QCD (Quantum Chromodynamics) |

certain). The theory describing the theory of the electromagnetic quantum field is called

electrodynamics (

(QCD)

In electromagnetic interactions, positive and negative electric charges (and magnetic charges – electricity and

magnetism are two forces that have been unified into electromagnetism, since they describe the same

phenomenon but in different reference frames according to special relativity). In the strong interaction, a

different type of charge is involved, not the electric charge. Whereas in electricity there are two types or

flavours of

referred to as

green and blue.

Colour charge is not to be confused with electric charge – photons mediate electric charge and gluons mediate

colour charge – they are separate phenomena. Colour charge also has nothing to do with real colours – the

designation of colours is just a conveneient nomenclature.

Quarks interact via the strong force, though they also experience the electromagnetic force since they have

both colour and electric charges. However, the strong force is many times stronger than the electromagnetic

force and is a short range force and so dominates nuclear interactions (though the electric charge on protons,

which is due to the electric charges on their constituent quarks are extremely important on the atomic and

molecular scales).

in much the same way that in QED two electrons may scatter off one-another by photon exchange. One key

difference between QCD and QED is that whereas in QED the photon carries no electric charge, in QCD the

interesting results.

A red quark (say an up quark: u,r) may scatter off a blue quark (say a strange quark: s,b) by gluon-exchange.

Now, because the gluon carries away colour charge with it, the colour charge of the quarks involved can be

changed. This is shown in the diagram below. Also shown is

because they carry colour charge, can scatter off one-another by gluon exchange!

As in QED, higher-order interactions are possible, though these become increasingly rarer at lower energies

as more and more complications are added. The two lowest order**vacuum polarisation corrections** are

shown below. These are so-called as they involve particle-antiparticle pair production and annihilation

occurring in the virtual state (remember that a vacuum becomes polarised as such particle-antiparticle pairs

are continuously being produced, and annihilated).

In one of these corrections, the virtual gluon transforms into a virtual quark-antiquark pair which annihilate

back into a virtual gluon. In the second, a gluon transforms into a gluon-gluon pair.

as more and more complications are added. The two lowest order

shown below. These are so-called as they involve particle-antiparticle pair production and annihilation

occurring in the virtual state (remember that a vacuum becomes polarised as such particle-antiparticle pairs

are continuously being produced, and annihilated).

In one of these corrections, the virtual gluon transforms into a virtual quark-antiquark pair which annihilate

back into a virtual gluon. In the second, a gluon transforms into a gluon-gluon pair.

Quarks are spin-½ fermions and each quark can have one of three colour charges: r, g or b. Different types

of quarks have different electric charges:

of quarks have different electric charges:

only exist as a discreet entity in a state of zero colour (or we can say the colour ‘white’). This state can be

achieved in one of three ways: by combining together three particles, one red, one green and one blue; by

combining three antiparticles, one anti-red, one anti-blue and one anti-green; or by combining a coloured particle

with a particle carrying its anticolour, e.g. a quark-antiquark pair in which the quark is red and the antiquark anti-

red. (AS colour and its anti-colour cancel out). This is why, at all but extremely high energies at least, a quark, or

a gluon, can not be seen in isolation – quarks only occur either as hadrons (three quarks, e.g. the proton (uud))

or as mesons (one quark and an antiquark).

Colour confinement is also the reason why the

be massless, the range of the force should be infinite. However, it is analogous to the electrical case – most

matter is approximately electrical neutral and so on the large scale no net electrical force exists between most

objects as the positive charges are cancelled out by an equal number of negative charges. Similarly, as nucleons

are overall colour neutral, they do not see one-another’s colour charge and the strong force does not extend

between nucleons (except via pion exchange; pions are a type of meson).

The higher-order vacuum polarisation effects already mentioned have interesting effects. On even smaller scales

than the large-scale cancellation of charges which weakens the effective electromagnetic force, QED predicts

that electric force will decline with distance, as it does, from a particle like an electron, due to screening of the

electron’s charge by some of the virtual photons it emits turning into virtual electron-positron pairs, which are

charged and so align to block out or cancel some of the effective charge seen at greater distances – an effect

called

more than countered by an effect present in the strong force, but not electromagnetism, because the gluons

carry colour charge and can also form gluon pairs. This latter effect is called

the strong force to grow stronger at longer ranges! This is counter-intuitive to our everyday experiences (with

gravity and elctromagnetism).

