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 Incoming links: Soft scalar-mass sensitivity

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 43 / 1 / b / Solution Created 2026-10-03 Updated 2026-10-07

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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

Λ

, one schematically finds

m_{H}^{2} = m_{H, bare}^{2} + δ m_{H}^{2}, δ m_{H}^{2} \sim \frac{c}{16 π ^{2}} Λ^{2} .

(1)

The 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} ≪ Λ^{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:

For a gauge boson, an explicit Proca mass term is forbidden by an unbroken gauge symmetry. This is gauge protection of a vector mass. In a Higgs phase, a vector mass is of order $gv$ , but the stability of $v$ then depends on the scalar mass that determines the symmetry-breaking scale. Gauge invariance does not, by itself, solve that scalar problem.
A chiral fermion mass is protected because setting it to zero restores an appropriate chiral symmetry. Perturbative corrections cannot generate a symmetry-forbidden mass; schematically $δ m_{f} \propto m_{f} lo g Λ$ . In the Standard Model, chiral gauge quantum numbers prohibit a bare fermion mass, and the Higgs mechanism permits masses through Yukawa couplings. This is chiral protection of a fermion mass, rather than an additive correction of order $Λ$ .
A squark or slepton is a scalar. Its bilinear $f^{†} f$ is allowed by gauge symmetries and by the ordinary chiral phase symmetry of its fermion partner. Consequently, without supersymmetry, those symmetries do not protect a small scalar mass against corrections of order $Λ^{2}$ .

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 form

δ m_{H}^{2} \propto ∣ y ∣^{2} \int^{Λ} \frac{d ^{4} p _{E}}{( 2 π ) ^{4}} [\frac{1}{p _{E}^{2} + m _{B}^{2}} - \frac{1}{p _{E}^{2} + m _{F}^{2}}] .

(2)

For 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

Λ^{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,

δ m_{H}^{2} \sim \frac{∣ y ∣ ^{2}}{16 π ^{2}} m_{soft}^{2} lo g \frac{Λ}{m _{soft}},

(3)

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.

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