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\subsection{Angular distribution}
\label{subsection:angular_dist}

The angular distributions used by the partial wave analysis to determine the spin quantum numbers of a meson decaying into $K^*(892)\bar{K}$ are defined following the formalism of A. Atkinson \citep{angular_dist}. The derivation begins by defining two sets of angles, $\Omega=(\theta,\phi)$ and $\Omega_H=(\theta_H,\phi_H)$, where $\Omega$ describes the $K^*(892)$ in the resonance helicity frame and $\Omega_H$ describes the $\bar{K}$ in the vector-meson helicity frame (See Figure \ref{fig:diag_ang}). According to the helicity formalism, the z-direction is defined as the unit vector along the direction of travel of the resonance in the center-of-mass frame (C.M.) of the system. Defining the z-direction as $\hat{z}$, the y-direction $\hat{y}$ is defined normal to the resonance and beam $\gamma$ directions. In other words, the y-direction is normal to the production plane defined by the meson resonance and the incident beam directions. To produce a right-handed system, the x-direction is defined by the cross product of $\hat{y}$ and $\hat{z}$. The initial coordinate system is thus defined as

\begin{align}
    \hat{z} &= \frac{\mathbf{p}_{K^*\bar{K},C.M.}}{|\mathbf{p}_{K^*\bar{K},C.M.}|}\\
    \hat{y} &= \frac{\mathbf{p}_{\gamma}\times\hat{z}}{|\mathbf{p}_{\gamma}\times\hat{z}|}\\
    \hat{x} &= \hat{y}\times\hat{z},
\end{align}

\noindent where $\mathbf{p}_{K^*\bar{K},C.M.}$ is the 3-momentum of the resonance in the C.M. frame and $\mathbf{p}_{\gamma}$ is the 3-momentum of the beam photon.

To determine the direction of travel of the $K^*(892)$ in this coordinate system, a boost along the z-direction is performed to move into the rest frame of the meson resonance. Using the 3-momentum vector of the $K^*(892)$ in the resonance rest frame $\mathbf{p}_{K^*,Res.}$, $\theta$ and $\phi$ can be determined by

\begin{align}
    &\cos\theta = \frac{\mathbf{p}_{K^*,Res.}\cdot\hat{z}}{|\mathbf{p}_{K^*,Res.}|}\\
    &\phi = \tan^{-1}\left(\frac{\mathbf{p}_{K^*,Res.}\cdot\hat{y}}{\mathbf{p}_{K^*,Res.}\cdot\hat{x}}\right).
\end{align}

\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{Omega_Omega_H_diagram.png}
    \caption[Diagram of the Decay Chain of a Meson to $K^\ast\bar{K}$ Showing the Angles Described by Solid Angles $\Omega$ and $\Omega_H$.]{Diagram of the decay chain of a meson to $K^\ast\bar{K}$ showing the angles described by solid angles $\Omega$ and $\Omega_H$.}
    \label{fig:diag_ang}
\end{figure}

To determine the second set of angles, describing the decay of the $K^*$, a new coordinate system is established for the helicity frame of the photoproduced meson resonance. Specifically, the z-direction $\hat{z}_H$ is defined as the direction of the $K^*(892)$ in the rest frame of the meson resonance. The decay plane of the resonance is establish by the z-directions of the resonance $\hat{z}$ and the $K^\ast(892)$ helicity frames $\hat{z}_H$, such that the y-direction $\hat{y}_H$ is perpendicular to this plane. As before, the x-direction $\hat{x}_H$ is defined to establish a right handed coordinate system. This coordinate system is defined as

\begin{align}
    \hat{z}_H &= \frac{\mathbf{p}_{K^*,Res.}}{|\mathbf{p}_{K^*,Res.}|}\\
    \hat{y}_H &= \frac{\hat{z}\times\hat{z}_H}{|\hat{z}\times\hat{z}_H|}\\
    \hat{x}_H &= \hat{y}_H\times\hat{z}_H.
\end{align}

\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{Phi_angle_diagram.png}
    \caption[$\Phi$ Defined Between a Normal to the Production Plane and the Incident Photon Beam Polarization Direction with Respect to the Floor.]{$\Phi$ defined between a normal to the production plane and the incident photon beam polarization direction with respect to the floor.}
    \label{fig:Phi_diag}
\end{figure}
\pagebreak
Similar to the determination of $\cos\theta$ and $\phi$, $\cos\theta_H$ and $\phi_H$ are defined as 

\begin{align}
    &\cos\theta_H = \frac{\mathbf{p}_{K,Vec.}\cdot\hat{z}_H}{|\mathbf{p}_{K,Vec.}|}\\
    &\phi_H = \tan^{-1}\left(\frac{\mathbf{p}_{K,Vec.}\cdot\hat{y}_H}{\mathbf{p}_{K,Vec.}\cdot\hat{x}_H}\right),
\end{align}

\noindent where $\mathbf{p}_{K,Vec.}$ is the 3-momentum of a kaon decaying from the $K^*$ in the rest frame of the vector meson. 

