lsection [5.8em] 2.3em

(In)equivalence of Metric-Affine and Metric Effective Field Theories

Gianfranco Pradisi and Alberto Salvio

Physics Department, University of Rome Tor Vergata,
via della Ricerca Scientifica, I-00133 Rome, Italy

I. N. F. N. - Rome Tor Vergata,
via della Ricerca Scientifica, I-00133 Rome, Italy

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Abstract

In a geometrical approach to gravity the metric and the (gravitational)
connection can be independent and one deals with metric-affine theories. We
construct the most general action of metric-affine effective field theories,
including a generic matter sector, where the connection does not carry
additional dynamical fields. Among other things, this helps in identifying the
complement set of effective field theories where there are other dynamical
fields, which can have an interesting phenomenology. Within the latter set, we
study in detail a vast class where the Holst invariant (the contraction of the
curvature with the Levi-Civita antisymmetric tensor) is a dynamical
pseudoscalar. In the Einstein-Cartan case (where the connection is metric
compatible and fermions can be introduced) we also comment on the possible
phenomenological role of dynamical dark photons from torsion and compute
interactions of the above-mentioned pseudoscalar with a generic matter sector
and the metric. Finally, we show that in an arbitrary realistic metric-affine
theory featuring a generic matter sector the equivalence principle always
emerges at low energies without the need to postulate it.

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###### Contents

  1. 1 Introduction
  2. 2 Ingredients
  3. 3 Theories with non-dynamical distorsion
    1. 3.1 General characterization
    2. 3.2 Theories with a falsely-dynamical distorsion
    3. 3.3 $f({\cal R})$ theories
  4. 4 Theories with dynamical distorsion
    1. 4.1 General characterization
    2. 4.2 Examples: Dynamical (pseudo)scalarons
      1. 4.2.1 Dynamical pseudoscalaron $\zeta^{\prime}$
      2. 4.2.2 Dynamical combination of $\zeta$ and $\zeta^{\prime}$
    3. 4.3 Examples: Poincaré gauge theories coupled to matter
      1. 4.3.1 Dark photons from torsion
      2. 4.3.2 Coupling the pseudoscalaron to matter
  5. 5 A note on the equivalence principle
  6. 6 Conclusions
  7. A Integrating out the distorsion

##  1 Introduction

General relativity (GR) is an extremely successful theory of gravity, which
agrees with all observations performed so far. Recent tests of GR include the
discovery of gravitational waves, whose production is consistent with
coalescing black holes [1], and the images of the black holes in the center of
the M87 and our galaxy produced by the Event Horizon Telescope [2, 3, 4, 5,
6].

Of course, GR has to be complemented by some matter fields. At least a set of
spin-1, spin-1/2 and spin-0 fields are needed to describe all we know about
non-gravitational physics, the Standard Model of particles (SM) and its
extensions that can account for the evidence of beyond-the-SM physics
(neutrino masses and mixings, dark matter, baryon asymmetry, etc.).

Moreover, a UV completion is also necessary because GR is known to be
nonrenormalizable by perturbative methods [7, 8] and to be within the regime
of validity of perturbation theory at energies much below the Planck scale.
However, at those low energies we can construct a consistent theory by adding
all possible operators along the lines of effective field theories [9] (see
also Refs. [10, 11] for reviews). The lower the dimensionality of a given
operator is the more relevant such operator is expected to be at low energies.

The main principle behind these constructions, including GR itself, is general
covariance (or the general relativity principle), which essentially states
that all laws of physics should be invariant under a general coordinate
transformation. This also implies that the field equations can be written in a
covariant form and renders the presence of tensors, such as the metric, and a
connection necessary. This geometrization of physics is commonly regarded as
one of the greatest achievements of Einstein’s theory. In GR and its effective
field theory (EFT) extensions, including ordinary matter fields (spin-1,
spin-1/2 and spin-0 fields), the connection is typically assumed to be the
Levi-Civita one, a functional of the metric. Theories of this sort are thus
called metric theories. However, from the geometrical point of view the metric
and the connection can be completely independent objects.

Therefore, a natural modification of gravity can be obtained by promoting the
connection to an independent degree of freedom, but preserving general
covariance. The resulting theories are called metric-affine (see Ref. [12] for
a recent discussion on this subject and references to other original articles
and Ref. [13] for a classic review). In general the difference between an
arbitrary connection and the Levi-Civita connection is a tensor, known as the
distorsion. The distorsion coincides with the contorsion when the theory is
metric compatible, i.e. when the covariant derivate of the metric vanishes,
which is required by the presence of fermions. The contorsion in turn is a
tensor that can be expressed in terms of the torsion and that vanishes if and
only if the torsion does. The metric-compatible theories are also known as
Poincaré gauge theories because they can always be formulated as theories with
a local Poincaré symmetry (see [14] for a recent review with many references
to original works).

One of the purposes of the present paper is to identify the general111For
previous less general studies see Refs. [15, 16, 17]. form of the action of a
metric-affine EFT that is equivalent to a metric EFT in the sense that does
not feature an independent dynamical distorsion: i.e. the distorsion can be
exactly integrated out and expressed in terms of the metric and the matter
fields that are not of gravitational origin (that do not come from the metric
and/or the distorsion). Indeed, even in a metric EFT additional gravitational
degrees of freedom besides the massless spin-2 graviton can emerge from the
metric because higher powers of the curvature tensors (that can involve higher
derivatives) are generically present.

The motivation for finding the general action described in the previous
paragraph is the fact that it helps us to tell whether a given metric-affine
theory does not feature an independent dynamical distorsion without performing
a direct calculation of the dynamical degrees of freedom. Also, with this
result in hand, one could automatically identify the complement set of metric-
affine EFTs that can potentially feature an independent dynamical distorsion.
This set of theories is particularly interesting as the new distorsion fields
can have interesting phenomenological consequences.

Another purpose of this paper is to discuss the validity of the equivalence
principle in these EFTs. The equivalence principle is often presented as the
starting point in formulating GR. However, in a metric EFT this principle is
generically broken by the higher-dimensional operators. Given that GR plus
minimally-coupled matter fields anyhow describe the low-energy limit of metric
EFTs the equivalence principle is always recovered at low energies in metric
theories. It is then natural to ask whether the same is true in general
metric-affine EFTs: is the equivalence principle always an emergent low energy
property in an arbitrary theory?

Let us now give an outline of the paper (a detailed summary of the results
will be given in the concluding section). In Sec. 2 we will present the key
ingredients that are needed to construct metric-affine EFTs. We will not limit
ourselves to the gravitational sector, but we will also include a general
matter content, namely an arbitrary number of scalars (or pseudoscalars),
gauge fields and fermions. The general action of theories with non-dynamical
distorsion will then be the topic of Sec. 3. After that, in Sec. 4, we will
discuss theories with dynamical distorsion, studying in detail some explicit
examples. The possible breaking of the equivalence principle and its possible
emergence at low energies in metric-affine theories will then be investigated
in Sec. 5. Finally, in the concluding Sec. 6 we offer a detailed summary of
the new results of the paper with some further discussions.

##  2 Ingredients

In this section we provide the main ingredients that are needed to construct
gravitational theories coupled to a generic matter sector. Most of the
material in this section is a review of well-known results, but it is all
needed to understand the subsequent sections. Here we will also take advantage
to fix our notation.

To describe gravity we start from the general relativity principle, which
states that all laws of physics should be invariant under general coordinate
transformations. To implement such principle we introduce a metric
$g_{\mu\nu}$ and a connection ${\cal A}_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ as independent fields. So we are in the
framework of metric-affine theories.

The metric would be needed even if gravity were absent, indeed writing the
flat metric222$\eta_{ab}$ represents the Minkowski metric and is needed to
recover all we know about non-gravitational physics. We use the mostly plus
signature convention $\\{\eta_{ab}\\}=$ diag$(-1,1,1,1)=\\{\eta^{ab}\\}.$
$ds^{2}=\eta_{ab}d\xi^{a}d\xi^{b}$ in general coordinates $x^{\mu}$ the metric
$g_{\mu\nu}$ appears: $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$. The general
transformation rule of the metric (obtained by requiring $ds^{2}$ invariant)
is

$g^{\prime}_{\alpha\beta}(x^{\prime})=\frac{\partial x^{\mu}}{\partial
x^{{}^{\prime}\alpha}}\frac{\partial x^{\nu}}{\partial
x^{{}^{\prime}\beta}}g_{\mu\nu}(x)$ (2.1)

and, generically, in the presence of gravity it is not possible to recover the
flat metric with a general coordinate transformation.

The connection, on the other hand, is needed in curved space to introduce
covariant derivatives of tensors, which are essential to write the field
equations (which contain derivatives) in a covariant form: the covariant
derivatives of a generic tensor $T_{\mu_{1}...\mu_{n}}^{\nu_{1}...\nu_{m}}$
with $n$ covariant indices and $m$ contravariant333As usual a covariant vector
is an object that transforms as $\frac{\partial}{\partial x^{\mu}}$ and a
contravariant vector is an object that transforms as $dx^{\mu}$ under general
coordinate transformations. A tensor with $n$ covariant indices and $m$
contravariant indices transforms as the direct product of $n$ covariant
vectors and $m$ contravariant vectors. Note that $g_{\mu\nu}$ is a tensor with
two covariant indices (see Eq. (2.1)). The inverse metric $g^{\mu\nu}$, i.e.
$g^{\mu\rho}g_{\rho\nu}=\delta^{\mu}_{\nu}$, (which exists, as shown below) is
a tensor with two contravariant indices because $\delta^{\mu}_{\nu}$ is
invariant. As usual here we raise and lower the spacetime indices through the
inverse metric $g^{\mu\nu}$ and $g_{\mu\nu}$, respectively. The flat indices
$a,b,...$ are raised and lowered with $\eta^{ab}$ and $\eta_{ab}$,
respectively. indices are

${\cal
D}_{\mu}T_{\mu_{1}...\mu_{n}}^{\nu_{1}...\nu_{m}}=\partial_{\mu}T_{\mu_{1}...\mu_{n}}^{\nu_{1}...\nu_{m}}+{\cal
A}_{\mu\leavevmode\nobreak\ \beta_{1}}^{\leavevmode\nobreak\
\,\nu_{1}}T_{\mu_{1}...\mu_{n}}^{\beta_{1}...\nu_{m}}+...+{\cal
A}_{\mu\leavevmode\nobreak\ \beta_{m}}^{\leavevmode\nobreak\
\,\nu_{m}}T_{\mu_{1}...\mu_{n}}^{\nu_{1}...\beta_{m}}-{\cal
A}_{\mu\leavevmode\nobreak\ \mu_{1}}^{\leavevmode\nobreak\
\,\alpha_{1}}T_{\alpha_{1}...\mu_{n}}^{\nu_{1}...\nu_{m}}-...-{\cal
A}_{\mu\leavevmode\nobreak\ \mu_{n}}^{\leavevmode\nobreak\
\,\alpha_{n}}T_{\mu_{1}...\alpha_{n}}^{\nu_{1}...\nu_{m}}.$ (2.2)

This calligraphic covariant derivative ${\cal D}$ is generically different
from the covariant derivative, which we denote $D$, computed with the Levi-
Civita (LC) connection

$\Gamma_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\
\,\rho}=\frac{1}{2}g^{\rho\tau}\left(\partial_{\mu}g_{\tau\sigma}+\partial_{\sigma}g_{\tau\mu}-\partial_{\tau}g_{\mu\sigma}\right).$
(2.3)

In order for the quantity in (2.2) to be a tensor with $m$ contravariant
indices and $n+1$ covariant indices ${\cal A}_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ should transform under general
coordinate transformations precisely as $\Gamma_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$. So

$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}\equiv{\cal
A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\
\,\rho}-\Gamma_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\
\,\rho},$ (2.4)

which we call the distorsion, transforms as a tensor. Theories where
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}=0$ are
called metric theories as the connection can be computed once the metric is
known in that case. The torsion $T_{\mu\nu\rho}$ is defined in terms of the
distorsion by

$T_{\mu\nu\rho}\equiv C_{\mu\nu\rho}-C_{\rho\nu\mu},$ (2.5)

which is antisymmetric with respect to the exchange $\mu\leftrightarrow\rho$.
The curvature associated with ${\cal A}_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ is defined by

${\cal F}_{\mu\nu\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\rho}\equiv\partial_{\mu}{\cal A}_{\nu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}-\partial_{\nu}{\cal
A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}+{\cal
A}_{\mu\leavevmode\nobreak\ \lambda}^{\leavevmode\nobreak\ \,\rho}{\cal
A}_{\nu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\lambda}-{\cal
A}_{\nu\leavevmode\nobreak\ \lambda}^{\leavevmode\nobreak\ \,\rho}{\cal
A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\lambda},$ (2.6)

which can be expressed in terms of $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ as

