Lecture 14: Mar 8th 2012
Reading for next week Griffiths chapter 7
No homework for next week.
Midterm Exam Due Thursday Mar 15th. Please work on the exam independently. Be
sure you have your copy!
Homework returned Tuesday.
1) Transition rates. Scattering cross sections and decays.
Phase space evaluation results.
Decay with no angular distribution
d/d = S|M|2pf/322m12
pf = (1/2m1)(m14 + m24 + m34 - m12m22 – m12m32 - m22m32)
Two to two scattering
d/d = (1/8)2 |M|2 S |pf|/|pi| 1/E2
2) This leaves understanding the matrix element.
At the most basic level the matrix element must represent the strength of the force and the
mass of the internal particles. Let’s consider a simple theory with no spins or other
complications. In this theory the basic vertex involves particles A, B and C.
In this toy theory as in the full Feynman theory we will construct matrix elements based
on a simple set of rules.
1) rules for defining –iM
2) For each vertex write a factor of the coupling constant –ig. g has units of
momentum in this toy theory. It will be dimensionless in real quantum field
theories. In the true Feynman theory the wave function of the particle will carry
units of momentum. g2 is proportional to the coupling constants, .
3) For each internal line or propagator write a factor of i/(q2-m2) where q is the four
momentum of the propagating particle and m the mass. Note that for the virtual
propagator q2m2 but the probability is enhanced when they are close. We can see
that a factor of 1 over mass squared is going to come in for a massive particle
such as a W in and small initial momentums such as a decay.
4) For each vertex write delta function to conserve momentum and energy (2)4
4(p1-p2-p3).
5) Integrate over the internal momentum with a factor of d4q/(2)4. Note the 4D
integration. The energy is not fixed by the mass of the particle in this case, since
the particle is virtual.
6) The first step will be to cancel out delta functions by performing an integral over
the internal momentum. There will a final delta function will state overall
conservation of momentum and energy. Remove this factor since it was already
used in evaluating the overall transition rate.
3) Lets treat decay as a simple one vertex process A  B+C. No propagator.
For a simple one vertex process.
-iM = [-ig] (2)4 4(pA-pB-pC)
remove the final delta function to preserve momentum and energy since it was already
stated in the transition rate.
M=g
g2
1/g2
Then
 = (S|p2|/8m12)|M|2
 = (|p2|/8m12)g2
 = 8m12/(|p2|g2)
Taking m, p and g in MeV.  is in units of Mev,  in units of 1/MeV and we have to
multiply by = 6.582x10-22 MeVs to extract the lifetime in seconds.
Note. In EM theory g can be identified with , g = (4)
For this decay the other side of the diagram is governed by the details of the bound state,
but to first approximation the bound particles are always interacting at high probability so
that side of the decay happens with a probability of 1.
4) Weak decay
The weak decay cannot be represented as a one vertex interaction since the probability of
the bound state interacting via the weak force is small. For instance for a meson to decay
it must be considered as a two to two process with the quark and anti-quark of the meson
annihilating each other and then the weak boson pair producing a lepton and antineutrino.
For this process the phase space will just be the phase space for one to two decay, meson
to lepton and antineutrino but the Matrix element must be calculated as a two to two
process. The toy theory is not rich enough to properly treat the bound state but this will
give us an estimate of true weak lifetimes. Note that due to the details of the bound states
weak lifetimes can vary by many orders of magnitude.
Consider pion decay to a electron and antineutrino. The masses of the electron and the
neutrino are small enough that they can be neglected.
 ~ |M|2/16m1
For the matrix element consider the simple two vertex annihilation process A+B  A+B
mediated by C with momentum q.
-iM =  [-ig] [i/(q2-m2)] [-ig] [1/(d4q/(2)4)] [(2)4 4(p1+p2-q)] [(2)4 4(q-p3-p4)]
Integrate over one delta function
-iM = [-ig] [i/((p1+p2)2-m2)] [-ig] (2)4 4(p1+p2-p3-p4)]
remove the second delta function
M = g2[1/((p1+p2)2-m2)]
Take the case where mW is large and we recover expected form of coupling constant for
the weak force.
M ~ g2/mw2
 ~ (g2/mW2)2/16m1
where anticipating electroweak unification g is identified with the EM  g = (4)
 ~ (/mW2)2/m1
 ~ W2/m1
Note again that the units don’t work out in our toy model unless you give the coupling
constants units. In this case alpha must have dimensions of MeV.
Invert and multiply by = 6.582x10-22 MeVs to get the lifetime
W = 10-5
m1 = m = 139.6 MeV
Therefore
 = 2.250x10-12 MeV: very small so we expect a relatively long lifetime
=
/ = 2.925x10-10 sec
Fairly similar to what we will eventually get for the weak force for pion decay to electron
and anti electron neutrino, though not surprising that it’s a few orders of magnitude off
the real value.
