Chapter 9 Standard model of particle physics

The standard model of particle physics is a theory that describes the fundamental particles and forces in the universe, excluding gravity (although we will mention gravity below). It is central to our understanding of how the universe works at the smallest scales. This chapter provides an introduction to the standard model, which we will explore in more detail in subsequent chapters.

9.1 Fundamental Forces and Interactions

We can distinguish four types of interaction:

1.

Gravitation
2.

Electromagnetic
3.

Strong nuclear
4.

Weak nuclear

9.1.1 Gravitation

For mass scales common in high-energy physics, gravitational interactions are negligible in comparison with other interactions. A quantum called the graviton (spin 2 boson) is thought to be the mediator of the gravitational interaction. As the gravitational force has infinite range, the graviton mass is zero. Gravity is not part of the standard model, as we do not have a working model of quantum gravity.

9.1.2 Weak nuclear

The first weak interactions to be observed were those of nuclear $\beta$ -decay. They were examples of the decay of bound protons or neutrons where the change in binding energy provides enough energy for the creation of electrons and neutrinos

	$\displaystyle p\rightarrow n+e^{+}+\nu_{e}$		(9.1)
	$\displaystyle n\rightarrow p+e^{-}+\bar{\nu_{e}}$

The weak interaction is a short-range interaction ( $R\sim 10^{-18}\,{\rm m}$ ). It is mediated by massive spin 1 $W$ and $Z$ bosons (see below).

9.1.3 Electromagnetic

The exchange particle for the electromagnetic interaction is the photon, a spin 1 boson. The interaction has infinite range, hence the photon must have zero mass.

9.1.4 Strong nuclear

We now regard nucleons (and mesons) as made up of quarks, with gluons as the exchange particles. They have zero rest mass and yet the strong nuclear force is extremely short-range. Why is the force not infinite range since the force carrier is massless? The reason is quarks (and gluons) carry a type of charge called colour. They are limited to short-range by colour confinement (see below).

9.1.5 Summary of conservation laws

See Tab. 9.1 for a summary of the conservation laws for each interaction. Charge conjugation ( $C$ ) is the operation that replaces all particles with their anti-particles in the same state, so that momenta, positions, etc. are unchanged. As with parity, applying charge conjugation twice must return everything back to the original state, so the eigenvalues of $\hat{C}$ are $C=\pm 1$ . The weak interaction violates $C$ and $P$ , but the full $C P T$ symmetry (i.e. the application of all 3 symmetries) is conserved. $C P$ violation is discussed in more detail in the Theoretical Particle Physics module.

Certain properties of particles are also conserved in their interactions. We will discuss these in detail in the next two lectures, but quantities conserved in all types of interaction are charge, baryon number and a separate lepton number for each generation of lepton.

Table 9.1: Conservation laws for strong, electromagnetic and weak interactions

Conservation law	Strong	Electromagnetic	Weak
Energy	Yes	Yes	Yes
Total Momentum	Yes	Yes	Yes
Total Angular momentum	Yes	Yes	Yes
Charge conjugation, C	Yes	Yes	No
Parity, P	Yes	Yes	No
CP	Yes	Yes	Almost
Time reversal, T	Yes	Yes	No?
CPT	No real	test	exists
Baryon Number	Yes	Yes	Yes
Lepton Number (3 laws)	Yes	Yes	Yes
Quark Number	Yes	Yes	No
Charge	Yes	Yes	Yes
Colour	Yes	Yes	Yes
Generation	Yes	Yes	No

9.2 Bosons and Fermions

In the standard model, we can classify particles into two types: bosons and fermions. Fermions have half-integer spin and as such obey the Pauli Exclusion Principle (PEP), which forbids more than 1 fermion from occupying the same quantum state. Fermions are ‘matter-like’ particles, and examples are leptons and quarks. Bosons have integer spin and do not obey the PEP. The force-carrying particles are examples of bosons.

