# Notes On Particle Physics

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```					TOPICS IN PHYSICS 2006-07                                                R Cherdack

Notes On Particle Physics

Introduction

People have realized that matter consisted of assemblies of basic building blocks
for many centuries although the triumph of the atomic theory is only a little more
than a century old. Current theory has it the building blocks of which the parts of
atoms and more exotic bits of matter consist are, depending on how you count,
somewhere between 24 and 63 in number. These notes will attempt to provide
some insight into the properties of these constituents, how they interact, and how
these interactions are studied.

Background
To study particle physics one has to have some facility with the nature of matter
and how it interacts with other matter. Particles of matter have properties that
determine the nature of the matter. Many of the physical properties are
conserved, i.e. appear the same, before and after an interaction.

Among the quantities associated with a particle is its mass. Mass can be thought
of as a form of energy and its equivalent in joules is Mc 2 where M is mass in kg
and c is the speed of light, 3E8 m/s [E8 means x10 8]. In particle physics the
mass is usually given in the equivalent energy in units of million electron volts
(Mev) or billion electron volts (GeV). For example, the mass of an electron is .511
MeV. An electron volt is the energy one electron gains crossing a potential
difference of one volt ( 1V = 1J/coul). Since the charge of an electron is -1.6E-19
C, 1 eV = 1.6E-19 eV. Mass is associated with the gravitational force.

Mass in current parlance is now considered to be the rest mass, the mass of a
particle in a frame of reference where it is stationary. Note that this includes any
mass associated with the potential energy it takes to assemble a particle. This
potential energy is often negative so that the mass of a particle is less than the
sum of masses of the particles from which it is formed. In the past, “relativistic
mass” was used to describe the inertia like property of matter and its kinetic
energy as seen from a frame in which it is moving. The terms “proper” or “rest”
mass were used for its stationary mass.

Total energy is conserved in particle interactions as is total momentum in each
direction. We will deal with this in relativistic terms later. For any assembly of
particles there is a frame of reference in which the total momentum equals zero,
the center of mass frame or com frame. This is a natural consequence of the
vector nature of momentum. [Note: What follows is a classical physics “proof”.
To make this true for relativistic cases requires a bit more formality since pi is

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TOPICS IN PHYSICS 2006-07                                                                                              R Cherdack

 Mvi and  depends on vi ]. One needs only to add all the momenta mi vi as
measured in any frame (the first frame) together and divide by the total mass.
This is the velocity of the center of mass frame as measured in the first frame.
Since the velocity of a particle in the com is its velocity in the first frame - the
velocity of the com frame itself, i.e. in the com frame vi incom = vi infirst -vcom

and in the center of mass frame the total momentum is

m v      i i in com         mi v i           in first    v com   miv i  miv com 
i                                i                                          i

m vi i in first
  m iv i          in first     miv com   miv iv i                   in first     mi   i

i                                i                     i                            i
mi
  m iv i          in first     m iv i        in first   0
i                                i

The com frame will be a useful one for considering many interactions. [For
relativistic speeds the center of mass frame is where the sum of the relativistic
   momenta is zero, i.e. where

1
       i com   miv i   in com        0,  i    com                               2
i                                                                   vi   in com
1           2
c

   Charge has already been alluded to above. It is a characteristic which in any
individual “whole” particle comes in integer multiples of the electron charge
including zero. It can be positive or negative, or in neutral particles, zero. Charge
is associated with electromagnetic force.

Size is a little more problematical. While the more massive particles associated
with the nucleus can be said to have dimensions of about 1E-15 m, electrons and
their relations seem to be true points. How one avoids infinite densities and
electrostatic energies (remember the energy to assemble a charge on a sphere
is proportional to 1/r which goes to infinity when r = 0) or singularities in space,
etc. is beyond this summary for now.

Other properties include angular momentum, spin, and some more exotic ones.

Particles are studied by looking at interactions. In many cases these are
collisions where two or more bodies come into close enough proximity to affect
each other‟s energy, momentum, and even composition in significant ways.
Decays, where one particle changes into another or more than one other particle,

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TOPICS IN PHYSICS 2006-07                                                  R Cherdack

are another source of information. The nature of the interacting particles, the
reaction products and the forces between them can be deduced from the
momentum, energy, trajectories, etc. before and following the interaction. The
rate at which interactions occur and the ratios of possible results are also
informative.

