Consistency of matter coupling in modified gravity

Let us consider the following generalized disformal transformation [22]:

with the coefficients $F_{i}$ being functions of $(\phi,X,Y,Z)$ and

Here and in what follows, we use the notations $\phi_{\mu}\coloneqq\partial_{\mu}\phi$ and $X_{\mu}\coloneqq\partial_{\mu}X$. Note that the right-hand side of (2) is the most general symmetric tensor of rank two constructed out of $\phi$, $\phi_{\mu}$, and $X_{\mu}$. Thanks to this structure, it is straightforward to construct the inverse metric $\bar{g}^{\mu\nu}$ such that $\bar{g}^{\mu\alpha}\bar{g}_{\alpha\nu}=\delta^{\mu}_{\nu}$ [22]. Moreover, as detailed in [22], one can systematically obtain the conditions under which the transformation is invertible [i.e., Eq. (2) can be uniquely solved for the unbarred metric at least locally in the configuration space] by focusing on the group structure of the transformations under the functional composition. In particular, the closedness under the functional composition requires that the coefficient functions $F_{i}$ satisfy

Namely, the kinetic term $\bar{X}$ of the scalar field associated with the barred metric, which is generically a function of $(\phi,X,Y,Z)$, must be a function only of $(\phi,X$):

With this requirement, one can construct the inverse transformation in a fully covariant manner. One can also see that, in order for the inverse transformation to exist, the following conditions should be satisfied on top of (5):

where we have defined $\bar{Y}\coloneqq\bar{g}^{\mu\nu}\phi_{\mu}\bar{X}_{\nu}$, $\bar{Z}\coloneqq\bar{g}^{\mu\nu}\bar{X}_{\mu}\bar{X}_{\nu}$ with $\bar{X}_{\mu}\coloneqq\partial_{\mu}\bar{X}$, and

In [37], the generalized disformal transformation (2) satisfying the invertibility conditions (5) and (6) was employed to construct a novel class of ghost-free scalar-tensor theories, which is dubbed as the generalized disformal Horndeski class (or the GDH class for short). A nontrivial thing in GDH theories (or any other ghost-free theories with degenerate higher-derivative interactions) is that the Ostrogradsky ghost could revive in general when matter fields are taken into account.^*2*2*2We expect that the mass of the revived Ostrogradsky ghost would scale as some inverse power of the matter energy density $\rho$. Therefore, from the EFT point of view, the ghost would be irrelevant at low energies for sufficiently small $\rho$. Interestingly, the authors of [37, 39] showed that a subclass of GDH theories generated by generalized disformal transformations of the following form allows for consistent bosonic matter coupling:

where ${\cal X}_{\mu}$ denotes the projection of $\partial_{\mu}X$ onto a constant-$\phi$ hypersurface and $f_{i}$’s are functions of $(\phi,X,\mathcal{Z})$ with^*3*3*3The quantity $W\coloneqq Y^{2}-XZ$ used in [37] plays essentially the same role as $\mathcal{Z}$ in the present paper.

Note that $f_{i}$’s here are related to $F_{i}$’s in (2) as

We shall discuss the issue of matter coupling in detail in §III.

In what follows, we employ the strategy of [22] to derive several formulae for the inverse metric and the inverse transformation associated with the generalized disformal metric (8), which are useful when we focus on this particular subclass of generalized disformal transformations. The expression for the inverse disformal metric $\bar{g}^{\mu\nu}$ is obtained by assuming

and then fixing each coefficient function $a_{i}(\phi,X,\mathcal{Z})$ so that $\bar{g}^{\mu\alpha}\bar{g}_{\alpha\nu}=\delta^{\mu}_{\nu}$. Written explicitly, we have

with $\mathcal{F}$ defined in (7), which can be expressed in terms of $f_{i}$’s as

As a result, we obtain

which is a function of $(\phi,X,\mathcal{Z})$ in general. In order for the disformal transformation to be invertible, we require that $\bar{X}$ does not depend on $\mathcal{Z}$, i.e., $\bar{X}=\bar{X}(\phi,X)$. This requirement poses a constraint among $f_{i}$’s, which allows us to express one of the $f_{i}$’s in terms of the others. For instance, $f_{1}$ is written in terms of the other functions as

We also assume $\bar{X}_{X}\neq 0$ so that we have $X$ as a function of $(\phi,\bar{X})$. Let us define

and then

Provided that $(\mathcal{Z}/\mathcal{F})_{\mathcal{Z}}\neq 0$, this equation allows us to express $\mathcal{Z}$ as a function of $(\phi,\bar{X},\bar{\mathcal{Z}})$. Now, the inverse disformal transformation can be written in the form

where $X$ and $\mathcal{Z}$ on the right-hand side should be respectively regarded as functions of $(\phi,\bar{X})$ and $(\phi,\bar{X},\bar{\mathcal{Z}})$, as explained earlier. The invertibility conditions are summarized as

Note that Eqs. (11)–(19) are consistent with those in [22] under the identification (10).

