---
title: BCJ Relations in Gauge Theory
url: https://www.emergentmind.com/topics/bern-carrasco-johansson-bcj-relations
type: topic
---

# BCJ Relations in Gauge Theory

The Bern–Carrasco–Johansson (BCJ) Relations are a system of linear identities among color-ordered tree-level scattering amplitudes, discovered in the context of gauge theory and string theory. They arise from a conjectured duality between color and kinematics (color–kinematics duality), which posits that numerators in cubic diagram expansions of gauge-theory amplitudes can be arranged to obey the same Jacobi identities as color factors. The BCJ relations drastically reduce the number of independent partial amplitudes and underpin the double-copy construction of gravitational amplitudes from gauge-theory inputs.

## 1. Algebraic Structure and Statement of the BCJ Relations

Let $A_{\text{full}}(1,\dots,n)$ denote the full tree-level gluon amplitude, decomposed as
\[
A_{\text{full}}(1,\dots,n)=\sum_{\sigma\in S_n/\mathbb{Z}_n}\mathrm{Tr}\bigl(T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(n)}}\bigr)A_n\bigl(\sigma(1,2,\dots,n)\bigr),
\]
where $A_n$ are the color-ordered partial amplitudes. The fundamental BCJ relation for $n$ legs reads
\[
\sum_{j=2}^{n-1}\left(k_1\cdot k_{2+\cdots+j}\right)A_n(2,3,\dots,j,1,j+1,\dots,n) = 0,
\]
with $k_{2+\cdots+j}=k_2+k_3+\cdots+k_j$ and Mandelstam invariants $s_{1,2\cdots j}=2k_1\cdot k_{2\cdots j}$ [1111.5759, 1608.05291, 1508.01432].

This linear system, together with cyclic invariance and Kleiss–Kuijf (KK) relations, reduces the basis of $n$-point tree-level amplitudes to $(n-3)!$ independent color orderings.

In scalar cubic theories, an analogous relation holds for color-ordered adjoint scalar amplitudes $A_S$:
\[
\sum_{i=2}^{n-1}s_{1i}\,A_S(2,3,\dots,i,1,i+1,\dots,n) = 0
\]
where $s_{ij}=2p_i\cdot p_j$ [1105.3503].

## 2. Color–Kinematics Duality and Origin of BCJ Relations

The color–kinematics duality asserts that gauge-theory amplitudes can be arranged such that the structure constants $c_i$ of cubic graphs and kinematic numerators $n_i$ satisfy identical Jacobi relations:
\[
c_i + c_j + c_k = 0 \;\Longrightarrow\; n_i + n_j + n_k = 0.
\]
At tree level, this duality directly implies the existence of BCJ amplitude relations among partial amplitudes, as linear consequences of the kinematic Jacobi identities [1111.5759, 1608.05291, 1508.01432].

The duality is not unique to gluons: scalar cubic theories also satisfy BCJ relations via the antisymmetry and Jacobi identities of their structure constants [1105.3503].

## 3. On-Shell Proofs and Recursion Techniques

The BCJ relations can be proved using on-shell recursion relations (BCFW shifts). The key steps involve:

- Performing a complex deformation of a pair of external momenta, for instance, $\hat{p}_1(z) = p_1 + z q,\,\hat{p}_n(z) = p_n - z q$ with $q^2=0$.
- Studying the large-$z$ scaling of deformed amplitudes, where for nonadjacent leg shifts, $A_n(z) \sim z^{-2}$, enabling vanishing boundary terms.
- Constructing a contour integral $\oint_{|z|=\infty}\frac{dz}{z}$ of the BCJ sum, which vanishes by power counting.
- Demonstrating that residues at all physical factorization poles cancel among terms in the BCJ sum, yielding a vanishing result and thus proving the relation [1111.5759, 1608.05291, 1005.1821, 1508.01432].

In scalar theories, nonadjacent BCFW shifts produce nontrivial boundary terms that are handled by diagrammatic grouping and cancellations via the Jacobi identity [1105.3503].

Supersymmetric theories, notably $\mathcal{N}=4$ SYM, allow all component BCJ relations to be recovered from superfield BCFW recursion due to the absence of boundary contributions and the polynomial nature of component expansion in Grassmann variables [1005.1821].

## 4. Structural Reduction: Cyclicity, KK, and KLT Double Copy

A hierarchy links the various amplitude relations:
- **Cyclicity:** cyclic invariance of $A_n$ reduces the basis from $n!$ to $(n-1)!$ amplitudes.
- **Kleiss–Kuijf (KK) relations:** further reduction to $(n-2)!$ via $U(1)$ decoupling identities.
- **BCJ relations:** final reduction to $(n-3)!$ via the fundamental BCJ linear constraints [1111.5759].

This minimal BCJ basis directly enables the Kawai–Lewellen–Tye (KLT) double-copy construction of gravity tree amplitudes from gauge theory:
\[
M_n = \sum_{\alpha,\beta\in S_{n-3}}A_n(1,\alpha,n-1,n)\,S[\alpha|\beta]\,A_n(1,\beta,n,n-1),
\]
where $S[\alpha|\beta]$ is the KLT/BCJ kernel composed of Mandelstam invariants [1111.5759, 1608.05291, 1111.0544].