Since a polarized photon beam is used in the experiment, a fifth angle $\Phi$ is established to describe the direction of the polarization with respect a vector normal to the production plane (See Figure \ref{fig:Phi_diag}). First, a photon beam polarization vector is established in the lab frame using the polarization angle $\phi_{pol}$ by

\begin{align}
    \mathbf{\epsilon}' &= \cos\phi_{pol} \hat{i} + \sin\phi_{pol} \hat{j}.
\end{align}

\noindent The initial coordinate system used to determine the $\Omega$ angles defines the y-direction as perpendicular to the production plane. The projection of $\hat{y}$ onto $\mathbf{\epsilon}'$ and the projection of $\mathbf{p}_{\gamma,C.M.}$ onto $\mathbf{\epsilon}'\times\hat{y}$ define the legs of the right triangle for determining $\Phi$ such that

\begin{align}
    \Phi &= \tan^{-1}\left(\frac{\frac{\mathbf{p}_{\gamma,C.M.}}{|\mathbf{p}_{\gamma,C.M.}|}\cdot(\mathbf{\epsilon}'\times\hat{y})}{\hat{y}\cdot\mathbf{\epsilon}'}\right),
\end{align}

\noindent where $\mathbf{p}_{\gamma,C.M.}$ is the 3-momentum of the incident photon beam in the C.M. frame of the system.

\subsection{Intensity function in helicity basis}

A cross section $\sigma$ for a given process is defined as the number of interaction per unit time per target particle divided by the incident flux. With respect to particular kinematic variables, the intensity for the production of meson states can be defined using the differential cross section. The intensity is

\begin{align}
    I(\Omega,\Omega_H,\Phi) &\equiv \frac{d\sigma}{dt~dm_{K^*\bar{K}}~d\Omega~d\Omega_H~d\Phi},
\end{align}

\noindent where $\Omega$ is the direction of the $K^*$ in the helicity frame of the system, $\Omega_H$ is the direction of the $K$ in the helicity frame of the photoproduced meson resonance, $\Phi$ the direction of the photon beam polarization with respect to the production plane, $m_{K^*\bar{K}}$ is the mass of the meson resonance, and $t$ is the Mandelstam momentum transfer (See Section \ref{subsection:angular_dist}). 

Using the formalism introduced by V. Mathieu \citep{Intensity_V}, which is expanded by M. Shepherd \citep{Intensity_S} and J. Stevens \citep{Intensity_J}, the intensity can be defined in helicity space. For polarized beams, the intensity is

\begin{align}
    I(\Omega,\Omega_H,\Phi) &= I^0(\Omega,\Omega_H) - P_\gamma I^1(\Omega,\Omega_H)\cos(2\Phi) - P_\gamma I^2\sin(2\Phi),
    \label{equation:I_exp}
\end{align}

\noindent where $P_\gamma$ is the degree of the linear polarization of the incident photon beam. Each $I$ in Equation  
\ref{equation:I_exp} can be defined in terms of decay amplitudes providing

\begin{align}
    I^0(\Omega,\Omega_H) &= \frac{\kappa}{2} \sum_\lambda A_\lambda(\Omega,\Omega_H)A^\ast_\lambda(\Omega,\Omega_H)\\
    I^1(\Omega,\Omega_H) &= \frac{\kappa}{2} \sum_\lambda A_{-\lambda}(\Omega,\Omega_H)A^\ast_\lambda(\Omega,\Omega_H)\\
    I^2(\Omega,\Omega_H) &= i\frac{\kappa}{2} \sum_\lambda \lambda A_{-\lambda}(\Omega,\Omega_H)A^\ast_\lambda(\Omega,\Omega_H),
\end{align}

