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% Copyright 2003 by Till Tantau <tantau@cs.tu-berlin.de>.
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%%%%%%%%%%%%%%%%%%%%

\newcommand{\glFC}[1]{#1-g$\ell$FC}
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\newcounter{Transparencia}

\title[$\delta a_\ell$ in g$\ell$FC-2HDM]{Electron and muon $g-2$ anomalies in general flavour conserving two Higgs doublets models}


\author[Miguel Nebot]{Miguel Nebot}

\institute[
\raisebox{-2pt}{\includegraphics[height=0.25cm]{Figs/logos/logo_ific.png}}\,\raisebox{-2pt}{\includegraphics[height=0.25cm]{Figs/logos/logo_uv.png}\ {\tiny\stepcounter{Transparencia}\arabic{Transparencia}}}
]{
{IFIC -- U. of Valencia}\\ \vspace{0.25cm}
\includegraphics[height=0.5cm]{Figs/logos/logo_ific.png}\quad \includegraphics[height=0.5cm]{Figs/logos/logo_uv.png}
\\ \vspace{0.25cm}
{\texttt{Miguel.Nebot@uv.es} }
}

\date{
Webinar -- October 16$^{th}$ 2020\\
{\colA International Institute of Physics}\\
{\colA Federal University of Rio Grande do Norte}
}

% \date{
% \begin{minipage}[l]{0.65\textwidth}
% {\colA\bf 5$^{\text{rd}}$ Lisbon Minischool on\\ Particle and\\ Astroparticle Physics}\\
% Costa da Caparica,\\ February 5$^{\textrm{th}}$ 2020
% \end{minipage}
% \begin{minipage}[r]{0.3\textwidth}
% \includegraphics[height=1.5cm]{Figs/logos/Poster_5th-School_Caparica_2.png}
% \end{minipage}
% }

\begin{document}
\addtocounter{Transparencia}{-1}
\maketitle




%%%%%
\begin{frame}
\frametitle{Outline}
\begin{enumerate}
{\colA 
\item Motivation
\item Flavour conserving 2HDMs \glFC{I} and \glFC{II}
\item New contributions $\delta a_\ell$
\item Constraints
\item Results
}
\end{enumerate}
\vspace{1cm}

Based on work done in collaboration with:\\
\hfill {\colA Francisco J. Botella \& Fernando Cornet-Gómez}\\
{\small
\hfill \href{https://inspirehep.net/literature/1799192}{\colA\underline{\ding{43}}}\ \texttt{arXiv:2006.01934}, PRD102 (2020)\\
\hfill \href{https://inspirehep.net/literature/1663949}{\colA\underline{\ding{43}}}\ \texttt{arXiv:1803.08521}, PRD98 (2018)
}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{Motivation}
Two ``anomalies'' in the anomalous magnetic moments of $e$ and $\mu$
\[\colR
\begin{aligned}
&\delta a_e\equiv a_e^{\rm Exp}-a_e^{\rm SM}=-(8.7\pm 3.6)\times 10^{-13}\\
&\delta a_\mu\equiv a_\mu^{\rm Exp}-a_\mu^{\rm SM}=+(2.7\pm 0.9)\times 10^{-9}
\end{aligned}
\]
\hfill where $a_\ell=(g_\ell-2)/2$
\begin{multline*}
\text{N.B. vertex }\bar\ell\ell A^\mu\ \raisebox{-6pt}{\includegraphics[width=0.8cm]{Figs/ffA.pdf}} \quad \gamma^\mu\,\to\,\Gamma^\mu=\gamma^\mu F_1(q^2)+i\frac{\sigma^{\mu\nu}q_\nu}{2m_\ell}F_2(q^2),\\ g_\ell=2[F_1(0)+F_2(0)]\ \Rightarrow\ {\colA a_\ell}=F_2(0)
\end{multline*}
{\colR Important}: \emph{opposite} signs!
\[
\frac{\delta a_e^{\rm Exp}}{\delta a_\mu^{\rm Exp}}\simeq -\left(\frac{m_e}{m_\mu}\right)^{\colR 1.508}
\]

\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\item Opposite sign eliminates many New Physics solutions
\item Not only the sign, e.g. if $\delta a_\ell\propto m_\ell^2$
\[
\frac{\delta a_e}{\delta a_\mu}=\left(\frac{m_e}{m_\mu}\right)^2=\left(\frac{m_e}{m_\mu}\right)^{\colA 0.492}\left(\frac{m_e}{m_\mu}\right)^{\colR 1.508}\simeq -0.072\, \frac{\delta a_e^{\rm Exp}}{\delta a_\mu^{\rm Exp}}
\]
\item If the origin of both anomalies is beyond SM, some sort of effective decoupling between $e$ and $\mu$ should be in place
\item 2 Higgs Doublets Models (2HDMs) incorporate new flavour structures that can implement that property
\begin{itemize}
\item not in symmetry-shaped 2HDMs of types I, II, X, Y (new couplings proportional to masses)
\item not in ``aligned 2HDMs'' (proportionality to masses again)
\item maybe in general flavour conserving 2HDMs (gFC-2HDMs)!
\end{itemize}
\end{itemize}
\end{frame}
%%%%%

