Coupled vector Gauss-Bonnet theories and hairy black holes

Let us first briefly revisit the case in which there is a scalar field $\phi$ coupled to the GB term of the form $\mu(\phi)\mathcal{G}$. Because of the antisymmetric property of $\delta^{\mu_{1}\cdots\mu_{k}}_{\nu_{1}\cdots\nu_{k}}$, the field equations of motion following from the coupling $\mu(\phi)\mathcal{G}$ are of second order in the metric tensor $g_{\mu\nu}$ and the scalar field $\phi$. On using the Riemann tensor and covariant derivatives of $\phi$, the GB term can be expressed in the form [76, 34]

$$\mathcal{G}=\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\nabla^{\delta}\left[\frac{\nabla^{\gamma}\nabla_{\rho}\phi\nabla_{\sigma}\phi}{X_{s}}\left({R^{\alpha\beta}}_{\mu\nu}-\frac{2}{3X_{s}}\nabla^{\alpha}\nabla_{\mu}\phi\nabla^{\beta}\nabla_{\nu}\phi\right)\right]\,,$$

(3)

where $\nabla^{\delta}$ is a covariant derivative operator and $X_{s}=-(1/2)\nabla_{\mu}\phi\nabla^{\mu}\phi$. Substituting this expression of $\mathcal{G}$ into the action

$${\cal S}_{\rm sGB}=\int{\rm d}^{4}x\sqrt{-g}\,\mu(\phi)\mathcal{G}\,,$$

(4)

and integrating (4) by parts, it follows that

$${\cal S}_{\rm sGB}=-\int{\rm d}^{4}x\sqrt{-g}\,\mu_{,\phi}(\phi)\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{1}{X_{s}}\nabla^{\gamma}\nabla_{\rho}\phi\nabla_{\sigma}\phi\nabla^{\delta}\phi\left({R^{\alpha\beta}}_{\mu\nu}-\frac{2}{3X_{s}}\nabla^{\alpha}\nabla_{\mu}\phi\nabla^{\beta}\nabla_{\nu}\phi\right)\,,$$

(5)

where $g$ is the determinant of $g_{\mu\nu}$, and we use the notations $F_{,\phi}\equiv\partial F/\partial\phi$ and $F_{,X_{s}}\equiv\partial F/\partial X_{s}$ for any arbitrary function $F$. We will expand the generalized Kronecker delta, integrate the action (5) by parts, and exploit the relation $[\nabla_{\mu},\nabla_{\nu}]\nabla^{\alpha}\phi={R^{\alpha}}_{\lambda\mu\nu}\nabla^{\lambda}\phi$ to eliminate contractions of the Riemann tensors. Up to boundary terms, the action (5) is equivalent to [34]

	$\displaystyle{\cal S}_{\rm sGB}$	$\displaystyle=$	$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,\biggl{[}G_{2s}-G_{3s}\square\phi+G_{4s}R+G_{4s,X_{s}}\left\{(\square\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)\right\}$		(6)
			$\displaystyle+G_{5s}G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi-\frac{1}{6}G_{5s,X_{s}}\left\{(\square\phi)^{3}-3(\square\phi)\,(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)+2(\nabla^{\mu}\nabla_{\alpha}\phi)(\nabla^{\alpha}\nabla_{\beta}\phi)(\nabla^{\beta}\nabla_{\mu}\phi)\right\}\biggr{]}\,,$

where $G_{\mu\nu}=R_{\mu\nu}-(1/2)g_{\mu\nu}R$ is the Einstein tensor, and

	$\displaystyle G_{2s}=-8\mu_{,\phi\phi\phi\phi}(\phi)X_{s}^{2}(3-\ln|X_{s}|)\,,\qquad G_{3s}=4\mu_{,\phi\phi\phi}(\phi)X_{s}(7-3\ln|X_{s}|)\,,$
	$\displaystyle G_{4s}=4\mu_{,\phi\phi}(\phi)X_{s}(2-\ln|X_{s}|)\,,\qquad G_{5s}=-4\mu_{,\phi}(\phi)\ln|X_{s}|\,.$		(7)

The action (6) belongs to a subclass of scalar Horndeski theories [77] with second-order Euler equations of motion. Originally, the equivalence of scalar-GB theories with Horndeski theories given by the coupling functions (7) was shown in Ref. [78] by using the field equations of motion. In Ref. [34], the same equivalence was proven at the level of the action (as explained above).

