The Standard Model and 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 $e$.

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?

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 $e = 1.6\times10^{-19}\ \text{C}$. 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?

A neutron is composed of one $\text{u}$ quark and two $\text{d}$ quarks (combination $\text{udd}$). Data: $q_u = +\frac{2}{3}\,e$ and $q_d = -\frac{1}{3}\,e$.

1. a) Calculate the total charge of the neutron by adding up the charges of its three quarks.
2. b) In light of the result, explain why the neutron is electrically neutral despite being composed of charged particles.
3. c) Does the neutron contain electric charge inside it? Reason your answer from your calculation.

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, $\pi^+$.

Delta baryons are combinations of three $\text{u}$ and $\text{d}$ quarks: $\Delta^{++}$ is $\text{uuu}$, $\Delta^{+}$ is $\text{uud}$, $\Delta^{0}$ is $\text{udd}$ and $\Delta^{-}$ is $\text{ddd}$. Data: $q_u = +\frac{2}{3}\,e$, $q_d = -\frac{1}{3}\,e$ and $e = 1.6\times10^{-19}\ \text{C}$.

1. a) Calculate the electric charge of each of the four delta baryons in terms of $e$.
2. b) Express these charges in coulombs.
3. c) Explain why the superscript of each particle ($++$, $+$, $0$, $-$) matches the result of your calculation.

A meson is composed of a quark and an antiquark. Consider only quarks and antiquarks of types $\text{u}$ and $\text{d}$. Data: $q_u = +\frac{2}{3}\,e$, $q_d = -\frac{1}{3}\,e$, and the charge of an antiquark is opposite to that of its quark.

1. a) Which quark-antiquark combination gives rise to a meson with charge $-e$? Verify it by adding the charges.
2. b) Which combinations give rise to neutral mesons? Verify it.
3. c) Justify, from the possible charges, why no meson can have charge $+2\,e$.

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.

Every second, about $10^{14}$ neutrinos coming from nuclear reactions in the Sun pass through our body without us noticing anything at all.

1. a) Based on their properties (charge, mass and type of interaction they feel), explain why neutrinos pass through the entire Earth barely interacting.
2. b) Which family of the Standard Model do neutrinos belong to, and how do they differ from quarks?
3. c) Reason why neutrino detectors, such as Super-Kamiokande in Japan, use gigantic water tanks installed deep underground.

Muons are unstable leptons, with a mean lifetime of only $\tau = 2.2\ \mu\text{s}$. They are produced in the upper atmosphere, at an altitude of about $15\ \text{km}$, when cosmic rays collide with air molecules, and they travel at practically the speed of light. Data: $c = 3\times10^{8}\ \text{m/s}$; consider a muon with speed $v = 0.999\,c$. Observe the diagram:

1. a) According to classical physics, calculate the distance the muon would travel, on average, before decaying.
2. b) Compare that distance with the $15\ \text{km}$ separating it from the ground and explain what classical physics would predict regarding the possibility of detecting muons at the Earth's surface.
3. c) Muons are indeed detected in abundance at the surface: explain qualitatively which relativistic effect (time dilation or, equivalently, length contraction) makes this possible.

In 1932, Carl Anderson discovered the positron in cosmic rays: a particle identical to the electron but with positive charge. It was the first antiparticle observed.

1. a) What properties do a particle and its antiparticle share, and in what property do they differ?
2. b) State the name and charge of the antiparticle of the proton.
3. c) What happens when a particle and its antiparticle meet?

In particle physics, energies are expressed in electronvolts and their multiples: $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$, $1\ \text{MeV} = 10^{6}\ \text{eV}$.

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.

According to relativity, every particle at rest possesses an energy $E_0 = m \times c^2$ associated with its mass. Data: $m_e = 9.11\times10^{-31}\ \text{kg}$, $c = 3\times10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$.

1. a) Calculate the rest energy of the electron in joules.
2. b) Convert the result to $\text{MeV}$ and check that it approximately matches the accepted value, $0.511\ \text{MeV}$.
3. c) Repeat the calculation for the proton ($m_p = 1.673\times10^{-27}\ \text{kg}$) and compare the result with $938\ \text{MeV}$.

