Phenomenology of Heavy Quark at the LHC

Abstract

In the Standard Model, the Higgs boson is responsible for giving particles mass through a process called the Higgs mechanism. However, the interactions between the Higgs boson and usual quarks are determined by Yukawa couplings, which means that the strength of their interactions depends on their mass. In contrast, vector-like quarks are expected to have a more “vector” coupling with the Higgs boson, which would make their interactions independent of their mass. This property could potentially make them easier to detect experimentally. In this chapter, we develop the theoretical framework of such exotic heavy quarks both in the standard model and beyond it such as the two Higgs doublets model (2HDM). This chapter introduces a Lagrangian which follows the most generic concept of the Standard Model which is consistent with the basics, and then we explore the consequences of identifying the eigenstates and interactions with SM-VLQ. We also provide a detailed description of the full interaction in the 2HDM mass basis. We shortly comment on the production of heavy quark pairs at the LHC.

1. Introduction

The introduction of vector quarks into the realm of particle physics represents a departure from the conventional characteristics of Standard Model (SM) quarks on multiple intriguing fronts. In stark contrast to their SM counterparts, vector quarks typically do not acquire masses through Yukawa couplings with a Higgs doublet, a feature that resonates with the experimental observations of the Higgs boson [1, 2]. This distinct trait opens up new avenues for exploration. What further sets vector quarks apart is their remarkable capacity for mixing with SM quarks. This mixing dynamic has profound implications, altering their couplings with essential bosons like the Z, W, and Higgs bosons [3]. Such mixing represents a straightforward mechanism for breaking the Glashow-Iliopoulos-Maiani mechanism, a tantalizing prospect that holds the potential for ushering in novel phenomena in low-energy physics [4]. However, the allure of vector quarks goes beyond these unique traits. Two primary theoretical motivations underpin their extensive study. Firstly, they play a pivotal role in the intricate dance of electroweak symmetry breaking, offering a promising avenue for explaining the observed light mass of the Higgs boson [5]. This becomes particularly compelling within theoretical frameworks that posit the Higgs as a pseudo Goldstone boson [6].

Secondly, vector-like quarks emerge organically as fermion resonances within theories that delve into the nuanced realm of partial flavor composition within the Standard Model. In both of these theoretical scenarios, substantial mixing between the novel vector quarks and the third family of SM quarks, often dubbed “top partners,” takes center stage [7, 8, 9, 10]. Notably, the presence of vector quarks is not confined to a single theoretical framework but extends its reach into diverse models. These encompass small Higgs and composite Higgs models, as well as their holographic counterparts [11]. Consequently, the study of vector quarks not only enriches our understanding of particle physics but also offers a tantalizing glimpse into the potential extensions and augmentations of the Standard Model, unearthing the profound intricacies that underlie the fundamental forces shaping the universe’s particle interactions.

Simultaneously, it is worth noting that testing quantum entanglement directly in proton-proton (pp) collisions at the Large Hadron Collider (LHC) poses a formidable challenge due to the macroscopic and complex nature of these interactions [12]. Quantum entanglement typically manifests at the microscopic scale, involving the correlation of properties between individual or pairs of particles. Nevertheless, researchers at the LHC can indirectly explore aspects related to quantum entanglement [13, 14, 15]. They may investigate the correlations and angular momentum conservation among particles produced in these high-energy collisions [16]. While not direct evidence of entanglement, these studies shed light on the quantum dynamics governing the particle interactions. Additionally, the search for new, exotic particles beyond the Standard Model is a primary mission of the LHC [17]. These hypothetical particles might exhibit quantum properties, including entanglement, which could be uncovered through their distinctive decay patterns and behaviors. Although the LHC primarily serves as a platform for fundamental particle physics research, it contributes to our broader comprehension of quantum mechanics and its role in the universe’s fundamental fabric [18, 19].

1.1 From weak eigenstates to mass eigenstates and mixing angles

Let us consider a general 2 × 2 Hermitian matrix of the form:

$$\mathcal{M}=\left(\begin{array}{cc}a& b\\ \overline{b}& M\end{array}\right),$$E1

Where a, b, c, and d are complex numbers. In order to diagonalize the above matrix with the aid of the unitary matrix $\mathcal{U},$ we seek a diagonal matrix ${\mathcal{M}}_{\mathrm{diag}}$ such that:

$${\mathcal{U}\mathcal{M}\mathcal{U}}^{†}={\mathcal{M}}_{\mathit{diag}}$$E2

Here ${\mathcal{U}}^{†}$ denotes the conjugate transpose of $\mathcal{U}.$ We adopt the following form for the unitary matrix $\mathcal{U}:$

$$\mathcal{U}=\left(\begin{array}{cc}\cos \theta & -sin\theta e^{\mathit{i\phi }}\\ sin\theta e^{-\mathit{i\phi }}& cos\theta \end{array}\right),$$E3

where θ and φ are real parameters. We determine these parameters by requiring that $\mathcal{U}$ diagonalizes $\mathcal{M}.$ Specifically, we choose θ and φ such that:

$$tan\left(2\theta \right)=\frac{2\mid b\mid }{M-a}\text{and}\phi =arg\left(b\right),$$E4

We proceed to substitute the matrices $\mathcal{U},{\mathcal{U}}^{†},$ and $\mathcal{M}$ into the equation for diagonalization, which we then simplify as follows:

$$\begin{array}{lll}{\mathcal{U}\mathcal{M}\mathcal{U}}^{†}& =& \left(\begin{array}{cc}c& -{\mathit{se}}^{\mathit{i\phi }}\\ {\mathit{se}}^{-\mathit{i\phi }}& c\end{array}\right)\left(\begin{array}{cc}a& b\\ \overline{b}& M\end{array}\right){\mathcal{U}}^{†}\\ & =& \left(\begin{array}{cc}\mathit{ca}-s\mid b\mid & \mathit{cb}-{\mathit{sMe}}^{\mathit{i\phi }}\\ {\mathit{sae}}^{-\mathit{i\phi }}+c\overline{b}& s\mid b\mid +\mathit{cM}\end{array}\right)\left(\begin{array}{cc}c& {\mathit{se}}^{\mathit{i\phi }}\\ -{\mathit{se}}^{-\mathit{i\phi }}& c\end{array}\right)\\ & =& \left(\begin{array}{cc}{c}^{2}a-\mathit{sc}\mid b\mid -\mathit{sc}\mid b\mid +s^{2}M& {\mathit{\text{scae}}}^{\mathit{i\phi }}-s^{2}b+c^{2}b-{\mathit{\text{scMe}}}^{\mathit{i\phi }}\\ {\mathit{\text{scae}}}^{-\mathit{i\phi }}+c^2\overline{b}-s^2\overline{b}-{\mathit{\text{scMe}}}^{-\mathit{i\phi }}& s^{2}a+\mathit{sc}\mid b\mid +\mathit{sc}\mid b\mid +c^{2}M\end{array}\right)\end{array}$$E5

Here, c = cos θ, s = sin θ, and $\overline{b}$ denotes the complex conjugate of b.

To simplify the (1,2) entry, we utilize the fact that $\mathcal{M}$ is Hermitian along with the following relationship derived from Eq. (1):

$$\frac{2\mathit{sc}}{c^2-s^2}=\frac{2\mid b\mid }{M-a}$$E6

This yields the following equations:

$$\begin{array}{lll}\left(1,2\right)& =& \mathit{sc}\left(a-M\right)e^{\mathit{i\phi }}+\left(c^2-s^2\right)b\\ & =& \frac{b}{M-a}\left(a-M\right)\left(c^2-s^2\right)e^{\mathit{i\phi }}+\left(c^2-s^2\right)b\\ & =& 0\end{array}$$E7

$$\left(2,1\right)={\left(1,2\right)}^{\ast }=0$$E8

$$\left(1,1\right)=c^{2}a+s^{2}M-2\mathit{sc}\mid b\mid$$E9

$$\left(2,2\right)=s^{2}a+c^{2}M+2\mathit{sc}\mid b\mid$$E10

We shall invoke the definition of the matrix $\mathcal{M},$ previously established as follows:

$$\mathcal{M}=\left(\begin{array}{cc}{y}_{33}& y_{34}\\ 0& M\end{array}\right)$$E11

Utilizing the aforementioned definition of $\mathcal{M},$ the products ${\mathcal{M}\mathcal{M}}^{†}$ and ${\mathcal{M}}^{†}\mathcal{M}$ can be computed as:

$${\mathcal{M}\mathcal{M}}^{†}=\left(\begin{array}{cc}{\left|y_{33}\right|}^2+{\left|y_{34}\right|}^2& y_{34}M\\ y_{34}M& M^2\end{array}\right)\text{where}{\mathcal{U}}_L=\left(\begin{array}{cc}\cos {\theta }_L& -sin{\theta }_L{e}^{i{\phi }_L}\\ sin{\theta }_L{e}^{-i{\phi }_L}& cos{\theta }_L\end{array}\right)$$E12

$${\mathcal{M}}^{†}\mathcal{M}=\left(\begin{array}{cc}{\left|y_{33}\right|}^2& y_{33}^{\ast }{y}_{34}\\ y_{33}{y}_{34}^{\ast }& M^2+{\left|y_{34}\right|}^2\end{array}\right)\text{where}{\mathcal{U}}_R=\left(\begin{array}{cc}\cos {\theta }_R& -sin{\theta }_R{e}^{i{\phi }_R}\\ sin{\theta }_R{e}^{-i{\phi }_R}& cos{\theta }_R\end{array}\right)$$E13

