Generalised Proca theories in teleparallel gravity


Generalised Proca theories of gravity represent an interesting class of vector–tensor theories where only three propagating degrees of freedom are present. In this work, we propose a new teleparallel gravity analog to Proca theories where the generalised Proca framework is extended due to the lower-order nature of torsion-based gravity. We develop a new action contribution and explore the example of the Friedmann equations in this regime. We find that teleparallel Proca theories offer the possibility of a much larger class of models in which do have an impact on background cosmology.

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  1. 1.

    R. Aldrovandi and J.G. Pereira, Teleparallel Gravity, vol. 173. Springer, Dordrecht, 2013.

  2. 2.

    R. Aldrovandi, P.B. Barros, J.G. Pereira, Spin and anholonomy in general relativity (2004)

  3. 3.

    E. Allys, P. Peter, Y. Rodriguez, Generalized Proca action for an Abelian vector field. JCAP (2016).

    Article  Google Scholar 

  4. 4.

    S. Bahamonde, K.F. Dialektopoulos, M. Hohmann, J. Levi Said, Post-Newtonian limit of teleparallel horndeski gravity. arXiv:2003.11554 [gr-qc]

  5. 5.

    S. Bahamonde, S. Capozziello, Noether symmetry approach in \(f(T, B)\) teleparallel cosmology. Eur. Phys. J. C 77(2), 107 (2017).

    ADS  Article  Google Scholar 

  6. 6.

    S. Bahamonde, C.G. Bhmer, M. Wright, Modified teleparallel theories of gravity. Phys. Rev. (2015).

    MathSciNet  Article  Google Scholar 

  7. 7.

    S. Bahamonde, M. Zubair, G. Abbas, Thermodynamics and cosmological reconstruction in \(f(T, B)\) gravity. Phys. Dark Univ. 19, 78–90 (2018).

    Article  Google Scholar 

  8. 8.

    S. Bahamonde, K.F. Dialektopoulos, J. Levi Said, Can Horndeski theory be recast using teleparallel gravity? Phys. Rev. D (2019).

    MathSciNet  Article  Google Scholar 

  9. 9.

    S. Bahamonde, J. Levi Said, M. Zubair, Solar system tests in modified teleparallel gravity. JCAP (2020).

    Article  Google Scholar 

  10. 10.

    S. Bahamonde, K.F. Dialektopoulos, V. Gakis, J. Levi Said, Reviving Horndeski theory using teleparallel gravity after GW170817. Phys. Rev. D 1, 1–2 (2020).

    MathSciNet  Article  Google Scholar 

  11. 11.

    L. Baudis, Dark matter detection. J. Phys. (2016).

    Article  Google Scholar 

  12. 12.

    J. Beltran Jimenez, L. Heisenberg, Derivative self-interactions for a massive vector field. Phys. Lett. B 757, 405–411 (2016).

    ADS  Article  MATH  Google Scholar 

  13. 13.

    D. Blixt, M. Hohmann, M. Krššk, C. Pfeifer, Hamiltonian Analysis In New General Relativity, in 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (2019)

  14. 14.

    D. Blixt, M. Hohmann, C. Pfeifer, Hamiltonian and primary constraints of new general relativity. Phys. Rev. D (2019).

    MathSciNet  Article  Google Scholar 

  15. 15.

    D. Blixt, M. Hohmann, C. Pfeifer, On the gauge fixing in the Hamiltonian analysis of general teleparallel theories. Universe (2019).

    Article  Google Scholar 

  16. 16.

    R. Briffa, S. Capozziello, J. Levi Said, J. Mifsud, and E.N. Saridakis, Constraining Teleparallel Gravity through Gaussian Processes. arXiv:2009.14582 [gr-qc]

  17. 17.

    D. Brizuela, J.M. Martin-Garcia, G.A. Mena Marugan, xPert: Computer algebra for metric perturbation theory. Gen. Rel. Grav. 41, 2415–2431 (2009).

  18. 18.

    Y.-F. Cai, S. Capozziello, M. De Laurentis, E.N. Saridakis, (T) teleparallel gravity and cosmology. Rept. Prog. Phys. (2016).

    Article  Google Scholar 

  19. 19.

    S. Capozziello, M. Capriolo, M. Transirico, The gravitational energy-momentum pseudotensor: the cases of \(f(R)\) and \(f(T)\) gravity. Int. J. Geom. Meth. Mod. Phys. (2018).

