calculated only once. The function F$_{i}$ is calculated on a grid of different values of x$_{1}$ , x$_{2}$, and µ$_{F}$ only once and interpolated between those points while evaluating the full cross section. This procedures significantly speeds up the calculations. 

Near threshold, the phase space for the emission of real gluons is limited and large logarithmic corrections to the above cross section calculation may arise from the incomplete cancellation of infrared divergences against the virtual gluon emission contributions. For jet production, these corrections are expected to contribute at very high x , where parton distributions are falling very steeply. For the inclusive jet cross section, these threshold corrections have been calculated to NNLO at next-to-leading-logarithmic accuracy [18] and are found to be small. However, the corrected cross section shows a substantial reduction of the scale µ$_{F}$ , µ$_{R}$ dependence. 

The partons radiate when they pass through a color field. In this type of radiation, two forms dominate, collinear and soft radiation. Collinear radiation is in the direction of parent parton, while soft radiation is just low energy emission. These soft and collinear radiation can be calculated in a leading-logarithmic approximation to all orders and this algorithm is a crucial component of event generators. In this formulism, a gluon radiates another gluon or converts into a q ¯ q pair according to DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) splitting functions [7] which depend on α$_{s}$ and a variable z which describes the energy sharing between two daughter partons. Similarly, a quark or an anti-quark can radiate a gluon which radiates further. This sequential radiation results in a shower of partons. The radiation process is continued until the parton virtual mass t is smaller than a mass scale t$_{0}$ . This simple procedure is augmented by angular ordering, i.e. each subsequent emission is required to have a smaller angle, to simulate color coherence effects. Color coherence leads to suppression of soft gluon radiation in certain regions of phase space. In the final state showers, the radiation is limited to a cone defined by color flow lines and the emission angle at each branch point is smaller than the previous emission angle. Partons with a virtual mass t ≤ t$_{0}$ ∼ 1 GeV /c 2 are combined into hadrons using a phenomenological hadronization model. The hadronization models have been tuned to reproduce the jet structure observed at e $^{+}$e − colliders. The showering process described above is used for radiation from outgoing partons and called final state radiation (FSR). For radiation from incoming partons, initial state radiation (ISR), the event generators use so-called backward evolution. First the momentum fractions x$_{1}$, x$_{2}$ of partons participating in the hard interaction are determined. 

