⚛️ high-energy theory

RG studies of scalar-field models of long-range interactions

This paper employs the functional renormalisation group to investigate nonlocal scalar-field models with long-range interactions, revealing how nonlocality alters infrared fixed-point structures, induces symmetry breaking, and confirms Sak's predictions for critical exponents across various interaction ranges.

Original authors: Alfio M. Bonanno, S. R. Haridev, Gaurav Narain

Published 2026-02-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alfio M. Bonanno, S. R. Haridev, Gaurav Narain

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how a crowd of people behaves. Usually, we assume people only interact with the person standing right next to them. This is like local physics: short-range, immediate connections.

But what if people could also shout across the entire stadium, influencing someone on the other side instantly? This is non-local physics (or long-range interactions). It's a bit weird, but it happens in the real world, from how magnets work to how the universe expands.

This paper is a deep dive into the "rules of the game" for a universe where these long-range whispers exist. The authors use a powerful mathematical tool called the Functional Renormalization Group (FRG) to see how these rules change as we zoom in and out of the energy scale.

Here is the story of their findings, broken down with some everyday analogies:

1. The Setup: The "Long-Range Whisper"

The authors are studying a simple model: a field of particles (like a calm lake) that usually only talks to its neighbors. But they add a special ingredient: a "long-range whisper" term.

The Analogy: Imagine a game of "Telephone." In a normal game, you whisper to the person next to you. In this new game, you can also whisper directly to someone 100 seats away. The authors want to know: Does this change the final message? Does it break the game? Or does it create a new, stable way of playing?

2. The First Discovery: Shifting the Goalposts

First, they treated the "whisper" strength as a fixed setting (like turning a dial on a radio).

What they found: The long-range whisper doesn't change the fundamental type of game being played (the "fixed points" remain the same), but it shifts the goalposts.
The Analogy: Think of a soccer match. The rules of soccer (the fixed points) are the same, but if you add a strong wind (the non-local interaction), the ball might drift. The players have to adjust their positions. The authors found that this "wind" can actually force the players to break symmetry—meaning it can push a calm, uniform crowd into a chaotic, organized riot (symmetry breaking) even when they wouldn't normally do so.

3. The Problem: The "Singularity" Trap

Next, they tried to let the "whisper" strength change as they zoomed out to the very lowest energies (the deep infrared).

The Problem: When the whisper is "positive" (pushing things apart), the math hits a wall. The flow of the game becomes "singular" (it breaks down) before they can reach the end.
The Analogy: Imagine driving a car toward a destination. If you drive with the "positive whisper," the road suddenly turns into a cliff edge before you get there. You can't reach the finish line. However, if the whisper is "negative" (pulling things together), the road is smooth, and you can drive all the way to the end, reaching a calm, stable state.

4. The Solution: The "Non-Local Gaussian" Safe Haven

To fix the broken road, they upgraded their math (adding "wavefunction renormalization," which is like adding better suspension to the car).

The Discovery: When they let the system settle at the very lowest energy, they found a single, stable destination: the Non-Local Gaussian Fixed Point.
The Analogy: No matter how you start the game, if you let it run long enough with these long-range rules, everything eventually settles into a specific, quiet pattern. It's like a pendulum that, instead of swinging back and forth, eventually stops at a unique, slightly tilted angle that only exists because of the long-range whispers. This is the "safe haven" of the theory.

5. The General Rule: The "Sak Prediction"

The authors then asked: "What if the whisper isn't just a simple shout, but a shout with a specific mathematical shape?" They tested different shapes (represented by a number called $\sigma$ ).

The Finding: They found a smooth transition.
- If the whisper is very strong (short-range-like), the game behaves like standard local physics.
- If the whisper is very weak (long-range), the game behaves like their new "Non-Local Gaussian" pattern.
- There is a specific "tipping point" (around $\sigma = 2$ ) where the game switches from one style to the other.
The Analogy: Think of a dimmer switch. As you turn the dial, the light doesn't just flicker; it smoothly transitions from a warm, local glow to a cool, long-range glow. The authors confirmed that their math perfectly matches predictions made by a physicist named Sak decades ago.

