Appendix

Three-jet Kinematics

The momenta for annihilation to three particles via a virtual photon are assigned according to fig. 54. In our calculations, at most one of the final particles has a non-zero mass, and, in this case, the kinematics for the process become very simple. Let us work in the rest frame of the `decaying' virtual photon, so that

Now define

Energy conservation demands

Take particle 3 to be massive, and write

Then

The expression for does not have such a simple form. However, for all and

If all the final particles are massless, this gives

When particle 3 has mass , define

Then

Now

so that the angle between the three-momenta of particles 1 and 2 is given by

The physical region for the process of fig. 54 is determined simply by demanding that

which implies

The physical region for a characteristic value of is illustrated in fig. 55. Solving the inequalities (A.13), one finds that

while

The expressions for and are simple only if all the final particles are massless, in which case

[ Figure 44 ] Definitions of the momenta for a process.

[ Figure 45 ] The physical region in the fractional energies for a process with three final-state particles. Particles 1 and 2 are taken to be massless, while particle 3 has a mass .

The momenta and of the incoming electron and positron clearly satisfy

in their c.m. frame (virtual photon rest frame). Momentum conservation requires the momenta of the final particles to lie in a plane. Let be the angle which the normal () to this plane makes with the direction of the electron's momentum. Then, if all the particles are massless,

where are the angles of the final particles' momenta relative to the axis formed by the intersection of the plane containing and the direction with the plane of the final particles.

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