Simplified Differential Equations Approach for Master Integrals

Abstract

In this talk we present the Simplified Differential Equations (SDE) Approach for Master Integrals (MI). Combined with the integrand reduction method for two-loop amplitudes it can pave the road for a fully automated NNLO calculation framework. Most recent achievements of the method, including double-box and pentabox MI calculations, are highlighted.

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

“…Here we consider an different approach, inspired by the integration technique used in [53], which is in turn inspired by well established integration techniques, see for instance [54,55]. Let us consider as an example the following Euler's expression B(2 + , 3 + 2 ) 2 F 1 ( , 2 + ; 5 + 3 ; z) = The third step is to truncate the sums at the desired order, work out the products and apply the shuffle algebra.…”

“…This problem can be solved by subtracting the main divergent part as follows (see [53]). Let's consider for simplicity an integrand with a divergence in x = 0 as for example x −1+ f (x) where f (x) is regular at x = 0.…”

The calculation of a massive planar pentabox with a differential equation method

“…The MI presented in this talk will be calculated with the SDE approach [26] and we refer to [29] for a detailed review 1 . We are interested in calculating the MI of two-loop QCD five-point amplitudes.…”

“…A very fruitful method for calculating and expressing MI in terms of GPs is the differential equations (DE) approach [15,16,17,18,19], which has been used in the past two decades to calculate various MI at two-loops [20,21,22,23,24,25]. In [26] a variant of the traditional DE approach to MI was presented, which was coined the Simplified Differential Equations (SDE) approach. In this talk we present an application of this method that first appeared in [27], concerning the calculation of planar massless MI relevant to five-point amplitudes with one off-shell leg, as well as the complete set of planar MI for five-point on-shell amplitudes.…”

Integration‐by‐parts identities in FDR

“…Thus, in a typical case, 10 4 − 10 6 integrals can be expressed in terms of O(10 − 100) MIs [14]. As for the actual computation of the MIs, IBP relations also serve as a starting point to write down linear systems of first-order differential equations in the kinematic invariants, that can be used for the determination of their analytic expressions [15][16][17][18][19][20]. Furthermore, methods exist to cast these systems in a canonical form in which all the analytical properties are explicitly exposed and the solution can be determined algebraically [21][22][23].…”