A General Picture of QCD

The underlying theory of strong interactions is Quantum Chromodynamics (QCD), which is responsible for all strong interaction phenomena. The QCD lagrange density is

 

$${\cal L}_{\rm QCD}~=~ \overline{q}_i^a\left[ i{\rm D}\hskip-0.65e... ...p0.2em- m_i\right]_a^b q_b^i \ -\ {1\over 4} G_{\mu\nu}^A G^{A\ \mu\nu} \ \ \ ,$$ (1)

where $q_i^a(x)$ is a quark field operator, where $i~=~u,d,s,c,b,t$ are the six standard model quarks with mass $m_i$ and $a$ is a colour index that runs from $a~=~1,2,3$ (or red, green, blue depending on your taste), e.g.

$$u^a\ =\ \left(\matrix{\textcolor{red}{u}\cr \textcolor{green}{u}\cr \textcolor{blue}{u}}\right) \ \ \ .$$ (2)

The covariant derivative that preserves local invariance under SU(3) color transformations is ${\rm D}\hskip-0.65em /\hskip0.2em~=~\gamma_\mu D^\mu$ where $D_\mu~=~\partial_\mu+ig_s T^A A^A_\mu$, where $A_\mu^A$ is a gluon field, $A~=~1,..,8$. The SU(3) field strength tensor is defined through ${1\over i g_s}\left[ D_\mu , D_\nu\right]~=~G_{\mu\nu}~=~G_{\mu\nu}^A T^A$, where $T^A$ are the SU(3) Gell-Mann matrices, and $g_s$ is the strong interaction coupling ``constant''.

This looks like a very simple theory, there are quarks $q_i^a$ which interact with each other via gluons $A_\mu^A$. This part looks exactly like electromagnetism, however there is a beast hidden in $G_{\mu\nu}^A G^{A\ \mu\nu}$, and that is the non-linear interaction between gluons. Unlike photons which do not interact with themselves (Maxwells equations are linear), the eight gluons can interact with each other through the three and four gluon interactions, a generic feature of a non-abelian theory (Yang-Mills theory). Through quantum fluctuations these interactions make the strong coupling $g_s$ depend upon the scale at which the particular process is occurring, i.e. it is not constant. Even more interesting is that $g_s$ gets larger for smaller momentum transfers (long-distances) and weaker for high momentum transfers (short-distances). This is called asymptotic freedom. Numerous experiments have been performed over the last few years and theoretical techniques developed to determine the value of $g_s$ in the region where it is small. I have ``borrowed'' (without permission from the ALEPH Collaboration website[2]) a plot of the coupling as a function of the ``scale'' at which the process is occurring. In fig. (1) the quantity $\alpha_s~=~g_s^2/(4\pi)$ is plotted.

  

Figure: The strong interaction coupling constant $\alpha _s (Q^2)$ as a function of $Q^2$.

Eventually, as the momentum-scale becomes smaller, i.e. we look at processes involving larger and larger distance scales, the coupling constant becomes large, too large for a perturbative treatment. The theory becomes non-perturbative and in this regime the quarks and gluons are no-longer perturbatively close to what the ``in'' and ``out'' states of a particular interaction process look like. In fact, no-one has ever observed an isolated quark or gluon....there is no evidence that free quarks or gluons exist. In fact the interaction between them become so strong that they ALWAYS bind into hadrons, such as protons, neutrons, pions, $\Delta$'s and so forth. These are states that are singlet under SU(3) colour transformations (it is worth spending a few minutes looking at the particle data book to see what we know about the observable hadrons). Thus, if we are interested in the low-momentum interactions of, say, nucleons then it is clear that we cannot use a perturbative approach in terms of quark and gluon fields and we have to develop a theory that is written entirely in terms of hadronic fields. A somewhat more technical description goes along the lines of the approximate $SU(2)_L\otimes SU(2)_R$ chiral symmetry of the lagrange density in eq. (1) is spontaneously broken to $SU(2)_V$, by an non-zerop value of $\langle 0\vert\overline{q}q\vert\rangle$. What all of this means will be covered in the next quarter. During this quarter we will study the phenomenology of the hadronic theory and look for ``exact'' results basd on the symmetries of QCD where they exist.

A feature that is obvious but one that we have not stressed is that the strong interactions between quarks and gluons doe NOT depend upon the flavor of the quark. The coupling between gluons and the $u$-quark is the same as the coupling between gluons and the $d$-quark. The two differences between the quarks are firstly their masses, and secondly their electric charges.