?Ref for Cutkosky cutting rules

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    Cutting Rules
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Discussion Overview

The discussion revolves around seeking references for understanding Cutkosky cutting rules in the context of quantum field theory (QFT), specifically for identifying the absorptive parts of one- or two-loop diagrams. Participants share their experiences with existing literature and express challenges in finding suitable resources.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests recommendations for references on Cutkosky cutting rules, noting that Peskin and Schroder does not meet their needs.
  • Another participant suggests "Analytic S Matrix" by Eden et al. as a potential resource.
  • A different participant mentions having reviewed numerous QFT and particle physics texts without finding significant information beyond Peskin and Schroder.
  • Concerns are raised about the perceived decline in popularity of Cutkosky rules among current academics, with one participant sharing their prior discussions with professors who also could not provide useful references.
  • A participant references an older book by Geoffrey Chew, "S-Matrix Theory of Strong Interactions," which discusses singularities and absorptive parts, and mentions its historical context within the S-Matrix approach.
  • Another participant acknowledges the complexity of the topic and the challenges in finding contemporary discussions on absorptive parts.

Areas of Agreement / Disagreement

Participants generally agree on the difficulty of finding adequate references on Cutkosky cutting rules, but multiple competing views exist regarding the relevance and availability of literature on the topic.

Contextual Notes

Some participants express uncertainty about the current academic focus on Cutkosky rules and the availability of resources, indicating a potential gap in contemporary literature.

mjsd
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I am wondering whether someone can suggest a good ref or two (preferrably with worked example) on how to use Cutkosky (or whatever it is called) cutting rules in OFT to help pick out the absorptive part of a 1- or 2-loops diagram. I have already tried Peskin and Schroder, which is good only up to a certain level.

thanks in advance
 
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Analytic S Matrix by Eden et al may be good book for you!
 
mjsd said:
I am wondering whether someone can suggest a good ref or two (preferrably with worked example) on how to use Cutkosky (or whatever it is called) cutting rules in OFT to help pick out the absorptive part of a 1- or 2-loops diagram. I have already tried Peskin and Schroder, which is good only up to a certain level.

thanks in advance

I have looked at about a dozen QFT and particle physics books I have and I have found nothing signifcant (i.e. it's even less than P&S), unfortunately.
 
it seems that these rules aren't that "popular" these days... apparently, before I posted this a while back I have already spoken to professors here at my uni...and because I couldn't find anything useful for my task back then... I asked the forum...

but thanks anyway... sometimes one just have to learn things the hard way... eg. trial and error
 
PRB147 said:
Analytic S Matrix by Eden et al may be good book for you!

thanks will check that out too
 
There's a very old Benjamin book, S-Matrix Theory of Strong Interactions, by Geoffrey Chew(1961), which in addition to discussing singularities and absorptive parts of diagrams, reprints two key papers on the subject -- Cutkosky's paper on singularities and Landau's paper on vertex analytic properties. Availability? See Amazon,...

The basic idea comes from the notion of pairs of Hilbert Transforms -- Dispersion relations if you will. Check out the Kramers-Kronig expression for dialectric constants, basically a Hilbert transform. An overly simplified approach is to note that

1/(X + ie) = - i delta(x) +P(1/x)

where P indicates the principal part, which is the basis for Hilbert Transforms.

More recent discussions can be found in Chap. 10 of Weinberg's QFT, and F. Gross's Relativistic Quantum Mechanics and Field Theory.

This stuff was big in the 1960s, part of the "anti-field theory" approach of Chew and the S-Matrix gang. But Gell-Man and his quarks,symmetries, and the field theory approaches won the day.

Sorry to be so sloppy, but it's been a while since I've thought about absorptive parts and the like.
Regards,
Reilly Atkinson
 
thanks, Reilly.

Sorry to be so sloppy, but it's been a while since I've thought about absorptive parts and the like.

that's a typical response I get from the older academics here...:smile:.. the young ones? usually say they've never used it nor looked into it closely.
 

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