%%%%%
% \begin{frame}
% \frametitle{}

% \end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{2HDMs}
\begin{itemize}
\item In 2HDMs the Yukawa sector is
\begin{multline*}
\mathscr L_{\rm Y}=
 -\wQLb{}\left(\SD{1}\matYukD{1}+\SD{2}\matYukD{2}\right)\wdR{}
 -\wQLb{}\left(\SDti{1}\matYukU{1}+\SDti{2}\matYukU{2}\right)\wuR{}\\
 -\wLLb{}\left(\SD{1}\matYukL{1}+\SD{2}\matYukL{2}\right)\wlR{}+\Hc
\end{multline*}
\hfill N.B. $\SDti{j}=i\sigma_2\SDc{j}$, neutrinos are massless
\item Expansion around vacuum appropriate for electroweak symmetry breaking
\[
\SD{j}=e^{i\theta_j}\begin{pmatrix}\varphi^+_j\\ \frac{\vev{j}+\rho_j+i\eta_j}{\sqrt 2}\end{pmatrix}
\]
\item Higgs basis, $\cb\equiv\cos\beta=\frac{\vev{1}}{\vev{}}$, $\sb\equiv \sin \beta =\frac{\vev{2}}{\vev{}}$
\[
\begin{pmatrix}\Hv\\ \Ho\end{pmatrix}=\HbROT\,
\begin{pmatrix}e^{-i\theta_1}\SD{1}\\ e^{-i\theta_2}\SD{2}\end{pmatrix},\quad \text{with}\quad 
\HbROT=\begin{pmatrix}\phantom{-}\cb & \sb\\ -\sb & \cb \end{pmatrix},\ \HbROTt=\HbROTinv
\]
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{2HDMs}
\begin{itemize}
\item Higgs basis
\[
\langle \Hv\rangle=\frac{\vev{}}{\sqrt 2}\left(\begin{matrix}0\\ 1\end{matrix}\right),\quad \langle \Ho\rangle=\left(\begin{matrix}0\\ 0\end{matrix}\right),\qquad \vev{}^2=\vev{1}^2+\vev{2}^2=\frac{1}{\sqrt{2}G_F} 
\]
\begin{equation*}
H_{1}=\begin{pmatrix} G^+\\ \frac{v+\nHH+iG^0}{\sqrt{2}}\end{pmatrix},\quad 
H_{2}=\begin{pmatrix} \cHp\\ \frac{\nHR+i\nHI}{\sqrt{2}}\end{pmatrix}
\end{equation*}
\begin{itemize}
\item would-be Goldstone bosons $G^{0}$, $G^{\pm}$
\item physical charged scalar $H^{\pm}$
\item neutral scalars $\{\nHH,\nHR,\nHI\}$, not the mass eigenstates
\end{itemize}
\item Yukawa couplings again
\begin{multline*}
\mathscr L_{\rm Y}=
 -\frac{\sqrt{2}}{v}\wQLb{}\left(\Hv\wmatMD+\Ho\wmatND\right)\wdR{}
 -\frac{\sqrt{2}}{v}\wQLb{}\left(\Hvti\wmatMU+\Hoti\wmatNU\right)\wuR{}\\
 -\frac{\sqrt{2}}{v}\wLLb{}\left(\Hv\wmatML+\Ho\wmatNL\right)\wlR{}+\Hc
\end{multline*}
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{2HDMs}
\begin{itemize}
\item Only the neutral component ($\downarrow$) of $\Hv$ has non-vanishing vev\\ \hfill $\Rightarrow$ $\wmatMf$ give the mass matrices, $f=u,d,\ell$
\item Usual bi-unitary changes into the different fermion mass bases
\begin{multline*}
\mathscr L_{\rm Y}= 
 -\frac{\sqrt{2}}{v}\QLb{}\left(\Hv{\colA\matMD}+\Ho{\colA\matND}\right)\dR{}
 -\frac{\sqrt{2}}{v}\QLb{}\left(\Hvti{\colA\matMU}+\Hoti{\colA\matNU}\right)\uR{}\\
 -\frac{\sqrt{2}}{v}\LLb{}\left(\Hv{\colA\matML}+\Ho{\colA\matNL}\right)\lR{}+\Hc
\end{multline*}
where
\begin{itemize}
\item {\colA$\matMf{f}$} are the diagonal fermion mass matrices
\item {\colA$\matNf{f}$} are the new flavour structures\\ \hfill (the ones that may explain the anomalies!)
\end{itemize}
\end{itemize}
\end{frame}
%%%%%


