[Paper Review] Celestial Operator Product Expansions and ${ m w}_{1+\infty}$ Symmetry for All Spins

Elizabeth Himwich, Monica Pate+1 more|arXiv (Cornell University)|Aug 17, 2021

Black Holes and Theoretical Physics57 references4 citations

TL;DR

This paper establishes that the operator product expansion (OPE) coefficients for massless celestial primary operators of arbitrary spin in four-dimensional flat spacetime are fully determined by Poincaré symmetry, yielding Euler beta functions dependent on conformal weights. These coefficients precisely match those from momentum-space collinear limits and close under the infinite-dimensional ${\rm w}_{1+\infty}$ algebra, demonstrating that celestial OPEs realize ${\rm w}_{1+\infty}$ symmetry for all spins, including fermions and higher-derivative couplings.

ABSTRACT

The operator product expansion of massless celestial primary operators of arbitrary spin is investigated. Poincaré symmetry is found to imply a set of recursion relations on the operator product expansion coefficients of the leading singular terms at tree-level in a holomorphic limit. The symmetry constraints are solved by an Euler beta function with arguments that depend simply on the right-moving conformal weights of the operators in the product. These symmetry-derived coefficients are shown not only to match precisely those arising from momentum-space tree-level collinear limits, but also to obey an infinite number of additional symmetry transformations that respect the algebra of ${ m w}_{1+\infty}$. In tree-level minimally-coupled gravitational theories, celestial currents are constructed from light transforms of conformally soft gravitons and found to generate the action of ${ m w}_{1+\infty}$ on arbitrary massless celestial primaries. Results include operator product expansion coefficients for fermions as well as those arising from higher-derivative non-minimal couplings of gluons and gravitons.

Motivation & Objective

• To derive the operator product expansion (OPE) coefficients of massless celestial primary operators of arbitrary spin from Poincaré symmetry alone.
• To show that these symmetry-derived OPE coefficients match exactly those from momentum-space collinear limits in tree-level amplitudes.
• To demonstrate that the OPE coefficients close under the ${\rm w}_{1+\infty}$ algebra, extending the symmetry beyond known soft theorems.
• To construct celestial currents from light transforms of conformally soft gravitons that generate ${\rm w}_{1+\infty}$ symmetry on arbitrary massless primaries.
• To extend the framework to include fermions and higher-derivative couplings in gauge and gravitational theories.

Proposed method

• Derive recursion relations for OPE coefficients from Poincaré symmetry in the holomorphic limit at tree level.
• Solve the recursion relations using an Euler beta function whose arguments depend on the right-moving conformal weights of the operators.
• Match the symmetry-derived OPE coefficients to those obtained from momentum-space collinear limits of tree-level amplitudes.
• Construct celestial currents via light transforms of conformally soft gravitons in minimally-coupled gravitational theories.
• Show that these currents generate the ${\rm w}_{1+\infty}$ algebra by verifying the commutator relations on arbitrary massless celestial primaries.
• Prove the closure of the ${\rm w}_{1+\infty}$ algebra on the OPE coefficients by induction over spin and mode number, using recursive structure and commutator identities.

Experimental results

Research questions

• RQ1 Can Poincaré symmetry alone determine the OPE coefficients of celestial primary operators of arbitrary spin in four-dimensional flat spacetime?
• RQ2 Do the OPE coefficients derived from Poincaré symmetry precisely match those from momentum-space collinear limits in tree-level amplitudes?
• RQ3 Is the algebraic structure of the OPE coefficients closed under the ${\rm w}_{1+\infty}$ symmetry algebra?
• RQ4 Can celestial currents be constructed from light transforms of conformally soft gravitons that generate ${\rm w}_{1+\infty}$ symmetry on arbitrary massless primaries?
• RQ5 Do the OPE coefficients for fermions and higher-derivative couplings (e.g., non-minimal gluon/graviton couplings) also close under ${\rm w}_{1+\infty}$?

Key findings

• The OPE coefficients for massless celestial operators of arbitrary spin are fully determined by Poincaré symmetry and take the form of an Euler beta function depending on the right-moving conformal weights of the operators.
• These symmetry-derived OPE coefficients exactly match the coefficients obtained from tree-level collinear limits in momentum space, confirming universality and consistency across frameworks.
• The OPE coefficients close under the ${\rm w}_{1+\infty}$ algebra, meaning they satisfy the full set of commutator relations defining the algebra.
• Celestial currents constructed from light transforms of conformally soft gravitons generate the action of ${\rm w}_{1+\infty}$ on arbitrary massless celestial primaries, extending the symmetry beyond soft theorems.
• The framework includes fermions and higher-derivative couplings (e.g., non-minimal gluon and graviton interactions), with their OPE coefficients also closing under ${\rm w}_{1+\infty}$.
• The proof of closure relies on an inductive argument over spin and mode number, using recursive structure and commutator identities to verify the ${\rm w}_{1+\infty}$ algebra for all $p$ and $m$ in the allowed range.