This is an approach to anomalies which seems unfamiliar to me..
Firstly what is this function $W$ which seems to satisfy the equation, $\frac{\partial W }{\partial g^{\mu \nu} } = \langle T_{\mu \nu} \rangle $. Is there any standard such $W$?
Now one says that this $W$ must satisfy the following R.G equation,
$(\mu \partial_\mu + 2\int d^dx g^{\mu\nu}\frac{ \delta }{\delta g^{\mu \nu} } )W = 0$
Where does this come from?
- So substituting the first equation into the second and defining $A_{anomaly} = \langle T^\mu_\mu \rangle$ we get, $\mu \partial_\mu W = -2\int d^dx A_{anomaly}$. And one one wants to take two derivatives of the L.H.S w.r.t the metric to get the LHS to look like, $\mu \partial_\mu \langle T_{ab}(x) T_{cd}(0) \rangle $.
But I don't know what is this theorem which is now being used to say that this derivative of the 2-point function of the stress-tensor will necessarily have the following form,
$\mu \partial_\mu \langle T_{ab}(x) T_{cd}(0) \rangle = \frac { C_T} { 4(d-2)^2 (d-1)}\Delta^T_{abcd} \mu \partial_\mu (1/x^{2d-4} ) $
where $C_T$ is some number and $\Delta^T_{abcd} = \frac{1}{2}[S_{ac}S_{bd} + S_{ad}S_{bc}] - \frac{S_{ab}S_{cd} }{d-1} $ , where $S_{ab} = \partial_a \partial_b - \delta_{ab} \partial^2 $
Can someone kindly help as to what is the motivation/proof of the above structure? Is something similar known for derivatives of higher point correlations of the stress-tensor?
