Astroparticle Physics

2.2 Problems 33metric. At very high temperatures, as existed at the time ofthe Big Bang, the symmetry was so perfect that all interactionscould be described by one universal force. The reasonthat increasingly large accelerators with higher energies arebeing constructed is to track down this universal descriptionof all forces.According to present beliefs, the all-embracing theory ofsupergravity (SUGRA) is embedded in the M theory, an 11-dimensional superstring theory. The smallest constituents ofthis superstring theory are p-dimensional objects (‘branes’)of the size of the Planck length L P = √¯hG/c3 (where Gis the gravitational constant, ¯h is Planck’s constant, and cis the velocity of light). Seven of the ten spatial dimensionsare compacted into a Calabi–Yau space. According to taste,the ‘M’ in the M theory stands for ‘membrane’, ‘matrix’,‘mystery’, or ‘mother (of all theories)’.universal forceMtheory2.2 Problems1. Which of the following reactions or decays are allowed?a) µ − → e − + γ ,b) µ + → e + + ν e +¯ν µ + e + + e − ,c) π 0 → γ + e + + e − ,d) π + → µ + + e − ,e) Λ → p + K − ,f) Σ + → n + π + ,g) K + → π + + π − + π + ,h) K + → π 0 + π 0 + e + + ν e .2. What is the minimum kinetic energy of a cosmic-raymuon to survive to sea level from a production altitudeof 20 km (τ µ = 2.197 03 µs, m µ = 105.658 37 MeV)?For this problem one should assume that all muons havethe given lifetime in their rest frame.3. Work out the Coulomb force and the gravitational forcebetween two singly charged particles of the Planck massat a distance of r = 1 fm!4. In a fixed-target experiment positrons are fired at a targetof electrons at rest. What positron energy is required toproduce a Z (m Z = 91.188 GeV)?