The search for light stops is of paramount importance, both in general as a promising
path to the discovery of beyond the standard model physics and more specifically as
a way of evaluating the success of the naturalness paradigm. While the LHC experiments
have ruled out much of the relevant parameter space, there are “stop gaps”, i.e.,
values of sparticle masses for which existing LHC analyses have relatively little
sensitivity to light stops. We point out that techniques involving on-shell constrained
Article funded by SCOAP3
0$. While one has the option of adding the further constraint that the squared masses of the relative particles be positive, we will not do so in this work.} for the relative particles, \begin{equation} \label{eq:relatives-equal} M_{R_1}^2 (\vec{q}_1,\tilde m)= M_{R_2}^2(\vec{q}_2, \tilde m), \end{equation} we are, in effect, imposing an additional constraint on the minimization over the invisible momenta $\vec{q}_i$. The applicability of the constraint~(\ref{eq:relatives-equal}) will be indicated by the second ${}_\sqcup$ index in~(\ref{M2notation}): as before, it will be equal to $C$ if the constraint~(\ref{eq:relatives-equal}) is applied and $X$ otherwise. Altogether, therefore, we have four possible $M_2$ variables: \begin{align} M_{2CC}(S;\tilde m) &\equiv \min_{\substack{\vec{q}_{1},\vec{q}_{2}\\ \vec{q}_{1T}+\vec{q}_{2T} = \mpt \\ M_{P_1}^2(\vec{q}_1,\tilde m) = M_{P_2}^2(\vec{q}_2,\tilde m) \\ M_{R_1}^2(\vec{q}_1,\tilde m) = M_{R_2}^2(\vec{q}_2,\tilde m)} } \left\{M_{P_1}(\vec{q}_{1},\tilde m) \right\}, \label{M2CCdef} \\ M_{2CX}(S;\tilde m) &\equiv \min_{\substack{\vec{q}_{1},\vec{q}_{2}\\ \vec{q}_{1T}+\vec{q}_{2T} = \mpt \\ M_{P_1}^2(\vec{q}_1,\tilde m) = M_{P_2}^2(\vec{q}_2,\tilde m)} } \left\{M_{P_1}(\vec{q}_{1},\tilde m) \right\}, \label{M2CXdef} \\ M_{2XC}(S;\tilde m) &\equiv \min_{\substack{\vec{q}_{1},\vec{q}_{2}\\ \vec{q}_{1T}+\vec{q}_{2T} = \mpt \\ M_{R_1}^2(\vec{q}_1,\tilde m) = M_{R_2}^2(\vec{q}_2,\tilde m)} } \left\{ \max\left[ M_{P_1}(\vec{q}_{1},\tilde m), M_{P_2}(\vec{q}_{2},\tilde m) \right] \right\}, \label{M2XCdef} \\% [2mm] M_{2XX}(S;\tilde m) &\equiv \min_{\substack{\vec{q}_{1},\vec{q}_{2}\\ \vec{q}_{1T}+\vec{q}_{2T} = \mpt } } \left\{ \max\left[ M_{P_1}(\vec{q}_{1},\tilde m), M_{P_2}(\vec{q}_{2},\tilde m) \right] \right\}. \label{M2XXdef} \end{align} \end{itemize} The definitions~(\ref{M2CCdef})--(\ref{M2XXdef}) should be contrasted to the analogous definition of the Cambridge $M_{T2}$ variable \beq M_{T2}(S;\tilde m) \equiv \min_{\substack{\vec{q}_{1T},\vec{q}_{2T}\\ \vec{q}_{1T}+\vec{q}_{2T} = \mpt } } \left\{ \max\left[ M_{TP_1}(\vec{q}_{1T},\tilde m), M_{TP_2}(\vec{q}_{2T},\tilde m) \right] \right\}, \label{MT2def} \eeq where only the transverse components $\vec{q}_{iT}$ are used, and the objective function (the function that is minimized) is the larger of the two \emph{transverse} masses of the parents. The main goal of this paper is to investigate and contrast the ability of the variables~(\ref{M2CCdef})--(\ref{MT2def}) to discriminate between the $t\bar{t}$ background of figure~\ref{fig:DecayTopologies}(a) and the three types of stop signal events of figure~\ref{fig:DecayTopologies}(b-d). We shall consider the respective variables for all three subsystems of figure~\ref{fig:DecaySubsystem}. Since we know the mass spectrum for the background event topology, we shall choose the test mass $\tilde m$ to be equal to the correct, SM value for the respective daughter particle. In particular, \begin{itemize} \item In subsystem $(ab)$, the parent particle is the top