Q-ball Formation through Affleck-Dine Mechanism

A Q-ball is a non-topological soliton with some conserved global charge in the scalar field theory [1]. A Qball solution exists if the energy minimum develops at non-zero φ with fixed charge Q. In terms of the effective potential of the field φ, V (φ)/φ take the minimum at φ 6= 0 [1,2]. The Q-ball naturally appears in the spectrum of the Minimal Supersymmetric Standard Model (MSSM) [3]. In particular, very large Q-balls could exist in the theory with a very flat potential [4], such as in the MSSM which has many flat directions, which consist of squarks and sleptons, carrying baryonic and/or leptonic charges [5]. It provides interesting attention to phenomenology and astrophysics [4,6,7]. Cosmologically, it is interesting that large Q-balls carrying baryonic charge (B-ball) can be promising candidate for the dark matter of the universe, and/or the source for the baryogenesis [8–10]. Moreover, they can explain why the energy density of baryons is as large as that of the dark matter (at least within a few orders of magnitude). If the effective potential of the field φ carrying the baryonic charge is very flat at large φ, as in the theory that the supersymmetry (SUSY) breaking occurs at low energy scales (gauge-mediated SUSY breaking), B-ball energy per unit charge decreases as the charge increases [4]. For large enough charge, such as B ∼ 10, the B-ball cannot decay into nucleon, and is completely stable, which implies that B-balls themselves can be the dark matter [8] with charges B = 10 − 10 [11], while baryons are created by the conventional AD mechanism. In the case of gravity-mediated SUSY breaking scenario, the B-balls can decay into quarks or nucleons, with the decay (evaporation) rate of the Q-ball proportional to the surface area [12], and if they decay after the electroweak phase transition, there are some advantage over the conventional AD baryogenesis [9,10]. For example, B-balls can protect the baryon asymmetry from the effects of lepton violating interactions above the electroweak scale when anomalous B + L violation is in thermal equilibrium. Another one is that Q-balls with B−L charge survive the sphaleron effects to create the same amounts of baryon and lepton numbers. In either case, it is necessary for Q-balls to have large charges, such as Q = 1022−1028 [9,10]. In this scenario, dark matter is LSP which arises from the Q-ball decay, and parameters of MSSM could be constrained by investigating the Q-ball cosmology [13]. Note that it is also possible to have stable Q-balls in the gravity-mediated SUSY breaking theory depending on the details of the features of the hidden sector [14]. Those large Q-balls are expected to be created through Affleck-Dine (AD) mechanism [15] in the inflationary universe [8–10]. The coherent state of the AD scalar field which consists of some flat direction in MSSM becomes unstable and instabilities develop. These fluctuations grows large to form Q-balls. The formation of large Q-balls has been studied only linear theory analytically [8–10] and numerical simulations was done in one dimensional lattices [8]. Both of them are based on the assumption that the Q-ball configuration is spherical so that we cannot really tell that the Q-ball configuration is actually accomplished. (Recently, some aspects of the dynamics of AD scalar and Q-ball formation were studied in Ref. [16], but the whole evolution was not investigated, which is important for the investigation of the Q-ball formation.) In this Letter, we study the dynamics of a complex scalar field with very flat potential numerically in one, two, and three dimensional lattices, without assuming spherical Q-ball configuration. On one dimensional lattices, it is equivalent to the system independent of other two dimensions, so that we are observing plane-like objects. We call them Q-walls. Likewise, string-like objects, which we call Qstrings, appear on two dimensional lattices. First we show where instabilities of a scalar field comes from. To be concrete, let us assume that the complex