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: $m_e = 9.11\times10^{-31}\ \text{kg}$, $c = 3\times10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$. 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.

In the Standard Model it is customary to express particle masses in $\text{MeV}/c^2$. Data: electron mass $m_e = 0.511\ \text{MeV}/c^2$, muon mass $m_\mu = 105.7\ \text{MeV}/c^2$, $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$ and $c = 3\times10^{8}\ \text{m/s}$.

1. a) Explain what it means to express a mass in $\text{MeV}/c^2$ and how it relates to equation $E_0 = m \times c^2$.
2. b) Calculate the mass of the muon in kilograms.
3. c) How many times more massive is the muon than the electron? This result, close to $207$, led to the muon being initially mistaken in the 1930s for the particle predicted by Yukawa, an identification later ruled out because of its weak interaction with nuclei.

Matter-antimatter annihilation converts $100\ \%$ of mass into energy, the highest efficiency permitted by physics. Data: $c = 3\times10^{8}\ \text{m/s}$; the combustion of petrol releases about $4.5\times10^{7}\ \text{J/kg}$.

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.

The mass of the proton is $938\ \text{MeV}/c^2$, but the masses of its three quarks (two $\text{u}$ and one $\text{d}$) barely add up to about $9\ \text{MeV}/c^2$, around $1\ \%$ of the total. Data: $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$. Observe the graph:

1. a) If the rest mass of the proton does not originate from the mass of its quarks, where does it fundamentally come from? Explain your answer using the equivalence $E_0 = m \times c^2$ and the energy associated with the strong interaction that binds the quarks.
2. b) Calculate in joules the energy equivalent to the mass of the proton.
3. c) Critically comment on this statement: "The mass of ordinary matter is, above all, trapped energy."

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: $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$, $c = 3\times10^{8}\ \text{m/s}$ and $m_p = 1.673\times10^{-27}\ \text{kg}$.

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.

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.

The following three particle reactions are proposed ($\bar{\text{p}}$ represents an antiproton, with charge $-e$; the antineutrino $\bar{\nu}_e$ and photons $\gamma$ have no charge):

• Reaction 1: $\text{p} + \bar{\text{p}} \rightarrow \gamma + \gamma$

• Reaction 2: $\text{n} \rightarrow \text{p} + \text{e}^-$

• Reaction 3: $\text{e}^- + \text{e}^- \rightarrow \text{e}^-$

1. a) Check for each reaction whether electric charge is conserved.
2. b) Reaction 2 conserves charge and yet is never observed as such in nature: an antineutrino is missing among the products. Explain which quantities would not be conserved without it (think about lepton number, spin, and the fact that the emitted electrons have a continuous energy spectrum).
3. c) In the proton-antiproton annihilation of reaction 1, assuming both particles are practically at rest, calculate the total energy available to the photons, in joules and in $\text{GeV}$. Data: $m_p = 1.673\times10^{-27}\ \text{kg}$, $c = 3\times10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$.

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 $E = m \times c^2$.
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.

The free proton is stable, but a free neutron decays in about $880\ \text{s}$ on average. Data (rest energies): $E_0(\text{n}) = 939.6\ \text{MeV}$, $E_0(\text{p}) = 938.3\ \text{MeV}$, $E_0(\text{e}^-) = 0.511\ \text{MeV}$; the rest energy of the antineutrino is practically zero. Observe the level diagram:

1. a) Verify with the data that the decay $\text{n} \rightarrow \text{p} + \text{e}^- + \bar{\nu}_e$ is energetically possible and calculate the energy released.
2. b) Reason, using the same data, why the spontaneous reverse process, $\text{p} \rightarrow \text{n} + \text{e}^+ + \nu_e$, cannot occur for a free proton.
3. c) Relate your conclusions to the fact that the stable matter in the universe is composed of protons and electrons, and not free neutrons.