\noindent where $A$ defines the decay amplitudes and $\lambda$ defines the photon beam helicity, of which $+,-$ terms are expected. Substituting the definitions of $I^0$, $I^1$, and $I^2$ in terms of the decay amplitudes into Equation \ref{equation:I_exp}, summing over helicity states, and applying Euler's formula, the intensity becomes

\begin{multline}
    I(\Omega,\Omega_H,\Phi) = \frac{\kappa}{2}[A_-(\Omega,\Omega_H)A^\ast_-(\Omega,\Omega_H) + A_+(\Omega,\Omega_H)A^\ast_+(\Omega,\Omega_H)\\ - P_\gamma(A_-(\Omega,\Omega_H)A^\ast_+(\Omega,\Omega_H) + A_+(\Omega,\Omega_H)A^\ast_-(\Omega,\Omega_H))e^{i2\Phi}].
\end{multline}

If the decay amplitudes are defined with rotations of $\Phi$ about the photon beam direction as $\tilde{A}_\pm(\Omega,\Omega_H,\Phi) = e^{\mp i\Phi}A_\pm(\Omega,\Omega_H,\Phi)$, the rotation is distributed through all decay amplitude terms. With further algebraic manipulation, the intensity is defined by

\begin{multline}
    I(\Omega,\Omega_H,\Phi) = \frac{\kappa}{4}[(1-P_\gamma)|\tilde{A}_+(\Omega,\Omega_H,\Phi) + \tilde{A}_-(\Omega,\Omega_H,\Phi)|^2\\ + (1+P_\gamma)|\tilde{A}_+(\Omega,\Omega_H,\Phi) - \tilde{A}_-(\Omega,\Omega_H,\Phi)|^2].
    \label{equation:intensity_hel}
\end{multline}

\noindent The amplitudes are expanded in terms of partial waves

\begin{align}
    A_\lambda &= \sum_{i}\sum_{m}T^i_{\lambda,m}\tilde{B},
    \label{equation:partial_wave}
\end{align}

\noindent where $T$ are the partial wave amplitudes and $\tilde{B}$ is the decay amplitude. 

\subsection{The decay amplitude}

The amplitude defined in Equation \ref{equation:partial_wave} used to determine the intensity can be defined in terms of two rotations on spin-space. Let $B$ be the matrix element associated with the two rotations of interest:

\begin{align}
    B &\equiv \langle j_1m_1j_2m_2|R^{(1)}R^{(2)}|Jm \rangle,
\end{align}

\noindent such that $R^{(1)}$ and $R^{(2)}$ act on $|j_1m_1\rangle$ and $|j_2m_2\rangle$, respectively. For the vector-pseudoscalar decay of interest, rotations with respect to the angles $\theta$, $\phi$, $\theta_H$, and $\phi_H$ are defined in the resonance and vector meson rest frames. As a result, $R^{(1)}$ is the rotation in the resonance rest frame, $R_r(\theta,\phi)$, and $R^{(2)}$ is the rotation in the vector meson rest frame, $R_v^{(2)}R_r^{(2)}$. Therefore, the matrix element can be written as

\begin{align}
    B &= \langle j_1m_1j_2m_2| R_r^{(1)}R_v^{(2)}R_r^{(2)} |Jm \rangle.
    \label{eq:decay_amp2}
\end{align}

\noindent If a complete set of spin projections with respect to $j_1$ and $j_2$ is introduced to Equation \ref{eq:decay_amp2}, $B$ can be written in terms of Wigner D-functions.

The Wigner D-functions are related to the spin-weighted spherical harmonics, operating to move between frames of the sequential steps of a meson decay using rotations on the spin space. The Euler angles $(\alpha,\beta,\gamma)$ for which the Wigner D-function is defined

\begin{align}
    D^J_{m'm} &= \langle Jm'|e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}|Jm\rangle \nonumber\\
    &= e^{-im'\alpha}d^J_{m'm}e^{-im\gamma},
\end{align}

\noindent where $d^J_{m'm}=\langle Jm'|e^{-i\beta J_y}|Jm \rangle$ or Wigner D-small. The convention of this work assumes $\gamma=0$ because a particle direction only needs the polar coordinates $(\alpha=\phi,\beta=\theta)$ to be completely defined \citep{PWA_Salgado}.

