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Next-to-Leading Order QCD Corrections and Massive Quarks in $e^+e^-\to 3$ Jets

Arnd Brandenburg, Peter Uwer

TL;DR

The paper develops a complete framework for next-to-leading order QCD corrections to e^+e^- → 3 jets with massive quarks by adapting phase space slicing to include quark masses. It delivers analytic virtual corrections to e^+e^- → Qar{Q}g and constructs a mass-aware subtraction scheme that cancels infrared singularities against real-emission contributions from four-parton final states. Numerical results for bottom-quark jets demonstrate significant mass effects on three-jet observables and improved perturbative stability at NLO, with explicit demonstrations of s_min independence. These developments enable precise QCD tests and potential extraction of heavy-quark masses from jet observables across center-of-mass energies.

Abstract

We present in detail a calculation of the next-to-leading order QCD corrections to the process $e^+e^-\to 3$ jets with massive quarks. To isolate the soft and collinear divergencies of the four parton matrix elements, we modify the phase space slicing method to account for masses. Our computation allows for the prediction of oriented three jet events involving heavy quarks, both on and off the Z resonance, and of any event shape variable which is dominated by three jet configurations. We show next-to-leading order results for the three jet fraction, the differential two jet rate, and for the thrust distribution at various c.m. energies.

Next-to-Leading Order QCD Corrections and Massive Quarks in $e^+e^-\to 3$ Jets

TL;DR

The paper develops a complete framework for next-to-leading order QCD corrections to e^+e^- → 3 jets with massive quarks by adapting phase space slicing to include quark masses. It delivers analytic virtual corrections to e^+e^- → Qar{Q}g and constructs a mass-aware subtraction scheme that cancels infrared singularities against real-emission contributions from four-parton final states. Numerical results for bottom-quark jets demonstrate significant mass effects on three-jet observables and improved perturbative stability at NLO, with explicit demonstrations of s_min independence. These developments enable precise QCD tests and potential extraction of heavy-quark masses from jet observables across center-of-mass energies.

Abstract

We present in detail a calculation of the next-to-leading order QCD corrections to the process jets with massive quarks. To isolate the soft and collinear divergencies of the four parton matrix elements, we modify the phase space slicing method to account for masses. Our computation allows for the prediction of oriented three jet events involving heavy quarks, both on and off the Z resonance, and of any event shape variable which is dominated by three jet configurations. We show next-to-leading order results for the three jet fraction, the differential two jet rate, and for the thrust distribution at various c.m. energies.

Paper Structure

This paper contains 8 sections, 78 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Outline of the calculation
  3. Kinematics and leading order results
  4. Virtual corrections
  5. The phase space slicing method for massive quarks
  6. Numerical results
  7. Summary
  8. Appendix

Figures (16)

  • Figure 1: Amplitude for the process $e^+(p_+)e^-(p_-) \rightarrow Z^\ast,\gamma^\ast\rightarrow Q(k_1) \bar{Q}(k_2)g(k_3)$
  • Figure 2: Loop diagrams.
  • Figure 3: Counterterm diagrams.
  • Figure 4: The three jet fraction $f_3^b$ at NLO as defined in the text at $\sqrt{s}=\mu=m_Z$ as a function of $y_{min}=s_{min}/(sy_{cut})$ for the JADE algorithm at a value of the jet resolution parameter $y_{cut}=0.03$ with $m^{\scriptsize{\hbox{${\overline{\hbox{MS}}}$}}}_b(\mu=m_Z)=3$ GeV and $\alpha_s(\mu=m_Z)=0.118$.
  • Figure 5: Same as Fig. \ref{['fig:sminja']}, but for the Durham algorithm.
  • ...and 11 more figures