Efficient Color-Dressed Calculation of Virtual Corrections

TL;DR

The paper addresses the challenge of including QCD color in NLO Monte Carlo calculations by extending color dressing to one-loop generalized unitarity. It develops a color-dressed LO recursion and extends it to color-dressed one-loop amplitudes, using partition-based unitarity cuts and residues that factorize into color-dressed tree amplitudes. It provides extensive numerical results for n-gluon scattering, demonstrating favorable exponential scaling and improved accuracy compared to color-ordered approaches, and analyzes Monte Carlo color-sampling convergence. The work suggests that color-dressed methods are particularly advantageous for high-multiplicity processes and can naturally accommodate quarks and electroweak particles, potentially enabling scalable, color-inclusive NLO event generation with parallelizable computation.

Abstract

With the advent of generalized unitarity and parametric integration techniques, the construction of a generic Next-to-Leading Order Monte Carlo becomes feasible. Such a generator will entail the treatment of QCD color in the amplitudes. We extend the concept of color dressing to one-loop amplitudes, resulting in the formulation of an explicit algorithmic solution for the calculation of arbitrary scattering processes at Next-to-Leading order. The resulting algorithm is of exponential complexity, that is the numerical evaluation time of the virtual corrections grows by a constant multiplicative factor as the number of external partons is increased. To study the properties of the method, we calculate the virtual corrections to $n$-gluon scattering.

Figures (23)

• Figure 1: A graphical representation of Eq. (\ref{['vertexblob']}) for $k=2$ and an off-shell gluon in QCD. Because of flavor conservation only one of the two vertices can contribute for any given partition.
• Figure 2: The first recursion step for the unordered gluon current with $u,\bar{d},s,\bar{s}$ quarks and a $W^-$ gauge boson in the final state. There are 15 contributions corresponding to all possible partitions of the final-state particles into two groups. Because of flavor conservation there are only 4 non-vanishing contributions for the "4+1" partitions (first term) and 2 non-vanishing contributions for the "3+2" partitions (second term).
• Figure 3: Top panel: comparison of Monte Carlo integrations for the various color-sampling schemes, including the standard deviation, to the exact color-summed result as a function of the number of evaluated phase-space points. One obtains $1.0034\pm0.0091$, $0.9989\pm0.0027$ and $0.9999\pm0.0022$ after $10^7$ steps for the "Naive", "Conserved" and "Non-Zero" sampling, respectively. Bottom panel: number of events required to reach a given relative accuracy on the numerical evaluation of the color-sampled amplitude. For the definition of $R_{\rm MC}(N_{\rm MC})$ and the values of the fit parameters determining the dashed curves, cf. the text.
• Figure 4: Top panel: comparison of Monte Carlo integrations for the various color-sampling schemes, including the standard deviation, to the exact color-summed result as a function of the number of evaluated phase-space points. One obtains $1.16\pm0.32$, $0.995\pm0.071$ and $0.913\pm0.037$ after $10^5$ steps for the "Naive", "Conserved" and "Non-Zero" sampling, respectively. Bottom panel: number of events required to reach a given relative accuracy on the numerical evaluation of the color-sampled amplitude. For the definition of $R_{\rm MC}(N_{\rm MC})$, cf. the text. The fit curves in terms of $\sigma/\mu(N_{\rm MC})$ are described by $14.0\,N_{\rm MC}^{-0.287}$, $2.84\,N_{\rm MC}^{-0.241}$ and $3.10\,N_{\rm MC}^{-0.331}$ for the "Naive", "Conserved" and "Non-Zero" sampling, respectively. The "Conserved" and "Non-Zero" approaches are slower by factors of $f=10.5$ and $f=13.3$, respectively (see text).
• Figure 5: Number of events required to reach a given relative accuracy on the numerical evaluation of the color-sampled amplitude. For the definition of $R_{\rm MC}(N_{\rm MC})$, cf. the text. The fit curves in terms of $\sigma/\mu(N_{\rm MC})$ are described by $8.04\,N_{\rm MC}^{-0.530}$, $3.25\,N_{\rm MC}^{-0.405}$ and $3.01\,N_{\rm MC}^{-0.344}$ for the "Naive", "Conserved" and "Non-Zero" sampling, respectively. The "Conserved" and "Non-Zero" approaches are slower by factors of $f=9.6$ and $f=10.8$, respectively (see text).