Classifying Operators in RG Flow

Hello! Let's continue our exploration of the Renormalization Group.

In our last lesson, we established that RG flows describe the evolution of a physical system from high to low energies, with CFTs acting as the scale-invariant fixed points of this evolution. We saw that a flow is initiated when we perturb a CFT with an operator, and that these flows are irreversible, as quantified by monotonicity theorems like the 2D c-theorem.

Today, we will precisely analyze the perturbations themselves. Our goal is to classify operators as relevant, irrelevant, or marginal based on their scaling dimensions and their effect on RG flow. This classification is the key to understanding the stability of fixed points and why fine-tuning is sometimes necessary to observe critical phenomena.

The Dimensionality of a Perturbation

Let's start by considering a CFT, described by an action $S_{C F T}$ , which we perturb by adding a scalar operator $O (x)$ with a coupling constant $λ$ :

S = S_{C F T} + λ \int d^{d} x O (x)

In natural units ( $ℏ = c = 1$ ), the action $S$ is a dimensionless quantity. Let's use this fact to determine the units, or mass dimension, of the coupling $λ$ .

The spacetime integral $\int d^{d} x$ has a mass dimension of $- d$ , since each coordinate $x^{μ}$ has a mass dimension of $- 1$ .
In the previous module, we established that an operator $O$ with a two-point function $⟨ O (x) O (0)⟩ \sim ∣ x ∣^{- 2Δ}$ has a mass dimension of $Δ$ .

For the perturbation term $λ \int d^{d} x O (x)$ to be dimensionless, the dimensions must cancel. Denoting the mass dimension of a quantity $Q$ as $[Q]$ , we have:

[λ] + [\int d^{d} x] + [O] = 0

[λ] - d + Δ = 0

This gives us the fundamental relation for the mass dimension of the coupling constant:

[λ] = d - Δ

This simple equation is incredibly powerful. It tells us how the "strength" of the perturbation inherently scales with energy. A coupling with a positive mass dimension will be large at high energies and small at low energies, and vice versa. However, in RG, we are interested in how a dimensionless coupling evolves. Let's define a dimensionless coupling $g$ by factoring out the energy scale $μ$ :

λ = g \cdot μ^{d - Δ}

The RG flow describes how $g$ changes as we change the energy scale $μ$ . The beta function for $g$ , at the classical level (ignoring quantum corrections), is given by how $g$ must change to keep $λ$ fixed if we change $μ$ :

β (g) = μ \frac{d g}{d μ} = - (d - Δ) g = (Δ - d) g

This tells us how $g$ changes as we increase the energy scale $μ$ (flow towards the UV). To see how it behaves when we flow to the IR (decreasing $μ$ ), it's often more intuitive to define a "flow parameter" $t = - ln (μ / μ_{0})$ , which increases as we go to the IR. The flow equation becomes $\frac{d g}{d t} = (d - Δ) g$ . This equation's solution reveals everything we need:

If $Δ < d$ , then $d - Δ > 0$ . The exponent is positive, so $g (t) = g_{0} e^{(d - Δ) t}$ grows as we flow to the IR ( $t \to \infty$ ).
If $Δ > d$ , then $d - Δ < 0$ . The exponent is negative, so $g (t) = g_{0} e^{(d - Δ) t}$ decays to zero as we flow to the IR.
If $Δ = d$ , then $d - Δ = 0$ . The coupling $g$ is constant at this level. Its fate depends on quantum corrections.

This behavior underpins the classification of operators.

Formal Classification and RG Stability

Let's now read a formal definition and see how it applies to a concrete example from statistical mechanics.

EPFL Lectures on Conformal Field Theory in D ⩾ 3 ...

The following text from Slava Rychkov's lecture notes provides a crisp definition of relevant, irrelevant, and marginal operators and illustrates their role within the context of the \lambda\phi^4 theory, which describes the Wilson-Fisher fixed point.

Please read the text block starting from 'In D < 4, both φ² and φ⁴ are relevant operators...' and ending just before 'The hierarchy problem...'. Pay close attention to the footnote that explicitly defines the three operator classes based on their dimension \Delta relative to the spacetime dimension D. Also, note the discussion on how the presence of a relevant operator at a fixed point leads to instability and requires fine-tuning.

Based on our derivation and the reading, we can now make the classification explicit:

