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

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 $100~\text{TeV}$?

• Burton Richter wrote: "[...] cross sections [$\sigma$] typically drop as $E^{-2}$." -- This seems to match formulas (47.1) - (47.12) of "The PDG Data Book", chap. 47: "Cross section formulae for specific processes", where $$\frac{1}{s}\simeq E^{-2}.$$ However, Figure 49.9 of chap. 49: "Plots of cross sections {...}" shows otherwise: $\sigma^{~p p}_{\text{tot}}$ rising with $\sqrt{s}$. This apparent discrepancy might be the root of my question... Sep 9, 2014 at 17:12
• Yeah, it's my understanding that the $pp$ cross section increase with energy is anomalous. Given that nobody knows what the cross section is actually going to be at 100 TeV, it's best to treat the numbers in the table as speculation.
– mng
Sep 10, 2014 at 3:16

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 $50~\text{TeV}$.

• DarioP: "However he [B. Richter, 1409.1196] gets the result multiplying by 7 both $N_1$ and $N_2$ while the total inelastic cross section $\sigma$ stays constant." -- That was my understanding, too. "the specific cross section of a resonance interesting to study. Those tends to scale with $E^{-2}$, [PDG Fig.] 49.5" -- Also matching PDG eqs. (47.1) - (47.12). "[...] The total inelastic cross section, $\sigma$, is assumed constant." -- Is there some justification for this difference? (Secondary processes?? ...) "$\sigma_{x,y}=$ [...]" -- Is there perhaps a derivation at PhysSE already? Sep 15, 2014 at 18:24
• Do not think about too complicate things, he is just getting some gross estimations, there is no need to take into account the dependency of $\sigma$ over $E$ also because he is interested in specific processes at high energy, so the total cross section is pretty useless. For the derivation of the beam sizes $\sigma_{x,y}$ you can look at any introductory book of accelerator physics like "An Introduction to Particle Accelerators" by E. Wilson. Sep 16, 2014 at 7:02