[Paper Review] Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization

Andrea Nützi, Michael Reiterer|arXiv (Cornell University)|Dec 16, 2018

Black Holes and Theoretical Physics16 references13 citations

TL;DR

This paper reinterprets Yang-Mills (YM) and General Relativity (GR) scattering amplitudes as $L_∞$ minimal model brackets derived from differential graded Lie algebras (dgLAs), providing a rigorous, gauge-independent construction via homological algebra. It establishes that tree-level amplitudes are uniquely determined by their factorization residues on prime divisors in momentum space, independent of Feynman diagrams or BCFW shifts, using Hartogs-type extension and homotopy transfer with optimal propagators.

ABSTRACT

Attached to both Yang-Mills and General Relativity about Minkowski spacetime are distinguished gauge independent objects known as the on-shell tree scattering amplitudes. We reinterpret and rigorously construct them as $L_\infty$ minimal model brackets. This is based on formulating YM and GR as differential graded Lie algebras. Their minimal model brackets are then given by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge independent when all internal lines are off-shell, not merely up to $L_\infty$ isomorphism, and we include a homological algebra proof of this fact. Using the homological perturbation lemma, we construct homotopies (propagators) that are optimal in bringing out the factorization of the residues of the amplitudes. Using a variant of Hartogs extension for singular varieties, we give a rigorous account of a recursive characterization of the amplitudes via their residues independent of their original definition in terms of Feynman graphs (this does neither involve so-called BCFW shifts nor conditions at infinity under such shifts). Roughly, the amplitude with $N$ legs is the unique section of a sheaf on a variety of $N$ complex momenta whose residues along a finite list of irreducible codimension one subvarieties (prime divisors) factor into amplitudes with less than $N$ legs. The sheaf is a direct sum of rank one sheaves labeled by helicity signs. To emphasize that amplitudes are robust objects, we give a succinct list of properties that suffice for a dgLa so as to produce the YM and GR amplitudes respectively.

Motivation & Objective

• To provide a rigorous, gauge-independent formulation of on-shell tree scattering amplitudes in YM and GR using $L_\infty$ minimal model brackets.
• To unify the treatment of YM and GR by formulating both as Maurer-Cartan equations in differential graded Lie algebras with momentum grading.
• To establish that amplitudes are uniquely determined by their factorization properties along codimension-one subvarieties (prime divisors) in complex momentum space.
• To eliminate reliance on Feynman diagrams or BCFW shifts by constructing a recursive characterization via Hartogs extension and homological perturbation.
• To prove that the sum of Feynman tree graphs is gauge-invariant even when individual graphs depend on gauge choices, using homotopy transfer and chain map arguments.

Proposed method

• Formalizing YM and GR as differential graded Lie algebras (dgLAs) with a momentum grading, where the Maurer-Cartan equation encodes the field equations.
• Constructing the $L_\infty$ minimal model brackets as sums of trivalent Feynman tree graphs, with vertices representing the dgLA bracket and edges representing homotopy contractions.
• Using the homological perturbation lemma to derive optimal propagators (homotopies) that reveal the factorization of amplitude residues.
• Applying a variant of Hartogs extension for singular varieties to recursively reconstruct $N$-point amplitudes from their residues on prime divisors corresponding to collinear or multiparticle singularities.
• Defining amplitudes as sections of a sheaf of rank-one sheaves over the variety of $N$ complex momenta, with helicity signs as labels.
• Proving gauge independence via chain map equivalence: the minimal model bracket is independent of the homotopy $h$ when all internal lines are off-shell, using $M_{h^\prime} - M_h = dE + Ed_{\text{tot}}$.

Experimental results

Research questions

• RQ1 How can on-shell tree scattering amplitudes in YM and GR be rigorously constructed as $L_\infty$ minimal model brackets in a gauge-independent way?
• RQ2 What is the geometric and homological characterization of these amplitudes independent of Feynman diagrams or BCFW recursion?
• RQ3 How do the residues of amplitudes along codimension-one subvarieties (e.g., collinear or multiparticle limits) factor into lower-point amplitudes?
• RQ4 What conditions on a dgLA ensure that its minimal model produces the correct YM or GR scattering amplitudes?
• RQ5 How can homotopy transfer and the homological perturbation lemma be used to derive optimal propagators that make factorization manifest?

Key findings

• The $L_\infty$ minimal model brackets of the dgLA formulation of YM and GR yield gauge-independent on-shell tree scattering amplitudes, even though individual Feynman tree graphs depend on gauge choices.
• Amplitudes are uniquely characterized as sections of a sheaf on the variety of $N$ complex momenta, with residues factoring into lower-point amplitudes along irreducible codimension-one subvarieties.
• The recursive reconstruction of $N$-point amplitudes is possible via Hartogs extension, without relying on BCFW shifts or boundary conditions at infinity.
• The homological perturbation lemma provides optimal propagators (homotopies) that make the factorization of residues manifest and reveal the underlying geometric structure.
• Gauge independence is rigorously proven via chain map equivalence: $M_{h^\prime} - M_h = dE + Ed_{\text{tot}}$ on off-shell internal lines, ensuring the sum of trees is gauge-invariant.
• The construction provides a succinct set of axioms for a dgLA to yield YM or GR amplitudes, based on momentum grading and the Maurer-Cartan equation.