Remarks on the Calculations of Charged Open String Amplitudes: the 1-loop Tadpole

In string theory, there are various physical situations where the world-sheet fields have a shifted moding. For instance, this is the case for the twisted closed string in Z_N orbifold or for the charged open string in a constant electro-magnetic field. Because of this feature, it is quite challenging to give explicit formulae describing the string interaction, even for the bosonic case. In this note, we focus on the case of the charged open bosonic string and construct the 1-loop tadpole which is an object generating all 1-point functions from the annulus in the presence of an external field. In the operator formalism, this represents one of the basic building blocks for the construction of a general loop amplitude.