Auxiliary field

In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian describing such a field $A$ contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):
$mathcal\{L\}\_\{aux\}=frac\{1\}\{2\}(A,A)+(f(varphi),A)$.
The equation of motion for $A$ is: $A(varphi)=-f(varphi)$ and the Lagrangian becomes:$mathcal\{L\}\_\{aux\}=-frac\{1\}\{2\}(f(varphi),f(varphi))$. Auxiliary fields do not propagate and hence the content of any theory remains unchanged by adding such fields by hand.If we have an initial Lagrangian $mathcal\{L\}\_\{0\}$ describing a field $varphi$ then the Lagrangian describing both fields is:
$mathcal\{L\}=mathcal\{L\}\_\{0\}(varphi)+mathcal\{L\}\_\{aux\}=mathcal\{L\}\_\{0\}(varphi)-frac\{1\}\{2\}(f(varphi),f(varphi))$.
Therefore, auxiliary fields can be employed to cancel quadratic terms in $varphi$ in $mathcal\{L\}\_\{0\}$ and linearize the action $mathcal\{S\}\; =\; int\{mathcal\{L\},d^nx\}.$

Examples of auxiliary fields are the complex scalar field F in a chiral superfield, the real scalar field D in a vector superfield, the scalar field B in BRST and the field in the Hubbard-Stratonovich transformation.

References

* Superspace, or One thousand and one lessons in supersymmetry [http://arxiv.org/abs/hep-th/0108200 arXiv:hep-th/0108200]