Introducing a complete set, the matrix element is 

\begin{align}
    B &= \langle j_1m_1j_2m_2| R_r^{(1)}R_v^{(2)}R_r^{(2)} \left[\sum_{m'_1}\sum_{m'_2}|j_1m'_1j_2m'_2\rangle \langle j_1m'_1j_2m'_2 |\right] Jm \rangle \nonumber\\
    &= \sum_{m'_1}\sum_{m'_2} \langle j_1m_1|R_r^{(1)}|j_1m'_1\rangle\langle j_2m_2|R_v^{(2)}R_r^{(2)}|j_2m'_2\rangle \langle j_1m'_1j_2m'_2|Jm\rangle,
\end{align}

\noindent where the first term in the sum is the Wigner D-function, $D_{m_1m'_1}^{j_1}(R_r)$. A second complete set can be introduced

\begin{align}
    B &= \sum_{m'_1}\sum_{m'_2} D_{m_1m'_1}^{j_1}(R_r) \langle j_2m_2|R_v^{(2)}\left[ \sum_{m''_2}|j_2m''_2\rangle \langle j_2m''_2|\right]R_r^{(2)}|j_2m'_2\rangle \langle j_1m'_1j_2m'_2|Jm\rangle\nonumber\\
    &= \sum_{m'_1}\sum_{m'_2}\sum_{m''_2} D_{m_1m'_1}^{j_1}(R_r) \langle j_2m_2|R_v^{(2)}|j_2m''_2\rangle \langle j_2m''_2|R_r^{(2)}|j_2m'_2\rangle \langle j_1m'_1j_2m'_2|Jm\rangle.
\end{align}

\noindent which introduces two more Wigner D-functions to the decay amplitude:

\begin{align}
    B &= \sum_{m'_1}\sum_{m'_2}\sum_{m''_2} D_{m_1m'_1}^{j_1}(R_r) D_{m''_2m'_2}^{j_2}(R_r)D^{j_2}_{m_2m''_2}(R_v) \langle j_1m'_1j_2m'_2|JM\rangle.
\end{align}

The two rotations of $j_1$ and $j_2$ in the resonance frame can be combined to a single rotation on $J'$ using the Clebsch-Gordon series, specifically

\begin{align}
    D_{m_1m'_1}^{j_1}(R_r)D_{m''_2m'_2}^{j_2}(R_r) &= \sum_{J'}\sum_{m'}\sum_{m''}\langle j_1m_1j_2m''_2|J'm'\rangle\langle j_1m'_1j_2m'_2|J'm''\rangle D_{m'm''}^{J'}(R_r)\nonumber,
\end{align}

\noindent which gives

\begin{multline}
    B = \sum_{m'_1}\sum_{m'_2}\sum_{m''_2}\sum_{J'}\sum_{m'}\sum_{m''} \langle j_1m_1j_2m''_2|J'm'\rangle\langle j_1m'_1j_2m'_2|J'm''\rangle\\\langle j_1m'_1j_2m'_2|Jm\rangle D_{m'm''}^{J'}(R_r)D_{m'_2m''_2}^{j_2}(R_v).
\end{multline}

The completeness relation 

\begin{align}
    \delta_{J'J}\delta_{m''m} &= \sum_{m'_1}\sum_{m'_2}\langle j_1m'_1j_2m'_2|J'm''\rangle\langle j_1m'_1j_2m'_2|Jm\rangle \nonumber
\end{align}

\noindent simplifies the decay amplitude to its final expression, such that the result only involves a rotation on spin space in the meson resonance and the vector meson's rest frame. The decay amplitude is 

\begin{align}
    B &= \sum_{m''_2}\sum_{m'}\langle j_1m_1j_2m''_2|Jm'\rangle D^J_{m'm}(R_r)D^{j_2}_{m_2m''_2}(R_v),
    \label{eqn:B_final}
\end{align}

\noindent which using the relation $[D^{(j)}_{mm'}(\gamma,\beta,\alpha)]^\ast=D^{(j)}_{m'm}(-\alpha,-\beta,-\gamma)$ is 

\begin{align}
    \tilde{B} &= \sum_{m''_2}\sum_{m'}\langle Jm'|j_1m_1j_2m''_2\rangle D^{J\ast}_{mm'}(R^p_r)D^{j_2\ast}_{m''_2m_2}(R^p_v),
    \label{eqn:B_final}
\end{align}

\noindent for the passive rotations $R^p$ used in the analysis. 

\subsection{Intensity function in reflectivity basis}

The intensity function $I(\Omega,\Omega_H,\Phi)$ developed to this point is built in terms of a helicity basis. However, helicity is not an eigenstate of parity. Therefore, the intensity function is transformed to the reflectivity basis, a quantum number representing the mirror reflection through the production plane. As a result, an explicit symmetry with parity is introduced, which assists in investigating the parity conserving strong interaction \citep{PWA_Salgado}.

