# Relation between $f(R)$ gravity and Tensor–vector–scalar (TeVeS) gravity

We know that there is a relation between f(R) gravity and scalar-tensor gravity. By applying the Legendre-Weyl transform, we can receive brans-dicke gravity from $f(R)$ gravity. If we start with the Lagrangian $$e^{-1}{\cal L}=f(\chi) + \frac {\partial f(\chi)}{\partial \chi }(R-\chi)$$ Variation with respect to $\chi$ gives rise to $$\frac {\partial^2 f}{\partial \chi^2 }(R-\chi)=0$$ Thus provided that $\frac {\partial^2 f}{\partial \chi^2 }\neq 0$, we have $\chi=R$, and hence receive the usual f(R). One can deﬁne $$\varphi=\frac{\partial f(\chi)}{\partial\chi}$$ and hence the $f(R)$ gravity is equivalent to the brans-dicke theory $$e^{-1}{\cal L}=\varphi R + f(\chi(\varphi))- \varphi\, \chi(\varphi)$$ With conformal transformantion $$g_{\mu\nu} \rightarrow\varphi g_{\mu\nu}$$ one can remove the non-canonical factor of $\varphi$ (jordan frame to einstein frame). But my questions are about Tensor–vector–scalar (TeVes) gravity...

The dynamical degrees of freedom of the TeVeS theory are a rank two tensor $g_{\mu\nu}$, a vector ﬁeld $A_\mu$ and a scalar ﬁeld $\varphi$. Ignoring coupling constants of the theory, the scalar field and vector field Lagrangian are $$h^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi\sqrt{-g}$$ $$[g^{\alpha\beta}g^{\mu\nu}(B_{\alpha\mu}B_{\beta\nu})+(g^{\mu\nu}A_\mu A_\nu-1)]\sqrt{-g}$$ respectively. with $h^{\alpha\beta}=g^{\alpha\beta}-A^\alpha A^\beta$ and $B_{\alpha\beta}=\partial_\alpha A_\beta-\partial_\beta A_\alpha$.

Is there any kind of transformation (like Legendre-Weyl for Scalar-Tensor and $f(R,\varphi)$) which can cause relation between TeVeS gravity and $f(R,\varphi,B_{\mu\nu}B^{\mu\nu})$?