Relaxing LHC constraints on the \(W_R\) mass
Author
Ozer Ozdal
Last Updated
3 years ago
License
Creative Commons CC BY 4.0
Abstract
XIth International Symposium:
Quantum Theory and Symmetries (QTS)
Université de Montréal
July 1-5, 2019
XIth International Symposium:
Quantum Theory and Symmetries (QTS)
Université de Montréal
July 1-5, 2019
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\title[]
{\bfseries{Relaxing LHC constraints on the $W_R$ mass}}
\subtitle{{\small based on \hyperlink{https://journals.aps.org/prd/pdf/10.1103/PhysRevD.99.035001}{Phys. Rev. D 99, 035001}}}
\author[]
{\underline{Özer Özdal}\inst{1} \and Mariana Frank\inst{1} \and Poulose Poulose\inst{2}}
\institute[Concordia University]
{
Concordia University\inst{1},
Indian Institute of Technology Guwahati\inst{2}
}
\date[WNPFC, 2018]
{XIth International Symposium: \\
Quantum Theory and Symmetries (QTS)\\
Université de Montréal\\
July 1-5, 2019}
% logo of my university
\titlegraphic{\includegraphics[width=3cm]{./figures/concordia-logo.png}
\hspace{7cm} \includegraphics[width=0.9cm]{./figures/Guwahati.png}}
%------------------------------------------------------------
%------------------------------------------------------------
%The next block of commands puts the table of contents at the
%beginning of each section and highlights the current section:
\AtBeginSection[]
{
\begin{frame}
\frametitle{Outline}
\tableofcontents[currentsection]
\end{frame}
}
%------------------------------------------------------------
\begin{document}
\frame{\titlepage} % Creates title page
%--------- table of contents after title page ------------
\begin{frame}
\frametitle{Outline}
\tableofcontents
\end{frame}
%---------------------------------------------------------
\section{Introduction}
%---------------------------------------------------------
\subsection{The Left-Right Symmetric Model}
%---------------------------------------------------------
\begin{frame}
\frametitle{The Left-Right Symmetric Model (LRSM)}
\begin{columns}
\begin{column}{0.5\textwidth}
\color{Black}
\centering
{\color{Black} \small \begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\mbox{}\;\;\;\;\;\mbox{}& \; Fields\; & \; $ SU(2)_L\times SU(2)_R\times U(1)_{B-L}$\; \\
\hline
\multirow{4}{*}{\begin{sideways}Matter\end{sideways}}& \(Q_{L_i}\) & \(({\bf 2},{\bf 1},+\frac{1}{3}) \) \\
&\(Q_{R_i}\) & \(({\bf 1},{\bf 2},-\frac{1}{3}) \) \\
&\(L_{L_i}\) & \(({\bf 2},{\bf 1},{-1}) \) \\
&\(L_{R_i}\) & \(({\bf 1},{\bf 2},{-1}) \) \\
\hline \hline
\multirow{3}{*}{\begin{sideways}Higgs\end{sideways}}&\(\Phi\) & \(({\bf 2},{\bf 2},0) \) \\
&\(\Delta_L\) & \(({\bf 3},{\bf 1},2) \) \\
&\(\Delta_R\) & \(({\bf 1},{\bf 3},2) \) \\
\hline
\end{tabular}
\end{center}
\end{table}}
\vspace{-0.3cm}
%$$Q_{EM}=T_{3L}+T_{3R}+\frac{B-L}{2}$$
\scriptsize
\centering
\begin{eqnarray*}
Q_{Li} = \begin{pmatrix}u_L \\ d_L\end{pmatrix}_i \, \sim
(\mathbf{2},\mathbf{1},\mathbf{1/3}) \, , ~&
Q_{Ri} =
\begin{pmatrix}u_R \\ d_R\end{pmatrix}_i \sim
(\mathbf{1},\mathbf{2},\mathbf{1/3}) \, , \qquad
\end{eqnarray*}
\begin{eqnarray*}
