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LECTURE 13
Elementary Particle Physics
(The chart on the classroom wall can be found at
http://www-pdg.lbl.gov/cpep/cpep
_sm_large.html.)
Four Fundamental Forces
As far as we know, there are just four fundamental forces in nature:
strong, weak, electromagnetic, and gravitational. When we think of
these forces in terms of quantum mechanics, we say that each of these
forces is mediated by the exchange of a particle. These mediating
particles are bosons (integer spin). You are familiar
with the electromagnetic force. The mediator is the photon and it has
spin 1. The photon is massless and has no electric charge.
The gravitational force is mediated by the graviton which has
spin 2, is massless, and has no electric charge. The fact that the
graviton and the photon are massless means that the electromagnetic
and gravitational forces are long range. The strong force holds
protons and neutrons together in the nucleus. Until about 1972,
the strong force between nucleons was thought to be mediated
by pions. This is still a good picture at low energies.
As we shall see,
the proton and the neutron are each made of 3 quarks; the strong force
holds these quarks together. The strong force between quarks is mediated by
the gluon which is massless and has spin 1 (like the photon).
Even though the gluon is massless, the strong force is short
range because the gluons attract each other and would rather clump up
and not spread over all space.
The weak force is responsible for
particle decays such as the decay of the neutron:
The weak force is mediated by spin 1 intermediate vector bosons:
W+, W-, and Z0. The superscripts denote
their electric charges. The W+ and W- each have a mass of
80,280 MeV/c2 while the Z0 has a mass of 91,188 MeV/c2.
The relative strengths of the 4 forces are
strong 10
electromagnetic 10-2
weak 10-13
gravitational 10-42
Leptons
The leptons are fermions that are insensitive to the strong force and
have spin 1/2. The most familiar lepton is the electron. There are 6
known leptons:
Lepton Symbol Mass (MeV/c2)) Charge
electron e 0.511 -1
electron neutrino
0
muon
105 -1
muon neutrino
0
tau
1771 -1
tau neutrino
0
Particles have corresponding antiparticles which have the opposite
spin and charge. Each of the leptons we listed has
an antilepton. For example there are antineutrinos. Antielectrons
are called positrons.
Antiparticles are denoted by a bar over their symbol. For example
is an electron antineutrino.
Helicity: If neutrinos are massless, then
they move at the speed of light.
Suppose we choose the direction of motion as the z axis, and look
at Sz, the component of spin along the z axis. If Sz=+1/2so that the z component of spin points along the direction of
motion, then the helicity is +1 and the neutrino is right-handed.
If Sz=-1/2so that the z component of spin points opposite to the direction of
motion, then the helicity is -1 and the neutrino is left-handed.
All neutrinos are left-handed and all antineutrinos are right-handed
(if neutrinos are massless).
Hadrons and Quarks
The lifetime of states which decay via the strong force is on the
order of 10-23 seconds. Particles are defined as
those entities which live much longer than this,
say 10-20 seconds or longer. All particles which are sensitive
to the strong force are called hadrons. Hadrons are divided into
2 classes: baryons and mesons. The baryons are fermions
and have half integer spin. The mesons are bosons and have integer spin.
Hadrons are made of quarks. A quark is a fermion with spin 1/2. A
baryon consists of 3 quarks, while a meson consists of a quark and
an antiquark. (All the hadrons and quarks have antimatter counterparts.)
There are 6 types or ``flavors'' of quarks: up, down, charm, strange,
top and bottom. The quarks have fractional charges.
Quark Symbol Mass (MeV/c*c) Charge
--------------------------------------------------------------------
up u 5 2/3
down d 10 -1/3
charm c 1600 2/3
strange s 180 -1/3
top t 180,000 2/3
bottom b 4500 -1/3
For example, a proton consists of uud. If we add the charges of
the constituent quarks, we get Q = +1. If we add the spins of the quarks,
S=1/2 is allowed. A neutron consists of udd. The sum of the charges
is Q = 0 and the spin is 1/2. A meson consists of ;
it has charge +1 and spin 0. A meson consists of ;
it has charge -1 and spin 0. A meson is the antiparticle of
a meson. A meson consists of
; it has charge and spin 0, and is its
own antiparticle.
Notice that there are 6 quarks and 6 leptons which is a nice symmetry.
Quantum Numbers
In the early days of particle physics people found a whole plethora
of particles (things living much longer than 10-23 seconds).
They tried to make sense of these particles by arranging them
in sensible ways.