The reflectivity amplitudes for a meson resonance decaying to a vector meson and pseudoscalar meson are related to the helicity amplitudes by

\begin{align}
    ^{(\epsilon)}T^i_m &= \frac{1}{2}(T^i_{+1,m}-\epsilon\tau_i(-1)^mT^i_{-1,-m}),
\end{align}

\noindent where $\epsilon$ is the reflectivity  quantum number and $\tau$ is the naturality of the particle exchange in the creation of the meson resonance. Both $\epsilon$ and $\tau$ have possible values of $\pm1$. This relationship is used to transform the decay amplitudes from the helicity basis to the reflectivity basis \citep{Intensity_V}. 

Beyond the transformation from the helicity basis to the reflectivity basis, it is necessary to convert from a sum over baryon helicities $(\lambda_1,\lambda_2)$ to a sum over nucleon spin flip. This introduces the $J^P$ quantum number necessary to identify spin-parity of particles. In order to convert the sum, the parity invariance is defined as

\begin{align}
    ^{(\epsilon)}T^i_{m;-\lambda_1-\lambda_2} &= \tau_i\epsilon(-1)^{\lambda_1-\lambda_2)}\left[^{(\epsilon)}T^i_{m;\lambda_1\lambda_2}\right],
\end{align}

\noindent where $[J_i]^{(\epsilon)}_{m,0} = ^{(\epsilon)}T^i_{m;+-}$ and $[J_i]^{(\epsilon)}_{m,1} = ^{(\epsilon)}T^i_{m;++}$ define the nucleon spin flip and non-flip amplitudes, respectively \citep{Intensity_V}. This information is used to build the final form of the intensity function 

\begin{multline}
    I(\Omega,\Omega_H,\Phi) = 2\kappa \sum_k [ (1-P_\gamma)[|\sum_{i_N,m}[J_i^N]^{(+)}_{m,k}Im(Z)+\sum_{i_U,m}[J_i^U]^{(-)}_{m,k}Im(Z)|^2 \\+ |\sum_{i_N,m}[J_i^N]^{(-)}_{m,k}Re(Z)+\sum_{i_U,m}[J_i^U]^{(+)}_{m,k}Re(Z)|^2] + (1+P_\gamma)[|\sum_{i_N,m}[J_i^N]^{(-)}_{m,k}Im(Z)\\+\sum_{i_U,m}[J_i^U]^{(+)}_{m,k}Im(Z)|^2 + |\sum_{i_N,m}[J_i^N]^{(+)}_{m,k}Re(Z)+\sum_{i_U,m}[J_i^U]^{(-)}_{m,k}Re(Z)|^2]],
\end{multline}

\noindent where $Z = e^{-i\Phi}\tilde{B}$.

%When summing over the possible baryon helicities, amplitudes with the same $\tau_i\epsilon$ product interfere.

\begin{figure}[t]
    \begin{subfigure}[t]{0.49\textwidth}
        \centering
        \includegraphics[width=\textwidth,keepaspectratio]{hKStarPlus_new.pdf}
        \caption{}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.49\textwidth}
        \centering
        \includegraphics[width=\textwidth,keepaspectratio]{hKStarMinus_new.pdf}
        \caption{}
    \end{subfigure}
    \caption[Invariant Mass of (a) $K^{+}\pi^0$ and (b) $K^{-}\pi^0$ from $\gamma p \rightarrow pK^{+}K^{-}\gamma\gamma$ Events. The Vertical Red Lines at $0.8~\mathrm{GeV}$ and $1.0~\mathrm{GeV}$ Indicating the Selection of Events with a $K^{*+}(892)$ and $K^{*-}(892)$ Meson.]
{Invariant mass of (a) $K^{+}\pi^0$ and (b) $K^{-}\pi^0$ from $\gamma p \rightarrow pK^{+}K^{-}\gamma\gamma$ events. 
The vertical red lines at $0.8~\mathrm{GeV}$ and $1.0~\mathrm{GeV}$ indicating the selection of events with a $K^{*+}(892)$ 
and $K^{*-}(892)$ meson.}
\end{figure}

\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{hKStarPlus_KMinus_new.pdf}
    \caption[Invariant Mass of $K^{*+}(892)K^-$ from $\gamma p \rightarrow pK^{+}K^{-}\gamma\gamma$ Events.]{Invariant mass of $K^{*+}(892)K^-$ from $\gamma p \rightarrow pK^{+}K^{-}\gamma\gamma$ events.} 
\end{figure}

