Deriving φ⁴ RG Flow: Momentum-Shell Integration

Recommended resources

The freely available resources this lesson will direct you to.

Hello! In our last lesson, we established the formal three-step momentum-shell renormalization group (RG) procedure for the Landau-Ginzburg-Wilson $ϕ^{4}$ Hamiltonian. This gave us a clear road map: integrate, rescale momenta, and rescale the field.

Today, we will execute that procedure. Our goal is to perform the momentum-shell integration and rescaling steps to derive the RG flow equations for $ϕ^{4}$ theory at one-loop order. This is where the conceptual framework becomes a powerful computational tool. We will perform the crucial integration step using perturbation theory, which will reveal how the parameters of our theory evolve as we zoom out to larger length scales. These evolution equations are the famous RG "beta functions," and their structure is the key to understanding critical phenomena.

Recap: The Setup

Recall our starting point, the LGW Hamiltonian in momentum space with a cutoff $Λ$ :

H [ϕ] = \int^{Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} [\frac{1}{2} (r + q^{2}) ∣ ϕ (q) ∣^{2}] + u \int^{Λ} \dots \int^{Λ} \frac{d ^{d} q _{1} \dots d ^{d} q _{4}}{( 2 π ) ^{4 d}} (2 π)^{d} δ (\sum q_{i}) ϕ (q_{1}) ϕ (q_{2}) ϕ (q_{3}) ϕ (q_{4})

To be consistent with standard literature and the excellent notes we will be using, let's adopt a slightly different but equivalent notation for the parameters: $r \to μ_{0}^{2}$ and $u \to g_{0}$ . The subscript '0' denotes the "bare" parameters at the cutoff scale $Λ$ .

Our task is to perform the RG transformation $R_{b}$ , which involves integrating out the "fast" modes $ϕ_{>}$ with momenta in the shell $Λ/ b < ∣ q ∣ < Λ$ , and seeing how this affects the effective Hamiltonian for the "slow" modes $ϕ_{<}$ .

1. Step 1: Integrating the Fast Modes

The core of the calculation is the integral over the fast modes:

e^{- H^{'} [ϕ_{<}]} = \int D ϕ_{>} e^{- H [ϕ_{<}, ϕ_{>}]} = e^{- H_{0} [ϕ_{<}]} \int D ϕ_{>} e^{- H_{0} [ϕ_{>}] - H_{in t} [ϕ_{<}, ϕ_{>}]}

where $H_{0}$ is the Gaussian part and $H_{in t}$ is the $g_{0} ϕ^{4}$ term. The integral can be rewritten using an expectation value over the Gaussian distribution of the fast modes:

H^{'} [ϕ_{<}] = H_{0} [ϕ_{<}] - ln ⟨ e^{- H_{in t} [ϕ_{<}, ϕ_{>}]} ⟩_{ϕ_{>}}

Since we cannot compute this exactly, we use a perturbative expansion in the coupling $g_{0}$ . This is the cumulant expansion you may have encountered before:

ln ⟨ e^{- V} ⟩ = - ⟨ V ⟩ + \frac{1}{2} (⟨ V^{2} ⟩ - ⟨ V ⟩^{2}) + O (V^{3})

Applying this to our problem, we will find corrections to the effective Hamiltonian $H^{'} [ϕ_{<}]$ at each order in $g_{0}$ . These corrections represent the effects of the fast modes we integrated out.

To perform this calculation, we need to handle averages of products of fields, like $⟨ ϕ_{>} ϕ_{>} ⟩$ and $⟨ ϕ_{>} ϕ_{>} ϕ_{>} ϕ_{>} ⟩$ . The tool for this is Wick's theorem, which states that for a Gaussian distribution, the average of a product of fields is the sum of all possible pairwise contractions (pairings). Each pair $⟨ ϕ_{>} (q) ϕ_{>} (- q)⟩$ gives a propagator $G_{0} (q) = \frac{1}{μ _{0}^{2} + q ^{2}}$ for a mode in the high-momentum shell.

The following resource provides an exceptionally clear derivation of this perturbative RG procedure. It will walk you through the calculation, introducing Feynman diagrams as a powerful way to organize the various terms.

[PDF] 3. The Renormalisation Group

This section from David Tong's notes, titled 'RG with Interactions', is the heart of our lesson. It meticulously carries out the perturbative integration, calculates the one-loop corrections to the couplings, and uses Feynman diagrams to visualize the process.

