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The Determination of $α_S$ at Hadron Colliders

W. T. Giele, E. W. N. Glover, J. Yu

TL;DR

This paper outlines a program to determine $\alpha_S$ and PDFs using hadron-collider data, emphasizing the wide range of momentum transfers accessible and the perturbative QCD framework. Using the 1-jet inclusive $E_T$ distribution from the CDF '88-'89 data, it demonstrates extracting $\alpha_S$ at various scales and tests its running against QCD predictions, while assessing theoretical uncertainties and PDF dependencies. The study shows consistency of $\alpha_S(M_Z)$ with DIS inputs and supports QCD running across 30–500 GeV; it also discusses methodological paths for reduced uncertainties with higher-luminosity Run 1 data and future colliders. Overall, the work establishes hadron colliders as a viable arena for precise $\alpha_S$ and PDF determinations within one experiment, with implications for global PDF fits and tests of the strong interaction gauge structure.

Abstract

Hadron colliders offer a unique opportunity to test perturbative QCD because, rather than producing events at a specific beam energy, the dynamics of the hard scattering is probed simultaneously at a wide range of momentum transfers. This makes the determination of $\al$ and the parton density functions (PDF) at hadron colliders particularly interesting. In this paper we restrict ourselves to extracting $\al$ for a given PDF at a scale which is directly related to the transverse energy produced in the collision. As an example, we focus on the single jet inclusive transverse energy distribution and use the published '88-'89 CDF data with an integrated luminosity of 4.2 pb$^{-1}$. The evolution of the coupling constant over a wide range of scales (from 30~GeV to 500~GeV) is clearly shown and is in agreement with the QCD expectation. The data to be obtained in the current Tevatron run (expected to be well in excess 100 pb$^{-1}$ for both the CDF and DØ experiments) will significantly decrease the experimental errors.

The Determination of $α_S$ at Hadron Colliders

TL;DR

This paper outlines a program to determine and PDFs using hadron-collider data, emphasizing the wide range of momentum transfers accessible and the perturbative QCD framework. Using the 1-jet inclusive distribution from the CDF '88-'89 data, it demonstrates extracting at various scales and tests its running against QCD predictions, while assessing theoretical uncertainties and PDF dependencies. The study shows consistency of with DIS inputs and supports QCD running across 30–500 GeV; it also discusses methodological paths for reduced uncertainties with higher-luminosity Run 1 data and future colliders. Overall, the work establishes hadron colliders as a viable arena for precise and PDF determinations within one experiment, with implications for global PDF fits and tests of the strong interaction gauge structure.

Abstract

Hadron colliders offer a unique opportunity to test perturbative QCD because, rather than producing events at a specific beam energy, the dynamics of the hard scattering is probed simultaneously at a wide range of momentum transfers. This makes the determination of and the parton density functions (PDF) at hadron colliders particularly interesting. In this paper we restrict ourselves to extracting for a given PDF at a scale which is directly related to the transverse energy produced in the collision. As an example, we focus on the single jet inclusive transverse energy distribution and use the published '88-'89 CDF data with an integrated luminosity of 4.2 pb. The evolution of the coupling constant over a wide range of scales (from 30~GeV to 500~GeV) is clearly shown and is in agreement with the QCD expectation. The data to be obtained in the current Tevatron run (expected to be well in excess 100 pb for both the CDF and DØ experiments) will significantly decrease the experimental errors.

Paper Structure

This paper contains 15 sections, 21 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical considerations
  3. Running coupling constants
  4. Extracting $\hbox{$\alpha_S$}$
  5. Theoretical Uncertainty
  6. The one-jet inclusive transverse energy distribution
  7. The QCD-fit
  8. The Best-fit
  9. The CDF data
  10. Determining $\hbox{$\alpha_S$}$
  11. Measurement of $\hbox{$\alpha_S$}^{(1)}(E_T)$
  12. Measurement of $\hbox{$\alpha_S$}^{(1)}(M_Z)$
  13. The QCD-fit to $\hbox{$\alpha_S$}^{(1)}(M_Z)$
  14. The Best-fit to $\hbox{$\alpha_S$}^{(1)}(M_Z)$
  15. Conclusions

Figures (6)

  • Figure 1: Comparison between the 1st, 2nd and 3rd order running of $\hbox{$\alpha_S$}(\mu_R)$ for $\hbox{$\alpha_S$}(M_Z)=0.118$. Fig. 1a shows $\hbox{$\alpha_S$}^{(n)}(\mu_R)$ from 10 GeV to 1000 GeV while Fig. 1b gives the relative change with respect to the 1st order running over the relevant energy range from 30 GeV to 500 GeV.
  • Figure 2: The uncertainty in $\hbox{$\alpha_S$}(E_T)$ due to variation in the renormalization scale (solid line). Also shown is the logarithmic tangent of eq. \ref{['logtan']} (dashed line) which has a simple analytic form.
  • Figure 3: The extracted $\hbox{$\alpha_S$}^{(1)}(M_Z, \mu_R=E_T)$ as a function of $E_T$ for the MRSA$^\prime$ parameterisation. The QCD-fit yields $\hbox{$\alpha_S$}^{(1)}(M_Z)= 0.121\pm 0.001\pm 0.008\pm 0.005$. The 68% confidence level Best-fits are shown as shaded bands.
  • Figure 4: The values of $\hbox{$\alpha_S$}^{(1)}(E_T)$ extracted from the published CDF data as a function of $E_T$ together with $\hbox{$\alpha_S$}^{(1)}(E_T)$ from the QCD- and Best-fits evolved from $M_Z$ to $E_T$ for the MRSA$^\prime$ parameterisation.
  • Figure 5: Comparison of the one-jet inclusive transverse energy distribution evaluated using $\hbox{$\alpha_S$}^{(1)}(E_T)$ from the QCD- and Best-fits evolved from $M_Z$ to $E_T$ for the MRSA$^\prime$ parameterisation.
  • ...and 1 more figures