... Bizdadea¹

E-mail address: bizdadea@central.ucv.ro

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... Cioroianu²

E-mail address: manache@central.ucv.ro

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... Danehkar³

Present address: School of Mathematics and Physics, Queen's University, Belfast BT7 1NN, UK. E-mail address: adanehkar01@qub.ac.uk

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... Saliu⁴

E-mail address: osaliu@central.ucv.ro

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...araru⁵

E-mail address: scsararu@central.ucv.ro

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... theories.⁶

Decomposition (87) does not include a component responsible for the self-interactions of the tensor field with the mixed symmetry $(2,1)$ since any such component has been proved in [62] to be trivial.

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... as⁷

In principle, one can add to $a_{2}^{\mathrm{int}}$ the terms $\left( \tilde{ M}_{2}\right) ^{\mu \nu \rho }\eta D_{\mu \nu \rho }+\tfrac{1}{... ...{\mu \nu }\sigma ^{\alpha \beta }\tilde{D}_{\mu \alpha }\tilde{ D}_{\nu \beta }$, where $\left( \tilde{M}_{2}\right) ^{\mu \nu \rho }$ is the Hodge dual of an expression similar to (85) with $W\left( \varphi \right) \rightarrow M_{2}\left( \varphi \right)$, and $\left( M_{3}\right) ^{\mu \nu }$ reads as in (85) with $W\left( \varphi \right) \rightarrow M_{3}\left( \varphi \right)$. Both $M_{2}$ and $M_{3}$ are some arbitrary, real, smooth functions depending on the undifferentiated scalar field. It can be shown that the above terms finally lead to trivial interactions, so they can be removed from the first-order deformation.

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... as⁸

In principle, one can add to $a_{3}^{\mathrm{int}}$ the term $\left( M_{1}\right) _{\mu \nu \rho }\tilde{D}^{\mu \nu }S^{\rho }$, where $\left( M_{1}\right) _{\mu \nu \rho }$ reads as in (84), with $W\left( \varphi \right) \rightarrow M_{1}\left( \varphi \right)$. It is possible to show that such a term outputs only trivial deformations.

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