D-term

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In theoretical physics, the D-term is the final term in the expansion of a vector superfield over fermionic coordinates. A superfield is a field that depends on all coordinates of the superspace, which is the coordinate space of a theory exhibiting supersymmetry.

A superspace can be expressed as a combination of ordinary space dimensions (x, y, z, ...,) and fermionic dimensions. 4D N = 1 global supersymmetry may be written using a superspace involving four extra fermionic coordinates ${\displaystyle \theta ^{1},\theta ^{2},{\bar {\theta }}^{1},{\bar {\theta }}^{2}}$, transforming as a two-component spinor and its conjugate. Every superfield may be expanded with respect to the new fermionic coordinates. The generic kind of superfields, typically a vector superfield, indeed depend on all these coordinates. The last term in the corresponding expansion, namely ${\displaystyle D\theta ^{1}\theta ^{2}{\bar {\theta }}^{1}{\bar {\theta }}^{2}}$, is called the D-term.

Manifestly supersymmetric Lagrangians may also be written as integrals over the whole superspace. A D-term obtained from a vector superfield solely by an integral over all of superspace is known as a Fayet–Iliopoulos D-term. Some special terms, such as the superpotential, may be written as integrals over ${\displaystyle \theta }$s only, which are known as F-terms, and should be contrasted with the present D-terms.