quantum field theory - 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:

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



2. Why is it needed?


3. 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?


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

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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).

Answer

1. 
   

   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.

   
   
   


2. 
   

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

   
   


3. 
   

   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)

   
   

$$L = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \frac{1}{2}(\partial^\mu \phi + m A^\mu)(\partial_\mu \phi + m A_\mu)$$

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