Fragmentation functions are of ten parametrized by the form...

Fragmentation functions are of ten parametrized by the form $$ D_{q}^{h}(z)=N \frac{(1-z)^{n}}{z} $$ where $n$ and $N$ are constants. Show that $$ N=(n+1)\langle z\rangle, $$ where $\langle z\rangle$ is the average fraction of the quark energy carried by hadrons of type $\mathrm{h}$ after fragmentation. Further, show that $$ n_{h} \sim \log \left(\frac{Q}{2 m_{h}}\right) $$ for the two-jet process of Fig. 11.4. That is, the multiplicity of hadrons $\mathrm{h}$ grows logarithmically with the annihilation energy.

Fragmentation functions are of ten parametrized by the form
$$
D_{q}^{h}(z)=N \frac{(1-z)^{n}}{z}
$$
where $n$ and $N$ are constants. Show that
$$
N=(n+1)\langle z\rangle,
$$
where $\langle z\rangle$ is the average fraction of the quark energy carried by hadrons of type $\mathrm{h}$ after fragmentation. Further, show that
$$
n_{h} \sim \log \left(\frac{Q}{2 m_{h}}\right)
$$
for the two-jet process of Fig. 11.4. That is, the multiplicity of hadrons $\mathrm{h}$ grows logarithmically with the annihilation energy.

Key Concepts

Fragmentation Functions

Fragmentation functions describe the probability that a high-energy quark or gluon evolves into a jet of hadrons. They specify the distribution of the energy fraction z carried by a particular hadron type in the fragmentation process, and are an essential tool in understanding how partonic energy is transferred into observable hadrons in high-energy collisions.

Normalization Condition

The normalization of fragmentation functions ensures that the integrated probability, when weighted appropriately by z, corresponds to physical observables such as the average energy fraction. In the given parametrization, the constant N is adjusted so that when the function is integrated over all possible z values, important quantities like the total energy fraction or the number of hadrons produced comply with experimental expectations and theoretical constraints.

Average Energy Fraction

The average energy fraction, denoted as ?z?, measures the mean share of the initial quark energy carried by each hadron produced in the fragmentation process. By integrating the product of z and the fragmentation function over z, one ensures that the normalization of the function corresponds to realistic energy partitioning. The relation N = (n+1)?z? emerges from this integration and normalization requirement, linking the shape parameters of the fragmentation function with the average energy carried by the hadrons.

Hadron Multiplicity and Logarithmic Growth

Hadron multiplicity refers to the number of hadrons produced in a high-energy process, such as in two-jet events from e?e? annihilation. The logarithmic dependence, expressed as n? ? log(Q/(2m?)), indicates that the number of produced hadrons grows slowly with the increase of the energy scale Q. This behavior arises from the underlying QCD dynamics and gluon radiation in the parton shower, which leads to a gradual increase in multiplicity with energy rather than an abrupt rise.

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