Expanding single trace YMS amplitudes with gauge invariant coefficients

This paper employs a bottom-up method based on soft theorems to recursively construct single-trace Yang-Mills-scalar amplitudes with manifest gauge invariance and permutation symmetry, yielding results equivalent to those derived via covariant color-kinematic duality.

The Gist

Imagine the universe as a giant, cosmic dance floor where particles like gluons (the carriers of the strong nuclear force) and scalars (simple, massless particles) are constantly colliding and scattering. Physicists call the mathematical description of these collisions "amplitudes." For decades, calculating these amplitudes has been like trying to solve a massive, tangled knot of string using only a specific, rigid set of rules (Feynman diagrams). It works, but it's messy, and the underlying beauty and symmetry of the dance often get hidden in the math.

This paper is about untangling that knot in a new, more elegant way. Here is the story of what the authors did, explained simply:

The Problem: The "Designated Driver" Flaw

In the past, physicists had a method to break down these complex particle collisions into simpler pieces. Think of it like translating a complex novel into a series of simple, short stories. However, the old translation method had a major flaw: it required picking one specific particle to be the "designated driver" (called a fiducial gluon).

• The Symmetry Break: In reality, all the gluon dancers are equal. But by picking one as the driver, the math treated them differently, breaking the natural symmetry of the group.
• The Gauge Invariance Issue: In physics, there's a rule called "gauge invariance." Imagine a song that sounds the same whether you play it in a major or minor key, or whether you shift the volume up or down. The physics shouldn't change just because you change how you describe the particle's "polarization" (its orientation). The old method hid this rule. If you tried to check if the math respected this rule, the answer wasn't obvious; it was buried under layers of complex algebra.

The authors wanted a new translation method that treated all gluons equally and made the "gauge invariance" rule obvious at every step.

The Solution: The "Soft Theorem" Detective Work

Instead of starting with a heavy textbook of rules (a Lagrangian) or equations of motion, the authors used a "bottom-up" approach. They acted like detectives using Soft Theorems.

• The Soft Theorem Analogy: Imagine a crowd of people shouting. If one person in the crowd suddenly whispers (becomes "soft"), the rest of the crowd's reaction follows a predictable pattern. The authors used this predictable pattern of "whispering" particles to reconstruct the whole crowd's behavior.
• The Process:
  1. Start Small: They began with the simplest possible dance: three particles (two scalars and one gluon). They figured out the rules for this tiny group using basic principles.
  2. Add Dancers (Scalars): They used the "whispering" rule for scalars to add more scalar particles to the dance, one by one, while keeping the number of gluons the same.
  3. The Magic Trick (BCJ Relations): At this stage, the math still had a slight asymmetry. The authors used a known mathematical relationship (the BCJ relation) to rearrange the terms. This was like shuffling a deck of cards to reveal a hidden pattern. Suddenly, the math became manifestly gauge invariant—meaning the rule that "the physics doesn't change with how you describe the orientation" was written clearly in the formula, not hidden.
  4. Add More Gluons: Finally, they used a "sub-leading" whispering rule for gluons to add more gluons to the dance. Because they started with a formula that already respected the symmetry, adding more gluons kept that symmetry intact.

The Result: A Perfectly Symmetric Recipe

The result is a new formula (an expansion) that breaks down complex particle collisions into a sum of simpler, pure scalar collisions.

• No Special Drivers: Unlike the old method, this new formula doesn't need to pick a "special" gluon. Every gluon is treated with the same respect, preserving the natural permutation symmetry (the idea that swapping two identical dancers doesn't change the dance).
• Clear Rules: The formula makes the gauge invariance obvious. You can look at the coefficients (the numbers multiplying the parts) and immediately see that they obey the physical rules, without needing to do a complex proof to verify it.
• The Cost: To get this perfect symmetry, the formula introduces some "spurious poles." Think of these as temporary, imaginary mathematical poles that appear in the calculation but cancel each other out in the end. They are a necessary trade-off to keep the symmetry visible.

Why It Matters

The authors show that this new method is equivalent to a previous discovery made by Clifford Cheung and James Mangan, who used a different, more traditional approach based on Lagrangians. The significance here is that the authors achieved the same result without using a Lagrangian or equations of motion. They built it entirely from the "on-shell" information—meaning they only used the properties of particles that are actually existing and moving, not hypothetical off-shell states.

In short, this paper provides a cleaner, more symmetric, and more intuitive way to calculate how particles scatter, revealing the hidden mathematical beauty of the universe's dance floor without relying on the heavy machinery of traditional field theory.

Technical Summary

Technical Summary: Expanding Single Trace YMS Amplitudes with Gauge Invariant Coefficients

Problem Statement
The paper addresses the construction of recursive expansions for single-trace tree-level Yang-Mills-scalar (YMS) amplitudes. While previous recursive expansions (e.g., in references [14, 15, 18]) successfully express YMS amplitudes in terms of bi-adjoint scalar (BAS) amplitudes, they suffer from two significant limitations:

1. Lack of Manifest Permutation Symmetry: These expansions require the selection of a "fiducial" external gluon. This choice breaks the manifest permutation symmetry among external gluons, making the equivalence of expansions based on different fiducial choices difficult to prove.
2. Obscure Gauge Invariance: In the iterative process of expanding YMS amplitudes down to pure BAS amplitudes, the gauge invariance for the polarization vector of the fiducial gluon is not manifest. Consequently, when the expansion is applied iteratively to all gluons, the resulting formula lacks manifest gauge invariance for any polarization. Since gauge invariance is a fundamental constraint in modern S-matrix programs, the authors seek a new expansion where gauge invariance is explicit in the coefficients for every external gluon.