9.3 Standard Model

The standard model of particle physics seeks to explain particle physics on the basis of the interactions between a small number of elementary particles which fall into three distinct types: two spin $1/2$ families of fermions – the leptons and the quarks, and one family of spin 1 bosons, the gauge bosons, which are responsible for carrying the force between them. In addition, the spin 0 Higgs boson is introduced to account for mass – without it, the particles would all have zero mass. These elementary particles are ‘point particles’, without internal structure or excited states.

9.3.1 Leptons

Leptons are spin 1/2 fermions, and come in 6 types or flavours. The most familiar example of a lepton is the electron ( $e^{-}$ ) which is bound in atoms by the electromagnetic force. The next most familiar lepton is the electron neutrino ( $\nu_{e}$ ) observed in $\beta$ -decay which occurs via the weak interaction.

In the standard model, the leptons have the properties given in Table 9.2. Note that, in the standard model, neutrinos have a mass which is identically zero. However, we know some of them must have non-zero mass from neutrino oscillation experiments, which takes us outside the standard model. We can only currently place lower bounds on these masses, and data suggests the sum of neutrino masses is $<1$ eV.

Table 9.2: Lepton properties.

	Lepton	Charge (e)	Mass (MeV)	Lifetime	Decays
First generation	e	-1	0.511003	$\infty$	-
	$\nu_{e}$	0	0	$\infty$	-
Second generation	$\mu$	-1	105.659	$2.2\times 10^{-7}$ s	e+ $\nu_{\mu}$ + $\bar{\nu_{e}}$
	$\nu_{\mu}$	0	0	$\infty$ ?	-
Third generation	$\tau$	-1	1784.2	$3\times 10^{-13}$ s	$\begin{array}[]{c}{\rm e}+\nu_{\tau}+\bar{\nu_{e}}\\ \mu+\nu_{\tau}+\bar{\nu_{\mu}}\end{array}$
	$\nu_{\tau}$	0	0	$\infty$ ?	-

9.3.2 Quarks

Quarks are also spin 1/2 fermions, and come in 6 flavours. Quarks and leptons can be subdivided into 3 generations, as shown in Tab. 9.3. The $\nu_{i}$ represent different neutrino flavours: the electron, muon and tau neutrino. The 6 flavours of quarks are labelled as up (u), down (d), strange (s), charmed (c), bottom (b) and top (t).

Table 9.3: Quark and lepton generations.

Generation	Quarks	Leptons
1st	up (u), down (d)	electron (e), electron neutrino ( $\nu_{\rm e}$ )
2nd	strange (s), charmed (c)	muon ( $\mu$ ), muon neutrino ( $\nu_{\mu}$ )
3rd	bottom (b), top (t)	tau ( $\tau$ ), tau neutrino ( $\nu_{\tau}$ )

Individual quarks are not directly observable. They are localized in particles the size of a proton. This means the concept of a quark mass is rather vague since quarks appear to be confined in clusters. The masses of the quarks also are different when they are constituents of baryons (that is collections of three quarks such as a proton) from their values when they are constituents of mesons (collections of two quarks). Even so, the relative magnitude of their masses is more readily defined and we find that the charmed quark is approximately five times heavier than the up, while the top is much heavier than the other varieties.

Table 9.4: Quark properties.

	Flavour	Charge (e)	Approximate mass (GeV)
First generation	$d$	-1/3	0.3
	$u$	2/3	0.3
Second generation	$s$	-1/3	0.5
	$c$	2/3	1.5
Third generation	$b$	-1/3	4.5
	$t$	2/3	174

Quarks have an additional property called colour. There are three different colours – red, green blue – and coloured objects cannot be observed. This leaves two possibilities

1.

States with 3 quarks of each of the colours (red + green + blue = white = colourless)
2.

States with 2 quarks with a colour-anticolour quark pair (e.g. red + anti-red = colourless)

In all there are 18 quarks = 6 flavours times 3 colours. The quarks have a charge of $-e/3$ or $2e/3$ . Any baryon which is made up of three quarks can have a charge of $-e$ , $0$ , $e$ or $2e$ . No baryon can exist with charge $-2e$ or $3e$ .

