Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 44 / 3 / i / Solution

For $0<g<g_{\ast }$, the positive beta function makes the running coupling increase toward the ultraviolet fixed point $g_{\ast }$. Assume a continuous locally Lipschitz beta function, a continuous anomalous dimension at the fixed point, and a finite nonzero reference value $C(r,g_{\ast })$. Then $g(t)\to g_{\ast }$ andThus the high-momentum factor has exponent $\gamma (g_{\ast })$ in $p^2$, and the propagator scales as $(p^2{)}^{-1+\gamma (g_{\ast })}$ up to slower corrections. If the fixed point is simple and attractive, ${\beta }^{\prime }(g_{\ast })<0$, the approach is exponential; suitable smoothness then gives $f(t)\sim A{e}^{2\gamma (g_{\ast })t}$ with finite $A$. The stated beta-function information alone supplies no numerical value for the anomalous dimension and does not exclude slower corrections at a nonsimple fixed point. This is two-point scaling at an ultraviolet fixed point.