%---------------------------------------------------------------------------------
%       Chapter 3: Top Charm and Higgs, and unexpected trio
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\chapter{Top, Charm, and Higgs, an Uncommon Trio} \label{htc-chapter}

\singlespacing

\epigraph{``\textit{What's in a name? That which we call a rose by any other name would smell as 
sweet}''}{--- William Shakespeare, \emph{Romeo and Juliet}}

\doublespacing % The whole thesis is supposed to be double spaced. The spacing will be switched locally when needed

This chapter describes our first analysis of Flavor Changing Neutral Currents and is summarized 
in our publication, {\color{red} ref: pphtc paper}. This chapter will serve as a more detailed analysis 
of this study, what it does, and why it is important. In regard to the famous line from one of 
Shakespeare's poems, scientists by and large have a very different opinion in regard to names and what
they mean and, moreover, what they can tell you about an object. Turns our wizards and occult societies 
have the same opinion but hopefully that doesn't reflect poorly on us.

    \section{What's in a name?}

    The title of the paper we put out summarizing this result was \emph{Flavor changing heavy Higgs 
    interactions at the LHC} which tells certain readers quite a lot about the project. Let us take 
    a moment to unpack that name a bit and get into what we're really looking at.

        \subsection{Flavor changing}

        As described earlier, the fundamental particle that make up matter as we know it involve quarks,
        leptons, and bosons. The quarks and leptons are divided into three generations and each type of 
        quark or lepton is referred to as a flavor. Generally, when a neutral boson decays into two 
        quarks it will decay into a quark/anti-quark pair, like the dominant decay mode of the Higgs, 
        $h^0 \to b \bar{b}$ {\color{red} ref: PDG}. In certain models like the 2HDM, however, we can get 
        mixed decays from these neutral particles, such as an interaction between a Higgs, a top, and a 
        charm. For a standard model Higgs, a Higgs with a mass of $125$ GeV, the only possible way for 
        this interaction to occur is through a top decay, $t \to c h^0$. If we relax the mass condition 
        on the Higgs and allow it to be heavier than $125$ or even $173$ GeV, then $H^0 \to t \bar{c}$ 
        and $H^0 \to c \bar{t}$ become possible. This is the heavy Higgs range.

        \subsection{Heavy Higgs interactions at the LHC}

        As mentioned above, heacy Higgs refers to Higgs bosons that have a mass larger than that of the 
        currently observed, Standard Model-like Higgs boson. If you recall, when we added a second 
        doublet to the Higgs sector then we actually generated four additional Higgs bosons which we 
        denoted: $H^0 \rm, \ A^0 \rm, \ H^{\pm}$. The CP-Even scalar, $H^0$, we assume to be heavier 
        than the one we have already discovered and so designate it as the heavy Higgs. The CP-odd 
        pseudo-scalar $A^0$ is our other neutral Higgs boson which we designate as the pseudo-scalar. 
        In cases where $m_{H^0} \text{ and } m_{A^0} > m_t = 173$ GeV, then the decays listed above, 
        $H^0 \ / \ A^0 \to c t$ are now possible. These are the decays that we choose to study. 
        %
        \begin{figure}[H]
            \begin{center}
                %
                \includegraphics[width=0.95\textwidth,keepaspectratio]{./figures/gghtc_diagram.eps}
                %
                \caption[Feynman diagram for the $\phi^0 \to tc$ process where we define 
                $\phi^0 = h^0 {\rm,} \ H^0 {\rm,} \ A^0$]{Feynman diagram for the $\phi^0 \to tc$ 
                process where we define $\phi^0 = h^0 {\rm,} \ H^0 {\rm,} \ A^0$. We look at both the 
                $W \to j j$ and the $W \to \ell \nu$ final states to maximize our signal strength.}
                \label{gghtc}
                %
            \end{center}
        \end{figure}
        %