    Article  MATH  Google Scholar 

  20. 20.

    T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Modified gravity and cosmology. Phys. Rept. 513, 1–189 (2012).

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    P. Creminelli, F. Vernizzi, Dark energy after GW170817 and GRB170817A. Phys. Rev. Lett. (2017).

    Article  Google Scholar 

  22. 22.

    C. de Rham, V. Pozsgay, New class of Proca interactions. Phys. Rev. D (2020).

    MathSciNet  Article  Google Scholar 

  23. 23.

    E. Di Valentino et al., Cosmology Intertwined II: The Hubble Constant Tension (2020)

  24. 24.

    E. Di Valentino et al., Cosmology Intertwined III: \(f \sigma _8\) and \(S_8\). arXiv:2008.11285 [astro-ph.CO]

  25. 25.

    C. Escamilla-Rivera, J. Levi Said, Cosmological viable models in \(f(T, B)\) theory as solutions to the \(H_0\) tension. Class. Quant. Grav. 37(16), 165002 (2020).

    ADS  Article  Google Scholar 

  26. 26.

    G. Farrugia, J. Levi Said, V. Gakis, E.N. Saridakis, Gravitational waves in modified teleparallel theories. Phys. Rev. D (2018).

  27. 27.

    A. Finch, J.L. Said, Galactic rotation dynamics in \(f(T)\) gravity. Eur. Phys. J. C 78(7), 560 (2018).

    ADS  Article  Google Scholar 

  28. 28.

    G.A.R. Franco, C. Escamilla-Rivera, J. Levi Said, Stability analysis for cosmological models in \(f(T, B)\) gravity. Eur. Phys. J. C 80(7), 677 (2020).

    ADS  Article  Google Scholar 

  29. 29.

    J. Gleyzes, D. Langlois, F. Piazza, F. Vernizzi, Essential building blocks of dark energy. JCAP 1308, 025 (2013).

    ADS  Article  Google Scholar 

  30. 30.

    A. Goldstein et al., An ordinary short gamma-ray burst with extraordinary implications: fermi-GBM detection of GRB 170817A. Astrophys. J. 848(2), L14 (2017).

    ADS  Article  Google Scholar 

  31. 31.

    A.G.-P. Gomez-Lobo, J.M. Martin-Garcia, Spinors: a Mathematica package for doing spinor calculus in General Relativity. Comput. Phys. Commun. 183, 2214–2225 (2012).

    ADS  Article  MATH  Google Scholar 

  32. 32.

    P. Gonzalez, Y. Vasquez, Teleparallel equivalent of lovelock gravity. Phys. Rev. D 92(12), 124023 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  33. 33.

    P. Gonzlez, S. Reyes, Y. Vsquez, Teleparallel equivalent of lovelock gravity. Generalizations and cosmological applications. JCAP (2019).

    Article  Google Scholar 

  34. 34.

    K. Hayashi, T. Shirafuji, New general relativity. Phys. Rev. D 19, 3524–3553 (1979). [Addendum: Phys. Rev. D 24, 3312–3314 (1982)]

  35. 35.

    F. Hehl, P. Von Der Heyde, G. Kerlick, J. Nester, General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne’eman, Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance. Phys. Rept. 258, 1–171 (1995).

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    L. Heisenberg, Generalised Proca theories, in 52nd Rencontres de Moriond on Gravitation, pp. 233–241 (2017). arXiv:1705.05387 [hep-th]

  38. 38.

    L. Heisenberg, Generalization of the Proca action. JCAP (2014).

    Article  Google Scholar 

  39. 39.

    L. Heisenberg, Scalar–vector–tensor gravity theories. JCAP (2018).

    Article  MATH  Google Scholar 

  40. 40.

    L. Heisenberg, A systematic approach to generalisations of General Relativity and their cosmological implications. Phys. Rept. 796, 1–113 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  41. 41.

    T. Kobayashi, M. Yamaguchi, J. Yokoyama, Generalized G-inflation: inflation with the most general second-order field equations. Prog. Theor. Phys. 126, 511–529 (2011).

    ADS  Article  MATH  Google Scholar 

  42. 42.

    T. Koivisto, M. Hohmann, L. Marzola, An axiomatic purification of gravity (2019)

  43. 43.

    K. Koyama, Cosmological tests of modified gravity. Rept. Prog. Phys. (2016).

    Article  Google Scholar 

  44. 44.

