Show that s+t+u=mA^2+mB^2+mC^2+mD^2, where mi is the rest...

Show that $$ s+t+u=m_A^2+m_B^2+m_C^2+m_D^2, $$ where $m_i$ is the rest mass of particle $i$. To display the kinematic (or physical) regions of processes related by crossing, we construct a two-dimensional plot which maintains the symmetry of $s, t, u$. The three axes $s, t, u=0$ are drawn (Fig. 4.7) to form an equilateral triangle of height $\sum m_i^2$. From any point inside and also outside (if attention is paid to the signs of $s, t, u)$ the triangle, the sum of the perpendicular distances to the axes is equal to the height of the triangle [see (4.44)]. It is easy to show that $s$ is the square of the total center-of-mass energy of the process $\mathrm{AB} \rightarrow \mathrm{CD}$ [see (3.12) or Exercise 4.6]. It is conventional to take the reaction under study to be the $s$ channel process. In our last example, this was $\mathrm{e}^{-} \mathrm{e}^{+} \rightarrow \mathrm{e}^{-} \mathrm{e}^{+}$scattering. The crossed reactions $\mathrm{A} \overline{\mathrm{D}} \rightarrow \mathrm{C} \overline{\mathrm{B}}$ and $\overline{\mathrm{D}} \mathrm{B} \rightarrow \mathrm{C} \overline{\mathrm{A}}$ are called the $u$ and $t$ channels, respectively, since $u$ and $t$ are equal to the square of the total center-of-mass energy in the respective channels (see Exercise 4.7).

Show that
$$
s+t+u=m_A^2+m_B^2+m_C^2+m_D^2,
$$
where $m_i$ is the rest mass of particle $i$.
To display the kinematic (or physical) regions of processes related by crossing, we construct a two-dimensional plot which maintains the symmetry of $s, t, u$. The three axes $s, t, u=0$ are drawn (Fig. 4.7) to form an equilateral triangle of height $\sum m_i^2$. From any point inside and also outside (if attention is paid to the signs of $s, t, u)$ the triangle, the sum of the perpendicular distances to the axes is equal to the height of the triangle [see (4.44)].

It is easy to show that $s$ is the square of the total center-of-mass energy of the process $\mathrm{AB} \rightarrow \mathrm{CD}$ [see (3.12) or Exercise 4.6]. It is conventional to take the reaction under study to be the $s$ channel process. In our last example, this was $\mathrm{e}^{-} \mathrm{e}^{+} \rightarrow \mathrm{e}^{-} \mathrm{e}^{+}$scattering. The crossed reactions $\mathrm{A} \overline{\mathrm{D}} \rightarrow \mathrm{C} \overline{\mathrm{B}}$ and $\overline{\mathrm{D}} \mathrm{B} \rightarrow \mathrm{C} \overline{\mathrm{A}}$ are called the $u$ and $t$ channels, respectively, since $u$ and $t$ are equal to the square of the total center-of-mass energy in the respective channels (see Exercise 4.7).

Key Concepts

Mandelstam Variables

Mandelstam variables (s, t, u) are invariant quantities used to describe the kinematics of scattering processes in relativistic physics. They are defined in a way that their values are independent of the particular inertial frame, and they are directly related to the energy and momentum of the particles involved in the scattering. These variables allow one to express certain conservation laws and relationships, such as s + t + u being equal to the sum of the squares of the rest masses of the particles, making them powerful tools in both theoretical calculations and the interpretation of experimental data.

Center-of-Mass Energy

The center-of-mass energy is the total energy available in the center-of-mass frame of a colliding system, and it is a key parameter in scattering experiments. In this frame, the energies and momenta of the particles are balanced, and the square of the center-of-mass energy is represented by one of the Mandelstam variables—typically s in the s-channel process. This energy scale is crucial because it sets the threshold for the production of new particles and determines the dynamics of the interactions.

Crossing Symmetry

Crossing symmetry is a fundamental concept in quantum field theory that relates different scattering processes by interchanging particles and antiparticles among the initial and final states. It implies that the amplitude for a particular physical process in one channel (such as the s-channel) can be analytically continued to describe processes in the t- or u-channels. This symmetry reflects the underlying invariances of the theory and provides deep connections between seemingly different reactions.

Kinematic Regions

Kinematic regions refer to the sets of values that the Mandelstam variables can take, which are constrained by energy-momentum conservation and the intrinsic properties of the particles involved. By graphically representing these regions—often through constructions like the equilateral triangle in the s, t, u space—one can visualize the physically allowed areas for various scattering channels. This helps in distinguishing regions where the processes are physically viable from those that are unphysical, thereby aiding in the analysis of scattering amplitudes and cross sections.

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