The mixing angles θ_L and θ_R can be expressed as follows:

$$\left\{\begin{array}{l}tan\left(2{\theta }_L\right)=\frac{2\mid y_{34}\mid M}{M^2-{\left|y_{33}\right|}^2-{\left|y_{34}\right|}^2}\text{where}{\phi }_L=\mathit{arg}\left(y_{34}\right)\\ tan\left(2{\theta }_R\right)=\frac{2\mid y_{33}{y}_{34}\mid }{M^2+{\left|y_{34}\right|}^2-{\left|y_{33}\right|}^2}\text{where}{\phi }_R=\mathit{arg}\left(y_{33}^{\ast }{y}_{34}\right)\end{array}\right.$$E14

Let us define a new matrix $\mathcal{M}$ with a distinct element configuration from the previous one:

$$\mathcal{M}=\left(\begin{array}{cc}{y}_{33}& 0\\ y_{43}& M\end{array}\right)$$E15

Upon calculating the products ${\mathcal{M}}^{\mathcal{M}}$ and ${\mathcal{M}}^{†}\mathcal{M},$ we obtain two matrices as a result:

$${\mathcal{M}\mathcal{M}}^{†}=\left(\begin{array}{cc}{\left|y_{33}\right|}^2& y_{33}{y}_{43}^{\ast }\\ y_{33}^{\ast }{y}_{43}& M^2+{\left|y_{43}\right|}^2\end{array}\right){\mathcal{M}}^{†}\mathcal{M}=\left(\begin{array}{cc}{\left|y_{33}\right|}^2+{\left|y_{43}\right|}^2& y_{43}^{\ast }M\\ y_{43}M& M^2\end{array}\right)$$E16

From the above expressions, we can extract the mixing angles θ_L and θ_R, as well as the phases φ_L and φ_R.

$$\left\{\begin{array}{l}tan\left(2{\theta }_L\right)=\frac{2\mid y_{33}{y}_{43}\mid }{M^2+{\left|y_{43}\right|}^2-{\left|y_{33}\right|}^2}\text{where}{\phi }_L=\mathit{arg}\left(y_{33}{y}_{43}^{\ast }\right)\\ tan\left(2{\theta }_R\right)=\frac{2\mid y_{43}\mid M}{M^2-{\left|y_{33}\right|}^2-{\left|y_{43}\right|}^2}\text{where}{\phi }_R=\mathit{arg}\left(y_{33}^{\ast }\right)\end{array}\right.$$E17

2. Framework of SM + VLQs singlet

In the basis of weak eigenstates (t, T), where t represents the standard model top quark, the matrix describing the masses of the top quark and vector-like quarks (VLQ) [20] can be expressed as follows

$$\mathcal{M}=\left(\begin{array}{cc}{y}_{33}\frac{v}{\sqrt{2}}& y_{34}\frac{v}{\sqrt{2}}\\ 0& M\end{array}\right)$$E18

The aforementioned mass matrix clearly indicates that the physical mass of the heavy top quark, m_T, differs from the mass parameter M due to the mixing of the weak eigenstates t and T. To obtain the physical states (t_L,R, T_L,R) in terms of the gauge eigenstates $\left({\tilde{t}}_{L,R},{\tilde{T}}_{L,R}\right),$ the mass matrix $\mathcal{M}$ can be diagonalized through a bi-unitary transformation, as described in Section 1.

$$\left(\begin{array}{c}{t}_{L,R}\\ T_{L,R}\end{array}\right)={\mathcal{U}}_{L,R}\left(\begin{array}{c}{\tilde{t}}_{L,R}\\ {\tilde{T}}_{L,R}\end{array}\right)$$E19

Where the rotation matrices ${\mathcal{U}}_L$ and ${\mathcal{U}}_R$ corresponding to left-handed and right-handed rotations, respectively, can be defined as follows:

$${\mathcal{U}}_{L,R}=\left(\begin{array}{cc}\cos \left({\theta }_{L,R}\right)& -sin\left({\theta }_{L,R}\right)e^{i{\phi }_{L,R}}\\ sin\left({\theta }_{L,R}\right)e^{-i{\phi }_{L,R}}& cos\left({\theta }_{L,R}\right)\end{array}\right)$$E20

In regards to the mixing angles, it is important to note that θ_L and θ_R are not considered independent parameters. This can be deduced from the bi-unitary transformations, which yield the following relationships:

$$tan2{\theta }_L=\left(\frac{2\mid y_{34}\mid \frac{v}{\sqrt{2}}M}{M^2-{\left|y_{33}\right|}^2\frac{v^2}{2}-\mid y_{34}\mid \frac{v^2}{2}}\right),tan2{\theta }_R=\left(\frac{\mid y_{33}{y}_{34}\mid v^2}{M^2+{\left|y_{34}\right|}^2\frac{v^2}{2}-\mid y_{33}\mid \frac{v^2}{2}}\right)$$E21

Here y₃₃ is assumed to be real, φ_L = arg(y₃₄) and ${\phi }_R=\mathit{arg}\left(y_{33}^{\ast }{y}_{34}\right).$.

Based on the aforementioned relation in Section 1, namely ${\mathcal{U}}_L{\mathcal{M}\mathcal{U}}_R^{†}={\mathcal{M}}_{\mathit{diag}},$ we can introduce the following notations:

$${\overline{u}}_L^0\mathcal{M}{u}_R^0={\overline{u}}_L^0{\mathcal{U}}_L^{†}{\mathcal{M}}_{\mathit{diag}}{\mathcal{U}}_R{u}_R^0\Rightarrow \left\{\begin{array}{l}{u}_{L,R}={\mathcal{U}}_{L,R}{u}_{L,R}^0\\ u_{L,R}^0={\mathcal{U}}_{L,R}^{†}{u}_{L,R}\end{array}\right.$$E22

2.1 Modified gauge boson couplings

The couplings of the W boson to the top quark (t) and its heavy partner (T) with the bottom quark (b) can be expressed in terms of the left-handed up-type quark mixing matrix ${\mathcal{U}}_L,$as follows (Figure 1).

Figure 1.

Typical Feynman diagram of the W boson couplings to the top quark (t) and its heavy partner (T) with the bottom quark (b).

$$\left({\overline{u}}_{L3}^0{\overline{u}}_{L4}^0\right)\left(10\right)b_L=\left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right){\mathcal{U}}_L\left(\begin{array}{l}1\\ 0\end{array}\right)b_L$$E23

where

$${\mathcal{U}}_L\left(\begin{array}{l}1\\ 0\end{array}\right)=\left(\begin{array}{c}{c}_L\\ s_L{e}^{-i{\phi }_L}\end{array}\right)$$E24

Using these expressions, we obtain the following couplings:

$$V_{\mathit{tb}}=c_L,\text{and}{V}_{\mathit{Tb}}=s_L{e}^{-i{\phi }_L}$$E25

The neutral current expression can be employed to describe the neutral couplings of the Z boson to pairs of top (t) and top partner (T) (Figure 2).

Figure 2.

Typical Feynman diagram of the Z boson couplings to pairs of top (t) and top partner (T).

$$\left(\begin{array}{cc}\overline{u}{L_3}^0& \overline{u}{L_4}^0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{l}{u}_{L_3}^0\\ u_{L_4}^0\end{array}\right)=\left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right){\mathcal{U}}_L\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right){\mathcal{U}}_L^{†}\left(\begin{array}{l}{t}_L\\ T_L\end{array}\right)$$E26

The computation of the product yields the following relationships:

$$\left(\begin{array}{cc}\overline{u}{L_3}^0& \overline{u}{L_4}^0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{l}{u}_{L_3}^0\\ u_{L_4}^0\end{array}\right)=\left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right){\mathcal{U}}_L\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right){\mathcal{U}}_L^{†}\left(\begin{array}{l}{t}_L\\ T_L\end{array}\right)$$E27

The aforementioned equations can be employed to define the following couplings:

$$X_{\mathit{tt}}=c_L^2,X_{\mathit{TT}}=s_L^2,X_{\mathit{Tt}}=c_L{s}_L{e}^{-i{\phi }_L}$$E28

2.2 Modified Higgs couplings

The neutral Higgs couplings to top (t) and top partner (T) pairs can be derived from the following equations (Figure 3):

Figure 3.

