The Matrix Element Method at Next-to-Leading Order

TL;DR

This work extends the Matrix Element Method from LO to NLO by enabling event-by-event NLO weights. It introduces a forward branching phase space generator to incorporate real radiation while preserving Born kinematics and uses adapted dipole subtraction to maintain a consistent, one-to-one mapping between virtual and real contributions. The approach yields a complete NLO MEM weight and, as a by-product, an unweighted NLO event generator for color-neutral final states. Validation against Z-boson mass measurements and a Higgs search in the ZZ→4ℓ channel demonstrates improved perturbative control and enhanced discriminating power at NLO. The framework paves the way for precise parameter extractions and systematic uncertainty control in electroweak processes, with potential extensions to top quark measurements and more complex final states.

Abstract

This paper presents an extension of the matrix element method to next-to-leading order in perturbation theory. To accomplish this we have developed a method to calculate next-to-leading order weights on an event-by-event basis. This allows for the definition of next-to-leading order likelihoods in exactly the same fashion as at leading order, thus extending the matrix element method to next-to-leading order. A welcome by-product of the method is the straightforward and efficient generation of unweighted next-to-leading order events. As examples of the application of our next-to-leading order matrix element method we consider the measurement of the mass of the Z boson and also the search for the Higgs boson in the four lepton channel.

Figures (9)

• Figure 1: The generation of the Born (and virtual) phase space from a given experimental event. The left hand side depicts a collision that results in the production of a colour neutral final state (represented here by four leptons in red) that do not balance in the transverse plane. The resulting imbalance ($X$, in blue) represents the remaining event which is not modelled in the Born matrix element. We apply a Lorentz transformation such that $X$ has no components in the transverse plane, with the remaining longitudinal and energy components absorbed into the colliding partons.
• Figure 2: Comparison between the lab and MEM frame predictions from the NLO calculation of MCFM (left) and Pythia (right) for the process $pp\rightarrow Z/\gamma^* \rightarrow \ell^+\ell^-$. In (a) we plot the invariant mass distribution of the two leptons and in (b) we show the $p_T$ of the positively charged lepton. In each plot the lab frame quantity is shown in black (dashed), while the MEM frame result is in red (solid).
• Figure 3: Comparison between MCFM (LO and NLO) and Pythia in different frames. On the left hand side $p_T^{\ell}$ is plotted in the MEM frame, whilst on the right hand side the lab frame equivalent is plotted. Predictions are normalised by the total cross section (or number of events in the Pythia case).
• Figure 4: Log-likelihoods obtained by a MEM analysis at LO (black) and NLO (red) for the measurement of $m_Z$ at the LHC using Pythia data. Errors represent MC integration uncertainty.
• Figure 5: Reconstructed $Z$ mass as a function of the upper bound on the transverse momentum of the dilepton system, $p^{\ell\ell}_T$. Errors represent the $1 \sigma$ deviation from the central value. Note that both LO and NLO calculations are performed at the same values of the cut, $p^{\ell\ell}_T$. In the plot the NLO points have been moved slightly to the right for clarity.