Using the modified matrix element method to constrain $L_{\mu }-L_{\tau }$ interactions

In this paper, we explore the discriminatory power of the matrix element method (MEM) in constraining the $L_{\mu }-L_{\tau }$ model at the LHC. The $Z^{\prime }$ boson associated with the spontaneously broken $U(1{)}_{L_{\mu }-L_{\tau }}$ symmetry only interacts with the second and third generation of leptons at tree level, and is thus difficult to produce at the LHC. We argue that the best channels for discovering this $Z^{\prime }$ are in $Z\to 4\mu$ and $2\mu +{\cancel{E}}_T$. Both these channels have a large number of kinematic observables, which strongly motivates the usage of a multivariate technique. The MEM is a multivariate analysis that uses the squared matrix element $|\mathcal{M}{|}^2$ to quantify the likelihood of the testing hypotheses. As the computation of the $|\mathcal{M}{|}^2$ requires knowing the initial and final state momenta and the model parameters, it is not commonly used in new physics searches. Conventionally, new parameters are estimated by maximizing the likelihood of the signal with respect to the background, and we outline scenarios in which this procedure is (in)effective. We illustrate that the new parameters can also be estimated by studying the $|\mathcal{M}{|}^2$ distributions, and, even if our parameter estimation is off, we can gain better sensitivity than cut-and-count methods. Additionally, unlike the conventional MEM, where one integrates over all unknown momenta in processes with ${\cancel{E}}_T$, we show an example scenario where these momenta can be estimated using the process topology. This procedure, which we refer to as the “modified squared matrix element,” is computationally much faster than the canonical matrix element method and maintains signal-background discrimination. Bringing the MEM and the aforementioned modifications to bear on the $L_{\mu }-L_{\tau }$ model, we find that with $300{\mathrm{fb}}^{-1}$ of integrated luminosity, we are sensitive to the couplings of $g_{Z^{\prime }}\gtrsim 0.002g_1$ and $M_{Z^{\prime }}<20\mathrm{GeV}$, and $g_{Z^{\prime }}\gtrsim 0.005g_1$ and $20\mathrm{GeV}<M_{Z^{\prime }}<40\mathrm{GeV}$, which is about an order of magnitude improvement over the cut-and-count method for the same amount of data.

Figure 1

The Feynman diagram for $Z\to 4\mu$ in the SM as well as the $Z^{\prime }$ contribution.

Figure 2

The plot of log likelihood ratio summed over 50000 events is shown above. The left plot is the distribution of SM events, where there is a spike at every reconstructed $M_{\mu \mu }$. The peak at $M_{Z^{\prime }}^a\sim 0\mathrm{GeV}$ is due to the photon propagator. For larger values of $M_{Z^{\prime }}^a$, there are fewer events with $M_{\mu \mu }=M_{Z^{\prime }}^a$, and the function looks smoother, which is the result of our basic cuts. In the right plot, the distribution of the signal events compared to the simulated background events is shown. In the signal events, which include both SM and $Z^{\prime }$ contributions, there is a peak at $M_{Z^{\prime }}^a=M_{Z^{\prime }}^{\mathrm{gen}}$ because of the contribution from the on-shell $Z^{\prime }$. It is important to mention that the optimal value of $M_{Z^{\prime }}^a$ is obtained by comparing the relative shapes of the signal plot with the background one. The scale on the y axis is irrelevant because it is highly sensitive to the number of events in our sample, and by increasing the sample size the numerical values on the y axis of the plots become more comparable. For our analysis, we manually brought the plots to similar y-axis values, to compare their shapes.

Figure 3

The distribution of the log likelihood ratio for the signal and background events summed over 50000 sample events. We have fixed $M_{Z^{\prime }}^a=M_{Z^{\prime }}^{\mathrm{gen}}=10\mathrm{GeV}$ while ${\epsilon }^{\mathrm{gen}}=0.01$. The signal and the background MC sample events increase as ${\epsilon }^a$ increases, providing us with no special ${\epsilon }^a$ value that optimizes the signal likelihood ratio. The numerical value on the y axis of the two plots as explained earlier is irrelevant and thus not shown.

Figure 4

The distribution of ${\mathcal{L}}^{-1}=|{\mathcal{M}}_{\mathrm{SM}}{|}^2/|{\mathcal{M}}_{Z^{\prime }+\mathrm{SM}}{|}^2$ for $M_{Z^{\prime }}^{\mathrm{gen}}=10\mathrm{GeV}$, and ${\epsilon }^{\mathrm{gen}}=0.05$. This plot shows that for $M_{Z^{\prime }}^a=10\mathrm{GeV}$, if we have ${\epsilon }^a=0.03$, the resonance region is not very well separated from the background region. If ${\epsilon }^a=0.5$, although the signal is far from 1, the background also spreads more. The middle value ${\epsilon }^a=0.05$ is the optimal one.

