I'm trying to calculate the conformal anomaly $c$ of a free scalar on a 2-sphere. I've seen other, indirect ways to do this, but since this is a free theory I feel like it should be possible to see it directly as the anomalous depenedence of the partition function on the radius of the sphere. However, I'm having a difficult time actually doing this.

In principle, one starts with the naive expression for the partition function, the inverse square root of the determinant of the Laplacian. This is obtained as a product of its eigenvalues, $l(l+1)$, raised to their multiplicity $2l+1$. We get:

$$Z = \prod_l (l(l+1)/\pi)^{-l-1/2}. $$

This diverges, so needs to be regulated. For example, we might impose a cutoff above momenta $\Lambda=l_{max}/R$. By dimensional analysis, the regulated partition function must be a function of $\Lambda R$, although ideally it would be cutoff independent. The point of the conformal anomaly is then that there is some dependence like:

$$\log Z \sim c \log (\Lambda R), $$

where, for the free scalar, $x=1$. This cannot be removed by a local counterterm, and so the partition function depends on the radius (it also depends on the cutoff, which, as far as I understand, is why we say the partition function is ill-defined).

This is how it should work, anyway. But inspecting the expression above, it seems like we'll actually get something like:

$$ \log Z \sim (\Lambda R)^2 \log (\Lambda R) .$$

Can anyone explain this discrepancy? Maybe someone could give a sketch of the correct way to do this calculation.

Moreover, why exactly do we think of this as a dependence on the radius? Can't we just talk about the cutoff in terms of the dimensionless quantitiy $l_{max}$, without mentioning $\Lambda$ or $R$ seperately? For example, $l_{max}$ is related to the number of points in a lattice regularization, with no reference to the size of the sphere.


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