- #1

LAHLH

- 409

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Hi,

I have a few questions on the content of this chapter.

A theory is renormalizable in general if all it's Lagrangian coefficents have +ve or 0 mass dimension. So far so good. Now Srednicki says he will use [tex] \phi^3[/tex] theory in d=6 as his example, to see how to construct a finite expression for scattering amplitude to arb high order if coupling 'g'.

Am I correct in thinking by scattering amplitude he just means the [tex] \langle f\mid i\rangle [/tex] as given in the LSZ? which then contains [tex]\langle 0\mid T\phi(x_1).....\phi(x^{'}_1)\mid 0\rangle[/tex]

So as he shows in CH10 we only care about the fully connected diagrams. So e.g. for 4 particles scattering we sum all diagrams with 4 sources with sources removed etc.

Then he goes onto:

So just as [tex] \Pi (k^2) [/tex] is the sum of all 1PI diagrams with two external sources (with the sources removed) (although this is not the exact propagator), and [tex] \b{V}_3 [/tex] is the sum of all 1PI diagrams with three external legs, so [tex] \b{V}_n [/tex] is the sum of all 1PI diagrams with n external legs.

He then constructs these upto E (the self energy and [tex] \b{V}_3 [/tex] plus all the higher n vertices from [tex] 4 \leq n \leq E [/tex].)

I just don't understand why he's doing this? what does he mean exactly by "process of interest", does he mean E particles scattering or something? If so why only sum vertices upto E, because surely you could still have vertices that have higher valence within your 1PI diagram, even if the diagram only has E legs.

Thanks alot for any explainations, really appreciate it.

PS how is [tex] \phi^3 [/tex] in d=6 even being used in this chapter as an example?

I have a few questions on the content of this chapter.

A theory is renormalizable in general if all it's Lagrangian coefficents have +ve or 0 mass dimension. So far so good. Now Srednicki says he will use [tex] \phi^3[/tex] theory in d=6 as his example, to see how to construct a finite expression for scattering amplitude to arb high order if coupling 'g'.

Am I correct in thinking by scattering amplitude he just means the [tex] \langle f\mid i\rangle [/tex] as given in the LSZ? which then contains [tex]\langle 0\mid T\phi(x_1).....\phi(x^{'}_1)\mid 0\rangle[/tex]

So as he shows in CH10 we only care about the fully connected diagrams. So e.g. for 4 particles scattering we sum all diagrams with 4 sources with sources removed etc.

Then he goes onto:

So just as [tex] \Pi (k^2) [/tex] is the sum of all 1PI diagrams with two external sources (with the sources removed) (although this is not the exact propagator), and [tex] \b{V}_3 [/tex] is the sum of all 1PI diagrams with three external legs, so [tex] \b{V}_n [/tex] is the sum of all 1PI diagrams with n external legs.

He then constructs these upto E (the self energy and [tex] \b{V}_3 [/tex] plus all the higher n vertices from [tex] 4 \leq n \leq E [/tex].)

I just don't understand why he's doing this? what does he mean exactly by "process of interest", does he mean E particles scattering or something? If so why only sum vertices upto E, because surely you could still have vertices that have higher valence within your 1PI diagram, even if the diagram only has E legs.

Thanks alot for any explainations, really appreciate it.

PS how is [tex] \phi^3 [/tex] in d=6 even being used in this chapter as an example?

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