%-----------------------------------------------------------------------------------------------------------------
%       Chapter 2: 2HDM and analytic methods
%-----------------------------------------------------------------------------------------------------------------
\chapter{Analytic Higgs Production}

\singlespacing

\epigraph{``\textit{Tiny, but fierce.}''}{
    --- Sanya, \emph{Changes, Jim Butcher}}

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

    \section{Higgs production modes}

    Higgs production at the LHC is generated predominately through either quark anti-quark annihilation or gluon 
    gluon fusion. Of course, the Higgs can couple to any particle that has mass but any other modes will generate 
    additional particles that may be detected if they are of a high enough energy. 
        
        \subsection{Dominant processes}

        The dominant processes used in our study of FCNH processes involved gluon gluon fusion through a heavy quark 
        loop, such as a top or bottom. Despite the gluon fusion production being a higher order process, the 
        abundance of gluons in the proton help to enhance its Higgs production rate, \Cref{ggfHiggs,ggfEffHiggs}, over 
        the tree level 
        $q \bar{q}$ process, \Cref{qqbHsChannel}.
        %
        \begin{figure}[H]
            \begin{center}
            %
                \subfigure[Feynman diagram for the gluon-gluon fusion production mode for the Higgs boson.]{
                    \includegraphics[height=3.5cm,keepaspectratio]{./figures/ggH_LoopDiagram_cropped.eps} \label{ggfHiggs}}
                %
                \quad
                %
                \subfigure[Feynman diagram for the effective gluon-gluon fusion production mode for the Higgs boson.]{
                    \includegraphics[height=3.5cm,keepaspectratio]{./figures/ggH_TreeDiagram_cropped.eps} \label{ggfEffHiggs}}
                %
                \caption[Feynman diagrams for the gluon-gluon fusion process showing the loop-level and tree-level]{
                Feynman diagrams for the gluon-gluon fusion process showing the loop-level and tree-level diagrams where time 
                is moving from left to right in the figures. Here the top and bottom contributions are listed as the couple 
                most strongly to the Higgs and will be your dominant loop particles.}
                \label{ggF}
            %
            \end{center}
            %
        \end{figure}
        %

        \subsection{Sub-dominant processes}

        The sub-dominant processes are generally disregarded in our calculations as they introduce additional final 
        state particles that we can exclude with careful selection criteria. However, they are worth introducing as 
        they help illustrate the coupling behavior of this Higgs and why we must treat it so carefully. All of the 
        diagrams are t-channel diagrams where the Higgs is produced in association with some number of jets, again, 
        if we require, in our selection criteria, only jets associated with the immediate Higgs decay we are interested
        in be present in our final state we can effectively remove these additional contributions. They could be 
        included later if we wanted additional channels in which to study Higgs production and decay but will introduce 
        additional background processes that may be difficult to reduce without ignoring too many signal events. Of 
        these additional decays, the most promising would be the $q \bar{q} \rightarrow H + j j$ decay mode as up quarks 
        carry the dominant fraction of the proton's initial momentum after gluons. Here we signify two light quarks 
        that are produced in association with the Higgs as jets, $j$.
        %
        \begin{figure}[H]
            \begin{center}
                %
                \subfigure[Feynman diagram for the t-channel production of the Higgs from a quark and anti-quark colliding.]{
                    \includegraphics[height=3.5cm,keepaspectratio]{./figures/qqH_tChannel_W_2.eps} \label{qqbHtChannel}}
                %
                \quad
                %
                \subfigure[Feynman diagram for the t-channel production of the Higgs from two quarks colliding.]{
                    \includegraphics[height=3.5cm,keepaspectratio]{./figures/qqH_tChannel_Z_2.eps} \label{qqHtChannel}}
                %
                \subfigure[Feynman diagram for the s-channel production of the Higgs from a $c \bar{c}$ or a $b \bar{b}$ 
                in the initial colliding protons]{
                    \includegraphics[height=3.5cm,keepaspectratio]{./figures/qqH_sChannel.eps} \label{qqbHsChannel}}
                %
                \caption[Feynman diagrams for the sub-dominant quark quark and quark anti-quark Higgs production modes.]{
                Feynman diagrams for the sub-dominant quark quark and quark anti-quark Higgs production modes. As before, 
                time is moving to the right in the figures. For the charm/bottom production modes we require the 5-flavor 
                scheme for the initial colliding protons.}
                %
                \label{qqbH}
                %
            \end{center}
        \end{figure}
        %

    \section{Effective theories}

    Effective theories are a broad classification of techniques used to approximate more complicated calculations or 
    corrections. In the context of high energy physics, effective theories most commonly refer to incorporating 
    higher order corrections into simple factors or, as is the case with this work, in reducing Loop-level calculations 
    to an approximate tree-level calculation. The transition can be seen in \Cref{ggF}, where \Cref{ggfHiggs} shows the 
    actual physical production process involving the virtual fermion loop and \Cref{ggfEffHiggs} shows the diagram for 
    the effective theory where we have condensed the loop into an approximate tree-level coupling.


