This is part 2 of a primer on the standard model. If you missed part 1, you can find it here.
The eightfold way
Let’s begin with the mesons (pions and kaons). They have charges of +1, 0 or -1, and strangenesses of +1, 0 or -1. We can draw a chart with strangeness on the vertical axis, against charge, and place each meson at its relevant position on the diagram. Usually the charge axis is drawn slanted and not horizontal. You could have the axes at right angles and you wouldn’t lose any information; you just wouldn’t make a pretty hexagon…
Note the inclusion of the eta ([latex]\eta[/latex]) meson for completeness – the [latex]\eta[/latex] had not been discovered at the time of Gell-Mann’s discovery of this pattern.
Baryons seem to be split into two groups, with one containing more massive particles than the other. If we plot a similar chart of the lighter baryons [latex](p, n, \Lambda, \Sigma, \Xi)[/latex], we get the pattern shown below.
The same hexagonal structure emerges with two particles in the centre of the hexagon instead of just one. This led to a search for an extra meson, and in 1961 the eta meson ([latex]\eta[/latex]) was discovered.
The similarity of patterns for mesons and baryons hints that there must be a deep underlying connection between them. It must have been pretty spooky to discover this!
As you might expect, the heavier baryons can be plotted on a similar diagram. This time something different happens. The hexagonal pattern emerges but with particles at the corners (let’s call it a triangle…!). The diagram below shows the lighter baryons greyed out, to show the correspondence with the heavier baryons.
When these three patterns were discovered they were called the eightfold path or eightfold way (after a Buddhist text). The patterns immediately begged an important question. If these particles are so fundamental (i.e. indivisible), WHY are they so obviously closely related?
A clue comes from Chemistry.
The Periodic Table in Chemistry shows that the atoms of the different elements are very closely related. The reason for this is that they are ‘made’ of the same constituents (protons, neutrons and electrons).
The eightfold way patterns are the Periodic Table of particle physics. The hadrons (baryons – including protons and neutrons – and mesons) are related because they are not fundamental – they are comprised of more basic units. Those basic units are called quarks.
The quark theory explains all the eightfold way patterns.
Quarks
In 1964 Gell-Mann and George Zweig independently noticed that the eightfold way patterns would arise automatically if hadrons (mesons and baryons) were made up of just three versions of a new particle, later named quarks. The three different flavours of quark (physicists sometimes really don’t make life easier with their choice of words – ‘flavours’ means versions) are called
- up ([latex]u[/latex]),
- down ([latex]d[/latex]), and
- strange ([latex]s[/latex]).
As well as these flavours, there are corresponding antiquarks (anti-up, anti-down and anti-strange).
The picture below shows the charges and masses of the up, down and strange quarks.
Antiquarks, written with a bar above the symbol (so [latex]\bar{u}[/latex], [latex]\bar{d}[/latex], [latex]\bar{s}[/latex]), have opposite charges to, and the same mass as, their corresponding quark. Notice the fractional charges! No other particles are known whose charges are fractions – that is, not whole number multiples of the electronic charge [latex]e[/latex]. All quarks and antiquarks have spin=1/2 (so they are fermions). Quarks are given a baryon number of +1/3 and antiquarks have a baryon number of -1/3.
Only the strange quark has ‘strangeness’ (value -1, and the anti-strange has a strangeness of 1). A particle is thus a strange particle if it contains 1 or more [latex]s[/latex] or [latex]\bar{s}[/latex] quarks – see the eightfold way patterns to check that non-zero strangeness corresponds to ‘having strange quark(s)’.
Building hadrons
There are two main ways of building a real particle from a cluster of quarks:
(a) quark plus anti-quark
These are the mesons, that is, the hadrons with integer spin (0 or 1). How do we know that a [latex]q\bar{q}[/latex] cluster is a meson? Well it follows from the spin of the quarks themselves. Each quark (and antiquark) has spin = 1/2, and two quarks can have their spins lined up in two ways:
- Spins parallel: ↑↑, overall spin = ½+½=1
- Spins antiparallel: ↑↓, overall spin = ½-½=0
Thus a [latex]q\bar{q}[/latex] cluster must have spin 0 or 1 and is a meson.
For example the cluster [latex]u\bar{d}[/latex] has charge 2/3 + 1/3 = +1 and has zero strangeness (because it does not have a strange quark). Looking at the eightfold way for mesons we can see that [latex]u\bar{d}[/latex] must be a [latex]\pi^+[/latex].
The different [latex]q\bar{q}[/latex] clusters possible, using the quarks [latex]u[/latex], [latex]d[/latex], and [latex]s[/latex], give the mesons in the eightfold way diagram!
(b) three quarks
These are the baryons (hadrons with half-integer spin). Why are these quark clusters baryons? Well, because the quarks may have spin alignments:
- ↑↑↓, overall spin=½+½-½=1/2
- ↑↑↑, overall spin=½+½+½=3/2
Thus a [latex]qqq[/latex] cluster must have spin 1/2 or 3/2 and is a baryon.
For example, the cluster [latex]uud[/latex] has charge 2/3+2/3-1/3=+1 and zero strangeness so it is a proton! Likewise, [latex]udd[/latex] has charge 2/3-1/3-1/3=0 and zero strangeness so it is a neutron! (Check the eightfold way pattern for baryons to see where this comes from.)
Let’s stop to think about that. We said before that you really are made of protons, neutrons and electrons. And that remains true. But the protons and neutrons are themselves composed of [latex]u[/latex] and [latex]d[/latex] quarks, and therefore so are you!
