Why is there sometimes an additional term in the orthogonality relations for the polarization vectors? When considering the polarization vectors of a massive spin-1 field, like an $A_\mu$ with Lagrangian density
$$ \tag{A} \mathscr{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{2}M^2 A_\mu A^\mu,$$
we expand the solutions of $A_\mu$ as
$$ \tag{B} A^\mu(x) = \sum_\lambda \int d \tilde{k}
[ \varepsilon^{\mu}(\lambda,k) a(\lambda,k) e^{-ikx} 
+ \varepsilon^{\mu}(\lambda,k)^* a^\dagger(\lambda,k) e^{ikx} ], $$
where the three polarization vectors $\{ \varepsilon(\lambda,k) \}_\lambda$ satisfy the completeness relations
$$ \tag{C} \sum_\lambda \varepsilon^\mu(\lambda,k) \varepsilon^{\nu*}(\lambda,k) = - \eta^{\mu\nu} + \frac{k^\mu k^\nu}{M^2}.  $$
The questions:


*

*Where does the second term $k^\mu k^\nu/M^2$ come from?

*Why is it needed?

*Using (C) and going to the massless $M \to 0$ limit we get a singularity, which would make the relation ill defined. So how should the massless limit be handled?

*In what other circumstances (massive/massless spin-X fields) is a similar correction to the completeness relations needed?



The above formulas can be found for example in Srednicki, (85.16), with a somewhat different notation (he uses opposite Minkowski space metric convention than me).
 A: *

*Observe that the "completeness" relation is just the projector onto the space spanned by the polarisation vectors. A priori, we know that $P_\epsilon^{\mu\nu} := \sum_\lambda \epsilon^\mu_\lambda(k)\epsilon^{*\nu}_\lambda(k)$ therefore projects onto the subspace orthogonal to the momentum, since $\zeta^\mu k_\mu = 0$ for any polarization vector $\zeta^\mu$. The most general symmetric 2-tensor depending on only the metric and the momentum is $P^{\mu\nu}_\epsilon = a\eta^{\mu\nu} + b k^\mu k^\nu$ with $a,b \in \mathbb{C}$ initially arbitrary. Applying the projector to the momentum yields $P^{\mu\nu}_\epsilon k_\mu = a k_\nu + b k^2 k_\nu \overset{!}{=} 0$, and using $k^2 = m^2$, the relation you quote is obtained.

*Because it is the condition that the vector is a polarization, and not any vector.

*The massless limit of the Proca action is indeed ill-defined. One can use the Stückelberg Lagrangian with an auxiliary scalar field $\phi$, which has a smooth massless limit as well as explicit gauge invariance for massive $\mathrm{U}(1)$ fields. (This is why massive photons are not a problem in QED)
$$ L = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \frac{1}{2}(\partial^\mu \phi + m A^\mu)(\partial_\mu \phi + m A_\mu)$$


*You need such a modification every time you have unphysical degrees of freedom, as we have with polarizations $\sim k$ in this case. 

