PHY304

Particle Physics

Dr C N Booth

Fundamental Interactions  -  1) Quantum ElectroDynamics
(QED)

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 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 a number of 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 z = ± B, where B is the Bohr magneton,

or, by analogy with atomic physics, writing

with

then

.

Without QED, the prediction is therefore that g =
2.  We can allow for the existence of extra physics by writing
g as
where a is known as the anomalous magnetic
moment.  The full QED calculation is very involved, but field theoretical
considerations lead us to the expectation that the highest order contribution to
a will be of the order of , .  We can even produce a qualitative explanation
of why these higher order processes might modify the observed magnetic
moment.  When a lepton, of spin   emits a photon, of spin 1, its spin, and hence
magnetic moment, must be flipped.  The average magnetic moment of a real
lepton might therefore be reduced, when compared with the expectation for a bare
particle.  Another effect gives a larger, positive contribution to
a.  A classical picture of the magnetic moment of a particle is that
it arises from a current loop around the centre of mass of the particle.
Emission and absorption of virtual photons leads to a jitter in the position of
the centre of the loop, so increasing .

The full QED calculation for the leading order contribution to a
gives .

Higher order terms depend on the lepton mass  (e.g. see Perkins section
6.8),

Experiment
A pioneering experiment to measure g-2
for the muon was performed by a team including the late Prof. Combley of this
department.  A polarised beam of muons was circulated in a circular orbit
in a uniform magnetic field.  The interaction between the magnetic field
and the magnetic moment exerts a torque on the muon, causing the spin direction
to precess at a rate which depends on the magnetic field  (Larmor and
Thomas precessions).  If g = 2, the momentum and spin
would turn at precisely the same rate, and so the polarisation would not change.

The actual polarisation of the muons was measured through their parity
non-conserving weak decays,
.This process is mediated by a W boson (as
will be discussed in the section of the course on weak
interactions), and it can easily be shown, by considering the helicities of
the neutrino and antineutrino, that, in the rest from of the +, the e+ direction tends to follow
the direction of the muon's spin.  In the lab frame, the highest energy
electrons have their decay momentum parallel to the muon's momentum, so they tag
times when the muon's spin is parallel to its momentum.  Exploiting these
facts, the precession rate, and so the value of a, can be measured.

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,

nicely accounting for the
above discrepancy.

Supplementary Material
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.

Back
to PHY304 Page