Incompatibility Of Simultaneous Non-Linear Realizations Of Scale Symmetry and Supersymmetry

Simultaneous nonlinear realizations of supersymmetry and softly broken scale and chiral symmetries are investigated. To guarantee Nambu-Goldstone realizations of the symmetries, the Goldstino decay constant is forced to vary as the explicit soft scale and chiral symmetry breaking parameters. Consequently, it must vanish in the chiral limit and the simultaneous nonlinear realizations of the super and scale symmetries proves inconsistent. Goldstone’s theorem [1] guarantees that associated with every spontaneously broken global symmetry there is a massless particle. Below the symmetry breaking scale, the dynamics of these Nambu-Goldstone degrees of freedom can be descibed by an action which realizes the spontaneously broken symmetry nonlinearly. This effective action encapsulates all the consequences of the symmetry current algebra and, through lowest non-trivial order in a derivative expansion, is unique up to reparametrization, independent of the underlying theory [2]-[3]. For the case of an internal global symmetry group G spontaneously broken to an invariant subgroup H , the Nambu-Goldstone fields, π , i = 1, ...dimG/H , act as coordinates of the coset manifold G/H . A particular choice of the coset coordinates is the standard realization parametrized as U(π) = e 2iTiπi Fπ , (1) where T is the fundamental representation of G and Fπ is the Nambu-Goldstone boson decay constant. While U transforms linearly under G, the Nambu-Goldstone fields, π, transform linearly only under H and nonlinearly under the spontaneously broken G generators. With this choice of coordinates, the G invariant effective Lagrangian is simply [4] L = F 2 π 4 Tr [ ∂μU †∂μU ] . (2) In addition to the spontaneous symmetry breaking, there also often appears some soft explicit symmetry breaking. For the case of chiral symmetry breaking, this explicit breaking takes the form of a soft mass term and the above effective Lagrangian is modified to L = F 2 π 4 Tr [ ∂μU †∂μU ] − uF 2 πTr [ mU † + Um ] . (3) Here m is the mass matrix characterizing the soft explicit breaking and u is the order parameter of the spontaneous symmetry breaking. For example, if the chiral symmetry is dynamically broken due to some underlying strong gauge interaction, then u = <ψ̄ψ> 2F 2 π , where ψ is a chiral fermion of the underlying theory. The above effective Lagrangian assumes that the theory is free of chiral anomalies. If such effects are also present, the effective action is further modified by the inclusion of a Wess-Zumino