${\cal F}_{\mu\nu\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\rho}=R_{\mu\nu\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\rho}+D_{\mu}C_{\nu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\
\,\rho}-D_{\nu}C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\
\,\rho}+C_{\mu\leavevmode\nobreak\ \lambda}^{\leavevmode\nobreak\
\,\rho}C_{\nu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\
\,\lambda}-C_{\nu\leavevmode\nobreak\ \lambda}^{\leavevmode\nobreak\
\,\rho}C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\lambda},$
(2.7)

where $R_{\mu\nu\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \rho}$
is the standard Riemann tensor444We use the conventions
$R_{\mu\nu\,\,\,\sigma}^{\quad\rho}\equiv\partial_{\mu}\Gamma_{\nu\,\sigma}^{\,\rho}-\partial_{\nu}\Gamma_{\mu\,\sigma}^{\,\rho}+\Gamma_{\mu\,\tau}^{\,\rho}\Gamma_{\nu\,\sigma}^{\,\tau}-\Gamma_{\nu\,\tau}^{\,\rho}\Gamma_{\mu\,\sigma}^{\,\tau},\quad
R_{\mu\nu}\equiv R_{\rho\mu\,\,\,\nu}^{\quad\rho},\quad R\equiv
g^{\mu\nu}R_{\mu\nu}.$ , i.e. ${\cal F}_{\mu\nu\leavevmode\nobreak\
\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \rho}$ evaluated at ${\cal A}_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}=\Gamma_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$. Starting from ${\cal
F}_{\mu\nu\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \rho}$
we can define a scalar

${\cal R}\equiv{\cal F}_{\mu\nu}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mu\nu}$ (2.8)

and a pseudoscalar (see [19, 20, 21])

${\cal R^{\prime}}\equiv\frac{1}{\sqrt{-g}}\epsilon^{\mu\nu\rho\sigma}{\cal
F}_{\mu\nu\rho\sigma},$ (2.9)

where $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric Levi-Civita
symbol with $\epsilon^{0123}=1$. We will refer to ${\cal R^{\prime}}$ as the
Holst invariant. The pseudoscalar ${\cal R^{\prime}}$ vanishes for
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}=0$ (that is
when the connection is the LC one) because of the cyclicity property
$R_{\mu\nu\rho\sigma}+R_{\nu\sigma\rho\mu}+R_{\sigma\mu\rho\nu}=0$, which is
the reason why in standard Riemannian geometry ${\cal R^{\prime}}$ is absent.
Therefore, ${\cal R^{\prime}}$ can be considered as a direct manifestation of
a connection that is independent of the metric. We will study its possible
dynamics in Sec. 4.2. By using (2.7) one obtains

$\displaystyle{\cal R}$ $\displaystyle=$ $\displaystyle
R+D_{\mu}C_{\nu}^{\leavevmode\nobreak\
\,\mu\nu}-D_{\nu}C_{\mu}^{\leavevmode\nobreak\
\,\mu\nu}+C_{\mu\leavevmode\nobreak\ \lambda}^{\leavevmode\nobreak\
\,\mu}C_{\nu}^{\leavevmode\nobreak\ \,\lambda\nu}-C_{\nu\leavevmode\nobreak\
\lambda}^{\leavevmode\nobreak\ \,\mu}C_{\mu}^{\leavevmode\nobreak\
\,\lambda\nu},$ (2.10) $\displaystyle{\cal R^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{-g}}\epsilon^{\mu\nu\rho\sigma}\left(D_{\mu}C_{\nu\rho\sigma}+C_{\mu\rho\lambda}C_{\nu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\lambda}\right).$ (2.11)

Note that we can decompose

${\cal F}_{\mu\nu\rho\sigma}=\frac{1}{16}g_{\mu\rho}g_{\nu\sigma}{\cal
R}-\frac{1}{4!\sqrt{-g}}\epsilon_{\mu\nu\rho\sigma}{\cal
R}^{\prime}+\tilde{\cal F}_{\mu\nu\rho\sigma},\qquad(\tilde{\cal
F}_{\mu\nu}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mu\nu}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon^{\mu\nu\rho\sigma}\tilde{\cal F}_{\mu\nu\rho\sigma}=0),$ (2.12)

where $\epsilon_{\mu\nu\rho\sigma}$ is the totally antisymmetric tensor with
$\epsilon_{0123}$ equal to the metric determinant $g$, such that
$g_{\mu\alpha}g_{\nu\beta}g_{\rho\gamma}g_{\sigma\delta}\epsilon^{\alpha\beta\gamma\delta}=\epsilon_{\mu\nu\rho\sigma}$.

All the ingredients introduced so far are sufficient to describe gravity only.
However, we want to include all the other interactions (electroweak, strong,
Yukawa interactions, etc.) so we also consider a generic number of real
scalars (or pseudoscalars) $\phi$, gauge fields $A^{I}_{\mu}$ corresponding to
an internal gauge group $G$ and fermions, which we represent here with Weyl
spinors $\psi$. Note that massive vector fields can be obtained as usual
through the Higgs or Stückelberg mechanisms.

In general the distorsion tensor does not have special properties. However, in
the presence of fermions one can show that it should be such that the
covariant derivative of the metric vanishes, or, in other words, the theory
should be metric compatible.

As we will recover now, this has to do with the fact that in a generic curved
spacetime fermion fields belong to the spinorial representation of a local
Lorentz group in the tangent space. Indeed in order to define them one
introduces a basis $\\{e_{a}\\}$ in the tangent space such that

$\eta_{ab}=e_{a}^{\mu}e_{b}^{\nu}g_{\mu\nu},$ (2.13)

where the “tetrads” $e_{a}^{\mu}$ are defined by expanding each $e_{a}$ in the
coordinate basis, $e_{a}=e_{a}^{\mu}\frac{\partial}{\partial x^{\mu}}$. We can
also define $e^{a}_{\mu}\equiv\eta^{ab}g_{\mu\nu}e^{\nu}_{b}$, which can be
considered as the components of some one-form fields $e^{a}$ in the one-form
basis $\\{dx^{\mu}\\}$, namely $e^{a}=e^{a}_{\mu}dx^{\mu}$. Using (2.13) one
finds that these quantities satisfy $e^{a}_{\mu}e^{\mu}_{b}=\delta^{a}_{b}$
and

$g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}.$ (2.14)

It follows that the inverse of the metric exists and is given by
$g^{\mu\nu}=e_{a}^{\mu}e_{b}^{\nu}\eta^{ab}$, which implies
$\eta^{ab}=e^{a}_{\mu}e^{b}_{\nu}g^{\mu\nu}$. The $e^{a}$ (and analogously the
$e_{a}$) are defined modulo local Lorentz transformations: if we redefine
$e^{{}^{\prime}a}=\Lambda^{a}_{\leavevmode\nobreak\ b}e^{b}$, where
$\Lambda^{a}_{\leavevmode\nobreak\ b}$ are the elements of a local Lorentz
transformation, we obtain the same metric
$g_{\mu\nu}=e^{{}^{\prime}a}_{\mu}e^{{}^{\prime}b}_{\nu}\eta_{ab}$. Let us
consider now a vector ${\cal V}$, which we take to be $G$-invariant for
simplicity, and expand it in the basis $\\{e_{a}\\}$, that is ${\cal V}={\cal
V}^{a}e_{a}$. The components ${\cal V}^{a}$ belong to the vector
representation of the local Lorentz group so their covariant derivative

${\cal D}_{\mu}{\cal V}^{a}=\partial_{\mu}{\cal V}^{a}+{\cal
A}_{\mu\leavevmode\nobreak\ b}^{\leavevmode\nobreak\ \,a}{\cal V}^{b}$ (2.15)

should feature a connection ${\cal A}_{\mu\leavevmode\nobreak\
b}^{\leavevmode\nobreak\ \,a}$ (known as the spin connection) whose values
belong to the Lorentz algebra: defining ${\cal A}_{\mu}^{\leavevmode\nobreak\
\,ab}\equiv{\cal A}_{\mu\leavevmode\nobreak\ b}^{\leavevmode\nobreak\
\,c}\eta^{bc}$, we can impose an antisymmetry with respect to the exchange of
the flat indices $a,b$:

${\cal A}_{\mu}^{\leavevmode\nobreak\ \,ab}=-{\cal
A}_{\mu}^{\leavevmode\nobreak\ \,ba}.$ (2.16)

The spin connection can be seen as the connection ${\cal
A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$ rewritten
using the tetrad basis and we can express one in terms of the other: this can
be done by considering the covariant derivative ${\cal D}{\cal V}$ and writing
the identities

${\cal D}_{\mu}{\cal V}^{\rho}\leavevmode\nobreak\
dx^{\mu}\otimes\frac{\partial}{\partial x^{\rho}}={\cal D}{\cal V}={\cal
D}_{\mu}{\cal V}^{a}\leavevmode\nobreak\ dx^{\mu}\otimes
e_{a}=e_{a}^{\rho}{\cal D}_{\mu}{\cal V}^{a}\leavevmode\nobreak\
dx^{\mu}\otimes\frac{\partial}{\partial x^{\rho}}$ (2.17)

which implies ${\cal D}_{\mu}{\cal V}^{\rho}=e_{a}^{\rho}{\cal D}_{\mu}{\cal
V}^{a}$. Using then (2.2) and ${\cal V}^{a}=e^{a}_{\lambda}{\cal V}^{\lambda}$
one finds

${\cal A}_{\mu\leavevmode\nobreak\ b}^{\leavevmode\nobreak\
\,a}=e^{a}_{\nu}{\cal A}_{\mu\leavevmode\nobreak\
\lambda}^{\leavevmode\nobreak\
\,\nu}e^{\lambda}_{b}-e^{\lambda}_{b}\partial_{\mu}e^{a}_{\lambda}.$ (2.18)

From this result one can show

${\cal D}_{\mu}e_{\nu}^{a}\equiv\partial_{\mu}e_{\nu}^{a}-{\cal
A}_{\mu\leavevmode\nobreak\ \nu}^{\leavevmode\nobreak\
\,\lambda}e^{a}_{\lambda}+{\cal A}_{\mu\leavevmode\nobreak\
b}^{\leavevmode\nobreak\ \,a}e^{b}_{\nu}=0$ (2.19)

and, therefore, using (2.14), the antisymmetry property (2.16) and the Leibniz
rule we obtain ${\cal D}_{\mu}g_{\alpha\beta}=0$.

The above-mentioned local Lorentz group is precisely the one with respect to
which fermions belong to the spinorial representation. Therefore, we recover
the well-known result that in the presence of fermions, when this local
Lorentz group is compulsory, the theory should be metric compatible. In the
absence of fermions, on the other hand, one can have ${\cal
D}_{\mu}g_{\alpha\beta}\neq 0$ and $T_{\mu\nu\rho}=0$, which is known as
Palatini gravity.

The gauge fields $A^{I}_{\mu}$, together with the connection ${\cal
A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$, allow us
to define a covariant derivative with respect to both general coordinate
transformations and elements of $G$, whose action on scalars and fermions
reads

${\cal D}_{\mu}\phi=\partial_{\mu}\phi+i\theta^{I}A^{I}_{\mu}\phi,\qquad{\cal
D}_{\mu}\psi=\partial_{\mu}\psi+it^{I}A^{I}_{\mu}\psi+\frac{1}{2}{\cal
A}^{ab}_{\mu}\sigma_{ab}\psi,$ (2.20)

where, recalling that we work with Weyl fermions,
$\sigma^{ab}\equiv\frac{1}{4}(\sigma^{a}\bar{\sigma}^{b}-\sigma^{b}\bar{\sigma}^{a})$,
also $\sigma^{i}\equiv-\bar{\sigma}^{i}$ (with $i=1,2,3$) are the Pauli
matrices and $\sigma^{0}\equiv\bar{\sigma}^{0}\equiv 1$ is the $2\times 2$
identity matrix. The gauge couplings are contained in the matrices
$\theta^{I}$ and $t^{I}$, which are the generators of $G$ in the scalar and
fermion representations, respectively.

We consider now the commutator of two covariant derivatives acting on a scalar
field $\phi$:

$[{\cal D}_{\mu},{\cal D}_{\nu}]\phi=\left[iF_{\mu\nu}^{I}\theta^{I}-({\cal
A}_{\mu\leavevmode\nobreak\ \nu}^{\leavevmode\nobreak\ \,\lambda}-{\cal
A}_{\nu\leavevmode\nobreak\ \mu}^{\leavevmode\nobreak\ \,\lambda}){\cal
D}_{\lambda}\right]\phi,$ (2.21)

where

$F_{\mu\nu}^{I}\equiv\partial_{\mu}A^{I}_{\nu}-\partial_{\nu}A^{I}_{\mu}-f^{KJI}A_{\mu}^{K}A_{\nu}^{J}$
(2.22)

and the $f^{KJI}$ are the structure constants of $G$. Note that both $[{\cal
D}_{\mu},{\cal D}_{\nu}]\phi$ and $({\cal A}_{\mu\leavevmode\nobreak\
\nu}^{\leavevmode\nobreak\ \,\lambda}-{\cal A}_{\nu\leavevmode\nobreak\
\mu}^{\leavevmode\nobreak\ \,\lambda}){\cal D}_{\lambda}\phi$ are tensors and
so, because of (2.21), also the $F_{\mu\nu}^{I}$ must be tensors. This shows
that the expression of the field strength tensor of $A^{I}_{\mu}$ in the
presence of a generic connection can be taken to be $F_{\mu\nu}^{I}$, namely
the same as the one in flat space even if the connection is not the LC one.

##  3 Theories with non-dynamical distorsion

Before going to the general characterization of theories with non-dynamical
distorsion it is useful to recall the structure of metric theories.

Einstein’s GR is the leading theory of this type in the low energy limit. Its
action is the standard Einstein-Hilbert one

$S_{\rm EH}=\int d^{4}x\sqrt{-g}\left(\frac{M_{P}^{2}}{2}R-\Lambda\right),$
(3.1)

where $M_{P}$ is the reduced Planck mass and $\Lambda$ is the cosmological
constant. We can also add higher curvature terms to $S_{\rm EH}$,

$\int
d^{4}x\sqrt{-g}\left(a_{2}R^{2}+b_{2}R_{\mu\nu}R^{\mu\nu}+\frac{a_{3}}{M_{P}^{2}}R^{3}+...+\frac{a_{4}}{M_{P}^{4}}(R_{\mu\nu}R^{\mu\nu})^{2}+...\right),$
(3.2)

where the $a_{i}$, $b_{i}$, etc. are freely adjustable dimensionless
coefficients.