For the linear scalar-GB coupling $\mu(\phi)=-\alpha\phi$, where $\alpha$ is a constant, we have $G_{5s}=4\alpha\ln|X_{s}|$ and $G_{2s}=G_{3s}=G_{4s}=0$. This falls into a subclass of shift-symmetric Horndeski theories where the field equations of motion are invariant under the shift $\phi\to\phi+c$. In the original form of the GB coupling $\phi\mathcal{G}$, the action is quasi-invariant under the shift $\phi\to\phi+c$, i.e., invariant up to a total derivative, while the Lagrangian in the Horndeski form is manifestly invariant under the shift. For $\mu(\phi)$ containing nonlinear functions of $\phi$, we generally have the $\phi$ dependence in $G_{2s},G_{3s},G_{4s},G_{5s}$. As we mentioned in Introduction, there are BH and neutron star solutions endowed with scalar hairs for such scalar-GB couplings.

If we want to construct theories in which a vector field $A_{\mu}$ is coupled to the GB term in some way, we need to construct a Lorentz-invariant scalar appearing in the Lagrangian. For instance, one may consider the coupling $A^{\mu}\nabla_{\mu}\mathcal{G}$. However, the equations of motion associated with this coupling contain derivatives higher than second order and hence such theories are generally prone to Ostrogradski instability. Another possible coupling would be $\mu_{\rm v}(X)\mathcal{G}$ where

$$X\equiv-\frac{1}{2}A_{\mu}A^{\mu}\,.$$

(8)

Again, this interaction may summon a ghostly DOF in the longitudinal sector of $A_{\mu}$ which can be understood by taking the decoupling of the longitudinal and transverse DOFs. The longitudinal mode becomes manifest by introducing the Stückelberg field according to the replacement $A_{\mu}\to g_{\rm v}A_{\mu}+\nabla_{\mu}\phi$. Then, the decoupling limit $g_{\rm v}\to 0$ gives $X\to X_{s}$. The interacting Lagrangian $\mu_{\rm v}(X)\mathcal{G}$ reduces to a coupling between the GB term and the derivative of $\phi$, not the scalar field itself, which should yield equations of motion with derivatives higher than second order.

As we already explained, the linear coupling $-\phi\mathcal{G}$ has a global shift symmetry and the vector-tensor theory can be obtained by localizing this global symmetry. This requires finding a vector field $\mathcal{J}^{\mu}$ whose divergence agrees with the GB term, $\nabla_{\mu}\mathcal{J}^{\mu}=\mathcal{G}$. After integration by parts, the coupling $-\phi\mathcal{G}$ becomes $\nabla_{\mu}\phi\mathcal{J}^{\mu}[\phi,g]$. As shown in Eq. (5), the action contains the derivatives of $\phi$ but not the field itself, for the linear coupling $\mu_{,\phi}=-\alpha$. The global shift symmetry can be localized by the replacement $\nabla_{\mu}\phi\to\nabla_{\mu}\phi+g_{\rm v}A_{\mu}$ with the help of the vector field $A_{\mu}$. The symmetry transformation is now $\phi\to\phi+\chi(x),A_{\mu}\to A_{\mu}-\nabla_{\mu}\chi(x)/g_{\rm v}$. The scalar field $\phi$ can be eliminated by setting the unitary gauge $\phi=0$. All in all, what we need is the replacement $\nabla_{\mu}\phi\to A_{\mu}$, where the gauge coupling $g_{\rm v}$ is absorbed into the definition of $A_{\mu}$. In this way, we can obtain a vector-tensor theory coupled to the integrated GB term.

However, there is an ambiguity in the above procedure. The second derivative $\nabla_{\mu}\nabla_{\nu}\phi$ is symmetric in its indices while the replaced quantity $\nabla_{\mu}A_{\nu}$ is not symmetric. We resolve this ambiguity by imposing the condition $\nabla_{\mu}\mathcal{J}^{\mu}=\mathcal{G}$ even after the replacement $\nabla_{\mu}\phi\to A_{\mu}$. The expression of $\mathcal{J}^{\mu}$ consistent with such requirement is given by

$\displaystyle\mathcal{J}^{\mu}\equiv\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\left[\frac{A^{\alpha}\nabla_{\nu}A^{\beta}}{X}\left(R^{\gamma\delta}{}_{\rho\sigma}-\frac{2}{3X}\nabla_{\rho}A^{\gamma}\nabla_{\sigma}A^{\delta}\right)\right]\,.$

(9)