        \subsection{Loop diagrams}

        Loop diagrams are present in many areas of particle physics as they allow for couplings between particles that 
        do not normally share any quantum numbers, and thus violate one or many conservation laws, and account for 
        corrections to physical processes due to the perturbative nature of the theory. A loop is any process that is 
        entirely virtual where the particles inside the loop are not required to be on the mass shell. Furthermore, 
        loops are processes that happen \emph{instantaneously} in the context of the decay process. What this means in 
        the context of performing calculations, since $\Delta E \Delta t \geq \sfrac{\bar{h}}{2}$ by quantum mechanics, 
        if our time is taken to be precisely known for the loop then we have no constraints on our energy. 

        Generally, loop diagrams are suppressed due to the presence of additional vertices and couplings associated with 
        those vertices, and the inclusion of the propagators associated with each virtual particle. That suppression is 
        outweighed in the case of Higgs production by a couple of factors. Gluons, for instance, carry a significant 
        fraction of the proton's momentum, more so than even the up quarks. In addition to that, the bottom quark couples 
        nearly four times as strongly as the next heaviest particle and the top quark nearly forty times as strongly as 
        the bottom. These two factors combined allow the gluon-gluon fusion process to dominate Higgs production at 
        hadronic colliders, such as the LHC.

            \subsubsection{Virtual particles}

            Virtual is a classification that refers to a particle or particle-like interaction that does not happen in resonance 
            with the mass shell of the particle. In the context of the diagrams above, this is only seen in the fermion loop in 
            the gluon gluon fusion process. Internally, to construct the transition matrix element from a virtual particle you 
            have to apply the completeness relation from quantum mechanics and integrate over the virtual particle's momentum, 
            which is unconstrained. Externally,  when unstable particles, like the top and Higgs, are in the final state their 
            decay products exhibit a well defined Breit-Wigner resonance at the mass value of the parent allowing us to identify 
            them. The decay products of virtual particles in the final state, however, will produce an invariant mass peak not 
            on the mass shell and we may need to look at additional metrics to get a clear picture of the interaction. 
            A good example of this is in the $H\to Z Z^{\ast} \to 4 \ell$ or beta decay of a neutron into a proton. The mass 
            difference between a neutron and a proton is $\approx 1.3$ MeV while the $W$ boson that mediates this decay has a 
            mass of $\approx 80$ GeV. The neutron-proton interaction simply does not have enough energy to produce a real $W$ 
            boson, however, the decay process itself must be mediated by the electroweak interaction. So the $W$ produced that 
            subsequently generates an electron and electron antineutrino is actually a virtual $W$, meaning it does not exist at 
            the mass shell. Mathematically this can be described by the equation $p^2 \neq M_{W}^2$, where $p$ is the four momentum 
            of the $W$ \cite{Peskin1995}. 

            \subsubsection{Propagators and energy limits}

            Part of the challenge with virtual particles, especially in the context of a loop as in the gluon fusion process,
            is evaluating the propagators with unknown limits. For a fermion the propagators is of the form,
            %
            \begin{align}
                %
                \Delta (p) &= \frac{i}{\slashed{p} - m + i\epsilon} \notag \\[1em]
                           &= \frac{i (\slashed{p} + m) }{p^2 - m^2 + i \epsilon} \label{fermiProp}.
                %
            \end{align}
            %
            With the uncertainty in the energy because of the instantaneous nature of the loop we can have either very small 
            or very large momenta in the denominator here. Further complications arise when we have multiple of these terms 
            appearing due to our loop having three virtual fermion lines, each with their own propagator. Even though these 
            particles are virtual and their lifespan is nearly instantaneous we still need the information about propagation 
            in these terms to correctly evaluate the physical decay. In low transverse momentum situations (low $P_T$, we can 
            approximate this behavior but not in high $P_T$ cases like heavy Higgs production. So to evaluate this process as 
            correctly as possible we need to generate an analytic solution, an analytic method for evaluating these loop 
            processes \cite{Brehmer2016}.