    M. Krssak, R. van den Hoogen, J. Pereira, C. Bhmer, A. Coley, Teleparallel theories of gravity: illuminating a fully invariant approach. Class. Quant. Grav. 36(18), 183001 (2019).

    ADS  Article  Google Scholar 

  45. 45.

    J. Levi Said, J. Mifsud, D. Parkinson, E.N. Saridakis, J. Sultana, K.Z. Adami, Testing the violation of the equivalence principle in the electromagnetic sector and its consequences in \(f(T)\) gravity. JCAP (2020).

    Article  Google Scholar 

  46. 46.

    LIGO Scientific, Virgo Collaboration, B.P. Abbott et al., GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. (2017).

  47. 47.

    D. Lovelock, The Einstein tensor and its generalizations. J. Math. Phys. 12, 498–501 (1971).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    J.M. Martín-García, xperm: fast index canonicalization for tensor computer algebra. CoRRabs/0803.0862 (2008). arXiv:0803.0862

  49. 49.

    J.M. Martin-Garcia, R. Portugal, L.R.U. Manssur, The invar tensor package. Comput. Phys. Commun. 177, 640–648 (2007).

    ADS  Article  MATH  Google Scholar 

  50. 50.

    J.M. Martin-Garcia, D. Yllanes, R. Portugal, The Invar tensor package: differential invariants of Riemann. Comput. Phys. Commun. 179, 586–590 (2008).

    ADS  Article  MATH  Google Scholar 

  51. 51.

    Y. Minami, E. Komatsu, New extraction of the cosmic birefringence from the Planck 2018 polarization data. Phys. Rev. Lett. (2020).

    Article  Google Scholar 

  52. 52.

    C. Misner, K. Thorne, and J. Wheeler, Gravitation. No. pt. 3 in Gravitation. W. H. Freeman, 1973.

  53. 53.

    M. Nakahara, Geometry, Topology and Physics, Second Edition. Graduate student series in physics. Taylor & Francis, London (2003)

  54. 54.

    T. Nutma, xTras: A field-theory inspired xAct package for mathematica. Comput. Phys. Commun. 185, 1719–1738 (2014).

    ADS  Article  MATH  Google Scholar 

  55. 55.

    T. Ortín, Gravity and Strings. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2004)

  56. 56.

    M. Ostrogradsky, Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850)

    Google Scholar 

  57. 57.

    A. Paliathanasis, de Sitter and Scaling solutions in a higher-order modified teleparallel theory. JCAP (2017).

    Article  Google Scholar 

  58. 58.

    L. Perenon, F. Piazza, C. Marinoni, L. Hui, Phenomenology of dark energy: general features of large-scale perturbations. JCAP (2015).

    Article  Google Scholar 

  59. 59.

    C. Pitrou, X. Roy, O. Umeh, xPand: An algorithm for perturbing homogeneous cosmologies. Class. Quant. Grav. (2013).

  60. 60.

    J. Sakstein, B. Jain, Implications of the neutron star merger GW170817 for cosmological scalar-tensor theories. Phys. Rev. Lett. (2017).

    Article  Google Scholar 

  61. 61.

    S. Weinberg, The cosmological constant problem. Rev. Mod. Phys. (1989).

    MathSciNet  Article  MATH  Google Scholar 

  62. 62.

    S. Weinberg, Frontmatter, vol. 1 (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

  63. 63.

    S. Weinberg, Cosmology (OUP, Oxford, 2008)

    Google Scholar 

  64. 64.

    R. Weitzenböock, Invariantentheorie (Noordhoff, Gronningen, 1923)

    Google Scholar 

  65. 65.

    R.P. Woodard, Avoiding dark energy with 1/r modifications of gravity. Lect. Notes Phys. 720, 403–433 (2007).

    ADS  Article  Google Scholar 

  66. 66.

    M. Wright, Conformal transformations in modified teleparallel theories of gravity revisited. Phys. Rev. (2016).

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JLS would also like to acknowledge funding support from Cosmology@MALTA which is supported by the University of Malta. The authors would like to acknowledge networking support by the COST Action CA18108. V.G would like to thank J. Beltran for useful and fruitful discussions.

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Correspondence to Viktor Gakis.

Appendix: Teleparallel Proca scalars

Appendix: Teleparallel Proca scalars

In this appendix, we will expand all the generators from Table 1 in all their possible index configurations. We will denote the generator or groups of generators with brackets like in the example after Table 1, . Note that the following sets of scalars are the full list of possible independent scalars.

