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Elementary Particle Physics - Lecture 1
Elementary particle physics describes the smallest constituents of matter and the forces that act
on them. Initially elementary particles were synonymous with the constituents of the atom: the
electron, the proton and the neutron.
With time a large number of similar types of particles, all unstable, were discovered. The
particle zoo in the 1960s contained hundreds of different particles, which put the term elementary
in doubt. We now know that of the constitutents of the atom, only the electron is elementary,
nucleons are composed of yet smaller objects.
In the current Standard Model there are twelve fundamental matter-particles, grouped in three
families, and three types of forces acting on them. We will soon study these in more detail, but
we will start by establishing a framework to understand interactions between particles in more
general terms.
A free particle which propagates without interacting is not very interesting, to learn more
about elemetary particles we need to study how they interact. To get started we need a notation
for these interactions, and some rules for what is possible and not, that’s what this course is really
about.
We are going to follow two paths: First we will connect to what you have learned in QM courses
and describe some simple examples which makes the connection into the world of elementary
particles. The correct framework to describe elementary particles is relativistic quantum field
theories, which is beyond the scope of this course, but we can learn quite a bit by extending some
of the things you know already.
Having recapitulated some results from the Quantum Mechanics course and extended them to
apply to elementary particles we will introduce the technique of Feynman diagrams. We follow this
order since Feynman diagrams look so deceptively simple that one can miss the point that they are
in fact a stenographic way of visualising the underlying very complex computations represented
by these diagrams. Armed with the powerful technique of Feynman diagrams we will be able to
understand a large variety of phenomena.
Units
First a quick word about which units you will meet in this course. We start by energy. By
macroscopic standards the energies we will encounter are very small. A convenient basic unit
for energy is therefore the electronvolt (eV), which is the energy gained by an electron when it
is accelerated by a potential difference of 1 Volt, in SI units: 1 eV = 1.6 · 10−19 J. From the
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electronvolt we derive the units keV, MeV, GeV and TeV that are successively three orders of
magnitude larger than the preceeding unit.
You will find that in atomic physics the energy scale is usually in the eV range, in nuclear
physics keV to MeV and particle physics GeV to TeV.
Mass is usually expressed in units of energy divided by c2 (easy to remember from the famous
E = mc2 ). Masses range from less than 1 eV for neutrinos, through the order of MeV to GeV
for most elementary particles up to a few hundred GeV for the heaviest nuclei - and the heaviest
elementary particles. You will sometimes encounter the atomic mass unit, u, which is defined to
be exactly 1/12 of the mass of a free 12 C atom in its ground-state, or if you wish 931.5 Mev /c2
Momentum is expressed in units of energy divided by c.
Many text on particle physics use the so called “natural units”, where c = h̄ = 1 in which case
Energy, mass and momentum have the same unit. We will not use this convention in this course.
Interacting particles
We normally use a notation very much like for chemical reactions to indicate particle reactions,
for example the process of an electron type neutrino colliding with a neutron to create a proton
and an electron: νe + n → p + e−
There are two important facts about reactions between elementary particles as described in
this way:
1 - There are conserved quantities, that is entities which have to have the same value after the
interaction as before. We will later see that the different types of interactions in some cases have
different conservation rules. There are however some universal conserved quantities like energy,
momentum and electrical charge that are preserved by all known interactions.
2 - If you have written down one reaction which meets the conservation criteria you can
construct another one by moving particles from one side of the reaction to the other - provided
you change it to its anti-particle. Thus in our example above you can move the neutrino to the
right-hand side if you change it to an anti-neutrino: n → p + e− + ν¯e . (Here we introduce the
notation that a bar over a particle symbol denotes its antiparticle) that describes the decay of the
neutron to a proton, an electron and an anti-electron-neutrino, which is also an allowed process.
Interactions as exchanges of particles
In relativistic quantum filed theory, which is the relevant framework for exact calculations of elementary particles, interactions between particles are described as an exchange of a force-carrier
between them, so for instance the scattering process e+ +p → e+ +p where a positron (antiparticle
to the electron) scatters from the proton can be described as the positron being scattered by a
photon emitted by the proton.
e+
e+
γ
p
p
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Range of forces
Now let’s consider a more general case, were particles A and B scatter against each-other by
exchanging a particle X with non-zero mass.
A
A
X
B
B
Looking at the interaction between A and X in the rest-frame of particle A we find that energy
and momentum cannot be simultaneously conserved, which seems to indicate that this process is
not possible, and that one cannot describe forces as exchange of massive particles.
However from the uncertainty relation we have ∆t·∆E ≥ h̄. This means that for processes with
duration ∆t we cannot observe non-conservation of energy which is less than h̄/∆t. The violation
of conservation of energy, when particle A emits a particle X, which is shown to be δE ≥ MX c2 ,
is therefore unobservable for processes which fulfills ∆t < h̄/Mx c2 . Thus, if the X-particle is
absorbed within this time we will not be able to observe a violation of energy-conservation. This
sets a maximum distance the X-particle can travel, which in turn define the maximum range for
the force transmitted by X to be R = h̄/MX c.
