SU(N) group-theory constraints on color-ordered five-point amplitudes at all loop orders

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

“…Four-point color-ordered amplitudes were also known to obey group-theory relations at two loops [12] and these were recently generalized to all loop orders [13]. All-loop-order group-theory constraints were also recently derived for five-point amplitudes [14]. Other recent work on loop-order relations among color-ordered amplitudes includes refs.…”

“…[13,14] and generalizing the known relations at tree level [10] and at one loop [11]. The other major goal of this paper is to describe how group-theory constraints on n-point color-ordered amplitudes naturally fall into sets associated with irreducible representations of the symmetric group S n .…”

“…At two loops and above, the full set of constraints on n-point color-ordered amplitudes has yet to be identified 6 , although all-loop group-theory relations for n = 4 and n = 5 have been derived using an iterative approach in refs. [13,14]. 4 For example, the number of independent color factors for tree-level n-point amplitudes is (n − 2)!…”

“…Again the form of the constraints repeats after two loops, so that odd-loop constraints for L ≥ 3 are equivalent to the one-loop constraints, and even-loop constraints for L ≥ 4 are equivalent to the two-loop constraints [14].…”

“…[13,14] by specifying a complete set of L-loop null vectors {r λ }. While this approach has the advantage of explicitness, it quickly becomes unwieldy as the dimensions of both the null space and the trace space grow with n. For example, listing the 200 coefficients of the 80 null vectors for two-loop six-point amplitudes would hardly be enlightening.…”