Abstract
In this article we present a method to generate analytic expressions for the integral coefficients of loop amplitudes using numerical evaluations only. We use highprecision arithmetics to explore the singularity structure of the coefficients and decompose them into parts of manageable complexity. To illustrate the usability of our method we provide analytical expressions for all helicity configurations of the colour-ordered six-point gluon amplitudes at one loop with a gluon in the loop.
Highlights
In this article we present a method to generate analytic expressions for the integral coefficients of loop amplitudes using numerical evaluations only
To illustrate the usefulness of our method, we present analytical expressions for colorordered six-gluon one-loop amplitudes with a gluon in the loop for all helicity configurations
The method is based on the iteration of the following steps: 1. evaluate E in singular limits to obtain the list of all factors in the least common denominator (LCD) and their exponents; 2. consider E in doubly singular limits to expose the dependency structure of the poles; 3. select a pole from the LCD and identify the set of necessary other factors needed in the denominator to fit its residue; 4. subtract the term obtained from E and reiterate from step 1
Summary
In this article we consider reconstructing an analytical expression for a rational quantity E for which we can calculate numerical values for arbitrary kinematics with arbitrary accuracy. Using complex momenta we can treat λi and λi as separate independent variables. This allows us to explore the singularity structure of our expression in a more controlled fashion. The method is based on the iteration of the following steps: 1. Evaluate E in singular limits to obtain the list of all factors in the least common denominator (LCD) and their exponents; 2. Consider E in doubly singular limits to expose the dependency structure of the poles; 3. Select a pole from the LCD and identify the set of necessary other factors needed in the denominator to fit its residue; 4. The following sections explain the elements of the method in more details
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