They also tried to understand why they saw the reactions they did and why
they didn't see other reactions. So they came up with selection rules
that said that certain quantities were conserved in a reaction or decay.
Here is a list of the conserved quantities:
- Energy and Momentum.
- Total Angular Momentum = sum of spin and orbital angular momentum.
- Electric Charge Q.
- Baryon Number B: Baryons have baryon number B=+1 and their
antiparticles have B=-1.
- Lepton Number L: Leptons have lepton number L=+1 and their
antiparticles have L=-1. So L=+1 for the electron and for
,
and L=-1 for the positron and for the electron antineutrino.
Until recently it was believed that electron
lepton number Le was conserved, as was muon lepton number and tau lepton number .
However, recent evidence for neutrino
oscillations indicates that one type of neutrino can become another
type of neutrino. It turns out that this can only happen if neutrinos
acquire mass. We will learn more about this later.
The 4 quantum numbers Q, B, and L are ``good'' in the sense
of being exactly conserved in every known reaction. (By the way, there
is no conservation of mesons. For example a can decay
into 2 photons.) So if we have a reaction that has the following
form
then the total Q, B, and L before the reaction will equal the
total Q, B, and L after the reaction.
We will now list
quantities that are conserved in strong interactions but not
conserved in electromagnetic and/or weak interactions.
- Strangeness S and Hypercharge Y: The strangeness quantum
number was invented to account for the unusually long decay lifetime
of certain baryons (e.g., ,
the 's, and the 's,)
and certain mesons (kaons). It turns out that these particles all have
at least one strange quark. If a particle has one strange quark, it has
S=-1. If it has 2 strange quarks, then its S=-2, etc.
If a particle has a strange antiquark, then it has S=+1, etc.
Hypercharge Y is defined as the sum of
the baryon number and the strangeness:
This is just a special case of the fact that strong and electromagnetic
interactions conserve quark flavor, but the weak interaction does not.
For example, the strong interaction is involved in the reaction
Notice that the total strangeness (S=0) is the same on both sides.
In decays, however, the nonconservation of strangeness is very conspicuous,
because for many particles this is the only way they can decay.
The ,
for instance, is the lightest strange baryon; if it is
to decay into lighter particles, strangeness cannot be conserved.
Thus the decay occurs via the weak interaction.
Its decay paths are
64% of the time,
and
36% of the time.
Similarly charmness, bottomness, and topness are conserved
in strong and electromagnetic interaction, but not in weak interactions.
Hypercharge was defined before charm, bottom, and top were discovered.
It should be upgraded, but it's easier just to think in terms of the
constituent quarks.
- Isotopic spin I and I3: In nuclear physics isotopic
spin I and its third component I3 were introduced so that the
neutron and the proton, with nearly equal masses, could be treated
as charge states of the same particle, the nucleon. Thus the neutron
and the proton were part of an isospin doublet with I=1/2. The
proton has I3=1/2 and the neutron has I3=-1/2. In particle
physics also the observed masses of the hadrons form clusters at
particular values; there are many families (n, p; ,
;
,
,
)
within which the masses are
nearly the same. Apparently, the strong interactions are nearly
independent of charge. It is convenient to regard these families as
isospin multiplets. We use the algebra of ordinary spin. So a multiplet
with isotopic spin I has a multiplicity 2I+1. The third component
I3 can be used to determine the charge of the particle using the
formula
This is true as long as the particle does not have charm, bottom, or top
quarks.
The up quark has I3=1/2, the down quark has I3=-1/2. They comprise
an I=1/2 doublet. The other quark flavors have I=0. Electromagnetic
interactions do not conserve I, e.g., the proton and the
neutron both have I=1/2 but they behave differently electromagnetically
(one has charge and the other doesn't).
- C, P, and T: Charge conjugation C changes a particle to
its antiparticle. In particular, it reverses the sign of the electric
charge and magnetic moment of the particle. Any particle that is its
own antiparticle has a C-parity associated with its wavefunction.
The C-parity of a state is odd if the wavefunction changes sign
under charge conjugation, and is even otherwise. The C-parity
of a photon is -1 because the electric field produced by
a charge (say an electron) changes sign if the particle is replaced
by its antiparticle (a positron). The C-parity of the spinless
is even because the
decays into 2 photons, the
C-parity of 2 photons is (-1)2=+1, and C-parity is conserved
in electromagnetic interactions.
As we have seen, the parity operation P reverses the direction of
any spatial vector ,
that is,
.
