N=(2,0) non-Abelian Proca-Stückelberg theory in six dimensions

Abstract

We formulate a $N=(2,0)$ supersymmetric non-Abelian Proca-St\uckelberg theory in six space-time dimensions (6D). As the foundation of our construction, we start with our recent work on $N=1$ supersymmetric Proca-St\uckelberg formulation in 4D with a Yang-Mills (YM) multiplet $({{A}_{\ensuremath{\mu}}}^{I},{\ensuremath{\lambda}}^{I})$ and a chiral multiplet $({\ensuremath{\varphi}}^{I},{\ensuremath{\chi}}^{I},{\ensuremath{\phi}}^{I})$, where the index $I=1,2,\dots{},\mathrm{dim}\text{ }G$ is for the adjoint representation of a non-Abelian group $G$, while ${\ensuremath{\varphi}}^{I}$ parametrizes the coordinates of the group manifold $G$. Since ${\ensuremath{\varphi}}^{I}$ and ${\ensuremath{\phi}}^{I}$ transform differently under $G$, the conventional global $R$ symmetry is lost. Next, we apply this mechanism to 6D with the two multiplets: a YM multiplet $({{A}_{\ensuremath{\mu}}}^{I},{\ensuremath{\lambda}}^{\underset{_}{\ensuremath{\alpha}}I})$ and a hypermultiplet (HM) $({\ensuremath{\phi}}^{iJ},{{\ensuremath{\chi}}_{\underset{_}{\ensuremath{\alpha}}}}^{I},{\ensuremath{\varphi}}^{I})$. The index $i=1,2,3$ is for the $\mathbf{3}$ of $Sp(1)$. The spinorial index $\underset{_}{\ensuremath{\alpha}}=(\ensuremath{\alpha},A)$ ($\ensuremath{\alpha}=1,\dots{},4$) is for the Majorana-Weyl spinor index for $D=5+1$ with $A=1,2$ for the $\mathbf{2}$ of $Sp(1)$. As opposed to the common notion that all four scalars in a HM in 6D must form the ($\mathbf{2},\mathbf{2}$) of global $Sp(1)\ifmmode\times\else\texttimes\fi{}Sp(1)$, we can use a scalar ${\ensuremath{\varphi}}^{I}$ in the ($\mathbf{1},\mathbf{1}$) of $Sp(1)\ifmmode\times\else\texttimes\fi{}Sp(1)$ as a Nambu-Goldstone boson absorbed into the longitudinal component of ${{A}_{\ensuremath{\mu}}}^{I}$, separated from the remaining three scalars ${\ensuremath{\phi}}^{iI}$ in the ($\mathbf{3},\mathbf{1}$) of $Sp(1)\ifmmode\times\else\texttimes\fi{}Sp(1)$. Similar to our recent result in 4D with broken automorphism $R$ symmetry, the new feature of our result is that all four scalars in the HM in 6D do not have to form the ($\mathbf{2},\mathbf{2}$) of $Sp(1)\ifmmode\times\else\texttimes\fi{}Sp(1)$.