I'm trying to reproduce the Sugawara construction calculation using this reference (page 14).
The normal-ordering of two local operators is defined as
N(XY)(w)=12πi∮wdxx−wX(z)Y(w).
Ok, that makes sense. The residue picks up the 0-th term in the OPE.
Now we proceed to calculate X(z)N(YZ)(w), which for some reason picks up two terms:
I don't understand how these two terms appear. The authors mention a "appropriate version of the Wick theorem" which to me looks like handwaving. WZW isn't a free theory and therefore the Wick theorem doesn't work.
On the other hand, the way I see it, we can use the XY OPE to write the product XY as a sum of local operators, and then we take the OPEs of all that operators with Z. Equivalently, we could have started with XZ and obtained a different sum of local operators, but taking their OPEs with Y is guaranteed to give the same result by associativity of OPE.
I've done the calculation on the whiteboard and obtained
Ja(z)N(JbJb)(w)=(k+h∨)Ja(w)(z−w)2+…,
which is the expected result expect for the factor of 2.
The irony is in that that factor of 2 is easily "explained" by the Wick theorem (there's 2 equivalent contractions), but I just can't see how that explanation can work in a nonlinear model like WZW, and also I can't see why my calculation is wrong.
Where am I wrong?
Update: example calculation for the Abelian case (fabc=0):
∮wdx2πiJ(z)J(w)J(x)x−w=∮wdx2πi1x−z(k(z−w)2+O(1))J(x)=kJ(w)(z−w)2
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