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\author{ {\fontspec{Trebuchet MS}M.Chrz\k{a}szcz, A.Mauri, N.Serra} (Universit\"{a}t Z\"{u}rich)}
\institute{UZH}
\title[Light inflaton model hunting guide]{Light inflaton model hunting guide}
\date{25 September 2014}


\begin{document}
\tikzstyle{every picture}+=[remember picture]

{
\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
\begin{frame}[c]%{\phantom{title page}}
\begin{center}
\begin{center}
	\begin{columns}
		\begin{column}{0.75\textwidth}
			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Light inflaton model hunting guide}
		\end{column}
                \begin{column}{0.02\textwidth}
                  {~}
                  \end{column}
                \begin{column}{0.23\textwidth}
                 % \hspace*{-1.cm}
                  \vspace*{-3mm}
                  \includegraphics[width=0.6\textwidth]{lhcb-logo}
                  \end{column}

	\end{columns}
\end{center}
	\quad
	\vspace{3em}
\begin{columns}
\begin{column}{0.44\textwidth}
\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}

\end{column}
\begin{column}{0.53\textwidth}
\includegraphics[height=1.3cm]{uzh-transp}
\end{column}
\end{columns}

\vspace{1em}
	\textcolor{black}{With A. Mauri, N.Serra}\normalsize\\
\vspace{0.5em}

	\textcolor{normal text.fg!50!Comment}{RD meeting, CERN\\ November 2, 2016}
\end{center}
\end{frame}
}


\begin{frame}[c]{The inflaton model}
	\begin{minipage}{\textwidth}
	\begin{small}
\ARROW The model is extremely simple:
\begin{align*}
V(H,S)=V_H + V_{\rm mix} +  V_S,
\end{align*}
where
\begin{align*}
V_H = -\mu^2 H^{\dag}H + \lambda_H (H^{\dag} H)^2\\
V_{\rm mix} = \frac{a_1}{2}\left(H^{\dag}H\right)S + \frac{a_2}{2}\left(H^{\dag}H\right)S^2\\
V_S = \frac{b_2}{2}S^2 + \frac{b_3}{3}S^3 + \frac{b_4}{4}S^4
\end{align*}
\ARROW Now the Lagrangian needs to be written in physical degrees of freedom:
\begin{itemize}
\item You start by minimizing the scalar potential.
\item Then you expand the group states (linear terms in h, s expansion vanish). 
\item Then you generate the mass is given(see backup for details):
\end{itemize}
\begin{align*}
\mathcal{L}_{mass}=-\frac{1}{2}
\begin{pmatrix}
h \\
s
\end{pmatrix}^T
\begin{pmatrix}
A & C \\
C & B
\end{pmatrix}
\begin{pmatrix}
s^{\prime}\\
h^{\prime}
\end{pmatrix}
\end{align*}

\end{small}
	\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{The inflaton model}
	\begin{minipage}{\textwidth}{~}\\
	\begin{small}
\ARROW You can diagonalize the matrix by orthogonal transformation:
\begin{align*}
\begin{pmatrix}
h \\
s
\end{pmatrix}=
\begin{pmatrix}
\sin \theta & \cos \theta \\
\cos \theta & \sin \theta
\end{pmatrix}
\begin{pmatrix}
s^{\prime}\\
h^{\prime}
\end{pmatrix}
\end{align*}
where
\begin{align*}
\cos \theta = 1 +\mathcal{O}(y^2),~~
\sin \theta = y + \mathcal{O}(y^3)~~
y = \frac{2C}{A-B}
\end{align*}
\ARROW For small mixing:
\begin{align*}
m^2_{S^{\prime}} \sim B - \frac{1}{4}(A-B)y^2
\end{align*}
\ARROW So now comes something that is the most important; all the interaction with the SM is done just by assuming it's Higgs and inserting: $h \to \sin \theta s$

\ARROW For example:
\begin{align*}
\mathcal{L}_Y = - m_f h \bar{\psi}_f \psi_f + {\rm h.c.}~~~~~\mapsto \mathcal{L}_{S^{\prime}ff}=  - m_f \sin \theta s \bar{\psi}_f \psi_f + {\rm h.c.}
\end{align*}


\end{small}
	\end{minipage}
		\vspace*{2.cm}
\end{frame}


\begin{frame}[c]{The inflaton model, so what?}
	\begin{minipage}{\textwidth}
	\begin{small}
\ARROW For a given mixing angle $\sin \theta$ and the inflaton mass we can calculate it's width:
\begin{align*}
\Gamma_{\ell \ell} = \frac{\sin^2 \theta}{8 \pi \nu^2}m_{\ell}^2 m_S \left(1-\frac{4m^2_{\ell}}{m^2_S} \right)^{\frac{3}{2}}\\
c \tau_S \approx 60 \times \left(\frac{0.01}{\sin \theta}\right)^2 \left(\frac{500}{m_S}\right)^3 \rm [mm]
\end{align*}

\ARROW Since in experiment we set limits in 2D space: $(m_S, c \tau_S)$ we can map it to: $(m_S, \sin \theta)$.\\
\ARROW Now the misunderstanding started with ''Light inflaton Hunter's Guide'' D. Gorbunov:
\begin{equation}                                                                                                                                                
{\rm Br} (\PB \to \chi X_s) ~\sim 10^{-6}\left(1-\frac{m^2_{\chi}}{m^2_b}\right) \left(\frac{\beta}{\beta_0}\right) \left(\frac{300{\rm MeV}}{m_{\chi}}\right)  
\end{equation}                                                                                                                                                  
and:
                                                                                                                                                   ''where Xs stands for strange meson channel mostly saturated by a sum of pseudoscalar and vector kaons.''



