# How do fixed points of the RGEs get perturbed in QFTs?

Coming from the bottom up, we can use the renormalization group equations to calculate if there are any fixed points and if yes, where they lie.

Fixed points correspond to scale invariant theories, i.e. theories where the beta functions vanish.

In solid-state physics the position of the fixed points are related to external parameters that can be controlled, like, for example, temperature and pressure. If the we tune these parameters to special values, namely the fixed point, we can observe the scale invariant behaviour.

What is controlling the positions of the fixed points in particle physics?

In other words, if we come from the top down, i.e. when we start with a scale invariant theory, how can the theory move away from the fixed point, if the theory is scale invariant? What is controlling the perturbation of the theory away from the fixed point and therefore the non-scale invariant behaviour?

As far as I know, in in particles physics one usually uses the normal RGEs to search for fixed points. However, there are no "external" control parameters and the behaviour is completely determined by the scale $\mu$. Thus the behaviour is quite different as in solid state physics. How can a theory by scale invariant at some scale $\mu_C$ and non-scale invariant at some lower scale?

## 3 Answers

In QFT, Renormalization is thought of a way to defined a theory in continuous space. There can be no grid spacing, i.e. no UV cut-off $\Lambda$.

The trick to do this is to choose a theory that lies exactly on the surface that is spawned by the relevant directions of an RG fixed point. This would be the critical surface if the RG flow was running backwards (i.e. if the cut-off was increased instead of lowered). Then the cut-off scale can be increased infinitely and the RG flow will just take us closer to the fixed point (which is only reached for $\Lambda\rightarrow \infty$).

This has cool implications:

• The theory is called renormalizable if the 'inverse critical manifold' has a finite dimension. In this case, you can pick a value for $\Lambda$ and a finite number of couplings to define a point on this manifold. Then the RG flow is well defined for any value of $\Lambda$ and you can start computing whatever you want to know.

• If your theory does not lie exactly on the 'inverse critical surface', then the RG flow eventually runs away as $\Lambda$ is increased beyond some value $\Lambda_0$. The large scales physics (for $\Lambda\ll \Lambda_0$ is however universal since the RG flow still passes close to the fixed point. This implies that we can make meaning full predictions about the observable world even if we don't really know what is happening at the smallest scales.

• You recover scale invariance on small scales since the RG flow gets infinitely close to the fixed point. On large scales however, the flow runs away from the fixed point and you can observe things like masses and correlation lengths.

You can think of it this way: It is an experimental fact that space is continuous. (Please don't mention 'beyond standard model physics', quantum gravity and all that stuff. It's another topic.) Then it must be possible to take the limit $\Lambda \rightarrow \infty$. It must therefore happen that the universe sits on some 'inverse critical manifold'. It's the job of the experimental physicist to measure where exactly. If the theory is renormalizable, then this will involve a finite number of measurements. Then the theoretician can start making predictions.

A theory cannot be scale invariant at some scale $\mu_C$ and non-scale invariant at some other scale: either a theory is completely scale invariant and there is no renormalization group flow, or it is not. In other words, scale invariance is a property of some quantum field theories, but not of others.

Quantum field theories that do not have this property still flow to a scale-invariant fixed point at low energy scales (or large distance scales), but this fixed point is completely determined by the theory and do not depend on "external" control parameters.

What particle physicists often do is to modify the particle content of a theory and try to understand what happens then. For instance in QCD, we know that with 6 types of quarks (like in the Standard Model), the low-energy physics displays confinement: only states that do not carry color are allowed. If on the contrary there were more types of quarks (say 16 of them), then the beta function could have a zero in the perturbative regime (see Banks-Zaks fixed point) and the low-energy theory would look very different from what we observe in nature, i.e. no confinement. "What happens when we change the number of QCD quarks between 6 and 16" are the kind of questions that some particle physicists are trying to answer.

In short, the positions of fixed points in particle physics are controlled by the particle content of the theory itself.

• Thanks for your answer. However, isn't there a contradiction if, on the one hand a scale invariant theory cannot be non-scale invariant at a lower scale (which sounds, of course, very reasonable) and on the other hand normal QFTs can flow to a fixed point and hence "become" scale invariant starting at this scale? – jak Jun 30 '17 at 4:50

There are two misconceptions in the statement of your question.

1. When you say "If the we tune these parameters to special values, namely the fixed point, we can observe the scale invariant behaviour" you are confusing critical point and fixed point which usually are two different points in theory space where the RG acts, see my previous answer: Critical 2d Ising Model
2. One does not perturb a fixed point by acting on it with the RG (which does nothing by definition) but by manually moving slightly away from it (and then acting with the RG), i.e., by adding a new term in the action. For example the $\phi^4$ model is obtained by adding a quartic term to the Gaussian fixed point.