The Standard Model and the fundamental particles

The Standard Model describes ordinary matter in terms of a few elementary particles: quarks and leptons. Quarks come in six types: up ($\text{u}$), down ($\text{d}$), strange ($\text{s}$), charm ($\text{c}$), bottom ($\text{b}$) and top ($\text{t}$), and the first two, with charges $q_u = +\frac{2}{3}\,e$ and $q_d = -\frac{1}{3}\,e$, form protons and neutrons.

1. a) How many types of quarks are there and which ones form the ordinary matter around us?
2. b) What does it mean for quarks to be "elementary" particles?
3. c) State the electric charge of quarks $\text{u}$ and $\text{d}$ expressed in units of the elementary charge $e$.

A proton is composed of three quarks: two $\text{u}$ quarks and one $\text{d}$ quark (combination $\text{uud}$). Data: $q_u = +\frac{2}{3}\,e$, $q_d = -\frac{1}{3}\,e$ and $e = 1,6\cdot10^{-19}\ \text{C}$. Observe the diagram:

1. a) Add up the charges of its three quarks and express the total charge of the proton in terms of $e$.
2. b) Calculate the value of that charge in coulombs.
3. c) Why is the proton said not to be an elementary particle?

Hadrons are classified into two main groups: baryons, composed of three quarks, and mesons, composed of a quark and an antiquark. Observe the diagram of a meson:

1. a) To which of the two groups do the proton and neutron belong?
2. b) The pion $\pi^+$ is a meson composed of a $\text{u}$ quark and a $\bar{\text{d}}$ antiquark. Knowing that $q_u = +\frac{2}{3}\,e$ and that an antiquark has the opposite charge to its quark, calculate the charge of the pion in terms of $e$.
3. c) Reason whether the result obtained is consistent with the name of the pion, $\pi^+$.

In addition to quarks, the Standard Model includes another family of matter particles: leptons. The electron, the muon and the tau, together with their three associated neutrinos, are leptons.

1. a) What property do leptons have in common that distinguishes them from quarks? (Think about whether or not they are formed of other particles and which interactions they feel.)
2. b) Which is the only charged lepton present in ordinary matter?
3. c) State two properties that distinguish neutrinos from the rest of the leptons.

In the Standard Model, in addition to matter particles (quarks and leptons), there are bosons. The photon, the $\text{W}$ and $\text{Z}$ bosons and gluons are exchange particles for interactions; the Higgs boson is also a boson, but it does not mediate any of the fundamental interactions.

1. a) Name the mediator boson of the electromagnetic interaction.
2. b) Which boson in the Standard Model is associated with particles having mass?
3. c) Explain in your own words the difference between the role of a mediator boson and that of a matter particle such as a lepton.

In a particle physics laboratory detector, six particles are recorded: electron, proton, neutron, photon, neutrino and Higgs boson.

1. a) Which of them are leptons?
2. b) Which are hadrons, that is, made of quarks?
3. c) Which are mediator bosons or associated with a field?

In particle physics, energies are expressed in electronvolts and their multiples: $1\ \text{eV} = 1,6\cdot10^{-19}\ \text{J}$, $1\ \text{MeV} = 10^{6}\ \text{eV}$.

1. a) The rest energy of the electron is $0,511\ \text{MeV}$: express it in joules.
2. b) The rest energy of the proton is $938\ \text{MeV}$: express it in joules.
3. c) How many times greater is the rest energy of the proton than that of the electron? Express the result in standard form.

In positron emission tomography (PET), a medical diagnostic technique, a radiotracer emitting positrons is injected into the patient. Each positron annihilates with a tissue electron, emitting two gamma photons in opposite directions. Data: $m_e = 9,11\cdot10^{-31}\ \text{kg}$, $c = 3\cdot10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1,6\cdot10^{-19}\ \text{J}$. Observe the diagram:

1. a) Assuming that the electron and the positron are practically at rest, calculate the total energy released in the annihilation, in joules.
2. b) How much energy does each of the two photons carry? Express it in $\text{MeV}$.
3. c) Using conservation of linear momentum, explain why the two photons travel in opposite directions and why this allows the PET scanner to locate the point where annihilation occurred.

Matter-antimatter annihilation converts $100\ \%$ of mass into energy, the highest efficiency permitted by physics. Data: $c = 3\cdot10^{8}\ \text{m/s}$; the combustion of petrol releases about $4,5\cdot10^{7}\ \text{J/kg}$.

1. a) Calculate the energy released when $1\ \text{g}$ of antimatter annihilates with $1\ \text{g}$ of ordinary matter. Remember that the mass of both disappears.
2. b) How many kilograms of petrol is that energy equivalent to? Express the result in standard form.
3. c) Despite this spectacular efficiency, antimatter will never be a practical energy source: reason why, thinking about where antimatter comes from and how much energy it costs to produce it.

In beta decay, a free neutron transforms according to the process $\text{n} \rightarrow \text{p} + \text{e}^- + \bar{\nu}_e$, where $\bar{\nu}_e$ is an electron antineutrino. Data: the neutron is $\text{udd}$, the proton is $\text{uud}$, $q_u = +\frac{2}{3}\,e$ and $q_d = -\frac{1}{3}\,e$. Observe the diagram of the process:

1. a) Check that electric charge is conserved in the process by adding up the charges before and after.
2. b) By comparing the compositions of the neutron and proton, what transformation does one of the quarks undergo during decay?
3. c) Given that the neutron is neutral and the proton has a charge $+e$, explain why it is essential that an electron is also emitted, and reason what evidence (conservation of energy and linear momentum) forced the postulation of the antineutrino as well.

An isolated quark has never been observed: quarks only exist forming combinations (baryons of three quarks or quark-antiquark mesons). This property of the strong interaction is called confinement. Data: $q_u = +\frac{2}{3}\,e$ and $q_d = -\frac{1}{3}\,e$.

1. a) Explain why the electric charge of any particle that can be freely observed is always an integer multiple of $e$, even though quarks have charges of $\pm\frac{1}{3}\,e$ or $\pm\frac{2}{3}\,e$. Check this with a baryon and a meson.
2. b) When trying to separate two quarks, the energy invested materialises as a quark-antiquark pair created from the vacuum, and instead of free quarks, two new hadrons appear. Relate this fact to confinement and to equation $E = m \cdot c^2$.
3. c) Compare the situation to snapping a magnet in half hoping to separate the north pole from the south pole: indicate how both phenomena are similar and how they differ.

Until 2012, the Higgs boson was the only piece of the Standard Model without experimental confirmation. Its existence explains why elementary particles have mass. Data: $1\ \text{eV} = 1,6\cdot10^{-19}\ \text{J}$, $c = 3\cdot10^{8}\ \text{m/s}$ and $m_p = 1,673\cdot10^{-27}\ \text{kg}$.

1. a) Qualitatively explain the role of the Higgs field: why does the photon, which does not interact with this field, have no mass, whereas the $\text{W}$ and $\text{Z}$ bosons, which do interact with it, are very massive?
2. b) The mass of the Higgs boson is $125\ \text{GeV}/c^2$: calculate its rest energy in joules and its mass in kilograms.
3. c) How many times more massive is the Higgs boson than a proton? Comment on why the world's most powerful accelerator was needed to produce it.