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qcd four - jet production in @xmath1 annihilation can be measured at lep and can be studied in its own right .
first of all , @xmath2 is the lowest order process which contains the non - abelian three - gluon - vertex at tree level and thus allows for an measurement of the colour factors @xmath3 , @xmath4 and @xmath5 of qcd .
this in turn may be used to put exclusion limits on light gluinos .
furthermore the qcd process is a background to w - pair production , when both w s decay hadronically and to certain channels of the search of the higgs boson like @xmath6 .
the one - loop matrix elements required for an nlo study of four - jet production are also an essential input for an nnlo calculation of three - jet production .
the latter one would be needed to reduce theoretical uncertainties in the extraction of the strong coupling at the z - pole .    in general leading - order calculations in qcd
give a rough description of the process under consideration , but they suffer from large uncertainties .
the arbitrary choice of the renormalization scale gives rise to an ambiguity , which is reduced only in an next - to - leading order calculation .
furthermore the internal structure of a jet and the sensitivity to the merging procedure of the jet algorithm are modelled only in an nlo analysis .
both uncertainties are related to the appearance of logarithms , ultraviolet in nature in the first case , infrared in the latter , which are calculated explicitly only in an nlo calculation .
a nlo calculation proceeds in two steps : first , one needs the relevant amplitudes , in our case @xmath7 and @xmath8 at tree level and @xmath7 at one loop . among these , the one - loop amplitudes are the most complicated ones and we will comment on their calculation in the next section .
the second step requires setting up a numerical monte carlo program which has to deal with infrared divergences .
we will focus on this point in the third section . in the last section we will give some numerical results .
we used a variety of modern techniques in order to calculate the one - loop amplitudes efficiently .
these include colour decomposition , where amplitudes are decomposed into simpler gauge - invariant partial amplitudes with definite colour structure , and the spinor helicity method , which consists in expressing all lorentz four - vectors and dirac spinors in terms of massless two - component weyl - spinors .
their use divides the task into smaller , more manageable pieces . also a decomposition inspired by supersymmetry proved to be useful , where the particles running around the loop are reexpressed in terms of supermultiplets . in a second step
the cut technique and factorization in collinear limits are used to constrain the analytic form of the partial amplitudes .    as an example
we explain in more detail the cut technique @xcite , which is based on unitarity . to obtain the coefficients of the basic box , triangle or bubble integrals
one considers the cuts in all possible channels .
each phase - space integral is rewritten with the help of the cutkosky rules as the imaginary part of a loop amplitude .
the power of this method lies within the fact , that on each side of the cut one has a full tree amplitude and not just a single feynman diagram .
this method allows one to reconstruct the one - loop amplitude up to terms without an imaginary part .
the remaining terms were obtained by examining the collinear limits .    for the reduction of tensor pentagon integrals we used a new reduction algorithm @xcite , based on the schouten identity and weyl spinors , which does not introduce artifical gram determinants in the denominator .
the one - loop amplitudes for the first subprocess @xmath9 were calculated in refs . @xcite and the amplitudes for the second subprocess @xmath1 @xmath10 in refs . @xcite .
the calculations of the two groups agree with each other .
the second major part of a general purpose nlo program for four jets is coding the one - loop amplitudes and the five parton tree - level amplitudes in a numerical monte carlo program . at leading order
the task is relatively simple : one parton corresponds to one jet . at nlo however , a jet can be modeled by two partons . at nlo
the cross section receives contributions from the virtual corrections and the real emission part .
only the sum of them is infrared finite , whereas when taken separately , each part gives a divergent contribution .
several methods to handle this problem exist , such as the phase - space slicing method @xcite , the subtraction method @xcite and the dipole formalism @xcite .
we have chosen the dipole formalism .
within the dipole formalism one subtracts and adds again a suitably chosen term : ^nlo & = & _ n+1 ( d^r - d^a ) + _ n ( d^v + _ 1 d^a ) the approximation term @xmath11 has to fullfill the following two requirements : first , @xmath11 must be a proper approximation to @xmath12 , with the same pointlike singular behaviour in @xmath13 dimensions as @xmath12 .
secondly , @xmath11 must be analytically integrable in @xmath13 dimensions over the one - parton subspace leading to the soft and collinear divergences .
let me now turn to the details of the monte carlo integration .
the heart of any monte carlo integration is the random number generator . among other things
, it should have a long period and should not introduce artifical correlations . as the default random number generator we use s_i & = & ( s_i-24 + s_i-55 ) 2 ^ 32 .
it was proposed by mitchell and moore and has a period of @xmath14 , where @xmath15 .
massless fourmomenta are generated with the help of the rambo - algorithm @xcite .
this algorithm generates events with a uniform weight .
adaptive importance sampling is implemented using the vegas - algorithm @xcite
. a naive implementation of the dipole formalism will give large statistical errors when performing a monte carlo integration over the real corrections with dipole factors subtracted . in order to improve the efficiency of the monte carlo integration we remap the phase space to make the integrand more flat . a simplified model for the term @xmath16 would be [ model ] f & = & _ 0 ^ 1 dx ( - ) where @xmath17 is assumed .
@xmath18 corresponds to the original real emission part with a soft or collinear singularity at @xmath19 , @xmath20 corresponds to the subtraction term of the dipole formalism .
( [ model ] ) can be rewritten as f & = & _ 0^ dx + _ ^0 dy ( f(e^y ) - g(e^y ) ) , where @xmath21 is an artificial parameter separating a numerically dangerous region from a stable region . using the taylor expansion for @xmath22
, one sees that the first term gives a contribution of order @xmath23 . in the second term
the @xmath24 behaviour has been absorbed into the integral measure by a change of variables @xmath25 , and the integrand tends to be more flat .
it should be noted that there is no approximation involved .
numerical programs for @xmath0 have been provided by four groups : menlo parc @xcite , debrecen @xcite , eerad2 @xcite and mercutio @xcite .
various cross - checks have been performed among these programs and they agree within statistical errors . here
we report on the numerical program `` mercutio '' , which was written in c++ .
the four - jet fraction is defined as @xmath26 the values obtained for the four - jet fraction for the durham algorithm with @xmath27 for various energies are given in table [ 4jetfrac ] .
.[4jetfrac ] the four - jet fraction at lo and nlo for the durham algorithm with @xmath27 and various energies . [ cols="^,^,^",options="header " , ]
the decrease with energy is mainly due to the running of the strong coupling .    with the numerical program for
@xmath2 one may also study the internal structure of three - jets events .
one example is the jet broadening variable defined as @xmath28 here @xmath29 is the momentum of particle @xmath30 transverse to the jet axis of jet @xmath31 , and the sum over @xmath30 extends over all particles in the jet @xmath31 .
the jet broadening variable is calculated for three - jet events defined by the durham algorithm and @xmath32 .
this choice is motivated by a recent analysis of the aleph collaboration @xcite .
figure [ figa ] shows the distribution of the jet broadening variable .
we report on the next - to - leading order qcd calculation for @xmath0 . 
 we explain some modern techniques which have been used to calculate the one - loop amplitudes efficiently . 
 we further report on the general purpose numerical program `` mercutio '' , which can be used to calculate any infrared safe four - jet quantity in electron - positron annihilation at next - to - leading order .