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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,

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In order to preform this measurement in practice, one can use any one of several variables, e.g., the transverse mass,

See Eq.

The event topologies considered in this paper: (a) single and (b) pair production of a

One possibility that has been suggested in the literature for determining the overall mass scale is to go beyond the leading order diagrams of Fig.

Given the difficulty of measuring the overall mass scale, it stands to reason that one should use the MEM

Strictly speaking, those analyses were using MEM-inspired optimal variables, whose distributions were used later to form the likelihood in a template method instead of as in Eq.

To our knowledge, the MEM has thus far been underutilized for the purposes of mass spectrum measurements of the sort we describe here. This is perhaps due to the practical challenges one usually encounters in the implementation of the MEM, e.g., the presence of instrumental and/or reducible physics backgrounds; the need to account for effects like the finite detector resolution, the underlying event, jet fragmentation; the challenge of integrating peaked integrand structures in phase space; incorporating higher order corrections to the matrix element, etc.,

We begin by considering single production of a resonance,

We focus on the photon-mediated

The (normalized) kinematic distributions of the leptons in the final state will depend on five model parameters:

Given this general setup, our main goal in this paper will be to attempt to measure

Unit-normalized lepton

In the MEM, the likelihood for a given event with a measured set of

In general

We will use

The likelihood for a set of

We first consider single

If we take

In our subsequent analyses we generate events with

We note that there are two relevant observables: the transverse momentum

One quantity, related to the difference of the squared masses of the

We postpone the question of the simultaneous determination of both

In Fig.

The mass scale is notoriously difficult to measure; even in this very simple topology it cannot be determined from kinematic endpoint measurements alone (unless we require hard ISR). Instead we have to rely on subtle effects. The two tools at our disposal are the distributions of the measured lepton

Unit-normalized distributions of the lepton transverse momentum

As a proof of principle, we perform a preliminary toy exercise to find the mass scale by simply fitting to

The minima of the

The width effects will manifest themselves in two places. First, there will be some smearing of the

The same effect would be observed in the

The same as Fig.

Results from a

We are now ready to measure the mass scale (

One-dimensional scan in

Simultaneous measurement of the mass

Having measured the two masses and the width, the only remaining task is to measure the chirality of the couplings.

In practice, these measurements will be done simultaneously; see the next section.

In analogy to Fig.We note that in the case when the BSM signal of Fig.

Left: fit to the chirality of the quark and lepton couplings to the heavy resonance. The input study point has

After the preliminary exercises shown in the previous sections, we now attempt to simultaneously measure the relevant parameters

As already shown in Eqs.

These expectations are confirmed in Fig.

Simultaneous measurement of all four parameters: the daughter particle mass

The sample mean and the standard deviation of the measured values (with the true values quoted in parentheses) are as follows:

For completeness, we also study the correlations among the different measurements

The Pearson correlation coefficient between variables

Correlations among the measured values of the parameters:

The top left panel in Fig.

Now let us consider pair production, as in the second diagram of Fig.

As in Fig.

In analogy to Fig.

The same as Fig.

As in the case of single production, for the measurement of the mass scale one cannot rely on endpoint measurements alone and needs to utilize the shapes of the relevant kinematic distributions. Figure

The same as Fig.

Figure

In analogy to single production, we now apply the MEM to measure simultaneously the mass scale, the width (Fig.

The same as the left panel in Fig.

Fit to the chirality of the lepton couplings in the case of pair production as in the second diagram of Fig.

The results displayed in Figs.

We have presented methods to measure masses, widths, and couplings, and, in particular, shown that (a) all of these parameters can be simultaneously measured using the MEM and (b) that the physics which gives sensitivity to each of these parameters is more transparent than it would be with many MVAs. While we have focused on the case of

K. M. would like to thank his CMS colleagues for useful discussions. A. B. and D. D. thank O. Mattelaer for technical help and advice. Work supported in part by the U.S. Department of Energy Grants No. DE-SC0010296 and No. DE-SC0010504. A. B. is grateful for the hospitality of the high energy theory group at the University of Florida and acknowledges support from Fulbright and Colciencias. D. D. acknowledges support from the University of Florida Informatics Institute in the form of a Graduate Student Fellowship. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation Grant No. PHY-1607611.