The fundamental interactions and particle accelerators

Modern physics recognises four fundamental interactions: gravitational, electromagnetic, weak nuclear and strong nuclear.

1. a) Order them from strongest to weakest on the scale of the atomic nucleus.
2. b) Which of them is, by a huge margin, the weakest?
3. c) Which of the four can keep two bodies bound together, forming a stable system? Give an example of each case.

Every phenomenon in nature is governed by one of the four fundamental interactions. Identify it in each case:

1. a) Protons and neutrons remain bound inside the nucleus despite the intense electric repulsion between the protons.
2. b) The electrons in an atom remain bound to the nucleus.
3. c) An unstable nucleus emits an electron in beta decay.
4. d) The Earth orbits around the Sun and objects fall to the ground.

The gravitational and electromagnetic interactions share an infinite range, but differ in essential aspects.

1. a) Which of the two can be attractive or repulsive and which is always attractive?
2. b) On which property of bodies does each one act?
3. c) If the electromagnetic force is far more intense than the gravitational force, why do we feel the weight exerted by the Earth on us but no electric force from it? Reason your answer considering the charges and masses of macroscopic bodies.

We will compare the electric and gravitational forces between two protons separated by any distance $r$. Data: $K = 9\times10^{9}\ \text{N}\times\text{m}^2/\text{C}^2$, $G = 6.67\times10^{-11}\ \text{N}\times\text{m}^2/\text{kg}^2$, $e = 1.6\times10^{-19}\ \text{C}$ and $m_p = 1.673\times10^{-27}\ \text{kg}$.

1. a) Write the expressions for the electric force $F_e = K\,\frac{e^2}{r^2}$ and the gravitational force $F_g = G\,\frac{m_p^2}{r^2}$, and check that their ratio does not depend on the distance $r$.
2. b) Calculate the numerical value of the ratio $\frac{F_e}{F_g}$ and express it in standard form.
3. c) Explain what conclusion can be drawn from this result regarding the role of gravity in particle physics.

The four fundamental interactions differ greatly in their range, that is, in the distance over which they are noticeable.

1. a) Which two interactions have an infinite range, decreasing with distance but never becoming entirely zero?
2. b) Which one acts only at distances on the order of the size of an atomic nucleus, about $10^{-15}\ \text{m}$?
3. c) Which one has the shortest range of all, on the order of $10^{-18}\ \text{m}$?

In the Standard Model, each interaction is transmitted through the exchange of mediating particles called bosons.

1. a) Which boson mediates the electromagnetic interaction?
2. b) Which bosons mediate the weak nuclear interaction?
3. c) Which bosons transmit the strong interaction between quarks?
4. d) What hypothetical particle, not yet detected, is postulated as the mediator of the gravitational interaction?

The strong nuclear interaction is the strongest of the four fundamental interactions and yet we never perceive it directly in everyday life.

1. a) What is its approximate range?
2. b) Using this datum, explain why its effects remain confined to the inside of atomic nuclei.
3. c) What would happen to nuclei with more than one proton if the strong interaction suddenly ceased to exist? What chemical elements could exist?

Neutrinos only participate in the weak nuclear interaction (and in the gravitational interaction, which is completely negligible on their scale). The range of the weak interaction is barely $10^{-18}\ \text{m}$.

1. a) Explain the consequence of such a short range on the probability of a neutrino interacting with an atom when passing through matter.
2. b) A nuclear reactor emits on the order of $10^{20}$ antineutrinos every second, but only a few per day are registered in a nearby detector: relate this fact to the properties of the weak interaction.
3. c) For this very reason, neutrinos are said to be "ideal messengers" from the interior of the Sun or a distant supernova: explain what this expression refers to.

The picture of boson exchange qualitatively explains the range of each interaction: the more massive the mediating boson, the shorter the range, because the virtual particle can only "exist" for a very brief time according to the uncertainty principle.

1. a) The photon has no mass: what range does this idea predict for the electromagnetic interaction?
2. b) The $\text{W}$ and $\text{Z}$ bosons have huge masses, of $80\ \text{GeV}/c^2$ and $91\ \text{GeV}/c^2$: what does this imply about the range of the weak interaction?
3. c) Gluons have no mass and yet the strong interaction has a short range ($10^{-15}\ \text{m}$): explain to which phenomenon, studied in this topic, this special case is due.

