= Solution
For a <massless supermultiplet>, put the momentum along the third axis. The matrix $2\sigma\cdot p$ in the <supersymmetry algebra> has rank one, so only one fermionic oscillator acts nontrivially. It relates states whose <helicities> differ by $\tfrac12$. Including the antiparticles required by <CPT completion of a supermultiplet> gives the following complete spectra.
A <chiral multiplet> contains a complex scalar $A$ and a <Weyl spinor> $\psi$. Its scalar particle and antiparticle have helicity $0$; its <fermion> particle and antiparticle have helicities $-\tfrac12$ and $+\tfrac12$. Therefore \b[$n_B=2=n_F$], with maximum absolute <helicity> $\tfrac12$.
A <supersymmetric vector multiplet> contains a massless gauge vector $A_\mu$, with helicities $+1,-1$, and a <Majorana spinor> <gaugino> $\lambda$, with helicities $+\tfrac12,-\tfrac12$. Again \b[$n_B=2=n_F$], now with maximum absolute <helicity> $1$. The chiral $F$ and vector $D$ <auxiliary fields> contribute no on-shell states.
\b[The Higgs mechanism is compatible with unbroken supersymmetry.] A <supersymmetric vacuum> may have nonzero scalar <vacuum expectation values> that break the internal <gauge symmetry> while obeying both <F-flatness> and <D-flatness>. For example, oppositely charged <chiral superfields> with $W=0$, no <Fayet–Iliopoulos term> and equal nonzero scalar magnitudes have $F=D=0$ but break the Abelian <gauge group>. In the <supersymmetric Higgs mechanism>, a massless <vector multiplet> combines with the <chiral multiplet> along the broken gauge direction. One real scalar becomes the longitudinal vector polarization; the other remains a physical real scalar, and the <gaugino> combines with the <Higgsino> into a <Dirac spinor>. The resulting massive <vector multiplet> has $3+1=4$ <bosonic> and $4$ <fermionic> states with a common mass, as in part (i). Any other uneaten <chiral multiplets> remain separately in the spectrum.
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