A periodic q̅ equation for form factors
Abstract
MHV form factors of the chiral stress-tensor multiplet in planar 𝒩=4 super Yang-Mills theory are dual to periodic null Wilson loops. We propose a periodic q̅ equation for the MHV and higher-helicity sectors at vanishing fermionic monodromy. The equation relates kinematic derivatives of NkMHV form factors to collinear integrals of Nk+1MHV 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 Q̅ equation, with coefficient Γcusp/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.
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BibTeX
@misc{he2026periodic,
title = {{A periodic $\bar q$ equation for form factors}},
author = {Song He and Jiahao Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30633},
eprint = {2610.30633},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30633},
}
RIS
TY - UNPB
TI - A periodic q̅ equation for form factors
AU - Song He
AU - Jiahao Liu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30633
UR - https://arxiv.org/abs/2610.30633
AB - MHV form factors of the chiral stress-tensor multiplet in planar 𝒩=4 super Yang-Mills theory are dual to periodic null Wilson loops. We propose a periodic q̅ equation for the MHV and higher-helicity sectors at vanishing fermionic monodromy. The equation relates kinematic derivatives of N^kMHV form factors to collinear integrals of N^k+1MHV 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 Q̅ equation, with coefficient Γ_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.
ER -