All-Loop Geometry for Four-Point Correlation Function

In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar N=4 super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an -loop geometry is attached. The loop geometry then consists of 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 loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of x²₁₂x²₃₄, x²₁₄x²₂₃ and x²₁₃x²₂₄. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to =3, which indeed yield correct loop integrands for the four-point correlator.