?Ref for Cutkosky cutting rules

[mjsd]

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

 

[PRB147]

Analytic S Matrix by Eden et al may be good book for you!

 

[nrqed]

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.

 

[mjsd]

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

 

[mjsd]

Analytic S Matrix by Eden et al may be good book for you!

thanks will check that out too

 

[reilly]

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

 

[mjsd]

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..... the young ones? usually say they've never used it nor looked into it closely.