Form Factors in N=4 Super Yang-Mills and Periodic Wilson Loops

TL;DR

This paper computes form factors of half-BPS operators in ${\cal N}=4$ SYM at tree level and one loop, focusing on the Sudakov case and general two-scalar plus multiple-gluon configurations, by applying recursion relations and unitarity. The tree-level results are holomorphic and resemble MHV amplitudes, while the one-loop corrections are expressed through two-mass easy box functions and triangles with bubble cancellations. A key finding is the agreement between these form factors and a specific periodic Wilson loop calculation at one loop, suggesting a novel weak-coupling duality between form factors and periodic Wilson loops. The work extends amplitude/correlation-function relations in ${\cal N}=4$ SYM and points to new directions, including non-BPS operators and higher-loop generalizations.

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 one-loop 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.

Figures (10)

• Figure 1: In Figure (a) we show the diagram calculating the cut in the $q^2$-channel of the Sudakov form factor \ref{['ff']}. The cross denotes a form factor insertion. A second diagram with legs 1 and 2 swapped has to be added and doubles up the result of the first diagram. The result of this cut is given by (twice) a cut one-mass triangle function, depicted in Figure (b).
• Figure 2: The $q^2$-cut of the one-loop form factor. Note that the complete cut is obtained by summing over cyclic permutations of $(1, \ldots , n)$.
• Figure 3: The cut of the one-loop form factor in the $s_{a+1 , b-1}$-channel.
• Figure 4: On the left, we represent a two-mass easy box function. The momenta $p_a$ and $p_b$ are null, whereas, in general the remaining momenta $P:= P_{a+1 \, b-1}$ and $Q:=-q + P_{b+1 \, a-1}$ are not null. The cases when either $P^2$ or $Q^2$, or both, are also null, correspond to the one-mass and zero-mass boxes, obtained as smooth limits from the expression \ref{['2mebst']} of the two-mass box function. On the right, we represent a two-mass triangle arising from the cuts considered in Section \ref{['2mt']}. The thick line represents the momentum carried by the operator.
• Figure 5: The $(q-p_a)^2$-cut of the one-loop form factor.