Invariance
of an interaction under P implies that the interaction is symmetric
under mirror reflection. The intrinsic parity of a particle is odd
or even depending on whether the wavefunction of the particle is odd
or even under P. The parity of a fermion is opposite to that of
its antiparticle while the partiy of a boson is the same as its
antiparticle. Quarks have
positive intrinsic parity, so antiquarks have negative parity.
The parity of a composite system in its ground state is the product of
the parities of its constituents. For excited states, there is an
extra factor of
where
is the orbital angular
momentum. Thus, in general, the mesons carry a parity of
.
So in the ground state, a pion has P=-1, 2 pions have P=+1 and
3 pions have P=-1. The intrinsic parity of a photon is -1.
T stands for time reversal.
Experimentally, both strong and electromagnetic interactions are found
to be invariant under the operations C, P, and T carried out
separately. The weak interaction is not invariant under P. (T. D. Lee
and C. N. Yang received the 1957 Nobel Prize for predicting this.) Most
reactions are found to be invariant to the combined operations of
CP, except for the very special but important case of kaon decay
and, more recently, Bo meson decay.
(Val Fitch and James Cronin won the 1980 Nobel Prize for discovering
CP violation, observed in Ko decay.) The CPT theorem states
that all interactions are invariant under the combined operation CPT.
Its proof is based on fundamental assumptions used in field theory.
As a consequence of the CPT theorem, CP violation implies T violation.
CP Violation
Let us discuss CP violation in more detail. We will discuss
the kaon system but similar considerations apply to the Bosystem. CP violation was discovered
by studying the neutral kaon system. Kaons are typically produced by the
strong interactions, in eigenstates of strangeness. Ko has
strangeness +1 and its antiparticle
has S=-1.
The weak interaction allows these particles to interconvert:
As a result the particles we normally observe in the laboratory are not
Ko and
, but rather some linear combination of
the two. In particular, we can form eigenstates of CP as follows.
The Ko's are pseudoscalars which means they are odd under parity:
Under charge conjugation,
So if we apply CP, we obtain
So we can construct normalized eigenstates of CP:
Thus
Neutral kaons decay via the weak interaction.
Assuming CP is conserved in the weak interactions, K1 can only decay
into a state with CP=+1, whereas K2 must decay into a state with
CP=-1. Typically neutral kaons decay into two or three pions. But
CP conservation requires that K1 decay into two pions (never 3);
K2 decays into 3 pions (never 2):
This is because both 2 and 3 pion states have C=+1 but has
P=+1 and has P=-1. The decay is much faster because
greater energy is released (corresponding to greater phase space for
the final states). In fact K1 has a lifetime of
sec while K2 has a lifetime of
sec. So if we start with a beam of Ko's
the K1 component will quickly decay away, and down the line we shall
have a beam of pure K2's. Near the source where the kaons are
made, we expect to see a lot of events, but farther along we expect
only decays. If at this point we observe a decay, we
shall know that CP has been violated. Such an experiment was reported
by James Cronin and Val Fitch in 1964. At the end of a beam 57 feet
long, they observed 45 two-pion events in a total of 22,700 decays. That's
a tiny fraction (1 part in 500), but it provided unmistakable evidence
of CP violation. Evidently the long-lived neutral kaon KL is not
a perfect eigenstate of CP after all, but contains a small admixture
of K1:
The coefficient is a measure of nature's departure from perfect
CP invariance; experimentally its magnitude is about
.
Further measurements have confirmed the violation of CP. In particular
although 34% of all KL's decay by the mode, some 39%
go to
Both reactions have CP=-1. Notice that CP takes (a) into (b),
so if CP were conserved, and KL were a pure eigenstate, (a) and
(b) would be equally probable. But experiments show that KL decays more
often into a positron than into an electron, by a fractional amount
. Here for the first time is a process that makes
an absolute distinction between matter and antimatter, and provides
an unambiguous, convention-free definition of positive charge:
it is the charge carried by the lepton preferentially produced in
the decay of the long-lived neutral K meson. CP violation
may be responsible for the matter-antimatter asymmetry observed in
the universe.
As we mentioned above, there is no evidence of CPT violation
which is based on very general assumptions like Lorentz invariance,
quantum mechanics and the idea that interactions are carried by fields.
If CPT is conserved and CP is violated, then time reversal invariance
(T) must be violated.