quark; the daughter particle is the neutrino, whose mass is taken to vanish ($\tilde m=0$). The other, ``relative'', particle is the $W^\pm$ boson. Then \emph{for background events}, all 5 variables~(\ref{M2CCdef})--(\ref{MT2def}) are bounded from above by the top mass: \beq M_{T2}(ab;0), M_{2XX}(ab;0), M_{2CX}(ab;0), M_{2XC}(ab;0), M_{2CC}(ab;0) \le m_t, \label{mtab} \eeq while for signal events, this bound can be violated. \item In subsystem $(a)$, the parent particle is again the top quark. The daughter particle is now the $W^\pm$ boson, with mass $\tilde m= m_W$. The relative particles are the two neutrinos, which in this case appear downstream outside the subsystem. For background events, the variables are again bounded by the top mass \beq M_{T2}(a;m_W), M_{2XX}(a;m_W), M_{2CX}(a;m_W), M_{2XC}(a;m_W), M_{2CC}(a;m_W) \le m_t. \label{mta} \eeq \item In subsystem $(b)$ the parent particles are the $W^\pm$ bosons, the daughter particles are the two neutrinos with mass $\tilde m=0$, and the relative particles are the top quarks appearing upstream outside the subsystem. The background events obey \beq M_{T2}(b;0), M_{2XX}(b;0), M_{2CX}(b;0), M_{2XC}(b;0), M_{2CC}(b;0) \le m_W. \label{mwb} \eeq \end{itemize} In principle, \emph{each} of the bounds~(\ref{mtab})--(\ref{mwb}) allows us to cut $100\%$ of the $t\bar{t}$ background events by removing events with values of the respective subsystem $M_2$ or $M_{T2}$ variable below the appropriate threshold (``high pass cut''). Therefore, as far as just the background is concerned, we have 15 alternative choices\footnote{Each of the five variables~(\ref{M2CCdef})--(\ref{MT2def}) can be applied for each of the three subsystems $(ab)$, $(a)$, and~$(b)$. The explicit definitions of the resulting 15 variables can be found in appendix~\ref{sec:appendix}.} for reducing it, and they should perform comparably well. The differences between the five variables~(\ref{M2CCdef})--(\ref{MT2def}) begin to emerge when we consider the effect of such a high pass cut on signal events. It has been shown~\cite{Cho-ml-2014naa} that the variables~(\ref{M2CCdef})--(\ref{MT2def}) obey the following hierarchy \begin{equation} \label{eq:inequality} M_{T2}(S; \tilde{m}) = M_{2XX}(S; \tilde{m}) = M_{2CX}(S; \tilde{m}) \leq M_{2XC}(S; \tilde{m}) \leq M_{2CC}(S; \tilde{m}) \end{equation} for any subsystem $S$ and any value of $\tilde m$. We should therefore expect the distributions of $M_{2XC}$ and $M_{2CC}$ to be more populated at higher values and in particular near their endpoints. As a result, signal events are more likely to pass if the cut is applied on the additionally constrained variables, $M_{2XC}$ and $M_{2CC}$, as opposed to the less constrained variables $M_{T2}$, $M_{2XX}$, and $M_{2CX}$. This expectation is borne out by the explicit studies below. The property~(\ref{eq:inequality}) offers an opportunity to increase the sensitivity of the LHC experiments to the presence of stop signals of the type described in figure~\ref{fig:DecayTopologies}(b-d). ]]>
m_t. \label{condition_ab} \eeq The corresponding region is delineated by the solid black line in figure~\ref{fig:map}, which shows a slice through the 3-dimensional mass parameter space for fixed $m_{\tilde \nu}=110$\,GeV\@. For convenience we choose to represent the remaining two degrees of freedom as the mass differences $m_{\tilde\chi^\pm}-m_{\tilde \nu}$ and $m_{\tilde t}-m_{\tilde\chi^\pm}$. The region satisfying the condition~(\ref{condition_ab}) is above and to the right of the solid black line in figure~\ref{fig:map}. If the SUSY mass spectrum happens to be in this region, the mass splitting between the stop and the sneutrino is sufficiently large to cause some number of signal events to ``leak" beyond the background endpoint~(\ref{mtab}). This means that the subsystem $(ab)$ variables are promising variables to cut on in order to separate signal from background. \begin{figure}[t] \centering \includegraphics[trim = 0 0 0 0, clip, scale=1]{figure4.png} ]]>
