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Single-minus gluon tree amplitudes are nonzero
Alfredo Guevara, Alexandru Lupsasca, David Skinner, Andrew Strominger, Kevin Weil
TL;DR
The paper addresses the apparent vanishing of single-minus tree-level gluon amplitudes by analyzing half-collinear kinematics and constructing a recursion for the supported amplitudes. It derives a simple formula in a restricted region and verifies that the construction satisfies several nontrivial consistency conditions.
Problem
Single-minus tree amplitudes are usually taken to vanish, but the paper identifies half-collinear configurations where this conclusion fails.
Method
The paper derives a Berends–Giele recursion for all n-point single-minus amplitudes and evaluates it in a partially restricted half-collinear region.
Results
In the restricted region, the paper obtains a simple formula for the amplitudes and verifies it through nontrivial consistency checks.
Takeaways & Limitations
Single-minus tree amplitudes are nonzero on the identified half-collinear support, where their stripped values are piecewise-constant integers.
Abstract
from arXiv · showhide
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.
A. Notation and useful identities
The paper fixes spinor-helicity conventions in (2,2) Klein signature, including reference-spinor choices, sign functions, propagator prescriptions, and Parke–Taylor factors used later.
- Massless momenta are represented by real spinors in (2,2) signature, with |i⟩=(1,z_i) and |i]=ω_i(1,ẑ_i).
- The notation uses z_ij=z_i−z_j and ẑ_ij=ẑ_i−ẑ_j for spinor contractions.
- Reference spinors define polarization vectors and can be chosen arbitrarily, subject to the stated non-singularity conditions.
- The paper uses the standard Feynman propagator 1/(p^2+iε) and normalizes delta functions so that ∫δ(x)dx=2π.
- The regularized Parke–Taylor factor incorporates the iε prescription, while away from ⟨k k+1⟩=0 it reduces to the unregularized form.
- The incomplete Parke–Taylor factor is the open-chain denominator of an off-shell Berends–Giele current, unlike the cyclic on-shell factor.
I. SINGLE-MINUS AMPLITUDES
Single-minus tree amplitudes can be supported on half-collinear kinematics, where the usual generic-kinematics vanishing argument fails and a recursion determines all n-point amplitudes.
- I. SINGLE-MINUS AMPLITUDES: A recursion relation derived from Berends–Giele methods determines all n-particle single-minus tree amplitudes.
- A. The half-collinear regime: The half-collinear regime sets all angle brackets ⟨ij⟩ to zero while remaining compatible with nonzero [ij] in (2,2) signature.
- A. The half-collinear regime: For generic kinematics, polarization-vector power counting leaves too few momentum factors to produce a nonzero single-minus amplitude.
- A. The half-collinear regime: The vanishing argument fails on the collinear locus because choosing the reference spinor |r⟩=|1⟩ makes positive-helicity polarizations singular.
- A. The half-collinear regime: The single-minus amplitude is supported when all ⟨ij⟩=0, with delta functions imposing collinearity and remaining momentum-conservation constraints.
- A. The half-collinear regime: The half-collinear regime also admits complex momenta, although the paper leaves their continuation for further study.
B. The recursion relation
The recursion constructs single-minus amplitudes from recursively defined preamplitudes, ordered partitions, vertex functions, and on-shell Parke–Taylor factors.
- B. The recursion relation: The recursion relation is the paper’s first main result and determines all n-particle single-minus tree amplitudes.It is equivalent to summing the corresponding Feynman diagrams but is slightly simpler.
- B. The recursion relation: For an ordered list S, the list momentum is summed over its constituent ẑλ_i, and the preamplitude is defined recursively by ordered partitions.
- B. The recursion relation: The construction starts with specified one- and two-element cases and extends to larger lists through partitions into at least three blocks.
- B. The recursion relation: Reference-spinor dependence in list momenta and sign functions drops out on the support of the collinear delta functions.
- B. The recursion relation: The stripped amplitude is obtained after determining the preamplitude, with the vertex function normalized by V_ẑλ1=1.
- B. The recursion relation: The on-shell Parke–Taylor object is related to the incomplete Parke–Taylor factor through LSZ reduction and a sign choice in the step-function argument.
C. Consistency checks
The recursively constructed stripped amplitudes satisfy several nontrivial identities, including relations associated with factorization and Weinberg’s soft theorem.
- C. Consistency checks: The stripped amplitudes satisfy a family of linear identities among different orderings.
- C. Consistency checks: Weinberg’s soft theorem is among the stated consistency checks for the single-minus amplitudes.
- C. Consistency checks: Explicit calculations verify that the recursion solution obeys all listed consistency properties, despite their not being evident from the recursion itself.
D. Concrete examples
Concrete examples show that single-minus amplitudes become unwieldy in general but simplify in the restricted region R1, where frequency-independent sign relations apply.
- B. Concrete examples: The examples simplify dramatically in R1, supporting a shorter formula for all n-point single-minus amplitudes in this restricted regime.The text presents the simplification as evidence for the existence of a general concise expression.
- A. Restricted kinematics within the half-collinear regime: R1 is defined by a frame with one negative frequency, ω1 < 0, and all other frequencies positive.This region further restricts half-collinear kinematics while remaining SO(2,2)-invariant through the existence of such a frame.