Hadron Spectroscopy
Back in the days when people just knew about 3 quark flavors
(u, d, and s), Gell-Mann pointed out that there was a systematic
way to put the 3 quarks together to get the known isospin multiplets.
This has been dubbed ``The Eightfold Way.''
Recall that when we discussed adding angular momenta, we said that
adding 2 spin-1/2 objects gives a singlet (S=0) and a triplet (S=1).
So when we combine a quark and an antiquark to make a meson, we
can have a singlet (pseudoscalar meson) or a triplet (vector meson).
(A pseudoscalar changes sign under a parity operation.)
In group theory notation we can write
where you can
regard the numbers as denoting the number of components. (Actually, it's
the dimension of the matrices in the representation of the group.
Spin-1/2 objects are represented by the Pauli
matrices which is a representation of the group called
``SU(2).'' , , and the identity
matrix are the elements of the group SU(2). In general,
spin is described by SU(2); this is true for any value of the spin,
not just spin-1/2.)
If one regards the
3 quark flavors as components of a multiplet (they are described by
the group SU(3)), it turns out that group theory
tells us that a quark and an antiquark can combine into an octet and
a singlet:
. That means that there is an octet
and a singlet of pseudoscalar mesons, and an octet and a singlet of
vector mesons. We can represent this pictorally by putting the 3
quark flavors at the corners of a triangle:
2.0 true in
3quarks.eps
where S is the strangeness quantum number.
Similarly the antiquarks are at the corners of an upside down triangle.
3antiquarks.eps
Now we can superpose the triangles to form an octet plus a singlet:
qqbar.eps
These give the spinless mesons:
mesons.eps
By superposing 3 quark triangles, one can get the baryon decuplet and
baryon octet. (see pages 401-403 in The New Physics).
In group theory language we would write
. The baryon decuplet has 10 baryons,
each with spin 3/2. The baryon octet has 8 baryons, each with spin 1/2.
This makes sense since we can get total spin 1/2 or 3/2 if we put together
3 quarks which each have spin 1/2. It turns out that
there is only one octet of baryons,
not two octets, because there is only one way to write a completely
antisymmetric wavefunction for the octet.
Color
In the decuplet there are baryons such as
which is made
of (uuu),
which consists of (ddd), and which consists of (sss). This appears to violate the Pauli
exclusion principle because we appear to be putting 3 identical
quarks in the same state. To get out of this problem, the
quarks were given an additional quantum number called ``color''.
Each quark flavor comes in 3 colors (red, blue, and green, say).
Antiquarks have anticolors.
To make a baryon, we simply take one quark of each color, then
the three u's in
are no longer identical (one is
red, one is green, and one is blue). Since the exclusion principle
only applies to identical particles, the problem evaporates.
The allowed quark combinations must follow the rule that all naturally occurring particles are colorless. By ``colorless''
I mean that either the total amount of each color is zero or all
three colors are present in equal amounts. (The latter case
mimics the fact that white light is made of many colors.) A meson
is colorless because it has a quark of one color (say, blue)
and an antiquark with the anticolor (antiblue). This rule
means that we won't find a particle made of 2 quarks or 4 quarks. It
also tells us that we won't find individual quarks in nature. The only
colorless combinations you can make are (mesons),
qqq (baryons), and
(antibaryons).
You could have 6 quarks, but we would interpret that as a bound state
of 2 baryons.
Actually you can't separate quarks because the strong interaction
between 2 quarks increases as the distance grows. So it takes too
much energy to separate quarks. With all that energy, quark-antiquark
pairs are created.
makequark.eps
The gluons each carry a color and an anticolor. Since
,
there is an octet of gluons:
,
,
, etc.
The color singlet (``1'') is given by
. The color singlet is not
a gluon but is represented by the colorless
mesons which mediate the strong interaction
at low energies. For example pions mediate the strong interaction between
nucleons.
Weak Interactions
Both quarks and leptons are affected by weak interactions. There
are two kinds of weak interactions: charged (mediated by )
and neutral (mediated by the Z0). Let's look at leptons first.
- Leptons: The fundamental charged vertex looks like this:
=2.0 true in
leptonW.eps
A negative lepton (e-, ,
or )
is converted into
the corresponding neutrino with the emission of a W- (or absorption
of a W+):
.
This implies that
is also allowed. We can
combine the primitive vertices together to make more complicated
reactions. For example,
is represented by the diagram:
=2.0 true in
MuNuScatt.eps
Such a neutrino-muon scattering event would be hard to set up in the
laboratory, but with a slight twist essentially the same diagram
describes the decay of the muon,
,
which happens
all the time:
=2.0 true in
muDecay.eps
Note that a particle traveling backwards in time is an antiparticle.
