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Write ${\mathbf{\sigma }}_i=\mathbf{m}+\delta {\mathbf{\sigma }}_i$ and neglect products of two fluctuations. ThenEach site has $q$ neighbours and each bond is counted once, so the mean-field approximation givesChoose the $x$ axis along $\mathbf{m}$. For the four-state clock model, the single-site partition function isConsequentlyand therefore

The mean-field approximation replacesby its first three terms. Since every site has $q$ neighbours,The sites then decouple, and their three possible weights sum toHence $c_1=1$, $c_2=2$, and

At fixed uniform $\varphi$, the nonconserved variable $p$ can relax freely, so mean-field approximation minimizeswith respect to $p$. The stationary solutions obeyThus $p=0$ when $A+c\varphi \ge 0$, while $p^2=-(A+c\varphi )/B$ when $A+c\varphi <0$. Substitution givesWriting this in the requested form yieldsbecause $A+c\varphi =c(\varphi -{\varphi }_o)$. The value ${\varphi }_o$ is the composition at which the coefficient of $p^2$ changes sign, so it is the mean-field threshold for polar ordering.

A self-consistency equation requires the average field assumed in a mean-field approximation to equal the expectation value calculated in that field.