Inelastic Axion Scattering

• Inelastic axion scattering is the process where axions or ALPs interact with matter or radiation, resulting in excited or transformed final states.
• It employs diverse mechanisms like axion-photon conversion, atomic excitation, and dark sector transitions, with cross sections scaling as 1/fₐ².
• The study of inelastic axion scattering informs experimental techniques, affecting cosmological constraints, dark matter searches, and signal discrimination in detectors.

Inelastic axion scattering is the set of processes in which axions or axion-like particles (ALPs) interact with matter or radiation in such a way that the final state differs from the initial state by excitation, transformation, or the emission/production of additional particles. This encompasses transitions where either the axion converts into other particles (e.g., a photon, electron-positron pair, or relativistic axions), scatters while exciting or ionizing atoms or nuclei, or transitions between different quantum states (e.g., between states in a dark sector or multi-component DM scenario). The paper of inelastic axion scattering is central to astrophysical, cosmological, and laboratory searches for axions, with implications for early-universe energetics, dark matter phenomenology, and new detection channels across a wide energy domain.

1. Fundamental Mechanisms of Inelastic Axion Scattering

Inelastic axion scattering comprises a diversity of reactions, driven by various axion couplings:

• Axion-photon conversion and production of fermion pairs: In the early universe, highly energetic axions produced by modulus decay undergo inelastic scattering with background photons, leading to processes such as $a + \gamma \to f + \bar{f}$, where $f$ is a quark or lepton. The cross section is strongly suppressed by the axion decay constant $f_a$, entering as $1/f_a^2$ in all leading expressions. The kinematics for such scatterings are determined by $E_{CoM}^2 = 2 E_a E_\gamma (1-\cos\theta)$, and the process accesses many final states kinematically forbidden at thermal energies (1304.1804).
• Inverse Primakoff and atomic inelasticity: In electronic systems, the inverse Primakoff process allows for both elastic and inelastic channels. While elastic conversion produces a photon and leaves the atom in its ground state ($a + A \to \gamma + A$), inelastic scattering can excite or ionize the atom ($a + A \to \gamma + A^*$ and $a + A \to \gamma + A^+ + e^-$). Here, response functions involving transition matrix elements encode the atomic excitation or ionization probabilities and cross sections (Abe et al., 2020, Wu et al., 2022).
• Axion-nucleon and axion-mediated dark sector transitions: In multi-component or axion-mediated dark matter models, inelastic axion scattering includes transitions such as $X_1 + N \to X_2 + N$ (where $X_1$ and $X_2$ are dark matter states of different mass), often mediated by non-QCD axions with off-diagonal couplings in the dark sector (Bae et al., 2023). These processes feature recoil spectra and kinematic thresholds analogous to inelastic WIMP scattering, with necessary conditions on the energy splitting and mediator mass.
• Multi-axion interactions and axion star phenomenology: In axion stars, inelastic reactions where several axions are transformed into photons or faster axions occur, with odd-number axion annihilations yielding photons and even-number axion fusion producing relativistic axions. The key channels are $(2j+1)a \to \gamma\gamma$ and $(2j)a \to 2a$ for integers $j \geq 1$ (Braaten et al., 2016). The rates depend on high powers of the axion number density and central density, relevant especially in dense axion star regimes.

2. Theoretical Formalism and Cross Section Structure

Computation of inelastic axion scattering cross sections and rates requires several core elements:

