Gauge invariance of hypermultiplet mass and adjoint couplings

Let $\mathcal{R}$ be the ring of even smooth chiral superfields on the specified flat superspace, with coefficients in the finite-dimensional supercommutative complex algebra of the superspace definition. Thus ${\bar{D}}_{\dot{\alpha }}f=0$ for $f\in \mathcal{R}$; its even elements commute. Fix an integer $n\ge 1$, identify a specified based complex vector space $E$ with ${\mathbb{C}}^n$ and its dual with row vectors, and take $Q\in {\mathcal{R}}^n$, $\tilde{Q}\in ({\mathcal{R}}^n{)}^{\ast }$ and $M\in M_n(\mathcal{R})$. Suppose an invertible even chiral matrix $U\in G{L}_n(\mathcal{R})$ acts by the explicitly supplied substitutions

$Q^{\mathrm{\prime }}=UQ,{\tilde{Q}}^{\mathrm{\prime }}=\tilde{Q}{U}^{-1},M^{\mathrm{\prime }}=UM{U}^{-1}\mathrm{.}$

Then the chiral polynomials $\tilde{Q}Q$ and $\tilde{Q}MQ$ are invariant. In particular $\sqrt{2}\tilde{Q}MQ+m\tilde{Q}Q$ is invariant for every constant $m\in \mathbb{C}$. Assigning the formal degrees $[Q]=[\tilde{Q}]=[M]=[m]=1$ makes this sum homogeneous of degree three. These are algebraic claims about the supplied action; they require no physical model or asserted supersymmetry realization. For the application $M={\rho }_{R\ast }(\mathrm{\Phi })$, the same conclusion holds whenever that represented adjoint field has the stated conjugation law.

Four-dimensional $N=1$ chiral superfield on flat superspace

Four-dimensional $N=1$ superspace Grassmann algebra and Berezin integration