\frac{m^2_t-m^2_W}{m_t}. \label{condition_a} \eeq The corresponding region extends above the red dashed line in figure~\ref{fig:map}. Finally, the condition for violating the background endpoints in subsystem $(b)$ follows from eqs.~(\ref{mwb}) and~(\ref{susymwb}): \beq \frac{\left(m^2_{\tilde t}-m^2_{\tilde \nu}\right)\left(m^2_{\tilde \chi^\pm}-m^2_{\tilde \nu}\right)}{m^2_{\tilde t}} > m_W^2. \label{condition_b} \eeq The region where this condition is satisfied is located to the right of the blue dot-dashed line in figure~\ref{fig:map}. For completeness, we shall also consider the variable $m_{bl}$, the invariant mass of the lepton and $b$ quark from a given branch of the decay. The endpoint of this quantity for signal events is given by \beq m_{b\ell}^{\rm max} = \sqrt{\frac{\left(m_{\tilde{t}}^2 - m_{\tilde{\chi}^\pm}^2\right)\left(m_{\tilde{\chi}^\pm}^2- m_{\tilde{\nu}}^2\right)}{m_{\tilde{\chi}^\pm}^2 }}, \label{susymbl} \eeq while background events will obey the bound \beq m_{b\ell} \le \sqrt{m_t^2 - M_W^2}. \label{mblmaxbknd} \eeq Therefore the condition for violating the background $m_{b\ell}$ endpoint is \beq \frac{\left(m_{\tilde{t}}^2 - m_{\tilde{\chi}^\pm}^2\right)\left(m_{\tilde{\chi}^\pm}^2- m_{\tilde{\nu}}^2\right)}{m_{\tilde{\chi}^\pm}^2 } > m_t^2 - M_W^2. \label{condition_bl} \eeq The corresponding region is found above and to the right of the diagonal magenta thin solid line in figure~\ref{fig:map}. \begin{table}[t] \centering \begin{tabular}{| c | c | c | c | c |} \hline ~\rule[13pt]{0pt}{0pt} & $m_{b\ell}$ & $M_{T2}(ab)$ & $M_{T2}(a)$ & $M_{T2}(b)$ \\ \textbf{Region} & endpoint & endpoint & endpoint & endpoint \\ ~ & violation & violation & violation & violation \\ \hline \textcolor{region1}{i} \!\!\! \rule[13pt]{0pt}{0pt} & No & Yes & Yes & Yes \\ \textcolor{region2}{ii} & Yes & Yes & Yes & Yes \\ \textcolor{region3}{iii} & Yes & Yes & Yes & No \\ \textcolor{region4}{iv} & No & Yes & Yes & No \\ \textcolor{region5}{v} & No & No & Yes & No \\ \textcolor{region6}{vi} & No & No & No & No \\ \textcolor{region7}{vii} & No & No & No & Yes \\ \textcolor{region8}{viii} & No & Yes & No & Yes \\ \textcolor{region9}{ix} & Yes & Yes & No & Yes \\ \hline \end{tabular} ]]>
10$\,GeV and $|\eta|<2.5$ ($|\eta|<2.4$) for electrons (muons). \item In order to reduce background from low mass resonances, in the $ee$ and $\mu\mu$ channels, we demand $m_{ee/\mu\mu}>20$\,GeV\@. Furthermore, to reduce the $Z+$jets background, events with dilepton masses within the $Z$-mass window are vetoed by requiring $|m_{ee/\mu\mu}-m_Z| > 15$\,GeV\@. \item To further suppress Drell-Yan, for the $ee$ and $\mu\mu$ channels, we apply a missing transverse energy cut of $\met>40$\,GeV\@. \item The event is required to have $\geq 2$ jets with $p_T > 30$\,GeV and $|\eta|<2.4$. It is also required that exactly two of these jets are $b$-tagged. \end{itemize} After these cuts, we are left with $t\bar{t}$ as the dominant (irreducible) background, and this will be the only background process we will consider here. The posterior cuts using $M_2$ variables are imposed for events already passing the above set of pre-selection cuts. \boldmath ]]>