Furthermore, we can also add to the theory a generic matter sector with action
$S_{\rm matter}=\int d^{4}x\sqrt{-g}\mathscr{L}_{\rm matter}$, where
$\mathscr{L}_{\rm matter}$ can contain (pseudo)scalar fields $\phi$, fermions
$\psi$ and gauge fields $A_{\mu}^{I}$. Besides the standard renormalizable
terms (which play the leading role in the low energy limit) $\mathscr{L}_{\rm
matter}$ can also contain higher-order terms built with $\phi$, $\psi$ and
$A_{\mu}^{I}$ as well as $g_{\mu\nu}$. All these terms, of course, must be
compatible with the given symmetries (general coordinate invariance, $G$ and
possibly some global symmetries). For example, we can add to $\mathscr{L}_{\rm
matter}$ terms of the form $(F_{\mu\nu}^{I}F^{I\mu\nu})^{2}$,
$({D}_{\mu}\phi{D}^{\mu}\phi)^{3}$, $R{D}_{\mu}\phi{D}^{\mu}\phi$ etc. with,
again, freely adjustable coefficients.

Adopting the EFT point of view, the higher the dimensionality of the generic
added term is the less relevance such term has at low energies. Using the same
reasoning, we do not add non-local terms too, because at sufficiently low
energies any non-locality will be described by a series of local terms.

###  3.1 General characterization

The purpose of this section is to identify the most general class of (local
effective field) theories of the type defined in Sec. 2 where the distorsion
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$ is not
dynamical. These theories are those whose action can be brought into the form

$S_{\rm eq}=\int d^{4}x\sqrt{-g}\left({\cal F}_{\mu\nu\rho\sigma}{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+\Sigma(\Phi,{\cal D}\Phi,C)\right),$ (3.3)

where $\Phi$ represents the set of fields that are independent of
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$, namely

$\Phi=\\{g_{\mu\nu},\phi,\psi,F_{\mu\nu}^{I},...\\},$ (3.4)

the dots are curvatures and covariant derivatives of the previous fields
constructed with the LC connection, ${\cal T}^{\mu\nu\rho\sigma}(\Phi)$ is a
rank-four contravariant tensor that depends on $\Phi$ only (not on its
derivatives) and $\Sigma(\Phi,{\cal D}\Phi,C)$ is a quantity that depends on
$\Phi$ and $C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$
only. Note that ${\cal T}^{\mu\nu\rho\sigma}(\Phi)$ and $\Sigma(\Phi,{\cal
D}\Phi,C)$ should also be invariant under gauge transformations of $G$ and
possibly some global symmetries, if any.

The reason why the distorsion is not dynamical for theories of the form (3.3)
is because the field equations of $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ are purely algebraic in
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$. Indeed,
the derivatives of the distorsion only appear in the first term proportional
to ${\cal T}^{\mu\nu\rho\sigma}$ and they are first derivatives, so, after an
integration by parts it is possible to make them act on ${\cal
T}^{\mu\nu\rho\sigma}$ instead. Therefore, in principle, these equations can
be solved exactly to find $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ as a functional of $\Phi$. Once this is
done, the theory with action $S_{\rm eq}$ can always be written as a metric
theory, whose general form has been described at the beginning of this section
3.

Note that the theory defined in (3.3) is the most general one with non-
dynamical distorsion. The reason is that even setting
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}=0$ one can
recover the most general metric theory: this is because, as we have specified,
the collective field $\Phi$ can also contain curvature tensors and covariant
derivatives of $\phi,\psi,\overline{\psi},F_{\mu\nu}^{I}$ constructed with the
LC connection. If one allows now for a non-vanishing distorsion, one can
anyhow express it in terms of $\Phi$ by using its field equations.

We can thus state that the theories with non-dynamical distorsion are those
whose action is linear in the curvature ${\cal F}_{\mu\nu\leavevmode\nobreak\
\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \rho}$ of the full connection ${\cal
A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$ with the
“coefficients” of the linear terms, i.e. the tensor ${\cal
T}^{\mu\nu\rho\sigma}(\Phi)$, being independent of the distorsion itself. This
class of theories can be regarded as equivalent formulations of the most
general metric theories with the given set of matter fields
$\\{\phi,\psi,A_{\mu}^{I}\\}$ (for examples of equivalent formulations of
specific metric theories see Refs. [22, 23, 24, 25]).

###  3.2 Theories with a falsely-dynamical distorsion

It is important to note that some theories, despite not appearing of the form
(3.3), can be brought into that form with appropriate redefinitions.

To illustrate this point let us consider as an example the case where the
action is

$S=\int d^{4}x\sqrt{-g}\left({\cal F}_{\mu\nu\rho\sigma}{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+\Delta(\Phi,\alpha(\Phi){\cal R}+\beta(\Phi){\cal
R}^{\prime})+\Sigma(\Phi,{\cal D}\Phi)\right),$ (3.5)

with

${\cal
T}^{\mu\nu\rho\sigma}(\Phi)=\alpha(\Phi)g^{\mu\rho}g^{\nu\sigma}+\beta(\Phi)\frac{\epsilon^{\mu\nu\rho\sigma}}{\sqrt{-g}}$
(3.6)

and $\Delta$ is a function of $\Phi$ and the specific combination
$\alpha(\Phi){\cal R}+\beta(\Phi){\cal R}^{\prime}$ only, where $\alpha$ and
$\beta$ are the same functions of $\Phi$ that appear in (3.6). Moreover, we
take $\Phi$ independent of the curvature and covariant derivatives built with
the LC connection and $\Sigma$ independent of ${\cal D}g_{\mu\nu}$; also we
take $\alpha$, $\beta$ and $\Delta$ independent of the metric and impose the
further constraint $1+\frac{\partial\Delta}{\partial z}(\Phi,z)>0$. In this
specific case, using (3.6), the action reads

$S=\int d^{4}x\sqrt{-g}\left(\alpha(\Phi){\cal R}+\beta(\Phi){\cal
R}^{\prime}+\Delta(\Phi,\alpha(\Phi){\cal R}+\beta(\Phi){\cal
R}^{\prime})+\Sigma(\Phi,{\cal D}\Phi)\right).$ (3.7)

Theories of this form actually belong to the class of (3.3) and, therefore,
feature a non-dynamical distorsion. In order to show that we introduce an
auxiliary field $z$ that allows us to write $S$ in the form

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left(\alpha(\Phi)(1+\frac{\partial\Delta}{\partial
z}(\Phi,z)){\cal R}+\beta(\Phi)(1+\frac{\partial\Delta}{\partial
z}(\Phi,z)){\cal R^{\prime}}\right.$ (3.8)
$\displaystyle\left.+\Delta(\Phi,z)-z\frac{\partial\Delta}{\partial
z}(\Phi,z)+\Sigma(\Phi,{\cal D}\Phi)\right).$

The action above is equivalent to the one in (3.7) because of the following
argument. First note that we can impose the condition
$\frac{\partial^{2}\Delta}{\partial z^{2}}\neq 0$ without loss of generality
given that around any point where $\frac{\partial^{2}\Delta}{\partial
z^{2}}=0$ we can write

$\Delta(\Phi,\alpha(\Phi){\cal R}+\beta(\Phi){\cal
R}^{\prime})\simeq\Delta_{0}(\Phi)+\Delta_{1}(\Phi)(\alpha(\Phi){\cal
R}+\beta(\Phi){\cal R}^{\prime})$ (3.9)

and the functions of $\Phi$ that we called here $\Delta_{0}$ and $\Delta_{1}$
can be absorbed in an appropriate redefinition of $\alpha$, $\beta$ and
$\Sigma$. Now, by using the field equation of $z$ computed using the action in
(3.8), we find

$\frac{\partial^{2}\Delta}{\partial z^{2}}(\Phi,z)(\alpha(\Phi){\cal
R}+\beta(\Phi){\cal R}^{\prime}-z)=0,$ (3.10)

which implies, using $\frac{\partial^{2}\Delta}{\partial z^{2}}\neq 0$, that
$z=\alpha(\Phi){\cal R}+\beta(\Phi){\cal R}^{\prime}$. By inserting this
result in (3.8) one recovers exactly (3.7).

The reason why these theories can be brought into the form (3.3) is because we
can absorb the dependence on $z$ in front of both ${\cal R}$ and ${\cal
R^{\prime}}$ in (3.8) through the metric rescaling

$g_{\mu\nu}\to\Omega^{2}g_{\mu\nu},$ (3.11)

where $\Omega^{2}$ depends only algebraically on $z$:

$\Omega^{2}(\Phi,z)=\frac{1}{1+\frac{\partial\Delta}{\partial z}(\Phi,z)}$
(3.12)

(here is where we use $1+\frac{\partial\Delta}{\partial z}(\Phi,z)>0$). After
this metric rescaling the spacetime derivatives of $z$ do not appear because
we do not change at the same time555The spin connection in (2.18) transforms
as ${\cal A}^{ab}_{\mu}\to{\cal
A}^{ab}_{\mu}-\eta^{ab}\partial_{\mu}\ln\Omega$ under the metric rescaling
(3.11) (see [26] for a related study) so that also the covariant derivative of
fermions in (2.20) is invariant. ${\cal A}_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$, $\phi$, $\psi$ and $A_{\mu}^{I}$ and,
as specified, we take $\Phi$ independent of the curvature and covariant
derivatives built with the LC connection and $\Sigma$ independent of ${\cal
D}g_{\mu\nu}$. Therefore, we can easily integrate out $z$ and express it in
terms of the other fields $\Phi$. So in this case $z$ is not dynamical and
there are no other degrees of freedom besides the metric and the matter fields
$\\{\phi,\psi,A_{\mu}^{I}\\}$.

###  3.3 $f({\cal R})$ theories

A particular form of Eq. (3.7) is

$S=\int d^{4}x\sqrt{-g}f({\cal R}),$ (3.13)

where $f$ is a function with $\frac{\partial f}{\partial{\cal R}}>0$ and
$\frac{\partial^{2}f}{\partial{\cal R}^{2}}\neq 0$. Therefore, we obtain that
also $f({\cal R})$ metric-affine theories666For a specific treatment of
$f({\cal R})$ see e.g. Refs. [27, 28, 29]. do not feature a dynamical
distorsion.

Also, as a consequence of the calculations we have performed in Sec. 3.2, the
$f({\cal R})$ metric-affine theories can actually be recast in the GR form
(3.1). Indeed, by defining the function $\Delta$ through

$\alpha{\cal R}+\Delta(\alpha{\cal R})\equiv f({\cal R}),$ (3.14)

where $\alpha$ is an arbitrary positive constant, we obtain (after the metric
rescaling in (3.11) and (3.12))

$S=\int d^{4}x\sqrt{-g}\left(\alpha{\cal
R}+\alpha^{2}\frac{f(\tilde{z})-\tilde{z}\frac{\partial
f}{\partial\tilde{z}}(\tilde{z})}{\frac{\partial
f}{\partial\tilde{z}}(\tilde{z})^{2}}\right),$ (3.15)

where $\tilde{z}\equiv z/\alpha$; the field $\tilde{z}$ is clearly non
dynamical and in principle we can solve its field equation and plug the
solution into the action to obtain

$S=\int d^{4}x\sqrt{-g}\left(\alpha{\cal R}-\Lambda\right),$ (3.16)

where

$\Lambda=\alpha^{2}\frac{\tilde{z}_{0}\frac{\partial
f}{\partial\tilde{z}_{0}}(\tilde{z}_{0})-f(\tilde{z}_{0})}{\frac{\partial
f}{\partial\tilde{z}_{0}}(\tilde{z}_{0})^{2}}$ (3.17)

and $\tilde{z}_{0}$ is a solution of the $\tilde{z}$ field equation. After
that, using the techniques in Appendix A (see also Ref. [25]), we can solve
the field equations of the distorsion and insert the solution into the action
to obtain precisely (3.1), with the identification $\alpha=M_{P}^{2}/2$.

This means, among other things, that $f({\cal R})$ metric-affine theories do
not have any other gravitational degrees of freedom besides the ordinary
graviton (see also [30] for a previous related discussion, and [31] for the
particular case $f({\cal R})\propto{\cal R}^{2}$). Instead, in $f(R)$ metric
theories the gravitational spectrum features, in additional to the ordinary
graviton, a dynamical scalar field: technically this happens because it is not
possible to rescale the metric as in (3.11) without changing the connection in
the metric case (where the connection is the LC one).

##  4 Theories with dynamical distorsion

###  4.1 General characterization

The general form (3.3) of theories with non-dynamical distorsion is useful,
among other things, because it helps us in identifying the class of theories
with a dynamical distorsion: they are those that can never be brought into the
form (3.3). Indeed, in this case kinetic terms for some components of the
distorsion necessarily appear. In general there can be other components of the
distorsion that remain non dynamical: we say that the distorsion is dynamical
when at least some components of this tensor feature kinetic terms.

In the following we provide some examples of (local effective field) theories
that cannot be brought into the form (3.3) and, in simple cases, compute
explicitly the kinetic terms for the dynamical components of the distorsion.

