addition, new charged particles are introduced when we try to explain anomalies in <sup>B</sup>-meson decay like <sup>B</sup> ! <sup>K</sup>ð Þ <sup>∗</sup> <sup>μ</sup>þμ� and <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> τν [7–16].

In this chapter, we review some candidates of new charged particles from BSM physics. After listing some examples of them, the applications to some flavor physics will be discussed focusing on some specific cases. We find it interesting to consider new charged particles which are related to flavor physics in both lepton and quark sectors.

#### 2. Some charged particles from beyond the standard model physics

In this section we review some examples of charged particles which are induced from BSM physics.

<sup>X</sup><sup>u</sup> <sup>¼</sup> <sup>V</sup><sup>u</sup> L Yu 1 ffiffi 2 <sup>p</sup> <sup>V</sup>u†

DOI: http://dx.doi.org/10.5772/intechopen.81404

Flavor Physics and Charged Particle

where LLi ¼ ν<sup>i</sup> ð Þ ; ℓ<sup>i</sup>

LY <sup>¼</sup> hij � <sup>1</sup>

<sup>þ</sup> <sup>h</sup><sup>∗</sup> ji 1 ffiffi 2 p ν † iLCδ�ℓ<sup>∗</sup>

scalar fields are given by

2.2 Vector-like leptons

cesses.

35

expanded as

T

iLCδþνjL � <sup>ℓ</sup><sup>T</sup>

jL <sup>þ</sup> <sup>ℓ</sup>†

In order to avoid the stringent constraints from rare Z ! ℓ�

ffiffi 2 <sup>p</sup> <sup>ℓ</sup><sup>T</sup>

LY ¼ fijL

c Li ð Þ iσ<sup>2</sup> LLj

<sup>R</sup> , <sup>X</sup><sup>d</sup> <sup>¼</sup> <sup>V</sup><sup>d</sup>

<sup>Δ</sup> <sup>¼</sup> <sup>δ</sup>þ<sup>=</sup> ffiffi

LY <sup>¼</sup> hijLT

L Yd 2 ffiffi 2 <sup>p</sup> <sup>V</sup>d†

A doubly charged scalar boson also appears from SUð Þ2 <sup>L</sup> triplet scalar field:

2

where v<sup>Δ</sup> is the VEV of the triplet scalar. Such a triplet scalar is motivated to generate neutrino mass known as Higgs triplet model or type-II seesaw mechanism [19–26]. We can write Yukawa interaction of triplet scalar and lepton doublets by

!

<sup>v</sup><sup>Δ</sup> <sup>þ</sup> <sup>δ</sup><sup>0</sup> <sup>þ</sup> <sup>i</sup>η<sup>0</sup> � �<sup>=</sup> ffiffi

Li

gation operator. In terms of the components, the Yukawa interaction can be

iLCδ��ℓ<sup>∗</sup>

h<sup>þ</sup> þ geee

iLCδþþℓjL <sup>þ</sup> <sup>ν</sup><sup>T</sup>

where <sup>C</sup>† ¼ �<sup>C</sup> is used. Another example of model including doubly charged scalar is Zee-Babu type model [27, 28] for neutrino mass generation at two-loop level. In such a type of model, one introduces singly and doubly charged scalars <sup>h</sup>�; <sup>k</sup>�� � � which are SUð Þ<sup>2</sup> <sup>L</sup> singlet. The Yukawa couplings associated with charged

c

where fij is antisymmetric under flavor indices. These Yukawa interactions can be used to generate neutrino mass with the nontrivial interaction in scalar potential:

Note that these charged scalars also contribute to lepton flavor violation pro-

The vector-like leptons (VLLs) are discussed in Ref. [29]. They are new charged particles without conflict of gauge anomaly problem and induce rich lepton flavor physics. To obtain mixing with the SM leptons, the representations of VLL under SUð Þ2 <sup>L</sup> � Uð Þ1 <sup>Y</sup> gauge symmetry can be singlet, doublet, and triplet under SUð Þ2 <sup>L</sup>.

consider the triplet representations 1ð Þ ; 3; �1 and 1ð Þ ; 3; 0 with hypercharges Y ¼ �1 and Y ¼ 0, respectively. The new Yukawa couplings thus can be written such that

�L<sup>Y</sup> <sup>¼</sup> <sup>L</sup>Y1Ψ1RH <sup>þ</sup> <sup>L</sup>Y2Ψ2RH<sup>~</sup> <sup>þ</sup> <sup>m</sup><sup>Ψ</sup>1TrΨ1<sup>L</sup>Ψ1<sup>R</sup> <sup>þ</sup> <sup>m</sup><sup>Ψ</sup>2TrΨ2<sup>L</sup>Ψ2<sup>R</sup> <sup>þ</sup> <sup>H</sup>:c:, (12)

iLCδþℓjL � �

jL � ν † iLCδ<sup>0</sup>ν<sup>∗</sup>

jL � � (9)

ReRkþþ þ gije

<sup>p</sup> <sup>δ</sup>þþ

2 <sup>p</sup> �δþ<sup>=</sup> ffiffi

<sup>R</sup> , <sup>X</sup><sup>ℓ</sup> <sup>¼</sup> <sup>V</sup><sup>ℓ</sup>

L Yℓ 2 ffiffi 2 <sup>p</sup> <sup>V</sup>ℓ†

2 p

Ciσ2ΔLLj þ h:c: , (8)

ffiffi 2 <sup>p</sup> <sup>ν</sup><sup>T</sup>

iLCδ�ν<sup>∗</sup>

kþþ þ h:c:, (10)

<sup>L</sup> with flavor index <sup>i</sup> and <sup>C</sup> <sup>¼</sup> <sup>i</sup>γ<sup>2</sup>γ<sup>0</sup> is the Dirac charge conju-

iLCδ<sup>0</sup>νjL � <sup>1</sup>

jL þ 1 ffiffi 2 <sup>p</sup> <sup>ℓ</sup>†

c Ri eRj

V ⊃μkþþh�h� þ c:c:: (11)

<sup>i</sup> ℓ<sup>∓</sup>

<sup>j</sup> decays, we here

<sup>R</sup> : (6)

, (7)

#### 2.1 Charged scalar bosons

Singly charged scalar appears from two-Higgs doublet model (2HDM) [17, 18] in which two SUð Þ2 <sup>L</sup> doublet Higgs fields are introduced:

$$H\_1 = \left( (v\_1 + \phi\_1 + i\eta\_1) / \sqrt{2} \right), \quad H\_2 = \left( (v\_2 + \phi\_2 + i\eta\_2) / \sqrt{2} \right), \tag{2}$$

where v1, <sup>2</sup> is the vacuum expectation values (VEVs) of Higgs fields. In general, one can write Yukawa interaction in terms of Higgs doublet fields as

$$\begin{aligned} -\mathcal{L}\_Y &= \overline{\mathcal{Q}}\_L Y\_1^d D\_R H\_1 + \overline{\mathcal{Q}}\_L Y\_2^d D\_R H\_2 + \overline{\mathcal{Q}}\_L Y\_1^u U\_R \tilde{H}\_1 + \overline{\mathcal{Q}}\_L Y\_2^u U\_R \tilde{H}\_2 \\ &+ \overline{L} Y\_1^\ell \ell\_R^\ell H\_1 + \overline{L} Y\_2^\ell \ell\_R^\ell H\_2 + H.c.,\end{aligned} \tag{3}$$

where all flavor indices are hidden, PR Lð Þ <sup>¼</sup> <sup>1</sup> � <sup>γ</sup><sup>5</sup> ð Þ=2; <sup>Q</sup><sup>i</sup> <sup>L</sup> <sup>¼</sup> ui <sup>L</sup>; di L � � and LL <sup>¼</sup> <sup>ν</sup><sup>i</sup> L;e<sup>i</sup> L � � are the SUð Þ<sup>2</sup> <sup>L</sup> quark and lepton doublets with flavor index <sup>i</sup>, respectively; <sup>f</sup> <sup>R</sup> (<sup>f</sup> <sup>¼</sup> U, D, <sup>ℓ</sup>) denotes the SUð Þ<sup>2</sup> <sup>L</sup> singlet fermion; <sup>Y</sup><sup>f</sup> <sup>1</sup>,<sup>2</sup> are the 3 � 3 Yukawa matrices; and <sup>H</sup><sup>~</sup> <sup>i</sup> <sup>¼</sup> <sup>i</sup>τ2H<sup>∗</sup> <sup>i</sup> with τ<sup>2</sup> being the Pauli matrix. There are two CPeven scalars, one CP-odd pseudoscalar, and two charged Higgs particles in the 2HDM, and the relations between physical and weak eigenstates can be given by

$$h = -s\_a \phi\_1 + c\_a \phi\_2,$$

$$H = c\_a \phi\_1 + s\_a \phi\_2,\tag{4}$$

$$H^{\pm}(A) = -s\_\beta \phi\_1^{\pm}(\eta\_1) + c\_\beta \phi\_2^{\pm}(\eta\_2),$$

where ϕ<sup>i</sup> η<sup>i</sup> ð Þ and η� <sup>i</sup> denote the real (imaginary) parts of the neutral and charged components of Hi, respectively; cαð Þ¼ s<sup>α</sup> cos αð Þ sin α , c<sup>β</sup> ¼ cos β ¼ v1=v, s<sup>β</sup> ¼ sin β ¼ v2=v, and vi are the vacuum expectation values (VEVs) of Hi and <sup>v</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 <sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> 2 p ≈ 246 GeV. In our notation, h is the SM-like Higgs, while H, A, and H� are new particles which appear in the 2HDM. In particular, Yukawa interactions with charged Higgs are given by

$$-\mathcal{L}\_{Y}^{H^{\pm}} = \sqrt{2} \,\overline{d}\_{L} V^{\dagger} \left[ -\frac{1}{\nu t\_{\beta}} \mathbf{m}\_{\mathbf{u}} + \frac{\mathbf{X}^{u}}{s\_{\beta}} \right] \mu\_{R} H^{-}$$

$$+ \sqrt{2} \overline{\mu}\_{L} V \left[ -\frac{t\_{\beta}}{\nu} \mathbf{m}\_{\mathbf{d}} + \frac{\mathbf{X}^{d}}{c\_{\beta}} \right] d\_{R} H^{+} \tag{5}$$

$$+ \sqrt{2} \overline{\nu}\_{L} \left[ -\frac{t\_{\beta}}{\nu} \mathbf{m}\_{\ell} + \frac{\mathbf{X}^{\ell}}{c\_{\beta}} \right] \ell\_{R} H^{+} + H.c.$$

where V is the CKM matrix and the matrix X<sup>f</sup> is defined by original Yukawa coupling and unitary matrix diagonalizing fermion mass

Flavor Physics and Charged Particle DOI: http://dx.doi.org/10.5772/intechopen.81404

2. Some charged particles from beyond the standard model physics

from BSM physics.

Charged Particles

H<sup>1</sup> ¼

LL <sup>¼</sup> <sup>ν</sup><sup>i</sup>

L;e<sup>i</sup> L

2.1 Charged scalar bosons

0

B@

�L<sup>Y</sup> <sup>¼</sup> QLY<sup>d</sup>

<sup>þ</sup> LY<sup>ℓ</sup>

Yukawa matrices; and <sup>H</sup><sup>~</sup> <sup>i</sup> <sup>¼</sup> <sup>i</sup>τ2H<sup>∗</sup>

where ϕ<sup>i</sup> η<sup>i</sup> ð Þ and η�

with charged Higgs are given by

�L<sup>H</sup>� <sup>Y</sup> <sup>¼</sup> ffiffi 2 <sup>p</sup> dLV† � <sup>1</sup>

> <sup>þ</sup> ffiffi 2 <sup>p</sup> uLV � <sup>t</sup><sup>β</sup>

<sup>þ</sup> ffiffi 2 <sup>p</sup> <sup>ν</sup><sup>L</sup> � <sup>t</sup><sup>β</sup>

coupling and unitary matrix diagonalizing fermion mass

<sup>v</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 <sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> 2

34

which two SUð Þ2 <sup>L</sup> doublet Higgs fields are introduced:

<sup>1</sup>DRH<sup>1</sup> <sup>þ</sup> QLY<sup>d</sup>

<sup>1</sup>ℓRH<sup>1</sup> <sup>þ</sup> LY<sup>ℓ</sup>

where all flavor indices are hidden, PR Lð Þ <sup>¼</sup> <sup>1</sup> � <sup>γ</sup><sup>5</sup> ð Þ=2; <sup>Q</sup><sup>i</sup>

tively; <sup>f</sup> <sup>R</sup> (<sup>f</sup> <sup>¼</sup> U, D, <sup>ℓ</sup>) denotes the SUð Þ<sup>2</sup> <sup>L</sup> singlet fermion; <sup>Y</sup><sup>f</sup>

H�ð Þ¼� A sβϕ�

components of Hi, respectively; cαð Þ¼ s<sup>α</sup> cos αð Þ sin α , c<sup>β</sup> ¼ cos β ¼ v1=v, s<sup>β</sup> ¼ sin β ¼ v2=v, and vi are the vacuum expectation values (VEVs) of Hi and

2 p

one can write Yukawa interaction in terms of Higgs doublet fields as

1

CA, H<sup>2</sup> <sup>¼</sup>

where v1, <sup>2</sup> is the vacuum expectation values (VEVs) of Higgs fields. In general,

<sup>2</sup>DRH<sup>2</sup> <sup>þ</sup> QLY<sup>u</sup>

� � are the SUð Þ<sup>2</sup> <sup>L</sup> quark and lepton doublets with flavor index <sup>i</sup>, respec-

<sup>1</sup> η<sup>1</sup> ð Þþ cβϕ�

p ≈ 246 GeV. In our notation, h is the SM-like Higgs, while H, A, and H� are new particles which appear in the 2HDM. In particular, Yukawa interactions

vt<sup>β</sup>

v md þ

where V is the CKM matrix and the matrix X<sup>f</sup> is defined by original Yukawa

� �

v m<sup>ℓ</sup> þ

mu þ

� �

� �

Xu sβ

> Xd cβ

Xℓ cβ

uRH�

dRH<sup>þ</sup>

ℓRH<sup>þ</sup> þ H:c:,

<sup>2</sup>ℓRH<sup>2</sup> þ H:c:,

even scalars, one CP-odd pseudoscalar, and two charged Higgs particles in the 2HDM, and the relations between physical and weak eigenstates can be given by

> h ¼ �sαϕ<sup>1</sup> þ cαϕ2, H ¼ cαϕ<sup>1</sup> þ sαϕ2,

Hþ 1 <sup>v</sup><sup>1</sup> <sup>þ</sup> <sup>ϕ</sup><sup>1</sup> <sup>þ</sup> <sup>i</sup>η<sup>1</sup> ð Þ<sup>=</sup> ffiffi

In this section we review some examples of charged particles which are induced

Singly charged scalar appears from two-Higgs doublet model (2HDM) [17, 18] in

0

B@

Hþ 2 <sup>v</sup><sup>2</sup> <sup>þ</sup> <sup>ϕ</sup><sup>2</sup> <sup>þ</sup> <sup>i</sup>η<sup>2</sup> ð Þ<sup>=</sup> ffiffi

<sup>1</sup>URH<sup>~</sup> <sup>1</sup> <sup>þ</sup> QLY<sup>u</sup>

<sup>i</sup> with τ<sup>2</sup> being the Pauli matrix. There are two CP-

<sup>2</sup> η<sup>2</sup> ð Þ,

<sup>i</sup> denote the real (imaginary) parts of the neutral and charged

2 p

<sup>2</sup>URH<sup>~</sup> <sup>2</sup>

<sup>L</sup>; di L � � and

<sup>1</sup>,<sup>2</sup> are the 3 � 3

<sup>L</sup> <sup>¼</sup> ui

1

CA, (2)

(3)

(4)

(5)

$$\mathbf{X}^{\mathbf{u}} = V\_L^{\boldsymbol{\mu}} \frac{Y\_1^{\boldsymbol{\mu}}}{\sqrt{2}} V\_R^{\boldsymbol{\mu}\dagger}, \quad \mathbf{X}^{\mathbf{d}} = V\_L^{\boldsymbol{\ell}} \frac{Y\_2^{\boldsymbol{\ell}}}{\sqrt{2}} V\_R^{\boldsymbol{\ell}\dagger}, \quad \mathbf{X}^{\boldsymbol{\ell}} = V\_L^{\boldsymbol{\ell}} \frac{Y\_2^{\boldsymbol{\ell}}}{\sqrt{2}} V\_R^{\boldsymbol{\ell}\dagger}. \tag{6}$$

A doubly charged scalar boson also appears from SUð Þ2 <sup>L</sup> triplet scalar field:

$$
\Delta = \begin{pmatrix}
\delta^+ / \sqrt{2} & \delta^{++} \\
(v\_\Delta + \delta^0 + i\eta^0) / \sqrt{2} & -\delta^+ / \sqrt{2}
\end{pmatrix},
\tag{7}
$$

where v<sup>Δ</sup> is the VEV of the triplet scalar. Such a triplet scalar is motivated to generate neutrino mass known as Higgs triplet model or type-II seesaw mechanism [19–26]. We can write Yukawa interaction of triplet scalar and lepton doublets by

$$L\_Y = h\_{\vec{\eta}} L\_{L\_i}^T \mathbf{C} i \sigma\_2 \Delta L\_{L\_{\vec{\eta}}} + h.c. \quad , \tag{8}$$

where LLi ¼ ν<sup>i</sup> ð Þ ; ℓ<sup>i</sup> T <sup>L</sup> with flavor index <sup>i</sup> and <sup>C</sup> <sup>¼</sup> <sup>i</sup>γ<sup>2</sup>γ<sup>0</sup> is the Dirac charge conjugation operator. In terms of the components, the Yukawa interaction can be expanded as

$$\begin{split} L\_{Y} &= h\_{\vec{\eta}} \Big( -\frac{1}{\sqrt{2}} \boldsymbol{\epsilon}\_{\text{i}\text{L}}^{T} \mathbf{C} \boldsymbol{\delta}^{+} \boldsymbol{\nu}\_{\text{j}\text{L}} - \boldsymbol{\epsilon}\_{\text{i}\text{L}}^{T} \mathbf{C} \boldsymbol{\delta}^{++} \boldsymbol{\ell}\_{\text{j}\text{L}} + \boldsymbol{\nu}\_{\text{i}\text{L}}^{T} \mathbf{C} \boldsymbol{\delta}^{0} \boldsymbol{\nu}\_{\text{j}\text{L}} - \frac{1}{\sqrt{2}} \boldsymbol{\nu}\_{\text{i}\text{L}}^{T} \mathbf{C} \boldsymbol{\delta}^{+} \boldsymbol{\ell}\_{\text{j}\text{L}} \\ &+ h\_{\vec{\eta}}^{\*} \Big( \frac{1}{\sqrt{2}} \boldsymbol{\nu}\_{\text{i}\text{L}}^{\dagger} \mathbf{C} \boldsymbol{\delta}^{-} \boldsymbol{\ell}\_{\text{j}\text{L}}^{\*} + \boldsymbol{\ell}\_{\text{i}\text{L}}^{\dagger} \mathbf{C} \boldsymbol{\delta}^{--} \boldsymbol{\ell}\_{\text{j}\text{L}}^{\*} - \boldsymbol{\nu}\_{\text{i}\text{L}}^{\dagger} \mathbf{C} \boldsymbol{\delta}^{0} \boldsymbol{\nu}\_{\text{j}\text{L}}^{\*} + \frac{1}{\sqrt{2}} \boldsymbol{\ell}\_{\text{i}\text{L}}^{\dagger} \mathbf{C} \boldsymbol{\delta}^{-} \boldsymbol{\nu}\_{\text{j}\text{L}}^{\*} \Big) \end{split} \tag{9}$$

where <sup>C</sup>† ¼ �<sup>C</sup> is used. Another example of model including doubly charged scalar is Zee-Babu type model [27, 28] for neutrino mass generation at two-loop level. In such a type of model, one introduces singly and doubly charged scalars <sup>h</sup>�; <sup>k</sup>�� � � which are SUð Þ<sup>2</sup> <sup>L</sup> singlet. The Yukawa couplings associated with charged scalar fields are given by

$$L\_Y = f\_{\vec{\eta}} \overline{L}\_{L\_i}^{\epsilon} (i\sigma\_2) L\_{\gamma} h^+ + \mathbf{g}\_{\epsilon\epsilon} \overline{e}\_R^{\epsilon} e\_R k^{++} + \mathbf{g}\_{\vec{\eta}} \overline{e}\_R^{\epsilon} e\_{\vec{R}\_{\vec{\eta}}} k^{++} + h.c.,\tag{10}$$

where fij is antisymmetric under flavor indices. These Yukawa interactions can be used to generate neutrino mass with the nontrivial interaction in scalar potential:

$$V \supset \mu k^{++} h^{-} h^{-} + c.c.\tag{11}$$

Note that these charged scalars also contribute to lepton flavor violation processes.