Elementary Particles.

The particles that make up matter in the universe can be classified into two
groups: leptons (light ones) and baryons (more massive ones). A third group of
“particles” are those which mediate (I think convey is a better word, but I will yield
to the usage of the texts) forces.

While there are a fair number, (or what may seem an unfair number to new
students of the subject) of particles, it is remarkable that the whole universe can
be pretty well described by this limited number of building blocks and that each
particular particle occurs in identical form whether in Basking Ridge or
Madagascar or Alpha Centurai or the remains of a supernova from 5 billion light
years away or five billion years ago. For example, electron is an electron.

Leptons are simple particles, but baryons, according the theory which has been
established over the last 40 years are comprised of various kinds of smaller
particles called quarks. Lets begin with the leptons.

Leptons
Leptons were originally thought to comprise just electrons. These are old familiar
friends with a mass of .911E-30 kg and a charge of – 1.6E-19 C. It was later
discovered that the electron had a counterpart with the same mass and angular
momentum (electrons seem to spin) but with an opposite charge of + 1.6E-19 C.
The new particle was named the positron and was the first anti particle
discovered.

It was noted that as some nuclei decayed, they emitted beta particles which were
identified as exactly the same as electrons. No problem with this, a neutron turns
itself into a proton by emitting an electron. However there were major problems if
the only object was emitted was an electron and the electron was emitted with a
range of energies. Why? If you think of a decay process as seen from the original
nucleus‟ frame of reference [just frame from now on], the initial momentum is
zero, since the nucleus has velocity of zero in its own frame. Therefore the
momentum of the resulting nucleus, the one after the decay must be equal and
opposite the momentum of the emitted electron. This of course means the
velocity of the electron must equal Mprod nucl/me x vnucl. This ratio is several
thousand. Since the momentum is the same for each decay product and kinetic
energy = pv/2 then the ratio of the kinetic energies is just the ratios of the

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TOPICS IN PHYSICS 2006-07                                                  R Cherdack

velocities and the electron has Mprodnulc/me as much energy as the product
nucleus.

For a given type of nucleus decay, the mass of the resulting nucleus is the same
for every decay and all electrons have the same mass. Thus, the energy from the
decay that should show up as kinetic energy is (Moriginal nucl –Mprod nucl – me) x c2.
This is the same for every same decay. The way the kinetic energy is split
between the product nucleus and the electron is fixed as M prodnulc/me. Therefore
the amount of (kinetic) energy the electron had, which was nearly all th nergy
released) should be same for all these same decay events. But it wasn‟t. Either
energy wasn‟t conserved or there was something being missed. Wolfgang Pauli,
and most of his colleagues opted for the second explanation. They were right.
About twenty or thirty years later, the electron neutrino was found. It is a neutral
(no charge) nearly massless, particle and is a member of the lepton family.

Naturally another similar particle had to be found, and the positron neutrino was
identified as the missing particle that occurred when nuclei had proton switch to
neutron and emitted positrons.

Unfortunately, things didn‟t top there. A more massive version of the electron
was found in analyzing data from cosmic rays and so the muon was found,
eventually accompanied by its neutrino and the antimatter equivalent and its
neutrino.

And just to put the icing on the cake a third generation of leptons was identified
from collisions in accelerators, giving us the tau and tau neutrino the associated
anti tau and its neutrino. Most particle physicists seem to think this is the
complete set of leptons and if we work hard and are lucky, we can get a glimpse
of why they believe this.

To summarize:

There are three generations of leptons the electron, muon, and tau. Each
generation has more mass (electron .911 E-30 kg[ or .511 MeV], muon 106 MeV,
tau 178 MeV). Each generation comprises a charged particle with charge –1.6E-
19, an antiparticle (which can be and is thought of as an antilepton) with charge
1.6 E-19 of same mass as the particle, and two neutral, nearly massless
neutrinos, one associated with each of the charged particles. By the way, foir
reasons that I hope will be clear, the neutrino associated with negatively charged
particle is an anti particle. In other words the neutrino that occurs when a lepton
electron is emitted is considered an anti-lepton, and the neutrino emitted when
the anti-lepton positron is emitted is considered a lepton.