###  4.2 Examples: Dynamical (pseudo)scalarons

As we have seen, the theories with non-dynamical distorsion are those whose
action can be brought into a form that is linear in the curvature of the full
connection with the “coefficients” of the linear terms being independent of
the distorsion itself. Therefore, generically, we can have a dynamical
distorsion by adding terms that are non linear in the curvature. So the first
examples of metric-affine theories with dynamical distorsion that we consider
have actions of the form

$S=\int d^{4}x\sqrt{-g}\left({\cal F}_{\mu\nu\rho\sigma}{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+\Delta(\Phi,{\cal R},{\cal
R}^{\prime})+\Sigma(\Phi,{\cal D}\Phi,C)\right),$ (4.1)

where $\Phi$, ${\cal T}^{\mu\nu\rho\sigma}(\Phi)$ and $\Sigma(\Phi,{\cal
D}\Phi,C)$ have been defined in Sec. (3.1) and $\Delta$ is a function of
$\Phi,{\cal R}$ and ${\cal R}^{\prime}$ only. Note that $\Delta(\Phi,{\cal
R},{\cal R}^{\prime})$ should also be invariant under gauge transformations of
$G$ and the global symmetries, if any. The function $\Delta$ can introduce the
non linearity in the curvature that is crucial to have a dynamical distorsion.
Indeed, barring specific choices of the action, such as those described in
Sec. 3.2, one has dynamical (pseudo)scalar degrees of freedom coming from the
distorsion in this case, as we now show.

Let us start with the case in which $\Delta$ does not depend on ${\cal
R}^{\prime}$, but can have a generic dependence on $\Phi$ and ${\cal R}$. This
case can be treated by introducing one auxiliary scalar field $\zeta$. The
action $S$ can be equivalently written as follows

$S=\int d^{4}x\sqrt{-g}\left({\cal F}_{\mu\nu\rho\sigma}{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+\Delta(\Phi,\zeta)+\frac{\partial\Delta}{\partial\zeta}(\Phi,\zeta)({\cal
R}-\zeta)+\Sigma(\Phi,{\cal D}\Phi,C)\right).$ (4.2)

To show this we observe that the field equation of $\zeta$ is

$({\cal R}-\zeta)\frac{\partial^{2}\Delta}{\partial\zeta^{2}}=0.$ (4.3)

and that we can require without loss of generality
$\frac{\partial^{2}\Delta}{\partial\zeta^{2}}\neq 0$. Indeed, around any point
with $\frac{\partial^{2}\Delta}{\partial\zeta^{2}}=0$ we can have at most a
linear dependence of $\Delta$ on ${\cal R}$, and we can, therefore, absorb
$\Delta$ in a redefinition of ${\cal T}^{\mu\nu\rho\sigma}$ and $\Sigma$ and
go back to the case of non-dynamical distorsion of Sec. 3. From (4.3) it
follows that the field equations fix $\zeta={\cal R}$ and (4.2) reduces to
(4.1). We can now write

$S=\int d^{4}x\sqrt{-g}\left({\cal
F}_{\mu\nu\rho\sigma}{\cal\bar{T}}^{\mu\nu\rho\sigma}(\Phi,\zeta)+\bar{\Sigma}(\Phi,\zeta,{\cal
D}\Phi,C)\right),$ (4.4)

where

$\displaystyle{\cal\bar{T}}^{\mu\nu\rho\sigma}(\Phi,\zeta)$
$\displaystyle\equiv$ $\displaystyle{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+g^{\mu\rho}g^{\nu\sigma}\frac{\partial\Delta}{\partial\zeta}(\Phi,\zeta),$
(4.5) $\displaystyle\bar{\Sigma}(\Phi,\zeta,{\cal D}\Phi,C)$
$\displaystyle\equiv$ $\displaystyle\Sigma(\Phi,{\cal
D}\Phi,C)+\Delta(\Phi,\zeta)-\zeta\frac{\partial\Delta}{\partial\zeta}(\Phi,\zeta).$
(4.6)

Therefore, we have come back to the previously studied case $\Delta=0$, but
with an extra scalar $\zeta$ in addition to the $\phi$ fields we started with.
In deriving the algebraic equations of $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$, derivatives of $\zeta$ generically
appear when we integrate by parts the terms coming from ${\cal
F}_{\mu\nu\rho\sigma}{\cal\bar{T}}^{\mu\nu\rho\sigma}(\Phi,\zeta)$ that
contain one derivative of the variation of $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$, see Eq. (2.7). This fact can produce a
kinetic term for $\zeta$, barring specific choices of the action. An example
of such specific choices is when ${\cal T}^{\mu\nu\rho\sigma}\propto
g^{\mu\rho}g^{\nu\sigma}$ as we have seen in Sec. 3.2.

Whether this new dynamical scalar $\zeta$ is a manifestation of the dynamics
of the distorsion is not clear. This is because ${\cal R}$, which is equal to
$\zeta$ by using the field equations, does not vanish when the distorsion is
zero (see Eq. (2.10)) and so a dynamical $\zeta$ could also correspond just to
an extra dynamical scalar from the metric.

Since this section is devoted to theories with a dynamical distorsion we then
consider the case where $\Delta$ depends on both $\cal R$ and $\cal
R^{\prime}$, but for now only through a linear combination

$\rho\equiv a(\Phi){\cal R}+b(\Phi)\cal R^{\prime}.$ (4.7)

Note that this situation is a generalization of the theories with a falsely-
dynamical distorsion that we have analyzed in Sec. 3.2, where $a=\alpha$,
$b=\beta$ and ${\cal T}^{\mu\nu\rho\sigma}$ was chosen to be of the specific
type (3.6). From the technical point of view this case can be treated
similarly, but, as we will see soon, generically there is one more dynamical
scalar here. Again we introduce an auxiliary field $z$ and we can show that
$S$ can be equivalently written as

$S=\int d^{4}x\sqrt{-g}\left({\cal F}_{\mu\nu\rho\sigma}{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+\Delta(\Phi,z)+\frac{\partial\Delta}{\partial
z}(\Phi,z)(\rho-z)+\Sigma(\Phi,{\cal D}\Phi,C)\right)$ (4.8)

if the non-restrictive condition $\frac{\partial^{2}\Delta}{\partial
z^{2}}\neq 0$ is imposed. At this point we can again write $S$ as in (4.4) but
with different redefined tensors:

$\displaystyle{\cal\bar{T}}^{\mu\nu\rho\sigma}(\Phi,z)$ $\displaystyle\equiv$
$\displaystyle{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+\left(g^{\mu\rho}g^{\nu\sigma}a(\Phi)+\frac{\epsilon^{\mu\nu\rho\sigma}}{\sqrt{-g}}b(\Phi)\right)\frac{\partial\Delta}{\partial
z}(\Phi,z),$ (4.9) $\displaystyle\bar{\Sigma}(\Phi,z,{\cal D}\Phi,C)$
$\displaystyle\equiv$ $\displaystyle\Sigma(\Phi,{\cal
D}\Phi,C)+\Delta(\Phi,z)-z\frac{\partial\Delta}{\partial z}(\Phi,z).$ (4.10)

So also here we have come back to the previously studied case $\Delta=0$, but
with a new scalar $z$. Again, barring specific choices of the action (e.g. the
ones of Sec. 3.2), the kinetic term of $z$ generically emerge when we solve
for the distorsion because of the term ${\cal
F}_{\mu\nu\rho\sigma}{\cal\bar{T}}^{\mu\nu\rho\sigma}(\Phi,\zeta)$, which
contains one derivative of $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$. When the kinetic term appears the
field $z$ shows its dynamical nature, but again it is not clear whether this
dynamics comes from the distorsion or from the metric because, using the field
equations, $z=\rho$ and Eqs. (4.7), (2.10) and (2.11) tell us that a part of
this dynamical field is sourced by the metric and a part is sourced by the
distorsion.

A class of theories where the distorsion is certainly dynamical can be found
by considering the generic case where the dependence of $\Delta$ on $\cal R$
and $\cal R^{\prime}$ is arbitrary. This case can be treated by introducing an
auxiliary scalar field $\zeta$ and an auxiliary pseudoscalar field
$\zeta^{\prime}$. The action can be equivalently written as follows

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left({\cal F}_{\mu\nu\rho\sigma}{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+\Delta(\Phi,\zeta,\zeta^{\prime})\right.$ (4.11)
$\displaystyle\left.+\frac{\partial\Delta}{\partial\zeta}(\Phi,\zeta,\zeta^{\prime})({\cal
R}-\zeta)+\frac{\partial\Delta}{\partial\zeta^{\prime}}(\Phi,\zeta,\zeta^{\prime})({\cal
R^{\prime}}-\zeta^{\prime})+\Sigma(\Phi,{\cal D}\Phi,C)\right).$

To show this we observe that the field equations of $\zeta$ and
$\zeta^{\prime}$ are, respectively,

$\displaystyle({\cal
R}-\zeta)\frac{\partial^{2}\Delta}{\partial\zeta^{2}}+({\cal
R^{\prime}}-\zeta^{\prime})\frac{\partial^{2}\Delta}{\partial\zeta^{\prime}\partial\zeta}$
$\displaystyle=$ $\displaystyle 0$ (4.12) $\displaystyle({\cal
R}-\zeta)\frac{\partial^{2}\Delta}{\partial\zeta^{\prime}\partial\zeta}+({\cal
R^{\prime}}-\zeta^{\prime})\frac{\partial^{2}\Delta}{\partial\zeta^{\prime
2}}$ $\displaystyle=$ $\displaystyle 0.$ (4.13)

Therefore, when the Hessian matrix of $\Delta$ (with respect to the variables
$\zeta$ and $\zeta^{\prime}$) is not singular these field equations imply
${\cal R}=\zeta$ and ${\cal R^{\prime}}=\zeta^{\prime}$ and (4.11) is
equivalent to (4.1). We can always require that the Hessian matrix of $\Delta$
is not singular without loss of generality because around any point where this
matrix is singular $\Delta$ depends at most linearly on a linear combination
of ${\cal R}$ and ${\cal R^{\prime}}$ (with a coefficient independent of the
other linearly independent combination) and we can go back to the previously
analysed cases with a redefinition of ${\cal T}^{\mu\nu\rho\sigma}$. Now we
can again write the action as in (4.4), but with the following redefined
tensors that this time depend on both $\zeta$ and $\zeta^{\prime}$:

$\displaystyle{\cal\bar{T}}^{\mu\nu\rho\sigma}(\Phi,\zeta,\zeta^{\prime})$
$\displaystyle\equiv$ $\displaystyle{\cal
T}^{\mu\nu\rho\sigma}(\Phi)+g^{\mu\rho}g^{\nu\sigma}\frac{\partial\Delta}{\partial\zeta}(\Phi,\zeta,\zeta^{\prime})+\frac{\epsilon^{\mu\nu\rho\sigma}}{\sqrt{-g}}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\Phi,\zeta,\zeta^{\prime}),$
(4.14) $\displaystyle\bar{\Sigma}(\Phi,\zeta,\zeta^{\prime},{\cal D}\Phi,C)$
$\displaystyle\equiv$ $\displaystyle\Sigma(\Phi,{\cal
D}\Phi,C)+\Delta(\Phi,\zeta,\zeta^{\prime})-\zeta\frac{\partial\Delta}{\partial\zeta}(\Phi,\zeta,\zeta^{\prime})-\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\Phi,\zeta,\zeta^{\prime}).$
(4.15)

So, again, we have come back to the previously studied case $\Delta=0$, but
with the new scalars $\zeta$ and $\zeta^{\prime}$ and when we derive the
algebraic equations of $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ derivatives of both $\zeta$ and
$\zeta^{\prime}$ appear in integrating by parts the terms coming from ${\cal
F}_{\mu\nu\rho\sigma}{\cal\bar{T}}^{\mu\nu\rho\sigma}(\Phi,\zeta,\zeta^{\prime})$.
So, generically, both $\zeta$ and $\zeta^{\prime}$ can be dynamical, barring
specific choices of the action777Note that whenever $\zeta$ and
$\zeta^{\prime}$ are non dynamical they can be integrated out and this leads
to an equivalent metric theory, which could have been obtained even without
the $\Delta$ term. .

The fields $\zeta$ and $\zeta^{\prime}$ have a purely geometrical origin. We
refer to them as the scalaron and the pseudoscalaron, respectively. The
pseudoscalaron is particularly interesting for our purposes because it
corresponds to a degree of freedom coming essentially from the distorsion:
using the field equations $\zeta^{\prime}={\cal R^{\prime}}$ and, according to
Eq. (2.11), ${\cal R^{\prime}}$ can be non zero only if the distorsion is not
zero. As discussed above $\zeta$ and $\zeta^{\prime}$ are generically
dynamical, but computing explicitly the corresponding kinetic and interaction
terms is of course very difficult and not very illuminating in the most
general case of (4.1). Therefore, from now on to study the (pseudo)scalaron we
focus on a less general class of theories. We take an action of the form

$S=\int d^{4}x\sqrt{-g}\left(\alpha(\Phi){\cal R}+\beta(\Phi){\cal
R^{\prime}}+\Delta(\Phi,{\cal R},{\cal R^{\prime}})+\Sigma(\Phi,{\cal
D}\Phi)\right),$ (4.16)

where $\alpha$ and $\beta$ are functions of $\Phi$. Also, for simplicity, we
take $\Phi$ independent of the curvature and covariant derivatives built with
the LC connection and $\Sigma$ independent of ${\cal D}g_{\mu\nu}$. This is
clearly a particular case of (4.1).

####  4.2.1 Dynamical pseudoscalaron $\zeta^{\prime}$

Let us now provide explicit examples of the most interesting case where
$\zeta^{\prime}$ is dynamical and explicitly compute its kinetic and potential
terms.

