= Solution
The geometric rms <beam emittance> measures transverse <phase space> area. In one plane,
$$
\epsilon_x=\sqrt{\langle x^2\rangle\langle x'^2\rangle-\langle xx'\rangle^2},\qquad\sigma_x^2=\epsilon_x\beta_x.
$$
The <betatron amplitude function> $\beta_x(s)$ describes how the beam envelope varies around the lattice. At a waist, $\langle xx'\rangle=0$ and $\langle x'^2\rangle=\epsilon_x/\beta_x$. 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 $\sqrt3$. Hence their product is smaller by three, and
$$
\boxed{\mathcal L_{\rm new}=3\mathcal L_{\rm old}}.
$$
The change is not a factor of nine: widths are proportional to the square root of the amplitude function. The transverse angular spread increases by $\sqrt3$. 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.
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