# Couplings and asymptotic freedom

Each of the four fundamental interactions can be characterized by a dimensionless parameter expressed in terms of universal constants, the coupling constant.
In the lagrangian of strong interactions, the coupling constant $\alpha_S$ is parameterized by the gauge coupling parameter,$g$, giving in natural units

$\alpha_S=\frac{g^2}{4\pi}$.

However, in a general field theory, the effective coupling constant is not a constant, but it depends on a momentum or distance scale due to renormalization effects. The effective coupling constant decreases at short distances, or equivalently, at high momenta. Such theories are said to be asymptotically free.

Asymptotic freedom is actually one of the main features of the QCD.

It is expressed by the dependence of the gauge bare coupling constant $g$ by the scale energy of the physical processes considered, described by the beta function. It is defined by the relation

$\beta(g)=\mu\frac{\partial g}{\partial \mu}=\frac{\partial g}{\partial ln \mu}$,

where $\mu$ is the energy scale of the given physical process.

In quantum field theories, a beta function encodes the running of a coupling parameter $g$. If the beta function vanishes, the theory is said to be scale-invariant.

In non-Abelian gauge theories, the beta function can be negative and therefore the coupling decreases logarithmically, as it is shown in the Figure below.

This was first found by F. Wilczek, D. Politzer and D. Gross, Nobel prize in Physics in 2004 “for the discovery of asymptotic freedom in the theory of strong interaction”.

As a consequence, quantitative calculations based on a perturbative sum of Feynman diagrams, works. Viceversa, perturbative calculations fail at high energies.

In QED, an electron can emit a single virtual photon or it can emit a photon that subsequently can annihilate in an electon-positron pair, and so on, provided that the energy remains conserved within the Heisenberg’s uncertainty principle.

Thus, around the bare charge of an electron, $e^+e^-$ pairs arise in the vacuum with positrons closer to the electron. This generates a cloud of polarized charges shielding the negative charge of the electron (charge screening).
The effect becomes more significant as one gets closer to the electron, for example with a charge probe.

In QCD, instead, the effect is the opposite and a charge is preferably
surrounded by other charges of the same colour. The more one approaches the charge, the weaker the charge appears (charge anti-screening), and the two charges become non-interacting (asymptotic freedom).

The asymptotic freedom can be analyzed in the framework of the renormalization group theory.
In field theories, quantum corrections obtained via the perturbative approach presents divergences that can be re-absorbed by renormalizing some appropriate parameters.
This is true for renormalizable theories such as QED and QCD.
The energy scale at which the divergences are absorbed is called the renormalization point.
One of the most important results of the renormalization group theory is the dependence of the coupling constant of the theory from the transferred momentum.

In particular, in the case of QCD, one gets for the leading order the following dependence on the transferred four-momentum $q$:

$\alpha(|q^2|)=\frac{12\pi}{\left(11n-2f\right)ln\left(\frac{|q^2|}{\Lambda^2_{QCD}}\right)}$,

where $n$ indicates the number of colours, $f$ the number of quark flavours and $\Lambda_{QCD}$
($\approx200\ MeV$) represents an intrinsic energy scale for the strong interaction, at which perturbative approach can be applied.
It is $11n > 2f$ in nature and, in consequence, the strength of strong interactions $\alpha_S$ decreases at small distances (or high energies).

Many interesting aspects of the QCD such as confinement from hadronic matter to QGP occur at low energies ($T \sim\Lambda_{QCD}$) and therefore they can not be treated perturbatively.
Asymptotic freedom of strong interactions permits to perturbatively study the theory only for high transferred momentum. For energies around $\Lambda_{QCD}$
the theory cannot be used and it is easy to see that
the equation above for the dependance on the transferred momentum presents a divergence for $q^2\rightarrow\Lambda^2_{QCD}$.