Torsion vector component \(v_{\mu }\)


Torsion axial component \(a_{\mu }\)

$$\begin{aligned}&\left\{ \epsilon aAFF,\epsilon aA\tilde{F}\tilde{F}\right\} \nonumber \\&I_{14}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \mu \nu \gamma }F{}^{\alpha }{}_{\mu }F{}_{\nu \gamma },\end{aligned}$$
$$\begin{aligned}&I_{15}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \mu \nu \gamma }F{}^{\beta }{}_{\mu }F{}_{\nu \gamma }, \end{aligned}$$
$$\begin{aligned}&I_{16}:=A_{\alpha }a{}^{\alpha }\epsilon {}^{\gamma \mu \nu \beta }F{}_{\gamma \mu }F{}_{\nu \beta }, \end{aligned}$$
$$\begin{aligned}&\left\{ \epsilon aAFFF,\epsilon aA\tilde{F}\tilde{F}F\right\} \nonumber \\&I_{17}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \beta \mu \nu }F{}_{\gamma \rho }F{}^{\gamma }{}_{\mu }F{}^{\rho }{}_{\nu },\end{aligned}$$
$$\begin{aligned}&I_{18}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\mu \rho \gamma \nu }F{}^{\alpha }{}_{\mu }F{}^{\beta }{}_{\rho }F{}_{\gamma \nu },\end{aligned}$$
$$\begin{aligned}&I_{19}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \rho \gamma \nu }F{}^{\alpha }{}_{\mu }F{}_{\gamma \nu }F{}^{\mu }{}_{\rho },\end{aligned}$$
$$\begin{aligned}&I_{20}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \rho \gamma \nu }F{}^{\beta }{}_{\mu }F{}_{\gamma \nu }F{}^{\mu }{}_{\rho },\end{aligned}$$
$$\begin{aligned}&I_{21}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\nu \rho \gamma \mu }F{}^{\alpha \beta }F{}_{\gamma \mu }F{}_{\nu \rho }, \end{aligned}$$
$$\begin{aligned}&I_{22}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \beta \nu \rho }F{}_{\nu \rho }F{}^{2}, \end{aligned}$$
$$\begin{aligned}&\left\{ \epsilon aAFFFF,\epsilon aA\tilde{F}\tilde{F}\tilde{F}\tilde{F},\epsilon aA\tilde{F}\tilde{F}FF\right\} \nonumber \\&I_{23}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \mu \rho \sigma }F{}^{\alpha }{}_{\mu }F{}^{\gamma }{}_{\sigma }F{}_{\nu \gamma }F{}^{\nu }{}_{\rho }, \end{aligned}$$
$$\begin{aligned}&I_{24}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \mu \rho \sigma }F{}^{\beta }{}_{\mu }F{}^{\gamma }{}_{\sigma }F{}_{\nu \gamma }F{}^{\nu }{}_{\rho }, \end{aligned}$$
$$\begin{aligned}&I_{25}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\rho \gamma \nu \sigma }F{}^{\alpha }{}_{\mu }F{}^{\beta }{}_{\rho }F{}^{\mu }{}_{\gamma }F{}_{\nu \sigma },\end{aligned}$$
$$\begin{aligned}&I_{26}:=A_{\alpha }a{}^{\alpha }\epsilon {}^{\mu \beta \nu \sigma }F{}_{\gamma \rho }F{}^{\gamma }{}_{\mu }F{}_{\nu \sigma }F{}^{\rho }{}_{\beta },\end{aligned}$$
$$\begin{aligned}&I_{27}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\mu \gamma \nu \sigma }F{}^{\alpha }{}_{\mu }F{}^{\beta }{}_{\rho }F{}_{\nu \sigma }F{}^{\rho }{}_{\gamma },\end{aligned}$$
$$\begin{aligned}&I_{28}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \gamma \nu \sigma }F{}^{\alpha }{}_{\mu }F{}^{\mu }{}_{\rho }F{}_{\nu \sigma }F{}^{\rho }{}_{\gamma }, \end{aligned}$$
$$\begin{aligned}&I_{29}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \gamma \nu \sigma }F{}^{\beta }{}_{\mu }F{}^{\mu }{}_{\rho }F{}_{\nu \sigma }F{}^{\rho }{}_{\gamma },\end{aligned}$$
$$\begin{aligned}&I_{30}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\sigma \gamma \nu \rho }F{}^{\alpha }{}_{\mu }F{}^{\beta \mu }F{}_{\nu \rho }F{}_{\sigma \gamma }, \end{aligned}$$
$$\begin{aligned}&I_{31}:=A_{\alpha }a{}^{\alpha }\epsilon {}^{\nu \rho \sigma \beta }F{}_{\nu \rho }F{}_{\sigma \beta }F{}^{2}, \end{aligned}$$
$$\begin{aligned}&I_{32}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \rho \sigma \gamma }F{}^{\alpha }{}_{\rho }F{}_{\sigma \gamma }F{}^{2}, \end{aligned}$$
$$\begin{aligned}&I_{33}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \rho \sigma \gamma }F{}^{\beta }{}_{\rho }F{}_{\sigma \gamma }F{}^{2}, \end{aligned}$$

