DIS: Deep inelastic scattering and the DGLAP evolution equation
PROBING CHARGE DISTRIBUTIONS: FORM FACTORS ...
There is a simple way to probe charge distributions: Let an electron beam scatter off the object and
compare the angular distribution of the scattered electrons with the known one for sdcattering off
an unpolarised point source. Written in terms of differential cross sections this reads
where Ω is the solid angle and F(q²) is known as the form factor. Also, Z is the charge number
of the target and E and k are the energy and the absolute value of the electron. By assumption the
electron is scattered off eleastically, thus E and k are identical for the incident and the outgoing
electron. Here, and in the following, q denotes the momentum transfer between the incoming electron and
the target,
For a static target, it is simple to see that the form factor is just the Fourier transform of the
charge distribution, i.e.
Taking, for convenience, the charge distribution to be normalised to one, implies that F(0)=1. For not
too large values of q, one may then expand the exponential in the Fourier transform, yielding
a connection to the mean square radius of the static targets charge distribution. This is because, if
the testing photon is soft, i.e. for small q, its long wavelength can resolve only the size of the
charge distribution but no details of its internal structure. It is simple to show then that for
exponetially decreasing charge distributions, the form factor assumes the form
Going now to a description of elastic electron-proton scattering that is more based on quantum field
theory, it is clear that the static limit for the target has to be released. The amplitude for this
process can then be written as the one-photon interaction of two currents, the eletron current j, and the
proton current J. Denoting the incoming and outgoing proton momenta by p and p', respectively, they read
where the square bracket reflects the fact that, since the proton is not an elementary particle. Thus,
it may not interact like a point particle; the square bracket is the most general interaction term
when assuming QED to be parity-conserving. The two form factors F are called the Dirac form factors.
κ is the anomalous magnetic moment of the (extended) proton and is measured to be 1.79.
The form factors obey
for neutrons, however,
and κ is measure to be -1.91.
However, using the form of the currents above and doing the algebra yields the elastic electron-proton
scattering cross section, aka the Rosenbluth formula
In order to eliminate interference terms from the cross section, linear combinations G of the
form factors F have been introduced, namely
known as the electric and magnetic form factor. Expressed through them the cross section reads
where τ=-q²/4M². In fact, there is one reference frame, the Breit-frame, where these
two form factors are closely related to the proton electric and magnetic moment distributions.
The Breit frame is defined by p = -p', i.e. it is something like the "photons rest frame".
Measurements show that the electric and magnetic form factors have roughly the same behaviour and
can be fitted by
The mean proton charge radius is thus given by
...AND STRUCTURE FUNCTIONS
Having thus measured the size of the charge distribution of the proton, it is tempting to increase the
momentum transfer carried by the photon to take a more detailed look. This sounds good, but there is
a catch: The larger the momentum transfer q², the larger the prbability for the proton to break up.
Then, instead of analysing elastic scattering one enters the region of inelastic electron-proton
scattering. This neccessitates to modify the picture developed so far. For not so large momentum
transfers, however, this modification will be small: Then, in most cases, just a Δ-resonance is
excited, which usually decays into a proton and apion or similar, i.e. one must consider reactions
like ep→ eΔ→ epπ. When the momentum transfer becomes even larger, however,
there won't be any excited resonance, instead there will be a violent break-up of the proton with
a potentially large number of particles emerging. In this case, it is nearly impossible to trace the
proton'S identity and a new way of formulating the problem theoretically must be found.
BJORKEN SCALING ...
... AND WHY IT IS VIOLATED
THE DGLAP EVOLUTION EQUATION