Typical Feynman diagram of the Higgs boson h couplings to pairs of top (t) and top partner (T).

$$\mathcal{M}=\left(\begin{array}{cc}\frac{y_{33}}{\sqrt{2}}& \frac{y_{34}}{\sqrt{2}}\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}\frac{y_{33}}{\sqrt{2}}& \frac{y_{34}}{\sqrt{2}}\\ 0& \frac{M^0}{v}\end{array}\right)-\left(\begin{array}{cc}0& 0\\ 0& \frac{M^0}{v}\end{array}\right)=\frac{M}{v}-\frac{M^0}{v}\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$$E29

Then we compute:

$$\left(\begin{array}{cc}\overline{u}{L_3}^0& \overline{u}{L_4}^0\end{array}\right)\mathcal{M}\left(\begin{array}{l}{u}_{R_3}^0\\ u_{R_4}^0\end{array}\right)=\left(\begin{array}{cc}\overline{u}{L_3}^0& \overline{u}{L_4}^0\end{array}\right)\frac{M}{v}\left(\begin{array}{l}{u}_{R_3}^0\\ u_{R_4}^0\end{array}\right)-\frac{M^0}{v}\left(\begin{array}{cc}\overline{u}{L_3}^0& \overline{u}{L_4}^0\end{array}\right)\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\left(\begin{array}{l}{u}_{R_3}^0\\ u_{R_4}^0\end{array}\right)$$E30

Further simplifying the above equation, we get:

$$\left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right)\frac{1}{v}\left(\begin{array}{cc}{m}_t& 0\\ 0& M\end{array}\right)\left(\begin{array}{l}{t}_R\\ T_R\end{array}\right)-\frac{M^0}{v}\left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right){\mathcal{U}}_L\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{+}\left(\begin{array}{l}{t}_R\\ T_R\end{array}\right)$$E31

$$\left(\begin{array}{cc}{c}_L& -s_L{e}^{i{\phi }_L}\\ s_L{e}^{-i{\phi }_L}& c_L\end{array}\right)\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{+}=\left(\begin{array}{cc}0& -s_L{e}^{i{\phi }_L}\\ 0& c_L\end{array}\right)\left(\begin{array}{cc}{c}_R& s_R{e}^{i{\phi }_R}\\ -s_R{e}^{-i{\phi }_R}& c_R\end{array}\right)$$E32

$$=\left(\begin{array}{cc}{s}_L{s}_R{e}^{i\left({\phi }_L-{\phi }_R\right)}& -s_L{c}_R{e}^{i{\phi }_L}\\ -c_L{s}_R{e}^{-i{\phi }_R}& c_L{c}_R\end{array}\right)$$E33

Applying the aforementioned transformation yields the following couplings:

$$y_{\mathit{ttH}}=c_L^2,$$E34

$$y_{\mathit{TTH}}=s_L^2,$$E35

$$y_{\mathit{TtH}}^L=\frac{m_t}{m_T}{s}_L{c}_L{e}^{\mathit{i\varphi }},y_{\mathit{TtH}}^R=s_L{c}_L{e}^{\mathit{i\varphi }}.$$E36

2.3 Framework of SM + VLQs doublet

Let us start with the Yukawa Lagrangian:

$$\mathcal{L}=-y_{4j}^u\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)u_{\mathit{Rj}}^0\tilde{\mathrm{\Phi }}-y_{4j}^d\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)d_{\mathit{Rj}}^0\mathrm{\Phi }+\mathit{hc}.$$E37

The matrix describing the masses the vector-like quarks (VLQ) can be expressed as follows:

$${\mathcal{M}}^u=\left(\begin{array}{cc}{y}_{33}^u\frac{v}{\sqrt{2}}& 0\\ y_{43}^u\frac{v}{\sqrt{2}}& {\mathcal{M}}^0\end{array}\right)\equiv \left(\begin{array}{cc}{W}_{33}^u& 0\\ W_{43}^u& {\mathcal{M}}^0\end{array}\right);{\mathcal{U}}_L^u{\mathcal{M}}^u{\mathcal{U}}_R^{u+}={\mathcal{M}}_{\mathit{diag}}^u$$E38

$${\mathcal{M}}^d=\left(\begin{array}{cc}{y}_{33}^d\frac{v}{\sqrt{2}}& 0\\ y_{43}^d\frac{v}{\sqrt{2}}& {\mathcal{M}}^0\end{array}\right)\equiv \left(\begin{array}{cc}{W}_{33}^d& 0\\ W_{43}^d& {\mathcal{M}}^0\end{array}\right);{\mathcal{U}}_L^d{\mathcal{M}}^d{\mathcal{U}}_R^{d+}={\mathcal{M}}_{\mathit{diag}}^d$$E39

The rotation matrices ${\mathcal{U}}_L^q$ and ${\mathcal{U}}_R^q$ corresponding to left-handed and right-handed rotations, respectively, can be defined as follows:

$${\mathcal{U}}_{L,R}^q=\left(\begin{array}{cc}{c}_{L,R}^q& -s_{L,R}^q{e}^{i{\phi }_{L,R}^q}\\ s_{L,R}^q{e}^{-i{\phi }_{L,R}^q}& c_{L,R}^q\end{array}\right)$$E40

$$\left\{\begin{array}{l}{u}_{L,R}={\mathcal{U}}_{L,R}^u{u}_{L,R}^0,u_{L,R}^0={\mathcal{U}}_{L,R}^{u†}{u}_{L,R}\\ d_{L,R}={\mathcal{U}}_{L,R}^d{d}_{L,R}^0,d_{L,R}^0={\mathcal{U}}_{L,R}^{d†}{d}_{L,R}\end{array}\right.$$E41

2.3.1 Charged currents

The Yukawa Lagrangian of charged currents may be expressed as follows:

$$\mathcal{L}=\frac{-g}{\sqrt{2}}\left[{\overline{u}}_{L4}^0{\gamma }^{\mu }{d}_{L4}^0+{\overline{u}}_{R4}^0{\gamma }^{\mu }{d}_{R4}^0+{\overline{u}}_{L3}^0{\gamma }^{\mu }{d}_{L3}^0\right]W_{\mu }^{+}+\mathit{hc}.$$E42

The Left-Handed (LH) terms can be represented as follows:

$$\left(\begin{array}{cc}{\overline{u}}_{L3}^0& {\overline{u}}_{L4}^0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right){\gamma }^{\mu }\left(\begin{array}{l}{d}_{L3}^0\\ d_{L4}^0\end{array}\right)=\left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right){\mathcal{U}}_L^u\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right){\mathcal{U}}_L^{d+}{\gamma }^{\mu }\left(\begin{array}{l}{b}_L\\ B_L\end{array}\right)$$E43

$$\left(\begin{array}{cc}{c}_L^u& -s_L^u{e}^{i{\phi }_L^u}\\ s_L^u{e}^{-i{\phi }_L^u}& c_L^u\end{array}\right)\left(\begin{array}{cc}{c}_L^d& c_L^d{e}^{i{\phi }_L^d}\\ -s_L^d{e}^{-i{\phi }_L^d}& c_L^d\end{array}\right)=\left(\begin{array}{cc}{c}_L^u{c}_L^d+s_L^u{s}_L^d{e}^{i\left({\phi }_L^u-{\phi }_L^d\right)}& c_L^u{s}_L^d{e}^{i{\phi }_L^d}-s_L^u{c}_L^d{e}^{i{\phi }_L^u}\\ s_L^u{c}_L^d{e}^{-i{\phi }_L^u}-c_L^u{s}_L^d{e}^{-i{\phi }_L^d}& s_L^u{s}_L^d{e}^{i\left({\phi }_L^d-{\phi }_L^u\right)}+c_L^u{c}_L^d\end{array}\right)$$E44

Applying the aforementioned transformation yields the following couplings:

$$y_{{\mathit{tbW}}^{+}}^L=c_L^u{c}_L^d+s_L^u{s}_L^d{e}^{i\left({\phi }_L^u-{\phi }_L^d\right)},$$E45

$$y_{{\mathit{TbW}}^{+}}^L=s_L^u{c}_L^d{e}^{-i{\phi }_L^u}-c_L^u{s}_L^d{e}^{-i{\phi }_L^d},$$E46

$$y_{{\mathit{tBW}}^{+}}^L=c_L^u{s}_L^d{e}^{i{\phi }_L^d}-s_L^u{c}_L^d{e}^{i{\phi }_L^u},$$E47

$$y_{{\mathit{TBW}}^{+}}^L=c_L^u{c}_L^d+s_L^u{s}_L^d{e}^{-i\left({\phi }_L^u-{\phi }_L^d\right)}.$$E48

The Right-Handed (RH) terms can be represented as follows:

$$\left(\begin{array}{cc}{\overline{u}}_{R3}^0& {\overline{u}}_{R4}^0\end{array}\right)\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\gamma }^{\mu }\left(\begin{array}{l}{d}_{R3}^0\\ d_{R4}^0\end{array}\right)=\left(\begin{array}{cc}{\overline{t}}_R& {\overline{T}}_R\end{array}\right){\mathcal{U}}_R^u\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{d+}{\gamma }^{\mu }\left(\begin{array}{l}{b}_L\\ B_L\end{array}\right)$$E49

$$\begin{array}{lll}\left(\begin{array}{cc}{c}_R^u& -s_R^u{e}^{i{\phi }_R^u}\\ s_R^u{e}^{-i{\phi }_R^u}& c_R^u\end{array}\right)\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{d+}& =& \left(\begin{array}{cc}0& -s_R^u{e}^{i{\phi }_R^u}\\ 0& c_R^u\end{array}\right)\left(\begin{array}{cc}{c}_R^d& s_R^d{e}^{i{\phi }_R^d}\\ -s_R^d{e}^{-i{\phi }_R^d}& c_R^d\end{array}\right)\\ & =& \left(\begin{array}{cc}{s}_R^u{s}_R^d{e}^{i\left({\phi }_R^u-{\phi }_R^d\right)}& -s_R^u{c}_R^d{e}^{i{\phi }_R^u}\\ -c_R^u{s}_R^d{e}^{-i{\phi }_R^d}& c_R^u{c}_R^d\end{array}\right)\end{array}$$E50

