Scalar-Loop YM-Scalar Building Blocks

• Scalar-loop YMS building blocks are defined as color-stripped one-loop integrands with scalar loops that decompose Yang–Mills amplitudes into gauge-invariant components.
• They serve as a bridge linking gluonic loop amplitudes, scalar loops, and double-copy constructions, ensuring the fulfillment of BCJ and Jacobi identities.
• These blocks are expanded onto a basis of bi-adjoint scalar integrands, enabling the explicit construction of one-loop BCJ numerators and gravitational amplitude uplift.

Scalar-loop Yang–Mills–scalar (YMS) building blocks are fundamental denominators and numerator structures in the modern algebraic and geometric understanding of one-loop gauge theory amplitudes. They play a pivotal role in decomposing and explicitly constructing Yang–Mills (YM) one-loop integrands and their color-kinematic dual numerators, serving as the canonical bridge between gluonic loop amplitudes, scalar loops, and the double-copy construction for gravitational amplitudes.

1. Definition and Structural Role

Scalar-loop Yang–Mills–scalar (YMS) building blocks are defined as the color-stripped, one-loop, color-ordered integrands of a theory in which the internal loop is made up of scalars (bi-adjoint or in the adjoint), and where external legs are a combination of gluons and color-adjoint or bi-adjoint scalars. Formally, these integrands are denoted as: $$I^{\text{YMS}}(\ell; j_1, ..., j_m \parallel g_1, ..., g_{n-m} \mid \rho)$$ where:

• $\ell$ is the loop momentum,
• $\rho$ is the cyclic ordering of all $n$ external legs,
• $\{j_1, ..., j_m\}$ is the subset of external bi-adjoint scalars,
• $\{g_1, ..., g_{n-m}\}$ is the subset of external gluons.

After further color-stripping the internal scalar loop, one obtains the "doubly" color-stripped integrand: $$I^{\text{YMS}}_{\text{DCS}}(\ell; j_1, ..., j_m \parallel g_1, ..., g_{n-m} \mid \rho)$$ In the modern two-step expansion for YM one-loop amplitudes, these YMS scalar-loop integrands provide an intermediate basis: the Yang–Mills integrand is expressed as a linear combination of YMS blocks, each of which is then expanded in terms of bi-adjoint scalar (BS) integrands, with coefficients that serve as explicit BCJ numerators (Du et al., 14 Nov 2025, Cao et al., 27 Dec 2024).

2. Universal Expansion and Gauge-Invariant Coefficients

The full $n$-gluon one-loop color-ordered integrand in dimension $D$ can always be expanded as: $$\mathcal{I}_n^{\text{YM}}(1, 2, ..., n; \ell) = \sum_{m=0}^n \;\sum_{\alpha \in S_{m-1}/\mathbb{Z}_2} \;B_{n,m}(\alpha)\; \,\mathcal{I}^{\text{scalar-loop}}_\alpha(1, ..., n; \ell)$$ where:

• $S_{m-1}/\mathbb{Z}_2$ denotes cyclically/reflection-inequivalent $m$-cycles,
• $B_{n,m}(\alpha)$ are gauge-invariant prefactors built from field strengths:

$$B_{n,m}^{\rm YM}(\alpha) = \text{tr}(f_{\alpha_1}f_{\alpha_2}\cdots f_{\alpha_m}),$$

and $B_{n,0} = D-2$ encodes pure scalar loop helicity sum.

• $\mathcal{I}^{\text{scalar-loop}}_\alpha$ is the scalar-loop YMS integrand with external order determined by $\alpha$.

This universal expansion, with explicit prefactors, holds for general gauge theories and matter content (Cao et al., 27 Dec 2024, Du et al., 14 Nov 2025).

3. Explicit Decomposition Formulas

In pure Yang–Mills, the master decomposition formula connecting the full integrand to scalar-loop YMS blocks is: $$I^{\text{YM}}(\ell; \rho) = (D-2)\,I^{\text{YMS}}(\ell; \emptyset \parallel \text{all gluons} \mid \rho) + \sum_{l=2}^n (-1)^l \!\!\!\sum_{j_1<\cdots<j_l \in G} \!\!\! \mathrm{Tr}(F_{j_1}\cdots F_{j_l})\; I^{\text{YMS}}(\ell; j_1\cdots j_l \parallel G\setminus\{j_1\cdots j_l\} \mid \rho)$$ with $$F_j^{\mu\nu} = k_j^\mu \epsilon_j^\nu - k_j^\nu \epsilon_j^\mu$$. This decomposition is rooted in the forward-limit and propagator-matrix approach and is exact for all $n$ (Du et al., 14 Nov 2025, Cao et al., 27 Dec 2024).

Subsequently, each doubly color-stripped YMS scalar-loop integrand is expanded in a basis of BS one-loop integrands: $$I^{\text{YMS}}_{\text{DCS}}(\ell; 1,\sigma \parallel G \mid \rho) = \sum_{\alpha \in \mathrm{shuffle}(\sigma, G)} C(\ell;1,\alpha) I^{\text{BS}}(\ell;1,\alpha \mid \rho)$$ where the coefficients $C(\ell;1,\alpha)$ are determined uniquely via algebraic consistency, as detailed below.

4. Algebraic Consistency and Explicit Coefficient Construction

The coefficients $C(\ell;1,\alpha)$ for expansion onto BS integrands must satisfy two key algebraic constraints:

• Loop-momentum-shift invariance: All $C(\ell;1,\alpha)$ are required to remain invariant under loop-momentum shifts corresponding to cyclic rotations of the external ordering (modulo labeling of the loop start), such that:

$C(\ell;1,\alpha)=C(\ell+k_1; \alpha,1)$

and similarly for further cyclic shifts.