L_{Li} = \begin{pmatrix}\nu_L \\ \ell_L\end{pmatrix}_i \, \sim
(\mathbf{2},\mathbf{1},\mathbf{-1}) \, , ~&
L_{Ri} =
\begin{pmatrix}\nu_R \\ \ell_R\end{pmatrix}_i \sim
(\mathbf{1},\mathbf{2},\mathbf{-1}) \, , \qquad
\end{eqnarray*}
\end{column}
\begin{column}{0.5\textwidth}
\centering
\scriptsize
\begin{displaymath}
\hspace{0.75cm} \xymatrix{ SU(3)_C \times SU(2)_L \times & \hspace{-2.0cm} \color{red} SU(2)_R \times U(1)_{B-L} \hspace{-1.5cm} \ar[d]^{\Delta_R} \\
SU(3)_C \times & \hspace{-1.cm} SU(2)_L \times \color{red} U(1)_{Y} \ar[d]^{\Phi} \\
SU(3)_C \times & \hspace{-1.cm} U(1)_{EM}
}
\end{displaymath}
\vspace{0.3cm}
\scriptsize
\begin{equation*}
\Phi \equiv \begin{pmatrix} \phi_1^0 & \phi_2^+ \\ \phi_1^- & \phi_2^0 \end{pmatrix} \sim (\mathbf{2},\mathbf{2},\mathbf{0})\,
\end{equation*}
\begin{eqnarray*}
\Delta_{L} \equiv \begin{pmatrix} \delta_{L}^+/\sqrt{2} & \delta_{L}^{++} \\ \delta_{L}^0 & -\delta_{L}^+/\sqrt{2} \end{pmatrix} \sim (\mathbf{3},\mathbf{1},\mathbf{2}) \\ ~~~ \Delta_{R} \equiv \begin{pmatrix} \delta_{R}^+/\sqrt{2} & \delta_{R}^{++} \\ \delta_{R}^0 & -\delta_{R}^+/\sqrt{2} \end{pmatrix} \sim (\mathbf{1},\mathbf{3},\mathbf{2})\,
\end{eqnarray*}
\end{column}
\end{columns}
\end{frame}
%---------------------------------------------------------
%\begin{frame}
%\frametitle{Field Configuration}
%\begin{eqnarray*}
% Q_{Li} = \begin{pmatrix}u_L \\ d_L\end{pmatrix}_i \, \sim
% (\mathbf{2},\mathbf{1},\mathbf{1/3}) \, , ~&
% Q_{Ri} =
% \begin{pmatrix}u_R \\ d_R\end{pmatrix}_i \sim
% (\mathbf{1},\mathbf{2},\mathbf{1/3}) \, , \qquad
%\end{eqnarray*}
%\begin{eqnarray*}
%L_{Li} = \begin{pmatrix}\nu_L \\ \ell_L\end{pmatrix}_i \, \sim
%(\mathbf{2},\mathbf{1},\mathbf{-1}) \, , ~&
%L_{Ri} =
%\begin{pmatrix}\nu_R \\ \ell_R\end{pmatrix}_i \sim
%(\mathbf{1},\mathbf{2},\mathbf{-1}) \, , \qquad
%\end{eqnarray*}
%\pause
%\begin{equation*}
%\Phi \equiv \begin{pmatrix} \phi_1^0 & \phi_2^+ \\ \phi_1^- & \phi_2^0 \end{pmatrix} \sim %(\mathbf{2},\mathbf{2},\mathbf{0})\,
%\end{equation*}
%\begin{equation*}
%\Delta_{L} \equiv \begin{pmatrix} \delta_{L}^+/\sqrt{2} & \delta_{L}^{++} \\ \delta_{L}^0 & -\delta_{L}^+/\sqrt{2} \end{pmatrix} \sim (\mathbf{3},\mathbf{1},\mathbf{2}) \, , ~~~ \Delta_{R} \equiv \begin{pmatrix} \delta_{R}^+/\sqrt{2} & \delta_{R}^{++} \\ \delta_{R}^0 & -\delta_{R}^+/\sqrt{2} \end{pmatrix} \sim (\mathbf{1},\mathbf{3},\mathbf{2})\,
%\end{equation*}
%\end{frame}
%---------------------------------------------------------
%---------------------------------------------------------
\begin{frame}
\frametitle{Symmetry Breaking}
\centering
$SU(2)_R\otimes U(1)_{B-L}$ $\longrightarrow$ $U(1)_Y$
\begin{gather*}
\quad \langle \Delta_{L}
\rangle = \begin{pmatrix} 0 & 0 \\ v_{L}e^{i\theta_L}/\sqrt{2} & 0
\end{pmatrix}, \quad \langle \Delta_{R} \rangle = \begin{pmatrix} 0 &
0 \\ \color{red} v_{R} \color{black} /\sqrt{2} & 0 \end{pmatrix}
\end{gather*}
\pause
\vspace{0.9cm}
$SU(2)_L\otimes U(1)_{Y}$ $\longrightarrow$ $U(1)_{EM}$
\vspace{0.2cm}
$v_R \gg (\kappa_1,~\kappa_2)\gg v_L$, \hspace{2cm} $\sqrt{\kappa_1^2+\kappa_2^2} = v = 246$ GeV