As mentioned earlier, we shall discuss the consistency of matter coupling in the presence of fermionic matter fields, whose action is written in terms of the tetrad $e^{a}_{\mu}$. Therefore, in this subsection, we develop the transformation law for the tetrad $\bar{e}^{a}_{\mu}=\bar{e}^{a}_{\mu}[e^{c}_{\alpha},\phi]$ so that $\bar{g}_{\mu\nu}=\eta_{ab}\bar{e}^{a}_{\mu}\bar{e}^{b}_{\nu}$, with $\bar{g}_{\mu\nu}$ being the generalized disformal metric studied in §II.1. Note that, instead of starting from the general form (2), we assume $\bar{g}_{\mu\nu}$ to be of the form (8) from the outset, since otherwise bosonic matter coupling revives the Ostrogradsky ghost [37, 39]. For this purpose, let us put the barred tetrad in the following form:^*4*4*4One can verify that (20) can be uniquely solved for the unbarred tetrad at least locally in the configuration space if the disformal transformation of the metric is invertible (see the Appendix).

with the coefficients $E_{I}$ ($I=0,1,\cdots,4$) being functions of $(\phi,X,\mathcal{Z})$. Then, we have

By equating this expression with (8), we have

where the first equation requires $f_{0}>0$. Note that we have only four equations for five unknown variables $E_{I}$, meaning that the system is underdetermined. Nevertheless, the ambiguity due to the underdeterminedness should be absorbed into a local Lorentz transformation and hence does not affect the action for a fermionic matter field, as we shall discuss below. By making use of this ambiguity, one can make $E_{3}=0$. We then obtain

Here, we assumed $E_{0}>0$ and chose a branch such that $\bar{e}^{a}_{\mu}=E_{0}e^{a}_{\alpha}$ (i.e., $E_{1}=E_{2}=E_{4}=0$) for the case of conformal transformations, where $f_{1}=f_{2}=f_{3}=0$ and $\bar{X}=X/f_{0}$. Now, the transformation law (20) for the tetrad is written only in terms of $\phi$, $\phi_{\mu}$, and ${\cal X}_{\mu}$, and hence the time derivative of the lapse function is absent under the unitary gauge. Note also that, as they should be, the barred tetrad satisfies $\bar{g}^{\mu\nu}\bar{e}^{a}_{\mu}\bar{e}^{b}_{\nu}=\eta^{ab}$ and its dual $\bar{e}^{\mu}_{a}$ (i.e., such that $\bar{g}^{\mu\nu}=\eta^{ab}\bar{e}^{\mu}_{a}\bar{e}^{\nu}_{b}$) is given by $\bar{e}^{\mu}_{a}=\bar{g}^{\mu\nu}\eta_{ab}\bar{e}^{b}_{\nu}$.

Let us now study the local Lorentz transformation. We assume that the transformation has the form

with $L_{I}$’s being functions of $(\phi,X,\mathcal{Z})$, such that

This equation poses the following constraints on $L_{I}$’s:

From the first equation, we can choose $L_{0}=1$. Then, the latter three equations should be satisfied by the remaining four functions, meaning that there exists a family of local Lorentz transformations characterized by one function of $(\phi,X,\mathcal{Z})$. Acting the local Lorentz transformation $\Lambda^{a}{}_{b}$ on the barred tetrad in (20), we have

where

As it should be, one can confirm that the left-hand sides of (22) are invariant under the local Lorentz transformation. Hence, one can make $\tilde{E}_{3}=0$ by choosing $L_{1}$ such that

Combining (26) and (29), we have the following solution for $L_{I}$’s:

Note that we chose a branch such that the local Lorentz transformation approaches identity (i.e., $L_{1}=L_{2}=L_{3}=L_{4}=0$) in the limit $E_{3}\to 0$. This guarantees that one can impose $E_{3}=0$ without loss of generality in obtaining (23).

