On form factors in N=4 sym

TL;DR

This work develops a perturbative analysis of form factors in planar N=4 SYM for both protected (half-BPS) and unprotected (Konishi) operators, and extends to general Δ0=n for half-BPS operators. Using N=1 superspace and D-algebra, the authors compute one- and two-loop form factors, discovering IR exponentiation governed by the cusp and collinear anomalous dimensions, with Γ_cusp^(1)=4 and Γ_cusp^(2)=-8 ζ2; for Konishi they find γ_K^(1)=3/(8π^2). Finite parts involve intricate multi-variable polylogarithms (Goncharov) and Appell functions, while a dual conformal invariant structure emerges, hinting at an underlying integrable framework and a possible Wilson loop–form factor duality. The results reinforce the deep connections between IR structure, dual conformal symmetry, and potential integrability in form factors of N=4 SYM, and point toward extending on-shell methods to these observables.

Abstract

In this paper we study the form factors for the half-BPS operators $\mathcal{O}^{(n)}_I$ and the $\mathcal{N}=4$ stress tensor supermultiplet current $W^{AB}$ up to the second order of perturbation theory and for the Konishi operator $\mathcal{K}$ at first order of perturbation theory in $\mathcal{N}=4$ SYM theory at weak coupling. For all the objects we observe the exponentiation of the IR divergences with two anomalous dimensions: the cusp anomalous dimension and the collinear anomalous dimension. For the IR finite parts we obtain a similar situation as for the gluon scattering amplitudes, namely, apart from the case of $W^{AB}$ and $\mathcal{K}$ the finite part has some remainder function which we calculate up to the second order. It involves the generalized Goncharov polylogarithms of several variables. All the answers are expressed through the integrals related to the dual conformal invariant ones which might be a signal of integrable structure standing behind the form factors.

Figures (7)

• Figure 1: Feynman diagram for the matrix element of the operator $\mathcal{O}$
• Figure 2: The relevant supergraphs. The internal black lines correspond to chiral propagators $\langle \bar{\Phi}_I^a\Phi_J^b \rangle$, wavy lines correspond to vector $\langle V^aV^b \rangle$ propagator (see App. A). $C0$ is the tree level diagram, and the rest are one-loop ones. External lines are $\Phi$ or $\bar{\Phi}$, and the lower bold line represents the insertion of the corresponding operator in modern notation. For the chiral operator $\mathcal{C}_{IJ}$ only the diagrams $C0$ and $C1$ contribute, while for non-chiral operators $\mathcal{V}_{I}^{J}$ and $~\mathcal{K}$ the other two ($B1$ and $B2$) are also relevant.
• Figure 3: The relevant supergraphs in the chiral case. $C1$ is the one-loop diagram, and the rest are two-loop ones. For the chiral operator $\mathcal{C}_{IJ}$ with two legs the last two diagrams $C7$ and $C8$ do not exist, they are only relevant for the operator $\mathcal{O}_n$ with $n\geq 3$. A grey circle is the one-loop effective vertex.
• Figure 4: The tree contribution to $\mathcal{O}^{(n)}_I$.
• Figure 5: The one-loop triangle diagram from the one-loop box diagram. The red dot should be taken to infinity, and the blue line (propagator in momentum space) should be contracted to a point.