#### 2.2 Vector-like leptons

The vector-like leptons (VLLs) are discussed in Ref. [29]. They are new charged particles without conflict of gauge anomaly problem and induce rich lepton flavor physics. To obtain mixing with the SM leptons, the representations of VLL under SUð Þ2 <sup>L</sup> � Uð Þ1 <sup>Y</sup> gauge symmetry can be singlet, doublet, and triplet under SUð Þ2 <sup>L</sup>. In order to avoid the stringent constraints from rare Z ! ℓ� <sup>i</sup> ℓ<sup>∓</sup> <sup>j</sup> decays, we here consider the triplet representations 1ð Þ ; 3; �1 and 1ð Þ ; 3; 0 with hypercharges Y ¼ �1 and Y ¼ 0, respectively. The new Yukawa couplings thus can be written such that

$$-\mathcal{L}\_Y = \overline{L}\mathbf{Y}\_1\Psi\_{1\mathbf{R}}H + \overline{L}\mathbf{Y}\_2\Psi\_{2\mathbf{R}}\tilde{H} + m\_{\Psi\_1}Tr\overline{\Psi}\_{1\mathbf{L}}\Psi\_{1\mathbf{R}} + m\_{\Psi\_2}Tr\overline{\Psi}\_{2\mathbf{L}}\Psi\_{2\mathbf{R}} + H.c.,\tag{12}$$

where we have suppressed the flavor indices; H is the SM Higgs doublet field, <sup>H</sup><sup>~</sup> <sup>¼</sup> <sup>i</sup>τ2H<sup>∗</sup> and the neutral component of Higgs field is <sup>H</sup><sup>0</sup> <sup>¼</sup> ð Þ <sup>v</sup> <sup>þ</sup> <sup>h</sup> <sup>=</sup> ffiffi 2 <sup>p</sup> . The representations of two VLLs are

$$\Psi\_1 = \begin{pmatrix} \Psi\_1^- / \sqrt{2} & \Psi\_1^0 \\ \Psi\_1^{--} & -\Psi\_1^- / \sqrt{2} \end{pmatrix}, \quad \Psi\_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} \Psi\_2^0 / \sqrt{2} & \Psi\_2^+ \\ \Psi\_2^- & -\Psi\_2^0 / \sqrt{2} \end{pmatrix}, \tag{13}$$

with Ψ<sup>þ</sup> <sup>2</sup> <sup>¼</sup> <sup>C</sup>Ψ� <sup>2</sup> and Ψ<sup>0</sup> <sup>2</sup> <sup>¼</sup> <sup>C</sup>Ψ<sup>0</sup> <sup>2</sup> . Since Ψ<sup>2</sup> is a real representation of SUð Þ2 <sup>L</sup>, the factor of 1= ffiffi 2 <sup>p</sup> in <sup>Ψ</sup><sup>2</sup> is required to obtain the correct mass term for Majorana fermion Ψ<sup>0</sup> <sup>2</sup> . Due to the new Yukawa terms of Y1, 2, the heavy neutral and charged leptons can mix with the SM leptons, after electroweak symmetry breaking (EWSB). Then the lepton mass matrices become 5 � 5 matrices and are expressed by

$$M\_{\ell} = \begin{pmatrix} \mathbf{m}\_{\ell} & \mathbf{Y}^{\ell}v \\ \mathbf{0} & \mathbf{m}\_{\Psi} \end{pmatrix}, \quad M\_{\nu} = \begin{pmatrix} \mathbf{m}\_{\nu} & \mathbf{Y}^{\nu}v \\ \mathbf{0} & \mathbf{m}\_{\Psi} \end{pmatrix}, \tag{14}$$

where <sup>ℓ</sup>0<sup>T</sup> <sup>¼</sup> <sup>e</sup>,μ,τ,τ<sup>0</sup>

Flavor Physics and Charged Particle

DOI: http://dx.doi.org/10.5772/intechopen.81404

leptons can be formulated by

2.3 Vector-like quarks

�L<sup>Y</sup>

37

�L<sup>h</sup>ℓℓ <sup>¼</sup> <sup>C</sup><sup>h</sup>

Ch ij <sup>¼</sup> <sup>m</sup>ℓ<sup>j</sup> v

the VLL contributions to h ! γγ, the couplings for hτ<sup>0</sup>

�LhΨΨ <sup>¼</sup> <sup>v</sup> <sup>∑</sup><sup>i</sup> <sup>Y</sup><sup>2</sup>

Higgs doublet and to the new Higgs singlet field are written as

<sup>V</sup>LTQ <sup>¼</sup> QLY1F1RH<sup>~</sup> <sup>þ</sup> QLY2F2RH <sup>þ</sup> <sup>~</sup>y1Tr <sup>F</sup>1LF1<sup>R</sup>

� � <sup>þ</sup> MF2Tr <sup>F</sup>2LF2<sup>R</sup>

2 p

mdia u � �

> mdia d � �

eigenstate before VLTQs are introduced; here all flavor indices are hidden,

þ MF1Tr F1LF1<sup>R</sup>

2 <sup>p</sup> <sup>X</sup> <sup>D</sup><sup>1</sup> �U1<sup>=</sup> ffiffi

!

<sup>F</sup><sup>1</sup> <sup>¼</sup> <sup>U</sup>1<sup>=</sup> ffiffi

Mu ¼

Md ¼

0

BB@

0

BB@

; τ<sup>00</sup> � � is the state of a physical charged lepton in lepton flavor

þ

v ∑<sup>i</sup> Y<sup>2</sup> 2i 2mΨ<sup>2</sup>

v2Y2iY2<sup>j</sup> m<sup>2</sup> Ψ<sup>2</sup>

τ<sup>0</sup> and hτ00

� �<sup>S</sup> <sup>þ</sup> <sup>~</sup>y2Tr <sup>F</sup>2LF2<sup>R</sup>

� � <sup>þ</sup> <sup>h</sup>:c:, (20)

2 p

2 p

1

CCA,

1

CCA,

: (21)

(22)

:

(18)

τ00 are expressed as

<sup>h</sup>τ″τ″: (19)

� �S

space. We use the notations of τ<sup>0</sup> and τ00 to denote the heavy-charged VLLs in mass basis. Using the parametrization of Eq. (16), the Higgs couplings to the SM-charged

If one sets me ¼ m<sup>μ</sup> ¼ 0, it is clear that in addition to the coupling hττ being modified, the tree-level flavor-changing couplings h- τ- μ and h- τ- e are also induced, and the couplings are proportional to <sup>m</sup>τ=v≈7:<sup>2</sup> � <sup>10</sup>�3. In order to study

> 1i 2mΨ<sup>1</sup>

hτ<sup>0</sup> τ<sup>0</sup> þ

Here we consider vector-like triplet quarks (VLTQs) that are discussed in Ref. [30] The gauge invariant Yukawa couplings of VLTQs to the SM quarks, to the SM

where QL is the left-handed SM quark doublet and it could be regarded as mass

, F<sup>2</sup> <sup>¼</sup> <sup>D</sup>2<sup>=</sup> ffiffi

<sup>3</sup>�<sup>3</sup> <sup>∣</sup> <sup>v</sup>Y1=<sup>2</sup> <sup>v</sup>Y2<sup>=</sup> ffiffi

02�<sup>3</sup> <sup>∣</sup> ð Þ <sup>m</sup><sup>F</sup> <sup>2</sup>�<sup>2</sup>

����� ∣ � � �� � � ��

02�<sup>3</sup> <sup>∣</sup> ð Þ <sup>m</sup><sup>F</sup> <sup>2</sup>�<sup>2</sup>

2

<sup>p</sup> �vY2=<sup>2</sup>

����� ∣ � � �� � � ��

<sup>3</sup>�<sup>3</sup> <sup>∣</sup> <sup>v</sup>Y1<sup>=</sup> ffiffi

The electric charges of U1, 2, D1, 2, X, and Y are found to be 2=3, �1=3, 5=3, and

�4=3, respectively. Therefore, U1, <sup>2</sup>ð Þ D1, <sup>2</sup> could mix with up (down) type SM quarks. Here MF1 2ð Þ is the mass of VLTQ, and due to the gauge symmetry, the VLTQs in the same multiplet state are degenerate. By the Yukawa couplings of Eq. (20), the 5 � 5 mass matrices for up and down type quarks are found by

2 <sup>p</sup> <sup>U</sup><sup>2</sup> <sup>Y</sup> �D2<sup>=</sup> ffiffi

!

<sup>H</sup><sup>~</sup> <sup>¼</sup> <sup>i</sup>τ2H<sup>∗</sup>, and <sup>F</sup>1 2ð Þ is the 2 � 2 VLTQ with hypercharge 2=3ð Þ �1=<sup>3</sup> . The representations of F1, <sup>2</sup> in SUð Þ2 <sup>L</sup> are expressed in terms of their components as follows:

v2Y1iY1<sup>j</sup> m<sup>2</sup> Ψ<sup>1</sup>

" # !

ijℓiLℓjRh þ H:c:,

<sup>δ</sup>ij � <sup>3</sup> 8

where the basis is chosen such that the SM lepton mass matrices are in diagonalized form, m<sup>ℓ</sup> is the SM charged lepton mass matrix, m<sup>Ψ</sup> ¼ diag mΨ<sup>1</sup> ; mΨ<sup>2</sup> ð Þ, and

$$\mathbf{Y}^{\ell} = \frac{1}{2} \begin{pmatrix} -Y\_{11} & Y\_{21} \\ -Y\_{12} & Y\_{22} \\ -Y\_{13} & Y\_{23} \end{pmatrix}, \quad \mathbf{Y}^{\nu} = \sqrt{2} \begin{pmatrix} Y\_{11} & Y\_{21}/2 \\ Y\_{12} & Y\_{22}/2 \\ Y\_{13} & Y\_{23}/2 \end{pmatrix}. \tag{15}$$

Note that the elements of <sup>Y</sup><sup>χ</sup> should be read as Yij <sup>¼</sup> ð Þ <sup>Y</sup><sup>i</sup> <sup>j</sup> , where the index i ¼ 1, 2 distinguishes the Yukawa couplings of the different VLLs and the index j ¼ 1; 2; 3 stands for the flavors of the SM leptons.

To diagonalize M<sup>ℓ</sup> and Mν, the unitary matrices V<sup>χ</sup> R,L with χ ¼ ℓ, ν so that Mdia <sup>χ</sup> <sup>¼</sup> <sup>V</sup><sup>χ</sup> LMχVχ† <sup>R</sup> are introduced. The information of <sup>V</sup><sup>χ</sup> <sup>L</sup> and <sup>V</sup><sup>χ</sup> <sup>R</sup> can be obtained from MχM† <sup>χ</sup> and M† <sup>χ</sup>M<sup>χ</sup> , respectively. According to Eq. (14), it can be found that the flavor mixings between heavy and light leptons in V<sup>χ</sup> <sup>R</sup> are proportional to the lepton masses. Since the neutrino masses are tiny, it is a good approximation to assume Vν <sup>R</sup> ≈1. If one further sets me ¼ m<sup>μ</sup> ¼ 0 in our phenomenological analysis, only τrelated processes have significant contributions among them. Unlike V<sup>χ</sup> R, the offdiagonal elements in flavor-mixing matrices V<sup>χ</sup> <sup>L</sup> are associated with Y1, <sup>2</sup>v=mΨ. In principle, the mixing effects can be of the order of 0:1 without conflict. In our example later, we examine these effects on h ! τμ. To be more specific, we choose parametrization that the unitary matrices in terms of Y1,<sup>2</sup> as

$$\mathbf{V}\_{L}^{\mathbb{X}} \approx \begin{pmatrix} \mathbf{1}\_{3 \times 3} - \boldsymbol{\varepsilon}\_{L}^{\mathbb{X}} \boldsymbol{\varepsilon}\_{L}^{\mathbb{X}} / 2 & -\boldsymbol{\varepsilon}\_{L}^{\mathbb{X}} \\ \boldsymbol{\varepsilon}\_{L}^{\mathbb{X}} & \mathbf{1}\_{3 \times 3} - \boldsymbol{\varepsilon}\_{L}^{\mathbb{X}} \boldsymbol{\varepsilon}\_{L}^{\mathbb{X}} / 2 \end{pmatrix}, \quad \mathbf{V}\_{R}^{\ell} \approx \begin{pmatrix} \mathbf{1}\_{3 \times 3} & -\boldsymbol{\varepsilon}\_{R}^{\ell} \\ \boldsymbol{\varepsilon}\_{R}^{\ell \dagger} & \mathbf{1}\_{3 \times 3} \end{pmatrix},\tag{16}$$

where V<sup>ν</sup> <sup>R</sup> ≈1 is used in our approximation, ε χ <sup>L</sup> <sup>≈</sup>vY<sup>χ</sup> =mΨ, and ε<sup>ℓ</sup> <sup>R</sup> <sup>≈</sup> <sup>v</sup>m† ℓY<sup>ℓ</sup>=m<sup>2</sup> Ψ. Combining the SM Higgs couplings and new Yukawa couplings of Eq. (12), the Higgs couplings to all singly charged leptons are obtained such as

$$-\mathcal{L}\_{h\ell'\ell'} = h\overline{\ell}\_L' V\_L' \begin{pmatrix} \mathbf{m}\_{\ell}/v & \mathbf{Y}^{\ell} \\ \mathbf{0} & \mathbf{0} \end{pmatrix} V\_R'^\dagger \ell'\_R + H.c.,\tag{17}$$

Flavor Physics and Charged Particle DOI: http://dx.doi.org/10.5772/intechopen.81404

where we have suppressed the flavor indices; H is the SM Higgs doublet field,

, <sup>Ψ</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

leptons can mix with the SM leptons, after electroweak symmetry breaking (EWSB). Then the lepton mass matrices become 5 � 5 matrices and are expressed

ffiffi 2 p

<sup>p</sup> in <sup>Ψ</sup><sup>2</sup> is required to obtain the correct mass term for Majorana

where the basis is chosen such that the SM lepton mass matrices are in diagonalized form, m<sup>ℓ</sup> is the SM charged lepton mass matrix, m<sup>Ψ</sup> ¼ diag mΨ<sup>1</sup> ; mΨ<sup>2</sup> ð Þ, and

CA, <sup>Y</sup><sup>ν</sup> <sup>¼</sup> ffiffi

1

i ¼ 1, 2 distinguishes the Yukawa couplings of the different VLLs and the index

masses. Since the neutrino masses are tiny, it is a good approximation to assume

principle, the mixing effects can be of the order of 0:1 without conflict. In our example later, we examine these effects on h ! τμ. To be more specific, we choose

> χ L

χ† L ε χ <sup>L</sup>=2

Combining the SM Higgs couplings and new Yukawa couplings of Eq. (12), the

mℓ=v Y<sup>ℓ</sup> 0 0 !

<sup>R</sup> ≈1. If one further sets me ¼ m<sup>μ</sup> ¼ 0 in our phenomenological analysis, only τ-

<sup>2</sup> . Due to the new Yukawa terms of Y1, 2, the heavy neutral and charged

, M<sup>ν</sup> <sup>¼</sup> <sup>m</sup><sup>ν</sup> <sup>Y</sup><sup>ν</sup>

2 p

0

B@

<sup>χ</sup>M<sup>χ</sup> , respectively. According to Eq. (14), it can be found that the

, V<sup>ℓ</sup>

χ <sup>L</sup> <sup>≈</sup>vY<sup>χ</sup>

> V<sup>ℓ</sup>† <sup>R</sup> ℓ<sup>0</sup>

Ψ0 <sup>2</sup> <sup>=</sup> ffiffi 2 <sup>p</sup> <sup>Ψ</sup><sup>þ</sup>

Ψ�

<sup>2</sup> �Ψ<sup>0</sup>

<sup>2</sup> . Since Ψ<sup>2</sup> is a real representation of SUð Þ2 <sup>L</sup>, the

v

Y<sup>11</sup> Y21=2 Y<sup>12</sup> Y22=2 Y<sup>13</sup> Y23=2

<sup>L</sup> and <sup>V</sup><sup>χ</sup>

<sup>R</sup> <sup>≈</sup> <sup>13</sup>�<sup>3</sup> �ε<sup>ℓ</sup>

!

=mΨ, and ε<sup>ℓ</sup>

ε ℓ† <sup>R</sup> 13�<sup>3</sup>

1

R,L with χ ¼ ℓ, ν so that

<sup>L</sup> are associated with Y1, <sup>2</sup>v=mΨ. In

R

<sup>R</sup> <sup>≈</sup> <sup>v</sup>m†

<sup>R</sup> þ H:c:, (17)

<sup>R</sup> are proportional to the lepton

0 m<sup>Ψ</sup> � �

!

2 <sup>p</sup> . The rep-

, (14)

CA: (15)

, where the index

<sup>R</sup> can be obtained

R, the off-

, (16)

ℓY<sup>ℓ</sup>=m<sup>2</sup> Ψ.

, (13)

2

<sup>2</sup> <sup>=</sup> ffiffi 2 p

<sup>H</sup><sup>~</sup> <sup>¼</sup> <sup>i</sup>τ2H<sup>∗</sup> and the neutral component of Higgs field is <sup>H</sup><sup>0</sup> <sup>¼</sup> ð Þ <sup>v</sup> <sup>þ</sup> <sup>h</sup> <sup>=</sup> ffiffi

1

<sup>1</sup> <sup>=</sup> ffiffi 2 p

<sup>2</sup> <sup>¼</sup> <sup>C</sup>Ψ<sup>0</sup>

<sup>M</sup><sup>ℓ</sup> <sup>¼</sup> <sup>m</sup><sup>ℓ</sup> <sup>Y</sup><sup>ℓ</sup><sup>v</sup>

0 m<sup>Ψ</sup>

�Y<sup>11</sup> Y<sup>21</sup> �Y<sup>12</sup> Y<sup>22</sup> �Y<sup>13</sup> Y<sup>23</sup>

Note that the elements of <sup>Y</sup><sup>χ</sup> should be read as Yij <sup>¼</sup> ð Þ <sup>Y</sup><sup>i</sup> <sup>j</sup>

<sup>R</sup> are introduced. The information of <sup>V</sup><sup>χ</sup>

related processes have significant contributions among them. Unlike V<sup>χ</sup>

!

resentations of two VLLs are

<sup>2</sup> <sup>¼</sup> <sup>C</sup>Ψ�

2

<sup>1</sup> <sup>=</sup> ffiffi 2 <sup>p</sup> <sup>Ψ</sup><sup>0</sup>

<sup>1</sup> �Ψ�

<sup>2</sup> and Ψ<sup>0</sup>

<sup>Y</sup><sup>ℓ</sup> <sup>¼</sup> <sup>1</sup> 2 0

B@

j ¼ 1; 2; 3 stands for the flavors of the SM leptons. To diagonalize M<sup>ℓ</sup> and Mν, the unitary matrices V<sup>χ</sup>

flavor mixings between heavy and light leptons in V<sup>χ</sup>

parametrization that the unitary matrices in terms of Y1,<sup>2</sup> as

<sup>L</sup> =2 �ε

<sup>L</sup> 13�<sup>3</sup> � ε

<sup>R</sup> ≈1 is used in our approximation, ε

<sup>ℓ</sup><sup>0</sup> ¼ hℓ 0 LV<sup>ℓ</sup> L

Higgs couplings to all singly charged leptons are obtained such as

!

diagonal elements in flavor-mixing matrices V<sup>χ</sup>

χ Lε χ†

ε χ†

�Lhℓ<sup>0</sup>

!

Ψ��

<sup>Ψ</sup><sup>1</sup> <sup>¼</sup> <sup>Ψ</sup>�

with Ψ<sup>þ</sup>

Charged Particles

factor of 1= ffiffi

fermion Ψ<sup>0</sup>

by

Mdia <sup>χ</sup> <sup>¼</sup> <sup>V</sup><sup>χ</sup>

Vν

36

from MχM†

Vχ

where V<sup>ν</sup>

LMχVχ†

<sup>χ</sup> and M†

<sup>L</sup> <sup>≈</sup> <sup>13</sup>�<sup>3</sup> � <sup>ε</sup>

where <sup>ℓ</sup>0<sup>T</sup> <sup>¼</sup> <sup>e</sup>,μ,τ,τ<sup>0</sup> ; τ<sup>00</sup> � � is the state of a physical charged lepton in lepton flavor space. We use the notations of τ<sup>0</sup> and τ00 to denote the heavy-charged VLLs in mass basis. Using the parametrization of Eq. (16), the Higgs couplings to the SM-charged leptons can be formulated by

$$\begin{aligned} -\mathcal{L}\_{h\ell\ell} &= \mathbf{C}\_{ij}^h \overline{\ell}\_{i\mathcal{L}} \ell\_{j\mathcal{R}} h + H.c., \\ \mathbf{C}\_{ij}^h &= \frac{m\_{\ell j}}{v} \left[ \delta\_{ij} - \frac{3}{8} \left( \frac{v^2 Y\_{1i} Y\_{2j}}{m\_{\Psi\_1}^2} + \frac{v^2 Y\_{2i} Y\_{2j}}{m\_{\Psi\_2}^2} \right) \right]. \end{aligned} \tag{18}$$

If one sets me ¼ m<sup>μ</sup> ¼ 0, it is clear that in addition to the coupling hττ being modified, the tree-level flavor-changing couplings h- τ- μ and h- τ- e are also induced, and the couplings are proportional to <sup>m</sup>τ=v≈7:<sup>2</sup> � <sup>10</sup>�3. In order to study the VLL contributions to h ! γγ, the couplings for hτ<sup>0</sup> τ<sup>0</sup> and hτ00 τ00 are expressed as

$$-\mathcal{L}\_{h\Psi\Psi} = \frac{\upsilon\sum\_{i}Y\_{1i}^{2}}{2m\_{\Psi\_{1}}}h\tau'\tau' + \frac{\upsilon\sum\_{i}Y\_{2i}^{2}}{2m\_{\Psi\_{2}}}h\tau''\tau''.\tag{19}$$

#### 2.3 Vector-like quarks

Here we consider vector-like triplet quarks (VLTQs) that are discussed in Ref. [30] The gauge invariant Yukawa couplings of VLTQs to the SM quarks, to the SM Higgs doublet and to the new Higgs singlet field are written as

$$\begin{split}-\mathcal{L}\_{\text{VLTQ}}^{Y} &= \overline{\mathcal{Q}}\_{\text{L}} \mathbf{Y}\_{1} F\_{1\text{R}} \bar{H} + \overline{\mathcal{Q}}\_{\text{L}} \mathbf{Y}\_{2} F\_{2\text{R}} H + \bar{\mathcal{y}}\_{1} Tr(\overline{F}\_{1\text{L}} F\_{1\text{R}}) S + \bar{\mathcal{y}}\_{2} Tr(\overline{F}\_{2\text{L}} F\_{2\text{R}}) S \\ &+ M\_{F\_{1}} Tr(\overline{F}\_{1\text{L}} F\_{1\text{R}}) + M\_{F\_{2}} Tr(\overline{F}\_{2\text{L}} F\_{2\text{R}}) + h.c.,\end{split} \tag{20}$$

where QL is the left-handed SM quark doublet and it could be regarded as mass eigenstate before VLTQs are introduced; here all flavor indices are hidden, <sup>H</sup><sup>~</sup> <sup>¼</sup> <sup>i</sup>τ2H<sup>∗</sup>, and <sup>F</sup>1 2ð Þ is the 2 � 2 VLTQ with hypercharge 2=3ð Þ �1=<sup>3</sup> . The representations of F1, <sup>2</sup> in SUð Þ2 <sup>L</sup> are expressed in terms of their components as follows:

$$F\_1 = \begin{pmatrix} U\_1/\sqrt{2} & X \\ D\_1 & -U\_1/\sqrt{2} \end{pmatrix}, \quad F\_2 = \begin{pmatrix} D\_2/\sqrt{2} & U\_2 \\ Y & -D\_2/\sqrt{2} \end{pmatrix}. \tag{21}$$

The electric charges of U1, 2, D1, 2, X, and Y are found to be 2=3, �1=3, 5=3, and �4=3, respectively. Therefore, U1, <sup>2</sup>ð Þ D1, <sup>2</sup> could mix with up (down) type SM quarks. Here MF1 2ð Þ is the mass of VLTQ, and due to the gauge symmetry, the VLTQs in the same multiplet state are degenerate. By the Yukawa couplings of Eq. (20), the 5 � 5 mass matrices for up and down type quarks are found by

$$\begin{aligned} M\_u &= \begin{pmatrix} \left(\mathbf{m}\_u^{\text{dia}}\right)\_{3\times3} & \left| & v\mathbf{Y}\_1/2 & v\mathbf{Y}\_2/\sqrt{2} \\ - & --- & \left| & -- & -- & -- \\ & \mathbf{0}\_{2\times3} & \left| & & (\mathbf{m}\_F)\_{2\times2} \\ & & & & (\mathbf{m}\_d)\_{3\times3} & \left| & v\mathbf{Y}\_1/\sqrt{2} & -v\mathbf{Y}\_2/2 \\ & -- & -- & - & \left| & -- & -- & -- \\ & & \mathbf{0}\_{2\times3} & \left| & & (\mathbf{m}\_F)\_{2\times2} \end{pmatrix} \right. \end{aligned} \right. \tag{22}$$

where mdia u � � <sup>3</sup>�<sup>3</sup> and <sup>m</sup>dia d � � <sup>3</sup>�<sup>3</sup> denote the diagonal mass matrices of SM quarks and diað Þ mF <sup>2</sup>�<sup>2</sup> <sup>¼</sup> mF<sup>1</sup> ; mF<sup>2</sup> ð Þ. Notice that a non-vanished vs could shift the masses of VLTQs. Since vs≪ v, we neglect the small effects hereafter. Due to the presence of Y1,2, the SM quarks, U1, 2, and D1, <sup>2</sup> are not physical states; thus one has to diagonalize Mu and Md to get the mass eigenstates in general. If vY<sup>i</sup> <sup>1</sup>, <sup>2</sup>≪ mF1, <sup>2</sup> , we expect that the off-diagonal elements of unitary matrices for diagonalizing the mass matrices should be of order of vY<sup>i</sup> <sup>1</sup>,2=mF1, <sup>2</sup> . By adjusting Y<sup>i</sup> <sup>1</sup>, 2, the off-diagonal effects could be enhanced and lead to interesting phenomena in collider physics.

the couplings of the LQs to the Higgs are independent parameters and irrelevant to the flavors, so by taking proper values for the parameters, the signal strength parameter for the Higgs to diphoton can fit the LHC data. For detailed analysis see

In this section, we review applications of charged particles to flavor physics by

Introduction of VLLs contributes to lepton flavor physics via Yukawa interactions discussed in previous section. Here we review the leptonic decay of the SM Higgs and LFV decay of charged lepton as an illustration based on Ref. [29].

