Physics 535 lectures notes: 2 – Sep 6th 2007
Homework: 1.1, 1.2, 1.3, 1.16(compare to particle data book online)
1) Review
Had understood the atom, and properties such as interference and diffraction using QM.
We had experimentally seen that light and particles such as electrons both had wavelike
and particle like properties. We knew that fast moving particles obeyed relativity.
However, attempts to create relativistic quantum theories led to problems like negative
energies.
We need a theory that:
a) Incorporated both quantum mechanics and relativity.
b) Explained the antiparticle, which had been observed.
c) Explained why particles and light had both wave and particle properties.
d) Explained the quantization of light, which before had been thought of as an
electromagnetic wave.
Quantum field theories such as Quantum Electrodynamics and could explain all these
features.
In quantum field theory the electric and magnetic forces or electromagnetic force are
fields (as is the electron). However the field is caused by many photons being
exchanged, which carry the information of the field. At the basic level everything is
governed by single exchanges of particles. The photon and the electron are the quanta of
these fields and the field can be excited such that the quanta are observable as real
particles. Note that the electromagnetic field can be excited to produce a photon in the
presence of electric charge or in other words electric charge is what the photon couples
to.
Diagrams of the interactions can be drawn which will always have a photon interacting
with two particle lines at a vertex. Depending on the orientation these diagrams can
either have absorption or emission of a photon or particle antiparticle pair annihilation or
creation to or from a photon.
Internal lines in the diagram or virtual – don’t have to maintain energy or momentum
conservation. However, the initial and final particles have to obey energy and
momentum conservation.
How does this give a 1/r2 force? The photons have energy maximum E and exist for a
maximum time of t such that Et~hbar. Or you can say they have maximum momentum p
and cross maximum distance r such that pr~hbar where r=tc. Then for force is F= dp/dt =
= dp/dr dr/dt = hbar c/r2. The strength of the interaction is governed by the charges
involved. F=CqQ/r2 or U=-CQ/r. The photons are excited by the presence of charges so
their number is going to be proportional to qQ times some constant that governs the
strength of individual electromagnetic interactions.
2) The weak interactions and the how protons were held together in the nucleus remained
mysteries.
Had to propose new interactions that would have their own quantum field theories.
Weak interaction. This interaction had much smaller probabilities or strengths of
interaction. Also needed a new particle to conserve momentum and energy. n->p+e-
Neutrino: electrically neutral and almost undetectable. Interacted with weak strength.
Strong interaction. This interaction had to have a much larger probability or strength of
interaction to hold protons together in the nucleus.
The reasons will be understood as we start to study the qualitative aspects of these
interactions.
We also observed a number of new mystery particles in cosmic ray experiments
muon: first seen in cosmic rays this particle had all the properties of the electron except it
was ~200 times more massive. Seemed to interact with the weak strength.
Pion: also >200 times more massive but appeared to interact with strong strength in
matter. Also came in neutral and charged versions.
Kaon: more massive than two pions and seen to decay to multiple pions with the weak
stregth. Also interacted in matter with strong strength.
Charged muons, pions and kaons also interacted with the electromagnetic strength but at
a much smaller level since the interaction effect (acceleration) is inversely proportional to
the mass.
3) The weak interaction and the SM
We knew that there were processes such as n->p+e- that happened over long time scales.
The free neutron lifetime is actually 15 minutes! These were long time scales compared
to for instance the time it takes for an atom in an excited state to decay to the ground state
electromagnetically emitting a photon. One way to explain this was with an interaction
that was extremely low probability or strength.
We also had a mystery in that momentum was not conserved in this interaction and the
electron energy could vary.
Also these didn’t appear to be any weak field that particles with weak charge were
repelled or attracted by.
The answer to these puzzles was to propose a new weak interaction. The weak
interaction involved a massive force carrier that when produced virtually could only exist
for a short time. This limited the strength and range of the interaction. Also as part of
the decay there had to be another particle that was produced. This particle was
massless(or almost massless!) and only interacted weakly.