To simplify the calculation of the kinetic and potential terms of
$\zeta^{\prime}$ here we also assume that $\Delta$ is independent of ${\cal
R}$ and that there are no matter fields $\\{\phi,\psi,A_{\mu}^{I}\\}$, so that
we can drop $\Sigma$ and write

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left(\alpha{\cal R}+\beta{\cal R^{\prime}}+\Delta({\cal
R^{\prime}})\right)$ (4.17) $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\alpha{\cal
R}+\left(\beta+\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta^{\prime})\right){\cal
R^{\prime}}+\Delta(\zeta^{\prime})-\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta^{\prime})\right]$

having required, again without loss of generality,
$\frac{\partial^{2}\Delta}{\partial\zeta^{\prime 2}}\neq 0$. The quantities
$\alpha$ and $\beta$ are real parameters here; we will shortly identify
$\alpha=M^{2}_{P}/2$ so we also have to assume $\alpha>0$; the ratio
$M_{P}^{2}/(4\beta)$ is also known as the Barbero-Immirzi parameter. In this
case, unlike those discussed in Sec. 3.2, it is not possible to have the
quantities in front of both ${\cal R}$ and ${\cal R^{\prime}}$ constant after
a metric rescaling and $\zeta^{\prime}$ becomes dynamical. Indeed, by using
(2.10) and (2.11) and integrating out $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ leads to (see Appendix A)

$S=\int d^{4}x\sqrt{-g}\left[\alpha
R-K(\zeta^{\prime})\frac{(\partial\zeta^{\prime})^{2}}{2}-U(\zeta^{\prime})\right],$
(4.18)

where we have defined

$K(\zeta^{\prime})=\frac{24M_{P}^{2}}{1+16B^{2}(\zeta^{\prime})}\left(\frac{\partial
B}{\partial\zeta^{\prime}}\right)^{2},\qquad
B(\zeta^{\prime})=\frac{\beta+\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta^{\prime})}{M_{P}^{2}},\qquad
U(\zeta^{\prime})=\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta^{\prime})-\Delta(\zeta^{\prime}).$
(4.19)

This is a standard Einstein-Hilbert action plus a kinetic and potential terms
for an ordinary matter field. So we have to identify

$\alpha=\frac{M_{P}^{2}}{2}.$ (4.20)

Note that $B$ has to depend non-trivially on $\zeta^{\prime}$ because of
$\frac{\partial^{2}\Delta}{\partial\zeta^{\prime 2}}\neq 0$. The second term
in (4.18) is a kinetic term of $\zeta^{\prime}$, which is therefore dynamical.
Note that $K(\zeta^{\prime})$ is always positive, so $\zeta^{\prime}$ is never
a ghost. We can render the kinetic term of this dynamical scalar canonical
through the field redefinition

$\omega(\zeta^{\prime})=\int_{0}^{\zeta^{\prime}}dx\sqrt{K(x)}.$ (4.21)

Indeed, calling $\zeta^{\prime}(\omega)$ the inverse function, which is
uniquely defined because $\frac{d\omega}{d\zeta^{\prime}}=\sqrt{K}>0$, and
inserting in (4.18) one obtains

$S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R-\frac{(\partial\omega)^{2}}{2}-U(\zeta^{\prime}(\omega))\right].$
(4.22)

The provided examples where $\zeta^{\prime}$ is dynamical are very interesting
because, as mentioned above, ${\cal R^{\prime}}$ is non-vanishing only when
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$ is present;
so in these cases the distorsion has a scalar dynamical component. Given the
relevance of this case we look for a general expression for the mass of
$\zeta^{\prime}$ (defined as the mass of the fluctuations of $\zeta^{\prime}$
around a Lorentz invariant solution). First note that a Lorentz invariant
stationary point of $S$ with respect to $\zeta^{\prime}$ has to be a
stationary point of
$\Delta-\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}$, that is
a solution of

$\zeta^{\prime}\frac{\partial^{2}\Delta}{\partial\zeta^{\prime 2}}=0.$ (4.23)

But $\frac{\partial^{2}\Delta}{\partial\zeta^{\prime 2}}\neq 0$ so the only
Lorentz invariant stationary point is $\zeta^{\prime}=0$. This can be
understood observing that the field equations fix ${\cal
R^{\prime}}=\zeta^{\prime}$ and Lorentz invariance requires
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}=0$, which
implies ${\cal R^{\prime}}=0$ according to Eq. (2.11). Note that Lorentz
invariance also requires
$\Delta-\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}=0$ and so,
using $\zeta^{\prime}=0$, one obtains $\Delta(0)=0$. To compute the mass of
$\zeta^{\prime}$ around $\zeta^{\prime}=0$ we can focus on the part of the
Lagrangian in (4.18) that is quadratic in $\zeta^{\prime}$,

$-\frac{24M_{P}^{2}}{(1+16B^{2}(0))}\left(\frac{\partial
B}{\partial\zeta^{\prime}}(0)\right)^{2}\frac{(\partial\zeta^{\prime})^{2}}{2}-\frac{1}{2}\frac{\partial^{2}\Delta}{\partial\zeta^{\prime
2}}(0)\zeta^{\prime 2}.$ (4.24)

So the squared mass of $\zeta^{\prime}$ is

$m^{2}_{\zeta^{\prime}}=\frac{(1+16B^{2}(0))\frac{\partial^{2}\Delta}{\partial\zeta^{\prime
2}}(0)}{24M_{P}^{2}\left(\frac{\partial
B}{\partial\zeta^{\prime}}(0)\right)^{2}}.$ (4.25)

We observe that $m^{2}_{\zeta^{\prime}}\neq 0$ as a consequence of
$\frac{\partial^{2}\Delta}{\partial\zeta^{\prime 2}}\neq 0$, which also
implies $\frac{\partial B}{\partial\zeta^{\prime}}\neq 0$, so that the
denominator in (4.25) never vanishes. The requirement that $\zeta^{\prime}$ is
not a tachyon leads to the condition
$\frac{\partial^{2}\Delta}{\partial\zeta^{\prime 2}}(0)>0$.

The potential $U(\zeta^{\prime}(\omega))$ can only be explicitly computed once
the function $\Delta$ is specified. Let us consider, for example888The ${\cal
R^{\prime}}^{2}$ term has appeared in different models in the literature, see
e.g. [32, 33, 34]. , $\Delta({\cal R^{\prime}})=c{\cal R^{\prime}}^{2}$ ,
where $c$ is a positive constant (so that $\zeta^{\prime}$ is not a tachyon).
In this case we obtain

$B(\zeta^{\prime})=\frac{\beta+2c\zeta^{\prime}}{M_{P}^{2}},\quad\frac{\partial
B}{\partial\zeta^{\prime}}=\frac{2c}{M_{P}^{2}},\quad
K(\zeta^{\prime})=\frac{96c^{2}}{M_{P}^{2}\left[1+\frac{16(2c\zeta^{\prime}+\beta)^{2}}{M_{P}^{4}}\right]},\quad
U(\zeta^{\prime})=c\zeta^{\prime 2}$ (4.26)

and so

$m^{2}_{\zeta^{\prime}}=\frac{(1+16\beta^{2}/M_{P}^{4})}{48c}M_{P}^{2}>0.$
(4.27)

In this simple quadratic case, by using the expression of $K$ in (4.26) one
obtains

$\omega(\zeta^{\prime})=\sqrt{\frac{3}{2}}M_{P}\left[\tanh^{-1}\left(\frac{4B(\zeta^{\prime})}{\sqrt{1+16B(\zeta^{\prime})^{2}}}\right)-\tanh^{-1}\left(\frac{4\beta}{\sqrt{M_{P}^{4}+16\beta^{2}}}\right)\right].$
(4.28)

By inverting this function one then finds

$\zeta^{\prime}(\omega)=\frac{1}{2c}\left(\frac{M_{P}^{2}\tanh
X(\omega)}{4\sqrt{1-\tanh^{2}X(\omega)}}-\beta\right),$ (4.29)

where

$X(\omega)\equiv\sqrt{\frac{2}{3}}\frac{\omega}{M_{P}}+\tanh^{-1}\left(\frac{4\beta}{\sqrt{16\beta^{2}+M_{P}^{4}}}\right)$
(4.30)

and the potential is

$U(\zeta^{\prime}(\omega))=c\zeta^{\prime}(\omega)^{2}=\frac{1}{4c}\left(\frac{M_{P}^{2}\tanh
X(\omega)}{4\sqrt{1-\tanh^{2}X(\omega)}}-\beta\right)^{2}.$ (4.31)

We see that the condition $c>0$, which ensures $m^{2}_{\zeta^{\prime}}>0$,
also ensures that the potential is bounded from below. The function
$\zeta^{\prime}(\omega)$ at large field values is (using
$(1-\tanh^{2}(x))\exp(2x)\to 4$ as $x\to\infty$)

$\zeta^{\prime}(\omega)=\frac{M_{P}^{2}}{16c}\text{sign}(\omega)\exp\left(\sqrt{\frac{2}{3}}\frac{|\omega|}{M_{P}}\right),\quad(|\omega|\gg
M_{P})$ (4.32)

and one obtains an exponential potential:

$U(\zeta^{\prime}(\omega))=\frac{M_{P}^{4}}{256c}\exp\left(\sqrt{\frac{8}{3}}\frac{|\omega|}{M_{P}}\right),\quad(|\omega|\gg
M_{P}).$ (4.33)

On the other hand, at small field values

$\zeta^{\prime}(\omega)=\frac{m_{\omega}\omega}{\sqrt{2c}},\quad
U(\zeta^{\prime}(\omega))=\frac{m_{\omega}^{2}\omega^{2}}{2}\quad(|\omega|\ll
M_{P}),$ (4.34)

where $m_{\omega}=m_{\zeta^{\prime}}$. For intermediate values of $\omega$ the
potential is shown in Fig. 1. We note that the behavior in the intermediate
region, unlike the one at large field values, depends crucially on the
Barbero-Immirzi parameter. The plots also show the invariance of the potential
under $\\{\omega,\beta\\}\to\\{-\omega,-\beta\\}$ which can be analytically
understood from Eqs. (4.30) and (4.31).

Figure 1: Potential of the canonically normalized pseudoscalaron (for
$\Delta({\cal R^{\prime}})=c{\cal R^{\prime}}^{2}$) multiplied by $c$. Left
plots: positive values of $\beta$. Right plots: negative values of $\beta$.

####  4.2.2 Dynamical combination of $\zeta$ and $\zeta^{\prime}$

In general, for actions of the form (4.16) a combination of $\zeta$ and
$\zeta^{\prime}$ can be dynamical. Although a dynamical combination of $\zeta$
and $\zeta^{\prime}$ is not an unambiguous sign of dynamical distorsion (as
$\zeta$ is sourced not only by the distorsion, but by the metric too, see Eq.
(2.10)), here we explicitly compute the kinetic and potential terms of such
dynamical combination in simple and illuminating cases. We do so in order to
compare them with the most interesting case where the distorsion field
$\zeta^{\prime}$ is clearly dynamical, which we have analyzed in Sec. 4.2.1.

As an example, we first consider the case where $\Delta$ depends on ${\cal R}$
and ${\cal R^{\prime}}$ only through a combination $a(\Phi){\cal
R}+b(\Phi){\cal R^{\prime}}$ that is linearly independent of
$\alpha(\Phi){\cal R}+\beta(\Phi){\cal R^{\prime}}$. This linear independence
is assumed in order not to fall into the cases examined in Sec. 3.2, which
have been proved not to contain extra degrees of freedom besides the metric,
and the matter fields $\\{\phi,\psi,A_{\mu}^{I}\\}$. Let us assume for
simplicity again that these matter fields are absent so that

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}(\alpha{\cal R}+\beta{\cal R^{\prime}}+\Delta(a{\cal R}+b{\cal
R^{\prime}}))$ (4.35) $\displaystyle=\int
d^{4}x\sqrt{-g}\left[\left(\alpha+a\frac{\partial\Delta}{\partial
z}(z)\right){\cal R}+\left(\beta+b\frac{\partial\Delta}{\partial
z}(z)\right){\cal R^{\prime}}+\Delta(z)-z\frac{\partial\Delta}{\partial
z}(z)\right],$

where in the second step we introduced the auxiliary field $z$ and we assumed,
again without loss of generality, $\frac{\partial^{2}\Delta}{\partial
z^{2}}(z)\neq 0$. The two functions in front of ${\cal R}$ and ${\cal
R^{\prime}}$ can only be proportional to each other when
$\frac{\partial\Delta}{\partial z}$ is constant (which is not compatible with
$\frac{\partial^{2}\Delta}{\partial z^{2}}(z)\neq 0$) and/or when $\\{a,b\\}$
and $\\{\alpha,\beta\\}$ are linearly dependent (which has been excluded in
this case). So it is not possible to remove both functions with a rescaling of
the metric $g_{\mu\nu}\to\Omega^{2}g_{\mu\nu}$. We can, however, convert the
function in front of ${\cal R}$ into $M_{P}^{2}/2$ by choosing

$\Omega^{2}(z)=\frac{M_{P}^{2}}{2(\alpha+a\frac{\partial\Delta}{\partial
z}(z))},$ (4.36)

whenever $\alpha+a\frac{\partial\Delta}{\partial z}(z)>0$, which we assume
from now on. After this metric rescaling

$S=\int d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}{\cal R}+M_{P}^{2}B(z){\cal
R^{\prime}}-U(z)\right],$ (4.37)

where

$B(z)=\frac{\beta+b\frac{\partial\Delta}{\partial
z}(z)}{2(\alpha+a\frac{\partial\Delta}{\partial z}(z))},\qquad
U(z)=\frac{M_{P}^{4}(z\frac{\partial\Delta}{\partial
z}(z)-\Delta(z))}{4(\alpha+a\frac{\partial\Delta}{\partial z}(z))^{2}}.$
(4.38)

By using again (2.10) and (2.11) and integrating out
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$ as we did
in Sec. 4.2.1 we obtain

$S=\int d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R-K(z)\frac{(\partial
z)^{2}}{2}-U(z)\right],$ (4.39)

where

$K(z)=\frac{24M_{P}^{2}}{1+16B^{2}(z)}\left(\frac{\partial B}{\partial
z}\right)^{2}.$ (4.40)

It is easy to show that $\frac{\partial B}{\partial z}\neq 0$ when
$\frac{\partial^{2}\Delta}{\partial z^{2}}(z)\neq 0$ and $\\{a,b\\}$ and
$\\{\alpha,\beta\\}$ are linearly independent. So $z$ has a non-vanishing
kinetic term and is thus a dynamical field in this case. Also, $K(z)$ is
always positive, so $z$ is never a ghost. Like we did before, we can render
the kinetic term of this dynamical scalar canonical through the redefinition
$\omega(z)$ in (4.21) and express the action in terms of $\omega$ like we did
in (4.22).

