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 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 μz = ± μB, where μB is the Bohr magneton,


or, by analogy with atomic physics, writing     μ = g μB s     with sz = ±½
then g = 2.

Without QED, the prediction is therefore that g = 2.  We can allow for the existence of extra physics by writing g as

g = 2 × (1 + a)
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 α, 1/137. We can even produce qualitative explanations of why these higher order processes might modify the observed magnetic moment. An electron is always surrounded by a cloud of virtual photons. The charge is carried by the electron, but the energy, and hence mass, is shared between electron and photons. The value of e/m for the "bare" electron is therefore slightly increased, and it is this which will be measured in the interaction with an external magnetic field. Alternatively, when a lepton, of spin ½, 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 μ.

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 section 6.5),


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 frame of the μ+, 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.


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