The n->p+e- interaction involved the neutron emitting a negatively charged Wintermediate boson which then split into the electron and antineutrino. Intermediate since
it was a virtual particle only produced for a short time within the interaction. In face the
W- is 81 times more massive than the proton.
Why a massive boson? tE~hbar. Or tmc2~hbar or range R = ct = hbar/mc. The force is
tricky to calculate. Since the energy is fixed by the mass of the boson you can’t take the
derivate like we did before for the electromagnetic force. Instead you have to solve the
wave equation for a massive particle propagating at relativistic speed and you find a
potential U=C/r e-r/R and a force F=C/(p2+m2). We will study the derivation of this later.
This scheme and the neutrino was more convincingly proved when the crossed
interaction n->p+e- was observed (took a flux of 5x1013 neutrinos) and the similar
interaction with and antineutrino was not observed.
The muon was involved in the picture also since you could observe the weak decay ->e where the neutrinos where muon and antielectron types.
We could tell there was a muon type of antineutrino since p+ -> +n
So the rules of these interactions where that they conserved lepton number (separately for
electrons and muons and their neutrinos) and the interacting was short range and low
probability or strength.
4) The quark model
The kaon was more massive than pion and seemed to involve a new charge since they
were at least produced in kaon antikaon pairs. Though they didn’t decay that way! The
decays had weak strength. They could also interact electromagnetically if sent through a
field and interacted strongly if fired into dense matter.
Since we had added the muon to the electron why not add a second generation of hadrons
as well.
Though this was all simpler if you adopted the quark model. There were compelling
reasons to adopt a quark model such as the presence of excited energy states that
otherwise had the same properties. Atoms could be in excited states as well because of
their structure involving a proton and an electron in quantized orbits. Even the nucleus
could be in excited states because of the structure of protons and neutrons.
Protons and neutrons were made up of uud and udd quarks respectively with charges u,
2/3, d -1/3. Pions were made up of uu+dd, ud or du quarks with the second quark being
and antiquark version to get charged and neutral pions. Kaons involved a strange quark
(charge -1/3) which was part of second generation of particles that was later rounded out
by the charm quark(charge 2/3).
These particles could decay via the weak force so evidently the quarks and leptons had a
weak or flavor charge in common. Though note that the strangeness or charmeness of the
second generation is not conserved. Kaons were first seen in K->pipi decays.
This zoo of particles soon expanded to include the tau lepton and neutrino and the bottom
and top particles. Top discovered in 1995 on the experiment I work at.
5) The strong force and the SM
However what bound the quarks together? They were charged and even closer together
so should experience massive electromagnetic attractive and repulsive forces.
Also we never saw free quarks!
The quantum field theory with these features was Quantum cromodynamics and the
gluon carried the strong force, which interacts with things that have strong or color
charge.
A more satisfying proof of this theory is from very high energy scattering experiments.
Deep inelastic scattering experiments have probed the structure of the proton and see the
quarks and infer gluons. Inelastic because the energy is high enough that we tend to blow
apart the proton.
A proof of the color charge is the existence of the Delta uuu particle which to obey Fermi
statistics must have each quark in a different color state.
The strong force was also limited in range, but by a difference mechanism. Instead of
having a massive force carrier the gluon carries color charge itself and can interact with
itself, which has the end result of limiting the range of the force. It also gives us other
features such as confinement of the quarks to groups of 2 or 3.
Calculating the potential you get V=-C/r +kr with the constant being near 1! This leads
to the interesting effect that it’s more probable to have interactions with lots of gluons
that with just one.
Because of the linear term as quarks separate they build up so much energy that
eventually you pop new quarks our of the vacuum converting the energy of the potential
to mass. The linear term physically can be thought of as a string of gluons stretching
between the quarks. When the string breaks it’s energy goes into making two new quarks
which will be near the old quarks. If it doesn’t break the quarks are pulled back into the
hadron, or confined.
6) Questions that led to the unification of the field theories into the SM.
Why did we have three quantum field theories have such different features such as:
the strength or probability of the interactions, the massiveness of some of the force
carrying bosons, and the ability of some of the bosons to interact with each other.
Next time SM in detail and experimental particle physics.