Analysis of symmetries in the causal loop-tree duality representations

Unveiling hidden symmetries within Feynman diagrams is crucial for achieving more efficient computations in high-energy physics. In this paper, we study the symmetries underlying the causal loop-tree duality representations through a graph-theoretic analysis. Focusing on the integrand-level representations of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi></a:math>-point functions at one loop, we examine their degeneracy and discover that different causal representations are interconnected through specific transformations arising from the symmetries of cut diagrams. Furthermore, the degeneracy is linked to algebraic constraints among the different causal thresholds. Our findings shed new light on the deeper structures of Feynman integrals and pave the way for significantly accelerating their calculation by interrelating different approaches.