Lie polynomials and a twistorial correspondence for amplitudes

Area of Science:

• Theoretical Physics
• Mathematical Physics

Background:

• The double copy relationship links gauge theories and gravity theories.
• Lie polynomials are algebraic structures with applications in various fields of mathematics and physics.

Purpose of the Study:

• To review Lie polynomials as a unifying mathematical framework for the double copy relationship.
• To establish a connection between Lie polynomials, moduli spaces, and twistorial geometry.
• To provide a new perspective on scattering amplitudes in gauge and gravity theories.

Main Methods:

• Reviewing Lie polynomials and their connection to the geometry and cohomology of moduli spaces.
• Introducing a twistorial correspondence between specific mathematical spaces.
• Applying the Penrose transform to develop formulae for scattering amplitudes.

Main Results:

• Lie polynomials naturally arise in the geometry of moduli spaces of points on the Riemann sphere.
• A twistorial correspondence is established, linking cotangent bundles to spaces of momentum invariants.
• The study provides a framework for Cachazo-He-Yuan (CHY) and ambitwistor-string formulae, relating them to the Penrose transform.
• A correspondence between CHY half-integrands and ABHY scattering forms is demonstrated.

Conclusions:

• Lie polynomials offer a fundamental mathematical structure underpinning the double copy relationship.
• The twistorial framework provides a natural geometric interpretation for scattering amplitudes in gauge and gravity theories.
• This work generalizes and offers a more invariant description of existing mathematical structures in the field.