Standard Model and Beyond · Problem 6.1

The Weak Interactions

Physics — Particle Physics · collider-luminosity

Problem

(a) Consider an idealized colliding beam experiment in which bunches of particles moving in the zz direction collide head-on with frequency ff. Show that the instantaneous luminosity defined on page 206 is given by L=fd1d2∫n(x,y)2dx dy≡fN1N2A,\mathfrak{L} = f d_1 d_2 \int n(x,y)^2 dx\, dy \equiv \frac{f N_1 N_2}{\mathcal{A}}, where d1,2d_{1,2} are the lengths of the bunches, and n(x,y)n(x,y) is the number density of each bunch as a function of the transverse directions, which is assumed to be the same for each beam and independent of zz within the bunch. N1,2N_{1,2} are the total number of particles in each bunch, and the effective transverse area A\mathcal{A} is defined by the second form. (b) Suppose n(x,y)n(x,y) is a constant nn over a transverse area AA and zero outside. Calculate A\mathcal{A}. (c) Show that A=4πσxσy\mathcal{A} = 4\pi\sigma_x\sigma_y for the more realistic Gaussian density profile n(x,y)∝e−12[(x/σx)2+(y/σy)2],n(x,y) \propto e^{-\frac{1}{2}[(x/\sigma_x)^2+(y/\sigma_y)^2]}, where σx,y\sigma_{x,y} are the RMS radii.

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