The fragmentation functions D(z) describe properties of...

The fragmentation functions $D(z)$ describe properties of partons and are therefore the same, no matter how the partons are produced. Consider the inclusive leptoproduction cross section $\sigma(\mathrm{ep} \rightarrow \mathrm{hX})$ and show that $$ \frac{1}{\sigma} \frac{d \sigma}{d z}(\mathrm{ep} \rightarrow \mathrm{hX})=\frac{\sum_{q} e_{q}^{2} f_{q}(x) D_{q}^{h}(z)}{\sum_{q} e_{q}^{2} f_{q}(x)} $$ where $f_{q}(x)$ are the proton structure functions of Chapter 9, see Fig. 11.6. The sum runs over the quarks and antiquarks that can be a parent of $h$.

The fragmentation functions $D(z)$ describe properties of partons and are therefore the same, no matter how the partons are produced. Consider the inclusive leptoproduction cross section $\sigma(\mathrm{ep} \rightarrow \mathrm{hX})$ and show that
$$
\frac{1}{\sigma} \frac{d \sigma}{d z}(\mathrm{ep} \rightarrow \mathrm{hX})=\frac{\sum_{q} e_{q}^{2} f_{q}(x) D_{q}^{h}(z)}{\sum_{q} e_{q}^{2} f_{q}(x)}
$$
where $f_{q}(x)$ are the proton structure functions of Chapter 9, see Fig. 11.6. The sum runs over the quarks and antiquarks that can be a parent of $h$.

Key Concepts

Inclusive Leptoproduction

Inclusive leptoproduction refers to the process in which a lepton scatters off a hadron and a specific hadron is observed in the final state, while all other products remain unobserved. The description of this process involves the convolution of PDFs, the hard scattering cross section, and fragmentation functions, with the sum over contributions weighted by factors such as the square of the quark’s electric charge.

Factorization Theorem

The factorization theorem in high-energy physics separates the short-distance, perturbatively calculable partonic processes from the long-distance, non-perturbative aspects such as PDFs and fragmentation functions. This separation allows us to express complex hard scattering processes as convolutions of universal functions and process-specific hard parts.

Fragmentation Functions

Fragmentation functions describe the probability that a parton (such as a quark or gluon) will hadronize into a specific hadron carrying a fraction z of the parton’s momentum. They encapsulate the process of hadronization in high-energy collisions and are universal, meaning they do not depend on the production mechanism of the initiating parton.

Parton Distribution Functions

Parton distribution functions (PDFs) provide the probability of finding a parton (like a quark or gluon) inside a hadron (such as a proton) with a specific momentum fraction x. These functions are essential in describing the internal structure of hadrons and play a crucial role in factorizing the initial state in high-energy scattering processes.

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