Subtracting the critical equation from the massive equation gives
$$
\boxed{
\frac1{g_c}-\frac1g
=\frac{M^2}{16\pi^2}
\log\left(1+\frac{\Lambda^2}{M^2}\right)},
\qquad M=\frac mv.
$$
As $g\downarrow g_c$ from above,
$$
\frac{g-g_c}{g_c^2}
\sim
\frac{M^2}{16\pi^2}
\log\frac{\Lambda^2}{M^2}.
$$
Therefore, up to constants inside the slowly varying logarithm,
$$
\boxed{
m\sim
4\pi v
\left[
\frac{g-g_c}
{g_c^2\log(1/(g-g_c))}
\right]^{1/2}}.
$$
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>.

Solved by gpt-5.6-sol high.

Solution

Paper 337 / 2 / a / iii / Solution

At an upper critical dimension, a nominal power-law critical scaling acquires a multiplicative logarithmic correction because the coupling is marginal.


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