To understand this effect fully one has to appreciate a key difference between electromagnetic and strong

force fields. These fields are illustrated below (the field lines show the lines of force):

force fields. These fields are illustrated below (the field lines show the lines of force):

Above: the electromagnetic field between two electric

charges (extending from the positive charge to the

negative charge). The force is conveyed by virtual

photons which do not interact with one-another

charges (extending from the positive charge to the

negative charge). The force is conveyed by virtual

photons which do not interact with one-another

Above: the field strong between an antiquark and a

Above: the field strong between an antiquark and a

quark. The force is conveyed by gluons which also

interact with one-another by strong-field interactions. This

gluon-gluon interaction pulls the field-lines closer together

between the quarks. If the distance between the quarks is

increased, by pulling them apart, the force of the

gluon-gluon interaction increases, pulling the lines even

closer together. The closer together the field-lines, the

stronger the force. Thus, the force increases with

separation.

Above: the field strong between an antiquark and a

quark. The force is conveyed by gluons which also

interact with one-another by strong-field interactions. This

gluon-gluon interaction pulls the field-lines closer together

between the quarks. If the distance between the quarks is

increased, by pulling them apart, the force of the

gluon-gluon interaction increases, pulling the lines even

closer together. The closer together the field-lines, the

stronger the force. Thus, the force increases with

separation.

Flavour (u,d,s, etc.) can be described by three quantum numbers, isospin (I), the third component of isospin

colour hypercharge (YC).

__Spin__

In our macroscopic world, a particle, like the Earth, can have intrinsic angular momentum by virtue of the fact it

is rotating or spinning on its axis. Such a particle may also have orbital angular momentum, by virtue of the

fact it moves in an orbit, as when the Earth orbits the Sun. Both these types of angular momentum are

important in physics – an electron has spin and it may have orbital angular momentum if it is confined in an

atom. However, the analogy is a crude one, since subatomic particles like electrons don’t follow well-defined

trajectories (not unless one invokes so-called ‘hidden variables’ for which there is currently no evidence of)

and an electron does not orbit the nucleus of an atom in any classical sense, it merely has a property that is

the quantum mechanical analogy of angular momentum. Here we focus on spin, as we are considering

separate non-composite particles and spin is an intrinsic value of particles.

In quantum mechanics angular momentum is quantised (meaning only certain values are possible for a given

system). An electron has only two possible spin values. A helpful (though not strictly correct) analogy is to

imagine these as clockwise or anticlockwise or up and down. Clearly, such statements as up and down or

clockwise and anticlockwise only have meaning if we define a direction. This direction is arbitrary, since the

electrons could be in any orientation (although a sample can be prepared to be all in the same orientation).

By convention we use the z-axis. The two possible values of spin for an electron, relative to the z-axis, are

found to be +½ħ and -½ħ, where ħ is Planck’s constant divided by 2p. These directional spin values are

designated by the quantum number Sz. Planck’s constant turns out to be a fundamental unit in quantum

mechanics. Ignoring the units (given by ħ) we can say that the spin quantum number, ms, for an electron can

have one of the two possible values of +½ or -½ (written together as ±½). Ignoring the sign we can say that

the spin of the electron, s, is ½.

colour hypercharge (YC).

is rotating or spinning on its axis. Such a particle may also have orbital angular momentum, by virtue of the

fact it moves in an orbit, as when the Earth orbits the Sun. Both these types of angular momentum are

important in physics – an electron has spin and it may have orbital angular momentum if it is confined in an

atom. However, the analogy is a crude one, since subatomic particles like electrons don’t follow well-defined

trajectories (not unless one invokes so-called ‘hidden variables’ for which there is currently no evidence of)

and an electron does not orbit the nucleus of an atom in any classical sense, it merely has a property that is

the quantum mechanical analogy of angular momentum. Here we focus on spin, as we are considering

separate non-composite particles and spin is an intrinsic value of particles.

In quantum mechanics angular momentum is quantised (meaning only certain values are possible for a given

system). An electron has only two possible spin values. A helpful (though not strictly correct) analogy is to

imagine these as clockwise or anticlockwise or up and down. Clearly, such statements as up and down or

clockwise and anticlockwise only have meaning if we define a direction. This direction is arbitrary, since the

electrons could be in any orientation (although a sample can be prepared to be all in the same orientation).

