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Register a new userNumerical Approach to Calculation of Feynman Loop Integrals
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In this paper, we describe a numerical approach to evaluate Feynman loop integrals. In this approach the key technique is a combination of a numerical integration method and a numerical extrapolation method. Since the computation is carried out in a fully numerical way, our approach is applicable to one-, two- and multi-loop diagrams. Without any analytic treatment it can compute diagrams with not only real masses but also complex masses for the internal particles. As concrete examples we present numerical results of a scalar one-loop box integral with complex masses and two-loop planar and non-planar box integrals with masses. We discuss the quality of our numerical computation by comparisons with other methods and also propose a self consistency check.
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We report on the progress in constructing contracted one-loop tensors. Analytic results for rank R=4 tensors, cross-checked numerically, are presented for the first time.
Higher-order radiative corrections play an important role in precision studies of the electroweak and Higgs sector, as well as for the detailed understanding of large backgrounds to new physics searches. For corrections beyond the one-loop level and involving many independent mass and momentum scales, it is in general not possible to find analytic results, so that one needs to resort to numerical methods instead. This article presents an overview over a variety of numerical loop integration techniques, highlighting their range of applicability, suitability for automatization, and numerical precision and stability. In a second part of this article, the application of numerical loop integration methods in the area of electroweak precision tests is illustrated. Numerical methods were essential for obtaining full two-loop predictions for the most important precision observables within the Standard Model. The theoretical foundations for these corrections will be described in some detail, including aspects of the renormalization, resummation of leading loop contributions, and the evaluation of the theory uncertainty from missing higher orders.
A purely numerical method, Direct ComputationMethod is applied to evaluate Feynman integrals. This method is based on the combination of an efficient numerical integration and an efficient extrapolation. In addition, high-precision arithmetic and parallelization technique can be used in this method if required. We present the recent progress in development of this method and show results such as one-loop 5-point and two-loop 3-point integrals.
We discuss briefly the first numerical implementation of the Loop-Tree Duality (LTD) method. We apply the LTD method in order to calculate ultraviolet and infrared finite multi-leg one-loop Feynman integrals. We attack scalar and tensor integrals with up to six legs (hexagons). The LTD method shows an excellent performance independently of the number of external legs.
We present a new algorithm for the reduction of one-loop emph{tensor} Feynman integrals with $nleq 4$ external legs to emph{scalar} Feynman integrals $I_n^D$ with $n=3,4$ legs in $D$ dimensions, where $D=d+2l$ with integer $l geq 0$ and generic dimension $d=4-2epsilon$, thus avoiding the appearance of inverse Gram determinants $()_4$. As long as $()_4 eq 0$, the integrals $I_{3,4}^D$ with $D>d$ may be further expressed by the usual dimensionally regularized scalar functions $I_{2,3,4}^d$. The integrals $I_{4}^D$ are known at $()_4 equiv 0$, so that we may extend the numerics to small, non-vanishing $()_4$ by applying a dimensional recurrence relation. A numerical example is worked out. Together with a recursive reduction of 6- and 5-point functions, derived earlier, the calculational scheme allows a stabilized reduction of $n$-point functions with $nleq 6$ at arbitrary phase space points. The algorithm is worked out explicitely for tensors of rank $Rleq n$.
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