Upon performing the calculations, the aforementioned transformation yields the expression for the following couplings:

$$y_{{\mathit{tbW}}^{+}}^R=s_R^u{s}_R^d{e}^i\left({\phi }_R^u-{\phi }_R^d\right),$$E51

$$y_{{\mathit{TbW}}^{+}}^R=-c_R^u{s}_R^d{e}^{-i{\phi }_R^d},$$E52

$$y_{{\mathit{tBW}}^{+}}^R=-s_R^u{c}_R^d{e}^{i{\phi }_R^u},$$E53

$$y_{{\mathit{TBW}}^{+}}^R=c_R^u{c}_R^d.$$E54

2.3.2 Neutral currents

The Yukawa Lagrangian of neutral currents may be expressed as follows:

$$\mathcal{L}=\frac{-g}{2c_W}\left[{\overline{u}}_{\mathit{Li}}^0{\gamma }^{\mu }{u}_{\mathit{Li}}^0-{\overline{d}}_{\mathit{Li}}^0{\gamma }^{\mu }{d}_{\mathit{Li}}^0+{\overline{u}}_{R4}^0{\gamma }^{\mu }{u}_{R4}^0-{\overline{d}}_{R4}^0{\gamma }^{\mu }{d}_{R4}^0\right]z_{\mu }+J_{\mathit{EM}}^{\mu },\text{with}i=3,4$$E55

The Left-Handed (LH) terms can be represented as follows:

$$\begin{array}{lll}\left(\begin{array}{cc}{\overline{u}}_{L3}^0& {\overline{u}}_{L4}^0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right){\gamma }^{\mu }\left(\begin{array}{l}{u}_{L3}^0\\ u_{L4}^0\end{array}\right)& =& \left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right){\mathcal{U}}_L^u\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right){\mathcal{U}}_L^u{\gamma }^{\mu }\left(\begin{array}{l}{t}_L\\ T_L\end{array}\right)\\ & =& \left(\begin{array}{cc}{\overline{t}}_L& {\overline{T}}_L\end{array}\right){\gamma }^{\mu }\left(\begin{array}{l}{t}_L\\ T_L\end{array}\right)\mathit{etc}\end{array}$$E56

Applying the aforementioned transformation yields the following couplings:

$$y_{\mathit{ttZ}}^L=y_{\mathit{bbZ}}^R=1,$$E57

$$y_{\mathit{TTZ}}^L=y_{\mathit{TTZ}}^R=1$$E58

$$y_{\mathit{tTZ}}^L=y_{\mathit{bBZ}}^L=0.$$E59

The Right-Handed (RH) terms can be represented as follows:

$$\begin{array}{lll}{\mathcal{U}}_R^u\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{u†}& =& \left(\begin{array}{cc}{c}_R^u& -s_R{e}^{i{\phi }^u}\\ s_R^u{e}^{-i{\phi }^u}& c_R^u\end{array}\right)\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)u_R^{u†}\\ & =& \left(\begin{array}{cc}0& -s_R^u{e}^{i{\phi }^u}\\ 0& c_R^u\end{array}\right)\left(\begin{array}{cc}{c}_R^u& s_R^u{e}^{i{\phi }^u}\\ -s_R^u{e}^{-i{\phi }^u}& c_R^u\end{array}\right)\\ & =& \left(\begin{array}{cc}{s}_R^{u^2}& -s_R^u{c}_R^u{e}^{i{\phi }^u}\\ -s_R^u{c}_R^u{e}^{-i{\phi }^u}& c_R^{u^2}\end{array}\right)\equiv X^u\end{array}$$E60

After performing the requisite calculations, the aforementioned transformation leads to the expression of the following couplings:

$$y_{\mathit{ttZ}}^R=s_R^{u^2}$$E61

$$y_{\mathit{TTZ}}^R=c_R^{u^2}$$E62

$$y_{\mathit{tTZ}}^R=-s_R^u{c}_R^u{e}^{i{\phi }_R^u}$$E63

$$y_{\mathit{bbZ}}^R=s_R^{d^2},$$E64

$$y_{\mathit{BBZ}}^R=c_R^{d^2},$$E65

$$y_{\mathit{bBZ}}^R=-s_R^d{c}_R^d{e}^{i{\phi }_R^d}.$$E66

2.3.3 Modified Higgs couplings

$$\mathcal{L}=-\left(\begin{array}{cc}{\overline{u}}_{L3}^0& {\overline{u}}_{L4}^0\end{array}\right)\left(\begin{array}{cc}{y}_{33}^u& 0\\ y_{34}^u& 0\end{array}\right)\frac{H}{\sqrt{2}}+\left(u\to d\right)+\mathit{hc}$$E67

$$\left(\begin{array}{cc}\frac{y_{33}^u}{\sqrt{2}}& 0\\ \frac{y_{43}^u}{\sqrt{2}}& 0\end{array}\right)=\frac{1}{v}\left(\begin{array}{cc}\frac{y_{33}^u}{\sqrt{2}}& 0\\ \frac{y_{43}^u}{\sqrt{2}}& M^0\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)=\frac{1}{v}{\mathcal{M}}^u\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$$E68

$$\begin{array}{lll}{\mathcal{U}}_L^u\frac{1}{v}{\mathcal{M}}^u\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right){\mathcal{U}}_R^{u+}& =& \frac{1}{v}{\mathcal{U}}_L^u{\mathcal{M}}^u{\mathcal{U}}_R^{u+}{\mathcal{U}}_R^u\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right){\mathcal{U}}_R^{u+}\\ & =& \frac{1}{v}{\mathcal{M}}_{\mathit{diag}}^u{\mathcal{U}}_R^u\left[1-\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\right]{\mathcal{U}}_R^{u+}\\ & =& \frac{1}{v}{\mathcal{M}}_{\mathit{diag}}^u\left(1-X^u\right)\end{array}$$E69

Upon completing the necessary calculations, the resulting transformation yields the following couplings:

$$y_{\mathit{ttH}}^R=c_R^{u^2}$$E70

$$y_{\mathit{TTH}}^R=s_R^{u^2}$$E71

$$y_{\mathit{tTH}}^L=s_R^u{c}_R^u{e}^{i{\varphi }_u},y_{\mathit{tTH}}^R=\frac{m_t}{m_T}\left(s_R^u{c}_R^u{e}^{i{\varphi }_u}\right)$$E72

$$y_{\mathit{bbH}}^R=c_R^{d^2},$$E73

$$y_{\mathit{BBH}}^R=s_R^{d^2},$$E74

$$y_{\mathit{bBH}}^R=s_R^d{c}_R^d{\mathit{ue}}^{i{\varphi }_d},y_{\mathit{bBH}}^R=\frac{m_t}{m_T}\left(s_R^d{c}_R^d{e}^{i{\varphi }_d}\right).$$E75

In Figure 4, we depict the branching ratios $\left(\mathcal{BR}s\right)$ of three decay channels, namely T → W⁺b, tZ, and th, as a function of the top quark partner mass m_T, at a fixed s_L for SM + T representation (left) and at a fixed $s_R^u$ and $s_R^d$ for SM + TB representation (right). It can be observed from both panels that the T → W⁺b decay mode exhibits dominance over other modes throughout the entire mass spectrum of T, ranging from 300 to 3000 GeV.

Figure 4.

BR(T → XY) as a function of m_T for SM + T (left) and SM + TB (right).

3. 2HDM + VLQs singlet: concepts and formulations

The most general renormalizable potential, of 2HDM + VLQ [21, 22], which is invariant under $S\mathcal{U}\left(2\right)\otimes \mathcal{U}\left(1\right)$ can be written as

$$\begin{array}{lll}V\left({\varphi }_1,{\varphi }_2\right)& =& m_1^2{\varphi }_1^{†}{\varphi }_1+m_2^2{\varphi }_2^{†}{\varphi }_2+\left(m_{12}^2{\varphi }_1^{†}{\varphi }_2+h.c\right)+{\lambda }_1{\left({\varphi }_1^{†}{\varphi }_1\right)}^2+{\lambda }_2{\left({\varphi }_2^{†}{\varphi }_2\right)}^2\\ & & +{\lambda }_3\left({\varphi }_1^{†}{\varphi }_1\right)\left({\varphi }_2^{†}{\varphi }_2\right)+{\lambda }_4\left({\varphi }_1^{†}{\varphi }_2\right)\left({\varphi }_2^{†}{\varphi }_1\right)+\left[{\lambda }_5{\left({\varphi }_1^{†}{\varphi }_2\right)}^2+h.c\right]\\ & & +{\varphi }_1^{†}{\varphi }_1\left[{\lambda }_6\left({\varphi }_1^{†}{\varphi }_2\right)+h.c\right]+{\varphi }_2^{†}{\varphi }_2\left[{\lambda }_7\left({\varphi }_1^{†}{\varphi }_2\right)+h.c\right]\end{array}$$E76

where ϕ_i, i = 1, 2 are complex $S\mathcal{U}\left(2\right)$ doublets

$${\varphi }_1=\left(\begin{array}{c}{G}^{+}\\ \frac{v+{\phi }_1^0+{\mathit{iG}}^0}{\sqrt{2}}\end{array}\right),{\varphi }_2=\left(\begin{array}{c}{H}^{+}\\ \frac{{\phi }_2^0+\mathit{iA}}{\sqrt{2}}\end{array}\right)$$E77

The physical CP-even scalars h and H are mixtures of ${\phi }_{1,2}^0:$

$$\left(\begin{array}{l}h\\ H\end{array}\right)=\left(\begin{array}{cc}{s}_{\mathit{\beta \alpha }}& s_{\mathit{\beta \alpha }}\\ c_{\mathit{\beta \alpha }}& -s_{\mathit{\beta \alpha }}\end{array}\right)\left(\begin{array}{l}{\phi }_1^0\\ {\phi }_2^0\end{array}\right)$$E78

where: s_βα = sin(β – α), c_βα = cos(β – α).

