Fundamental forces

In the Newtonian view of the universe, mysterious action at a distance is commonplace. Like electric charges
mysteriously repel, unlike charges attract, due to electrostatic or Coulomb forces. Masses attract one
another across space by means of gravity. Newton himself objected to the apparent absurdity of this
mysterious action at a distance.

In modern quantum physics this is not quite the case and these forces are instead modeled as being
mediated by the exchange of particles. A positively charged proton exchanges photons (particles or quanta
of electromagnetic energy / light) with a negatively charged electron, producing the
Coulomb force of
electrostatic attraction.

This theory of forces mediated by the exchange of particles was developed by Yukawa. Yukawa examined
the attractive
nuclear force between protons and neutrons in the atomic nucleus, which holds the nucleus
together. The neutron-proton interaction diminishes rapidly for distances greater than about 2 fm (fm =
femtometre, 1 fm = 10^-15 m) comparable to the nuclear radius. Let’s set the
range parameter, a = 2 fm.
Yukawa helped develop
quantum field theory which states that forces between particles are mediated by
the exchange of
virtual particles (virtual quanta).

These particles are virtual (said to be off mass shell) since they have an ephemeral existence in which their
energy manifests as momentum rather than as directly measurable mass – the particle scan not be directly
observed whilst in the virtual state, though at least some of these particles can also occur in non-virtual
states (said to be on mass shell) as ordinary particles. These particles are produced from nothing by the
force field and so would violate energy-conservation if they were detectable as ordinary particles, instead
they are allowed to temporarily violate energy conservation so long as this is within the
energy-time
uncertainty principle
. This principle states that the more precisely the energy of a system is measured
(measuring actually changes the system, so a more precise measurement forces the system to have a more
restricted value) the less precisely the time duration of that state is. Mathematically: In short this means that a system can borrow (i.e. create from nothing) a tiny amount of energy for a long
period of time, or a large amount of energy for a short period of time. This means that if a system creates a
highly energetic virtual particle, then that particle is shorter-lived, and since no such particle is assumed to
travel faster than light, this means  that the particle can not travel as far before it disappears and so the force
has a shorter range.

According to the theory of special relativity, the energy of a particle is given by:  Where: c = speed of light = 2.998 m/s, m = particle mass, p = particle momentum (a vector with both
magnitude and direction). For a stationary particle, at rest (p = 0), this gives the familiar equation: E = mc^2,
or m = E/c^2, which tells us that mass can be treated as a form of energy, with a tiny amount of mass being
equivalent to a massive amount of energy.

Yukawa estimated the mathematical form of the nuclear potential (the nuclear force-field), solved the
differential equation satisfied by waves (particles, see wave-particle duality) in this potential, specifically for
traveling or propagating waves, corresponding to the virtual particles in motion, and obtained the energy of
these waves as: Where: a is our range parameter.

Comparing this to the expression for relativistic energy (equation 2) we find that: For the range parameter, a. This gives us the mass or energy of the virtual particle (when stationary at rest)
that corresponds to a maximum range, a. For the nuclear force we have a = 2 fm, which gives us a mass
equivalent to an energy of about 100 MeV. In particle physics, the masses of particles are usually expressed
as energy equivalents in units of the electronvolt, eV. One eV is the energy gained by an electron when it is
accelerated by an electric force field or potential difference of one volt (V) and 1eV = 1.602×10^−19  joules (J)
and 1 mega-electronvolt, 1 MeV is one million eV. Note that the eV is a tiny amount of energy, but we are

So, Yukawa told us that we were looking for a particle with mass, m = 100 MeV, approximately, as the force
carrier for the nuclear force. In fact, it is now known that the pions with m ~ 100 MeV are the nuclear force-
carriers. Yukawa succeeded in making an astounding prediction that was correct!

Feynman Diagrams

(1) The Electromagnetic Interaction

Consider the diagrams below which illustrate the interaction of two electrons, repelling one-another by
exchanging a virtual particle, which is a photon for the electric 9or more properly electromagnetic) force:
Time travels from left to right in these diagrams. The electrons are deflected or scattered off one-another as
they move together, by exchanging a virtual photon, depicted as the wavy line. Particles, like the electron,
always have arrows pointing from left to right, since they travel forward in time. Mathematically, anti-particles
can be thought of as either as positive energy anti-particles travelling forwards in time, or as negative-energy
particles travelling backwards in time, and so have arrows in the reverse direction by convention and to
distinguish them. We have no anti-particles in the above diagram, but the anti-electron is the positron, which is
much like an electron but with positive charge and it’s symbol is e+. There are two versions of the diagram,
both given above, with different time-orderings, depending which of the two electrons emitted the particle.
Now, since the particle is virtual and either electron can emit it, or indeed both, then it is more accurate to
depict the virtual particle as a combination of the two possible virtual photons travelling in either direction.

