deletions | additions       diff --git a/We_start_by_applying_the__.tex b/We_start_by_applying_the__.tex  index b2d4fe5..ec0e47f 100644  --- a/We_start_by_applying_the__.tex  +++ b/We_start_by_applying_the__.tex 
...
\end{multline}  Simplifying  \begin{multline}  I\propto (2\pi)^2f(\mu_d-\mu)  |t_{1}|^2 |t_{2}|^2 \Biggr\{ \int f(\mu_d-\mu)\int  [1-f(\epsilon-\mu)] P_{1}(\mu_d-\epsilon)d\epsilon \\ -[1-f(\mu_d-\mu)]\int f(\epsilon-\mu) P_{1}(\epsilon-\mu_d) d\epsilon \Biggr\}   \end{multline}  Remarkably, this integral, and therefore the current $I$ is in general nonzero with an asymmetric environment.