9.3.3 Force carriers

The particles associated with the interactions between leptons and quarks are bosons of integer spin. For example, a photon couples to charged particles and has a spin of 1. A gluon couples to coloured particles (quarks only) and has a spin of 1. The $Z^{0}$ and $W^{\pm}$ couple to both leptons and quarks and have a spin of 1. Gluons, photons and the massive $W^{\pm}$ and $Z^{0}$ particles are all bosons. There are however differences:

•

Photons and $Z^{0}$ particles carry no charge or colour.
•

$W^{+}$ and $W^{-}$ carry charge but no colour.
•

Gluons carry colour but no charge.

Colour is like a new kind of ‘charge’ but it is more complex than the electrical charge.

Both photons and gluons are particles with zero rest mass so they lead to an interaction which varies as $q^{2}/r$ which has infinite range. The $Z^{0}$ and $W$ particles have mass (about 80 GeV) and an interaction which varies as

\frac{q^{2}}{r}e^{-rM_{W}}

(9.2)

which has a finite range. A finite mass corresponds to a finite range to the interaction, a zero mass to an infinite range.

For a long time, it was a mystery how the $W$ and $Z^{0}$ particles could have a mass. This mystery has been solved (in some sense) by the theory of Weinberg and Salam, but at the expense of introducing a new particle called the Higgs boson. It is the Higgs boson that allows particles to have a mass. The discovery of the Higgs boson (or potentially bosons) has led to initial confirmation that the present picture of elementary particle physics is correct. It has a mass of around 125 GeV.

The details of the mediators in the standard model are given in Table 9.5.

Table 9.5: Gauge particles.

Mediator	Charge (e)	Mass	Force
Gluon (g)	0	0	strong
Photon ( $\gamma$ )	0	0	electromagnetic
$W^{\pm}$	$\pm$ 1	80.6 $\pm$ 0.04 GeV	weak
$Z^{0}$	0	91.177 $\pm$ 0.031 GeV	weak

9.4 Identical Particles

All particles of one type are identical. What do we mean by particles being identical? According to quantum field theory, there is one field, for the whole Universe, for each type of particle. Particle states are excitations of this field. For example, all electrons are identical since they all arise from the same field. Switching any two particles of the same type therefore results in the same overall wavefunction, up to a minus sign. For any two fermions quantum field theory says the overall wave function picks up a minus sign (it is anti-symmetric) under the interchange of the two particles, i.e.

\psi({\bf r}_{1},{\bf r}_{2})=-\psi({\bf r}_{2},{\bf r}_{1})\,.

(9.3)

This property gives rise to the Pauli exclusion principle. For bosons, the wave function is symmetric under identical particle exchange. This means

\psi({\bf r}_{1},{\bf r}_{2})=\psi({\bf r}_{2},{\bf r}_{1})\,.

(9.4)

Thus for example the tau particles are all indistinguishable from each other but are distinguishable from muons because they are more massive. Similarly, all electron neutrinos are identical.

9.5 Hadrons

Another class of particles, the hadrons, are also observed in nature. These include the neutron ( $n$ ) and the proton ( $p$ ) as well as the pions ( $\pi^{+}$ , $\pi^{-}$ , $\pi^{0}$ ). However, these are not elementary particles but are made up of quarks bound together by the strong interaction.

Particles composed of quarks are known as hadrons. There are two types of hadron

Baryons: These contain 3 quarks (e.g. proton, neutron). The only stable baryon is the proton, all others decay.
Mesons: These contain 2 quarks, as a quark/anti-quark pair. There are no stable mesons, they quickly decay into leptons and photons. Examples of mesons include the pion and kaon, e.g. $\pi^{+}=u\bar{d}$ and $K^{-}=\bar{u}s$ .

9.6 Summary

In summary, the standard model has the following assumptions

1.

The basic constituents of matter are quarks and leptons, both with spin 1/2.
2.

Quarks come in 6 flavours and 3 colours. Leptons come in 6 flavours.
3.

They interact via the spin 1 (gauge) bosons.
4.

Quarks and leptons are subdivided into 3 generations.
5.

The spin 0 Higgs boson is responsible for giving particles mass.