From Eq. (18), it can be seen that the modified Higgs couplings to the SM leptons are proportional lepton masses. By comparison with other lepton channels, it can be seen that the ττ mode is the most significant one, and thus we estimate the influence on <sup>h</sup> ! <sup>τ</sup>þτ�. Using the values that satisfy BR hð Þ ! μτ <sup>≈</sup>10�4, the devia-

<sup>Γ</sup><sup>S</sup><sup>M</sup> <sup>h</sup> ! <sup>τ</sup>þτ� ð Þ <sup>¼</sup> <sup>1</sup> � <sup>6</sup>v<sup>2</sup>Y<sup>2</sup>

pp ! h ! τþτ� in our estimation is μττ ≈0:88, where the measurements from ATLAS and CMS are 1:44þ0:<sup>42</sup> �0:<sup>37</sup> [33] and 0:<sup>91</sup> � <sup>0</sup>:27 [34], respectively. Although the current data errors for the ττ channel are still large, the precision measurement of

the rare tau decays and to the flavor-conserving muon anomalous magnetic moment. We first investigate the muon g � 2, denoted by Δaμ. The lepton flavorchanging coupling hμτ can provide contribution to Δa<sup>μ</sup> through the Higgs-mediated loop diagrams. However, as shown in Eq. (18), the induced couplings are associated with m<sup>ℓ</sup>j=vℓLiℓRj; only the right-handed tau lepton has a significant contribution.

 

If the SM Higgs production cross section is not changed, the signal strength for

In the following, we investigate the contributions of new couplings in Eq. (18) to

decay also. Since the couplings are suppressed by mτ=v and mμ=v, the BR for τ ! 3μ is of the order of 10�14. We also estimate the process <sup>τ</sup> ! μγ via the <sup>h</sup>-mediation.

<sup>μ</sup> <sup>¼</sup> ð Þ� <sup>28</sup>:<sup>8</sup> � <sup>8</sup>:<sup>0</sup> <sup>10</sup>�<sup>10</sup> [2]. A similar situation happens in <sup>τ</sup> ! <sup>3</sup><sup>μ</sup>

3 8m<sup>2</sup> Ψ

 

2

<sup>μ</sup>mτ= vm<sup>2</sup> h

<sup>16</sup>π<sup>2</sup> <sup>m</sup>τμσμνð Þ CLPL <sup>þ</sup> CRPR <sup>τ</sup>Fμν, (26)

≈0:88: (25)

so that the value of

tion of Γ h ! τþτ� ð Þ from the SM prediction can be obtained as

κττ � <sup>Γ</sup> <sup>h</sup> ! <sup>τ</sup>þτ� ð Þ

μττ can test the effect or give strict limits on the parameters.

The induced Δa<sup>μ</sup> is thus suppressed by the factor of m<sup>2</sup>

The effective interaction for τ ! μγ is expressed by

<sup>L</sup><sup>τ</sup>!μγ <sup>¼</sup> <sup>e</sup>

Δa<sup>μ</sup> is two orders of magnitude smaller than current data

3.1.2 τ ! μγ process in vector-like lepton model

Δa<sup>μ</sup> ¼ a

39

exp <sup>μ</sup> � <sup>a</sup><sup>S</sup><sup>M</sup>

3. Examples of applying charged particles to flavor physics

Table 1 in Ref. [31].

Flavor Physics and Charged Particle

DOI: http://dx.doi.org/10.5772/intechopen.81404

considering VLLs and LQs as examples.

3.1 Flavor physics from vector-like lepton

3.1.1 Modification to h ! τþτ� branching ratio

#### 2.4 Scalar leptoquarks

In this subsection we consider leptoquarks (LQs) which are discussed for example in Refs. [31, 32]. The three LQs are Φ7=<sup>6</sup> ¼ ð Þ 2; 7=6 , Δ1=<sup>3</sup> ¼ ð Þ 3; 1=3 , and <sup>S</sup><sup>1</sup>=<sup>3</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>1</sup>=<sup>3</sup> under SUð Þ<sup>2</sup> <sup>L</sup>; <sup>U</sup>ð Þ<sup>1</sup> <sup>Y</sup> � � SM gauge symmetry, where the doublet and triplet representations can be taken as

$$\Phi\_{7/6} = \begin{pmatrix} \phi^{5/3} \\ \phi^{2/3} \end{pmatrix}, \quad \Delta\_{1/3} = \begin{pmatrix} \delta^{1/3}/\sqrt{2} & \delta^{4/3} \\ \delta^{-2/3} & -\delta^{1/3}/\sqrt{2} \end{pmatrix}, \tag{23}$$

where the superscripts are the electric charges of the particles. Accordingly, the LQ Yukawa couplings to the SM fermions are expressed as

$$\begin{split} -L\_{LQ} &= \left[ \overline{\mathbf{u}} \, \mathbf{V} \mathbf{k} P\_R \ell \phi^{5/3} + \overline{d} \, \mathbf{k} P\_R \ell \phi^{2/3} \right] + \left[ -\overline{\mathbf{f}} \, \overline{\mathbf{k}} P\_R \mathbf{u} \phi^{-5/3} + \overline{\mathbf{v}} \, \overline{\mathbf{k}} P\_R \mathbf{u} \phi^{-2/3} \right] \\ &+ \left[ \overline{\mathbf{u}^\*} \, \mathbf{V}^\* \mathbf{y} P\_L \nu \delta^{-2/3} - \frac{1}{\sqrt{2}} \overline{\mathbf{u}^\*} \, \mathbf{V}^\* \mathbf{y} P\_L \ell \delta^{1/3} - \frac{1}{\sqrt{2}} \overline{\mathbf{g}^\*} \, \mathbf{y} P\_L \nu \delta^{1/3} - \overline{d}^\* \, \mathbf{y} P\_L \ell \delta^{4/3} \right], \\ &+ \left( \overline{\mathbf{u}^\*} \, \mathbf{V}^\* \mathbf{\tilde{y}} P\_L \ell - \overline{d}^\* \, \mathbf{\tilde{y}} P\_L \nu + \overline{u}^\* \mathbf{w} P\_R \ell \right) \mathbf{S}^{1/3} + h.c., \end{split} \tag{24}$$

where the flavor indices are hidden, <sup>V</sup> � <sup>U</sup><sup>u</sup> LU<sup>d</sup>† <sup>L</sup> denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix, Uu,d <sup>L</sup> are the unitary matrices used to diagonalize the quark mass matrices, and U<sup>d</sup> <sup>L</sup> and U<sup>u</sup> <sup>R</sup> have been absorbed into <sup>k</sup>, <sup>k</sup>~, <sup>y</sup>, <sup>y</sup>~, and w. In this setup, we treat the neutrinos as massless particles and their flavor mixing effects are rotated away as an approximation. There is no evidence for any new CP violation, so in the following, we treat the Yukawa couplings as real numbers for simplicity.

The scalar LQs can also couple to the SM Higgs field via the scalar potential, and the cross section for the Higgs to diphoton can be modified in principle. However,


Table 1.

List of examples of charged particles from new physics discussed in this review showing SUð Þ� 3 SUð Þ� 2 Uð Þ1 <sup>Y</sup> representations and applications to phenomenology.

where mdia u � �

Charged Particles

ces should be of order of vY<sup>i</sup>

<sup>S</sup><sup>1</sup>=<sup>3</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>1</sup>=<sup>3</sup> under SUð Þ<sup>2</sup> <sup>L</sup>; <sup>U</sup>ð Þ<sup>1</sup> <sup>Y</sup>

triplet representations can be taken as

<sup>Φ</sup>7=<sup>6</sup> <sup>¼</sup> <sup>ϕ</sup><sup>5</sup>=<sup>3</sup>

<sup>þ</sup> uc <sup>V</sup>∗yPLνδ�2=<sup>3</sup> � <sup>1</sup>

Kobayashi-Maskawa (CKM) matrix, Uu,d

nalize the quark mass matrices, and U<sup>d</sup>

representations and applications to phenomenology.

where the flavor indices are hidden, <sup>V</sup> � <sup>U</sup><sup>u</sup>

�

numbers for simplicity.

Table 1.

38

ϕ<sup>2</sup>=<sup>3</sup> !

LQ Yukawa couplings to the SM fermions are expressed as

2.4 Scalar leptoquarks

<sup>3</sup>�<sup>3</sup> and <sup>m</sup>dia

d � �

nalize Mu and Md to get the mass eigenstates in general. If vY<sup>i</sup>

and diað Þ mF <sup>2</sup>�<sup>2</sup> <sup>¼</sup> mF<sup>1</sup> ; mF<sup>2</sup> ð Þ. Notice that a non-vanished vs could shift the masses of VLTQs. Since vs≪ v, we neglect the small effects hereafter. Due to the presence of Y1,2, the SM quarks, U1, 2, and D1, <sup>2</sup> are not physical states; thus one has to diago-

that the off-diagonal elements of unitary matrices for diagonalizing the mass matri-

In this subsection we consider leptoquarks (LQs) which are discussed for exam-

� � SM gauge symmetry, where the doublet and

!

h i

ffiffi 2

= ffiffi 2 <sup>p</sup> <sup>δ</sup><sup>4</sup>=<sup>3</sup> <sup>δ</sup>�2=<sup>3</sup> �δ<sup>1</sup>=<sup>3</sup><sup>=</sup> ffiffi

<sup>1</sup>,2=mF1, <sup>2</sup> . By adjusting Y<sup>i</sup>

could be enhanced and lead to interesting phenomena in collider physics.

ple in Refs. [31, 32]. The three LQs are Φ7=<sup>6</sup> ¼ ð Þ 2; 7=6 , Δ1=<sup>3</sup> ¼ ð Þ 3; 1=3 , and

, <sup>Δ</sup>1=<sup>3</sup> <sup>¼</sup> <sup>δ</sup><sup>1</sup>=<sup>3</sup>

�LLQ <sup>¼</sup> <sup>u</sup>VkPRℓϕ<sup>5</sup>=<sup>3</sup> <sup>þ</sup> <sup>d</sup>kPRℓϕ<sup>2</sup>=<sup>3</sup> � � þ �ℓk~PRuϕ�5=<sup>3</sup> <sup>þ</sup> <sup>ν</sup>k~PRuϕ�2=<sup>3</sup>

ffiffi 2 <sup>p</sup> uc <sup>V</sup><sup>∗</sup>

<sup>þ</sup> uc <sup>V</sup>∗y~PL<sup>ℓ</sup> � dc <sup>y</sup>~PL<sup>ν</sup> <sup>þ</sup> ucwPRℓÞS<sup>1</sup>=<sup>3</sup> <sup>þ</sup> <sup>h</sup>:c:,

where the superscripts are the electric charges of the particles. Accordingly, the

<sup>y</sup>PLℓδ<sup>1</sup>=<sup>3</sup> � <sup>1</sup>

LU<sup>d</sup>†

<sup>L</sup> and U<sup>u</sup>

The scalar LQs can also couple to the SM Higgs field via the scalar potential, and the cross section for the Higgs to diphoton can be modified in principle. However,

and w. In this setup, we treat the neutrinos as massless particles and their flavor mixing effects are rotated away as an approximation. There is no evidence for any new CP violation, so in the following, we treat the Yukawa couplings as real

Particle type SUð Þ<sup>3</sup> ; SUð Þ <sup>2</sup> ; <sup>U</sup>ð Þ<sup>1</sup> <sup>Y</sup><sup>Þ</sup> � Examples of application

Vector-like lepton 1ð Þ ; 3; �1 , 1ð Þ ; 3; 0 Lepton flavor violation Vector-like quark 3ð Þ ; 3; 2=3 , 3ð Þ ; 3; �1=3 Quark flavor physics

Charged scalar 1ð Þ ; 3; 1 , 1ð Þ ; 2; 1=2 Neutrino mass, lepton flavor violation

Scalar leptoquark 3ð Þ ; 2; 7=6 , 3ð Þ ; 3; 1=3 , 3ð Þ ; 1; 1=3 Meson decay, lepton flavor violation

List of examples of charged particles from new physics discussed in this review showing SUð Þ� 3 SUð Þ� 2 Uð Þ1 <sup>Y</sup>

� �

<sup>3</sup>�<sup>3</sup> denote the diagonal mass matrices of SM quarks

<sup>1</sup>, <sup>2</sup>≪ mF1, <sup>2</sup> , we expect

, (23)

,

(24)

<sup>1</sup>, 2, the off-diagonal effects

2 p

<sup>p</sup> dc <sup>y</sup>PLνδ<sup>1</sup>=<sup>3</sup> � <sup>d</sup><sup>c</sup> <sup>y</sup>PLℓδ<sup>4</sup>=<sup>3</sup>

<sup>L</sup> denotes the Cabibbo-

<sup>R</sup> have been absorbed into <sup>k</sup>, <sup>k</sup>~, <sup>y</sup>, <sup>y</sup>~,

<sup>L</sup> are the unitary matrices used to diago-

the couplings of the LQs to the Higgs are independent parameters and irrelevant to the flavors, so by taking proper values for the parameters, the signal strength parameter for the Higgs to diphoton can fit the LHC data. For detailed analysis see Table 1 in Ref. [31].

#### 3. Examples of applying charged particles to flavor physics

In this section, we review applications of charged particles to flavor physics by considering VLLs and LQs as examples.

#### 3.1 Flavor physics from vector-like lepton

Introduction of VLLs contributes to lepton flavor physics via Yukawa interactions discussed in previous section. Here we review the leptonic decay of the SM Higgs and LFV decay of charged lepton as an illustration based on Ref. [29].

### 3.1.1 Modification to h ! τþτ� branching ratio

From Eq. (18), it can be seen that the modified Higgs couplings to the SM leptons are proportional lepton masses. By comparison with other lepton channels, it can be seen that the ττ mode is the most significant one, and thus we estimate the influence on <sup>h</sup> ! <sup>τ</sup>þτ�. Using the values that satisfy BR hð Þ ! μτ <sup>≈</sup>10�4, the deviation of Γ h ! τþτ� ð Þ from the SM prediction can be obtained as

$$\kappa\_{\tau\tau} \equiv \frac{\Gamma(h \to \tau^+ \tau^-)}{\Gamma^{\text{SM}}(h \to \tau^+ \tau^-)} = \left| 1 - \frac{6\nu^2 Y\_3^2}{8m\_{\Psi}^2} \right|^2 \approx 0.88.\tag{25}$$

If the SM Higgs production cross section is not changed, the signal strength for pp ! h ! τþτ� in our estimation is μττ ≈0:88, where the measurements from ATLAS and CMS are 1:44þ0:<sup>42</sup> �0:<sup>37</sup> [33] and 0:<sup>91</sup> � <sup>0</sup>:27 [34], respectively. Although the current data errors for the ττ channel are still large, the precision measurement of μττ can test the effect or give strict limits on the parameters.

### 3.1.2 τ ! μγ process in vector-like lepton model

In the following, we investigate the contributions of new couplings in Eq. (18) to the rare tau decays and to the flavor-conserving muon anomalous magnetic moment. We first investigate the muon g � 2, denoted by Δaμ. The lepton flavorchanging coupling hμτ can provide contribution to Δa<sup>μ</sup> through the Higgs-mediated loop diagrams. However, as shown in Eq. (18), the induced couplings are associated with m<sup>ℓ</sup>j=vℓLiℓRj; only the right-handed tau lepton has a significant contribution. The induced Δa<sup>μ</sup> is thus suppressed by the factor of m<sup>2</sup> <sup>μ</sup>mτ= vm<sup>2</sup> h so that the value of Δa<sup>μ</sup> is two orders of magnitude smaller than current data Δa<sup>μ</sup> ¼ a exp <sup>μ</sup> � <sup>a</sup><sup>S</sup><sup>M</sup> <sup>μ</sup> <sup>¼</sup> ð Þ� <sup>28</sup>:<sup>8</sup> � <sup>8</sup>:<sup>0</sup> <sup>10</sup>�<sup>10</sup> [2]. A similar situation happens in <sup>τ</sup> ! <sup>3</sup><sup>μ</sup> decay also. Since the couplings are suppressed by mτ=v and mμ=v, the BR for τ ! 3μ is of the order of 10�14. We also estimate the process <sup>τ</sup> ! μγ via the <sup>h</sup>-mediation. The effective interaction for τ ! μγ is expressed by

$$\mathcal{L}\_{\tau \to \mu \gamma} = \frac{e}{16\pi^2} m\_\tau \overline{\mu} \sigma\_{\mu \nu} (\mathcal{C}\_L P\_L + \mathcal{C}\_R P\_R) \tau F^{\mu \nu}, \tag{26}$$

#### Figure 1.

Contours for BRð Þ <sup>τ</sup> ! μγ (dashed) as a function of <sup>Y</sup> and <sup>m</sup>Ψ, where the constraint from <sup>Γ</sup><sup>Z</sup> <sup>i</sup>nv (solid) is included. (The plot is taken from ref. [29]).

where CL ¼ 0 and the Wilson coefficient CR from the one loop is obtained as

$$\mathbf{C}\_{\mathcal{R}} \approx \frac{\mathbf{C}\_{23}^{h} \mathbf{C}\_{33}^{h}}{2m\_{h}^{2}} \left( \ln \frac{m\_{h}^{2}}{m\_{\tau}^{2}} - \frac{4}{3} \right). \tag{27}$$

where ℓ ¼ ð Þ e; μ , and these measurements can test the violation of lepton flavor

more than 2:5σ deviation from the SM prediction. Furthermore, a known anomaly is the muon anomalous magnetic dipole moment (muon g � 2), where its latest mea-

explained by introducing LQs and we review possible scenarios in the following.