we write:

$$\widetilde{{\varphi }_1}=\left(\begin{array}{c}\frac{v+{\phi }_1^0-{\mathit{iG}}^0}{\sqrt{2}}\\ -G^{-}\end{array}\right),\widetilde{{\varphi }_2}=\left(\begin{array}{c}\frac{{\phi }_2^0-\mathit{iA}}{\sqrt{2}}\\ -H^{-}\end{array}\right)$$E79

where $\widetilde{{\varphi }_i}=i{\tau }_2{\varphi }_i^{\ast }\left(\mathrm{i}=1,2\right),$.

we add the Lagrangian of Yukawa interaction for the heavy Quark $T_{L,R}^0$ and $B_{L,R}^0:$

$$\begin{array}{ll}-& {\mathcal{L}}_Y^{\mathit{{\rm I}{\rm I}}}\supset y_t\overline{Q_L^0}\widetilde{{\varphi }_2}{T}_R^0+{\lambda }_t\overline{Q_L^0}\widetilde{{\varphi }_1}{T}_R^0+M_T\overline{T_L^0}{T}_R^0+y_b\overline{Q_L^0}\left({\sigma }_1\widetilde{{\varphi }_2}\right)B_R^0\\ & +{\lambda }_b\overline{Q_L^0}\left({\sigma }_1\widetilde{{\varphi }_1}\right)B_R^0+M_B\overline{B_L^0}{B}_R^0+\mathit{Hc}\end{array}$$E80

where $\overline{Q_L^0}=\left(\overline{t_L^0},\overline{b_L^0}\right),$ and ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.

3.1 Neutral Higgs couplings

• For the Light-Light couplings of Top quarks to the triplets Higgs h, H, and A (Figure 5)

  $$y_{\mathit{htt}}=\left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)c_L^2$$E81
  $$y_{\mathit{Htt}}=\left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)c_L^2$$E82
  $$y_{\mathit{Att}}=-cot\beta c_L^2$$E83

• For the heavy-heavy couplings of Top quarks to the triplets Higgs h, H, and A

  $$y_{\mathit{hTT}}=\left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)s_L^2$$E84
  $$y_{\mathit{HTT}}=\left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)s_L^2$$E85
  $$y_{\mathit{ATT}}=-cot\beta s_L^2$$E86

• For the Light-heavy left and right couplings of Top quarks to the triplets Higgs h, H, and A

$$\begin{array}{lll}{y}_{\mathit{htT}}^L& =& \left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)\frac{m_t}{m_T}{c}_L{s}_L{e}^{\mathit{i\varphi }}{y}_{\mathit{htT}}^R=\left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)c_L{s}_L{e}^{\mathit{i\varphi }}\\ y_{\mathit{HtT}}^L& =& \left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)\frac{m_t}{m_T}{c}_L{s}_L{e}^{\mathit{i\varphi }}{y}_{\mathit{HtT}}^R=\left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)c_L{s}_L{e}^{\mathit{i\varphi }}\\ y_{\mathit{AtT}}^L& =& -cot\beta \frac{m_t}{m_T}{c}_L{s}_L{e}^{\mathit{i\varphi }}{y}_{\mathit{AtT}}^R=-cot\beta c_L{s}_L{e}^{\mathit{i\varphi }}\end{array}$$E87

Figure 5.

Typical Feynman diagram of the Higgs bosons h, H and a couplings to pairs of top (t) and top partner (T).

3.2 Charged Higgs couplings

The couplings of the charged Higgs bosonH^± to the top quark (t) and its heavy partner (T) with the bottom quark (b) can be expressed as follows (Figure 6):

Figure 6.

Typical Feynman diagram of the charged Higgs boson H⁺ couplings to the top quark (t) and its heavy partner (T) with the bottom quark (b).

$$y_{{\mathit{tbH}}^{+}}^L=c_L,y_{{\mathit{tbH}}^{+}}^R=1$$E88

$$y_{{\mathit{TbH}}^{+}}^L=s_L,y_{{\mathit{tbH}}^{+}}^R=0$$E89

4. 2HDM + VLQs doublet case

4.1 Neutral Higgs couplings

Following the methodology outlined in the preceding section, the Higgs coupling can be determined within the framework of the 2HDM + TB [23, 24] as follows:

• For the Light-Light couplings of Top quarks to the triplets Higgs h, H, and A

  $$y_{\mathit{htt}}=\left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)c_R^{u^2}$$E90
  $$y_{\mathit{Htt}}=\left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)c_R^{u^2}$$E91
  $$y_{\mathit{Att}}=-cot\beta c_R^{u^2}$$E92

• For the heavy-heavy couplings of Top quarks to the triplets Higgs h, H, and A

  $$y_{\mathit{hTT}}=\left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)s_R^{u^2}$$E93
  $$y_{\mathit{HTT}}=\left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)s_R^{u^2}$$E94
  $$y_{\mathit{ATT}}=-cot\beta s_R^{u^2}$$E95

• For the Light-heavy left and right couplings of Top quarks to the triplets Higgs h, H, and A

$$\begin{array}{lll}{y}_{\mathit{htT}}^L& =& \left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)s_R^u{c}_R^u{e}^{i{\varphi }_u}{y}_{\mathit{htT}}^R=\left(s_{\mathit{\beta \alpha }}+c_{\mathit{\beta \alpha }}cot\beta \right)\frac{m_t}{m_T}{s}_R^u{c}_R^u{e}^{i{\varphi }_u}\\ y_{\mathit{HtT}}^L& =& \left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)s_R^u{c}_R^u{e}^{i{\varphi }_u}{y}_{\mathit{HtT}}^R=\left(c_{\mathit{\beta \alpha }}-s_{\mathit{\beta \alpha }}cot\beta \right)\frac{m_t}{m_T}{s}_R^u{c}_R^u{e}^{i{\varphi }_u}\\ y_{\mathit{AtT}}^L& =& -cot\beta s_R^u{c}_R^u{e}^{i{\varphi }_u}{y}_{\mathit{AtT}}^R=-cot\beta \frac{m_t}{m_T}{s}_R^u{c}_R^u{e}^{i{\varphi }_u}\end{array}$$E96

4.2 Charged Higgs couplings

Let us start with the Yukawa Lagrangian (Figure 7):

Figure 7.

Typical Feynman diagram of the charged Higgs boson H⁺ couplings to the top quark (t) and its heavy partner (T) with the bottom quark (b) and its heavy partner (B).

$$\begin{array}{lll}\mathcal{L}& =& -y_{\mathit{ij}}^{u\ast }{\overline{q}}_{\mathit{Li}}{u}_{\mathit{Rj}}\mathrm{ϵ}{H}_u^{\ast }+y_{\mathit{ij}}^{d\ast }{\overline{q}}_{\mathit{Li}}{d}_{\mathit{Rj}}\mathrm{ϵ}{H}_d^{\ast }-y_{4j}^{u\ast }{\overline{Q}}_L{u}_{\mathit{Rj}}\mathrm{ϵ}{H}_u^{\ast }+y_{4j}^{d\ast }{\overline{Q}}_L{d}_{\mathit{Rj}}\mathrm{ϵ}{H}_d^{\ast }+h.c.\\ & =& -y_{\mathit{ij}}^{u\ast }\left(\begin{array}{cc}{\overline{u}}_{\mathit{Li}}^0& {\overline{d}}_{\mathit{Li}}^0\end{array}\right)\left(\begin{array}{l}{H}_u^{0\ast }\\ -H_u^{+}\end{array}\right)u_{\mathit{Rj}}^0+y_{\mathit{ij}}^{d\ast }\left(\begin{array}{cc}{\overline{u}}_{\mathit{Li}}^0& {\overline{d}}_{\mathit{Li}}^0\end{array}\right)\left(\begin{array}{l}{H}_d^{-}\\ -H_d^{0\ast }\end{array}\right)u_{\mathit{Rj}}^0\\ & & -y_{4j}^{u\ast }\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)\left(\begin{array}{l}{H}_u^{0\ast }\\ -H_u^{+}\end{array}\right)u_{\mathit{Rj}}^0+y_{4j}^{d\ast }\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)\left(\begin{array}{l}{H}_d^{-}\\ -H_d^{0\ast }\end{array}\right)u_{\mathit{Rj}}^0\\ & =& -y_{\mathit{ij}}^{u\ast }\left(\begin{array}{cc}{\overline{u}}_{\mathit{Li}}^0& {\overline{d}}_{\mathit{Li}}^0\end{array}\right)\left(\begin{array}{c}\frac{1}{\sqrt{2}}\left[cos\alpha h^0+sin\alpha H_1^0-i\left(sin\beta G^0+cos\beta P_1^0\right)\right]\\ -\left(sin\beta G^{+}+cos\beta H^{+}\right)\end{array}\right)u_{\mathit{Rj}}^0\\ & & +y_{\mathit{ij}}^{d\ast }\left(\begin{array}{cc}{\overline{u}}_{\mathit{Li}}^0& {\overline{d}}_{\mathit{Li}}^0\end{array}\right)\left(\begin{array}{c}-\left(cos\beta G^{-}+sin\beta H^{-}\right)\\ -\frac{1}{\sqrt{2}}\left[-sin\alpha h^0+cos\alpha H_1^0+i\left(cos\beta G^0-sin\beta P_1^0\right)\right]\end{array}\right)u_{\mathit{Rj}}^0\\ & & -y_{4j}^{u\ast }\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)\left(\begin{array}{c}\frac{1}{\sqrt{2}}\left[cos\alpha h^0+sin\alpha H_1^0-i\left(sin\beta G^0+cos\beta P_1^0\right)\right]\\ -\left(sin\beta G^{+}+cos\beta H^{+}\right)\end{array}\right)u_{\mathit{Rj}}^0\\ & & +y_{4j}^{d\ast }\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)\left(\begin{array}{c}-\left(cos\beta G^{-}+sin\beta H^{-}\right)\\ -\frac{1}{\sqrt{2}}\left[-sin\alpha h^0+cos\alpha H_1^0+i\left(cos\beta G^0-sin\beta P_1^0\right)\right]\end{array}\right)u_{\mathit{Rj}}^0\end{array}$$E97

