On-shell diagrams, Graßmannians and integrability for form factors

Abstract

We apply on-shell and integrability methods that have been developed in the context of scattering amplitudes in N = 4 SYM theory to tree-level form factors of this theory. Focussing on the colour-ordered super form factors of the chiral part of the stresstensor multiplet as an example, we show how to systematically construct on-shell diagrams for these form factors with the minimal form factor as further building block in addition to the three-point amplitudes. Moreover, we obtain analytic representations in terms of Graßmannian integrals in spinor helicity, twistor and momentum twistor variables. While Yangian invariance is broken by the operator insertion, we find that the form factors are eigenstates of the integrable spin-chain transfer matrix built from the monodromy matrix that yields the Yangian generators. Constructing them via the method of R operators allows to introduce deformations that preserve the integrable structure. We finally show that the integrable properties extend to minimal tree-level form factors of generic composite operators as well as certain leading singularities of their n-point loop-level form factors.

On soft theorems and form factors in N = 4 $$ \mathcal{N}=4 $$ SYM theory

“…After this paper was submitted to arXiv, the paper [69] appeared where some topics considered in this paper where further developed and generalized.…”

Grassmannians and form factors with q 2 = 0 in N $$ \mathcal{N} $$ =4 SYM theory

“…Special case of form factors of operators corresponding to "defect insertions" was considered in [49]. Integrability properties of 1/2-BPS form factors where investigated in [50][51][52][53][54] and in an important paper [55] with an explicit construction based on quantum inverse scattering method. Soft theorems in the context of form factors where considered in [54].…”

“…The study of Grassmannian representations for form factors was initiated in [54,55]. The more general case of form factors with q 2 = 0 was successfully considered in [55].…”

“…The more general case of form factors with q 2 = 0 was successfully considered in [55]. In principle one should be able to derive Grassmannian representation for q 2 = 0 case from the results of [55] by taking appropriate soft limit with respect to one of two spinor variables parameterizing operator's off-shell momentum q.…”

Syzygies probing scattering amplitudes

“…It can also be used to fix the ambiguity of the integrands of the non-planar amplitudes. Another application is to simplify the Grassmannian integral form of the N = 4 Super Yang-Mills non-planar amplitude [15][16][17][18][19][20][21]. Furthermore, syzygies can potentially be used to construct the loop-level scattering amplitudes from unitarity cuts [22,23] and to probe the amplitude relations beyond the KK-relation [24] and the BCJ-relation [25] in Yang-Mills theory.…”