No Primordial Magnetic Field from Domain Walls

It is pointed out that, contrary to some claims in the literature, the domain walls cannot be a source of a correlated at large scales primordial magnetic field, even if the fermionic modes bound on the wall had ferromagnetic properties. In a particular model with massive (2+1) dimensional fermions bound to a domain wall, previously claimed to exhibit a ferromagnetic behavior, it is explicitly shown that the fermionic system in fact has properties of a normal diamagnetic with the susceptibility vanishing at high temperature. The existence of magnetic field correlated at a galactic scale [1] is believed to require a strong primordial field correlated at cosmological distances at some stage in the early universe. This phenomenon would find a natural explanation, if the primordial field was created by extended objects, having a cosmological size. One class of extended objects that might have existed in the early universe is provided by domain walls, assuming that a model containing these walls successfully avoids general constraints [2] on undesirable cosmological consequences of such extended defects. There has been several claims in the literature [3, 4, 5, 6] that the modes of fermion field bound to a domain wall produce, in certain models, a magnetic field B, which could provide a much needed explanation of a primordial magnetic field in the early universe. The purpose of this note is to point out that, as enticing as this explanation could be, it cannot be physically correct. Namely, it is almost trivial to show that, irrespective of the details of the dynamics, a correlated over the entire area of a flat domain wall magnetic field should be greatly suppressed by an inverse power of a cosmological size, even if the wall exhibited a ferromagnetic behavior. Furthermore, in the models that are sufficiently well formulated [3, 4], the walls are in fact diamagnetic, contrary to the previous claims. In this letter the model of Ref.[4, 5] with massive fermionic modes with broken parity is considered, and it is shown that the fermion system on the wall is a normal diamagnetic, while the previous claim of ferromagnetism is based on an incorrect formula [8] for the B dependent free energy of the fermionic vacuum. 1 In order to prove the statement about the suppression of the magnetic field that a domain wall might spontaneously create, let us introduce a large normalization box with the sides of length L, and consider a flat domain wall, spanning the box along the x, y plane, thus z being the coordinate axis perpendicular to the wall. From symmetry, a correlated magnetic field can have only the z component Bz, which does not depend on the coordinates x and y parallel to the wall. Then the Maxwell’s equation div B = 0 dictates that Bz also cannot depend on the coordinate z, i.e. Bz has to be constant: Bz = B. The energy of such constant magnetic field is proportional to the volume L of the box, while, the energy, associated with dynamics on the wall is proportional to the area L of the wall, so that the total energy of the system is written as E(B) = L B 2 + L f(B) (1) with f(B) being the surface energy density in the presence of the magnetic field, determined by specific dynamics on the wall. Clearly, for any finite function f(B) the value of B The claim of [3] was explicitly shown [7] to be based on an incorrect calculation of the energy of the gas of massless (2+1) dimensional fermions in magnetic field.