Susy Theories and QCD: Numerical Approaches

TL;DR

The paper surveys numerical on-shell and unitarity methods for precision NLO QCD predictions at the LHC, emphasizing a universal, automate-friendly framework that constructs loop amplitudes from on-shell tree data. It details the decomposition of one-loop amplitudes into cut-containing and rational parts, and presents loop-integrand parametrizations (including the OPP approach) and D-dimensional extensions to capture rational terms. Through illustrative triple- and quadruple-cut examples and a thorough treatment of spinor-helicity, color decomposition, and supersymmetric relations, the work demonstrates robust, stable techniques capable of handling multi-jet final states like W/Z+jets. The methodology is scalable and adaptable to beyond-Standard-Model contexts, supporting high-precision collider phenomenology and informing improvements in subtraction, showering, and observable design.

Abstract

We review on-shell and unitarity methods and discuss their application to precision predictions for LHC physics. Being universal and numerically robust, these methods are straight-forward to automate for next-to-leading-order computations within Standard Model and beyond. Several state-of-the-art results including studies of W/Z+3-jet and W+4-jet production have explicitly demonstrated the effectiveness of the unitarity method for describing multi-parton scattering. Here we review central ideas needed to obtain efficient numerical implementations. This includes on-shell loop-level recursions, the unitarity method, color management and further refined tricks.

Figures (13)

• Figure 1: The measured $E_T$ distribution of the softest observed jet in inclusive $W\,\!+3$-jet production, compared to the NLO prediction PRLW3BHW3jDistributions. In the upper panels the NLO distribution is the solid (black) histogram, and CDF data points are the (red) points, whose inner and outer error bars, respectively, denote the statistical and total uncertainties (excluding the luminosity error) on the measurements. The LO predictions are shown as dashed (blue) lines. The lower panel shows the distribution normalized to the full NLO prediction. The scale-dependence bands are shaded (gray) for NLO and cross-hatched (brown) for LO. Reprinted fig. \ref{['W3TevatronFigure']} with permission from W3jDistributions p.27. Copyright (2009) by the American Physical Society.
• Figure 2: A comparison of the $p_T$ distributions of the softest observable jets in $W^-$+n-jet ($n=1,2,3,4$) production, respectively. The setup describes the LHC proton-proton collisions at $\sqrt{s}=7$ TeV as published in W4jets. In the upper panels the NLO distribution is the solid (black) histogram and the LO predictions are shown as dashed (blue) lines. The lower panels show the LO distribution and LO and NLO scale-dependence bands normalized to the central NLO prediction.
• Figure 3: Representative diagrams of matrix elements for $W\,\!+4$-jet production: (a) the eight-point loop amplitudes $qg\rightarrow e\bar{\nu} q' ggg$, and (b) the nine-point tree-level amplitudes $qg\rightarrow e\bar{\nu} q' gggg$ needed for the real contribution. The $e\bar{\nu}$ pair couples to the quarks via a W boson.
• Figure 4: Representative Feynman diagrams for primitive amplitudes with distinct states propagating in the loop: (a) shows a gluon loop of the primitive amplitude $A_4^{\rm g}$, (b) a fermion loop of $A_4^{\rm f}$ and (c) a mixed fermion/gluon loop of $A_4^{\rm LR}$. For the mixed amplitudes (c) we keep track of the routing of the fermion line around the loop; '$L\,R$' indicates that the first fermion lines turns left and the second right when entering the loop.
• Figure 5: Feynman diagram representation of basis of scalar integrals appearing in the eq. (\ref{['IntegralBasis']}): (a) box diagram associated to the 4-point integral, (b) triangle diagram associated to the 3-point integral, and (c) bubble diagram associated to the 2-point integral. Each corner may have one or more external lines attached to it.