So in the diagram the neutrino
traveling backwards in time
is the antineutrino
.
The fundamental neutral vertex is:
=2.0 true in
leptonZ.eps
In this case
can be any lepton, including neutrinos. The Z0
mediates neutral weak processes such as neutrino-electron scattering
(
):
=2.0 true in
MueScatt.eps
Neutrino scattering is extremely difficult to observe experimentally.
Presumably scattering between 2 electrons can occur with the exchange
of a Z0, but this is masked by the much stronger electromagnetic
interaction involving the exchange of a photon. Experiments at DESY
(in Hamburg) studied the reaction
at very high energy and found unmistakable evidence of a contribution
from the Z0. So neutral weak processes do occur.
Note that the leptonic weak vertices connect members of the same
generation: e- converts to a
with emission of a
W-, or
with the emission of a Z0,
but e- never goes to
nor does
go to
.
In this way electron number, muon number, and tau
number conservation are enforced at the leptonic weak vertices.
- Quarks:
For quarks the fundamental charged vertex is:
=2.0 true in
quarkW.eps
A quark with charge -1/3 (i.e., d, s, or b) converts into the
corresponding quark with charge +2/3 (u, c, or t, respectively),
with the emission of a W-. The outgoing quark carries the same
color as the ingoing one. But the flavor changes because flavor is
not conserved in weak interactions.
The far end of the W line can couple to leptons (a ``semileptonic''
process), or to other quarks (a purely hadronic process). The most
important semileptonic process is
:
=2.0 true in
dNuUe.eps
Because of quark confinement, this process would never occur in nature
as it stands. However, if we combine the quark with a u and a dquark, we obtain the beta decay of the neutron
(
):
=2.0 true in
neutronDecay.eps
The weak interaction is not confined to a quark generation.
This allows strangeness changing weak interactions, such as the
decay of the lambda (
), which
involves the conversion of a strange quark into an up-quark.
=2.0 true in
lambdaDecay.eps
The quark generations are ``skewed'' for the purposes of the weak
interactions. Instead of
the weak force couples pairs
where
,
,
and
are linear
combinations of the physical quarks d, s, and b:
This
matrix is called the Kobayashi-Maskawa matrix.
It is close to being the unit matrix; the diagonal elements are
close to unity and the off-diagonal elements are small. This
matrix allows mixing between quark generations. Vud measures
the coupling of u to d, Vus the coupling of u to s,
and so on.
Neutrino Oscillations
Until recently it was believed that neutrinos are massless. However
there is now evidence that neutrinos may indeed carry a small
amount of mass. Among these experiments are those being carried out
by Hank Sobel (UCI) and his collaborators at Super Kamiokande in
Japan. How do you measure the mass of a neutrino? You can't stop it
and weigh it. It turns out that if neutrinos have mass, they can
transform into another neutrino flavor after a while. So as the
neutrino goes along in space, it changes its flavor, e.g., from
to
. These flavor
changes are called neutrino oscillations and it is these oscillations
that experiments are trying to observe. To see why having mass corresponds
to neutrino oscillations, let us assume that neutrinos have finite
mass (
). Further let us suppose that the neutrino
eigenstates of the weak interaction Hamiltonian HW are not the
same as the neutrino mass eigenstates. Then
are the eigenstates of HW and
are the mass eigenstates. In other words their energy is given by
As a result, Schroedinger's equation
implies that these mass eigenstates will have a time dependence given by
.
These two representations are related by a unitary transformation:
As a simple example, consider mixing two neutrinos.
(We choose and
because that's what Super
Kamiokande is looking at.)
We can take
and plug it into Schroedinger's
equation
to see how
and
will evolve in time. We find
The probability
of detecting at time tis given by
where at
is pure . Plugging in
eq. (19), we find that
Using
, we find that
Here we are assuming that the neutrino momentum
.
Then
So the probability of detecting oscillates in time
if neutrinos have mass. Super K has found that
eV2 at the
90% confidence level.
There are other implications for massive neutrinos. Recall that
all massless neutrinos are lefthanded and all massless antineutrinos
are righthanded. If neutrinos have mass, it also means that there are
some righthanded neutrinos and some lefthanded antineutrinos, though
none have been found so far.
Next: About this document ...
Clare Yu
2002-12-05