# Diagramatic perturbative expansion of QCD

1. Aug 14, 2008

### Sunset

Hi!

Has anybody seen the perturbative expansion of the generating functional of QCD $$Z[J,\xi,\xi*,\eta,\eta*]$$ expressed with Feynman diagrams? I mean, there should be an expansion, containing external sources denoted by something like
-------o abbreviation for $$i \int d^4 x J$$
-------# abbreviation for $$i \int d^4 x \eta$$
and so on...

I haven't found any book showing this.

Is it maybe simply the sum of all possible graphs with their combinatorical prefactors that can be constructed from the Feynman rules?

Best regards Martin

2. Aug 14, 2008

### Avodyne

Yep. When sources are attached, there is an extra combinatoric factor of 1/n! for each type of source, where n is the number of that type of source that appears in the diagram.

3. Aug 14, 2008

### Sunset

Hi Avodyne!

This would be really cool... but I want to make sure, that we mean the same thing:

$$\eta$$ is the quark source
J the gauge field (gluon) source
$$\xi$$ the ghost source

quark line is a straight line
ghost line is dottet
gluon line is twidled

$$i \int d^4 x \eta$$ I draw as a dot
$$i \int d^4 x \xi$$ as triangle
$$i \int d^4 x J$$ as box

So Z is equal the little bitmap I attached (up to prefactors and understood that there are infinite many more diagrams i.e. all possible ones ) ? (I didn't take care of colors and flavors, just assume there is only one flavor and one color, if one takes into account more colors then more diagrams...) One would have to draw ALL possible graphs, that means disconnected graphs as the last one, too.

If that is true , is the generating functional W=lnZ also exactly the sum of ALL possible connected diagrams?

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4. Aug 14, 2008

### Sunset

Oh wait, I did a mistake. Understood, each external point should have a source (forgot to draw them)

5. Aug 14, 2008

### Sunset

I corrected that one

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6. Aug 14, 2008

### Avodyne

For some reason I'm not able to view your 2nd picture. In the first, the quark loop with a single gluon attached is zero. And one has to get the combinatoric factors right. But then, yes, Z is just the sum of all possible diagrams (connected and disconnected), and W=log(Z) is the sum of just the connected diagrams.

This is all explained pretty well for phi^3 theory in the book by Srednicki (google to find a free draft copy online).

7. Aug 15, 2008

### Sunset

ok, great!

I wasn't sure if really everything goes the same way as in phi^n theory.

Thanks a lot,

Martin