QCD

The strong interactions among quarks and gluons are described by Quantum Chromodynamics (QCD), the non-abelian gauge theory based on the gauge group $SU(3)_C$. Each quark flavour corresponds to a colour triplet in the fundamental representation of $SU(3)$ and the gauge fields needed to maintain the gauge symmetry, the gluons, are in the adjoint representation of dimension 8. Gauge invariance, realized á la Wigner-Weyl, ensures that gluons are massless. The QCD Lagrangian may be written as

$${\cal{L}}_{QCD} = - \frac{1}{4} F^a_{\mu\nu} F^{\mu\nu}_a + \bar{\psi}_i (i \gamma^{\mu} D_{\mu} - m) \psi_{i}$$ (364)

where

$$F^a_{\mu\nu} = \partial_{\mu}G^a_{\nu} - \partial_{\nu}G^a_{\mu} + g f^{abc} G_{b\mu} G_{c\nu}$$ (365)

stands for the gluon field tensor, $\psi_i$ are the quark fields and the covariant derivative is defined by

$$D_{\mu} = \partial_{\mu} - i g T_a G^a_{\mu}$$

The strong coupling is represented by $g$ and indices are summed over $a = 1,...,8$ and over $i=1,2,3$. Finally, $T_a = \lambda_a/2$ and $f_{abc}$ are the $SU(3)$ generators and structure constants, respectively, which are related by $[T_a,T_b] = i f_{abc} T^c$.

Figure 21: Basic processes in perturbative QCD

Like in Quantum Electrodynamics (QED), the procedure employed to deal consistently with the divergences that occur in the computation of strong interactions beyond the tree level, shows that the actual strong coupling depends on the energy scale of the process. But in opposition to QED, this renormalized strong coupling is small at high energy (momentum), going to zero logarithmically. QCD has the property of asymptotic freedom. Consequently, in this regime perturbation theory is valid and tests against experimental data can be performed in terms of hadrons. Figure 21 summarize the basic QCD perturbative processes appearing in different circumstances.

Experiments with $e^- e^+$ colliders provide clean results for QCD tests. Recently, a huge amount of experimental data came from the HERA electron-proton collider and also from the Tevatron at Fermilab. In both cases, there is a hadronic remnant that make the analysis a little more involved. All this experimental evidence support the existence of quarks being colour triplets of spin $1/2$ and of gluons being vector octets. Moreover, the presence of the QCD coupling has manifested itself in different measurements, as well as the above mentioned property of asymptotic freedom. This information comes mainly from the study of the so called two- and three-jets events mentioned in the phenomenological introduction.

When a given process needs a higher order in perturbation theory to be known, it is necessary to compute not only the renormalized strong coupling constant but also the appropriate corrections to the relevant cross-sections. As is usual in Quantum Field Theory, a regularization-renormalization procedure is in order, just to absorb divergences into the definition of physical quantities. This prescription requires the introduction of a new scale $\mu$, fixing the renormalization point, and all renormalized quantities begin to depend on it. Nevertheless, different prescriptions must end with the same predictions for observables.

In order to illustrate how the general procedure works, ending with the Renormalization Group equations that guarantee that physical observables do not depend on the scale $\mu$, let us show what happens with Green functions. Just to remember the procedure, let us begin with a single particle irreducible Green function $\Gamma$. In general, to control divergences, one has to introduce an ultra-violet cut-off $\Lambda$, or the equivalent dimensional regularization parameter, in the loop momentum integral defining the $\Gamma$. In a renormalizable theory, as QCD is, a renormalized Green function is defined as

$$\Gamma_R(p_i, g, \mu) = Z_{\Gamma}(g_0,\Lambda/\mu) \Gamma_U(p_i,g_0,\Lambda)$$

