Question about derivation of tensor in Di Francesco's CFT

Question

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two dimensions. He writes that the most general form of the tensor is $$S_{\mu \nu \rho \sigma} = (x^2)^{-4} \left\{ A_1 g_{\mu \nu} g_{\rho \sigma} (x^2)^2 + A_2 (g_{\mu \rho}g_{\nu \sigma} + g_{\mu \sigma}g_{\nu \rho})(x^2)^2 + A_3(g_{\mu \nu}x_{\rho}x_{\sigma} + g_{\rho \sigma}x_{\mu}x_{\nu})x^2 + A_4 x_{\mu}x_{\nu}x_{\rho}x_{\sigma}\right\}$$ This I understand and have obtained this result myself. What I don't understand however, is why he has neglected the following term since it seems to satisfy all the constraints presented on P.108: $$S_{\mu \nu \rho\sigma} = A_5 (x^2)^{-3} (g_{\mu \sigma} x_{\rho}x_{\nu} + g_{\mu \rho}x_{\sigma}x_{\nu} + g_{\nu \sigma}x_{\rho}x_{\mu} + g_{\nu \rho}x_{\sigma}x_{\mu})$$ In another thread I posted here, I wondered whether this could be reduced to terms already present in the form Di Francesco gave, but I was quickly reassured this to not be the case. So, if anyone is familiar with his book and would be willing to clarify this it would be great. I asked a professor at my university and he was not sure either why it has been neglected, so I thought I would pose the question here. Many thanks.

This post imported from StackExchange Physics at 2014-09-12 20:19 (UCT), posted by SE-user CAF

Comments

You are perfectly right. I made a calculus error in my remark, so I hide it...

I find too : A1 = 3A - A5, A2 = -A, A3 =  - 4 A + 2 A5, A4 =  8 A -8 A5 .

And yes, the double trace is zero, whatever A5 is.

So, my remark was just completely wrong..., and there is still a lot of mystery

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So, in some sense, the final interesting result of the book (mean and standard deviation of \(T^\mu_\mu\) are zero), is always valid, even if you add a \(A_5\)term.

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Ok, thanks for checking. So the result is that whether or not we add on the $A_5$ term, the trace of the Schwinger function in two dimensions vanishes for all $x$? So, the $A_5$ term is in some sense superfluous since its addition does not change the result?