Color-kinematics duality and double copy

Rewrite a tree amplitude as a sum over cubic graphs. Four-point contact vertices can be distributed into graph numerators by multiplying and cancelling inverse propagators. Color factors satisfy antisymmetry and Jacobi identities with signs fixed by oriented cubic vertices. Dual numerators satisfy the corresponding signed relations. They are kinematic functions of polarizations and momenta, rather than another literal color Lie algebra attached to each external particle.

At four massless points choose orientations so the three color factors sum to zero and require the same relation for the three numerators. Denominators are s, t and u in the all-incoming convention, with their sum zero. Many numerator choices can represent the same amplitude. A generalized gauge shift changes the representation while leaving the color-dressed sum unchanged; it is not necessarily the same operation as changing an external polarization reference.

Consider shifting each numerator by alpha times its channel denominator. The gauge kernel changes by alpha times the color Jacobi sum and hence remains fixed. Because s plus t plus u vanishes, this particular shift also preserves the numerator Jacobi relation. The same shift in a double-copy kernel cancels when the replacement numerators satisfy the corresponding Jacobi identity. This is a direct algebraic explanation for the representation consistency in the example.

The physical construction requires more than Jacobi-looking numbers: input numerators must represent the appropriate gauge-theory amplitudes, with correct cuts, factorization and external states. Tree-level dual constructions are established in important theory classes. At loops, a proposed representation also requires its propagator routing, symmetry factors, state sums, regulator and cut coverage. A successful low-point algebra exercise does not establish local dual numerators for every theory and loop order.

Tensoring two gluon polarization spaces contains more than the graviton polarization alone. Pure Yang–Mills double copy generally produces gravity coupled to a dilaton and an antisymmetric tensor. Selecting external gravitons does not automatically remove those other states from loop sums. Obtaining a specified pure-gravity theory can require projections or compensating matter constructions whose consistency must be checked, rather than silently discarding unwanted internal states.

Likewise a compact gravity integrand does not by itself prove ultraviolet finiteness or a nonperturbative quantum-gravity completion. Integration, counterterms and infrared prescriptions remain necessary. Questions about locality, useful power counting, matter representations and regulator compatibility sharpen the domain of a duality claim. They should be posed for an explicit theory and amplitude class. The dependable statement is a relation between verified representations and their state content, not an unrestricted identity between every gauge process and Einstein gravity.