[Paper Review] Universality of Nonperturbative Effects in Event Shapes

Christopher Lee, George Sterman|arXiv (Cornell University)|Mar 9, 2006

Machine Learning in Materials Science16 citations

TL;DR

This paper establishes the universality of nonperturbative power corrections in event shape distributions—such as thrust, C-parameter, and angularities—using soft-collinear effective theory (SCET) and energy flow operators. It proves that the leading power corrections to the mean values of these observables are universally related to a single nonperturbative parameter 𝒜, without relying on the single-gluon approximation, by exploiting Lorentz boost invariance of ultrasoft Wilson lines and the structure of shape functions.

ABSTRACT

Nonperturbative effects in event shape distributions can be characterized by shape functions derived in the eikonal approximation or, equivalently, from soft-collinear effective theory. The use of energy flow operators and the boost invariance of the Wilson lines of soft gluons in the shape functions leads to a proof of universality for power corrections to the mean values of event shapes, without invoking the single gluon approximation.

Motivation & Objective

• To establish the universality of nonperturbative power corrections in event shape distributions such as thrust, C-parameter, and angularities.
• To derive these universal relations without relying on the single-gluon approximation, which has been used in prior derivations.
• To demonstrate that the leading power correction to the mean value of any event shape is proportional to a single universal nonperturbative parameter 𝒜.
• To show that this universality arises from the Lorentz invariance of ultrasoft Wilson lines and the structure of energy flow operators in soft-collinear effective theory (SCET).

Proposed method

• Uses shape functions defined via matrix elements of Wilson lines in the ultrasoft sector, expressed as $ S_e(e) = \frac{1}{N_C} \text{Tr} \sum_{X_u} |\langle X_u| Y_n \bar{Y}_{\bar{n}} |0\rangle|^2 \delta(e - e(X_u)) $.
• Introduces a transverse energy flow operator $ \mathcal{E}_T(\eta) $ that acts on final states to project the transverse momentum weighted by pseudorapidity.
• Applies Lorentz boost invariance to show that the matrix element of $ \mathcal{E}_T(\eta) $ is independent of the rapidity $ \eta $, enabling the extraction of universal coefficients.
• Performs a Taylor expansion of the delta function in the shape function to compute the first moment, isolating the leading power correction.
• Uses the invariance of the matrix element under boosts to show that the coefficient of the correction depends only on the integral of the observable’s kernel over pseudorapidity.
• Derives the universal coefficients $ c_{\tau_a} = \frac{2}{1-a} $ and $ c_C = 3\pi $ from the rapidity integrals of the respective event shape kernels.

Experimental results

Research questions

• RQ1 Can the universality of nonperturbative power corrections in event shapes be derived without assuming single soft gluon emission?
• RQ2 What is the role of Lorentz invariance in the structure of shape functions for event shapes in soft-collinear effective theory?
• RQ3 How do energy flow operators and Wilson lines in the ultrasoft sector lead to universal power corrections in event shape distributions?
• RQ4 Why is the nonperturbative parameter 𝒜 independent of the rapidity scale in the matrix element?
• RQ5 Can the standard universality relations for mean values of event shapes be derived directly from the shape function formalism without approximations?

Key findings

• The first moment of the event shape distribution receives a universal power correction proportional to $ \mathcal{A} $, the nonperturbative matrix element of the transverse energy flow operator.
• The coefficient of the correction for angularities is $ c_{\tau_a} = \frac{2}{1-a} $, derived from the integral $ \int d\eta \, e^{-|\eta|(1-a)} = \frac{2}{1-a} $.
• For the C-parameter, the coefficient is $ c_C = 3\pi $, obtained from $ \int d\eta \, \frac{3}{\cosh \eta} = 3\pi $.
• The derivation of these relations does not require the single-gluon approximation, relying instead on Lorentz invariance and the structure of SCET shape functions.
• The nonperturbative parameter $ \mathcal{A} $ is independent of the rapidity $ \eta $, due to the boost invariance of the Wilson line matrix elements.
• The shape function formalism allows the derivation of the shift $ \frac{d\sigma}{de}(e) \to \frac{d\sigma}{de}(e - c_e \mathcal{A}) $ in the perturbative distribution, confirming the universality of power corrections.