[Paper Review] The Coulomb-branch S-matrix from massless amplitudes

Kiermaier, Michael|arXiv (Cornell University)|May 26, 2011

Electromagnetic Scattering and Analysis61 references14 citations

TL;DR

This paper proposes a systematic method to compute tree-level S-matrices on the Coulomb branch of N=4 SYM using soft-scalar limits of massless on-shell amplitudes at the origin of moduli space. By carefully choosing soft-scalar momenta, it cancels soft and collinear divergences and constructs a massive CSW-like expansion that reproduces known all-n results efficiently, enabling exact computation of massive amplitudes from massless data.

ABSTRACT

We present a systematic method to extract the entire tree-level S-matrix on the Coulomb branch of N=4 SYM from soft-scalar limits of on-shell amplitudes at the origin of moduli space. Massive amplitudes in the spontaneously-broken theory can thus be computed from on-shell amplitudes in the massless, unbroken theory. To check this correspondence, we first prove that soft and collinear divergences in the required massless amplitudes cancel for a judicious choice of soft-scalar momenta. We then explicitly verify our proposal in examples with arbitrarily many external legs and to all orders in the mass. As a byproduct, the construction leads to a massive CSW-like expansion that reproduces several known all-n results for Coulomb-branch amplitudes in an effortless way. We briefly discuss the extension of our method to loop integrands.

Motivation & Objective

• To systematically extract tree-level S-matrices on the Coulomb branch of N=4 SYM from massless on-shell amplitudes at the origin of moduli space.
• To resolve the frame-dependence and momentum mismatch issues in naive soft-scalar limits by introducing a light-like reference vector q.
• To cancel soft and collinear divergences in soft-scalar amplitudes through a judicious choice of soft-scalar momenta directions.
• To construct a massive CSW-like expansion that efficiently reproduces known all-n results for Coulomb-branch amplitudes.
• To extend the method to loop integrands in a large-N limit.

Proposed method

• Uses soft-scalar limits of massless on-shell amplitudes at the origin of moduli space to generate Coulomb-branch amplitudes.
• Introduces a light-like reference vector q to define massive momentum decompositions and frame-dependent polarizations.
• Applies a vev-scalar symmetrization procedure to simplify amplitudes and handle color-ordered structures.
• Employs a momentum shift via βi = -m²i/(4q·pi) to map massless momenta to massive ones in the soft limit.
• Derives a CSW-like expansion by summing over soft scalar insertions with specific momentum scaling, leading to exact massive propagators.
• Proves cancellation of soft and collinear divergences in the sum over soft scalar insertions by reparameterizing the sum and using combinatorial identities.

Experimental results

Research questions

• RQ1 Can tree-level S-matrices on the Coulomb branch of N=4 SYM be systematically reconstructed from massless on-shell amplitudes at the origin of moduli space?
• RQ2 How can soft and collinear divergences in soft-scalar limits be canceled for arbitrary numbers of external legs and masses?
• RQ3 What is the precise mapping between massive momenta and massless momenta in the soft limit, and how is frame dependence encoded?
• RQ4 Can a CSW-like expansion be constructed for massive amplitudes that reproduces known all-n results efficiently?
• RQ5 Does the correspondence between massless amplitudes and Coulomb-branch amplitudes extend to loop integrands in a large-N limit?

Key findings

• Soft and collinear divergences in the required massless amplitudes cancel for a specific choice of soft-scalar momenta directions, enabling a well-defined limit.
• The proposed method successfully computes full n-point Coulomb-branch amplitudes from soft limits, valid to all orders in the mass.
• The construction yields a massive CSW-like expansion that reproduces known all-n results for Coulomb-branch amplitudes in a simple and efficient manner.
• The sum over soft scalar insertions maps massless propagators 1/(P⊥)² to massive ones 1/P² via a combinatorial reorganization of the sum.
• The derivation confirms that the sum over soft scalar insertions effectively replaces εri → -βi, leading to the correct massive momentum flow.
• The method provides a systematic framework to compute massive amplitudes from massless data, with potential extension to loop integrands in the large-N limit.