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Automated Evaluation of One-Loop Six-Point Processes for the LHC

Thomas Reiter

TL;DR

The work develops a comprehensive, automated framework for computing one-loop QCD amplitudes with many external legs, addressing both colour and spinor structures. It fuses colour-flow decomposition, irreducible representations, and spinor helicity methods with dimensional regularisation to produce numerically stable tensor and scalar loop reductions. Central innovations include recursive recoupling formalisms (Sun representations, Garnir relations, LR rule), a spinor-helicity projection suite, and tensor reduction by subtraction that isolates IR/UV sectors and avoids Gram-determinant instabilities. The approach is demonstrated in the broader context of LHC-relevant processes, enabling automated generation of Fortran90 code for massless QCD amplitudes and paving the way for scalable NLO predictions for complex multi-jet final states.

Abstract

In the very near future the first data from LHC will be available. The searches for the Higgs boson and for new physics will require precise predictions both for the signal and the background processes. Tree level calculations typically suffer from large renormalization scale uncertainties. I present an efficient implementation of an algorithm for the automated, Feynman diagram based calculation of one-loop corrections to processes with many external particles. This algorithm has been successfully applied to compute the virtual corrections of the process $u\bar{u}\to b\bar{b}b\bar{b}$ in massless QCD and can easily be adapted for other processes which are required for the LHC.

Automated Evaluation of One-Loop Six-Point Processes for the LHC

TL;DR

The work develops a comprehensive, automated framework for computing one-loop QCD amplitudes with many external legs, addressing both colour and spinor structures. It fuses colour-flow decomposition, irreducible representations, and spinor helicity methods with dimensional regularisation to produce numerically stable tensor and scalar loop reductions. Central innovations include recursive recoupling formalisms (Sun representations, Garnir relations, LR rule), a spinor-helicity projection suite, and tensor reduction by subtraction that isolates IR/UV sectors and avoids Gram-determinant instabilities. The approach is demonstrated in the broader context of LHC-relevant processes, enabling automated generation of Fortran90 code for massless QCD amplitudes and paving the way for scalable NLO predictions for complex multi-jet final states.

Abstract

In the very near future the first data from LHC will be available. The searches for the Higgs boson and for new physics will require precise predictions both for the signal and the background processes. Tree level calculations typically suffer from large renormalization scale uncertainties. I present an efficient implementation of an algorithm for the automated, Feynman diagram based calculation of one-loop corrections to processes with many external particles. This algorithm has been successfully applied to compute the virtual corrections of the process in massless QCD and can easily be adapted for other processes which are required for the LHC.

Paper Structure

This paper contains 138 sections, 9 theorems, 610 equations, 20 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Principles of qcd
  2. The Lagrangian Density of qcd
  3. The Effective Coupling and Asymptotic Freedom
  4. Flavour Symmetry
  5. Representations of qcd Amplitudes
  6. Colour-Flow Decomposition
  7. sun Diagrammatics
  8. Colour Decomposition
  9. Other Colour Bases
  10. Recoupling Relations
  11. Irreducible Representations of sun
  12. Diagrammatics for the Symmetric Group
  13. Garnir Relations and Young's Natural Representation
  14. Construction of the Representation Matrices
  15. gln-Dimensions
  16. ...and 123 more sections

Key Result

Theorem 1

There is only one independent, non-vanishing choice of the permutation that connects the symmetrisers of a projector $P_Y$ on their right to the antisymmetrisers on their left. Rearranging any non-vanishing choice of the internal lines of the projector to any other non-vanishing choice of that permu

Figures (20)

  • Figure 1: Results from the analysis of electro-weak precision data as of March 2008 Barate:2003sz:2005emashowpull. Left: global fit of 18 sm observables. Right: the current best fit for the mass of a sm Higgs boson. The yellow area is experimentally excluded. Combined analysis of precision measurements predict a light sm Higgs boson.
  • Figure 2: Sensitivity of the ATLAS experiment to the discovery of a sm Higgs boson for an intermediate mass range for integrated luminosities of 30 (left) and 100. The plots show $S/\sqrt{B}$ where $S$ is the number of signal events and $B$ is the number of background events. ATLAS:1999fr
  • Figure 3: $5\sigma$ discovery contours in the ($M_{A^0}$, $\tan\beta$) parameter plane for the channels $gg\rightarrow b\bar{b}H\rightarrow b\bar{b}b\bar{b}$ ($H\in\{h^0,H^0,A^0\}$) and $gg\rightarrow H^0\rightarrow (h^0h^0)/(A^0A^0)\rightarrow b\bar{b}b\bar{b}$ for an integrated luminosity of 30 for ATLAS and cms individually, combining their statistic. Dai:1996rn
  • Figure 4: Schematic picture of the amplitude $q\bar{q}\rightarrow q^\prime\bar{q}^\prime$.
  • Figure 5: Definition of the momenta at an arbitrary $N$-point integral
  • ...and 15 more figures

Theorems & Definitions (14)

  • Definition 1: Young Tableau
  • Definition 2: (Semi)-Standard Tableau
  • Definition 3: Young Projector
  • Theorem 1: Uniqueness of $P_Y$
  • Theorem 2: Properties of $P_\lambda$
  • Theorem 3: Garnir Relations
  • Definition 4: Skew Tableau
  • Definition 5: Lattice word
  • Theorem 4: lr rule
  • Theorem 5
  • ...and 4 more