High-energy annihilation
Frank Rieger gives the high-energy cross section as
σ_e+e- = π r_0^2 / γ * (ln (2γ) - 1),
where
γ = 1 / sqrt(1 - v^2 / c^2)
is the Lorentz factor of the electron and the positron in the center of mass frame, and r_0 is the classical radius of the electron.
Imagine that there would be a "potential"
V = -B e^2 / r^2
between the electron and the
positron, where
B = 2.5 * 10^-5 J m^2 / C^2.
We set the numerical value of B as 2.8 * 10^-15 times the Coulomb constant, so that V is equal to the Coulomb potential at the distance of the electron classical radius r_0.
To achieve scattering to a "significant" angle, a fast electron of total energy E must go deep into the potential well, so that
E = -V.
What is the cross section for such a process?
The total energy of the electron is
E = γ m_e c^2.
Let us solve r from -V = E, that is,
B e^2 / r^2 = γ m_e c^2.
We get
r = sqrt(B e^2 / (γ m_e c^2))
= 2.8 * 10^-15 m * 1 / sqrt(γ)
as the radius of the closest approach.
The impact parameter b is probably 2 r for a 1 / r^2 potential. The cross section is
σ = 4 π r^2
= 4 π r_0^2 / γ.
If γ = 100, the formula of Frank Rieger gives the cross section
(π r_0^2 / 100) * (ln(200) - 1),
where ln(200) - 1 = 4.3.
Our formula agrees quite well with Rieger at the 50 MeV scale, but at other energies, the factor
ln(2 γ) - 1
causes some difference.
From where does the logarithm term come in the Rieger formula? It makes the cross section larger when we increase the energy E a lot. The potential must be steeper than 1 / r^2. Maybe something like
~ ln(r) / r^2
for r << 1?
Low-energy annihilation of a pair
Frank Rieger gives the low-energy cross section as
σ_e+e- = 1 / β * π r_0^2,
where the speed of the particles is β c, and r_0 is the classical electron radius.
Sidney A. Coon et al. (2002) in their Section 2.4 in formula (51) state the classical scattering angle for the 1 / r^2 potential as a function of the angular momentum L.
If we in the formula cut the initial velocity v of the particle to a half, we can keep L constant by doubling the impact parameter b. This means that the cross section grows 4-fold if we halve the initial velocity v. Thus, the 1 / r^2 potential is classically not steep enough for the Rieger formula.
The potential in pair annihilation is a black hole potential?
The 1 / r^2 potential has circular orbits at every r (incidentally, the energy of every orbit is the same). To decrease the cross section, we need a potential which does not have circular orbits arbitrarily deep.
Such a potential exists around a Schwarzschild black hole. Close to the event horizon, a ray of light has to be sent almost vertically for it to reach infinity. Otherwise, the potential makes the light to curve back to the horizon.
The lowest circular orbit is at 3/2 times the Schwarzschild radius.
The annihilation of a pair is somehow related to falling into a black hole. The infinitely deep Coulomb potential might form a singularity if the energy were not sent away as photons.
Let us check what are the cross sections associated with a Schwarzschild black hole.
Chris Doran et al. (2005) in their formula (3) have the black hole capture cross section in terms of the initial velocity v of the particle. The cross section is ~ 1 / v^2. Thus, it is not like the annihilation cross section of a pair.
Making the potential flat at large distances cuts the dependence of the cross section on a small initial velocity
We can cut the dependence of the cross section on the initial velocity v simply by making the potential flat at larger distances. The Yukawa potential is very flat far away, and for small v, the cross section is essentially constant.
Thus, there probably exists a potential for which the classical cross section is linear in v. We did not yet find that potential.
Scattering in the 1 / r^2 potential differs in classical mechanics from quantum mechanics
Boris Kayser (1974) writes that the cross section for scattering from a 1 / r^2 potential is quite different from classical mechanics if calculated with the Born approximation.
The classical cross section is linear with the potential strength while the Born approximation claims it is quadratic.
Kayser explains that classical scattering happens in a "strong coupling regime", and therefore, the Born approximation does not apply.
Which is the right way to treat annihilation? Feynman diagrams probably use the Born approximation, and empirical data seems to support the Feynman way.
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