Lepton energy scale and resolution corrections based on the minimization of an analytical likelihood: IJazZ2.0

Abstract

We present a novel method to determine lepton energy scale and resolution corrections by means of an analytical likelihood maximization applied to Drell-Yan $Z \to \ell\ell$ events. The approach relies on an exact analytical treatment of the energy smearing, avoiding random-number-based convolution techniques. This formulation results in a fully differentiable likelihood enabling the use of automatic differentiation algorithms, and thus a substantial reduction in computational cost. The method, implemented in the \ijazz software, allows the simultaneous extraction of scale and resolution parameters across multiple lepton categories defined by detector or kinematic variables. We validate the technique using toy Monte Carlo studies and realistic Pythia-based simulations, demonstrating unbiased parameter recovery and accurate uncertainty estimates. Particular attention is given to categorizations involving lepton transverse momentum, for which a relative-$p_T$ strategy is introduced to mitigate biases induced by category migration and kinematic correlations. The method is further adapted to photon-energy scale measurement in $Z \to μ^-μ^+γ$ decays. Compared to conventional approaches, the analytical method improves numerical stability, robustness of the minimization, and computational performance, making it well suited for large-scale precision calibration tasks at the LHC.

Figures (11)

• Figure 1: Left: the original MC distribution (Breit--Wigner) from 10,000 generated events. Right: the smeared MC distribution obtained using a random smearing technique, with $n_\mathrm{smear}$ trials per original MC event, together with the analytical smearing prediction (dashed line). It can be seen that the analytical smearing accurately reproduces the expected distribution, which is approximated by $n_\mathrm{smear}=10{,}000$; the two distributions are indistinguishable.
• Figure 2: Validation of the method using a naive MC simulation. A subset of events is decalibrated and smeared according to known functions (injected curves). These parameters are accurately recovered by the fit (points). The top panels show the absolute values of the parameters, while the bottom panels display the ratio of fitted to injected values. The left (right) plots correspond to the response (${r_{\ell}}\xspace_b$) and resolution ($\sl_b$) parameters, respectively.
• Figure 3: adaptive binning: The bin width is adapted so that the total number of events in the simulation is the same in each bin, the total number of bins in each category depends on the available MC statistics in this category. The points corresponds to the fitted data (in this case a toy MC dataset) while the histogram refers to the fitted prediction from the smeared simulation.
• Figure 4: Validation of the statistical uncertainties due to limited MC statistics for the response parameters (left) and smearing parameters (right). Triangles show the standard deviation of 100 measurements using the same data but different simulations, while stars show the MC uncertainty predicted by Eq. \ref{['eq:mc_err']}. The dashed line indicates the statistical uncertainty from the data alone; since data and MC sample sizes are equal, MC and data uncertainties are of the same order.
• Figure 5: Measured scale (left) and smearing (right) parameters using the Pythia-based DY simulation. The dashed line corresponds to the injected parameters, while the shaded bands indicate the contribution from limited MC statistics to the total uncertainty. A $p_T$-dependent categorisation introduces small biases in the measured scale and larger biases in the smearing parameters.