Let us determine now the mass of $z$ (defined as the mass of the fluctuations
of $z$ around a Lorentz invariant solution). By construction on a solution of
the field equation $z=a{\cal R}+b{\cal R^{\prime}}$, as it can be easily
checked from (4.35), so in a Lorentz invariant stationary point $z=0$ (see
Eqs. (2.10) and (2.11)). Note that Lorentz invariance also requires $U(0)=0$
and so, using the second expression in (4.38), also $\Delta(0)=0$ and

$\frac{\partial U}{\partial z}(0)=0\qquad\frac{\partial^{2}U}{\partial
z^{2}}(0)=\frac{M_{P}^{4}\frac{\partial^{2}\Delta}{\partial
z^{2}}(0)}{4\left(\alpha+a\frac{\partial\Delta}{\partial z}(0)\right)^{2}}.$
(4.41)

Expanding the action in (4.39) at quadratic order in $z$ we then easily obtain
the squared mass of $z$:

$m_{z}^{2}=\frac{M_{P}^{2}(1+16B^{2}(0))\frac{\partial^{2}\Delta}{\partial
z^{2}}(0)}{96\left(\alpha+a\frac{\partial\Delta}{\partial
z}(0)\right)^{2}\left(\frac{\partial B}{\partial z}(0)\right)^{2}}.$ (4.42)

Given the assumption we have made, $m_{z}^{2}$ is always finite and non
vanishing. It is also positive for $\frac{\partial^{2}\Delta}{\partial
z^{2}}(0)>0$, which is then the condition in order for $z$ not to be a
tachyon.

The potential of $z$ can only be computed once we specify the function
$\Delta$. As an example, we take now a quadratic function like we did in Sec.
4.2.1, $\Delta(z)=cz^{2}$, where $c$ is a positive constant (so that $z$ is
not a tachyon). In this case we obtain

$\displaystyle B(z)=\frac{\beta+2bcz}{2\alpha+4acz},\quad
U(z)=\frac{cM_{P}^{4}z^{2}}{4(\alpha+2acz)^{2}},$ (4.43)
$\displaystyle\frac{\partial B}{\partial z}=\frac{c(\alpha
b-a\beta)}{(\alpha+2acz)^{2}},\quad K(z)=\frac{24c^{2}M_{P}^{2}(\alpha
b-a\beta)^{2}}{(\alpha+2acz)^{4}\left[1+\frac{4(\beta+2bcz)^{2}}{(\alpha+2acz)^{2}}\right]},\quad
m^{2}_{z}=\frac{M_{P}^{2}\left(\alpha^{2}+4\beta^{2}\right)}{48c(\alpha
b-a\beta)^{2}}>0.$

Note that the quantity $\alpha b-a\beta$ never vanishes because $\\{a,b\\}$
and $\\{\alpha,\beta\\}$ have been assumed to be linearly independent. In this
case the potential $U(z)$ is asymptotically flat at large $z$, unlike the
$U(\zeta^{\prime})$ considered in Sec. 4.2.1 at large $\zeta^{\prime}$.
However, expressing $z$ in terms of $B$ through the first equation in (4.43)
to find $U$ as a function of $B$ we obtain

$U=\frac{M_{P}^{4}(2\alpha B-\beta)^{2}}{16c(\alpha b-a\beta)^{2}},$ (4.44)

which is, surprisingly, the same potential as the one in (4.26) once we
express $\zeta^{\prime}$ in terms of $B$ and we redefine the parameters
appropriately. Given that the kinetic term of $B$ is also the same (see the
first expression in (4.19) and (4.40)) this scalar-tensor theory is precisely
the same as the one of Sec. 4.2.1, which features a dynamical distorsion.

Let us consider now another example. A combination of $\zeta$ and
$\zeta^{\prime}$ can be dynamical for actions of the form (4.16) also when the
Hessian matrix of $\Delta$ (with respect to ${\cal R}$ and ${\cal
R^{\prime}}$) is not singular. This example, as we will see, is a bit more
complicated to analyze, but it can be considered as a more generic case:
$\Delta$ can be expected to depend on both ${\cal R}$ and ${\cal R^{\prime}}$
rather than on a specific linear combination of them. To illustrate how a
kinetic term can emerge we take again the simple case where there are no
matter fields $\\{\phi,\psi,A_{\mu}^{I}\\}$ so that, introducing the two
auxiliary fields $\zeta$ and $\zeta^{\prime}$, we can write

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\left(\alpha+\frac{\partial\Delta}{\partial\zeta}(\zeta,\zeta^{\prime})\right){\cal
R}+\left(\beta+\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta,\zeta^{\prime})\right){\cal
R^{\prime}}\right.$ (4.45)
$\displaystyle\left.+\Delta(\zeta,\zeta^{\prime})-\zeta\frac{\partial\Delta}{\partial\zeta}(\zeta,\zeta^{\prime})-\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta,\zeta^{\prime})\right].$

By performing again a local rescaling of the metric
$g_{\mu\nu}\to\Omega^{2}g_{\mu\nu}$ with

$\Omega^{2}=\frac{M_{P}^{2}}{2\left(\alpha+\frac{\partial\Delta}{\partial\zeta}(\zeta,\zeta^{\prime})\right)}$
(4.46)

(having assumed
$\alpha+\frac{\partial\Delta}{\partial\zeta}(\zeta,\zeta^{\prime})>0$) we
obtain

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}{\cal
R}+M_{P}^{2}B(\zeta,\zeta^{\prime}){\cal
R^{\prime}}-U(\zeta,\zeta^{\prime})\right],$ (4.47)

where this time

$B(\zeta,\zeta^{\prime})=\frac{\beta+\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta,\zeta^{\prime})}{2(\alpha+\frac{\partial\Delta}{\partial\zeta}(\zeta,\zeta^{\prime}))},\quad
U(\zeta,\zeta^{\prime})=\frac{M_{P}^{4}}{4\left(\alpha+\frac{\partial\Delta}{\partial\zeta}(\zeta,\zeta^{\prime})\right)^{2}}\left(\zeta\frac{\partial\Delta}{\partial\zeta}(\zeta,\zeta^{\prime})+\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\zeta,\zeta^{\prime})-\Delta(\zeta,\zeta^{\prime})\right).$

Note that $B$ generically depends on both $\zeta$ and $\zeta^{\prime}$. By
using (2.10) and (2.11) and integrating out $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ as we did in Sec. 4.2.1 we obtain

$S=\int
d^{4}x\sqrt{-g}\left\\{\frac{M_{P}^{2}}{2}R-K(B(\zeta,\zeta^{\prime}))\frac{(\partial
B)^{2}}{2}-U(\zeta,\zeta^{\prime})\right\\},$ (4.48)

where

$K(B)=\frac{24M_{P}^{2}}{(1+16B^{2})}.$ (4.49)

Therefore, the field $B(\zeta,\zeta^{\prime})$ is the dynamical combination of
$\zeta$ and $\zeta^{\prime}$. Since $K(B)$ is always positive, $B$ is never a
ghost.

In order to compute the potential of $B$ we need to integrate out the other
independent combination of $\zeta$ and $\zeta^{\prime}$ that is not dynamical.
We can do so by imposing that $U$ is stationary with respect to variations of
$\zeta$ and $\zeta^{\prime}$ with constant $B(\zeta,\zeta^{\prime})$. Calling
$b$ such constant value, when $\zeta$ is varied $\zeta^{\prime}$ must equal
$\zeta^{\prime}_{b}(\zeta)$, which is the function of $\zeta$ such that
$B(\zeta,\zeta^{\prime}_{b}(\zeta))=b$. Assuming that
$\zeta^{\prime}_{b}(\zeta)$ is a single-valued differentiable function, the
condition that $U$ is stationary with respect to variations of $\zeta$ and
$\zeta^{\prime}$ with constant $B(\zeta,\zeta^{\prime})$ can be expressed as
follows

$\left.\frac{\partial U}{\partial\zeta}+\frac{\partial
U}{\partial\zeta^{\prime}}\frac{d\zeta^{\prime}_{b}}{d\zeta}\right|_{b=B(\zeta,\zeta^{\prime})}=0.$
(4.50)

Imposing this constraint on $\zeta$ and $\zeta^{\prime}$ integrates out the
other non-dynamical scalar and allows us to express $U$ in terms of $B$ only.
The resulting action is

$S=\int d^{4}x\sqrt{-g}\left\\{\frac{M_{P}^{2}}{2}R-K(B)\frac{(\partial
B)^{2}}{2}-U(B)\right\\}.$ (4.51)

Once again, we can render the kinetic term of this dynamical scalar canonical
through the redefinition $\omega(B)$ in (4.21) and express the action in terms
of $\omega$ like we did in (4.22).

We cannot determine explicitly $U(B)$ until we specify the function $\Delta$.
As an example let us consider the case where $\Delta$ is a generic quadratic
function of $\zeta$ and $\zeta^{\prime}$, namely

$\Delta(\zeta,\zeta^{\prime})=c\zeta^{2}+c^{\prime}\zeta^{\prime
2}+c_{m}\zeta\zeta^{\prime},$

whose Hessian matrix is not singular for $4cc^{\prime}\neq c_{m}^{2}$. In this
case

$B(\zeta,\zeta^{\prime})=\frac{\beta+c_{m}\zeta+2c^{\prime}\zeta^{\prime}}{2(\alpha+2c\zeta+c_{m}\zeta^{\prime})},\qquad
U(\zeta,\zeta^{\prime})=\frac{M_{P}^{4}\left(c\zeta^{2}+c^{\prime}\zeta^{\prime
2}+c_{m}\zeta\zeta^{\prime}\right)}{4(\alpha+2c\zeta+c_{m}\zeta^{\prime})^{2}}$
(4.52)

and one finds (for $bc_{m}\neq c^{\prime}$)

$\displaystyle\zeta^{\prime}_{b}(\zeta)$ $\displaystyle=$
$\displaystyle\frac{\beta-2\alpha b+(c_{m}-4bc)\zeta}{2(bc_{m}-c^{\prime})},$
(4.53) $\displaystyle\left.\frac{\partial U}{\partial\zeta}+\frac{\partial
U}{\partial\zeta^{\prime}}\frac{d\zeta^{\prime}_{b}}{d\zeta}\right|_{b=B(\zeta,\zeta^{\prime})}$
$\displaystyle=$
$\displaystyle\frac{M_{P}^{4}\left(4cc^{\prime}-c_{m}^{2}\right)(\alpha\zeta+\beta\zeta^{\prime})}{4(\alpha+2c\zeta+c_{m}\zeta^{\prime})^{2}[2c^{\prime}\alpha-
c_{m}\beta+(4cc^{\prime}-c_{m}^{2})\zeta]}.$ (4.54)

Since the non singularity of the Hessian matrix of $\Delta$ requires
$4cc^{\prime}\neq c_{m}^{2}$, integrating out the non-dynamical scalar through
Eq. (4.50) then gives $\alpha\zeta=-\beta\zeta^{\prime}$. This condition,
together with $B=B(\zeta,\zeta^{\prime})$ allows us to express both $\zeta$
and $\zeta^{\prime}$ in terms of $B$ and the potential of this dynamical
scalar reads

$U(B)=\frac{M_{P}^{4}(2\alpha B-\beta)^{2}}{16\left[\beta(\beta c-\alpha
c_{m})+\alpha^{2}c^{\prime}\right]}.$ (4.55)

Again this is the same potential as the one in (4.26) once we express
$\zeta^{\prime}$ there in terms of $B$ and redefine the parameters
appropriately. Like in the previous example, also the kinetic term of $B$ is
the same (see the first expression in (4.19) and (4.49)) so this scalar-tensor
theory is again precisely the same as the one of Sec. 4.2.1, that features a
dynamical distorsion. We then see that this theory is much more general than
what we could have imagined from the analysis of Sec. 4.2.1.

One can of course find cases where both $\zeta$ and $\zeta^{\prime}$ are
dynamical: for example one can introduce, like in (4.1), a dependence of
$\Sigma$ on $C$, which is not invariant but transforms inhomogeneously under
$g_{\mu\nu}\to\Omega^{2}g_{\mu\nu}$ for a spacetime-dependent $\Omega$. But,
as observed before, it is only $\zeta^{\prime}$ that is directly linked to the
distorsion. Since we are interested in a dynamical distorsion we do not
explore these further possibilities here and leave them for future work.