This is depicted below: Such a diagram is sometimes described as non-time ordered, but we can still think of time travelling from left
to right, it’s just that the photon-exchange occurs in the middle and can proceed in either direction.
This is not the only possible interaction when electrons scatter off one-another. It is possible for two or more
photons to be exchanged, for example: This is an example of a second-order process, with two events happening, the former diagrams being first-
order processes. It is also possible to have third-order and higher-order processes. Fortunately for our
calculations, it turns out that the first-order processes dominate and the higher-order processes can usually
be ignored, though some of them produce important physical effects and they may become more important
at very high energies.

Any two (or more) particles with electric charge can interact via the electromagnetic force. Below, an electron
scatters off a muon, m– (a muon is like a type of less stable heavy electron): One consequence of this quantum field theory is that the force fields that give rise to long-range forces are
quantised – they are made up of distinct packets of energy or quanta, manifest as virtual particles. They
also occupy the whole of space though the number of force-carrying particles in the field in any region can
vary. Within even a ‘perfect’ vacuum, virtual particles can phase in and out of existence, so-called
quantum
fluctuations
, so long as they do not violate the energy-time uncertainty principle. When two non-charged
metal plates are placed very close to one-another in a vacuum, there is a force of attraction or repulsion
between them. This force is due to the appearance of virtual particles between the plates, virtual particles
that can then interact with the atoms in the plates. If the two plates are very close together, the force
becomes large. At a separation of 10 nm (10 millionths of a mm) the force becomes equivalent to about one
atmosphere of pressure (pressure is force per unit area). This force, due to the
vacuum energy is called
the
Casimir effect.

The
Lamb Shift is a modification to the energy of the electron in atoms when in certain orbitals, due to the
interactions between the electron and the vacuum, due to quantum fluctuations. The
Darwin term is a
correction to the s-orbital energy levels of hydrogen atoms and is due to the electron undergoing
jitter-
motion (zwitterbewegung)
and is again due to vacuum fluctuations. Specifically, electron-positron
pairs
are spontaneously formed in the vacuum, but they soon disappear again since the positron is
antimatter, it is an anti-electron, and so the electron and positron will soon collide and cancel one-another
out again. However, sometimes, the positron cancels with another ‘real’ electron, such as the electron
orbiting an atom, leaving the virtual electron behind which can now become real by replacing the lost
electron without energy-time violation. This causes the electron to apparently keep disappearing and then
reappearing somewhere else, because it is not really the same electron (though distinguishing one electron
from another is something as a fallacy anyway).

Quantum fluctuations also occur when one particle momentarily turns into two particles and then back into
one particle, without violating energy-time conservation. An electron constantly emits and reabsorbs
photons and electron-positron pairs (indirectly) by the following processes:  Screening

In the diagram below (in which we have averaged the time-ordering again) two electrons scatter off one
another (repel one-another) by exchanging a virtual photon, but the photon turns into an electron-positron
pair whilst in transit (this conserves the electric charge since a photon has no electric charge and an
electron-positron pair also have zero total charge). The electron and its anti-particle promptly annihilate to
produce a photon which is absorbed by the other electron. This is a higher-order process. Since electrons emit virtual photons all the time, the net effect is that an electron becomes surrounded by a
cloud or ‘sea’ of virtual electron-positron pairs and the positive charges of the positrons partially block or
screen the charge on the electron. This reduces the effective charge and effective electromagnetic force,
though this effect diminishes as one approaches closer to the electron, until the ‘naked’ electron is exposed
and the full force of its charge can be felt. One consequence of this screening is that the s electron in a
hydrogen atom experiences a stronger attractive force to the nucleus, since s electrons approach closer to
the nucleus than do p, d or f electrons and so they are more tightly bound because of screening.

Electron-positron Annihilation

We have already seen some consequences of electron-positron pair-production. Let us look more closely at
the process of electron-positron annihilation. This is the first-order annihilation process: Remember that the arrows do not show direction of travel as such, but simply point to the right for a particle
and to the left for an anti-particle. The electron and positron have opposite electric charges and so are
attracted towards one-another. As they come together and annihilate, they emit photons – all their energy
is totally converted into the photons. Note that these photons are real and detectable, and so the pair
annihilate in a burst of light.

Two photons may also be emitted in a first-order event: Two-jet events Sometimes the photon produced by annihilation converts into a quark-antiquark pair (electric charge is
conserved again) and these quarks then undergo
fragmentation, converting into a number of hadrons
Hadrons are particles made up of quarks or quarks and antiquarks and come in two
types:
mesons are made of a quark and an antiquark, baryons of three quarks. These jet events dominate
at high energies (in the centre of mass reference frame at energies of 15-40 GeV). Three-jet events have
also been observed, in which either the quark or antiquark emits a gluon, the gluon, quark and antiquark

Many other reactions are possible, such as the reaction (notice the conservation of electric charge): The diagram below shows the 8 basic vertex processes that can occur in interactions between an electron,
a positron and a photon. These vertices are only half-processes (virtual processes) since, in order to
prevent energy-time violation, any virtual particles produced must be annihilated at another vertex – at least
two connected vertices are required to make a real process.

