%---------------------------------------------------------------------------------
%       Chapter 4: Tau Lepton and Muon meet the Higgs, an unexpected trio
%---------------------------------------------------------------------------------
\chapter{When the Higgs Meets the Tau and the Muon}

\singlespacing

\epigraph{``\textit{You cannot hide, I see you!}''}{--- Sauron, \emph{The Lord of the Rings}, J.R.R. Tolkien}

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

The Higgs interaction with the tau, $\tau$, and the muon, $\mu$, was our first experimental indication that 
FCNC processes may be visible at the LHC. The excess announced by CMS during Run I was only a $2.4 \sigma$ 
significance, so not nearly enough to constitute a discovery, but was enough to give us a hint at where we 
should look. In addition, this would be the first FCNC process kinematically accessible by a SM-like Higgs 
with a mass below the top mass, $m_t$. The SM-like or light Higgs, $h^0$, still has a \cosba factor that 
reduces the visibility of a \tamu signal so it is also prudent to consider the heavier Higgs states, $H^0$ 
and $A^0$, as they are proportional to the complimentary \sinba term. 

\section{Leptonic FCNC processes}

The overall process of a Higgs field coupling to a tau and muon, $\phi \to \tamu$ where $\phi = H^0, \ A^0, \ h^0$, 
is not fundamentally any different than the Higgs field coupling to the top and charm, \Cref{2HDMLagrangian}. It is 
still the off-diagonal elements of the $\rho^{F}$, or in this case $\rho_{\tau \mu}$, that generate these flavor 
changing states. One important difference between the $htc$ coupling and the $h\tamu$ coupling is the \tamu state 
is a viable decay for the SM Higgs process and we do not need to wait for a more massive particle to see such an 
interaction. Unfortunately, such a decay is suppressed as we approach the alignment limit by the \cosba term. In 
addition, since the $\tau$ and $\mu$ are so light their coupling to the Higgs field is not as strong as the heavy 
quark states, charm and top. \Cref{BRFHtamu,BRFhtamu} illustrate how favorable such a decay is in the context of 
the other possible channels. In \Cref{BRFHtamu} we scan over possible values of the heavy Higgs particle much as was 
done in the $htc$ study. In \Cref{BRFhtamu}, however, since the mass of the Higgs boson is fixed by observation at 
$125.2 \pm 0.24$ GeV \footnote{http://pdglive.lbl.gov/Particle.action?node=S035{\&}init=0}, we scan over possible $\rho_{\tamu}$ values instead.
%
\begin{figure}[H]
    \begin{center}
    %
        \subfigure[$M_{H}$ branching ratios obtained using 2HDMC.]{
            \includegraphics[width=0.85\textwidth,keepaspectratio]{./figures/H0_ratios_okular_new.png} \label{BRFHtamu}}
        %
        \quad
        %
        \subfigure[$M_{h}$ branching ratios obtained using 2HDMC.]{
            \includegraphics[width=0.85\textwidth,keepaspectratio]{./figures/lh0_ratios_okular_new.png} \label{BRFhtamu}}
    %
    \caption[This figure shows the branching ratios of (a) the heavy Higgs scalar, $H^{0}$, for a range of Higgs 
    masses]{This figure shows the branching ratios of (a) the heavy Higgs scalar, $H^{0}$, for a range of Higgs 
        masses. Included are curves for several test values of the $\rho_{\tau \mu} = 0.2, \ 0.01,$ and 
        $2.489 \times 10^{-2}$ (CSA). In (b) the branching fractions of the SM-like Higgs are shown as a function 
        of the $\rho_{\tau \mu}$ coupling as the mass was fixed at $125.1$ GeV. Here, in (a), preference has been 
        given to show alternative Higgs decays that may serve as background to $\phi^{0} \to \tau \mu$ signal, 
        where $\phi= H^0, \ A^0, \ h^0$.}
    \label{H0BranchingRatios}
    %
    \end{center}
\end{figure}
%
Normally, all branching fractions will add up to unity collectively, in \Cref{BRFHtamu}, the $H^{0} \to t t$, 
the $H^{0} \to tc$, $H^{0} \to bb$, and the $H^{0} \to h^0 h^0$ are not shown, which is why the plot only shows 
values up to 0.1 or 10\%. Due to this choice, all decay processes involved here do not encompass the main decay 
modes of $H^{0}$ so even if $\rho_{tamu}$ were to be high, it would still be considered a rare process. The 
advantage of the lepton sector, however, is it does not have as much background from QCD processes.
%

\section{Interplay between the $\phi \to tc$ and $\phi \to \tau \mu$}

In addition to the background processes generated from the heavy Higgs scalar, $H^0$, we also have to take into 
account the $\phi \to tc$ flavor changing processes. In some ways this is an addendum to our work in \cite{Altunkaynak2015}
with the added complexity associated with decaying tau particles. Since we are using a general 2HDM still, FCNC 
processes, if present, are at the tree-level and add another decay channel to the particle states with masses 
$\geq m_{t} = 173.2$ GeV. Including additional decay modes in the heavier Higgs states causes their decay width, 
the sum of all of their decay modes, to change as well. In order to accurately make calculations and predictions 
for $\phi \to \tamu$ signal we then have to include information about $\rho_{tc}$. As no observations of a 
$H^0, A^0 \to tc$ signal has been reported yet, we have no experimental benchmark values to report. As reference 
we chose to use a conservative value chosen as a case study in \cite{Altunkaynak2015} of $\rho_{tc} = 0.1$. 

