We derive an explicit expression for the pure vector superfield in gauge theories in the Wess-Zumino gauge from which the supersymmetric action is directly obtained from a so-called -term. By a pure vector superfield, it is meant that its theta independent part transforms as a Lorentz vector. The pure vector superfield is not to be confused with the well known (scalar)-vector superfield [1-4] obtained by imposing a reality condition on the general scalar superfield, whose theta independent part is a scalar, and neither is to be confused with the well known spinor superfield [1-4], whose theta independent part is a spinor, both containing a vector field, and the supersymmetric action is obtained from the latter from a so-called -term. Although the derivation is somehow tedious, the theta dependent part of the pure vector superfield turns out to be not complicated.

In the celebrated Wess-Zumino gauge, and in a four component representation, the (scalar)-vector superfield takes the form [5,6]

with the following residual gauge transformation

where the gauge function is given by

One may define a pure vector superfield [5,6] as follows

where is the charge conjugation matrix in the chiral representation. Under the supergauge transformation Equation (2),

where we recall that is left-chiral and hence is right-chiral. Accordingly, they are, respectively, annihilated by the supercovariant derivatives

where. That is,

We may rewrite the transformation rule in Equation (5) as

Due to the first equality in Equation (6), we may replace the product in the second term on the extreme right-hand side of Equation (7) by their anticommutator. This anti-commutator may be obtained from by multiplying it by

leading to

and showing that it transforms as a non-abelian gauge field.

Using the relations,

Equation (4) may be equivalently re-expressed as

In the Wess-Zumino gauge,

Applying the supercovariant derivative to it and using, in the process, the expansion of the product together with the orthogonality relations between the product of any of two of, , , give

Multiplying the latter equation by

from the left, leads to

Now we apply to the above equation, and use, in the process, the following properties,

to obtain

where

The identities

and, lead to the following expressions for, and,

Before giving the final expression for, we note it may be now re-written as

since, hence

as is easily checked, where note that the quadratic term from the exponential generates also a term from the -independent term within the square brackets in Equation (23). Since the exponential term represents the translation operator of the argument of the component fields by, our final expression for, in the Wess-Zumino supergauge, becomes simply

Using the relation, with the (Hermitian) matrices as the generators of the underlying group, Equation (24) is equivalently expressed as

where

and denotes the adjoint representation of the generators.

Although the derivation is somehow tedious, the final expression of the pure vector superfield and its theta dependent part are not complicated. The explicit expression for the pure vector superfield allows the construction of the supersymmetric action corresponding directly to the so-called -term, as is readily checked, rather than from the -term constructed out of a spinor superfield, as is usually done. We hope that this novel expression of a pure vector superfield, derived here, will be useful in supersymmetric (vector) gauge theories and justifies this analysis.

The author would like to thank his colleagues at the Institute for the interest they have shown in this work.