Lecture 4: Feb 2nd 2012
Reading today: Griffiths Chapter 2
Reading next time: Griffiths Chapter 2, Perkins Chapter 2 handed out in class.
Homework: Griffiths: 2.2, 2.4, 2.5, Perkins 2.4
Due Feb 9th
The Standard model forces
1) Electromagnetic Force
Quantum Field Theory: Quantum Electrodynamics
Field Quanta, force carrier: Photon, m=0
Charge: Electric charge. Couples to all things with electric charge
Diagrams: three-line vertex with electron (positron), electron (positron) and photon
Conserves: Energy and momentum, electric charge, other charges, baryon number, lepton
number, quark generations (first, second and third generation), spin and angular
momentum, P Parity, C charge congregation, T time reversal, CP, CPT
Potential: V = -CqQ/r = -/r
Range: infinite
Strength:  = e2/4hbar = 1/137
Cross section: 10-33 m2
Lifetime: 10-20 sec
In addition to various measurements of the strength interactions are often characterized
by various discrete properties such as inverting the charge or coordinates.
Charge conjugation invariance: when inverting all the charges. This interaction should
occur with the same rate
Parity: inversion of all the spatial coordinates
Time: reverse time to make the backwards reaction
Example interactions:
Basic vertex, electron absorbing a photon: Photon electron-electron vertex Probability .
Each vertex contributes a factor of  to the probability of the interaction.
Electron positron annihilation and creation: Probability 2
Electron (positron) Electron (positron) scattering: Probability 2
Note for an electron and a positron the above two have the same result! To get the total
probability we must sum them.
Full probability dependence given by Rutherford scattering relation: 2/q4
Electron-Electron scattering will have a different probability though still at order 2.
Brehmstrahlung (must be near an atom, virtual electron interacts with the atom one more
factor of alpha): Probability 3
Pair production (must be near nucleus, virtual electron interacts with nucleus):
Probability 3
Note on the single vertex process. Not a real process must actually happen in the field of
a atom where it can absorb an additional photon to be relativistically allowed.
Each vertex contributes a factor of  to the probability of the interaction.
In this case the interaction with the atom does not contribute a factor of alpha since the
electron involved in such a process is orbiting very close to the nucleus and the
probability of interaction with the nucleus in a reasonable time period is essentially one.
These examples drive much of the function of particle detectors
An interesting property: The virtual photons radiating from a charged particle can split
into electrons and positrons and then annihilate back to a photon. These charges will
orient with the positron toward the original charge (if negative) and the electron away
from the charge. This effect, which is like a dielectric medium, tends to screen the
original charge. As a result the effective coupling constant changes, becoming larger
depending on how close you get to the original charge and smaller further away. We
would say the coupling constant runs.
2) Strong Force
Quantum Field Theory: Quantum Chromo Dynamics
Field Quanta, force carrier: 8 gluons , m=0
Charge: red green and blue color charge, 8 gluons carry different combinations of color
anti-color
Couples to: quarks and gluons
Diagrams: three line vertex with quark (anti quark), quark (anti quark) and gluon, or three
gluons
Conserves: Energy and momentum, electric charge, color charge, other charges(lepton
number quark flavor/generation), baryon number, spin and angular momentum, C, P, T,
CP, CPT
Potential: V =- s/r + kr
Range 10-15 m – by confinement of color neutral combinations
Strength: s = 1.3 (1.3 at 0.217GeV low energy, 0.3 at 1 GeV mid energy, 0.1 at 100
GeV high energy) (EM 1/137)
Cross section: 10-30 m2 (EM 10-33)
Lifetime: 10-23 sec (EM 10-20 sec)
Examples
Strong quark annihilation…
Note quark annihilation would also have a contribution from the electromagnetic force
but it is so small compared to the strong contribution that it can usually be ignored.
Scattering via a gluon
These interactions explain the behavior of hadrons in dense matter particle detectors.
Interesting effects.
Virtual gluons can split into quarks or gluons and annihilate back to gluons. This effect
tends to anti-screen. A bare quark would radiate a strong color field attracting other
quarks. Until you have 3 (red green blue) or 2 (color anti-color) you don’t have a stable
color neutral system. In fact as small distances inside hadrons the strength of the strong
force is very small. Small enough that perturbation theory works and in fact quark inside
a nucleon are essentially free – asymptotic freedom.
As you get very close to the quark(takes very small wavelengths or high energy) the
coupling constant get very small. Near EM level. A hint of Unification?
If momentum is transferred to one of the quarks in this system the kr part of the potential
will pull back the quark back. In terms of a potential this can be thought of as due to the
strong color field. If the quark has enough momentum the gluon string between the
quarks will build up enough energy to form two new quarks. The new configuration of
two new quark and shorter gluon strings has a lower potential.
Note: nucleons are held together by the residual color charge just as molecules are held
together by the residual electric charge outside of the otherwise electrically neutral atoms.
3) Weak Force
Quantum Field Theory: Quantum Flavor dynamics
Field Quanta, force carrier: W+ and W- m=81GeV and Z m=91GeV
Charge: Weak flavor charge. Lepton number, quark flavor
Couples to: All things with flavor charge (all the quarks and leptons)
Diagrams: W, three line vertex with electron neutrino and W or quark quark and W.
Three line vertex with lepton (anti)-lepton or quark (anti)-quark and Z.