5) Scattering example A+A  B + B
cross section from the phase space evaluation
d/d = (1/8)2 |M|2 S |pf|/|pi| 1/E2
For the matrix element consider the simple two vertex annihilation process A+A  B+B
scattered by particle C.
-iM =  [-ig] [i/(q2-m2)] [-ig] [1/(d4q/(2)4)] [(2)4 4(p1-p3-q)] [(2)4 4(p2+q-p4)]
Integrating and applying the first delta function gives q = p1-p3
M1 = [g2/(( p1-p3)2-m2)] [(2)4 4(p1+p2-p3-p4)]
The second delta function states energy and momentum conservation so we remove it
since it was already stated in the phase space evaluation.
M1 = [g2/(( p1-p3)2-m2)]
However in this case there is also an ambiguity in which whether particle 3 of type B
came from particle 1 or 2 of type A. This means that there are two possible diagrams that
can contribute when we sum over the final states.
-iM =  [-ig] [i/(q2-m2)] [-ig] [1/(d4q/(2)4)] [(2)4 4(p1-p4-q)] [(2)4 4(p2+q-p3)]
The only difference here is that particles 3 and 4 have been interachanged.
M2 = [g2/(( p1-p4)2-m2)]
Total M = [g2/(( p1-p3)2-m2)] + [g2/(( p1-p4)2-m2)]
Now evaluate for masses of A and B of m and a massless particle C.
( p1-p3)2 = p12 + p32 – 2p1.p3 = E12 + E32 – p12 – p32 – 2E1E3 + p1.p13
But in CM E1=E3 and | p1| = | p3| = p
( p1-p3)2 = -2p2 (1-cosθ)
( p1-p4)2 = -2p2 (1+cosθ)
M = -g2/ (p2sin2θ)
d/d = (1/8)2 |M|2 S |pf|/|pi| 1/E2
d/d = (1/8)2 g4/ (p4sin4θ) ½ 1/E2
where is the total energy of both initial or final particles.
5) We will discuss renormalization in detail in chapter 7
The key issue is that if you compute higher order diagrams in our toy theory or in real
quantum electrodynamics we find that there are divergent terms proportional to the
logarithm of the momentum of the propagator particle as you integrate to infinity.
Consider that these diagrams typically have 4 internal propagators that must be integrated
over and 4 vertices that introduce delta functions. One delta function conserves overall
momentum and energy conservation so only three of the propagators integrals can be
performed using the delta functions leaving one that must be integrated. This integral,
which can involve momentums, from zero to infinity leads to the divergent terms.
1) Features of quantum field theories
For the toy theory we developed we used scalar particles and presented a set of rules that
would allow the calculation of lifetimes and scattering cross sections. This simple theory
was actually quite successful at calculating the order of magnitude of decay lifetimes and
simple scattering problems. The results were consistent with the qualitative features of
the forces and interactions. The rules used diagrammatic representation of interactions,
coupling constants, propagators for virtual particles, and from the form of the propagator
produced effective modified coupling constants for heavy force carriers.
However, to more fully understand these quantum field theories we have to develop a
treatment from basic concepts. The results of this treatment should have the following
features.
a) The system should be described by a wave equation, which we will develop by
quantizing the total energy to derive the Hamiltonian.
b) The particles involved should be described by wave functions that are the
solutions or eigenstates of a wave equation.
c) These solutions should intrinsically include the ideas of antiparticles, spin, total
angular momentum, parity, charge conjugation and other conserved properties.
These properties will be described by operators and the wave functions will be
simultaneous eigenstates of these operators and the Hamiltonian. They operators
will commute with the Hamiltonian.
d) We will treat scattering and decay perturbatively. The default wave functions will
be wave functions in free space with no potential. The potential of the fields will
perturb the system, scattering the particles from the initial to the final state
<i|V|f>. <i|V|f> is the matrix element.
e) To find V we will quantize the classical potential of the fields using the
electromagnetic field as an example. The quantized potential will give us the
propagator.
f) <i|V|f> will be equivalent to a lowest order Feynman diagram. Higher order
terms will involve multiple propagators. The propagator will be identified with
the force carrying photon in the case of electromagnetic scattering. Note that the
propagator of the photon will also intrinsically include spin information.
g) Particles, electrons, can also be thought of a probability fields (or potentials) and
they will have a propagator for them as well.
h) With this formulation the interactions can be represented diagrammatically with a
set of rules to translate the diagram into the matrix element <i|V|f>. Higher order
diagrams are just multiple transitions and will happen at lower probability for
cases where the coupling constants are small.
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