In the center-of-mass frame for the process AB →CD, show that d Q=(1)/(4 π^2) (pf)/(4 √(s)) d Ω

In the center-of-mass frame for the process AB →CD, show...

In the center-of-mass frame for the process AB $\rightarrow$ CD, show that $$ \begin{gathered} d Q=\frac{1}{4 \pi^{2}} \frac{p_{f}}{4 \sqrt{s}} d \Omega \\ F=4 p_{i} \sqrt{s} \end{gathered} $$ and hence that the differential cross section is $$ \left.\frac{d \sigma}{d \Omega}\right|_{c m}=\frac{1}{64 \pi^{2} s} \frac{p_{f}}{p_{i}}|\Omega|^{2} $$ where $d \Omega$ is the element of solid angle about $\mathbf{p}_{C}, s=\left(E_{A}+E_{B}\right)^{2},\left|\mathbf{p}_{A}\right|=$ $\left|\mathbf{p}_{B}\right|=p_{i}$ and $\left|\mathbf{p}_{C}\right|=\left|\mathbf{p}_{D}\right|=p_{f}$.

In the center-of-mass frame for the process AB $\rightarrow$ CD, show that
$$
\begin{gathered}
d Q=\frac{1}{4 \pi^{2}} \frac{p_{f}}{4 \sqrt{s}} d \Omega \\
F=4 p_{i} \sqrt{s}
\end{gathered}
$$
and hence that the differential cross section is
$$
\left.\frac{d \sigma}{d \Omega}\right|_{c m}=\frac{1}{64 \pi^{2} s} \frac{p_{f}}{p_{i}}|\Omega|^{2}
$$
where $d \Omega$ is the element of solid angle about $\mathbf{p}_{C}, s=\left(E_{A}+E_{B}\right)^{2},\left|\mathbf{p}_{A}\right|=$ $\left|\mathbf{p}_{B}\right|=p_{i}$ and $\left|\mathbf{p}_{C}\right|=\left|\mathbf{p}_{D}\right|=p_{f}$.

Key Concepts

Invariant Amplitude and Kinematic Invariants

The invariant amplitude is a Lorentz invariant quantity that encodes all the dynamics of the scattering process. Its square gives the probability density for the scattering event, independent of the reference frame. Alongside, kinematic invariants like the Mandelstam variable s (which represents the square of the total energy in the center-of-mass frame) are used to succinctly capture the energy-momentum relations dictated by conservation laws. These invariants are essential in expressing the differential cross section in a form that is independent of the observer’s frame of reference.

Differential Cross Section

The differential cross section quantifies the probability per unit solid angle for a scattering process to occur in a particular direction. It is expressed in terms of the square of the invariant amplitude, the phase space factor, and the flux factor. This quantity is fundamental in scattering experiments as it provides direct insight into the angular distribution and strength of the interactions between particles.

Center-of-Mass Frame

The center-of-mass frame is a special frame of reference in which the total momentum of the system is zero. This frame significantly simplifies the kinematics of scattering processes because the energies and momenta of the particles are symmetrically distributed. Many formulas, including those for the phase space and flux factors, take on simpler forms in the center-of-mass frame, making it a natural choice for theoretical analyses and experimental interpretations.

Phase Space Integration

Phase space integration is the process of summing over all the allowed final state configurations of a scattering or decay process. In scattering theory, the phase space element (dQ) provides the measure for how the final state particles are distributed in momentum or energy, consistent with energy-momentum conservation. This integration is crucial to relate the microscopic interaction (embodied in the invariant amplitude) to observable quantities like cross sections.

Flux Factor

The flux factor represents the effective rate at which the incident particles overlap and interact in a scattering process. It accounts for the relative velocity and density of the incoming particles and is used to normalize the scattering amplitude so that the computed cross section has the correct probability interpretation. In many calculations, especially in the center-of-mass frame, the flux factor takes a simplified form, making it easier to relate the invariant amplitude to measurable quantities.

Transcript

00:01 Given point p, we're trying to show that the total solid angle around point p for whatever shape we are integrating over is equal to 4 pi when the point p is within that shape.

00:28 So we first need to demonstrate that it would be helpful if we can quantify 4 pi.

00:37 By an integration over a sphere.

00:41 Since that would be quite simple, we don't have to worry about a change in radius.

00:47 So we're going to have to demonstrate that the total solid angle integrating over a sphere is equal to if we integrated it over a cube or any other shape.

01:07 So how are we going to do that? let's first project a solid angle and let's have it projected onto a very tiny region of the surface area of some shape.

01:31 And we'll name this region ds.

01:37 However, solid angles are with respect to the perpendicular area.

01:53 Over the radius squared.

01:56 So we really need to find the area perpendicular to the point p given ds.

02:15 And in this case, i'll put this as da since it could be infinitesimal and we'll just note this also on our solid angle equation.

02:28 How we can equate this is by noticing that the perpendicular particular area with respect to p projected by ds can be projected by simplitizing by the cosine of the angle between the normal of the plane of ds and a line moving away from point p.

02:57 But there's also something we can now note is if we just divide by r squared for both sides...

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