\begin{gather*}
\langle \Phi\rangle = \begin{pmatrix} \kappa_1/\sqrt{2} & 0 \\ 0 &
\kappa_2e^{i\alpha}/\sqrt{2} \end{pmatrix}
\end{gather*}
\end{frame}
%--------------------------------------------------------
\begin{frame}
\frametitle{LRSM Lagrangian}
\begin{eqnarray*}
\mathcal{L}_{\rm{LRSM}}= \mathcal{L}_{\rm{\rm kin}}+\mathcal{L}_{Y}-V(\Phi,\Delta_L, \Delta_R) \,
\end{eqnarray*}
\pause
\vspace{0.5cm}
\begin{eqnarray*}
\mathcal{L}_{\rm kin}=i\sum\bar{\psi}\gamma^\mu D_\mu\psi \\
\end{eqnarray*}
\vspace{-0.99cm}
\scriptsize
\begin{eqnarray*}
=\bar{L}_L\gamma^{\mu}\left(i\partial_{\mu}+g_{L}\frac{\vec{\tau}}{2}\cdot\vec{W}_{L\mu}-\frac{g_{B-L}}{2}B_{\mu}\right)L_L
+\bar{L}_R\gamma^{\mu}\left(i\partial_{\mu}+ g_{R}\frac{\vec{\tau}}{2}\cdot\vec{W}_{R\mu}-\frac{g_{B-L}}{2}B_{\mu}\right)L_R \\
+\bar{Q}_L\gamma^{\mu}\left(i\partial_{\mu}+g_{L}\frac{\vec{\tau}}{2}\cdot\vec{W}_{L\mu}+\frac{g_{B-L}}{6}B_{\mu}\right) Q_L
+\bar{Q}_R\gamma^{\mu}\left(i\partial_{\mu}+g_{R}\frac{\vec{\tau}}{2}\cdot \vec{W}_{R\mu}+\frac{g_{B-L}}{6}B_{\mu}\right)Q_R
\end{eqnarray*}
\vspace{0.5cm}
\pause
\small
\begin{eqnarray*}
\mathcal{L}_Y&=&-\Big[Y_{L_L} {\bar L}_{L} \Phi L_{R} +{\tilde Y}_{L_R} {\bar L}_R \Phi L_L+
Y_{Q_L} {\bar Q}_L {\tilde \Phi} Q_R
+{\tilde Y}_{Q_R} {\bar Q}_R {\tilde \Phi} Q_L \\
&+&\color{red} h_{L}^{ij} \color{black} \overline {L}^c_{L_i} i \tau_2 \Delta_L L_{L_j}
+\color{red} h_{R}^{ij} \color{black}\overline {L}^c_{R_i}i \tau_2 \Delta_R L_{R_j} +\rm{h.c.} \Big]\, ,
\end{eqnarray*}
\end{frame}
%--------------------------------------------------------
%--------------------------------------------------------
\begin{frame}
\frametitle{LRSM Higgs Potential}
\tiny
\begin{eqnarray*}
\!\!\! \! V(\phi,\Delta_L,\Delta_R)& = &\color{red}-\mu_{1}^2 \color{black}\left({\rm Tr}\left[\Phi^\dagger\Phi\right]\right)-\color{red} \mu_{2}^2 \color{black}\left({\rm Tr}\left[\tilde{\Phi}\Phi^\dagger\right]+\left({\rm Tr}\left[\tilde{\Phi}^\dagger\Phi\right]\right)\right)-\color{red}\mu_{3}^2 \color{black}\left({\rm Tr}\left[\Delta_L\Delta_L^{\dagger}\right]+{\rm Tr}\left[\Delta_R\Delta_R^{\dagger}\right]\right) \nonumber \\
&+&\color{red} \lambda_1 \color{black} \left(\left({\rm Tr}\left[\Phi\Phi^\dagger\right]\right)^2\right)
+\color{red} \lambda_2 \color{black} \left(\left({\rm Tr}\left[\tilde{\Phi}\Phi^\dagger\right]\right)^2
+\left({\rm Tr}\left[\tilde{\Phi}^\dagger\Phi\right]\right)^2\right)
+\color{red} \lambda_3 \color{black} \left({\rm Tr}\left[\tilde{\Phi}\Phi^\dagger\right]{\rm Tr}\left[\tilde{\Phi}^\dagger\Phi\right]\right)\nonumber\\
&+&\color{red} \lambda_4 \color{black}\left({\rm Tr}\left[\Phi\Phi^{\dagger}\right]\left({\rm Tr}\left[\tilde{\Phi}\Phi^\dagger\right]
+{\rm Tr}\left[\tilde{\Phi}^\dagger\Phi\right]\right)\right) %\nonumber \\
+\color{red} \rho_1 \color{black} \left(\left({\rm Tr}\left[\Delta_L\Delta_L^{\dagger}\right]\right)^2
+\left({\rm Tr}\left[\Delta_R\Delta_R^{\dagger}\right]\right)^2\right)\nonumber\\
&+&\color{red} \rho_2 \color{black} \left({\rm Tr}\left[\Delta_L\Delta_L\right]{\rm Tr}\left[\Delta_L^{\dagger}\Delta_L^{\dagger}\right]
