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

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.

• The paper demonstrates that the chiral algebra bootstrap determines all-loop form factors in N=4 self-dual Yang-Mills, eliminating the need for explicit Feynman diagram calculations.
• It employs twistor space, supersymmetry, and associativity to fix analytic OPE structures, ensuring only single poles appear in loop-level computations.
• Results confirm the vanishing of one-minus and all-plus amplitudes across loops and provide explicit formulas for full-color, nonplanar observables, paving the way for advanced applications.

Form Factors in $\mathscr{N}=4$ Self-Dual Yang-Mills via the Chiral Algebra Bootstrap

Introduction and Context

This paper applies the chiral algebra bootstrap (CAB) to compute form factors in self-dual 4-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SDSYM), focusing especially on non-planar, all-multiplicity observables. The CAB relates perturbative gauge theory data to two-dimensional chiral algebra correlators on the celestial sphere, effectively encoding loop-level splitting functions through chiral operator product expansions (OPEs). The approach leverages twistor space, Koszul duality, supersymmetry, and associativity to fix the analytic structures of these observables without explicit Feynman diagram calculations.

Chiral Algebra Bootstrap Framework

Twistor Space and SDSYM

SDSYM admits a geometric reformulation as a holomorphic Chern-Simons theory on $\mathbb{CP}^{3|4}$ twistor superspace, with the gauge superconnection expanded in fermionic coordinates. The field content matches the $\mathcal{N}=4$ multiplet restricted to the self-dual sector, and the action is invariant under supersymmetric gauge transformations. The CAB is guaranteed to yield a well-defined associative chiral algebra reflecting quantum integrability, with OPEs directly encoding all collinear splitting functions.

Chiral Algebra: Structure and Constraints

Operators in the chiral algebra correspond to massless eigenstates on the celestial sphere, classified by energy and Lorentz boosts, forming highest-weight representations. The OPEs are fixed by kinematic and symmetry requirements, including supersymmetry and twistor-induced conformal structure. Remarkably, these constraints suffice to fix the analytic OPE data at all loop orders, and the spectrum of allowed poles is severely restricted—no double or higher-order poles exist due to combined dilatation symmetry.

Key tree-level OPEs are presented explicitly, involving currents, gauginos, scalars, and their supersymmetric representations. The all-loop generalization is constructed via associativity recursion, with coefficients determined through symmetry and Jacobi identity arguments.

Absence of Double Poles at Loop Level

A central technical result is the all-loop proof that SDSYM chiral algebra OPEs admit only single poles. The proof employs the associativity constraint and symmetry analysis, showing that possible double-pole coefficients vanish identically for any Lie algebra. This rigorous exclusion matches previous conjectures and one-loop proofs, highlighting the simplicity and power of the CAB in supersymmetric contexts.

Computation of Form Factors: $\text{tr}(B^m)$ Insertions

Tree-Level Results

The CAB approach reproduces known tree-level MHV sector form factors for operators such as $\text{tr}(B^2)$, matching Nair's MHV superamplitude. By exploiting symmetry constraints and OPE residues, the analytic form of $n$-point MHV form factors is derived, including explicit results for up to four points and arbitrary multiplicity.

Loop-Level Analysis

One-Loop and Two-Loop Results

Strong selection rules emerge from supersymmetry and the CAB: one-minus and all-plus amplitudes vanish at all loop orders, confirmed both via explicit computation and Ward identities. For two-loop form factors, operator renormalization is handled by supplementing with $tr(F^2)$ counterterm insertions; all-plus correlators vanish once operator momentum is set to zero, consistent with the non-existence of all-plus amplitudes in $\mathcal{N}=4$ SYM.

Higher Loop Structure and Operator Mixing

The computational framework generalizes to form factors involving $tr(B^3)$, relevant for Higgs amplitudes and maximal transcendentality in QCD/ SYM comparisons. Explicit expressions are provided for three-minus, two-minus, one-minus, and all-plus helicity sectors at tree, one, two, and three loops, respectively. Nonzero contributions arise only where the Feynman rules and symmetry constraints permit, and higher-loop amplitudes vanish identically due to grading arguments tied to supersymmetric selection rules.

Numerical and Analytic Claims

• All-multiplicity, nonplanar form factors for $\mathcal{N}=4$0 and $\mathcal{N}=4$1 are computed up to two loops, with explicit analytic formulae shown.
• Vanishing of one-minus and all-plus amplitudes is established for all loops, matching supersymmetric Ward identities.
• The CAB method efficiently computes full-color, non-planar, all-loop form factors without direct Feynman diagram evaluations except for normalization fixes.

Implications and Future Directions

Practical Implications

This work consolidates the CAB as a powerful tool for analytic computation of nonplanar, full-color form factors in quantum-integrable gauge theories, especially in supersymmetric contexts where nontrivial cancellation and selection rules heavily constrain amplitude structures. Its recursive structure and manifest symmetry incorporation will facilitate high-multiplicity, high-loop computations relevant for phenomenology (e.g., Higgs production amplitudes and comparison with QCD maximal transcendentality).

Theoretical Implications

The results deepen the geometric and algebraic connections between four-dimensional gauge theory, twistor constructions, celestial chiral algebras, and bootstrap methods. The proof of absence of double poles at all loops—rooted in twistorial symmetry and associativity—offers insight into the structure of quantum field theory OPEs in integrable systems. The techniques generalize naturally to theories with less or no supersymmetry, where more intricate pole structures and combinatorics can arise.

Speculation on Future Developments

• Extension of CAB computations to nonsupersymmetric gauge theories, including operator mixing and anomaly-induced effects.
• Application to full amplitudes, including nonplanar corrections beyond the self-dual sector of SYM and QCD.
• Integration of CAB with other bootstrap programs (e.g., flux-tube, hexagon, and celestial amplitudes) to further elucidate dualities and correspondences.
• Programmatic computation of form factors at arbitrary loop order using symbolic and algebraic geometry algorithms.

Conclusion

The chiral algebra bootstrap provides a highly efficient and algebraically robust framework for deriving form factors in self-dual $\mathcal{N}=4$2 Yang-Mills theory, yielding analytic expressions at all loop orders and multiplicities. Symmetry-driven constraints and associativity recursion entirely determine OPE singularity structures, with vanishing results for certain helicity sectors. The results lay the foundation for advanced applications in both supersymmetric and non-supersymmetric gauge theories, and bridge geometric, algebraic, and analytic approaches to quantum field theory amplitudes.