# Question about derivation of tensor in Di Francesco's CFT

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.

• As explained here, the $A_5$ term would correspond to some kind of a "trace anomaly" of the energy-momentum tensor. – Dilaton Sep 16 '14 at 10:54
• Hi Dilaton, thanks for commenting here, I do not remember posting my question on overflow. I could not understand the argument very well, what does it mean to say a 'trace anomaly'? – CAF Sep 16 '14 at 11:14
• For clarity, it was uncovered that there is indeed no trace anomaly and that the vanishing of the trace still holds everywhere even when we include the addition of $A_5$. This may be the reason why the author neglected its addition, viewing it as perhaps a superfluous term or maybe there is more to it.... So the question is still open for answers to anybody else who is interested. Thanks. – CAF Sep 18 '14 at 13:21