Party density function

A parton density (function) (abbreviated PDF ) is a function that indicates the distribution of that portion of the momentum of a hadron (in particular a proton or neutron ) that a parton ( quark or gluon ) carries. ${\displaystyle f_{i}}$

Parton densities were first introduced in the Parton model by Richard P. Feynman in the 1960s . This model is used to describe very high-energy collisions between hadrons and leptons or other hadrons and treats hadrons as a kind of cloud of free partons.

In the context of the modern theory of strong interaction , quantum chromodynamics  (QCD), the parton densities describe the non- perturbative part of the cross-section of a lepton-hadron or hadron-hadron collision . In the QCD, the parton densities are well-defined, universal functions; H. for a given hadron they always have the same value, regardless of the process under consideration. For example, parton densities obtained from the experimental data of a process such as deep inelastic electron-proton scattering can be used to predict the event rates of another process such as Higgs boson production in proton-proton collisions at the Large Hadron Collider  (LHC) become.

The assumption that the partons behave like free particles in the hadron is assumed ad hoc in the parton model . In QCD, this property is called asymptotic freedom and is explained by the fact that the coupling constant of the strong interaction becomes smaller and smaller at small distances.

definition

The PDF indicates the probability density of finding the corresponding parton with the momentum fraction of the hadron on an energy scale . Strictly speaking, this definition only applies in a reference system in which the hadron carries an infinite momentum frame . ${\displaystyle f_{i/h}(x,Q)}$ ${\displaystyle Q}$ ${\displaystyle i}$ ${\displaystyle x}$ ${\displaystyle h}$

The abbreviated notation can sometimes be found in the literature . For example, the Upquark PDF describes this . ${\displaystyle i(x,Q)}$ ${\displaystyle u(x,Q)}$

properties

Due to the model description on which the PDFs are based, one can make some limiting statements about the PDFs from the assumptions. On the one hand, these statements can be used to determine the PDFs more precisely, and on the other hand, the weaknesses of the model can be checked using PDFs that do not assume these assumptions.

Due to the composition of the hadrons from partons and the normalization to the total hadronic momentum, it is to be expected that the total parton momentum, taking into account their distribution, will again form the total hadron momentum. This results in the following completeness relation , which is in particular also independent of the observed energy scale Q:

${\displaystyle \sum _{i}\int _{0}^{1}xf_{i}(x,Q){\text{d}}x=1}$

Proton PDFs

Since protons are composed of two upquarks and one downquark (the valence quarks of the proton) and the weak interaction is negligible, other quarks can only be created by pair production from gluons. It is therefore to be expected that the Quark PDFs for these Quarks will match the Antiquark PDFs:

${\displaystyle f_{i}(x,Q)\equiv f_{\bar {i}}(x,Q)}$

It is also to be expected that the following three identities apply to the valence quarks (index s for strange quarks):

${\displaystyle \int _{0}^{1}(f_{u}(x,Q)-f_{\bar {u}}(x,Q)){\text{d}}x=2}$

${\displaystyle \int _{0}^{1}(f_{d}(x,Q)-f_{\bar {d}}(x,Q)){\text{d}}x=1}$

${\displaystyle \int _{0}^{1}(f_{s}(x,Q)-f_{\bar {s}}(x,Q)){\text{d}}x=0}$

These are basically synonymous with the statement that the proton consists of two upquarks and one downquark.

Dependence on the energy scale

In the QCD, the parton densities themselves cannot be calculated in terms of perturbation theory, but the QCD makes definitive predictions of how the parton densities change between lower and higher energy scales. These predictions are provided by the DGLAP equations , which are named after the first letters of the authors' surnames: Dokshitzer , Gribow , Lipatow, and Altarelli and Parisi . It is sufficient to know the course of the parton densities for a low scale  Q ₀ , then the values can be calculated for all higher scales Q > Q ₀ . This process is called evolution .

The energy scale  Q is usually given in the unit Giga electron volt  (GeV); where Q is a value that characterizes the process to be calculated, e.g. B. the virtuality of an exchanged particle .

The dependence of the parton densities on the energy scale Q is a violation of the Bjorken scale invariance and distinguishes the predictions of the QCD from the simpler Quark-Parton model. The experimental measurement of this Q dependency is an important quantitative test of the predictions of QCD.

Available PDF sets

Well-known PDF collaborations are:

• CTEQ ( The Coordinated Theoretical-Experimental Project on QCD )
• MSTW ( Martin-Stirling-Thorne-Watt Parton Distribution Functions )
• NNPDF ( Neural Network Parton Distribution Functions )

A comprehensive list of available PDF sets can be found at LHAPDF.

literature

• The Coordinated Theoretical-Experimental Project on QCD (CTEQ). Handbook of Perturbative QCD.
• ME Peskin, DV Schroeder: An introduction to quantum field theory (= Frontiers in Physics ). Addison-Wesley Advanced Book Program, 1995, ISBN 0-201-50397-2 .
• John Collins: Parton distribution functions (definition) . In: Scholarpedia . tape 7 , no. 7 , 2012, p. 10851 , doi : 10.4249 / scholarpedia.10851 .
• Joël Feltesse: Introduction to Parton Distribution Functions . In: Scholarpedia . tape 5 , no. 11 , 2010, p. 10160 , doi : 10.4249 / scholarpedia.10160 .

Individual evidence

1. ^ The CTEQ Meta-Page
2. ↑ Hepforge: MSTW PDFs
3. ↑ Hepforge: NNPDF
4. ↑ Hepforge: LHAPDF