By convention we use the z-axis. The two possible values of spin for an electron, relative to the z-axis, are

found to be +½ħ and -½ħ, where ħ is Planck’s constant divided by 2p. These directional spin values are

designated by the quantum number Sz. Planck’s constant turns out to be a fundamental unit in quantum

mechanics. Ignoring the units (given by ħ) we can say that the spin quantum number, ms, for an electron can

have one of the two possible values of +½ or -½ (written together as ±½). Ignoring the sign we can say that

the spin of the electron, s, is ½.

and protons are very similar particles with the following properties:

They have the same spin and parity and almost identical masses, but differ in electric charge. These

properties are due to their quark constituents – d and u quarks are similar except for their differing electric

charges. This suggests we have a certain symmetry between the u and d quarks and between the proton

and neutron, but this symmetry is only approximate. If the u and d quarks had identical masses and the

forces acting upon each are identical, then the masses of the proton and neutron would be identical. In fact

the d quark is slightly heavier than the u quark and although the strong forces acting on each are identical,

they have different electric charges and so the electromagnetic forces acting on them differ. Nevertheless,

this approximate symmetry, which is called**isospin symmetry** (as the particles in each group have identical

spins), is useful in classifying particles.

A particle’s isospin is described by two quantum numbers:

properties are due to their quark constituents – d and u quarks are similar except for their differing electric

charges. This suggests we have a certain symmetry between the u and d quarks and between the proton

and neutron, but this symmetry is only approximate. If the u and d quarks had identical masses and the

forces acting upon each are identical, then the masses of the proton and neutron would be identical. In fact

the d quark is slightly heavier than the u quark and although the strong forces acting on each are identical,

they have different electric charges and so the electromagnetic forces acting on them differ. Nevertheless,

this approximate symmetry, which is called

spins), is useful in classifying particles.

A particle’s isospin is described by two quantum numbers:

The use of the third component means that we are defining an isospin axis, similar to what was done for

spin, except we are talking about a mathematical isospin space not actual space.

First some definitions of particle classes:

**Hadron**: a particle made up of quarks bound together by the strong force.

**Baryon**: a type of hadron made up of three quarks (3q) bound together by the strong force, e.g. proton, p

(uud, i.e. 2 u quarks and 1 d quark). An antibaryon is a particle made up of three antiquarks bound together

by the strong force, e.g. antiproton (anti-up, anti-up, anti-down).

**Meson**: a type of hadron made of a quark and an anti-quark bound together by the strong force, e.g.

positive pion, p+ ( up, anti-down).

To obtain the spin of a composite particle, we have to add together the spins of its substituents, including

their possible signs. For example, a pion has spin zero (+½ add –½ = 0) as the spins couple in such a way

that one quark/anti-quark has spin-up and the other spin-down. The spin of a proton or of a neutron, both

baryons made of 3q, the total spin has a magnituide of ½ (+½ add –½ add ±½ = ½, since the direction of

spin of a single particle is arbitrary).

There is a group of 8 baryons, formed only from u, d and s quarks, with spin = ½, forming an**isospin group**

called an**octet** (a **multiplet** of 8) as shown below:

spin, except we are talking about a mathematical isospin space not actual space.

First some definitions of particle classes:

(uud, i.e. 2 u quarks and 1 d quark). An antibaryon is a particle made up of three antiquarks bound together

by the strong force, e.g. antiproton (anti-up, anti-up, anti-down).

positive pion, p+ ( up, anti-down).

To obtain the spin of a composite particle, we have to add together the spins of its substituents, including

their possible signs. For example, a pion has spin zero (+½ add –½ = 0) as the spins couple in such a way

that one quark/anti-quark has spin-up and the other spin-down. The spin of a proton or of a neutron, both

baryons made of 3q, the total spin has a magnituide of ½ (+½ add –½ add ±½ = ½, since the direction of

spin of a single particle is arbitrary).

There is a group of 8 baryons, formed only from u, d and s quarks, with spin = ½, forming an

called an

This is one of a number of **multiplets** and they are plotted on a graph of hypercharge versus the third

component of isospin; so, for example, the proton and neutron both have a hypercharge of +1, but the

proton has a third component of isospin = +½, the neutron of –½. Thus, although all the particles in this

octet have the same spin, using the isospin quantum numbers we can label each particle uniquely,

except that two particles in the centre of the diagram have the same isospin quantum numbers (both

zero): the neutral sigma particle and the neutral lambda particle. Notice that these particles have the

same quark compositions, but they are different particles, so there must be another quantum number to

distinguish these states.

component of isospin; so, for example, the proton and neutron both have a hypercharge of +1, but the

proton has a third component of isospin = +½, the neutron of –½. Thus, although all the particles in this

octet have the same spin, using the isospin quantum numbers we can label each particle uniquely,

except that two particles in the centre of the diagram have the same isospin quantum numbers (both

zero): the neutral sigma particle and the neutral lambda particle. Notice that these particles have the

same quark compositions, but they are different particles, so there must be another quantum number to

distinguish these states.

