Renormalon
Encyclopedia
In quantum field theory
, renormalon is a term suggested by 't Hooft
to signify a certain type of singularities and contributions. The renormalon singularity is a possible type of singularities arising in the complex Borel plane, as counterpart of instanton
singularities. Associated with such singularities, renormalon contributions are discussed in the context of quantum chromodynamics
(QCD) and usually have the power-like form
as functions of the momentum 
(here 
is the momentum cut-off). They are cited against the usual logarithmic effects like 
.
. According to the Lipatov method,
-th order contribution of perturbation theory into any quantity can be evaluated at large 
in the saddle-point approximation for functional integrals and is determined by instanton
configurations. This contribution behaves usually as
in dependence on 
and is frequently associated with approximately the same (
) number of Feynman diagram
s. Lautrup has noted that there exist individual diagrams giving approximately the same contribution.
In principle, it is possible that such diagrams are automatically taken into account in Lipatov’s calculation, because its interpretation in terms of diagrammatic technique is problematic. However, 't Hooft put forward a conjecture that Lipatov's and Lautrup's contributions are related with different types of singularities in the Borel plane, the former with instanton ones and the latter with renormalon ones. Existence of instanton singularities is beyond any doubt, while existence of renormalon ones was never proved rigorously in spite of numerous efforts. Among the essential contributions one should mention the application of the operator product expansion
, as was suggested by Parisi.
Recently a proof was suggested for absence of renormalon singularities in
theory and a general criterion for their existence was formulated in terms of the asymptotic behavior of the Gell-Mann - Low function 
. Analytical results for asymptotics of 
in 
theory and QED indicate the absence of renormalon singularities in these theories.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, renormalon is a term suggested by 't Hooft
Gerardus 't Hooft
Gerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...
to signify a certain type of singularities and contributions. The renormalon singularity is a possible type of singularities arising in the complex Borel plane, as counterpart of instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
singularities. Associated with such singularities, renormalon contributions are discussed in the context of quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(QCD) and usually have the power-like form
as functions of the momentum (here is the momentum cut-off). They are cited against the usual logarithmic effects like .Brief history
Perturbation series in quantum field theory are usually divergent as was firstly indicated by DysonDyson
-People:*Andre Dyson , an American football player*Charles W. Dyson , a U.S. Navy rear admiral*Esther Dyson , consultant and philosopher in emerging digital technology, the daughter of Freeman Dyson...
. According to the Lipatov method,
-th order contribution of perturbation theory into any quantity can be evaluated at large in the saddle-point approximation for functional integrals and is determined by instantonInstanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
configurations. This contribution behaves usually as
in dependence on and is frequently associated with approximately the same () number of Feynman diagramFeynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s. Lautrup has noted that there exist individual diagrams giving approximately the same contribution.
In principle, it is possible that such diagrams are automatically taken into account in Lipatov’s calculation, because its interpretation in terms of diagrammatic technique is problematic. However, 't Hooft put forward a conjecture that Lipatov's and Lautrup's contributions are related with different types of singularities in the Borel plane, the former with instanton ones and the latter with renormalon ones. Existence of instanton singularities is beyond any doubt, while existence of renormalon ones was never proved rigorously in spite of numerous efforts. Among the essential contributions one should mention the application of the operator product expansion
Operator product expansion
- 2D Euclidean quantum field theory :In quantum field theory, the operator product expansion is a Laurent series expansion of two operators...
, as was suggested by Parisi.
Recently a proof was suggested for absence of renormalon singularities in
theory and a general criterion for their existence was formulated in terms of the asymptotic behavior of the Gell-Mann - Low function . Analytical results for asymptotics of in theory and QED indicate the absence of renormalon singularities in these theories.External links
- Renormalons - a review of renormalons from the physics literature