Conserves: Energy and momentum, electric charge, lepton number, color charge, baryon
number, CPT
Violates: quark generation (W only, no flavor changing neutral current involving the
neutral Z), C, P, CP(at a very small level)
Potential: V = -W/r = -/M(W,Z)2r. Another hint of Unification
Range: 10-18
Strength: W = 10-5(EM 1/137)
Cross section: 10-44 m2(EM 10-33)
Lifetime: 10-8 sec or longer EM 10-20 sec)
Examples:
Same types of diagrams as for the other two forces but now the W (not the Z) changes
particle types.
The weak force involves a large number in interactions with all the quark and leptons
since they all have flavor. When determining what sort of interactions are possible or
probable the guiding rules are that the interactions should conserve momentum and
energy, electric charge and various types of weak flavor when appropriate.
Some example interactions are
d-> uW- -> ue- anti e: seen for neutrons, n(ddu) -> p(duu) e- antie
- -> W-  -> e- anite 
d anti u -> W- -> e- anti e: seen for charged pions - -> e- anti e
s -> uW- -> ue- anti e: seen for charged kaons, for K-(s anti u) -> 0(u anit u) e- antie
changes generations.
Flavor changing between the quark generations can be explained by the idea that the
three down type particles d, s, and b and the three up type particles u, c, and t are
eigenstates of the strong force, but that the eigenstates of the weak force are slightly
different for the down type particles d’, s’ and b’ which can be expressed as a linear
combination of the d, s and b strong eigenstates. These coefficients can be summed up in
the 3x3 CKM rotation matrix.
Consider the b quark. When the b quark is in a meson or baryon coupled to other particles
by gluons so it is definitely in a strong eigenstate. Therefore, it is in some superposition
of d’, s’, and b’. When it interacts with the W+ it might interact as any of the three with
a probability governed by the components of the linier combination it is in (CKM matrix)
The W particle would convert b’ to t’ type eigenstates(particles) b(-1/3) -> W- t(2/3) with
highest probability, but this decay is forbidden since the t quark is much more massive
than the b quark If it interacts as a d’ what you will see is -> W- u or if a s’ what you
will see is -> W- c. Therefore there is a finite probability of decaying to a c or a u quark.
t is forbidden by energy conservation. According to the CKM matrix constants c is much
more likely.
This is the only way a particle from second and third generation can decay to the lowest
energy generation. These particles all have long lifetime and in fact hadrons with b
quarks have very long lifetimes since the b transitions to the other generations have small
probabilities. The superposition of b’, s’, d’ for the b quark is mostly b’.
The neutral Z particles have scattering and annihilation or pair production interactions
much like the photons. Though these interactions can also involve neutrinos. Flavor
changing between the generations like B->Zd is not possible for the Z particle. This can
be understood mathematically. The W an interaction can be written as <u|WUCKM|s>,
where a s strong eigenstate has some probability of being in a d’ weak eigenstate and
being converted to a u quark. The matrix U is unitary because it conserves probability.
For the Z, <d|U+ZU|s>, where U+ is the complex transform of U. However, U doesn’t act
on the Z since it isn’t a quark and U+ZU = Z, and <d|Z|s> has zero probability since d and
s are orthogonal eigenstates.
Note FCNC, flavor changing neutral currents does happen at higher order at very low
probability through higher order diagrams.
Note that the weak boson couple to each other also but the interactions are very low
probability so you don’t see them very often. They don’t lead to interesting new effects
like in QCD.
4) Unification
A milestone toward the standard model was the Z particle, the third quanta of the weak
force. This particle was neutral and had similar interactions to the electromagnetic force
such as e+e- -> Z -> e+e-. However it took a long time to find this particle since no one
expected it! Any type of interaction involving the Z had an equivalent higher probability
EM interaction, except for the neutrino interactions, which are very difficult to observe.
Later it was seen that as you observed interactions such as particle-particle annihilation at
higher and higher energies the constants that governed the electromagnetic and weak
interactions changed and eventually unified at a high value. Essentially at energies well
above the Z mass(91 times the proton mass) the mass of the carrier becomes irrelevant to
the interaction. This was the first strong hint that there had to be some unifying theory
behind all the diverse quantum field theories that had existed.
Another way to put this is the both interactions have the same coupling constant and it’s
just the extra factor of the mass of the gauge boson in the denominator that makes that
gives the difference between the strengths of the interaction. Doing that calculation using
the measured strength of the weak and electromagnetic interactions in similar
configurations involving photons and Zs gives mZ=91GeV. Exactly what was later
measured.
Also a milestone for theory since it was predicted before it was found!
In addition the strong force is seen to weaken at higher energies so it may also unify.
One mystery in this picture is still present. Why do the W and Z have large mass. To
explain this you need to introduce an additional particle that by coupling with particles
can give them mass. This is the Higgs particle. The Higgs also gives mass to all the
other massive particles.
In fact the b, s and d states can be considered eigenstates of the Higgs interaction and the
strong interaction. As before the b’, s’ and d’ states are still the Weak eigenstates. The
Higgs particle interacts with any particle that has mass. Since these particles have mass
they are in eigenstates of the Higgs particle (for instance when in a configuration where
they have definite mass, such as a hadron). The neutrios, if they had no mass would be
pure eigenstates of the Weak force. But since they do have some mass you can have
interactions change lepton flavor generation just like with the quarks.
These quantum field theories including the unified electroweak theory with the Higgs
particle to explain the mass of the W and Z make up the standard model (SM).