%%%%%
\begin{frame}
\frametitle{2HDMs}
{\small 
\[
\begin{aligned}
\mathscr L_{\rm Y}=&  -\wQLb{}\left(\SD{1}{\colA\matYukD{1}}+\SD{2}{\colA\matYukD{2}}\right)\wdR{}  -\wQLb{}\left(\SDti{1}{\colA\matYukU{1}}+\SDti{2}{\colA\matYukU{2}}\right)\wuR{}\\
&\quad -\wLLb{}\left(\SD{1}{\colA\matYukL{1}}+\SD{2}{\colA\matYukL{2}}\right)\wlR{}+\Hc\\
=& -\frac{\sqrt{2}}{v}\QLb{}\left(\Hv{\colA\matMD}+\Ho{\colA\matND}\right)\dR{} -\frac{\sqrt{2}}{v}\QLb{}\left(\Hvti{\colA\matMU}+\Hoti{\colA\matNU}\right)\uR{}\\
 &\quad-\frac{\sqrt{2}}{v}\LLb{}\left(\Hv{\colA\matML}+\Ho{\colA\matNL}\right)\lR{}+\Hc
\end{aligned}
\]
}
\begin{itemize}
\item Natural Flavour Conservation: only one Yukawa matrix $\neq 0$ in each sector (e.g. $\mathbb{Z}_2$ symmetry, types I, II, X, Y)  $\Rightarrow$ {\colA$\matNf{f}=\pm t_\beta^{\mp 1}\matMf{f}$}
\item ``Aligned'' 2HDM: {\colA$\matNf{f}=\zeta_f\matMf{f}$}\\ \hfill RGE: unstable quark sector, stable lepton sector
\item General flavour conserving: {\colA diagonal $\matNf{f}$}\\ \hfill RGE: unstable quark sector, stable lepton sector
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The \glFC{I} and \glFC{II} models}
Finally
\begin{itemize}
\item Model \glFC{I} is defined by
\begin{equation*}
{\colA\matNU}=\tbinv\matMU,\quad {\colA\matND}=\tbinv\matMD,\quad {\colR\matNL}=\text{diag}({\colR\nl{e}},{\colR\nl{\mu}},{\colR\nl{\tau}})
\end{equation*}
The couplings $\matNU,\matND$ are the same as in 2HDMs of types I or X
\item Model \glFC{II} is defined by
\begin{equation*}
{\colA\matNU}=\tbinv\matMU,\quad {\colA\matND}=-\tb\matMD,\quad {\colR\matNL}=\text{diag}({\colR\nl{e}},{\colR\nl{\mu}},{\colR\nl{\tau}})
\end{equation*}
The couplings ${\colA\matNU},{\colA\matND}$ are the same as in 2HDMs of types II or Y
\hfill [N.B. $\tb\equiv \tan\beta$ and $\tbinv\equiv\cot\beta$]
\item ${\colR\matNL}$ is diagonal, arbitrary and stable at one loop level under RGE (it remains diagonal): the 
effective decoupling among the new $e$ and $\mu$ couplings required to explain the $g-2$ anomalies\\ \hfill $\leftrightarrow$ independence of ${\colR\nl{e}}$ and ${\colR\nl{\mu}}$
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The \glFC{I} and \glFC{II} models}
Completing the model
\begin{itemize} 
\item since the quark sector is a type I or type II 2HDM, adopt a $\ZZ$ symmetric scalar potential
{\small
\begin{multline*}
\!\!\!\!\!\!\!\!\mathcal V(\SD{1},\SD{2})= 
\mu_{11}^2\SDd{1}\SD{1}+\mu_{22}^2\SDd{2}\SD{2}+\left(\mu_{12}^2\SDd{1}\SD{2}+\Hc\right)
+\lambda_1(\SDd{1}\SD{1})^2+\lambda_2(\SDd{2}\SD{2})^2\\
+2\lambda_3(\SDd{1}\SD{1})(\SDd{2}\SD{2})+2\lambda_4(\SDd{1}\SD{2})(\SDd{2}\SD{1})+\left(\lambda_5(\SDd{1}\SD{2})^2+\Hc\right)
\end{multline*}
}
$\mu_{12}^2\neq 0$ $\Rightarrow$ softly broken $\ZZ$ symmetry
\item Mass matrix of the neutral scalars $\mNSc$, diagonalised by a $3\times 3$ real orthogonal matrix $\ROTmat$
\begin{equation*}
\ROTmatT\,\mNSc\,\ROTmat=\text{diag}(\mh^2,\mH^2,\mA^2)\,,\quad \ROTmatinv=\ROTmatT
\end{equation*}
\item Physical neutral scalars $\{\nh,\nH,\nA\}$:
\begin{equation*}
\begin{pmatrix}\nh\\ \nH\\ \nA\end{pmatrix}=\ROTmatT\begin{pmatrix}\nHH\\ \nHR\\ \nHI\end{pmatrix}
\end{equation*}
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The \glFC{I} and \glFC{II} models}
\begin{itemize}
\item Flavour conserving Yukawa couplings of the neutral scalars
\begin{equation*}
\mathscr L_{N}=-\sum_{S=\nh,\nH,\nA}\,\sum_{f=u,d,\ell}\sum_{j=1}^3\frac{m_{f_j}}{\vev{}}\,S\,\bar f_j(a_{f_j}^S+ib_{f_j}^S\gamma_5)f_j
\end{equation*}
\item Further simplifications
\begin{enumerate}
\item there is no CP violation in the scalar sector,
\item the new Yukawa couplings are real, $\im{\nl{\ell}}=0$
\end{enumerate}
In the scalar sector, this corresponds to
\begin{equation*}
\ROTmat=\begin{pmatrix}\sba & -\cba & 0\\ \cba & \sba & 0\\ 0 & 0 & 1\end{pmatrix},\qquad \left\{\begin{matrix}\sba\equiv \sin(\alpha-\beta)\\ \cba\equiv \cos(\alpha-\beta)\end{matrix}\right.
\end{equation*}
$\alpha-\frac{\pi}{2}$: mixing angle in the change of basis $\{\rho_j,\eta_j\}$ $\to$ $\{G^0,\nh,\nH,\nA\}$
\item Alignment limit ($\nh$ has the SM Higgs couplings): $\sba\to 1$
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The \glFC{I} and \glFC{II} models}
\begin{equation*}
\mathscr L_{N}=-\sum_{S=\nh,\nH,\nA}\,\sum_{f=u,d,\ell}\sum_{j=1}^3\frac{m_{f_j}}{\vev{}}\,S\,\bar f_j(a_{f_j}^S+ib_{f_j}^S\gamma_5)f_j
\end{equation*}
Quark couplings
% \begin{table}[h!tb]
\begin{center}
\begin{tabular}{c c|c|c|c|c|}
\cline{3-6}
                                        & \phantom{\Large $I^a$}  & $a_u^S$ & $b_u^S$ & $a_d^S$ & $b_d^S$  \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{\glFC{I}}} & $\nh$ & $\sba+\cba\tbinv$ & 0       & $\sba+\cba\tbinv$ & 0             \\ \cline{2-6} 
\multicolumn{1}{|c|}{}                  & $\nH$ & $-\cba+\sba\tbinv$ & 0       & $-\cba+\sba\tbinv$ & 0             \\ \cline{2-6} 
\multicolumn{1}{|c|}{}                  & $\nA$ & 0       & $-\tbinv$ & 0       &  $+\tbinv$   \\ \hline \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{\glFC{II}}} & $\nh$ & $\sba+\cba\tbinv$ & 0       & $\sba-\cba\tb$ & 0             \\ \cline{2-6} 
\multicolumn{1}{|c|}{}                  & $\nH$ & $-\cba+\sba\tbinv$ & 0       & $-\cba-\sba\tb$ & 0           \\ \cline{2-6} 
\multicolumn{1}{|c|}{}                  & $\nA$ & 0       &   $-\tbinv$  & 0       &    $-\tb$    \\ \hline
\end{tabular}
% \caption{Fermion couplings to neutral scalars.\label{TAB:neutralcouplings}}
\end{center}
% \end{table}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The \glFC{I} and \glFC{II} models}
\begin{equation*}
\mathscr L_{N}=-\sum_{S=\nh,\nH,\nA}\,\sum_{f=u,d,\ell}\sum_{j=1}^3\frac{m_{f_j}}{\vev{}}\,S\,\bar f_j(a_{f_j}^S+ib_{f_j}^S\gamma_5)f_j
\end{equation*}
Lepton couplings
% \begin{table}[h!tb]
\begin{center}
\begin{tabular}{c c|c|c|}
\cline{3-4}
                                        & \phantom{\Large $I^a$} & $a_{\ell}^S$ & $b_{\ell}^S$ \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{\glFC{I}}} & $\nh$ & $\sba+\cba\noml{\ell}$ & 0       \\ \cline{2-4} 
\multicolumn{1}{|c|}{}                  & $\nH$ & $-\cba+\sba\noml{\ell}$ & 0       \\ \cline{2-4} 
\multicolumn{1}{|c|}{}                  & $\nA$ & 0       &   $\noml{\ell}$ \\ \hline \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{\glFC{II}}} & $\nh$ & $\sba+\cba\noml{\ell}$ & 0       \\ \cline{2-4} 
\multicolumn{1}{|c|}{}                  & $\nH$ & $-\cba+\sba\noml{\ell}$ & 0       \\ \cline{2-4} 
\multicolumn{1}{|c|}{}                  & $\nA$ & 0       &   $\noml{\ell}$ \\ \hline
\end{tabular}
% \caption{Fermion couplings to neutral scalars.\label{TAB:neutralcouplings}}
\end{center}
% \end{table}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The \glFC{I} and \glFC{II} models}
\begin{itemize}
\item Absence of CP violation $\Leftrightarrow$ $a_{f}^{S}b_{f}^{S}=0$\\
$\Rightarrow$ absence of new contributions to electric dipole moments, in particular to the electron EDM\\ \hfill (quite constrained  $\abs{d_e}<1.1\times 10^{-29}$ e$\cdot$cm)
\end{itemize}