Now it can be seen that the neutral sigma and neutral lambda particles, although both uds, are distinctly separate

quantum states, as they have different isospins.

Also included in the above table are the mass (in units of MeV/c2) electric charge (Q), the spin, isospin values for

all three isospin quantum numbers, strangeness (S) which is -1 multiplied by the number of strange quarks and the

parity (P). It should also be noted that of all these particles, only the proton is stable in the isolated state. Free

neutrons decay with a half-life of about 925 seconds, but are much more stable inside most nuclei. The other

particles are extremely unstable, lasting on average for less than a nanosecond! This explains why matter contains

primarily protons and neutrons. The protons and neutrons, or nucleons, are also the lightest two members of this

family and it is generally the case that in a family of particles, the lighter members are the more stable.

Now we said that isospin is a measure of flavour. Leptons, like the electron, are particles that are not made of

quarks and so have no flavour and no values for isospin (not even zero). [The photon is an oddity as it can exist in

two states, one with isospin, I = 0, the other with isospin, I = 1.] We would expect, therefore, that isospin is due to

the quarks making up the hadrons. Indeed, the quarks have the following isospin properties:

quantum states, as they have different isospins.

Also included in the above table are the mass (in units of MeV/c2) electric charge (Q), the spin, isospin values for

all three isospin quantum numbers, strangeness (S) which is -1 multiplied by the number of strange quarks and the

parity (P). It should also be noted that of all these particles, only the proton is stable in the isolated state. Free

neutrons decay with a half-life of about 925 seconds, but are much more stable inside most nuclei. The other

particles are extremely unstable, lasting on average for less than a nanosecond! This explains why matter contains

primarily protons and neutrons. The protons and neutrons, or nucleons, are also the lightest two members of this

family and it is generally the case that in a family of particles, the lighter members are the more stable.

Now we said that isospin is a measure of flavour. Leptons, like the electron, are particles that are not made of

quarks and so have no flavour and no values for isospin (not even zero). [The photon is an oddity as it can exist in

two states, one with isospin, I = 0, the other with isospin, I = 1.] We would expect, therefore, that isospin is due to

the quarks making up the hadrons. Indeed, the quarks have the following isospin properties:

Let’s check some of these values using the formulas for Y and I3 given in the box above:

1) For the d quark: Y = ⅓ + 0 + 0 +0 + 0 = ⅓; I3 = Q – Y/2 = –⅓ - 1/6 = –2/6 – 1/6 = –3/6 = –½

2) For the u quark: Y = Y = ⅓ + 0 + 0 +0 + 0 = ⅓; I3 = Q – Y/2 = ⅔ - 1/6 = ½.

These quarks can be grouped into 3 pairs, called generations. The first generation contains the d and u

quarks (the two most commonly encountered in matter under normal conditions). For an isospin, I = ½, there

are only two possible values of I3, –½ and ½, corresponding to the d quark and the u quark respectively.

Again notice that the most commonly encountered quarks are the lightest.

1) For the d quark: Y = ⅓ + 0 + 0 +0 + 0 = ⅓; I3 = Q – Y/2 = –⅓ - 1/6 = –2/6 – 1/6 = –3/6 = –½

2) For the u quark: Y = Y = ⅓ + 0 + 0 +0 + 0 = ⅓; I3 = Q – Y/2 = ⅔ - 1/6 = ½.

These quarks can be grouped into 3 pairs, called generations. The first generation contains the d and u

quarks (the two most commonly encountered in matter under normal conditions). For an isospin, I = ½, there

are only two possible values of I3, –½ and ½, corresponding to the d quark and the u quark respectively.

Again notice that the most commonly encountered quarks are the lightest.

However, the first three are incorrect, though the latter six do occur. There are eight gluon types in total and

the other two are:

the other two are:

This might seem odd at first, however, there are good mathematical reasons for this choice. First of all recall

that we are constructing a mathematical model of a reality – as long as the maths works that is all that

matters. In quantum mechanics particles can exist in a mixture of states, such as when an electron in an

atom is in a mixed s/p state, until the particle is measured and then it collapses into just one of the possible

states (s or p). To form all the necessary mixtures we need the basis states. Think of a point in 3D space, we

can define this point using Cartesian (x,y,z) axes, for which we require 3 values (other systems of forbidden

due to the technical requirement for the wavefunction to be asymmetric) – not spatial dimensions in the real

sense, simply 9 mathematical dimensions or degrees of freedom. However, we can not use the nine we

assumed at first, because the nine basis states must be orthogonal. To achieve orthogonality we have to

use the two more complex mixtures given above. With these 8 we can then define any combination of colour

and anticolour.

