Completing Axion Double Level Crossings

This paper refines the theory of axion double level crossings in multi-axion systems, demonstrating that these phenomena—characterized by a high-temperature crossing followed by a QCD-induced crossing—are common in $Z_{\mathcal N}$ axion models but are constrained by the number of axions ($\mathcal N$) in both light and heavy scenarios, with significant implications for axion cosmology.

The Gist

The Big Picture: A Cosmic Dance of Invisible Particles

Imagine the universe is filled with a vast, invisible orchestra of particles called axions. Most of us know about the "famous" axion (the QCD axion), which was invented to solve a specific puzzle in physics. But string theory suggests there isn't just one; there is a whole "String Axiverse"—a massive crowd of these particles, some very heavy, some incredibly light, all dancing together.

This paper is about what happens when these axions interact, specifically focusing on a phenomenon called "Double Level Crossings."

The Analogy: The Elevator and the Staircase

To understand "level crossing," imagine two elevators moving in a building:

1. Elevator A represents the famous QCD axion. Its speed and position change drastically as the universe cools down (like an elevator stopping at different floors depending on the time of day).
2. Elevator B represents the other "Axion-Like Particles" (ALPs). They move at a steady, constant pace.

Level Crossing happens when, as the universe cools, these two elevators pass each other. For a brief moment, they are at the exact same height. In physics, when particles "cross levels" like this, they can swap identities or change their energy dramatically. It's like two dancers swapping partners mid-song.

"Double Level Crossing" is the twist in this story. Usually, you might expect the elevators to cross once. But this paper shows that in certain scenarios, they cross twice.

1. First Crossing: They pass each other high up in the building (at high temperatures, early in the universe's history).
2. Second Crossing: They pass each other again right at a specific, critical moment (when the universe cools to a specific temperature called $T_{QCD}$).

The Plot Twist: The "N" Factor

The authors introduce a character named $N$ (a number representing how many "mirror worlds" or hidden dimensions exist in their model). Think of $N$ as the volume knob on a radio.

• The Light Scenario (Small Decay Constants): Imagine the ALPs are lightweights. If you turn the volume knob ($N$) up just right, you get a beautiful "Double Crossing" dance.
  • The Catch: If you turn the volume up too high (making $N$ too huge), the dance breaks. The first crossing disappears, and you only get a single crossing. It's like turning the music up so loud the speakers blow out.
• The Heavy Scenario (Large Decay Constants): Imagine the ALPs are heavyweights. Here, if the volume knob ($N$) is turned too low, the dance also fails. You need a minimum amount of "volume" to get the double crossing to happen.

The "Toy Examples" (The Lab Rats)

The authors ran simulations (their "toy examples") to prove this. They set up three scenarios:

1. The Single Crossing: A small $N$. The axions cross once, then stop.
2. The Double Crossing: A medium $N$. The axions cross once, then cross again right at the critical moment. This is the "sweet spot" the paper is excited about.
3. The Multiple Crossings: A large $N$. The axions cross multiple times, but if $N$ gets too big, the pattern gets messy and the double crossing fails again.

Why Should We Care? (The Cosmological Implications)

Why does this matter? Because axions are a leading candidate for Dark Matter—the invisible stuff holding galaxies together.

• The Energy Swap: When these "level crossings" happen, the axions can swap energy. If they cross twice, it changes how much "stuff" (energy) ends up in the universe today.
• The Goldilocks Zone: This paper tells us that the universe has to be "just right." If the number of hidden worlds ($N$) is too big or too small, the axions won't do their double dance. If they don't dance, the amount of Dark Matter we calculate might be wrong.
• New Rules: The authors had to rewrite the rulebook. Previously, scientists thought the "Light" axions were just those lighter than the main axion. Now, they realize the rule depends on $N$. A particle can be "light" even if it's heavier than the main axion, as long as $N$ is large enough.

The Takeaway

This paper is like a choreographer realizing that a dance routine only works if the music volume is set perfectly.

• Too quiet (Small $N$ in Heavy scenario): The dance doesn't start.
• Too loud (Large $N$ in Light scenario): The dancers trip and fall.
• Just right: You get a spectacular Double Level Crossing.

By understanding these crossings, physicists can better predict how much Dark Matter exists in our universe and where to look for it in future experiments. It's a small adjustment in the math that could have huge consequences for our understanding of the cosmos.

Technical Summary

1. Problem Statement

The paper addresses the cosmological evolution of axions within the "String Axiverse" framework, where the QCD axion coexists with a multitude of ultra-light Axion-Like Particles (ALPs).

• The Core Issue: In scenarios involving mass mixing between a $Z_N$ axion (a candidate for Dark Matter arising from mirror worlds) and multiple ALPs, the system undergoes "level crossings" as the universe cools through the QCD phase transition temperature ($T_{QCD}$).
• The Gap: Previous studies established that a "double level crossing" (two distinct crossings) can occur in a two-axion system (one $Z_N$ axion and one ALP). However, the behavior of systems with $N > 2$ axions (one $Z_N$ axion and multiple ALPs) was not fully characterized. Specifically, the conditions under which multiple double level crossings occur, and how the number of axions ($N$) and decay constants affect these transitions, remained unclear.
• Physical Context: Level crossings are analogous to the MSW effect in neutrino oscillations. They can induce adiabatic or non-adiabatic transitions in axion energy density, significantly altering the relic abundance of Dark Matter and other cosmological observables.

2. Methodology

The authors employ a theoretical framework based on the low-energy effective Lagrangian of multi-axion mass mixing.