    \section{Strangers in a strange land} \label{htc-roadmap}

    Now, one might wonder, if a decay of $t \to c h^0$ is possible, why haven't we seen it yet, and 
    furthermore, what is hoped to be gained by this study. To answer the first question we need to 
    look at how particles couple in the SM. For starters, let me remind you we are dealing with fields 
    and as such all of the operators and couplings that govern how these fields operate are tensors of 
    some form or another. As it happens, all of our experimental evidence so far supports the construction 
    of the yukawa couplings in the Higgs sector, in the sense that they are all diagonal. The interaction 
    Lagrangian for the Yukawa couplings is as follows,
    %
    \begin{align}
        %
        \mathscpt{L}_{Y} = -\frac{1}{\sqrt{2}} \overline{U} \kappa_{u} h^0 U \label{SMyukawa} \ {\rm,}
        %
    \end{align}
    %
    where $\psi_{L,R}$ refer to the fermion fields, $h^0$ is the SM Higgs field, and the $\kappa_u$ term is
    defined as:
    %
    \begin{align}
        %
        \kappa_u \equiv \begin{pmatrix} \kappa_{uu} & 0 & 0 \\ 0 & \kappa_{cc} & 0 \\ 0 & 0 & \kappa_{tt} \end{pmatrix}.
        \label{kappaUp}
        %
    \end{align}
    %
    In this coupling matrix the terms along the diagonal are the same flavor couplings and the off-diagonal elements 
    describe the strength of the flavor mixing couplings. Since they are all zero in the SM these couplings are forbidden 
    at the tree level. They can exist as loop processes, \cref{SMgghtc}, but those are higher order processes and are less 
    favorable in general as well as being further suppressed by the GIM mechanism, {\color{red} ref: GIM mechanism}.
    %
    \begin{figure}[H]
        \begin{center}
            %
            \includegraphics[width=0.75\textwidth,keepaspectratio]{./figures/gghtc_SM_diagram.eps}
            %
            \caption[Feynman diagram for the $h^0 \to tc$ loop process thath allows for FCNCs in the SM.]{Feynman diagram 
            for the $h^0 \to tc$ loop process that allows for FCNCs in the SM. Not shown are the decay modes of the top 
            in this diagram but they would be the same as in the previous figure.}
            \label{SMgghtc}
            %
        \end{center}
    \end{figure}
    %

    With this in mind, we can safely say that any signatures of a FCNC process would indicate something beyond the SM. 
    Furthermore, by looking at these processes in the context of a general 2HDM we can gather information about how an 
    extended Higgs sector operates. This is specifically important as many of the predominant extensions of the SM rely 
    on such an extension, {\color{red} ref: SUSY paper, other 2HDM models}. The advantage of a general treatment, 
    however, is that we do not require some of the more restrictive assumptions in more complete models. While this does 
    limit our ability to analyze and predict other beyond the SM (BSM) properties and effects it allows our findings to 
    be more versatile and adaptable to the different models we may want to look at.

    When we adopt a general 2HDM we gain several key advantages. First, we can now see FCNCs at the tree level as shown 
    in \cref{gghtc} due to the fact that we now have a new Lagrangian that describes our interactions,
    %
    \begin{align}
        %
	    {\cal L}_Y =& -\frac{1}{\sqrt{2}} \sum_{F = U,D,L}
	     \bar{F}\left[  \left( \kappa^Fs_{\beta-\alpha}+\rho^Fc_{\beta-\alpha} \right) h^0 +
	     \left( \kappa^Fc_{\beta-\alpha}-\rho^Fs_{\beta-\alpha} \right)H^0 \right] F \notag \\
	     %
	    & +\frac{1}{\sqrt{2}} \sum_{F = U,D,L}
	     \bar{F} \left [ i \, {\rm sgn}(Q_F)\rho^F A^0 \right ] P_RF \label{2HDMYukawa} \\
	     %
	    & -\bar{U} \left( V \rho^D P_R - \rho^{U\dagger} V P_L \right) D H^+
	      -\bar{\nu} \left( \rho^L P_R \right) L H^+ + {\rm H.c.} \ , \notag
        %
    \end{align}
    %
    where $U, \ D, \text{ and } L$ are the up-type quarks, down-type quarks, and leptons respectively; 
    $c_{\beta-\alpha} \equiv \cosba$; and $s_{\beta-\alpha} \equiv \sinba$. As can be seen in the Lagrangian above, 
    we have coupling terms to both $h^0$, our SM-like Higgs, and the new Higgs bosons, $H^0 \text{ and } A^0$. In order 
    for the experimentally verified SM to hold we need this new Lagrangian to reproduce SM results while retaining the 
    ability to produce these new effects. The limit in which \cref{2HDMYukawa} becomes \cref{SMyukawa} is called the 
    alignment limit. In the context of our model, we reach this limit as $\sinba \to 1$ by allowing the $\kappa^F$ 
    matrix to be equal to the SM $\kappa$ matrix. This also means that all of our FCNC information is stored in the 
    $\rho^F$ matrices.

