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Nf-contribution to the virtual correction for electroweak vector boson production at NNLO

Dario Kermanschah, Matilde Vicini

TL;DR

This work tackles the long-standing challenge of evaluating two-loop hadronic scattering amplitudes by integrating over loop momentum and phase space simultaneously. It combines local IR and UV counterterms with threshold subtractions, and analytically integrates loop energies using loop-tree duality and related representations to yield finite integrands in $d=4$ which are amenable to Monte Carlo evaluation. The authors compute the $N_f$-dependent part of the finite remainder of NNLO virtual corrections for processes producing up to three massive electroweak vector bosons in $pp$ collisions, providing new results and benchmarks. The approach demonstrates the feasibility and flexibility of a purely numerical framework for complex multi-loop, multi-leg, multi-scale problems, and points to a path toward full NNLO predictions including real-emission contributions.

Abstract

Multi-loop scattering amplitudes are difficult to evaluate due to singularities of the integrals involved, especially with increasing number of loops, external legs, and mass scales. For the first time for hadronic collisions at two loops, we enable the combined numerical integration over loop momentum and phase space by tackling infrared, ultraviolet and threshold singularities simultaneously using local subtractions. We demonstrate the feasibility of our approach by calculating previously unknown perturbative corrections for processes of interest to the Large Hadron Collider, namely the Nf-part of the finite remainder of the phase-space integrated virtual corrections at next-to-next-to-leading order (NNLO) in QCD for the production of up to three massive electroweak vector bosons in proton-proton collisions.

Nf-contribution to the virtual correction for electroweak vector boson production at NNLO

TL;DR

This work tackles the long-standing challenge of evaluating two-loop hadronic scattering amplitudes by integrating over loop momentum and phase space simultaneously. It combines local IR and UV counterterms with threshold subtractions, and analytically integrates loop energies using loop-tree duality and related representations to yield finite integrands in which are amenable to Monte Carlo evaluation. The authors compute the -dependent part of the finite remainder of NNLO virtual corrections for processes producing up to three massive electroweak vector bosons in collisions, providing new results and benchmarks. The approach demonstrates the feasibility and flexibility of a purely numerical framework for complex multi-loop, multi-leg, multi-scale problems, and points to a path toward full NNLO predictions including real-emission contributions.

Abstract

Multi-loop scattering amplitudes are difficult to evaluate due to singularities of the integrals involved, especially with increasing number of loops, external legs, and mass scales. For the first time for hadronic collisions at two loops, we enable the combined numerical integration over loop momentum and phase space by tackling infrared, ultraviolet and threshold singularities simultaneously using local subtractions. We demonstrate the feasibility of our approach by calculating previously unknown perturbative corrections for processes of interest to the Large Hadron Collider, namely the Nf-part of the finite remainder of the phase-space integrated virtual corrections at next-to-next-to-leading order (NNLO) in QCD for the production of up to three massive electroweak vector bosons in proton-proton collisions.
Paper Structure (24 sections, 100 equations, 6 figures, 10 tables)

This paper contains 24 sections, 100 equations, 6 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Framework
  4. Method
  5. Scattering kinematics and singular surfaces
  6. Negative invariants $\left(p=p_1-p_S\right)$
  7. $p=p_1$ or $p=p_2$
  8. Positive invariants $\left(p=p_S \right)$
  9. Construction of IR- and UV- finite amplitudes
  10. The one-loop amplitude
  11. The $N_f$-contribution to the two-loop amplitude
  12. Treatment of thresholds
  13. Integration of loop energies
  14. Subtraction of threshold singularities
  15. Implementation
  16. ...and 9 more sections

Figures (6)

  • Figure 1: The one-loop Feynman diagrams for $q(p_1)\bar{q}(p_2) \to \gamma(p_3) \gamma (p_4)$ for one fixed ordering in the final state photons. The momentum labels consistent with the local IR and UV counterterms reported in section \ref{['subsec:babis_ct']}.
  • Figure 2: One diagram contributing to the $N_f$-part of the two-loop amplitude for $q(p_1)\bar{q}(p_2) \to \gamma(p_3) \gamma (p_4)$. All other diagrams can be constructed from the one-loop amplitude by inserting a fermion bubble into the gluon propagator.
  • Figure 3: Diagrammatic representation of momentum conservation in one- and two-loop diagrams. On-shell momenta must satisfy the respective conditions in eqs. \ref{['eq:one-loop-threshold']} and \ref{['eq:two-loop-threshold']}, for the specified (or equivalently reversed) time direction.
  • Figure 4: Thresholds of the one-loop and the $N_f$-part of the two-loop partial amplitude for $q(p_1)\bar{q}(p_2)\to V(q_1)V(q_3)V(q_2)$, where the massive electroweak vector bosons $V$ are attached in the depicted order. With matching colours we highlight in (\ref{['subfig:cutkosky']}) the Cutkosky cuts, in (\ref{['subfig:ellipsoids']}) the singular surfaces for one particular configuration in phase space in the COM frame with example external boson masses $m_{\{1,2,3\}}=\hat{E}_\text{CM}/\{8,10,12\}$, and in eq:threshold_firsteq:threshold_last their respective definitions.
  • Figure 5: Convergence of the Monte Carlo integration for the contribution $F^{(2,N_f)}_\text{FFS}$ to the process $q\bar{q}\to\gamma^*\gamma^*\gamma^*$ at the first phase-space point in listing \ref{['lst:4']}.
  • ...and 1 more figures