10 - H1 - Desy

1.2 Basics of electron - proton scattering 7
1.2.2 The strong interaction and renormalisation
The QPM fails to explain why quarks are never observed in the final state, although they
appear to be quasi free in deep-inelastic scattering experiments. The model also fails to
describe the scaling violation observed at lower and higher x values. The quantum chromodynamics
(QCD) theory instead was very successful in describing the measurements.
The QCD provides the quarks with additional quantum number, the colour charge red,
green or blue. The mediating gauge bosons of QCD are eight massless, spin one particles
with no electrical charge. Instead, each gluon carries a combination of colour-anticolour
charge allowing for gluon self-interaction (i.e. splitting g → gg is possible). Gluon selfcoupling
is a reason for an anti-screening effect, which causes the strong coupling constant
α s to behave differently than the electromagnetic coupling α. At small momentum scales
(large distances) α s becomes very large and partons interact strongly. This forces the
partons to be bound into colour neutral states, called hadrons. The mentioned effect is
called confinement. With increasing momentum scale, α s becomes very small, until the
parton is effectively observed as a free particle. This effect is usually called asymptotic
freedom.
In perturbative QCD (pQCD) particle scattering cross sections are given as a power series
in α s . Quark and gluon loop diagrams, such as shown in figure 1.4 start contributing
beyond leading order (LO). The calculation of such loop contributions incorporates an
integration over all particle momenta p in the loop. In the limit p → ∞ ultraviolet
(UV) divergences occur which can be absorbed by introducing an arbitrary dimensionless
renormalisation scale µ R . Above µ R scale all loops are absorbed into the coupling constant.
The strong coupling constant α s at scale µ R > Λ QCD can be written as
α s (µ 2 R ) =
12π
(1.11)
(33 − 2N f ) · ln(µ 2 R /Λ2 QCD
),
where N f is the number of quark flavours with m q < µ R . The parameter Λ QCD determines
the scale at which α s becomes large, so that power series in α s do not longer converge
and perturbation theory is not applicable anymore. Experimentally Λ QCD was found to
be of order ∼ 200 MeV. The scale dependence of the strong coupling constant α s can be
observed in figure 1.5, the effect often referred to as the running α s .
2 A4?A4 " G; 2G h !6K
2 A4?A4 " G; 2G h !6K
a) b)
Figure 1.4: Higher order loop corrections to the scattering process. Gluon loop (a) and
fermion loop (b).
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