If we select only the H^± particle, the result is:

$$\begin{array}{lll}\mathcal{L}& \supset & y_{\mathit{ij}}^{u\ast }\left(\begin{array}{cc}{\overline{u}}_{\mathit{Li}}^0& {\overline{d}}_{\mathit{Li}}^0\end{array}\right)\left(\begin{array}{c}0\\ \cos \beta H^{+}\end{array}\right)u_{\mathit{Rj}}^0\\ & & +y_{\mathit{ij}}^{d\ast }\left(\begin{array}{cc}{\overline{u}}_{\mathit{Li}}^0& {\overline{d}}_{\mathit{Li}}^0\end{array}\right)\left(\begin{array}{c}\sin \beta H^{-}\\ 0\end{array}\right)u_{\mathit{Rj}}^0\\ & & +y_{4j}^{u\ast }\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)\left(\begin{array}{c}0\\ \cos \beta H^{+}\end{array}\right)u_{\mathit{Rj}}^0\\ & & +y_{4j}^{d\ast }\left(\begin{array}{cc}{\overline{u}}_{L4}^0& {\overline{d}}_{L4}^0\end{array}\right)\left(\begin{array}{c}\sin \beta H^{-}\\ 0\end{array}\right)u_{\mathit{Rj}}^0\\ & \supset & y_{\mathit{ij}}^{u\ast }cos\beta H^{\pm }{\overline{d}}_{\mathit{Li}}^0{u}_{\mathit{Rj}}^0+y_{\mathit{ij}}^{d\ast }sin\beta H^{\pm }{\overline{u}}_{\mathit{Li}}^0{d}_{\mathit{Rj}}^0\\ & & +y_{4j}^{u\ast }cos\beta H^{\pm }{\overline{d}}_{L4}^0{u}_{\mathit{Rj}}^0+y_{4j}^{d\ast }sin\beta H^{\pm }{\overline{u}}_{L4}^0{d}_{\mathit{Rj}}^0\\ & \supset & y_{\mathit{ij}}^{u\ast }cos\beta H^{\pm }{\overline{d}}_{\mathit{Li}}^0{u}_{\mathit{Rj}}^0+y_{4j}^{u\ast }cos\beta H^{\pm }{\overline{d}}_{L4}^0{u}_{\mathit{Rj}}^0\\ & & +y_{\mathit{ij}}^{d\ast }sin\beta H^{\pm }{\overline{u}}_{\mathit{Li}}^0{d}_{\mathit{Rj}}^0+y_{4j}^{d\ast }sin\beta H^{\pm }{\overline{u}}_{L4}^0{d}_{\mathit{Rj}}^0\end{array}$$E98

Subsequently, we can express the Yukawa Lagrangian of the charged Higgs boson as follows:

$$\mathcal{L}=\left(\begin{array}{cc}{{\overline{u}}^0}_{L3}& {{\overline{u}}^0}_{L4}\end{array}\right)\left(\begin{array}{cc}{y}_{33}^d& 0\\ y_{43}^d& 0\end{array}\right)\left(\begin{array}{l}{d}_{R3}^0\\ d_{R4}^0\end{array}\right)sin\beta H^{\pm }+\left(\begin{array}{cc}{{\overline{d}}^0}_{L3}& {{\overline{d}}^0}_{L4}\end{array}\right)\left(\begin{array}{cc}{y}_{33}^u& 0\\ y_{43}^u& 0\end{array}\right)\left(\begin{array}{l}{u}_{R3}^0\\ u_{R4}^0\end{array}\right)cos\beta H^{\pm }$$E99

$$\left(\begin{array}{cc}{y}_{33}^d& 0\\ y_{43}^d& 0\end{array}\right)=\frac{\sqrt{2}}{v}\left[\left(\begin{array}{cc}\frac{y_{33}^{d}v}{\sqrt{2}}& 0\\ \frac{y_{43}^{d}v}{\sqrt{2}}& \frac{y_{44}^{d}v}{\sqrt{2}}\end{array}\right)-\left(\begin{array}{cc}0& 0\\ 0& M^0\end{array}\right)\right]=\frac{\sqrt{2}}{v}{\mathcal{M}}^d-\frac{M^0\sqrt{2}}{v}\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$$E100

$$\begin{array}{lll}{\mathcal{U}}_L^u\left[\frac{\sqrt{2}}{v}{\mathcal{M}}^d-\frac{M^0\sqrt{2}}{v}\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\right]{\mathcal{U}}_R^{d†}& =& \frac{\sqrt{2}}{v}{\mathcal{U}}_L^u\left[{\mathcal{U}}_L^{d†}{\mathcal{U}}_L^d{\mathcal{M}}^d{\mathcal{U}}_R^{d†}{\mathcal{U}}_R^d-M^0\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\right]{\mathcal{U}}_R^{d†}\\ & =& \frac{\sqrt{2}}{v}{\mathcal{U}}_L^u{\mathcal{M}}_{\mathit{diag}}^d\underset{=1}{\underbrace{{\mathcal{U}}_R^d{\mathcal{U}}_R^{d†}}}-M^0\frac{\sqrt{2}}{v}{\mathcal{U}}_L^u\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{d†}\\ & =& \frac{\sqrt{2}}{v}\left(A\times {\mathcal{M}}_{\mathit{diag}}^d-m_{B}B\right)\end{array}$$E101

with

$$\begin{array}{lll}A& =& {\mathcal{U}}_L^u{\mathcal{U}}_L^{d†}\\ & =& \left(\begin{array}{cc}{c}_L^u& -s_L^u{e}^{i{\varphi }_u}\\ s_L^u{e}^{-i{\varphi }_u}& c_L^u\end{array}\right)\left(\begin{array}{cc}{c}_L^d& s_L^d{e}^{i{\varphi }_d}\\ -s_L^d{e}^{-i{\varphi }_d}& c_L^d\end{array}\right)\\ & =& \left(\begin{array}{cc}{c}_L^u{c}_L^d+s_L^u{s}_L^d{e}^{i\left({\varphi }_u-{\varphi }_d\right)}& c_L^u{s}_L^d{e}^{i{\varphi }_d}-c_L^d{s}_L^u{e}^{i{\varphi }_u}\\ c^d{s}_L^u{e}^{-i{\varphi }_u}-c_L^u{s}_L^d{e}^{-i{\varphi }_d}& s_L^u{s}_L^d{e}^{i\left({\varphi }_d-{\varphi }_u\right)}+c_L^u{c}_L^d\end{array}\right)\end{array}$$E102

$$\begin{array}{lll}B& =& {\mathcal{U}}_L^u\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{d†}\\ & =& \left(\begin{array}{cc}{c}_L^u& -s_L^u{e}^{i{\varphi }_u}\\ -s_R^d{e}^{-i{\varphi }_u}& c_L^u\end{array}\right)\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right){\mathcal{U}}_R^{d†}\\ & =& \left(\begin{array}{cc}0& -s_L^u{e}^{i{\varphi }_u}\\ 0& c_L^u\end{array}\right)\left(\begin{array}{cc}{c}_R^d& s_R^d{e}^{i{\varphi }_d}\\ -s_R^d{e}^{i{\varphi }_d}& c_R^d\end{array}\right)\\ & =& \left(\begin{array}{cc}{s}_L^u{s}_R^d{e}^{i\left({\varphi }_u-{\varphi }_d\right)}& -c_R^d{s}_L^u{e}^{i{\varphi }_u}\\ -c_L^u{s}_R^d{e}^{-i{\varphi }_d}& c_L^u{c}_R^d\end{array}\right)\end{array}$$E103

Now we can define the Matrix M^u as follows:

$$\begin{array}{l}{M}^u=A\times {\mathcal{M}}_{\mathit{diag}}^d-m_{B}B\\ =\left(\begin{array}{cc}\left[c_L^u{c}_L^d+s_L^u{s}_L^d{e}^{i\left({\varphi }_u-{\varphi }_d\right)}\right]m_b-{\mathit{mBs}}_L^u{s}_R^d{e}^{i\left({\varphi }_u-{\varphi }_d\right)}& \left[c_L^u{s}_L^d{e}^{i{\varphi }_d}-c_L^d{s}_L^u{e}^{i{\varphi }_u}\right]m_B+m_B{c}_R^d{s}_L^u{e}^{i{\varphi }_u}\\ \left[c_L^d{s}_L^u{e}^{-}i{\varphi }_u-c_L^u{s}_L^d{e}^{-i{\varphi }_d}\right]m_b+m_B{c}_L^u{s}_R^d{e}^{-i{\varphi }_d}& \left[s_L^u{s}_L^d{e}^{i\left({\varphi }_d-{\varphi }_u\right)}+c_L^u{c}_L^d\right]m_B-m_B{c}_L^u{c}_R^d\end{array}\right)\end{array}$$E104

