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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 307 / 3 / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

At one loop, plot each inverse squared coupling

g_{a}^{- 2}

against

lo g μ

. The three Standard Model lines have slopes that do not pass through one common point. Above the superpartner threshold, additional scalar and fermion vacuum-polarization diagrams change the slopes: for example, an

S U (2)

gauge-boson two-point function receives a loop from a left-handed Standard Model fermion doublet and an additional loop from its scalar superpartner, while the

S U (2)

gauge multiplet adds a gaugino loop alongside the gauge-boson and ghost loops. With the Minimal supersymmetric Standard Model field content, the three resulting straight lines meet to good accuracy. This is supersymmetric gauge coupling unification.

The differential equation

\frac{d g _{a}}{d l o g μ} = β_{a} g_{a}^{3}

(1)

implies

\frac{d}{d l o g μ} g_{a}^{- 2} = - 2 β_{a} .

(2)

Hence

g_{a}^{- 2} (μ) = g_{a}^{- 2} (μ_{0}) - 2 β_{a} lo g \frac{μ}{μ _{0}},

(3)

g_{a} (μ) = \frac{g _{a} ( μ _{0} )}{1 - 2 β _{a} g _{a}^{2} ( μ _{0} ) lo g ( μ / μ _{0} )} .

(4)

Let

L = lo g (M_{G U T} / M_{Z})

. Equality of

g_{1}

and

g_{2}

at the unification scale gives

g_{1}^{- 2} (M_{Z}) - 2 β_{1} L = g_{2}^{- 2} (M_{Z}) - 2 β_{2} L,

(5)

M_{G U T} = M_{Z} exp [\frac{g _{1}^{- 2} ( M _{Z} ) - g _{2}^{- 2} ( M _{Z} )}{2 ( β _{1} - β _{2} )}] .

(6)

Similarly,

g_{3}^{- 2} (M_{Z}) - g_{2}^{- 2} (M_{Z}) = 2 (β_{3} - β_{2}) L,

(7)

while

g_{2}^{- 2} (M_{Z}) - g_{1}^{- 2} (M_{Z}) = 2 (β_{2} - β_{1}) L .

(8)

Eliminating

L

yields

g_{3}^{- 2} (M_{Z}) = g_{2}^{- 2} (M_{Z}) + A [g_{2}^{- 2} (M_{Z}) - g_{1}^{- 2} (M_{Z})],

(9)

with

A = \frac{β _{3} - β _{2}}{β _{2} - β _{1}} .

(10)

For the GUT-normalized MSSM one-loop coefficients proportional to

(33/5, 1, - 3)

, this is

A = 5/7

Current coupling measurements approximately satisfy this MSSM relation and point to

M_{G U T}

of order

1 0^{16}

GeV, much more accurately than nonsupersymmetric one-loop running. Exact equality is not expected because two-loop evolution and threshold corrections from split superpartner and GUT-scale masses shift the lines; the absence so far of directly observed superpartners also prevents the threshold spectrum from being fixed experimentally.

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