We study two-body scattering as an SU(N)-equivariant map acting on tensor-product representation spaces and analyze the entanglement generated by the S-matrix. This representation-theoretic perspective separates group structure from dynamics: the decomposition of R⊗R′ fixes the invariant operator algebra and therefore the qualitative entangling power of the process. For particles in the fundamental representation, EndSU(N)(N⊗N) = Span{𝕀, 𝕊}, so only the identity and swap directions preserve separability, whereas generic combinations generate entanglement. Adjoint-adjoint scattering involves a larger invariant algebra involving d-tensors and is intrinsically entangling. In Yang-Mills theory one can use color-kinematics duality to show that the color kernel lies on a fixed ray of this operator space, yielding a universal maximum of the outgoing entanglement for scattering at right angles, <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>E</mi> <mo>⋆</mo> <mfenced close=")" open="("> <mn>2</mn> </mfenced> </msubsup> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ {E}_{\star}^{(2)}=\frac{3}{4} $</EquationSource> </InlineEquation> for SU(2) and <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>E</mi> <mo>⋆</mo> <mfenced close=")" open="("> <mn>3</mn> </mfenced> </msubsup> <mo>≃</mo> <mn>0.91</mn> </math></EquationSource> <EquationSource Format="TEX">$ {E}_{\star}^{(3)}\simeq 0.91 $</EquationSource> </InlineEquation>, independent of kinematics. Dimension-six operators preserve this universality, while dimension-eight deformations populate new color sectors and shift <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>E</mi> <mo>⋆</mo> <mfenced close=")" open="("> <mi>N</mi> </mfenced> </msubsup> </math></EquationSource> <EquationSource Format="TEX">$ {E}_{\star}^{(N)} $</EquationSource> </InlineEquation>, suggesting that entanglement in color space functions as a tomographic probe of effective operators. In helicity space, requiring maximally entangled inputs to scatter into maximally entangled outputs uniquely selects the Yang-Mills quartic coupling and enforces the color Jacobi identity, restating the on-shell Ward constraints as conditions on entanglement preservation. Our results suggest that the information-theoretic viewpoint unifies algebraic, geometric, and dynamical aspects of scattering.

Entanglement, Yang-Mills, and the scattering matrix as an SU(N)-equivariant kernel

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2026年6月19日

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Kun-Feng Lyu,
Rahul Muraleedharan,
Kuver Sinha

摘要

We study two-body scattering as an SU(N)-equivariant map acting on tensor-product representation spaces and analyze the entanglement generated by the S-matrix. This representation-theoretic perspective separates group structure from dynamics: the decomposition of R⊗R′ fixes the invariant operator algebra and therefore the qualitative entangling power of the process. For particles in the fundamental representation, End_SU(N)(N⊗N) = Span{𝕀, 𝕊}, so only the identity and swap directions preserve separability, whereas generic combinations generate entanglement. Adjoint-adjoint scattering involves a larger invariant algebra involving d-tensors and is intrinsically entangling. In Yang-Mills theory one can use color-kinematics duality to show that the color kernel lies on a fixed ray of this operator space, yielding a universal maximum of the outgoing entanglement for scattering at right angles, $E_{⋆}^{(2)} = \frac{3}{4}$ $ {E}_{\star}^{(2)}=\frac{3}{4} $ for SU(2) and $E_{⋆}^{(3)} ≃ 0.91$ $ {E}_{\star}^{(3)}\simeq 0.91 $ , independent of kinematics. Dimension-six operators preserve this universality, while dimension-eight deformations populate new color sectors and shift $E_{⋆}^{(N)}$ $ {E}_{\star}^{(N)} $ , suggesting that entanglement in color space functions as a tomographic probe of effective operators. In helicity space, requiring maximally entangled inputs to scatter into maximally entangled outputs uniquely selects the Yang-Mills quartic coupling and enforces the color Jacobi identity, restating the on-shell Ward constraints as conditions on entanglement preservation. Our results suggest that the information-theoretic viewpoint unifies algebraic, geometric, and dynamical aspects of scattering.