12 Scattering Theory
Bound state perturbation theory, discussed in the preceding chapter, has allowed us to understand
the energy shifts and modifications to quantum states as corrections to the Hamiltonian are succes-
sively taken into account. Various outcomes of calculations may be compared with the outcomes of
various experiments, and used to check the theory. But, the parameters of the bound state problem
are essentially fixed: we studied various discrete energy levels of various systems. The one way in
which we can bring in external parameters is by applying external electric and/or magnetic fields.
Perturbation theory of the continuous spectrum is physically even more important. The reason
is that in the continuous spectrum, the momentum or the energy of the incoming “probe” can
always be adjusted at will, and thus automatically providesus with a tunable free parameter. This
is calledscattering theory. External electric and/or magnetic fields may also be present and will
then allow us to probe the system with even more parameters and thus with even more detail. The
free momentum of the incoming probe allows us to compare the outcome of theoretical calculations
with the outcome of experiments for entire ranges of momentum and energy values. We can now
compare entire functions of momentum as instead of isolateddata points in the case of the discrete
spectrum. Furthermore, scattering allows us to introduce supplementary energy into the system,
and thus to access a range of excitations of the system beyondits ground state.
A large class of scattering problems may be approximated by scattering in the presence of a
time-independent or static source. In two body problems, this may be done by going to the center
of mass frame, and reducing the problem to a one-body problemin a potential, using translation
invariance.
Fixed-target experimentsare one example of such scattering processes. They are the oldest
experimental set-ups for the systematic study of scattering. Here, a light probe (usually an electron,
proton, neutron or neutrino) is used and scattered off a heavytarget which is considered fixed, in
as much as recoil effects of the target are negligible. An alternative experimental set-up is provided
byparticle colliders. In a collider experiment, two beams traveling in opposite directions are made
to interact in a localized interaction region. Most frequently, the beams are symmetric, in the
sense that they are composed of particle of the same mass and adjusted to have the same energy.
The most popular set-ups aree+e−colliders (SLAC), andp+p−colliders (CERN). The resulting
scattering products will then be evenly distributed in a 4πsolid angle.^10
At high energiesE(much larger than the massMof the target), (symmetric) collider experi-
ments are much more efficient, due to relativistic effects. Thisis because what really matters for
the strength of the interaction is the center of mass energy of the process. In a collider experiment,
two beams of opposite momenta and equal energyEproduce a center of mass energyEcm= 2E.
But in a fixed target experiment, with a beam of particles of massmand energyEincident on a
target of particles of massM, the center of mass energy is given byEcm^2 = 2EMc^2 + (M^2 −m^2 )c^4 ,
(^10) For certain experiments, there is actually an advantage to having an asymmetric set-up, with particle of
unequal masses and/or unequal energies. This set-up will produce scattering products carrying a momentum
peaked around a definite forward momentum which allows experimentalists to use a detector of less than 4π
solid angle coverage.