In the following, we will show that this vector field satisfies the relation $\nabla_{\mu}\mathcal{J}^{\mu}=\mathcal{G}$. In doing so, we use the relation $\nabla_{[\mu}R^{\alpha\beta}{}_{\nu\rho]}=0$ and the antisymmetric property of the generalized Kronecker delta. We also exploit the following equality

$$\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\nabla_{\mu_{1}}\left(\frac{A^{\nu_{1}}}{X^{2}}\right)\nabla_{\mu_{2}}A^{\nu_{2}}\nabla_{\mu_{3}}A^{\nu_{3}}\nabla_{\mu_{4}}A^{\nu_{4}}=-\frac{1}{2X^{3}}\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}}A_{\mu_{1}}A^{\nu_{1}}\nabla_{\mu_{2}}A^{\nu_{2}}\nabla_{\mu_{3}}A^{\nu_{3}}\nabla_{\mu_{4}}A^{\nu_{4}}\nabla_{\mu_{5}}A^{\nu_{5}}=0\,,$$

(10)

together with the expansion of $\delta^{\mu_{1}\cdots\mu_{d}}_{\nu_{1}\cdots\nu_{d}}$ in $d$ dimensions:

$$\delta^{\mu_{1}\cdots\mu_{d}}_{\nu_{1}\cdots\nu_{d}}=\sum_{k=1}^{d}(-1)^{d+k}\delta^{\mu_{d}}_{\nu_{k}}\delta^{\mu_{1}\cdots\mu_{k}\cdots\mu_{d-1}}_{\nu_{1}\cdots\bar{\nu}_{k}\cdots\nu_{d}}\,,$$

(11)

where $\bar{\nu}_{k}$ means that this index is omitted. Notice that the 5-dimensional generalized Kronecker delta $\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}}$ vanishes in 4-dimensional spacetime, whose property was used in the second equality of Eq. (10). Then, it follows that

$$\nabla_{\mu}\mathcal{J}^{\mu}={\cal F}_{1}+{\cal F}_{2}+{\cal F}_{3}\,,$$

(12)

where

	$\displaystyle{\cal F}_{1}$	$\displaystyle=$	$\displaystyle\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{A^{\alpha}}{X}\nabla_{\mu}\nabla_{\nu}A^{\beta}R^{\gamma\delta}{}_{\rho\sigma}\,,$		(13)
	$\displaystyle{\cal F}_{2}$	$\displaystyle=$	$\displaystyle\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\nabla_{\mu}\left(\frac{A^{\alpha}}{X}\right)\nabla_{\nu}A^{\beta}R^{\gamma\delta}{}_{\rho\sigma}\,,$		(14)
	$\displaystyle{\cal F}_{3}$	$\displaystyle=$	$\displaystyle-\frac{2}{3}\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{A^{\alpha}}{X^{2}}\nabla_{\mu}\left(\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}\nabla_{\sigma}A^{\delta}\right)\,.$		(15)

Since $\nabla_{\mu}\nabla_{\nu}A^{\beta}-\nabla_{\nu}\nabla_{\mu}A^{\beta}={R^{\beta}}_{\lambda\mu\nu}A^{\lambda}$, Eq. (13) reduces to

$${\cal F}_{1}=\frac{1}{2}\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{A^{\alpha}A^{\lambda}}{X}{R^{\beta}}_{\lambda\mu\nu}R^{\gamma\delta}{}_{\rho\sigma}=\frac{1}{4}\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}{R^{\alpha\beta}}_{\mu\nu}R^{\gamma\delta}{}_{\rho\sigma}=\mathcal{G}\,,$$

(16)

where, in the second equality, we used the relation

$$\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\left(\frac{2A^{\nu_{1}}A^{\lambda}}{X}R^{\nu_{2}}{}_{\lambda\mu_{1}\mu_{2}}R^{\nu_{3}\nu_{4}}{}_{\mu_{3}\mu_{4}}-R^{\nu_{1}\nu_{2}}{}_{\mu_{1}\mu_{2}}R^{\nu_{3}\nu_{4}}{}_{\mu_{3}\mu_{4}}\right)=\frac{1}{2X}\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}}A_{\mu_{1}}A^{\nu_{1}}R^{\nu_{2}\nu_{3}}{}_{\mu_{2}\mu_{3}}R^{\nu_{4}\nu_{5}}{}_{\mu_{4}\mu_{5}}=0\,.$$

(17)

