Singularity-free Feynman integral bases

Standard integration-by-parts (IBP) reduction methods typically yield Feynman integral bases where the reduction of some integrals gives rise to coefficients singular as the dimensional regulator <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>ε</a:mi> <a:mo stretchy="false">→</a:mo> <a:mn>0</a:mn> </a:mrow> </a:math> . These singular coefficients can also appear in scattering amplitudes, obscuring their structure, and rendering their evaluation more complicated. We investigate the use of bases in which the reduction of any integral is free of singular coefficients. We present two general algorithms for constructing such bases. The first is based on sequential <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>D</d:mi> <d:mo>=</d:mo> <d:mn>4</d:mn> </d:math> IBP reduction. It constructs a basis iteratively by projecting onto the finite part of the set of IBP relations. The second algorithm performs Gaussian elimination within a local ring forbidding division by <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>ε</f:mi> </f:math> while permitting division by polynomials in <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>ε</h:mi> </h:math> finite at <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>ε</j:mi> <j:mo>=</j:mo> <j:mn>0</j:mn> </j:math> . We study the application of both algorithms to a pair of two-loop examples, the planar and nonplanar double-box families of integrals. We also explore the incorporation of finite Feynman integrals into these bases. In one example, the resulting basis provides a simpler and more compact representation of a scattering amplitude.