⚛️ phenomenology

Small-instanton effects in an atlas of KSVZ axion models

This paper investigates how small-instanton contributions in various KSVZ axion models with vector-like quarks can significantly enhance the axion mass and modify the standard relationship between axion mass, decay constant, and photon coupling, thereby opening new parameter spaces for axion searches.

Original authors: Ning Chen, Saurabh K. Shukla

Published 2026-08-11

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CC BY 4.0

Original authors: Ning Chen, Saurabh K. Shukla

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 the universe as a giant, cosmic puzzle where every piece has a specific job. Some pieces are the familiar actors we know, like the protons and electrons that make up your body. But there are also invisible, ghostly pieces that scientists suspect are hiding in the shadows, trying to solve a mystery that has stumped physicists for decades: the "Strong CP Problem." Think of the Strong CP Problem as a glitch in the universe's code. According to the rules of the strong force (the glue holding atomic nuclei together), there should be a tiny, invisible knob that controls how matter behaves. If this knob were turned even a fraction of a degree, the universe would be a very different, and likely lifeless, place. Yet, when we look at the real world, the knob is perfectly centered at zero. Why?

To fix this glitch, physicists proposed a new, invisible particle called the "axion." You can think of the axion as a cosmic thermostat. If the universe's knob tries to drift away from zero, the axion automatically adjusts itself to push it back, keeping everything stable. This axion is also a leading candidate for "dark matter," the mysterious stuff that holds galaxies together but doesn't emit light. However, there's a catch: we don't know exactly how heavy this axion is or how strongly it interacts with light. If it's too heavy or interacts too weakly, our current experiments might miss it entirely. This is where the story of "small instantons" comes in. In the quantum world, space isn't empty; it's a bubbling foam of tiny, fleeting events called "instantons." These are like microscopic whirlpools that briefly twist the fabric of space-time. For a long time, scientists thought these whirlpools were too small and weak to matter much for the axion. But what if they aren't?

This paper by Ning Chen and Saurabh K. Shukla takes a fresh look at these tiny quantum whirlpools, specifically in a popular version of the axion theory called the KSVZ model. The authors ask a simple but profound question: What happens if we pack the universe with extra, heavy particles (called vector-like quarks) that interact with these instantons? They use a set of mathematical "rules of thumb" (known as dimensional analysis) to calculate how these extra particles change the behavior of the instantons. Their findings suggest that if we have the right number of these extra particles, the tiny whirlpools can become surprisingly powerful. Instead of being a tiny background noise, they can become the main driver of the axion's mass.

The paper explores three different scenarios: having just one type of heavy particle, having many identical copies of that particle, or having a mix of different heavy particles. They found that in certain cases, especially when there are many copies of these heavy particles, the instantons become "UV dominated." In plain English, this means the instantons are so sensitive to the highest energy scales in the universe (near the Planck scale) that their contribution to the axion's mass can become huge. In fact, for some specific combinations of particles, the axion could become millions of times heavier than we previously thought, shifting its mass from the tiny "micro-electronvolt" range up to the "mega-electronvolt" (MeV) range.

Crucially, the authors show that this doesn't just change the axion's weight; it changes the relationship between its weight and how it talks to light (photons). Usually, scientists expect a strict rule: if you know the axion's mass, you can predict exactly how it interacts with light. But if these small-instanton effects are real, that rule breaks. The axion could be much heavier without becoming any more "visible" to our detectors in the way we expect. This opens up a whole new "map" of possibilities for where to look for the axion. The paper suggests that current experiments might be looking in the wrong place, focusing on the old, narrow path while the axion could be hiding in a vast, unexplored region of the parameter space. While the paper doesn't prove that these heavy particles exist, it rigorously demonstrates that if they do, the consequences for our search for the axion would be dramatic, potentially explaining why we haven't found it yet and pointing us toward new, more massive targets for future experiments.

=== DRAFT ===
Technical Summary: Small-instanton effects in an atlas of KSVZ axion models

Problem Statement
The strong-CP problem arises from the absence of a symmetry in the Standard Model (SM) that enforces the smallness of the physical QCD vacuum angle, $\bar{\theta}$ . While the Peccei-Quinn (PQ) mechanism introduces a dynamical axion field to relax $\bar{\theta}$ to zero, the properties of this axion—specifically its mass ( $m_a$ ) and coupling to photons ( $g_{a\gamma\gamma}$ )—are traditionally determined by non-perturbative QCD effects. However, short-distance (UV) instantons can also contribute to the axion potential. If these small-instanton contributions are significant, they can alter the topological susceptibility, thereby modifying the standard relation between the axion mass, decay constant, and photon coupling. The authors investigate the magnitude of these effects across a comprehensive "atlas" of Kim-Shifman-Vainshtein-Zakharov (KSVZ) axion models, which extend the SM with heavy vector-like quarks (VQs).

