Dispersion-theory approach to the correlation function for critical scattering

A dispersion-theory approach provides an exact representation of the two-point correlation function $G(\overrightarrow{\mathrm{q}},t)=C{t}^{-\gamma }g\left(q^2{\xi }^2\right)$, where $t=\frac{(T-T_c)}{T_c}$ and $\xi$ is the correlation length, which describes the critical scattering near a second-order phase transition. The threshold property of the spectral weight function $F\left(\frac{x}{3}\right)\propto \mathrm{Im}{g}^{-1\mathrm{}}(-x^2)$ is used to motivate an approximation to the scaling function $g\left(x^2\right)$ based on the asymptotically exact Fisher-Langer approximant $g_{\mathrm{FL}}\left(x^2\right)=\left(\frac{C_1}{x^{2-\eta }}\right)(1+\frac{C_2}{x^{\frac{(1-\alpha )}{\nu }}}+\frac{C_3}{x^{\frac{1}{\nu }}})$, where $\alpha$, $\eta$, $\nu$ are the usual critical exponents. The approximation consists of truncating the spectral function associated with $g_{\mathrm{FL}}^{-1\mathrm{}}\left(x^2\right)$ in a manner designed to simulate the known threshold property of the exact spectral function, namely $F\left(\frac{x}{3}\right)=0\mathrm{}$ for $x\le 3$. The new approximant is checked on the two-dimensional Ising model and shown to agree with the exact result to better than 0.03% for all $x$. Near four dimensions, the agreement with exact results is also excellent. A phenomenological approach to the intermediate three-dimensional case is presented, and shown to be in good agreement with high-temperature series and $\epsilon$-expansion results. The dispersion-theory approach is also shown to provide a convenient framework for calculations within the $\epsilon$ expansion and is used to compute $g\left(x^2\right)$ to $O\left({\epsilon }^3\right)$, a new result.