# Classification of generalised higher-order Einstein-Maxwell Lagrangians

Aimeric Colléaux  Astrophysics Research Center, The Open University of Israel,
Ra’anana, Israel  David Langlois  Université Paris Cité, CNRS, Astroparticule
et Cosmologie, F-75013 Paris, France  Karim Noui  Laboratoire de Physique des
deux Infinis IJCLab, Université Paris Saclay, CNRS, France

###### Abstract

We classify all higher-order generalised Einstein-Maxwell Lagrangians that
include terms linear in the curvature tensor and quadratic in the derivatives
of the electromagnetic field strength tensor. Using redundancies due to the
Bianchi identities, dimensionally dependent identities and boundary terms, we
show that a general Lagrangian of this form can always be reduced to a linear
combination of only 21 terms, with coefficients that are arbitrary functions
of the two scalar invariants derived from the field strength. We give an
explicit choice of basis where these 21 terms include 3 terms linear in the
Riemann tensor and 18 terms quadratic in the derivatives of the field
strength.

##  I Introduction

The construction of scalar-tensor theories and vector-tensor theories with
non-minimal coupling to the spacetime curvature in 4 dimensions was explored
systematically by Horndeski almost half a century ago Horndeski:1974wa ;
Horndeski:1976gi . In doing so, he imposed the apparently reasonable
restriction that the equations of motion should be second-order. This
condition was believed, until a few years ago, to be indispensable to avoid
the presence of extra degrees of freedom, known as Ostrogradki ghosts, which
are expected to lead to problematic instabilities111In the point of view of
Effective Field Theories (EFT) approach, these extra degrees of freedom might
not be problematic as their masses could be too large to be reached..

However, it was later realised that imposing second-order equations of motion
turns out to be too restrictive. Indeed, as shown explicitly in the context of
scalar-tensor theories, it is possible to construct Lagrangians that lead to
higher-order (third or fourth order) equations of motion but do not contain
any extra degree of freedom, provided they satisfy some degeneracy conditions
Langlois:2015cwa ; Langlois:2015skt 222See Gleyzes:2014dya ; Gleyzes:2014qga
for earlier works pointing to viable theories beyond the Horndeski family,
although it was not yet realised that degeneracy was the key ingredient to
avoid extra degrees of freedom.. These theories, dubbed DHOST theories in
BenAchour:2016cay , were fully classified up to quadratic order (in second
derivatives of the scalar field) in Langlois:2015cwa  and up to cubic order
later in BenAchour:2016fzp  (see Langlois:2018dxi  and Kobayashi:2019hrl  for
reviews). This type of construction was also considered in the case of vector-
tensor theories of gravity, dubbed generalised Proca theories
Heisenberg:2014rta ; Allys:2015sht ; BeltranJimenez:2016rff , as well as in
higher order extensions of Einstein-Maxwell theory from disformal
transformations Gumrukcuoglu:2019ebp ; DeFelice:2019hxb . In the present work,
we focus our attention on $U(1)$ vector-tensor theories non-minimally coupled
to gravity, including derivatives of the field strength in their Lagrangian.

In the flat spacetime limit, it was recently shown that such Lagrangians
cannot lead to second-order equations of motion Deffayet:2013tca .
Historically, the first theory belonging to this class, i.e. quadratic in
derivatives of the field strength, has been proposed many decades ago by Bopp
and Podolsky in Bopp1 ; Podolsky:1942zz ; Bopp2 ; Podolsky:1944zz  as an
improvement of the UV behaviour of classical electrodynamics. For instance,
the Coulomb potential is non-singular at the origin in this model. More
general Lagrangians quadratic in the derivative of the field strength can also
be found in the context of quantum electrodynamics (QED), where the derivative
expansion of the one-loop effective action, obtained by integrating out the
electron in an external slowly varying electromagnetic field precisely
generate such effective self-interactions, generalising the famous Euler-
Heisenberg action Lee:1989vh ; Cangemi:1994by ; Gusynin:1995bc ;
Gusynin:1998bt ; Navarro-Salas:2020oew ; Karbstein:2021obd .

As far as effective quantum interactions between gravity and electromagnetism
are concerned, the one-(electron)-loop effective action in the presence of
weak external gravitational and electromagnetic fields has famously been found
by Drummond and Hathrel Drummond:1979pp . Its generalisations for strong
external and slowly varying fields have been investigated in
Bastianelli:2007me ; Bastianelli:2008cu  and precisely fall into the family of
Lagrangians we are going to study. Interesting implications of these QED-
induced non-minimal couplings are ranging from superluminality deRham:2020zyh
to gravitational birefringence and their effect on gravitational waves
Ejlli:2018hke ; Ejlli:2020fpt .

