Declustering Algebra
Start with \(P_{\rm{comb}}^\mu\) : look up z, \(\theta_A\), \(m_A\), and \(m_B\). \(m_{comb}\) ignored (is this actually true in the boosting ?)
Boost to \(P_{\rm{comb}}^z = 0\). Rotate to \(P_{\rm{comb}}^y = 0\).
![BROKEN LINK: DeclusteringAlgebraDrawing.excalidraw|200]
\[z_A \equiv \frac{|\vec{A}|\cos\theta_A}{P_T^{comb}}\]
\(P_{comb}^\mu = \begin{pmatrix} E^{comb} \\ P_T^{comb} \\ 0 \\ 0 \end{pmatrix}\) Where (for the moment) we are assuming \(E^{comb} = P_T^{comb}\)
Now we need to get \(\vec{P}_A\) and \(\vec{P}_B\) (From now on will just write \(\vec{A}\) and \(\vec{B}\))
\(\vec{A} = \begin{pmatrix} |\vec{A}|\cos \theta_A \\ 0 \\ -|\vec{A}|\sin \theta_A \end{pmatrix}\) and \(\vec{B} = \begin{pmatrix} |\vec{B}|\cos \theta_B \\ 0 \\ +|\vec{A}|\sin \theta_B \end{pmatrix}\)
So we need \(|\vec{A}|, \theta_A\) \(|\vec{B}|, \theta_B\) .
We will look up \(z_A\) and \(\theta_A\) which fix \(|\vec{A}|\) and \(\theta_A\). Will constrain the other two unknowns from \(\vec{P}_{comb} = \vec{A} + \vec{B}\)
1) Constraint \(P_{comb}^z = 0\)
\(|\vec{A}| \sin\theta_A = |\vec{B}|\sin \theta_B\) or \(\sin\theta_B = \frac{|\vec{A}|}{|\vec{B}|} \sin \theta_A\)
(equation A)
2) Constraint \(P_{comb}^T = P_A^T + P_B^T\)
\[ P_T^{comb} = |\vec{A}|\cos\theta_A + |\vec{B}|cos\theta_B \] \[ = z_A P_T^{comb} + |\vec{B}|cos\theta_B \] or
$$
| \vec{B} | = \frac{(1-z)PTcomb}{cosθB} |
$$ (equation B)
Combining A & B
\[ \sin\theta_B = \frac{p_T^{comb} z_A}{\cos\theta_A}\frac{\sin\theta_A}{(1-z_A)p_T^{comb}} \cos \theta_B \]
OR
\[ \tan \theta_B = \frac{z}{1-z} \tan\theta_A \]
Putting it Together
\(\vec{A} = \begin{pmatrix} |\vec{A}|\cos \theta_A \\ 0 \\ -|\vec{A}|\sin \theta_A \end{pmatrix}\) = \(\begin{pmatrix} z_A P_T^{comb} \\ 0 \\ -z_A P_T^{comb} \tan\theta_A \end{pmatrix}\) because \(|\vec{A}| = \frac{z_A P_T^{comb}}{\cos\theta_A}\)
and
\(\vec{B} = \begin{pmatrix} (1-z_A) P_T^{comb} \\ 0 \\ \frac{(1-z_A)P_T^{comb}}{\cos\theta_B} \sin\theta_B \end{pmatrix}\) = \(\begin{pmatrix} (1-z_A) P_T^{comb} \\ 0 \\ (1-z_A)P_T^{comb} (\frac{z_A}{1-z_A}) \tan\theta_A \end{pmatrix}\)
now have the new vectors in terms of the looked up variables \(z_A\) and \(\theta_A\).
Follow-ups
Links:
Clustering Algebra [BROKEN LINK: Splitting Templates] Jet DeClustering
202410191153