    Once again, the model we are using is a general 2HDM so there are no additional simplifying assumptions we can 
    make about the couplings or constraints. For this study, valid parameters that must be considered are the value of 
    $\cosba$ and the diagonal parameters of the $\rho^F$ matrix: 
    $\rho_{tt}{\rm, } \ \rho_{bb}{\rm,} \ \rho_{cc}\text{, and } \rho_{\tau \tau}$. In addition, to to the freedom 
    granted by not imposing additional symmetries we are also not required to be Hermitian in the Yukawa part of the 
    Lagrangian. In practice, this means that the off diagonal terms, $\rho_{tc} \text{ and } \rho_{ct}$, can be very 
    different from one another. In our case this is actually a very helpful scenario as $\rho_{ct}$ is heavily 
    constrained by the CKM matrix, {\color{red} ref: CKM paper}, but $\rho_{tc}$ is not bound by the same constraints. 
    In the context of the neutral Higgs bosons, since they are scalars and pseudo-scalars they have no chirality 
    preference which allows us to adopt a funny \emph{effective} coupling combining the two:
    %
    \begin{align}
        %
        \widetilde{\rho}_{tc} = \frac{1}{\sqrt{2}} \sqrt{ \rho^2_{tc} + \rho^2_{ct} }   \label{EFFrhotc} \ .
        %
    \end{align}
    %

        \subsection{Good news and bad news} \label{htc-teaser}

        The scene is set for our foray into FCNCs as a method for studying the presence of a second Higgs doublet in 
        the most general sense. Now, how does one study a FCNC process and how do you know if what your seeing is 
        correct or not? 
        
        The benchmark of the signal we are looking for is some top and charm pair coming from a single resonant particle. 
        Tops are not very long lived and so they are likely to decay and the dominant mode for that decay is into a $b$ 
        quark and a $W$ boson. The $W$ boson, like the top, does not generally stay around for too long and also decays. 
        About $30\%$ of the time the $W$ will decay into a lepton and a neutrino, say an $e$ or a $mu$, but another $67\%$ 
        of the time it will decay hadronically into quark/anti-quark pair like a $u \bar{d}$ for instance. At detectors, 
        these decay products are all that survive and are visible in the context of analyzing your process. The main 
        challenge with any decay process you want to study is then with picking out the signal of interest when there are 
        other processes producing the same final set. We can control this issue in a couple of ways, one of which is 
        selecting decay modes that are unique in some way and thus do not have a lot of background contamination. On that 
        note, we actually chose to study the leptonic decay mode of the $W$ for our signal as any signal involving $3+$ 
        jets generally has a large background from $t\bar{t}$ production and general QCD processes. The other method of 
        controlling background contamination is to use what are called selection rules. How selection rules work, is they 
        exploit physical properties of your signal that your background is either incapable of producing or exhibits these
        properties at a much lower rate. 