In the present article, our goal is to classify all possible Lagrangians that
are linear in the curvature and quadratic in derivatives of the field strength
tensor, without investigating at this stage the presence and nature of
potential extra degrees of freedom. A first exploration of the presence of
extra degrees of freedom will be considered for the flat case in a future
work. Even limiting the Lagrangians to terms linear in the curvature and
quadratic in the field strength derivatives, the number of possible terms
appears enormous. Fortunately, there exists a number of redundancies, which we
exploit systematically in order to drastically reduce the number of terms that
need to be taken into account in a generic Lagrangian. These redundancies
follow from a) the index symmetries of the various possible scalar terms; b)
the Bianchi identities resulting from the $U(1)$ symmetry of the vector field;
c) the Bianchi identities of the Riemann tensor ; d) the dimensional dependent
identities (DDIs); e) the boundary terms.

Taking into account the first four redundancies, which are purely algebraic,
we show that the most general Lagrangian of our interest can be written as a
linear combination of 4 Riemann-linear terms and 22 field-strength-quadratic
terms, whose coefficients are arbitrary functions of the two scalar invariants
constructed from the field-strength tensor. We finally find five independent
boundary terms of the appropriate form, which enables us to further reduce the
number of terms in the minimal Lagrangian to 21.

The structure of this article is the following. In the next section, we
present the general action describing the theories we are interested in,
introduce some useful notations, and explain how we will proceed to eliminate
redundant terms. In section III, we examine the Riemann-dependent terms and
show that they can be reduced to only 4 terms. We then turn, in section IV to
the major part of this work, where we show that all the terms quadratic in the
derivatives of the field strength can be represented by only 22 terms. After
this purely algebraic analysis, we discuss, in section V, how boundary terms
can also reduce the minimal number of terms in the most general Lagrangian: we
identify 5 relevant boundary terms. As summarised in section VI, this means
that the most general Lagrangian can be written as a linear combination of
only 21 terms, up to arbitrary coefficients that are scalar functions of the
two electromagnetic invariants. We conclude in the final section. More
technical aspects on our derivations are provided in an appendix.

##  II Higher-order Einstein-Maxwell theories

We consider the most general family of four-dimensional abelian vector-tensor
theories that are linear in the curvature tensor333Our convention for the
Riemann tensor is such that
$(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})v_{c}=R_{abc}{}^{d}v_{d}$. and
quadratic in the covariant derivative of the Faraday tensor, or field
strength,

$F_{ab}=\nabla_{a}A_{b}-\nabla_{b}A_{a}$ (1)

where $A_{a}$ is the gauge potential vector. The action describing these
theories can be written in the form

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left(L_{0}+\frac{1}{4}A^{abcd}R_{abcd}+B^{abc,def}\nabla_{a}F_{bc}\nabla_{d}F_{ef}\right),$
(2)

where the scalar ${L}_{0}$ and the tensors $A^{abcd}$ and $B^{abc,def}$
contain all possible terms that can be built from combinations of the metric
$g_{ab}$, of the Levi-Civita tensor $\varepsilon_{abcd}$ and of the Faraday
tensor $F_{ab}$. Of course, the tensors $A^{abcd}$ and $B^{abc,def}$ can be
simplified to take into account the symmetries of the Riemann tensor or of the
term quadratic in the covariant derivative of $F_{ab}$: for instance, any term
in $A^{abcd}$ that is symmetric with respect to the indices $a$ and $b$ is
irrelevant since it will disappear after contraction with the Riemann tensor.

Let us first recall that one can construct two scalars from the field strength
tensor (1), which correspond to the two well-known Lorentz invariants
available in standard electromagnetism, namely

$\displaystyle f_{2}\equiv F^{ab}F_{ab}\,,\qquad
f_{*}\equiv{}^{*}\\!F_{ab}\,F^{ab},$ (3)

where

${}^{*}\\!F_{ab}\equiv\frac{1}{2}\varepsilon_{abcd}F^{cd}$ (4)

is the dual field strength.

In order to use compact notations, it will be very useful to define the
tensors $F_{I}^{ab}$ where the index $I$ corresponds to the number of
contracted Faraday tensors, i.e.

