PHY304 | Particle Physics | Dr C N Booth |

The quark model originally arose from the analysis of symmetry patterns
using group theory. The octets, nonets, decuplets etc. could easily
be explained with coloured quarks and the application of the Pauli exclusion
principle. It is found that the quark model explains a large number
of features of the observed particles and their interactions. However,
we must consider whether quarks are mathematical abstractions, or whether
there is evidence for point-like, fractionally charged, coloured constituents.

Note that since each quark can exist in 3 colours, the sum must be over
both colours and flavours. At low energies, where u, d and s quarks
can be produced, the value of *R* is then predicted to be

The ratio R of the cross-section for ^{+}e^{−} → hadrons,^{+}e^{−} → μ^{+} μ^{−}.R is constant above R, assuming that the primary process is formation of a
quark-antiquark pair, is 11/3 if pairs of u, d, s, c, b quarks are
excited and they have three colour degrees of freedom. The data come
from many storage-ring experiments. |

For **elastic** scattering, when the nucleon remains a nucleon, *W*
= *M**q*^{2} = 2*M*ν*x* as ,
then *x* = 1.**scale**".
Now if we consider the nucleon
to be made up of stationary point-like particles of mass *m*_{q},
then
will be a constant for elastic scattering off *these* particles, which
will fix .
However, unlike the nucleon, the quark will *not* be at rest, having
considerable momentum within the proton (due to the uncertainty principle).
When we perform a Lorentz transform from the rest frame of the quark to that of the proton,
integrating over the distribution of quark momenta leads to the form factor of the proton.
However, as long as the quarks are point-like
the form factor should only depend on *q*^{2} through the
dimensionless ratio *x*, and the cross section shows "**scale invariance**".
This is indeed observed, as shown in
the figure.
(Here *F*_{2} is proportional to the form factor.)

F_{2}(q^{2}, ν) as a function of q^{2}
at x = 0.25. For this choice of x, it can be seen that there is practically no
dependence on q^{2}, that is there is exact "scaling".
(Data from the Stanford Linear Accelerator Center.) |

Scattering from an extended object like the proton (rather than from point-like constituents
with it) would produce a very different distribution.
As calculated in an early homework, the form factor for a structure-less proton
drops rapidly with *q*^{2}, reaching very small values for
*q*^{2} above 1 or 2 (GeV/*c*)^{2}.

However, this is not the whole story!
It is found that quarks
carry only half the momentum of a moving nucleon, the rest being carried by
electrically neutral gluons, which are invisible to the virtual photon.
The gluons also produce virtual q
pairs, and if the probing photon has high enough energy (or *q*^{2})
it can also scatter these into real (positive energy) states.
So at high enough energies, the structure functions do indeed start to depend
on *q*^{2}, and scaling is violated!

Electron-positron scattering data from HERA. For

**Supplementary Reading Material**

For further non-technical discussion of the evidence
for quarks, you might like to consult *The Ideas of Particle Physics*,
as follows:

- Cross-section ratio
*R*- chapter 34 and sections 35.1 - 35.2 - Deep inelastic scattering & scaling - chapters 25 to 27
- Scaling violation - section 32.2