\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The \glFC{I} and \glFC{II} models}
Yukawa couplings of the charged scalar
\begin{multline*}
\mathscr L_{Ch}=-\frac{1}{\sqrt{2}\vev{}}\sum_{f=q,l}\sum_{j,k=1}^3\left\{\cHm\ferXb{f}{-\frac{1}{2}}{,j}(\alpha_{jk}^f+i\beta_{jk}^f\gamma_5)\ferX{f}{\frac{1}{2}}{,k}\right.\\ \left.+\cHp\ferXb{f}{\frac{1}{2}}{,k} (\alpha_{jk}^{f*}+i\beta_{jk}^{f*}\gamma_5)\ferX{f}{-\frac{1}{2}}{,j}\right\}
\end{multline*} 
\hfill with $q_{+\frac{1}{2},j}=u_{j}$, $q_{-\frac{1}{2},j}=d_{j}$, $l_{+\frac{1}{2},j}=\nu_{j}$, $l_{-\frac{1}{2},j}=\ell_{j}$
% 
% \begin{table}[h!tb]
\begin{center}
% \begin{tabular}{Sc|Sc|Sc|Sc|Sc|}
\begin{tabular}{c|c|c|}
\cline{2-3}
                                  & $\alpha_{ij}^q$                         & $\beta_{ij}^q$  \\ \hline
\multicolumn{1}{|c|}{\glFC{I}}  & $\Vc{ji}\,\tbinv (m_{u_j}-m_{d_i})$      & $\Vc{ji}\, \tbinv (m_{u_j}+m_{d_i})$ \\ \hline
\multicolumn{1}{|c|}{\glFC{II}} & $ \Vc{ji}\,(\tbinv m_{u_j}+\tb m_{d_i})$ & $\Vc{ji}\,(\tbinv m_{u_j}-\tb m_{d_i})$ \\ \hline
\end{tabular}
\end{center}
% \newline\vspace{0.25cm}
\begin{center}
\begin{tabular}{c|c|c|}
\cline{2-3}
                                  &  $\alpha_{ij}^l$         & $\beta_{ij}^l$         \\ \hline
\multicolumn{1}{|c|}{\glFC{I}}    & $-\nrl{\ell_i}\delta_{ij}$ & $\nrl{\ell_i}\delta_{ij}$ \\ \hline
\multicolumn{1}{|c|}{\glFC{II}}   & $-\nrl{\ell_i}\delta_{ij}$ & $\nrl{\ell_i}\delta_{ij}$ \\ \hline
\end{tabular}