The electrons you are made of are not, so far as we know, composed of subunits in this way. They appear to be truly fundamental. Electrons are leptons, and do not feel the strong interaction. Those particles that do experience the strong interaction (the hadrons) are the particles made of quarks…
This means that the whole of the material universe (atoms, molecules, people, buildings, planets, stars etc) is made from leptons and up and down quarks and nothing else! Strange particles are not just strange – they are very rare and have very short lifetimes.
As we have already seen, baryons can exist with:
- spin=3/2 (quark spins parallel ↑↑↑), or
- spin=1/2 (quark spins antiparallel ↑↑↓).
Spin parallel is a higher energy state than antiparallel so spin 3/2 baryons are heavier than spin 1/2 baryons (remember the equivalence of energy and mass: [latex]E=mc^2[/latex]).
We can now understand why there are two separate eight-fold way patterns for baryons:
- the light baryons have spin=1/2
- the heavy baryons have identical quark structures but are in spin parallel configuration and therefore have greater energy and hence greater mass. We say that the heavy baryons are excited states of the light baryons. In the eightfold way for baryons, the orange particles are the excited states of the grey (lighter) baryons.
For example the proton has the same quark composition as the [latex]\Delta^+[/latex] baryon, but their spins are different. The proton is much more stable because it is in a lower energy state due to having spins antiparallel. In fact, the proton is the only stable baryon, because it has the least energy. Even neutrons will decay given the chance, which is exactly what happens in the type of radioactivity called beta decay.
Beyond [latex]u[/latex], [latex]d[/latex] and [latex]s[/latex]
Earlier in this article we talked of ‘over 100’ new particles, but so far we have not accounted for that many. Why? Because so far we have only ‘built’ hadrons using [latex]u[/latex], [latex]d[/latex], and [latex]s[/latex] quarks. In Gell-Mann’s time, that was all that was required to explain the particles known.
But after the quark model was introduced, particle physicists smashed particles together with ever-increasing energy (that’s how they roll – the physicists, that is), and this produced more and more particles with greater and greater mass.
All the combinations of [latex]u[/latex], [latex]d[/latex] and [latex]s[/latex] had been exhausted. But it turns out that there are three other types of quark (plus their anti-quarks). They are called, in order of increasing mass/energy and therefore order of discovery, charm ([latex]c[/latex]), bottom ([latex]b[/latex]) and top ([latex]t[/latex]). Think of the number of baryon and meson combinations you can make if allowed to use these too!
For example, the [latex]J/\Psi[/latex] meson, that was discovered in 1974, provided evidence of the charm quark. And a few years later, the bottom quark was found via the discovery of the upsilon meson. You can find these particles in the expanded timeline below, along with a few others that we haven’t mentioned.
Quarks and leptons are related…
So there are six quarks (plus anti-quarks), ‘arranged’ in related pairs called generations. The first generation are the up and down quarks – and are therefore what everyday stuff is made of. Mass increases as we go ‘up the generations’.
It turns out that there are also six leptons, as shown in green in the timeline above. Remember that leptons are fundamental – not composite – particles that do not experience the strong interaction. They are also arranged in three generations. This suggests a deep connection. The hadrons are connected to each other because they are all made of quarks, but leptons are fundamental – so their connection to hadrons is more mysterious. There are symmetries at play that are well-explained in many books, but which are too complicated to be explained here.
The three ‘generations’ of lepton are:
- the electron [latex]e[/latex] and its neutrino [latex]\nu_e[/latex]
- the muon [latex]\mu[/latex] and its neutrino [latex]\nu_{\mu}[/latex]
- the tau [latex]\tau[/latex] and its neutrino [latex]\nu_{\tau}[/latex]
The neutrino is a particle with no charge and much less mass than an electron. Neutrinos interact so weakly with matter that billions of them fly straight through your body every second, and have no effect on you.
Leptons have a lepton number of +1, antileptons have a lepton number of -1, and particles which are not leptons have a lepton number of 0. Lepton number is another quantity that is always conserved in reactions.
The standard model
We are nearly at a point to claim familiarity with all the fundamental particles in the standard model, as shown at the top of part 1 of this article. Here it is again. Notice that the baryons and mesons are not on this diagram, because they are not fundamental.
The quarks are shown in white, arranged into their three generations. The leptons likewise, in green. Notice the connections between the two – the second and third generations are identical to the first, except in mass. The masses increase from left to right. Matter consists of quarks and leptons, and everyday matter consists of the grey first generation, precisely because they are the lowest mass generations – if you manage to make a second generation particle, it will very quickly decay into first generation particles.
Quarks and leptons are all fermions, so matter is made of fermions… And for each of the 6 types of quark (white) and six types of lepton (green), there exists a corresponding antiparticle.
We spoke briefly of the pion as an exchange particle within a nucleus that helps to hold it together. Quarks need holding together too, in the protons, neutrons and other hadrons. The strong interaction is transmitted within hadrons by exchange bosons called gluons, shown as [latex]g[/latex]. There are exchange particles for the other forces, and they are all bosons (shown in blue). The photon mediates the electromagnetic force, and [latex]W[/latex] and [latex]Z[/latex] bosons do the same for the weak interaction. Presumably, gravity (the fourth and final force in nature) also has an exchange particle, and it has been termed the graviton, but it sits uncomfortably outside the standard model, until physicists can find a way to reconcile the theories of quantum mechanics and relativity. And in yellow is the Higgs boson, found in 2012 at CERN. Whereas matter is composed of fermions, forces are transmitted by bosons…
This concludes this primer – now go and read further. If you’re bold, you’ll find out about the mysterious symmetries we referred to! Two excellent reads are those referred to in the text: The Cosmic Onion by Frank Close, and The Particle at the End of the Universe by Sean Carroll.