Perturbative quantum chromodynamics (alsoperturbative QCD) is a subfield of particle physics in which the theory of strong interactions,Quantum Chromodynamics (QCD), is studied by using the fact that the strong coupling constant${\displaystyle {\alpha }_s}$ is small in high energy or short distance interactions, thus allowingperturbation theory techniques to be applied. In most circumstances, making testable predictions with QCD is extremely difficult, due to the infinite number of possible topologically-inequivalent interactions. Over short distances, the coupling is small enough that this infinite number of terms can be approximated accurately by a finite number of terms. Although only applicable at high energies, this approach has resulted in the most precise tests of QCD to date^{[citation needed]}.

An important test of perturbative QCD is the measurement of the ratio of production rates for${\displaystyle e^{+}{e}^{-}\to \text{hadrons}}$ and${\displaystyle e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}}$. Since only the total production rate is considered, the summation over all final-state hadrons cancels the dependence on specific hadron type, and this ratio can be calculated in perturbative QCD.

Most strong-interaction processes can not be calculated directly with perturbative QCD, since one cannot observe freequarks andgluons due tocolor confinement. For example, the structurehadrons has anon-perturbative nature. To account for this, physicists^[who?] developed theQCD factorization theorem, which separates thecross section into two parts: the process dependent perturbatively-calculable short-distanceparton cross section, and the universal long-distance functions. These universal long-distance functions can be measured with global fit to experiments and includeparton distribution functions,fragmentation functions,multi-parton correlation functions,generalized parton distributions,generalized distribution amplitudes and many kinds ofform factors. There are several collaborations for each kind of universal long-distance functions. They have become an important part of modernparticle physics.

Quantum chromodynamics is formulated in terms of theLagrangian density

The matter content of the Lagrangian is aspinor field${\displaystyle \psi }$ and agauge field${\displaystyle A_{\mu }}$, also known as the gluon field.

The spinor field has spin indices, on which thegamma matrices${\displaystyle {\gamma }^{\mu }}$ act, as well as colour indices on which thecovariant derivative${\displaystyle D_{\mu }}$ acts. Formally the spinor field${\displaystyle \psi (x)}$ is then a function of spacetime valued as a tensor product of a spin vector and a colour vector.

Quantum chromodynamics is agauge theory and so has an associatedgauge group${\displaystyle G}$, which is a compactLie group. A colour vector is an element of some representation space of${\displaystyle G}$.

The gauge field${\displaystyle A_{\mu }}$ is valued in theLie algebra${\displaystyle \mathfrak{g}}$ of${\displaystyle G}$. Similarly to the spinor field, the gauge field also has a spacetime index${\displaystyle \mu }$, and so is valued as a co-vector tensored with an element of${\displaystyle \mathfrak{g}}$. In Lie theory, one can always find a basis${\displaystyle t^a}$ of${\displaystyle \mathfrak{g}}$ such that${\displaystyle \text{tr}(t^a{t}^b)={\delta }^{ab}}$. In differential geometry${\displaystyle A_{\mu }}$ is known as a connection.

The gauge field does not appear explicitly in the Lagrangian but through the curvature${\displaystyle F_{\mu \nu },}$ defined$${\displaystyle F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }+ig[A_{\mu },A_{\nu }].}$$This is known as thegluon field strength tensor or geometrically as thecurvature form. The parameter${\displaystyle g}$ is thecoupling constant for QCD.

By expanding${\displaystyle F_{\mu \nu }}$ into${\displaystyle F_{\mu \nu }^a}$ and usingFeynman slash notation, the Lagrangian can then be written schematically in a more elegant form

While this expression is mathematically elegant, with manifest invariance to gauge transformations, for perturbative calculations it is necessary to fix a gauge. The gauge-fixing procedure was developed byFaddeev andPopov. It requires the introduction ofghost fields${\displaystyle c(x)}$ which are valued in${\displaystyle \mathfrak{g}.}$ After the gauge fixing procedure the Lagrangian is written

Where${\displaystyle \xi }$ is the gauge-fixing parameter. Choosing${\displaystyle \xi =1}$ is known asFeynman gauge.

After expanding out the curvature and covariant derivatives, the Feynman rules for QCD can be derived throughpath integral methods.

The techniques for renormalization of gauge theories and QCD were developed and carried out by't Hooft. For a small number of particle types (quark flavors), QCD has a negativebeta function and therefore exhibitsasymptotic freedom.

Showing that QCD is renormalizable at one-loop order requires the evaluation ofloop integrals, which can be derived from Feynman rules and evaluated usingdimensional regularization.

• Factorization of Hard Processes in QCD

• Peskin, M. E., Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Westview Press.

• Quantum chromodynamics

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