where $p_i$ stands for the external particle momenta, $g_0$ and $g$ are the bare and the renormalized couplings, respectively. This $\Gamma_R$ is then finite in the limit $\Lambda \rightarrow \infty$ but it depends on the scale at which the value of the renormalized quantities are fixed, the prescription parameter $\mu$. The function $Z_{\Gamma}$ is a product of renormalization factors. Due to the fact that the unrenormalized $\Gamma_U$ is obviously independent of $\mu$, one has to demand

$$\frac{d \Gamma_U}{d \mu} = 0$$

and consequently, the Renormalization Group Equation (RGE)

$$\left( \mu \frac{\partial}{\partial \mu} + \beta \frac{\partial}{\partial g} + \gamma \right) \Gamma_R(p_i, g, \mu) = 0$$ (366)

has to be verified. Here $\gamma$ is the anomalous dimension, depending on the particular Green function under consideration, and the beta-function is universal

$$\gamma = \frac{\mu}{Z_{\Gamma}} \frac{\partial Z_{\Gamma}}... ... \beta(g) = \mu \frac{\partial g}{\partial \mu}$$ (367)

If there is only one large momentum scale $Q$, or $Q^2$ as it is standard to quote, as it is the case here, one can express all $p_i$ in terms of a fixed fraction $x_i$ of $Q$. Then, defining the so called evolution variable

$$t = \frac{1}{2} \ln \left(\frac{Q^2}{\mu^2} \right)$$ (368)

it is possible to introduce the momentum dependent, or running coupling through the integral

$$t = \int^{g(t)}_{g(0)} \frac{d g^{\prime}}{\beta(g^{\prime})}$$

and the general solution of the RGE reads

$$\Gamma(t,g(0),x_i) = \Gamma(0,g(t),x_i) exp \left[ \int^... ...prime} \frac{\gamma(g^{\prime})} {\beta(g^{\prime})} \right]$$

This solution explicitly shows that the $Q$-scale dependence of $\Gamma$ arises entirely through the running coupling $g(t)$. Introducing now the usual notation

$$\alpha_s = \frac{g^2}{4\pi}$$ (369)

one can expand the beta-function in a power series in $\alpha_s$

$$\beta(\alpha_s) = \mu \frac{\partial \alpha_s}{\partial \mu}... ...2 \pi} \alpha^2_s - \frac{\beta_1}{4 \pi^2} \alpha^3_s - ...$$ (370)

where it results that

$$\beta_0 = 11 - \frac{2}{3} N_f \beta_1 = 51 - \frac{19}{3} N_f$$ (371)

Here $N_f$ indicates the number of flavours that can be excited (with mass less than $\mu$) at the scale $\mu$.

It is clear that the solution of the differential equation for $\alpha_s$ introduces a constant, called $\Lambda_{QCD}$, which has to be fixed by using experimental data. The resulting $\alpha_s$ can be written as

$$\alpha_s(\mu, \Lambda_{QCD}) = \frac{4 \pi}{\beta_0 \ln(\m... ...(\mu^2/\Lambda_{QCD}^2)]} {\ln(\mu^2/\Lambda_{QCD}^2)}\right\}$$ (372)

This expression for the running coupling shows clearly the property of asymptotic freedom of QCD, i.e., the coupling vanishes when the scale becomes asymptotic, namely $\mu \rightarrow \infty$. Consequently, in this momentum regime, perturbation theory is valid.

A very clear quantitative test of perturbative QCD is provided by the measurement of $\alpha_s$ in different processes at different scales $Q^2$. In Figure 22 there is a summary of the various determinations of $\alpha_s$.

Figure 22: QCD running coupling

The present world average for the coupling at the $Z^0$ mass is

$$\alpha_s(M_Z) = 0.119 \pm 0.002$$

which implies

$$\Lambda_{QCD}^{\overline{MS}} = 220 +78-63 MeV$$

corresponding to five flavours excited and in the conventional $\overline{MS}$ prescription commonly used.

This small paragraph devoted to the factor $3$ of $3!$ which represents the Standard Model of the Electroweak and Strong Interactions is only worth as a preface to the corresponding course on QCD.