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.
A precise calibration of the photon and lepton energy scales is a key requirement for precision measurements at high-energy collider experiments. In particular, the measurement of the Higgs boson mass in decay channels involving photons or charged leptons, such as H → γγ and H → ZZ * → 4ℓ, is directly sensitive to small biases in the energy or momentum scale. Since electrons and photons share a largely common electromagnetic response in the detector, the energy-scale calibration of electrons is propagated to photons, with additional corrections accounting for differences in shower development and upstream material effects.
Decays of the Z boson into charged-lepton pairs, Z → ℓℓ with ℓ ≡ e, µ, provide a primary in-situ reference for lepton energy or momentum calibration, owing to their large production rate and the precisely known Z-boson mass [1]. The calibration is typically performed by constraining the reconstructed dilepton invariant-mass distribution to a reference line shape. However, the invariant mass depends simultaneously on the response of both leptons and is affected by the natural width of the Z boson, final-state radiation, and detector resolution effects. These features introduce correlations between scale and resolution parameters, complicating their simultaneous extraction in differential calibration schemes.
The calibration strategy is based on a comparison between the lepton-energy scale measured in data and that predicted by a detailed detector simulation. Residual differences between data and simulation are interpreted as corrections to the energy scale and resolution and are extracted using Z → ℓℓ decays. Within the framework we propose, these residual mismodeling effects are assumed to induce approximately Gaussian distortions of the di-lepton invariant-mass distribution, referred to as smearing hereafter.
The core of the method consists of describing the smearing effects using an analytical approach, enabling the construction of a fully analytic likelihood. Automatic differentiation techniques [2] are then used to efficiently compute exact gradients with respect to all calibration parameters. This method is implemented in a software tool named IJazZ2.0 (I Just AnalyZe the Z), which is freely available via a PyPI distribution [3] and is described in section 2. The treatment of statistical uncertainties arising from the finite size of the simulated Z → e + e -sample is discussed in section 3. Additional considerations related to energyresponse linearity are presented in section 4, and the extension of the method to photon energy-scale measurements using Z → µ + µ -γ decays is described in section 5.
The aim of IJazZ2.0 is to measure the differences between Monte Carlo (MC) simulation and data in terms of lepton energy response and resolution. These calibration parameters are usually measured with respect to a set of variables ⃗ X describing the lepton properties and detector conditions, such as the polar and azimuthal angles θ and ϕ, the pseudorapidity η ≡ln[tan(θ/2)], and shower-shape observables characterizing the lateral and longitudinal development of the electromagnetic cluster in the calorimeter. The transverse momentum p T is omitted for the moment as it requires a dedicated treatment as it is already part of the invariant-mass calculation.
The di-lepton invariant mass is defined as:
where ∆η and ∆ϕ denote the differences in pseudorapidity and azimuthal angle between the two leptons, and p T,1 and p T,2 are their transverse momenta. The angular quantities ∆η and ∆ϕ are typically extremely well measured in collider experiments; therefore, in the present approach, they are assumed to be perfectly known and not to introduce any bias or smearing in the reconstructed invariant mass. We note r ℓ ( ⃗ X) and σ ℓ ( ⃗ X), respectively the data/MC relative energy scale and data/MC energy smearing. To be more specific, in this method, we correct the energy from the simulation to match the one in data. Therefore, r ℓ ( ⃗ X) is a correction to be applied to the lepton energy in the simulation:
where E mc corrS is the scale-corrected energy while E mc raw is the original lepton energy. Usually and conversely, one can correct afterward the lepton energy in the data back to the simulation level with the formula:
Concerning the energy resolution, it is assumed to be always better (i.e. smaller) in the simulation, thus the energy resolution in the simulation needs to be degraded (smeared) to its corresponding level in data. Because the simulation already includes most of the effects due to the detector response (energy loss, material…), the modest degradation due to the imperfect modelling of the simulation is assumed to follow a normal distribution. The lepton-energy degradation is done in the simulation with random number trials from a normal distribution. Therefore, for each lepton in the simulation, its energy is drawn from the probabilistic distribution (including scale and smearing
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