Since $h^0 \to tc$ decay is considered inaccessible, the choice of $\rho_{tc}$ does not have an effect on the 
SM, or light Higgs, decay width. The $h^0 \to \tamu$ decay, however, is accessible and has a profound impact on 
the SM Higgs width and must be treated with some care. In addition to constraints placed on the coupling by looking 
at BABAR \cite{TheBABARCollaboration2009} and Belle II \cite{Aushev2010} data for $\tau \to \mu \gamma$ that say 
$\rho_{\tamu} \leq 0.26$, we also must consider how much deviation from the reported SM Higgs width is acceptable. 
%
\begin{table}[H]
    \centering
    \resizebox{\textwidth}{!}{%
    \begin{tabular}{|c|c|c|c|c|}
    \hline
    $\rho_{\tau \mu}$ & $2.489 \times 10^{-2}$ & $0.1$                  & $0.2$                  & $0.26$                 \\ \hline
    $\Gamma_{h}$      & $3.991 \times 10^{-3}$ & $4.45 \times 10^{-3}$  & $5.95 \times 10^{-3}$  & $7.324 \times 10^{-3}$ \\ \hline
    $g_{h \tau \mu}$  & $1.76 \times 10^{-4}$  & $7.10 \times 10^{-3}$  & $1.40 \times 10^{-2}$  & $1.83 \times 10^{-2}$  \\ \hline
    \end{tabular}%
    }
    \caption{This table shows the SM Higgs width, $\Gamma_h$ calculated from 2HDMC, as a function of $\rho_{\tau \mu}$ with 
    $\cos(\beta - \alpha) = 0.1$ and $\rho_{tc} = 0.1$. The $g_{h \tau \mu}$ terms show the coupling to the Higgs field, 
    accounting for all other coefficients, as shown in \Cref{GhCalculation}.}
    \label{GhTable}
\end{table}
%
The table shown above highlights the impact that changing $\rho_{\tamu}$ has on the $h^{0}$ decay width, $\Gamma_{h}$. Also reported in that table
are values for the adjusted Higgs coupling which takes into account the value of $\rho_{\tamu}$ and $\cosba$ in the Lagrangian,
%
\begin{align}
    %
    g_{h \tau \mu} \equiv \frac{ \rho_{\tau \mu} \cos(\beta - \alpha) }{ \sqrt{2} } \, . \label{GhCalculation}
    %
\end{align}
%
Based on the constraints of \cite{TheBABARCollaboration2009,Aushev2010} and the results shown in \Cref{GhTable} case studies were chosen 
for $\rho_{\tamu}= 2.489 \Exp{-3}, / 0.1, \text{ and } 0.2$ keeping us within 1\%, 12\%, and 40\% of the reported width 
respectively. A discrepancy from the observed SM width of 40\% is quite extensive and is not viewed as a realistic assumption 
but will serve to provide upper-limits on coupling strengths for this study. 

\section{Decaying the $\tau$} \label{tamu-tauDecay}

Generally speaking, particle decays are very formulaic. You generate Feynman diagrams to explain the physics of the process, obtain the Feynman 
rules for each vertex, compute the trace, evaluate the phase space, and compute the integral. This process does not change much with the tau, but 
it does get slightly more complicated. Like the top, the tau is the heaviest lepton with a mass or around 
$1.77$ GeV\footnote{http://pdglive.lbl.gov/Particle.action?node=S035{\&}init=0}. This means a couple things. First, the decay of a tau into another 
lepton is possible. Second, it can decay into a hadron, specifically a meson that contains a quark anti-quark pair. 

If the tau decays hadronically, the subsequent meson will most likely cause a shower in the hadronic calorimeter which will 
cause the tau to look like a funny jet, aptly name a tau-jet. Dealing with jets in a detector simulation usually is not very hard, it gets tagged as 
a jet and from a theory standpoint we don't need to know much more than that. With a tau-jet, however, the particulars of the jet can depend heavily 
on which meson the tau decayed through, either a $\pi$, $\rho$, or $a1$. All three decay modes then have to be taken into account to accurately model 
the hadronic tau decay mode. 

The leptonic decay mode is more straightforward, the tau decays into a virtual $W$ which then decays into either a $\mu \nu_{\mu}$ or an $e \nu_{e}$ 
with nearly the same frequency, $10.6\%$ and $10.7\%$ respectively. In this study, to reduce contamination, we are mainly interested in the 
electron/positron final state as that then gives us to different flavor leptons in the final state. With that in mind, we do take special care to 
use the properties of the electron tau decay instead of a more generic lepton decay that could be either an $e$ or a $\mu$.

In addition to the challenges associated with the decay of the tau, production of the tau presents its own challenges, especially in this $\tamu$ 
process. The SM-like Higgs has a mass of $125.1$ GeV and we expect the heavy Higgs and pseudoscalar Higgs to be even heavier still. The muon and 
tau by contrast have a collective mass of about $1.883$ GeV. Because of this large mass difference, the tau and muon generated from the event are 
highly boosted, meaning they are traveling more or less along the same path as the parent particle, instead of scattering at new angles. Likewise, the 
tau decay products are also going to be highly boosted which causes problems when the event is reconstructed. In order to get a picture or what happened 
we have to employ the collinear approximation \cite{Hagiwara1990} where we use information about the parent particle 
and the momentum deposited in the transverse plane after the tau decay to effectively construct the momentum of parent tau. For the muon this is not 
necessary since it is already a final state particle, but the tau will decay and so we need to know the four momentum it had to accurately reconstruct 
the event. 

The collinear approximation essentially says that the decay product will simply carry some fraction of the parent particle's momentum but be in the same 
direction, hence the col in collinear. Mathematically this means
%
\begin{align}
    %
    P(\ell,j) = \chi_{\ell,j} \cdot P(\tau) \label{collinearP} \, .
    %
\end{align}
%
Likewise, there will be some missing transverse momentum from a tau neutrino, $\nu_{\tau}$, that will carry the rest of the parent momentum
%
\begin{align}
    %
    \slashed{E}_{T} = (1 - \chi_{\ell,j}) P_{T}(\tau) \label{collinearMET} \, ,
    %
\end{align}
%
where $\chi_{\ell,j}$ represents the momentum fraction the lepton or hadron $(j)$ has from the parent tau, $P_{T}(\tau)$ denotes the transverse momentum 
of the tau, and $P_{\ell,j}$ represents the momentum of the decay product of the tau. In the situation above, it is the $P(\tau)$ and the $\chi_{\ell,j}$ 
that are not known but $P(\ell,j)$ and $\MET$ are final state values that would be measured at the collider. We then rely on the tried and true system of 
equations method where we can combine these two equations to effectively reconstruct the momentum of the parent tau
%
\begin{align}
    %
    P(\tau) &= \frac{P(\ell,j)}{\chi_{\ell,j}} \,  \label{tauP} , \\ 
    %
    &\text{ where} \notag \\
    %
    \chi_{\ell,j} =& \frac{P_{T}(\ell,j)}{P_{T}(\ell,j) + \MET} \, .    \label{ChiLepHad}
    %
\end{align}
%
This allows us then to know the most probable four momentum for the parent tau and will let us more accurately produce the reconstructed Higgs that 
produced the tau and muon pair.

