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Electroweak two-loop Sudakov logarithms for on-shell fermions and bosons

W. Beenakker, A. Werthenbach

TL;DR

This work computes virtual electroweak Sudakov double logarithms at one and two loops for arbitrary on-shell/on-resonance SM particles using the temporal Coulomb gauge. The authors show that Sudakov corrections reside entirely in external-leg self-energies, derive an all-order Goldstone equivalence in this gauge, and establish that the SM behaves dynamically like an unbroken theory in the Sudakov limit. A key result is that two-loop Sudakov corrections exponentiate the one-loop terms, δZ^(2) = 1/2 (δZ^(1))^2, once all diagram topologies are included. The analysis highlights the importance of mass gaps and gauge cancellations in the SM and provides explicit one- and two-loop formulas for fermionic and bosonic external states across sectors, with implications for high-energy collider phenomenology and Bloch-Nordsieck considerations.

Abstract

We calculate the virtual electroweak Sudakov (double) logarithms at one- and two-loop level for arbitrary on-shell/on-resonance particles in the Standard Model. The associated Sudakov form factors apply in a universal way to arbitrary non-mass-suppressed electroweak processes at high energies, although this universality has to be interpreted with care. The actual calculation is performed in the temporal Coulomb gauge, where the relevant contributions from collinear-soft gauge-boson exchange are contained exclusively in the self-energies of the external on-shell/on-resonance particles. In view of the special status of the time-like components in this gauge, a careful analysis of the asymptotic states of the theory is required. From this analysis we derive an all-order version of the Goldstone-boson Equivalence Theorem without the need for finite compensation factors. By exploiting conditions obtained from non-renormalization requirements, which are a consequence of our choice of gauge, we show that the Sudakov corrections can be extracted through a combination of energy derivatives and projections by means of external sources. We observe that the Standard Model behaves dynamically like an unbroken theory in the Sudakov limit, in spite of the fact that the explicit particle masses are needed at the kinematical (phase-space) level while calculating the Sudakov form factors.

Electroweak two-loop Sudakov logarithms for on-shell fermions and bosons

TL;DR

This work computes virtual electroweak Sudakov double logarithms at one and two loops for arbitrary on-shell/on-resonance SM particles using the temporal Coulomb gauge. The authors show that Sudakov corrections reside entirely in external-leg self-energies, derive an all-order Goldstone equivalence in this gauge, and establish that the SM behaves dynamically like an unbroken theory in the Sudakov limit. A key result is that two-loop Sudakov corrections exponentiate the one-loop terms, δZ^(2) = 1/2 (δZ^(1))^2, once all diagram topologies are included. The analysis highlights the importance of mass gaps and gauge cancellations in the SM and provides explicit one- and two-loop formulas for fermionic and bosonic external states across sectors, with implications for high-energy collider phenomenology and Bloch-Nordsieck considerations.

Abstract

We calculate the virtual electroweak Sudakov (double) logarithms at one- and two-loop level for arbitrary on-shell/on-resonance particles in the Standard Model. The associated Sudakov form factors apply in a universal way to arbitrary non-mass-suppressed electroweak processes at high energies, although this universality has to be interpreted with care. The actual calculation is performed in the temporal Coulomb gauge, where the relevant contributions from collinear-soft gauge-boson exchange are contained exclusively in the self-energies of the external on-shell/on-resonance particles. In view of the special status of the time-like components in this gauge, a careful analysis of the asymptotic states of the theory is required. From this analysis we derive an all-order version of the Goldstone-boson Equivalence Theorem without the need for finite compensation factors. By exploiting conditions obtained from non-renormalization requirements, which are a consequence of our choice of gauge, we show that the Sudakov corrections can be extracted through a combination of energy derivatives and projections by means of external sources. We observe that the Standard Model behaves dynamically like an unbroken theory in the Sudakov limit, in spite of the fact that the explicit particle masses are needed at the kinematical (phase-space) level while calculating the Sudakov form factors.

Paper Structure

This paper contains 16 sections, 194 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Coulomb gauge
  3. The Coulomb gauge: asymptotic states and external wave-function factors in the Sudakov limit
  4. The charged-boson sector
  5. The neutral-boson sector
  6. Electroweak one-loop Sudakov logarithms
  7. The fermionic self-energy at one-loop level
  8. The bosonic self-energies at one-loop level
  9. General one-loop Sudakov logarithms
  10. Electroweak two-loop Sudakov logarithms
  11. The fermionic self-energy at two-loop level
  12. The bosonic self-energies at two-loop level
  13. General two-loop Sudakov logarithms
  14. Conclusions
  15. One- and two-loop integrals
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Diagrams that can give rise to Sudakov logarithms in the one-loop gauge-boson self-energy
  • Figure :