= Solution
For a beam with rms <geometric emittance> $\epsilon_y$, its vertical size at the collision point is $\sigma_y^\star=\sqrt{\epsilon_y\beta_y^\star}$. Reducing $\beta_y^\star$ therefore concentrates the beam, increasing the number of <particles> encountering the opposite beam per transverse area. For identical Gaussian beams colliding head-on,
$$
\mathcal L=\frac{n_b f_{\rm rev}N_1N_2}{4\pi\sigma_x^\star\sigma_y^\star}.
$$
Keep bunch populations, horizontal size, collision frequency and emittances fixed. If both vertical beta values are decreased by a factor $f>1$, meaning $\beta_{y,\rm new}^\star=\beta_{y,\rm old}^\star/f$, both vertical beam sizes decrease by $1/\sqrt f$, and
$$
\boxed{\frac{\mathcal L_{\rm new}}{\mathcal L_{\rm old}}=\sqrt f.}
$$
The <collider luminosity>[luminosity] is inversely proportional to the common vertical size, not its square, because only one transverse dimension has changed. The same ratio follows from the unequal-beam overlap denominator $\sqrt{\sigma_{y1}^2+\sigma_{y2}^2}$.
A <final-focus quadrupole> system on either side of the interaction point forms a waist there. Strong <quadrupole magnet>[quadrupole] doublets or triplets, with suitable matching sections, focus both planes. At a symmetric waist $\beta'=0$ and the nearby drift gives $\beta(s)=\beta^\star+s^2/\beta^\star$, so small $\beta^\star$ requires large beam envelopes and strong optical gradients in the adjacent magnets. Chromatic correction is needed for off-momentum <particles>. The formula assumes negligible crossing-angle and hourglass reductions; a bunch length comparable to $\beta^\star$ changes the effective overlap.
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