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In [[physics]],  a '''renormalon''' (a term suggested by [[Gerardus 't Hooft|'t Hooft]]<ref>'t Hooft G, in: The whys of subnuclear physics (Erice, 1977),  ed. A Zichichi, Plenum Press, New York, 1979.</ref>) 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]], the associated [[Borel summation|Borel transform]] of the series can have singularities as a function of the complex transform parameter.<ref name=Beneke1999>{{cite journal|last=Beneke|first=M.|title=Renormalons|journal=Physics Reports|date=August 1999|volume=37|issue=1-2|pages=1-142|doi=10.1016/S0370-1573(98)00130-6|url=http://www.sciencedirect.com/science/article/pii/S0370157398001306|accessdate=20 April 2013}}</ref> The renormalon is a possible type of singularity arising in this complex ''Borel plane'', and is a counterpart of an [[instanton]] singularity. Associated with such singularities, '''renormalon''' contributions are discussed in the context of [[quantum chromodynamics]] (QCD)<ref name=Beneke1999 /> and usually have the power-like form <math>\left(\Lambda/Q\right)^p</math> as functions of the momentum <math>Q</math> (here <math>\Lambda</math> is the momentum cut-off). They are cited against the usual logarithmic effects like <math>\ln\left(\Lambda/Q\right)</math>.


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==Brief history==
 
Perturbation series in quantum field theory are usually divergent as was firstly indicated by [[Freeman Dyson]].<ref>F.J.Dyson, Phys.Rev. 85, 631 (1952)</ref> According to the Lipatov method,<ref>L.N.Lipatov, Zh.Eksp.Teor.Fiz. 72, 411(1977) [Sov.Phys. JETP 45, 216 (1977)].</ref> <math>N</math>-th order contribution of perturbation theory into any quantity can be evaluated at large <math>N</math> in the saddle-point approximation for functional integrals and is determined by [[instanton]] configurations. This contribution behaves usually as <math>N!</math> in dependence on <math>N</math> and is frequently associated with approximately the same (<math>N!</math>) number of [[Feynman diagram]]s. Lautrup<ref>B.Lautrup, Phys.Lett. B 69, 109 (1977).</ref> 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.<ref>G.Parisi, Phys.Lett. B 76, 65 (1978); Nucl.Phys. B 150, 163 (1979).</ref>
 
Recently a proof was suggested for absence of renormalon singularities in <math>\phi^4</math> theory and a general criterion for their existence was formulated<ref>I. M. Suslov, JETP 100, 1188 (2005), http://arxiv.org/abs/hep-ph/0510142.</ref> in terms of the asymptotic behavior of the Gell-Mann - Low function <math>\beta(g)</math>. Analytical results for asymptotics of <math>\beta(g)</math> in <math>\phi^4</math> theory<ref>I. M. Suslov, JETP 107, 413 (2008), http://arxiv.org/abs/1010.4081; JETP 111, 450 (2010), http://arxiv.org/abs/1010.4317.</ref> and QED<ref>I. M. Suslov, JETP 108, 980 (2009), http://arxiv.org/abs/0804.2650.</ref> indicate the absence of renormalon singularities in these theories.
 
==References==
{{reflist|2}}
 
[[Category:Quantum chromodynamics]]

Revision as of 08:42, 2 December 2013

In physics, a renormalon (a term suggested by 't Hooft[1]) 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, the associated Borel transform of the series can have singularities as a function of the complex transform parameter.[2] The renormalon is a possible type of singularity arising in this complex Borel plane, and is a counterpart of an instanton singularity. Associated with such singularities, renormalon contributions are discussed in the context of quantum chromodynamics (QCD)[2] and usually have the power-like form (Λ/Q)p as functions of the momentum Q (here Λ is the momentum cut-off). They are cited against the usual logarithmic effects like ln⁡(Λ/Q).

Brief history

Perturbation series in quantum field theory are usually divergent as was firstly indicated by Freeman Dyson.[3] According to the Lipatov method,[4] N-th order contribution of perturbation theory into any quantity can be evaluated at large N in the saddle-point approximation for functional integrals and is determined by instanton configurations. This contribution behaves usually as N! in dependence on N and is frequently associated with approximately the same (N!) number of Feynman diagrams. Lautrup[5] 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.[6]

Recently a proof was suggested for absence of renormalon singularities in ϕ4 theory and a general criterion for their existence was formulated[7] in terms of the asymptotic behavior of the Gell-Mann - Low function β(g). Analytical results for asymptotics of β(g) in ϕ4 theory[8] and QED[9] indicate the absence of renormalon singularities in these theories.

References

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  1. ↑ 't Hooft G, in: The whys of subnuclear physics (Erice, 1977), ed. A Zichichi, Plenum Press, New York, 1979.
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  3. ↑ F.J.Dyson, Phys.Rev. 85, 631 (1952)
  4. ↑ L.N.Lipatov, Zh.Eksp.Teor.Fiz. 72, 411(1977) [Sov.Phys. JETP 45, 216 (1977)].
  5. ↑ B.Lautrup, Phys.Lett. B 69, 109 (1977).
  6. ↑ G.Parisi, Phys.Lett. B 76, 65 (1978); Nucl.Phys. B 150, 163 (1979).
  7. ↑ I. M. Suslov, JETP 100, 1188 (2005), http://arxiv.org/abs/hep-ph/0510142.
  8. ↑ I. M. Suslov, JETP 107, 413 (2008), http://arxiv.org/abs/1010.4081; JETP 111, 450 (2010), http://arxiv.org/abs/1010.4317.
  9. ↑ I. M. Suslov, JETP 108, 980 (2009), http://arxiv.org/abs/0804.2650.
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