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Second Order QCD Corrections to $Γ(t \to Wb)$

K. G. Chetyrkin, R. Harlander, T. Seidensticker, M. Steinhauser

TL;DR

The paper presents a novel approach to compute O(α_s^2) QCD corrections to the top-quark decay t→Wb, including finite W-boson mass effects. By expanding the top propagator in z=q^2/M_t^2 and applying Padé approximants (with optional conformal mapping) to reach z=1, the authors obtain reliable results even when exact three-loop integrals are unavailable. They validate their method by reproducing known massless-W results and performing cross-checks with b→ulν and μ decays, finding small but non-negligible M_W^2/M_t^2 and M_W^4/M_t^4 contributions. The final numerical corrections suggest good perturbative stability and demonstrate the method’s potential for broader applications in heavy-quark phenomenology.

Abstract

Corrections of $O(α_s^2)$ to the decay of the top quark into a W boson and a bottom quark are calculated. The method is based on an expansion of the top quark propagator for small external momentum, q, as compared to the top quark mass, M_t. The physical point q^2 = M_t^2 is reached through Padé approximations. The described method allows to take effects induced by a finite W boson mass into account. The numerical relevance of the result is discussed. Important cross-checks against recent results for the decay rate $b \to u l \barν$ and the two-loop QED corrections to $μ$-decay are performed.

Second Order QCD Corrections to $Γ(t \to Wb)$

TL;DR

The paper presents a novel approach to compute O(α_s^2) QCD corrections to the top-quark decay t→Wb, including finite W-boson mass effects. By expanding the top propagator in z=q^2/M_t^2 and applying Padé approximants (with optional conformal mapping) to reach z=1, the authors obtain reliable results even when exact three-loop integrals are unavailable. They validate their method by reproducing known massless-W results and performing cross-checks with b→ulν and μ decays, finding small but non-negligible M_W^2/M_t^2 and M_W^4/M_t^4 contributions. The final numerical corrections suggest good perturbative stability and demonstrate the method’s potential for broader applications in heavy-quark phenomenology.

Abstract

Corrections of to the decay of the top quark into a W boson and a bottom quark are calculated. The method is based on an expansion of the top quark propagator for small external momentum, q, as compared to the top quark mass, M_t. The physical point q^2 = M_t^2 is reached through Padé approximations. The described method allows to take effects induced by a finite W boson mass into account. The numerical relevance of the result is discussed. Important cross-checks against recent results for the decay rate and the two-loop QED corrections to -decay are performed.

Paper Structure

This paper contains 12 sections, 35 equations, 1 figure, 13 tables.

Table of Contents

  1. Introduction
  2. Method of the calculation
  3. First order QCD corrections
  4. Second order QCD corrections
  5. General remarks
  6. The limit $M_W=0$
  7. Subleading terms in $M_W$
  8. Quadratic terms in $M_W$
  9. Quartic terms in $M_W$
  10. Results at ${\cal O}(\alpha_s^2)$
  11. Estimate for $b\to ul\bar{\nu}$ and $\mu \to e\nu_\mu\bar{\nu}_e$
  12. Conclusions

Figures (1)

  • Figure 1: Sample diagrams for the top quark self energy.