###  4.3 Examples: Poincaré gauge theories coupled to matter

The distorsion, in the most general case, does not only include scalars and
pseudoscalars, but also higher rank tensors, which, in the most general case
also lead to spin-3, spin-2 and spin-1 particles (see Ref. [12] for a detail
discussion and a summary of previous works). Here we consider the case of
Poincaré gauge theories, also known as Einstein-Cartan theories (see [35, 36]
for detailed reviews): the gravitational fields are represented by the tetrads
and the connection, which, as we have seen in Sec. 2, has to be metric
compatible, i.e. ${\cal D}_{\rho}g_{\mu\nu}=0$. From the physical point of
view this is not a restrictive choice because, as we have seen in section 2,
in order to have fermions it is necessary to introduce the tetrads and have a
metric-compatible connection. In this case the distorsion coincides with what
is known as the contorsion, which can be expressed in terms of the torsion:

$C_{\mu\nu\rho}=\frac{1}{2}(T_{\mu\nu\rho}+T_{\nu\mu\rho}-T_{\mu\rho\nu}),$
(4.56)

which is antisymmetric in the second and third indices. From this equation and
(2.5) we see that the contorsion vanishes if and only if the torsion does. As
we have seen in Sec. 2, the tetrads are defined modulo local Lorentz
transformations, which together with the local translations always present in
any generally covariant theory, leads to local Poncaré symmetry (hence the
name Poincaré gauge theories).

As shown in [37] (see also Refs. [38, 39] for subsequent studies), in Poincaré
gauge theories in the absence of matter fields (i.e. without
$\\{\phi,\psi,A_{\mu}^{I}\\}$) the metric and the connection generically
contain three spin-2 fields (one of which correspond to the ordinary massless
graviton), plus four spin-1 and three spin-0 fields (including the fields
$\zeta$ and $\zeta^{\prime}$ discussed in Sec. 4.2). The spin-3 field present
in the most general case is removed by the condition of metric compatibility.
Subsequently, it was shown that the stability of these theories can only occur
if the additional spin-2 fields (besides the ordinary graviton) are massive at
least in the absence of matter fields [40]. The argument was based on an
expansion of the action at the quadratic level in the fluctuations around the
flat (Minkowski) spacetime.

If one introduces ordinary matter fields $\\{\phi,\psi,A_{\mu}^{I}\\}$ this
result does not change as we now show. To see this let us first introduce some
scalar or pseudoscalar fields $\phi$. Since we want to exclude the presence of
massless spin-2 fields we take these scalars to be massless because otherwise
it would not be possible to construct a quadratic mixing term between them and
the massless components of the contorsion. The only possible independent
scalar or pseudoscalar terms involving the contorsion and $\phi$ at the
quadratic level and with only one derivative are then

$C_{\mu\nu}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mu}\partial^{\nu}\phi,\qquad\epsilon^{\mu\nu\rho\sigma}C_{\mu\nu\rho}\partial_{\sigma}\phi,$
(4.57)

which, of course, can only be constructed with those $\phi$ fields that are
invariant under the gauge group $G$. The terms in (4.57) are mixing terms
between $\phi$ and a vector field $C_{\mu\nu}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mu}$ and a pseudovector field
$\epsilon^{\mu\nu\rho\sigma}C_{\mu\nu\rho}$. So they do not affect the spin-2
sector. Actually the quadratic terms in (4.57) even vanish in the massless
sector as one can always decompose the above mentioned vector and pseudovector
fields into spin-1 fields that are transverse and spin-0 fields whose
d’Alembertian is anyhow zero in the massless case. Non-vanishing scalar or
pseudoscalar terms with more than one derivative cannot be constructed either
as they would unavoidably contain (because $C_{\mu\nu\rho}$ is antisymmetric
in the second and third indices) a d’Alembertian acting on $\phi$, which
vanishes because $\phi$ are massless fields.

Similarly, considering gauge fields, one can construct quadratic terms that
involve both the contorsion and an Abelian gauge field $A_{\mu}$, such as

$C_{\mu\nu\rho}\partial^{\mu}F^{\nu\rho},\quad
C_{\nu\mu\rho}\partial^{\mu}F^{\nu\rho},\quad
C_{\alpha\nu}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\alpha}\partial_{\mu}F^{\mu\nu},\quad\epsilon^{\mu\nu\rho\sigma}C_{\mu\nu\rho}\partial_{\alpha}F^{\alpha}_{\leavevmode\nobreak\
\leavevmode\nobreak\
\sigma},\quad\epsilon^{\mu\nu\rho\sigma}C_{\mu\nu\alpha}\partial^{\alpha}F_{\rho\sigma},\quad...\,,$
(4.58)

where $F_{\mu\nu}$ is the field strength of $A_{\mu}$. But it is always
possible to choose the gauge in a way that fields with a non-vanishing spin
are described by transverse tensors so, recalling that the d’Alembertian of
any massless field vanishes, these terms do not modify the spin-2 sector. Of
course, with fermions it is not possible to construct terms involving
$C_{\mu\nu\alpha}$ that change the quadratic action because fermions always
come in pair.

We conclude that, even in the presence of matter fields, the argument of [40]
holds and the two extra spin-2 fields besides the ordinary graviton must be
massive to have a stable theory.

####  4.3.1 Dark photons from torsion

The vector $v_{\nu}\equiv C_{\alpha\nu}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \alpha}$, and the pseudovector
$p^{\sigma}\equiv\frac{\epsilon^{\mu\nu\rho\sigma}}{\sqrt{-g}}C_{\mu\nu\rho}$,
that we have already discussed in the previous section, contain spin-1
particles, which can play the role of dark photons of gravitational origin.
Dark photons have interesting phenomenology (see e.g. [41, 42]) as they can
act as portals to dark sectors.

Note that after integrating by parts the third and fourth terms in (4.58) one
obtains mixing kinetic terms between the vector $v_{\mu}$ and an Abelian gauge
field $A_{\mu}$ and between the pseudovector $p_{\mu}$ and $A_{\mu}$,

$v_{\mu\nu}F^{\mu\nu},\quad p_{\mu\nu}F^{\mu\nu},$ (4.59)

where

$v_{\mu\nu}\equiv\partial_{\mu}v_{\nu}-\partial_{\nu}v_{\mu},\qquad
p_{\mu\nu}\equiv\partial_{\mu}p_{\nu}-\partial_{\nu}p_{\mu}$ (4.60)

are the field strengths of $v_{\mu}$ and $p_{\mu}$. If $F_{\mu\nu}$ is the
electromagnetic field strength the terms (4.59) are mixing terms between the
photon and the torsion dark photons. These mixing terms give the possibility
of detecting the effect of the dark photons when they are massive [41]. In the
massless case interaction terms between the dark photons and the SM fields are
necessarily higher dimensional (non-renormalizable) operators [43] that might,
however, induce observable effects depending on the size of their
coefficients. Such higher dimensional operators are allowed in our EFT
approach (generically, the couplings of $v_{\mu}$ and $p_{\mu}$ in the metric
theory depends on the initial metric-affine action [44]).

One sees that theories where the connection carries extra degrees of freedom
(besides the metric) generically lead to the existence of (and thus motivate)
dark photons. In total there are two dark photons with negative parity and two
with positive parity: $v_{\mu}$, $p_{\mu}$ and other two spin-1 fields (one
with positive parity and another one with negative parity) that come from the
other independent components of the torsion, as it can be easily shown by
using the results of [37].

One might think that the torsion spin-1 fields cannot couple to the
(pseudo)scalars $\phi$ because the torsion is part of the full connection (and
(pseudo)scalars are invariant under proper orthochronous Poincaré
transformations). However, in the most general Poincaré gauge theory we could
also include these spin-1 fields in the covariant derivative of $\phi$ by
adding to the action appropriate terms: considering, as an example, $v_{\mu}$
such a term would be

$\displaystyle\int d^{4}x\sqrt{-g}\left(-\left[({\cal
D}_{\mu}+iv_{\mu})\phi\right]^{\dagger}({\cal D}^{\mu}+iv^{\mu})\phi+{\cal
D}_{\mu}\phi^{\dagger}{\cal D}^{\mu}\phi\right)$ $\displaystyle\int
d^{4}x\sqrt{-g}\left(iv_{\mu}\left[\phi^{\dagger}{\cal
D}^{\mu}\phi-\left({\cal
D}^{\mu}\phi\right)^{\dagger}\phi\right]-v_{\mu}v^{\mu}\phi^{\dagger}\phi\right)$
(4.61)

and analogous terms for the other spin-1 fields. It is clear that these terms
depend on $\phi$, ${\cal D}_{\mu}\phi$ and $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}$ and can, therefore, be included in a
function like $\Sigma(\Phi,{\cal D}\Phi,C)$ in Eqs. (3.3) and (4.1).

We do not study here cases where the torsion spin-2 fields are dynamical due
to standard difficulties when one attempts an extension to a fully covariant
theory in the presence of additional spin-2 fields besides the graviton, see
e.g. Ref. [40].

####  4.3.2 Coupling the pseudoscalaron to matter

One of the most interesting component of the distorsion, that can be
dynamical, is the pseudoscalaron $\zeta^{\prime}$, which we have discussed in
Sec. 4.2. This field is also present in Poincaré gauge theories, because in
Sec. 4.2 we have not used that ${\cal D}_{\rho}g_{\mu\nu}\neq 0$.

In order to illustrate how the pseudoscalaron couples to a generic matter
sector let us take an action of the form

$S=\int d^{4}x\sqrt{-g}\left[\alpha(\phi){\cal R}+\beta(\phi){\cal
R^{\prime}}+\Delta(\phi,{\cal R^{\prime}})+\Sigma(\Phi,{\cal D}\Phi)\right],$
(4.62)

where $\alpha$, $\beta$ and $\Delta$ are generic functions of the
(pseudo)scalars $\phi$, the function $\Delta$ has an additional dependence on
${\cal R^{\prime}}$, which has been added to introduce the pseudoscalaron (see
Sec. 4.2.1), and

$\displaystyle\Sigma(\Phi,{\cal D}\Phi)$ $\displaystyle=$
$\displaystyle-\frac{{\cal D}_{\mu}\phi_{k}\,{\cal
D}^{\mu}\phi_{k}}{2}-V(\phi)-\frac{1}{4}F_{\mu\nu}^{I}F^{\mu\nu I}$ (4.63)
$\displaystyle+\frac{1}{2}(\bar{\psi}_{j}i\not{\cal
D}\psi_{j}-M_{ij}\psi_{i}\psi_{j}-Y^{k}_{ij}\psi_{i}\psi_{j}\phi_{k}+\hbox{h.c.}),$

represents the matter Lagrangian, where $V$ is the potential. The coefficients
$Y^{a}_{ij}$ and $M_{ij}$ are generic Yukawa couplings and fermion mass
parameters. As usual, since we work with Weyl fermions, $\not{\cal
D}\psi_{j}=\bar{\sigma}^{\mu}{\cal D}_{\mu}\psi_{j}$, where
$\bar{\sigma}^{\mu}=e^{\mu}_{a}\bar{\sigma}^{a}$ and $\bar{\psi}$ represent
the (transpose) hermitian conjugate of $\psi$. All terms are contracted in a
gauge-invariant way with respect to both the local Poincaré group and the
gauge group $G$. This action is clearly a particular case of (4.16). This
form, despite not being the most general one, is suggested by the structure of
the SM (although it also covers, among others, any of its renormalizable
extensions) and by the geometrical interpretation of the torsion as part of
the full connection: in (4.62) we only use the covariant derivative ${\cal D}$
rather than the one, $D$, constructed with the Levi-Civita connection or, in
other words, $\Sigma$ does not explicitly depend on the contorsion
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$. The matter
Lagrangian in (4.63) is general enough to accomodate not only all the SM
fields but also additional fields needed to describe the current evidence of
beyond-the-SM physics (neutrino masses and mixings, dark matter, baryon
asymmetry, etc.).