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TOPICS IN PHYSICS 2006-07                                                   R Cherdack

Quarks
Quarks are the constituents of baryons. Baryons are broken down into two
groups, hadrons consisting of three quarks, and mesons consisting of two.
Baryons are either all quark or all anti-quark in their makeup. Mesons are always
one quark and one anti-quark.

Quarks are never seen outside of the baryons and are probably better thought of
as components of particles rather than as particles in their own right. We deduce
the existence of quarks by the way the baryons interact with other baryons and
with leptons in collisions and decays.

Quarks, also come in three generations as leptons do, each with four particles,
which is comforting to those seeking order in particle physics. However the
generations are somewhat different from those for leptons. Each generation
comprises two quarks and two anti-quarks. The generations of quarks are:
1: down, up; anti-down, anti-up
2: strange, charm; anti-strange, anti-charm
3: bottom, top; anti-bottom, anti-top

All quarks have a charge of either -1/3 x 1.6 E-19 C or + 2/3 x 1.6E-19C. All anti-
quarks have the opposite signed charge of their corresponding quark. For
example, the down, strange, and bottom quarks all have –1/3 x 1.6 E-19 C. While
the up, charm, and top quarks all have 2/3 x 1.6E-19C, the anti-down, anti-
strange, and anti-bottom all have +1/3 x 1.6E-19C.

As with the leptons, each generation is more massive than the preceding one.

A quark‟s very nature is considered a quantity or characteristic, sometimes
described as flavor. For example, particles with one strange quark in their
makeup have a certain strangeness.

Quarks have another property in some ways similar to charge. It is designated as
color but it has nothing to do with color as is usually meant, i.e. a characteristic of
light. This is a really unfortunate choice of nomenclature but we will have to live
with it. Every quark can come in red, blue or green. Every anti-quark can come
as anti-red, anti-blue, or anti-green. However no complete particle can have
color. Complete baryons are either combinations of three quarks (hadrons) or a
quark and an anti-quark (mesons). For a baryon to exist, it must have one quark
of each color, which can be thought of as adding to absence of color (just as red,
green, and blue, add to white as seen by your eye – but don‟t dwell on the
analogy). Mesons achieve the same lack of color by having a color and anti color
quark 9e.g one green quark and one anti-green anti-quark).

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TOPICS IN PHYSICS 2006-07                                                      R Cherdack

Special Relativity Review
Special relativity is the principle that physics is the same in all inertial frames (i.e,
not accelerating frames). Since the equations governing electromagnetic waves (
Maxwell‟s Equations) yield a velocity of c for any electromagnetic wave, the
speed of light must be same to any observer in any inertial frame regardless of
the speed of the observer with respect to the source or to any other observer
moving with constant velocity. This yielded the following transforms where v is
the velocity of frame K‟ (t‟,x‟,y‟,z‟) in frame K (t,x,y,z) and here is assumed to be
in the x direction.

1

2
v
1
2
c
vx
t'  (t    )
c2
x'  (x  vt)
y' y
z' z

Making use of these transforms and the fact that the energy of a photon is hf and
   the momentum is hf/c, we found that the energy associated with motion (kinetic
energy) of an object with mass M was M(-1)c2 and the momentum was Mv. The
quantity M(-1)c2 is equal to Mv2/2, our old equation for KE, when v is much
smaller than c.

Another important fact is that for any motion, the quantity:

1
d 2  dt 2           2
(dx 2  dy 2  dz 2 )
c

is the same in every inertial frame, i.e., is invariant.
   Clearly in the frame of a moving particle, it has mass M, its rest mass, velocity =
0, and therefore kinetic energy and momentum both equal zero. In a stationary
frame, usually called the lab frame, the particle has velocity v. Its kinetic energy
is M(-1)c2 and its total energy is the sum of its kinetic energy and the energy
equivalent of its mass i.e Elab = M(-1)c2+Mc2 = Mc2.