% \caption{Fermion couplings to $\cH$.\label{TAB:chargedcouplings}}
\end{center}
% \end{table}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\begin{itemize}
\item Full prediction
\begin{equation*}
{\colA a_\ell^{\rm Th}}=a_\ell^{\rm SM}+{\colR\delta a_\ell}
\end{equation*}
$a_\ell^{\rm SM}$: SM contribution; ${\colR\delta a_\ell}$ the corrections due to the model
\item To solve the discrepancies, the aim is
\[
{\colR\delta a_e}\simeq \delta a_e^{\rm Exp},\qquad {\colR\delta a_\mu}\simeq \delta a_\mu^{\rm Exp}
\]
\hfill within models \glFC{I} and \glFC{II}
\item Introduce $\Delta_\ell$
\begin{equation*}
\delta a_\ell=K_\ell\,\Delta_\ell,\qquad K_\ell=\frac{1}{8\pi^2}\left(\frac{m_\ell}{\vev{}}\right)^2=\frac{1}{8\pi^2}\left(\frac{gm_\ell}{2M_W}\right)^2
\end{equation*}
 $K_\ell$ collect typical factors arising in one loop contributions
\[
K_e\simeq 5.5\times 10^{-14},\qquad K_\mu\simeq 2.3\times 10^{-9}
\]
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\[
\begin{aligned}
&K_e\simeq 5.5\times 10^{-14},\qquad K_\mu\simeq 2.3\times 10^{-9}\\
&\ \Rightarrow\quad {\colR\Delta_e}\simeq-16,\qquad {\colR\Delta_\mu}\simeq 1
\end{aligned}
\]
\begin{itemize}
\item Contributions at one loop and at two loops (Barr-Zee type) can be relevant
\end{itemize}
\begin{center}
\includegraphics[width=0.3\textwidth]{1loop-Neutral.pdf}\quad
\includegraphics[width=0.3\textwidth]{1loop-Charged.pdf}\quad
\includegraphics[width=0.3\textwidth]{BarrZee-Neutral.pdf}
\end{center}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\begin{itemize}
\item To gain some insight we start with the leading terms in $(m_\ell/\mS)^2$ of the one loop contributions ($\mathrm{S}=\nh,\nH,\nA)$) in the alignment limit $\sba\to 1$
\begin{equation*}
\Delta_{\ell}^{(1)}\simeq {\colR\nl{\ell}^{2}}\left(\frac{{\colA I_{\ell\nH}}}{\mH^{2}}-\frac{{\colA I_{\ell\nA}}-2/3}{\mA^{2}}-\frac{1}{6{\mcH}^{2}}\right)
\end{equation*}
where
\begin{equation*}%
{\colA I_{\ell\mathrm{S}}}=-\frac{7}{6}-2\ln \left(\frac{m_{\ell}}{\mS}\right)
\end{equation*}
\hfill [N.B. Same in both models \glFC{I} \& \glFC{II}]
\item We do not consider light scalars/pseudoscalars,\\ \hfill we assume $\mh<\mH,\mA$
\item Typical values for $\mS\in[0.2;\, 2.0]$ TeV
\begin{equation*}%
{\colA I_{e\mathrm{S}}}\in [24.6;29.2]\,,\qquad {\colA I_{\mu\mathrm{S}}}\in [13.9;18.5]
\end{equation*}%
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\[
\begin{aligned}
&\Delta_{\ell}^{(1)}\simeq {\colR \nl{\ell}^{2}}\left(\frac{{\colA I_{\ell\nH}}}{\mH^{2}}-\frac{{\colA I_{\ell\nA}}-2/3}{\mA^{2}}-\frac{1}{6{\mcH}^{2}}\right)\\
&{\colA I_{e\mathrm{S}}}\in [24.6;29.2]\,,\qquad {\colA I_{\mu\mathrm{S}}}\in [13.9;18.5]
\end{aligned}
\]
\begin{itemize}
\item Dominant contributions mediated by $\nH$ and $\nA$ (logarithmically enhanced):
${\colR\Delta_e}\simeq -16$ can only come from $\nA$: 
\[
{\colR\Delta_e}\simeq -[\nrle]^2I_{e\nA}/\mA^2\ \text{requires}\ {\colA[\nrle]^2\sim \mA^2}
\]
$\Rightarrow$ violate perturbativity requirements for Yukawa couplings or contraints from resonant dilepton searches
\item We \emph{do not} expect an explanation of $\delta a_e$ in terms of one loop contributions
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\[
\begin{aligned}
&\Delta_{\ell}^{(1)}\simeq \nl{\ell}^{2}\left(\frac{I_{\ell\nH}}{\mH^{2}}-\frac{I_{\ell\nA}-2/3}{\mA^{2}}-\frac{1}{6{\mcH}^{2}}\right)\\
&I_{e\mathrm{S}}\in [24.6;29.2]\,,\qquad I_{\mu\mathrm{S}}\in [13.9;18.5]
\end{aligned}
\]
\begin{itemize}
\item Dominant contributions mediated by $\nH$ and $\nA$ (logarithmically enhanced):
${\colR\Delta_\mu}\simeq 1$ can only come from $\nH$: 
\[
{\colR\Delta_\mu}\simeq [\nrlm]^2I_{\mu\nH}/\mH^2\ \text{requires}\ {\colA[\nrlm]^2\sim [\mH/4]^2}
\]
$\Rightarrow$ a not too heavy $\nH$ (in order to have reasonably perturbative $\nl{\mu}$) and $\mA>\mH$ in order to avoid cancellations
\item An explanation of $\delta a_\mu$ in terms of one loop contributions\\ \hfill \emph{might be possible}
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\begin{itemize}
\item Dominant two loop contributions: Barr-Zee diagrams
\item In the same approximation (leading $m_\ell/\mS$ terms, $\sba\to 1$)
\begin{equation*}
\Delta_{\ell}^{(2)}=-\left( \frac{2\alpha}{\pi}\right) \left(
\frac{\nl{\ell}}{m_{\ell}}\right) F
\end{equation*}
$F$ depends on
\begin{itemize} 
\item masses of the fermions in the closed loop,
\item couplings of those fermions to $\nH$ and $\nA$,
\item $\mH$ and $\mA$
\end{itemize}
\begin{equation*}
\begin{aligned}
F_{\rm I} &=\frac{\cot\beta}{3}\left[4(f_{t\nH}+g_{t\nA}) + (f_{b\nH}-g_{b\nA})\right] + \frac{\nrlt}{m_{\tau}}(f_{\tau\nH}-g_{\tau\nA})\,,    \\
F_{\rm II} &=\frac{\cot\beta }{3}\left[4(f_{t\nH}+g_{t\nA}) -\tan^{2}\beta (f_{b\nH}-g_{b\nA})\right] + \frac{\nrlt}{m_{\tau}}(f_{\tau\nH}-g_{\tau\nA})
\end{aligned}
\end{equation*}
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\begin{equation*}
f_{f\mathrm{S}}\equiv f\left(\frac{m_{f}^{2}}{\mS^{2}}\right)\,,\quad g_{f\mathrm{S}}\equiv g\left(\frac{m_{f}^{2}}{\mS^{2}}\right)
\end{equation*}
\begin{center}
% \begin{figure}[!htb]
\includegraphics[width=0.5\textwidth]{2D_log2loopfunctions-vs-logz-crop.pdf}\qquad
% \includegraphics[width=0.3\textwidth]{2D_log2loopf-vs-mS-crop.pdf}\qquad
% \includegraphics[width=0.3\textwidth]{2D_log2loopg-vs-mS-crop.pdf}
% \caption{Loop functions.\label{fig:2loopfunctions}}
% \end{figure}
\end{center}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\begin{equation*}
f_{f\mathrm{S}}\equiv f\left(\frac{m_{f}^{2}}{\mS^{2}}\right)\,,\quad g_{f\mathrm{S}}\equiv g\left(\frac{m_{f}^{2}}{\mS^{2}}\right)
\end{equation*}
\begin{center}
% \begin{figure}[!htb]
% \includegraphics[width=0.25\textwidth]{2D_log2loopfunctions-vs-logz-crop.pdf}\qquad
\includegraphics[width=0.4\textwidth]{2D_log2loopf-vs-mS-crop.pdf}\qquad
\includegraphics[width=0.4\textwidth]{2D_log2loopg-vs-mS-crop.pdf}
% \caption{Loop functions.\label{fig:2loopfunctions}}
% \end{figure}
\end{center}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
\begin{itemize}
\item Relevant aspects
\begin{itemize}
\item $f(z)\simeq g(z)$ in the range of interest
\item the largest values correspond to the heavier fermion
\item the values of $f$ and $g$ for the top quark contributions vary between $0.1$ and $1$
\end{itemize}
\item Considering the {\colA dominant top quark terms}, for {\colA$\tb\simeq 1$} and {\colA$\mH\simeq \mA$}, one can realize that for {\colA$\mH\sim 1-2$ TeV}, {\colR$\delta a_e$} can be explained with {\colA$\nrle\sim 3-7$ GeV} ($\nrle>0$ gives $\delta a_e<0$)
\item To obtain {\colR$\delta a_\mu$} from the same type of contribution
\[
% \delta a_{\mu}=\frac{m_{\mu}\,\nrlm}{m_{e}\,\nrle}\delta a_{e}
%\frac{m_{\mu}\,\nrlm}{\delta a_{\mu}}=\frac{m_{e}\,\nrle}{\delta a_{e}}\ \Rightarrow\ 
{\colR\nrlm}={\colA\frac{\delta a_{\mu}\,m_{e}\,\nrle}{\delta a_{e}\,m_{\mu}}}\simeq {\colR-15\nrle}
\]
Different signs of $\delta a_e$ and $\delta a_\mu$ $\to$ freedom to have\\ \hfill opposite $\nrle$ and $\nrlm$\\
Same assumptions {\colA$\tb\sim 1$}, {\colA$\mA\sim\mH\sim 1-2$ TeV}\\ \hfill $\to$ {\colA$\nrlm\in-[45;105]$ GeV}\\
Argument applies to both models \glFC{I} and \glFC{II}% since $4(f_{t\nH}+g_{t\nA})$ is dominant in both $F_{\rm I}$ and $F_{\rm II}$
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
Beyond {\colA$\tb\sim 1$}
\begin{itemize}
\item {\colA$\tb\ll 1$} excluded in 2HDMs of types I and II by flavour constraints $\Rightarrow$ excluded in \glFC{I} and \glFC{II} as well
\item What about {\colA$\tb\gg 1$} and {\colR$\delta a_\ell$}?
\item The factor {\colA$F$ }
\begin{equation*}
\Delta_{\ell}^{(2)}=-\left( \frac{2\alpha}{\pi}\right) \left(
\frac{\nl{\ell}}{m_{\ell}}\right) {\colA F}
\end{equation*}
\hfill is quite model dependent
\item We consider for reference {\colA$\tb\sim 1$} and {\colA$\mA\sim\mH\sim 1-2$ TeV}, which can reproduce the anomalies, and analyse how to maintain that prediction if, for definiteness, {\colR$\tb\mapsto\tb=50$}
\end{itemize}
\end{frame}
%%%%%