Correspondence between scattering from potential and scattering through
particle exchange
In the Quantum Mechanics course you described scattering of particles as the result of interactions
with a potential. We now want to look at a simplified example where we can view scattering
from both perspectives and validate that these descriptions agree. We assume that particle A is
much heavier than particle B (otherwise we cannot really think of the potential as being timeindependent).
If the particle X, mediating the force, is spin-0 it obeys the static Klein-Gordon equation:
∇2 φ(x) =
M2X c4
φ(x)
h̄2
If MX = 0 we recognise this as the equation for the electrostatic potential. For two particles
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with charge e an -e we the get potential energy as V(r) = −eφ(r) = − 4π
0 r
For MX 6= 0 we can verify through insertion that a solution to the Klein-Gordon equation
g2 e−r/R
above is given by V(r) = − 4π
r , where R is the range as defined above, and g is a parameter determining the strength of the interaction - the coupling - equivalent to electric charge for
electromagnetism .
This potential - the Yukawa potential - is the potential experienced when two particles are
scatttered through the exchange of massive spinless particles.
Probabilities of interactions
The probability for an interaction can be computed through the following steps:
1 - Compute the transition amplitude or matrix element for the interaction.
2 - Square the amplitude
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3 - Multiply with phase space factors, essentially proportional to the density of avalable final states.
This procedure is called “Fermi’s Golden rule, often written as Γi→f =
2π
2
h̄ |hf |H|ii| ρ.
The tricky bit is to compute the transition amplitude hf |H|ii. We will not
R derive how this is
done here, but in the wave-function formalism this is essentially the integral Ψ∗f HΨi dV or if - as
here - we are considering the scattering from Rone plane wave to another through the interaction
with a potential V(r) this becomes M (q) = d3 r V(r) exp(iq · r/h̄) where q is the momentum
transfer: q = qf - qi .
2 2
h̄
If the scattering potential is the Yukawa potential this integral can be evaluated to M (q 2 ) = q2−g
2 c2 .
+MX
This is a special case - remember that we derived it assuming that A is stationary - in which case
the absolute value of q doesn’t change since this is an example of elastic scattering. In the more
general case, where we allow B to recoil, the complete calculation is a bit more complicated,
g2 h̄2
2
so we just quote the result: M (q 2 ) = q2 −M
denotes the four-momentum transfer:
2 c2 where q
X
q2 = (Ef − Ei )2 /c2 − (qf − qi )2
This expression for the amplitude is derived under some specific assumptions: both particles couple with similar strength to the exchange-particle, the exchange particle is spin-0, the
interaction is described as the exchange of only one particle and so on.
Theoretical physicists made an enormous effort to use relativistic quantum field theory to
compute similar expressions for cases where these assumptions were not fulfilled. These expressions
share some common features, like the coupling strength entering once for each particle-mediator
interaction that takes place, and a “propagator term” inversely proportional to the difference
between the square of the momentum transfer and the square of the mediator mass.
Next lecture we will introduce Feynman graphs, which is a simple set of rules to construct
graphical representations of particle interactions. The rules for how these can be made ensures
that we can only draw graphs for processes that are possible. In addition the rules establish a oneto-one correspondence between the graphs and the underlying calculations for the computation of
the amplitude for a process.
Cross-sections
imagine that we aim a beam of highly energetic protons towards a target also consisting of protons
(can be realised as a tank of liquid hydrogen). If the target is thin, a large number of protons in
the incoming beam will just pass through the target, while a few will react with target particles.
The rate W, interactions per unit time, is then proportional to the number N of target-particles
per unit area as viewed by the beam and the flux J, the number of beam-particles per unit area
and unit time that enter the target: W = J · N · σ. The constant of proportionality σ is called
cross-section and is specific for each given process. We define a new quantity, Luminosity, L for
the producd J · N, such that the physical obaservble - the interaction rate - is written W = L · σ.
In this expression L encodes the knowledge about the experimental setup, while the crosssection, σ, holds the knowledge about the underlying interaction that takes place. Evaluation of
σ always rely on computation of the transition amplitude, which often becomes a complicated
sum of terms when spin-effects are taken into account. We will not enter the details of these
computations, but it is important to understand how the physically measurable quantities relate
to the transition amplitude.
Summmary
The salient points of this lecture are:
• Interactions between particles can be described as either one of the particles interacting with
a potential which source is the other particle or as an exchange of a mediating force-particle
between the two particles.
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• If the exchange-particle has non-zero mass the range of the force is limited to R =
h̄
mX ·c
• The probability for an interaction to take place is proportional to M 2 ρ where M2 is the
square of the matrix element, or transition amplitude, and ρ is the so called phase-space
factor, proportional to the number of available final states.
• The form of the matrix elements depends on the details of the process studied, but can in
general be computed in the framework of relativistic quantum field theories.
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