Please read Section 3.4, 'RG with Interactions' (up to, but not including, 3.4.4 'Beta Functions'). Pay close attention to: How the interaction term is expanded and which terms contribute to the average (3.4.1). The first-order correction (order g_0) which renormalizes the mass term \mu^2. This is the simplest loop diagram. The introduction of Wick's theorem and how it's used to calculate the second-order correction (order g_0^2) that renormalizes the coupling constant g. The correspondence between the calculated terms and the Feynman diagrams presented. This visual language is invaluable.

Summary of the Integration Results

After working through the reading, you will have seen how integrating out the fast modes $ϕ_{>}$ generates corrections to the parameters in the effective Hamiltonian for the slow modes $ϕ_{<}$ . To one-loop order, the new parameters $μ^{'2}$ and $g^{'}$ (before rescaling) are:

Correction to $μ^{2}$ : The diagram corresponding to this calculation is a simple loop, representing two $ϕ_{>}$ legs from a single vertex being contracted.

This diagram shows two external $ϕ_{<}$ lines connected to a vertex. The other two lines from the vertex form an internal loop of $ϕ_{>}$ modes. This loop integral contributes a correction to the $ϕ_{<}^{2}$ term, i.e., to the mass-squared $μ^{2}$ .

The calculation yields a correction to the quadratic term:
$μ^{'2} = μ_{0}^{2} + 12 g_{0} \int_{Λ/ b}^{Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{q ^{2} + μ _{0}^{2}}$
Correction to $g$ : The correction to the coupling constant appears at second order in the expansion. The key diagram has four external $ϕ_{<}$ lines and an internal loop of two $ϕ_{>}$ propagators.

This diagram shows four external $ϕ_{<}$ lines connected to two vertices. The vertices are joined by two internal $ϕ_{>}$ propagators, forming a loop. The integral over the momenta of these internal lines gives the correction to the four-point vertex, i.e., to the coupling $g$ .

This leads to a correction to the quartic term:
$g^{'} = g_{0} - 36 g_{0}^{2} \int_{Λ/ b}^{Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{( q ^{2} + μ _{0}^{2} ) ^{2}}$

Test your understanding!

Why does the correction to the coupling constant $g$ only appear at order $g_{0}^{2}$ , while the correction to the mass $μ^{2}$ appears at order $g_{0}$ ? Think about the number of vertices and fields involved.

Show answer

The interaction is $g_{0} ϕ^{4}$ . A correction to $μ^{2}$ modifies the $ϕ^{2}$ term. We can get a $ϕ^{2}$ term from a single interaction vertex by contracting two of its four field legs, leaving two external legs. This requires one vertex, hence one factor of $g_{0}$ .

A correction to $g$ modifies the $ϕ^{4}$ term. To get a $ϕ^{4}$ term with a loop correction, we need at least two interaction vertices (two factors of $g_{0}$ ). We take four external field legs (two from each vertex) and contract the remaining four legs (two from each vertex) with each other to form an internal loop structure. This requires two vertices, hence the correction is of order $g_{0}^{2}$ .

2. Steps 2 & 3: Rescaling and Deriving the Flow Equations

Now we have the effective parameters $μ^{'2}$ and $g^{'}$ for a theory with cutoff $Λ/ b$ . To complete the RG step, we rescale momentum and the field to return to a theory with cutoff $Λ$ .

Momentum Rescaling: $q \to \tilde{q} = b q$
Field Rescaling: $ϕ (\tilde{q}) \to \tilde{ϕ} (\tilde{q}) = ζ^{- 1} ϕ (\tilde{q} / b)$

As derived in the previous lesson (and the supporting notes), to keep the kinetic term canonical, we must choose $ζ = b^{(d + 2) /2}$ . Applying these rescalings to the $μ^{'2} ϕ^{2}$ and $g^{'} ϕ^{4}$ terms gives the fully renormalized parameters $\tilde{μ}^{2}$ and $\tilde{g}$ :

\tilde{μ}^{2} = b^{2} μ^{'2} = b^{2} (μ_{0}^{2} + 12 g_{0} \int_{Λ/ b}^{Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{q ^{2} + μ _{0}^{2}})

\tilde{g} = b^{4 - d} g^{'} = b^{4 - d} (g_{0} - 36 g_{0}^{2} \int_{Λ/ b}^{Λ} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{( q ^{2} + μ _{0}^{2} ) ^{2}})

These are the finite recursion relations. To get the differential flow equations (the beta functions), we consider an infinitesimal transformation, where $b = e^{δ l}$ with $δ l ≪ 1$ . In this limit, the integral is over a thin shell of momentum space. Let's define the flow parameter as $l = ln b$ . The beta functions are then $β_{μ} = d μ^{2} / d l$ and $β_{g} = d g / d l$ .

