||Dr C N Booth
Evidence in Support of the Quark Model
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.
Electron-positron annihilation to produce a particle-antiparticle pair proceeds
though a time-like virtual photon.
(The difference between time-like and space-like virtual particles will be
explored in the lecture.)
At energies much greater than twice the rest mass of a quark, the amplitude
for pair production of a quark-antiquark pair is proportional to the product
of the charge on an electron, e, and the charge on the quark, say
The cross-section is thus proportional to z2e4.
The quarks are, of course, not observed themselves, but seen in the combinations
known as hadrons. The ratio, R, of the cross-section for production
of hadrons divided by that for production of μ+μ−
pairs at the same energy is just given by
R = Σi zi2
where the sum is over all quarks which can take part in the production.
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
compatible with the value shown in the first figure. As the c and
b thresholds are crossed, the value of R goes through wide excursions
in resonance regions, before settling down to values expected to be
In fact, above 3.6 GeV the heaviest lepton, the τ,
is also produced, and this decays predominantly hadronically, adding another
unit to R - see the second figure. (Above the b threshold,
the longer path-length of the τ
allows these decays to be removed).
|The ratio R of the cross-section for e+e− → hadrons,
divided by that for e+e− → μ+ μ−.
The fact that R is constant above 10 GeV CMS energy
is proof of the point-like nature of hadron constituents. The predicted
value of 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.
Deep Inelastic Lepton Scattering
The values of R above indicate that hadrons are indeed made of fractionally
charged, coloured objects.
We now look for evidence that these are
really point-like particles, in the inelastic scattering of electrons or
muons off nucleons via the exchange of a virtual photon. This occurs
when the struck proton or neutron absorbs energy in breaking up to form
a hadronic system, of invariant mass W.
W is not always
easy to measure directly, but may be deduced by considering the four-momentum
transfer, q, in the scattering.
We have already seen that,
to a good approximation,
q2 = 2EiEf (1 − cosθ)
where Ei and Ef are the initial and
final lepton energies and θ
is the lepton scattering angle. From the hadronic state’s point of
view, it is easy to show that
q2 = M2 + 2 Mν − W2
where M is the mass of the nucleon and ν
is the change in energy of the hadronic state (and hence minus that of
Therefore q2 and ν
For elastic scattering, when the nucleon remains a nucleon, W
= M and q2 = 2Mν.
Equivalently, if we define x as ,
then x = 1. q and ν
are no longer independent, but are said to "scale".
Now if we consider the nucleon
to be made up of stationary point-like particles of mass mq,
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 q2 through the
dimensionless ratio x, and the cross section shows "scale invariance".
This is indeed observed, as shown in
(Here F2 is proportional to the form factor.)
F2(q2, ν) as a function of q2
at x = 0.25. For this choice of x, it can be seen that there is practically no
dependence on q2, 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 q2, reaching very small values for
q2 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 q2)
it can also scatter these into real (positive energy) states.
So at high enough energies, the structure functions do indeed start to depend
on q2, and scaling is violated!
Electron-positron scattering data from HERA.
For x greater than about 0.1, there is little dependence of F2
on q2 for a given x, and scaling holds.
For small x, F2 increases with q2 due to
scattering off virtual quark-antiquark pairs, and scaling is violated.
Note the variable on the ordinate is defined to displace the points vertically according to their
x value, to prevent the lines lying on top of each other!
Supplementary Reading Material
For further non-technical discussion of the evidence
for quarks, you might like to consult The Ideas of Particle Physics,
(The above chapter references are for edition 3.
For edition 2, consult chapters 35, 36.1-36.2, 26-28 and 33.2.)
- 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
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