• Model Formulation:
  • They define a Lagrangian containing one $Z_N$ axion ($\theta_N$) and $N$ ALPs ($\Theta_i$).
  • The mass mixing is governed by a domain wall number matrix ($n_{ij}$) and axion masses ($m$) and decay constants ($f$).
  • Two distinct scenarios are analyzed based on the relative magnitude of the ALP decay constants ($f_{Ai}$) versus the $Z_N$ axion decay constant ($f_{aN}$):
    1. Light Axion Scenario: $f_{Ai} < f_{aN}$ (suppresses axion energy density).
    2. Heavy Axion Scenario: $f_{Ai} > f_{aN}$ (enhances axion energy density).
• Matrix Analysis:
  • The authors construct the mass mixing matrix ($M^2$) for both scenarios.
  • They decompose the large $(N+1) \times (N+1)$ matrix into effective $2 \times 2$ sub-matrices to calculate mass eigenvalues ($m_{h,i}, m_{l,i}$) and identify crossing points.
• Temperature Evolution:
  • They model the temperature dependence of the $Z_N$ axion mass ($m_{aN}(T)$), which transitions from a high-temperature power-law behavior to a zero-temperature constant mass at $T_{QCD}$.
  • They derive analytical expressions for the first level crossing temperatures ($T_{\times i}$) for each ALP pair.
• Numerical Illustration:
  • The authors generate "toy models" with $n_{ALP} = 2$ to visualize the evolution of mass eigenvalues as a function of temperature for varying values of $N$.

3. Key Contributions

A. Generalization to Multi-Axion Systems

The paper extends the concept of double level crossings from a 2-axion system to a general $N$-axion system. It demonstrates that double level crossings are a prevalent phenomenon in the mass mixing of $Z_N$ axions and multiple ALPs, not just a rare 2-body effect.

B. Redefinition of Light and Heavy Scenarios

A major theoretical contribution is the redefinition of the criteria for "light" and "heavy" axion scenarios.

• Previous Definition: Light ($f_{Ai} < f_{aN}$) and Heavy ($f_{Ai} > f_{aN}$).
• New Definition: The authors derive that the existence of the first level crossing imposes stricter constraints involving $N$:
  • Light Scenario: $f_{Ai} < \frac{f_{aN}}{N}$
  • Heavy Scenario: $f_{Ai} > \frac{f_{aN}}{N}$
• Implication: The new "heavy" condition is weaker, allowing $f_{Ai}$ to be smaller than $f_{aN}$ (provided $N$ is large enough), while the "light" condition is stronger, requiring $f_{Ai}$ to be significantly smaller than $f_{aN}$ as $N$ increases.

C. Discovery of Multiple Double Level Crossings

In the light axion scenario, the authors show that as $N$ increases, the system can undergo multiple instances of double level crossings.

• Mechanism: A "double crossing" consists of a first crossing at high temperature ($T_{\times i}$) followed by a second crossing at $T_{QCD}$ induced by the $Z_N$ mass transition.
• Non-Monotonic Behavior: While larger $N$ generally increases the frequency of these events, the authors find that excessively large $N$ can actually prevent double level crossings in the light scenario (by pushing the first crossing temperature too high or violating the mass hierarchy). Conversely, in the heavy scenario, excessively small $N$ can prevent them.

4. Results

• Toy Model Analysis ($n_{ALP}=2$):
  • Small $N$: Results in a "Single Level Crossing" (effectively one crossing event despite two mathematical intersections).
  • Intermediate $N$: Results in "Single & Double Level Crossings." One pair of eigenvalues crosses twice (double), while another crosses once.
  • Large $N$: Results in "Multiple Double Level Crossings," where multiple pairs of eigenvalues undergo the double-crossing phenomenon.
• Parameter Space Constraints:
  • The study maps the viable parameter space in the $\{N, f_{Ai}\}$ plane.
  • It establishes that for a double level crossing to occur, the first crossing temperature $T_{\times i}$ must be close to $T_{QCD}$. If $N$ is too large (light scenario) or too small (heavy scenario), $T_{\times i}$ deviates significantly, and the "double" nature of the crossing is lost.
• Cosmological Implications:
  • The parameter $N$ acts as a critical constraint on the viable parameter space for the $Z_N$ axion mass ($m_{aN}$) and decay constant ($f_{aN}$).
  • The presence of multiple ALPs with hierarchical masses imposes additional constraints on $N$.
  • These findings suggest that future experiments (e.g., CASPEr-electric) searching for ultra-light ALPs with specific decay constants could indirectly constrain the $Z_N$ axion model and the number of mirror worlds ($N$).

5. Significance

• Refinement of Axion Cosmology: The paper provides a more rigorous understanding of how axion energy density evolves in complex multi-field scenarios, moving beyond simple two-field approximations.
• Dark Matter Constraints: By clarifying the conditions for level crossings, the work refines predictions for the relic density of $Z_N$ axions and ALPs, which are prime Dark Matter candidates.
• Experimental Guidance: The redefined conditions ($f_{Ai} \gtrless f_{aN}/N$) and the sensitivity of level crossings to $N$ offer new targets for experimental searches. If specific ALP couplings are detected, they could rule out or support specific values of $N$ in the $Z_N$ axion model.
• Theoretical Consistency: The work resolves ambiguities in previous literature regarding the definition of light/heavy axion scenarios in the presence of multiple fields, ensuring that cosmological calculations of axion production are based on physically consistent parameters.

In summary, this work establishes that double level crossings are a robust feature of the String Axiverse with $Z_N$ symmetry, but their realization is highly sensitive to the number of axion species ($N$) and their decay constants, necessitating a re-evaluation of the parameter space for axion Dark Matter.