    \section{Analytic solutions: a `simple' alternative}

     The technical aspects of calculating analytic cross sections lie in the calculation of the Transition Matrix Element,
     often referred to as $M$ that has to be squared in order to produce the well-defined phase probability that we 
     know of as cross section. This piece is where all of the physics from our theory takes place, it tells us how 
     we can go from some initial state to some final state and describes any intermediate interactions and particles 
     that facilitate this \emph{transition}. In the case of the fermion loop that supports the coupling of the Higgs 
     to gluons, you encounter the following trace as described in \Cref{loopMomentum} and \Cref{loopTrace},
     %
     \pagebreak
     %
     \begin{align}
         %
         N^{\mu \nu} = {\rm Tr} \left\{ [ \slashed{\ell} + \slashed{k}_1 + \slashed{k}_2 + m_{q} ] \gamma^{\nu} 
         [ \slashed{\ell} + \slashed{k}_1 + m_{q} ] \gamma^{\mu} [ \slashed{\ell} + m_{q} ] \right\} \ .
         \label{loopTrace}
         %
     \end{align}
     %
     \begin{figure}[H]
         \begin{center}
             %
             \includegraphics[width=0.55\textwidth,keepaspectratio]{./figures/ggh_momentum.eps}
             %
             \caption[Feynman diagram for the gluon-gluon fusion production mode of the Higgs. Here the designations]{
                 Feynman diagram for the gluon-gluon fusion production mode of the Higgs. Here the 
                 designations $\ell{\rm,} \ k_1{\rm,} \ k_2$ refer to the momentum allocation for the fermion loop, 
                 initial state particles, and $m_{q}$ is the mass of the fermion, either a top or bottom in this case.}
             %
             \label{loopMomentum}
             %
         \end{center}
     \end{figure}
     %
     %\vspace{-1.5cm}

     Here, as discussed before in regard to the energy uncertainty, we have to integrate over the unknown fermion 
     momentum, $\ell$. Including the propagators associated with each fermion line we have to then evaluate the 
     integral,
     %
     \begin{align}
         %
         T^{\mu \nu} &= \notag \\
         \int \frac{d^{N} \ell}{(2 \pi)^{N}} &\frac{-2 f N^{\mu \nu}}{
             \left[ (\ell + k_1 + k_2)^2 - m_q^2 + i\epsilon \right]\left[ (\ell + k_1)^2 - m^2_q + i\epsilon 
             \right]\left[ \ell^2 - m^2_q + i\epsilon \right]} \ ,
         \label{loopIntegral}
         %
     \end{align}
     %
     where we acquire a factor of $2$ from being able to go clockwise or counterclockwise around the loop 
     and a $-1$ from the nature of the closed fermion loop and account for the couplings and color factors in the 
     $f$ term. The $N$ above denotes the dimensionality of the integral which we leave as arbitrary initially, 
     $N = 4 + \epsilon$, and then take the limit of our solution as $\epsilon \to 0$. If we evaluate the trace, 
     \Cref{loopTrace}, we then obtain tensors as suggested by \cite{Passarino1979},
     %
     \begin{align}
         %
         C_0 &= \int \frac{d^{N} \ell}{(2 \pi)^{N}} \frac{1}{D_1 D_2 D_3} \notag \\[4pt]
         C^{\mu}_1 &= \int \frac{d^{N} \ell}{(2 \pi)^{N}} \frac{\ell^{\mu}}{D_1 D_2 D_3} \label{tensorLoops} \\[4pt]
         C^{\mu \nu}_2 &= \int \frac{d^{N} \ell}{(2 \pi)^{N}} \frac{\ell^{\mu} \ell^{\nu}}{D_1 D_2 D_3} \notag \\[6pt]
         D_1 \equiv \! \left[ (\ell + k_1 + k_2)^2 - m_q^2 + i\epsilon \right] \! {\rm,} \hspace{3pt} 
         &D_2 \equiv \! \left[ (\ell + k_1)^2 - m^2_q + i\epsilon \right] \! {\rm,} \hspace{3pt}
         D_3 \equiv \! \left[ \ell^2 - m^2_q + i\epsilon \right] \notag\!. 
         %
     \end{align}
     %
     While $C_0$ is a scalar integral and can be evaluated explicitly, $C_1 \text{ and } C_2$ are not and have to be 
     to be reduced to scalar integrals through tensor reduction techniques. After applying the tensor reduction 
     techniques, $C_1 \text{ and }C_{2}$ can be written in terms of $C_0$ and the overall loop part of the matrix element 
     for the production mode can be written as,
     %
     \begin{align}
         %
         |T|^2 = -4 f^2 \left[ 2 + (s - 4m^2_q) C_0 \right]
         \label{ReducedLoopInt}
         %
     \end{align}
     %

