✎ advanced_tools:borel_summation [The Physics Travel Guide]

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====== Borel Summation ======

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
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<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
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For a nice discussion, see section 11.4 in Rajaramans "Solitons and Instantons" and page 6 in https://arxiv.org/pdf/hep-th/0207046.pdf
<tabbox Researcher> 
<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
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--> Why isn't the Borel transformation applicable in QCD?#

See section 20.7 in Weinberg's QFT book Vol. 2.

" In quantum chromodynamics it is the renormalons that obstruct the use of the Borel transformation to sum the perturbation series [...] [T]here are instanton solutions in non Abelian gauge theories like quantum chromodynamics, but these also yield relatively harmless singularities of $B(z)$ on the negative real axis. the real problem in quantum chromodynamics is with a different class of singularities, known as renormalons. "

(Renormalons are due to the running of the gauge couplings and originate in diagrams that yield contributions that grow like $n!$. According to Weinberg, they are "associated with terms in the operator product expansion".)

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