Calculating Track Thrust with Track Functions

TL;DR

The paper develops a framework to predict track-based event shapes in $e^+e^-$ collisions using track functions within SCET, enabling perturbative QCD calculations for observables that are not IRC-safe. By performing NLL$'$ resummation and incorporating leading non-perturbative power corrections, the authors show that track thrust distributions closely resemble calorimeter thrust distributions due to cancellations between non-perturbative parameters. They derive a comprehensive factorization theorem for track thrust, compute the relevant hard, jet, and soft functions, and validate the approach by comparing to ALEPH/DELPHI data and by using Pythia to calibrate track-function inputs. The results indicate that track-based measurements can achieve competitive precision and offer a path to applying track-function techniques to other track-based observables in future collider studies.

Abstract

In e+e- event shapes studies at LEP, two different measurements were sometimes performed: a "calorimetric" measurement using both charged and neutral particles, and a "track-based" measurement using just charged particles. Whereas calorimetric measurements are infrared and collinear safe and therefore calculable in perturbative QCD, track-based measurements necessarily depend on non-perturbative hadronization effects. On the other hand, track-based measurements typically have smaller experimental uncertainties. In this paper, we present the first calculation of the event shape track thrust and compare to measurements performed at ALEPH and DELPHI. This calculation is made possible through the recently developed formalism of track functions, which are non-perturbative objects describing how energetic partons fragment into charged hadrons. By incorporating track functions into soft-collinear effective theory, we calculate the distribution for track thrust with next-to-leading logarithmic resummation. Due to a partial cancellation between non-perturbative parameters, the distributions for calorimeter thrust and track thrust are remarkably similar, a feature also seen in LEP data.

Figures (13)

• Figure 1: Illustration of the track thrust measurement in an $e^+ e^-$ event with jets initiated by a $q \bar{q}$ pair. Solid lines indicate charged particles and dashed lines indicate neutral particles. For track thrust, the thrust axis $\hat{t}$ is determined by the charged particles alone. The event is divided into hemispheres $A$ and $B$ by a plane perpendicular to the thrust axis.
• Figure 2: ALEPH (top) and DELPHI (bottom) measurements of calorimeter and track thrust. Error bars correspond to the statistical and systematic uncertainties added in quadrature. The experimental uncertainties associated with the track-based measurements are noticeably smaller.
• Figure 3: Top: NLL$'$ distributions for calorimeter and track thrust including the leading non-perturbative correction $\Omega_1^\tau$. Next-to-leading logarithmic resummation is included together with ${\cal O}(\alpha_s)$ fixed-order matching contributions. The NLL$'$ calculation exhibits the same qualitative similarity between calorimeter and track thrust as seen in LEP data. Bottom: comparing our analytic results to the DELPHI measurement. There is good quantitative agreement in the tail region where our NLL$'$ calculation is most accurate. The theoretical uncertainties are from scale variation alone, and do not include the (correlated) uncertainties in $\alpha_s$ or $\Omega_1^\tau$, nor uncertainties in our track function extraction.
• Figure 4: Perturbative QCD calculation of the quark (top) and gluon (middle and bottom) track functions at NLO from Eqs. \ref{['eq:defTqQCD']} and \ref{['eq:defTgQCD']} with partonic intermediate states. The NLO track function gets contributions from both branches of the collinear splitting. We do not display virtual diagrams, which vanish in pure dimensional regularization, or diagrams corresponding to Wilson line emissions.
• Figure 5: Distributions for calorimeter and track thrust from Eq. \ref{['eq:NLO']} at $\mathcal{O}(\alpha_s)$. The NLO track functions are extracted from Pythia 8.150 Sjostrand:2006zaSjostrand:2007gs using the procedure in Ref. Chang:2013rca.