\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{hKStarMinus_KPlus_new.pdf}
    \caption[Invariant Mass of $K^{*-}(892)K^+$ from $\gamma p \rightarrow pK^{+}K^{-}\gamma\gamma$ Events.]{Invariant mass of $K^{*-}(892)K^+$ from $\gamma p \rightarrow pK^{+}K^{-}\gamma\gamma$ events.}
\end{figure}

\begin{table}[H]
    \centering
    \begin{tabular}{c|c|c|c|c|c}
        $J$ & PID & PDG center & PDG width & Fit center & Fit width\\
            &     &   (MeV)    &   (MeV)   &     (MeV)  &    (MeV) \\     
        \hline
        $0$ & $\eta(1405)$ & $1408.8\pm2.0$    & $50.1\pm2.6$  & $1406\pm2$ & $49.46\pm7.07$ \\
        $0$ & $\eta(1475)$ & $1475\pm4$    & $90\pm9$  & $1475\pm10$ & $105.8\pm2.24$\\
        $1$ & $f_1(1420)$  & $1426.3\pm0.9$  & $54.5\pm2.6$  & $1436\pm11$ & $48.40\pm4.17$\\
        $1$ & $f_1(1510)$  & $1518\pm5$    & $73\pm25$ & $1503\pm5$ & $71.78\pm12.76$\\
        $2$ & $f_2(1430)$  & ${\sim}1430$ & NA & $1438\pm4$ & $62.36\pm5.70$\\
        $2$ & $f_2(1525)$  & $1517.4\pm2.5$    & $86\pm5$  & $1538\pm4$ & $90.03\pm13.73$\\
    \end{tabular}
    \caption[Comparison of the Fit Parameters for Figures \ref{fig:J=0_mesons}, \ref{fig:J=1_mesons}, and \ref{fig:J=2_mesons} With Respect To the PDG\footnote{Fit width column is determined by $\Gamma=2.355\sigma$ for full width at half maximum.}.]{Comparison of the fit parameters for Figures \ref{fig:J=0_mesons}, \ref{fig:J=1_mesons}, and \ref{fig:J=2_mesons} with respect to the PDG\footnote{Fit width column is determined by $\Gamma=2.355\sigma$ for full width at half maximum.}.}
\end{table}

\begin{figure}
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{hMassAcc1.pdf}
    \caption[Invariant Mass Distribution for $K^\ast\bar{K}$ With $J=0$ As Determined By PWA. Red Line Shows Total Fit To the Data. Green Is a Gaussian for $\eta(1405)$ and Blue Is a Gaussian for the $\eta(1475)$. Yellow Line Models a Smooth Background Contribution With a Third-degree Polynomial.]{Invariant mass distribution for $K^\ast\bar{K}$ with $J=0$ as determined by PWA. Red line shows total fit to the data. Green is a Gaussian for $\eta(1405)$ and blue is a Gaussian for the $\eta(1475)$. Yellow line models a smooth background contribution with a third-degree polynomial.}
    \label{fig:J=0_mesons}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{hMassAcc2.pdf}
    \caption[Invariant Mass Distribution for $K^\ast\bar{K}$ With $J=1$ As Determined By PWA. Red Line Shows Total Fit To the Data. Magenta Line Shows a Gaussian for $f_1(1420)$ and Green Shows a Gaussian for the $f_1(1510)$. The Cyan Line Shows the Background Modeled By an Exponential.]{Invariant mass distribution for $K^\ast\bar{K}$ with $J=1$ as determined by PWA. Red line shows total fit to the data. Magenta line shows a Gaussian for $f_1(1420)$ and green shows a Gaussian for the $f_1(1510)$. The cyan line shows the background modeled by an exponential.}
    \label{fig:J=1_mesons}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{hMassAcc3.pdf}
    \caption[Invariant Mass Distribution for $K^\ast\bar{K}$ With $J=1$ As Determined By PWA. Red Line Shows Total Fit To the Data. Green Line Shows a Gaussian for $f_2(1430)$ and Blue Shows a Gaussian for the $f'_2(1525)$. The Magenta Line Shows the Background Modeled By a Gaussian for the $f_2(1260)$.]{Invariant mass distribution for $K^\ast\bar{K}$ with $J=1$ as determined by PWA. Red line shows total fit to the data. Green line shows a Gaussian for $f_2(1430)$ and blue shows a Gaussian for the $f'_2(1525)$. The magenta line shows the background modeled by a Gaussian for the $f_2(1260)$.}
    \label{fig:J=2_mesons}
\end{figure}

\bibliography{dis}

\end{document}
