A periodic $\bar q$ equation for form factors

Abstract

MHV form factors of the chiral stress-tensor multiplet in planar $\mathcal N=4$ super Yang-Mills theory are dual to periodic null Wilson loops. We propose a periodic $\bar q$ equation for the MHV and higher-helicity sectors at vanishing fermionic monodromy. The equation relates kinematic derivatives of $\mathrm N^k\mathrm{MHV}$ form factors to collinear integrals of $\mathrm N^{k+1}\mathrm{MHV}$ form factors with one additional external particle, providing a recursive relation in loop order. In periodic momentum supertwistors, it has the same collinear-integral form as the two dotted components of the amplitude $\bar Q$ equation, with coefficient $Γ_{\rm cusp}/4$, despite the absence of ordinary dual conformal invariance at fixed operator momentum. For parity-even scalar MHV remainders, combining the equation with its parity image supplies all directions of the kinematic differential. We verify the equation analytically for the one-loop four-point NMHV form factor and reconstruct the two-loop three-point MHV remainder at function level. Further two-loop MHV tests give numerical agreement with derivatives of the known four-point remainder at a fixed kinematic configuration, and for five points, all six chiral derivatives agree exactly at symbol level for a one-parameter family of kinematics. The structure of the collinear source further gives all-loop predictions for MHV last entries, consistent with known results and form a space of dimension $n(2n-5)$. The periodic Wilson-loop picture also motivates an all-loop derivation. Using the form-factor OPE, we extract the one-loop correction to the fermion energy from the logarithmic source and recover its vanishing at zero momentum.