 Solution

First interpret the displayed critical correlation function as the connected correlation functionAbove the transition at zero field the subtraction is zero; in an ordered phase it is essential. For a dimensionless source coupled to the total magnetization $M=\int d^{d}xm(x)$, differentiating $Z$ twice gives $\chi =V^{-1}(⟨M^2⟩-⟨M{⟩}^2)$. Translation invariance then proves the correlation-function susceptibility sum ruleInsert the scaling form, use spherical coordinates and set $s=r/\xi$:For $\eta <2$, finite nonzero $g(0)$ and an integrable large-$s$ tail, the last integral tends to a finite nonzero constant. Microscopic separations add an analytic response background. Thus the singular magnetic susceptibility obeys ${\chi }_s\asymp {\xi }^{2-\eta }$, and at zero field $\xi \asymp |t{|}^{-\nu }$ gives the Fisher scaling relationA dimensional magnetic field would add a nonsingular inverse-temperature prefactor. Integrating the full unsubtracted ordered-phase correlation function instead would yield a volume-divergent term $V⟨m{⟩}^2$, which is not the intrinsic susceptibility appearing in this exponent identity.