If the s channel process is e^- μ^- →e^- μ^-, show that the boundaries of the physical regions o

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If the s channel process is e^- μ^- →e^- μ^-, show that the...

If the $s$ channel process is $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$, show that the boundaries of the physical regions of this and the crossed channel reactions are given by
$$
t=0, \quad s u=\left(M^{2}-m^{2}\right)^{2}
$$

Key Concepts

Mandelstam Variables

Mandelstam variables (s, t, and u) are kinematic invariants used to describe the energy and momentum transferred in particle scattering processes. They are defined in such a way that they remain unchanged under Lorentz transformations, making them especially useful in high energy physics for simplifying and comparing scattering amplitudes across different frames of reference.

Scattering Channels

Scattering channels refer to the different ways in which initial and final-state particles can interact in a reaction, typically categorized as the s-channel, t-channel, and u-channel. Each channel corresponds to a different rearrangement of particle momenta, often linked by the invariant Mandelstam variables, and analyzing them helps in understanding the underlying dynamics of the scattering process.

Physical Region Boundaries

The physical region of a scattering process is defined by the constraints imposed by conservation laws and the masses of the involved particles. This region determines the allowed kinematic configurations (values of Mandelstam variables) where the process can physically occur. The boundaries reveal important threshold effects and limit points, such as where momentum transfer vanishes or reaches critical values.

Crossing Symmetry

Crossing symmetry is the principle that relates different scattering processes by interchanging incoming and outgoing particles. This symmetry implies that the amplitude for one reaction can be analytically continued to describe another reaction (a crossed channel), and it plays a crucial role in determining how the allowed kinematic regions of one channel map to those of another.

Kinematic Constraints

Kinematic constraints in scattering processes arise from energy-momentum conservation and the mass-shell conditions of the particles involved. These constraints lead to specific relationships among the Mandelstam variables, such as the condition t = 0 or the relation between s and u expressed by s u = (M^2 - m^2)^2, which delineate the boundaries of the physical regions for the reactions.

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