Eliminating spurious poles from gauge-theoretic amplitudes (0905.1473v1)

Abstract: This note addresses the problem of spurious poles in gauge-theoretic scattering amplitudes. New twistor coordinates for the momenta are introduced, based on the concept of dual conformal invariance. The cancellation of spurious poles for a class of NMHV amplitudes is greatly simplified in these coordinates. The poles are eliminated altogether by defining a new type of twistor integral, dual to twistor diagrams as previously studied, and considerably simpler. The geometric features indicate a supersymmetric extension of the formalism at least to all NMHV amplitudes, allowing the dihedral symmetry of the super-amplitude to be made manifest. More generally, the definition of `momentum-twistor' coordinates suggests a powerful new approach to the study of scattering amplitudes.

• The paper introduces a momentum-twistor framework to systematically remove spurious poles in NMHV gauge amplitudes.
• It demonstrates that reparameterizing with momentum-twistors simplifies amplitude expressions and clarifies dual conformal symmetry.
• The work has practical implications for more efficient high-loop calculations and suggests extensions beyond the NMHV sector.

An Exploration of Spurious Pole Elimination in Gauge-Theoretic Amplitudes

The paper "Eliminating spurious poles from gauge-theoretic amplitudes" by Andrew Hodges addresses a complex problem within the field of gauge theory, specifically the representation of scattering amplitudes. The primary focus of the paper is the elimination of spurious poles, which are troubling singularities appearing in the intermediate steps of certain scattering amplitude calculations but are absent in the final physical quantities. These issues stem from the representations provided by the Britto-Cachazo-Feng-Witten (BCFW) recursion relations. Hodges introduces a novel framework utilizing twistor theory—specifically, momentum-twistors—to simplify and eventually eliminate these poles, thereby streamlining calculations and enhancing the symmetry manifestations in amplitudes.

Introduction to Spurious Poles and NMHV Amplitudes

The paper starts by addressing the persistent problem of spurious poles in non-maximally helicity-violating (NMHV) amplitudes in gauge theories. The BCFW recursion relations, despite their efficacy in simplifying amplitude computations, yield intermediate expressions that contain these unphysical singularities. Hodges uses the split-helicity six-field NMHV amplitude as a case paper, illustrating how different pivot choices in the recursion can result in distinct, complex expressions. These expressions include terms where poles appear to cancel only after extensive algebraic manipulations, suggesting an artificial complication in the computation process.

Momentum-Twistor Coordinates and Dual Conformal Symmetry

To tackle these complications, Hodges proposes the adoption of momentum-twistor coordinates, a reparametrization inspired by dual conformal symmetry concepts. This approach allows one to encode the entire kinematic setup of a scattering process within twistor space. The essence of this innovation lies in transforming momentum conservation constraints into geometric conditions on twistor variables, effectively absorbing them and simplifying the resulting expressions.

Momentum-twistors are leveraged to express various contributing terms of an amplitude, revealing that seemingly complex terms simplify under this coordinate system. The most critical revelation is how momentum-twistors expose and help eliminate spurious poles by redefining them as removable singularities within projective twistor integrals, linked intimately to the geometry of twistor spaces.

Integral Representations and Geometric Insights

The work introduces integrals over projective twistor space that manifestly display the cancellation of spurious poles. In essence, these integrals find a geometric interpretation where spurious poles correspond to spurious boundaries in twistor space. Hodges employs examples such as tetrahedral contours in projective twistor space to illustrate how these boundaries can be removed or redefined, thereby eliminating the unphysical poles.

The paper extends this analysis to NMHV amplitudes with varying numbers of external legs, such as seven fields and beyond, using novel polyhedral structures in twistor space. The integrals map the amplitude computation problem to a geometric problem of finding the correct polyhedral decomposition in twistor space, with dihedral symmetry emerging naturally from these multipartite divisions.

Implications and Theoretical Extensions

By presenting this innovative framework, Hodges not only improves amplitude calculations but also suggests a new perspective on the underlying symmetry structures in gauge theories. This work implicates both practical advancements, potentially simplifying high-loop calculations, and theoretical insights by providing a clearer manifestation of dual conformal symmetry.

Moreover, the paper indicates promising pathways for the extension of this approach beyond the NMHV sector, hinting at applicability in non-linear and loop-level scenarios. The momentum-twistor formalism offers a fertile ground for further inquiry into both supersymmetric extensions and beyond-the-standard-model QFTs, given its elegance and flexibility.

Conclusion

Overall, the paper marks a significant step forward in handling spurious poles in gauge-theoretic amplitudes, showcasing the utility of twistor theory in modern theoretical physics. Moving beyond conventional spinor approaches, Hodges' work emphasizes a geometric handling of complex algebraic problems, establishing a foundational technique that holds potential to influence future developments in the computation of scattering amplitudes. The integration of dual conformal symmetry with momentum-twistors presents an exciting avenue for further exploration, heralding new breakthroughs in our understanding of fundamental quantum field theories.