According to the interactions in Eq. (24), we first formulate the four-Fermi

and pseudoscalar currents, respectively. Taking the Fierz transformations, the

cPLbℓjPLν<sup>i</sup> <sup>þ</sup> <sup>~</sup>y3<sup>i</sup>

PLν<sup>i</sup> þ ∑ a

ν<sup>ℓ</sup><sup>0</sup> and b ! sℓ0þℓ0� decays. For the b ! cℓ<sup>0</sup>

ν<sup>ℓ</sup><sup>0</sup> decays can be expressed as follows:

V2<sup>a</sup>~yaj

where the indices i, j are the lepton flavors and the LQs in the same representation are taken as degenerate particles in mass. It can be seen that the interaction structure obtained from the triplet LQ is the same as that from the W-boson one. The doublet LQ generates an ð Þ� S � P ð Þ S � P structure as well as a tensor structure. However, the singlet LQ can produce currents of ð Þ� V � A ð Þ V � A ,

ð Þ� S � P ð Þ S � P , and tensor structures. Nevertheless, we show later that the singlet LQ makes the main contribution to the RD and RD<sup>∗</sup> excesses. Note that it is difficult to explain RD,D<sup>∗</sup> by only using the doublet or/and triplet LQs when the RK excess

With the Yukawa couplings in Eq. (24), the effective Hamiltonian for the

<sup>s</sup>γ<sup>μ</sup> ð Þ PLb <sup>ℓ</sup>jγμPRℓ<sup>j</sup>

<sup>s</sup>γ<sup>μ</sup> ð Þ PLb <sup>ℓ</sup>jγμPLℓ<sup>j</sup>

where the Fierz transformations have been applied. By Eq. (31), it can be clearly seen that the quark currents from both the doublet and triplet LQs are left-handed; however, the lepton current from the doublet (triplet) LQ is right(left)-handed. When one includes Eq. (31) in the SM contributions, the effective Hamiltonian for

� �,

� �,

~ ki<sup>2</sup> and ~y3<sup>i</sup>

w2<sup>j</sup> 2m<sup>2</sup> S þ k3j ~ ki<sup>2</sup> 2m<sup>2</sup> Φ

> ~y3i 2m<sup>2</sup> S

! <sup>1</sup>

4

PLνi,

cγμPLbℓjγ<sup>μ</sup>

�0:<sup>07</sup> � 0:05 [39], which indicate a

ν<sup>ℓ</sup><sup>0</sup> processes,

(30)

(31)

w2<sup>j</sup> are ð Þ� S � P ð Þ S � P ,

cσμνPLbℓjσμνPLν<sup>i</sup>

<sup>μ</sup> <sup>¼</sup> ð Þ� <sup>28</sup>:<sup>8</sup> � <sup>8</sup>:<sup>0</sup> <sup>10</sup>�<sup>10</sup> [2]. These anomalies would be

~y2<sup>j</sup> are ð Þ� S � P ð Þ S þ P , where S and P denote the scalar

universality. The averaged results from the heavy flavor averaging group are RD ¼ 0:403 � 0:040 � 0:024 and RD<sup>∗</sup> ¼ 0:310 � 0:015 � 0:008 [35], and the SM predictions are around RD ≈0:3 [36, 37] and RD<sup>∗</sup> ≈0:25, respectively. Further tests of lepton flavor universality can be made using the branching fraction ratios: RKð Þ <sup>∗</sup> <sup>¼</sup> BR B ! <sup>K</sup>ð Þ <sup>∗</sup> <sup>μ</sup>þμ� � �=BR B ! <sup>K</sup>ð Þ <sup>∗</sup> <sup>e</sup>þe� � �. The current LHCb measurements

are RK <sup>¼</sup> <sup>0</sup>:745þ0:<sup>090</sup> �0:<sup>074</sup> � <sup>0</sup>:036 [38] and RK<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:69þ0:<sup>11</sup>

exp <sup>μ</sup> � <sup>a</sup>S<sup>M</sup>

3.2.1 Effective interactions for semileptonic B-decay

the induced current-current interactions from k3<sup>j</sup>

y2<sup>j</sup> and ~y3<sup>i</sup>

w2<sup>j</sup> 2m<sup>2</sup> S þ k3j ~ ki<sup>2</sup> 2m<sup>2</sup> Φ

!

yajy3<sup>i</sup> 4m<sup>2</sup> Δ

the b ! sℓ0þℓ0� decays is written as

41

cγμPLbℓjγ<sup>μ</sup>

and other strict constraints are satisfied at the same time.

<sup>b</sup> ! <sup>s</sup>ℓ0þℓ0� decays mediated by <sup>ϕ</sup><sup>2</sup>=<sup>3</sup> and <sup>δ</sup><sup>4</sup>=<sup>3</sup> can be expressed as

<sup>H</sup><sup>b</sup>!<sup>s</sup> <sup>¼</sup> <sup>k</sup>3jk2<sup>j</sup> 2m<sup>2</sup> Φ

> � y3j y2j 2m<sup>2</sup> Δ

surement is Δa<sup>μ</sup> ¼ a

Flavor Physics and Charged Particle

DOI: http://dx.doi.org/10.5772/intechopen.81404

interactions for the b ! cℓ<sup>0</sup>

Hamiltonian for the b ! cℓ<sup>0</sup>

and those from y3<sup>i</sup>

<sup>H</sup><sup>b</sup>!<sup>c</sup> ¼ �~y3<sup>i</sup>

� ∑ a V2<sup>a</sup>

Accordingly, the BR for τ ! μγ is expressed as

$$\frac{BR(\tau \to \mu\gamma)}{BR(\tau \to e\overline{\nu}\_{\epsilon}\nu\_{\tau})} = \frac{3a\_{\epsilon}}{4\pi G\_F^2} |C\_R|^2. \tag{28}$$

We present the contours for BRð Þ τ ! μγ as a function of coupling Y and m<sup>Ψ</sup> in Figure 1, where the numbers on the plots are in units of 10�12. It can be seen that the resultant BRð Þ <sup>τ</sup> ! μγ can be only up to 10�12, where the current experimental upper bound is BRð Þ <sup>τ</sup> ! μγ < 4:<sup>4</sup> � <sup>10</sup>�<sup>8</sup> [2].

#### 3.2 B-meson flavor physics with leptoquarks

This section is based on Ref. [32]. Several interesting excesses in semileptonic B decays have been observed in experiments such as (i) the angular observable P<sup>0</sup> <sup>5</sup> of <sup>B</sup> ! <sup>K</sup><sup>∗</sup>μþμ� [7], where a 3<sup>σ</sup> deviation due to the integrated luminosity of 3.0 fb �<sup>1</sup> was found at the LHCb [8, 9], and the same measurement with a 2:6σ deviation was also confirmed by Belle [10] and (ii) the branching fraction ratios RD,D<sup>∗</sup> , which are defined and measured as follows:

$$R\_D = \frac{\overline{B} \to D\pi\nu}{\overline{B} \to D\ell\nu} = \begin{cases} 0.375 \pm 0.064 \pm 0.026 & \text{Belle[11]},\\ 0.440 \pm 0.058 \pm 0.042 & \text{BBar[12,13]}, \end{cases}$$

$$R\_{D^\*} = \frac{\overline{B} \to D^\*\pi\nu}{\overline{B} \to D^\*\ell\nu} = \begin{cases} 0.302 \pm 0.030 \pm 0.011 & \text{Belle[14]},\\ 0.270 \pm 0.035 \pm \_{-0.028}^{+0.028} & \text{Belle[15]},\\ 0.332 \pm 0.024 \pm 0.018 & \text{BBar[12,13]},\\ 0.336 \pm 0.027 \pm 0.030 & \text{LHCl[16]}. \end{cases} \tag{29}$$

40

#### Flavor Physics and Charged Particle DOI: http://dx.doi.org/10.5772/intechopen.81404

where ℓ ¼ ð Þ e; μ , and these measurements can test the violation of lepton flavor universality. The averaged results from the heavy flavor averaging group are RD ¼ 0:403 � 0:040 � 0:024 and RD<sup>∗</sup> ¼ 0:310 � 0:015 � 0:008 [35], and the SM predictions are around RD ≈0:3 [36, 37] and RD<sup>∗</sup> ≈0:25, respectively. Further tests of lepton flavor universality can be made using the branching fraction ratios: RKð Þ <sup>∗</sup> <sup>¼</sup> BR B ! <sup>K</sup>ð Þ <sup>∗</sup> <sup>μ</sup>þμ� � �=BR B ! <sup>K</sup>ð Þ <sup>∗</sup> <sup>e</sup>þe� � �. The current LHCb measurements are RK <sup>¼</sup> <sup>0</sup>:745þ0:<sup>090</sup> �0:<sup>074</sup> � <sup>0</sup>:036 [38] and RK<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:69þ0:<sup>11</sup> �0:<sup>07</sup> � 0:05 [39], which indicate a more than 2:5σ deviation from the SM prediction. Furthermore, a known anomaly is the muon anomalous magnetic dipole moment (muon g � 2), where its latest measurement is Δa<sup>μ</sup> ¼ a exp <sup>μ</sup> � <sup>a</sup>S<sup>M</sup> <sup>μ</sup> <sup>¼</sup> ð Þ� <sup>28</sup>:<sup>8</sup> � <sup>8</sup>:<sup>0</sup> <sup>10</sup>�<sup>10</sup> [2]. These anomalies would be explained by introducing LQs and we review possible scenarios in the following.

#### 3.2.1 Effective interactions for semileptonic B-decay

According to the interactions in Eq. (24), we first formulate the four-Fermi interactions for the b ! cℓ<sup>0</sup> ν<sup>ℓ</sup><sup>0</sup> and b ! sℓ0þℓ0� decays. For the b ! cℓ<sup>0</sup> ν<sup>ℓ</sup><sup>0</sup> processes, the induced current-current interactions from k3<sup>j</sup> ~ ki<sup>2</sup> and ~y3<sup>i</sup> w2<sup>j</sup> are ð Þ� S � P ð Þ S � P , and those from y3<sup>i</sup> y2<sup>j</sup> and ~y3<sup>i</sup> ~y2<sup>j</sup> are ð Þ� S � P ð Þ S þ P , where S and P denote the scalar and pseudoscalar currents, respectively. Taking the Fierz transformations, the Hamiltonian for the b ! cℓ<sup>0</sup> ν<sup>ℓ</sup><sup>0</sup> decays can be expressed as follows:

$$\mathcal{H}\_{b\to c} = \left(-\frac{\bar{\mathcal{V}}\_{3i}\mu\_{\mathcal{V}}}{2m\_{\mathcal{S}}^{2}} + \frac{k\_{3}\bar{\mathcal{k}}\_{i2}}{2m\_{\Phi}^{2}}\right)\overline{c}\mathcal{P}\_{L}b\,\overline{\mathcal{e}}\_{j}\mathcal{P}\_{L}\nu\_{i} + \left(\frac{\bar{\mathcal{V}}\_{3i}\mu\_{\mathcal{V}}}{2m\_{\mathcal{S}}^{2}} + \frac{k\_{3}\bar{\mathcal{k}}\_{i2}}{2m\_{\Phi}^{2}}\right)\frac{1}{4}\overline{c}\sigma\_{\mu\nu}\mathcal{P}\_{L}b\,\overline{\mathcal{e}}\_{j}\sigma^{\mu\nu}\mathcal{P}\_{L}\nu\_{i} $$
 
$$ -\sum\_{a} \mathcal{V}\_{2a}\frac{\mathcal{V}\_{aj}\mathcal{V}\_{3i}}{4m\_{\Delta}^{2}}\overline{c}\mathcal{V}\_{\mu}\mathcal{P}\_{L}b\,\overline{\mathcal{e}}\_{j}\mathcal{V}\_{\nu}\nu\_{i} + \sum\_{a} \mathcal{V}\_{2a}\bar{\mathcal{V}}\_{aj}\frac{\bar{\mathcal{V}}\_{3i}}{2m\_{\mathcal{S}}^{2}}\overline{c}\mathcal{V}\_{\mu}\mathcal{P}\_{L}b\,\overline{\mathcal{e}}\_{j}\gamma^{\mu}\mathcal{P}\_{L}\nu\_{i} \,\tag{30}$$

where the indices i, j are the lepton flavors and the LQs in the same representation are taken as degenerate particles in mass. It can be seen that the interaction structure obtained from the triplet LQ is the same as that from the W-boson one. The doublet LQ generates an ð Þ� S � P ð Þ S � P structure as well as a tensor structure. However, the singlet LQ can produce currents of ð Þ� V � A ð Þ V � A , ð Þ� S � P ð Þ S � P , and tensor structures. Nevertheless, we show later that the singlet LQ makes the main contribution to the RD and RD<sup>∗</sup> excesses. Note that it is difficult to explain RD,D<sup>∗</sup> by only using the doublet or/and triplet LQs when the RK excess and other strict constraints are satisfied at the same time.

With the Yukawa couplings in Eq. (24), the effective Hamiltonian for the <sup>b</sup> ! <sup>s</sup>ℓ0þℓ0� decays mediated by <sup>ϕ</sup><sup>2</sup>=<sup>3</sup> and <sup>δ</sup><sup>4</sup>=<sup>3</sup> can be expressed as

$$\begin{split} \mathcal{H}\_{b \longrightarrow s} &= \frac{k\_{3\bar{\jmath}} k\_{2\bar{\jmath}}}{2m\_{\Phi}^{2}} (\overline{\varsigma}\gamma^{\mu} P\_{L} b) \left( \overline{\ell}\_{j}^{\*} \chi\_{\mu} P\_{R} \ell\_{j}^{\prime} \right), \\ &- \frac{\mathcal{Y}\_{3\bar{\jmath}} \mathcal{Y}\_{2\bar{\jmath}}}{2m\_{\Delta}^{2}} (\overline{\varsigma}\gamma^{\mu} P\_{L} b) \left( \overline{\ell}\_{j}^{\*} \chi\_{\mu} P\_{L} \ell\_{j}^{\prime} \right), \end{split} \tag{31}$$

where the Fierz transformations have been applied. By Eq. (31), it can be clearly seen that the quark currents from both the doublet and triplet LQs are left-handed; however, the lepton current from the doublet (triplet) LQ is right(left)-handed. When one includes Eq. (31) in the SM contributions, the effective Hamiltonian for the b ! sℓ0þℓ0� decays is written as

where CL ¼ 0 and the Wilson coefficient CR from the one loop is obtained as

ln <sup>m</sup><sup>2</sup> h m<sup>2</sup> τ � 4 3

<sup>¼</sup> <sup>3</sup>α<sup>e</sup> 4πG<sup>2</sup> F j j CR 2

We present the contours for BRð Þ τ ! μγ as a function of coupling Y and m<sup>Ψ</sup> in Figure 1, where the numbers on the plots are in units of 10�12. It can be seen that the resultant BRð Þ <sup>τ</sup> ! μγ can be only up to 10�12, where the current experimental

This section is based on Ref. [32]. Several interesting excesses in semileptonic B decays have been observed in experiments such as (i) the angular observable P<sup>0</sup>

0:375 � 0:064 � 0:026 Belle 11 ½ �,

0:440 � 0:058 � 0:042 BaBar 12 ½ � ; 13 ,

0:302 � 0:030 � 0:011 Belle 14 ½ �,

<sup>0</sup>:<sup>270</sup> � <sup>0</sup>:035�þ<sup>0</sup>:<sup>028</sup> �0:<sup>025</sup> Belle 15 ½ �,

0:332 � 0:024 � 0:018 BaBar 12 ½ � ; 13 ,

0:336 � 0:027 � 0:030 LHCb 16 ½ �,

<sup>B</sup> ! <sup>K</sup><sup>∗</sup>μþμ� [7], where a 3<sup>σ</sup> deviation due to the integrated luminosity of 3.0 fb �<sup>1</sup> was found at the LHCb [8, 9], and the same measurement with a 2:6σ deviation was also confirmed by Belle [10] and (ii) the branching fraction ratios RD,D<sup>∗</sup> , which are

� �

: (27)

<sup>i</sup>nv (solid) is

: (28)

<sup>5</sup> of

(29)

CR <sup>≈</sup> <sup>C</sup><sup>h</sup>

BRð Þ τ ! μγ BRð Þ τ ! eνeντ

Accordingly, the BR for τ ! μγ is expressed as

upper bound is BRð Þ <sup>τ</sup> ! μγ < 4:<sup>4</sup> � <sup>10</sup>�<sup>8</sup> [2].

defined and measured as follows:

RD<sup>∗</sup> <sup>¼</sup> <sup>B</sup> ! <sup>D</sup><sup>∗</sup>τν <sup>B</sup> ! <sup>D</sup><sup>∗</sup>ℓ<sup>ν</sup>

40

RD <sup>¼</sup> <sup>B</sup> ! <sup>D</sup>τν B ! Dℓν

Figure 1.

Charged Particles

included. (The plot is taken from ref. [29]).

3.2 B-meson flavor physics with leptoquarks

¼

(

¼

8 >>>>>><

>>>>>>:

23C<sup>h</sup> 33 2m<sup>2</sup> h

Contours for BRð Þ <sup>τ</sup> ! μγ (dashed) as a function of <sup>Y</sup> and <sup>m</sup>Ψ, where the constraint from <sup>Γ</sup><sup>Z</sup>

Charged Particles

$$\mathcal{H}\_{b\to s} = \frac{G\_F \alpha\_{em} V\_{tb} V\_{ts}^\*}{\sqrt{2}\pi} \left[ H\_{1\mu} L^{\mu} + H\_{2\mu} L^{5\mu} \right],\tag{32}$$

where Cbox ¼ mBs f

Flavor Physics and Charged Particle

the b ! cℓ<sup>0</sup>

where Ð

using F† αβ � � ab

½ �� dX <sup>Ð</sup>

Fk<sup>~</sup> k � �

Fw<sup>~</sup><sup>y</sup> � �

<sup>B</sup><sup>þ</sup> ! <sup>K</sup>þνν < 1:<sup>6</sup> � <sup>10</sup>�<sup>5</sup>

<sup>r</sup><sup>ℓ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> Cν SM

43

can bound the parameters of ~y3<sup>i</sup>

late the BR for B<sup>þ</sup> ! Kþνν as

<sup>Δ</sup>ð Þ <sup>m</sup>1; <sup>m</sup><sup>2</sup> ab <sup>≈</sup> xm<sup>2</sup>

instead of Fαβ � �

ab ¼ ð Þ Vk <sup>3</sup><sup>b</sup>

ab <sup>¼</sup> <sup>w</sup>3<sup>b</sup> Vy<sup>~</sup> � �

2 Bs

excess, the rough magnitude of LQ couplings is ∣y3<sup>i</sup> y2<sup>i</sup>

BRð Þ¼ ℓ<sup>b</sup> ! ℓaγ

and the current measurement is Δmexp

DOI: http://dx.doi.org/10.5772/intechopen.81404

νℓ<sup>0</sup> decays.

=3, f Bs ≈0:224 GeV is the decay constant of Bs-meson [40],

ð Þ aR ab � � � � 2

dxdydzð Þ 1 � x � y � z , ð Þ aL ab can be obtained from ð Þ aR ab by

þ 2 3

þ 2 3

� �,

<sup>Δ</sup>a<sup>μ</sup> <sup>≃</sup> � <sup>m</sup>μð Þ aL <sup>þ</sup> aR <sup>a</sup>¼b¼<sup>μ</sup>: (39)

, and the SM result is around 4 � <sup>10</sup>�6. Thus, <sup>B</sup><sup>þ</sup> ! <sup>K</sup>þνν

ts ffiffi 2

<sup>p</sup> <sup>α</sup>e<sup>m</sup>

2π sin <sup>2</sup>θ<sup>W</sup>

. The four-Fermi interaction structure, which is

BR<sup>S</sup><sup>M</sup> <sup>B</sup><sup>þ</sup> ! <sup>K</sup><sup>þ</sup> ð Þ νν , (40)

X xð Þ<sup>t</sup> , (41)

� �,

our parameter values, it can be shown that the resulting ΔmBs agree with the current experimental data. However, ΔmBs can indeed constrain the parameters involved in

In addition to the muon g � 2, the introduced LQs can also contribute to the lepton flavor violating processes ℓ<sup>0</sup> ! ℓγ, where the current upper bounds are BRð Þ <sup>μ</sup> ! <sup>e</sup><sup>γ</sup> < 4:<sup>2</sup> � <sup>10</sup>�<sup>13</sup> and BRð Þ <sup>τ</sup> ! <sup>e</sup>ð Þ <sup>μ</sup> <sup>γ</sup> < 3:3 4ð Þ� :<sup>4</sup> <sup>10</sup>�<sup>8</sup> [2], and they can strictly constrain the LQ couplings. To understand the constraints due to the

> 48π3αemCba G2 Fm<sup>2</sup> ℓb

> > ð

d X½ � mt Fk<sup>~</sup>

ab, and the function Fk<sup>~</sup>

x Δ mt ð Þ ; m<sup>Φ</sup> ab

x Δ mt ð Þ ; mS ab

Note that Vk3<sup>b</sup> ≈k3<sup>b</sup> and Vy~3<sup>a</sup> ≈~y3<sup>a</sup> are due to Vub,cb≪Vtb ≈1. From Eq. (24), we can see that the doublet and singlet LQs can simultaneously couple to both left- and right-handed charged leptons, and the results are enhanced by mt. Other LQ contributions are suppressed by m<sup>ℓ</sup> due to the chirality flip in the external lepton legs, and thus they are ignored. Based on Eq. (37), the muon g � 2 can be obtained as

As mentioned earlier, the singlet LQ does not contribute to b ! sℓ0þℓ� at the tree level, but it can induce the b ! sνν process, where the current upper bound is

induced by the LQ, is the same as that induced by the W-boson, so we can formu-

<sup>1</sup> � <sup>r</sup><sup>ℓ</sup> j j0 <sup>2</sup> !