6. The High-Speed Case: The "Lifshitz" Zone

Finally, they looked at extreme cases where the "whisper" is so strong it acts like a higher-dimensional force (specifically, $\sigma = 4$ ).

The Result: This corresponds to a state called "Isotropic Lifshitz criticality."
The Analogy: This is like the game changing from 2D chess to 3D chess. The rules get more complex, but the authors found that their method still works, predicting the behavior exactly as other experts had guessed.

The Big Takeaway

This paper is a map for navigating a strange, non-local universe.

Non-locality matters: Long-range interactions change where things settle, even if they don't change the fundamental rules.
Stability exists: Despite the weirdness of long-range whispers, there is a stable, predictable state (the Non-Local Gaussian Fixed Point) that the universe settles into at low energies.
The Tool Works: The "Functional Renormalization Group" is a robust tool that can handle these complex, non-local interactions, proving it's useful for understanding everything from condensed matter physics to the mysteries of gravity and the early universe.

In short: The universe might be whispering to itself across vast distances, and thanks to this paper, we now have a better idea of how those whispers shape the final shape of reality.

1. Problem Statement

The paper investigates the Renormalization Group (RG) flow and fixed-point structure of scalar field theories featuring non-local interactions. Specifically, the authors analyze a model where the standard local action is augmented by a non-local term of the form $\phi (-\nabla^2)^{\sigma/2} \phi$ .

Context: Non-locality arises naturally in effective field theories (when integrating out light degrees of freedom) and in quantum gravity frameworks (e.g., string theory, non-local gravity models like Woodard-Deser).
Core Question: How does the introduction of non-locality (specifically long-range interactions) alter the infrared (IR) behavior, phase transition patterns, and fixed-point structure of scalar field theories compared to standard local theories?
Specific Focus: The study centers on the case $\sigma = -2$ (corresponding to a $1/k^2$ propagator modification, or $\phi \frac{1}{-\nabla^2} \phi$ ) and generalizes to arbitrary $\sigma$ , including higher-derivative cases ( $\sigma > 2$ ).

2. Methodology

The authors employ the Functional Renormalization Group (FRG) approach, specifically utilizing the Wetterich flow equation for the effective average action $\Gamma_k$ .

Model Definition:
The non-local Lagrangian density is defined as:
$\mathcal{L} = \frac{1}{2}(\partial\phi)^2 + \frac{1}{2}m^2\phi^2 + \frac{\lambda_2}{2}\phi \frac{1}{-\nabla^2}\phi + U(\phi)$
where $\lambda_2$ is the non-local coupling.
Approximations:
1. Local Potential Approximation (LPA): The wavefunction renormalization $Z_k$ is set to 1 (or treated as constant), and the non-local coupling $\lambda_2$ is initially treated as an external, non-flowing parameter.
2. LPA' (LPA Prime): The analysis is extended to include scale-dependent wavefunction renormalization ( $Z_k$ ), allowing for the calculation of the anomalous dimension $\eta$ .
3. Generalization: The model is generalized to interactions of the form $\phi (-\nabla^2)^{\sigma/2} \phi$ .
Analytical Techniques:
- Spike Analysis: A numerical method used to solve the non-linear flow equations for the potential $u_k(\phi)$ by identifying "spikes" (singularities) in the solution space, which correspond to fixed points.
- Polynomial Truncation: Expanding the potential in powers of the field ( $\phi^{2n}$ ) to derive flow equations for couplings.
- Local Equivalent Theory: The non-local theory is mapped to an equivalent local theory involving two scalar fields ( $\phi$ and an auxiliary field $\psi$ ) to validate the FRG results.
- Effective Dimensionality: For $\sigma > 2$ , the authors use an effective dimension approach ( $D_{eff} = 2d/\sigma$ ) to map the problem to standard local theories.

