Power corrections and resummation of radiative corrections in the single dressed gluon approximation - the average thrust as a case study

TL;DR

The paper tackles infrared power corrections in Minkowskian QCD observables through renormalon resummation, proposing a dressed skeleton expansion with a leading single-gluon term and a scheme-invariant skeleton coupling. It develops and compares several regularization schemes (APT, PV, infrared cutoff) and extends the analysis to two-loop running, linking regularization choices to power corrections. The average thrust in e+e- is used as a case study, where the leading 1/Q power correction arises from soft-gluon dynamics, and resummed perturbation theory plus a 1/Q term yields a precise alpha_s extraction around 0.11, with stability across loop orders. The work suggests that resummation of running-coupling effects, when combined with power corrections, provides a more faithful description of data and motivates applying the approach to other observables, though care must be taken regarding non-inclusive effects and skeleton-coupling universality.

Abstract

Infrared power corrections for Minkowskian QCD observables are analyzed in the framework of renormalon resummation, motivated by analogy with the skeleton expansion in QED and the BLM approach. Performing the ``massive gluon'' renormalon integral a renormalization scheme invariant result is obtained. Various regularizations of the integral are studied. In particular, we compare the infrared cutoff regularization with the standard principal value Borel sum and show that they yield equivalent results once power terms are included. As an example the average thrust < T > in e+e- annihilation is analyzed. We find that a major part of the discrepancy between the known next-to-leading order calculation and experiment can be explained by resummation of higher order perturbative terms. This fact does not preclude the infrared finite coupling interpretation with a substantial 1/Q power term. Fitting the regularized perturbative sum plus a 1/Q term to experimental data yields alpha_s^{MSbar}(M_Z) = 0.110 \pm 0.002.

Figures (10)

• Figure 1: Average of $1-{\rm thrust}$ as a function of the center of mass energy Q, according to the available experimental data experiment and the leading order (LO) and next-to-leading order (NLO) perturbative QCD calculation in the $\overline{\rm MS}$ scheme with $\mu_R=Q$, given $\alpha_s^{\hbox{$\overline{\hbox{\tiny MS}}\,$}}({\rm M_Z})=0.117$.
• Figure 2: Phase-space for the emission of a virtual gluon with $\mu^2\,=\,\epsilon\,Q^2\,=\,0.1\,Q^2$ in the plane of the quark and anti-quark energy fractions ($x_{1,2}$). Continuous lines represent phase-space limits: the upper (curved) line corresponds to the softest gluons (\ref{['large_epsilon']}) while the lower (linear) line corresponds to the hardest (\ref{['small_epsilon']}). Dashed lines represent the separation of phase-space according to which particle carries the largest momentum and thus determines the thrust axis (cf. eq. (\ref{['thrust']})): in the upper left region $T=x_2$, in the upper right region $T=x_1$ and in the lower region $T=\sqrt{x_3^2-4\epsilon}$.
• Figure 3: The characteristic function ${\cal F}(\epsilon)$ for the average thrust (\ref{['F_def']}) as a function of $\log_{10}(\epsilon)$, where $\mu^2=\epsilon Q^2$ is the "gluon mass". ${\cal F}(\epsilon)$ is represented by a thick black line reaching zero in the limit $\epsilon\longrightarrow 1$. It is compared with its asymptotic expansion at small $\epsilon$ (\ref{['F_ana']}), for $n=\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2}$, where $n$ is the highest order term ${\cal O}(\epsilon^n)$ taken into account in each approximating curve.
• Figure 4: The derivative of the characteristic function for the average thrust $\dot{{\cal F}}(\epsilon)$ as a function of $\log_{10}(\epsilon)$, where $\mu^2=\epsilon Q^2$ is the "gluon mass". The separate contributions from kinematic configurations where one of the quarks or the gluon carries the largest momentum are shown as well. The dashed line represents the leading ($n=\frac{1}{2}$) term in the small $\epsilon$ expansion of this function, $\dot{{\cal F}}(\epsilon)\simeq 4\sqrt{\epsilon}$.
• Figure 5: Average of $1-{\rm thrust}$ as a function of Q: experimental data is compared with naive perturbative QCD results (LO and NLO in $\overline {\rm MS}$ with $\mu_R=Q$) and with the resummed perturbative series in the APT or principal value Borel sum regularizations (the two coincide). All the theoretical calculations are based on $\alpha_s^{\hbox{$\overline{\hbox{\tiny MS}}\,$}}({\rm M_Z})=0.117$. Both one-loop (upper dashed) and two-loop (dot-dash) resummation results ($R_{\hbox{\tiny APT}}$) are presented, where the running coupling $\bar{a}_{\hbox{\tiny PT}}$ is in the "gluon bremsstrahlung" scheme. In the two-loop case we also show eq. (\ref{['inc_NLO']}) (lower continuous line; just below the upper dashed line) which includes the full ${\cal O}(\alpha_s^2)$ term and eq. (\ref{['inc_lambda']}) (upper continuous line) which includes in addition a fitted $\lambda/Q$ term. The lower dashed line is the absolute value of the imaginary part of the one-loop Borel sum, which reflects the magnitude of the renormalon ambiguity.