        \subsection{Tools of the trade} \label{htc-cuts}

        As mentioned before, the two big tools we have in our toolbox is carefully selecting our decay modes and controlling 
        how we analyze the data to favor our signal over the background. As phenomenologists we also want to make sure our 
        data sets and analysis reflect the techniques and methods used at the detectors by experimentalists. In a typical 
        detector collision there are roughly $40,000$ events {\color{red} ref: Detector stuff}, of which only about $1\%$ 
        of that is actually readout and collected due to simple limitations of space and readout speed. When a collision 
        occurs the collision and decay information passes through very sophisticated trigger systems the allow the LHC 
        collaborations to pick out \emph{interesting} behavior. Part of the selection rules we use then are to match the 
        LHC triggers used by ATLAS and CMS {\color{red} ref: ATLAS and CMS TDR}. For our particular process we implemented 
        the cuts listed in \cref{cuts-htc}.
        %
        \begin{table}[H]
        \centering
        \begin{tabular}{|c|c|}
        \hline
            ${\rm L} = 25\tablefootnote{$\sqrt{s} = 8$ TeV}{\rm,} \ 30\tablefootnote{$\sqrt{s} = 13 {\rm,} \ 14$ TeV} \text{ fb}^{-1}$ & 
            ${\rm L} = 300 \text{ or } 3000 \text{fb}^{-1}$ \tablefootnote{$\sqrt{s} = 14$ TeV} \\ \hline \hline
        $\slashed{E}_{T} > 20$ GeV & $\slashed{E}_{T} > 40$ GeV \\ \hline
        $p_{T}(j) > 20$ GeV & $p_{T}(j) > 30$ GeV \\ \hline
        $\epsilon_{b} = 60\%$, $\epsilon_{c} = 14\%$, $\epsilon_{j} = 1\%$ & 
            $\epsilon_{b} = 50\%$, $\epsilon_{c} = 14\%$, $\epsilon_{j} = 1\%$ \\ \hline
        \multicolumn{2}{|c|}{$p_{T}(\ell) > 20$ GeV} \\ \hline
        \multicolumn{2}{|c|}{$|\eta_{\ell}|{\rm,} \ |\eta_{j}| < 2.5$} \\ \hline
        \multicolumn{2}{|c|}{$\left| m_{b \ell \nu} - m_{t} \right| < 0.20 m_{t}$} \\ \hline
        \multicolumn{2}{|c|}{$\left| m_{b \ell \nu c} - m_{\phi} \right| < 0.20 m_{\phi}$} \\ \hline
        \multicolumn{2}{|c|}{$0.85 p_{c} < p_{T}(c) < 1.10 p_{c}$} \\ \hline
        \end{tabular}%
            \caption[We require exactly 1 lepton to meet the $p_{T}(\ell)$ cut above and exactly 1 b-tagged jet and 1 non b-tagged jet to 
            meet the $p_{T}(j)$ cut listed. The charm momentum for the signal, $p_c$, and background is found according to]{We require 
            exactly 1 lepton to meet the $p_{T}(\ell)$ cut above and exactly 1 b-tagged jet and 1 non b-tagged jet to meet the $p_{T}(j)$ 
            cut listed. The charm momentum, $p_c$, for the signal and background is found according to \cref{PcharmCut} and $\slashed{E}_{T}$ 
            is calculated by adding up all the visible particle momenta.} 
        \label{cuts-htc}
        \end{table}
        %
        The efficiencies listed above ($\epsilon_{b,c,j}$) are the frequencies at which the particle of interest is tagged as a b-jets. We 
        refer to these as tagging, $\epsilon_{b}$, and mistagging, $\epsilon_{c,j}$ efficiencies. The top three rows are specific to the 
        integrated luminosity of the detector; $25 \text{ and } 30 fb^{-1}$ or $300, \text{ and } 3000 \text{fb}^{-1}$; while the rest are 
        applied for all luminosities.
        %
        \begin{align}
            %
            p_{c} = \frac{m_{\phi}}{2} \left[ 1 - \frac{m^2_{t}}{m^2_{\phi}} \right]
            \label{PcharmCut}
            %
        \end{align}

