Sharpening \(m_{T2}\) cusps: the mass determination of semi-invisibly decaying particles from a resonance

We revisit mass determination techniques for the minimum symmetric event topology, namely \(X\) pair production followed by \(X \to \ell N\), where \(X\) and \(N\) are unknown particles with the masses to be measured, and \(N\) is an invisible particle, concentrating on the case where \(X\) is pair produced from a resonance. We consider separate scenarios, with different initial constraints on the invisible particle momenta, and present a systematic method to identify the kinematically allowed mass regions in the \((m_N, m_X)\) plane. These allowed regions exhibit a cusp structure at the true mass point, which is equivalent to the one observed in the \(m_{T2}\) endpoints in certain cases. By considering the boundary of the allowed mass region we systematically define kinematical variables which can be used in measuring the unknown masses, and find a new expression for the \(m_{T2}\) variable as well as its inverse. We explicitly apply our method to the case that \(X\) is pair produced from a resonance, and as a case study, we consider the process \(pp \to A \to \tilde \chi_1^+ \tilde \chi_1^-\), followed by \(\tilde \chi_1^\pm \to \ell^{\pm} \, \tilde \nu_{\ell}\), in the Minimal Supersymmetric Standard Model and show that our method provides a precise measurement of the chargino and sneutrino masses, \(m_X\) and \(m_N\), at \(14 \, \mathrm{TeV}\) LHC with \(300 \, \mathrm{fb}^{-1}\) luminosity.