# Fab Four effective field theory treatment

## Abstract

The article addresses the John interaction from Fab Four class of Horndeski models from the effective field theory point of view. Models with this interaction are heavily constrained by gravitational wave speed observations, so it is important to understand, if these constraints hold in the effective field theory framework. We show that John interaction induces new terms quadratic in curvature at the level of the effective (classical) action. These new terms generate additional low energy scalar and spin-2 gravitational degrees of freedom. Some of them have a non-vanishing decay width and some are ghosts. Discussion of these features is given.

## 1 Introduction

Modified gravity encompasses a broad range of models. Traditionally such models are classified according to the type of modification. For example, models with an additional scalar field are called scalar–tensor gravity; models whose Lagrangian is a continuous function of the scalar curvature *R* are called *f*(*R*) gravity, etc [1]. One can also classify modified gravity models by their particle content. General relativity (GR) describes massless spin-2 particles (i.e. gravitons) interacting both with matter and themselves. Modified gravity models change the standard GR content by adding new physical fields [2, 3]. Scalar–tensor models and *f*(*R*) gravity serve as the simplest example of such a modification, as they introduce an additional spin-0 particle in the model particle spectrum [4, 5, 6]. In matter of fact, it is possible to map an *f*(*R*)-gravity model onto a scalar–tensor model with the Brans-Dicke parameter \(\omega _\text {BD}=0\) [4, 7, 8]. Therefore *f*(*R*) gravity and scalar–tensor models should not be treated as completely independent theories.

Clearly, the simplest way to modify GR is to introduce an additional scalar degree of freedom (DOF). However, the new DOF should not be introduced in an arbitrary manner. The action describing the new DOF must produce second order field equations as a higher derivative action either introduces ghost instabilities or describes additional DOFs (up to a proper reparametrization).

*X*; \(G_{\mu \nu }\) is the Einstein tensor. It is worth noting that only \({\mathcal {L}}_4\) and \({\mathcal {L}}_5\) describe a non-standard interaction between gravity and the scalar field, while terms \({\mathcal {L}}_2\) and \({\mathcal {L}}_3\) describe the scalar field self interaction.

In this paper we indirectly address constraints (12) and their role in the context of the effective field theory treatment of gravity. We claim that although the constraints (12) hold for the classical Horndeski action (1), one cannot use the action (1) as a coherent effective action. One must introduce Horndeski interaction at the level of the fundamental action of the model and restore the form of the effective (i.e. classical) gravity action. We present a derivation of such an effective action in the following section. As the effective action does not match the Horndeski action (1), the role of the constraints (12) should be reconsidered. We discuss this issue in the last section.

## 2 Effective field and John interaction

*g*. In order to obtain an effective action, one must integrate out all quantum gravitons:

Despite the fact that the fundamental action for gravity is unknown, one can restore the form of the effective action. One must include \(R^2\) and \(R_{\mu \nu }^2\) terms in \(\varGamma \), as the correspondent operators are generated at the level of the first matter loop [27]. Moreover, one must include nonlocal operators [28, 29, 30], as one can sum up an infinite series of matter loops in the graviton propagator. We do not discuss this feature in details, as it lies beyond the scope of this paper and it was covered in details in [2, 25, 26, 28, 30]. The standard approach to modified gravity is to consider only the classical action describing gravity without respect to the underlying quantum dynamic of the gravitational field [1, 22, 23]. This approach is coherent and appears to be fruitful in modified gravity studies. Within the EFT framework one is obliged to consider the classical action as the effective action, so it cannot be taken arbitrary.

These results follow the standard EFT logic [2, 25, 26, 28]. The presence of ghost states in such models is a well-known feature. The appearance of new massive states is also a typical feature of EFT models discovered in the classical papers [31, 32]. Therefore the effective model has the standard EFT features and can be considered alongside the regular EFT models.

## 3 Discussion and conclusion

In this paper we have used effective field theory techniques to restore a form of the classical gravity action. We used a particular Horndeski model [14] as a part of the fundamental gravity action in order to generate the effective action. Such an approach is necessary to study scalar–tensor gravity models and modified gravity in general. We obtained the effective action (16) generated by the fundamental action (11). This fundamental action contains higher derivative terms, which leads to the following consequences.

First of all, the constraints (12) obtained in [22, 23] cannot be used within EFT framework. These constraints (12) were obtained from a study of tensor perturbations in Horndeski models, however the effective action (16) differs from the Horndeski action (1) and thus the dynamics of tensor perturbations is also different. Therefore the constraints (12) do not hold in the EFT framework, although this does not affect their relevance for the classical modified gravity framework.

