|PHY304||Particle Physics||Dr C N Booth|
In the model of interactions we have proposed, a charge, for example, interacts by emitting and absorbing virtual photons. We now examine the possibility of observing consequences of this which are not predicted by standard quantum mechanics. Our model of an electron interacting with an external electromagnetic field involves it in absorbing a virtual photon, and thus changing its momentum. However, other internal interactions can occur. An electron, of momentum p, may emit a virtual photon of momentum k, and hence continue with a reduced momentum p-k until it reabsorbs the virtual photon. Similarly a photon of momentum k may convert into a virtual electron positron pair, with the electron and positron sharing the original momentum, until the virtual pair recombine and produce the photon again. Indeed, more complicated cases may occur, involving combinations of emission of virtual photons with virtual pair productions. Each coupling of a photon to a fermion line, known as a vertex, involves a factor in the amplitude, so α in the cross-section, where
The above internal interactions are known as self-energy terms. The first contributes to the apparent mass of the electron, while the second contributes to the effective charge (and is responsible for the process known as "vacuum polarisation"). We can sum over all diagrams, and integrate over all internal loop momenta, in an attempt to calculate the effective mass and charge. However, naive attempts to do this result in values that are infinitely large! A little thought shows that it is not reasonable to put in the measured vales of m and e as the bare parameters of the theory, and then calculate some modified, effective values. Whenever we do an experiment with an electron, it is surrounded by its cloud of virtual photons and pairs - it is meffective that we measure in experiments! A rigorous mathematical process known as renormalisation allows us to use the physical mass and charge and (for most processes) ignore internal loops in the particle lines. External loops (e.g. those coupling incoming and outgoing particles) will, however, be expected to have a finite, observable effect.
There are several properties which exhibit the quantum nature of
the electromagnetic interaction.
One of these is the magnetic moment
of charged leptons, in particular the muon.
The Dirac equation for
a point-like electron or muon, which arises from a relativistic quantum mechanical treatment
of the particle but without the use of a field theoretical approach to
describe interactions, predicts that a component of the lepton's intrinsic
magnetic moment must be
|then||g = 2.|
Without QED, the prediction is therefore that
The full QED calculation for the leading order contribution to a gives 0.5α/π.
Higher order terms depend on the lepton mass (e.g. see Perkins
The actual polarisation of the muons was measured through their parity non-conserving weak decays,
|Experimentally||a = (1165924. ± 9. )x10-9.|
|A QED calculation, including higher orders, predicts||a = (1165851.7 ± 2.3)x10-9,|
|leaving a discrepancy of||(72.3 ± 9.3)x10-9.|
Though the above theoretical result is very close to the experimental value, there is still a significant difference - something must be missing! This is the effect of other (strongly interacting) particles, which can also contribute to vacuum polarisation but are not included in the QED calculation.
|A calculation of this additional term yields||(70.2 ± 8.0)x10-9,|
(The above results are taken from the CERN
For further non-technical discussion of Feynman diagrams, virtual processes and renormalisation, you might like to consult The Ideas of Particle Physics, sections 4.6 to 4.10.