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arxiv: 2604.21015 · v1 · submitted 2026-04-22 · ✦ hep-th

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Form factors of mathscr{N}=4 self-dual Yang-Mills from the chiral algebra bootstrap

Jaazib Charanya , Anthony Morales , Natalie M. Paquette
Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:23 UTC · model grok-4.3

classification ✦ hep-th
keywords chiral algebra bootstrapself-dual Yang-Millsform factorsholomorphic collinear splitting functionsN=4 super Yang-Millstwistor spaceKoszul dualityoperator product expansion
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The pith

The chiral algebra bootstrap yields all-loop holomorphic collinear splitting functions and new two-loop form factors for self-dual N=4 super Yang-Mills.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies the chiral algebra bootstrap to self-dual N=4 super Yang-Mills by feeding in twistor space data and Koszul duality, then using supersymmetry and associativity to fix the complete set of holomorphic collinear splitting functions at every loop order. These splitting functions, which are the singularities that build form factors, also serve as the celestial chiral algebra OPEs of the theory. The same associativity constraint supplies a direct proof that loop-level OPEs contain no double poles. The authors then evaluate several form factors up to two loops, recovering known cases and producing new expressions that insert powers of the anti-self-dual field strength. These objects compute supersymmetric versions of Higgs amplitudes restricted to the self-dual sector.

Core claim

We use a combination of twistor space input, Koszul duality, supersymmetry, and associativity to obtain the all-loop holomorphic collinear splitting functions for SDSYM. We also use associativity to provide a simple proof of the conjecture that there are no double-poles in the loop-level OPEs for this theory. We conclude by computing several form factors, including both a reproduction of several known results and novel form factors up to two loops involving insertions of powers of the anti-self-dual field strength. These form factors compute a supersymmetric version of Higgs amplitudes in the self-dual sector.

What carries the argument

The chiral algebra bootstrap, in which holomorphic collinear splitting functions (equivalently celestial chiral algebra OPEs) recursively generate loop-level form factors from initial data supplied by twistor space and Koszul duality.

If this is right

  • All-loop holomorphic collinear splitting functions for SDSYM are now determined explicitly.
  • Loop-level OPEs in the theory contain no double poles.
  • Form factors up to two loops, including new ones with anti-self-dual field strength insertions, are available in closed form.
  • These form factors equal supersymmetric Higgs amplitudes in the self-dual sector.
  • The method works for other quantum-integrable self-dual gauge theories once the same initial data are supplied.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bootstrap could be run at higher loops to produce three-loop and beyond form factors without new input data.
  • Explicit OPEs obtained here supply concrete data for testing conjectures in celestial holography.
  • Comparison of these self-dual form factors with full-theory Higgs amplitudes would quantify the contribution of the anti-self-dual sector.
  • Absence of double poles may be linked to the infinite-dimensional symmetry algebra that renders the theory integrable.

Load-bearing premise

The chiral algebra bootstrap program applies directly to self-dual N=4 Yang-Mills, with all singularities of the form factors generated exactly by the holomorphic collinear splitting functions.

What would settle it

An independent twistor-space or unitarity-based computation of any two-loop form factor containing an insertion of the anti-self-dual field strength that differs from the expression obtained here would falsify the bootstrap result.

read the original abstract

The chiral algebra bootstrap (CAB) is a novel bootstrap program for form factors in quantum-integrable self-dual gauge theories, some of which in turn are helicity amplitudes in the corresponding gauge theories. The singularities that recursively generate a given (loop-level) form factor are holomorphic collinear splitting functions, equivalently celestial chiral algebra OPEs, of the self-dual theory. In this note, we apply the chiral algebra bootstrap to the simple example of self-dual 4d $\mathscr{N}=4$ super Yang-Mills (SDSYM). We use a combination of twistor space input, Koszul duality, supersymmetry, and associativity to obtain the all-loop holomorphic collinear splitting functions for SDSYM. We also use associativity to provide a simple proof of the conjecture that there are no double-poles in the loop-level OPEs for this theory. We conclude by computing several form factors, including both a reproduction of several known results and novel form factors up to two loops involving insertions of powers of the anti-self-dual field strength. These form factors compute a supersymmetric version of Higgs amplitudes in the self-dual sector. Detailed sample computations are provided to familiarize the reader with the CAB method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper applies the chiral algebra bootstrap (CAB) to self-dual N=4 super Yang-Mills (SDSYM). It combines twistor space input, Koszul duality, supersymmetry, and associativity to derive the all-loop holomorphic collinear splitting functions (equated to celestial chiral algebra OPEs). Associativity is used to prove the absence of double poles in the loop-level OPEs. The work concludes with explicit computations of several form factors up to two loops, reproducing known results and providing novel ones involving insertions of powers of the anti-self-dual field strength, interpreted as supersymmetric Higgs amplitudes in the self-dual sector. Detailed sample computations are included to illustrate the method.

