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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 64 , Issue 1 : 210012(2021) https://doi.org/10.1007/s11433-020-1607-3

Higher derivative scalar-tensor monomials and their classification

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  • ReceivedApr 24, 2020
  • AcceptedAug 3, 2020
  • PublishedNov 24, 2020
PACS numbers

Abstract


Acknowledgment

I would like to thank M. Crisostomi for the discussion. This work was supported by the National Natural Science Foundation of China (Grant No. 11975020).


Supplement

Appendix

A brief review of the Hovrava gravity and the Einstein-Aether theory

There exist higher derivative scalar-tensor theories that are ghostfree when the scalar field is assumed to be timelike. These theories are highly related to the Lorentz breaking gravity theories such as the Hovrava gravity [23] (see also ref. [48] for a recent review) and the Einstein-Aether theory [49].

The Hovrava gravity was originally proposed to improve the renormalizability of the general relativity by introducing the higher order spatial derivatives only while keeping the temporal derivatives up to the second order. The basic operators are the spatial curvature ${}^{3}\!~R_{ij}$, the extrinsic curvature $K_{ij}$, the acceleration $a_{i}$ as well as their spatial derivatives $\nabla_{i}$. Hovrava assumed that the full spacetime diffeomorphism is broken to $t\rightarrow~\xi^{0}(t)$ and $x^{i}\rightarrow~\xi^{i}(t,x^{k})$ so that the spatial diffeomorphism remains. It is easy to show that the basic operators have the dimensions \begin{equation}[{}^{3}\!R_{ij}] = 2, [K_{ij}] = 3, [a_{i}] = 1, [\nabla_{i}] = 1, \tag{184}\end{equation} under such coordinate transformations. With these basic building blocks, the total Lagrangian of the Hovrava gravity is a summation of the following scalar operators order by order [50]: \begin{eqnarray*}\text{dim 6:} & & K_{ij}K^{ij},K^{2},\,{}^{3}\! R \,{}^{3}\!R_{ij}\,{}^{3}\! R^{ij},\left(^{3}\! R\right)^{3},{}^{3}\! R_{j}^{i}\,{}^{3}\! R_{k}^{j}\,{}^{3}\! R_{i}^{k},\left(\nabla_{k}\,{}^{3}\! R_{ij}\right)^{2} \\ & & \left(a_{i}a^{i}\right)^{2}\,{}^{3}\! R,\left(a_{k}a^{k}\right)a_{i}a_{j}{}^{3}\! R^{ij},\left(a_{i}a^{i}\right)^{3},a^{i}\nabla_{j}\nabla^{j}a_{i},\cdots \\ \text{dim 5:} & & K_{ij}\,{}^{3}\!R^{ij},\varepsilon^{ijk}\,{}^{3}\!R_{il}\nabla_{j}\,{}^{3}\!R_{k}^{l},\varepsilon^{ijk}a_{i}a_{l}\nabla_{j}\,{}^{3}\!R_{k}^{l},a_{i}a_{j}K^{ij},\cdots \\ \text{dim 4:} & & \left(^{3}\!R\right)^{2},{}^{3}\! R_{ij}\,{}^{3}\!R^{ij},\left(a_{i}a^{i}\right)^{2},\left(a_{i}a^{i}\right){}^{3}\! R,a_{i}a_{j}\,{}^{3}\!R^{ij},\cdots \\ \text{dim 3:} & & \varepsilon^{ijk}\left(\Gamma_{il}^{m}\partial_{j}\Gamma_{km}^{l}+\frac{2}{3}\Gamma_{il}^{n}\Gamma_{jm}^{l}\Gamma_{kn}^{m}\right), \\ \text{dim 2:} & & ^{3}\!R,a_{i}a^{i}, \\ \text{dim 1:} & & \text{None}. \end{eqnarray*} Several extensions to the original version of the Hovrava gravity have been proposed. In particular, the so-called “healthy extension” [24], which abandons the projectability condition and includes all the possible operators listed above, attracted much attention. The healthy extension of the Hovrava gravity can be viewed as the (unitary) gauge-fixed version of the higher derivative scalar-tensor theories with a timelike scalar field [40,43], in which the Lorentz invariance is spontaneously broken by the existence of the timelike scalar field. On the other hand, since the higher order spatial derivatives are allowed in the Hovrava gravity, the covariant formulation of the healthy extension of the Hovrava gravity will also contain higher order covariant derivatives.

This is also similar to the Einstein-Aether theory, in which the Lorentz symmetry is broken by the existence of a preferred frame defined by a timelike unit vector field $u_{\mu}$. The general action of the Einstein-Aether theory takes the form [51]: \begin{equation}\mathcal{L}=R+M_{\phantom{\alpha\beta}\mu\nu}^{\alpha\beta}\nabla_{\alpha}u^{\mu}\nabla_{\beta}u^{\nu}+\lambda\left(u_{\mu}u^{\mu}+1\right), \tag{185}\end{equation} with \begin{equation}M_{\phantom{\alpha\beta}\mu\nu}^{\alpha\beta}\equiv a_{1}g^{\alpha\beta}g_{\mu\nu}+a_{2}g_{\mu}^{\alpha}g_{\nu}^{\beta}+a_{3}g_{\nu}^{\alpha}g_{\mu}^{\beta}+a_{4}u^{\alpha}u^{\beta}g_{\mu\nu}. \tag{186}\end{equation} The Lagrange multiplier $\lambda$ ensures that the vector field $u_{\mu}$ is timelike. When further assuming that $u_{\mu}$ is hypersurface orthogonal, i.e., $u_{\mu}~\propto~\nabla_{\mu}\phi$ with $\phi$ a scalar field, the resulting theory can be regarded as the projectable version of the infrared limit of the Hovrava gravity when the potential for the scalar field is constant [52]. In this sense, the scalar version of the Einstein-Aether theory also provides a generally covariant formulation for the projectable Hovrava gravity.


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  • Figure 3

    Diagrammatic representations of the monomials of $d=1$.

  • Figure 4

    (Color online) Diagrammatic representations of the monomials in the complete basis of $d=2$.

  • Figure 5

    (Color online) Diagrammatic representations of the 8 parity preserving unfactorizable monomials of $d=3$.

  • Figure 6

    (Color online) Diagrammatic representation of the parity violating monomial of $d=3$.

  • Figure 7

    (Color online) Diagrammatic representations of the 29 unfactorizable parity preserving monomials of $d=4$.

  • Figure 8

    (Color online) Diagrammatic representations of the 15 unfactorizable parity violating monomials of $d=4$.

  • Table 1  

    Table 1Classification of scalar-tensor monomials up to $d=4$

    $d$Irreducible Reducible
    $\left(c_{0};d_{2},d_{3}\right)$ $\left(c_{0},c_{1},c_{2};d_{2},d_{3},d_{4}\right)$
    1 $\left(0;1,0\right)$
    2 $\left(0;2,0\right)$$\left(1;0,0\right)$ $\left(0,0,0;0,1,0\right)$
    3 $\left(0;3,0\right)$$\left(0;1,1\right)$$\left(1;1,0\right)$ $\left(0,0,0;0,0,1\right)$$\left(0,1,0;0,0,0\right)$
    4 $\left(0;4,0\right)$$\left(0;2,1\right)$$\left(0;0,2\right)$$\left(1;2,0\right)$$\left(2;0,0\right)$$\left(1;0,1\right)$ $\left(0,0,0;1,0,1\right)$$\left(0,1,0;1,0,0\right)$$\left(0,0,1;0,0,0\right)$
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