<sup>S</sup><sup>M</sup> <sup>¼</sup> GFVtbV<sup>∗</sup>

2:

ℓ<sup>0</sup> ! ℓγ decays, one expresses their branching ratios (BRs) such as

with Cμ<sup>e</sup> ≈1, Cτ<sup>e</sup> ≈0:1784, and Cτμ ≈ 0:1736. ð Þ aR ab is written as

ð Þ <sup>4</sup><sup>π</sup> <sup>2</sup>

~ ka<sup>3</sup> 5 3

> 3a 1 3

<sup>1</sup> <sup>þ</sup> ð Þ <sup>y</sup> <sup>þ</sup> <sup>z</sup> <sup>m</sup><sup>2</sup>

~y2i

<sup>3</sup> <sup>∑</sup> ℓ0

BR B<sup>þ</sup> ! <sup>K</sup><sup>þ</sup> ð Þ νν <sup>≈</sup> <sup>1</sup>

<sup>þ</sup> <sup>y</sup>3ℓ0y2ℓ<sup>0</sup> 4m<sup>2</sup> Δ

� �, C<sup>ν</sup>

~y3ℓ0~y2ℓ<sup>0</sup> 2m<sup>2</sup> S

ð Þ aR ab <sup>≈</sup> <sup>3</sup>

Bs <sup>¼</sup> <sup>1</sup>:<sup>17</sup> � <sup>10</sup>�<sup>11</sup> GeV [2]. To satisfy the RKð Þ <sup>∗</sup>

þ ð Þ aL ab � � � � <sup>2</sup> � � (36)

<sup>k</sup> is given by

1 � x Δð Þ mΦ; mt ab

1 � x Δð Þ mS; mt ab (38)

<sup>k</sup> � Fw<sup>~</sup><sup>y</sup> <sup>Þ</sup>ab, � (37)

<sup>∣</sup> � <sup>∣</sup>k3ik2<sup>i</sup><sup>∣</sup> � <sup>5</sup> � <sup>10</sup>�3. Using

where the leptonic currents are denoted by Lð Þ<sup>5</sup> <sup>μ</sup> ¼ ℓγμ γ<sup>5</sup> ð Þℓ, and the related hadronic currents are defined as

$$\begin{aligned} H\_{1\mu} &= \mathcal{C}\_{9}^{\ell} \overline{\varsigma} \gamma\_{\mu} P\_{L} b - \frac{2m\_{b}}{q^{2}} \mathcal{C}\_{7} \overline{\varsigma} i \sigma\_{\mu\nu} q^{\nu} P\_{R} b, \\ H\_{2\mu} &= \mathcal{C}\_{10}^{\ell} \overline{\varsigma} \gamma\_{\mu} P\_{L} b. \end{aligned} \tag{33}$$

The effective Wilson coefficients with LQ contributions are expressed as

$$\begin{split} \mathbf{C}^{\ell}\_{9(10)} &= \mathbf{C}^{\text{SM}}\_{9(10)} + \mathbf{C}^{\text{LQ},\ell'}\_{9(10)}, \\ \mathbf{C}^{\text{LQ},\ell\_{j}}\_{9} &= -\frac{1}{4\epsilon\_{\text{SM}}} \left( \frac{k\_{3\circ}k\_{\text{\mathcal{Y}}}}{m\_{\Phi}^{2}} - \frac{\mathcal{Y}\_{3\circ}\mathcal{Y}\_{\mathcal{Y}}}{m\_{\Delta}^{2}} \right), \\ \mathbf{C}^{\text{LQ},\ell\_{j}}\_{10} &= -\frac{1}{4\epsilon\_{\text{SM}}} \left( \frac{k\_{3\circ}k\_{\text{\mathcal{Y}}}}{m\_{\Phi}^{2}} + \frac{\mathcal{Y}\_{3\circ}\mathcal{Y}\_{\mathcal{Y}}}{m\_{\Delta}^{2}} \right), \end{split} \tag{34}$$

where <sup>c</sup>S<sup>M</sup> <sup>¼</sup> VtbV<sup>∗</sup> tsαemGF<sup>=</sup> ffiffi 2 <sup>p</sup> <sup>π</sup> � � and Vij is the CKM matrix element. From Eq. (34), it can be seen that when the magnitude of CLQ,ℓ<sup>j</sup> <sup>10</sup> is decreased, <sup>C</sup>LQ,ℓ<sup>j</sup> <sup>9</sup> can be enhanced. That is, the synchrony of the increasing/decreasing Wilson coefficients of C<sup>N</sup><sup>P</sup> <sup>9</sup> and C<sup>N</sup><sup>P</sup> <sup>10</sup> from new physics is diminished in this model. In addition, the sign of CLQ,ℓ<sup>0</sup> <sup>9</sup> can be different from that of <sup>C</sup><sup>L</sup>Q,ℓ<sup>0</sup> <sup>10</sup> . As a result, when the constraint from Bs ! <sup>μ</sup>þμ� decay is satisfied, we can have sizable values of CLQ,<sup>μ</sup> <sup>9</sup> to fit the anomalies of RK and angular observable in <sup>B</sup> ! <sup>K</sup>∗μþμ�. Although the LQs can contribute to the electromagnetic dipole operators, since the effects are through one-loop diagrams and are also small, the associated Wilson coefficient C<sup>7</sup> is mainly from the SM contributions.

#### 3.2.2 Constraints from ΔF ¼ 2, radiative lepton flavor violating, B<sup>þ</sup> ! Kþνν, Bs ! μþμ�, and Bc ! τν processes

Before we analyze the muon g � 2, RDð Þ <sup>∗</sup> , and RKð Þ <sup>∗</sup> problems, we examine the possible constraints due to rare decay processes. Firstly, we discuss the strict constraints from the ΔF ¼ 2 processes, such as F � F oscillation, where F denotes the neutral pseudoscalar meson. Since K � K, D � D, and Bd � Bd mixings are involved, the first-generation quarks and the anomalies mentioned earlier are associated with the second- and third-generation quarks. Therefore, we can avoid the constraints by assuming that <sup>k</sup>1ℓ<sup>0</sup> <sup>≈</sup> <sup>~</sup> kℓ0 <sup>1</sup> ≈y1ℓ<sup>0</sup> ≈~y1ℓ<sup>0</sup> ≈ w1<sup>i</sup> ≈0 without affecting the analyses of RDð Þ <sup>∗</sup> and RKð Þ <sup>∗</sup> . Thus, the relevant ΔF ¼ 2 process is Bs � Bs mixing, where ΔmBs ¼ 2∣ BsjHjBs � �∣ is induced from box diagrams, and the LQ contributions can be formulated as

$$\begin{split} \Delta m\_{B\_{i}} &\approx \frac{\mathbf{C}\_{\text{box}}}{\left(4\pi\right)^{2}} \left[ \frac{\mathfrak{Z}}{4} \left(\frac{\sum\_{i=1}^{3} \mathcal{V}\_{3} \mathcal{V}\_{2i}}{m\_{\Delta}}\right)^{2} + \left(\frac{\sum\_{i=1}^{3} k\_{3} k\_{2i}}{m\_{\Phi}}\right)^{2} \right] \\ &+ \frac{\mathbf{C}\_{\text{box}}}{\left(4\pi\right)^{2}} \left[ \left(\frac{\sum\_{i=1}^{3} \bar{\mathcal{V}}\_{3i} \bar{\mathcal{V}}\_{2i}}{m\_{\mathcal{S}}}\right)^{2} + 2 \frac{\left(\sum\_{i=1}^{3} \mathcal{V}\_{3i} \bar{\mathcal{V}}\_{2i}\right) \left(\sum\_{i=1}^{3} \bar{\mathcal{V}}\_{3} \mathcal{V}\_{2i}\right)}{m\_{\mathcal{S}}^{2} - m\_{\Delta}^{2}} \ln\left[\frac{m\_{\mathcal{S}}}{m\_{\Delta}}\right] \right], \end{split} \tag{35}$$

Flavor Physics and Charged Particle DOI: http://dx.doi.org/10.5772/intechopen.81404

<sup>H</sup><sup>b</sup>!<sup>s</sup> <sup>¼</sup> GFαemVtbV<sup>∗</sup>

where the leptonic currents are denoted by Lð Þ<sup>5</sup>

<sup>H</sup>1<sup>μ</sup> <sup>¼</sup> <sup>C</sup><sup>ℓ</sup>

<sup>H</sup>2<sup>μ</sup> <sup>¼</sup> <sup>C</sup><sup>ℓ</sup>

Cℓ

CLQ,ℓ<sup>j</sup>

CLQ,ℓ<sup>j</sup>

tsαemGF<sup>=</sup> ffiffi

Eq. (34), it can be seen that when the magnitude of CLQ,ℓ<sup>j</sup>

<sup>9</sup> can be different from that of <sup>C</sup><sup>L</sup>Q,ℓ<sup>0</sup>

hadronic currents are defined as

Charged Particles

where <sup>c</sup>S<sup>M</sup> <sup>¼</sup> VtbV<sup>∗</sup>

<sup>9</sup> and C<sup>N</sup><sup>P</sup>

from the SM contributions.

assuming that <sup>k</sup>1ℓ<sup>0</sup> <sup>≈</sup> <sup>~</sup>

<sup>Δ</sup>mBs <sup>≈</sup> <sup>C</sup>box ð Þ <sup>4</sup><sup>π</sup> <sup>2</sup>

> þ Cbox ð Þ <sup>4</sup><sup>π</sup> <sup>2</sup>

ΔmBs ¼ 2∣ BsjHjBs

formulated as

42

Bs ! μþμ�, and Bc ! τν processes

kℓ0

5 4

4

4

∑3 <sup>i</sup>¼<sup>1</sup>y3<sup>i</sup> y2i m<sup>Δ</sup> !<sup>2</sup>

∑3 <sup>i</sup>¼<sup>1</sup>~y3<sup>i</sup> ~y2i mS !<sup>2</sup>

cients of C<sup>N</sup><sup>P</sup>

sign of CLQ,ℓ<sup>0</sup>

ffiffi 2

<sup>9</sup>sγμPLb � <sup>2</sup>mb

The effective Wilson coefficients with LQ contributions are expressed as

4cS<sup>M</sup>

4cS<sup>M</sup>

be enhanced. That is, the synchrony of the increasing/decreasing Wilson coeffi-

anomalies of RK and angular observable in <sup>B</sup> ! <sup>K</sup>∗μþμ�. Although the LQs can contribute to the electromagnetic dipole operators, since the effects are through one-loop diagrams and are also small, the associated Wilson coefficient C<sup>7</sup> is mainly

3.2.2 Constraints from ΔF ¼ 2, radiative lepton flavor violating, B<sup>þ</sup> ! Kþνν,

and RKð Þ <sup>∗</sup> . Thus, the relevant ΔF ¼ 2 process is Bs � Bs mixing, where

!<sup>2</sup> 2

þ 2

<sup>þ</sup> <sup>∑</sup><sup>3</sup>

∑3 <sup>i</sup>¼<sup>1</sup>y3<sup>i</sup> ~y2i � �

Before we analyze the muon g � 2, RDð Þ <sup>∗</sup> , and RKð Þ <sup>∗</sup> problems, we examine the possible constraints due to rare decay processes. Firstly, we discuss the strict constraints from the ΔF ¼ 2 processes, such as F � F oscillation, where F denotes the neutral pseudoscalar meson. Since K � K, D � D, and Bd � Bd mixings are involved, the first-generation quarks and the anomalies mentioned earlier are associated with the second- and third-generation quarks. Therefore, we can avoid the constraints by

9 10 ð Þ <sup>þ</sup> <sup>C</sup>LQ,ℓ<sup>0</sup> 9 10 ð Þ ,

> k3jk2<sup>j</sup> m<sup>2</sup> Φ

k3jk2<sup>j</sup> m<sup>2</sup> Φ þ y3j y2j m<sup>2</sup> Δ

� y3j y2j m<sup>2</sup> Δ

� �

� �

<sup>10</sup>sγμPLb:

9 10 ð Þ <sup>¼</sup> <sup>C</sup>S<sup>M</sup>

<sup>9</sup> ¼ � <sup>1</sup>

<sup>10</sup> ¼ � <sup>1</sup>

2

from Bs ! <sup>μ</sup>þμ� decay is satisfied, we can have sizable values of CLQ,<sup>μ</sup>

<sup>q</sup><sup>2</sup> <sup>C</sup>7siσμνq<sup>ν</sup>

ts

<sup>p</sup> <sup>π</sup> <sup>H</sup>1<sup>μ</sup>L<sup>μ</sup> <sup>þ</sup> <sup>H</sup>2<sup>μ</sup>L5<sup>μ</sup> � �, (32)

PRb,

,

,

<sup>10</sup> is decreased, <sup>C</sup>LQ,ℓ<sup>j</sup>

<sup>10</sup> . As a result, when the constraint

<sup>p</sup> <sup>π</sup> � � and Vij is the CKM matrix element. From

<sup>10</sup> from new physics is diminished in this model. In addition, the

<sup>1</sup> ≈y1ℓ<sup>0</sup> ≈~y1ℓ<sup>0</sup> ≈ w1<sup>i</sup> ≈0 without affecting the analyses of RDð Þ <sup>∗</sup>

3 5

∑3 <sup>i</sup>¼<sup>1</sup>~y3<sup>i</sup> y2i � �

ln mS m<sup>Δ</sup> 3 5, (35)

� �∣ is induced from box diagrams, and the LQ contributions can be

<sup>i</sup>¼<sup>1</sup>k3ik2<sup>i</sup> m<sup>Φ</sup>

� � <sup>2</sup>

m<sup>2</sup> <sup>S</sup> � <sup>m</sup><sup>2</sup> Δ

<sup>μ</sup> ¼ ℓγμ γ<sup>5</sup> ð Þℓ, and the related

(33)

(34)

<sup>9</sup> can

<sup>9</sup> to fit the

where Cbox ¼ mBs f 2 Bs =3, f Bs ≈0:224 GeV is the decay constant of Bs-meson [40], and the current measurement is Δmexp Bs <sup>¼</sup> <sup>1</sup>:<sup>17</sup> � <sup>10</sup>�<sup>11</sup> GeV [2]. To satisfy the RKð Þ <sup>∗</sup> excess, the rough magnitude of LQ couplings is ∣y3<sup>i</sup> y2<sup>i</sup> <sup>∣</sup> � <sup>∣</sup>k3ik2<sup>i</sup><sup>∣</sup> � <sup>5</sup> � <sup>10</sup>�3. Using our parameter values, it can be shown that the resulting ΔmBs agree with the current experimental data. However, ΔmBs can indeed constrain the parameters involved in the b ! cℓ<sup>0</sup> νℓ<sup>0</sup> decays.

In addition to the muon g � 2, the introduced LQs can also contribute to the lepton flavor violating processes ℓ<sup>0</sup> ! ℓγ, where the current upper bounds are BRð Þ <sup>μ</sup> ! <sup>e</sup><sup>γ</sup> < 4:<sup>2</sup> � <sup>10</sup>�<sup>13</sup> and BRð Þ <sup>τ</sup> ! <sup>e</sup>ð Þ <sup>μ</sup> <sup>γ</sup> < 3:3 4ð Þ� :<sup>4</sup> <sup>10</sup>�<sup>8</sup> [2], and they can strictly constrain the LQ couplings. To understand the constraints due to the ℓ<sup>0</sup> ! ℓγ decays, one expresses their branching ratios (BRs) such as

$$BR(\mathcal{E}\_b \to \mathcal{E}\_a \chi) = \frac{48\pi^3 a\_{\rm em} C\_{\rm ba}}{G\_F^2 m\_{\mathcal{E}\_b}^2} \left( \left| (a\_R)\_{ab} \right|^2 + \left| (a\_L)\_{ab} \right|^2 \right) \tag{36}$$

with Cμ<sup>e</sup> ≈1, Cτ<sup>e</sup> ≈0:1784, and Cτμ ≈ 0:1736. ð Þ aR ab is written as

$$\delta(\boldsymbol{a}\_{\boldsymbol{R}})\_{ab} \approx \frac{3}{\left(4\pi\right)^{2}} \int d[\boldsymbol{X}] \, m\_{l} \left(\boldsymbol{F}\_{k\bar{b}} - \boldsymbol{F}\_{w\bar{\mathcal{V}}}\right)\_{ab\mathcal{V}} \tag{37}$$

where Ð ½ �� dX <sup>Ð</sup> dxdydzð Þ 1 � x � y � z , ð Þ aL ab can be obtained from ð Þ aR ab by using F† αβ � � ab instead of Fαβ � � ab, and the function Fk<sup>~</sup> <sup>k</sup> is given by

$$\left(F\_{b\bar{k}}\right)\_{ab} = \left(\mathbf{Vk}\right)\_{3b}\bar{k}\_{a3}\left(\frac{5}{3}\frac{\varkappa}{\Delta(m\_t, m\_\Phi)\_{ab}} + \frac{2}{3}\frac{1-\varkappa}{\Delta(m\_\Phi, m\_t)\_{ab}}\right),$$

$$\left(F\_{w\bar{\jmath}}\right)\_{ab} = w\_{3b}\left(\mathbf{V\bar{\jmath}}\right)\_{3a}\left(\frac{1}{3}\frac{\varkappa}{\Delta(m\_t, m\_S)\_{ab}} + \frac{2}{3}\frac{1-\varkappa}{\Delta(m\_S, m\_t)\_{ab}}\right),\tag{38}$$

$$\Delta(m\_1, m\_2)\_{ab} \approx xm\_1^2 + (y+z)m\_2^2.$$

Note that Vk3<sup>b</sup> ≈k3<sup>b</sup> and Vy~3<sup>a</sup> ≈~y3<sup>a</sup> are due to Vub,cb≪Vtb ≈1. From Eq. (24), we can see that the doublet and singlet LQs can simultaneously couple to both left- and right-handed charged leptons, and the results are enhanced by mt. Other LQ contributions are suppressed by m<sup>ℓ</sup> due to the chirality flip in the external lepton legs, and thus they are ignored. Based on Eq. (37), the muon g � 2 can be obtained as

$$
\Delta a\_{\mu} \simeq -m\_{\mu} (a\_L + a\_R)\_{a=b=\mu^\*}.\tag{39}
$$

As mentioned earlier, the singlet LQ does not contribute to b ! sℓ0þℓ� at the tree level, but it can induce the b ! sνν process, where the current upper bound is <sup>B</sup><sup>þ</sup> ! <sup>K</sup>þνν < 1:<sup>6</sup> � <sup>10</sup>�<sup>5</sup> , and the SM result is around 4 � <sup>10</sup>�6. Thus, <sup>B</sup><sup>þ</sup> ! <sup>K</sup>þνν can bound the parameters of ~y3<sup>i</sup> ~y2i . The four-Fermi interaction structure, which is induced by the LQ, is the same as that induced by the W-boson, so we can formulate the BR for B<sup>þ</sup> ! Kþνν as

$$BR(B^{+} \to K^{+} \nu \overline{\nu}) \approx \frac{1}{3} \left( \sum\_{\ell'} |\mathbf{1} - r\_{\ell'}|^2 \right) BR^{\text{SM}}(B^{+} \to K^{+} \nu \overline{\nu}),\tag{40}$$

$$r\_{\ell'} = \frac{1}{C\_{\text{SM}}^v} \left( \frac{\tilde{\mathcal{V}}\_{\text{3\ell}'} \tilde{\mathcal{V}}\_{\text{2\ell}'}}{2m\_{\text{S}}^2} + \frac{\mathcal{V}\_{\text{3\ell}'} \mathcal{V}\_{\text{2\ell}'}}{4m\_{\text{\Delta}}^2} \right), \quad C\_{\text{SM}}^v = \frac{G\_{\text{F}} V\_{tb} V\_{ts}^\*}{\sqrt{2}} \frac{a\_{\text{em}}}{2\pi \sin^2 \theta\_W} X(\mathbf{x}\_t), \tag{41}$$

where xt <sup>¼</sup> <sup>m</sup><sup>2</sup> <sup>t</sup> =m<sup>2</sup> <sup>W</sup> and X xð Þ<sup>t</sup> can be parameterized as X xð Þ<sup>t</sup> <sup>≈</sup>0:65x0:<sup>575</sup> <sup>t</sup> [41]. According to Eq. (31), the LQs also contribute to Bs ! μþμ� process, where the BRs measured by LHCb [42] and prediction in the SM [43] are BR Bs ! <sup>μ</sup>þμ� ð Þexp <sup>¼</sup> <sup>3</sup>:<sup>0</sup> � <sup>0</sup>:6þ0:<sup>3</sup> �0:2 � � � <sup>10</sup>�<sup>9</sup> and BR Bs ! <sup>μ</sup>þμ� ð ÞS<sup>M</sup> <sup>¼</sup> ð Þ� <sup>3</sup>:<sup>65</sup> � <sup>0</sup>:<sup>23</sup> <sup>10</sup>�9, respectively. The experimental data are consistent with the SM prediction, and in order to consider the constraint from Bs ! μþμ�, we use the expression for the BR as [44].

$$\frac{\text{BR}(B\_{\text{s}} \to \mu^{+}\mu^{-})}{\text{BR}(B\_{\text{s}} \to \mu^{+}\mu^{-})} = \left| \mathbf{1} - \mathbf{0}.\text{24}C\_{10}^{LQ,\mu} \right|^{2}.\tag{42}$$

V p2; <sup>ε</sup> <sup>j</sup>qγμbjB p<sup>1</sup>

Flavor Physics and Charged Particle

V p2; <sup>ε</sup> <sup>j</sup>qγμγ5bjB p<sup>1</sup>

V p2; <sup>ε</sup> <sup>j</sup>qσμνbjB p<sup>1</sup>

taken as [47]

B ! Dℓ<sup>0</sup>

Table 2.

45

can be given by

<sup>¼</sup> <sup>i</sup>εμνρσε<sup>ν</sup><sup>∗</sup>p<sup>ρ</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81404

<sup>¼</sup> <sup>2</sup>mVA<sup>0</sup> <sup>q</sup><sup>2</sup> ð Þ <sup>ε</sup><sup>∗</sup> � <sup>q</sup>

<sup>¼</sup> εμνρσ <sup>ε</sup><sup>ρ</sup><sup>∗</sup> <sup>p</sup><sup>1</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>

tions of motion, i.e., <sup>i</sup>∂μqγμ<sup>b</sup> <sup>¼</sup> mb � mq

and the other form factors are taken to be

for the B ! D form factors were described by [48].

<sup>B</sup> ! D B ! <sup>D</sup><sup>∗</sup>

<sup>B</sup> ! K B ! <sup>K</sup><sup>∗</sup>

σ<sup>2</sup> 0.41

B ! P,V transition form factors, as parameterized in Eqs. (46) and (47).

