SUSY solves the hierarchy problem by cancelling the quadratic divergences in the Higgs mass. For example, the top quark loop contributes delta m_H^2 ~ -3 y_t^2 Lambda^2 / (8 pi^2) to the Higgs mass-squared. How does the top squark (stop) cancel this?
AThe stop has opposite electric charge and its loop contribution has opposite sign
BThe stop is a scalar with the same Yukawa coupling y_t, and its loop contributes delta m_H^2 ~ +3 y_t^2 Lambda^2 / (8 pi^2) — the opposite sign arises because fermion loops and boson loops contribute with opposite signs to the Higgs self-energy; if the stop and top have the same coupling and mass, the cancellation is exact
CThe stop absorbs the top quark's contribution through mixing
DThe stop loop is suppressed by a factor of 1/Lambda^2 that cancels the Lambda^2 from the top loop
In supersymmetry, for every fermion loop that contributes +Lambda^2, there is a boson (superpartner) loop that contributes -Lambda^2, and vice versa. The relative sign comes from the minus sign for closed fermion loops in Feynman diagrams. If SUSY is exact (superpartners have the same mass as their partners), the cancellation is perfect and there are no quadratic divergences at all. When SUSY is broken (superpartners are heavier), the cancellation is imperfect and gives delta m_H^2 ~ (m_stop^2 - m_top^2) * y_t^2 * ln(Lambda) / (8 pi^2). For the hierarchy problem to be solved 'naturally,' the stop mass should be within a factor of ~10 of the top mass, suggesting m_stop below ~1-2 TeV.
Question 2 Short Answer
In the MSSM with R-parity conservation, the lightest supersymmetric particle (LSP) is stable. If the LSP is the lightest neutralino (a mixture of the superpartners of the photon, Z, and Higgs bosons), it is a natural dark matter candidate. Why?
Think about your answer, then reveal below.
Model answer: The lightest neutralino has the right properties for a WIMP (weakly interacting massive particle) dark matter candidate: it is electrically neutral (it doesn't emit or absorb light), it is stable (R-parity prevents its decay into SM particles), it interacts via the weak force (giving detectable cross sections in direct detection experiments and the right annihilation rate in the early universe), and its mass can naturally be in the 100 GeV - 1 TeV range. The 'WIMP miracle' is that a particle with weak-scale mass and weak-scale couplings naturally produces a thermal relic abundance consistent with the observed dark matter density Omega_DM ~ 0.26 — the annihilation cross section sigma*v ~ 3 x 10^{-26} cm^3/s is in the right ballpark for a weakly interacting particle with mass ~100 GeV.
While the WIMP miracle is suggestive, it is not proof. Direct detection experiments (XENON, LZ, PandaX) have excluded much of the natural WIMP parameter space without a detection. The remaining allowed regions include well-tempered neutralinos, co-annihilation scenarios, and resonance regions. If the LSP is a gravitino or axino instead of a neutralino, the phenomenology changes entirely.
Question 3 Multiple Choice
The MSSM has 105 new free parameters beyond the Standard Model's 19. Despite this, SUSY is considered a predictive framework. How is this possible?
ABecause most of the parameters are unmeasurable
BBecause specific SUSY-breaking mechanisms (gravity mediation, gauge mediation, anomaly mediation) relate the 105 parameters to a small number of inputs at a high scale — for example, the constrained MSSM (CMSSM) reduces the parameters to just 5 (m_0, m_{1/2}, A_0, tan beta, sign(mu)), which then predict the entire superpartner spectrum through renormalization group evolution
CBecause the 105 parameters are all very small
DBecause experiments can only measure a few parameters at a time
The large number of MSSM parameters reflects our ignorance of how SUSY is broken. Specific breaking mechanisms impose relations among the soft SUSY-breaking parameters at the mediation scale, reducing the parameter count dramatically. Gravity mediation gives 5 parameters (CMSSM/mSUGRA), gauge mediation gives 6, anomaly mediation gives 3-4. Each scenario predicts a distinctive pattern of superpartner masses and mixing angles. LHC searches have excluded the simplest versions of each scenario in significant regions of parameter space, but more complex (and less constrained) scenarios remain viable.