CFTs and RG Flow

The Space of Theories and RG Flow

Imagine a space where every point represents a distinct quantum field theory, defined by a specific set of coupling constants in its Lagrangian. The Renormalization Group (RG) provides a way to trace a path—a "flow"—through this space. This flow isn't arbitrary; it describes how the theory's parameters change as we move from high energies (short distances, UV) to low energies (long distances, IR).

The fundamental operation behind RG is coarse-graining, or integrating out high-energy degrees of freedom. To build a solid intuition for this, let's start with a conceptual video.

As the video illustrates, the RG flow maps out trajectories in the space of theories. The key insight is that these flows often converge. Even if we start from very different microscopic theories (different points in the parameter space), they might flow to the same theory at long distances. This is the deep reason for universality.

In quantum field theory, this "zooming out" corresponds to changing the energy scale $\mu$. The RG flow is mathematically described by the beta functions, ${\beta }_i$, which tell us how the coupling constants $g_i$ of the theory change with the energy scale:

A fixed point of the RG flow is a theory where all beta functions vanish, ${\beta }_i(g^{\ast })=0$. At such a point, the couplings no longer change with the energy scale. The theory is scale-invariant. Conformal Field Theories are the fixed points of the RG flow.

We can visualize this flow in the space of theories:

Driving the Flow: Perturbing a CFT

How does a flow begin? It starts when we perturb a CFT. As we saw in the last two lessons, this means adding a local operator ${\mathcal{O}}_i$ to the action of the CFT:

This perturbation breaks the scale invariance of the original CFT (unless the operator is exactly marginal), giving the coupling $g_i$ a non-zero beta function and initiating an RG flow. The CFT we started from is called the UV fixed point. The flow will then proceed towards the IR fixed point, which is the theory that describes the physics at very low energies.

The nature of this flow is determined by the scaling dimension ${\Delta }_i$ of the perturbing operator ${\mathcal{O}}_i$. We can briefly classify the possibilities (we will study this in detail in the next lesson):

• Relevant Operator (${\Delta }_i<d$): The perturbation becomes more important at low energies. It drives the theory away from the original CFT to a new IR fixed point. This is the most common scenario, describing, for instance, what happens when you tune the temperature away from $T_c$.
• Irrelevant Operator (${\Delta }_i>d$): The perturbation becomes less important at low energies. The theory flows back to the original CFT in the IR.
• Marginal Operator (${\Delta }_i=d$): The effect of the perturbation is independent of the scale, at least to first order. It can either deform the theory to a nearby CFT (creating a conformal manifold) or initiate a slow "logarithmic" flow.

So, a typical RG flow is a trajectory connecting a UV CFT to an IR CFT. A crucial question arises: are these flows reversible? Can we flow from CFT A to CFT B, and also from B to A?

The Irreversibility of RG Flow: Monotonicity Theorems

The answer is no. RG flow is an irreversible process, akin to the increase of entropy in thermodynamics. We lose information as we integrate out high-energy modes. This intuition is made precise by a set of powerful results known as monotonicity theorems or c-theorems. These theorems identify a quantity, which we can think of as a measure of the "number of degrees of freedom," that must monotonically decrease along any RG flow from a UV fixed point to an IR fixed point.

The c-Theorem in Two Dimensions

The most famous of these is Zamalodchikov's c-theorem for 2D QFTs. It states that for any RG flow connecting a UV CFT to an IR CFT, their respective central charges must satisfy:

The central charge $c$, which we will study in depth in Module 4, counts the effective number of massless degrees of freedom. The c-theorem provides a fundamental constraint on the possible dynamics of 2D quantum field theories.

The proof of this theorem is a beautiful piece of QFT logic. Let's study the argument presented in Zohar Komargodski's lecture notes, which relies on the properties of the stress-energy tensor.

The core of the argument is the construction of a function $C(r)$, where $r$ is the distance scale, such that:

1. $C(r\to 0)=c_{UV}$
2. $C(r\to \infty )=c_{IR}$
3. $\frac{dC}{dr}\le 0$

This is established by analyzing the two-point function of the trace of the stress-energy tensor, $⟨T(x)T(0)⟩$. In a CFT, the trace $T_{\mu }^{\mu }$ is zero (classically), but in a general QFT that is flowing between two CFTs, it is non-zero. The positivity of this correlator in a unitary theory leads directly to the inequality.

The logic in these notes can be further generalized. By introducing a background "dilaton" field $\tau (x)$ that couples to the trace of the stress tensor, $T_{\mu }^{\mu }$, one can show that the low-energy effective action for the dilaton must have a kinetic term whose coefficient is proportional to $c_{UV}-c_{IR}$. Unitarity requires this kinetic term to be positive, thus proving the theorem again from a different, powerful perspective related to effective field theory. This is detailed in section 5 of the same notes.

Monotonicity in Higher Dimensions

What about dimensions other than two? The central charge $c$ is specific to 2D. However, similar theorems exist in higher dimensions, where other quantities play the role of $c$.

• The a-Theorem in 4D: In four dimensions, the trace of the stress-energy tensor can have a quantum anomaly proportional to two terms, the Euler density $E_4$ and the square of the Weyl tensor $W^2$, with coefficients $a$ and $c$ respectively. It is conjectured (and proven under certain assumptions) that the $a$-anomaly coefficient is monotonic:

  $$a_{UV}>a_{IR}$$

  The coefficient $a$ can be related to the partition function on a 4-sphere, $Z_{S^4}$.

• The F-Theorem in 3D: In three dimensions, there is no trace anomaly. The monotonic quantity is believed to be the finite part of the free energy on a 3-sphere, denoted $F=-\log |Z_{S^3}|$. The F-theorem states:

  $$F_{UV}>F_{IR}$$

  Remarkably, $F$ is also equal to the universal part of the entanglement entropy across a circular boundary, providing a deep link between RG flow and quantum information.

These theorems provide a powerful, organizing principle for the space of all quantum field theories, establishing an irreversible, directional arrow for the renormalization group flow.

Conclusion

In this lesson, we have formalized the concept of moving between different physical regimes as a flow in the space of quantum field theories.

Key Takeaways:

• RG Flow: The Renormalization Group describes how a theory's parameters evolve from high energies (UV) to low energies (IR). This evolution is a trajectory, or "flow," in the abstract space of all possible theories.
• Fixed Points: Conformal Field Theories are the scale-invariant fixed points of the RG flow, where the beta functions of all couplings vanish.
• Flow Trajectories: An RG flow is typically a path from a UV CFT to an IR CFT, initiated by perturbing the UV theory with a relevant operator.
• Irreversibility: RG flow is irreversible. This is captured by monotonicity theorems (like the c-, a-, and F-theorems), which state that certain quantities measuring the "number of degrees of freedom" must decrease along any flow from UV to IR.

In our next lesson, we will complete this picture by focusing on the drivers of the flow. We will formally classify operators as relevant, irrelevant, or marginal based on their scaling dimensions and analyze their precise effect on RG flow.