[Paper Review] Recent developments for multi-leg QCD amplitudes with massive particles

Rutger H. Boels, Christian Schwinn+1 more|ArXiv.org|Dec 20, 2007

Particle physics theoretical and experimental studies21 references4 citations

TL;DR

This paper extends modern on-shell methods—supersymmetric Ward identities, on-shell recursion relations (BCFW), and CSW rules—for calculating multi-leg QCD amplitudes with massive quarks and scalars. It derives exact relations between massive quark and scalar amplitudes via SUSY, generalizes BCFW recursion to massive particles using appropriate spinor shifts, and constructs CSW vertices for massive scalars via canonical transformations, enabling efficient tree-level amplitude computations in massive QCD and beyond.

ABSTRACT

We review the extension of modern techniques for the calculation of helicity amplitudes in QCD to massive particles. The focus is on the use of supersymmetric Ward identites that relate amplitudes with massive quarks to those with massive scalars, the application of on-shell recursion relations to amplitudes with massive quarks and an extension of the CSW rules to massive scalars.

Motivation & Objective

• To extend modern on-shell techniques like BCFW recursion and CSW rules to multi-leg QCD amplitudes involving massive quarks and scalars.
• To establish supersymmetric Ward identities (SWIs) that relate amplitudes of massive quarks to those of massive scalars in unbroken SUSY Yang-Mills theory.
• To generalize the BCFW recursion relation to massive particles by defining appropriate spinor shifts that avoid spurious poles and ensure convergence at infinity.
• To extend the CSW diagrammatic approach to include massive scalars by using canonical transformations to eliminate non-MHV vertices and generate a tower of MHV vertices.
• To provide a systematic framework for computing tree-level amplitudes with massive particles, paving the way for loop-level and electroweak extensions.

Proposed method

• Uses supersymmetric Ward identities (SWIs) in unbroken SUSY Yang-Mills theory to relate amplitudes of massive quarks to those of massive scalars via SUSY transformations of external states.
• Applies a modified BCFW shift to external massive quark states by decomposing massive momenta into light-like vectors and shifting spinors with reference spinors from the opposite momentum, ensuring no spurious poles.
• Employs a canonical transformation method on the light-cone gauge Lagrangian to eliminate non-MHV vertices (e.g., $\mathcal{L}^{(3)}_{\bar{\phi}A_z\phi}$) and generate a tower of MHV vertices for massive scalars.
• Derives new CSW vertices for massive scalars, including a specific form $V_{\text{CSW}}(\bar{\xi}_1, g^+_2, \dots, \xi_n) = \mathrm{i}2^{n/2-1} \frac{-m^2 \langle{1n}\rangle}{\langle{12}\rangle \dots \langle{n1}\rangle}$, which accounts for the mass term.
• Uses the resulting CSW vertices to prove BCFW recursion relations for massive scalar amplitudes without auxiliary shifts, directly extending the diagrammatic argument from massless cases.
• Validates the approach by showing that the $z \to \infty$ behavior of amplitudes vanishes under allowed shifts, such as $(g_i^+, g_j^+)$, $(g_i^+, Q_j^+)$, and $(Q_i^-, Q_j^-)$, ensuring recursion stability.

Experimental results

Research questions

• RQ1 How can supersymmetric Ward identities be used to relate amplitudes of massive quarks to those of massive scalars in QCD?
• RQ2 What modifications are required to apply the BCFW recursion relation to amplitudes with massive quarks, and which helicity configurations are allowed?
• RQ3 Can the CSW rules be extended to include massive scalars, and how are the resulting MHV vertices derived?
• RQ4 What is the role of canonical transformations in eliminating non-MHV vertices and generating a consistent tower of MHV vertices for massive scalars?
• RQ5 How can the $z \to \infty$ behavior of shifted amplitudes be controlled to ensure the validity of on-shell recursion for massive particles?

Key findings

• The paper derives a direct SUSY Ward identity relating massive quark amplitudes to scalar amplitudes: $\langle{1q}\rangle A(\bar{Q}_1^+, g_2^+, \dots, Q_n^-) = \langle{nq}\rangle A(\bar{\phi}_1^+, g_2^+, \dots, \phi_n^-)$, enabling computation of quark amplitudes from scalar ones.
• For massive quarks, the BCFW shift is generalized by decomposing massive momenta into light-like vectors and using reference spinors from the opposite momentum, ensuring no spurious poles and valid recursion.
• The $z \to \infty$ limit of amplitudes vanishes for allowed helicity shifts, including $(g_i^+, g_j^+)$, $(g_i^+, Q_j^+)$, and $(Q_i^-, Q_j^-)$, confirming the applicability of BCFW recursion to massive particles.
• A new CSW vertex for massive scalars is derived as $V_{\text{CSW}}(\bar{\xi}_1, g^+_2, \dots, \xi_n) = \mathrm{i}2^{n/2-1} \frac{-m^2 \langle{1n}\rangle}{\langle{12}\rangle \dots \langle{n1}\rangle}$, which arises from the mass term under canonical transformation.
• The CSW diagrammatic approach is extended to massive scalars by constructing a twistor action and using canonical transformations to generate a complete set of MHV vertices, including the new mass-dependent term.
• The framework allows direct proof of BCFW recursion for massive scalar amplitudes without auxiliary shifts, demonstrating the consistency and power of the extended on-shell methods.