Martin & Shaw, appendix A
Four consequences of the Lorentz transformation:
Until recently (2001) the CESR electron-positron storage ring at Cornell was operated at a center-of-mass (CM) energy of 10.6GeV, sufficient to produce the Upsilon(4s) particle in the collision of an electron and positron. The electron and positron beams were, of necessity, of the same energy -- they counter-circulated in the same magnetic field, and the Upsilon(4s) was produced at rest in the lab frame.
The Upsilon(4s) decays rapidly to a B mesons-anti-B meson pair, B+B- for our purposes,
Since there is no external influence on this system, the total 4-momentum after the decay must equal the 4-momentum before the decay.
Now it is easy to determine the relativistic quantities $beta; and $gamma; for the B mesons. Since p=γβm, and E=γm, we have:
The study of B meson decays has yielded numerous insights into the action of the strong and weak forces. B mesons decay in a proper time of about 1.5ps (the proper time is the time as measured in the rest frame of the particle). Idenfitying the decay is aided if the B meson travels from its production point before decaysing, and for some measurements, this motion is required so that the time for the decay to occur can be determined. The distance travelled (in the lab) is the speed of the particle (in the lab) multiplied by the decay time (in the lab):
An alternative accelerator design was proposed that would result in larger decay lengths in the lab, enhancing the physics that can be done. This design is known as an asymmetric e+e- collider. Two of them were built, one in Japan at the KEK laboratory, and the other at the Stanford Linear Accelerator Center (SLAC) in California. The SLAC accelerator is called PEP-II, since it reuses much of the infrastructure of the original PEP storage ring. At PEP-II, the electron beam has energy Ee- = 9.0GeV, and the positron beam has energy Ee+ = 3.1GeV. Both energies are thousands of times the mass of the electron, me = 0.511MeV, so that to good approximation, the magnitude of the electron and positron momenta are equal to their energies (in units where c=1). Therefore, the CM energy of an electron-positron collision is
When the Υ(4s) decays to a B and anit-B meson, the B mesons move relatively slowly in the Υ(4s) rest frame. Each B meson carries about half the momentum of the Υ(4s) in the lab frame,