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usage of of the blending \\technique in HEP!\\ -\ARROW Each $\tau$ source ha it's own ''targer'' classifier.\\ +\ARROW Each $\tau$ source has it's own ''targer'' classifier.\\ \ARROW 5 classifiers are then ''blended'' together.\\ \ARROW Gain $6\%$ on signal efficiency.\\ \begin{exampleblock}{} @@ -1273,11 +1273,11 @@ {\tiny{ \eqa{\label{dist} \frac{d^4\Gamma}{dq^2\,d\!\cos\theta_K\,d\!\cos\theta_l\,d\phi}&=&\frac9{32\pi} \bigg[ -J_{1s} \sin^2\theta_K + J_{1c} \cos^2\theta_K + (J_{2s} \sin^2\theta_K + J_{2c} \cos^2\theta_K) \cos 2\theta_l\nn\\[1.5mm] -&&\hspace{-2.7cm}+ J_3 \sin^2\theta_K \sin^2\theta_l \cos 2\phi + J_4 \sin 2\theta_K \sin 2\theta_l \cos\phi + J_5 \sin 2\theta_K \sin\theta_l \cos\phi \nn\\[1.5mm] -&&\hspace{-2.7cm}+ (J_{6s} \sin^2\theta_K + {J_{6c} \cos^2\theta_K}) \cos\theta_l -+ J_7 \sin 2\theta_K \sin\theta_l \sin\phi + J_8 \sin 2\theta_K \sin 2\theta_l \sin\phi \nn\\[1.5mm] -&&\hspace{-2.7cm}+ J_9 \sin^2\theta_K \sin^2\theta_l \sin 2\phi \bigg]\,, +\frac{3}{4} (1- F_L) \sin^2\theta_K + F_L \cos^2\theta_K \nn\\[1.5mm] &&\hspace{-2.7cm} + \left(\frac{1}{4}(1-F_L) \sin^2\theta_K - F_L \cos^2\theta_K\right) \cos 2\theta_l\nn\\[1.5mm] +&&\hspace{-2.7cm}+ S_3 \sin^2\theta_K \sin^2\theta_l \cos 2\phi + S_4 \sin 2\theta_K \sin 2\theta_l \cos\phi + S_5 \sin 2\theta_K \sin\theta_l \cos\phi \nn\\[1.5mm] +&&\hspace{-2.7cm}+ (S_{6s} \sin^2\theta_K + {S_{6c} \cos^2\theta_K}) \cos\theta_l ++ S_7 \sin 2\theta_K \sin\theta_l \sin\phi + S_8 \sin 2\theta_K \sin 2\theta_l \sin\phi \nn\\[1.5mm] +&&\hspace{-2.7cm}+ S_9 \sin^2\theta_K \sin^2\theta_l \sin 2\phi \bigg]\,, \nonumber} }} $\color{JungleGreen}{\Rrightarrow}$ This is the most general expression of this kind of decay. @@ -2208,6 +2208,11 @@ } \only<2>{ \begin{center} +\includegraphics[width=0.6\textwidth]{images/charmloop2.png} +\end{center} +} +\only<3>{ +\begin{center} \includegraphics[width=0.6\textwidth]{images/charmloop3.png} \end{center} } @@ -2216,21 +2221,6 @@ \vspace*{2.1cm} \end{frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame}{If not NP?} -{~} - \begin{minipage}{\textwidth} -\begin{itemize} -\item How about our clean $P_i$ observables? -\item The QCD cancel as mentioned only at leading order. -\item Comparison to normal observables with the optimised ones. -\end{itemize} -\includegraphics[width=0.9\textwidth]{images/C9_S_P.png} - - - \end{minipage} - \vspace*{2.1cm} -\end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -2381,6 +2371,15 @@ \end{frame} +\begin{frame} +\begin{center} +\includegraphics[width=0.9\textwidth]{images/criv.png} +\end{center} + + +\ARROW Stolen from A.Crivelin +\end{frame} + @@ -2674,6 +2673,266 @@ \vspace*{2.1cm} \end{frame} +\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} + +\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} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}{If not NP?} +{~} + \begin{minipage}{\textwidth} +\begin{itemize} +\item How about our clean $P_i$ observables? +\item The QCD cancel as mentioned only at leading order. +\item Comparison to normal observables with the optimised ones. +\end{itemize} +\includegraphics[width=0.9\textwidth]{images/C9_S_P.png} + + + \end{minipage} + \vspace*{2.1cm} +\end{frame} + + + + \backupend \end{document}