For the computation of ${\cal F}_{2}$, we exploit the following property

	$\displaystyle\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\left[\nabla_{\mu_{1}}\left(\frac{A^{\nu_{1}}}{X}\right)\nabla_{\mu_{2}}A^{\nu_{2}}R^{\nu_{3}\nu_{4}}{}_{\mu_{3}\mu_{4}}-\frac{A^{\nu_{1}}A^{\alpha}}{X^{2}}R^{\nu_{2}}{}_{\alpha\mu_{1}\mu_{2}}\nabla_{\mu_{3}}A^{\nu_{3}}\nabla_{\mu_{4}}A^{\nu_{4}}\right]$
	$\displaystyle=-\frac{1}{2X^{2}}\delta^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}_{\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}}A_{\mu_{1}}A^{\nu_{1}}R^{\nu_{2}\nu_{3}}{}_{\mu_{2}\mu_{3}}\nabla_{\mu_{4}}A^{\nu_{4}}\nabla_{\mu_{5}}A^{\nu_{5}}=0\,.$		(18)

Then, we have

$${\cal F}_{2}=\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{A^{\alpha}A^{\lambda}}{X^{2}}{R^{\beta}}_{\lambda\mu\nu}\nabla_{\rho}A^{\gamma}\nabla_{\sigma}A^{\delta}\,.$$

(19)

Expanding the covariant derivative $\nabla_{\mu}$ in ${\cal F}_{3}$ and using the commutation relation of $\nabla_{\mu}\nabla_{\nu}A^{\beta}$, we obtain

$${\cal F}_{3}=-\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{A^{\alpha}A^{\lambda}}{X^{2}}{R^{\beta}}_{\lambda\mu\nu}\nabla_{\rho}A^{\gamma}\nabla_{\sigma}A^{\delta}=-{\cal F}_{2}\,.$$

(20)

On using Eqs. (16) and (20), Eq. (12) yields

$$\nabla_{\mu}\mathcal{J}^{\mu}=\mathcal{G}\,.$$

(21)

Thus, the divergence of $\mathcal{J}^{\mu}$ is equivalent to the GB term.

We consider a scalar quantity $A_{\mu}\mathcal{J}^{\mu}$ as a candidate for the ghost-free Lagrangian of the vector field $A_{\mu}$ coupled to the integrated GB term. As a generalization, we also multiply an arbitrary function $f(X)$ of $X$ with the scalar product $A_{\mu}\mathcal{J}^{\mu}$. The interaction part of the action in such theories is given by

$\displaystyle{\cal S}_{\rm vGB}=\int{\rm d}^{4}x\sqrt{-g}\,f(X)A_{\mu}\mathcal{J}^{\mu}\,,$

(22)

which is composed of two parts:

	$\displaystyle{\cal S}_{{\rm vGB}1}$	$\displaystyle=$	$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{f(X)}{X}A_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}{R^{\gamma\delta}}_{\rho\sigma}\,,$		(23)
	$\displaystyle{\cal S}_{{\rm vGB}2}$	$\displaystyle=$	$\displaystyle-\int{\rm d}^{4}x\sqrt{-g}\,\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\frac{2f(X)}{3X^{2}}A_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}\nabla_{\sigma}A^{\delta}\,.$		(24)

On using the properties $\nabla_{\mu}X=-A_{\nu}\nabla_{\mu}A^{\nu}$ and $\nabla_{\mu}\ln X=-A_{\nu}\nabla_{\mu}A^{\nu}/X$, we find that Eqs. (23) and (24) reduce, respectively, to

	$\displaystyle{\cal S}_{{\rm vGB}1}$	$\displaystyle=$	$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,\biggl{[}G_{5}(X)G_{\mu\nu}\nabla^{\mu}A^{\nu}$		(25)
			$\displaystyle\qquad\qquad\qquad-\frac{4f(X)}{X}\left(A^{\rho}\nabla_{\sigma}A^{\sigma}-A^{\sigma}\nabla_{\sigma}A^{\rho}\right)\nabla_{\nu}\nabla_{\rho}A^{\nu}-\frac{4f(X)}{X}\left(A^{\sigma}\nabla_{\sigma}A^{\nu}-A^{\nu}\nabla_{\sigma}A^{\sigma}\right)\nabla_{\nu}\nabla_{\rho}A^{\rho}\biggr{]}\,,$
	$\displaystyle{\cal S}_{{\rm vGB}2}$	$\displaystyle=$	$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\,\biggl{[}-\frac{2f(X)}{3X}\delta^{\mu\nu\rho}_{\alpha\beta\gamma}\nabla_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}-3f_{5}(X)\delta^{\mu\nu\rho}_{\alpha\beta\gamma}A^{\alpha}A_{\lambda}\nabla_{\mu}A^{\lambda}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}$		(26)
			$\displaystyle\qquad\qquad\qquad+\frac{4f(X)}{X}\left(A^{\rho}\nabla_{\sigma}A^{\sigma}-A^{\sigma}\nabla_{\sigma}A^{\rho}\right)\nabla_{\nu}\nabla_{\rho}A^{\nu}+\frac{4f(X)}{X}\left(A^{\sigma}\nabla_{\sigma}A^{\nu}-A^{\nu}\nabla_{\sigma}A^{\sigma}\right)\nabla_{\nu}\nabla_{\rho}A^{\rho}\biggr{]}\,,$

