On the Kinematic Reconstruction of Deep Inelastic Scattering at HERA: the $Σ$ Method

TL;DR

The paper introduces the Σ method for reconstructing deep inelastic scattering kinematics at HERA, enabling precise measurements of F2(x,Q^2) across the full kinematic range, including challenging low-x, low-Q^2 regions, with reduced radiative corrections. Through detailed comparisons with the traditional electron, double-angle, and mixed methods, Σ demonstrates superior x-resolution at high y and resilience to initial-state radiation, while still functioning well at low x and high Q^2. It also explores extensions such as the eΣ method and kinematic fitting, and analyzes QED radiative corrections, proposing ISR-independent variants (IDA, IΣ) as complementary approaches. The work provides practical guidance for achieving high-precision F2 extractions at HERA and highlights how method choices impact radiative corrections and systematic uncertainties.

Abstract

We review and compare the reconstruction methods of the inclusive deep inelastic scattering variables used at HERA. We introduce a new prescription, the $Σ$ method, which allows to measure the structure function of the proton $F_2(x,Q^2)$ in a large kinematic domain, and in particular in the low x-low$Q^2$ region, with small systematic errors and small radiative corrections. A detailed comparison between the $Σ$ method and the other methods is shown. Extensions of the $Σ$ method are presented. The effect of QED radiation on the kinematic reconstruction and on the structure function measurement is discussed.

Figures (6)

• Figure 1: Monte Carlo comparison of $y$ reconstruction for several hadronic methods w.r.t. to $y_e$, at high (0.2-0.5) and very high $y$ (0.5-0.8). Both bias and resolution improve with $y$ for the $\Sigma$ method although these become worse for the hadrons-only method. $y_{DA}$ has larger biases and resolutions than $y_{\Sigma}$, as can be read from the fit results written inside the figure.
• Figure 2: Comparison $x_{method} / x$ at low $Q^2$ ($Q^2 > 7~GeV^2$) for the $\Sigma$, mixed , DA and $e$ methods. From top to bottom, each row represent a bin in $y$: very high (0.5-0.8), high (0.2-0.5), medium (0.1-0.2), low (0.05-0.1), very low (0.01-0.05).
• Figure 3: Comparison $Q^2_{method} / Q^2$ at low $Q^2$ ($Q^2 > 7~GeV^2$) for the $\Sigma$ , DA and $e$ methods. From top to bottom, each row represent a bin in $y$: very high (0.5-0.8), high (0.2-0.5), medium (0.1-0.2), low (0.05-0.1), very low (0.01-0.05).
• Figure 4: Comparison $x_{method} / x$ at high $Q^2$ ($Q^2 > 200~GeV^2$) for the $\Sigma$, mixed , DA and $e$ methods. From top to bottom, each row represent a bin in $y$: very high (0.5-0.8), high (0.2-0.5), medium (0.1-0.2), low (0.05-0.1), very low (0.01-0.05).
• Figure 5: Radiative corrections (excluding vacuum polarisation correction) as a function of $x$ in two bins of $Q^2$ for the $\Sigma$, $e\Sigma$, DA, $e$ and mixed method, requesting two minimal scattered electron energy: $E > 4~GeV$ (upper curve of each method) and $E> 8~GeV$ (lower curve).