If it's complex how could one calculate the 2 points function of A(+) and A? Thanks
Gauge fields are real. They are in the adjoint representations of the gauge groups, which are real representations. On a related note, you can show that if you have a collection of fieldstrengths that can only be expressed in the Lagrangian as complex combinations, then you cannot define vector potentials corresponding to these complex fieldstrengths. Therefore, you cannot have gauge fields in a theory if the corresponding fieldstrengths in the theory are complex.
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