Similarly, the matrix M^d can be defined in the same manner as M^u.

$$M^d=\left(\begin{array}{cc}\left[c_L^d{c}_L^u+s_L^d{s}_L^u{e}^{i\left({\varphi }_d-{\varphi }_u\right)}\right]m_t-{\mathit{mTs}}_L^d{s}_R^u{e}^{i\left({\varphi }_d-{\varphi }_u\right)}& \left[c_L^d{s}_L^u{e}^{i{\varphi }_u}-c_L^u{s}_L^d{e}^{i{\varphi }_d}\right]m_T+m_T{c}_R^u{s}_L^d{e}^{i{\varphi }_d}\\ \left[c_L^u{s}_L^d{e}^{-i{\varphi }_d}-c_L^d{s}_L^u{e}^{-i{\varphi }_u}\right]m_t+m_T{c}_L^d{s}_R^u{e}^{-i{\varphi }_u}& \left[s_L^d{s}^u{d}_L{e}^{i\left({\varphi }_u-{\varphi }_d\right)}+c_L^d{c}^u\right]m_T-m_T{c}_L^d{c}_R^u\end{array}\right)$$E105

Finally, the charged Higgs couplings can be defined as follows:

• For the coupling H^±tb:

  $$y_{H^{+}\mathit{tb}}^L=c_L^d{c}_L^u+\frac{s_L^d}{s_L^u}\left(s_L^{u^2}-s_R^{u^2}\right)e^{i\left({\varphi }_u-{\varphi }_d\right)}$$E106
  $$y_{H^{+}\mathit{tb}}^R=\frac{m_b}{m_t}\left[c_L^u{c}_L^d+\frac{s_L^u}{s_L^d}\left(s_L^{d^2}-s_R^{d^2}\right)e^{i\left({\varphi }_u-{\varphi }_d\right)}\right]$$E107

• For the coupling H⁺Tb:

  $$y_{H^{+}\mathit{Tb}}^L=c_L^d{c}_L^u+\frac{s_L^d}{s_L^u}\left(s_L^{u^2}-s_R^{u^2}\right)e^{-i{\varphi }_d\ast }$$E108
  $$y_{H^{+}\mathit{Tb}}^R=\frac{m_b}{m_T}\left[c_L^u{c}_L^d+\frac{s_L^u}{s_L^d}\left(s_L^{d^2}-s_R^{d^2}\right)e^{-i{\varphi }_d}\right]$$E109

• For the coupling H⁺tB

  $$y_{H^{+}\mathit{tB}}^L=\frac{m_t}{m_B}\left[c_L^u{c}_L^d{e}^{i{\varphi }_d}+\frac{c_L^d}{s_L^u}\left(s_L^{d^2}-s_R^{d^2}\right)e^{i{\varphi }_u}\right]$$E110
  $$y_{H^{+}\mathit{tB}}^R=c_L^d{c}_L^u+\frac{s_L^d}{s_L^u}\left(s_L^{u^2}-s_R^{u^2}\right)e^{{\varphi }_u}$$E111

• For the coupling H^±TB:

  $$y_{H^{+}\mathit{tb}}^L=s_L^d{s}_L^u{e}^{i\left({\varphi }_d-{\varphi }_u\right)}+\frac{c_L^d}{c_L^u}\left(s_R^{u^2}-s_L^{u^2}\right)$$E112
  $$y_{H^{+}\mathit{tb}}^R=\frac{m_B}{m_T}\left[s_L^u{s}_L^d{e}^{i\left({\varphi }_d-{\varphi }_u\right)}+\frac{c_L^u}{c_L^d}\left(s_R^{d^2}-s_L^{d^2}\right)e^{i\left({\varphi }_u-{\varphi }_d\right)}\right]$$E113

Figure 8 presents the branching ratios $\left(\mathcal{BR}s\right)$ for all open decay channels, including T → W⁺b, tZ, th, H⁺b, Ht, and At, as a function of the mass of the top quark partner m_T. The left panel illustrates the singlet representation 2HDM + T, with fixed values for m_h, m_H, m_A, tan β, and s_L, while the right panel depicts the 2HDM + TB representation with the same fixed 2HDM parameters in addition to $s_R^u$ and $s_R^d.$

Figure 8.

BR(T → XY) as a function of m_T for 2HDM + T (left) and 2HDM + TB (right).

5. Summary of gauge interactions

5.1 Light–light interactions

$$\begin{array}{lll}{\mathcal{L}}_W& =& -\frac{g}{\sqrt{2}}\overline{t}{\gamma }^{\mu }\left(y_{\mathit{tb}}^L{P}_L+y_{\mathit{tb}}^R{P}_R\right){\mathit{bW}}_{\mu }^{+}+H.c.,\\ {\mathcal{L}}_Z& =& -\frac{g}{2c_W}\overline{t}{\gamma }^{\mu }\left(y_{\mathit{tt}}^L{P}_L+y_{\mathit{tt}}^R{P}_R-2Q_t{s}_W^2\right){\mathit{tZ}}_{\mu }\\ & & -\frac{g}{2c_W}\overline{b}{\gamma }^{\mu }\left(-y_{\mathit{Zbb}}^L{P}_L-y_{\mathit{bb}}^R{P}_R-2Q_b{s}_W^2\right){\mathit{bZ}}_{\mu }+H.c.,\\ {\mathcal{L}}_h& =& -\frac{{\mathit{gm}}_t}{2M_W}{y}_{\mathit{tt}}^h\overline{t}{\mathit{th}}^0-\frac{{\mathit{gm}}_b}{2M_W}{y}_{\mathit{bb}}^h\overline{b}{\mathit{bh}}^0+H.c.,\\ {\mathcal{L}}_H& =& -\frac{{\mathit{gm}}_t}{2M_W}{y}_{\mathit{tt}}^H\overline{t}{\mathit{tH}}^0-\frac{{\mathit{gm}}_b}{2M_W}{y}_{\mathit{bb}}^H\overline{b}{\mathit{bH}}^0+H.c.,\\ {\mathcal{L}}_A& =& -i\frac{{\mathit{gm}}_t}{2M_W}{y}_{\mathit{tt}}^A\overline{t}{\gamma }_5\mathit{tA}+i\frac{{\mathit{gm}}_b}{2M_W}{y}_{\mathit{bb}}^A\overline{b}{\gamma }_5\mathit{bA}+H.c.,\\ {\mathcal{L}}_{H^{+}}& =& -\frac{{\mathit{gm}}_t}{\sqrt{2}{M}_W}\overline{t}\left(cot\beta y_{\mathit{tb}}^L{P}_L+tan\beta y_{\mathit{tb}}^R{P}_R\right){\mathit{bH}}^{+}+H.c.\end{array}$$E114

5.2 Heavy-heavy interactions

$$\begin{array}{lll}{\mathcal{L}}_W& =& -\frac{g}{\sqrt{2}}\overline{Q}{\gamma }^{\mu }\left(y_{Q{Q}^{\prime }}^L{P}_L+y_{Q{Q}^{\prime }}^R{P}_R\right)Q^{\prime }{W}_{\mu }^{+}+H.c.,\\ {\mathcal{L}}_Z& =& -\frac{g}{2c_W}\overline{Q}{\gamma }^{\mu }\left(\pm y_{\mathit{QQ}}^L{P}_L\pm y_{\mathit{QQ}}^R{P}_R-2Q_Q{s}_W^2\right){\mathit{QZ}}_{\mu }\\ {\mathcal{L}}_h& =& -\frac{{\mathit{gm}}_Q}{2M_W}{y}_{\mathit{QQ}}^h\overline{Q}{\mathit{Qh}}^0+H.c.,\\ {\mathcal{L}}_H& =& -\frac{{\mathit{gm}}_Q}{2M_W}{y}_{\mathit{QQ}}^H\overline{Q}{\mathit{QH}}^0+H.c.,\\ {\mathcal{L}}_A& =& \pm i\frac{{\mathit{gm}}_Q}{2M_W}{y}_{\mathit{QQ}}^A\overline{Q}{\gamma }_5\mathit{QA}+H.c.,\\ {\mathcal{L}}_{H^{+}}& =& -\frac{{\mathit{gm}}_Q}{\sqrt{2}{M}_W}\overline{Q}\left(cot\beta y_{\mathit{QQ}}^L{P}_L+tan\beta y_{\mathit{QQ}}^R{P}_R\right){\mathit{QH}}^{+}+H.c.\end{array}$$E115

