NLO parton showers and subtractive techniques

Motivated by asking how to combine parton showers with nonleading QCD matrix elements, we discuss a subtractive technique based on gauge-invariant Wilsonline operators and how this can be used to treat the soft region. For precision phenomenology at future high energy colliders, it will be valuable to construct Monte Carlo event generators in which next-to-leading-order (NLO) QCD corrections can be incorporated in parton shower algorithms. There is at present no systematic method for doing this. Procedures based on subtractive methods have recently been proposed [1–3]. An important step in the implementation of this program is to show how to decompose Feynman graphs into sums of terms over different regions, with the terms arranged so as to correspond to factors in a factorization formula suitable for the Monte Carlo application. An example of such a decomposition is given in Ref. [2] for the photon-gluon fusion process in leptoproduction. In this case soft gluons do not enter at the leading power, so that leading regions do not overlap. To handle general cases, however, one needs to treat graphs with soft gluons and hence with overlapping leading regions. Fully numerical [4] or semi-analytical [5] subtraction methods have been devised to calculate NLO quantities that are infrared safe. These methods are not directly applicable in event generators that simulate the fully exclusive structure of the hadronic final states, since here the quantities being computed are not infrared safe in perturbation theory. In particular, one cannot use a cancellation of soft gluon contributions between real and virtual graphs. The technique we discuss in this talk is constructed so that the following properties are satisfied [3]: 1) Talk at the Linear Collider Workshop LCWS2000, Fermilab, 24-28 October 2000. 2) Work supported in part by the US Department of Energy. (a) The integrand for the hard scattering coefficient is to be an integrable function, even when the corrections are applied to a process that is not infrared safe. (b) The terms in the expansion of each Feynman graph should arise from matrix elements of gauge-invariant operators. (c) In particular, the necessary cut-offs on rapidity integrations should be defined gauge-invariantly. This involves the use of Wilson lines along non-lightlike directions [6]. (d) The evolution equations [7] with respect to these cut-offs should be simple, in the sense that there should be no power-law remainder terms. In what follows we illustrate this technique using a simple example, one-gluon emission graphs in leptoproduction. We consider a generic observable associated with the reaction γ(q) + q(p) → g(k) + q(k). We denote this by σ[φ], where φ is a weight function that contains the definition of the particular observable under consideration as a function of the final states as well as nonperturbative parts of the cross section, including the parton density. We denote by α and β the gluon’s fractional lightcone momenta, α = k/(xP), β = k/(Q/(2xP)) (with x the Bjorken variable, Q the photon virtuality, P the hadron’s plus momentum), and by φ the gluon’s azimuthal angle. We represent σ[φ] as