        In the formula and table above, $m_{\phi}$ is the mass of the Higgs boson taking part in the decay. In this study, either a heavy 
        Higgs boson, $H^0$, or the pseudo-scalar Higgs, $a^0$. The cuts in \cref{cuts-htc} are chosen to both represent what has been done 
        at the detectors before through trigger levels and searches that constrain our signal, {\color{red} ref: 53-55 from htc paper}, 
        and from our own analysis of signal versus background events. In addition to background processes that mimic the final state that 
        we are trying to detect there are also built-in restrictions due to the nature of the particle themselves that can hinder such a 
        search.  


        \subsection{Rules of the Realm} \label{htc-constraints}
        
        The hindrance in the heavier Higgs states has to do with how likely they are to decay into the state we want in the first place. 
        \Cref{htc-BranchingFracs} shows the dominant decay modes of both the heavy Higgs scalar and the pseudoscalar. When the masses 
        of these new particles are around that of the SM-like or light Higgs boson our $tc$ decay mode actually dominates but becomes 
        less significant as the mass increases above twice $m_{t}$. In our case, the $t\bar{t}$ background can be reduced with the cuts 
        we apply for this study seeing as at minimum a $t\bar{t}$ even would produce a final state with either 2 b-jets and 2 leptons, or 
        2 b-jets, 2 light jets, and an isolated lepton, both of which violate our event selection criteria. This means for us, then that 
        we do not have to worry about much contamination from ancillary Higgs decay processes but we still have significant SM backgrounds 
        we need to analyze, the strongest of which is the $W \ +$ jets background.
        %
        \pagebreak
        %
        \begin{figure}[H]
            \begin{center}
            %
                \subfigure[$M_{H}$ branching ratios obtained using 2HDMC.]{
                    \includegraphics[width=0.80\textwidth,keepaspectratio]{./figures/H0_BrFacs_htc.pdf} \label{BRF_Htc}}
                %
                \quad
                %
                \subfigure[$M_{A}$ branching ratios obtained using 2HDMC.]{
                    \includegraphics[width=0.80\textwidth,keepaspectratio]{./figures/A0_BrFacs_htc.pdf} \label{BRF_Atc}}
            %
            \caption[Branching fractions of (a) the heavier Higgs scalar, $H^0$ and (b) the Higgs pseudoscalar, $A^0$ as a function of 
            Higgs mass, $m_{\phi}$, with $\cosba = 0.1$, $\rhotc = 0.24$, and $\rho_{ii} = \kappa_{i}$ for the diagonal couplings]{
            Branching fractions of (a) the heavier Higgs scalar, $H^0$ and (b) the Higgs pseudoscalar, $A^0$ as a function of Higgs mass, 
            $m_{\phi}$, with $\cosba = 0.1$, $\rhotc = 0.24$, and $\rho_{ii} = \kappa_{i}$ for the diagonal couplings.}
            \label{htc-BranchingFracs}
            %
            \end{center}
        \end{figure}
        %
        \pagebreak
        %

        The branching ratios depicted in \cref{htc-BranchingFracs} show a best case scenario for a $\rhotc$ coupling to be at the ATLAS 
        sensitivity level {\color{red} ref: 18 from htc paper} that can still be consistent with the alignment limit. Some of the 
        striking features are the sharp breaks around 300 GeV and the non-existence of an $H^0 \to h^0 h^0$ decay mode before then. In 
        the case of the sharp break around 300 GeV this corresponds to decays into real $t \bar{t}$ pairs being kinematically favorable 
        once your resonant mass reaches $2*m_{t} \approx 350$ GeV. The exclusion of the $H^0 \to h^0 h^0$ decay at $M_{H^0} < 300$ GeV 
        is partly due to kinematic accessibility as with the top but also involves the strength of the mixing between the heavy Higgs 
        state, $H^0$, and the light Higgs, $h^0$, controlled by the mixing angle, $\alpha$.