Purely tensorial component \(t_{\alpha \beta \gamma }\)

$$\begin{aligned}&\left\{ tAFF,tA\widetilde{F}\widetilde{F}\right\} \nonumber \\&I_{38}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }t{}^{\alpha \mu \gamma }, \end{aligned}$$
$$\begin{aligned}&I_{39}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \mu }t{}^{\beta \gamma \mu },\end{aligned}$$
$$\begin{aligned}&I_{40}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }t{}^{\gamma \mu \alpha },\end{aligned}$$
$$\begin{aligned}&\left\{ tAFFF,tA\widetilde{F}\widetilde{F}F\right\} \nonumber \\&I_{41}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }F{}^{\mu }{}_{\nu }t{}^{\alpha \gamma \nu },\end{aligned}$$
$$\begin{aligned}&I_{42}:=A_{\alpha }F{}_{\beta \gamma }F{}^{2}t{}^{\alpha \beta \gamma }, \end{aligned}$$
$$\begin{aligned}&I_{43}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \nu }F{}^{\gamma }{}_{\mu }t{}^{\beta \nu \mu },\end{aligned}$$
$$\begin{aligned}&I_{44}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}_{\mu \nu }t{}^{\gamma \mu \nu }, \end{aligned}$$
$$\begin{aligned}&I_{45}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \nu }F{}^{\gamma }{}_{\mu }t{}^{\mu \nu \beta },\end{aligned}$$
$$\begin{aligned}&\left\{ tAFFFF,tA\widetilde{F}\widetilde{F}\widetilde{F}\widetilde{F},tA\widetilde{F}\widetilde{F}FF,tAFFFF\right\} \nonumber \\&I_{46}:=A_{\alpha }F{}_{\beta }{}^{\mu }F{}^{\beta }{}_{\gamma }F{}_{\mu \nu }F{}^{\nu }{}_{\rho }t{}^{\alpha \rho \gamma },\end{aligned}$$
$$\begin{aligned}&I_{47}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }F{}^{2}t{}^{\alpha \mu \gamma },\end{aligned}$$
$$\begin{aligned}&I_{48}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \nu }F{}^{\gamma }{}_{\mu }F{}^{\nu }{}_{\rho }t{}^{\beta \mu \rho },\end{aligned}$$
$$\begin{aligned}&I_{49}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \mu }F{}^{2}t{}^{\beta \gamma \mu }, \end{aligned}$$
$$\begin{aligned}&I_{50}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}_{\mu \rho }F{}^{\mu }{}_{\nu }t{}^{\gamma \rho \nu },\end{aligned}$$
$$\begin{aligned}&I_{51}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}^{\gamma }{}_{\mu }F{}_{\nu \rho }t{}^{\mu \nu \rho },\end{aligned}$$
$$\begin{aligned}&I_{52}:=A_{\alpha }F{}_{\beta }{}^{\mu }F{}^{\beta }{}_{\gamma }F{}_{\mu \nu }F{}^{\nu }{}_{\rho }t{}^{\gamma \rho \alpha }, \end{aligned}$$
$$\begin{aligned}&I_{53}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }F{}^{2}t{}^{\gamma \mu \alpha },\end{aligned}$$
$$\begin{aligned}&I_{54}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}_{\mu \rho }F{}^{\mu }{}_{\nu }t{}^{\nu \rho \gamma }. \end{aligned}$$

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Nicosia, GP., Said, J.L. & Gakis, V. Generalised Proca theories in teleparallel gravity. Eur. Phys. J. Plus 136, 191 (2021).

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