\section{SM physics background} \label{tamu-Background}

As with any signal search in particle physics, there is more than one way our select final state can be produced. The challenge then is tuning your analysis 
to pick out your signal over the background without making too many assumptions. The $\phi \to \tamu$ signal is no exception and also provides us with some 
interesting challenges. For the signal in this case, since we are looking at both the hadronic and leptonic decay modes of the tau, we have to include the 
backgrounds associated with each process. In addition, as is indicated in \Cref{BRFHtamu}, there are several possible decay products of a heavy 
Higgs scalar that need to be accounted for. 

    \subsection{Hadronic $\tau$ backgrounds}

    For the hadronic decay of the tau we are looking for a muon and associated jet that is highly boosted. This same final state can also be produced from 
    an intermediate $Wj$ process where the $W$ decays to a muon and a muon neutrino, $\mu \nu_{\mu}$ that comes from either a single top or an electroweak 
    particle like a photon, $\gamma$, or a $Z$. If we first produce a pair of taus, $\tau \tau$, and we allow one to decay hadronically into a jet and the 
    other to decay leptonically into a muon, then this will also produce similar kinematics to our signal. A very similar background process exists if we 
    consider a $WW$ intermediate process where one $W$ decays into a $\mu \nu_{\mu}$ and the other decays into a tau that then decays hadronically and so 
    will be tagged as a jet. 
    %
    \begin{table}[H]
    \centering
    \resizebox{\textwidth}{!}{%
        \begin{tabular}{|c|c|c|c|}
        \hline
        $\sigma(p p \to X)$                   & $\sqrt{s} = 8$ TeV          & $\sqrt{s} = 13$ TeV          & $\sqrt{s} = 14$ TeV          \\ \hline
        $p p \to W W \to \tau \mu$ hadronic   & 0.9145 fb                   & 1.5188 fb                    & 1.6245 fb                    \\ \hline
        $p p \to \tau \tau$ hadronic          & 37.2235 fb                  & 58.7502 fb                   & 62.8659 fb                   \\ \hline
        $p p \to W j \to \mu \nu_{\mu} \, j$    & $4.4804 \times 10^{2}$ fb   & $8.0711 \times 10^{2}$ fb    & $8.8093 \times 10^{2}$ fb    \\ \hline
        \end{tabular}%
    }
    \caption{This table shows the hadronic background cross sections for the $\phi \to \tamu$ signal with $M_{H} = 125.1$ GeV. Here we have already applied 
        our selection rules for the signal that will be further discusses in \Cref{tamu-EventSelection}.}
    \label{hadronic-backgrounds}
    \end{table}
    %
    \begin{figure}[H]
        %
        \begin{center}
            %
            \includegraphics[width=0.95\textwidth,keepaspectratio]{./figures/13TeV_rhotc010_HadronicPlot.pdf}
            %
            \caption[Here the SM background for the hadronic decay process at $\sqrt{s} = 13$ TeV is shown as it compares to the signal cross section]{
                Here the SM background for the hadronic decay process at $\sqrt{s} = 13$ TeV is shown as it compares to the signal cross section after full 
                selection cuts. The black curves show the cross section as a function of $M_{H} \in [125,500]$ GeV for the various case studies of 
                $\rho_{\tau \mu}$ that were selected: $2.489 \times 10^{-2}, \ 0.10, \ 0.20$; for $\rho_{tc} = \cosba = 0.1$.} 
            \label{hadronicSummary}
            %
        \end{center}
        %
    \end{figure}

    \subsection{Leptonic $\tau$ backgrounds}

    In the leptonic sector we have to contend with even more background processes than we did than with the hadronic decay mode. Furthermore, we have to be cautious 
    as to which type of lepton the tau decays into. If the tau decays into another muon then we have backgrounds from $ZZ$ and $Z\gamma$ final states, if we 
    do not we can avoid those backgrounds. Shown below, \Cref{leptonic-backgrounds}, is a table showing the dominant background processes for the different 
    flavor $(e \mu)$ lepton final state and same flavor lepton final states will be added later in this work. For the $e \mu$ final state the dominant 
    background processes are $WW$, $\tau \tau$, and $WW \to \tamu$. In each case, one particle decays into the muon we expect and the other decays into an 
    electron. 
    %
    \begin{table}[H]
    \centering
    \resizebox{\textwidth}{!}{%
        \begin{tabular}{|c|c|c|c|}
        \hline
        $\sigma(p p \to X)$                             & $\sqrt{s} = 8$ TeV          & $\sqrt{s} = 13$ TeV          & $\sqrt{s} = 14$ TeV          \\ \hline
        $p p \to W W \to \tau \mu$ leptonic             & 0.2949 fb                   & 0.4853 fb                    & 0.5200 fb                    \\ \hline
        $p p \to \tau \tau$ leptonic                    & 10.22  fb                   & 16.18  fb                    & 17.33  fb                    \\ \hline
        $p p \to W W \to \mu \nu_{\mu} e \nu_{e}$       & 2.108  fb                   & 3.682  fb                    & 3.967  fb                    \\ \hline
        \end{tabular}%
    }
    \caption{This table shows the leptonic background cross sections for the $\phi \to \tamu$ signal with $M_{H} = 125.1$ GeV. Here we have already applied 
        our selection rules for the signal that will be further discusses in \Cref{tamu-EventSelection}.}
    \label{leptonic-backgrounds}
    \end{table}
    %
    \begin{figure}[H]
        %
        \begin{center}
            %
            \includegraphics[width=0.95\textwidth,keepaspectratio]{./figures/13TeV_rhotc010_LeptonicPlot.pdf}
            %
            \caption[Here the SM background for the leptonic decay process at $\sqrt{s} = 13$ TeV is shown as it compares to the signal cross section]{
                Here the SM background for the leptonic decay process at $\sqrt{s} = 13$ TeV is shown as it compares to the signal cross section after full 
                selection cuts. The black curves show the cross section as a function of $M_{H} \in [125,500]$ GeV for the various case studies of $\rho_{\tau \mu}$ 
                that were selected: $2.489 \times 10^{-2}, \ 0.10, \ 0.20$; for $\rho_{tc} = \cosba = 0.1$.} 
            \label{leptonicSummary}
            %
        \end{center}
        %
    \end{figure}

