One-Loop BCJ Numerators in Gauge and Gravity

• One-loop BCJ numerators are kinematic functions assigned to cubic diagrams that satisfy Jacobi identities to manifest color-kinematics duality.
• The integrand oxidation algorithm constructs these numerators iteratively while ensuring proper loop-momentum dependence and adherence to symmetry constraints.
• They underpin double-copy constructions in gravity amplitudes by canceling spurious loop dependencies and revealing deep algebraic structures.

A one-loop BCJ numerator is a kinematic function assigned to each cubic one-loop diagram in gauge theory such that it satisfies the same algebraic Jacobi identities as the corresponding color factor, thereby rendering color-kinematics (Bern–Carrasco–Johansson, BCJ) duality manifest at the integrand level. These numerators are central to understanding the double-copy structure of gravity amplitudes and to the modern bootstrapping of perturbative gauge and gravity amplitudes, especially in theories with high supersymmetry. Their systematic construction, algebraic properties, and role in loop-level amplitudes are the subject of an extensive body of work, with explicit algorithms and nontrivial checks through high multiplicity. This article collects the foundational principles, explicit constructions, and key implications of one-loop BCJ numerators, with focus on four-dimensional and ten-dimensional maximally supersymmetric Yang–Mills theories.

1. Fundamental Structure and Duality

One-loop amplitudes in gauge theory admit a decomposition in terms of cubic diagrams,

$$\mathcal{A}_n^{\ell=1} = i\,g^n \sum_{\Gamma_i}\! \int\!\frac{d^D\ell}{(2\pi)^D}\, \frac{1}{S_i} \frac{n_i(\ell)\, c_i}{\prod_k P_{k i}(\ell)}\,,$$

where $c_i$ is the color factor for diagram $\Gamma_i$, $P_{k i}$ are the scalar propagators, $S_i$ is a symmetry factor, and $n_i(\ell)$ is the kinematic numerator. BCJ duality demands the existence of numerators $n_i(\ell)$ obeying the same Jacobi relations as the $c_i$: $$c_i + c_j + c_k = 0 \implies n_i(\ell) + n_j(\ell) + n_k(\ell) = 0.$$ Upon satisfying this duality ("color-dual numerators"), one can immediately write a gravity amplitude by the double-copy procedure,

$$\mathcal{M}_n^{\ell=1} = i(\kappa/2)^n \sum_{\Gamma_i} \int \frac{d^D\ell}{(2\pi)^D} \frac{1}{S_i} \frac{n_i(\ell)\ \tilde{n}_i(\ell)}{\prod_k P_{k i}(\ell)}.$$

This duality at one loop generalizes the manifest BCJ color-kinematic structure observed at tree level, but at loop level the explicit establishment of these properties is highly nontrivial and sensitive to loop-momentum routing ambiguities, contact (rational) terms, and power-counting constraints (Bjerrum-Bohr et al., 2013).

2. Explicit Construction: Integrand Oxidation and Algebraic Structure

The central method for constructing one-loop BCJ numerators in $\mathcal{N}=4$ SYM is the “integrand oxidation” algorithm (Bjerrum-Bohr et al., 2013). Rather than postulating a large ansatz and solving for all cubic numerators at once, the algorithm proceeds inductively:

• Initialization at boxes: Box (four-point) numerators are first constructed, typically as loop-momentum-independent rational functions in spinor or tensor invariants, and are fully symmetric under corner permutations.
• Stepwise oxidation: Given all $m$-gon numerators, $n_m$, one constructs the $(m{+}1)$-gon numerators $n_{m+1}$ by a sequence of adjacent Jacobi identities. Commuting one leg around the loop yields a finite difference equation,

$$n_{m+1}(a,b,\ldots;\ell) - n_{m+1}(a,b,\ldots;\ell-p_a) = x_{m+1}(a,b,\ldots)$$

where $x_{m+1}$ is built from known lower-point numerators. This structure limits the maximal loop-momentum power in $n_{m+1}$ to at most $m{-}3$.

• Global constraints: Vanishing of all triangle and bubble coefficients (by maximal supersymmetry), together with cyclicity and reflection symmetries, uniquely fix the polynomial dependence on $\ell$.
• Jacobi/integrand correspondence: The web of Jacobi identities is in direct correspondence with the system of algebraic integrand reductions, i.e., partial-fraction decompositions down to pentagons via trace-based coefficients (Bjerrum-Bohr et al., 2013).

For example, at five points (pentagon), the box numerators are $\ell$-independent; the pentagon numerator is also $\ell$-independent. At six points, the pentagon is now linear in $\ell$; the hexagon has both constant and linear $\ell$ dependence. The iterative hierarchy persists to higher multiplicities.