## 5. Extensions and Generalizations

### (a) Scalar, Fermion, and QCD Amplitudes

BCJ relations apply to amplitudes with external matter (quarks, scalars), provided at least one gluon is present. In QCD, the relations take the form
\[
\sum_{i=2}^{n-1}\left(\sum_{j=1}^i s_{2j}\right)A_n(1,3,\dots,i,2,i+1,\ldots,n-1,n) = 0,
\]
where 2 is the chosen gluon [1508.01432, 1608.05291].

### (b) String Theory and Amplitude Monodromy

Open string amplitudes at four points satisfy "stringy" BCJ relations:
\[
\sin(\pi k_2\cdot k_4)\,A_{st} = \sin(\pi k_1\cdot k_2)\,A_{tu},
\]
reflecting monodromy properties of the integration domain and generalizing the field-theory BCJ relation to all energies. In the $\alpha'\to0$ limit, this reduces to the Yang–Mills BCJ identity [1601.03813].

### (c) Gauge Invariance and Symmetry-Based Derivations

Recent proofs demonstrate that BCJ relations are enforced by a symmetry of gauge-theory amplitudes: invariance under momentum-dependent shifts of color factors, termed "color-factor symmetry." This perspective unifies string, field-theory, and diagrammatic proofs and is applicable to amplitudes with arbitrary massless/massive external states in any representation, so long as gluons are present [1608.05291].

### (d) Beyond Flat Space: AdS and Polytopal Generalizations

Color–kinematics duality, and hence BCJ relations, generalize to anti-de Sitter (AdS) correlators by replacing kinematic invariants with differential operators acting on embedding-space contact diagrams. The resulting BCJ-type relations among boundary correlators reduce, in the flat-space limit, to standard field-theory BCJ equations [2106.10822].

Generalizations also arise in twisted de Rham cohomology and positive-geometry approaches, where the number of independent BCJ basis elements matches the number of bounded chambers of accordiohedral polytopes, extending the usual $(n-3)!$ counting of the CHY/associahedral case [2009.10114].

## 6. Loop-Level Generalizations

The extension of BCJ relations to one-loop and higher-loop integrands is an area of active research. For one-loop amplitudes, integrand-level BCJ-type relations relate various partial integrands and are crucial for manifestly gauge-invariant KLT-type double-copy formulas at one loop [2312.01580, 1601.00235, 1612.00417, 2509.25129, 1507.06288]. The key structural property persists, with kinematic numerators and color factors entering one-loop graphs such that algebraic Jacobi identities among color factors induce corresponding relations among kinematic numerators and partial amplitudes.

## 7. Algebraic and Hopf-Algebraic Interpretations

Recent advances use the formalism of kinematic Hopf algebras to organize BCJ numerators and amplitude relations. The algebra is built from non-commutative generators labeled by flavor and kinematical structure, with the extended quasi-shuffle product encoding all symmetries, Jacobi identities, and amplitude relations. This structure makes the combinatorics and gauge invariance of BCJ numerators manifest and provides closed-form expressions for numerators satisfying all necessary constraints [2208.05519].

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**References:**
- [1111.5759] An Introduction to On-shell Recursion Relations
- [1105.3503] BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory
- [1608.05291] Color-factor symmetry and BCJ relations for QCD amplitudes
- [1508.01432] Proof of the fundamental BCJ relations for QCD amplitudes
- [1005.1821] U(1)-decoupling, KK and BCJ relations in $\mathcal{N}=4$ SYM
- [1206.5970] Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
- [2208.05519] Kinematic Hopf algebra for amplitudes and form factors
- [1601.03813] The String BCJ Relations Revisited and Extended Recurrence relations of Nonrelativistic String Scattering Amplitudes
- [2106.10822] BCJ Amplitude Relations for Anti-de Sitter Boundary Correlators in Embedding Space
- [2009.10114] On Polytopes and Generalizations of the KLT Relations
- [2312.01580] One-loop Bern-Carrasco-Johansson Numerators on Quadratic Propagators from the Worldsheet
- [1612.00417] New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level
- [2509.25129] Loop-Level Double Copy Relations from Forward Limits
- [1510.08846] Berends-Giele recursions and the BCJ duality in superspace and components
- [1111.0544] BCJ and KK Relations from BRST Symmetry and Supergravity Amplitudes

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**Summary Table of Structural Reductions:**

| Symmetry/Relation          | Number of Independent Orders  | Prototype Relation Form                                                    |
|---------------------------|-------------------------------|----------------------------------------------------------------------------|
| Cyclicity                 | $(n-1)!$                      | $ A_n(1,2,\dots,n)=A_n(2,3,\dots,1) $                                     |
| KK (Kleiss–Kuijf)         | $(n-2)!$                      | $ A_n(1,\{\alpha\},n,\{\beta\}) = (-1)^{|\beta|}\sum_{\sigma\in \text{OP}}A_n(1,\sigma,n) $ |
| BCJ                       | $(n-3)!$                      | $ \sum_{j=2}^{n-1}(k_1\cdot k_{2\cdots j})A_n(2,3,\dots,j,1,j+1,\dots,n)=0 $      |

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The BCJ relations are foundational to the modern analytic S-matrix program, directly underlie the color–kinematics duality and the double-copy construction of gravitational amplitudes, and have been generalized to multiple classes of quantum field theories and string models. The algebraic, geometric, and recursion-related structures emerging from these relations unify many aspects of gauge and gravity amplitudes.

Source: https://www.emergentmind.com/topics/bern-carrasco-johansson-bcj-relations