Kinematics and Cross Section for a Two-body Process

In this section we begin the description of kinematics selection and cross-section calculation. The example is for the case of a process, with final-state masses assumed to be vanishing. Later on we will expand to finite fixed masses, and to resonances.

Consider two incoming beam particles in their c.m. frame, each with
energy
. The total squared c.m. energy is then
. The two partons that enter the hard
interaction do not carry the total beam momentum, but only fractions
and , respectively, i.e. they have four-momenta

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There is no reason to put the incoming partons on the mass shell, i.e. to have time-like incoming four-vectors, since partons inside a particle are always virtual and thus space-like. These space-like virtualities are introduced as part of the initial-state parton-shower description, see section , but do not affect the formalism of this section, wherefore massless incoming partons is a sensible ansatz. The one example where it would be appropriate to put a parton on the mass shell is for an incoming lepton beam, but even here the massless kinematics description is adequate as long as the c.m. energy is correctly calculated with masses.

The squared invariant mass of the two partons is defined as

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In addition to and , two additional variables are needed
to describe the kinematics of a scattering
.
One corresponds to the azimuthal angle of the scattering
plane around the beam axis. This angle is always isotropically
distributed for unpolarized incoming beam particles, and so need not
be considered further. The other variable can be picked as
, the polar angle of parton 3 in the c.m. frame of the
hard scattering. The conventional choice is to use the variable

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If the two outgoing particles have masses and ,
respectively, then the four-momenta in the c.m. frame of the hard
interaction are given by

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Then and are modified to

with

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The cross section for the process
may be written as

The choice of scale is ambiguous, and several alternatives are
available in the program. For massless outgoing particles the default
is the squared transverse momentum

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The expresses the differential cross section for a scattering, as a function of the kinematical quantities , and , and of the relevant masses. It is in this function that the physics of a given process resides.

The performance of a machine is measured in terms of its
luminosity , which is directly proportional to the
number of particles in each bunch and to the bunch crossing
frequency, and inversely proportional to the area of the bunches at
the collision point. For a process with a as given by
eq. (), the differential event rate is given
by
, and the number of events collected
over a given period of time

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