\end{small}
	\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{The inflaton model, so what?}
	\begin{minipage}{\textwidth}
	\begin{small}
\ARROW In the interpretation of the $\PB \to \PKstar \xi$ they followed Gorbunov and assumed that the $33\%$ of $X_s$ are a $\PKstar$ and used the above formula.\\
\ARROW We followed a different approach and calculated the exclusive widths:
\begin{align*}
\mathcal{M}=-\frac{1}{2}c_h \theta \langle K \vert \bar{s}b \vert B \rangle = -\frac{1}{2} c_h \theta \frac{M_B^2-M_K^2}{m_b-m_s} f_0(m_s^2)
\end{align*}
\begin{align*}
\Gamma_{\PB \to \PKstar \chi} = \frac{c_h \theta^2}{64 \pi} \lambda^{1/2}(1,\frac{M_K^2}{M_B^2}, \frac{M_S^2}{M_B^2})f_0(m_S^2) \frac{(M_B^2 - M_K^2)}{M_B(m_b- m_s)^2}
\end{align*}

\ARROW Now after we calculate this our self we found the solution in the literature \href{http://journals.aps.org/prd/pdf/10.1103/PhysRevD.83.054005}{ B.Batel et. al}


\end{small}
	\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{The inflaton model, so what?}
	\begin{minipage}{\textwidth}
	\begin{small}
	{~}\\
\ARROW Now if we look how many of the $X_s$ are $\PK$ and $\PKstar$ we define the variables:
\begin{align*}
r_K = \frac{\PB \to \PK \chi}{\PB \to X_s \chi}~~~~r_{\PKstar} = \frac{\PB \to \PKstar \chi}{\PB \to X_s \chi}
\end{align*}
\includegraphics[width=0.45\linewidth]{images/rk_graph.png}
\includegraphics[width=0.45\linewidth]{images/rkstar.png}  \\
{~}\\
\ARROW So the assumption about $33~\%$ was generous assumption.

\end{small}
	\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{Cross check of the calculations}
	\begin{minipage}{\textwidth}
	\begin{small}
	{~}\\
\ARROW So we have cross-checked this calculations with old Higgs papers. \\
\ARROW In the 80s they thought that Higgs might he light enough that it can be produced in $\PB$ decays.\\
\ARROW From Haber, et al.:
\begin{center}
\includegraphics[width=0.45\textwidth]{images/higgs.png}
\end{center}
\ARROW So everything is consistent.
\end{small}
	\end{minipage}
		\vspace*{2.cm}
\end{frame}


\begin{frame}{The result}
	\begin{minipage}{\textwidth}


\begin{center}
\includegraphics[angle=-90,width=0.9\textwidth]{{images/Inflaton_parameter_space_log}.pdf}
\end{center}


	\end{minipage}
		\vspace*{2.cm}
\end{frame}




\backupbegin

\begin{frame}\frametitle{Backup}

\ARROW ves:
\begin{align*}
\langle H \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}
0\\
\nu
\end{pmatrix}\\
\langle S \rangle = x.
\end{align*}


\ARROW Minimalization:
\begin{align*}
0 = -\mu^2 + \lambda_H \nu^2 + a_1 x^2  + \frac{1}{2} x^2\\
0 = b_2 + b_3 x+ b_4 x^2 + \frac{a_1\nu^2}{4 x_0} + \frac{a_2 \nu_0^2}{2}
\end{align*}

\end{frame}




\begin{frame}\frametitle{Backup}

\ARROW generate the mass:
\begin{align*}
\mathcal{L}_{mass}=-\frac{1}{2}
\begin{pmatrix}
h \\
s
\end{pmatrix}^T
\begin{pmatrix}
A & C \\
C & B
\end{pmatrix}
\begin{pmatrix}
s^{\prime}\\
h^{\prime}
\end{pmatrix}
\end{align*}

\begin{align*}
\cos \theta = 2(1+y^{-2}(1-\sqrt{1+y^2}))^{-0.5}\\
\sin \theta = 2(1+y^{-2}(1+\sqrt{1+y^2}))^{-0.5}\\
y=\frac{2C}{A-B}
\end{align*}
\ARROW Mass eigenvalues:
\begin{align*}
m^2_{h^{\prime}/s^{\prime}} = \frac{1}{2}\left[A + B \pm (A-B)\sqrt{1+y^2}\right]
\end{align*}

\end{frame}



\begin{frame}\frametitle{Naturalnes}

\ARROW If the $\sin \theta \ll 1$
\begin{align*}
m^2_{h^{\prime}} \sim 2 \lambda_H \nu^2\\
m^2_{s^{\prime}} \sim b_3 x + 2b_4 x^2 -\frac{a_1\nu^2}{4 x}
\end{align*}
\ARROW Since Higgs is $125\GeV$ then $m_S \sim \mathcal{O}(1)$

\end{frame}

\backupend

\end{document}
