Conformal weights of the vertex operator

[maverick280857]

Hi,

I'm trying to prove that the conformal weight of the bosonic vertex operator $:e^{ik\cdot X}:$ is $\left(\frac{\alpha'k^2}{4},\frac{\alpha'k^2}{4}\right)$.

I've done some algebra but I think I am making some mistake with a factor of 2 somewhere because I get a 1/2 instead of a 1/4. My attempt is detailed in the attachment.

Can someone please tell me what is wrong here? Its probably a dumb thing, but I don't see it.

Thanks!

 

[maverick280857]

To the moderator: this should probably be moved to the "Beyond the Standard Model" subforum. Apologies for the inconvenience!