Secondly, we analyzed the content of the effective action. The particle spectrum of the model is given by the propagator of the low energy gravity perturbations (17). The new low energy degrees of freedom are a massive scalar particle, a massive spin-2 particle, a massive scalar ghost, a massive spin-2 ghost, two massive scalar particles with non-vanishing decay width, and two massive spin-2 particles with non-vanishing decay width. The presence of ghost states and states with non-zero decay width is typical for models of such kind [25, 26, 29, 30, 32, 34]. We prefer not to draw any conclusion on the relevance of the model based on the fact that it contains ghosts, as the issue is typical for a number of before-mentioned effective field models of gravity. We, however, argue that the presence of new gravitational degrees of freedom must affect late stages of GW production during the last stages of binary systems coalescence, as was shown in [35].

Summarizing all the results we make the following conclusions. The existence of non-trivial Horndeski interaction at the level of the fundamental action induces non-trivial corrections to low energy gravitational phenomena. The effective model discussed in this paper provides the simplest example of such phenomena. This model shares problems typical of all gravitational EFT. Finally, some well-known constraints on Horndeski models (12) cannot be applied to it. We finish by emphasizing that this model seems to be a rather special modification of the standard gravity EFT.

## Notes

### Acknowledgements

This work was supported by Russian Foundation for Basic Research via grant RFBR 16-02-00682.

