Genus drop in hyperelliptic Feynman integrals
Robin Marzucca, Andrew J. McLeod, Ben Page, Sebastian Pögel, Stefan Weinzierl
Abstract
The maximal cut of the nonplanar crossed box diagram with all massive internal propagators was long ago shown to encode a hyperelliptic curve of genus 3 in momentum space. Surprisingly, in Baikov representation, the maximal cut of this diagram only gives rise to a hyperelliptic curve of genus 2. To show that these two representations are in agreement, we identify a hidden involution symmetry that is satisfied by the genus 3 curve, which allows it to be algebraically mapped to the curve of genus 2. We then argue that this is just the first example of a general mechanism by means of which hyperelliptic curves in Feynman integrals can drop from genus <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>g</a:mi></a:math> to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo stretchy="false">⌈</c:mo><c:mi>g</c:mi><c:mo>/</c:mo><c:mn>2</c:mn><c:mo stretchy="false">⌉</c:mo></c:math> or <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">⌊</g:mo><g:mi>g</g:mi><g:mo>/</g:mo><g:mn>2</g:mn><g:mo stretchy="false">⌋</g:mo></g:math>. We find an algorithm to test for the presence of genus drop, and highlight further instances of this mechanism in Feynman integrals. Published by the American Physical Society 2024