1pσ 2

mB þ mV

<sup>σ</sup>

� <sup>A</sup><sup>2</sup> <sup>q</sup><sup>2</sup> ð Þ <sup>ε</sup><sup>∗</sup> � <sup>q</sup>

þ2 <sup>ε</sup><sup>∗</sup> � <sup>q</sup> <sup>q</sup><sup>2</sup> <sup>p</sup><sup>ρ</sup> 1pσ

<sup>2</sup>V q<sup>2</sup> ð Þ mB þ mV

where <sup>V</sup> <sup>¼</sup> <sup>D</sup><sup>∗</sup> <sup>K</sup><sup>∗</sup> ð Þ when <sup>q</sup> <sup>¼</sup> c sð Þ, <sup>ε</sup><sup>0123</sup> <sup>¼</sup> 1, σμνγ<sup>5</sup> <sup>¼</sup> ð Þ <sup>i</sup>=<sup>2</sup> εμνρσσρσ, and <sup>ε</sup><sup>μ</sup> is the polarization vector of the vector meson. Here we note that the form factors associated with the weak scalar/pseudoscalar currents can be obtained through the equa-

numerical estimations, the <sup>q</sup>2-dependent form factors <sup>F</sup>þ, FT, <sup>V</sup>, <sup>A</sup>0, and <sup>T</sup><sup>1</sup> are

f q<sup>2</sup> <sup>¼</sup> <sup>f</sup>ð Þ <sup>0</sup>

f q<sup>2</sup> <sup>¼</sup> <sup>f</sup>ð Þ <sup>0</sup>

A detailed discussion of the form factors can be referred to [47]. The

The values of fð Þ 0 , σ1, and σ<sup>2</sup> for each form factor are summarized in Table 2.

next-to-next-leading (NNL) effects obtained with the LCQCD Some Rule approach

According to the form factors in Eqs. (44) and (45), and the interactions in Eqs. (30) and (32), we briefly summarize the differential decay rates for the semileptonic B decay processes, which we use for estimating RDð Þ <sup>∗</sup> and RK. For the

ν<sup>ℓ</sup><sup>0</sup> decay, the differential decay rate as a function of the invariant mass q<sup>2</sup>

F<sup>þ</sup> F<sup>0</sup> FT V A<sup>0</sup> A<sup>1</sup> A<sup>2</sup> T<sup>1</sup> T<sup>2</sup> T<sup>3</sup> f(0) 0.67 0.67 0.69 0.76 0.69 0.66 0.62 0.68 0.68 0.33 σ<sup>1</sup> 0.57 0.78 0.56 0.57 0.58 0.78 1.40 0.57 0.64 1.46

f(0) 0.36 0.36 0.35 0.44 0.45 0.36 0.32 0.39 0.39 0.27 σ<sup>1</sup> 0.43 0.70 0.43 0.45 0.46 0.64 1.23 0.45 0.72 1.31 σ<sup>2</sup> 0.27 0.36 0.38 0.62 0.41

,

p<sup>1</sup> þ p<sup>2</sup> 

<sup>q</sup><sup>2</sup> <sup>q</sup><sup>μ</sup> <sup>þ</sup> ð Þ mB <sup>þ</sup> mV <sup>A</sup><sup>1</sup> <sup>q</sup><sup>2</sup> <sup>ε</sup><sup>∗</sup>

<sup>T</sup><sup>1</sup> <sup>q</sup><sup>2</sup> ð Þþ <sup>ε</sup><sup>ρ</sup><sup>∗</sup>q<sup>σ</sup> <sup>m</sup><sup>2</sup>

<sup>2</sup> <sup>T</sup><sup>2</sup> <sup>q</sup><sup>2</sup> � <sup>T</sup><sup>1</sup> <sup>q</sup><sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

qb and <sup>i</sup>∂<sup>μ</sup> <sup>q</sup>γμγ ð Þ¼� <sup>5</sup><sup>b</sup> mb <sup>þ</sup> mq

<sup>1</sup> � <sup>q</sup><sup>2</sup>=M<sup>2</sup> <sup>1</sup> � <sup>σ</sup>1q<sup>2</sup>=M<sup>2</sup> <sup>þ</sup> <sup>σ</sup>2q<sup>4</sup>=M<sup>4</sup> , (46)

<sup>1</sup> � <sup>σ</sup>1q<sup>2</sup>=M<sup>2</sup> <sup>þ</sup> <sup>σ</sup>2q<sup>4</sup>=M<sup>4</sup> : (47)

<sup>μ</sup> � <sup>m</sup><sup>2</sup>

,

<sup>q</sup><sup>2</sup> <sup>T</sup><sup>2</sup> <sup>q</sup><sup>2</sup> � <sup>T</sup><sup>1</sup> <sup>q</sup><sup>2</sup>

<sup>B</sup> � <sup>m</sup><sup>2</sup> V <sup>q</sup><sup>2</sup> <sup>q</sup><sup>μ</sup>

<sup>B</sup> � <sup>m</sup><sup>2</sup> V

> m<sup>2</sup> <sup>B</sup> � <sup>m</sup><sup>2</sup> V

<sup>T</sup><sup>3</sup> <sup>q</sup><sup>2</sup> ,

<sup>μ</sup> � <sup>ε</sup><sup>∗</sup> � <sup>q</sup> <sup>q</sup><sup>2</sup> <sup>q</sup><sup>μ</sup> 

(45)

qγ5b. For

In addition to the <sup>B</sup>� ! <sup>D</sup>ð Þ <sup>∗</sup> τν decay, the induced effective Hamiltonian in Eq. (30) also contributes to the Bc ! τν process, where the allowed upper limit is BR B� <sup>c</sup> ! τν � � < 30% [45]. According to previous results given by [45], we express the BR for Bc ! τν as

$$BR(B\_{\varepsilon} \to \pi \overline{\nu}\_{\tau}) = \tau\_{B\_{\varepsilon}} \frac{m\_{\rm Rc} m\_{\pi}^2 \mathfrak{f}\_{B\_{\varepsilon}}^2 G\_F^2 |V\_{cb}|^2}{8\pi} \left(1 - \frac{m\_{\tau}^2}{m\_{B\_{\varepsilon}}^2}\right)^2 \left|1 + e\_L + \frac{m\_{B\_{\varepsilon}}^2}{m\_{\tau}(m\_b + m\_{\varepsilon})} e\_P\right|^2,\tag{43}$$

where f Bc is the Bc decay constant, and the εL,P in our model is given as

$$\begin{split} \varepsilon\_{L} &= \frac{\sqrt{2}}{4G\_{F}V\_{cb}} \left[ -\sum\_{a} V\_{2a} \frac{\mathcal{Y}\_{a3}\mathcal{Y}\_{33}}{4m\_{\Delta}^{2}} + \sum\_{a} V\_{2a} \frac{\tilde{\mathcal{Y}}\_{a3}\tilde{\mathcal{Y}}\_{33}}{2m\_{S}^{2}} \right], \\ \varepsilon\_{P} &= \frac{\sqrt{2}}{4G\_{F}V\_{cb}} \left[ \frac{\tilde{\mathcal{Y}}\_{33}w\_{23}}{2m\_{S}^{2}} - \frac{k\_{33}\tilde{k}\_{32}}{2m\_{\Phi}^{2}} \right]. \end{split}$$

Using <sup>τ</sup>Bc <sup>≈</sup>0:<sup>507</sup> � <sup>10</sup>�<sup>12</sup> s, mBc <sup>≈</sup>6:275 GeV, <sup>f</sup> Bc <sup>≈</sup>0:434 GeV [46], and Vcb <sup>≈</sup>0:04, the SM result is BR<sup>S</sup><sup>M</sup>ð Þ Bc ! τντ <sup>≈</sup>2:1%. One can see that the effects of the new physics can enhance the Bc ! τντ decay by a few factors at most in our analysis.

#### 3.2.3 Observables: RDð Þ <sup>∗</sup> and RKð Þ <sup>∗</sup>

The observables of RDð Þ <sup>∗</sup> and RKð Þ <sup>∗</sup> are the branching fraction ratios that are insensitive to the hadronic effects giving clearer test of lepton universality in Bmeson decay, but the associated BRs still depend on the transition form factors. In order to calculate the BR for each semileptonic decay process, we parameterize the transition form factors for B ! P by

$$\begin{split} \langle P(p\_2) | q \eta^\mu b | \overline{B}(p\_1) \rangle &= F\_+(q^2) \left( \left( p\_1 + p\_2 \right)^\mu - \frac{m\_B^2 - m\_P^2}{q^2} q^\mu \right) + \frac{m\_B^2 - m\_P^2}{q^2} q^\mu F\_0(q^2), \\ \langle P(p\_2) | q \sigma\_{\mu\nu} b | \overline{B}(p\_1) \rangle &= -i \left( p\_{1\mu} p\_{2\nu} - p\_{1\mu} p\_{2\mu} \right) \frac{2F\_T(q^2)}{m\_B + m\_P}, \end{split} \tag{44}$$

where P can be the D qð Þ ¼ c or K qð Þ ¼ s meson and the momentum transfer is given by q ¼ p<sup>1</sup> � p2. For the B ! V decay where V is a vector meson, the transition form factors associated with the weak currents are parameterized such that

Flavor Physics and Charged Particle DOI: http://dx.doi.org/10.5772/intechopen.81404

where xt <sup>¼</sup> <sup>m</sup><sup>2</sup>

�0:2

the BR for Bc ! τν as

BR Bð Þ¼ <sup>c</sup> ! τντ τBc

<sup>3</sup>:<sup>0</sup> � <sup>0</sup>:6þ0:<sup>3</sup>

Charged Particles

BR B�

analysis.

P p<sup>2</sup>

P p<sup>2</sup>

44

� �jqγ<sup>μ</sup>bjB p<sup>1</sup>

� �jqσμνbjB p<sup>1</sup>

<sup>t</sup> =m<sup>2</sup>

<sup>W</sup> and X xð Þ<sup>t</sup> can be parameterized as X xð Þ<sup>t</sup> <sup>≈</sup>0:65x0:<sup>575</sup>

10

� � � 2

1 þ ε<sup>L</sup> þ

~ya3~y<sup>33</sup> 2m<sup>2</sup> S

,

According to Eq. (31), the LQs also contribute to Bs ! μþμ� process, where the BRs measured by LHCb [42] and prediction in the SM [43] are BR Bs ! <sup>μ</sup>þμ� ð Þexp <sup>¼</sup>

� � � <sup>10</sup>�<sup>9</sup> and BR Bs ! <sup>μ</sup>þμ� ð ÞS<sup>M</sup> <sup>¼</sup> ð Þ� <sup>3</sup>:<sup>65</sup> � <sup>0</sup>:<sup>23</sup> <sup>10</sup>�9, respectively. The experimental data are consistent with the SM prediction, and in order to consider the constraint from Bs ! μþμ�, we use the expression for the BR as [44].

BR Bs ! <sup>μ</sup>þμ� ð ÞS<sup>M</sup> <sup>¼</sup> <sup>1</sup> � <sup>0</sup>:24CLQ ,<sup>μ</sup>

In addition to the <sup>B</sup>� ! <sup>D</sup>ð Þ <sup>∗</sup> τν decay, the induced effective Hamiltonian in Eq. (30) also contributes to the Bc ! τν process, where the allowed upper limit is

where f Bc is the Bc decay constant, and the εL,P in our model is given as

� ∑ a V2<sup>a</sup>

~y33w<sup>23</sup> 2m<sup>2</sup> S

<sup>c</sup> ! τν � � < 30% [45]. According to previous results given by [45], we express

τ m<sup>2</sup> Bc � � � �

!<sup>2</sup>

ya3y<sup>33</sup> 4m<sup>2</sup> Δ þ ∑ a V2<sup>a</sup>

� <sup>k</sup>33<sup>~</sup> k<sup>32</sup> 2m<sup>2</sup> Φ

" #

Using <sup>τ</sup>Bc <sup>≈</sup>0:<sup>507</sup> � <sup>10</sup>�<sup>12</sup> s, mBc <sup>≈</sup>6:275 GeV, <sup>f</sup> Bc <sup>≈</sup>0:434 GeV [46], and Vcb <sup>≈</sup>0:04, the SM result is BR<sup>S</sup><sup>M</sup>ð Þ Bc ! τντ <sup>≈</sup>2:1%. One can see that the effects of the new physics can enhance the Bc ! τντ decay by a few factors at most in our

The observables of RDð Þ <sup>∗</sup> and RKð Þ <sup>∗</sup> are the branching fraction ratios that are insensitive to the hadronic effects giving clearer test of lepton universality in Bmeson decay, but the associated BRs still depend on the transition form factors. In order to calculate the BR for each semileptonic decay process, we parameterize the

� �<sup>μ</sup> � <sup>m</sup><sup>2</sup>

� � <sup>2</sup>FT <sup>q</sup><sup>2</sup> ð Þ

where P can be the D qð Þ ¼ c or K qð Þ ¼ s meson and the momentum transfer is given by q ¼ p<sup>1</sup> � p2. For the B ! V decay where V is a vector meson, the transition

form factors associated with the weak currents are parameterized such that

� �

<sup>B</sup> � <sup>m</sup><sup>2</sup> P <sup>q</sup><sup>2</sup> <sup>q</sup><sup>μ</sup>

mB þ mP

,

þ m<sup>2</sup> <sup>B</sup> � <sup>m</sup><sup>2</sup> P <sup>q</sup><sup>2</sup> <sup>q</sup><sup>μ</sup>

� �

:

� � �

BR Bs ! μþμ� ð Þ

mBcm<sup>2</sup> τ f 2 Bc G2 <sup>F</sup>j j Vcb 2 <sup>8</sup><sup>π</sup> <sup>1</sup> � <sup>m</sup><sup>2</sup>

ffiffi 2 p 4GFVcb

ffiffi 2 p 4GFVcb

ε<sup>L</sup> ¼

ε<sup>P</sup> ¼

3.2.3 Observables: RDð Þ <sup>∗</sup> and RKð Þ <sup>∗</sup>

transition form factors for B ! P by

� � � � <sup>¼</sup> <sup>F</sup><sup>þ</sup> <sup>q</sup><sup>2</sup> ð Þ <sup>p</sup><sup>1</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>

� � � � ¼ �i p1<sup>μ</sup>p2<sup>ν</sup> � <sup>p</sup>1<sup>ν</sup>p2<sup>μ</sup>

<sup>t</sup> [41].

: (42)

m<sup>2</sup> Bc mτð Þ mb þ mc

εP

F<sup>0</sup> q<sup>2</sup> � �,

(44)

� � � �

2 ,

(43)

$$
\begin{split}
\langle\mathcal{V}(p\_{2},\epsilon)|\overline{q}\gamma\_{\mu}b|\overline{B}(p\_{1})\rangle &= i\epsilon\_{\mu\rho\sigma}\epsilon^{\nu\rho}p\_{1}^{\rho}p\_{2}^{\sigma}\frac{2V(q^{2})}{m\_{\text{B}}+m\_{V}},\\ \langle\mathcal{V}(p\_{2},\epsilon)|\overline{q}\gamma\_{\mu}\gamma\_{5}b|\overline{B}(p\_{1})\rangle &= 2m\_{V}A\_{0}(q^{2})\frac{\varepsilon^{\nu}\cdot q}{q^{2}}q\_{\mu} + (m\_{B}+m\_{V})A\_{1}(q^{2})\left(\varepsilon\_{\mu}^{\*}-\frac{\varepsilon^{\nu}\cdot q}{q^{2}}q\_{\mu}\right) \\ &- A\_{2}(q^{2})\frac{\varepsilon^{\nu}\cdot q}{m\_{B}+m\_{V}}\left((p\_{1}+p\_{2})\_{\mu}-\frac{m\_{B}^{2}-m\_{V}^{2}}{q^{2}}q\_{\mu}\right),\\ \langle\mathcal{V}(p\_{2},\epsilon)|\overline{q}\sigma\_{\mu}b|\overline{B}(p\_{1})\rangle &= \varepsilon\_{\mu\rho\sigma}\left[\varepsilon^{\nu\rho}\left(p\_{1}+p\_{2}\right)^{\sigma}T\_{1}(q^{2})+\varepsilon^{\rho\kappa}q^{\sigma}\frac{m\_{B}^{2}-m\_{V}^{2}}{q^{2}}\left(T\_{2}(q^{2})-T\_{1}(q^{2})\right)\right] \\ &+2\frac{\varepsilon^{\nu}\cdot q}{q^{2}}p\_{1}^{\rho}p\_{2}^{\sigma}\Bigl(T\_{2}(q^{2})-T\_{1}(q^{2})+\frac{q^{2}}{m\_{B}^{2}-m\_{V}^{2}}T\_{3}(q^{2})\Bigr)\Biggr],\end{split} \tag{45}$$

where <sup>V</sup> <sup>¼</sup> <sup>D</sup><sup>∗</sup> <sup>K</sup><sup>∗</sup> ð Þ when <sup>q</sup> <sup>¼</sup> c sð Þ, <sup>ε</sup><sup>0123</sup> <sup>¼</sup> 1, σμνγ<sup>5</sup> <sup>¼</sup> ð Þ <sup>i</sup>=<sup>2</sup> εμνρσσρσ, and <sup>ε</sup><sup>μ</sup> is the polarization vector of the vector meson. Here we note that the form factors associated with the weak scalar/pseudoscalar currents can be obtained through the equations of motion, i.e., <sup>i</sup>∂μqγμ<sup>b</sup> <sup>¼</sup> mb � mq qb and <sup>i</sup>∂<sup>μ</sup> <sup>q</sup>γμγ ð Þ¼� <sup>5</sup><sup>b</sup> mb <sup>þ</sup> mq qγ5b. For numerical estimations, the <sup>q</sup>2-dependent form factors <sup>F</sup>þ, FT, <sup>V</sup>, <sup>A</sup>0, and <sup>T</sup><sup>1</sup> are taken as [47]

$$f(q^2) = \frac{f(\mathbf{0})}{\left(1 - q^2/\mathcal{M}^2\right)\left(1 - \sigma\_1 q^2/\mathcal{M}^2 + \sigma\_2 q^4/\mathcal{M}^4\right)},\tag{46}$$

and the other form factors are taken to be

$$f\left(q^2\right) = \frac{f(\mathbf{0})}{1 - \sigma\_1 q^2 / \mathbf{M}^2 + \sigma\_2 q^4 / \mathbf{M}^4}.\tag{47}$$

The values of fð Þ 0 , σ1, and σ<sup>2</sup> for each form factor are summarized in Table 2. A detailed discussion of the form factors can be referred to [47]. The next-to-next-leading (NNL) effects obtained with the LCQCD Some Rule approach for the B ! D form factors were described by [48].

According to the form factors in Eqs. (44) and (45), and the interactions in Eqs. (30) and (32), we briefly summarize the differential decay rates for the semileptonic B decay processes, which we use for estimating RDð Þ <sup>∗</sup> and RK. For the B ! Dℓ<sup>0</sup> ν<sup>ℓ</sup><sup>0</sup> decay, the differential decay rate as a function of the invariant mass q<sup>2</sup> can be given by


Table 2.

B ! P,V transition form factors, as parameterized in Eqs. (46) and (47).

$$\begin{split} \frac{d\Gamma\_{D}^{\ell'}}{dq^{2}} &= \frac{G\_{F}^{2}|V\_{cb}|^{2}\sqrt{\lambda\_{D}}}{256\pi^{3}m\_{B}^{3}} \left(1 - \frac{m\_{\ell'}^{2}}{q^{2}}\right)^{2} \left[\frac{2}{3}\left(2 + \frac{m\_{\ell'}^{2}}{q^{2}}\right) \left|\mathcal{X}\_{+}^{\ell'}\right|^{2} + \frac{2m\_{\ell'}^{2}}{q^{2}} \left|\mathcal{X}\_{0}^{\ell'} + \frac{\sqrt{q^{2}}}{m\_{\ell'}}\mathcal{X}\_{S}^{\ell'}\right|^{2}\right] \\ &+ 16\left(\frac{2}{3}\left(1 + \frac{2m\_{\ell'}^{2}}{q^{2}}\right) \left|\mathcal{X}\_{T}^{\ell'}\right|^{2} - \frac{m\_{\ell'}}{\sqrt{q^{2}}}\mathcal{X}\_{T}^{\ell'}\mathcal{X}\_{0}^{\ell'}\right)\right], \end{split} \tag{48}$$

From Eq. (52), the measured ratio RK in the range <sup>q</sup><sup>2</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup>

RK ¼

RK<sup>∗</sup> is similar to RK, and thus we only show the result for RK.

