[Paper Review] Scale Optimisation and Event Shapes in Deep-Inelastic Scattering

Michael Dinsdale|ArXiv.org|Dec 5, 2005

Particle physics theoretical and experimental studies13 references3 citations

TL;DR

This paper investigates scale optimization in deep-inelastic scattering (DIS) event shape means using the Principle of Minimal Sensitivity (PMS) to reduce reliance on power corrections. By introducing separate renormalization scales for quark- and gluon-initiated subprocesses (PMS₂), the study finds a significant reduction in required power corrections at high Q, though this leads to poor low-energy fits, suggesting optimized scales may mimic higher-order perturbative effects rather than non-perturbative ones.

ABSTRACT

We study the effect of optimising the renormalisation and factorisation scales on perturbative calculations of event shape means defined in the Breit frame of ep DIS. Unlike in the case of e^+e^- event shape means, this has only a small effect on the NLO QCD predictions and a large power correction is still required to fit the data. However, if separate renormalization scales are introduced for the quark- and gluon-initiated sub-processes the optimisation has a much larger effect and greatly reduces the size of the required power corrections. Unfortunately, there are then problems fitting the low energy data for some observables.

Motivation & Objective

• To assess whether scale optimization via the Principle of Minimal Sensitivity (PMS) reduces the need for power corrections in DIS event shape means.
• To compare two PMS prescriptions: PMS₁ (common scale for quark and gluon processes) and PMS₂ (separate scales for quark- and gluon-initiated subprocesses).
• To evaluate whether optimized scales in DIS, as in $e^+e^-$, can reduce apparent non-perturbative effects by better approximating higher-order perturbative corrections.
• To test if PMS₂, which improves high-energy agreement, introduces unphysical behavior at low energies, affecting data fit quality.

Proposed method

• The Principle of Minimal Sensitivity (PMS) is applied to determine optimal renormalization and factorization scales in NLO QCD calculations for DIS event shapes.
• Two scale optimization schemes are used: PMS₁, where a single scale μ is used for both quark and gluon subprocesses, and PMS₂, where separate scales are assigned to quark- and gluon-initiated contributions.
• The MRST2001E parton distribution functions (PDFs) are used in the calculations, with scale dependence explicitly tracked in the NLO cross-sections.
• Theoretical predictions are compared to H1 collaboration data for event shape means in the Breit frame, with power corrections parametrized by a single non-perturbative parameter $\overline{\alpha}_0$.
• Fits are performed across multiple observables (e.g., $\tau_c$, $\rho_E$, $y_{fJ}$) to extract $\overline{\alpha}_0$ values under different scale choices.
• The convergence and stability of the perturbative series are analyzed, particularly comparing PMS₂ to standard $\overline{MS}$-scheme results and to $e^+e^-$ results where scale optimization reduced power corrections.

Experimental results

Research questions

• RQ1 Can scale optimization via PMS reduce the size of required power corrections in DIS event shape means, similar to what was observed in $e^+e^-$ annihilation?
• RQ2 Does using separate renormalization scales for quark- and gluon-initiated subprocesses (PMS₂) lead to better perturbative convergence than a common scale (PMS₁)?
• RQ3 How does the performance of PMS₂ compare to PMS₁ and the standard $\mu = M = Q$ choice in fitting low- and high-energy data?
• RQ4 To what extent do the observed improvements in high-energy agreement under PMS₂ suggest that apparent power corrections are actually higher-order perturbative effects?
• RQ5 Can the PMS approach resolve the tension between perturbative predictions and data without relying on large non-perturbative power corrections?

Key findings

• PMS₁ improves the fit quality of power corrections compared to the standard $\mu = M = Q$ choice, but the reduction in required $\overline{\alpha}_0$ is modest.
• PMS₂ significantly reduces the required power correction size at high $Q$, with $\overline{\alpha}_0$ values dropping to around 0.5–0.6, comparable to $e^+e^-$ results.
• Despite improved high-energy agreement, PMS₂ predictions deviate strongly from data at low $Q$, showing rapid growth that worsens the fit quality.
• The $\overline{\alpha}_0$ values for PMS₂ are smaller than for PMS₁ and the standard choice, indicating a stronger suppression of non-perturbative effects.
• The low-energy failure of PMS₂ suggests that while it better approximates higher-order corrections, it may not be reliable for extrapolation to low scales.
• The results imply that PMS₂ may more accurately capture higher-order perturbative effects, potentially explaining why $e^+e^-$ data required no power corrections when optimized, though this remains speculative without NNLO data.