    \subsection{$\phi \to XX$ backgrounds}

    In both the hadronic decays and the leptonic decays we have another layer of complexity added to our system due to the kinematic regime we are working in. 
    Specifically we have additional backgrounds that, like out signal, are mediated by the SM Higgs paricle. The SM Higgs can decay into a $WW\mbox{*}$, a 
    $ZZ\mbox{*}$, and a $\tau^+ \tau^-$ as well as to the \tamu signal we are searching for. Our advantage is, however, that the rate at which it produces 
    these additional particles is much lower than other modes that do not produce the same final state particles we are interested in. The disadvantage that 
    we must overcome is that selecting events based on their ability to reconstruct a Higgs resonance may not be as fruitful for these stats as they do involve 
    a Higgs resonance. Below, \Cref{Higgs-backgrounds}, shows calculated values of these Higgs mediated backgrounds.
    %
    \begin{table}[H]
    \centering
    \resizebox{\textwidth}{!}{%
        \begin{tabular}{|c|c|c|c|}
        \hline
        $\sigma(p p \to X)$                             & $\sqrt{s} = 8$ TeV          & $\sqrt{s} = 13$ TeV          & $\sqrt{s} = 14$ TeV          \\ \hline
        $p p \to h^0 \to \tau \tau$ leptonic            & 0.4633  fb                  & 1.004  fb                    & 1.123  fb                    \\ \hline
        $p p \to h^0 \to \tau \tau$ hadronic            & 2.117  fb                   & 4.581  fb                    & 5.127  fb                    \\ \hline
        $p p \to h^0 \to W W \to \mu \nu_{\mu} e \nu_{e}$ & $4.049 \times 10^{-2}$ fb  & $8.881 \times 10^{-2}$ fb   & $9.810 \times 10^{-2}$ fb   \\ \hline
        \end{tabular}%
    }
    \caption{This table shows Higgs mediated background cross sections for the $\phi \to \tamu$ signal with $M_{H} = 125.1$ GeV. Here we have already applied 
        our selection rules for the signal that will be further discusses in \Cref{tamu-EventSelection}.}
    \label{Higgs-backgrounds}
    \end{table}

\section{Event selection} \label{tamu-EventSelection}

The background cross sections depicted in \Cref{hadronic-backgrounds,leptonic-backgrounds,Higgs-backgrounds} were all calculated using carefully developed 
selection rules that were designed to disfavor the background and favor the signal. In any search for an exotic decay mode the best efficiency is obtained 
by doing your best to exploit the unique physics associated with your signal. In our case, one of the most telling selection rules is the transverse energy 
distribution of the final state particles. On this same note, this work has based many of our cuts on the original CMS and ATLAS publications 
\cite{Collaboration2015,Aad:2016blu}, where they apply additional cuts on the angular distibution of the final decay products to isolate the unique properties of the signal.
Some of the more novel cuts that were applied in particular to this study are described in more detail below.

    \subsection{Reconstructing the Higgs}

    Looking at the process $p p \to \phi \to \tamu$, another key identifying feature is the presence of a Higgs resonance to generate the \tamu final state. To 
    verify that the event we are looking at is indeed our signal this reconstruction must be checked and can also be used to try and discriminate against background 
    processes. As described in \Cref{tamu-tauDecay}, our first step is to correctly and fully reconstruct the tau particle using the collinear approximation. For the 
    signal this allows us to more accurately characterize the central resonance and verify that is was a Higgs. For the background it instead purposefully 
    mischracterizes one of the decay products and will not reconstruct a signal as clean as the $\phi \to \tamu$ signal.
    %
    \begin{figure}[H]
        %
        \centering
        %
        \includegraphics[width=0.85\textwidth,keepaspectratio]{./figures/Mtamu_comparison.pdf}
        %
        \caption[This plot shows the reconstructed $\tamu$ resonance for our signal and the leptonic $WW$ background. Here it can be seen that the Higgs signal has a 
        sharp peak]{This plot shows the reconstructed $\tamu$ resonance for our signal and the leptonic $WW$ background. Here it can be seen that the Higgs signal has 
        a sharp peak right at the Higgs mass as expected and the $WW$ process has a much broader shape with a peak much closer to the mass of its dominant resonance, 
        the $Z$. The above values are shown for a $\sqrt{s} = 8$ TeV and with only basic cuts applied $(p_{T} > 10 \text{ GeV and } |\eta| < 2.5)$.}
        \label{mTaMu-ref}
        %
    \end{figure}
    %
    This mismatch between the shapes and locations of the peaks is what is exploited with such a cut on the reconstructed invariant mass, $M_\text{inv}$, of the 
    $\tamu$ resonance. It can be somewhat less significant for the $\phi \to XX$ backgrounds but is invaluable for the other electroweak backgrounds.

    \subsection{Importance of transverse mass}

    As mentioned before, many of our cuts and selection rules are based on information or restrictions imposed by the experimental collaborations, such as $p_{T}$ and 
    $|\eta|$ requirements for triggers and separation requirements to satisfy isolation cuts in a very naive way. There are some cuts that are also theoretically motivated 
    and can help the experimentalists select data more efficiently. One such cut for us is a cut on the transverse mass, $M_{T}$. Since the tau decay will involve the 
    creation of a neutrino we can expect to have some missing transverse energy in our final state particles. 
    %
    \begin{figure}[H]
        %
        \centering
        %
        \includegraphics[width=0.85\textwidth,keepaspectratio]{./figures/Mtrans_comparison.pdf}
        %
        \caption[This plot shows the transverse mass of the $\mu$ + \MET and $e$ + \MET for our signal and the leptonic $WW$ background. Here it can be seen that the 
        $WW$ has symmetric final state partiles whle the signal is more asymmetric]{This plot shows the transverse mass of the $\mu$ + \MET and $e$ + \MET for our signal 
        and the leptonic $WW$ background. Here it can be seen that the $WW$ has symmetric final state partiles whle the signal is more asymmetric. This plot shows the 
        reconstructed $\tamu$ resonance for our signal and the leptonic $WW$ background. Here it can be seen that the Higgs signal has a sharp peak right at the Higgs mass 
        as expected and the $WW$ process has a much broader shape with a peak much closer to the mass of its dominant resonance, the $Z$. The above values are shown for a 
        $\sqrt{s} = 8$ TeV and with only basic cuts applied $(p_{T} > 10 \text{ GeV and } |\eta| < 2.5)$.}
        \label{mTaMu-ref}
        %
    \end{figure}
    %
    We can construct a picture of the transverse mass of a particle, $q$, in the final state through looking at its transverse components,
    %
    \begin{align}
        %
        M^2_{T} = \left( p_{T}(q) + \slashed{E}_{T} \right)^2 - \left( q_{x} + \slashed{E}_{x} \right)^2 - \left( q_{y} + \slashed{E}_{y} \right)^2
        \label{transMass}
        %
    \end{align}
    %

    We can use this to our advantage by applying cuts on the transverse mass of certain particles. Following the lead of CMS, we specifically chose to cut on the 
    transverse mass of a $\mu \nu$ pair and the transverse mass of an $e \nu$ pair. If the final state particle, say the muon, came from a single particle decay we can 
    expect the transverse mass of that particle have a sharp drop around the mass of the parent particle \cite{WWMT}. Our signal, on the other hand, should exhibit 
    this behavior only in the electron transverse mass and not the muon.

