A covariant description of the interactions of axion-like particles and hadrons

This paper presents a covariant framework for calculating the decay rates of axion-like particles that couple to both gluons and quarks by identifying redefinition-invariant coupling combinations, thereby providing physical expressions applicable across a broad range of ALP masses.

The Gist

Imagine the universe is a giant, bustling kitchen. Inside this kitchen, there are tiny, invisible chefs called quarks and gluons that cook up all the matter we see (protons, neutrons, etc.). For a long time, physicists thought they had the recipe for everything, but there's a missing ingredient in the recipe book that causes a weird glitch in the universe's symmetry.

Enter the Axion-Like Particle (ALP). Think of an ALP as a "ghost chef." It's a hypothetical, invisible particle that might be the missing ingredient that fixes the glitch. But here's the problem: we don't know exactly how this ghost chef interacts with the real ingredients. Does it whisper to the quarks? Does it shake hands with the gluons? Or does it do both?

This paper is like a universal translator and a master cookbook written by Reuven Balkin and his team. They created a new, foolproof way to predict what happens when these ghost chefs (ALPs) try to cook with the real ingredients, no matter how they interact.

Here is the breakdown of their work using everyday analogies:

1. The Problem: The "Translation" Mess

In physics, you can describe the same situation using different "languages" or "bases." Imagine describing a car accident. You could say, "The red car hit the blue car," or "The blue car was hit by the red car." Both are true, but the words change.

For a long time, physicists calculating how ALPs decay (break apart) were using different "languages." Sometimes, depending on which language they chose, they got different answers for how fast the ALP would disappear. This is like two mechanics looking at the same broken engine and giving you two different repair times just because they used different manuals. That's bad science; the answer should be the same regardless of the language.

The Paper's Solution: The authors built a universal translator. They identified specific combinations of numbers (couplings) that stay the same no matter which "language" you speak. Now, they can calculate the ALP's behavior without getting confused by the translation errors.

2. The Toolkit: The Chiral Lagrangian (The Recipe Book)

To predict how an ALP decays, you need a recipe book. In the world of subatomic particles, this book is called the Chiral Lagrangian.

• Low Mass (The Slow Cook): If the ALP is light (like a feather), it behaves like a gentle simmer. The authors use "Chiral Perturbation Theory" (χPT), which is like a precise recipe for slow-cooking. It works great for light particles.
• High Mass (The Pressure Cooker): If the ALP is heavy (like a brick), it behaves like a high-pressure explosion. Here, they use "Perturbative QCD," which is more like a rough estimate based on raw ingredients.
• The Middle Ground (The Danger Zone): The tricky part is the middle mass (between 1 and 3 GeV). It's too heavy for the slow-cook recipe but too light for the pressure-cook estimate. This is where previous recipes failed.

The Paper's Innovation: They created a hybrid cooking method.

• They took the slow-cook recipe and added "form factors." Think of these as adjustment knobs. As the ALP gets heavier, they turn the knobs to adjust the recipe based on real-world data (like looking at how other similar particles actually behave in experiments).
• They also created a "bridge" to smoothly connect the slow-cook and pressure-cook methods so there are no gaps in the recipe.

3. The "Ghost" Mixing

One of the coolest parts of the paper is how they handle mixing.
Imagine the ALP is a chameleon. When it interacts with the kitchen, it doesn't just stay a ghost; it briefly turns into a regular particle (like a pion or an eta meson) before turning back.

• The authors realized that if you just look at the "ghost" part, you miss the "chameleon" part.
• They developed a mathematical way to combine the direct ghost interactions with the chameleon transformations into a single, solid number. This ensures they don't double-count or miss anything.

4. The Results: What Happens When the Ghost Cooks?

Using their new universal translator and hybrid recipe, they simulated three different types of "Ghost Chefs" (Benchmark Models):

1. The Gluon Chef: This ghost only talks to gluons. It mostly turns into pairs of photons (light) or specific combinations of pions.
2. The Dark Pion Chef: This ghost has a specific relationship with quarks. It tends to break apart into strange particles (like kaons) at higher masses.
3. The Strange Chef: This ghost loves the "strange" quark. It has a very different menu of decay products compared to the others.

Why does this matter?
If we build a machine to catch these ghost chefs (like the experiments at CERN or future colliders), we need to know exactly what to look for.

• If the ghost is light, we look for flashes of light (photons).
• If it's medium-heavy, we look for specific clouds of pions and kaons.
• If it's heavy, we look for different patterns.

