[Paper Review] Generalised unitarity for dimensionally regulated amplitudes within FDF

William J. Torres Bobadilla|arXiv (Cornell University)|Jan 21, 2016

Black Holes and Theoretical Physics21 references3 citations

TL;DR

This paper presents a four-dimensional formulation (FDF) within dimensional regularization to compute one-loop helicity amplitudes and study Colour-Kinematics duality in off-shell diagrams. Using generalized unitarity and FDF's explicit four-dimensional spinors and polarizations, it derives analytic expressions for Higgs plus five-gluon amplitudes and shows that C/K duality is violated off-shell due to subgraph contributions, restoring only on-shell. The method enables precise NLO computations in QCD.

ABSTRACT

We review the Four-Dimensional-Formulation variant of the Four-Dimensional-Helicity scheme, by showing two applications of this regularisation scheme. The first one is the computation of one-loop helicity amplitudes, for which we present preliminary results for the analytic expressions of the one-loop Higgs plus five- gluon amplitudes. In the second part, we study the Colour-Kinematics duality for off-shell diagrams in gauge theories coupled to matter, showing in a diagrammatic way that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to definite set of violating terms due to the contributions of sub-graphs only.

Motivation & Objective

• To extend the Four-Dimensional-Helicity (FDH) scheme via the Four-Dimensional-Formulation (FDF) for precise one-loop amplitude computations in dimensional regularization.
• To compute analytic one-loop amplitudes for Higgs plus five-gluon production using generalized unitarity and FDF.
• To investigate the validity of Colour-Kinematics duality in off-shell diagrams constructed with axial-gauge Feynman rules.
• To identify the origin of C/K duality violation in off-shell amplitudes and show its restoration on-shell.
• To establish a diagrammatic framework for C/K duality in higher-point and multi-loop amplitudes using FDF.

Proposed method

• FDF implements dimensional regularization by splitting momenta and gamma matrices into four-dimensional and $-2ar{ heta}$-dimensional parts using projection tensors $g^{ ueta}$ and $ ilde{g}^{ ueta}$.
• Fermion spinors in FDF satisfy modified Dirac equations: $u(ar{ u})\bar{u}(\bar{\nu}) = \not{\ell} + i\mu\gamma^5 + m$, ensuring correct propagator numerators.
• Polarization vectors for $d$-dimensional gluons are decomposed into four-dimensional transverse and $-2\epsilon$-dimensional components, with the latter contributing to massive vector propagators.
• Generalized unitarity is applied by cutting the integrand and reconstructing coefficients via on-shell recursion, using FDF's explicit four-dimensional helicity states.
• The Jacobi identity for kinematic numerators is analyzed diagrammatically in axial gauge, with off-shellness introducing anomalous terms from subgraph contributions.
• The decomposition of kinematic numerators into terms proportional to $p_i^2$ and $p_i^2 p_j^2$ (Eq. 23) reveals the explicit structure of C/K-violating terms.

Experimental results

Research questions

• RQ1 Can the FDF scheme be used to compute one-loop amplitudes for Higgs plus five-gluon production with full analyticity?
• RQ2 How does the Colour-Kinematics duality behave in off-shell diagrams when using axial-gauge Feynman rules in dimensional regularization?
• RQ3 What is the origin of C/K duality violation in off-shell amplitudes, and how is it related to subgraph structures?
• RQ4 Does the C/K duality hold in multi-loop amplitudes when particles are off-shell, and if not, under what conditions is it restored?
• RQ5 Can the generalized unitarity method be effectively combined with FDF to compute higher-point amplitudes in QCD?

Key findings

• The FDF scheme successfully reproduces known one-loop amplitudes such as $gg\to gg$, $q\bar{q}\to gg$, $gg\to Hg$, $gg\to ggg$, and $gg\to gggg$.
• Analytic expressions for the one-loop Higgs plus five-gluon amplitude are derived using generalized unitarity and FDF, marking a significant step toward full Higgs + 3-jet computations.
• C/K duality is violated in off-shell diagrams due to contributions from subgraphs involving internal propagators, which introduce anomalous terms.
• The C/K-violating terms are fully identified as arising from the decomposition of kinematic numerators into terms proportional to $p_i^2$ and $p_i^2 p_j^2$, as shown in Eq. (23).
• The C/K duality is restored in the on-shell limit, confirming consistency with known on-shell duality structures.
• The diagrammatic analysis confirms that C/K duality at multi-loop level requires on-shell conditions for all four particles in the Jacobi identity, with violations localized in subgraphs.