+{\rm Tr}\left[\Delta_R\Delta_R\right]{\rm Tr}\left[\Delta_R^{\dagger}\Delta_R^{\dagger}\right]\right) +
\color{red} \rho_3 \color{black} \left({\rm Tr}\left[\Delta_L\Delta_L^{\dagger}\right]{\rm Tr}\left[\Delta_R\Delta_R^{\dagger}\right]
\right)\nonumber \\
&+&\color{red} \rho_4 \color{black} \left({\rm Tr}\left[\Delta_L\Delta_L\right]{\rm Tr}\left[\Delta_R^{\dagger}\Delta_R^{\dagger}\right]
+{\rm Tr}\left[\Delta_L^{\dagger} \Delta_L^{\dagger}\right]{\rm Tr}\left[\Delta_R\Delta_R\right]\right)
%\nonumber\\
+\color{red} \alpha_1 \color{black} {\rm Tr}\left[\Phi\Phi^{\dagger}\right]\left({\rm Tr}\left[\Delta_L\Delta_L^{\dagger}\right]
+{\rm Tr}\left[\Delta_R\Delta_R^{\dagger}\right]\right) \nonumber \\
&+&\color{red} \alpha_2 \color{black} \left({\rm Tr}\left[\Phi\tilde{\Phi}^{\dagger}\right]{\rm Tr}\left[\Delta_R\Delta_R^{\dagger}\right]
+{\rm Tr}\left[\Phi^{\dagger}\tilde{\Phi}\right]{\rm Tr}\left[\Delta_L\Delta_L^{\dagger}\right]\right)
%\nonumber \\
+\color{red} \alpha_2^{*} \color{black} \left({\rm Tr}\left[\Phi^{\dagger}\tilde{\Phi}\right]{\rm Tr}\left[\Delta_R\Delta_R^{\dagger}
\right]+{\rm Tr}\left[\tilde{\Phi}^{\dagger}\Phi\right]{\rm Tr}\left[\Delta_L\Delta_L^{\dagger}\right]\right) \nonumber \\
&+& \color{red} \alpha_3 \color{black} \left({\rm Tr}\left[\Phi\Phi^{\dagger}\Delta_L\Delta_L^{\dagger}\right]
+{\rm Tr}\left[\Phi^{\dagger}\Phi\Delta_R\Delta_R^{\dagger}\right]\right)%\nonumber\\
+\color{red} \beta_1 \color{black}\left({\rm Tr}\left[\Phi\Delta_R\Phi^{\dagger}\Delta_L^{\dagger}\right]
+{\rm Tr}\left[\Phi^{\dagger}\Delta_L\Phi\Delta_R^{\dagger}\right]\right)\nonumber\\
&+&\color{red} \beta_2 \color{black} \left({\rm Tr}\left[\tilde{\Phi}\Delta_R\Phi^{\dagger}\Delta_L^{\dagger}\right]
+{\rm Tr}\left[\tilde{\Phi}^{\dagger}\Delta_L\Phi\Delta_R^{\dagger}\right]\right)%\nonumber\\
+\color{red} \beta_3 \color{black} \left({\rm Tr}\left[\Phi\Delta_R\tilde{\Phi}^{\dagger}\Delta_L^{\dagger}\right]
+{\rm Tr}\left[\Phi^{\dagger}\Delta_L\tilde{\Phi}\Delta_R^{\dagger}\right]\right)
\end{eqnarray*}
\end{frame}
%--------------------------------------------------------
%--------------------------------------------------------
\begin{frame}
\frametitle{Gauge Sector}
\begin{equation*}
\left(\begin{array}{c}Z_R^\mu\\B^\mu\end{array} \right) =
\left(\begin{array}{cc}\cos\phi&-\sin\phi\\\sin\phi&\cos\phi\end{array}\right) \left(\begin{array}{c}W_R^{3\mu}\\V^\mu\end{array} \right)
\end{equation*}
\pause
\begin{equation*}
\left(\begin{array}{c}Z_L^\mu\\B^\mu\\Z_R^\mu\end{array} \right) =
\left(\begin{array}{ccc}\cos\theta_W&-\sin\theta_W \sin\phi&-\sin\theta_W\cos\phi\\
\sin\theta_W&\cos\theta_W\sin\phi&\cos\theta_W\cos\phi\\
0&\cos\phi&-\sin\phi\end{array}\right) \left(\begin{array}{c}W_L^{3\mu}\\W_R^{3\mu}\\V^\mu\end{array} \right)
\end{equation*}
\begin{eqnarray*}
M_{A}&=&0 \nonumber \\
M^2_{Z_{1,2}}&=&\frac14 \Big[ \left[g_L^2 v^2+2v_R^2 (g_R^2+g_{B-L}^2)\right] \\ &\mp& \sqrt{\left[g_L^2 v^2+2v_R^2 (g_R^2+g_{B-L}^2)\right]^2-4g_L^2(g_R^2+2g_{B-L}^2)v^2v_R^2} \Big] \,.
