Past exam of the quantum field theory course of the University of Oxford / 2018 / 3 / d

For a fixed point $g^{\ast }$ with $\beta (g^{\ast })=0$, put $g=g^{\ast }+h$. To first order in the displacement,Hence ${\beta }^{\prime }(g^{\ast })<0$ means ultraviolet attraction, since $h\to 0$ as $\mu \to \infty$; ${\beta }^{\prime }(g^{\ast })>0$ means infrared attraction, since $h\to 0$ as $\mu \to 0$. If the derivative vanishes, the linear test is inconclusive and higher terms determine the direction of flow.

For $\delta =4-D>0$ and $b=3/(16{\pi }^2)$, the retained beta function is $-\delta g+b{g}^2$. Its two zeros and slopes areThus the positive interacting Wilson-Fisher fixed point isand is infrared attractive in the coupling direction. Couplings between zero and $g^{\ast }$ increase towards $g^{\ast }$ when the scale decreases; those just above $g^{\ast }$ decrease towards it. The Gaussian fixed point $g=0$ is ultraviolet attractive for $D<4$.

The conclusion is local and perturbative. For $4-D\ll 1$, $b{g}^{\ast }=4-D$ is small, making the expansion controlled; extrapolation far below four dimensions requires more information. It is also a statement about the displayed coupling flow, not stability against every possible deformation of the full theory. In particular, the mass term is a relevant perturbation of the critical theory and must be tuned to reach the massless critical surface. Calling this fixed point infrared stable must not erase that additional relevant direction.