How does the number of events per bunch collision scale (as function of energy, luminosity ...)

Question

Looking at Table 1 of Burton Richter's recent article High Energy Colliding Beams; What Is Their Future? I'm wondering how the number of events per bunch collision ("$N_b$") scales for the collider designs being compared.

As the article notes (p. 6)

[...] the new luminosity required is very roughly proportional to the square of the energy because cross sections [$\sigma$] typically drop as $E^{-2}$. A seven-fold increase in energy from that of HL-LHC to a 100-TeV collider therefore requires a fifty-fold increase in luminosity [$\mathcal L$].

The examples of Table 1 illustrate such scaling, where the 50-fold increase in luminosity seems entirely due to the number of "Particles per Bunch" (along with both beam currents) being increased by a factor of $\approx 7$, while other relevant "beam parameters" ("Bunch spacing", $\beta^{\ast}$, $\epsilon_n$) are kept constant.

Question
Why, in these examples, does the number of events per bunch collision also show a 50-fold increase instead of staying (roughly) constant as $$ N_b \sim \sigma \times \mathcal L$$ ?

p.s.
Since the article and the examples, as far as I understand them, deal with a seven-fold increase in energy, is there possibly some mistake in the first row of values in Table 1, i.e. the "Beam energy" of the "LHC-100" examples being $50~\text{TeV}$ rather than $100~\text{TeV}$?

Answer 1

At the beginning of page 6 he talks about the specific cross section of a resonance interesting to study. Those tends to scale with $E^{-2}$, look at figure 49.5 in the PDG and compare the $J/\Psi$, the $\Upsilon$ and the $Z$. This is the reason why he would like to increase the luminosity with $E^2$. The total inelastic cross section, $\sigma$, is assumed constant.

So $N_b \sim \sigma\times\mathcal{L}$ goes up as $\mathcal{L}$ goes up, remember its expression:

$$\mathcal{L}=\frac{N_1N_2fn_b}{4\pi\sigma_x\sigma_y}$$ where $N$ are the number of particles in the colliding bunches, $f$ is the revolution frequency, $n_b$ is the total number of bunches in the ring and $\sigma_{x,y}$ are the horizontal and vertical beam sizes:

$$\sigma_{x,y} = \sqrt{\frac{\beta^*_{x,y}\varepsilon_{n_{x,y}}}{\gamma}}$$ where $\gamma$ is the relativistic factor. There is an important distinction between the normalized emittance: $\varepsilon_n$, which is preserved during the acceleration, and the geometrical emittance: $\varepsilon=\varepsilon_n/\gamma$ which determines the beam size.

The shrinking of the beam with the energy (adiabatic damping) gives a dependency of $\mathcal{L}$ over $E$. He explicitly neglects this giving the reason in the paragraph between page 6 and 7, which is not clear to me, comments are welcomed! However he gets the result multiplying by $7$ both $N_1$ and $N_2$ while the total inelastic cross section $\sigma$ stays constant.

PS. Yes, that should be $50~\text{TeV}$.