Constructing QCD one-loop amplitudes

Abstract

In the context of constructing one-loop amplitudes using a unitarity bootstrap approach we discuss a general systematic procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted after examining the behavior of the cut integrand as the unconstrained parameters of a specifically chosen parameterization of the cut loop momentum approach infinity. Measurements of new physics at the forthcoming experimental program at CERN's Large Hadron Collider (LHC) will require a precise understanding of processes at next-to-leading order (NLO). This places increased demands for the computation of new one-loop amplitudes. This in turn has spurred recent developments towards improved calculational techniques. Direct calculations using Feynman diagrams are in general inefficient. Developments of more efficient techniques have usually centered around unitarity techniques [1], where tree amplitudes are effectively 'glued' together to form loops. The most straightforward application of this method, in which the cut loop momentum is in D = 4, allows for the computation of 'cut-constructible' terms only, i.e. (poly)logarithmic containing terms and any related constants. QCD amplitudes contain, in addition to such terms, rational pieces which cannot be derived using such cuts. These 'missing' rational parts can be extracted using cut loopmore » momenta in D = 4-2 {var_epsilon}. The greater difficulty of such calculations has restricted the application of this approach, although recent developments [3, 4] have provided new promise for this technique. Recently the application of on-shell recursion relations [5] to obtaining the 'missing' rational parts of one-loop processes [6] has provided an alternative very promising solution to this problem. In combination with unitarity methods an 'on-shell bootstrap' approach provides an efficient technique for computing complete one-loop QCD amplitudes [7]. Additionally other new methods have also proved fruitful for calculating rational terms [8]. Such developments have again refocused attention on the optimization of the derivation of the cut-constructable pieces of the amplitude. Deriving cut-constructible terms for any one-loop amplitude reduces to the computation of coefficients of a set of scalar bubble, scalar triangle and scalar box integral functions. Box coefficients may be found with very little work, directly from the quadruple cut of the relevant box function [9]. A unique box coefficient contributes to each distinct quadruple cut. Unfortunately triangle and bubble coefficients cannot be derived in quite so direct a manner. Multiple scalar integral coefficients appear inside a two-particle cut or triple cut. It is therefore necessary to disentangle the relevant bubble or triangle coefficients from any other coefficients sharing the same cut [1, 4, 10, 11]. The large number of NLO processes of interest for the LHC suggests that a completely automated computational procedure is highly desired. To this end we discuss, in this proceeding, a recently proposed method [12, 13] for the direct, efficient and systematic extraction of bubble and triangle coefficients which is well suited to automation.« less