superfield

A superfield is a field defined on superspace, an extension of ordinary spacetime by anticommuting Grassmann coordinates. It provides a compact description of supersymmetry by grouping bosonic and fermionic degrees of freedom into a single object that transforms linearly under supersymmetry.

The most common types are chiral superfields and vector superfields. A chiral superfield typically contains a complex scalar, a Weyl fermion, and an auxiliary field, while a vector superfield describes gauge degrees of freedom and contains a gauge field, a gaugino, and an auxiliary field. The auxiliary fields do not propagate but are needed to close the SUSY algebra off-shell.

In four-dimensional N=1 theories, a chiral superfield Phi satisfies a constraint involving superspace derivatives and has an expansion Phi(y, theta) = phi(y) + sqrt(2) theta psi(y) + theta^2 F(y), with y^mu = x^mu + i theta sigma^mu bar theta. The corresponding vector superfield V is real and can be written in a convenient gauge (the Wess-Zumino gauge) with components including the gauge field A_mu, the gaugino lambda, and an auxiliary field D. The gauge-field strength is encoded in a chiral field W_alpha = -1/4 bar D^2 D_alpha V.

Supersymmetric actions are built as superspace integrals: kinetic terms for chiral fields come from ∫ d^4θ Phi^† Phi, superpotential terms from ∫ d^2θ W(Phi) plus its hermitian conjugate, and gauge kinetic terms from ∫ d^2θ Tr(W_alpha W^alpha) plus h.c. This formalism makes supersymmetry manifest and simplifies model construction.

Historically, superfields were introduced in the 1970s by Wess and Zumino and by Salam and Strathdee, and they remain central to modern SUSY model building and string theory.