For the process \(1+2 \rightarrow 3+4\), the Mandelstam variables \(s, t\) and \(u\) are defined as \(s=\left(p_{1}+p_{2}\right)^{2}\), \(t=\left(p_{1}-p_{3}\right)^{2}\) and \(u=\left(p_{1}-p_{4}\right)^{2}\). Show that $$ s+u+t=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2} $$

In particle physics, a scattering process describes an interaction where particles collide and scatter into new directions or transform into different particles. Understanding this concept is crucial for analyzing high-energy physics experiments. Typically, we represent a scattering process as \(1+2 \rightarrow 3+4\). Here:

• Particle 1 and Particle 2 are the initial state particles that interact.
• Particle 3 and Particle 4 are the final state particles resulting from the interaction.

The scattering process allows us to explore fundamental forces and particle interactions in experiments using particle colliders. Learning about the energy and momentum exchanged during these interactions helps us to understand the fundamental forces and the properties of particles. This is where the Mandelstam variables come in handy, as they encapsulate the essential information about the scattering event.

Four-momentum is a foundational concept in relativity and particle physics that combines energy and momentum into a single four-vector. It is denoted as \(p = (E/c, \mathbf{p})\) where:

• \(E\) is the total energy of the particle.
• \(\mathbf{p}\) is the three-momentum vector of the particle.

The four-momentum vector is crucial for relativistic physics because it remains consistent across all reference frames. This makes calculations more straightforward and ensures physical laws hold true in all scenarios. In a scattering process:

• \(p_1\) and \(p_2\) represent the four-momenta of the initial-state particles.
• \(p_3\) and \(p_4\) represent the four-momenta of the final-state particles.

Each magnitude of these four-momenta squares equals the mass squared of the respective particle, that is, \(p_i^2 = m_i^2\) where \(i\) can be 1, 2, 3, or 4. This property is utilized in the given exercise to derive the sum of Mandelstam variables.

Invariant mass is an essential property in high-energy physics, representing the mass of a system that remains unchanged regardless of the reference frame. It is a scalar quantity derived from four-momentum vectors. When considering two particles, their invariant mass is given by: \[ M^2 = (p_1 + p_2)^2 \] Usually, the sum of Mandelstam variables \(s, t, \) and \(u\) relates to the invariant masses of the particles involved. In the context of the scattering process \(1 + 2 \rightarrow 3 + 4\), the Mandelstam variables are:

• \(s = (p_1 + p_2)^2\), which represents the invariant mass squared of the initial state particles.
• \(t = (p_1 - p_3)^2\), indicating the momentum transfer between initial particle 1 and final particle 3.
• \(u = (p_1 - p_4)^2\), indicating the momentum transfer between initial particle 1 and final particle 4.

By understanding and summing these variables, one can confirm that:\[ s + t + u = m_1^2 + m_2^2 + m_3^2 + m_4^2 \] where each \(m_i^2\) corresponds to the squared invariant mass of particle \(i\).