5.3 Light-heavy interactions

$$\begin{array}{lll}{\mathcal{L}}_W& =& -\frac{g}{\sqrt{2}}\overline{Q}{\gamma }^{\mu }\left(y_{\mathit{Qq}}^L{P}_L+y_{\mathit{Qq}}^R{P}_R\right){\mathit{qW}}_{\mu }^{+}\\ & & -\frac{g}{\sqrt{2}}\overline{q}{\gamma }^{\mu }\left(y_{\mathit{qQ}}^L{P}_L+y_{\mathit{qQ}}^R{P}_R\right){\mathit{QW}}_{\mu }^{+}+H.c.\\ {\mathcal{L}}_Z& =& -\frac{g}{2c_W}\overline{q}{\gamma }^{\mu }\left(\pm y_{\mathit{qQ}}^L{P}_L\pm y_{\mathit{qQ}}^R{P}_R\right){\mathit{QZ}}_{\mu }+H.c\\ {\mathcal{L}}_h& =& -\frac{{\mathit{gm}}_T}{2M_W}\overline{t}\left(Y_{\mathit{htT}}^L{P}_L+y_{\mathit{htT}}^R{P}_R\right){\mathit{Th}}^0\\ & & -\frac{{\mathit{gm}}_B}{2M_W}\overline{b}\left(y_{\mathit{hbB}}^L{P}_L+y_{\mathit{hbB}}^R{P}_R\right){\mathit{Bh}}^0+H.C.\\ {\mathcal{L}}_H& =& -\frac{{\mathit{gm}}_T}{2M_W}\overline{t}\left(y_{\mathit{HtT}}^L{P}_L+y_{\mathit{HtT}}^R{P}_R\right){\mathit{TH}}^0\\ & & -\frac{{\mathit{gm}}_B}{2M_W}\overline{b}\left(y_{\mathit{HbB}}^L{P}_L+y_{\mathit{HbB}}^R{P}_R\right){\mathit{Bh}}^0+H.C.\\ {\mathcal{L}}_A& =& i\frac{{\mathit{gm}}_T}{2M_W}\overline{t}\left(y_{\mathit{AtT}}^L{P}_L-Y_{\mathit{AtT}}^R{P}_R\right)\mathit{TA}\\ & & -i\frac{{\mathit{gm}}_B}{2M_W}\overline{b}\left(y_{\mathit{AbB}}^L{P}_L-y_{\mathit{AbB}}^R{P}_R\right)\mathit{BA}+H.C.\\ {\mathcal{L}}_{H^{+}}& =& -\frac{{\mathit{gm}}_T}{\sqrt{2}{M}_W}\overline{T}\left(cot\beta y_{\mathit{Tb}}^L{P}_L+tan\beta y_{\mathit{Tb}}^R{P}_R\right){\mathit{bH}}^{+}\\ & & -\frac{{\mathit{gm}}_B}{\sqrt{2}{M}_W}\overline{t}\left(cot\beta y_{\mathit{tB}}^L{P}_L+tan\beta y_{\mathit{tB}}^R{P}_R\right){\mathit{BH}}^{+}+H.c,\end{array}$$E116

6. Vector-like quarks pair production at the LHC

The production process of vector-like quark pairs (VLQs) is an essential mechanism for investigating the properties of these hypothetical particles and for understanding the behavior of high energy interactions. A key benefit of VLQ pair production is its model independence, since it depends only on the masses of the VLQs and the energy of the colliding particles. At hadron colliders [25, 26], the extra-heavy quarks are able to get paired production because of their unique gauge couplings with the gluons.

In Figure 9, a typical Feynman diagram of pair-produced vector-like quarks $Q\left(\equiv T,B,X,Y\right)$ is shown.

Figure 9.

Typical Feynman diagram of pair-produced vector-like quarks $Q\left(\equiv T,B,X,Y\right)$.

The expression for the total inclusive $t\overline{t}$ production cross-section can be written as follows, as described in [27]:

$${\sigma }_{\mathrm{tot}}\left(\mathit{pp}\to T\overline{T}\right)=\sum _{\mathit{ij}}{\int }_0^{{\beta }_{\max }}\mathit{d\beta }{\mathrm{\Phi }}_{\mathit{ij}}\left(\beta ,{\mu }_F^2\right){\widehat{\sigma }}_{\mathit{ij}}\left(\beta ,m^2,{\mu }_F^2,{\mu }_R^2\right)$$E117

Here, the indices i and j enumerate all possible initial state particles; β_max is defined as $\sqrt{1-4m^2/s},$ where $\sqrt{s}$ is the centre-of-mass energy of the hadron collider; and $\beta =\sqrt{1-\rho },$ where $\rho \equiv 4m^2/s,$ is the relative velocity of the final state top quarks with pole mass m and partonic centre-of-mass energy $\sqrt{s}$.

The function Φ is the partonic flux and is given by:

$${\mathrm{\Phi }}_{\mathit{ij}}\left(\beta ,{\mu }_F^2\right)=\frac{2\beta }{1-{\beta }^2}\mathcal{L}\mathit{ij}\left(\frac{1-\beta max}{1-\beta },{\mu }_F^2\right)$$E118

Here, $\mathcal{L}$ is the usual partonic luminosity, defined as:

$$\mathcal{L}\left(x,{\mu }_F^2\right)=x\left(f_i\otimes f_j\right)\left(x,{\mu }_F^2\right)$$E119

The renormalization and factorization scales are denoted by μ_R and μ_F, respectively. Setting μ_F = μ_R = m, the NNLO (next-to-next-to-leading order) partonic cross-section can be expanded as follows:

$${\widehat{\sigma }}_{\mathit{ij}}\left(\beta \right)=\frac{{\alpha }_s^2}{m^2}\left({\sigma }_{\mathit{ij}}^{\left(0\right)}+{\alpha }_s{\sigma }_{\mathit{ij}}^{\left(1\right)}+{\alpha }_s^2{\sigma }_{\mathit{ij}}^{\left(2\right)}+\mathcal{O}\left({\alpha }_s^3\right)\right)$$E120

In the above equation, α_s represents the renormalized $\overline{\mathrm{MS}}$ coupling, considering the presence of N_L = 5 active flavors at scale ${\mu }_R^2=m^2.$ Moreover, the functions ${\sigma }_{\mathit{ij}}^{\left(n\right)}$ solely depend on the value of β.

In Figure 10, we present the production cross-section at the next-to-next-to-leading order (NNLO) of vector-like quark T pairs produced at the Large Hadron Collider (LHC), as a function of their masses m_Q. The curves are depicted in green and blue, corresponding to centre of mass energies of 14 and 13 TeV, respectively.

Figure 10.

Production cross section for pair-produced vector-like quarks T at the LHC. In green (blue) we show the NNLO cross section for 14 TeV (13 TeV) Centre of mass energy computed by top++ [28].

7. Conclusions

In this chapter, we focus on the Lagrangian framework that adheres to the fundamental principles of the standard model, exploring its consequences by determining eigenstates and interactions with SM-VLQ. We provide a comprehensive examination of the full interaction within the mass basis of 2HDM. Additionally, we touch upon the pair production of heavy quarks at the LHC, operating at energies of 13 and 14 TeV, underlining the importance of these endeavors in advancing our understanding of the fundamental forces and particles shaping the universe.

Simultaneously, the pursuit of heavy quarks, which share quantum characteristics like spin and electric charge with their standard model counterparts, is highly motivated. These quarks are abundantly produced within the LHC, and their discovery would mark a monumental advancement in particle physics. Although current experimental searches for vector-like quarks at CERN’s LHC have not yielded conclusive evidence, the quest continues, holding the potential to usher in a new era of particle physics research. In this context, researchers at the LHC grapple with the challenge of directly testing quantum entanglement in proton-proton collisions, which is complicated by the macroscopic nature of these interactions. Quantum entanglement, typically a phenomenon observed at the microscopic scale, involves correlations between particles or pairs of particles. While the LHC’s experiments may not provide direct proof of entanglement, they indirectly explore related aspects by studying correlations and angular momentum conservation among particles generated during collisions.

Appendix: heavy quark decay widths

The partial widths for T decays, including all possible mixing terms, are

$$\begin{array}{lll}\mathrm{\Gamma }\left(T\to W^{+}b\right)& =& \frac{g^2}{64\pi }\frac{m_T}{M_W^2}\lambda {\left(m_T,m_b,M_W\right)}^{1/2}\left\{\left({\left|y_{\mathit{Tb}}^L\right|}^2+{\left|y_{\mathit{Tb}}^R\right|}^2\right)\right.\\ & & \left.\times \left[1+r_W^2-2r_b^2-2r_W^4+r_b^4+r_W^2{r}_b^2\right]-12r_W^2{r}_b|y_{\mathit{Tb}}^L|y_{\mathit{Tb}}^{R\ast }\right\},\\ \mathrm{\Gamma }\left(T\to \mathit{Zt}\right)& =& \frac{g}{128\pi c_W^2}\frac{m_T}{M_Z^2}\lambda {\left(m_T,m_t,M_Z\right)}^{1/2}\left\{\left({\left|y_{\mathit{tT}}^L\right|}^2+{\left|y_{\mathit{tT}}^R\right|}^2\right)\right.\\ & & \left.\times \left[1+r_Z^2-2r_t^2-2r_Z^4+r_t^4+r_Z^2{r}_t^2\right]-12r_Z^2{r}_t|y_{\mathit{tT}}^L|y_{\mathit{tT}}^{R\ast }\right\},\\ \mathrm{\Gamma }\left(T\to \mathit{Ht}\right)& =& \frac{g^2}{128\pi }\frac{m_T}{M_W^2}\lambda {\left(m_T,m_t,M_H\right)}^{1/2}{\left|y_{\mathit{tT}}^H\right|}^2\left(1+6r_t^2-r_H^2+r_t^4-r_t^2{r}_H^2\right),\end{array}$$E121

and for the 2HDM-T quark they are completely analogous, with r_x ≡ m_x/m_Q, where x = t, b, W, Z, H and Q is the heavy quark, and

$$\lambda \left(x,y,z\right)\equiv \left(x^4+y^4+z^4-2x^2{y}^2-2x^2{z}^2-2y^2{z}^2\right).$$E122