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A Precise Determination of $α_s$ from the C-parameter Distribution

André H. Hoang, Daniel W. Kolodrubetz, Vicent Mateu, Iain W. Stewart

Abstract

We present a global fit for $α_s(m_Z)$, analyzing the available C-parameter data measured at center-of-mass energies between $Q=35$ and $207$ GeV. The experimental data is compared to a N$^3$LL$^\prime$ + $\mathcal{O}(α_s^3)$ + $Ω_1$ theoretical prediction (up to the missing 4-loop cusp anomalous dimension), which includes power corrections coming from a field theoretical nonperturbative soft function. The dominant hadronic parameter is its first moment $Ω_1$, which is defined in a scheme which eliminates the $\mathcal{O}(Λ_{\rm QCD})$ renormalon ambiguity. The resummation region plays a dominant role in the C-parameter spectrum, and in this region a fit for $α_s(m_Z)$ and $Ω_1$ is sufficient. We find $α_s(m_Z)=0.1123\pm 0.0015$ and $Ω_1=0.421\pm 0.063\,{\rm GeV}$ with $χ^2/\rm{dof}=0.988$ for $404$ bins of data. These results agree with the prediction of universality for $Ω_1$ between thrust and C-parameter within 1-$σ$.

A Precise Determination of $α_s$ from the C-parameter Distribution

Abstract

We present a global fit for , analyzing the available C-parameter data measured at center-of-mass energies between and GeV. The experimental data is compared to a NLL + + theoretical prediction (up to the missing 4-loop cusp anomalous dimension), which includes power corrections coming from a field theoretical nonperturbative soft function. The dominant hadronic parameter is its first moment , which is defined in a scheme which eliminates the renormalon ambiguity. The resummation region plays a dominant role in the C-parameter spectrum, and in this region a fit for and is sufficient. We find and with for bins of data. These results agree with the prediction of universality for between thrust and C-parameter within 1-.

Paper Structure

This paper contains 21 sections, 43 equations, 19 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Theory Review
  3. Factorized Cross Section Formula
  4. Profile Functions
  5. Hadron-mass effects
  6. Experimental Data
  7. Fit Procedure
  8. Results
  9. Impact of Resummation and Power Corrections
  10. Perturbative Uncertainty from the Scan
  11. Experimental Fit Uncertainty
  12. Individual Theory Scan Errors
  13. Effects of $\mathbf{\Omega_1}$ hadron-mass resummation
  14. Dataset dependence
  15. Final Results
  16. ...and 6 more sections

Figures (19)

  • Figure 1: Bands for the profile functions for the renormalization scales $\mu_H$, $\mu_J(C)$, $\mu_S(C)$ when varying the profile parameters.
  • Figure 2: Plots of the $r$ dependence of $g_C(r)\, \Omega_1(R_\Delta,\mu_\Delta,r)$ for different values of $\theta(R_\Delta,\mu_\Delta)$. We normalize to $\Omega_1^C(R_\Delta,\mu_\Delta)$, since it is simply an overall multiplicative factor.
  • Figure 3: Difference between the default cross section and the cross section varying only one parameter. We vary $\alpha_s (m_Z)$ by $\pm\, 0.001$ (solid red), $2\,\Omega_1$ by $\pm \,0.1$ (dashed blue) and $\Omega_2^C$ by $\pm\,0.5$ (dotted green). The three plots correspond to three different center of mass energies: (a) $Q=35$ GeV, (b) $Q=91.2$ GeV, (c) $Q=206$ GeV.
  • Figure 4: The evolution of the value of $\alpha_s(m_Z)$ adding components of the calculation. An additional $\sim 8\%$ uncertainty from not including power corrections is not included in the two left points.
  • Figure 5: The first two panels show the distribution of best-fit points in the $\alpha_s(m_Z)$-$2\Omega_1$ and $\alpha_s(m_Z)$-$2\overline\Omega_1$ planes. Panel (a) shows results including perturbation theory, resummation of large logs, the soft nonperturbative function and $\Omega_1$ defined in the Rgap scheme with renormalon subtractions. Panel (b) shows the results as in panel (a), but with $\overline\Omega_1$ defined in the $\overline{\textrm{MS}}$ scheme without renormalon subtractions. In both panels the dashed lines corresponds to an ellipse fit to the contour of the best-fit points to determine the theoretical uncertainty. The respective total (experimental + theoretical) 39% CL standard uncertainty ellipses are displayed (solid lines), which correspond to $1$-$\sigma$ (68% CL) for either one-dimensional projection. The big points represent the central values in the random scan for $\alpha_s(m_Z)$ and $2\,\Omega_1$. Likewise, the two panels at the bottom show the distribution of best-fit points in the $\alpha_s(m_Z)$-$\chi^2/{\rm dof}$ plane. Panel (c) shows the $\chi^2/{\rm dof}$ values of the points given in panel (a), whereas panel (b) shows the $\chi^2/{\rm dof}$ values of the points given in panel (b).
  • ...and 14 more figures