All-loop geometry for four-point correlation functions

Abstract

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. Published by the American Physical Society 2024

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