%%%%%
\begin{frame}
% \frametitle{TheD new contributions to $\delta a_\ell$}
\begin{equation*}
% \begin{aligned}
F_{\rm I}=\frac{{\colA\cot\beta}}{3}\left[4(f_{t\nH}+g_{t\nA}) + (f_{b\nH}-g_{b\nA})\right] + \frac{\nrlt}{m_{\tau}}(f_{\tau\nH}-g_{\tau\nA})
% F_{\rm I} &=\frac{\cot\beta}{3}\left[4(f_{t\nH}+g_{t\nA}) + (f_{b\nH}-g_{b\nA})\right] + \frac{\nrlt}{m_{\tau}}(f_{\tau\nH}-g_{\tau\nA})\,,    \\
% F_{\rm II} &=\frac{\cot\beta }{3}\left[4(f_{t\nH}+g_{t\nA}) -\tan^{2}\beta (f_{b\nH}-g_{b\nA})\right] + \frac{\nrlt}{m_{\tau}}(f_{\tau\nH}-g_{\tau\nA})
% \end{aligned}
\end{equation*}
\begin{itemize}
\item In \glFC{I}, the {\colA$\cot\beta$} suppression can be compensated with \emph{smaller} {\colA$\mH$}, {\colA$\mA$} and \emph{larger} {\colA$\nrle$}: e.g. {\colA$\mA\sim\mH\sim 200$ GeV} gives a factor of {\colR 10} with respect to {\colA$\mA\sim\mH\sim 1-2$ TeV}, ${\colA\nrle}\mapsto {\colR 5}{\colA\nrle}$ required to fully compensate the factor of {\colR$50$}
\item That is, {\colR$\delta a_e$} can be reproduced by the two loop contributions in the {\colA$\tb\gg 1$} regime with light {\colA$\nH$}, {\colA$\nA$} and {\colA$\nrle\sim 15-35$ GeV}
\item What about {\colR$\delta a_\mu$}?\\
 {\colA$\nrlm\mapsto 5\nrlm$} gives {\colA$\nrlm\in -[225; 505]$ GeV},\\ \hfill in conflict with perturbativity requirements!\\
Fortunately, for light {\colA$\mH$}, e.g. {\colA$\mH\in[200;400]$ GeV} and {\colA$\abs{\nrlm}\sim \mH/4\in [50;100]$ GeV}, the one loop contributions\\ \hfill can reproduce {\colR$\delta a_\mu$} !
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{The new contributions to $\delta a_\ell$}
Summarizing, \emph{two} types of solutions
\begin{enumerate}
\item {\colR``Solution \solA''}:  scalars with masses in the {\colA 1--2 TeV} range, {\colA$\tb\sim 1$}, and both anomalies produced by {\colA two loop Barr-Zee} contributions.\\
{\colA$\nrle$} in the {\colA few GeV range}, {\colA$\nrlm\sim -15\nrle$}\\
Solution a priori present in \emph{both} \glFC{I} and \glFC{II}
\item {\colR``Solution \solB''}: {\colA$\tb\gg 1$}, lighter $\nH$, {\colA$\mH\in[200;400]$ GeV}, and a heavier $\nA$. {\colR$\delta a_e$} is obtained with {\colA two loop} contributions while {\colR$\delta a_\mu$} is {\colA one loop} controlled. Contrary to solution {\colR\solA}, there is \emph{no} linear relation among {\colA$\nrlm$} and {\colA$\nrle$}, and in fact both signs of {\colA$\nrlm$} can work.\\
 With this simplified analysis, this second kind of solution seems to be available in the \glFC{I} model, but it is not clear if that is the case in the \glFC{II} model too. From the full numerical analysis, the answer is {\bf no}.
\end{enumerate}
\end{frame}
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\begin{frame}
\frametitle{Analysis}
Full numerical analysis
\begin{itemize}
\item Markov chain MonteCarlo
\item Likelihood $\mathcal L=e^{\chi^2/2}$
\item Usual form $\chi^2_{\mathcal O}=\left(\frac{\mathcal O_{\rm Th}-\mathcal O_{\rm Exp}}{\sigma_{\rm Exp}}\right)^2$
\begin{itemize}
\item Observable $\mathcal O$, \item prediction $\mathcal O_{\rm Th}$,\item measurement $\mathcal O_{\rm Exp}\pm\sigma_{\rm Exp}$
\end{itemize}
\item + correlations, asymmetric uncertainties, etc
\item N.B. not for all constraints (e.g. bounds)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Constraints}
Shopping list
\begin{itemize}
\item $\delta a_\ell$ constraints
\item Scalar sector
\item Fermion sector
\item Higgs signal strengths
\item $\cH$ mediated contributions
\begin{itemize}
\item Lepton flavour universality
\item $b\to s\gamma$, $B_q^0$--$\bar B_q^0$ mixing
\end{itemize}
\item $e^+e^-\to \mu^+\mu^-,\tau^+\tau^-$ at LEP
\item LHC searches
\begin{itemize}
\item searches of dilepton resonance: $\sigma(pp\to S)_{[\rm ggF]}\times\BR{S\to \ell^+\ell^-}$, $S=\nH,\nA$ and $\ell=\mu,\tau$
\item searches of charged scalars: $\sigma(pp\to\cH tb)\times\BR{\cH\to f}$, $f=\tau\nu,tb$
\end{itemize}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Constraints: $\delta a_\ell$}

\begin{itemize}
\item The anomalies
\begin{equation*}
\delta a_e^{\rm Exp}=-(8.7\pm 3.6)\times 10^{-13}\,,\quad \delta a_\mu^{\rm Exp}=(2.7\pm 0.9)\times 10^{-9}\,.
\end{equation*}
\item The natural $\delta a_\ell$ constraint
\begin{equation*}
\chi^2_0(\delta a_e,\delta a_\mu)=\left(\frac{\delta a_e-c_e}{\sigma_e}\right)^2+\left(\frac{\delta a_\mu-c_\mu}{\sigma_\mu}\right)^2\,,
\end{equation*}
\item We impose a stronger requirement
\begin{equation*}
\chi^2(\delta a_e,\delta a_\mu)= \left\{\begin{matrix} 0,& \text{if }\chi^2_0(\delta a_e,\delta a_\mu)\leq\frac{1}{4},\\ 10^6\times(\chi^2_0(\delta a_e,\delta a_\mu)-\frac{1}{4}),& \text{if } \chi^2_0(\delta a_e,\delta a_\mu)>\frac{1}{4}.\end{matrix}\right.
\end{equation*}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Constraints: $\delta a_\ell$}
\begin{center}
\includegraphics[width=0.5\textwidth]{2D_typeI-re-SBS-anomlm-vs-anomle-crop.pdf}\\
Allowed $\delta a_\mu$ vs. $\delta a_e$ region
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Constraints}
\begin{itemize}
\item Scalar sector
\begin{itemize}
\item potential bounded from below
\item perturbativity and perturbative unitarity of $2\to 2$ high energy scattering
\item electroweak precision (oblique parameters $S$, $T$)
\end{itemize}
\item Fermion sector: perturbative Yukawa couplings
\begin{equation*}
\chi^2_{\rm Pert}(\nl{\ell})=\left\{\begin{matrix}0,\text{ for }\abs{\nl{\ell}}\leq n_0,\\ \left(\frac{\abs{\nl{\ell}}-n_0}{\sigma_{n_0}}\right)^2,\text{ for }\abs{\nl{\ell}}> n_0.\end{matrix}\right.
\end{equation*}
with $n_0=95$ GeV and $\sigma_{n_0}=1$ GeV
\item Higgs signal strengths:
\begin{itemize}
\item production $\times$ decay signal strengths of the usual channels
\item large lepton couplings: also include $\nh\to\mu^+\mu^-,e^+e^-$ information
\end{itemize}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Constraints}
\begin{itemize}
\item $\cH$ mediated contributions
\begin{itemize}
\item Lepton flavour universality: purely leptonic decays $\ell_j\to\ell_k\nu\bar\nu$, decays with light pseudoscalar mesons $K,\pi\to e\nu,\mu\nu$ and $\tau\to K\nu,\pi\nu$
\item $b\to s\gamma$, $B_q^0$--$\bar B_q^0$ mixing
\end{itemize}
\item $e^+e^-\to \mu^+\mu^-,\tau^+\tau^-$ at LEP\\ \hfill (cross sections up to $\sqrt{s}=208$ GeV)
\item LHC searches
\begin{itemize}
\item searches of dilepton resonance: $\sigma(pp\to S)_{[\rm ggF]}\times\BR{S\to \ell^+\ell^-}$, $S=\nH,\nA$ and $\ell=\mu,\tau$
\item searches of charged scalars: $\sigma(pp\to\cH tb)\times\BR{\cH\to f}$, $f=\tau\nu,tb$
\end{itemize}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{itemize}
\item Importance of LHC searches:\\ \hfill we separate results with/without them
\item Scalar potential with exact $\mathbb{Z}_2$ 
\begin{itemize}
\item gives scalar masses below $1$ TeV (no solution \solA)
\item does not allow $\tb\gg 1$ (no solution \solB)
\end{itemize}
and thus we introduce soft breaking $\mu_{12}^2\neq 0$
\item Results shown for model \glFC{I},\\ in \glFC{II}
\begin{itemize}\item same solution \solA\ regions\item solution \solB\ regions absent\end{itemize}
\end{itemize}