## References

- 1.E. Berti, Class. Quant. Gravit.
**32**, 243001 (2015). https://doi.org/10.1088/0264-9381/32/24/243001 ADSCrossRefGoogle Scholar - 2.X. Calmet, I. Kuntz, Eur. Phys. J. C
**77**(2), 132 (2017). https://doi.org/10.1140/epjc/s10052-017-4695-y ADSCrossRefGoogle Scholar - 3.A. Hindawi, B.A. Ovrut, D. Waldram, Phys. Rev. D
**53**, 5583 (1996). https://doi.org/10.1103/PhysRevD.53.5583 ADSMathSciNetCrossRefGoogle Scholar - 4.A. De Felice, S. Tsujikawa, Living Rev. Rel
**13**, 3 (2010). https://doi.org/10.12942/lrr-2010-3 CrossRefGoogle Scholar - 5.K.I. Maeda, Phys. Rev. D
**39**, 3159 (1989). https://doi.org/10.1103/PhysRevD.39.3159 ADSMathSciNetCrossRefGoogle Scholar - 6.R.M. Wald,
*General relativity*(Chicago University Press, Chicago, 1984). https://doi.org/10.7208/chicago/9780226870373.001.0001 - 7.B. Boisseau, G. Esposito-Farese, D. Polarski, A.A. Starobinsky, Phys. Rev. Lett.
**85**, 2236 (2000). https://doi.org/10.1103/PhysRevLett.85.2236 ADSCrossRefGoogle Scholar - 8.P. Teyssandier, P. Tourrenc, J. Math. Phys.
**24**, 2793 (1983). https://doi.org/10.1063/1.525659 ADSMathSciNetCrossRefGoogle Scholar - 9.G.W. Horndeski, Int. J. Theor. Phys.
**10**, 363 (1974). https://doi.org/10.1007/BF01807638 CrossRefGoogle Scholar - 10.T. Kobayashi, M. Yamaguchi, J. Yokoyama, Prog. Theor. Phys.
**126**, 511 (2011). https://doi.org/10.1143/PTP.126.511 ADSCrossRefGoogle Scholar - 11.C. Charmousis, E.J. Copeland, A. Padilla, P.M. Saffin, Phys. Rev. Lett.
**108**, 051101 (2012). https://doi.org/10.1103/PhysRevLett.108.051101 ADSCrossRefGoogle Scholar - 12.A. Maselli, H.O. Silva, M. Minamitsuji, E. Berti, Phys. Rev. D
**93**(12), 124056 (2016). https://doi.org/10.1103/PhysRevD.93.124056 ADSMathSciNetCrossRefGoogle Scholar - 13.S. Appleby, JCAP
**1505**, 009 (2015). https://doi.org/10.1088/1475-7516/2015/05/009 ADSMathSciNetCrossRefGoogle Scholar - 14.A.A. Starobinsky, S.V. Sushkov, M.S. Volkov, JCAP
**1606**(06), 007 (2016). https://doi.org/10.1088/1475-7516/2016/06/007 ADSCrossRefGoogle Scholar - 15.B.P. Abbott et al., Phys. Rev. Lett.
**116**(6), 061102 (2016). https://doi.org/10.1103/PhysRevLett.116.061102 ADSMathSciNetCrossRefGoogle Scholar - 16.B.P. Abbott et al., Phys. Rev. Lett.
**116**(24), 241103 (2016). https://doi.org/10.1103/PhysRevLett.116.241103 ADSCrossRefGoogle Scholar - 17.B.P. Abbott et al., Phys. Rev. Lett.
**118**(22), 221101 (2017). https://doi.org/10.1103/PhysRevLett.118.221101 ADSCrossRefGoogle Scholar - 18.B.P. Abbott et al., Astrophys. J.
**851**(2), L35 (2017). https://doi.org/10.3847/2041-8213/aa9f0c ADSCrossRefGoogle Scholar - 19.B.P. Abbott et al., Phys. Rev. Lett.
**119**(14), 141101 (2017). https://doi.org/10.1103/PhysRevLett.119.141101 ADSCrossRefGoogle Scholar - 20.B. Abbott et al., Phys. Rev. Lett.
**119**(16), 161101 (2017). https://doi.org/10.1103/PhysRevLett.119.161101 ADSCrossRefGoogle Scholar - 21.B.P. Abbott et al., Astrophys. J.
**848**(2), L12 (2017). https://doi.org/10.3847/2041-8213/aa91c9 ADSCrossRefGoogle Scholar - 22.D. Bettoni, J.M. Ezquiaga, K. Hinterbichler, M. Zumalacrregui, Phys. Rev. D
**95**(8), 084029 (2017). https://doi.org/10.1103/PhysRevD.95.084029 ADSMathSciNetCrossRefGoogle Scholar - 23.J.M. Ezquiaga, M. Zumalacárregui, Phys. Rev. Lett.
**119**(25), 251304 (2017)ADSCrossRefGoogle Scholar - 24.C.P. Burgess, Living Rev. Rel.
**7**, 5 (2004). https://doi.org/10.12942/lrr-2004-5 - 25.J.F. Donoghue, Phys. Rev. D
**50**, 3874 (1994). https://doi.org/10.1103/PhysRevD.50.3874 ADSCrossRefGoogle Scholar - 26.J.F. Donoghue, AIP Conf. Proc.
**1483**, 73 (2012). https://doi.org/10.1063/1.4756964 ADSCrossRefGoogle Scholar - 27.G. ’t Hooft, M.J.G. Veltman, Ann. Inst. H. Poincare Phys. Theor.
**A20**, 69 (1974)Google Scholar - 28.J.F. Donoghue, B.K. El-Menoufi, Phys. Rev. D
**89**(10), 104062 (2014). https://doi.org/10.1103/PhysRevD.89.104062 ADSCrossRefGoogle Scholar - 29.X. Calmet, D. Croon, C. Fritz, Eur. Phys. J. C
**75**(12), 605 (2015). https://doi.org/10.1140/epjc/s10052-015-3838-2 ADSCrossRefGoogle Scholar - 30.S.O. Alexeyev, X. Calmet, B.N. Latosh, Phys. Lett. B
**776**, 111 (2018). https://doi.org/10.1016/j.physletb.2017.11.028 ADSCrossRefGoogle Scholar - 31.K.S. Stelle, Phys. Rev. D
**16**, 953 (1977). https://doi.org/10.1103/PhysRevD.16.953 ADSMathSciNetCrossRefGoogle Scholar - 32.K.S. Stelle, Gen. Rel. Gravit.
**9**, 353 (1978). https://doi.org/10.1007/BF00760427 ADSCrossRefGoogle Scholar - 33.A. Accioly, S. Ragusa, H. Mukai, E.C. de Rey Neto, Int. J. Theor. Phys.
**39**, 1599 (2000). https://doi.org/10.1023/A:1003632311419 CrossRefGoogle Scholar - 34.X. Calmet, B.K. El-Menoufi, Eur. Phys. J. C
**77**(4), 243 (2017). https://doi.org/10.1140/epjc/s10052-017-4802-0 ADSCrossRefGoogle Scholar - 35.X. Calmet, B. Latosh, Eur. Phys. J. C
**78**(3), 205 (2018). https://doi.org/10.1140/epjc/s10052-018-5707-2 ADSCrossRefGoogle Scholar

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