Relevant Operator: $Δ < d$
- The coupling grows towards the IR.
- Effect on Flow: A relevant perturbation is a powerful one. It drastically alters the low-energy physics, driving the system away from the original UV fixed point toward a new IR fixed point. From the perspective of the UV fixed point, relevant operators correspond to unstable directions in the space of theories. To remain at the fixed point (i.e., to observe criticality), one must "fine-tune" the couplings of all relevant operators to zero. In the context of the Ising model, the operator corresponding to the energy density has $Δ < d$ , and its coupling is proportional to $(T - T_{c})$ . To stay at the critical point, we must tune its coupling to zero by setting $T = T_{c}$ .
Irrelevant Operator: $Δ > d$
- The coupling shrinks to zero towards the IR.
- Effect on Flow: An irrelevant perturbation becomes negligible at low energies. If you perturb a CFT with an irrelevant operator, the RG flow will simply take you back to the original fixed point in the IR. These operators modify the UV physics but do not change the universal long-distance behavior. They correspond to stable directions of the RG flow. This is the mathematical reason for universality: many different microscopic theories (differing by their irrelevant couplings) flow to the same IR fixed point.
Marginal Operator: $Δ = d$
- The coupling is classically dimensionless and does not change under RG flow at the tree level. Its fate is decided by quantum (loop) corrections.
- Effect on Flow: There are two main possibilities:
  1. Truly (or Exactly) Marginal: If the beta function is zero to all orders in perturbation theory, the operator deforms the CFT into a new, distinct CFT. Such operators generate a continuous family of fixed points, known as a conformal manifold. The value of the marginal coupling simply parameterizes your position on this manifold. A classic example is the gauge coupling in $N = 4$ Super Yang-Mills theory.
  2. Marginally Relevant/Irrelevant: If quantum corrections generate a non-zero beta function, the coupling will "slowly" flow, either growing (marginally relevant) or shrinking (marginally irrelevant) logarithmically with the energy scale. The coupling in 4D QCD is a famous example of a marginally irrelevant coupling, leading to asymptotic freedom (the theory becomes free in the UV).

This framework provides a complete picture of the local geometry of RG flows around a fixed point.

The View from Two Dimensions

This classification is universal, but it's worthwhile to state it for the $d = 2$ case, which is central to your interests in string theory.

4. Introducing Conformal Field Theory

These concepts apply in any dimension. Let's look at the specific case of two-dimensional CFTs, which are central to string theory. David Tong's notes provide a clear summary.

Please read the subsection beginning 'One can consider perturbing a conformal field theory...', focusing only on the three bullet points that classify perturbations based on the operator dimension \Delta relative to d=2. This succinctly recaps our discussion in the 2D context.

Test your understanding!

Imagine you are studying a 4-dimensional CFT. You discover three scalar primary operators: $O_{1}$ with $Δ_{1} = 3.8$ , $O_{2}$ with $Δ_{2} = 4.0$ , and $O_{3}$ with $Δ_{3} = 5.2$ . You perturb the theory with a linear combination $S_{p er t} = \int d^{4} x (g_{1} O_{1} + g_{2} O_{2} + g_{3} O_{3})$ .

Classify each operator.
What must you do to the initial couplings $g_{i}$ in the UV to ensure your theory flows to the original CFT in the IR?

Show answer

Classification:
- $O_{1}$ : $Δ_{1} = 3.8 < 4$ , so it is a relevant operator.
- $O_{2}$ : $Δ_{2} = 4 = 4$ , so it is a marginal operator.
- $O_{3}$ : $Δ_{3} = 5.2 > 4$ , so it is an irrelevant operator.
Flow to the original CFT:
To ensure the RG flow returns to the original fixed point in the IR, we must eliminate any perturbations that grow or do not decay.
- The irrelevant coupling $g_{3}$ will naturally decay to zero in the IR, so we don't need to do anything about it. This is a stable direction.
- The relevant coupling $g_{1}$ will grow exponentially in the IR, driving the theory away from the fixed point. To prevent this, we must fine-tune its initial value to zero: $g_{1} = 0$ .
- The marginal coupling $g_{2}$ is more subtle. If it is truly marginal, its value simply selects a different CFT on a conformal manifold. If we want to return to the original CFT, we must also set $g_{2} = 0$ . If it is marginally relevant or irrelevant, it will cause a slow drift. To be safe and ensure stability, one must typically set the coefficients of all relevant and marginal operators to zero. The crucial step is the fine-tuning of the relevant coupling $g_{1}$ .

Conclusion

We have now established a powerful and predictive framework for understanding the consequences of perturbing a conformal field theory.

Key Takeaways:

The behavior of a perturbation $λ \int d^{d} x O (x)$ under RG flow is determined by the scaling dimension $Δ$ of the operator relative to the spacetime dimension $d$ .
Relevant operators ( $Δ < d$ ) correspond to unstable directions. Their couplings grow in the IR, driving the theory to a new fixed point. Reaching an unstable fixed point requires fine-tuning their couplings to zero.
Irrelevant operators ( $Δ > d$ ) correspond to stable directions. Their couplings vanish in the IR, meaning the long-distance physics is insensitive to them. This is the foundation of universality.
Marginal operators ( $Δ = d$ ) are classically scale-invariant. They can lead to continuous families of CFTs (conformal manifolds) or induce slow, logarithmically running flows.

In the previous lesson, we saw that systems flow towards common IR fixed points. Today, we've identified the mechanism: the irrelevance of most operators washes out the microscopic details, while the relevance of a few key operators determines the phase structure and the need for fine-tuning.

In our next lesson, we will build on this to describe the principle of universality classes and their classification by CFTs. We'll see how entire families of different physical systems can be grouped into classes that are all described by the same CFT at their critical point.

Recommended resources

The Dimensionality of a Perturbation

Formal Classification and RG Stability

The View from Two Dimensions

Conclusion