The above discussion clarifies that the tetrad transformation law associated with the generalized disformal transformation (8) is given by (20) with (23). A caveat is that there still remain ambiguities, even at the starting point (20): One may note that one could incorporate yet higher derivatives, e.g.,

into the tetrad transformation law (20), while keeping the metric transformation law (8). We expect that such terms would also be absorbed into a (generalized) local Lorentz transformation. Since it is more convenient for our purpose to use a simpler form of tetrad transformations, we adopt (20) with (23) for the following arguments rather than studying more involved forms.

Let us first study the case of bosonic matter fields coupled to GDH theories. We assume that the scalar field $\phi$ in the gravitational sector has a timelike gradient and adopt the unitary gauge where $\phi=\phi(t)$. Also, we introduce the Arnowitt-Deser-Misner (ADM) variables as

where $N$ is the lapse function, $N^{i}$ is the shift vector, and $h_{ij}$ is the induced metric.

It is important to note that the Lagrangian of ordinary bosonic matter fields does not contain derivatives of the metric to which they are minimally coupled. For instance, the action of a canonical scalar field $\sigma$ is given by

where no derivative acts on $g_{\mu\nu}$. The same is true for the following action of a canonical gauge field $A^{A}_{\mu}$:

where $g_{*}$ denotes the gauge coupling constant and $f^{ABC}$ is the structure constant associated with the gauge group. One could of course consider a matter field whose Lagrangian contains derivatives of the metric (e.g., cubic Galileon as studied in [38, 40]), but we are mostly interested in ordinary matter fields like those described by the actions (34) and (35) or at most their nonlinear generalizations (e.g., k-essence scalar field [41] and nonlinear electrodynamics [42, 43]).

Now, let us consider matter fields that are coupled to the generalized disformal metric, as we mentioned above in (32). As clarified in [37, 39], under the unitary gauge, the generalized disformal metric (2) in general contains the time derivative of the lapse function $N$ and so does the matter action $S_{\rm m}[\bar{g}_{\mu\nu},\Psi]$, which makes $N$ dynamical when coupled to Horndeski gravity. This is the origin of the revival of the Ostrogradsky ghost in the presence of matter fields. On the other hand, restricted to the subclass of generalized disformal transformations given by (8), the matter coupling is consistent, i.e., the Ostrogradsky ghost does not revive. To see this, recall that the transformation (8) is constructed out of $\phi$, $\phi_{\mu}$, and ${\cal X}_{\mu}$. Under the unitary gauge, the quantity ${\cal X}_{\mu}$ corresponds to the three-dimensional covariant derivative of $X=-\dot{\phi^{2}}/N^{2}$ on a constant-$t$ hypersurface. Therefore, the matter action does not yield the kinetic term of $N$, and hence the matter coupling is consistent. In the language of the original frame, GDH theories that are generated via the generalized disformal transformation of the form (8) allow for consistent bosonic matter coupling.^*6*6*6The condition for consistent matter coupling becomes tighter in scalar-tensor theories with a nondynamical scalar field (e.g., the cuscuton [44] or its extension [45, 46]), as pointed out in [37].

Having discussed the consistency of bosonic matter coupling, let us consider a fermionic matter field. For concreteness, we consider a free massless Dirac spinor $\psi$ in a curved spacetime described by the following action [47]:^*7*7*7Note that the discussion here would apply to the case of massive Dirac fields as well as Majorana fields.

where $e\coloneqq\det e^{a}_{\mu}$, ${\rm c.c.}$ denotes the complex conjugate, and $\gamma^{a}$ denotes the gamma matrices in the Minkowski spacetime such that $\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\eta^{ab}\mathbbm{1}$, with $\mathbbm{1}$ being the identity matrix in the spinor indices. Note that we put hats on local Lorentz indices ($a,b,\cdots=\{\hat{0},\hat{1},\hat{2},\hat{3}\}$). Here, we do not use the notation $\bar{\psi}$. Usually, it denotes $\psi^{\dagger}{\rm i}\gamma^{\hat{0}}$ in the literature, but in the present paper we reserve $\bar{\psi}$ to denote a transformation of the spinor field [see Eq. (47)]. The covariant derivative acting on the Dirac field is defined by

Here, $\gamma_{ab}\coloneqq(\gamma_{a}\gamma_{b}-\gamma_{b}\gamma_{a})/2$ and the (torsion-free) spin connection $\omega_{\mu}{}^{ab}$ is defined by

where $\Gamma^{\lambda}_{\mu\nu}$ is the Christoffel symbol associated with the metric. Equivalently, in terms of differential forms, the spin connection 1-form $\omega^{ab}\coloneqq\omega_{\mu}{}^{ab}{\rm d}x^{\mu}$ is defined so that

with $e^{a}\coloneqq e^{a}_{\mu}{\rm d}x^{\mu}$ being the tetrad 1-form.

