Single-minus gluon tree amplitudes are nonzero

Abstract: Single-minus tree-level $n$-gluon scattering amplitudes are reconsidered. Often presumed to vanish, they are shown here to be nonvanishing for certain "half-collinear" configurations existing in Klein space or for complexified momenta. We derive a piecewise-constant closed-form expression for the decay of a single minus-helicity gluon into $n-1$ plus-helicity gluons as a function of their momenta. This formula nontrivially satisfies multiple consistency conditions including Weinberg's soft theorem.

• The paper rigorously demonstrates that single-minus gluon tree amplitudes are nonzero in half-collinear kinematics using a closed-form formula and recursive methods.
• The methodology exploits specialized spinor-helicity configurations and piecewise-constant, integer-valued recursion, ensuring consistency with gauge theory constraints.
• Implications extend to self-dual Yang–Mills, gravitational amplitudes, and celestial holography, challenging traditional helicity selection rules.

Single-minus Gluon Tree Amplitudes: Nonvanishing in “Half-Collinear” Kinematics

Context and Motivation

The computation of scattering amplitudes in Yang–Mills theory is of central importance to both mathematical physics and phenomenological applications. Traditionally, tree-level $n$-gluon color-ordered amplitudes with only one minus-helicity gluon (and $n-1$ plus-helicity gluons)—the so-called "single-minus" amplitudes—have been assumed to vanish for real, generic kinematic configurations due to power-counting arguments and restrictions from helicity selection rules. Parke–Taylor’s formula provides a compact expression for the maximally helicity violating (MHV) amplitudes, which have precisely two minus-helicity gluons [Parke:1986gb], and forms the backbone of efficient calculations and recursive structures in gauge theory.

This paper (2602.12176) revisits the single-minus tree-level amplitudes and rigorously demonstrates their nonvanishing character for certain “half-collinear” configurations, which exist in complexified momentum space or in Klein space (signature (2,2)). The result is a closed-form, piecewise-constant expression for the decay of a single minus-helicity gluon into $n-1$ plus-helicity gluons, with rich implications for both Yang–Mills theory and special sectors such as self-dual Yang–Mills (SDYM).

Technical Results

Kinematic Regime and Nonvanishing Amplitudes

The authors define the “half-collinear” regime as the locus in spinor-helicity variables where all inner products $\langle i\, j\rangle$ vanish for $i, j$ running over the external gluons. In (2,2) signature or for complex momenta, this regime allows nonzero square brackets $[ij]$, in contrast to Minkowski space. Within this specialized kinematic region, the usual argument for vanishing of single-minus amplitudes breaks down: the inability to consistently choose reference spinors owing to singular polarization vectors invalidates traditional power-counting restrictions.

Consequently, the tree-level single-minus amplitude $\mathcal{A}_n(1^-, 2^+, \dots, n^+)$ is supported precisely in the half-collinear regime, enforced by a set of $\delta$-functions on spinor variables.

Piecewise-Constant Amplitudes and Recursion

The stripped single-minus amplitude $A_{1\dots n}$, carrying no helicity weight, admits a recursive structure equivalent to the Berends–Giele recursion [Berends:1987me], which is consistent with Feynman diagram expansion. The key recursion is:

$$A_{1\dots n} = -\sum_{\mathrm{o.p.}}\widehat{\mathrm{PT}}_{S_1\dots S_A} \prod_{a=1}^A \bar{A}_{S_a}$$

where $\widehat{\mathrm{PT}}$ is an “on-shell” Parke–Taylor factor, $\bar{A}_{S_a}$ are preamplitudes defined in terms of ordered partitions, and the sum is over all ordered partitions.

In explicit calculations up to $n=6$, the amplitude $A_{1\dots n}$ is piecewise-constant, integer-valued ($+1$, $-1$, $0$), with chamber structure controlled by the sign functions of kinematic invariants and their discontinuities across codimension-one walls.

Closed-form Formula for Restricted Kinematics

For the region $R$—where $\omega_1 < 0$, $\omega_a > 0$ ($a > 1$) in some $\mathrm{SO}(2,2)$ frame—the amplitude simplifies to a compact product formula:

$$A_{1\dots n}\big|_R = \frac{1}{2^{n-2}} \prod_{m=2}^{n-1} \left[ \operatorname{sgn}_{m,m+1} + \operatorname{sgn}_{1,2\dots m} \right]$$

Here, each sign factor $\operatorname{sgn}_{ij}$ is dictated by the sign of kinematic brackets. This formula was conjectured by a generative LLM and proven by internal OpenAI tools. It satisfies all key consistency conditions, including Weinberg's soft theorem, cyclicity, Kleiss–Kuijf relations, and $\mathsf{U}(1)$ decoupling, none of which are manifest from direct inspection.

Implications for Self-Dual Yang–Mills

Single-minus amplitudes also appear in the SDYM sector, potentially addressing a long-standing tension: SDYM’s classical solution space is nontrivial, but tree-level amplitudes were previously thought to yield only trivial two- and three-point results. The nonvanishing $n$-point single-minus amplitudes in the half-collinear regime now provide candidate quantum support for the classical complexity.

Extensions and Consistency

The newly constructed framework generalizes directly to graviton amplitudes (via color–kinematics duality) and admits supersymmetrization. The results are expected to transform under celestial holography’s $S$-algebra and extended symmetry structures, with Mellin transforms relating to Lauricella functions. The appendices provide rigorous derivations via time-ordered perturbation theory and master identities for the underlying sign functions.

Numerical Results and Contradictory Claims

The amplitude formula is piecewise constant and integer-valued, with explicit examples provided up to six external gluons. The claim that single-minus tree-level amplitudes are nonvanishing for special kinematics directly contradicts the conventional wisdom that such amplitudes vanish generically, and provides explicit construction and proof for all $n$ via recursive and closed-form formulas.

Practical and Theoretical Implications

Practically, the results suggest new building blocks for recursive computation in Yang–Mills, especially in analytic continuations and specialized sector calculations. Theoretically, they encourage a reevaluation of the helicity selection rules in complexified or signature $(2,2)$ spaces, open avenues for improved understanding of amplitude chamber structure, and offer potential resolutions to puzzles in SDYM and celestial holography.

Extensions to gravitational and supersymmetric amplitudes promise deeper insights into symmetry and algebraic structures underlying scattering, with immediate applicability to quantum field theory formulations beyond standard Minkowski signature.

Speculation on Future Developments

Future work may uncover simpler expressions for single-minus amplitudes in broader kinematic regimes, elucidate their inner structural role in Yang–Mills and gravitational theories, and integrate the chamber behavior into geometric and twistor-space approaches. It is plausible that further exploration will yield more refined analytic continuations and links to dualities in celestial holography or amplitude stratification.

Conclusion

This paper provides the first rigorous demonstration that single-minus $n$-gluon tree-level amplitudes are nonzero for half-collinear or complexified kinematics, with explicit piecewise-constant integer-valued formulas and recursion. The results refute prior assumptions of vanishing, introduce new analytic tools for amplitude computation, and yield immediate implications for SDYM and celestial holography. The chamber structure and simplification of the formula in special regions foreshadow deeper structural understanding of nontrivial amplitude sectors, with potential ramifications for quantum gauge theories and gravity.