OPE of Lorentz current with tachyon vertex

Question

This is a question related to chapter 2 in Polchinski's string theory book. On page 43 Polchinski calculates the Noether current from spacetime translations and then calculates its OPE with the tachyon vertex, see equations (2.3.13) and (2.3.14)

$$j_a^{\mu} = \frac{i}{\alpha'}\partial_a X^{\mu}, \tag{2.3.13}$$ $$ j^{\mu}(z) :e^{i k\cdot X(0,0)}:\quad \sim\ \frac{k^{\mu}}{2 z} :e^{i k\cdot X(0,0)}:\tag{2.3.14} $$

I wanted to do a similar calculation but for spacetime Lorentz transformations. First I calculated the Noether current, I get $$ L^{\mu\nu}(z)~=~ :X^{\mu} \partial X^{\nu}: ~-~ (\mu \leftrightarrow \nu).$$ Next I calculated the OPE using Wick's formula (in the form of equation 2.2.10). My result is $$ L^{\mu\nu}(z) :e^{i k\cdot X(0)}: \quad \sim\ -\frac{\alpha'}{2} \ln |z|^2\ i k^{\mu} :\partial X^{\nu} e^{i k\cdot X(0)}: ~-~\frac{\alpha'}{2} \frac{1}{z}\ i k^{\nu} :X^{\mu} e^{i k\cdot X(0)}: ~-~ (\mu \leftrightarrow \nu).$$ I think this answer is incorrect because of the logarithm in the right hand side. So my questions are

1. Is $ L^{\mu\nu}(z)$ defined above indeed the Noether current from spacetime Lorentz transformations?

2. Is the OPE $ L^{\mu\nu}(z) :e^{i k\cdot X(0)}:$ above correct?

3. Is there a link where this calculation is performed so that I can check my result?

Answer 1

I do not know if you have not written, or didn't do the double contractions. But apart from that, it is correct.

I haven't checked the calculation of $L$, but notice that your $L^{\mu\nu}$ is not holomorphic. And so the logarithmic term is expected.