$\displaystyle F^{a}_{2\,b}\equiv F^{a}_{\ c}F^{c}_{\ b}\,,\qquad
F^{a}_{3\,b}\equiv F^{a}_{\ c}F^{c}_{\ d}F^{d}_{\ b}\,,\quad{\rm etc}\,,$ (5)

and, for later convenience, we also generalize this notation to include the
metric and the field strength, so that

$F^{a}_{0\,b}\equiv g^{a}_{\ b}\,,\qquad F^{a}_{1\,b}\equiv F^{a}_{\ \ b}\,.$
(6)

It is worth noting that, due to the Cayley-Hamilton relation in four
dimensions, any $F_{K}^{ab}$ with $K\geq 4$ can always be written in terms of
the first four tensors of the tower, namely $F_{I}^{ab}$ with $0\leq I\leq 3$.
Indeed, the identity

$\displaystyle\delta^{a}_{[b}F^{e_{1}}{}_{e_{1}}F^{e_{2}}{}_{e_{2}}F^{e_{3}}{}_{e_{3}}F^{e_{4}}{}_{e_{4}]}=0\,,$
(7)

where the brackets denote antisymmetrisation over the corresponding indices,
leads to the Cayley-Hamilton relation

$F^{a}_{4\,b}+\frac{1}{2}f_{2}F^{a}_{2\,b}-\frac{1}{4}(f_{4}-\frac{1}{2}f_{2}^{2})\delta^{a}_{b}=0\,,$
(8)

where we have introduced

$f_{4}\equiv F_{2}^{ab}F_{2ab}=F^{a}_{4a}\,.$ (9)

This implies that $F^{ab}_{4}$ can be rewritten as a linear combination of
lower weight444The weight, throughout this paper, denotes the number of $F$,
contracted or not. tensors $F^{ab}_{2}$ and $g_{ab}$, with coefficients that
depend on the two scalars $f_{2}$ and $f_{4}$.

By induction, all tensors $F^{ab}_{I}$ with $I$ even can similarly be
expressed as linear combinations of the two tensors $F^{ab}_{2}$ and $g_{ab}$
with coefficients that are functions of the two scalars $f_{2}$ and $f_{4}$.
Moreover, by contracting (8) by $F^{b}_{\ c}$, one finds that $F^{ab}_{5}$ can
be written in terms of $F^{ab}_{3}$ and $F^{ab}$ and of the scalars $f_{2}$
and $f_{4}$. Again by induction, the same applies to any tensor $F^{ab}_{I}$
with $I$ odd.

As mentioned earlier, we allow the Lagrangian terms in (2) to include a
dependence on the Levi-Civita tensor $\varepsilon_{abcd}$. Since

$\varepsilon_{abcd}\,\varepsilon^{efgh}=-\delta^{efgh}_{abcd}\,,$ (10)

where $\delta$ on the right hand side is the generalised Kronecker symbol
(which corresponds to an antisymmetrised product of usual Kronecker symbols),
any tensor that is constructed with an even number of $\varepsilon_{abcd}$,
which we call an even-parity tensor, can be equivalently expressed without
resorting to $\varepsilon_{abcd}$. For odd-parity tensors, i.e. which contains
an odd number of $\varepsilon$, they can be rendered even by multiplication
with any odd scalar, the simplest being $f_{*}$ introduced earlier. For
example, multiplying $f_{*}$ by itself, one gets the even scalar

$f_{*}^{2}=4\left(f_{4}-\frac{1}{2}f_{2}^{2}\right)\equiv 4g_{4}\,.$ (11)

More generally, given an odd scalar $S=\varepsilon_{abcd}S^{abcd}$, where the
tensor $S^{abcd}$ is of even parity, and a generic function $\Phi$, we can
write

$\displaystyle\Phi\left(f_{2},f_{4}\right)S=\widetilde{\Phi}\left(f_{2},f_{4}\right)\delta^{efgh}_{abcd}F_{ef}F_{gh}S^{abcd}\equiv\widetilde{\Phi}\left(f_{2},f_{4}\right)\widetilde{S}$
(12)

where $\widetilde{S}$ is a parity even scalar and555Note that we chose the
negative branch of the square-root.

$\displaystyle\widetilde{\Phi}\left(f_{2},f_{4}\right)=\frac{\Phi\left(f_{2},f_{4}\right)}{4\sqrt{g_{4}}}\,.$
(13)

As a consequence, we can safely ignore any dependence on $\varepsilon_{abcd}$
in the rest of our analysis.

From our discussion above, we conclude that all possible Lagrangian terms can
be written as combinations of the form

$\displaystyle\mathcal{R}_{IJ}\equiv
F^{ab}_{I}F^{cd}_{J}\left(\alpha_{IJ}R_{abcd}+\beta_{IJ}R_{acbd}\right)\;,\;\;\;\;\mathcal{F}_{IJK}=\sum_{i}\gamma^{(i)}_{IJK}F^{ab}_{I}F^{cd}_{J}F^{ef}_{K}\left(\nabla
F\nabla F\right)^{(i)}_{abcdef}\,,\;\;\;$ (14)

where $I,J,K\in\\{0,1,2,3\\}$ and the coefficients $\alpha,\beta,\gamma$ are
arbitrary scalar functions of $f_{2}$ and $f_{4}$. The last term on the right-
hand side of the second relation denotes all possible combinations of the six
free indices in $\nabla F\nabla F$.

