Derivation of M SV Corrected
If SV is reconstructed and \(\vec{p}_{\rm{SV}}\) i snot aligned w/the displacement its a sign that there were missing particles.
Can correct for the "minimal" amount of missing particle pass such that \(\vec{p}_{\rm{SV}} \propto \hat{d}\) where \(\hat{d}\) is the the displacement vector.
Let \(\hat{x}\) along align with \(\hat{d}\)
\[ p = \begin{pmatrix} p \cos \theta \\ p \sin \theta \end{pmatrix} \]
Easy way and the Hard way
Hard Way
Define
\[ \vec{p}_{\rm{miss}} = \begin{pmatrix} c \\ - p \sin \theta \end{pmatrix} \] Where c is the parameter to be found. Then calculate \(M_{\rm{comb}}^2(c)\) (assuming the missing particle is massless) and find c that minimizes \(M_{\rm{comb}}^2(c)\) (Lots of algrebra…)
Easy way
Boost to MC frame. For minimim mass SV the missing \(\vec{p}\) is opposite the \(\vec{p}\)
In CM frame. \(\vec{p} = P \sin \theta\) and \(\vec{p}_{\rm{miss}} = - P \sin \theta\)
In CM frame, \(E_{\rm{comb}} = M_{\rm{comb}} = E_{reco} + E_{miss} = \sqrt{M_{SV} + P^2 \sin^2\theta} + P \sin \theta\)
Follow-ups
Links:
202408310955