Revisiting Q-ball Interactions with Matters

Technical Summary: Revisiting Q-ball Interactions with Matters

Problem Statement
Q-ball dark matter (DM) is a candidate for macroscopic dark matter, characterized as a non-topological solitonic configuration stabilized by global charge and energy conservation. A crucial mechanism for detecting Q-ball DM involves its interaction with ordinary matter, particularly the scattering of nucleons (or constituent quarks) off the Q-ball. Previous literature, specifically Ref. [17], conjectured that quarks incident on a Q-ball are reflected as anti-quarks with a probability of order unity. This process was predicted to lead to significant energy release via the annihilation of reflected anti-quarks with surrounding matter, potentially leaving unique signatures in paleo-detectors (ancient minerals).

However, the authors identify two critical physical effects overlooked in previous analyses:

1. Energy Cost (Chemical Potential): The reflection of a quark as an anti-quark requires the addition of a squark to the Q-ball to conserve baryon number. This incurs an energy cost equal to the chemical potential ($\omega$) of the Q-ball. If the incident energy cannot cover this cost, the reflection process is kinematically forbidden.
2. Electromagnetic Charge-up: If a proton is reflected as an anti-proton, the Q-ball acquires an electromagnetic charge. This charge alters the scattering cross-section via Coulomb interactions, potentially suppressing further interactions.

Methodology
The authors revisit the scattering process of quarks off a Q-ball by incorporating the time-oscillating nature of the Q-ball background and the associated chemical potential $\omega$.

• Theoretical Framework: The study utilizes a toy model describing the scattering of colorless quarks off a Q-ball, based on supersymmetric extensions of the Standard Model. The Lagrangian includes couplings between squarks ($\phi$), quarks ($\psi$), and gluinos ($\lambda$).
• Scattering Analysis:
  • Infinite Wall Approximation (Section 2.2): The authors first analyze scattering on an infinitely large Q-ball wall. They solve the Dirac equation for fermions in a background where the scalar field oscillates as $\phi = \phi_0 e^{-i\omega t}$. The wave functions inside and outside the Q-ball are matched at the boundaries, accounting for the energy shift ($E \to E-2\omega$) required for the reflection of a left-handed quark into a right-handed anti-quark.
  • Finite-Size Spherical Scattering (Section 2.3): The analysis is generalized to a finite-size, spherically symmetric Q-ball. The wave functions are expanded in terms of spinor spherical harmonics. The authors solve for the scattering coefficients ($B_\pm$) representing reflected quarks and anti-quarks by matching boundary conditions at the Q-ball radius $R$.
• Nucleon Implications (Section 3): The results from quark scattering are applied to nucleon scattering. The authors evaluate the kinetic energy of galactic nucleons ($v \sim 10^{-3}$) against the energy cost ($2\omega$) and the Coulomb barrier effects for charged Q-balls.

Key Contributions and Results

1. Kinematic Suppression of Anti-quark Reflection:
   The study demonstrates that the reflection of an incident quark as an anti-quark is only possible if the incident energy $E$ exceeds the energy cost $2\omega$ (specifically, $E > 2\omega$ for the process to pay the chemical potential cost).

   • For macroscopic Q-balls with radii $R \sim \mathcal{O}(10)$ fm, the chemical potential is $\omega \simeq 20$ MeV.
   • The kinetic energy of a nucleon in the Milky Way halo is $E_{kin} \simeq 0.47$ keV.
   • Consequently, for galactic Q-ball DM, the condition $E_{kin} < 2\omega$ is not met. Therefore, nucleons cannot be reflected as anti-nucleons. The process is kinematically forbidden.

2. Partial Reflection and Excitation:
   While full reflection as an anti-nucleon is impossible, the authors suggest that constituent quarks within a nucleon (with momenta $\sim \Lambda_{QCD} \approx 200$ MeV) might satisfy the energy condition. However, due to the linear confinement potential of QCD, single anti-quark emission is prohibited. Instead, this interaction is interpreted as an excitation of the Q-ball via nucleon absorption, followed by the emission of a nucleon or pions. This process does not result in the clean anti-matter signature previously hypothesized.

3. Electromagnetic Charge and Coulomb Suppression:
   The paper analyzes the scenario where a Q-ball acquires an electromagnetic charge ($Z_Q$) through absorption and subsequent emission processes.

   • A positively charged Q-ball repels protons via the Coulomb force.
   • The scattering cross-section is suppressed by a Coulomb barrier factor, approximated as $\exp(-2\pi Z_Q \alpha / v)$.
   • For typical galactic velocities, this suppression is significant (e.g., $\sim 10^{-20 Z_Q}$), drastically reducing the interaction rate compared to neutral Q-balls.
   • The authors also consider bound state formation with electrons, which can discharge the Q-ball, but note that for high charges, the Coulomb barrier still suppresses scattering unless the Q-ball radius is comparable to the Bohr radius of the bound state.

4. S-Matrix Structure:
   The authors derive the S-matrix for the scattering process. In the limit of large Q-ball radius and normal incidence, the S-matrix shows that for $E > 2\omega$, a left-handed quark is reflected as a right-handed anti-quark with probability of order unity. However, for $E < 2\omega$, the transmission of the anti-quark component is exponentially suppressed, and the scattering is dominated by elastic reflection or other channels that do not involve charge conjugation.

Significance and Claims
The paper claims to correct a previous misconception regarding the detectability of Q-ball DM via anti-matter signatures.

• Refutation of Previous Hypothesis: The authors assert that the conjecture in Ref. [17]—that Q-balls reflect nucleons as anti-nucleons leading to observable energy release—is invalid for galactic Q-ball DM due to the energy cost imposed by the chemical potential.
• Revised Detection Prospects: The study implies that direct detection experiments (including paleo-detectors) searching for Q-ball DM via anti-matter tracks or annihilation signatures must reconsider their sensitivity, as the primary interaction channel (nucleon $\to$ anti-nucleon reflection) is kinematically blocked.
• Charge Accumulation: The paper highlights that Q-balls interacting with matter in the Milky Way may accumulate electromagnetic charge, leading to a scattering cross-section that is heavily suppressed by the Coulomb barrier, further distinguishing their observational signatures from neutral Q-balls.

The authors conclude that a detailed understanding of the flat direction dependence (e.g., $udd$ vs. $uude$) is necessary to determine the fraction of Q-balls that might be electromagnetically charged and to accurately predict experimental signals.