Symmetry factor in 3-points Green functions of QCD

Oct 2019
1
0
Montpellier
Hello Everyone ,

I am new to this forum and not a native speaker, so please be indulgent (Angel). So here is my problem:

I started my Ph.D last months in QCD and I am trying to determine the symmetry factor of a 3 point Green function in QCD and all of my attempt to find back the expected value of "1" has failed. The diagram is attached to this post. I have two references "Peskin&Schroeder p.530" and "Bailin&Love; Introduction to gauge field theory, p.139", that tells me this results. I am applying the same technique that is developped in page 92 of Peskin&Schroeder's book and as you'll see, I can't find anything else than 1/3 for the overall symmetry factor. Here is my calculations:

So this diagram correspond to the Wick contractions shown in the second file attached (sorry I didn't know how to make everything appear directly in the text).

The first factorial 3 comes from the Taylor expansion of order 3 and the second one from the QCD symmetrized 3-point gluon vertex. Now we can proceed to the evaluation of the overall symmetry factor:

1) \(\displaystyle (1/3!)^2\) as we just said
2) \(\displaystyle 2!\) from the interchange of the last two similar vertex
3) There is 3 possibility for placing the contraction on the first gluon field at point x, then 2 possibility for the second gluon field at point x => \(\displaystyle 3*2=3!\)
4) all contractions are then unique and no other symmetry factor arise

Finally we find : \(\displaystyle (1/3!)^2 * 2! * 3! = 1/3\)

So basically, where am I wrong ? I should find 1 instead of one third...
Thank you all for your help, I've already spent a lot of time upon this problem.
 

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