This paper gives experimentalists the exact map of what to expect for any type of ALP, whether it talks to gluons, quarks, or both. It removes the guesswork and ensures that if we find a ghost, we can correctly identify its "flavor" and understand its role in the universe's recipe.

Summary

In short, this paper is a standardized instruction manual for the universe's most elusive ghost chefs. It fixes the translation errors that confused physicists for years, creates a seamless bridge between low-energy and high-energy cooking, and tells us exactly what kind of "food" (decay products) we should expect to find in our detectors, helping us solve the mystery of dark matter and the strong CP problem.

Technical Summary

1. Problem Statement

Axion-like particles (ALPs) with masses in the GeV range ($O(\text{GeV})$) are predicted by various extensions of the Standard Model (SM), including solutions to the strong CP problem and dark matter candidates. Calculating their decay rates and production cross-sections in this mass regime is theoretically challenging due to the transition between two valid theoretical frameworks:

• Low Mass ($m_a \lesssim 1$ GeV): Chiral Perturbation Theory ($\chi$PT) is reliable for exclusive decay channels.
• High Mass ($m_a \gtrsim 2-3$ GeV): Perturbative QCD (pQCD) provides inclusive rate estimates.
• The Intermediate Gap ($1 \text{ GeV} \lesssim m_a \lesssim 3 \text{ GeV}$): Neither $\chi$PT nor pQCD is fully sufficient. Previous attempts to bridge this gap often relied on specific assumptions about ALP couplings (e.g., coupling only to gluons or only to quarks) or specific UV completions.

A critical issue identified in previous literature is basis dependence. Physical observables, such as decay rates, must be invariant under quark-field redefinitions (axial rotations). However, earlier calculations often depended on arbitrary field-redefinition parameters, leading to unphysical results (e.g., missing contributions in $K^+ \to \pi^+ a$ decays). There was a lack of a general, covariant framework that handles arbitrary couplings to both quarks and gluons while maintaining basis invariance across the entire GeV mass range.

2. Methodology

The authors develop a covariant framework that ensures physical observables are independent of the choice of quark-field basis. The methodology involves three main pillars:

A. Identification of Field-Redefinition Invariants

Starting from the effective Lagrangian at the GeV scale involving ALP couplings to light quarks ($q=u,d,s$) and gluons, the authors perform an axial rotation of the quark fields:
$$q' = e^{-i\kappa_q \frac{a}{f_a}\gamma_5} q$$
They demonstrate that the coupling parameters ($c_g, c_\gamma, c_q, \beta_q$) transform under this rotation. By analyzing these transformations, they identify five independent invariants that govern all physical processes:

1. $\tilde{\beta}_q \equiv \beta_q + c_q$
2. $\tilde{c}_g \equiv c_g - \langle c_q \rangle$
3. $\tilde{c}_\gamma \equiv c_\gamma - N_c \langle c_q Q Q \rangle$
   (Where $\langle \dots \rangle$ denotes the trace in flavor space).
   Any physical observable must depend solely on these invariants, not the individual coupling parameters.

B. Embedding into the Chiral Lagrangian

The authors embed the ALP into the $U(3)_L \times U(3)_R$ chiral Lagrangian. They construct the Lagrangian terms (kinetic, mass, vector-meson, Wess-Zumino, and photon terms) ensuring they respect the local symmetry and the identified invariants.

• Mixing: They derive the mixing between the ALP and neutral pseudoscalar mesons ($\pi^0, \eta, \eta'$). Crucially, they define an invariant ALP matrix $\tilde{T}_a$ which combines the mixing angles and direct couplings. This matrix replaces the basis-dependent mixing parameters in all interaction vertices.
• Interactions: They derive explicit interaction vertices for:
  • $VPP$ (Vector-Pseudoscalar-Pseudoscalar)
  • $PVV$ (Pseudoscalar-Vector-Vector)
  • $VPPP$ and $PPPP$ (Multi-pseudoscalar processes)
  • Scalar and Tensor resonance contributions ($S \to PP$, $f_2 \to \pi\pi$).

