%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Jacobian for 2$\times$2 Mass Covariance Matrix}
\label{Jacob2x2COV}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We wish to obtain the $2\times 2$ covariance matrix for 
$m_{\rm jj}$ and $m_{\ell\nu}$ after the constrained kinematic fit. 
This may be accomplished by a performing Jacobian transformation 
from the fit parameter space
$\{E_{\rm j1},\theta_{\rm j1},E_{\rm j2},\theta_{\rm j2},E_{\ell}\}$ 
to the mass space $\{m_{\rm jj}, m_{\ell\nu}\}$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{ \bf V(y)} \approx  {\bf J^T V(x) J}
\label{eqn-CovJacobianTransformation}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
J_{ij} = \frac{\partial y_i}{\partial x_j}\Bigg\arrowvert_{x_j = x_j^{fit}} 
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
with 
${\bf x} \equiv (E_{\rm j1}, \theta_{\rm j1}, E_{\rm j2}, \theta_{\rm j2}, 
E_\ell)^T$ and ${\bf y}= (m_{\rm jj}, m_{\ell\nu})^T$.  The masses
written in terms of the fit parameters take the form\footnote{Here, we assume
three-momentum conservation and a massless neutrino.}:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
m_{\rm jj} &=& \sqrt{ \left(E_{\rm j1}+E_{\rm j2}\right)^2-
                      \left|E_{\rm j1}{\bf \beta_{\rm j1} +} 
                            E_{\rm j2}{\bf \beta_{\rm j2}}\right|} \\ 
*[4mm]
m_{\ell\nu}&=& \sqrt{ \left(E_\ell+|E_{\rm j1}{\bf \beta_{\rm j1} +} 
                                    E_{\rm j2}{\bf \beta_{\rm j2} +}
                                    E_\ell{\bf \beta_\ell}|\right)^2 -
                      \left|E_{\rm j1}{\bf \beta_{\rm j1} +} 
                            E_{\rm j2}{\bf \beta_{\rm j2}}\right|}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where ${\bf \beta_i} = (\beta^x_i\sin\theta_i\cos\phi_i, \:\:
                        \beta^y_i\sin\theta_i\sin\phi_i, \:\:
                        \beta^z_i\cos\theta_i)={\bf p_i}/E_i$.  In the 
constrained kinematic fit, $|{\bf \beta_i}|=|{\bf p_i}/E_i|=\:${\em constant}.
Hence, the elements of the transformation (or, derivative) matrix ${\bf J}$ 
may then be written as:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{\partial m_{\rm jj}}{\partial E_{\rm j1}} &=& \frac{1}{m_{\rm jj}} 
\Biggl[ E_{h} - {\bf \beta}_{\rm j1} \cdot {\bf p}_{h} \Biggr] 
\\
*[4mm]
\frac{\partial m_{\rm jj}}{\partial E_{\rm j2}} &=& \frac{1}{m_{\rm jj}} 
\Biggl[ E_{h} - {\bf \beta}_{\rm j2} \cdot {\bf p}_{h} \Biggr] 
\\
*[4mm]
\frac{\partial m_{\rm jj}}{\partial E_\ell} &=& 0
\\
*[4mm]
\frac{\partial m_{\ell\nu}}{\partial E_\ell} &=& \frac{1}{m_{\ell\nu}}
{\bf \beta}_\ell \cdot {\bf \Delta}
\\
*[4mm]
\frac{\partial m_{\rm jj}}{\partial \theta_{\rm j1}} &=& - \frac{p_{\rm j1}^z}
{m_{\rm jj}} 
( p_{h}^x \cos \phi_{\rm j1} + p_{h}^y \sin \phi_{\rm j1} - p_{h}^z \tan 
                                                             \theta_{\rm j1} ) 
\\
*[4mm]
\frac{\partial m_{\rm jj}}{\partial \theta_{\rm j2}} &=& - \frac{p_{\rm j2}^z}
{m_{\rm jj}} 
( p_{h}^x \cos \phi_{\rm j2} + p_{h}^y \sin \phi_{\rm j2} - p_{h}^z \tan 
                                                             \theta_{\rm j2} ) 
\\
*[4mm]
\frac{\partial m_{\ell\nu}}{\partial \theta_{\rm j1}} &=&  \frac{p_{\rm j1}^z}
{m_{\ell\nu}} 
( \Delta^x \cos \phi_{\rm j1} + 
  \Delta^y \sin \phi_{\rm j1} - 
  \Delta^z \tan \theta_{\rm j1} ) \\
*[4mm]
\frac{\partial m_{\ell\nu}}{\partial \theta_{\rm j2}} &=&  \frac{p_{\rm j2}^z}
{m_{\ell\nu}} 
( \Delta^x \cos \phi_{\rm j2} + 
  \Delta^y \sin \phi_{\rm j2} - 
  \Delta^z \tan \theta_{\rm j2} ) 
%\nonumber \\
%*[4mm]
%\frac{\partial m_{\ell\nu}}{\partial \theta_3} &=&  \frac{p_3^z}
%{m_{\ell\nu}} 
%( p_{3}^x \cos \phi_3 + p_{3}^y \sin \phi_3 - p_{3}^z \tan \theta_3 ) 
%\nonumber 
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where ${\bf p}_h = {\bf p}_{\rm j1}+{\bf p}_{\rm j2}$, 
and ${\bf \Delta}={\bf p}_{\ell}-{\bf p}_{h}$.