- A. Restricted kinematics within the half-collinear regime: In R1, certain sign functions become independent of the frequencies, reducing sign factors to functions of the z-variables.The simplification is expressed by the relations sgij = sg z̃ij and sg1j = sg z̃j1 for i,j ≥ 2.
- A. Restricted kinematics within the half-collinear regime: Frequency dependence is not eliminated from composite expressions such as sg2,34.Thus the R1 simplification applies to certain sign functions, not uniformly to every expression.
B. Concrete examples
The paper conjectures and then proves a compact R1 formula for single-minus amplitudes, using simplified examples and a time-ordered perturbation theory argument with consistency checks.
- B. Concrete examples: Momentum conservation and spinor identities dramatically reduce the long amplitude expressions in R1.The reduced expressions are obtained after applying the R1 sign relations together with momentum conservation and spinor identities.
- B. Concrete examples: The simplified examples suggest a shorter formula extending to all n-particle amplitudes in region R1.This motivates the conjecture that extends the observed pattern to arbitrary multiplicity.
- C. General formula: The proposed solution obeys the soft theorem, cyclicity, Kleiss–Kuijf, and U(1) decoupling constraints despite R1 itself singling out particle 1.Cyclicity is restored by extending the result to analogous regions Rk where particle k has negative frequency.
- C. General formula: In R1, each factor takes values that make the amplitude piecewise constant, with jumps across codimension-one walls where relevant brackets change sign.The product form exposes these chamber walls directly.
- C. General formula: The proof uses time-ordered perturbation theory in three stages: establish a vanishing condition, collapse the recursion, and reduce the remaining vertex function to the final formula.The argument is organized around the recursion in R1 and the final reduction of the vertex function.
- C. General formula: A weighted-variance identity guarantees a cut whose step-function factor vanishes, forcing the product-form recursion to vanish in the required case.The relevant sign ratio is controlled by positive frequency products and the sign of a weighted-average difference.
2. Collapsing the recursion
The recursion collapses because, for the relevant configurations, only the all-singleton partition contributes, yielding the stated reduced form. The result extends to every consecutive subset S of size at least two.
- For every consecutive S ⊂ {2, . . . , n} with |S| ≥ 2, the authors establish a more general result.
- The derivation is described as reminiscent of the largest-time equation in time-ordered perturbation theory.
- Applying the recursion to A2···n1 leaves only the all-singleton partition (2|3| · · · |n) as a nonzero contribution.
- The recursion collapse is proved to yield equation (41).
3. Evaluating ¯V˜λ2···˜λn
The vertex is reorganized using sign functions, momentum conservation, and antisymmetry of the bracket. Combining the resulting relations reproduces the paper’s final expression.
- The vertex V̄_{λ̃2···λ̃n} is reorganized in terms of sign functions.
- Momentum conservation and antisymmetry of the bracket are used to rewrite the relevant expression.
- The relation between the sg and Θ functions helps complete the vertex evaluation.
- Combining equations (51) and (54) exactly recovers the final result (39).
Appendix A: The master identity
Appendix A derives a generalized distributional identity using Fourier transformation and time-domain manipulations. Fourier transforming back recovers the master identity, with Θ(x) related to the sign function.
- The appendix generalizes a well-known identity involving delta functions using manipulations from time-ordered perturbation theory.
- For n = 3, the generalized identity produces a sign-function expression multiplied by three delta functions.
- The generalized identity is proved by taking its Fourier transform and working in the time domain, where the b_i are treated as energies.
- Fourier transforming in the time variables recovers the master identity after using Θ(x) = 1 + sg(x).
1. Berends–Giele Recursion
The appendix derives Berends–Giele recursion for off-shell Yang–Mills currents and uses ordered partitions, Parke–Taylor factors, contact terms, and collinear delta functions to obtain the on-shell single-minus amplitude.
- 1. Berends–Giele Recursion: The planar coefficients obey Berends–Giele recursion, which is equivalent to summing Feynman diagrams with one off-shell leg.
- 1. Berends–Giele Recursion: When the remaining on-shell legs have plus helicity, the recursion agrees with the self-dual Yang–Mills recursion.
- 1. Berends–Giele Recursion: The form-factor recursion is solved with two-dimensional preamplitudes by replacing one vertex with an incomplete Parke–Taylor factor.
- 1. Berends–Giele Recursion: For a fixed ordered partition, the recursion’s j-sum produces a Parke–Taylor term for the combined partition and a contact term δV.
- 1. Berends–Giele Recursion: The contact contribution stitches partition blocks together through a V-vertex, while the final term restores the missing one-block partition.
3. LSZ Reduction
The single-minus amplitude is obtained by putting the final leg on shell and stripping universal momentum-conservation support. Algebraic manipulation then yields the final result (21).
- 3. LSZ Reduction: Putting the “last” leg on shell yields the single-minus amplitude from the off-shell construction.The on-shell limit is evaluated using the master identity in Appendix A.
- 3. LSZ Reduction: Universal momentum-conservation support is stripped before extracting the final amplitude expression.
- 3. LSZ Reduction: After algebraic simplification, the construction produces the final result (21).