Double copy beyond tree level and perturbation theory

6 problems

• Double copy for scattering in anti-de Sitter space

  Partially resolved

  The squaring trick is proven for tree-level scattering in flat spacetime. Whether it works in anti-de Sitter space, a curved spacetime used in holography, is unclear.

• Exact classical double copy for generic vacuum spacetimes

  Partially resolved

  Some exact black-hole solutions of Einstein's equations are squares of simple electromagnetic solutions. Whether every solution of Einstein's equations has such a gauge-theory partner is unknown.

• Color-kinematics dual form of the five-loop $N=4$ super-Yang-Mills amplitude

  Open

  The squaring trick needs the gauge-theory answer arranged so that its momentum-dependent parts obey the same algebra as its color (charge) parts. At five loops nobody has found such an arrangement for four-gluon scattering in the most symmetric gauge theory.

• Kinematic Lie algebra of full Yang-Mills theory

  Partially resolved

  Color factors come from a known algebra, the rules for combining color charges. The matching algebra for the momentum-dependent parts is known only for a simplified self-dual sector of the theory.

• Is four-dimensional $N=8$ supergravity ultraviolet finite at seven loops?

  Open

  $N=8$ supergravity is the most symmetric theory of gravity in four dimensions and might lack the infinities that ruin ordinary quantum gravity. Known symmetries first allow an infinity at seven loops, a calculation not yet done.

• Local color-kinematics duality for one-loop pure Yang-Mills amplitudes

  Partially resolved

  Even at one loop it is unclear whether gluon amplitudes of ordinary Yang-Mills theory, with no supersymmetry, can be put in the special form using simple polynomial numerators.