where

$$G_{5}(X)\equiv 4\int{\rm d}\tilde{X}\frac{f(\tilde{X})}{\tilde{X}}+8f(X)\,,\qquad f_{5}(X)\equiv-\frac{2f_{,X}}{3X}\,.$$

(27)

Taking the sum of ${\cal S}_{{\rm vGB}1}$ and ${\cal S}_{{\rm vGB}2}$, terms on the second lines of Eqs. (25) and (26) cancel each other. On using the property

$$f_{5}\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}A_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}\nabla_{\sigma}A^{\delta}=\frac{4}{3}f_{,X}\delta^{\mu\nu\rho}_{\alpha\beta\gamma}\nabla_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}-3f_{5}\delta^{\mu\nu\rho}_{\alpha\beta\gamma}A^{\alpha}A_{\lambda}\nabla_{\mu}A^{\lambda}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}\,,$$

(28)

we obtain the reduced action

$${\cal S}_{\rm vGB}=\int{\rm d}^{4}x\sqrt{-g}\left[G_{5}(X)G_{\mu\nu}\nabla^{\mu}A^{\nu}-\frac{1}{6}G_{5,X}(X)\delta^{\mu\nu\rho}_{\alpha\beta\gamma}\nabla_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}+f_{5}(X)\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}A_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}\nabla_{\sigma}A^{\delta}\right]\,.$$

(29)

This belongs to a subclass of beyond GP theories proposed in Ref. [84] and thus it is free from the Ostrogradski ghost.

In theories where the function $f(X)$ is constant, i.e.,

$$f(X)=\alpha={\rm constant}\,,$$

(30)

we have

$$G_{5}(X)=4\alpha\ln|X|\,,\qquad f_{5}(X)=0\,.$$

(31)

Note that we dropped a constant term $8\alpha$ in $G_{5}(X)$, as it does not contribute to the field equations of motion. Since the last term on the right hand side of Eq. (29) vanishes, the action ${\cal S}_{\rm vGB}$ belongs to a subclass of GP theories [79, 80, 82]. We recall that, from Eq. (7), the linear scalar-GB coupling $\mu(\phi)\mathcal{G}$ with $\mu(\phi)=-\alpha\phi$ corresponds to $G_{5s}(X_{s})=4\alpha\ln|X_{s}|$ with $G_{2s}(X_{s})=G_{3s}(X_{s})=G_{4s}(X_{s})=0$. The coupling (31) is the vector-tensor analogue to the linear scalar-GB coupling in Horndeski theories.

In the following, we will focus on GP theories given by the functions (31). Varying the action (29) with respect to $A_{\mu}$, it follows that

$\displaystyle\frac{\delta{\cal S}_{\rm vGB}}{\delta A_{\mu}}=\alpha\left(\mathcal{J}^{\mu}+\mathcal{J}_{F}^{\mu}\right)\,,$

(32)

where $\mathcal{J}^{\mu}$ is defined by Eq. (9), and

$\displaystyle\mathcal{J}_{F}^{\mu}\equiv\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}A_{\nu}F^{\alpha\beta}\left(\frac{1}{X^{2}}\nabla^{\gamma}A_{\rho}\nabla^{\delta}A_{\sigma}-\frac{1}{2X}R^{\gamma\delta}{}_{\rho\sigma}\right)\,,$

(33)

with $F^{\alpha\beta}\equiv\nabla^{\alpha}A^{\beta}-\nabla^{\beta}A^{\alpha}$. The sum of $\mathcal{J}^{\mu}$ and $\mathcal{J}_{F}^{\mu}$ can be expressed in a compact form

$$\mathcal{J}^{\mu}+\mathcal{J}_{F}^{\mu}=\delta^{\mu\nu\rho\sigma}_{\alpha\beta\gamma\delta}\left[\frac{A^{\alpha}\nabla^{\beta}A_{\nu}}{X}\left(R^{\gamma\delta}{}_{\rho\sigma}-\frac{2}{3X}\nabla^{\gamma}A_{\rho}\nabla^{\delta}A_{\sigma}\right)\right]\,.$$

