In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an $\ell$-loop geometry is attached. The loop geometry then consists of $\ell$ lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the loop geometry can be viewed as fibration over a tree geometry. The fibration naturally dissects the base into chambers, in which the degree-$4\ell$ loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of $x_{1,2}^2{x}_{3,4}^2$, $x_{1,4}^2{x}_{2,3}^2$, and $x_{1,3}^2{x}_{2,4}^2$. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to $\ell =3$, which indeed yield correct loop integrands for the four-point correlator.

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• Gauge theories
• Large-N expansion in field theory
• Scattering amplitudes
• Supersymmetric field theories
• Conformal field theory