The differential cross section is measured in certain frame should depend on the frame geometry. and the angle from one frame is different from other. and it is related by:
Lets define the scattering situation in center of momentum frame with a dash and the before scattering is capital letter, after scattering is small letter.
From the particle 2, we can define the Lorentz transform from center of momentum frame to lab frame. Using hyperbolic expression of Lorentz transform, it is just a negative rotation in the hyperbolic space.
Thus, the relation of the particle 1 after the scattering is :
Thus, the angle in lab and CM frame is related by:
for elastic scattering, and . and in low speed, and , then is simply the mass ratio. it reduced to classical form:
the cosine of the lab frame is :
In classical limit:
There fore, the different cross sections are related :
notices that the and is related, they are not independent. and there is a singular point.
This graph is m1 = 1, elastic scattering from P = 0 ( red) , 1 ( orange ), 2 ( green), 3 (blue), 4 ( purple) , 5 ( black). In P= 0 case, it is not mean there is no collision, it is just a classical curve from small momentum.