Affleck-Dine Q-ball Formation

• Affleck-Dine Mechanism with Q-ball Formation is a theoretical framework where scalar fields with global charges fragment into non-topological solitons during early cosmic evolution.
• It employs lattice simulations to reveal a power-law relationship (Q ∝ |Φ|^4) between the initial field amplitude and Q-ball charge, illuminating the role of thermal effects and cosmic expansion.
• The transition from a thermal logarithmic to a gravity-mediated potential causes Q-balls to evolve from thick-wall to thin-wall profiles, affecting baryogenesis and dark matter production.

The Affleck-Dine (AD) mechanism with Q-ball formation describes how the dynamics of scalar fields carrying conserved global charges in the early universe can lead to the emergence of non-topological solitons—Q-balls—during cosmological evolution. In the standard supersymmetric context, the AD mechanism enables the generation of large asymmetries (such as baryon or lepton number) when scalar fields called AD fields (flat directions in the supersymmetric potential) develop large vacuum expectation values during and after inflation. Q-ball formation is triggered as the coherent AD condensate fragments due to instabilities in its potential, particularly when thermal and radiative corrections render the effective mass squared flatter than quadratic. The subsequent evolution, transformation, and decay of these Q-balls crucially impacts cosmological baryogenesis, dark matter production, and the possible generation of observable gravitational waves.

1. Q-Ball Formation in the Thermal Logarithmic Potential

The emergence of Q-balls in the context of the AD mechanism centers on the early universe dynamics of a complex scalar field $\Phi$ that carries a global $U(1)$ charge. In scenarios where thermal effects dominate, the scalar potential takes a logarithmic form: $$V_T(\Phi) \sim T^4 \log\left(\frac{|\Phi|^2}{T^2}\right)$$ and, in lattice simulations, a refined version

$$V_T(\Phi) = T^4 \log\left(1 + \frac{|\Phi|^2}{T^2}\right)$$

is used. Despite the fact that the background temperature $T$ decreases with cosmic expansion (e.g., $T \propto a^{-3/8}$ in an inflaton oscillation-dominated epoch), rapid growth of spatial fluctuations allows Q-ball formation to proceed nearly as if the potential were static.

A key numerical result from lattice simulations is the scaling relation between the Q-ball charge $Q$ and the AD field amplitude at the onset of oscillation $|\Phi|$: $$Q = \beta' \left(\frac{|\Phi|}{T_{\rm osc}}\right)^4, \qquad \beta' \simeq 2 \times 10^{-3}$$ where $T_{\rm osc}$ is the temperature at the onset of oscillation. The scaling $Q \propto |\Phi|^4$ reflects the fact that the charge is proportional to the volume within a horizon containing the fluctuating Q field, and the growth rate of field perturbations in a logarithmic potential supports this strong power-law dependence.

2. Time Evolution of Q-Ball Properties Under Cosmic Expansion

After formation, Q-balls experience evolution in their internal profiles due to the expansion of the universe and the changing background temperature. The field solution can be written as

$$\Phi(\vec{x}, t) = \frac{1}{\sqrt{2}} \phi(\vec{x}) e^{i\omega t}$$

and the charge $Q$ is conserved. Taking into account the scaling of gradient energy, thermal potential, and rotational energy, the properties of a $D$-dimensional Q-ball scale with the cosmic scale factor $a$ as: $$\begin{aligned} R &\propto a^{- \frac{4 - \gamma}{D+1}} \ \omega &\propto a^{- \frac{D-3 + \gamma}{D+1}} \ \phi_c &\propto a^{\frac{D-3 - (D-1)\gamma/2}{D+1}} \end{aligned}$$ where $R$ is the comoving Q-ball size, $\omega$ the rotation frequency, $\phi_c$ the central field value, and $\gamma$ characterizes the scaling of the thermal potential ($V_T \propto a^{-\gamma}$, with $\gamma=3/2$ for inflaton oscillation and $\gamma=4$ for radiation). The comoving Q-ball shrinks, while the central field value and rotation frequency exhibit non-trivial power-law evolution determined by the interplay of spatial dimensionality and the background thermal history.

3. Transition to Gravity-Mediated Domains and Q-Ball Profile Transformation

As the universe cools, the thermal logarithmic potential $V_T$ yields in importance to the gravity-mediated mass term: $$V_m = m_{3/2}^2 |\Phi|^2 \left[1 + K \log \left(\frac{|\Phi|^2}{M_*^2}\right)\right]$$ where $m_{3/2}$ is the gravitino mass, $K$ is a radiative correction parameter, and $M_*$ a renormalization scale. Notably, for $K > 0$, the pure gravity-mediated potential does not support Q-ball solutions.