Significance. If the central derivations hold, this advances the CAB program by furnishing all-loop splitting functions for SDSYM and a consistency proof for OPE singularities, while delivering concrete form factor results that connect twistor methods to celestial algebras. The reproduction of known results alongside new two-loop computations, plus the sample calculations, strengthens the case for the bootstrap's applicability to self-dual integrable theories.

major comments (2)
  1. [Main derivation and abstract] The derivation of the all-loop holomorphic collinear splitting functions rests on the claim that twistor space input combined with Koszul duality supplies complete initial data that, together with supersymmetry and associativity, uniquely determines the functions at all orders. This assumption is load-bearing for both the all-loop claim and the novel two-loop form factors; the manuscript should explicitly address whether loop-dependent corrections or additional constraints could arise beyond the provided data (see the discussion following the abstract and the main derivation section).
  2. [Proof of no double poles] The associativity-based proof that there are no double poles in the loop-level OPEs is presented as straightforward, but the specific associativity relations invoked and their application across loop orders require more detail to confirm they close without external input or hidden assumptions.
minor comments (3)
  1. The notation for splitting functions and OPEs would benefit from a dedicated table or explicit low-order examples early in the text to improve readability for readers new to CAB.
  2. [Introduction] References to prior CAB literature and twistor-space results should be expanded in the introduction to clarify the precise novelty of the SDSYM application.
  3. [Sample computations] The sample computations section is helpful but could include more intermediate steps for one of the novel two-loop form factors to aid verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the load-bearing aspects of the derivation. We agree that additional explicit discussion will strengthen the presentation and will revise the manuscript accordingly. Below we address each major comment in turn.

read point-by-point responses

  1. Referee: [Main derivation and abstract] The derivation of the all-loop holomorphic collinear splitting functions rests on the claim that twistor space input combined with Koszul duality supplies complete initial data that, together with supersymmetry and associativity, uniquely determines the functions at all orders. This assumption is load-bearing for both the all-loop claim and the novel two-loop form factors; the manuscript should explicitly address whether loop-dependent corrections or additional constraints could arise beyond the provided data (see the discussion following the abstract and the main derivation section).

    Authors: The twistor-space input together with Koszul duality furnishes the complete tree-level and one-loop data for the holomorphic collinear splitting functions. Supersymmetry then restricts the possible higher-loop structures, while associativity of the resulting celestial chiral algebra is used to fix all coefficients uniquely, as verified by the explicit two-loop form-factor computations that reproduce known results without introducing new parameters. We will add a short clarifying paragraph immediately after the abstract and in the main derivation section (around the discussion of the bootstrap closure) that explicitly states why no loop-dependent corrections or external constraints arise beyond the supplied initial data, supported by the consistency of the all-loop expressions with the associativity conditions. revision: yes

  2. Referee: [Proof of no double poles] The associativity-based proof that there are no double poles in the loop-level OPEs is presented as straightforward, but the specific associativity relations invoked and their application across loop orders require more detail to confirm they close without external input or hidden assumptions.

    Authors: The proof proceeds from the associativity of the OPEs in the celestial chiral algebra, specifically the requirement that the triple collinear limits satisfy the Jacobi identity at each loop order. We will expand the relevant subsection to display the explicit associativity relations (including the relevant three-point and four-point OPE associators) and walk through their application order by order, showing that they close on the all-loop splitting functions already derived and do not require additional external input. This elaboration will make transparent that the absence of double poles follows directly from the bootstrap data without hidden assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends from external inputs to new results.

full rationale

The paper obtains all-loop holomorphic collinear splitting functions from twistor space input, Koszul duality, supersymmetry, and associativity, then applies these to compute form factors (reproducing known results and generating novel ones up to two loops). Associativity additionally supplies a consistency proof for the absence of double poles in OPEs. No quoted step equates a derived quantity to its own inputs by construction, renames a fit as a prediction, or reduces the central claim to a self-citation chain; the computations build outward from the stated initial data in a self-contained manner.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions about the structure of the chiral algebra OPEs and the applicability of the bootstrap; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The singularities that recursively generate a given (loop-level) form factor are holomorphic collinear splitting functions, equivalently celestial chiral algebra OPEs, of the self-dual theory.
    This is the foundational premise of the CAB program stated in the abstract.
  • domain assumption Twistor space input and Koszul duality supply the initial data needed to bootstrap the splitting functions when combined with supersymmetry and associativity.
    Explicitly listed as part of the combination used to obtain the all-loop results.

pith-pipeline@v0.9.0 · 5524 in / 1572 out tokens · 35039 ms · 2026-05-09T23:23:47.600764+00:00 · methodology

discussion (0)

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Reference graph

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