Intuitively, considering say the red/anti-red state, one could ask whether or not the order in which the

colour/anti-colour occurs is important – we might expect the mixing of these same-colour states to be more

complicated.

The gluon is predicted to be massless and has no electric charge and a spin of 1.

__Examples of the Strong Interaction__

__1. Collision of an electron beam with a positron beam:__

that we are constructing a mathematical model of a reality – as long as the maths works that is all that

matters. In quantum mechanics particles can exist in a mixture of states, such as when an electron in an

atom is in a mixed s/p state, until the particle is measured and then it collapses into just one of the possible

states (s or p). To form all the necessary mixtures we need the basis states. Think of a point in 3D space, we

can define this point using Cartesian (x,y,z) axes, for which we require 3 values (other systems of forbidden

due to the technical requirement for the wavefunction to be asymmetric) – not spatial dimensions in the real

sense, simply 9 mathematical dimensions or degrees of freedom. However, we can not use the nine we

assumed at first, because the nine basis states must be orthogonal. To achieve orthogonality we have to

use the two more complex mixtures given above. With these 8 we can then define any combination of colour

and anticolour.

Intuitively, considering say the red/anti-red state, one could ask whether or not the order in which the

colour/anti-colour occurs is important – we might expect the mixing of these same-colour states to be more

complicated.

The gluon is predicted to be massless and has no electric charge and a spin of 1.

Above: a positron and electron annihilate one-another, producing a high-energy photon which decays into a

quark/antiquark pair. The quark and antiquark may bind together to form a meson, but usually each will

decay into a jet of hadrons of varied types. It is hard to tell which jet was produced by the quark and which by

the anti-quark. From the angular distribution of these jets, it can be deduced that the quark has spin ½.

quark/antiquark pair. The quark and antiquark may bind together to form a meson, but usually each will

decay into a jet of hadrons of varied types. It is hard to tell which jet was produced by the quark and which by

the anti-quark. From the angular distribution of these jets, it can be deduced that the quark has spin ½.

At lower energies, certain resonances occur, such as the one above which leads to the production of the very

short-lived neutral rho meson. The composition of the rho meson is:

short-lived neutral rho meson. The composition of the rho meson is:

The mechanism of this decay can be detected by processes like that shown below: a u quark in the rho

meson has emitted a gluon, which decays into a down and an anti-down quark. By colour confinement, these

quarks assemble into the two pions.

meson has emitted a gluon, which decays into a down and an anti-down quark. By colour confinement, these

quarks assemble into the two pions.

At low and intermediate energies, resonances also occur which decay much more slowly, by a suppressed

strong decay. An example is shown below: a J/psi meson is produced from the high-energy photon. This

meson consists of a charm and anti-charm quark (it is the first excited state of charmonium). There are no

convenient mesons for this particle to decay directly into by a strong reaction, so it takes a more

roundabout route – it emits 3 gluons, each of which decays into a quark/antiquark pair and the pairs

arrange to form the pion triplet:

strong decay. An example is shown below: a J/psi meson is produced from the high-energy photon. This

meson consists of a charm and anti-charm quark (it is the first excited state of charmonium). There are no

convenient mesons for this particle to decay directly into by a strong reaction, so it takes a more

roundabout route – it emits 3 gluons, each of which decays into a quark/antiquark pair and the pairs

arrange to form the pion triplet:

The suppression of this delayed strong force decay is called **Zweig suppression**. A process is Zweig

suppressed if it requires several gluons to do it.

suppressed if it requires several gluons to do it.

of another proton:

Here the d and u quarks scatter by gluon exchange, being knocked out from their protons, and since quarks

can not exist as individual particles (at least not under these conditions) the two quarks decay into hadron jets.

Some two-jet events may be due to other events, for example involving more gluons.

QCD is a mathematical theory and calculations can be performed to calculate the cross-sections for such

collisions – that is the likelihood of each possible event. The mathematics is very involved, however, and

beyond the scope of this article.

can not exist as individual particles (at least not under these conditions) the two quarks decay into hadron jets.

Some two-jet events may be due to other events, for example involving more gluons.

QCD is a mathematical theory and calculations can be performed to calculate the cross-sections for such

collisions – that is the likelihood of each possible event. The mathematics is very involved, however, and

beyond the scope of this article.

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