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.

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: $K = 9\cdot10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2$, $G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$, $e = 1,6\cdot10^{-19}\ \text{C}$ and $m_p = 1,673\cdot10^{-27}\ \text{kg}$.

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 $10^{-18}\ \text{m}$?

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 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).

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

1. a) Explain, using the equivalence $E = m \cdot c^2$, 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?

In the LHC each proton reaches an energy of $6,5\ \text{TeV}$ ($1\ \text{TeV} = 10^{12}\ \text{eV}$). Data: $1\ \text{eV} = 1,6\cdot10^{-19}\ \text{J}$; 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.

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: $1\ \text{eV} = 1,6\cdot10^{-19}\ \text{J}$. 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 $E = m \cdot c^2$.
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).