\end{eqnarray*}
\end{frame}
%--------------------------------------------------------
%---------------------------------------------------------
\subsection{Charged Sector}
%---------------------------------------------------------
\begin{frame}
\frametitle{Charged Sector}
\centering
\begin{equation*}
\left(\begin{array}{c}W_1\\W_2\end{array} \right) =
\left(\begin{array}{cc}\cos\xi&-\sin\xi\\\sin\xi&\cos\xi\end{array}\right) \left(\begin{array}{c}W_L\\W_R\end{array} \right) \,
\end{equation*}
\vspace{0.5cm}
\pause
In the limit of $(\kappa_1, \kappa_2)\ll v_R$ and $g_R\sim g_L$ we have \\
$\displaystyle \sin\xi \approx \frac{\kappa_1\kappa_2}{v_R^2},~\sin^2\xi\approx 0,~~\cos\xi\approx 1 $, leading to
\begin{eqnarray*}
M^2_{W_1}= \frac14 g_L^2v^2\, , \qquad M_{W_2}^2=\frac{1}{4}\left[2g_R^2v_R^2+g_R^2v^2+2g_Rg_L
\frac{\kappa^2_1\kappa^2_2}{v_R^2} \right]
\end{eqnarray*}
\end{frame}
%--------------------------------------------------------
%---------------------------------------------------------
\section{W$_R$ Mass Limits at the LHC}
%---------------------------------------------------------
%--------------------------------------------------------
\begin{frame}
\frametitle{W$_R$ Mass Limits at the LHC}
\centering
\begin{equation*}
W_R \to t\bar{b} \hspace{5cm} W_R \to jj
\end{equation*}
\includegraphics[scale=0.20]{./figures/Wprimetb_CMS.png}
\includegraphics[scale=0.20]{./figures/Wprimejj_ATLAS.png}
\end{frame}
%--------------------------------------------------------
\begin{frame}
\frametitle{W$_R$ Mass Limits at the LHC}
\centering
\begin{equation*}
W_R \to \ell \nu_R \to \ell \ell W_R^\star \to \ell \ell q q^\prime, ~~\ell = e \hspace{0.15cm} {\rm or} \hspace{0.15cm} \mu \, .
\end{equation*}
\includegraphics[scale=0.20]{./figures/WReejj_CMS.png}
\includegraphics[scale=0.20]{./figures/WRmumujj_CMS.png}
\end{frame}
%--------------------------------------------------------
%---------------------------------------------------------
\section{Motivation}
%---------------------------------------------------------
\begin{frame}
\frametitle{Motivation for g$_L \neq$ g$_R$}
\centering
\begin{equation*}
\frac{1}{e^2}=\frac{1}{g_L^2}+ \frac{1}{g_R^2}+\frac{1}{g_{B-L}^2}\, , \hspace{4cm} \frac{1}{g_Y^2}= \frac{1}{g_R^2}+\frac{1}{g_{B-L}^2}\,
\end{equation*}
Setting $\displaystyle \sin \phi=\frac{g_{B-L}}{\sqrt{g_R^2+g_{B-L}^2}}$ and
$\displaystyle \sin \theta_W=\frac{g_{Y}}{\sqrt{g_L^2+g_{Y}^2}}$, we get
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{equation*}
\tan \theta_W=\frac{g_R \sin \phi}{g_L} \le\frac{g_R}{g_L} \, ,
\end{equation*}
\begin{alertblock}{\centering Theoretical constraint on g$_R$ gauge coupling}
\centering
$ {g_L} \tan \theta_W \le g_R \, $
\end{alertblock}
\end{column}
\begin{column}{0.4\textwidth}
\centering
\includegraphics[scale=0.30]{./figures/WRmassvsZRmass.png}
\end{column}
\end{columns}
\end{frame}
%--------------------------------------------------------
%--------------------------------------------------------
%---------------------------------------------------------
\section{Results}
%---------------------------------------------------------
\begin{frame}
\frametitle{Analysis}
\tiny
\begin{table}{
\setlength\tabcolsep{6pt}
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{|l|c||l|c|}
\hline
Observable & Constraints & Observable & Constraints \\
\hline
$ \Delta{B_s} $ & [10.2-26.4] &
$ \Delta{B_d} $ & [0.294-0.762] \\
$ \Delta{M_K} $ & $<$ 5.00 $\times 10^{-15}$ &
$ \frac{\Delta{M_K}}{\Delta{M_K^{SM}}} $ & [0.7-1.3] \\
$ \epsilon_K $ & $<$ 3.00 $\times 10^{-3}$ &