Figure 5

The distribution of ${\mathcal{L}}^{-1}=|{\mathcal{M}}_{\mathrm{SM}}{|}^2/|{\mathcal{M}}_{Z^{\prime }+\mathrm{SM}}{|}^2$ for $M_{Z^{\prime }}^{\mathrm{gen}}=10\mathrm{GeV}$, and ${\epsilon }^{\mathrm{gen}}=0.05$ is shown, where we are assuming $M_Z^a=25\mathrm{GeV}$, and we vary ${\epsilon }^a$. Here, a relatively large ${\epsilon }^a$ is needed to separate the signal from background. However, with increasing ${\epsilon }^a$, we also get broader peaks and therefore our sensitivity is not good as when $M_{Z^{\prime }}^a=M_{Z^{\prime }}^{\mathrm{gen}}$. In this particular example ${\epsilon }^a=0.5$ is the optimal choice.

Figure 6

The bound on $L_{\mu }-L_{\tau }$ from different experiments. The purple line is the LHC-8 bounds, the black line is the CCFR bound, and the solid gray line is the bound from the BABAR experiment [75]. The dashed blue line is the bound from LHC-14 with luminosity $300{\mathrm{fb}}^{-1}$ up to $3\sigma$ with the cut-and-count method [66]. The red dashed line in the figure shows the reach with MEM, but assuming we know the mass of $Z^{\prime }$, and with ${\epsilon }^a=0.05$. This is the best sensitivity we could get using MEM. The dotted green line is assuming $M_{Z^{\prime }}^a=25\mathrm{GeV}$, the dashed brown line is $M_{Z^{\prime }}^a=10\mathrm{GeV}$, and the dotted-dashed dark brown line is $M_{Z^{\prime }}^a=2\mathrm{GeV}$ while choosing the optimal value for ${\epsilon }^a$. The MEM bounds are also with luminosity $300{\mathrm{fb}}^{-1}$ up to $3\sigma$.

Figure 7

The main Feynman diagram that contributes to the signal. The sources of missing energy are shown in red.

Figure 8

The main irreducible backgrounds to the $\text{dimuon}+\mathrm{MET}$ process. The sources of missing energy are shown in red. The left-most diagram is the ${{\tau }^{+}{\tau }^{-}|}_{\text{dimuon}}$, which is the main background to our signal. If the neutrinos are the same (e.g., ${\nu }_{\ell }={\nu }_{\mu }$), the right three diagrams interfere, and thus we must use $(pp\to {\mu }^{+}{\mu }^{-}{\nu }_{\ell }{\overline{\nu }}_{\ell }{)}_{\mathrm{SM}}$ to describe them. The diagram with $Z^{\prime }$ (shown in Fig. 7) may also interfere with the right three diagrams and thereby, the signal is described by $(pp\to {\mu }^{+}{\mu }^{-}{\nu }_{\ell }{\overline{\nu }}_{\ell }{)}_{\mathrm{SM}+Z^{\prime }}$.

Figure 9

The weights $|{\mathcal{M}}_{\tau \tau }^{\mathrm{mod}}{|}^2$ for various MC sample events. The dotted purple line shows the weights for MC generated ${{\tau }^{+}{\tau }^{-}|}_{\text{dimuon}}$ (10 million generated events). The solid red line shows the MC $(pp\to {\mu }^{+}{\mu }^{-}{\nu }_{\ell }{\overline{\nu }}_{\ell })$ background (1 million events), and the dashed blue line is MC $(pp\to {\mu }^{+}{\mu }^{-}{\nu }_{\ell }{\overline{\nu }}_{\ell })$ including $Z^{\prime }$ (1 million events). Because the assumptions were chosen based on ${{\tau }^{+}{\tau }^{-}|}_{\text{dimuon}}$ topology and were not reasonable in other processes, we have $|{\mathcal{M}}_{\tau \tau }^{\mathrm{mod}}{|}^2<0$ for part of the distribution of other processes.

Figure 10

The weights $|{\mathcal{M}}_{\tau \tau }^{\mathrm{mod}}{|}^2$ for $(pp\to {\mu }^{+}{\mu }^{-}{\nu }_{\ell }{\overline{\nu }}_{\ell })$ background (SM only, solid line) and signal ($\mathrm{SM}+Z^{\prime }$, dashed line) in the negative $|{\mathcal{M}}_{\tau \tau }^{\mathrm{mod}}{|}^2$ region.

Figure 11

The left plot represents the invariant mass of a muon and the associated neutrino $M_{\nu \mu }$, and the right plot is the separation between the two neutrinos $\mathrm{\Delta }R({\nu }_{\ell },{\overline{\nu }}_{\ell })$. These plots show the distribution of the events after requiring $|{\mathcal{M}}_{\tau \tau }^{\mathrm{mod}}{|}^2<-4$, and demonstrate that the signal events belong to on-shell $W$ production with $Z^{\prime }$ (near) on shell as well.

Figure 12

The new bounds according to our study using MEM compared with the cut-and-count method [66]. The $(g-2{)}_{\mu }$ band is also shown in brown. For $M_{Z^{\prime }}>2m_{\mu }$, the MEM bound depends on whether we know $M_{Z^{\prime }}$ or not. The red line shown here is our best bound. For $M_{Z^{\prime }}<2m_{\mu }$, our analysis is independent of $M_{Z^{\prime }}$. The bounds from the MEM and cut-and-count method are with luminosity of $300{\mathrm{fb}}^{-1}$ and are up to $3\sigma$.