The three Standard Model gauge couplings (alpha_1, alpha_2, alpha_3) run with energy scale due to quantum corrections. When extrapolated to high energies using the SM particle content, they approach each other but do not quite meet at a single point. Adding SUSY particles changes the running. What happens?
AThe couplings diverge faster and never meet
BWith the MSSM particle content, the three couplings meet to good approximation at a single point (the GUT scale M_GUT ~ 2 x 10^{16} GeV), with a unified coupling alpha_GUT ~ 1/24 — this 'gauge coupling unification' is one of the strongest indirect arguments for both SUSY and grand unification
CThe couplings become equal at the Planck scale
DAll three couplings become zero (asymptotic freedom)
The running of the gauge couplings is governed by the beta functions, which depend on the particle content of the theory. In the SM, the three couplings approach each other at ~10^{13-15} GeV but miss by several percent. Adding SUSY particles (which contribute to the beta functions above the SUSY scale ~1 TeV) modifies the running just enough to achieve unification at M_GUT ~ 2 x 10^{16} GeV within experimental uncertainties. This is not guaranteed to happen -- it requires the right particle content -- and is considered a non-trivial success of the MSSM.
Question 2 Short Answer
The simplest GUT, Georgi-Glashow SU(5), places the left-handed down quark and the left-handed lepton in the same multiplet. This means there exist gauge bosons (X and Y) that can transform quarks into leptons, mediating proton decay. The predicted proton lifetime is approximately tau_p ~ M_X^4 / (alpha_GUT^2 * m_p^5). Why hasn't proton decay been observed?
Think about your answer, then reveal below.
Model answer: The proton lifetime depends on the fourth power of the heavy gauge boson mass M_X. For M_GUT ~ 10^{16} GeV and alpha_GUT ~ 1/24, the predicted lifetime is tau_p ~ 10^{34-36} years. The minimal SU(5) model predicts tau(p -> e+ pi0) ~ 10^{31} years, which has been excluded by Super-Kamiokande (limit: tau > 2.4 x 10^{34} years). Minimal non-SUSY SU(5) is therefore ruled out. SUSY GUTs predict longer lifetimes (10^{34-36} years) and different dominant decay channels (p -> K+ nu-bar), which are within the reach of next-generation experiments like Hyper-Kamiokande and DUNE. The current limit on p -> K+ nu-bar is tau > 5.9 x 10^{33} years from Super-K.
Proton decay is the smoking-gun prediction of grand unification. Its non-observation has eliminated the simplest models but not the general idea. The predicted lifetimes in SUSY GUTs and SO(10) models are tantalizingly close to current experimental limits, making proton decay searches one of the most important experiments in fundamental physics.
Question 3 Multiple Choice
SO(10) is considered a more attractive GUT group than SU(5) because a single 16-dimensional spinor representation of SO(10) contains all 15 SM fermions of one generation plus one additional state. What is this extra state?
AA fourth color of quark
BA right-handed neutrino — SO(10) naturally includes a right-handed neutrino in each generation, which enables the seesaw mechanism for generating small neutrino masses through a heavy Majorana mass term at the GUT scale
CA mirror fermion with opposite chirality
DA supersymmetric partner
The 16 of SO(10) decomposes under SU(5) as 10 + 5-bar + 1, where the 10 and 5-bar contain the 15 known SM fermion states and the 1 is a gauge singlet: the right-handed neutrino nu_R. This is exactly the field needed for the seesaw mechanism: a Yukawa coupling gives a Dirac mass m_D ~ v (electroweak scale), and a Majorana mass M_R ~ M_GUT gives light neutrino masses m_nu ~ m_D^2/M_R ~ 0.01 eV, naturally explaining why neutrino masses are so tiny. SO(10) thus connects grand unification, neutrino masses, and possibly leptogenesis into a single framework.