Lepton and photon energy scale and resolution corrections based on the minimization of an analytical likelihood: IJazZ2.0
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 Gaussian energy smearing model, 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.