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 Incoming links: Upper-critical-dimension logarithmic correction

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 337 / 2 / a / iii / Solution Created 2026-09-24 Updated 2026-09-25

Subtracting the critical equation from the massive equation gives

\frac{1}{g _{c}} - \frac{1}{g} = \frac{M ^{2}}{16 π ^{2}} lo g (1 + \frac{Λ ^{2}}{M ^{2}}), M = \frac{m}{v} .

g ↓ g_{c}

from above,

\frac{g - g _{c}}{g _{c}^{2}} \sim \frac{M ^{2}}{16 π ^{2}} lo g \frac{Λ ^{2}}{M ^{2}} .

Therefore, up to constants inside the slowly varying logarithm,

m \sim 4 π v [\frac{g - g _{c}}{g _{c}^{2} lo g ( 1/ ( g - g _{c} ))}]^{1/2} .

The square-root mean-field power is modified by a logarithm. It is therefore not a pure quantum-critical power law; this is an upper-critical-dimension logarithmic correction.

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