In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a renormalon (a term suggested by
't Hooft) is a particular source of divergence seen in perturbative approximations to quantum field theories (QFT). When a formally divergent series in a QFT is summed using
Borel summation
In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several vari ...
, the associated
Borel transform of the series can have singularities as a function of the complex transform parameter.
The renormalon is a possible type of singularity arising in this complex ''Borel plane'', and is a counterpart of an
instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
singularity. Associated with such singularities, renormalon contributions are discussed in the context of
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)
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
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
. 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
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
configurations. This contribution behaves usually as
in dependence on
and is frequently associated with approximately the same (
) number of
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
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
In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vertex ...
, 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.
References
Quantum chromodynamics
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