    \section{Translating physics into code}

    Obtaining an analytic solution is one step in finally calculating the cross section and starting to look at what we 
    can find out about these Flavor Changing Neutral Currents. The next step is no less daunting and involves developing 
    computer calculations and code to generate these responses. Analyzing a signal and developing discovery potential plots 
    requires us to calculate the signal and dominant physics background, and apply selection criteria to both to try and 
    minimize contamination. Ultimately this comes down to a numerical integration technique with some dimensionality set 
    by the complexity of the calculation. In our case, there is not much to be gained by using higher level languages 
    such as C++ and many of the libraries and programs we can use to help piece together the information are written in 
    Fortran so that is the chosen language for the main code that we developed. 

        \subsection{Code development and computational efficiency}

        The cross section code I developed with Baris Altunkaynak utilizes subroutines and refined functions to correctly 
        interpret the physics and develop the integrals in such a way that we are always integrating under the limits of 
        zero to one. In addition to analytically calculating the transition matrix element we also develop routines to 
        calculate phase space for pair decays and calculate appropriate Jacobian factors. 

        The development of this code interface has allowed for better portability between systems which is ideal 
        considering we utilize both our own personal hardware and the OU Supercomputing Center for Education and 
        Research for many of our calculations. An additional advantage is the security of having one static `cut' file 
        where we can easily apply the same analysis to the signal and background codes by simply pointing it to the 
        proper file to include. This improved my personal code integrity and allowed me to easily check calculations 
        against other group members or software as described in the next section.

        \subsection{Consistency check}

        In order to check the accuracy of our calculations we compared results to those obtained from MadGraph, version 5, 
        for preliminary or basic results, such as standard model Higgs production. We were able to achieve agreement within 
        1\% of what MadGraph predicted, which we viewed as a more than acceptable margin. MadGraph was then used to produce 
        matrix elements for our background processes and interfaced these with our cross section code for consistency. Doing 
        so also allows us to more easily control our selection criteria without having to rely on the limited options and 
        algorithms employed by MadGraph. Once code techniques are verified and selection criteria are selected we can then 
        move on to production and produce the data presented in this dissertation \cite{Alwall2014}. For analysis 
        purposes we rely on programs such as Mathematica and ROOT to analyze and compare our signal data to our background 
        data. ROOT analysis allows us to perform more realistic detector simulations but also requires us to use C++ 
        language for calculation.

        \subsection{Theory versus phenomenology} \label{pheno}

        In addition to high energy experiment and high energy theory there is a unique branch or classification of researchers 
        known as phenomenologists. Phenomenology it a blend of the theoretical side of high energy physics and the experimental 
        side and this is the category in which our research group is classified. Traditional high energy theory focuses mainly 
        on the refining and development of new models to try and explain the world around us. In many scenarios this involves 
        research that is more mathematically driven where you turn the knobs and dials of nature and see what all you can make 
        within the parameter space you are given. String theory and some aspects of supersymmetry (SUSY) can fall into this 
        category if the research's main focus is on model building.

        Experimentalists on the other hand work directly with the data and often look at several theories simultaneously to 
        see which one fits and describes the data best. Alongside theory, these two groups play a sort of cat and mouse style 
        game where sometimes theory leads the discovery and sometimes the data leads to new theories. An example is the recent 
        hint of a di-photon event excess at the LHC 
        \cite{CMScollaboration2016,CMScollaboration2015,ATLAScollaboration2016,ATLAScollaboration2015}, which spawned hundreds of new papers 
        on the theory side trying to explain what might cause this excess. It turns out as more data was collected it was 
        a statistical anomaly and not a new discovery but it still sparked a lot of interest. Contrary to that, discoveries of 
        the top quark and Higgs boson were well heralded by theory which helped to direct the experimental searches and analysis 
        to look for those particular signatures, \cite{Aad2012,Chatrchyan2012,Campagnari:1996ai}. 