$\frac{\epsilon_K}{\epsilon^{SM}_K} $ & [0.7-1.3] \\
BR$(B^0 \to X_s \gamma) $ & $ [2.99,3.87]\times10^{-4} $ &
$\frac{BR(B^0 \to X_s \gamma)}{BR(B^0 \to X_s \gamma)_{SM}} $ & [0.7-1.3] \\
$M_{h} $ & $ [124,126] $ GeV &
$M_{H_{1,2}^{\pm\pm}} $ & $>$ 535 GeV \\
$M_{H_4,A_2,H_2^{\pm}} $ & $> 4.75 \times M_{W_R}$ &
& \\
\hline
\end{tabular}
\caption{\label{tab:constraints} Current experimental bounds imposed for consistent solutions.}}
\end{table}
\vspace{-0.5cm}
\begin{columns}
\begin{column}{0.4\textwidth}
\tiny
\begin{table}
% \setlength\tabcolsep{20pt}
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{|c|c|}
\hline
Parameter & Scanned range \\
\hline
$v_{R}$ & $[2.2, ~20]$~TeV \\
$V_{\rm CKM}^{R}$:~~$c^R_{12},~c^R_{13},~c^R_{23}$ & $[-1,~ 1]$ \\
diag$(h_R^{ij})$ & $[0.001,~ 1]$ \\
\hline
\end{tabular}
\caption{\label{tab:scan_lim} Scanned parameter space.}
\end{table}
\end{column}
\begin{column}{0.6\textwidth}
\tiny
\vspace{-0.75cm}
\begin{equation*}
M_{\nu_{R}}^{ij}= h_{R}^{ij}v_{R}\
\end{equation*}
\begin{equation*}
V_{\rm CKM}^R = \begin{bmatrix}
c^R_{12}c^R_{13} & s^R_{12}c^R_{13} & s^R_{13}e^{i\delta_R} \\
-s^R_{12}c^R_{23}-c^R_{12}s^R_{23}s_{13}e^{i\delta_R} & c^R_{12}c^R_{23}-s^R_{12}s^R_{23}s^R_{13}e^{i\delta_R} & s^R_{23}c^R_{13} \\
s^R_{12}s^R_{23}-c^R_{12}c^R_{23}s^R_{13}e^{i\delta_R} & -c^R_{12}c^R_{23}-s^R_{12}c^R_{23}s^R_{13}e^{i\delta_R} & c^R_{23}c^R_{13}
\end{bmatrix}
\end{equation*}
\end{column}
\end{columns}
\end{frame}
%---------------------------------------------------------
\subsection{Scenario I: $M_{\nu_R} > M_{W_R} $}
%---------------------------------------------------------
\begin{frame}
\frametitle{ Scenario I: $M_{\nu_R} > M_{W_R} $}
\centering
\scriptsize
\begin{columns}
\begin{column}{0.5\textwidth}
%\underline{W$_R$ $\to$ t$\bar{b}$} \hspace{4.5cm} \underline{W$_R$ $\to$ jj} \\
\centering
\underline{$g_L \neq g_R = 0.37$, tan $\beta$ = 0.01, $V_{\rm CKM}^L = V_{\rm CKM}^R$} \\
\vspace{0.25cm}
$\left.\begin{tabular}{l}
BR($W_R \to W_L h$) \\
BR($W_R \to W_L Z_L$) \\
\end{tabular}\right\}$ invisible
BR($W_R \to t {\bar b}$) $\sim$ 32\% - 33\% \\
\vspace{0.75cm}
\underline{$g_L \neq g_R = 0.37$, tan $\beta$ = 0.5, $V_{\rm CKM}^L = V_{\rm CKM}^R$} \\
\vspace{0.25cm}
BR($W_R \to W_L h$) $\sim$ 1.95\% \\
BR($W_R \to W_L Z_L$) $\sim$ 2.0\% \\
BR($W_R \to t {\bar b}$) $\sim$ 31.0\% - 31.8\%
\vspace{0.75cm}
\underline{$g_L \neq g_R = 0.37$, tan $\beta$ = 0.5, $V_{\rm CKM}^L \ne V_{\rm CKM}^R$} \\
\vspace{0.25cm}
\small
BR($W_R \to t {\bar b}$) $\sim$ 20\% for high $M_{W_R}$ (4 TeV) \\
\hspace{2.05cm} $\sim$ 29\% for low $M_{W_R}$ (1.5 TeV)
\end{column}
\begin{column}{0.5\textwidth}
\centering
\includegraphics[scale=0.25]{./figures/WR_heavyneutrino.png} \\
\includegraphics[scale=0.25]{./figures/WRtojj_heavyneutrino.png}
\end{column}
\end{columns}
\end{frame}
%---------------------------------------------------------
\subsection{Scenario II: $ M_{\nu_R} < M_{W_R} $}
%---------------------------------------------------------
\begin{frame}
\frametitle{Scenario II: $ M_{\nu_R} < M_{W_R} $}
\centering
%\underline{W$_R$ $\to$ t$\bar{b}$} \hspace{4.5cm} \underline{W$_R$ $\to$ jj} \\
\begin{columns}
\begin{column}{0.5\textwidth}
%\underline{W$_R$ $\to$ t$\bar{b}$} \hspace{4.5cm} \underline{W$_R$ $\to$ jj} \\
\centering
\scriptsize
\underline{$g_L = g_R$ ,tan $\beta$ = 0.01, $V_{\rm CKM}^L = V_{\rm CKM}^R$} \\
\vspace{0.25cm}
\scriptsize
BR($W_R \to \nu_R\ell$) $\sim$ 5.8\% (each family) \\
BR($W_R \to t {\bar b}$) $\sim$ 26.5\% - 27.3\% \\
\vspace{0.5cm}
\underline{$g_L \neq g_R = 0.37$, tan $\beta$ = 0.01, $V_{\rm CKM}^L = V_{\rm CKM}^R$} \\