Thus we have a way to express two conserved quantities, E (Total energy =Mc2)
and p (momentum = Mv), as measured in the lab frame. You can see where the
term relativistic mass for M, now no longer in use, came from.

Note that E2-p2c2 is invariant in switching between any inertial frames and that it
equals M2c4. We will make use of this and the com frame concept when
analyzing collisions and decays.

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TOPICS IN PHYSICS 2006-07                                                  R Cherdack

Interactions General (Forces)

Particles interact. It would be a pretty dull universe if they didn „t. For one thing,
physicists would be out of work, but since they wouldn‟t exist, this would be the
least of their worries. Force is one word used to describe interactions. The
current situation is that there are 3 or 4 forces used to describe these
interactions. The or comes from whether you want to treat the weak and
electromagnetic forces as separate or as the electroweak force. We‟ll see how
you feel about this in a few weeks. Lets treat them as speaarate force for now.
So the forces are:
gravity
weak
electromagnetic
strong

Gravity

Gravity is the mutual attraction of masses and can be described as the mutual
attraction of masses suing FG = GM1M2/r122. The “mediating” particle associated
with the gravitational force (i.e. the particles bodies exchange with one another to
convey the gravitational force) is called the graviton, and they are still trying to
find one. Gravity can also be thought of the curvature to space time caused by
mass. Unless you want to spend the next few weeks on general relativity, we will
let this go for now, with the comment that gravity is many orders of magnitude
weaker than the any of the other forces when dealing with particles currently
obtainable.

Electromagnetic Force

The electromagnetic force is the one we encounter in atomic and molecular
reactions which are governed by the mutual repulsion of electrons and their
attraction to the protons in nuclei. The property associated with electromagnetic
force is charge, which is a conserved quantity. In any interaction the total cahrge
is conserved. For example:

n  p   e   e
0

starts with zero total charge ands with +1 –1 + 0 = 0 total charge.
   The particle thought of as mediating the electromagnetic force is the photon.
Charged particles affect each other‟s energy and momentum by exchanging
photons. Photons have energy given by hf, Planck‟s constant x frequency, and

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TOPICS IN PHYSICS 2006-07                                                  R Cherdack

momentum given by hf/c = h/. These must be taken into account when
determining the possibility of an interaction and its implications

Photons can result when charged particles accelerate and in particle interactions
or decays that result in a reduction in mass. Photons are neutral and therefore
cannot transmit charge from particle to another. They also have no rest mass
and so to exist, must travel at the speed of light. It is said that photons do not
interact with each other since they are not charged particles. [However, I am
unconvinced since light does exhibit interference which could be interpreted as
photon - photon interaction. I will try to sort this out before the year ends.]

Weak Force

The weak force is a force associated with the transformations known as decays
and some scattering reactions. The property of matter associated with the weak
force is not given a special identity like mass, charge, or color, but all leptons and
quarks have it.

For leptons, the interactions associated with weak force do not produce any
changes in generations, so the number the electron generation particles, and the
number of muon generation particles, and the number of tau generation particles
is each constant. Keep in mind that an anti-lepton carries a lepton number of –1
for its generation. This is why the neutrino associated with a lepton is considered
an anti-particle.

For quarks, the weak force can mediate interactions which lead to changes in
quark generations such as a strange quark becoming an anti-up and a down.

The mediating particles for the weak force are the W +, the W -, and the Z. See
your charts for the properties of these “virtual” particles. Because the W particles
are charged, quarks can exchange charges with other quarks so that an up can
be transformed to a down by absorbing a W - or emitting a W +.

Strong Force

The strong force is the force between quarks. It is associated with the color of
quarks and is closely related to the fact that no real particle can exist without
color neutrality.

The particle that is thought to mediate the strong force is the gluon. Gluons carry
color. It is massless and presumably travels at the speed of light. Gluons carry no
charge. It does carry colors and unlike the photon gluon gluon interactions are
possible.

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TOPICS IN PHYSICS 2006-07                                       R Cherdack

The strong force operates between quarks and does not cause changes in
quarks other than color.

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