All-loop geometry for four-point correlation functions

In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>4</a:mn></a:math> super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mo>ℓ</d:mo></d:math>-loop geometry is attached. The loop geometry then consists of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mo>ℓ</f:mo></f:math> lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the can be viewed as fibration over a . The fibration naturally dissects the base into chambers, in which the degree-<h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mn>4</h:mn><h:mo>ℓ</h:mo></h:math> loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msubsup><j:mi>x</j:mi><j:mrow><j:mn>1</j:mn><j:mo>,</j:mo><j:mn>2</j:mn></j:mrow><j:mn>2</j:mn></j:msubsup><j:msubsup><j:mi>x</j:mi><j:mrow><j:mn>3</j:mn><j:mo>,</j:mo><j:mn>4</j:mn></j:mrow><j:mn>2</j:mn></j:msubsup></j:math>, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msubsup><l:mi>x</l:mi><l:mrow><l:mn>1</l:mn><l:mo>,</l:mo><l:mn>4</l:mn></l:mrow><l:mn>2</l:mn></l:msubsup><l:msubsup><l:mi>x</l:mi><l:mrow><l:mn>2</l:mn><l:mo>,</l:mo><l:mn>3</l:mn></l:mrow><l:mn>2</l:mn></l:msubsup></l:math>, and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi>x</n:mi><n:mrow><n:mn>1</n:mn><n:mo>,</n:mo><n:mn>3</n:mn></n:mrow><n:mn>2</n:mn></n:msubsup><n:msubsup><n:mi>x</n:mi><n:mrow><n:mn>2</n:mn><n:mo>,</n:mo><n:mn>4</n:mn></n:mrow><n:mn>2</n:mn></n:msubsup></n:math>. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mo>ℓ</p:mo><p:mo>=</p:mo><p:mn>3</p:mn></p:math>, which indeed yield correct loop integrands for the four-point correlator.