\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.45\textwidth]{2D_typeI-re-SBS-Renlm-vs-Renle-crop.pdf}\\
{\colA$\nrlm$} versus {\colA$\nrle$}
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-Renle-vs-logtb-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-Renlm-vs-logtb-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-Renlt-vs-logtb-crop.pdf}\\
$\nl{\ell}$ couplings versus $\tb$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-Renle-vs-MH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-Renlm-vs-MH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-Renlt-vs-MH-crop.pdf}\\
$\nl{\ell}$ couplings versus $\mH$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-MH-vs-MChH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-MA-vs-MChH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-MA-vs-MH-crop.pdf}\\
Masses of the new scalars
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.4\textwidth]{2D_typeI-re-SBS-MChH-vs-logtb-crop.pdf}\quad
\includegraphics[width=0.4\textwidth]{2D_typeI-re-SBS-MH-vs-logtb-crop.pdf}\\
Masses of the new scalars versus $\tan\beta$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.4\textwidth]{2D_typeI-re-SBS-ppAlmlm-vs-MA-crop.pdf}\quad
\includegraphics[width=0.4\textwidth]{2D_typeI-re-SBS-ppHlmlm-vs-MH-crop.pdf}\\
$[pp]_{\rm ggF}\to\mathrm{S}\to\mu^+\mu^-$ versus $\mS$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRHlele-vs-MH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRHlmlm-vs-MH-crop.pdf}\\
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRHltlt-vs-MH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRHqtqt-vs-MH-crop.pdf}\\
Dominant decay channels of $\nH$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRAHZ-vs-MA-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRAlmlm-vs-MA-crop.pdf}\\
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRAltlt-vs-MA-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRAqtqt-vs-MA-crop.pdf}\\
Dominant decay channels of $\nA$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRChHHW-vs-MChH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRChHlmnu-vs-MChH-crop.pdf}\\
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRChHltnu-vs-MChH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-BRChHqtqb-vs-MChH-crop.pdf}\\
Dominant decay channels of $\cH$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-ppChHlmln-vs-MChH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-ppChHltln-vs-MChH-crop.pdf}\quad
\includegraphics[width=0.3\textwidth]{2D_typeI-re-SBS-ppChHqtqb-vs-MChH-crop.pdf}\\
$pp\to\cH(tb)\to\ell\nu,tb$ versus $\mcH$
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Results}
\begin{center}
\includegraphics[width=0.5\textwidth]{2D_typeI-re-SBS-ILC-lelelmlm-vs-Ecm_examples-crop.pdf}\\
$e^+e^-\to\mu^+\mu^-$ for $\sqrt{s}\in [0.2;1.0]$ TeV,\\ examples of solution \solB\ in model \glFC{I}
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Conclusions}
\begin{itemize}
\item General 2HDMs without SFCNC in the lepton sector are a robust framework (stable under RGE)
\item Lepton flavour universality is broken beyond $\propto$ mass 
\item Two models, I-g$\ell$FC \& II-g$\ell$FC, to address the $\delta a_\ell$ anomalies
\item Quark \& scalar sector as type I, II 2HDMs, softly broken $\mathbb{Z}_2$
\item Two types of solutions in agreement with constraints
\begin{enumerate}
\item ``Heavy'', present in both models
\begin{itemize}
\item new scalars have masses in the 1--2.5 TeV range,
\item $\vev{1}\sim\vev{2}$,
\item both $\delta a_\ell$ from two loop Barr-Zee contributions
\end{itemize}
\item ``Light'', present in I-g$\ell$FC, not in II-g$\ell$FC
\begin{itemize}
\item new scalars have masses below 1 TeV,
\item $\vev{1}\ll\vev{2}$,
\item $\delta a_e$ from two loop Barr-Zee contributions, $\delta a_\mu$ from one loop
\end{itemize}
\end{enumerate}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{}
\begin{center}
{\Huge\colA Thank you!}
\end{center}
\end{frame}
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\begin{frame}
% \frametitle{}
\begin{center}
{\Huge\colA Backup}
\end{center}
\end{frame}
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\begin{frame}
\frametitle{Yukawa couplings}
Neutral scalars
\begin{multline*}
\mathscr L_{\nS\bar ff}=
-\frac{\nS}{v}\,\bar f\,\left[\ROT{1s}\matMf{f}+\ROT{2s}\frac{\matNf{f}+\matNfd{f}}{2}+i\epsilon_{(f)}\ROT{3s}\frac{\matNf{f}-\matNfd{f}}{2}\right]\,f\\
-\frac{\nS}{v}\,\bar f\,\gamma_5\,\left(\ROT{2s}\frac{\matNf{f}-\matNfd{f}}{2}+i\epsilon_{(f)}\ROT{3s}\frac{\matNf{f}+\matNfd{f}}{2}\right)\,f
\end{multline*}
where $s=1,2,3$ in correspondence with $\nS=\nh,\nH,\nA$; $f=u,d,\ell$; in terms proportional to $\ROT{3s}$, $\epsilon_{(d)}=\epsilon_{(\ell)}=-\epsilon_{(u)}=1$