\vspace{0.25cm}
\scriptsize
BR($W_R \to \nu_R\ell$) $\sim$ 6.7\% (each family) \\
BR($W_R \to t {\bar b}$) $\sim$ 25.7\% - 26.5\%
\vspace{0.5cm}
\underline{$g_L \neq g_R = 0.37$, tan $\beta$ = 0.5, $V_{\rm CKM}^L = V_{\rm CKM}^R$} \\
\vspace{0.25cm}
\scriptsize
BR($W_R \to \nu_R\ell$) $\sim$ 6.7\% (each family) \\
BR($W_R \to W_L h$) $\sim$ 1.95\% \\
BR($W_R \to W_L Z_L$) $\sim$ 2.0\% \\
BR($W_R \to t {\bar b}$) $\sim$ 24.8\% - 25.6\%
\vspace{0.5cm}
\underline{$g_L \neq g_R = 0.37$, tan $\beta$ = 0.5, $V_{\rm CKM}^L \neq V_{\rm CKM}^R$} \\
\vspace{0.25cm}
\scriptsize
BR($W_R \to t {\bar b}$) $\sim$ 15.7\% for high $M_{W_R}$ (4 TeV) \\
\hspace{2cm} $\sim$ 24.7\% for low $M_{W_R}$ (1.5 TeV)
\end{column}
\begin{column}{0.5\textwidth}
\centering
\includegraphics[scale=0.25]{./figures/WRtb_halfneut.png} \\
\includegraphics[scale=0.25]{./figures/WRtojj_neuthalfWR.png}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Scenario II: $ M_{\nu_R} < M_{W_R} $}
\centering
\begin{columns}
\begin{column}{0.5\textwidth}
\centering
\tiny
\vspace{-0.75cm}
\begin{eqnarray*}
W_R \to \ell \nu_R \to \ell \ell \color{red} W_R^\star \color{black} \to \ell \ell q q^\prime, ~~\ell = e \hspace{0.15cm} {\rm or} \hspace{0.15cm} \mu \, . \\
W_R \to \ell \nu_R \to \ell \ell \color{red} W_L \color{black} \to \ell \ell q q^\prime,~~ \ell = e \hspace{0.15cm} {\rm or} \hspace{0.15cm} \mu \,.
\end{eqnarray*}
\includegraphics[scale=0.32]{./figures/gLeqgR_BRWR3500_tanb.png}
\vspace{-0.4cm}
\begin{eqnarray*}
\overline{\nu}W_L^{+\mu}\ell &\longrightarrow& \frac{i}{\sqrt{2}} \gamma^\mu \left(g_L P_L K_L\cos\xi -g_R P_R K_R\sin \xi \right) \, \\
\overline{\nu}W_R^{+\mu}\ell &\longrightarrow& \frac{i}{\sqrt{2}} \gamma^\mu \left(g_R P_R K_R\cos\xi -g_L P_L K_L\sin \xi \right) \,
\end{eqnarray*}
\end{column}
\begin{column}{0.5\textwidth}
\centering
\includegraphics[scale=0.32]{./figures/BR_PHIW3500_tanb.png} \\
\begin{minipage}{0.3\textwidth}
\tiny
$K_L$ and $K_R$ are PMNS mixing matrices in the left and right leptonic sectors, defined as
%
\begin{eqnarray*}
K_L=V_L^{\nu\dagger}V_L^\ell, \\
K_R=V_R^{\nu\dagger}V_R^\ell.
\end{eqnarray*}
\end{minipage}
\begin{minipage}{0.6\textwidth}\raggedleft
\includegraphics[width=\linewidth]{./figures/WRtolljj_Feynman.png}
\end{minipage}
\noindent
\end{column}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Scenario II: $ M_{\nu_R} < M_{W_R} $}
\centering
\underline{eejj final state} \hspace{4.5cm} \underline{$\mu$$\mu$jj final state} \\
\includegraphics[scale=0.32]{./figures/halfneut_eejj.png}
\includegraphics[scale=0.32]{./figures/halfneut_mumujj.png}
\end{frame}
%---------------------------------------------------------
\subsection{Correlating $W_R$ and $\nu_R$ mass bounds}
%---------------------------------------------------------
\begin{frame}
\frametitle{Correlating $W_R$ and $\nu_R$ mass bounds}
\centering
\begin{columns}
\begin{column}{0.33\textwidth}
\includegraphics[scale=0.25]{./figures/WRNRexcl_gLeqgR_eeqq_CMS.png} \\
\includegraphics[scale=0.25]{./figures/WRNRexcl_gLeqgR_mumuqq_CMS.png}
\end{column}
\pause
\begin{column}{0.33\textwidth}
\includegraphics[scale=0.25]{./figures/WRNRexcl_largetan_eeqq.png} \\
\includegraphics[scale=0.25]{./figures/WRNRexcl_largetan_mumuqq.png}
\end{column}
\pause
\begin{column}{0.33\textwidth}
\includegraphics[scale=0.25]{./figures/WRNRexcl_difCKMR_eeqq.png} \\
\includegraphics[scale=0.25]{./figures/WRNRexcl_difCKMR_mumuqq.png}
\end{column}
\end{columns}
\end{frame}
\section{Conclusion}
\begin{frame}
\frametitle{Conclusion}
\begin{table}[]