In the 1920s, beta decay experiments showed a puzzling result: the emitted electrons did not all emerge with the same energy, as a two-body process would require, but with a continuous energy spectrum. Observe the graph:

1. a) Explain which conservation principles (energy and momentum) appeared to be violated if only the electron were emitted in the decay.
2. b) What hypothesis did Wolfgang Pauli formulate in 1930 to save these principles, and what properties (charge, mass, interaction with matter) did the proposed particle need to have?
3. c) That particle, the neutrino, was not experimentally detected until 1956: relate this 26-year delay to the range of the weak interaction and the properties attributed to it by Pauli.

In the 1960s, Glashow, Weinberg and Salam demonstrated that the electromagnetic and weak interactions are two manifestations of a single interaction, the electroweak interaction, which earned them the Nobel Prize.

1. a) Explain what it means to "unify" two interactions and in which energy regime (very high or everyday) they would manifest as a single one.
2. b) If the $\text{W}$ and $\text{Z}$ bosons have mass and the photon does not, explain why, at everyday energies, the two interactions appear completely different in range and strength.
3. c) The gravitational interaction has not yet been unified with the others: comment on what fundamental difficulty separates it from the other three (think about the existence or absence of a quantum theory of gravity and the detection of the graviton).

Unit challenge. Become a CERN researcher by designing a report on collisions and subatomic particle detection.

1. Look up the mass of the $Z$ boson ($M_Z \approx 91.2\text{ GeV}/c^2$) on the official CERN Open Data portal. In a short factsheet, identify its fundamental interaction, mediating boson, and the fundamental particles (quarks or leptons) into which it can decay according to the Standard Model.

2. Model scattering in colliders: roll a marble down a home-made ramp toward a fixed cylindrical obstacle and measure the lateral deflection $y$ after the collision at a fixed distance $L$ beyond the obstacle using a ruler ($10$ repetitions), estimating an uncertainty of $\Delta y = \pm 0.1\text{ cm}$.

3. Build a table with your values ($y_i \pm \Delta y$) and create a frequency distribution plot based on the estimated deflection angle $\theta_i \approx \arctan(y_i/L)$.

4. Write a reasoned proposal connecting your test to the operation of modern accelerators: explain how the LHC bends and detects charged leptons using magnetic fields ($F = q \times v \times B$) to reconstruct events such as that of the $Z$ boson.

Evaluation criteria:

• Rigorous identification of Standard Model particles and interactions using CERN data.

• Methodological correctness in taking measurements, uncertainty calculation, table, and plot.

• Quality of the report in linking the home experiment with accelerator and detector physics.

To study elementary particles, physicists build increasingly large and powerful accelerators.

1. a) Explain, using the equivalence $E = m \times c^2$, why producing very massive particles in the laboratory requires huge collision energies.
2. b) Explain, using the de Broglie relation $\lambda = \frac{h}{p}$, why "seeing" smaller and smaller structures requires increasingly fast projectile particles.
3. c) What energy advantage does colliding two beams head-on in opposite directions have over firing a single beam at a fixed target?

The LHC (Large Hadron Collider) at CERN, near Geneva, is a $27\ \text{km}$ circumference ring excavated about $100\ \text{m}$ underground, between Switzerland and France. Observe the diagram of the ring:

1. a) Which particles does the LHC mainly collide, and from which chemical element are they obtained?
2. b) What is the purpose of the thousands of superconducting magnets lining the ring?
3. c) Name the best-known scientific milestone of the LHC and the year it was announced.

In the LHC each proton reaches an energy of $6.5\ \text{TeV}$ ($1\ \text{TeV} = 10^{12}\ \text{eV}$). Data: $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$; the kinetic energy of a mosquito in flight is about $10^{-6}\ \text{J}$.

1. a) Express the energy of an LHC proton in joules.
2. b) Compare it with the kinetic energy of the mosquito and comment on how a single subatomic particle can concentrate an energy of that magnitude.
3. c) In each head-on collision, the energies of the two protons add together, giving $13\ \text{TeV}$ in total. If all that energy were converted into the rest mass of new protons ($E_0 = 938\ \text{MeV}$ each), what is the maximum number of protons that could be created? Express the result in standard form.