The crossover point at which the gravity-mediated term dominates at the Q-ball center occurs when

$$V_T(\phi_c) < V_m(\phi_c) \implies \phi_c > \phi_{\rm eq} \sim \frac{T^2}{m_{3/2}}$$

As the universe expands ($a$ increases, $T$ drops), this condition is inevitably reached. Contrary to the naive expectation that gravity-mediation would immediately destabilize or destroy the Q-ball, lattice simulations and analytic studies reveal that Q-balls persist, but their internal profiles undergo a continuous transformation: the thick-wall (gauge-mediation-like) Q-ball morphs smoothly into a thin-wall configuration. This is controlled by the radial equation

$$\frac{d^2\phi}{dr^2} + \frac{2}{r} \frac{d\phi}{dr} + [\omega^2 \phi - dV/d\phi] = 0$$

with standard localized boundary conditions.

As $m_{3/2}/M_F$ increases (where $M_F$ is the typical scale at Q-ball formation), the solution develops a sharper, thin-wall-like profile, the rotation frequency $\omega$ rises, and the field amplitude at the center increases in order to conserve the global charge $Q$ as the potential changes. This transformation process is a generic dynamical outcome of coupled cosmic expansion and temperature evolution.

4. Cosmological and Baryogenesis Implications

The persistence and transformation of Q-balls have significant implications for early Universe baryogenesis, dark matter production, and potential gravitational wave sources. Initially, when dominated by the thermal logarithmic term, the Q-ball's charge is fixed by the initial field amplitude via the strong $Q \propto |\Phi|^4$ scaling. Even as the thermal background vanishes and gravity-mediation would not, by itself, allow Q-balls (for $K > 0$), the already-formed soliton transitions to a new internal configuration rather than being erased. This metamorphosis ensures that the Q-balls survive to lower temperatures ($T \sim m_{3/2}$), after which they decay or disperse, leaving a nontrivial inhomogeneous field distribution.

This evolution affects both baryon and possible dark matter production in the AD scenario. The ability of Q-balls to transition, rather than disintegrate, broadens the parameter space for successful baryogenesis and enables the possibility of long-lived relics or late-decaying Q-balls that could themselves be dark matter candidates or produce secondary cosmological signatures such as gravitational waves during phase transitions or decays.

5. Numerical Lattice Simulations and Analytical Scaling Relations

Lattice simulations are essential in confirming both the formation process and the subsequent dynamical evolution of Q-balls in time-dependent potentials. Quantitative results for the charge-amplitude relation, profile transformation, and scaling laws for soliton properties are robust across different simulation setups.

The table below summarizes the central analytical and simulation-derived relations:

Property	Scaling/Formula	Remarks
Q-ball charge	$Q = \beta' (|\Phi|/T_{\rm osc})^4$, $\beta' \simeq 2\times10^{-3}$	Strong dependence on initial field amplitude
Field profile transformation	Smooth transition thick-wall $\rightarrow$ thin-wall	Enabled by combined potential, persists even for $K>0$
Time evolution (scaling laws)	$R(a),\,\omega(a),\,\phi_c(a)$ (see Section 2)	Determined by background evolution
Condition for mass term dominance	$\phi_c > T^2 / m_{3/2}$	Onset of gravity-mediation effects

These results highlight the predictive power of numerical and analytic approaches in describing Affleck-Dine field dynamics under realistic early universe conditions.

6. Consequences for Affleck–Dine Baryogenesis and Model-Building

The AD scenario with Q-ball formation, especially in a time-dependent background that couples thermal and gravity-mediated potentials, provides a reservoir of late-time inhomogeneities and extended field configurations. The robustness of Q-ball survival and transformation implies that the usual constraints on the sign and size of $K$ in the gravity-mediation potential are relaxed when combined with realistic thermal histories: Q-balls will form and persist so long as the initial potential supports the thick-wall regime and the charge is set during this epoch.

This persistence modifies decay rates, the reprocessing of charge into the visible sector, and the interplay with dark matter candidates—crucially affecting predictions for the baryon-to-dark matter ratio, potential gravitational wave backgrounds, and the allowed parameter space for supersymmetric cosmology.

7. Summary

• The thermal logarithmic potential $V_T \sim T^4 \log(1+|\Phi|^2/T^2)$ produces fast Q-ball formation during Affleck–Dine dynamics, with $Q$ strongly dependent on the field amplitude at oscillation onset.
• Scaling laws for Q-ball size, field value, and rotation frequency dictated by cosmic expansion provide predictions for their subsequent evolution.
• As thermal effects decay, a gravity-mediated mass term can dominate, but Q-balls transition smoothly from thick-wall to thin-wall profiles, surviving even in regimes that would classically preclude Q-ball solutions.
• This dynamical pathway allows Q-balls to play a critical role in baryogenesis, dark matter scenarios, and potentially observable cosmological relics.
• The framework established by lattice simulation and analytic scaling robustly captures these phenomena, informing both phenomenological and model-building aspects of the Affleck–Dine mechanism with Q-ball formation.