• Interaction Lagrangians: Typical terms include $c_f(m_f/f_a) a\,\bar{\psi}\gamma^5\psi$ for axion-fermion couplings; $g_{a\gamma\gamma} a F_{\mu\nu}\tilde{F}^{\mu\nu}$ for axion-photon interactions; and $(g_s^2/(32\pi^2 f))\, a\, G\tilde{G}$ for gluonic couplings. For dark sector transitions, couplings are often taken as $$(\xi_{ij}/f)(\partial_\mu a)\bar{\chi}_i\gamma^5\gamma^\mu\chi_j$$ (Bae et al., 2023).
• Thresholds and kinematic constraints: The center-of-mass energy must surpass final-state mass thresholds, e.g., $E_a E_\gamma \ge 2 m_q^2$ for $a + \gamma \to q + \bar{q}$ (1304.1804). In inelastic nucleus scattering, the minimum velocity required is raised by the excitation energy, $$v_{min} = \sqrt{(m_T E_R/(2\mu_T^2)) + {E^*}/{\sqrt{2m_T E_R}}}$$ (Arcadi et al., 2019). For $X_1 \to X_2$ dark sector transitions, the recoil spectrum is restricted by both $\delta$ and detector thresholds (Bramante et al., 2016).
• Matrix elements and response functions: For atomic inelasticity, the cross section integrates over the sum of squared transition matrix elements between initial and final electronic/nuclear states, encapsulated in the "incoherent scattering function" $S(q, Z)$ or similar response functions $\mathscr{R}_L$ and $\mathscr{R}_T$ (Abe et al., 2020, Wu et al., 2022).
• Energy and parametric scaling: Low-energy cross sections (keV-scale axions, atomic inelasticity) are highly sensitive to atomic or nuclear structure, scaling with powers of $\langle r^2\rangle$; for high energies (supernova axions), the dependence simplifies and the inelastic contribution becomes subdominant, suppressed as $\sim 1/Z$ (Abe et al., 2020).
• Suppression mechanisms: All tree-level induced cross sections are suppressed by appropriate high powers of $f_a$. For processes mediated by heavy (QCD) axions, the signal is out of reach for current direct detection; non-QCD axions (smaller $f$) open the parameter space to observable signatures (Bae et al., 2023).

3. Cosmological and Astrophysical Relevance

Inelastic axion scattering impacts several cosmological and astrophysical observables:

• Big Bang Nucleosynthesis (BBN): During BBN, inelastic axion-photon scattering produces energetic quark and lepton pairs, initiating electromagnetic/hadronic cascades that alter light element abundances. The energy deposition per photon, weighted by axion number density and scattering rate, constrains combinations of $f_a$, modulus mass $m_\Phi$, and reheating temperature $T_{reh}$. Overproduction of $^4$He yields the most stringent limits (1304.1804).
• Nonthermal dark matter production: Axion scattering at high center-of-mass energies can generate supersymmetric partner particles long after chemical freeze-out, serving as a nonthermal production mechanism for stable dark matter (e.g., LSPs in R-parity SUSY). The rare but energetically favorable up-scattering can yield non-negligible relic abundances, especially for axions produced by heavy field decay (1304.1804).
• Relic backgrounds: The majority of axions produced in early-universe scenarios survive with energies redshifted to $\mathcal{O}(100\,\text{eV})$, forming a Cosmic Axion Background (CAB) with flux ${\sim}10^6$ cm$^{-2}$ s$^{-1}$, which is isotropically distributed. Its spectrum and isotropy differ substantially from the solar axion flux, motivating experimental searches for this relic population (1304.1804).
• Axion star phenomenology: Inelastic multi-axion reactions play a key role in axion star dynamics and potential signatures. In dilute axion stars, radiative loss is dominated by $a \to \gamma\gamma$, with extremely small rates. In dense stars, $4a \to 2a$ dominates, leading to bursts of relativistic axions and potentially to detectable monochromatic radio signals at odd harmonics of $m_a$ (Braaten et al., 2016).

4. Laboratory and Detector Implications

Experimental searches leverage both elastic and inelastic axion scattering channels:

• Direct detection via nuclear recoil: Inelastic scattering where the nucleus is left in an excited state, as characterized by increased $v_{min}$ and the presence of monoenergetic de-excitation photons, enables background discrimination in, e.g., Xe-based TPCs sensitive to both S1 and S2 signals (Arcadi et al., 2019). In models with multi-component DM or axion-mediated up-scattering, the kinematic recoil window is dictated by the mass splitting $\delta$, nuclear form factors, and the target mass (Bramante et al., 2016, Bae et al., 2023).
• Atomic inelasticity in electron couplings: For axion or ALP interactions with electrons, atomic excitation or ionization must be incorporated. Modern calculations use relativistic Hartree–Fock wavefunctions to compute scattering form factors and response functions accurately. Experimental data from XENONnT and TEXONO have been used to set competitive bounds on ALP couplings in the 1 eV–10 keV mass range (Wu et al., 2022), probing parameter space previously inaccessible to laboratory searches.
• Signal topology and discrimination: In dual-phase xenon detectors, inelastic nuclear channels create summed signals from both the recoil and the de-excitation photon, shifting S2/S1 ratios and enabling improved rejection of elastic and background events (Arcadi et al., 2019). For atomic excitation/ionization, the emission of secondary photons or electrons is similarly distinctive.
• Role of heavy vs light targets: Sensitivity to inelastic transitions, especially for large mass splitting $\delta$ in dark sector models (and by analogy, in axion-induced processes), is enhanced in detectors with heavy nuclei (I, W, Xe). These maximize the reduced mass and accessible recoil window (Bramante et al., 2016).