By performing steps similar to those made around Eq. (4.17), the action in
(4.62) can be equivalently rewritten as follows

$S=\int d^{4}x\sqrt{-g}\left[\alpha(\phi){\cal
R}+M_{P}^{2}B(\phi,\zeta^{\prime}){\cal
R^{\prime}}+\Delta(\phi,\zeta^{\prime})-\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\phi,\zeta^{\prime})+\Sigma(\Phi,{\cal
D}\Phi)\right]$ (4.64)

having required again, without loss of generality,
$\frac{\partial^{2}\Delta}{\partial\zeta^{\prime 2}}\neq 0$. Here the function
$B(\phi,\zeta^{\prime})$ is

$B(\phi,\zeta^{\prime})=\frac{\beta(\phi)+\frac{\partial\Delta}{\partial\zeta^{\prime}}(\phi,\zeta^{\prime})}{M_{P}^{2}}.$
(4.65)

In (4.64) the pseudoscalaron $\zeta^{\prime}$ appears explicitly, but the
other torsion components are not dynamical like in Sec. 4.2.1. We can again
integrate out the torsion by using the method of Appendix A to find

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\alpha(\phi)R-\frac{1}{4}F_{\mu\nu}^{I}F^{\mu\nu
I}-\frac{D_{\mu}\phi_{k}\,D^{\mu}\phi_{k}}{2}-U(\phi,\zeta^{\prime})\right.$
(4.66) $\displaystyle\hskip
62.59596pt+\frac{1}{2}(\bar{\psi}_{j}i\not{D}\psi_{j}-M_{ij}\psi_{i}\psi_{j}-Y^{k}_{ij}\psi_{i}\psi_{j}\phi_{k}+\hbox{h.c.})$
$\displaystyle\hskip
62.59596pt\left.-\frac{\alpha(\phi)V_{\mu}V^{\mu}-\frac{\alpha(\phi)}{4}\partial_{\mu}\alpha(\phi)\partial^{\mu}\alpha(\phi)-2M_{P}^{2}B(\phi,\zeta^{\prime})\partial_{\mu}\alpha(\phi)V^{\mu}}{\frac{2}{3}M_{P}^{4}(B^{2}(\phi,\zeta^{\prime})+\frac{\alpha(\phi)^{2}}{4M_{P}^{4}})}\right],$

where the full potential is

$U(\phi,\zeta^{\prime})=V(\phi)-\Delta(\phi,\zeta^{\prime})+\zeta^{\prime}\frac{\partial\Delta}{\partial\zeta^{\prime}}(\phi,\zeta^{\prime})$
(4.67)

and $V_{\mu}$ is defined by

$V_{\mu}\equiv
M_{P}^{2}\partial_{\mu}B(\phi,\zeta^{\prime})+\frac{1}{8}\bar{\psi}_{j}\bar{\sigma}_{\mu}\psi_{j}.$
(4.68)

Note that $U(\phi,\zeta^{\prime})$ contains some interactions of
$\zeta^{\prime}$ with the $\phi$ fields, e.g. the Higgs. The last line in Eq.
(4.66) contains other interactions of $\zeta^{\prime}$ as well as its kinetic
term, which emerges from the $V_{\mu}V^{\mu}$ term. Note that in this class of
theories the pseudoscalaron interacts with $\phi$ and the fermions $\psi$, but
not with the gauge fields $A^{I}_{\mu}$: this is because the starting action
(4.62) does not feature couplings between the torsion and $A^{I}_{\mu}$. The
pseudoscalaron here interacts with $\phi$ through the function $\alpha$ and
the potential $U$ and also has two- and four-fermion interactions.

A commonly encountered case is
$\alpha(\phi)=M_{P}^{2}/2+\xi_{kl}\phi_{k}\phi_{l}$, where $\xi_{kl}$ are real
coefficients, sometimes called non-minimal couplings. In this case one
recovers the standard Einstein-Hilbert action for gravity at small field
values, when $\alpha\simeq M_{P}^{2}/2$. One can easily compute the
interactions in terms of the $\xi_{kl}$ (including those involving the
pseudoscalaron) by expanding $\alpha(\phi)$ in powers of
$\xi_{kl}\phi_{k}\phi_{l}/M_{P}^{2}$.

##  5 A note on the equivalence principle

In any modification or extension of GR it is natural to ask whether (and to
what extent) the equivalence principle holds. It is particularly interesting
to answer this question in the context of metric-affine theories as these are
gravitational theories constructed starting from the geometrical principle of
general covariance.

Let us first recall what the equivalence principle states: for any fixed
spacetime point $X$, it is possible to choose a reference frame (called
locally inertial frame) where the laws of physics are those without gravity in
a small enough neighbourhood of $X$.

A first thing one may note is that the equivalence principle is ambiguous if
one does not specify what is meant by “the laws of physics without gravity”.
In order to eliminate this ambiguity, given our current description of
fundamental non-gravitational forces, we understand that the physics without
gravity is described by a theory with ordinary matter, such as the one present
in the SM and its common extensions. This can feature (pseudo)scalars, gauge
fields and fermions, which are enough to account for all matter we observe and
address the evidence of beyond-the-SM physics. Massive (pseudo)vector fields,
for example, can be modeled by gauge fields and (pseudo)scalars using the
Stückelberg or Higgs mechanism. Also note that pseudoscalars and pseudovectors
are present in the QCD spectrum and appear in popular SM extensions, such as
those featuring an axion. Therefore, the scalar $\zeta$ and pseudoscalar
$\zeta^{\prime}$, which we defined in Sec. 4.2, as well as the vector
$v_{\mu}$ and pseudovector $p_{\mu}$ encountered in Sec. 4.3.1 are particular
examples of ordinary matter fields. Since, starting from general covariance,
gravity is described by the metric $g_{\mu\nu}$ and the connection ${\cal
A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$, as
discussed in Sec. 2, we conclude that the equivalence principle tells us that
in the locally inertial frame $g_{\mu\nu}(X)=\eta_{\mu\nu}$ and the effect of
${\cal A}_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}(X)$
is indistinguishable from that of such ordinary matter.

Another part of the equivalence principle that calls for a clarification are
the words “small enough”. Following the argument in [18], we interpret them as
the requirement that not only $g_{\mu\nu}(X)=\eta_{\mu\nu}$, but also
$\partial_{\rho}g_{\mu\nu}(X)=0$ in the locally inertial frame. With this
interpretation the equivalence principle also tells us that
$\Gamma_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}(X)=0$
and the effect of $C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\
\,\rho}(X)$ is indistinguishable from that of ordinary matter in the locally
inertial frame. So any physical effect of this $C_{\mu\leavevmode\nobreak\
\sigma}^{\leavevmode\nobreak\ \,\rho}(X)$ that cannot be accounted for by
ordinary matter may be interpreted as a violation of the equivalence principle
(see also Ref. [45] for a related discussion).

It is important to note that a violation of this principle can even occur in a
metric theory, through the presence of higher dimensional terms in the action,
which start to be relevant at high energies. An example is the term $\int
d^{4}x\sqrt{-g}\,RF_{\mu\nu}^{I}F^{I\mu\nu}$: in a spacetime where $R\neq 0$
locally, such as the de Sitter spacetime of cosmological relevance, this term
would lead to an observable modification of electrodynamics due to gravity
even in arbitrarily small neighbourhood of $X$. This is not surprising because
the equivalence principle is a classical local statement but at very small
distances, i.e. at very high energies, we expect quantum gravity effects to
show up and these can lead to higher dimensional terms in the EFT description,
such as the one we have just mentioned. The (classical) equivalence principle
is expected to fail in a quantum gravity framework, while general covariance
can survive [46].

On the other hand, as we have seen in Sec. 4.3, starting from the general
relativity principle, the dynamical components of the distorsion that can be
massless are only spin-1 and spin-0 fields for realistic theories (that must
be stable and feature fermions and whose connection is, therefore, metric
compatible). So at low enough energies the effect of
$C_{\mu\leavevmode\nobreak\ \sigma}^{\leavevmode\nobreak\ \,\rho}$ is
indistinguishable from that of ordinary matter not only at $X$ in the locally
inertial frame, but in any frame and at any point. Furthermore, in the low
energy limit metric-affine theories coupled to spin-0, spin-1/2 and spin-1
fields are described by the Einstein-Hilbert term computed with the LC
connection, Eq. (3.1), plus the renormalizable action of the matter fields
$\\{\phi^{\prime},\psi,A_{\mu}^{{}^{\prime}I}\\}$ (where $\phi^{\prime}$ and
$A_{\mu}^{{}^{\prime}I}$ include $\phi$ and $A_{\mu}^{I}$ plus all spin-0 and
spin-1 massless dynamical fields from the torsion), which do satisfy the
equivalence principle. This result does not change if one also considers other
fields with spin 3/2 or higher than or equal to two: the only massless
particles with spin higher than or equal to two that can interact with gravity
in a Minkowski background are gravitons and massless spin 3/2 particles should
interact exactly as gravitinos in supergravity [47, 48]. But supersymmetry
must be broken at low energies in order for the theory to be realistic and as
soon as this happens the gravitino acquires a mass.

Therefore, we see that, although general covariance does not imply the
equivalence principle at all energies, the latter in general emerges at low
energies from the former in realistic theories.

##  6 Conclusions

We conclude by providing a detailed summary of the new results of this paper
with some further discussions.

  * •

After an introduction and some background material in Secs. 1 and 2, in Sec. 3
we have constructed the most general action of metric-affine EFTs that are
equivalent to metric ones, namely those theories with a non-dynamical
distorsion. We have included a generic matter sector featuring an arbitrary
number of spin-1, spin-1/2 and spin-0 fields. We have pointed out, however,
that in some specific cases the action can be brought in that form with
appropriate redefinitions although it might not look so initially. The bottom
line of that section is that the actions with non-dynamical distorsion are
those that can be recast in a form linear in the curvature of the full
connection with the “coefficients” of the linear terms being independent of
the distorsion itself. This class is very vast and includes as a particular
case, among many others, $f({\cal R})$ theories.

  * •

In Sec. 4 we have studied some examples of theories that have instead a
dynamical distorsion.

We have investigated in detail a vast class where the parity-odd Holst
invariant ${\cal R}^{\prime}$ is a dynamical pseudoscalar field
(pseudoscalaron). This field is supported by the distorsion (it vanishes when
the distorsion does) and can, therefore, be regarded as a genuine distorsion
field. The pseudoscalaron can coexist with a dynamical scalaron ${\cal R}$ and
a generic matter sector. In the simplest cases we have been able to compute
explicitly the pseudoscalaron kinetic term, mass and potential.

In the same section, we have also discussed general Poincaré gauge theories
coupled to matter, where the connection is metric compatible and fermions can
be introduced. We have extended a previous result by Neville in a pure
gravitational theory [40] to the presence of a generic matter sector, showing
that the spin-2 fields from the torsion cannot be massless compatibly with the
stability requirements and thus cannot appear at low enough energies. Also, we
have commented on the possible phenomenology of torsion spin-1 fields, which
can play the role of dark photons. At the end of Sec. 4 we have computed
interactions of the pseudoscalaron with a generic matter sector and, of
course, the metric.

These results can be used in the future to study the role of the
pseudoscalaron in the early and late universe as well as the possible
scattering, production mechanisms and decays of this torsion field.

  * •

Sec. 5 presents a proof that in generic realistic, and thus metric compatible,
metric-affine EFTs the equivalence principle (appropriately defined) always
emerges at low energies, although it is generically violated at high energies.
This was possible by means of the extension of Neville’s result to a general
matter sector, which we presented in Sec. 4.3: the massless dynamical torsion
fields can only have spin 1 or spin 0 and can, therefore, be represented by
ordinary matter fields; so at low enough energies the theory can be described
by the Einstein-Hilbert action complemented by minimally-coupled ordinary
matter fields, which satisfy the equivalence principle.

### Acknowledgments

We thank Emanuele Orazi, Raffaele Savelli and Ilya Shapiro for useful
discussions and Roberto Percacci for useful comments on the preprint. This
work has been partially supported by the grant DyConn from the University of
Rome Tor Vergata.

##  Appendix A Integrating out the distorsion

Let us discuss here how the distorsion can be integrated out (i.e. how to
determine the effective action after solving the distorsion field equations)
for actions of the form (4.64). In the case where the connection is metric
compatible, i.e. ${\cal D}_{\rho}g_{\mu\nu}=0$, this procedure has been
performed in [17] and [33] and we have explicitly checked their results.

We here show that, starting from999If one changes the starting action in a way
that a metric compatible connection is no longer a solution of the connection
field equations, like in e.g. [49], obviously one cannot show the same.
(4.64), the action of the effective metric theory that is obtained by
integrating out the distorsion is the same even if the connection is not
necessarily metric compatible. To this purpose we actually demonstrate
something more: if the field equations of the distorsion admit more than one
solution the effective action obtained by substituting the distorsion with any
solution of its field equations is uniquely determined (i.e. such effective
action does not depend on which solution for the distorsion we choose).

After using (2.10)-(2.11) and performing some integrations by parts,
Lagrangians of the form (4.64) can be written as follows:

$\frac{1}{2}x_{i}A_{ij}x_{j}-J_{i}x_{i}+Q,$ (A.1)

where the $x_{i}$ represent the components of the distorsion and $A_{ij}$,
$J_{i}$ and $Q$ are real distorsion-independent coefficients, which can
depend, however, on the other fields (the metric and the matter fields). We
take $A_{ij}=A_{ji}$ without loss of generality. In the matrix formalism (A.1)
reads

$\frac{1}{2}x^{T}Ax-J^{T}x+Q$ (A.2)

and the field equations of the distorsion are then

$Ax=J.$ (A.3)

This is a standard linear inhomogeneous equation with $A$ and $J$ real and
$A^{T}=A$. If there are eigenvectors $x^{(n)}$ of $A$ with zero eigenvalues,
$Ax^{(n)}=0$, there are solutions of Eq. (A.3) if and only if $J^{T}x^{(n)}=0$
for all $n$. On the other hand, if the $x^{(n)}$ do not exist, i.e. $\det
A\neq 0$, there are no conditions on $J$ for the existence of solutions. Let
us assume now, as we have already mentioned, that there are solutions of
(A.3), so that $J^{T}x^{(n)}=0$ for all $n$ if some $x^{(n)}$ exist. The
general solution of (A.3) can then be written

$x=A^{-1}J+\sum_{n}c_{n}x^{(n)},$ (A.4)

where the $c_{n}$ are arbitrary real coefficients that label all possible
solutions. Note that $A^{-1}J$ is well defined because $J$ is orthogonal to
all $x^{(n)}$. We now plug this solution into the Lagrangian in (A.2) to
obtain

$Q-J^{T}\sum_{n}c_{n}x^{(n)}-\frac{1}{2}J^{T}A^{-1}J=Q-\frac{1}{2}J^{T}A^{-1}J,$
(A.5)

where we used $J^{T}x^{(n)}=0$ for all $n$. We see that if at least a solution
of the field equations of the distorsion exists then the action is uniquely
determined: this is because the dependence on the $c_{n}$ has disappeared.

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