(34)

Let us consider the case in which the Maxwell term $F\equiv-(1/4)F^{\alpha\beta}F_{\alpha\beta}$ and the mass term $X$ are present in addition to the action (29) with $f(X)=\alpha$. The action in such a subclass of GP theories is given by

$${\cal S}=\int{\rm d}^{4}x\sqrt{-g}\left[\frac{1}{g_{\rm v}^{2}}F+\eta X+G_{5}(X)G_{\mu\nu}\nabla^{\mu}A^{\nu}-\frac{1}{6}G_{5,X}(X)\delta^{\mu\nu\rho}_{\alpha\beta\gamma}\nabla_{\mu}A^{\alpha}\nabla_{\nu}A^{\beta}\nabla_{\rho}A^{\gamma}\right]\,,$$

(35)

where $g_{\rm v}$ and $\eta$ are constants and $G_{5}(X)=4\alpha\ln|X|$. Varying this action with respect to $A_{\mu}$, it follows that

$$\frac{1}{g_{\rm v}^{2}}\nabla_{\nu}F^{\mu\nu}+\eta A^{\mu}=\alpha(\mathcal{J}^{\mu}+\mathcal{J}_{F}^{\mu})\,.$$

(36)

While the GB term does not explicitly show up in Eq. (36), it appears by taking the divergence of Eq. (36) as

$$\eta\nabla_{\mu}A^{\mu}=\alpha\mathcal{G}+\alpha\nabla_{\mu}\mathcal{J}_{F}^{\mu}\,,$$

(37)

where we used the relation $\nabla_{\mu}\nabla_{\nu}F^{\mu\nu}=0$ and Eq. (21).

Taking the decoupling limit $g_{\rm v}\to 0$ with the replacement $A^{\mu}\to g_{\rm v}A^{\mu}+\nabla^{\mu}\phi$, we have $\mathcal{J}_{F}^{\mu}\to 0$ and hence Eqs. (36) and (37) reduce to the Maxwell equation, $\nabla_{\nu}F^{\mu\nu}=0$, and the equation of motion for the scalar field, $\eta\nabla_{\mu}\nabla^{\mu}\phi=\alpha\mathcal{G}$, respectively. This latter scalar field equation also follows by varying the Lagrangian $L=\eta X_{s}-\alpha\phi\mathcal{G}$ with respect to $\phi$, so the linearly coupled scalar-GB theory is recovered by taking the above decoupling limit. In shift-symmetric Horndeski theories, using the expression of $\mathcal{G}$ in Eq. (3) shows that the scalar field equation can be expressed in the form $\nabla_{\mu}j^{\mu}=0$, where $j^{\mu}$ is a conserved current. When this equation is solved for $j^{\mu}$, there is an integration constant corresponding to boundary/initial conditions of the system.

In vector-tensor theories the equation of motion for the vector field $A^{\mu}$ is given by Eq. (36), which does not contain an integration constant. Although Eq. (37) corresponds to the differential version of Eq. (36), one cannot choose an arbitrary integration constant when integrating Eq. (37). This property is different from that in shift-symmetric Horndeski theories discussed above. If we apply GP theories to the isotropic and homogeneous cosmological background, the temporal vector component $A_{0}$ is always related to the Hubble expansion rate $H$ [90, 91, 92]. This is known as a tracker solution, in which case we do not have a freedom of changing initial conditions of the vector field. In scalar-tensor theories, on the other hand, it is possible to choose initial conditions away from the tracker because of the existence of the integration constant said above [93, 94]. As a result, one can distinguish between shift-symmetric Horndeski theories and GP theories from the background cosmological dynamics.

If we apply linear scalar-GB theory to the static and spherically symmetric background in vacuum, there are hairy BH solutions satisfying the boundary condition $X_{s}=0$ on the horizon [24, 25]. This boundary condition fixes the integration constant mentioned above [36]. In vector-GB theory, the similar boundary condition, like $X=0$, cannot be necessarily imposed because of the absence of an arbitrary constant in Eq. (36). This implies that the BH solution in vector-GB theory should be different from that in linear scalar-GB theory. In Sec. III, we will investigate the property of hairy BH solutions in vector-GB theory.