Are there two deadweight welfare losses associated with a Pigovian tax? I am constructing a diagram for an indirect tax (specific) imposed to correct a negative externality of consumption. 


There is a deadweight welfare loss from the externality (represented in blue) because, although it is reduced,the tax does not achieve to shift the supply curve to a point where MSB=MSC (Qm).

However, the tax itself also should produce a deadweight welfare loss (represented in green). Is it correct that there are two deadweight welfare losses in this situation or is there no deadweight welfare loss associated with a Pivogian tax?
The Pigovian tax is responsible for neither of the deadweight losses in your diagram.

The Pigovian tax has partially, but not wholly, corrected a deadweight loss that was caused by the negative externality.

There is a deadweight loss associated with Pigovian taxes: that is the administrative cost of collecting the tax. This is not pictured in your diagram.