Akin to gauge field, why GR's lagrangian is not $R_{abcd}R^{abcd}$? What's the mathematical or physical meaning of $R_{abcd}R^{abcd}$? For gauge field theory, the Lagrangian of the gauge field is $$\mathcal{L}=-\frac{1}{4}\mathrm{tr}(\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu})=-\frac{1}{8}F_{a\ \mu\nu}F^{a \ \mu\nu}$$
The field  strength $F^a_{\phantom\a \mu\nu}$ where $\mu\nu$ is the coordinate index and
$a$  is the fiber index.
So analogous to the gauge field, $R_{abcd}$ where $ab$ is the fiber index and $cd$ is the
coordinate index. And akin to the Lagrangian of gauge field, the Lagrangian of gravity should 
be $R_{abcd}R^{abcd}$. While it is actually the Einstein-Hilbert action $R$. My questions are:


*

*What's the mathematical or physical meaning of $R_{abcd}R^{abcd}$?

*Why gravity's Lagrangian is not $R_{abcd}R^{abcd}$? If some field's Lagrangian is $R_{abcd}R^{abcd}$, what's the physical properties of this field?

 A: The Lagrangian for GR is
$$
L \propto \int R \sqrt{-g} \, d^4 x
$$
where $R$ is the Ricci scalar
$$
R = R^\mu_\mu = R^{\mu \nu}_{\mu \nu}
$$
So, this is a scalar which is related linearly to all the components of the Riemann tensor, and is a second-order differential of the metric $g$ of the form
$$
R \sim g \partial^2 g + (\partial g)^2
$$
This is typical for a Lagrangian. Your proposal involves the square of the Riemann tensor, and so is a non-linear second-order differential of $g$ with terms like $(\partial g)^4$ and $(\partial^2 g)^2$.
The concrete answer is that the Lagrangian shown above leads to Einstein's equations, and your suggestion doesn't.
A: They are not analogous. $R_{abcd}$ is just Riemann Tensor and $R_{abcd}R^{abcd}$ is Riemann Tensor squared. 


*

*Mathematically they must be squared since having a Single term Riemann Tensor / Ricci Tensor in gravitational action doesn't make sense. Physically speaking they are modification of Einstein Hilbert action. 

*They are curvature not field one shall not confuse between the two. I encourage you to read Gauss-Bonnet gravity for a example of such theories. 
