Wilson Loop Form Factors: A New Duality

Abstract

We find a new duality for form factors of lightlike Wilson loops in planar N = 4 super-Yang-Mills theory. The duality maps a form factor involving an n-sided lightlike polygonal super-Wilson loop together with m external on-shell states, to the same type of object but with the edges of the Wilson loop and the external states swapping roles. This relation can essentially be seen graphically in Lorentz harmonic chiral (LHC) superspace where it is equivalent to planar graph duality. However there are some crucial subtleties with the cancellation of spurious poles due to the gauge fixing. They are resolved by finding the correct formulation of the Wilson loop and by careful analytic continuation from Minkowski to Euclidean space. We illustrate all of these subtleties explicitly in the simplest non-trivial NMHV-like case. and the identification of coordinates with momenta (1.3) to their supersymmetric analogs. The super-Wilson loop form factors are considered in the planar limit and in the lowestorder perturbative approximation (Born level). The introduction of Grassmann variables (θ i on the Wilson loop contour and η j for the on-shell states) allows us to probe the duality for more complicated configurations of particle helicities. By analogy with the amplitudes, we call the contributions at the lowest level in the Grassmann expansion MHV-like, at the next level NMHV-like, etc. At MHV level we confirm the result of [20]. The NMHV level is much more complicated, the form factor being a non-trivial rational function of the kinematical data. Yet, we show that the duality still works, in a rather simple and suggestive way, by just matching planar Feynman diagrams. This allows us to argue that it should hold for the complete supersymmetric object (at all Grassmann levels) and also beyond the Born approximation.The key to understanding the duality is the appropriate superspace formulation of the Wilson loop and its form factor. In the conventional approach the chiral supersymmetric Wilson loop [4,5,6] is formulated in terms of constrained on-shell super-connections [21,22], which makes the Feynman diagram technique highly inefficient. In this paper we prefer to use the Lorentz harmonic chiral (LHC) superspace approach [23]. It provides an offshell formulation of the chiral N = 4 SYM theory in terms of unconstrained prepotentials, best suited for supersymmetric quantisation. LHC superspace is an alternative to the twistor formulation [24,25], closer in spirit to traditional field theory (see also [26]). The main idea is to consider the interacting theory as a perturbation of the self-dual sector. The twistor formulation has been successfully used to justify the so-called MHV rules for the computation of the amplitude [27], to prove the duality between supersymmetric Wilson loops and amplitudes [5], to compute off-shell correlation functions of the N = 4 stress-tensor multiplet [28]. More recently, the LHC formalism was applied to finding the non-chiral completion of the correlators [29] and to the calculation of form facto...

The twistor Wilson loop and the amplituhedron

“…This produces a pole in the WLD. There is an interesting three-way cancellation of this type of spurious pole between three related diagrams (see [34,38] for previous work also describing this mechanism). An example set of diagrams is shown in Figure 6.…”

“…Here, we provide a brief description of planar Wilson loops in N = 4 Super Yang Mills in super twistor space and define the WLDs that arise. We do not derive these here, for their derivation see [32][33][34].…”

Four-dimensional ambitwistor strings and form factors of local and Wilson line operators

“…The general practice when studying form factors is to consider the case of local gauge invariant color singlet operators. However, one may also consider gauge invariant (the representation under global gauge transformation is not necessary singlet) non-local operators, for example Wilson loops (lines) or their products [53][54][55][56][57][58][59][60][61][62][63][64][65]. An insertion of Wilson line operator will then correspond to the off-shell or reggeized gluon in such formulation.…”

“…Also the connected prescription formulae were extended to Standard Model amplitudes [72]. Besides, there are several results for the Grassmannian integral representation of form factors of operators from stress-tensor operator supermultiplet [73][74][75][76] and Wilson line operator insertions [63,64], see also [65] for a recent interesting discussion of duality for Wilson loop form factors 1 .…”

Ambitwistor strings and reggeon amplitudes inN=4SYM

“…Also some formulae were extended to Standard Model amplitudes [53]. Besides, there are several results for the Grassmannian integral representation of form factors of operators from stress-tensor operator supermultiplet [54][55][56][57] and reggeon amplitudes (form factors 1 of Wilson line insertions) [58,59], see also [60] for a recent interesting duality for Wilson loop form factors. A very close research direction is the twistor and Lorentz harmonic chiral superspace formulation of form factors and correlation functions developed in [61][62][63][64][65][66][67], see the discussion in conclusion.…”