On the Primordial Magnetic Field from Domain Walls

In this paper we discuss once more the zero mode contribution to the vacuum energy density. We show that a careful treatment of the zero modes leads to the conclusion that domain walls may be ferromagnetic, and could generate a magnetic field of cosmological interest. Electronic address: [email protected] Electronic address: [email protected] In Ref. [1] we suggested that the spontaneous generation of uniform magnetic condensate in QED3 [2], [3], [4] could give rise to ferromagnetic domain walls at the electroweak phase transition. Moreover, we suggested that these domain walls generate a magnetic field at the electroweak scale which can be relevant for the generation of the primordial magnetic field [5]. In this paper we focus on some relevant points of our technique with respect to other approaches. In particular in Ref. [6], for example, the author points out that massive (2 + 1) dimensional fermions bound to a domain wall behave diamagnetically rather than ferromagnetically. This delicate question concernes the magnetic field contribution to the fermionic vacuum free energy discussed for the first time in Ref. [2]. The aim of the present paper is twofold. First, we show that the results of [6] derive from an incorrect treatment of the fermionic zero mode contribution to the zero temperature vacuum energy density. Secondly, we show that the finite temperature calculation of Ref. [6] agrees with our previous findings [4], [7]. Moreover, if one takes into account the correct definition of an Abelian magnetic field in presence of a varying scalar field condensate, then it turns out that the classical magnetic energy is proportional to the area of the wall. This last result supports our previous proposal in Ref. [1]. The crucial point of the question is the zero temperature vacuum energy density due to massive planar fermions in presence of a constant magnetic field: Evac = −eB 2 π ∞