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The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
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Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
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Several restrictions limit the use of Friedel–Crafts reactions. First, the halogen in the alkyl halide must be attached to an sp3-hybridized carbon for the Friedel–Crafts reactions to occur. Vinyl or aryl halides do not react since the carbocations formed are unstable under the reaction conditions. Second, Friedel–Crafts alkylation is susceptible to carbocation rearrangement, and the major products obtained have a rearranged carbon skeleton. In contrast, the acylium ion is...
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Setting Limits on Supersymmetry Using Simplified Models
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Large orders and strong-coupling limit in functional renormalization.

Mikhail N Semeikin1, Kay Jörg Wiese1

  • 1Sorbonne Université, ENS, PSL, Laboratoire de Physique de l'Ecole Normale Supérieure, CNRS-, Université Paris Cité, 24 rue Lhomond, 75005 Paris, France.

Physical Review. E
|December 23, 2025
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Summary
This summary is machine-generated.

We establish Borel summability for the functional renormalization group (FRG) in zero dimensions. In the strong-coupling limit, microscopic disorders flow to a universal fixed point, relevant for elastic systems.

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Area of Science:

  • Theoretical Physics
  • Condensed Matter Physics

Background:

  • The functional renormalization group (FRG) is a powerful tool for studying quantum field theories.
  • Understanding the large-order behavior of FRG is crucial for its reliable application.
  • Disordered systems, particularly elastic systems, present significant theoretical challenges.

Purpose of the Study:

  • To investigate the large-order behavior of the functional renormalization group (FRG).
  • To establish conditions for Borel summability in a zero-dimensional model.
  • To analyze the flow of microscopic disorders in the strong-coupling limit.

Main Methods:

  • Mathematical analysis of FRG equations.
  • Application of Borel summation techniques.
  • Contour integral representation of FRG derivatives and series.
  • Investigation of the strong-coupling limit.

Main Results:

  • Borel summability is established for a wide range of microscopic couplings in a zero-dimensional model.
  • The Borel transform and the original FRG series are expressed as integrals.
  • All short-ranged microscopic disorders are shown to flow to the same universal fixed point in the strong-coupling limit.

Conclusions:

  • The study provides a rigorous mathematical framework for FRG in the presence of disorder.
  • The identified universal fixed point offers insights into the behavior of disordered elastic systems.
  • These findings advance the applicability of FRG to complex physical phenomena.