[PDF] 3. The Renormalisation Group

We now have all the pieces. This next section of the notes, 'Beta Functions', puts them together to derive the differential flow equations.

Please read Section 3.4.4, 'Beta Functions'. This short section shows how to convert the finite recursion relations we just found into differential equations by considering an infinitesimal transformation. Focus on the final result, equations (3.45), which are the one-loop beta functions for \phi^4 theory at d=4.

After performing the infinitesimal limit, we arrive at the celebrated RG flow equations at one-loop order. Letting $ϵ = 4 - d$ , they are:

\frac{d μ ^{2}}{d l} = 2 μ^{2} + A g

\frac{d g}{d l} = ϵ g - B g^{2}

where $A$ and $B$ are positive constants that depend on the dimension and are derived from the loop integrals evaluated at the cutoff $Λ$ . For example, $B = 36 K_{d} Λ^{- ϵ}$ where $K_{d}$ is a geometric factor. The precise values are not as important as the signs and the dependence on $g$ and $ϵ$ .

3. The Punchline: Wilson-Fisher Fixed Point

We have successfully derived the flow equations. Now, let's briefly analyze them to see the profound result. We are looking for fixed points, where the couplings stop flowing: $d μ^{2} / d l = 0$ and $d g / d l = 0$ .

Let's focus on the flow of the coupling constant $g$ :

β_{g} (g) = \frac{d g}{d l} = ϵ g - B g^{2}

This equation has two fixed points where $β_{g} (g) = 0$ :

Gaussian Fixed Point: $g^{*} = 0$ . This is the free, non-interacting theory.
Wilson-Fisher Fixed Point: $g^{*} = ϵ / B$ .

This second fixed point is the crucial discovery.

For $d > 4$ , $ϵ < 0$ . The only physical fixed point (with $g \geq 0$ ) is $g^{*} = 0$ . The flow $d g / d l \approx ϵ g$ pushes $g$ to zero, so the interaction is irrelevant. Mean-field theory is correct.
For $d < 4$ , $ϵ > 0$ . Now a new fixed point appears at a positive value $g^{*} = ϵ / B$ ! The flow $d g / d l \approx ϵ g$ near $g = 0$ is positive, pushing the coupling away from the Gaussian fixed point. The interaction is relevant. The system flows to the new, interacting Wilson-Fisher fixed point.

This new fixed point, which exists only for dimensions below 4, controls the universal critical behavior of systems like the Ising model. Its properties, not those of the Gaussian fixed point, give the correct critical exponents.

You can read about this exciting conclusion in the final resource for today.

[PDF] 3. The Renormalisation Group

The final piece of the puzzle is to see how our derived beta functions lead to a new, non-trivial fixed point. This section from the notes introduces the epsilon expansion and reveals the Wilson-Fisher fixed point.

Please read the beginning of Section 3.5, 'The Epsilon Expansion', up to the end of 3.5.1 'The Wilson-Fisher Fixed Point'. It uses the beta functions we just derived to find the new fixed point and analyzes its stability.

Conclusion

This has been a calculation-heavy but deeply rewarding lesson. We have transitioned from the abstract RG procedure to concrete, quantitative predictions.

Key Takeaways:

The integration of high-momentum modes in $ϕ^{4}$ theory is performed perturbatively in the coupling constant $g$ .
At one-loop order, corrections to the mass-squared $μ^{2}$ arise at order $g$ , while corrections to the coupling $g$ arise at order $g^{2}$ .
Combining the integration with the rescaling steps yields the RG flow equations, or beta functions: $\frac{d g}{d l} = (4 - d) g - B g^{2}$ .
For dimensions $d < 4$ , the flow equation for $g$ reveals a new, non-trivial, and stable Wilson-Fisher fixed point at $g^{*} \propto (4 - d)$ .
This interacting fixed point, not the free-theory Gaussian fixed point, governs the physics of critical phenomena in realistic dimensions ( $d = 2, 3$ ), explaining why mean-field theory fails.

Preview of the Next Lesson:

We have found the Wilson-Fisher fixed point and established its importance. In the next lesson, we will begin our study of the Epsilon Expansion. We will analyze the RG flow near $d = 4$ to calculate the critical exponents ( $ν$ , $η$ , etc.) as a series in $ϵ = 4 - d$ . This will provide the first non-mean-field estimates for the exponents that describe real physical systems at criticality.