Emergence of Unitarity and Locality from Hidden Zeros at One-Loop Order

Recent investigations into the geometric structure of scattering amplitudes have revealed the surprising existence of “hidden zeros”: secret kinematic loci where tree-level amplitudes in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>Tr</a:mi><a:mo stretchy="false">(</a:mo><a:msup><a:mrow><a:mi>ϕ</a:mi></a:mrow><a:mrow><a:mn>3</a:mn></a:mrow></a:msup><a:mo stretchy="false">)</a:mo></a:mrow></a:math> theory, the nonlinear sigma model (NLSM), and Yang-Mills theory vanish. In this Letter, we propose the extension of hidden zeros to one-loop order in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Tr</e:mi><e:mo stretchy="false">(</e:mo><e:msup><e:mi>ϕ</e:mi><e:mn>3</e:mn></e:msup><e:mo stretchy="false">)</e:mo></e:math> theory and the NLSM using the “surface integrand” technology introduced by Arkani-Hamed We demonstrate that, under the assumption of locality, one-loop integrands in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>Tr</i:mi><i:mo stretchy="false">(</i:mo><i:msup><i:mi>ϕ</i:mi><i:mn>3</i:mn></i:msup><i:mo stretchy="false">)</i:mo></i:math> are unitary if and only if they satisfy these loop hidden zeros. We also present strong evidence that the hidden zeros themselves contain the constraints from locality, leading us to conjecture that the one-loop <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>Tr</m:mi><m:mo stretchy="false">(</m:mo><m:msup><m:mi>ϕ</m:mi><m:mn>3</m:mn></m:msup><m:mo stretchy="false">)</m:mo></m:math> integrand can be fixed by hidden zeros from a generically nonlocal, nonunitary . Near the one-loop zeros, we uncover a simple factorization behavior and conjecture that NLSM integrands are fixed by this property, also assuming neither locality nor unitarity. This Letter represents the first extension of such uniqueness results to loop integrands, demonstrating that locality and unitarity emerge from other principles even beyond leading order in perturbation theory.