Discovering the Axiverse via Fifth Forces

This paper argues that searches for fifth forces must account for the collective effects of multiple axion-like particles predicted by string theory, demonstrating that the resulting spin-dependent and spin-independent potentials can be used to distinguish between different axiverse scenarios.

The Gist

Imagine the universe is like a giant orchestra. For decades, physicists have been listening for a specific instrument—a tiny, ghostly particle called an axion—hoping to hear its solo. Most experiments are tuned to listen for just one axion, like trying to find a single violin in a concert hall.

However, this paper suggests that if our universe is built on the complex rules of string theory, the orchestra isn't playing a solo. It's playing a massive, chaotic symphony with thousands, or even millions, of these axion instruments all at once. The authors call this collection of hidden particles the "Axiverse."

Here is the simple breakdown of their discovery:

1. The "Fifth Force" Problem

We know four fundamental forces: gravity, electromagnetism, and the strong and weak nuclear forces. Physicists are always hunting for a "Fifth Force." Usually, they look for this force by seeing how particles with "spin" (like tiny spinning tops) interact with each other.

The paper argues that if you have a whole crowd of axions (the Axiverse), you can't just assume they act like a single violin. Instead, they act like a choir.

• The Spin-Dependent Force: If you have a choir of axions, the force they exert on spinning particles gets stronger. If there are $N$ axions, the force gets $N$ times stronger.
• The Spin-Independent Force: This is the real surprise. When axions interact with non-spinning matter (like a regular rock or a table), they usually do so in pairs. If you have a choir of $N$ axions, the number of possible pairs is huge. The force gets stronger by a factor of $N^2$ (N squared).
  • Analogy: If one person claps, it's loud. If 100 people clap, it's louder. But if 100 people are clapping in pairs with each other, the noise explodes. A crowd of 100 axions creates a "Fifth Force" that is 10,000 times stronger than a single axion could ever create.

2. The "Ripple in the Pond" Analogy

Imagine dropping a single stone into a pond. It creates one perfect, circular ripple that fades out smoothly. This is what current experiments expect to see: a clean, single ripple caused by one axion.

Now, imagine dropping hundreds of stones of different sizes into the pond at once.

• The big stones create ripples that travel far.
• The tiny stones create ripples that fade away almost immediately.
• The result isn't one smooth circle. It's a messy, complex pattern of overlapping waves.

The authors show that the "Fifth Force" from an Axiverse looks exactly like this messy pond. Instead of a smooth curve, the force changes shape in "steps" or "kinks" as you move closer to or further from the source. These kinks happen because different axions have different masses (weights), and each mass creates a ripple that fades out at a specific distance.

3. How to Spot the Axiverse

The paper proposes that we don't need to find the axions one by one. Instead, we can look at the shape of the force itself.

• If we measure the force between two objects and see a smooth, single curve, it's likely just one axion (or no axion).
• If we see a jagged, complex curve with specific "steps" or a sudden spike in strength, it's a smoking gun that we are dealing with a whole Axiverse.

The authors tested this idea using three different "musical scores" (theoretical models):

1. Kaluza-Klein ALPs: A model based on extra dimensions.
2. Kaluza-Klein Maxions: A variation where the axions are "maximally" different from standard expectations.
3. Type IIB String Theory: A complex model from string theory with thousands of axions.

In all cases, they found that the "Fifth Force" becomes incredibly strong and takes on a unique, multi-layered shape that a single axion could never mimic.

4. The "Earth as a Microphone"

The paper notes that if these axions are incredibly light (so light that their "ripples" are as big as the Earth itself), the entire Earth acts like a giant microphone, picking up the signal from all these axions at once. This would create a detectable gradient (a slope) in the force field around us, which experiments like torsion balances (super-sensitive scales) could potentially measure.

The Bottom Line

This paper tells us that if the universe is full of these hidden axion particles, we shouldn't just look for a single "note." We should look for the symphony. By measuring the strength and the specific "wiggles" in the Fifth Force, we might be able to count how many axions exist and even figure out which version of string theory describes our universe, all without needing a particle accelerator the size of the solar system.

Technical Summary

Technical Summary: Discovering the Axiverse via Fifth Forces

Problem Statement
Extensions of the Standard Model (SM), particularly string theory, predict the existence of a vast number of light, axion-like particles (ALPs), a scenario termed the "axiverse." While experimental searches for axions are extensive, they predominantly target a single, light state. Current experimental methodologies for detecting fifth forces often assume a non-relativistic potential induced by the exchange of a single axion field. However, in an axiverse scenario containing $N_a$ different axions, the collective contribution of these states to fifth forces is non-trivial. Specifically, standard searches may fail to account for the constructive interference of multiple light states or the distinct radial profiles generated by a spectrum of axion masses. Furthermore, while spin-dependent forces (linear exchange) scale with $N_a$, spin-independent forces (quadratic exchange) scale with $N_a^2$, potentially offering a significantly enhanced signal that is currently overlooked in single-field analyses.

Methodology
The authors derive the general non-relativistic potentials for fifth forces induced by the exchange of $N_a$ distinct axion fields.