The resolution with which a particle beam can explore a structure is limited by its de Broglie wavelength: only details larger than $\lambda$ can be "seen". For ultrarelativistic particles, $\lambda \approx \frac{h \times c}{E}$ holds. Data: $h = 6.63\times10^{-34}\ \text{J}\times\text{s}$, $c = 3\times10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$.

1. a) Calculate $\lambda$ for an LHC proton with energy $6.5\ \text{TeV}$.
2. b) The size of a proton is about $10^{-15}\ \text{m}$: compare this with your result and reason whether the beam can resolve the internal structure of the proton.
3. c) Estimate, using the same formula, the $\lambda$ of the electrons in an electron microscope accelerated with $100\ \text{kV}$ (energy of $100\ \text{keV}$) and explain why that instrument cannot be used to study quarks.

In a circular accelerator, a charged particle forced to curve its trajectory emits radiation (synchrotron radiation) and loses energy. For the same energy, these losses are enormously greater the smaller the mass of the particle. Data: $m_e = 0.511\ \text{MeV}/c^2$ and $m_p = 938\ \text{MeV}/c^2$.

1. a) Calculate how many times more massive the proton is than the electron and reason why the LHC accelerates protons rather than electrons to reach the highest energies.
2. b) What advantage would an electron collider have, however? Think about the internal structure of each particle.
3. c) Linear accelerators greatly reduce the problem of synchrotron radiation: explain why, considering the trajectory of the particles.

In the LHC, the beams cross about $4\times10^{7}$ times per second, but only a tiny fraction of the collisions contain genuinely interesting processes. The Higgs boson, for example, is produced in approximately one in every $10^{10}$ collisions.

1. a) Estimate how many Higgs bosons are produced per second and how long it takes, on average, to produce one.
2. b) Reason why months of data collection were needed to accumulate a sufficient sample of Higgs bosons and announce the discovery.
3. c) Detectors cannot store all events and must filter them in real time, discarding almost all of them: explain what practical problem (volume of data) makes this necessary.

To keep the LHC protons in their circular orbit, superconducting dipole magnets are used. The effective magnetic radius of the ring is $r = 2.8\ \text{km}$. For an ultrarelativistic charged particle, $p \approx \frac{E}{c}$ holds and the radius of the trajectory in a perpendicular magnetic field is $r = \frac{p}{q \times B}$. Data: $e = 1.6\times10^{-19}\ \text{C}$, $c = 3\times10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$.

1. a) Calculate the momentum of a $7\ \text{TeV}$ proton in $\text{kg}\times\text{m/s}$.
2. b) Rearrange the expression for the radius to solve for the magnetic field, and calculate the value required to curve these protons in the ring.
3. c) The maximum field achievable with a conventional electromagnet with an iron core is about $2\ \text{T}$: compare this with your result and justify why the LHC requires superconducting magnets cooled with liquid helium to $1.9\ \text{K}$.

The Higgs boson has a mean lifetime of barely $10^{-22}\ \text{s}$: it decays long before it can reach any detector. Data: $c = 3\times10^{8}\ \text{m/s}$.

1. a) Calculate the maximum distance it would cover in that time even if it travelled at practically the speed of light, and conclude why it is impossible to observe it directly.
2. b) How then is such a particle "detected"? Explain the role of its decay products (for example, two photons or four leptons) and of the laws of conservation of energy and momentum in reconstructing its mass.
3. c) The Higgs signal was a small statistical excess over a huge background of similar events: reason why the 2012 announcement required accumulating millions of collisions and applying a statistical criterion as demanding as "five sigma".

In a collision of a beam against a fixed target, a large part of the energy is spent conserving the momentum of the system and is not available to create new particles. Data: $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$. Observe both setups:

1. a) Explain, using conservation of momentum, why in a head-on collision of two identical beams (zero total momentum) all the energy remains available to create mass according to $E = m \times c^2$.
2. b) The LHC collides $6.5\ \text{TeV}$ protons with $6.5\ \text{TeV}$ protons: what is the centre-of-mass energy of each collision? Express it in $\text{TeV}$ and in joules.
3. c) Reason what practical drawbacks the collider mode has compared to a fixed target (think about the probability of two tiny beams colliding and the technological precision required).