3. Key Contributions and Results

A. Behavior in the LPA (Non-flowing Coupling)

Fixed Point Shift: When treating $\lambda_2$ as an external parameter, the non-local coupling shifts the position of the Wilson-Fisher (WF) fixed point but does not change the critical exponents. The critical exponents remain identical to the standard local theory.
Symmetry Breaking: The non-local parameter induces a change in phase transition patterns. As $\lambda_2$ increases, the system can transition from a symmetric phase to a broken symmetry phase, effectively inducing symmetry breaking.
Convexity and Singularities: When analyzing the convexity of the effective potential as $k \to 0$ (deep IR) without polynomial truncation, the authors find that for $\lambda_2 > 0$ , the flow equation becomes singular at a finite RG time $t$ . This implies that the flow cannot be continued to the deep infrared for positive non-local couplings within this truncation, unlike the local case ( $\lambda=0$ ).

B. Behavior in LPA' (Flowing Coupling and $Z_k$ )

Non-Local Gaussian Fixed Point (NL GFP): When the flow of the non-local coupling is included via wavefunction renormalization, the analysis reveals that the only IR-stable fixed point is the Non-Local Gaussian Fixed Point.
- In this fixed point, the non-local coupling dominates, and the theory flows to a trivial (Gaussian) state in the deep IR.
- This was confirmed via both spike analysis and polynomial truncation (up to $N=13$ ).
Stability: The NL GFP is shown to be IR-stable (all eigenvalues of the stability matrix are positive, except for the trivial direction).

C. Generalization to Arbitrary $\sigma$

The authors analyze the dependence of IR properties on the exponent $\sigma$ in the interaction $\phi (-\nabla^2)^{\sigma/2} \phi$ :

Regime $\sigma < d/2$ : The IR-stable fixed point is the Non-Local Gaussian Fixed Point.
Regime $d/2 < \sigma < \sigma^*$ : A new Non-Local Non-Gaussian Fixed Point emerges from the NL GFP at $\sigma = d/2$ . This fixed point is IR-stable in this intermediate range.
Regime $\sigma^* \leq \sigma \leq 2$ : The non-local fixed point merges with the standard local Wilson-Fisher fixed point at $\sigma^* = 2 - \eta$ $σ^{*} = 2 - η$ (where $\eta$ $η$ is the anomalous dimension). For $\sigma \geq \sigma^*$ $σ \geq σ^{*}$ , the standard local fixed points become IR-stable.
- This transition aligns with Sak's prediction regarding the crossover between long-range and short-range universality classes.
Higher Derivatives ( $\sigma > 2$ ): For $\sigma = 4$ (Isotropic Lifshitz criticality), the authors use the effective dimension method ( $D_{eff} = 2d/4 = d/2$ ). They find that the fixed point structure matches previous literature, identifying $d=4$ and $d=8$ as the lower and upper critical dimensions for the isotropic Lifshitz point.

D. Validation via Local Theory

The authors re-derived the flow equations using a local two-field formulation ( $\phi$ and $\psi$ ). The results matched the non-local formulation up to a constant factor, which is attributed to the Jacobian normalization in the path integral of the auxiliary field. This validates the consistency of the non-local FRG approach.

4. Significance

Theoretical Framework: The paper establishes that the Functional Renormalization Group is a robust tool for studying non-local field theories, capable of handling arbitrary spacetime dimensions and non-locality exponents ( $\sigma$ ).
Resolution of IR Issues: It provides a detailed map of how non-local interactions modify the IR landscape, identifying a distinct "Non-Local Gaussian" phase that dominates the deep infrared for strong long-range interactions ( $\sigma < d/2$ ).
Phase Transitions: The work demonstrates that non-locality is not merely a perturbation but can fundamentally alter phase transition mechanisms, potentially inducing symmetry breaking where none existed in the local limit.
Applications: The findings have implications for:
- Cosmology: Understanding non-local gravity models and dark energy alternatives.
- Condensed Matter: Describing systems with strong long-range interactions (e.g., dipolar systems).
- Quantum Gravity: Offering insights into UV completion and the resolution of IR divergences in gravity theories.

Conclusion

The study concludes that while non-locality preserves the universality class of the Wilson-Fisher fixed point for certain ranges of $\sigma$ , it introduces a distinct Non-Local Gaussian Fixed Point that governs the deep infrared for strong long-range interactions. The transition between non-local and local universality classes occurs smoothly at $\sigma = d/2$ and $\sigma = 2-\eta$ , confirming theoretical predictions and providing a comprehensive phase diagram for non-local scalar field theories.