vela said:##\vec{p}_\text{f}## is the three-momentum of particle C.
Try differentiating ##-m_c^2 = p_c^2-E_c^2## to calculate ##dE_c/dp_c##. I think the problem with your attempt is that you're assuming ##dv_c/dp_c = 0## and ##dv_d/dp_c = 0##.
A two-body relativistic cross section is a measurement of the probability of a scattering event between two particles in a relativistic system. It takes into account the energy and momentum of both particles and is used to study interactions between particles at high energies.
A two-body relativistic cross section is derived using the principles of quantum mechanics and special relativity. The starting point is the scattering amplitude, which describes the probability of a scattering event. The cross section is then obtained by integrating the squared amplitude over all possible final states of the particles.
The value of a two-body relativistic cross section is primarily affected by the energy and momentum of the particles involved in the scattering event. The masses and charges of the particles also play a role, as well as any other interactions or forces present in the system.
A two-body relativistic cross section is an important tool in experimental physics as it allows scientists to study the properties of particles and their interactions at high energies. It is used to analyze data from particle accelerators and other high-energy experiments, and can provide valuable insights into the fundamental forces and particles of the universe.
While a two-body relativistic cross section is a powerful tool, it does have some limitations. It assumes that only two particles are involved in the scattering event and does not account for any additional particles or interactions that may occur. It also relies on theoretical models and assumptions, so the results may not always perfectly match experimental data.