A solution to this problem was previously found by Clifford Cheung and James Mangan [27] using a "covariant color-kinematic duality" approach based on Lagrangians and equations of motion. The primary motivation of this work is to reconstruct this result using a "bottom-up" method that relies solely on on-shell information, without invoking an action or equations of motion.

Methodology
The authors employ a recursive construction based on the universal soft behaviors of massless particles. The methodology proceeds in the following steps:

1. Bootstrap of 3-Point Amplitudes: The authors first construct the 3-point single-trace YMS amplitude with one external gluon and two scalars. By imposing constraints on mass dimension, Lorentz invariance, and the absence of poles, they uniquely determine the kinematic part.
2. Inversion of Soft Theorems for Scalars: To increase the number of external scalars while keeping the number of gluons fixed, the authors invert the leading-order soft theorem for external scalars. This allows them to insert scalars into the amplitude recursively.
3. Utilization of BCJ Relations: The initial expansion derived from soft theorems for a single gluon is transformed using the Bern-Carrasco-Johansson (BCJ) relations. This step is crucial for rewriting the expansion into a form where the coefficient vanishes if the polarization vector is replaced by the corresponding momentum, thereby manifesting gauge invariance. This introduces a spurious pole (e.g., $k_n \cdot k_p$) but ensures the coefficient is gauge invariant.
4. Inversion of Sub-leading Soft Theorems for Gluons: To increase the number of external gluons, the authors invert the sub-leading soft theorem for external gluons. Unlike the leading order, the sub-leading soft operator acts on both momenta and polarization vectors. This step inserts new gluons into the amplitude in a pattern that preserves the manifest gauge invariance established in the previous step.
5. Recursive Generalization: The process is demonstrated explicitly for amplitudes with two and three external gluons. By analyzing the sub-leading soft behavior and enforcing permutation symmetry among the gluons, the authors derive a general recursive formula.

Key Contributions and Results
The paper derives a new recursive expansion for single-trace YMS amplitudes $A_{YS}(1, \dots, n; \{p_i\}_m | \sigma_{n+m})$ with $m$ external gluons and $n$ external scalars. The result is given by:

$$A_{YS}(1, \dots, n; \{p_i\}_m | \sigma_{n+m}) = \sum_{\vec{\alpha}} \frac{k_n \cdot F_{\vec{\alpha}} \cdot Y_{\vec{\alpha}}}{k_n \cdot k_{p_1 \dots p_m}} A_{YS}(1, \{2, \dots, n-1\} \shuffle \vec{\alpha}, n; \{p_i\}_m \setminus \alpha | \sigma_{n+m})$$

Where:

• $\vec{\alpha}$ represents ordered subsets of the external gluons.
• $F_{\vec{\alpha}}$ is a product of field strength tensors $f_i^{\mu\nu} = k_i^\mu \epsilon_i^\nu - \epsilon_i^\mu k_i^\nu$.
• $Y_{\vec{\alpha}}$ is a combinatory momentum sum of external scalars to the left of the first element of $\vec{\alpha}$.
• The denominator $k_n \cdot k_{p_1 \dots p_m}$ ensures permutation symmetry among the gluons.

Key Features of the Result:

• Manifest Gauge Invariance: The coefficients contain the tensor $f_i^{\mu\nu}$, which vanishes under the replacement $\epsilon_i \to k_i$. Thus, gauge invariance is explicit for every external gluon polarization.
• Manifest Permutation Symmetry: The formula does not require a fiducial gluon; the sum over ordered subsets $\vec{\alpha}$ and the symmetric denominator ensure that the expansion is invariant under the permutation of external gluons.
• On-Shell Construction: The derivation relies entirely on soft theorems, BCJ relations, and on-shell constraints, avoiding the use of Lagrangians or off-shell ansatzes.

Significance and Claims
The authors claim that this work provides a "bottom-up" reconstruction of the expansion found in [27], validating the covariant color-kinematic duality result through purely on-shell methods. The significance lies in:

1. Resolving Gauge Invariance Issues: It provides a formula where the gauge invariance of all polarizations is manifest, a property not present in previous recursive expansions that relied on fiducial gluons.
2. Iterative Applicability: The expansion can be applied iteratively to reduce YMS amplitudes to pure BAS amplitudes without ever losing manifest gauge invariance, although this process generates a variety of spurious poles.
3. Extension to Gravity: Through the double copy structure, the authors note that their result can be directly extended to single-trace Einstein-Yang-Mills (EYM) amplitudes, yielding a gauge-invariant expansion for graviton-gluon scattering.

The paper concludes by noting that while the sub-leading soft theorem is sufficient to detect all terms in the expansion (as argued in [28]), the leading order soft theorem is insufficient because terms involving field strength tensors vanish at leading order. The authors suggest that this recursive method could potentially be applied to find gauge-invariant expansions for pure Yang-Mills amplitudes, which would lead to manifestly gauge-invariant BCJ numerators.