\end{frame}
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\begin{frame}
\frametitle{Yukawa couplings}
Charged scalars
\begin{multline*}
\mathscr L_{\cH ud}=
\frac{\cHm}{\sqrt{2} v}\bar\Fd\left[\CKMdag\matNU-\matNDd\CKMdag+\gamma_5\left(\CKMdag\matNU+\matNDd\CKMdag\right)\right]\Fu\\
+\frac{\cHp}{\sqrt{2} v}\bar\Fu\left[\matNUd\CKM-\CKM\matND+\gamma_5\left(\matNUd\CKM+\CKM\matND\right)\right]\Fd
\end{multline*}
and
\begin{equation*}\label{eq:ChYuk:lep}
\mathscr L_{\cH \ell\nu}=-\frac{\sqrt{2}}{v}\cHp\,\nLb{}\,\PMNSdag\matNL\,\lR{}-\frac{\sqrt{2}}{v}\cHm\,\lRb{}\,\matNLd\PMNS\,\nL{}
\end{equation*}
$\CKM$ and $\PMNS$ are, respectively, the CKM and PMNS mixing matrices (massless neutrinos assumed, one can set $\PMNS\to\id$)
\end{frame}
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\begin{frame}
\frametitle{Loop contributions to $\delta a_\ell$}
\begin{center}
\includegraphics[width=0.3\textwidth]{1loop-Neutral.pdf}\quad
\includegraphics[width=0.3\textwidth]{1loop-Charged.pdf}\quad
\includegraphics[width=0.3\textwidth]{BarrZee-Neutral.pdf}
\end{center}
\end{frame}
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\begin{frame}
\frametitle{One loop contributions to $\delta a_\ell$}
Yukawa interactions of the form 
\begin{equation*}
\mathscr L_{S\ell\ell}=-\frac{m_\ell}{\vev{}}S\bar\ell(a_\ell^S+ib_\ell^S\gamma_5)\ell
\end{equation*}
give one loop contributions
\begin{equation*}
\Delta a_\ell^{(1)}=\frac{1}{8\pi^2}\frac{m_\ell^2}{\vev{}^2}\sum_{S}\left\{[a_\ell^S]^2\left(2I_2(x_{\ell S})-I_3(x)\right)-[b_\ell^S]^2I_3(x_{\ell S})\right\},
\end{equation*}
with $x_{\ell S}\equiv {m_\ell^2}/{m_{S}^2}$ and
\begin{equation*}
\begin{aligned}
&I_2(x)=1+\frac{1-2x}{2x\sqrt{1-4x}}\ln\left(\frac{1+\sqrt{1-4x}}{1-\sqrt{1-4x}}\right)+\frac{1}{2x}\ln x\\
&I_3(x)=\frac{1}{2}+\frac{1}{x}+\frac{1-3x}{2x^2\sqrt{1-4x}}\ln\left(\frac{1+\sqrt{1-4x}}{1-\sqrt{1-4x}}\right)+\frac{1-x}{2x^2}\ln x
\end{aligned}
\end{equation*}
\end{frame}
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\begin{frame}
\frametitle{One loop contributions to $\delta a_\ell$}
For $x\ll 1$,
\begin{equation*}
\begin{aligned}
&I_2(x)\simeq x\left(-\frac{3}{2}-\ln x\right)+x^2\left(-\frac{16}{3}-4\ln x\right)+\mathcal O(x^3)\\
&I_3(x)\simeq x\left(-\frac{11}{6}-\ln x\right)+x^2\left(-\frac{89}{12}-5\ln x\right)+\mathcal O(x^3)
\end{aligned}
\end{equation*}
For $m_\ell\ll m_S$,
\begin{equation*}
\Delta a_\ell^{(1)}=\frac{1}{8\pi^2}\frac{m_\ell^2}{m_S^2}\frac{m_\ell^2}{\vev{}^2}\left\{-[a_\ell^S]^2\left(\frac{7}{6}+\ln\left(\frac{m_\ell^2}{m_S^2}\right)\right)+[b_\ell^S]^2\left(\frac{11}{6}+\ln\left(\frac{m_\ell^2}{m_S^2}\right)\right)\right\}
\end{equation*}
\end{frame}
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\begin{frame}
\frametitle{One loop contributions to $\delta a_\ell$}
Yukawa interactions of the form 
\begin{equation*}
\mathscr L_{C\ell\nu}=-C^-\bar\ell(a_\ell^C+ib_\ell^C\gamma_5)\nu-C^+\bar\nu(a_\ell^{C\ast}+ib_\ell^{C\ast}\gamma_5)\ell
\end{equation*}
give one loop contributions
\begin{equation*}
\Delta a_\ell^{(1)}=-\frac{1}{8\pi^2}\sum_{C}\left\{\abs{a_\ell^C}^2+\abs{b_\ell^C}^2\right\}\,H(x_{\ell C})
\end{equation*}
where $x_{\ell C}=m_\ell^2/m_{C^\pm}^2$, and
\begin{equation*}
H(x)=-\frac{1}{2}+\frac{1}{x}+\frac{1-x}{x^2}\ln(1-x)\,,\quad H(x)\simeq \frac{x}{6}+\frac{x^2}{12}+\mathcal O(x^3)\text{ for } x\ll 1
\end{equation*}
\end{frame}
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\begin{frame}
\frametitle{Two loop contributions to $\delta a_\ell$}
For quarks
\begin{equation*}
\mathscr L_{S\bar ff}=-\frac{m_f}{\vev{}}S\bar f(\alpha_f^S+i\beta_f^S\gamma_5)f
\end{equation*}
The two loop Barr-Zee contributions to the anomalous magnetic moment of lepton $\ell$
\begin{equation*}
\Delta a_\ell^{(2)}=-\frac{\alpha^2}{4\pi^2s_W^2}\frac{m_\ell^2}{M_W^2}\sum_{f}\sum_{S}N_c^fQ_f^2
\left\{
a_\ell^S\alpha_f^Sf(z_{fS})-b_\ell^S\beta_f^Sg(z_{fS})
\right\}
\end{equation*}
$f$: fermions in the closed fermion loop ($N_c^f$ colour, $Q_f$ electric charge, $z_{fS}=m_f^2/m_S^2$)\\
$S$: neutral scalars connecting the closed fermion loop with the external lepton line
\end{frame}
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\begin{frame}
\frametitle{Two loop contributions to $\delta a_\ell$ loop contributions to $\delta a_\ell$}
Loop functions
\begin{equation*}
\begin{aligned}
&f(z)=\frac{z}{2}\int_0^1\!\!\!\! dx\,\frac{1-2x(1-x)}{x(1-x)-z}\,\ln\left(\frac{x(1-x)}{z}\right)\\
&g(z)=\frac{z}{2}\int_0^1\!\!\!\! dx\,\frac{1}{x(1-x)-z}\,\ln\left(\frac{x(1-x)}{z}\right)
\end{aligned}
\end{equation*}
\end{frame}
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