F-term

In theoretical physics, one often analyzes theories with supersymmetry in which F-terms play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using a superspace. This superspace involves four extra fermionic coordinates \( \theta ^{1},\theta ^{2},{\bar {\theta }}^{1},{\bar {\theta }}^{2} \), transforming as a two-component spinor and its conjugate.

Every superfield—i.e. a field that depends on all coordinates of the superspace—may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called chiral superfields, that only depend on the variables \( \theta \) but not their conjugates. The last term in the corresponding expansion, namely \( F\theta ^{1}\theta ^{2} \), is called the F-term. Applying an infinitesimal supersymmetry transformation to a chiral superfield results in yet another chiral superfield whose F-term, in particular, changes by a total derivative. This is significant because then \( {\displaystyle \int {d^{4}x\,F(x)}} \) is invariant under SUSY transformations as long as boundary terms vanish. Thus F-terms may be used in constructing supersymmetric actions.

Manifestly-supersymmetric Lagrangians may also be written as integrals over the whole superspace. Some special terms, such as the superpotential, may be written as integrals over θ {\displaystyle \theta } \theta s only. They are also referred to as F-terms, much like the terms in the ordinary potential that arise from these terms of the supersymmetric Lagrangian.