\centering
\small
\begin{tabular}{c|c|c|c}
\hline \hline
&\multicolumn{2}{c|}{ }&\\
Scenario I: $M_{\nu_R} > M_{W_R} $ & \multicolumn{2}{c|}{Lower limits for $ M_{W_R}$ (GeV)}&Exclusion \\
\cline{2-3}
&&&channel \\
&Expected&Observed&\\
\hline \hline
$ g_L = g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 3450 & 3600 & $W_R \to tb$ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 2700 & 2700 & $W_R \to tb$ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 2675 & 2675 & $W_R \to tb$ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L \neq V_{\rm CKM}^R$& 1940 & 2360& $W_R \to tb$ \\ \hline \hline
$ g_L = g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 3625 & 3620 & $W_R \to jj$ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 2700 & 2555 & $W_R \to jj$ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 2650 & 2500 & $W_R \to jj$ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L \neq V_{\rm CKM}^R$ & 2010 & 2000 & $W_R \to jj$ \\ \hline \hline
\end{tabular}
\caption{Lower limits for $M_{W_R}$ in GeV, when $M_{\nu_R} > M_{W_R}$.}
\label{tab:ExclusionBoundsI}
% \end{center}
\end{table}
\end{frame}
\begin{frame}
\frametitle{Conclusion}
\begin{table}[]
\centering
\small
\begin{tabular}{c|c|c|c}
\hline \hline
&\multicolumn{2}{c|}{ }&\\
Scenario II: $M_{\nu_R} < M_{W_R} $ & \multicolumn{2}{c|}{Lower limits for $M_{W_R}$ (GeV)}&Exclusion \\[1mm]
\cline{2-3}
&&&channel \\
&Expected&Observed&\\ [1mm] \hline \hline
$ g_L = g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 4420 & 4420 & $W_R \to qqee $ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 3800 & 3800 & $W_R \to qqee $ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 3720 & 3725 & $W_R \to qqee $ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L \neq V_{\rm CKM}^R$ & 3300 & 3100 & $W_R \to qqee $ \\ \hline \hline
$ g_L = g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 4500 & 4420 & $W_R \to qq \mu\mu $ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.01, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 3950 & 3800 & $W_R \to qq\mu \mu $ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L= V_{\rm CKM}^R$ & 3900 & 3750 & $W_R \to qq \mu \mu $ \\ \hline
$ g_L \neq g_R $, $\tan\beta$ = 0.5, $V_{\rm CKM}^L\neq V_{\rm CKM}^R$ & 3400 & 3350 & $W_R \to qq\mu \mu $ \\ \hline \hline
\end{tabular}
\caption{Lower limits for $M_{W_R}$ in GeV when $M_{\nu_R} < M_{W_R} $.}
\end{table}
\end{frame}
\begin{frame}
\frametitle{Conclusion}
\begin{table}[]
\begin{center}
%\large
\begin{tabular}{c||c|c}
\hline
& $\bold{BM\ I :}~ M_{\nu_R} > M_{W_R} $ & $\bold{BM\ II :}~ M_{\nu_R} < M_{W_R} $ \\ \hline \hline
$m_{W_R}$ [GeV] & 2557 & 3689 \\ \hline
$m_{\nu_R}$ [GeV] & 16797 & 1838 \\ \hline
$\sigma$(pp $\to$ $W_R $) [fb] @13 TeV & 48.7 & 3.98 \\ \hline
$\sigma$(pp $\to$ $W_R $) [fb] @27 TeV & 478.0 & 77.3 \\ \hline \hline
BR($W_R \to t\overline{b}$) [\%] & 26.3 & 19.9 \\ \hline
BR($W_R \to jj$) [\%] & 58.6 & 45.8 \\ \hline
BR($W_R \to \nu_R \ell $) [\%] & - & 6.5 (each family) \\ \hline
BR($W_R \to h_1 W_L $) [\%] & 1.8 & 1.5 \\ \hline
BR($W_R \to W_L Z $) [\%] & 2.0 & 1.6 \\ \hline \hline
BR($\nu_R \to \ell qq^\prime $) [\%] & - & 65.3 \\ \hline
BR($\nu_R \to W_ L \ell$) [\%] & 1.1$\times 10^{-4}$ & 33.1 \\ \hline
BR($\nu_R \to W_ R \ell$) [\%] & 99.9 & - \\ \hline \hline
\end{tabular}
\caption{Related Branching Ratios and Cross Sections for {\bf BM I} and {\bf BM II}.}
\label{tab:xSectionBenchmark}
\end{center}
\end{table}
\end{frame}
\begin{frame}
\frametitle{}
\centering
{\Huge \textit{Thank you!}}
\end{frame}
\end{document}