5. Parameter Dependence and Model Implications

Central parameters governing inelastic axion scattering include:

• Axion decay constant ($f_a$): All inelastic scattering rates scale as $1/f_a^2$ or higher. Lower $f_a$ (non-QCD axions or ALPs) enhance interaction strength and observational feasibility, while QCD axions ($f_a \gtrsim 10^9$ GeV) remain out of reach for most experiments (Bae et al., 2023).
• Reheating temperature ($T_{reh}$): Determines the density of thermal photon targets for axion-photon scattering, entering both cosmological bounds and relic flux calculations (1304.1804).
• Mass splittings ($\delta$): For inelastic scattering involving excitation or conversion to higher-mass states (either in nuclei, atoms, or dark matter), the phase space and recoil kinematics are sharply set by $\delta$. As $\delta$ increases, the minimum required kinetic energy increases and signal rates are suppressed, favoring optimized analysis windows (extending to hundreds of keV) and heavy target materials (Bramante et al., 2016).
• Mixing and model structure: In clockwork SUSY axion models, the arrangement (normal/inverted) of axino and axion spectra determines whether inelastic scattering rates are exponentially suppressed or can reach detectable levels. The effective cross section integrates clockwork mixing matrix elements, axion/axino mass spectrum, and mediator properties (Bae et al., 2023).
• Atomic structure: For low-energy axion-electron/atomic scattering, precise knowledge of the atomic form factor $F(q)$ is necessary—naive screened Coulomb potentials overestimate cross sections by orders of magnitude relative to more accurate relativistic calculations (Abe et al., 2020).

6. Experimental Prospects and Future Directions

Advancements in sensitivity, analysis window breadth, and target technology are expanding the parameter space accessible to inelastic axion scattering:

• Detector enhancements: Future large-scale and ultra-low threshold detectors (e.g., DARWIN) promise greater exposure and ability to distinguish inelastic signal topologies (coinicident photon/electron), potentially probing ALP-photon couplings down to $g_{a\gamma\gamma} \lesssim 10^{-12}$ GeV$^{-1}$ and ALP masses as high as $\mathcal{O}$(MeV) (Wu et al., 2022).
• Reanalysis of high-recoil data: Extending the analysis range in Xe-based experiments from the typical 1–30 keV up to 100–500 keV would significantly improve sensitivity to large $\delta$ inelastic transitions—critical for both inelastic dark matter and inelastic axion/ALP signals (Bramante et al., 2016).
• Directional detection and substructure studies: Because inelastic axion scattering is sensitive to the local axion flux, velocity distribution, and dark matter substructures, future work may exploit effective angular velocity signatures, directional modulation, and gradient-induced effects, in analogy with the techniques outlined for elastic axion-electron interactions (Yang et al., 23 Jul 2025).
• Astrophysical and cosmological signatures: Inelastic axion channels contribute to early-universe energetics, dark radiation signals, and dense astrophysical phenomena (e.g., radio lines from axion stars, nonthermal supernova bursts), providing multiple observables for both cosmology and astrophysics (1304.1804, Braaten et al., 2016, Abe et al., 2020).
• Model discrimination: Inelastic transitions offer additional degrees of freedom for distinguishing between possible underlying particle physics models, often enabling more efficient rejection of incorrect operator hypotheses or interaction structures in both laboratory and astrophysical contexts (Arcadi et al., 2019).

7. Quantum Information Interpretation: Scattering Entropy and Entanglement

Inelastic axion scattering is inherently associated with quantum entanglement between outgoing channels. The scattering entropy formalism quantifies this: for a composite final state $$|\Psi\rangle = |u_1\rangle\otimes|G\rangle + |u_2\rangle\otimes|X\rangle$$ (elastic and inelastic channels, respectively), the entropy is $S_{pt} = -P_G\log_2 P_G - P_X\log_2 P_X$, where $P_G$ and $P_X$ are the normalized elastic and inelastic probabilities (Miller, 2023). In the black disk limit (maximal inelasticity), $S_{pt}=1$. This provides a theoretical underpinning for interpreting the quantum dynamics of axion scattering in highly absorptive, strongly coupled, or multi-channel scenarios.

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Inelastic axion scattering encompasses a rich set of processes central to constraints on axion properties, dark matter detection, BBN physics, and quantum information aspects of scattering. Its theoretical modeling connects multi-scale particle physics, nuclear/atomic structure, early-universe cosmology, and the design of advanced detector technologies.