1. Theoretical Framework: They define the interaction Lagrangian involving linear (spin-dependent) and quadratic (spin-independent) couplings between axions and nucleons (protons and neutrons). The quadratic interactions arise from shift-symmetry breaking.
2. Potential Calculation:
   • Spin-Dependent ($V_1$): Calculated as the sum of single-axion exchanges. In the limit of light axions ($m_i < 1/r$), the potential scales linearly with the number of axions ($N_a$).
   • Spin-Independent ($V_2$): Calculated via the one-loop exchange of axion pairs. The authors evaluate the Laplace transform of the spectral density function for the exchange of axions with masses $m_i$ and $m_j$. This includes full mass dependence and interference terms.
3. Benchmark Scenarios: To illustrate the phenomenological impact, the authors model three specific axiverse scenarios:
   • Kaluza-Klein (KK) ALP: Based on a KK axiverse with constant decay constants and a specific mass spectrum.
   • KK Maxion: A scenario where QCD axion parameters are maximally far from the canonical band, with decay constants rescaled to isolate the effect of increasing axion number.
   • Type IIB String Theory: Utilizing the Kreuzer-Skarke database to model axions arising from the Hodge number $h_{1,1}$ of Calabi-Yau compactifications. Masses and decay constants are derived from normalized divisor volumes.
4. Analysis: The authors compare the resulting effective potentials against the standard single-axion potential ($N_a=1$), analyzing the scaling behavior with distance $r$ and the number of axions $N_a$.

Key Contributions

• Generalized Potential Formulation: The paper provides a rigorous derivation of the spin-independent potential for multiple axion exchanges, including the specific functional form involving Bessel functions and interference terms that arise from mass spectra.
• Scaling Laws: The authors demonstrate that in the axiverse limit (where $m_i < 1/r$), the spin-independent force is enhanced by a factor of $N_a^2$, whereas the spin-dependent force is enhanced by $N_a$.
• Distinctive Radial Profiles: The study shows that the presence of multiple axions with different masses results in a potential that is a superposition of different radial profiles. This creates a "step-like" structure or a modified slope in the potential $V(r)$ that cannot be replicated by a single-range force law.
• String Theory Specifics: For Type IIB string theory scenarios, the authors show that the total potential strength depends sensitively on the Hodge number $h_{1,1}$, scaling approximately as $N_a^8$ for spin-dependent and $N_a^{16}$ for spin-independent potentials due to the correlation between the number of axions and their decay constants.

Results

• Short-Distance Behavior: For distances $r$ smaller than the Compton wavelength of the heaviest axion, the potential follows the characteristic $1/r^3$ behavior of two-particle exchange. As $r$ increases, heavier axions "turn off," causing the potential to deviate from the single-axion profile.
• Enhancement Magnitude: In the KK ALP scenario with $N_a=100$, the effective potential scales as $V^{(100)}_2(r)/V^{(1)}_2(r) \sim N_a^2/r$, significantly exceeding single-field predictions.
• String Theory Fluctuations: While Type IIB string theory models exhibit statistical fluctuations due to random sampling of divisor volumes, the averaged potentials show a near-perfect $r^{-3}$ dependence in the range $r \in [10^{-5}, 10^{-1}]$ m.
• Spin-Dependent vs. Independent: The authors find that for Type IIB string theory, the spin-independent potential grows faster with increasing $h_{1,1}$ than the spin-dependent potential. Extrapolating their fits, the two potentials would become equal at approximately $h_{1,1} \sim 10^5$, suggesting that for very large axiverse numbers, the spin-independent force could dominate.
• Step-Like Features: The ratio of the full potential to the potential generated only by light axions reveals step-like increases corresponding to the "turning off" of heavier axions. These features are distinct from the smooth exponential decay of a single massive field.

Significance and Claims
The paper claims that searches for fifth forces offer a unique and powerful strategy to discover the axiverse, distinct from traditional axion dark matter searches which often target specific mass resonances.

• Differentiation: The shape of the non-relativistic potential (specifically the mixture of radial profiles and step-like features) can be used to distinguish between different axiverse scenarios and potentially count the number of axion states.
• Sensitivity: Fifth force searches are presented as the most sensitive laboratory probes for axion couplings to nucleons in the mass range $10^{-8} \text{ eV} \le m_a \le 10^{-2} \text{ eV}$, particularly when enhanced by the $N_a^2$ factor in the spin-independent channel.
• Model Discrimination: By evaluating contributions associated with specific string theory mass spectra, the authors assert that effective potentials can be constructed to distinguish between different UV completion models (e.g., different Hodge numbers in string theory).
• Limitations: The authors modestly note that while the perturbative expansion holds for the scenarios considered (up to $N_a \sim 10^{25}$), higher-order loop contributions become relevant for extremely large $N_a$. Additionally, they acknowledge that current black hole superradiance constraints are limited to narrow mass ranges and rely on individual mass eigenstates, whereas fifth force searches probe the collective effect of the spectrum.

The paper concludes that the distinctive patterns in fifth forces generated by multiple axion fields provide a viable pathway to probing the structure of the axiverse, offering constraints and discovery potential that complement and, in specific regimes, surpass current single-field experimental limits.