What does mixing under RG flow mean?

[Pacopag]

what does "mixing under RG flow" mean?

I'm trying to understand the language people use when they talk about RG flow. Here's what I understand:
We have the RG flow equations governing the couplings' dependence on energy scale
[tex]\mu {dg_i \over d\mu}=\beta(g_j)[\tex],
and we can find the "fixed points" by solving
[tex]\beta(g_j)[\tex]=0[\tex].
We can linearize to find "stable" and "unstable" trajectories in coupling space, emerging from these fixed points.

What I'm not so sure of is what "RG flow" means. I'm guessing that if we move away from the fixed point, then vary the energy scale, we will move along whatever trajectory we are sitting on. Is this what RG flow means?

My real question is this: What does it mean for two operators to "mix under RG flow"?
Thanks for any help.

 

[fzero]

What I'm not so sure of is what "RG flow" means. I'm guessing that if we move away from the fixed point, then vary the energy scale, we will move along whatever trajectory we are sitting on. Is this what RG flow means?

Basically, yes. We imagine a coordinate system whose axes correspond to the coupling constants of the theory. Solutions to the beta function equations generally produce a manifold of points that correspond to the allowed configurations of coupling constants. RG flows are trajectories on the RG manifold. These paths are parameterized by the energy scale. The RG flow is analogous to the way a classical system will roll down a potential energy function to seek a stable equilibrium. RG fixed points are like local minima of the potential energy.

My real question is this: What does it mean for two operators to "mix under RG flow"?
Thanks for any help.

If we consider correlation functions of composite operators constructed from the fundamental fields of the theory, we can use perturbation theory to compute them. In the loop expansion, we will encounter divergences that must be removed through a renormalization scheme. This will involve adding counterterms to cancel the divergences. These counterterms involve the correlation functions of operators of the same or lower scaling dimension as the operator we started with. For instance, if we have a scalar field theory, we might be considering the correlator

$$\langle 0 | \phi(x) \phi(y) \phi^3(z) |0\rangle.$$

Counterterms that could appear in the renormalization are expressions like

$$\langle 0 | \phi(x) \phi(y) ( m \phi^2(z)) |0\rangle$$

or

$$\langle 0 | \phi(x) \phi(y) ( \Box \phi(z)) |0\rangle .$$

One concludes that the renormalized correlation function is computed with the renormalized operator

$$[ \phi^3 ] = Z_0 \phi^3 + Z_1 m \phi^2 + Z_2 \Box \phi + \cdots.$$

This is what we mean by operator mixing under the RG group. In general, only operators with the same symmetries can mix.

 

[Pacopag]

Wow! That's an excellent answer. I wish I could buy you a beer for that one. Thanks very much.