Performing integration over field variable

[aaaa202]

I have this exercise, where I am to integrate out the fluctuations in the order parameter δρ, but I don't really know how to do this. The integral is over r and imaginary time. Should I try to complete the square, so the correlation term between δρ and Θ drops out? In that case which shift of variables should I make?

 

[nrqed]

I have this exercise, where I am to integrate out the fluctuations in the order parameter δρ, but I don't really know how to do this. The integral is over r and imaginary time. Should I try to complete the square, so the correlation term between δρ and Θ drops out? In that case which shift of variables should I make?

Yes, just complete the square. The shift of variable will simply be $\delta \rho \rightarrow \delta \rho - i \partial_r \theta /g$.