Introduction

Studies of scaling violations in structure functions in Deep Inelastic Scattering (DIS) have helped to establish Quantum Chromodynamics (QCD) as the theory of strong interactions and have led to measurements of the strong coupling constant, $\alpha_{s}^{}$. Similar scaling violations are predicted in the fragmentation functions of quarks and gluons [1]. The fragmentation function D^h(x_p, Q²) for a parton represents the probability that it fragments to form a hadron h carrying a fraction x_p of the parton's momentum. As with structure functions, they cannot be calculated in perturbative QCD but can be evolved as a function of the appropriate energy scale.

In DIS and e⁺e^- annihilation hadroproduction experiments, as the energy scale increases, $\alpha_{s}^{}$ decreases and the phase space for gluon radiation increases, so that the primary quarks in an interaction will tend to lose more of their energy. Therefore we expect the x_p spectrum, where x_p = 2p/$\sqrt{s}$ and $\sqrt{s}$ is the centre-of-mass energy, to change: the high momentum end will be depopulated and the radiated gluons will tend to populate the low x_p region. Thus, if x_p is measured as a function of centre-of-mass energy, we expect to see a scaling violation due to QCD effects.

Such studies have been performed at LEP [2,3] where, by combining lower energy data from PETRA, scaling violations in the density of single particles, ${\frac{1}{\sigma_{tot}}}$${\frac{d\sigma}{dx_p}}$ have been fitted as a function of $\sqrt{s}$, to extract a value of $\alpha_{s}^{}$. In DIS the single particle density is a convolution of the hard scattering cross-section with a parton density and a fragmentation function: $\sigma$  =  $\sigma_{\mbox{{\tiny hard}}}^{}$  $\otimes$  f  $\otimes$  D. The strong coupling constant enters through each of these components.