Biased Discrete Symmetry Breaking and Fermi Balls

The spontaneous breaking of an approximate discrete symmetry is considered, with the resulting protodomains of true and false vacuum being separated by domain walls. Given a strong, symmetric Yukawa coupling of the real scalar field to a generic fermion, the domain walls accumulate a gas of fermions, which modify the domain wall dynamics. The splitting of the degeneracy of the ground states results in the false vacuum protodomain structures eventually being fragmented into tiny false vacuum bags with a Fermi gas shell (Fermi balls), that may be cosmologically stable due to the Fermi gas pressure and wall curvature forces, acting on the domain walls. As fermions inhabiting the domain walls do not undergo number density freeze out, stable Fermi balls exist only if a fermion anti-fermion asymmetry occurs. Fermi balls formed with a new Dirac fermion that possesses no standard model gauge charges provide a novel cold dark matter candidate. It is well known that spontaneous breaking of a discrete symmetry can produce topological structures composed of different domains separated by topological defects [1, 2, 3]. In the simplest such physical scenario, the topological defects produced are domain walls [1] (transition regions between spatial domains that possess topologically different vacuum orientations), which within the context of cosmological models, have been applied to phenomena ranging from energetically soft topological defects [4] and structurons [5, 6] for the formation of large scale structure [7, 8], to significant deviations from thermal equilibrium at the QCD scale [9], neutrino balls [10, 11], and an origin for cosmological Gamma Ray Bursts [12]. In this paper, the interaction of domain walls with a fermion sector is considered, which suggests the possible production of composite microscopic cosmological relics referred to henceforth as Fermi balls. These Fermi balls, under certain conditions, provide an unusual source for cold dark matter, and may be relics of the seeds for possible structure formation in the cold dark matter scenario. The simplest model exhibiting topological structure is that of a real scalar field φ with a Lagrange density of the form L = 1 2 ∂μφ∂ φ− λ 2 8 (φ − φ20)2 (1) Clearly, equation (1) possess a Z2 symmetry (invariance under φ → −φ), which if spontaneously broken results in a vacuum expectation value (VEV) for φ that has two possible values; < φ >= ±φ0. These two VEV’s correspond to topologically distinct vacuum orientations (distinct values of the order parameter); here the notion of topologically distinct vacua implies that one vacuum orientation cannot be continuously deformed into the other. Yet due to the Z2 symmetry being exact, neither VEV is preferred, so the determination of the VEV in a particular spatial region is set by random fluctuations in φ. Thus the spontaneous symmetry breaking results in a randomly generated network of spatial domains of both vacuum orientations that are separated by transition regions called domain walls ( topological defects). The form of the domain wall solution is a topological soliton of class π0 [13], and is easily obtained from the equation of motion for φ. The simplest such solution is that of a planar domain wall in the xy plane at z = 0 with the boundary conditions φ(z → ±∞) = ±φ0, and has the form < φ >= φ0 tanh( zδ ). Here δ = 2 λφ0 is the wall thickness. Typically, δ is assumed to be small compared to the average radius of curvature of the walls (the thin wall approximation ), so that the domain walls can be treated as two dimensional