Systematic approach to ℓ -loop planar integrands from the classical equation of motion

In this paper, we present a recursive method for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mo>ℓ</a:mo> </a:math> -loop planar integrands in colored quantum field theories. We start with the classical equation of motion and then pick out the comb component, which will help us to define the loop kernels. Then we construct the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mo>ℓ</c:mo> </c:math> -loop integrands based on some recursion rules for the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mo>ℓ</e:mo> </e:math> -loop kernels. Finally, we reach a recursion formula for the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mo>ℓ</g:mo> </g:math> -loop planar integrands. Our method can be easily generalized to general quantum field theories, even non-Lagrangian theories, to obtain the planar part of the whole <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mo>ℓ</i:mo> </i:math> -loop integrands.