Vibrational modes of Q-balls

We study linear perturbations of classically stable Q-balls in theories admitting analytic solutions. Although the corresponding boundary value problem is non-Hermitian, the analysis of perturbations can also be performed analytically in certain regimes. We show that in theories with the flat potential, large Q-balls possess soft excitations. We also find a specific vibrational mode for Q-balls with a near-critical charge, where the perturbation theory for excitations can be developed. Comparing with the results on stability of Q-balls provides additional checks of our analysis.


Setup
Spectrum in the flat potential Spectrum in the polynomial potential Conclusions Backups

Setup
We will be interested in the models of one complex scalar field in 3+1 dimensions with a potential of a special form, The ansatz, energy and charge of a Q-ball We will study small perturbations on top of these configurations.
The potential will be chosen so that to allow for analytical treatment of both the solitons and their perturbations.
Note that the problem of finding a spectrum of bound states of a Q-ball is not an eigenvalue problem for an Hermitian operator.
Setup Spectrum in the flat potential Spectrum in the polynomial potential Conclusions Backups Q-balls in the flat potential Consider a potential consisting of two parabolic branches joined at some point |ϕ| = v . Require the presence of a flat direction, Parabolic potential with the flat direction The set of Q-balls split on two branches. One of them (with ω < ωc ) contains classically stable solutions. Another (with ω > ωc ) corresponds to unstable "Q-clouds" (Alford, M. G.'88) The critical frequency ωc ≈ 0.960m corresponds to the soliton with the minimal possible energy and charge. Perturbations of Q-balls in the flat potential An appropriate ansatz governing the dynamics of small oscillations on top of the classically stable Q-balls reads as follows (M. N. Smolyakov'18) Perturbation ansatz where the parameter γ is taken to be real and positive, ψ The functions g and h are determined by the potential. In our case Hence, equations are disentangled everywhere except the single point R such that Overview of Q-balls Setup Spectrum in the flat potential Spectrum in the polynomial potential Conclusions Backups Perturbations of Q-balls in the flat potential To study bound states, one imposes Boundary conditions and also γ + ω < m. Perturbations of Q-balls in the flat potential Features of the spectrum: At ω → 0, one has Q → ∞. Hence, large Q-balls possess soft modes. In this limit, the spectrum linearizes, The number of bound states of large Q-balls is proportional to its size 3 .
At intermediate frequencies the Q-balls do not support bound states.
Close to the stability bound ω = ωc one vibrational spherically-symmetric mode reappears. For it γ ∼ √ ωc − ω This mode continues analytically into the instability region where it becomes the decay mode.
Setup Spectrum in the flat potential Spectrum in the polynomial potential Conclusions Backups Perturbations of Q-balls in the flat potential The structure of the spectrum with a non-zero orbital momentum is similar to that with l = 0: Figure: The discrete spectrum of linear perturbations of classically stable Q-balls in the flat potential, at l = 1 (the left panel) and l = 2 (the right panel).
Note the absence of vibrational modes near the cusp point ω = ωc .

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Overview of Q-balls Setup Spectrum in the flat potential Spectrum in the polynomial potential Conclusions Backups

Polynomial potential
In order to allow Q-balls in a theory of one scalar field with a polynomial potential, it is necessary to include non-renormalizable self-interactions in the latter. Here we consider the simplest bounded below potential of the sixth degree, The frequencies of Q-balls are confined in the region ω min < ω < m = ω 2 min + δ v 4 The thin-wall approximation is applicable near the lower limit. It is controlled by the small parameter = ω − ω min Setup Spectrum in the flat potential Spectrum in the polynomial potential Conclusions Backups

Q-balls in the thin-wall regime
In the thin-wall regime, the properties of a Q-ball are well captured by few quantities -the distance R to the wall and the magnitude f 0 of the field in the interior region.
In order to justify the description of a soliton in terms of a finite set of variables, a suitable thin-wall ansatz must be adopted.
To study perturbations on top of a Q-ball, it suffices to choose the simplest ansatz: Thin-wall ansatz With this ansatz the energy and the charge of the Q-ball are Minimizing E while keeping Q fixed, one gets Spectrum in the flat potential Spectrum in the polynomial potential Conclusions Backups Perturbations in the thin-wall regime The equations for perturbations are the same as before. The functions f and g are now given by The equations are disentangled in the exterior of the Q-ball, r > R. In the interior, r < R, one obtains separate equations for the rotated vector Ξ = (ξ 1 , ξ 2 ) T such that where U diagonalizes the non-diagonal part of the linearized equations.
The resulting solutions are joined at r = R. This gives the spectrum of allowed values of γ. It is important to note that the near-critical regime of these (in general, relativistic) solitons can be analyzed by the means of the perturbation theory with respect to the relative frequency γ of an excitation. Decay mode of Q-clouds in the flat potential The decay mode is captured by the following spherically-symmetric ansatz, Defineγ asγ 2 ≡ γ 2 for ω < ωc ,γ 2 ≡ −γ 2 for ω ≥ ωc Perturbation theory near the cusp point Whenever γ is small, one can make use of the perturbation theory with respect to γ. Then, the linear perturbations of a Q-ball take a simple form Similarly, for the decay mode we have In this expression, the first term represents the Goldstone mode corresponding to the global U(1)-symmetry of the theory.