Dual conformal invariance for form factors

Abstract

Form factors of the stress-tensor multiplet operator in N = 4 supersymmetric Yang-Mills reveal surprisingly simple structures similar to those appearing in scattering amplitudes. In this paper we show that, as for the case of amplitudes, they also enjoy dual conformal symmetry. We compute the dual conformal anomaly at one loop for an arbitrary number of particles and generic helicities, which matches the expression of the dual conformal anomaly of amplitudes, and perform explicit checks for MHV and NMHV one-loop form factors. In the NMHV case the realisation of dual conformal symmetry requires a delicate cancellation of offending terms arising from three-mass triangles, which we explicitly check in the case of the four-point NMHV form factor.

A three-point form factor through five loops

“…2 Form factors of a variety of operators have been calculated in N = 4 sYM theory using modern techniques originally developed in the context of scattering amplitudes. These include recursion relations [14][15][16][17], on-shell diagrams, polytopes and Graßmannians [18][19][20][21][22][23], twistor space actions [24][25][26][27], a connected prescription [28][29][30], color-kinematics duality [31][32][33], and a dual description via the AdS/CFT correspondence [14,[34][35][36][37][38]. These results are currently limited to two loops for n ≥ 3.…”

“…The FFOPE is based on the dual description of the form factor in terms of a periodic Wilson loop [14,34,35,37,38]. Similar to scattering amplitudes, this dual Wilson loop is defined via dual points x i , where x i+1 − x i = p i .…”

Form factor recursion relations at loop level

“…where the ∼ sign indicates that the identity holds up to an appropriate shift by some integer multiple of a period. It is important to rewrite the results introduced so far in terms of region variables for the purpose of establishing recursion relations at loop level discussed in this paper, and to associate to each diagram a well defined behaviour under dual conformal transformations described in detail in the companion paper [46]. In terms of region variables, one can rewrite (3.6) and (3.9) as…”

“…However q is not light-like, hence the argument of[48] does not apply here. Actually we will show in the companion paper[46] that the full dual conformal symmetry is preserved by form factor diagrams.…”

Complete form factors in Yang-Mills from unitarity and spinor helicity in six dimensions

“…Modern amplitude techniques were applied to form factors of TrF 2 , which compute the leading contribution to Higgs þ multigluon amplitudes in the effective Lagrangian approach, including MHV diagrams [31,32] at tree level [33,34] and one loop [35], and a combination of one-loop MHV diagrams and recursion relations [36]. Recent work [37][38][39][40][41] addressed the computation of the four-dimensional cut-constructible part of Higgs+multi-gluon scattering from operators of mass dimension seven using generalized unitarity [42,43] applied to form factors [39,40,[44][45][46][47][48][49][50][51][52][53][54][55]. The key point of this work is that we extend dimensional reconstruction to any form factor of operators involving vector fields, which requires the subtraction of form factors of an appropriate class of scalar operators that we identify.…”