# Closed-Time-Path & leptogenesis

I'm trying to reproduce some of the results from this paper by B. Garbrecht but I'm having some difficulties getting his Eq. (110),

$$\mathrm{i} \delta S_{N i j}^{<,>}=\mathrm{i}\delta S_{N i}\left(\mathrm{i}\Sigma_{N i j}^{>}-\mathrm{i}\Sigma_{N i j}^{T}\right) \mathrm{i} S_{N j}^{T}-\mathrm{i} S_{N i}^{\bar{T}}\left(\mathrm{i}\Sigma_{N i j}^{>}-\mathrm{i}\Sigma_{N i j}^{\bar{T}}\right) \mathrm{i} \delta S_{N j}$$

from Eq. (109),

$$\mathrm{i} S_{N i j}^{\mathrm{wv}>}=\mathrm{i} S_{N i}^{>} \mathrm{i}\Sigma_{N i j}^{<} \mathrm{i} S_{N j}^{>}+\mathrm{i} S_{N i}^{\bar{T}} \mathrm{i}\Sigma_{N i j}^{>} \mathrm{i} S_{N j}^{T}-\mathrm{i} S_{N i}^{>} \mathrm{i}\Sigma^{T} \mathrm{i} S_{N j}^{T}-\mathrm{i} S_{N i}^{\bar{T}} \mathrm{i}\Sigma_{N i j}^{\bar{T}} \mathrm{i} S_{N j}^{>}\ .$$

The text gives the following suggestion:

In this expression, we have dropped terms that contain products of on-shell $$\delta$$-functions pertaining to $$N_i$$ and $$N_j$$ with $$i\neq j$$ that cannot be simultaneously satisfied.

Any comments would be very welcome, thanks.

The two things that are happening here are that one throws out terms where both of the $$S_i$$ are on-shell (since $$i\neq j$$, then the product of the $$\delta$$-functions yield zero) and also that the $$T$$ and $$\bar T$$ propagators have on-shell out-of-equilibrium contributions.
For example, in particular, in the second term of Eq. (109), one replaces one of the propagators by $$\delta S$$ and one remains an off-shell $$T$$ or $$\bar T$$ propagator, resulting in two terms in Eq. (110). On the other hand, the first term does not contribute at all.