Spontaneous Supersymmetric Generation of an Indeterminate Mass Scale

If a global continuous symmetry of a supersymmetric field theory is spontaneously broken while preserving the supersymmetry, the resulting theory has a massless superfield. One of its two bosonic degrees of freedom is the familiar phase rotation of the usual massless Nambu-Goldstone boson, but the other is a scale transformation. An indeterminate mass scale is thus generated. However, any breaking of the supersymmetry would resolve this ambiguity. The physical mass scales of a renormalizable quantum field theory are expected to be determined by its explicit parameters, such as in quantum electrodynamics, or by the structure of its interactions, such as in quantum chromodynamics. In either case, even if the vacuum has nontrivial topology, the physical mass scales of the theory are uniquely determined. Consider the textbook example of the U(1) scalar model, with the potential V1 = m φφ+ 1 2 λ(φφ). (1) If m < 0, then the global U(1) symmetry is broken by the vacuum expectation value (VEV) of the scalar field φ, i.e. 〈φ〉 ≡ v = √ −m2 λ e. (2) Redefining φ ≡ v + e (