Abstract
We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter x and the dimension d, in the whole kinematic range of x. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the overline{mathrm{MS}} and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy.
Highlights
An interesting approach to obtain numerical results of loop integrals has been presented in ref. [14] where an imaginary mass is added to all propagators
We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter x and the dimension d, in the whole kinematic range of x
In this paper we present a numerical method to compute multiloop integrals with two mass scales, i.e., one dimensionless parameter
Summary
We describe a method to obtain numerical results of Feynman integrals. In general there are other exceptional points depending on the physical application under consideration In such cases one often proceeds as follows: one establishes the differential equations with respect to the variable x for the master integrals [7, 8]. The method, which is described in the following, does not have such limitations It is, to a large extent, insensitive to the complexity of the differential equations since only expansions around certain kinematical points are considered. Proceeding this way one, in principle, ends up with an over-determined system of linear equations which in general has no solution due to the numerical errors introduced by truncating the expansions To circumvent this issue we proceed as follows: we start with the simplest master integrals and fix the constant for the leading pole. For the integral families considered in this paper we did not encounter such additional singularities
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