    \subsection{Asymmetric selection rules}

    Due to the nature all but the $WW \to \tamu$ background the processes, we would expect their decays to be symmetric. This is a property that is not shared in the 
    $\phi \to \tamu$ signal. Before the tau decays, the two particles are symmetric and, in fact, their momenta are nearly on top of each other as seen in, \Cref{PTtamu-fig}. 
    %
    \begin{figure}[H]
        %
        \centering
        %
        \includegraphics[width=0.85\textwidth,keepaspectratio]{./figures/PT_example.pdf}
        %
        \caption{Here the $p_{T}$ for the $\tau$ and $\mu$ are plotted before any selection cuts are applied. This signature is similar to what is seen in many of the 
        background distributions.}
        \label{PTtamu-fig}
        %
    \end{figure}
    %
    However, once the tau decays into either a jet or a lepton, there is additional energy lost in the form of a neutrinos that will offset the transverse spectrum of the 
    visible particles. Based on this property we have applied some interesting asymmetric cuts on the $P_{T}$ and $\eta$ of the muon and jet or $e$ from the tau decay as 
    highlighted in the table below, \Cref{AsymmetricCuts-tbl}.
    %
    \begin{table}[H]
    \centering
    %\resizebox{\textwidth}{!}{%
        \begin{tabular}{|c|c|c|}
        \hline
        Cut Parameter                              & Leptonic: $\tau \to e \nu_{e} \nu_{\tau}$ & Hadronic: $\tau \to j \nu_{\tau}$ \\ \hline
        $p_{T} (\mu)$                              & $> 50$ GeV                                & $> 45 $ GeV                       \\ \hline
        $p_{T}(\tau)$                              & $> 10$ GeV                                & $> 35$ GeV                        \\ \hline %\hline
        $M_{T}(\mu)$                               & $> 40$ GeV                                & -                                 \\ \hline
        $M_{T}(\tau)$                              & $< 65$ GeV                                & $< 50$ GeV                        \\ \hline %\hline
        $\Delta \phi_{p^{\mu}_{T} - p^{\tau}_{T}}$ & $> 2.7$                                   & $> 2.7$                           \\ \hline
        $\Delta \phi_{p^{\tau}_{T} - \MET}$        & $< 0.5$                                   & -                                 \\ \hline \hline
        $|\eta_{\tau}|$                            & $< 2.3$                                   & $< 2.3$                           \\ \hline
        $|\eta_{\mu}|$                             & $< 2.1$                                   & $< 2.1$                           \\ \hline
        \end{tabular}%
    %}
    \caption{This table summarizes the cuts applied to the final state particles under the conditions were the tau decays hadronically and is tagged as a jet and when it 
        decays leptonically into an electron. The $\eta$ cut depicted above is not asymmetric due to the nature of the decay but is chosen to match the trigger efficiency 
        of CMS.}
    \label{AsymmetricCuts-tbl}
    \end{table}
    %
    %\pagebreak
    %