Ð q2 max q2 min dq<sup>2</sup>

Ð q2 max q2 min dq<sup>2</sup>

After discussing the possible constraints and observables of interest, we now present the numerical analysis to determine the common parameter region where the RDð Þ<sup>∗</sup> and RKð Þ <sup>∗</sup> anomalies can fit the experimental data. Before presenting the numerical analysis, we summarize the relevant parameters, which are related to the

The parameters related to the radiative LFV, ΔB ¼ 2, and B<sup>þ</sup> ! Kþνν processes

, ∑ i y3i ~y2i � � <sup>∑</sup>

order to avoid the μ ! eγ and τ ! ℓγ constraints and obtain a sizable and positive

current data. In order to further suppress the number of free parameters and avoid

<sup>10</sup> so that Bs ! μþμ� can fit the experimental data. As mentioned

<sup>1</sup> ≈y1<sup>i</sup> ≈~y1<sup>i</sup> ≈ w1<sup>i</sup> � 0. When we omit these small coupling constants, the

k23; RK : k32k22, y32y22; RDð Þ<sup>∗</sup> : k32k22, y32y22, ~y3ℓ0w2ℓ0;

, y32y<sup>22</sup> � �<sup>2</sup>

2

~y2<sup>i</sup> are ignored due to the constraint from B<sup>þ</sup> ! Kþνν. The typical

k23k31, ~y32w31, w32~y31;

k33k3a, ~y33w3a, w33~y3<sup>a</sup>;

k13, 33, k31, 33, w3<sup>i</sup>) as a small value. From the upper limit of

; Bs ! μþμ� : k32k22, y32y22;

i ~y3i y2i � �,

~y2<sup>i</sup> < 0:03, and thus the resulting ΔmBs is smaller than the

. From Eqs. (54) and (55), we can see that in

dΓKμμ=dq<sup>2</sup>

k23, ~y32w32; RK : k3<sup>ℓ</sup>k2<sup>ℓ</sup>, y3<sup>ℓ</sup>y2<sup>ℓ</sup>;

can be estimated by

Flavor Physics and Charged Particle

DOI: http://dx.doi.org/10.5772/intechopen.81404

3.2.4 Numerical analysis

are defined as

specific measurements as follows:

muon <sup>g</sup> � <sup>2</sup> : <sup>k</sup>32<sup>~</sup>

RDð Þ <sup>∗</sup> : <sup>k</sup>3ℓ0<sup>~</sup>

<sup>μ</sup> ! <sup>e</sup><sup>γ</sup> : <sup>k</sup>32<sup>~</sup>

<sup>τ</sup> ! <sup>ℓ</sup>a<sup>γ</sup> : <sup>k</sup>33<sup>~</sup>

ΔmBs : ∑

y2i , ~y3<sup>i</sup> ~y2i

B<sup>þ</sup> ! Kþνν : ~y3<sup>i</sup>

where z3iz2<sup>i</sup> ¼ k3ik2i, y3<sup>i</sup>

B<sup>þ</sup> ! Kþνν, we obtain ~y3<sup>i</sup>

muon <sup>g</sup> � <sup>2</sup> : <sup>k</sup>32<sup>~</sup>

Δaμ, we can set (~

value of CLQ,<sup>μ</sup>

where ~y3<sup>i</sup>

<sup>k</sup>1ℓ<sup>0</sup> <sup>≈</sup> <sup>~</sup> kℓ0

47

kℓ0 <sup>2</sup>, ∑ a

k13, ~

ka3, ~

~y2i , y3<sup>i</sup> y2i

i z3iz2<sup>i</sup> � �<sup>2</sup>

large fine-tuning of couplings, we employ the scheme with kij ≈ ~

above, to avoid the bounds from the K, Bd, and D systems, we also adopt

correlations of the parameters in Eqs. (54) and (55) can be further simplified as

values of these parameters for fitting the anomalies in the b ! sμþμ� decay are y32ð Þ k<sup>32</sup> , y22ð Þ� k<sup>22</sup> 0:07, so the resulting ΔmBs is smaller than the current data, but

sign of yij can be selected to obtain the correct sign for <sup>C</sup>LQ,ℓ<sup>j</sup>

Bs ! μþμ� : k32k22, y32y22; ΔmBs : ð Þ k32k<sup>22</sup>

<sup>m</sup>in; q<sup>2</sup>

<sup>d</sup>ΓKee=dq<sup>2</sup> : (53)

<sup>V</sup>2<sup>a</sup> <sup>y</sup>3ℓ0ya<sup>ℓ</sup><sup>0</sup> ; <sup>~</sup>y3ℓ0~yaℓ0Þ, <sup>~</sup>y3ℓ0w2ℓ<sup>0</sup> : � (54)

<sup>m</sup>ax � � <sup>¼</sup> ½ � <sup>1</sup>; <sup>6</sup> GeV<sup>2</sup>

(55)

kji ≈ ∣yij∣, where the

<sup>9</sup> and to decrease the

, (56)

where the Xℓ<sup>0</sup> α n o functions and LQ contributions are

Xℓ<sup>0</sup> <sup>þ</sup> <sup>¼</sup> ffiffiffiffiffi λD <sup>p</sup> <sup>1</sup> <sup>þ</sup> <sup>C</sup>ℓ<sup>0</sup> V � �F<sup>þ</sup> <sup>q</sup><sup>2</sup> � �, Xℓ<sup>0</sup> <sup>0</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> <sup>B</sup> � <sup>m</sup><sup>2</sup> D � � <sup>1</sup> <sup>þ</sup> <sup>C</sup>ℓ<sup>0</sup> V � �F<sup>0</sup> <sup>q</sup><sup>2</sup> � � Xℓ<sup>0</sup> <sup>S</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> <sup>B</sup> � <sup>m</sup><sup>2</sup> D mb � mc Cℓ<sup>0</sup> S ffiffiffiffi q2 q F<sup>0</sup> q<sup>2</sup> � �, Xℓ<sup>0</sup> <sup>T</sup> ¼ � ffiffiffiffiffiffiffiffiffi q2λ<sup>D</sup> p mB þ mD Cℓ<sup>0</sup> <sup>T</sup> FT <sup>q</sup><sup>2</sup> � � C<sup>ℓ</sup><sup>0</sup> <sup>V</sup> ¼ ffiffi 2 p 8GFVcb ∑ a V2<sup>a</sup> ~y3ℓ0~ya<sup>ℓ</sup><sup>0</sup> m<sup>2</sup> S � <sup>y</sup>3ℓ0ya<sup>ℓ</sup><sup>0</sup> 2m<sup>2</sup> Δ � �, C<sup>ℓ</sup><sup>0</sup> <sup>S</sup> ¼ � ffiffi 2 p 4GFVcb ~y3ℓ0w2ℓ<sup>0</sup> 2m<sup>2</sup> S � <sup>k</sup>3ℓ0<sup>~</sup> kℓ0 2 2m<sup>2</sup> Φ !, C<sup>ℓ</sup><sup>0</sup> <sup>T</sup> ¼ ffiffi 2 p 16GFVcb ~y3ℓ0w2ℓ<sup>0</sup> 2m<sup>2</sup> S þ <sup>k</sup>3ℓ0<sup>~</sup> kℓ0 2 2m<sup>2</sup> Φ !, <sup>λ</sup><sup>H</sup> <sup>¼</sup>m<sup>4</sup> <sup>B</sup> <sup>þ</sup> <sup>m</sup><sup>4</sup> <sup>H</sup> <sup>þ</sup> <sup>q</sup><sup>4</sup> � <sup>2</sup> <sup>m</sup><sup>2</sup> Bm<sup>2</sup> <sup>H</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> Hq<sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> m2 B � �: (49)

We note that the effective couplings C<sup>ℓ</sup><sup>0</sup> <sup>S</sup> and C<sup>ℓ</sup><sup>0</sup> <sup>T</sup> at the mb scale can be obtained from the LQ mass scale via the renormalization group (RG) equation. Our numerical analysis considers the RG running effects with

C<sup>ℓ</sup><sup>0</sup> <sup>S</sup> =C<sup>ℓ</sup><sup>0</sup> T � � μ¼mb = C<sup>ℓ</sup><sup>0</sup> <sup>S</sup> =C<sup>ℓ</sup><sup>0</sup> T � � <sup>μ</sup>¼Oð Þ <sup>T</sup>eV � <sup>2</sup>:0 at the mb scale [49]. The <sup>B</sup> ! <sup>D</sup><sup>∗</sup>ℓ<sup>0</sup> νℓ0 decays involve D<sup>∗</sup> polarizations and more complicated transition form factors, so the differential decay rate determined by summing all of the D<sup>∗</sup> helicities are

$$\frac{d\Gamma\_{D^\*}^{\ell'}}{dq^2} = \sum\_{h=L\_2+\iota\_-} \frac{d\Gamma\_{D^\*}^{\ell'h}}{dq^2} = \frac{G\_F^2 |V\_{cb}|^2 \sqrt{\lambda\_{D^\*}}}{256\pi^3 m\_B^3} \left(1 - \frac{m\_{\ell'}^2}{q^2}\right)^2 \sum\_{h=L\_2+\iota\_-} V\_{D^\*}^{\ell'h}(q^2),\tag{50}$$

where λD<sup>∗</sup> is found in Eq. (53) and the detailed V<sup>ℓ</sup><sup>0</sup> h D∗ n o functions are shown in the appendix. According to Eqs. (48) and (50), RM (<sup>M</sup> <sup>¼</sup> D, D<sup>∗</sup>) can be calculated by

$$R\_M = \frac{\int\_{m\_r^2}^{q\_{\text{max}}^2} dq^2 \left(d\Gamma\_M^{\epsilon}/dq^2\right)}{\int\_{m\_\ell^2}^{q\_{\text{max}}^2} dq^2 \left(d\Gamma\_M^{\epsilon}/dq^2\right)}\tag{51}$$

where q<sup>2</sup> <sup>m</sup>ax ¼ ð Þ mB � mM <sup>2</sup> and Γ<sup>ℓ</sup> <sup>M</sup> <sup>¼</sup> <sup>Γ</sup><sup>e</sup> <sup>M</sup> <sup>þ</sup> <sup>Γ</sup><sup>μ</sup> M � �=2. For the <sup>B</sup> ! <sup>K</sup>ℓþℓ� decays, the differential decay rate can be expressed as [50].

$$\begin{split} \frac{d\Gamma\_{K\ell\ell}(q^2)}{dq^2} &\approx \frac{|c\_{\rm SM}|^2 m\_B^3}{3\cdot 2^8 \pi^3} \left(1 - \frac{q^2}{m\_B^2}\right)^{3/2} \\ &\times \left[ \left| C\_9' F\_+(q^2) + \frac{2m\_b C\_7}{m\_B + m\_\ell} F\_T(q^2) \right|^2 + \left| C\_{10}' F\_+(q^2) \right|^2 \right]. \end{split} \tag{52}$$

From Eq. (52), the measured ratio RK in the range <sup>q</sup><sup>2</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup> <sup>m</sup>in; q<sup>2</sup> <sup>m</sup>ax � � <sup>¼</sup> ½ � <sup>1</sup>; <sup>6</sup> GeV<sup>2</sup> can be estimated by

$$R\_K = \frac{\int\_{q\_{\rm min}^2}^{q\_{\rm max}^2} dq^2 d\Gamma\_{K\mu\mu}/dq^2}{\int\_{q\_{\rm min}^2}^{q\_{\rm max}^2} dq^2 d\Gamma\_{K\epsilon}/dq^2}.\tag{53}$$

RK<sup>∗</sup> is similar to RK, and thus we only show the result for RK.

#### 3.2.4 Numerical analysis

dΓℓ<sup>0</sup> D dq<sup>2</sup> <sup>¼</sup> <sup>G</sup><sup>2</sup>

Charged Particles

Xℓ<sup>0</sup> <sup>þ</sup> <sup>¼</sup> ffiffiffiffiffi λD <sup>p</sup> <sup>1</sup> <sup>þ</sup> <sup>C</sup>ℓ<sup>0</sup>

Xℓ<sup>0</sup> <sup>S</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup>

C<sup>ℓ</sup><sup>0</sup> <sup>V</sup> ¼

C<sup>ℓ</sup><sup>0</sup> <sup>S</sup> ¼ �

C<sup>ℓ</sup><sup>0</sup> <sup>S</sup> =C<sup>ℓ</sup><sup>0</sup> T � �

by

46

<sup>λ</sup><sup>H</sup> <sup>¼</sup>m<sup>4</sup>

dΓ<sup>ℓ</sup><sup>0</sup> D∗ dq<sup>2</sup> <sup>¼</sup> <sup>∑</sup>

where q<sup>2</sup>

<sup>F</sup>j j Vcb

<sup>þ</sup><sup>16</sup> <sup>2</sup> 3 1 þ

<sup>B</sup> � <sup>m</sup><sup>2</sup> D

mb � mc

ffiffi 2 p 8GFVcb

<sup>B</sup> <sup>þ</sup> <sup>m</sup><sup>4</sup>

μ¼mb

= C<sup>ℓ</sup><sup>0</sup> <sup>S</sup> =C<sup>ℓ</sup><sup>0</sup> T � �

<sup>h</sup>¼L, <sup>þ</sup>, �

<sup>m</sup>ax ¼ ð Þ mB � mM

dq<sup>2</sup> <sup>≈</sup> j j <sup>c</sup>S<sup>M</sup>

� <sup>C</sup><sup>ℓ</sup>

� � �

<sup>d</sup>Γ<sup>K</sup>ℓℓ <sup>q</sup><sup>2</sup> ð Þ

the differential decay rate can be expressed as [50].

2 m3 B <sup>3</sup> � <sup>28</sup>π<sup>3</sup> <sup>1</sup> � <sup>q</sup><sup>2</sup>

where the Xℓ<sup>0</sup>

256π3m<sup>3</sup> B

α n o

V � �

> ffiffiffiffi q2 q

~y3ℓ0w2ℓ<sup>0</sup> 2m<sup>2</sup> S

<sup>H</sup> <sup>þ</sup> <sup>q</sup><sup>4</sup> � <sup>2</sup> <sup>m</sup><sup>2</sup>

We note that the effective couplings C<sup>ℓ</sup><sup>0</sup>

dΓ<sup>ℓ</sup><sup>0</sup> h D∗ dq<sup>2</sup> <sup>¼</sup> <sup>G</sup><sup>2</sup>

ical analysis considers the RG running effects with

~y3ℓ0~ya<sup>ℓ</sup><sup>0</sup> m<sup>2</sup> S

Cℓ<sup>0</sup> S

∑ a V2<sup>a</sup>

ffiffi 2 p 4GFVcb

<sup>2</sup> ffiffiffiffiffi λD p

<sup>1</sup> � <sup>m</sup><sup>2</sup> ℓ0 q2 � �<sup>2</sup> 2

<sup>F</sup><sup>þ</sup> <sup>q</sup><sup>2</sup> � �, Xℓ<sup>0</sup>

F<sup>0</sup> q<sup>2</sup> � �, Xℓ<sup>0</sup>

� �

� <sup>k</sup>3ℓ0<sup>~</sup> kℓ0 2 2m<sup>2</sup> Φ

Bm<sup>2</sup> <sup>H</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

<sup>F</sup>j j Vcb

where λD<sup>∗</sup> is found in Eq. (53) and the detailed V<sup>ℓ</sup><sup>0</sup>

RM ¼

<sup>2</sup> and Γ<sup>ℓ</sup>

<sup>9</sup>F<sup>þ</sup> <sup>q</sup><sup>2</sup> ð Þþ <sup>2</sup>mbC<sup>7</sup>

256π<sup>3</sup>m<sup>3</sup> B

> Ð q2 max m<sup>2</sup> τ

> Ð q2 max m<sup>2</sup> ℓ

<sup>M</sup> <sup>¼</sup> <sup>Γ</sup><sup>e</sup>

m<sup>2</sup> B

mBþmK FT <sup>q</sup><sup>2</sup> ð Þ

� �

� �<sup>3</sup>=<sup>2</sup>

!

� <sup>y</sup>3ℓ0ya<sup>ℓ</sup><sup>0</sup> 2m<sup>2</sup> Δ

Xℓ<sup>0</sup> T � � �

� � � 2 � <sup>m</sup>ℓ<sup>0</sup> ffiffiffiffi <sup>q</sup><sup>2</sup> <sup>p</sup> <sup>X</sup>ℓ<sup>0</sup>

!#

2m<sup>2</sup> ℓ0 q2 � � 3 2 þ m<sup>2</sup> ℓ0 q2 � �

functions and LQ contributions are

<sup>0</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup>

<sup>T</sup> ¼ �

,

, C<sup>ℓ</sup><sup>0</sup> <sup>T</sup> ¼

Hq<sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

from the LQ mass scale via the renormalization group (RG) equation. Our numer-

decays involve D<sup>∗</sup> polarizations and more complicated transition form factors, so the differential decay rate determined by summing all of the D<sup>∗</sup> helicities are

> <sup>2</sup> ffiffiffiffiffiffiffi λD<sup>∗</sup> p

the appendix. According to Eqs. (48) and (50), RM (<sup>M</sup> <sup>¼</sup> D, D<sup>∗</sup>) can be calculated

dq<sup>2</sup> dΓ<sup>τ</sup>

dq<sup>2</sup> dΓ<sup>ℓ</sup>

<sup>M</sup> <sup>þ</sup> <sup>Γ</sup><sup>μ</sup> M

M=dq<sup>2</sup> � �

� � � 2 <sup>þ</sup> <sup>C</sup><sup>ℓ</sup>

� �:

4

Xℓ<sup>0</sup> þ � � �

<sup>T</sup> Xℓ<sup>0</sup> 0

<sup>B</sup> � <sup>m</sup><sup>2</sup> D � � <sup>1</sup> <sup>þ</sup> <sup>C</sup>ℓ<sup>0</sup>

> ffiffiffiffiffiffiffiffiffi q2λ<sup>D</sup> p

mB þ mD

m2 B

<sup>S</sup> and C<sup>ℓ</sup><sup>0</sup>

<sup>μ</sup>¼Oð Þ <sup>T</sup>eV � <sup>2</sup>:0 at the mb scale [49]. The <sup>B</sup> ! <sup>D</sup><sup>∗</sup>ℓ<sup>0</sup>

<sup>1</sup> � <sup>m</sup><sup>2</sup> ℓ0 q2 � �<sup>2</sup>

∑ <sup>h</sup>¼L, <sup>þ</sup>, �

h D∗ n o V<sup>ℓ</sup><sup>0</sup> h

M=dq<sup>2</sup> � � (51)

� �=2. For the <sup>B</sup> ! <sup>K</sup>ℓþℓ� decays,

<sup>10</sup>F<sup>þ</sup> <sup>q</sup><sup>2</sup> ð Þ � � �

� 2

:

(52)

� � � 2 þ 2m<sup>2</sup> ℓ0 <sup>q</sup><sup>2</sup> <sup>X</sup>ℓ<sup>0</sup> 0 þ

,

2 2

V � �

> Cℓ<sup>0</sup> <sup>T</sup> FT <sup>q</sup><sup>2</sup> � �

ffiffi 2 p 16GFVcb F<sup>0</sup> q<sup>2</sup> � �

~y3ℓ0w2ℓ<sup>0</sup> 2m<sup>2</sup> S þ <sup>k</sup>3ℓ0<sup>~</sup> kℓ0 2 2m<sup>2</sup> Φ

<sup>T</sup> at the mb scale can be obtained

!

� � � � �

ffiffiffiffi q<sup>2</sup> p mℓ<sup>0</sup> Xℓ<sup>0</sup> S

� � � � �

(48)

,

(49)

νℓ0

<sup>D</sup><sup>∗</sup> <sup>q</sup><sup>2</sup> � �, (50)

functions are shown in

After discussing the possible constraints and observables of interest, we now present the numerical analysis to determine the common parameter region where the RDð Þ<sup>∗</sup> and RKð Þ <sup>∗</sup> anomalies can fit the experimental data. Before presenting the numerical analysis, we summarize the relevant parameters, which are related to the specific measurements as follows:

$$\begin{aligned} \text{muon} \ g - 2: &k\_{32}\tilde{k}\_{23}, \tilde{\jmath}\_{32} w\_{32}; \quad R\_K: &k\_{3\ell}k\_{2\ell}, \jmath\_{3\ell}\jmath\_{2\ell}; \\ R\_{D^{(\*)}}: &k\_{3\ell'}\tilde{k}\_{\ell'2}, \sum\_{a} V\_{2a}(\jmath\_{3\ell'}\jmath\_{a\ell'}, \bar{\jmath}\_{3\ell'}\bar{\jmath}\_{a\ell'}), \bar{\jmath}\_{3\ell'}w\_{2\ell'}. \end{aligned} \tag{54}$$

The parameters related to the radiative LFV, ΔB ¼ 2, and B<sup>þ</sup> ! Kþνν processes are defined as

$$\begin{aligned} \mu &\rightarrow e\gamma : k\_{32}\ddot{k}\_{13}, \ddot{k}\_{23}\dot{k}\_{31}, \ddot{\jmath}\_{32}w\_{31}, w\_{32}\ddot{\jmath}\_{31}; \\ \tau &\rightarrow \ell\_{a}\gamma : k\_{33}\ddot{k}\_{a3}, \ddot{k}\_{33}\dot{\wp}\_{3a}, \ddot{\jmath}\_{33}w\_{3a}, w\_{33}\ddot{\jmath}\_{3a}; \\ B^{+} &\rightarrow K^{+}\nu\ddot{\nu} : \ddot{\jmath}\_{3}\ddot{\jmath}\_{2i}, \ddot{\jmath}\_{3i}\dot{\jmath}\_{2i}; \quad B\_{s} \rightarrow \mu^{+}\mu^{-} : k\_{32}k\_{22}, \ddot{\jmath}\_{32}\dot{\jmath}\_{2i}; \\ \Delta m\_{B\_{i}} &: \left(\sum\_{i} \varpi\_{3i}\varpi\_{2i}\right)^{2}, \left(\sum\_{i} \jmath\_{3i}\ddot{\nu}\_{2i}\right) \left(\sum\_{i} \ddot{\nu}\_{3}\wp\_{2i}\right), \end{aligned} \tag{55}$$

where z3iz2<sup>i</sup> ¼ k3ik2i, y3<sup>i</sup> y2i , ~y3<sup>i</sup> ~y2i . From Eqs. (54) and (55), we can see that in order to avoid the μ ! eγ and τ ! ℓγ constraints and obtain a sizable and positive Δaμ, we can set (~ k13, 33, k31, 33, w3<sup>i</sup>) as a small value. From the upper limit of B<sup>þ</sup> ! Kþνν, we obtain ~y3<sup>i</sup> ~y2<sup>i</sup> < 0:03, and thus the resulting ΔmBs is smaller than the current data. In order to further suppress the number of free parameters and avoid large fine-tuning of couplings, we employ the scheme with kij ≈ ~ kji ≈ ∣yij∣, where the sign of yij can be selected to obtain the correct sign for <sup>C</sup>LQ,ℓ<sup>j</sup> <sup>9</sup> and to decrease the value of CLQ,<sup>μ</sup> <sup>10</sup> so that Bs ! μþμ� can fit the experimental data. As mentioned above, to avoid the bounds from the K, Bd, and D systems, we also adopt <sup>k</sup>1ℓ<sup>0</sup> <sup>≈</sup> <sup>~</sup> kℓ0 <sup>1</sup> ≈y1<sup>i</sup> ≈~y1<sup>i</sup> ≈ w1<sup>i</sup> � 0. When we omit these small coupling constants, the correlations of the parameters in Eqs. (54) and (55) can be further simplified as

$$\begin{aligned} \text{muon} \ g - 2: &k\_3 \ddot{k}\_{23}; \ \mathsf{R}\_K: \mathsf{k}\_{32} \mathsf{k}\_{22}, \ \mathsf{y}\_{32} \mathsf{y}\_{22}; \ \mathsf{R}\_{D^{(\*)}}: \mathsf{k}\_{32} \mathsf{k}\_{22}, \ \mathsf{y}\_{32} \mathsf{y}\_{22}, \ \mathsf{y}\_{3'} \mathsf{w}\_{2'}; \\ \mathsf{B}\_i &\rightarrow \mu^+ \mu^-: \mathsf{k}\_{32} \mathsf{k}\_{22}, \ \mathsf{y}\_{32} \mathsf{y}\_{22}; \ \Delta m\_{\mathsf{B}\_i}: \left(\mathsf{k}\_{32} \mathsf{k}\_{22}\right)^2, \left(\mathsf{y}\_{32} \mathsf{y}\_{22}\right)^2, \end{aligned} \tag{56}$$

where ~y3<sup>i</sup> ~y2<sup>i</sup> are ignored due to the constraint from B<sup>þ</sup> ! Kþνν. The typical values of these parameters for fitting the anomalies in the b ! sμþμ� decay are y32ð Þ k<sup>32</sup> , y22ð Þ� k<sup>22</sup> 0:07, so the resulting ΔmBs is smaller than the current data, but

these parameters are too small to explain RDð Þ <sup>∗</sup> . Thus, we must depend on the singlet LQ to resolve the RD and RD<sup>∗</sup> excesses, where the main free parameters are now ~y3ℓ0w2ℓ<sup>0</sup> .

