About electroweak domain walls in Majoron models

Some time ago it was claimed in “Spontaneous breaking of lepton number and cosmological domain wall problem” [Lazarides ] that nonperturbative instantons of the weak interaction SU ( 2 ) W lead to the formation of domain walls in Majoron models owing to the anomaly of the spontaneously broken global lepton number L symmetry U ( 1 ) L with respect to SU ( 2 ) W . We point out that it has long been known that this effect can be completely rotated away unless there is a source of explicit B + L breaking present, where B denotes the baryon number. We further estimate the tiny instanton-induced Majoron mass from B + L breaking and analyze the cosmological impact of such domain walls. In general, this scenario does not lead to a cosmological catastrophe, and we demonstrate that the tiny instanton-induced mass can act as a bias term to collapse walls induced by a larger source of lepton number breaking. Alternatively, this electroweak Majoron could act as dynamical dark energy.