QCD
Asymptotic freedom
- The effect of hadrons as bound states of
quarks (and gluons?)
- Asymptotic freedom: The "what" and the "why"
In the previous section some phenomenological models for the quark-antiquark
potential in the static limit have been discussed. The natural question arises how
this behaviour of the interaction strength, large at large distances and small at
small distances, can be understood from first principles, i.e. starting from the
Lagrangian. A simple way to look at it is to consider higher-order corrections to
fermion-fermion scattering: in QED this would be electron-electron scattering, in QCD
quark-quark scattering is a good candidate. At first order in the perturbation theory
closed loops appear inside the photon/gluon propagator, respectively
(Test question: which ones?). When inspecting them it becomes apparent that they diverge
as the momentum running inside the loop goes to infinity. This effect is called
ultraviolet divergence. In quantum field theory, this is dealt with by
regularising the divergencies and by construction of appropriate ultraviolet counter terms
that are added to the original Lagrangian and render it ultraviolet finite. However, these
counter terms induce some freedom to the theory, since finite pieces may be treated in
different ways. As a consequence, renormalised, i.e. physical, quantities, such as coupling
constanrts must be measured in order to fix them at one point, where they have been measured.
If then this quantity is used at a different scale, the renormalisation procedure
induces a change of the quantity, which thus runs.
In QED, at first order, the coupling constant is usually fixed in the Thomson limit,
i.e. for static poles at large distances. Thus the reference is
At first order, the running of the fine structure constant is given by
This is fun! With increasing energy, the coupling constant increases. And in fact,
a careful experimental anylsis shows that at the Z pole 1/α∼128, in agreement with
theory.
In QCD the behavior of the coupling constant is strikingly different. There,
where
Here, it is clear that the coupling constant diverges for Q→0. In fact, it diverges
earlier, namely when the denominator vanishes. This is known as the Landau pole, in QCD
located at scales of the order of a few hundred MeV. However, due to this divergence,
there is no well-defined measurement of the QCD coupling conatant for low energies/long
distances. In this regime bound state effects take over, and it is impossible to
extract a single quark-gluon coupling.
That's why other reference scales are needed in order to fix the coupling constant, once
its scaling behaviour is known. At this point it is important to note that it does not
matter which point μ is chosen as a reference point; at any given order of the
calculation, the difference between two different choices is at higher orders. This
allows to systematically reduce the dependence on the scale choices in any calculation.
In turn, suitable scale choices in a caluclation can be used in order to minimize
the dependence of higher order terms on the actual result.
Another point is worth discussing here: Due to the freedom in choosing a reference
scale for the coupling constant one may try to introduce a "real" parameter Λ in terms
of which the coupling constant can be calculated:
where Λ∼ 250 MeV. Obviously, Λ describes the Landau pole. The following graph
shows a compilation of different measured values of the QCD coupling constant and the
blue band indicates the scaling behaviour as given by a careful calculation.
- How to measure &alphas