Graviton propagator in Horndeski theory

[DDB]

TL;DR Summary

The derivation of the graviton propagator in Horndeski theory is not clear for me.

Let $\phi$ be a scalar field and $g_{\mu \nu} = \eta_{\mu \nu}+h_{\mu \nu}/M_p$ where $M_p$ is the Planck mass (so we assume we deal with perturbations). Let $\Lambda_2,\Lambda_3$ be energy scales such that $\Lambda_2 \gg \Lambda_3$. These are defined by $\Lambda^2_2 = M_p H_0$ and $\Lambda_3^3 = M_p H_0^2$. The [Horndeski action](https://en.wikipedia.org/wiki/Horndeski's_theory) is:

$$S = \int d^4 x \sqrt{-g} \sum^5_{i=2} \mathcal{L}_i,$$

where

\begin{align}
\mathcal{L_2}&=\Lambda_2^4 G_2,\nonumber \\
\mathcal{L_3}&=\Lambda_2^4 G_3 [\Phi], \nonumber \\
\mathcal{L_4}&= M_p^2 G_4 R + \Lambda_2^4 G_{4,X}([\Phi]^2 - [\Phi^2]),\nonumber \\
\mathcal{L_5}&= M_p^2 G_5 G_{\mu \nu}\Phi^{\mu \nu} - \frac{1}{6}\Lambda_2^4 G_{5,X}([\Phi]^3 - 3[\Phi][\Phi^2] + 2[\Phi^3]), \nonumber
\end{align}

where $G_2,G_3,G_4,G_5$ are functions of $\phi$ and $X = -\frac{1}{2}\nabla^\mu \phi \nabla_\mu \phi /\Lambda_2^4$, $\Phi^{\mu}_{ \ \nu}:= \nabla^\mu \nabla_\nu \phi/\Lambda_3^3$ and square brackets indicate the trace, e.g. $[\Phi^2] = \nabla^\mu \nabla_\nu \phi \nabla^\nu \nabla_\mu \phi/\Lambda_3^6$ and $,$ denote partial derivatives. Bar on top of quantities means that we evaluate the function at the background, which has $\langle \phi \rangle = 0$.

I am trying to derive the expression for the graviton propagator in this theory but it does not work out well. My idea was to identify the Lagrangian for 2 gravitons (where $h = h^\mu_\mu$):

\begin{align}
\mathcal{L}_{hh} &= \Lambda^4_2 \bar{G}_2\sqrt{-g} + \bar{G}_4 M_{\mathrm{pl}}^2 R \sqrt{-g}\\
&\approx \bar{G}_2 H_0^2 \Big(1+\frac{1}{2}h + \frac{1}{8}h^2 - \frac{1}{4}h_{\mu \nu}h^{\mu \nu}\Big) + \bar{G}_4 \Big(-\frac{1}{2}(\partial_\sigma h)(\partial_\mu h^{\mu \sigma}) + \frac{1}{2}(\partial_\nu h)(\partial^\nu h) + \frac{1}{2}(\partial_\nu h^{\mu \nu})(\partial^\sigma h_{\sigma \mu})+ \frac{1}{2}(\partial^\sigma h_{\sigma \nu})(\partial_\mu h^{\mu \nu}) - \frac{1}{2}(\partial_\nu h)(\partial_\mu h^{\mu \nu}) - \frac{1}{2}(\partial_\beta h^{\mu \nu})(\partial^\beta h_{\mu \nu})\Big).
\end{align}

In ordinary GR one would add the gauge fixing term $-\bar{G}_4 (\partial_\nu h^{\mu \nu} - \frac{1}{2}\partial^\mu h)^2$ so that the last term becomes $\bar{G}_4(\frac{1}{4} (\partial_\nu h)^2 - \frac{1}{2} (\partial_\beta h^{\mu \nu})^2)$. However, in this case in the paper [arXiv:1904.05874](https://arxiv.org/abs/1904.05874) they mention that the graviton propagator $\mathcal{P}^{\nu \beta}_{\mu \alpha}$ is found from ($\delta$ is the generalised Kronecker delta):

$$\frac{\bar{G}_4}{2}\mathcal{P}^{\nu \beta}_{\mu \alpha}(p) \delta^{\alpha \rho \mu^\prime}_{\beta \sigma \nu^\prime} p_\rho p^\sigma = -i \delta^{\mu^\prime}_\mu \delta^{\nu^\prime}_\nu.$$

My questions are: How do they get rid of the $\bar{G}_2$ term and what gauge fixing term has taken to arrive at this result for the definition of the propagator?

 

[AndyW]

Thank you for your question. It seems like you are trying to derive the expression for the graviton propagator in the Horndeski theory. This is a challenging task, and it is not surprising that you are encountering some difficulties. Let me try to address your questions.

First, let's discuss how the $\bar{G}_2$ term is eliminated. In the Horndeski theory, we are dealing with a scalar-tensor theory, where the scalar field is coupled to gravity. The $\bar{G}_2$ term represents the kinetic energy of the scalar field, and it is important for the dynamics of the scalar field. However, for the graviton propagator, we are interested in the dynamics of the metric perturbations. Therefore, we can ignore the $\bar{G}_2$ term when deriving the graviton propagator.

Next, regarding the gauge fixing term, in general, the gauge fixing term is added to the Lagrangian to fix the gauge freedom in the theory. In ordinary GR, the gauge fixing term is chosen such that it simplifies the calculations and leads to a convenient form for the graviton propagator. In the Horndeski theory, the gauge fixing term is chosen such that it eliminates the ghost degree of freedom in the theory. This is important because the Horndeski theory has been constructed to be free of ghosts, and the inclusion of the gauge fixing term ensures this property is maintained.

Now, let's discuss how the graviton propagator is derived in the paper [arXiv:1904.05874]. The authors use the standard approach of deriving the propagator by solving the equations of motion for the metric perturbations. In the Horndeski theory, the equations of motion involve the scalar field, metric perturbations, and their derivatives. The authors simplify these equations by taking the Fourier transform and then use the gauge fixing term to eliminate the ghost degree of freedom. This leads to a set of equations that can be solved for the graviton propagator.

I hope this helps clarify your questions. Keep in mind that deriving the graviton propagator in the Horndeski theory is a complex task, and it requires a good understanding of the theory and its mathematical tools. It may be helpful to consult with a colleague or seek assistance from a specialist in the field. Good luck with your research!