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(→q,t)=Ct^-γg(q²ξ²), where t=(T-Tc)Tc and ξ is the correlation length, which describes the critical scattering near a second-order phase transition. The threshold property of the spectral weight function F(x/3)∝Img^-1(-x²) is used to motivate an approximation to the scaling function g(x²) based on the asymptotically exact Fisher-Langer approximant ${g}FL({x}²)=({{C}₁}{{x}^{2{-}{η}}})(1+{{C}₂}{{x}^{{(1{-}{α})}{{ν}}}}+{{C}₃}{{x}^{{1}{{ν}}}})$, where ${α}$, ${η}$, ${ν}$ are the usual critical exponents. The approximation consists of truncating the spectral function associated with ${g}FL^{{-}1}({x}²)$ in a manner designed to simulate the known threshold property of the exact spectral function, namely $F(x/3)=0$ for $x{≤}3$. 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 ε-expansion results. The dispersion-theory approach is also shown to provide a convenient framework for calculations within the ε expansion and is used to compute g(x²) to O(ε³), a new result.