Massive and massless spin-1 particles with gauge symmetry without Stueckelberg fields

Abstract

In order to generate mass for an abelian spin-1 vector field while preserving gauge invariance we couple it to a symmetric tensor. The derivative coupling includes up to three derivatives. We show that unitarity, causality and absence of Stueckelberg (compensating) fields single out a unique model up to trivial field redefinitions. The model contains one massive and one massless spin-1 particle. It is shown by means of a master action to be dual to the direct sum of a Maxwell plus a Maxwell–Proca theory.

Appendices

Appendix A

Following closely the notation of [24] we display here the rank-4 projection operators $P_{JJ}^{(s)}$ and transition operators $P_{WS}^{(0)},P_{SW}^{(0)}$ which appear in (14):

(65)

(66)

(67)

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where the rank-2 spin-1 and spin-0 projection operators are given by

$$ \theta_{\alpha\beta} = ( \eta_{\alpha\beta} - \omega_{\alpha \beta} ), \qquad\omega_{\alpha\beta} = \frac{\partial _{\alpha} \partial _{\beta}}{\Box }. $$

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In order to check the operator G ⁻¹ given in (14) the reader can use the algebra

$$ \bigl( P_{IJ}^{(s)} \bigr)_{\lambda\mu\alpha\beta} \bigl( P_{LK}^{(r)} \bigr)^{\alpha\beta\gamma\delta} = \delta^{sr} \delta_{JL} \bigl( P_{IK}^{(s)} \bigr)_{\lambda\mu}^{\quad \gamma\delta}. $$

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Appendix B

Assuming c ₄≠0 here we show that the spin-2 pole of G ⁻¹ at c ₄□−d=0 represents a ghost. The relevant piece of A(k) regarding the residue at k ²=−m ²=−d/c ₄, see (14), (20), (19), and (29) is given by

$$ A(k) = - \frac{i}{2} \frac{J_{\mu\nu}^* ( P_{SS}^{(2)} )_{\mu\nu\gamma\delta}J^{\gamma\beta}}{c_4(k^2 + m^2)}. $$

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Using the general decomposition of a symmetric tensor in momentum space: $J_{\mu\nu} = k_{\mu}J_{\nu} + k_{\nu}J_{\mu} + J_{\mu\nu}^{T}$ with $k^{\mu}J_{\mu\nu}^{T} = 0 = k^{\nu}J_{\mu\nu}^{T}$ we write

$$ J^*J \equiv J_{\mu\nu}^* \bigl( P_{SS}^{(2)} \bigr)_{\mu\nu\gamma\delta}J^{\gamma\beta} = \bigl( J_{\mu\nu}^T \bigr)^* J_T^{\mu\nu} - \frac{ J^*_T J_T }{D-1} $$

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where $J_{T} = (J_{T})^{\mu}_{\,\,\,\, \mu}$. Consequently, the imaginary part of the residue of A(k) at k ²=−m ² is given by

$$ R_m^{(s=2)} = - \frac{J^* J }{2 c_4}. $$

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It can be shown that $J_{T}^{*}J_{T} > 0$ at any dimension D>2 and $J_{T}^{*}J_{T} = 0$ at D=2. Since d>0, see (38), the absence of tachyons (m ²=d/c ₄>0) requires c ₄>0. Thus, R _m <0 and we end up with a spin-2 ghost. Likewise one can check that spin-0 poles of G ⁻¹ at K=0 lead at least to one ghost.