# Probability for scattering event

I am reading Schwartz QFT. On page 61 in eq (5.20) he gives an expression that describes the probability for a $$2\to n$$ scattering event to happen:

$$dP=\frac{T}{V}\frac{1}{(2E_1)(2 E_2)}\left|\mathcal{M}\right|^2 \, d\Pi_{\rm LIPS}\tag{5.20}$$

where $$T$$ and $$V$$ are finite time and space volumes and

$$d\Pi_{\rm LIPS}=(2\pi)^4\delta^{(4)}\left(p_1+p_2-\sum_{i=1...n}p_i\right)\times \prod_{i=1...n}\frac{d^3p_i}{(2\pi)^3}\frac{1}{2E_i}.\tag{5.21}$$

Now I consider the example for a $$2\to 1$$ event. Writing everything out yields:

$$dP=\frac{T}{V}\frac{1}{(2E_1)(2 E_2)(2E_f)}\left|\mathcal{M}\right|^2 \, (2\pi)\delta^{(4)}\left(p_1+p_2-p_f\right)d^3p_f$$

where $$p_f$$ is the final state momentum. Performing the integral over $$p_f$$ gives:

$$P=\frac{T}{V}\frac{1}{(2E_1)(2 E_2)(2E_f)}\left|\mathcal{M}\right|^2 \, (2\pi)\delta^{(1)}\left(E_1+E_2-E_f\right)$$

with $$E_f=\sqrt{E_1^2+E_2^2}$$. I am confused that in this expressions $$T$$ and $$V$$ still appear. My feeling tells me that they should drop out since one takes the limit $$t,V\to \infty$$ in the end.

According to Schwartz, the factor $$\frac{T}{V}$$ drops out when we go from differential probability $$dP$$ to the differential cross-section $$d\sigma~=~ \frac{V}{T} \frac{1}{|\vec{v}_1-\vec{v}_2|}dP.\tag{5.6}$$