VARYING CENTRE-OF MASS ENERGIES
As already mentioned above, there are situations when the energies of the particles in the
initial state are not fixed which trigger the reaction. In particular, this is true for
experiments with colliding hadrons, such as protons. When these hadrons are "prepared" in
the accelerator, their respective energies are pretty much fixed; in most cases, the
energy-smearing due to machine effects is completely negligible, and the energies can
to an excellent approximation be considered as the same for all particles in a beam.
However, hadrons have a substructure, they consist of constituents, usually denoted as
partons. These constituents can be thought of as the quarks and gluons forming the hadron
in question. Their individual four-momenta add up to yield the four-momentum of their
parent hadron. The binding energy of the partons can be though of as being of the order
of the hadron mass, roughly 1 GeV. In fact, this is a consequence of the structure of
strong interactions, which will be discussed in a later chapter. If, however, a reaction
occurs with characteristic energy scales much larger than the binding energy, the partons
inside the hadron are resolved - in other words, it is them, who interact individually,
rather than the collective interaction of all partons inside a hadron. To describe this,
a few assumptions can be made, which will turn out later, to be valid to a good approximation:
Details on this will be discussed in the QCD section.
- partons are massless;
- they move in parallel with their parent hadron, i.e. taking the beam to be
parallel to the z-axis translates into the partonic four-momentum having neither
an x- nor a y-component;
- their energy fraction w.r.t. the hadron energy is described by a quantity called
the Bjorken-x (a refinement of this will be presented later), summing it over all
partons inside a hadron yields one,
- the probability of finding a parton of a certain kind inside a given hadron is
determined through a function of the Bjorken-x and the characteristic resolution
scale of the partons inside a hadron, f(x,Q). It is called a parton distribution
function. This function is universal for the hadron and parton species and in that
At this point it should at least be noted that also for point-like incoming particles,
such as electrons, whose energy in principle is fixed, a similar reasoning can apply.
This is because charged particles are usually accompanied with a coherent cloud of
virtual photons, which, as a consequence of quantum mechanics, are produced, exist for
some limited time and recombine with the electron. When an interaction occurs this
coherence is broken, and the electron interacts with duly reduced energy. Of course, using
the same Bjorken-x idea of above, the probability of finding an electron-"parton" inside
an electron peaks at x=1. Again, this will be discussed later.
In both cases, in order to calculate a cross section, the partonic initial state is not
fixed any longer, implying that an integral over all configurations has to be performed.
Usually this translates into the following
where the first σ is the inclusive one, integrated over all parton configurations,
and the second one is the corresponding exclusive cross section for two partons with
fixed energies. It thus depends on the scale of the partonic subsystem.
CONNECTING THE BJORKEN-X WITH S AND Y
However, at this point it is worthwhile to consider the effects of this intrinsic structure
of the beam particles on kinematics. According to the assumptions above, and taking all
partons to be massless, their momenta can be written as
Considering for simplicity symmetric collisions, where both incoming beams have the same
energy, the respective centre-of mass systems are given by the hadron and the parton momenta.
Denoting the hadron momenta with P, neglecting their mass and denoting the parton momenta
with p, the corresponding c.m. energies squared read
Of course, the parton centre-of mass system is described by a boost along the z-axis w.r.t.
the hadron c.m. system. It thus has a rapidity, given by
Therefore, the two Bjorken-x of the two partons can be expressed through the centre-of mass
energy of the partonic system and through its rapidity w.r.t. the hadron c.m. system.
This connection reads
SPECIFYING FOR 2→2 PROCESSES
Often, the scattering processes in question are of the type 2→ 2, see
above for details. Assuming the hadronic c.m. system
to be the laboratory system, the tranverse momenta of the outgoing particles just compensate
each other, i.e., in two dimensions,
Then, the absolute value of the transverse momenta and their respective rapidities
in the laboratory system are connected to the Bjorken-x through
This can be expressed through their average and difference,
In these coordinates,
and the invariant mass of the produced system reads