Form factors in $ \mathcal{N} = 4 $ super Yang-Mills and periodic Wilson loops

Abstract

We calculate form factors of half-BPS operators in N = 4 super Yang-Mills theory at tree level and one loop using novel applications of recursion relations and unitarity. In particular, we determine the expression of the one-loop form factors with two scalars and an arbitrary number of positive-helicity gluons. These quantities resemble closely the MHV scattering amplitudes, including holomorphicity of the tree-level form factor, and the expansion in terms of two-mass easy box functions of the oneloop result. Next, we compare our result for these form factors to the calculation of a particular periodic Wilson loop at one loop, finding agreement. This suggests a novel duality relating form factors to periodic Wilson loops.

Lagrangian insertion in the light-like limit and the super-correlators/super-amplitudes duality

“…Each Lagrangian 5 We will be more specific about what we mean by these diagrams later. 6 Except of course through the outside of the polygon but as mentioned this will be interpreted as part of the other amplitude in the duality. insertion will supply a single derivative so the diagram will be proportional to:…”

“…Generalized unitarity can also be applied to objects containing local gauge-invariant operators such as correlation functions [5] and form factors [6,7,[26][27][28][29][30][31][32][33][34][35][36]. Since generalized unitarity is a momentum space method, the local operators will have to be Fourier transformed.…”

“…Finally the result is Fourier transformed back into position space [5]. This approach has many merits: form factors of some operators have been shown to have simple structures reminiscent of the ones found in scattering amplitudes [6,7], and, although the work cited here deals with N = 4 super-Yang-Mills, it could easily be applied to other theories. Unfortunately, correlation functions are best expressed in position space so some of the symmetries may not be apparent until after the Fourier transform.…”

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

“…all kinematical invariants one can encounter in form factor computation it is necessary to consider different closed contours [36,53,66,67] Γ k where momentum q is inserted at different positions among p i momenta (see figure 14). It is convenient to parametrize these contours with different sets of coordinates y k i :…”

“…So, one can think of y k i as of points on large periodical contour x i [36,53,66,67]. Sum over k runs through all inequivalent contours Γ k .…”