[Paper Review] MHV Lagrangians for Yang--Mills and QCD

James H. Ettle|ArXiv.org|Aug 14, 2008

Particle physics theoretical and experimental studies115 references3 citations

TL;DR

This paper constructs a canonical field transformation that yields a Yang–Mills Lagrangian with vertices proportional to Parke–Taylor MHV amplitudes, enabling algebraic reconstruction of off-shell light-cone Yang–Mills amplitudes. It introduces completion vertices to evade the S-matrix equivalence theorem, allowing computation of previously inaccessible tree-level and one-loop amplitudes, including the $(+++ +)$ one-loop amplitude, and extends the formalism to massless QCD with fundamental fermions using dimensional regularization.

ABSTRACT

Over the past few decades, it has been realised that gauge theory scattering amplitudes have structures much simpler than the traditional Feynman graph driven approach would suggest. In particular, Parke and Taylor found a particularly simple expression for the tree-level amplitudes with two gluons of different helicity than the others (the so-called MHV amplitudes). Cachazo, Svrcek and Witten (CSW) devised rules for constructing tree-level amplitudes by sewing lower-valence MHV amplitudes together with scalar propagators. It was shown by Mansfield in 2005 that a canonical change of the field variables could be constructed that resulted in a lagrangian whose vertices were proportional to MHV amplitudes, continued off-shell by CSW's prescription, the so-called Canonical MHV Lagrangian. We derive the explicit form of this transformation and use this to show that the vertices are indeed the Parke--Taylor amplitudes for up to five gluons. Noting that CSW's MHV rules cannot be used to construct the tree-level (-++) or one-loop (++++) amplitudes, we extend our work to augment the MHV rules with so-called completion vertices. These permit construction of these missing amplitudes by means of evasion of the S-matrix equivalence theorem. Indeed, together they reconstruct off-shell light-cone Yang-Mills amplitudes algebraically. We also give a prescription for dimensional regularisation of the Canonical MHV Lagrangian. Finally, we construct a canonical MHV lagrangian with massless fermions in the fundamental representation using a similar methodology.

Motivation & Objective

• To derive the explicit form of the canonical field transformation that yields a Lagrangian with MHV vertices matching Parke–Taylor amplitudes for up to five gluons.
• To resolve the limitations of CSW rules by introducing completion vertices that evade the S-matrix equivalence theorem, enabling computation of missing amplitudes such as the tree-level $(-+++)\,$ and one-loop $(++++)$ amplitudes.
• To extend the MHV Lagrangian formalism to massless QCD with fundamental fermions using a similar canonical transformation approach.
• To provide a prescription for dimensional regularization of the canonical MHV Lagrangian to handle ultraviolet and infrared divergences in loop calculations.
• To demonstrate that the MHV Lagrangian can reconstruct off-shell amplitudes algebraically, offering a new paradigm for perturbative gauge theory computations.

Proposed method

• Derives the explicit form of the canonical field transformation that maps the standard Yang–Mills Lagrangian to one with MHV vertices, using a series expansion in terms of $\mathcal{A}$ and $\bar{\mathcal{A}}$ fields.
• Uses the CSW prescription to continue MHV vertices off-shell, ensuring they match known Parke–Taylor amplitudes for up to five external gluons.
• Introduces completion vertices—additional interaction terms—that restore gauge invariance and allow construction of amplitudes otherwise forbidden by the S-matrix equivalence theorem.
• Applies the field transformation to the counterterm in a four-dimensional light-cone regulator to recover the one-loop $(++++)$ amplitude, validating the method beyond tree level.
• Constructs a $D$-dimensional version of the canonical MHV Lagrangian to enable dimensional regularization, preserving the MHV structure in $4-2\epsilon$ dimensions.
• Extends the formalism to massless QCD by including fundamental fermions via a similar canonical transformation, deriving MHV vertices for quark-gluon and quark-antiquark processes.

Experimental results

Research questions

• RQ1 How can a canonical field transformation be explicitly derived to yield a Yang–Mills Lagrangian with vertices matching Parke–Taylor MHV amplitudes?
• RQ2 Why do CSW rules fail to compute certain amplitudes like the tree-level $(-+++)$ and one-loop $(++++)$ amplitudes, and how can this be resolved?
• RQ3 Can completion vertices be systematically introduced to evade the S-matrix equivalence theorem and reconstruct missing amplitudes algebraically?
• RQ4 How can the canonical MHV Lagrangian be consistently regularized in dimensional regularization for loop calculations?
• RQ5 Can the MHV Lagrangian formalism be extended to include massless fermions in the fundamental representation of QCD?

Key findings

• The canonical field transformation successfully reproduces Parke–Taylor MHV amplitudes for up to five gluons, confirming the correctness of the MHV Lagrangian at tree level.
• Completion vertices enable the algebraic reconstruction of the tree-level $(-+++)$ amplitude, which is otherwise forbidden by the S-matrix equivalence theorem.
• The one-loop $(++++)$ amplitude is recovered by applying the field transformation to the counterterm in a four-dimensional light-cone regulator, demonstrating the method's viability at loop level.
• The $D$-dimensional MHV Lagrangian is constructed with $4-2\epsilon$-dimensional vertices, providing a consistent framework for dimensional regularization of the canonical MHV action.
• The MHV Lagrangian for massless QCD with fundamental fermions is derived, yielding explicit vertices for processes such as two quarks and two gluons, four quarks, and two quarks with three gluons.
• The formalism demonstrates that completion vertices are necessary for non-vanishing amplitudes when standard CSW rules fail, and that these vertices are essential for reconstructing amplitudes with non-vanishing cuts at one loop.