Operator product expansion involving derivatives

Question

I have questions regarding the equation (2.2.4) in Polchinski Vol 1:

$$ X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2) = -\frac{\alpha'}{2}\eta^{\mu\nu} \ln|z_{12}|^2 + \sum_{k=1}^\infty \frac{1}{k!}\left[(z_{12})^k:X^\nu \partial^k X^\mu(z_2,\bar{z}_2): + (\bar{z}_{12})^k:X^\nu \bar{\partial}^k X^\mu(z_2,\bar{z}_2):\right]. \tag{2.2.4}$$

Here $z_{12} = z_1-z_2$, and the :: means normal-ordering:

$$ :X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2): = X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2) + \frac{\alpha'}{2}\eta^{\mu\nu} \ln|z_{12}|^2 .\tag{2.1.21b}$$

Now, how can we accommodate the normal-ordered string $$:X^\nu \partial^k X^\mu(z_2,\bar{z}_2):$$ in the definition above? I can imagine there are at least two complications:

1. a derivative is involved, and

2. the product product is now at one single point $(z_2,\bar{z}_2)$, where would the log term $\ln|z_{12}|^2$ go?

Answer 1

1. First of all, it should stressed that the lhs. of eq. (2.2.4) and the first terms on the rhs. of eq. (2.1.21b) are radially ordered, i.e. there are an implicitly written radial ordering symbol ${\cal R}$ in these eqs.

2. Eq. (2.1.21b) is the definition of the conformal normal ordering in terms of the radial ordering.

3. It is assumed that the normal ordered term $$:X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2): ~=~ \sum_{k=1}^\infty \frac{1}{k!}\left[(z_{12})^k:X^\nu \partial^k X^\mu(z_2,\bar{z}_2): ~+~ (\bar{z}_{12})^k:X^\nu \bar{\partial}^k X^\mu(z_2,\bar{z}_2):\right]$$ is analytic. The rhs. is merely a defining notation for theses Taylor coefficients.

4. Note that Polchinski operates with various notions of normal ordering, cf. p. 59-60. It turns out that the conformal normal ordering and the creation/annihilation normal ordering agree in the matter (but not the ghost) sector of the string.

5. Recall that the string $X(z,\bar{z})$ is a Fourier expansion in creation/annihilation/oscillator modes $\alpha_n$, $\tilde{\alpha}_m$, etc. If the normal ordering $:~:$ is the creation/annihilation normal ordering, i.e. with creation (annihilation) operators ordered to the left (right), respectively, then normal ordering $:~:$ is independent of world-sheet (WS) coordinates, and we could then in principle calculate all the coefficients in the Taylor expansion (2.2.4) in terms of creation & annihilation modes.