Cosmological constraint on the scale of the supersymmetric singlet Majoron

In a supersymmetric theory with a spontaneously broken global symmetry $G$, if the scale of sypersymmetry breaking, $M_s$, is smaller than the scale $M_G$ of the global symmetry, the Nambu-Goldstone boson $\chi$ is accompanied by two massive superpartners (a fermionic ${\Psi }_{\chi }$ and a scalar boson ${\sigma }_{\chi }$) with a mass of the order of $M_s$. Cosmological considerations imply stringent constraints on the couplings of ${\Psi }_{\chi }$ and ${\sigma }_{\chi }$. Application of these considerations to the supersymmetric singlet Majoron model leads to an upper limit on the scale $V_{\mathrm{BL}}$ of global ${\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}_{B-L}$ symmetry to be ≤ ${10}^{14}$ GeV, for reasonable values of parameters in the theory.