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# ACAT 2019

10-15 March 2019
Steinmatte conference center
Europe/Zurich timezone

## Toward an efficient evaluation of two-loop massive scalar integralss

11 Mar 2019, 18:00
20m
Steinmatte Room C

#### Steinmatte Room C

Oral Track 3: Computations in Theoretical Physics: Techniques and Methods

### Speaker

Mr Jean-Philippe Guillet (LAPTh CNRS)

### Description

A key ingredient in an automated evaluation of two-loop multileg processes is a
fast and numerically stable evaluation of scalar Feynman integrals. In this respect, the calculation of two-loop three- and four-point functions in the general complex mass case so far relies on multidimensional numerical integration through sector decomposition whereby a reliable result has a high computing cost, whereas the derivation of a fully analytic result remains beyond reach. It would therefore be useful to perform part of the Feynman
parameter integrations analytically in a systematic way to let only a reduced
number of integrations to be performed numerically. Such a working program has been initiated for the calculation of massive two-loop $N$-point functions
using analytically computed building blocks. This approach is based on the implementation of two-loop scalar $N$-point functions in four dimensions $^{(2)}I_{N}^{4}$ as double integrals in the form:
$$^{(2)}I_{N}^{4} \sim \sum \int_{0}^{1} d \rho \int_{0}^{1} d \xi \, P(\rho,\xi) \; ^{(1)}\widetilde{I}_{N+1}^{4}(\rho,\xi)$$ where the building blocks $^{(1)}\widetilde{I}{N+1}^{4}(\rho,\xi)$ involved in the integrands are similar to "generalised" one-loop $(N+1)$-point Feynman-type integrals, and where $P(\rho,\xi)$ are weighting functions. The $^{(1)}\widetilde{I}{N+1}^{4}(\rho,\xi)$ are "generalised" in the sense that the integration domain spanned by the Feynman parameters defining them is no longer the usual simplex ${ 0 \leq z_{j} \leq 1, j = 1,\cdots,N+1; \sum_{j=1}^{N+1} z_{j}=1}$ at work for the one-loop $(N+1)$-point function, but another domain (e.g. a cylinder with triangular basis) which depends on the topology of the two-loop $N$-point function considered. The generalisation concerns also the underlying kinematics, which, besides external momenta, depends on two extra Feynman parameters $\rho$ and $\xi$. The parameter space spanned by this kinematics is larger than the one spanned in one-loop $(N+1)$-particle processes at colliders.The only two remaining integrations over $\rho,\xi$ to be performed numerically represent a substantial gain w.r.t. a fully numerical integration of the many Feynman parameter two-loop integrals. As a first step in this direction, the method developped has been successfully applied to the usual one-loop four-point function for arbitrary masses and kinematics as a proof of concept'', showing its ability to circumvent the subtleties of the various analytic continuations in the kinematical variables in a systematic way, in a series ofthree articles. The target work, namely its practical implementation to compute the building blocks $^{(1)}\widetilde{I}_{N+1}^{4}(\rho,\xi)$ is to be elaborated and presented
in a future series of articles.

### Primary authors

Mr Jean-Philippe Guillet (LAPTh CNRS) Mr Eric Pilon (LAPTh CNRS) Mr Yoshimitsu Shimizu (KEK) Mr Sadek Zidi (Jijel University, Algeria)