Methodology
The authors employ a semiclassical instanton calculation combined with Dimensional Analysis (NDA) to estimate the small-instanton contribution to the axion potential. The methodology proceeds as follows:

Model Construction: The study considers KSVZ models extended by a complex PQ scalar $\Phi$ and heavy VQs. The SM fermions are PQ-neutral. The authors analyze three distinct spectral configurations:
- A single VQ representation.
- Multiple identical copies of a single VQ representation.
- Sets of distinct VQ representations.
  The VQ representations are drawn from a classification in Refs. [53, 54], restricted to those that remain perturbative up to the Planck scale ( $M_{Pl}$ ) under two-loop gauge running.
Zero-Mode Saturation: In an instanton background, fermion zero modes must be saturated to generate a non-zero amplitude. The authors calculate the number of zero modes ( $2|N|$ ) for each VQ representation, where $N$ is the color anomaly coefficient. Saturation is achieved via:
- Mass insertions ( $m$ ), which saturate one pair of zero modes.
- Scalar-Yukawa loops, which saturate two pairs.
  The condition for saturation is $2|N| = 2k + r$ , where $k$ is the number of Yukawa loops and $r$ is the number of mass insertions.
Instanton-Size Integral and UV Sensitivity: The contribution to the axion susceptibility ( $\chi$ ) is determined by integrating over the instanton size $\rho$ . The integral's behavior is governed by the exponent $\Delta = b_3 + r - 4$ , where $b_3$ is the QCD beta-function coefficient.
- If $\Delta < 0$ , the integral is UV-dominated (sensitive to the cutoff $M_{UV}$ ).
- If $\Delta > 0$ , it is IR-dominated.
  The authors identify that increasing the number of VQ copies or using specific distinct sets can increase $r$ (or allow more Yukawa loops), potentially lowering $\Delta$ and rendering the contribution UV-sensitive.
Perturbativity Constraints: To ensure the validity of the semiclassical expansion, the authors impose that all gauge couplings (SU(3) $_c$ , SU(2) $_W$ , U(1) $_Y$ ) and the VQ Yukawa coupling remain perturbative ( $\alpha < 1$ ) up to $M_{Pl}$ . This restricts the maximum number of identical VQ copies allowed for each representation.
Phenomenological Mapping: Assuming the small-instanton potential is aligned with the QCD potential (minima coincide), the total susceptibility is $\chi_{total} = \chi_{QCD} + \chi_{SI}$ . This modifies the axion mass ( $m_a^2 = \chi_{total}/f_a^2$ ) while the photon coupling remains determined by the anomaly ratio $E/N$ and $f_a$ .

Key Contributions and Results

Atlas of UV Sensitivity: The paper systematically maps 14 viable VQ representations (excluding one that fails perturbativity at one copy) to determine their instanton-induced susceptibility.
- Single VQ: UV sensitivity is generally confined to representations with large color representations (e.g., color-15 singlets, octets, sextet doublets). The strongest scaling for a single VQ is $(M_{Pl}/\Lambda)^{31/3}$ .
- Identical Copies: Adding identical copies increases the zero-mode count, allowing for more Yukawa loops and mass insertions. This can render previously IR-dominated spectra UV-sensitive. The strongest scaling found is $(M_{Pl}/\Lambda)^{13}$ for two copies of the octet doublet $Q=(8, 2, -1/2)$ , while 25 copies of the triplet singlet $Q=(3, 1, -1/3)$ yield a scaling of $(M_{Pl}/\Lambda)^{38/3}$ .
- Distinct Sets: Combinations of distinct VQs (up to 7 representations) can also achieve $(M_{Pl}/\Lambda)^{13}$ scaling.
Axion Mass Enhancement: The small-instanton contribution increases the axion mass relative to the standard QCD prediction for a fixed decay constant $f_a$ .
- For a single VQ, the maximum mass shift reaches $m_a \approx 3.43$ eV.
- For identical copies, the mass can be enhanced to $m_a \approx 71.2$ MeV (for 2 copies of $R_{13}$ ).
- For distinct sets, the maximum is $m_a \approx 65.3$ MeV.
  These results demonstrate that small-instanton effects can shift the axion mass from the micro-eV scale (standard QCD axion) to the MeV scale within perturbative KSVZ frameworks.
Modification of the $(m_a, g_{a\gamma\gamma})$ Plane: The authors show that the standard QCD relation $g_{a\gamma\gamma} \propto m_a$ is modified. Because the small-instanton term increases $m_a$ without changing the anomaly coefficients ( $E/N$ ) or $f_a$ directly, the physical axion mass increases while the photon coupling remains "anchored" to the original $f_a$ .
- This decoupling allows the axion to populate regions of the $(m_a, g_{a\gamma\gamma})$ plane that are inaccessible to standard QCD axions.
- Specifically, for a fixed $f_a$ , the coupling $g_{a\gamma\gamma}$ is suppressed by a factor of $(1 + \chi_{SI}/\chi_{QCD})^{-1/2}$ relative to the standard QCD band prediction for a given mass.
- The study identifies specific trajectories in the parameter space (e.g., for $R_{11}$ , $R_{13}$ , and distinct sets $S_2$ – $S_7$ ) that lie outside the conventional QCD axion band, offering new targets for experimental searches.

Significance and Claims
The paper claims to provide a comprehensive "atlas" of small-instanton effects in KSVZ models, demonstrating that these UV contributions are not merely negligible corrections but can dominate the axion potential under specific, perturbatively consistent conditions.

The authors emphasize that the resulting axion mass is not solely a function of the VQ representation but depends critically on the complete UV matter content and the specific interactions (mass insertions vs. Yukawa loops) used to saturate zero modes. By showing that the small-instanton susceptibility can compete with or exceed the QCD susceptibility, the work opens new regions in the axion parameter space. The authors conclude that these effects modify the standard mass-coupling relation, creating "reachable" regions for axion searches that lie outside the traditional QCD axion band, thereby motivating a broader search strategy that accounts for UV-sensitive instanton contributions.

The paper remains modest regarding its scope, noting that it assumes aligned potentials, neglects renormalizable VQ-SM fermion mixing, and does not include supersymmetric extensions or specific UV completions that might alter the zero-mode saturation mechanisms.

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