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For rms geometric beam emittance $ϵ$, the transverse beam variance is ${\sigma }^2(s)=ϵ\beta (s)$. At a beam waist, the angular variance is $ϵ/\beta$. Reducing both transverse amplitude functions by a factor $a$ at fixed emittances increases the ideal head-on collider luminosity by $a$; practical finite-bunch-length and crossing-angle effects can limit this gain.

The equilibrium horizontal beam emittance set by synchrotron-radiation excitation and damping in an uncoupled storage ring. The radiation-integral relation is ${\epsilon }_{x0}=C_q{\gamma }^2{I}_5/(J_x{I}_2)$; similar cell optics give a cubic dependence on bending angle.

For a truly isolated Hamiltonian bunch, Liouville's theorem preserves the fine-grained volume in its full $6N$-dimensional phase space. Hamiltonian manipulation cannot increase its full phase-space density by compressing all canonical position and momentum directions simultaneously. Merely focusing the beam to a smaller transverse spot generally increases angular or momentum divergence; it does not by itself reduce beam emittance.

The distinction between full volume and projected spreads matters. Coupling transverse and longitudinal degrees of freedom can exchange emittance or move spread into other coordinates and correlations, so a projected transverse quantity may decrease without reducing the complete phase volume. That is redistribution, not cooling of every degree of freedom. In linear uncoupled optics, preservation of each transverse canonical area makes the spot-size/divergence tradeoff particularly direct.

Actual cooling allows the beam to exchange energy and entropy with other systems. In electron cooling, a colder comoving electron population receives the protons' random kinetic energy and is subsequently renewed or cooled. In stochastic cooling, a pickup measures deviations and a downstream kicker applies feedback corrections; the electronics, measurement noise and environment belong to the enlarged system. CERN's description identifies exactly this pickup-and-kicker arrangement.^[ref]

Thus these methods relax the idealization that the proton bunch alone is isolated. Including the cooling apparatus does not require the complete closed dynamics to violate Liouville's theorem. For protons, direct radiative damping is much weaker than for electrons, so stochastic or electron cooling is a more relevant practical suggestion than relying on strong synchrotron damping.

The geometric rms beam emittance measures transverse phase space area. In one plane,The betatron amplitude function ${\beta }_x(s)$ describes how the beam envelope varies around the lattice. At a waist, $⟨x{x}^{\prime }⟩=0$ and $⟨x^{\prime 2}⟩={ϵ}_x/{\beta }_x$. This betatron amplitude function is unrelated to either the relativistic velocity ratio $v/c$ or a renormalization-group beta function.

With the beam emittances and bunch populations held fixed, reducing each interaction-point amplitude function to one third makes both transverse rms widths smaller by $\sqrt{3}$. Hence their product is smaller by three, andThe change is not a factor of nine: widths are proportional to the square root of the amplitude function. The transverse angular spread increases by $\sqrt{3}$. In a real machine, stronger final focusing, finite bunch length, crossing angle, aperture and beam-beam limits can reduce the nominal gain; the question's ideal symmetric head-on approximation neglects these effects.

Conventional RF cavities accelerate beams with controlled phase, energy spread and transverse beam emittance. Their material surfaces limit sustainable gradients to order $10$–$100\mathrm{MV}{\mathrm{m}}^{-1}$, depending on technology and operating conditions. Mature RF systems already reached multi-GeV and much higher beam energies in 2015. The 2013 ILC baseline proposed $250\mathrm{GeV}$ electrons colliding with $250\mathrm{GeV}$ positrons, approximately $31.5\mathrm{MV}{\mathrm{m}}^{-1}$ superconducting cavities and a roughly $31\mathrm{km}$ facility. These were design parameters, not an operating collider.^[ref]

In plasma wakefield acceleration, an intense laser displaces plasma electrons and excites a longitudinal charge-density wave. A suitably injected electron bunch gains energy from that wave, with fields of order tens to hundreds of $\mathrm{GV}{\mathrm{m}}^{-1}$. A concrete result available before this examination was $4.2\mathrm{GeV}$ from a $9\mathrm{cm}$ capillary, with $6\%$ rms energy spread, $6\mathrm{pC}$ bunch charge and $0.3\mathrm{mrad}$ rms divergence. These values characterize that experiment, not every laser-plasma beam; divergence alone is not an emittance measurement.^[ref]

A collider concept stages many separately driven plasma modules, with fresh laser pulses, interstage capture and refocusing, controlled injection, and a final collision focus. Dephasing and pump depletion limit a single stage. Although the accelerating sections could be much shorter than ILC linacs, a useful collider also requires high repetition rate and wall-plug efficiency, reproducible low energy spread, preserved normalized emittance, stable timing, high bunch charge and acceptable beam-beam effects. Positron acceleration is an additional challenge because an electron-optimized plasma wake need not focus positrons. Large gradient alone does not demonstrate the luminosity or beam quality of an RF collider.