Since $\omega_{\mu}{}^{ab}$ involves derivatives of the tetrad/metric, one may think that derivatives of the lapse function and/or shift vector show up in the matter Lagrangian, which could make the fermionic matter coupling inconsistent even in Horndeski theories. Nevertheless, this is not the case as we explain below. For the ADM metric (33), the tetrad 1-form can be explicitly written as

where ${}^{(3)}\!e^{\hat{i}}_{k}$ denotes the triad such that $h_{kl}=\delta_{\hat{i}\hat{j}}{}^{(3)}\!e^{\hat{i}}_{k}{}^{(3)}\!e^{\hat{j}}_{l}$. From (40), we have

where the time derivative (denoted by a dot) acts only on the triad associated with the induced metric. Comparing these equations with (39), it is clear that the time derivative of $N$ or $N^{i}$ does not appear in the spin connection.

On the other hand, the time derivative of the triad shows up through

This means that, the spinor action would yield the time derivative of $N$ after the disformal transformation. Indeed, from (20), the transformation law for the triad is obtained as

where the right-hand sides involve the lapse function (or its spatial derivatives through ${\cal X}_{k}$ and $\mathcal{Z}$). As a result, the spin connection associated with the barred tetrad yields $\dot{N}$, which could make the spinorial matter coupling inconsistent.

In order to investigate the degeneracy structure in detail, let us focus on terms in the spinor action (36) that involve the time derivative of the spinor and the triad:

with $h\coloneqq\det h_{kl}=(\det{}^{(3)}\!e^{\hat{i}}_{k})^{2}$. It should be noted that the right-hand side does not depend on $N$ or $N^{i}$. Substituting the barred triad into the above equation, we obtain

where we have employed (43) and the following formula:

One can absorb the overall factor $E_{0}^{2}(E_{0}+\mathcal{Z}E_{4})$ by redefining the spinor $\psi$ as

where, as mentioned above, we use $\bar{\psi}$ to denote the new spinor field rather than $\psi^{\dagger}{\rm i}\gamma^{\hat{0}}$. Then, (45) takes the form

Note that, except for the last term, the right-hand side has the same form as (44).

Let us study how $\dot{\bar{\psi}}$ and $\dot{N}$ show up in the EOMs. The variation of the action with respect to $\bar{\psi}^{\dagger}$ involves the following terms:

Hence, the EOM for $\bar{\psi}^{\dagger}$, together with its Hermitian conjugate, yields^*8*8*8We choose a representation of the gamma matrices such that $\gamma^{\hat{i}}$’s are Hermitian matrices.

Likewise, the EOM for $N$ contains $\dot{\bar{\psi}}$, $\dot{\bar{\psi}}^{\dagger}$, and $\dot{N}$ in general. To avoid an unwanted extra DOF, these equations should be degenerate, i.e., there should exist an appropriate linear combination of the EOMs for $N$ and $\bar{\psi}$ that gives a constraint equation where any of $\dot{\bar{\psi}}$, $\dot{\bar{\psi}}^{\dagger}$, or $\dot{N}$ does not appear. However, we see that this is not the case: By using (50), one can erase $\dot{\bar{\psi}}$ and $\dot{\bar{\psi}}^{\dagger}$ simultaneously from the EOM for $N$, but $\dot{N}$ still remains, which cannot be removed by use of (49) without reintroducing $\dot{\bar{\psi}}$. This problem originates from the last term in (48), which is proportional to $E_{4}^{2}$. This means that there exists an unwanted extra DOF so long as $E_{4}\neq 0$, i.e., $f_{3}\neq 0$ [see Eq. (23)]. Conversely, if $f_{3}=0$, then (48) has the same form as (44), and hence the spinor can be coupled to gravity without reviving the Ostrogradsky ghost.

To summarize, the consistency of the spinorial matter coupling requires an additional condition $f_{3}=0$ compared to the bosonic case. Thus, we identified a class of GDH theories to which both bosonic and fermionic matter fields can be consistently coupled. Written explicitly, the generalized disformal metric after imposing the consistency of matter coupling takes the form

where $f_{i}$’s are functions of $(\phi,X,\mathcal{Z})$. Here, in order for the transformation to be invertible, the function $f_{1}$ should be related to $f_{0}$ and $f_{2}$ as

through some function $\bar{X}=\bar{X}(\phi,X)$. We note that the class of generalized disformal transformations of the form (51) covers conventional disformal transformations (i.e., those with at most first-order derivative of $\phi$).