Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 43 / 1

Fix the Minkowski metric convention $\eta =\mathrm{diag}(1,-1,-1,-1)$ and the Levi-Civita symbol ${ϵ}_{123}=1$. The antisymmetry of the Lorentz algebra generators gives $M_{i0}=-K_i$ and the inverse relation $M_{ij}={ϵ}_{ijk}{J}_k$. Inserting one temporal index in each generator immediately yieldsTwo Lorentz boosts therefore generate a rotation through their commutator; the minus sign distinguishes this algebra from the rotation algebra in four-dimensional Euclidean space.

For a spatial generator and a boost, the same Lorentz algebra givesContracting with ${ϵ}_{iab}/2$ givesThus the three Lorentz boosts transform as a spatial vector under rotations.

Finally, the all-spatial bracket becomesUse $M_{ab}={ϵ}_{abr}{J}_r$ in the double contraction with ${ϵ}_{iab}{ϵ}_{jcd}/4$. The epsilon contraction identity reduces it to $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$. One can check the sign directly: $J_1=M_{23}$ and $J_2=M_{31}$ give $[J_1,J_2]=i{M}_{12}=i{J}_3$; cyclic permutations give the other nonzero brackets. The requested coefficients areThese are rotation and boost commutators with a fixed metric signature. The PDF does not explicitly specify the signature. Keeping its generator convention but choosing $\eta =\mathrm{diag}(-1,1,1,1)$ reverses every displayed algebra coefficient: the pairs become $(1,0),(0,-1),(-1,0)$. More generally, if $s={\eta }_{00}=\pm 1$, the three nonzero coefficients are $A_1=-s$, $B_2=s$ and $A_3=s$. Specifying the Minkowski metric is therefore necessary to make the numerical signs unambiguous.

The hierarchy problem begins with a large separation of scales: the weak scale is much smaller than possible unification or gravitational scales. A fundamental scalar mass is especially sensitive to a much heavier scale. In an effective description with ultraviolet cutoff $\Lambda$, one schematically findsThe coefficient $c$ contains gauge, scalar and Yukawa couplings, with different signs for bosonic and fermionic loops. The explicit quadratic cutoff dependence is regulator-dependent, but a heavy physical particle coupled to the Higgs produces a threshold correction of order its mass squared. That heavy-threshold sensitivity is the physical difficulty.

The technical hierarchy problem asks whether a small scale, once chosen, remains stable under such radiative corrections. Maintaining $m_H^2\ll {\Lambda }^2$ by cancelling unrelated bare and loop contributions, and readjusting the cancellation at successive orders, is the naturalness concern. It is different from explaining why the small scale was chosen in the first place. Supersymmetry primarily supplies protection against the technical instability; a complete explanation of the origin of supersymmetry-breaking scales needs additional dynamics.

The relevant comparison of mass types is:

In exact supersymmetry, each supermultiplet has matched bosonic and fermionic degrees of freedom, with their interaction strengths related. Opposite loop signs then give supersymmetric cancellation of quadratic divergences. This is not just equality of state counts: the supersymmetric relation between quartic and Yukawa couplings is also essential. A schematic paired-loop contribution has the high-momentum formFor equal partner masses the displayed contributions cancel. With a small splitting, the difference falls as $(m_F^2-m_B^2)/p_E^4$, so the remaining ultraviolet sensitivity is logarithmic, proportional to the splitting rather than to ${\Lambda }^2$.

Realistic partners need not have exactly equal masses. Soft supersymmetry breaking permits scalar squared masses, gaugino masses and suitable trilinear interactions without restoring the unwanted quadratic sensitivity. Typically,up to coefficients and threshold details. Squarks and sleptons can therefore be heavier than their chiral fermion partners while the scalar sector remains stable against the much larger ultraviolet scale. Nevertheless, very large soft masses, particularly in the Higgs-coupled sector, leave large finite or logarithmic corrections and require tuning. This soft scalar-mass sensitivity remains after the quadratic divergence has cancelled. Supersymmetry solves the quadratic radiative instability when breaking is suitably soft; it does not make arbitrarily heavy superpartners natural or explain the entire hierarchy by itself.