• BCJ and Jacobi identities: The form of $C(\ell;1,\alpha)$ is determined so that upon expansion into subcurrents, all antisymmetry and Jacobi relations among trivalent cubic vertices are respected. This is analogous to ensuring color-kinematics duality for the kinematic numerators of scalar-loop graphs.

Through these constraints, the expansion is rendered unique with all gauge freedom fixed (Du et al., 14 Nov 2025).

Explicit Low-Point Expressions

For up to three gluons (with remaining external particles as scalars), the explicit forms are

• One gluon $p$:

$$C(\ell;1,\alpha)=\epsilon_p \cdot X_p(\ell;1,\alpha), \quad X_p^\mu = \ell^\mu + \sum_{\text{scalars left of }p} k_i^\mu$$

• Two gluons $p,q$:

$$C(\ell;1,\alpha) = \left[ \epsilon_p^\mu \epsilon_q^\nu - \frac{\epsilon_p\cdot\epsilon_q}{k_p\cdot k_q} k_p^\mu k_q^\nu\right] X_p^\mu X_q^\nu$$

• Three gluons $p,q,s$:

$$C(\ell;1,\alpha)= \Big\{ \epsilon_p^\mu \epsilon_q^\nu \epsilon_s^\rho - \frac{\epsilon_p\cdot\epsilon_q}{k_p\cdot k_q} k_p^\mu k_q^\nu \epsilon_s^\rho - \frac{\epsilon_p\cdot\epsilon_s}{k_p\cdot k_s} k_p^\mu \epsilon_q^\nu k_s^\rho - \frac{\epsilon_q\cdot\epsilon_s}{k_q\cdot k_s} \epsilon_p^\mu k_q^\nu k_s^\rho \Big\} X_p^\mu X_q^\nu X_s^\rho$$

All $C(\ell;1,\alpha)$ satisfy the requisite shift-invariance and BCJ properties (Du et al., 14 Nov 2025).

5. Construction via Differential Operators and Berends-Giele Currents

Scalar-loop YMS building blocks admit alternative but equivalent characterization through differential operations on the pure scalar-loop part: $$\mathcal{I}^{\text{scalar-loop}}_\alpha(1,...,n;\ell) = \mathcal{D}_\alpha^{(m)} \; \mathcal{I}_{\emptyset}(1,...,n;\ell)$$ Here, $\mathcal{I}_{\emptyset}$ is the all-gluon scalar-loop integrand, and the action of $\mathcal{D}_\alpha^{(m)}$ involves derivatives with respect to $\ell\cdot\epsilon_i$ and $\epsilon_b\cdot k_a$ with prescribed planar adjacency, encoding the replacement of gluons with scalars and the insertion of associated field-strengths (Cao et al., 27 Dec 2024, Chen et al., 2023).

In the language of Berends-Giele currents, the scalar-loop building block is retrieved by applying the dimension-raising operator (differentiation with respect to $D$ for spacetime dimension) on the gluonic pre-integrand, and trace operators which transmute gluon polarizations into scalars: $$I_S^{(1)}(\phi_X, g_{\setminus X} \mid \ell) = \left[\prod_{i \in X} \mathcal{T}[i\,n]\right] D[I^{(1)}_{\text{YM}}(1,...,n|\ell)]$$ (Chen et al., 2023)

6. Factorization, Leading Singularities, and Cut Consistency

Scalar-loop YMS blocks are constructed to ensure correct factorization across tree and loop cuts:

• Each propagator structure corresponds to a pinch of an underlying combinatorial or surface variable, e.g., cutting a loop-winding arc reduces the one-loop amplitude to a product of two tree amplitudes linked by a $1/\ell^2$ propagator (Arkani-Hamed et al., 2023).
• Leading singularity tests up to two loops systematically match the residues and symbols of known gauge-theory amplitudes in all cases examined (Arkani-Hamed et al., 2023).

7. Expansion into Bi-adjoint Scalar Integrands and Double-Copy Implications

After expressing the full YM integrand in terms of scalar-loop YMS building blocks, each block is further expanded over the basis of bi-adjoint scalar (BS) one-loop integrands: $$I^{\text{YMS}}_{\text{DCS}}(\ell; 1,\sigma \parallel G \mid \rho) = \sum_{\alpha \in \mathrm{shuffle}(\sigma, G)} C(\ell;1,\alpha) I^{\text{BS}}(\ell;1,\alpha \mid \rho)$$ This expansion isolates the BCJ numerators for Yang–Mills and controls the structure necessary for double-copy constructions. Compatibility with KLT double-copy at one loop requires imposing further physical conditions, notably concerning the loop-momentum parametrization and ensuring mutual locality of the double-copy summands. When welded tree-level YMS numerators satisfy the same cyclic and shift symmetries, they reproduce the unique one-loop BCJ numerators, making the scalar-loop YMS building blocks the core geometric and combinatorial object underlying one-loop color-kinematics dual representations and their gravity (double-copy) uplift (Du et al., 14 Nov 2025).

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References:

• Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories (Du et al., 14 Nov 2025)
• On one-loop amplitudes in gauge theories (Cao et al., 27 Dec 2024)
• Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory (Arkani-Hamed et al., 2023)
• Differential Operators and Unifying Relations for 1-loop Feynman Integrands from Berends-Giele Currents (Chen et al., 2023)
• One-loop amplitudes in Einstein-Yang-Mills from forward limits (Porkert et al., 2022)