After discussing the constraints and the correlations among various processes, we present the numerical analysis. There are several LQs in this scenario, but we use mLQ to denote the mass of all LQs. From Eqs. (37), (39), and (56), we can see that the muon <sup>g</sup> � 2 depends only on <sup>k</sup>32<sup>~</sup> k<sup>23</sup> and mΦ. Here we illustrate Δa<sup>μ</sup> as a function of k32~ k<sup>23</sup> in Figure 2(a), where the solid, dashed, and dotted lines denote the results for m<sup>Φ</sup> ¼ 1:5, 5, and 10 TeV, respectively, and the band is the experimental value with 1σ errors. Due to the mt enhancement, k32~ k<sup>23</sup> � 0:05 with m<sup>Φ</sup> � 1 TeV can explain the muon g � 2 anomaly.

According to the relationships shown in Eq. (56), RK, Bs ! μþμ�, and ΔmBs depend on the same parameters, i.e., k32k<sup>22</sup> and y32y22. We show the contours for these observables as a function of k32k<sup>22</sup> and y32y<sup>22</sup> in Figure 2(b), where the data with 1σ errors and mLQ ¼ 1:5 TeV are taken for all LQ masses. Based on these results, we see that <sup>Δ</sup>mBs <sup>&</sup>lt; <sup>Δ</sup>m<sup>e</sup>xp Bs in the range of ∣k32k22∣, ∣y32y22∣ < 0:05, where RK and BR Bs ! μþμ� ð Þ can both fit the experimental data simultaneously. In addition, we show CLQ ,<sup>μ</sup> <sup>9</sup> ¼ �½ � <sup>1</sup>:5; �0:<sup>5</sup> in the same plot. We can see that <sup>C</sup>LQ ,<sup>μ</sup> <sup>9</sup> � �1, which is used to explain the angular observable P<sup>0</sup> 5, can also be achieved in the same common region. According to Figure 2(b), the preferred values of k32k<sup>22</sup> and y32y<sup>22</sup> where the observed RK and Bs ! <sup>μ</sup>þμ� and the <sup>C</sup>LQ,<sup>μ</sup> <sup>9</sup> ¼ �½ � 1:5; �0:5 overlap are around <sup>k</sup>32k22; <sup>y</sup>32y<sup>22</sup> � �ð Þ <sup>0</sup>:001; <sup>0</sup>:<sup>004</sup> and � ð Þ <sup>0</sup>:025; <sup>0</sup>:<sup>03</sup> . The latter values are at the percentage level, but they are still not sufficiently large to explain the treedominated RD and RD<sup>∗</sup> anomalies.

After studying the muon g � 2 and RK anomalies, we numerically analyze the ratio of BR <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> τντ to BR <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> <sup>ℓ</sup>ν<sup>ℓ</sup> , i.e., RDð Þ <sup>∗</sup> . The introduced doublet and triplet LQs cannot efficiently enhance RDð Þ<sup>∗</sup> , so in the following estimations, we only focus on the singlet LQ contributions, where the four-Fermi interactions shown in Eq. (30) come mainly from the scalar- and tensor-type interaction structures. Based

on Eqs. (48), (50), and (51), we show the contours for RD and RD<sup>∗</sup> as a function of <sup>~</sup>y33w<sup>23</sup> and <sup>~</sup>y32w<sup>22</sup> <sup>~</sup>y31w21<sup>Þ</sup> in Figure 3(a) and (b), where the horizontal dashed and

Contours for (a) RD and (b) RD<sup>∗</sup> , where the solid lines denote the data with 1σ and 2σ errors, respectively. The

 . Contours for (c) RD and (d) RD<sup>∗</sup> , where the solid and dashed lines denote the situations with and without tensor operator contributions, respectively. In this case, we take mLQ ¼ 1:5 TeV. (These plots

 , we obtain the limits <sup>∣</sup>~y3<sup>ℓ</sup>w2<sup>ℓ</sup>∣≤1:5 and <sup>~</sup>y33w23>0. In order to clearly demonstrate the influence of tensor-type interactions, we also calculate the

Figure 3(c) and (d), where the solid and dashed lines denote the cases with and

different responses to the tensor operators, where the latter is more sensitive to the tensor interactions. RD and RD<sup>∗</sup> can be explained simultaneously with the tensor couplings. In order to understand the correlation between BR Bð Þ <sup>c</sup> ! τντ and RDð Þ <sup>∗</sup> , we show the contours for BR Bð Þ <sup>c</sup> ! τντ and RDð Þ <sup>∗</sup> as a function of w23~y<sup>33</sup> and mS in Figure 4, where ~y32w<sup>22</sup> ≈~y31w<sup>21</sup> ≈0 are used, and the gray area is excluded by

<sup>T</sup> ¼ 0. The contours obtained for RD and RD<sup>∗</sup> are shown in

, whereas the vertical dotted lines are the

<sup>T</sup> , respectively. According to these plots, we can see that RD and RD<sup>∗</sup> have

vertical dotted lines in both plots denote BR<sup>e</sup>xp <sup>B</sup>� ! <sup>D</sup> <sup>ℓ</sup>νℓ; τντ <sup>ð</sup> ½ �Þ ¼ <sup>½</sup>2:27� <sup>0</sup>:11; <sup>0</sup>:<sup>77</sup> � <sup>0</sup>:25�% and BR<sup>e</sup>xp <sup>B</sup>� ! <sup>D</sup><sup>∗</sup> <sup>ℓ</sup>νℓ; τντ <sup>ð</sup> ½ �Þ ¼ ½ � <sup>5</sup>:<sup>69</sup> � <sup>0</sup>:19; <sup>1</sup>:<sup>88</sup> � <sup>0</sup>:<sup>20</sup> %, respectively, and mLQ ¼ 1:5 TeV is used, and the data with 2σ errors are taken. For

simplicity, we take ~y31w<sup>21</sup> ≈~y32w22. When considering the limits from

horizontal dashed lines in both plots denote the BR<sup>e</sup>xp <sup>B</sup><sup>þ</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> <sup>ℓ</sup>ν<sup>ℓ</sup>

BR <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> <sup>ℓ</sup><sup>0</sup>

Figure 3.

BR<sup>e</sup>xp <sup>B</sup><sup>þ</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> τντ

are taken from ref. [32]).

Flavor Physics and Charged Particle

DOI: http://dx.doi.org/10.5772/intechopen.81404

without C<sup>ℓ</sup><sup>0</sup>

49

situation by setting C<sup>ℓ</sup><sup>0</sup>

νℓ0

#### Figure 2.

(a) Δa<sup>μ</sup> as a function of k32~ k<sup>23</sup> with m<sup>Φ</sup> ¼ 1:5, 5, 10 TeV, where the band denotes the experimental data with <sup>1</sup><sup>σ</sup> errors. (b) Contours for RK, Bs ! <sup>μ</sup>þμ�, <sup>Δ</sup>mBs , and <sup>C</sup>LQ,<sup>μ</sup> <sup>9</sup> as a function of k32k<sup>22</sup> and y32y22, where the ranges of RK and Bs ! <sup>μ</sup>þμ� are the experimental values with <sup>1</sup><sup>σ</sup> errors and mLQ <sup>¼</sup> <sup>1</sup>:<sup>5</sup> TeV. For <sup>C</sup>LQ, <sup>μ</sup> <sup>9</sup> , we show the range for CLQ,<sup>μ</sup> ¼ �½ � <sup>1</sup>:5; �0:<sup>5</sup> . (These plots are taken from Ref. [32]).

#### Flavor Physics and Charged Particle DOI: http://dx.doi.org/10.5772/intechopen.81404

#### Figure 3.

these parameters are too small to explain RDð Þ <sup>∗</sup> . Thus, we must depend on the singlet LQ to resolve the RD and RD<sup>∗</sup> excesses, where the main free parameters are now

After discussing the constraints and the correlations among various processes, we present the numerical analysis. There are several LQs in this scenario, but we use mLQ to denote the mass of all LQs. From Eqs. (37), (39), and (56), we can see that

k<sup>23</sup> in Figure 2(a), where the solid, dashed, and dotted lines denote the results

for m<sup>Φ</sup> ¼ 1:5, 5, and 10 TeV, respectively, and the band is the experimental value

According to the relationships shown in Eq. (56), RK, Bs ! μþμ�, and ΔmBs depend on the same parameters, i.e., k32k<sup>22</sup> and y32y22. We show the contours for these observables as a function of k32k<sup>22</sup> and y32y<sup>22</sup> in Figure 2(b), where the data with 1σ errors and mLQ ¼ 1:5 TeV are taken for all LQ masses. Based on these

and BR Bs ! μþμ� ð Þ can both fit the experimental data simultaneously. In addition,

common region. According to Figure 2(b), the preferred values of k32k<sup>22</sup> and y32y<sup>22</sup>

at the percentage level, but they are still not sufficiently large to explain the tree-

After studying the muon g � 2 and RK anomalies, we numerically analyze the

triplet LQs cannot efficiently enhance RDð Þ<sup>∗</sup> , so in the following estimations, we only focus on the singlet LQ contributions, where the four-Fermi interactions shown in Eq. (30) come mainly from the scalar- and tensor-type interaction structures. Based

� �ð Þ <sup>0</sup>:001; <sup>0</sup>:<sup>004</sup> and � ð Þ <sup>0</sup>:025; <sup>0</sup>:<sup>03</sup> . The latter values are

<sup>9</sup> ¼ �½ � <sup>1</sup>:5; �0:<sup>5</sup> in the same plot. We can see that <sup>C</sup>LQ ,<sup>μ</sup>

k<sup>23</sup> and mΦ. Here we illustrate Δa<sup>μ</sup> as a function

Bs in the range of ∣k32k22∣, ∣y32y22∣ < 0:05, where RK

5, can also be achieved in the same

, i.e., RDð Þ <sup>∗</sup> . The introduced doublet and

k<sup>23</sup> with m<sup>Φ</sup> ¼ 1:5, 5, 10 TeV, where the band denotes the experimental data with

ranges of RK and Bs ! <sup>μ</sup>þμ� are the experimental values with <sup>1</sup><sup>σ</sup> errors and mLQ <sup>¼</sup> <sup>1</sup>:<sup>5</sup> TeV. For <sup>C</sup>LQ, <sup>μ</sup>

<sup>9</sup> as a function of k32k<sup>22</sup> and y32y22, where the

<sup>9</sup> , we

<sup>9</sup> ¼ �½ � 1:5; �0:5 overlap are

k<sup>23</sup> � 0:05 with m<sup>Φ</sup> � 1 TeV can

<sup>9</sup> � �1, which

~y3ℓ0w2ℓ<sup>0</sup> .

Charged Particles

of k32~

the muon <sup>g</sup> � 2 depends only on <sup>k</sup>32<sup>~</sup>

explain the muon g � 2 anomaly.

results, we see that <sup>Δ</sup>mBs <sup>&</sup>lt; <sup>Δ</sup>m<sup>e</sup>xp

dominated RD and RD<sup>∗</sup> anomalies.

is used to explain the angular observable P<sup>0</sup>

where the observed RK and Bs ! <sup>μ</sup>þμ� and the <sup>C</sup>LQ,<sup>μ</sup>

to BR <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> <sup>ℓ</sup>ν<sup>ℓ</sup>

we show CLQ ,<sup>μ</sup>

around k32k22; y32y<sup>22</sup>

ratio of BR <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> τντ

Figure 2.

48

(a) Δa<sup>μ</sup> as a function of k32~

<sup>1</sup><sup>σ</sup> errors. (b) Contours for RK, Bs ! <sup>μ</sup>þμ�, <sup>Δ</sup>mBs , and <sup>C</sup>LQ,<sup>μ</sup>

show the range for CLQ,<sup>μ</sup> ¼ �½ � <sup>1</sup>:5; �0:<sup>5</sup> . (These plots are taken from Ref. [32]).

with 1σ errors. Due to the mt enhancement, k32~

Contours for (a) RD and (b) RD<sup>∗</sup> , where the solid lines denote the data with 1σ and 2σ errors, respectively. The horizontal dashed lines in both plots denote the BR<sup>e</sup>xp <sup>B</sup><sup>þ</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> <sup>ℓ</sup>ν<sup>ℓ</sup> , whereas the vertical dotted lines are the BR<sup>e</sup>xp <sup>B</sup><sup>þ</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> τντ . Contours for (c) RD and (d) RD<sup>∗</sup> , where the solid and dashed lines denote the situations with and without tensor operator contributions, respectively. In this case, we take mLQ ¼ 1:5 TeV. (These plots are taken from ref. [32]).

on Eqs. (48), (50), and (51), we show the contours for RD and RD<sup>∗</sup> as a function of <sup>~</sup>y33w<sup>23</sup> and <sup>~</sup>y32w<sup>22</sup> <sup>~</sup>y31w21<sup>Þ</sup> in Figure 3(a) and (b), where the horizontal dashed and vertical dotted lines in both plots denote BR<sup>e</sup>xp <sup>B</sup>� ! <sup>D</sup> <sup>ℓ</sup>νℓ; τντ <sup>ð</sup> ½ �Þ ¼ <sup>½</sup>2:27� <sup>0</sup>:11; <sup>0</sup>:<sup>77</sup> � <sup>0</sup>:25�% and BR<sup>e</sup>xp <sup>B</sup>� ! <sup>D</sup><sup>∗</sup> <sup>ℓ</sup>νℓ; τντ <sup>ð</sup> ½ �Þ ¼ ½ � <sup>5</sup>:<sup>69</sup> � <sup>0</sup>:19; <sup>1</sup>:<sup>88</sup> � <sup>0</sup>:<sup>20</sup> %, respectively, and mLQ ¼ 1:5 TeV is used, and the data with 2σ errors are taken. For simplicity, we take ~y31w<sup>21</sup> ≈~y32w22. When considering the limits from BR <sup>B</sup> ! <sup>D</sup>ð Þ <sup>∗</sup> <sup>ℓ</sup><sup>0</sup> νℓ0 , we obtain the limits <sup>∣</sup>~y3<sup>ℓ</sup>w2<sup>ℓ</sup>∣≤1:5 and <sup>~</sup>y33w23>0. In order to clearly demonstrate the influence of tensor-type interactions, we also calculate the situation by setting C<sup>ℓ</sup><sup>0</sup> <sup>T</sup> ¼ 0. The contours obtained for RD and RD<sup>∗</sup> are shown in Figure 3(c) and (d), where the solid and dashed lines denote the cases with and without C<sup>ℓ</sup><sup>0</sup> <sup>T</sup> , respectively. According to these plots, we can see that RD and RD<sup>∗</sup> have different responses to the tensor operators, where the latter is more sensitive to the tensor interactions. RD and RD<sup>∗</sup> can be explained simultaneously with the tensor couplings. In order to understand the correlation between BR Bð Þ <sup>c</sup> ! τντ and RDð Þ <sup>∗</sup> , we show the contours for BR Bð Þ <sup>c</sup> ! τντ and RDð Þ <sup>∗</sup> as a function of w23~y<sup>33</sup> and mS in Figure 4, where ~y32w<sup>22</sup> ≈~y31w<sup>21</sup> ≈0 are used, and the gray area is excluded by

such as charged scalar boson, vector-like leptons, vector-like quarks, and

Flavor Physics and Charged Particle

DOI: http://dx.doi.org/10.5772/intechopen.81404

Author details

Takaaki Nomura

51

leptoquarks. After showing some properties and interactions of these particles, we reviewed some applications to flavor physics in which lepton flavor physics with vector-like lepton and B-meson physics with leptoquarks are focused on as an illumination. We have seen rich phenomenology that would be induced from such new charged particles, and they will be also tested in the future experiments.

School of Physics, Korea Institute for Advanced Study, Seoul, Republic of Korea

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: nomura@kias.re.kr

provided the original work is properly cited.

#### Figure 4. Contours for BR Bð Þ <sup>c</sup> ! τντ and RDð Þ <sup>∗</sup> as a function of w23~y<sup>23</sup> and mS. (The plot is taken from ref. [32]).

BR B� <sup>c</sup> ! τν � � < 0:3. We can see that the predicted BR Bð Þ <sup>c</sup> ! τντ is much smaller than the experimental bound.

Finally, we make some remarks regarding the constraint due to the LQ search at the LHC. Due to the flavor physics constraints, only the S<sup>1</sup>=<sup>3</sup> Yukawa couplings ~ytτ, ~ybντ , and wc<sup>τ</sup> can be of <sup>O</sup>ð Þ<sup>1</sup> . These couplings affect the <sup>S</sup><sup>1</sup>=<sup>3</sup> decays but also their production. Therefore, in addition to the S<sup>1</sup>=<sup>3</sup> -pair production, based on the Oð Þ1 Yukawa couplings, the single S<sup>1</sup>=<sup>3</sup> production becomes interesting. In the pp collisions, the single <sup>S</sup><sup>1</sup>=<sup>3</sup> production can be generated via the gb ! <sup>S</sup>�1=<sup>3</sup> ντ and gc ! <sup>S</sup>�1=<sup>3</sup> τ<sup>þ</sup> channels. Using CalcHEP 3.6 [51, 52] with the CTEQ6 parton distribution functions [53], their production cross sections with ∣w23∣ � ∣~ybντ <sup>∣</sup> � ffiffi 2 <sup>p</sup> and mLQ <sup>¼</sup> 1000 GeV at ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV can be obtained as 3.9 fb and 2.9 fb, respectively, whereas the S<sup>1</sup>=<sup>3</sup> -pair production cross section is <sup>σ</sup> pp ! <sup>S</sup>�1=<sup>3</sup> <sup>S</sup><sup>1</sup>=<sup>3</sup> � �≈2:4 fb. If we assume that S�1=<sup>3</sup> predominantly decays into tτ, bντ, and cτ with similar BRs, i.e. BR S�1=<sup>3</sup> ! <sup>f</sup> � � � <sup>1</sup>=3, then the single <sup>S</sup><sup>1</sup>=<sup>3</sup> production cross section <sup>σ</sup> <sup>S</sup>�1=<sup>3</sup> X � � times BR S�1=<sup>3</sup> ! <sup>f</sup> � � with <sup>X</sup> and <sup>f</sup> as the possible final states can be estimated as around 1 fb. The LQ coupling w<sup>23</sup> involves different generations, so the constraints due to the collider measurements may not be applied directly. However, if we compare this with the CMS experiment [54] based on a single production of the secondgeneration scalar LQ, we find that the values of σ � BR at mLQ � 1000 GeV are still lower than the CMS upper limit with few fb. The significance of this discovery depends on the kinematic cuts and event selection conditions, but this discussion is beyond the scope of this study, and we leave the detailed analysis for future research.

#### 4. Conclusions

We have reviewed some charged particles which appear from physics beyond the Standard Model of particle physics. Some possible candidates of them are listed

#### Flavor Physics and Charged Particle DOI: http://dx.doi.org/10.5772/intechopen.81404

such as charged scalar boson, vector-like leptons, vector-like quarks, and leptoquarks. After showing some properties and interactions of these particles, we reviewed some applications to flavor physics in which lepton flavor physics with vector-like lepton and B-meson physics with leptoquarks are focused on as an illumination. We have seen rich phenomenology that would be induced from such new charged particles, and they will be also tested in the future experiments.

### Author details

Takaaki Nomura School of Physics, Korea Institute for Advanced Study, Seoul, Republic of Korea

\*Address all correspondence to: nomura@kias.re.kr

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

BR B�

Figure 4.

Charged Particles

~ybντ

gc ! <sup>S</sup>�1=<sup>3</sup>

mLQ <sup>¼</sup> 1000 GeV at ffiffi

whereas the S<sup>1</sup>=<sup>3</sup>

BR S�1=<sup>3</sup> ! <sup>f</sup> � �

BR S�1=<sup>3</sup> ! <sup>f</sup> � �

research.

50

4. Conclusions

than the experimental bound.

production. Therefore, in addition to the S<sup>1</sup>=<sup>3</sup>

<sup>c</sup> ! τν � � < 0:3. We can see that the predicted BR Bð Þ <sup>c</sup> ! τντ is much smaller

Contours for BR Bð Þ <sup>c</sup> ! τντ and RDð Þ <sup>∗</sup> as a function of w23~y<sup>23</sup> and mS. (The plot is taken from ref. [32]).

, and wc<sup>τ</sup> can be of <sup>O</sup>ð Þ<sup>1</sup> . These couplings affect the <sup>S</sup><sup>1</sup>=<sup>3</sup> decays but also their

Yukawa couplings, the single S<sup>1</sup>=<sup>3</sup> production becomes interesting. In the pp colli-


assume that S�1=<sup>3</sup> predominantly decays into tτ, bντ, and cτ with similar BRs, i.e.

fb. The LQ coupling w<sup>23</sup> involves different generations, so the constraints due to the collider measurements may not be applied directly. However, if we compare this with the CMS experiment [54] based on a single production of the second-

generation scalar LQ, we find that the values of σ � BR at mLQ � 1000 GeV are still lower than the CMS upper limit with few fb. The significance of this discovery depends on the kinematic cuts and event selection conditions, but this discussion is beyond the scope of this study, and we leave the detailed analysis for future

We have reviewed some charged particles which appear from physics beyond the Standard Model of particle physics. Some possible candidates of them are listed

τ<sup>þ</sup> channels. Using CalcHEP 3.6 [51, 52] with the CTEQ6 parton distri-

� <sup>1</sup>=3, then the single <sup>S</sup><sup>1</sup>=<sup>3</sup> production cross section <sup>σ</sup> <sup>S</sup>�1=<sup>3</sup>

with X and f as the possible final states can be estimated as around 1

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV can be obtained as 3.9 fb and 2.9 fb, respectively,

<sup>S</sup><sup>1</sup>=<sup>3</sup> � �

sions, the single <sup>S</sup><sup>1</sup>=<sup>3</sup> production can be generated via the gb ! <sup>S</sup>�1=<sup>3</sup>

bution functions [53], their production cross sections with ∣w23∣ � ∣~ybντ

Finally, we make some remarks regarding the constraint due to the LQ search at the LHC. Due to the flavor physics constraints, only the S<sup>1</sup>=<sup>3</sup> Yukawa couplings ~ytτ,


ντ and

<sup>∣</sup> � ffiffi 2 <sup>p</sup> and

≈2:4 fb. If we

X � �

times

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ffiffi

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Section 3

Electron Positron Plasma

57

Section 3
