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The Nature of the Weak Interaction
----------------------------------
The Feynman path integral representation gives the probability amplitude
of going from point A to point B (distance s) in terms of the propagator.
The propagator is a function of space-time distance and is also crucially
dependent on the mass of the particle. Now, in experiments we typically
measure momentum. The laws of conservation of momentum hold and
therefore the difference in momentum between the particles on the LHS
of the diagram must equal the momentum of the emitted W boson. From
QM we know that distance and momentum space are related through the
Fourier Transform. From this, it is possible to specify the propagator
in terms of momentum space. The probability amplitude, δ(p), that a
particle with mass, m will travel s in momentum space is of the form:
δ(p) = 1/(p^{2} - m^{2} + iε)
The small imaginary term, iε, is put there to ensure the correct
boundary conditions are met and handle the singularity at p^{2} = m^{2}. The
limit ε -> 0 is taken at the end of the calculation.
Therefore, for low momenta, δ = -1/m^{2}
Thus, the probability of traveling distance s falls off very rapidly
with mass.
The combined probability amplitude of the process shown in the diagram
can thus be written as:
g_{W}^{2}/m_{W}^{2} = G_{F} ... the FERMI G
The net result is the weak interactions are weak because of the large
mass of the W boson. The probability of the above process occurring is
very roughly:
g_{W}^{4}/m_{W}^{4}
where g_{W} is of the order of the fine structure constant equal to
(1/4πε_{0})(e^{2}/hc) or about 1/137 and m_{W} is the mass of the W boson.
This is a very small number! Therefore, most of the time this process
doesn't happen - the W boson is more likely to be continuously
emitted and absorbed as shown in the diagram below.
Note: m = 0 for a virtual photon. For low k processes which are very
typical in QED, the probability amplitude of the process is very much
larger than it is for the W boson.
The mass associated with the virtual W boson, m_{W}, is not the same as
the mass of the real W particle detected in the lab (~80 GeV). The
latter case involves the Higgs mechanism - virtual particles do not.
The mass of the virtual boson is of the order of 100 times the proton
mass or about 10GeV. However, it depends on probabilities and
comes from the fact the W carries some 4-momentum from the quarks
to the electron/neutrino.
The question that arises is 'how does the W boson, which is much
heavier than either n or p, get its energy?'. The answer to this is
related to the difference between virtual and real particles (see
below). In a nutshell, the virtual particle gets its mass by
APPARENTLY violating conservation laws within the bounds of the
energy-time uncertainty principle. A somewhat poor analogy is
that the W boson borrows energy from the field but has to repay
it before time runs out!
Charge Conservation
-------------------
The relationship between charge, weak isospin and weak hypercharge
is given by:
Q = I_{3} + Y_{W}/2
Consider the LHS:
Q_{d} = (-1/2) + (1/6) = (-1/3)
Q_{u} = (1/2) + (1/6) = (2/3)
Therefore, Q changes by -1
Consider the RHS:
Q_{ν} = (-1/2) + (1/2) = (0)
Q_{e} = (-1/2) + (-1/2) = (-1)
Therefore, Q changes by -1
Virtual versus Real Particles
-----------------------------
In quantum field theory real particles are viewed as being observable
excitations of underlying quantum fields. Virtual particles are also
viewed as excitations of the underlying fields, but are not observable.
They are 'temporary' in the sense that they appear in calculations, but
are not detected as single particles. Virtual particles are essentially
mathematical constructs that only exist for extremely short periods
of time and correspond to the propagators appearing in the Feynman
diagrams.
Where δ(s) is the PROPAGATOR and g is a coupling constant.
A virtual particle does not necessarily carry the same mass as the
corresponding real particle and, because it is 'short-lived', the
energy-time uncertainty principle allows it to APPEAR not to conserve
energy and momentum. In other words, a virtual particle does not
precisely obey the formula m^{2}c^{4} = E^{2} - p^{2}c^{2} for a period of time
corresponding to its very short lifetime. Another way of saying this
is that there can be fluctuations in energy, ΔE within some time Δt,
that obeys ΔEΔt ~ h. This is expressed by the phrase 'off mass shell'.
Actual particles, of course, never violate conservation laws.
The longer a virtual particle appears to 'live', the more closely it
adheres to the 'mass-shell' relationship associated with a real
particle.
Virtual particles are the force carriers. Higher energy (mass) particles
cannot travel as far as lower-energy virtual particles. This explains why
the electric, strong, and weak forces diminish with distance. If we
assume the particles in the above diagram are electrons, the electron
on the right could shoot out a photon. Having given up some of its
energy and momentum, that electron would recoil. Next the emitted
photon would run into the other electron. Upon absorbing the photon’s
energy and momentum, this electron would careen off toward the upper
left. This is the basis of the repulsive force.
* This same phenomenon can also take place with tunnelling. An
alpha particle tunnelling out of a nucleus SEEMINGLY has to violate
conservations laws for a very short period of time to surmount the
potential barrier that keeps it in the nucleus.
Because virtual particles only exist for short periods of time,
observing them requires a large amount of energy. For example, the
W particle emission and absorbtion process illustrated above take
place at enormously high frequencies. In order to see it in the lab
requires the generation of extremely high energy photons (E = hf).
Unfortunately, as a consequence of QM where observation disturbs
the system, the amount of energy that needs to supplied actually
ends up creating the particle that you are trying to observe. Thus,
in a process like the above, it is possible to observe real W bosons
but you have to hit a proton or neutron with an energy that ultimately
results in the ejection of the W particle. This is a much different
process than the weak interaction.
Resonances
----------
When the incoming real particles have enough energy a real instead
of a virtual particle can be produced. The process becomes 'on shell'
and is greatly enhanced - a RESONANCE. Resonances are described by
the BREIT-WIGNER probability distribution:
f(E) ∝ 1/(E^{2} - m^{2} + imΓ)
Where
Γ is the RESONANCE WIDTH (or decay width) = h/τ and τ is the lifetime
of the particle and m is the mass f the particle.
The distribution arises from the propagator of an unstable particle
which has a denominator of the form p^{2} - m^{2} + iε.
Production of on-shell particles in colliders can be accomplished in
different ways. One way is to collide electrons and positrons (LEP).
Consider W and Z bosons generated in this way. Now, these bosons
only live for a short period of time before they decay into different
products with probabilities defined by their BRANCHING RATIOS. At
70%, the most likely decay process for the W and Z bosons is that
they will decay into quark antiquark pairs (hadrons). Thus, we can
write.
_ _
ee -> Z or W -> qq (JETS)
Even if the bosons decay before they can reach the detectors, their
presence can be detected by keeping track of the total rate of these
final states.
The resonances for these both look like:
The Z particle is strongly peaked and comes in at ~ 91 GeV. In this
particular experiment the W bosons were produced in pairs. The
combined mass of the W^{+}W^{-} combination is not strongly peaked and is
between 160 GeV and 200 GeV (80 - 100 GeV per W).
Typically all decay processes (called CHANNELS) are considered in
experiments. Not all channels will exhibit resonances. More accurate
measurements of the W^{±} masses have been made by colliding protons with
antiprotons (Tevatron) and observing the Lepton channels. These
experiments have shown the W^{±} mass to be very close to 80.4 GeV.