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

step 1 In this problem, we're looking at molecular orbitals. And first, we're going to bring two 2s orb

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 $$ t=0, \quad s u=\left(M^2-m^2\right)^2 $$ where $m$ and $M$ are the electron and muon masses, respectively. Construct the Mandelstam plot.

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
$$
t=0, \quad s u=\left(M^2-m^2\right)^2
$$
where $m$ and $M$ are the electron and muon masses, respectively. Construct the Mandelstam plot.

Key Concepts

Mandelstam Variables

The Mandelstam variables (s, t, u) are invariant quantities defined in relativistic scattering processes. They are constructed from the four-momenta of the particles involved and represent the squared center-of-mass energy (s), the momentum transfer (t), and a third variable (u) that completes a relation among them. These variables allow for expressing kinematic constraints and scattering amplitudes in a Lorentz-invariant manner, essential for analyzing particle interactions.

Physical Regions in Scattering

The physical region in a scattering process corresponds to the set of values of the Mandelstam variables that can be realized with real momenta subject to energy-momentum conservation and the on-shell conditions of the particles. Boundaries of these regions are defined by kinematic limits, such as when the momentum transfer becomes zero or when the invariants satisfy specific equality relations, ensuring the process remains physically allowed.

Crossing Symmetry

Crossing symmetry is the principle that relates different scattering processes by interchanging initial and final state particles. This symmetry implies that the analytic structure of the scattering amplitude in one channel can be continued into other channels (such as the t or u channels), thereby unifying the description of processes in different kinematic regions. It serves as a powerful tool for understanding the full analytic behavior and constraints on scattering amplitudes.

Mandelstam Plot

A Mandelstam plot is a graphical representation that illustrates the allowed physical regions for scattering processes in terms of two Mandelstam variables, typically s versus t or s versus u. The plot shows the boundaries imposed by kinematic constraints, such as the condition t = 0 and relations among s, t, and u. It provides insight into how different channel processes (direct and crossed) are connected through the analytic continuation of the kinematic variables.

Transcript

Please give Ace some feedback