Application of perturbative methods to quantum chromodynamics, the study of the strong interaction
Perturbative quantum chromodynamics (also perturbative 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 is small in high energy or short distance interactions, thus allowing perturbation 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 and . 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.
The matter content of the Lagrangian is a spinor field and a gauge field, also known as the gluon field.
The spinor field has spin indices, on which the gamma matrices act, as well as colour indices on which the covariant derivative acts. Formally the spinor field is then a function of spacetime valued as a tensor product of a spin vector and a colour vector.
Quantum chromodynamics is a gauge theory and so has an associated gauge group, which is a compact Lie group. A colour vector is an element of some representation space of .
The gauge field is valued in the Lie algebra of . Similarly to the spinor field, the gauge field also has a spacetime index , and so is valued as a co-vector tensored with an element of . In Lie theory, one can always find a basis of such that . In differential geometry is known as a connection.
By expanding into and using Feynman slash notation, the Lagrangian can then be written schematically in a more elegant form
Gauge fixed Lagrangian
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 by Faddeev and Popov. It requires the introduction of ghost fields which are valued in After the gauge fixing procedure the Lagrangian is written
Where is the gauge-fixing parameter. Choosing is known as Feynman gauge.
After expanding out the curvature and covariant derivatives, the Feynman rules for QCD can be derived through path integral methods.
Renormalization
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 negative beta function and therefore exhibits asymptotic freedom.
One-loop renormalization
Showing that QCD is renormalizable at one-loop order requires the evaluation of loop integrals, which can be derived from Feynman rules and evaluated using dimensional regularization.