        We also must take into account the strength of the background processes that mimic our final state. One of our best discriminating
        tools in this case is applying the cuts that were highlighted in \cref{cuts-htc}. In the case of this search the Higgs mass window 
        had the largest effect at reducing the background when coupled with the cut on the charm momentum. The $p_{T}$ and $|\eta|$ cuts 
        for more for eliminating contamination from low energy electroweak processes and $t\bar{t}$ pair production. The most significant 
        background was still in the $W \ +$ jets channel which could be an electroweak process or could be produced through a single top. 
        Additional contamination from $t\bar{t}$ processes where one of the b-jets was misidentified, one $W$ decays leptonically, and 
        the other $W$ decays into two soft jets that do not pass our $p_{T}$ cuts. The tight requirements for the $t\bar{t}$ channel 
        allowed us to effectively eliminate or at least reduce the otherwise overwhelming production rate to a process on the order of our
        signal event. \Cref{htc-bkgnd} shows the cross sections of the signal and background processes as a function of heavy Higgs mass 
        for the $H^0 \to t c$ decay. 
        %
        \pagebreak
        %
        \begin{landscape}
        %
        \begin{figure}[H]
            \begin{center}
            %
                \subfigure[Cross section and dominant SM background at $\sqrt{s} = 8$ TeV.]{
                    \includegraphics[height=0.66\textheight,keepaspectratio]{./figures/sigma8.eps} \label{BRF_Htc}}
                %
                \quad
                %
                \subfigure[Cross section and dominant SM background at $\sqrt{s} = 14$ TeV.]{
                    \includegraphics[height=0.6525\textheight,keepaspectratio]{./figures/sigma14.eps} \label{BRF_Atc}}
            %
                \caption[The cross section of the heavy Higgs scalar, $H^0$ (solid, black) for 
                $\sigma(p p \to H^0 \to t \bar{c} + c \bar{t} \to b j_{c} \ell \nu + X)$ at the LHC as a function 
                of mass, $m_{H}$]{The cross section of the heavy Higgs scalar, $H^0$ (solid, black) for 
                $\sigma(p p \to H^0 \to t \bar{c} + c \bar{t} \to b j_{c} \ell \nu + X)$ at the LHC as a function 
                of mass, $m_{H}$. Results are shown for bench mark values of $\rhotc = 0.24, \ 1.0$ for 
                $\sqrt{s} = 8, \ 14$ TeV with a fixed $\cosba = 0.1$. For th background processes shown (dashed 
                lines), we have applied all selection rules, appropriate $K$-factors, and tagging efficiencies. }
            \label{htc-bkgnd}
            %
            \end{center}
        \end{figure}
        %
        \end{landscape}


    \section{Proclamations and predictions} \label{htc-results}

    The test of any good theory is how well it makes predictions and what those predictions can tell us. As discussed in \cref{pheno}, 
    a measure that both experimentalists and phenomenologists use to determine a discovery is statistical significance, \cref{StatSig2},
    %
    \begin{align}
        %
        N_{SS} = \frac{N_{S}}{\sqrt{N_{B}}} \geq 5 \ {\rm.} 
        %
        \label{StatSig2}
        %
    \end{align}
    %
    After all cuts and tagging efficiencies our signal is still roughly two orders of magnitude below our background but that
    is not as concerning due to the proportionality of $N_{S}$ to $\sqrt{N_B}$. From this information we are able to then find 
    projections of what values of \rhotc are accessible through searches at the LHC. Also, keep in mind, even if we do not see anything 
    necessarily does not mean it is not there. It is well accepted that the dominant branching ratio of the SM-like, light Higgs, $h^0$ 
    is to a $b\bar{b}$ pair which is just now reaching the point where it might be accessible through direct searches 
    {\color{red} ref: $h^0$ branching fractions from PDG and any direct $b\bar{b}$ searches currently published}. In many of these 
    cases we simply need more data, higher integrated luminosities, to drive down our statistical uncertainties and enhance these rare 
    signal events.