\section{Results and predictions}

Currently, for this study, we look at both the heavy Higgs decay process and the SM-like light Higgs decay as possible signatures for the \tamu flavor changing process. All 
cuts have been applied and cross sections are calculated for heavy Higgs masses from $150$ GeV to $500$ GeV for $\sqrt{s} = 8, \ 13, \ 14$ TeV using the newly updated CT14 
PDFs \cite{Hou2016}. For the signal, $M_{\phi} / 2$ is used as the factorization and renormalization scale and the dynamic choice of $\sqrt{s}$ is chosen for all the 
background processes since they lack a defined core resonance and are mediated, generally, by a $\gamma$ or a $Z$. In the case of the $Z$ mediated decays we could adopt a 
similar factorization scale and renormalization scale to that used for the Higgs signal but such a choice would not be appropriate for the $\gamma$ mediated process. All 
background events are calculated using the same fortran code package developed for the signal process with the exception of importing MG5 matrix elements in lieu of full 
analytic calculations. For SM processes, MG5 is well tested and used as a benchmark comparison for any study we do to as high a level as we can trust the comparison and 
only breaks down in the exotic regime or, in the case of $H^{0}$, when the effective theory breaks down. But, for well known and tested processes, the use of MG5 matrix 
elements offers an extra level of confidence in our calculation and the ability to run full detector simulations on the background predictions. The following results 
were obtained for both leptonic and hadronic tau decay states to be compared against the background cross sections presented in 
\Cref{hadronic-backgrounds,leptonic-backgrounds,Higgs-backgrounds}.
%
\begin{landscape}
%
\begin{table}[H]
\centering
\resizebox{1.2\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$M_{H}$                                 & 125 GeV                                 & 150 GeV                                 & 200 GeV                                 & 300 GeV                                 & 400 GeV                                 & 500 GeV                                 \\ \hline
top Loop K-factor                       & 1.89743                                 & 1.90633                                 & 1.92299                                 & 1.9447                                  & 1.91845                                 & 1.87137                                 \\ \hline
bottom loop K-fator                     & 1.04143                                 & 1.06498                                 & 1.10215                                 & 1.15453                                 & 1.19169                                 & 1.22052                                 \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$5.950 \times 10^{-3}$} & \multirow{2}{*}{$2.410 \times 10^{-1}$} & \multirow{2}{*}{$3.350 \times 10^{-1}$} & \multirow{2}{*}{$7.625 \times 10^{-1}$} & \multirow{2}{*}{$4.662$}                & \multirow{2}{*}{$1.229 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.146 \times 10^{-3}$} & \multirow{2}{*}{$6.377 \times 10^{-2}$} & \multirow{2}{*}{$1.167 \times 10^{-1}$} & \multirow{2}{*}{$4.080 \times 10^{-1}$} & \multirow{2}{*}{$4.189$}                & \multirow{2}{*}{$1.170 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.176 \times 10^{-3}$} & \multirow{2}{*}{$8.361 \times 10^{-3}$} & \multirow{2}{*}{$4.284 \times 10^{-2}$} & \multirow{2}{*}{$2.972 \times 10^{-1}$} & \multirow{2}{*}{$4.041$}                & \multirow{2}{*}{$1.151 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$5.428$}                & \multirow{2}{*}{$6.380 \times 10^{2}$}  & \multirow{2}{*}{$4.202 \times 10^{2}$}  & \multirow{2}{*}{$1.750 \times 10^{2}$}  & \multirow{2}{*}{$3.742 \times 10^{1}$}  & \multirow{2}{*}{$8.919$}                \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$7.790$}                & \multirow{2}{*}{$6.040 \times 10^{2}$}  & \multirow{2}{*}{$3.184 \times 10^{2}$}  & \multirow{2}{*}{$8.190 \times 10^{1}$}  & \multirow{2}{*}{$1.044 \times 10^{1}$}  & \multirow{2}{*}{$2.238$}                \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$7.725$}                & \multirow{2}{*}{$2.883$}                & \multirow{2}{*}{$5.428 \times 10^{-1}$} & \multirow{2}{*}{$7.036 \times 10^{-2}$} & \multirow{2}{*}{$6.773 \times 10^{-3}$} & \multirow{2}{*}{$1.424 \times 10^{-3}$} \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
\end{tabular}%
}
    \caption{Cross section data calculated for $h^{0}/H^{0} \to \tamu \to \mu e  + \MET + X$ for $\sqrt{s} = 8$ TeV. In all cases the full suite of selection rules have been applied and the total width of each Higgs resonance 
    is reported as calculated using 2HDMC with $\rho_{tc} = \cosba = 0.1$ and $\rho_{\tamu}$ as indicated.}
\label{leptonic8TeV-tbl}
\end{table}
%
\begin{table}[H]
\centering
\resizebox{1.2\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$M_{H}$                                 & 125 GeV                                 & 150 GeV                                 & 200 GeV                                 & 300 GeV                                  & 400 GeV                                 & 500 GeV                                 \\ \hline
top Loop K-factor                       & 1.89743                                 & 1.90633                                 & 1.92299                                 & 1.9447                                   & 1.91845                                 & 1.87137                                 \\ \hline
bottom loop K-fator                     & 1.04143                                 & 1.06498                                 & 1.10215                                 & 1.15453                                  & 1.19169                                 & 1.22052                                 \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$5.950 \times 10^{-3}$} & \multirow{2}{*}{$2.410 \times 10^{-1}$} & \multirow{2}{*}{$3.350 \times 10^{-1}$} & \multirow{2}{*}{$7.625 \times 10^{-1}$}  & \multirow{2}{*}{$4.662$}                & \multirow{2}{*}{$1.229 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                          &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.146 \times 10^{-3}$} & \multirow{2}{*}{$6.377 \times 10^{-2}$} & \multirow{2}{*}{$1.167 \times 10^{-1}$} & \multirow{2}{*}{$4.080 \times 10^{-1}$}  & \multirow{2}{*}{$4.189$}                & \multirow{2}{*}{$1.170 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                          &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.176 \times 10^{-3}$} & \multirow{2}{*}{$8.361 \times 10^{-3}$} & \multirow{2}{*}{$4.284 \times 10^{-2}$} & \multirow{2}{*}{$2.972 \times 10^{-1}$}  & \multirow{2}{*}{$4.041$}                & \multirow{2}{*}{$1.151 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                          &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.173 \times 10^{1}$}  & \multirow{2}{*}{$1.426 \times 10^{3}$}  & \multirow{2}{*}{$9.992 \times 10^{2}$}  & \multirow{2}{*}{$4.642 \times 10^{2}$}   & \multirow{2}{*}{$1.097 \times 10^{2}$}  & \multirow{2}{*}{$2.740 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                          &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.566 \times 10^{1}$}  & \multirow{2}{*}{$1.350 \times 10^{3}$}  & \multirow{2}{*}{$7.571 \times 10^{2}$}  & \multirow{2}{*}{$2.172 \times 10^{2}$}   & \multirow{2}{*}{$3.061 \times 10^{1}$}  & \multirow{2}{*}{$7.210$}                \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                          &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.669 \times 10^{1}$}  & \multirow{2}{*}{$6.446$}                & \multirow{2}{*}{$1.291$}                & \multirow{2}{*}{$1.866 \times 10^{-1s}$} & \multirow{2}{*}{$1.986 \times 10^{-2}$} & \multirow{2}{*}{$4.587 \times 10^{-3}$} \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                          &                                         &                                         \\ \hline
