Classical Electrodynamics · Problem 11.25

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · colliding-beam-kinematics

Problem

In colliding beam machines such as the Tevatron at Fermilab or the numerous e+e−e^+e^- storage rings, counterrotating relativistic beams of particles are stored and made to collide more or less head-on in one or more interaction regions. Let the particles in the two beams have masses m1m_1 and m2m_2 and momenta p1p_1 and p2p_2, respectively, and let them intersect with an angle θ\theta between the two beams. (a) Show that, to order (m/p)2(m/p)^2 inclusive, the square of the total energy in the cm frame is W2=4p1p2cos⁡2θ2+(p1+p2)(m12p1+m22p2)W^2 = 4p_1 p_2 \cos^2\frac{\theta}{2} + (p_1 + p_2)\left(\frac{m_1^2}{p_1} + \frac{m_2^2}{p_2}\right) (b) Show that the cm inertial frame has a velocity in the laboratory given by βcm=(p1+p2)sin⁡θ/2(E1+E2)sin⁡α\beta_{\rm cm} = \frac{(p_1 + p_2)\sin\theta/2}{(E_1 + E_2)\sin\alpha} where tan⁡α=(p1+p2p1−p2)tan⁡θ2\tan\alpha = \left(\frac{p_1 + p_2}{p_1 - p_2}\right)\tan\frac{\theta}{2} The angle α\alpha is defined in the figure. (c) Check that the results of part b agree with those of Problem 11.23b. (d) If the crossing angle is θ=20°\theta = 20° and the colliding protons have p1=p2=100p_1 = p_2 = 100 GeV/cc, is the laboratory frame a reasonable approximation to the cm frame? Consider, for example, a proton-proton inelastic collision involving pion production and examine the collinearity of two pions produced with equal and opposite momenta of 10 GeV/cc in the cm frame.

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