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Register a new userA new method for evaluating scalar one-loop Feynman integrals in general space-time dimension
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Authors
Khiem Hong Phan
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In this paper, we propose a new method for evaluating scalar one-loop Feynman integrals in generalized D-dimension. The calculations play an important building block for two-loop and higher-loop corrections to the processes at future colliders such as the Large Hadron Collider (LHC) and the International Linear Collider (ILC). In this method, scalar one-loop N-point functions will be presented as the one-fold Mellin-Barnes representation of (N-1)-point ones with shifting space-time dimension. This representation offers a clear advantage that we can construct recursively the analytic expressions for N-point functions from the basic ones which are one-point functions. The compact formulae for scalar one-loop two-point functions with massive internal lines and three-point, four-point functions with massless internal lines are given as examples in this article. In particular, they are written in terms of generalized hypergeometric series such as Gauss, Appell F 1 functions. We also perform a sample numerical check for the analytical expressions in this report by comparing with LoopTools and AMBRE/MB. We find that the numerical results from this work are in good agreement with LoopTools at $epsilon^0$ -expansion and AMBRE/MB at higher-order of $epsilon$-expansion, at higher D-dimension.
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The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension $d$ has been solved for the basis of scalar one- to four-point functions with indices one. In 2003 the solution of difference equations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions $_2F_1$, vertices need additionally Kamp{e} de F{e}riet-Appell functions $F_1$, and box integrals also Lauricella-Saran functions $F_S$. In this study, alternative recursive Mellin-Barnes representations are used for the representation of $n$-point functions in terms of $(n-1)$-point functions. The approach enabled the first derivation of explicit solutions for the Feynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and box Feynman integrals and derive a numerical approach for the necessary Appell functions $F_1$ and Saran functions $F_S$ at arbitrary kinematical arguments.
Representations are derived for the basic scalar one-loop vertex Feynman integrals as meromorphic functions of the space-time dimension $d$ in terms of (generalized) hypergeometric functions $_2F_1$ and $F_1$. Values at asymptotic or exceptional kinematic points as well as expansions around the singular points at $d=4+2n$, $n$ non-negative integers, may be derived from the representations easily. The Feynman integrals studied here may be used as building blocks for the calculation of one-loop and higher-loop scalar and tensor amplitudes. From the recursion relation presented, higher n-point functions may be obtained in a straightforward manner.
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.
A complete one-loop matching calculation for real singlet scalar extensions of the Standard Model to the Standard Model effective field theory (SMEFT) of dimension-six operators is presented. We compare our analytic results obtained by using Feynman diagrams to the expressions derived in the literature by a combination of the universal one-loop effective action (UOLEA) approach and Feynman calculus. After identifying contributions that have been overlooked in the existing calculations, we find that the pure diagrammatic approach and the mixed method lead to identical results. We highlight some of the subtleties involved in computing one-loop matching corrections in SMEFT.
We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.
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