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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.