Perturbative quantum chromodynamics (also perturbative QCD) is a subfield of particle physics in which the theory of strong interactions,
Quantum Chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
(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 .
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.
Most strong-interaction processes can not be calculated directly with perturbative QCD, since one cannot observe free
quarks and
gluons due to
color confinement. For example, the structure
hadrons has a
non-perturbative nature. To account for this, physicists developed the
QCD factorization theorem, which separates the
cross section into two parts: the process dependent perturbatively-calculable short-distance
parton cross section, and the universal long-distance functions. These universal long-distance functions can be measured with global fit to experiments and include
parton distribution functions,
fragmentation functions,
multi-parton correlation functions,
generalized parton distributions
In particle physics, the parton model is a model of hadrons, such as protons and neutrons, proposed by Richard Feynman. It is useful for interpreting the cascades of radiation (a parton shower) produced from quantum chromodynamics (QCD) processes ...
,
generalized distribution amplitudes and many kinds of
form factors. There are several collaborations for each kind of universal long-distance functions. They have become an important part of modern
particle physics.
Mathematical formulation of QCD
Quantum chromodynamics is formulated in terms of the
Lagrangian density
Expressions in the Lagrangian
Matter content
The matter content of the Lagrangian is a
spinor field
and a
gauge field
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, 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
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...
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
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
and so has an associated
gauge group , which is a compact
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
. A colour vector is an element of some representation space of
.
The gauge field
is valued in the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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.

The gauge field does not appear explicitly in the Lagrangian but through the curvature
defined
This is known as the
gluon field strength tensor
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.
The strong interaction is one of the fundamental interactions of nature, and the quantum fie ...
or geometrically as the
curvature form. The parameter
is the
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
for QCD.
By expanding
into
and using
Feynman slash notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form),
: \ \stackrel\ \gamma^1 A_ ...
, the Lagrangian can then be written schematically in a more elegant form
Gauge fixed Lagrangian
While this expression is mathematically elegant, with manifest gauge symmetry, for perturbative calculations it is necessary to fix a gauge. The gauge-fixing procedure was developed by
Faddeev and
Popov
Popov (; masculine), or Popova (; feminine), is a common Russian, Bulgarian, Macedonian and Serbian surname. Derived from a Slavonic word ''pop'' (, "priest"). The fourth most common Russian surname, it may refer to:
* Alek Popov (born 1966), ...
. It requires the introduction of
ghost fields which are valued in
After the gauge fixing procedure the Lagrangian is written
{{Equation box 1
, indent =:
, equation = :
, border
, border colour = #50C878
, background colour = #ECFCF4
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. It is known that QCD, for a small number of spinors, 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
__NOTOC__
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in the evaluation of Fe ...
.
External links
Factorization of Hard Processes in QCD
References
* Peskin, M. E., Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Westview Press.
Quantum chromodynamics