\end{tabular}%
}
    \caption{Cross section data calculated for $h^{0}/H^{0} \to \tamu \to \mu e  + \MET + X$ for $\sqrt{s} = 13$ TeV. In all cases the full suite of selection rules have been applied and the total width of each Higgs resonance 
    is reported as calculated using 2HDMC with $\rho_{tc} = \cosba = 0.1$ and $\rho_{\tamu}$ as indicated.}
\label{leptonic13TeV-tbl}
\end{table}
%
\begin{table}[H]
\centering
\resizebox{1.2\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$M_{H}$                                 & 125 GeV                                 & 150 GeV                                 & 200 GeV                                 & 300 GeV                                 & 400 GeV                                 & 500 GeV                                 \\ \hline
top Loop K-factor                       & 1.89743                                 & 1.90633                                 & 1.92299                                 & 1.9447                                  & 1.91845                                 & 1.87137                                 \\ \hline
bottom loop K-fator                     & 1.04143                                 & 1.06498                                 & 1.10215                                 & 1.15453                                 & 1.19169                                 & 1.22052                                 \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$5.950 \times 10^{-3}$} & \multirow{2}{*}{$2.410 \times 10^{-1}$} & \multirow{2}{*}{$3.350 \times 10^{-1}$} & \multirow{2}{*}{$7.625 \times 10^{-1}$} & \multirow{2}{*}{$4.662$}                & \multirow{2}{*}{$1.229 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.146 \times 10^{-3}$} & \multirow{2}{*}{$6.377 \times 10^{-2}$} & \multirow{2}{*}{$1.167 \times 10^{-1}$} & \multirow{2}{*}{$4.080 \times 10^{-1}$} & \multirow{2}{*}{$4.189$}                & \multirow{2}{*}{$1.170 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.176 \times 10^{-3}$} & \multirow{2}{*}{$8.361 \times 10^{-3}$} & \multirow{2}{*}{$4.284 \times 10^{-2}$} & \multirow{2}{*}{$2.972 \times 10^{-1}$} & \multirow{2}{*}{$4.041$}                & \multirow{2}{*}{$1.151 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.311 \times 10^{1}$}  & \multirow{2}{*}{$1.602 \times 10^{3}$}  & \multirow{2}{*}{$1.132 \times 10^{3}$}  & \multirow{2}{*}{$5.332 \times 10^{2}$}  & \multirow{2}{*}{$1.276 \times 10^{2}$}  & \multirow{2}{*}{$3.224 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.751 \times 10^{1}$}  & \multirow{2}{*}{$1.517 \times 10^{3}$}  & \multirow{2}{*}{$8.576 \times 10^{2}$}  & \multirow{2}{*}{$2.495 \times 10^{2}$}  & \multirow{2}{*}{$3.560 \times 10^{1}$}  & \multirow{2}{*}{$8.483$}                \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.867 \times 10^{1}$}  & \multirow{2}{*}{$7.242$}                & \multirow{2}{*}{$1.462$}                & \multirow{2}{*}{$2.144 \times 10^{-2}$} & \multirow{2}{*}{$2.310 \times 10^{-2}$} & \multirow{2}{*}{$5.397 \times 10^{-3}$} \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
\end{tabular}%
}
    \caption{Cross section data calculated for $h^{0}/H^{0} \to \tamu \to \mu e  + \MET + X$ for $\sqrt{s} = 14$ TeV. In all cases the full suite of selection rules have been applied and the total width of each Higgs resonance 
    is reported as calculated using 2HDMC with $\rho_{tc} = \cosba = 0.1$ and $\rho_{\tamu}$ as indicated.}
\label{leptonic14TeV-tbl}
\end{table}
%
\begin{table}[H]
\centering
\resizebox{1.2\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$M_{H}$                                 & 125 GeV                                 & 150 GeV                                 & 200 GeV                                 & 300 GeV                                 & 400 GeV                                 & 500 GeV                                 \\ \hline
top Loop K-factor                       & 1.89743                                 & 1.90633                                 & 1.92299                                 & 1.9447                                  & 1.91845                                 & 1.87137                                 \\ \hline
bottom loop K-fator                     & 1.04143                                 & 1.06498                                 & 1.10215                                 & 1.15453                                 & 1.19169                                 & 1.22052                                 \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$5.950 \times 10^{-3}$} & \multirow{2}{*}{$2.410 \times 10^{-1}$} & \multirow{2}{*}{$3.350 \times 10^{-1}$} & \multirow{2}{*}{$7.625 \times 10^{-1}$} & \multirow{2}{*}{$4.662$}                & \multirow{2}{*}{$1.229 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.146 \times 10^{-3}$} & \multirow{2}{*}{$6.377 \times 10^{-2}$} & \multirow{2}{*}{$1.167 \times 10^{-1}$} & \multirow{2}{*}{$4.080 \times 10^{-1}$} & \multirow{2}{*}{$4.189$}                & \multirow{2}{*}{$1.170 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.176 \times 10^{-3}$} & \multirow{2}{*}{$8.361 \times 10^{-3}$} & \multirow{2}{*}{$4.284 \times 10^{-2}$} & \multirow{2}{*}{$2.972 \times 10^{-1}$} & \multirow{2}{*}{$4.041$}                & \multirow{2}{*}{$1.151 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.486 \times 10^{1}$}  & \multirow{2}{*}{$1.820 \times 10^{3}$}  & \multirow{2}{*}{$1.255 \times 10^{3}$}  & \multirow{2}{*}{$5.401 \times 10^{2}$}  & \multirow{2}{*}{$1.168 \times 10^{2}$}  & \multirow{2}{*}{$2.649 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$1.984 \times 10^{1}$}  & \multirow{2}{*}{$1.723 \times 10^{3}$}  & \multirow{2}{*}{$9.511 \times 10^{2}$}  & \multirow{2}{*}{$2.528 \times 10^{2}$}  & \multirow{2}{*}{$3.259 \times 10^{1}$}  & \multirow{2}{*}{$6.972$}                \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$2.114 \times 10^{1}$}  & \multirow{2}{*}{$8.227$}                & \multirow{2}{*}{$1.622$}                & \multirow{2}{*}{$2.172 \times 10^{-1}$} & \multirow{2}{*}{$2.115 \times 10^{-2}$} & \multirow{2}{*}{$4.436 \times 10^{-3}$} \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
\end{tabular}%
}
    \caption{Cross section data calculated for $h^{0}/H^{0} \to \tamu \to \mu j_{\tau}  + \MET + X$ for $\sqrt{s} = 8$ TeV. In all cases the full suite of selection rules have been applied and the total width of each Higgs resonance 
    is reported as calculated using 2HDMC with $\rho_{tc} = \cosba = 0.1$ and $\rho_{\tamu}$ as indicated.}
\label{hadronic8TeV-tbl}
\end{table}
%
\begin{table}[H]
\centering
