Monodromy-like relations for finite loop amplitudes

Abstract

Abstract:We investigate the existence of relations for finite one-loop amplitudes in YangMills theory. Using a diagrammatic formalism and a remarkable connection between tree and loop level, we deduce sequences of amplitude relations for any number of external legs.

Note on symmetric BCJ numerator

“…(1.2) This duality between the color and kinematic factors was later found to be present in a variety of Yang-Mills theories [2][3][4][5][6][7][8][9][10] and, perhaps most surprisingly, was shown to be valid at least for the first few loop levels [2,3,[5][6][7][8][9][11][12][13][14][15][16][17][18][19]. The apparently symmetrical structure also suggests mirror versions of the existing color decomposition formulations.…”

“…, n − 1) by construction. Since all permutations can be generated by successive permutations between the (n − 1) consequtive pairs, we can reduce our checking to the following two permutations: (12) and (n − 1)n. Now let us consider the numerators of dual-DDM form n 213...(n−1)n and n 123...n(n−1) . We have two ways to get the same n α from basis numerators: one is by relabeling n 123...(n−1)n and another one, by using Jacobi relation and antisymmetry.…”

Abelian Z-theory: NLSM amplitudes and α ′ -corrections from the open string

“…The "KK-like" relations among the ζ 2 -orders of open superstring amplitudes [112,117,118] for instance are known to annihilate permutation sums at multiplicities n ≥ 5 and therefore…”

Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes at all loop orders

“…Other recent work on loop-order relations among color-ordered amplitudes includes refs. [15][16][17][18][19][20].…”