Martin & Shaw, appendix A

Four consequences of the Lorentz transformation:

- The relativity of simultaneity
- Lorentz contraction
- Time dilation
- Velocity addition

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,

Υ(4s) -> B^{+}B^{-}

Let's find the momentum of the B mesons.
Since there is no external influence on this system, the total 4-momentum after the decay must equal the 4-momentum before the decay.

p_{4s} = p_{B+} + p_{B-}

This is really just a compact way of expressing the conservation of
energy and 3-momentum:
pvec_{B+} + pvec_{B-} = 0
(mom. cons.)

E_{B+} + E_{B-} =
E_{4s} = M_{4s} (energy cons.)

Momentum conservation tells us that the B momenta must be equal in
magnitude and opposite in direction.
In the following, pE

E_{B}^{2} = p_{B}^{2} + M_{B}^{2}.

Therefore,
M_{4s} = sqrt(p_{B}^{2} +
M_{B}^{2}) + sqrt(p_{B}^{2} +
M_{B}^{2})

= 2sqrt(p_{B}^{2} +
M_{B}^{2}).

Squaring and solving for the B momentum yields:
= 2sqrt(p

p_{B} = sqrt(M_{4s}^{2}/4 - M_{B}^{2}).

The Υ(4s) mass is M
p_{B} = 0.341GeV

E_{B} = 5.290GeV

A selected list of particle masses is presented in Appendix E, and a
long listing of particle properties can be found at pdg.lbl.gov.
E

Now it is easy to determine the relativistic quantities $beta; and $gamma; for the B mesons. Since p=γβm, and E=γm, we have:

β = p/E = 0.0645

γ = E/m = 1.002

γβ = p/m = 0.0646.

The B mesons produced at CESR move at about 6.5% of the speed of light.
γ = E/m = 1.002

γβ = p/m = 0.0646.

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):

l = β c t^{lab}.

The decay time in the lab is the relativistic transform of the proper
decay time, t
l = γβ c t^{proper} = 29μm.

This is a rather small distance, not easily resolved in the detector
used at CESR known as CLEO.
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 E_{e-} =
9.0GeV, and the positron beam has energy E_{e+} =
3.1GeV.
Both energies are thousands of times the mass of the electron,
m_{e} = 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

W = 2sqrt(E_{e-}E_{e+}) = 10.6GeV

or roughly the mass of the Υ(4s).
But now, the CM momentum (the momentum of the Υ(4s) in the
lab) is
pvec_{CM} =
pvec_{e-}-pvec_{e+} =
E_{e-}-E_{e+} = 5.9GeV.

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,

p_{B}^{lab} = 5.9GeV, and

E_{B}^{lab} =
sqrt([p_{B}^{lab}]^{2} +
m_{B}^{2}) = 5.9GeV.

Using the same procedure as above, we can calculate the decay length
of the B meson in the lab:
E

l = γβct^{proper} =
(p_{B}^{lab}/m_{B})ct^{proper} =
500μm

This distance is an order of magnitude greater, and now readily
measured with the silicon vertex detectors.
Copyright © Robert Harr 2005