Confusion with central charge in CFT and improvement of energy-momentum tensor

Question

In theory of free scalar field $$ S = \int d^2x \;\partial_\mu \phi \partial^\mu \phi $$ $$ \langle \phi(z) \phi(w)\rangle \propto \ln(z-w) $$

Exist family of energy-momentum tensors (new term correspond to non-minimal coupling to gravity $\alpha R \phi^2$, example with linear coupling $R\phi$): $$ T_{\mu\nu} = \partial_\mu \phi \partial_\nu \phi + \alpha (\partial_\mu\partial_\nu - \eta_{\mu\nu}\Box) \phi^2 $$ Obviously $\partial^\mu T_{\mu\nu} = 0$, but $T_\mu^\mu \neq 0$ if $\alpha \neq 0$. And $$ T = T_{zz} =\partial_z \phi \partial_z \phi + \alpha (\partial_z\partial_z)\phi^2 = (1+2\alpha)\partial_z \phi \partial_z \phi + 2\alpha \phi \partial_z^2\phi $$

So this shift will affect standard OPE: $$ T(z)T(w) = \frac{1/2}{(z-w)^4} + \frac{2T(w)}{(z-w)^2} + \frac{\partial T(w)}{(z-w)} + \dots $$

To (I'm interested only in first term): $$ T(z)T(w) = \frac{1/2 (1+6\alpha+10\alpha^2)}{(z-w)^4} + \frac{-1/2 \ln(z-w)}{(z-w)^4} + \frac{2T(w)}{(z-w)^2} (1+??) + \frac{\partial T(w)}{(z-w)}(1+??) + \dots $$

And if one choose for example $\alpha = 1$, one obtain central charge $c = 17$ for free scalar field.. What about ln term in OPE?

Could someone to clarify and interpretate this confusion? Why we say that $c=1$ for free scalar filed?

Answer 1

The free scalar field can have any complex central charge, due to the linear modification of the energy-momentum tensor that you mention. The case $c=1$ is a bit special, in particular it allows compactification with an arbitrary radius, while for general $c$ the radius is quantized. See Section 4.1 of my review article for more details: https://arxiv.org/abs/1406.4290

In the context of string theory, the free scalar with $c\neq 1$ is sometimes called the linear dilaton theory.