\resizebox{1.2\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$M_{H}$                                 & 125 GeV                                 & 150 GeV                                 & 200 GeV                                 & 300 GeV                                 & 400 GeV                                 & 500 GeV                                 \\ \hline
top Loop K-factor                       & 1.89743                                 & 1.90633                                 & 1.92299                                 & 1.9447                                  & 1.91845                                 & 1.87137                                 \\ \hline
bottom loop K-fator                     & 1.04143                                 & 1.06498                                 & 1.10215                                 & 1.15453                                 & 1.19169                                 & 1.22052                                 \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$5.950 \times 10^{-3}$} & \multirow{2}{*}{$2.410 \times 10^{-1}$} & \multirow{2}{*}{$3.350 \times 10^{-1}$} & \multirow{2}{*}{$7.625 \times 10^{-1}$} & \multirow{2}{*}{$4.662$}                & \multirow{2}{*}{$1.229 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.146 \times 10^{-3}$} & \multirow{2}{*}{$6.377 \times 10^{-2}$} & \multirow{2}{*}{$1.167 \times 10^{-1}$} & \multirow{2}{*}{$4.080 \times 10^{-1}$} & \multirow{2}{*}{$4.189$}                & \multirow{2}{*}{$1.170 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.176 \times 10^{-3}$} & \multirow{2}{*}{$8.361 \times 10^{-3}$} & \multirow{2}{*}{$4.284 \times 10^{-2}$} & \multirow{2}{*}{$2.972 \times 10^{-1}$} & \multirow{2}{*}{$4.041$}                & \multirow{2}{*}{$1.151 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$3.209 \times 10^{1}$}  & \multirow{2}{*}{$4.070 \times 10^{3}$}  & \multirow{2}{*}{$2.986 \times 10^{3}$}  & \multirow{2}{*}{$1.434 \times 10^{3}$}  & \multirow{2}{*}{$3.429 \times 10^{2}$}  & \multirow{2}{*}{$8.540 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$4.284 \times 10^{1}$}  & \multirow{2}{*}{$3.853 \times 10^{3}$}  & \multirow{2}{*}{$2.263 \times 10^{3}$}  & \multirow{2}{*}{$6.710 \times 10^{2}$}  & \multirow{2}{*}{$9.564 \times 10^{1}$}  & \multirow{2}{*}{$2.247 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$4.565 \times 10^{1}$}  & \multirow{2}{*}{$1.839 \times 10^{1}$}  & \multirow{2}{*}{$3.858$}                & \multirow{2}{*}{$5.764 \times 10^{-1}$} & \multirow{2}{*}{$6.206 \times 10^{-2}$} & \multirow{2}{*}{$1.430 \times 10^{-2}$} \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
\end{tabular}%
}
    \caption{Cross section data calculated for $h^{0}/H^{0} \to \tamu \to \mu j_{\tau}  + \MET + X$ for $\sqrt{s} = 13$ TeV. In all cases the full suite of selection rules have been applied and the total width of each Higgs resonance 
    is reported as calculated using 2HDMC with $\rho_{tc} = \cosba = 0.1$ and $\rho_{\tamu}$ as indicated.}
\label{hadronic13TeV-tbl}
\end{table}
%
\begin{table}[H]
\centering
\resizebox{1.2\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$M_{H}$                                 & 125 GeV                                 & 150 GeV                                 & 200 GeV                                 & 300 GeV                                 & 400 GeV                                 & 500 GeV                                 \\ \hline
top Loop K-factor                       & 1.89743                                 & 1.90633                                 & 1.92299                                 & 1.9447                                  & 1.91845                                 & 1.87137                                 \\ \hline
bottom loop K-fator                     & 1.04143                                 & 1.06498                                 & 1.10215                                 & 1.15453                                 & 1.19169                                 & 1.22052                                 \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$5.950 \times 10^{-3}$} & \multirow{2}{*}{$2.410 \times 10^{-1}$} & \multirow{2}{*}{$3.350 \times 10^{-1}$} & \multirow{2}{*}{$7.625 \times 10^{-1}$} & \multirow{2}{*}{$4.662$}                & \multirow{2}{*}{$1.229 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.146 \times 10^{-3}$} & \multirow{2}{*}{$6.377 \times 10^{-2}$} & \multirow{2}{*}{$1.167 \times 10^{-1}$} & \multirow{2}{*}{$4.080 \times 10^{-1}$} & \multirow{2}{*}{$4.189$}                & \multirow{2}{*}{$1.170 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\Gamma_{H}$ GeV                        & \multirow{2}{*}{$4.176 \times 10^{-3}$} & \multirow{2}{*}{$8.361 \times 10^{-3}$} & \multirow{2}{*}{$4.284 \times 10^{-2}$} & \multirow{2}{*}{$2.972 \times 10^{-1}$} & \multirow{2}{*}{$4.041$}                & \multirow{2}{*}{$1.151 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$3.588 \times 10^{1}$}  & \multirow{2}{*}{$4.573 \times 10^{3}$}  & \multirow{2}{*}{$3.383 \times 10^{3}$}  & \multirow{2}{*}{$1.647 \times 10^{3}$}  & \multirow{2}{*}{$3.988 \times 10^{2}$}  & \multirow{2}{*}{$3.224 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.2$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$4.790 \times 10^{1}$}  & \multirow{2}{*}{$4.329 \times 10^{3}$}  & \multirow{2}{*}{$2.563 \times 10^{3}$}  & \multirow{2}{*}{$7.708 \times 10^{2}$}  & \multirow{2}{*}{$1.112 \times 10^{1}$}  & \multirow{2}{*}{$2.644 \times 10^{1}$}  \\
$\rho_{\tau \mu} = 0.1$                 &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
$\sigma_{H}$ (fb)                       & \multirow{2}{*}{$5.105 \times 10^{1}$}  & \multirow{2}{*}{$2.067 \times 10^{1}$}  & \multirow{2}{*}{$4.370$}                & \multirow{2}{*}{$6.622 \times 10^{-1}$} & \multirow{2}{*}{$7.219 \times 10^{-2}$} & \multirow{2}{*}{$1.682 \times 10^{-3}$} \\
$\rho_{\tau \mu} = 2.489 \times 10^{-2}$ &                                         &                                         &                                         &                                         &                                         &                                         \\ \hline
\end{tabular}%
}
    \caption{Cross section data calculated for $h^{0}/H^{0} \to \tamu \to \mu j_{\tau}  + \MET + X$ for $\sqrt{s} = 14$ TeV. In all cases the full suite of selection rules have been applied and the total width of each Higgs resonance 
    is reported as calculated using 2HDMC with $\rho_{tc} = \cosba = 0.1$ and $\rho_{\tamu}$ as indicated.}
\label{hadronic14TeV-tbl}
\end{table}
%
\end{landscape}


The work for this project is ongoing, as we still need to look at the pseudoscalar signal and extend the mass range for the Heavy Higgs process out to $\approx 1$ TeV as 
was done with the htc project. The final results will then be submitted for publication. With the additional data and further refinement of the search, namely in the context 
of more points and a higher fidelity scan as a function of Higgs mass, discovery potentials will be created to visually represent our findings. Preliminary findings suggest 
that the end of Run II should provide us with enough data to have seen enough evidence of a \tamu flavor changing process involving a light, SM-like Higgs boson with a coupling 
as low as our base value of $\rho_{\tamu} = 2.489  \Exp{-2}$ while we will need at least $1800 \text{ fb}^{-1}$ before we can see such a signal in the heavy Higgs sector making 
such a signal not visible until after the high luminosity upgrade of the LHC.