3. Explicit Examples: MHV, NMHV, and Self-dual Sectors

Through seven points, one-loop BCJ numerators have been worked out in explicit form for both MHV and NMHV sectors:

• MHV sector: All box numerators are $\ell$-independent rational functions of spinor brackets and traces (e.g., $n_4([1,2],3,4,5) = i\delta^8(Q)\frac{\langle12\rangle^2\langle34\rangle\langle45\rangle\langle53\rangle}{\operatorname{tr}(\slashed{p}_1\slashed{p}_2\slashed{p}_3\slashed{p}_4)}$). Pentagons and hexagons have polynomial structure in $\ell$ as determined by the oxidation constraints (Bjerrum-Bohr et al., 2013).
• NMHV sector: Starting from the known BDDK box coefficients, the same integrand oxidation, together with reflection and Jacobi relations, yields a linear system for the unknown numerator components. Loop-momentum dependence cancels precisely at pentagons and lower, leaving a unique color-dual representation that matches known NMHV results (Bjerrum-Bohr et al., 2013).
• Self-dual/amplitude relations: For all-plus amplitudes, dimension-shifting formulas relate the self-dual partial amplitude to the $D\to D+4$ shifted MHV amplitude; the relevant box numerators are identified by matching self-dual box cuts, and can be written in terms of D-dimensional unitarity cuts (Bjerrum-Bohr et al., 2013).

The hexagon and heptagon numerators at six and seven points involve power-counting-consistent quadratic (and cubic for higher $n$) $\ell$ dependence; all contact terms are fixed by global constraints and Jacobi plus reflection symmetries.

4. Algebraic Relations, Integrand Reduction, and Jacobi Consistency

The algebraic structure of one-loop BCJ numerators is tightly connected to partial-fraction relations in integrand reduction (Bjerrum-Bohr et al., 2013):

• The reduction of $m$-gon integrals to pentagons yields coefficients expressible as traces over external momenta, directly paralleling the structure of the corresponding Jacobi identities:

$\frac{1}{\prod_{i=1}^m P_i} = \sum \frac{r(\alpha_1|\ldots|\alpha_5)}{P_{j_1}P_{j_2}P_{j_3}P_{j_4}P_{j_5}},\quad r(12|3|4|5|6) = -\frac{\mathrm{tr}(\slashed{p}_3\slashed{p}_4\slashed{p}_5\slashed{p}_6)\mathrm{tr}(\slashed{p}_1\ldots\slashed{p}_6)}{\mathrm{tr}(\slashed{p}_1\ldots\slashed{p}_6)}$

• The cancellation of higher-power $\ell$ terms in each numerator against the contributions from lower-polygon oxidation is ensured by the unique solution to the Jacobi plus global-constraint system.
• These algebraic relations guarantee that only pentagon (and lower) integrals remain after integrand reduction and that they are independent of any spurious $\ell$ dependence.

5. Applications: Double Copy and Gravity Integrands

The BCJ numerators constructed in this framework form the essential input for double-copy constructions of $\mathcal{N}=8$ supergravity amplitudes (Bjerrum-Bohr et al., 2013). The gravity integrand is obtained as

$$\mathcal{M}_n^{\ell=1} = i\left(\frac{\kappa}{2}\right)^n \sum_{\Gamma_i} \int \frac{d^D\ell}{(2\pi)^D} \frac{1}{S_i} \frac{n_i(\ell)\, n_i(\ell)}{\prod_k P_{k i}},$$

where the "square" is in the space of color-dual numerators. Remarkably, all loop-momentum dependence cancels after integrand reduction down to pentagons and lower, reflecting the integrand-level color-kinematics duality. In six-point cases, all quadratics in loop momentum disappear upon summing pentagon-squared terms, leaving only scalar (momentum-independent) box integrals. This structure directly reproduces the ultraviolet and infrared features of $\mathcal{N}=8$ supergravity (Bjerrum-Bohr et al., 2013).

6. Generalizations, Loop-Level Symmetries, and Algorithmic Construction

• Loop-level symmetries: The one-loop BCJ numerators obey not only Jacobi but also cyclic and reflection symmetries, as required for integrand-level consistency and compatibility with color-ordering conventions (Bjerrum-Bohr et al., 2013).
• Algorithmic approaches: The bottom-up “oxidation” algorithm, the partial-fraction/trace polynomial relations, and the systematic avoidance of triangle/bubble integrals yield a blueprint for constructing BCJ numerators at arbitrary multiplicity and for different helicity sectors.
• Connection to self-dual theory: In the MHV sector, box numerators are tightly connected to the self-dual (all-plus) one-loop amplitude, and the explicit D-dimensional box cuts provide formulas for these numerators (e.g., $$n_4([[1,2],3],4,5,6) = -\frac{1}{2}\delta^8(Q)s_{12}(s_{23}C_{123|4|5|6} - s_{31}C_{213|4|5|6})$$) (Bjerrum-Bohr et al., 2013).

7. Impact and Outlook

One-loop BCJ numerators, as systematically constructed in the oxidation and integrand-reduction framework, reveal that color-kinematics duality is not simply an artifact of tree-level amplitudes but can be extended (with strict algebraic constraints) to the full loop-integrand level. Their existence and uniqueness—under global constraints and power-counting—provides a direct path to constructing manifestly color-kinematics-dual representations of amplitudes in maximally supersymmetric gauge and gravity theories, with important consequences for ultraviolet properties, infrared structure, and the landscape of connections to string theory via double copy.

Recent work has generalized these methods, providing deeper insight into the algebraic and geometric origins of BCJ numerators, their worldsheet/ambi-twistor construction, and the forward-limit relations to tree-level data. These developments establish the role of one-loop BCJ numerators as central algebraic objects in the modern theory of scattering amplitudes (Bjerrum-Bohr et al., 2013).