Transversality of derivative-exponential primary vertices ($p\cdot \zeta =0,p^{\mu }{\zeta }_{\mu \nu }=p^{\nu }{\zeta }_{\mu \nu }=0$)

For free embedding bosons, the operator product expansion of the stress tensor with ${\zeta }_{\mu }:\partial X^{\mu }{e}^{ipX}:$ has a cubic pole $-i{\alpha }^{\prime }(p\cdot \zeta ):e^{ipX}:/2$. Its vanishing is precisely the primary condition, with weight $1+{\alpha }^{\prime }{p}^2/4$. The two-derivative closed-string vertex has the two displayed transversality conditions from its holomorphic and antiholomorphic cubic poles. Being primary does not require $p^2=0$; that extra condition follows when a physical integrated vertex must have conformal weights $(1,1)$.