Measuring the mass, width, and couplings of semi-invisible resonances with the matrix element method

We demonstrate the use of the matrix element method (MEM) for the measurement of masses, widths, and couplings in the case of single or pair production of semi-invisibly decaying resonances. For definiteness, we consider the two-body decay of a generic resonance to a visible particle from the Standard Model (SM) and a massive invisible particle. It is well known that the mass difference can be extracted from the endpoint of a transverse kinematic variable like the transverse mass, $M_T$, or the Cambridge $M_{T2}$ variable, but measuring the overall mass scale is a very difficult problem. We show that the MEM can be used to obtain not only the absolute mass scale, but also the width of the resonance and the tensor structure of its couplings. Apart from new physics searches, our results can be readily applied to the case of SM $W$ boson production at the CERN Large Hadron Collider (LHC), where one can repeat the measurements of the $W$ properties in a general and model-independent framework.

Figure 1

The event topologies considered in this paper: (a) single and (b) pair production of a $W$-like resonance decaying leptonically.

Figure 2

Unit-normalized lepton $P_T$ distributions for single $W$ production in the limit of ${\mathrm{\Gamma }}_W=0$. The invisible particle mass is $M_{\nu }=500\mathrm{GeV}$; the mass of the parent particle is varied as shown.

Figure 3

Unit-normalized distributions of the lepton transverse momentum $P_T$ (top) and longitudinal momentum $P_z$ (bottom), for different values of $M_W$ and ${\mathrm{\Gamma }}_W=0$ obtained using analytical expressions and, in the case of $P_z$, pdfs from LHAPDF [57]. The mass, $M_{\nu }$, of the invisible particle has been fixed from the measurement (8).

Figure 4

${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}$. fit to $M_{\nu }$ from $P_{\ell z}$ templates in $W^{+}$ production (left) and $W^{-}$ production (right).

Figure 5

The same as Fig. 2, but for fixed $M_W=1000\mathrm{GeV}$, and several values of the width ${\mathrm{\Gamma }}_W$ (as indicated in the figure).

Figure 6

Results from a ${\chi }^2$ fit to the one-dimensional $P_{\ell T}$ distribution for a data sample of 10 000 events. The fitted parameters are $M_{\nu }$ and ${\mathrm{\Gamma }}_W/M_W$, with $M_W$ computed from (8). The study point ($\times$) has $M_W=1000\mathrm{GeV}$, $M_{\nu }=500\mathrm{GeV}$, and ${\mathrm{\Gamma }}_W=50\mathrm{GeV}$. The dashed line marks the flat direction (11). The color bar indicates the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}$. (we used 100 bins).

Figure 7

One-dimensional scan in $M_{\nu }$ for a fixed width of $5\%\times M_W$ and $M_W$ given by Eq. (9) (left) and a one-dimensional scan in ${\mathrm{\Gamma }}_W$ with $M_W$ and $M_{\nu }$ fixed (right), for the $W^{+}$ sample, with the likelihood calculated using 10 000 events.

Figure 8

Simultaneous measurement of the mass $M_W$ and the width, ${\mathrm{\Gamma }}_W$, of the heavy resonance with the MEM for $W^{+}$ production (left) and $W^{-}$ production (right). The $\times$ ($+$) marks the input values (the result from the fit). The dashed line represents the relation (11). Contours represent $-2\ln (\mathcal{L}/{\mathcal{L}}_{\max })$ calculated with 1000 events.

Figure 9

Left: fit to the chirality of the quark and lepton couplings to the heavy resonance. The input study point has $M_W=1000\mathrm{GeV}$, ${\mathrm{\Gamma }}_W=50\mathrm{GeV}$, ${\phi }_{\ell }=0$, and ${\phi }_q=0$. Contours represent $-2\ln (\mathcal{L}/{\mathcal{L}}_{\max })$ calculated with 1000 events. Right: lepton $P_z$ distributions for different chiralities, obtained from analytic expressions and LHAPDF pdfs [57] for $M_W=1000\mathrm{GeV}$ and ${\mathrm{\Gamma }}_W=0$.

Figure 10

Simultaneous measurement of all four parameters: the daughter particle mass $M_{\nu }$ (blue), the parent particle mass parameter $\mu$ defined in (8) (orange), the parent particle width parameter $\gamma$ defined in (14) (green), and the relative chirality ${\phi }_{\mathrm{rel}}$ defined in (13) (black). We show normalized distributions of the measured values for each parameter over 100 samples of 1000 events each, using MEM. The input values for our study point were $M_{\nu }=500\mathrm{GeV}$, $M_W=1000\mathrm{GeV}$, ${\mathrm{\Gamma }}_W=50\mathrm{GeV}$ and ${\cos }^2{\phi }_{\mathrm{rel}}=0.5$, which translates into $2\mu =750\mathrm{GeV}$ and $\gamma =62.5\mathrm{GeV}$.

Figure 11

Correlations among the measured values of the parameters: $M_{\nu }$ and $M_W$ (top left), $M_{\nu }$ and $2\mu$ (top right), $M_{\nu }$ and ${\mathrm{\Gamma }}_W$ (bottom left), and $M_{\nu }$ and $\gamma$ (bottom right).

Figure 12

The same as Fig. 5, but for the case of pair production, where we study the $M_{T2}$ distribution instead of $P_T$. In the lower panel we show the bin-by-bin ratio of the number of events for different widths, normalized to the case of ${\mathrm{\Gamma }}_W=50\mathrm{GeV}$.

Figure 13

The same as Fig. 3, but for the case of pair production. We showcase several variables whose distributions are sensitive to the overall mass scale: the dilepton invariant mass, $m_{\ell \ell }$ (upper left), the transverse lepton momentum, $P_{\ell T}$ (upper right), the larger of the two longitudinal lepton momenta (in absolute value), $\max (|P_{\ell z}|,|P_{\overline{\ell }z}|)$ (lower left), and the other longitudinal lepton momentum with its sign chosen as $\mathrm{sgn}(P_{\ell z}{P}_{\overline{\ell }z})\min (|P_{\ell z}|,|P_{\overline{\ell }z}|)$ (lower right).

Figure 14

The same as the left panel in Fig. 8, but for pair production, i.e., the second diagram in Fig. 1. Contours represent $-2\ln (\mathcal{L}/{\mathcal{L}}_{\max })$ calculated with 500 events. The $\times$ ($+$) marks the input values (the result from the fit).

Figure 15

Fit to the chirality of the lepton couplings in the case of pair production as in the second diagram of Fig. 1 using 600 events. In the left panel, the couplings were chosen to be purely chiral, ${\phi }_{\ell }=0$, while in the right panel they were vectorlike: ${\phi }_{\ell }=45°$.