Dimensionally Regulated One-Loop Integrals

We describe methods for evaluating one-loop integrals in 4−2 dimensions. We give a recursion relation that expresses the scalar n-point integral as a cyclicly symmetric combination of (n− 1)point integrals. The computation of such integrals thus reduces to the calculation of box diagrams (n = 4). The tensor integrals required in gauge theory may be obtained by differentiating the scalar integral with respect to certain combinations of the kinematic variables. Such relations also lead to differential equations for scalar integrals. For box integrals with massless internal lines these differential equations are easy to solve. Submitted to Physics Letters B Research supported by the Texas National Research Laboratory Commission grant FCFY9202. Research supported by the Department of Energy under grant DE-AC03-76SF00515. †Laboratory of the Direction des Sciences de la Matière of the Commissariat à l’Energie Atomique of France. Many processes of interest at current and future e+e− and hadron colliders involve large numbers of final state particles. Radiative corrections to these processes are needed for precise tests of the standard model. It is therefore useful to have techniques for evaluating one-loop integrals where the number of external legs is large. As the loop integrals appearing in radiative corrections are often infrared and/or ultraviolet divergent, it is desirable to regulate them by performing them in 4− 2 dimensions. In this Letter, we derive a relation between the n-point and (n− 1)-point one-loop integrals, which for n > 4 allows the recursive determination of the general n-point scalar integral in D = 4−2 , as a linear combination of box integrals (n = 4), provided only that the external momenta are restricted to lie in four dimensions, and neglecting O( ) corrections. The required box integrals can generally be evaluated in closed form through O(1), that is to say, with all poles in manifest, and with all functions of the kinematic invariants expressed in terms of logarithms and dilogarithms. (A compact expression for the general infrared-finite box integral has recently been given by Denner, Nierste, and Scharf [1]; the infrared-divergent box integrals with all internal lines massless are collected in ref. [2].) Therefore, the higher-point integrals can now be represented in the same closed form. In a separate paper [2], we apply these techniques to determine explicitly the pentagon integral with all external lines massless, or with one external mass. Various authors [3,4,5,6] have discussed the computation of pentagon and higher-point integrals that can be evaluated in D = 4 (i.e. that are infrared finite). In particular, Melrose [3] and independently van Neerven and Vermaseren [5] have expressed the D = 4 pentagon integral as a linear combination of five D = 4 box integrals, and the relation we find for n = 5 may be thought of as the dimensionally-regulated version of their equations. References [3,5] also express the fourdimensional n-point scalar integral for n ≥ 6 (with external momenta restricted to D = 4) as a sum of six (n − 1)-point integrals; the derivation in ref. [5] extends straightforwardly to (4− 2 )dimensional loop-momenta as well. For n > 6 these relations are of a somewhat different type than the relations that we find. We have been informed that Ellis, Giele, and Yehudai [7] have recently evaluated the D = 4− 2 pentagon integrals by an independent technique. In gauge theories, tensor integrals appear in which the n-point integral may contain up to n powers of the loop momentum in the numerator of the integrand. It is possible to perform a Brown-Feynman [8] or Passarino-Veltman [9] reduction of the integrand, solving a system of algebraic equations to reduce the tensor integrals to a linear combination of scalar integrals [10]. The framework developed here provides another method for computing tensor integrals. Feynman parametrization converts tensor integrals into integrals where polynomials in the Feynman param-