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In Peskin and Schroeder's book (P&S), on the botton of page 106, the authors say that the total cross section transforms as its only non-invariant factor, namely:

$$ {1 \over E_{A} E_{B} |v_A - v_B|} $$

Where $E_i$ and $v_i$ are energies and velocities of the incoming particles ($i=A,B$). The authors then conclude that the cross section itself is not invariant. They actually go as far as rationalizing that it transforms as an area should (invariant to boosts in one direction but not on the other two).

This is at odds with many other sources (here is an example, look for equation 3.18 and 3.19), where an invariant cross section is obtained and that very factor is found out to be:

$$ {1 \over F} = {1 \over \sqrt{(p_A.p_B)^2-m_A^2 m_B^2}} = {1 \over \sqrt{|E_A\vec{p}_B-E_B\vec{p}_A|^2-|\vec{p}_A\times\vec{p}_B|^2}} $$

Where $F$ is the so called Møller's invariant flux factor ($p_i$ are the four momenta, and $m_i$ the masses). And the conclusion here is that the cross section is Lorentz invariant.

Of course, the second expression reduces to the first in any frame that $\vec{p}_A\times\vec{p}_B = 0$ (in particular the center of mass frame or the usual colinear bean "laboratory frame").

I am under the impression that P&S assume such a frame in more than one step of the calculation and that's why their result is frame dependent, but that means that their conclusion is wrong. Am I missing something?

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Peskin and Schroeder assume the laboratory frame, as is evident from the top of page 106:

The difference $|v_A-v_B|$ is the relative velocity of the beams as viewed from the laboratory frame.

As a result, as you (and they) have also pointed out, the cross section is not Lorentz invariant. They furthermore explain that it is only invariant with respect to boosts along the $z$-direction. There is no contradicton in their derivation an conclusion.

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