Principle of minimal singularity for Green’s functions
Wenliang Li
Abstract
Analytic continuations of integer-valued parameters can lead to profound insights, such as angular momentum in Regge theory, the number of replicas in spin glasses, the number of internal degrees of freedom, the spacetime dimension in dimensional regularization, and Wilson’s renormalization group. In this work, we consider a new kind of analytic continuation of correlation functions, inspired by two recent approaches to underdetermined Dyson-Schwinger equations in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>D</a:mi></a:math>-dimensional spacetime. If the Green’s functions <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>G</c:mi></c:mrow><c:mrow><c:mi>n</c:mi></c:mrow></c:msub><c:mo>=</c:mo><c:mo stretchy="false">⟨</c:mo><c:msup><c:mrow><c:mi>ϕ</c:mi></c:mrow><c:mrow><c:mi>n</c:mi></c:mrow></c:msup><c:mo stretchy="false">⟩</c:mo></c:mrow></c:math> admit analytic continuation to complex values of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>n</g:mi></g:math>, the two different approaches are unified by a novel principle for self-consistent problems: Singularities in the complex plane should be minimal. This principle manifests as the merging of different branches of Green’s functions in the quartic theories. For <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>D</i:mi><i:mo>=</i:mo><i:mn>0</i:mn></i:math>, we obtain the closed-form solutions of the general <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>g</k:mi><k:msup><k:mi>ϕ</k:mi><k:mi>m</k:mi></k:msup></k:math> theories, including the cases with complex coupling constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>g</m:mi></m:math> or noninteger power <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>m</o:mi></o:math>. For <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>D</q:mi><q:mo>=</q:mo><q:mn>1</q:mn></q:math>, we derive rapidly convergent results for the Hermitian quartic and non-Hermitian cubic theories by minimizing the complexity of the singularity at <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>n</s:mi><s:mo>=</s:mo><s:mi>∞</s:mi></s:math>. Published by the American Physical Society 2024