    Using the satistical significance criteria and analyzing our signal over a range of different viable \rhotc values allows us then 
    to predict what \rhotc should produce a $5\sigma$ result based on the anticipated luminosity values a the LHC; 
    $30, \ 300, \text{ and } 3000 \text{fb}^{-1}${\color{red} ref: luminosity goals}; and the possible mass of the heavier Higgs state. 
    For these calculations we assume a degeneracy in all of the heavy Higgs masses allowing $m_{H^0} \approx m_{a^0}$,
    %
    \pagebreak
    %
    \begin{figure}[H]
        \begin{center}
        %
            \subfigure{
                \includegraphics[width=0.465\textwidth,keepaspectratio]{./figures/LHCreach1a.eps}}
            %
            \quad
            %
            \subfigure{
                \includegraphics[width=0.465\textwidth,keepaspectratio]{./figures/LHCreach2a.eps}}
        %
            \subfigure{
                \includegraphics[width=0.465\textwidth,keepaspectratio]{./figures/LHCreach1b.eps}}
            %
            \quad
            %
            \subfigure{
                \includegraphics[width=0.465\textwidth,keepaspectratio]{./figures/LHCreach2b.eps}}
        %
            \caption[Discovery potential for a $5\sigma$ significance in the $m_{\phi} - \rhotc$ plane for the 
            $\sigma(p p \to H^0 \to t \bar{c} + c \bar{t} \to b j_{c} \ell \nu + X)$ signal at LHC center of mass energies, 
            $\sqrt{s} = 13 \ (14)$ TeV]{Discovery potential for a $5\sigma$ significance in the $m_{\phi} - \rhotc$ plane for the 
            $\sigma(p p \to H^0 \to t \bar{c} + c \bar{t} \to b j_{c} \ell \nu + X)$ signal at LHC center of mass energies, 
            $\sqrt{s} = 13 \ (14)$ TeV, the dashed (solid) contours. (a) and (c) are discovery potentials for the heavy, $H^0$, Higgs 
            alone while (b) and (d) show combined results for both $H^0$ and $A^0$. In (a) and (b) $\cosba = 0.1$ while we relax that 
            constraint in (c) and (d) and allow $\cosba = 0.2$. The shaded regions above the contours indicate the favorable parameter 
            space for \rhotc given the $5\sigma$ discovery criteria we used to construct the curves.}
        \label{htc-discovery}
        %
        \end{center}
    \end{figure}
    %
    What we find with these discovery potentials is that we have a considerable parameter space in which we can hope to find evidence 
    of flavor changing neutral currents involving the production and decay of heavy Higgs states that accompany extending the Higgs 
    sector of the SM with a second Higgs doublet. This is done in a general two Higgs doublet model where we specifically do not place 
    any hard constraints on how the two doublets should interact and only require that we can reproduce observed SM results in the 
    alignment limit as $\sinba \to 1$. Much of the parameter space will be swept out by the end of Run 2 but there is some viable 
    parameter space that may not be accessible until after the high luminosity upgrade is completed and new data is collected. 

    Our basis for this research was in some way rooted by the excess of FCNCs reported by CMS in Run 1 in the $h^{0} \to \tau \mu$ 
    channel {\color{red} ref: CMS $\tau \mu$ paper}. However, 2HDMs are prominent in many extensions of the SM including supersymmetry 
    which requires a Type-II 2HDM where one Higgs doublet preferentially couples to up-type quarks and the other to down-type quarks. 
    If we discover these events, that may then give us some insight into the structure of such a 2HDM which can help narrow down what 
    BSM theories might be governing the global picture. We also chose to look at the $\phi^0 \to t c$ case due to its strong coupling 
    to the Higgs field and relatively straight forward analysis. While $h^{0} \to \tau \mu$ events have been reported at some 
    significance the analysis and process of producing event data for those decays takes considerable more effort as described in the 
    next chapter on our analysis or leptonic FCNCs with $\tau \mu$ events.