The diagram below illustrates the 8 possible basic processes that may occur in positron and electron
interactions with a photon:  Quantum Electrodynamics (QED)

QED is the quantum field theory of the electromagnetic interaction – the interaction between electrically
charged particles that is mediated by virtual photons. We have looked at some qualitative results of this
theory, however, it is also a quantitative theory. The theory makes use of a set of equations which allow us
to calculate such things as the
reaction rates or decay rates for unstable particles (that decay by the
electromagnetic interaction) and reaction
cross-sections for scattering or collision processes, such as the
reaction between an electron and a proton, or the scattering of a beam of electrons from a target of protons
at lower energies. A cross-section gives us the likelihood of two particles colliding and interacting and is
measured in units called
barns (as in ‘couldn’t hit the side of a barn’!).

At higher energies other forces come into play with protons, since they are made-up of quarks and quarks
as well as participating in electromagnetic interactions, participate in reactions governed by other quantum
fields, in particular the strong nuclear force which is covered by a quantum field theory called
quantum
chromodynamics (QCD)
and whose interactions are mediated by the exchange of virtual gluons.

Fundamental Forces

The electromagnetic interaction is one example of an elementary or fundamental force. There are four such
known forces: electromagnetic, strong, weak and gravity. Each of these forces is mediated, in the quantum
field theory, by the exchange of virtual particles (all bosons with spin-1, called gauge bosons) as follows:

Electromagnetic: photon, field theory is QED
Strong: gluon, field theory is QCD
Weak: W+, W– and Z0 particles, field theory is the electroweak theory
Gravity: graviton?, theory is general relativity, no satisfactory quantum theory

Gravity is a bit of a problem and not well understood. Some have predicted that it is mediated by particles
called gravitons, that have yet to be discovered. We will come back to gravity later. Since particles are
generally tiny, and gravity is relatively very weak, it is nearly always negligible in particle interactions. Types of Elementary Particle

• Fermions are particles of half-integer spin (spin = quantised rotational angular momentum).
Fermions also obey a set of rules given by Fermi-Dirac statistics. Both compound and elementary
particles can be fermions or bosons.

• Bosons are particles of integer spin and obey Bose-Einstein statistics.

• All particles, whether elementary (i.e. undivisible) or compound (i.e. composed of two or more
elementary particles) are either fermions or bosons.

• Elementary particles or fundamental particles are those that  do not appear to be composed of
smaller particles. These include the leptons, the quarks and the gauge bosons.

• Elementary particles that are fermions are the leptons, quarks and their antimatter equivalents.

• Elementary particles that are bosons are all the gauge bosons. Gauge bosons are the particles that
mediate the elementary forces: the photon, the gluon, the W+, W–and Z0 bosons and possibly the
graviton. The gauge bosons all have a spin of 1.

The table below illustrates the fundamental forces that dominate different classes of fundamental particle (in
addition to gravity which affects all particles): The Standard Model

The elementary particles: the leptons, quarks and the gauge bosons together with the quantum field
theories that explain and predict their behaviour and interactions make-up a system called the standard
model. Whether or not this model includes all fundamental forces and all elementary particles remains an
open question, however, the model has so far stood the test of time and explained all or most experimental
results to date. Quarks

There are six principle quarks: up (u), down (d), strange (s), charmed (c), top (t) and bottom (b) and their six
antimatter equivalents. Each of these 12 varieties also exists in one of three states that corresponds to a novel
property similar in some ways to electric charge, but relevant to the strong force, and this is called the
colour or
colour charge. This is not a real colour as quarks are invisible and colourless, but it is used for convenience
and the three charges are
red (r), green (g) and blue (b). With 12 quarks/antiquarks in three possible
flavours, we have (12 x 3) 36 different quarks. The quarks are also grouped into three generations:            Quantum Electrodynamics (QED)  The Nuclear Force

So what about the nuclear force that binds protons and neutrons together, which we mentioned at the beginning of this
article? The nuclear force, not to be confused with the strong force, is not a fundamental force. Protons and neutrons
are not elementary particles, they are
one quark/antiquark. Baryons are made of three quarks, a proton is uud and a neutron is ddu. The force between these
particles is mediated by particle exchange, but not by exchanging elementary bosons, instead
pions are exchanged.
Pions are
mesons, a type of hadron composed of a quark and an antiquark. Pions are bosons with zero spin.
These pion exchanges can be represented by Feynman
diagrams, however, the diagrams here simplify the process. In
reality the pions exchange gluons by the strong force with a
nucleon at each vertex, so fundamentally, the nuclear force is
brought about by the strong force. The gluons are exchanged
between quarks in the pions and the nucleons. Note that there
are three pion types and charged pions change nucleon type,
the neutral pion does not 