 Solution

For a beam with rms geometric emittance ${ϵ}_y$, its vertical size at the collision point is ${\sigma }_y^{\star }=\sqrt{{ϵ}_y{\beta }_y^{\star }}$. 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,Keep bunch populations, horizontal size, collision frequency and emittances fixed. If both vertical beta values are decreased by a factor $f>1$, meaning ${\beta }_{y,\mathrm{new}}^{\star }={\beta }_{y,\mathrm{old}}^{\star }/f$, both vertical beam sizes decrease by $1/\sqrt{f}$, andThe 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 doublets or triplets, with suitable matching sections, focus both planes. At a symmetric waist ${\beta }^{\prime }=0$ 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.