C. Bridging the Mass Gap (Data-Driven Approach)

To extend the results from the $\chi$PT regime ($m_a < 1$ GeV) to the pQCD regime ($m_a > 3$ GeV), the authors adopt and generalize the data-driven approach from Ref. [38]:

• Form Factors: They introduce hadronic form factors $F_Y(\mu)$ for exclusive amplitudes to account for high-energy corrections. These are constrained by $e^+e^-$ scattering data and QCD power counting (scaling as $\mu^{4-n}$).
• Matching Condition: For the intermediate region ($1.2 \text{ GeV} < m_a < 3 \text{ GeV}$), they construct a matching condition for the invariant matrix $\tilde{T}_a$. They equate the exclusive $\chi$PT rates with the inclusive pQCD rates (partonic $a \to gg, \bar{q}q$) to determine the behavior of $\tilde{T}_a$ at high energies, ensuring a smooth transition between the hadronic and partonic descriptions.

3. Key Contributions

1. Basis-Independent Formalism: The primary contribution is the rigorous derivation of ALP-hadron interactions that are manifestly invariant under quark-field redefinitions. This resolves ambiguities in previous calculations where results depended on arbitrary basis choices.
2. General Coupling Framework: Unlike previous works restricted to specific coupling scenarios (e.g., gluon-only or quark-only), this framework handles arbitrary linear combinations of quark and gluon couplings.
3. Comprehensive Decay Channels: The paper provides explicit formulas for a wide range of decay modes, including:
   • $a \to \gamma\gamma, \gamma V, VV$
   • $a \to VPP$ (e.g., $\omega\pi\pi, \gamma\pi\pi$)
   • $a \to PPP$ (e.g., $3\pi, \eta\pi\pi, K\bar{K}\pi$)
   • Contributions from scalar ($\sigma, f_0, a_0, \kappa$) and tensor ($f_2$) resonances.
4. Improved Resonance Treatment: The authors refine the treatment of the $\kappa$ (or $K^*_0(700)$) resonance, replacing the naive Breit-Wigner distribution with an empirical amplitude derived from $B$-factory data ($\eta_c \to KK\pi$) to better account for the $K^*_0(1430)$ interference and unitarity issues.

4. Results

The authors applied their framework to three benchmark models to illustrate the phenomenological impact of different coupling structures:

• Model 1 (Gluon Dominance): $\tilde{c}_g = 1$, others zero.
  • Dominant decays: $\eta\pi\pi$ and $\gamma\pi\pi$ for $m_a \sim 1$ GeV.
  • At high masses, the ALP matrix becomes the identity, forbidding certain decays like $a \to V(PP)P$.
• Model 2 (Dark Pions): Couplings aligned with $Z$-boson couplings ($c_q = \text{diag}(1/2, -1/2, -1/2)$).
  • Shows distinct branching ratios compared to Model 1, particularly in the $K\bar{K}\pi$ channel for $m_a > 1.5$ GeV.
• Model 3 (Strange Dominance): $\tilde{\beta}_q = \text{diag}(0,0,1)$.
  • Highlights the sensitivity of decay modes to specific quark flavor couplings.

Key Findings:

• The dominant decay mode shifts with mass: $\gamma\gamma$ for $m_a < 0.5$ GeV; $3\pi$ and $\pi\pi\gamma$ for $0.5-1$ GeV; $\eta\pi\pi$ for $1-1.5$ GeV; and $K\bar{K}\pi$ for $m_a > 1.5$ GeV (in models with significant quark couplings).
• The framework successfully reproduces known $\chi$PT results at low masses and matches inclusive pQCD rates at high masses.
• The "unaccounted for" rate (difference between exclusive sum and inclusive rate) is minimized by the matching procedure, though uncertainties of 20–30% remain due to unknown strong phases and higher-order resonances.

5. Significance

This work provides the standard theoretical tool for analyzing GeV-scale ALPs in current and future experiments (e.g., Belle II, LHCb, NA62, and future fixed-target experiments).

• Phenomenological Robustness: By eliminating basis dependence, it ensures that experimental limits and discovery potentials derived from these calculations are physically consistent and not artifacts of a specific mathematical convention.
• Model Agnostic: It allows experimentalists to interpret data for any UV completion that generates arbitrary quark/gluon couplings, rather than being restricted to specific benchmark models.
• Bridging the Gap: The data-driven matching procedure offers a reliable method to estimate rates in the difficult 1–3 GeV mass window, a region where many new physics searches are currently active.

In summary, the paper establishes a rigorous, covariant, and comprehensive framework for ALP-hadron interactions, resolving previous theoretical inconsistencies and enabling precise predictions for the next generation of ALP searches.