Subatomic Physics

306 The Electromagnetic Interaction
to be simultaneous, Møller and Bhabha scattering can be separated cleanly from
Rutherford scattering. A second disadvantage of the approach just outlined is not
so easily overcome: The energy available in the c.m. to explore the structure of the
electromagnetic interaction is small because of the small electron rest mass. We
have studied this problem in Section 2.7; in Eq. (2.32) we found the total energy
available in the c.m.,
W ≈ (2E0mec 2 ) 1/2 . (10.83)
With E0 = 10 GeV, the total energy available in the c.m. becomes
W ≈ 100 MeV.
Even at 10-GeV incident energy there is not enough c.m. energy to even produce a
muon pair. The path around this difficulty has already been shown in Section 2.8;
it is the use of colliding beams. As e + e − collisions have yielded some of the most
beautiful results and promise to continue to do so, we will discuss a few of the
experiments and data in the following sections.
One interesting concept occurs in connection with Bhabha scattering. The virtual
photons in the photon-exchange and in the annihilation diagram (Fig. 10.11(c)
and (d)) have very different properties. Both photons are virtual and do not satisfy
the relation E = pc. Consider both reactions in the c.m. In the exchange diagram,
the incoming and the outgoing electrons have the same energies but opposite
momenta. Consequently, energy and momentum of the virtual photon are given by
Eγ = Ee − E ′ e =0, p γ = p e − p ′ e =+2p e. (10.84)
If we define a “mass” for the virtual photon through the relation E 2 =(pc) 2 +
(mc 2 ) 2 , we find (11)
(mc 2 ) 2 = −(2pec) 2 < 0. (10.85)
The virtual photon in the exchange diagram carries only momentum—no energy.
The square of its mass is negative. Such a photon is called spacelike. In the annihilation
diagram, the situation is reversed,
Eγ = E e − + E e + =2E, p γ = p e − + p e + =0. (10.86)
The virtual photon carries only energy—no momentum. The square of its mass is
given by
(mc 2 ) 2 =(2E) 2 > 0; (10.87)
it is positive and the photon is called timelike. In electron-positron scattering, both
spacelike and timelike photons enter. The agreement of experiment with theory
indicates that these concepts are correct, even if they sound strange at first.
11 The “mass” defined here is related to the four-momentum transfer, q, bym 2 =(q/c) 2 . It is
equal to the actual particle mass only for free particles.