        Phenomenology is a unique approach where you play both roles as a theorist and experimentalist. Occasionally 
        phenomenologists will analyze data from the LHC to support a study they are conducting. In our work we often try to 
        implement the selection strategies used by the CMS and ATLAS collaborations to adequately model an experimental analysis. 
        Instead of looking at data and trying to find the physics or building and perfecting models that have yet to be tested, 
        we directly put models to the test in mock experimental settings so we can make predictions. At times this requires us to 
        do a little model building ourselves but at the end of the day our goal is to still run simulations and try and look at 
        the data like an experimentalist so we can tell them where to put their focus or offer suggestions on types of analyses 
        they can conduct on their data to verify or disprove certain models. This requires us to have knowledge of the 
        detectors, how they operate, and we need to be able to break down or expand the BSM theories and SM extensions that we 
        wish to test. 

        Many of the studies we perform involve the utilization of several programs, many of which are modified to fit our needs, 
        and developing our own framework for calculation. We also adopt several analysis criteria and methodologies from our 
        experimentalist colleagues. One such criteria is the $5\sigma$ significance notion where we do not treat something as a 
        discovery unless it has the proper statistical significance. Statistical significance, in this case, is determined by 
        looking at how far above the standard background the signal + background event data lies. If the signal `bump' minus 
        $2.5$ times it's uncertainty lies at the edge of, or outside, the background plus $2.5$ times its uncertainty, it is said 
        to have a $5 \sigma$ significance and counts as a discovery. Mathematically this can be represented by the following:
        %
        \begin{align}
            %
            \sigma_{S} &\geq \frac{n}{L} \left[ n + 2 \sqrt{L \sigma_{B}} \right] \label{StatSigCrossX} \\
            %
            \text{or, in } &\text{terms of number of events,} \notag \\
            %
            N_{SS} &= \frac{N_{S}}{\sqrt{N_{B}}} = \frac{ L \sigma_{S} }{ \sqrt{L \sigma_{B}} } \geq 5 \ .\label{StatSig1}
            %
        \end{align}
        %
        In \Cref{StatSigCrossX}, $\sigma_{S, \ B}$ is the cross section of the signal + background and background only, respectively, 
        and $n$ is the scale parameter that selects how many standard deviations we are away from each datum. So, if we want a 
        $5\sigma$ significance then we set $n=2.5$ in to constitute a discovery. In \Cref{StatSig1} ths is translated to a more 
        familiar measure of the number of signal and background events, $N_{S, \ B}$ respectively. The conversion from cross section 
        to number of events is straightforward, $N = \sigma \times L$ where, again, $\sigma$ is the cross section (probability) 
        and $L$ is the integrated luminosity (amount of data collected).

        Often times, as mentioned before, phenomenologists rely on experimental results to inform the studies we undertake. For 
        instance, the idea to look at FCNCs as possible indicators of new physics beyond the SM was hinted by an excess of 
        $h^0 \to \tau \mu$ events reported by CMS during their Run 1 analysis \cite{Collaboration2015}. If these FCNC 
        processes exist, they should be not only in the leptonic sector but also in the hadronic sector as they are both fermions and 
        couple to the Higgs field in similar ways. Since that interaction is directly proportional to mass,
        %
        \begin{align}
            %
            \mathscpt{L}_{Y} = - \frac{\lambda}{\sqrt{2}} &\sum_{F = U,D,L} \overline{F}_L \Phi^0 F_R \label{generalYukawa} \\
            %
            &\text{where} \notag \\
            %
            \lambda &\propto \frac{m_{F}}{v} \label{YukawaCoupling} \ {\rm,}
            %
        \end{align}
        %
        where $v$ is the Higgs vacuum expectation value, experimentally $246$ GeV \cite{Olive:2016xmw}, then and FCNCs involving 
        a Higgs interaction are likely to be seen most readily in an interaction between the top, charm, and Higgs. Since the top has 
        a mass larger than that of the SM-like, light Higgs boson, $h^{0} \to t c$ is not kinematically favorable. However, if we 
        adopt a general 2HDM as discussed in \Cref{2HDMbreakdown}, then a heavy Higgs, $H^0$, or pseudoscalar Higgs, $A^0$, decaying 
        into a top charm pair is kinematically favorable for any masses more than twice $m_t$, or about 350 GeV. Furthermore, in a 
        general 2HDM the heavier Higgs states FCNC couplngs are proportional to \sinba while the SM FCNC coupling is proportional to 
        \cosba and current experimental results favor $\sinba \to 1$. Our first approach to these rare decays then is to investigate 
        the heavy Higgs sector and see if we can place constraints on the strength of the FCNC processes and make some predictions for 
        what the LHC might see from that sector at higher energies and luminosities.


        

