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The fermion-loop scheme for finite-width effects in e^+ e^- annihilation into four fermions

W. Beenakker, A. Denner, S. Dittmaier, J. Hoogland, R. Kleiss, G. J. van Oldenborgh, C. G. Papadopoulos, G. Passarino

TL;DR

The paper develops a gauge-invariant framework (fermion-loop scheme) to incorporate finite-width effects of W and Z bosons in e^+e^- → 4f processes by resumming fermionic 1PI corrections and renormalizing with complex pole masses and running couplings. It provides explicit expressions for fermionic self-energies and triple gauge-boson vertices, constructs renormalized amplitudes for LEP1/LEP2, and verifies Ward identities to ensure gauge invariance. A simple effective Born prescription is introduced to implement the scheme in practice, and numerical comparisons with other gauge-invariance-preserving schemes are shown across LEP2, NLC, and beyond. The results highlight percent-level differences from non-FL schemes at LEP2 and higher energies, underscoring the importance of preserving SU(2) gauge invariance and laying groundwork for including bosonic corrections in future work.

Abstract

We describe the gauge-invariant treatment of the finite-width effects of W and Z bosons in the fermion-loop scheme and its application to the six-fermion (LEP2) processes e^- e^+ -> four fermions, with massless external fermions. The fermion-loop scheme consists in including all fermionic one-loop corrections in tree-level amplitudes and resumming the self-energies. We give explicit results for the unrenormalized fermionic one-loop contributions to the gauge-boson self-energies and the triple gauge-boson vertices, and perform the renormalization in a gauge-invariant way by introducing complex pole positions and running couplings. A simple effective Born prescription is presented, which allows for a relatively straightforward implementation of the fermion-loop scheme in LEP1 and LEP2 processes. We apply this prescription to typical LEP2 processes, i.e., e^- e^+ -> μ^- \barν_μu \bar{d}, e^- e^+ -> s \bar{c} u \bar{d}, and e^- e^+ -> e^- \barν_e u \bar{d}, and give numerical comparisons with other gauge-invariance-preserving schemes in the energy range of LEP2, NLC and beyond.

The fermion-loop scheme for finite-width effects in e^+ e^- annihilation into four fermions

TL;DR

The paper develops a gauge-invariant framework (fermion-loop scheme) to incorporate finite-width effects of W and Z bosons in e^+e^- → 4f processes by resumming fermionic 1PI corrections and renormalizing with complex pole masses and running couplings. It provides explicit expressions for fermionic self-energies and triple gauge-boson vertices, constructs renormalized amplitudes for LEP1/LEP2, and verifies Ward identities to ensure gauge invariance. A simple effective Born prescription is introduced to implement the scheme in practice, and numerical comparisons with other gauge-invariance-preserving schemes are shown across LEP2, NLC, and beyond. The results highlight percent-level differences from non-FL schemes at LEP2 and higher energies, underscoring the importance of preserving SU(2) gauge invariance and laying groundwork for including bosonic corrections in future work.

Abstract

We describe the gauge-invariant treatment of the finite-width effects of W and Z bosons in the fermion-loop scheme and its application to the six-fermion (LEP2) processes e^- e^+ -> four fermions, with massless external fermions. The fermion-loop scheme consists in including all fermionic one-loop corrections in tree-level amplitudes and resumming the self-energies. We give explicit results for the unrenormalized fermionic one-loop contributions to the gauge-boson self-energies and the triple gauge-boson vertices, and perform the renormalization in a gauge-invariant way by introducing complex pole positions and running couplings. A simple effective Born prescription is presented, which allows for a relatively straightforward implementation of the fermion-loop scheme in LEP1 and LEP2 processes. We apply this prescription to typical LEP2 processes, i.e., e^- e^+ -> μ^- \barν_μu \bar{d}, e^- e^+ -> s \bar{c} u \bar{d}, and e^- e^+ -> e^- \barν_e u \bar{d}, and give numerical comparisons with other gauge-invariance-preserving schemes in the energy range of LEP2, NLC and beyond.

Paper Structure

This paper contains 23 sections, 100 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Renormalization
  3. Resumming the propagators
  4. Definition of renormalized parameters
  5. Renormalized propagators and fermion--gauge-boson vertex functions
  6. The process $e^+e^-\rightarrow f\bar{f}$
  7. The triple gauge-boson vertex
  8. The unrenormalized triple gauge-boson vertex
  9. Ward identities for the triple gauge-boson vertex
  10. Basic matrix elements for LEP2 processes
  11. Gauge invariance of renormalized LEP2 amplitudes
  12. Application of the fermion-loop scheme to typical LEP2 processes
  13. Amplitudes
  14. Input parameters
  15. Schemes
  16. ...and 8 more sections

Figures (3)

  • Figure 1: The effect of gauge-breaking terms in the CC20 process at $\sqrt{s}=175\;\mathrm{GeV}$ as a function of the virtuality $q_\gamma^2$ of the photon without ISR and without cuts, as predicted by WWF.
  • Figure 2: Total cross-sections for the CC10 process (default setup), as predicted by WTO, and deviations of the various schemes with respect to the fixed-width scheme in percent.
  • Figure 3: Feynman diagrams for the fermion-loop corrections to the $BW^+W^-$ vertex ($B=\gamma,Z$). All particles are assumed to be incoming.