Modulo the arbitrary scalar coefficients, the number of terms of the above
form is finite but still remains high. As we will see below, this number can
be further reduced by systematically taking into account various identities.
The first category of identities are the Bianchi identities, both for the
Riemann tensor,

$R_{[abc]d}=0\,,$ (15)

and for the field strength

$\nabla_{[a}F_{bc]}=0\,.$ (16)

Other useful identities are the dimensionally dependent identities (DDIs)
Edgar:2001vv . These identities simply come from the fact that, in a
4-dimensional spacetime, all tensors that are antisymmetric with respect to 5
indices, or more, automatically vanish, so that

$\Lambda^{\dots}_{[a_{1}a_{2}a_{3}a_{4}a_{5}]\dots}=0\,,$ (17)

where the dots stand for possible additional indices, which are arbitrary. The
Cayley-Hamilton identity (8) is one particular example. In this way, one can
construct scalar DDIs by contracting all indices, i.e. writing

$\Lambda^{a_{1}a_{2}a_{3}a_{4}a_{5}}_{[a_{1}a_{2}a_{3}a_{4}a_{5}]}=0\,,$ (18)

where $\Lambda$ is an arbitrary 10-index tensor. Note that any quantity
involving more than five antisymmetrised indices can be reduced to this form
by Laplace’s formula for determinants. Moreover, if the tensor $\Lambda$
contains a Kronecker delta, this would produce an overall $(d-4)$ factor,
making the corresponding identity trivial.

Finally, we will also use the possibility to relate different terms up to
boundary terms, since the latter are irrelevant in the equations of motion.

##  III Riemann-dependent terms

We start by the classification of all the terms that are linear in the Riemann
tensor, of the form

$\mathcal{R}_{IJ}\equiv
F^{ab}_{I}F^{cd}_{J}\left(\alpha_{IJ}R_{abcd}+\beta_{IJ}R_{acbd}\right)\equiv\mathcal{R}^{(\alpha)}_{IJ}+\mathcal{R}^{(\beta)}_{IJ}\,.$
(19)

It can be checked that the sum of the indices $I+J$ has to be even, otherwise
one of the indices, say $I$, is even and the other one is odd, which implies
that $\mathcal{R}_{IJ}$ vanishes due the symmetries of the Riemann tensor
since $F_{I}$ is symmetric and $F_{J}$ antisymmetric. Moreover the terms
$\mathcal{R}^{(\alpha)}_{IJ}$ vanish when either $I$ or $J$ is even, again
because of the symmetries of the Riemann tensor. The non-trivial terms that
remain are thus $\mathcal{R}^{(\beta)}_{00}$, $\mathcal{R}^{(\beta)}_{02}$,
$\mathcal{R}^{(\beta)}_{22}$, $\mathcal{R}_{11}$, $\mathcal{R}_{13}$ and
$\mathcal{R}_{33}$.

Taking into account the first Bianchi identities for the Riemann tensor (15),
we obtain the relation

$\,,\qquad F^{ab}_{I}F^{cd}_{J}\left(R_{abcd}-2R_{acbd}\right)=0\,,\qquad{\rm
for}\ I\ {\rm and}\ J\ {\rm odd}$ (20)

which shows that all the $\mathcal{R}^{(\beta)}_{IJ}$ can be reexpressed as
$\mathcal{R}^{(\alpha)}_{IJ}$ when both $I$ and $J$ are odd. We are now left
with
$\mathcal{R}^{(\beta)}_{00},\mathcal{R}^{(\beta)}_{02},\mathcal{R}^{(\beta)}_{22},\mathcal{R}^{(\alpha)}_{11},\mathcal{R}^{(\alpha)}_{13}$
and $\mathcal{R}^{(\alpha)}_{33}$.

Finally, we investigate whether some of these terms could be related by scalar
DDIs of the form (18), when the tensor $\Lambda$ contains the Riemann tensor
and products of the field strength. Due to the Bianchi identities (15), the
antisymmetrisation must involve at most two indices of the Riemann tensor,
with the other indices coming from the $F_{I}$. In order to reach a total of
10 indices, this means that the tensor $\Lambda$ must contain at least three
$F_{I}$, which leads us to relations of the form666Note that the terms
$F^{a_{1}}_{I[a_{1}}F^{a_{2}}_{Ja_{2}}F^{a_{3}}_{Ka_{3}}F^{a_{4}}_{La_{4}}R^{a_{5}}_{a_{5}]}$
can have the colour indices $(IJKL)=(1111)$ or $(2211)$. In both cases the
result is trivial.

$\displaystyle\mathscr{R}_{IJK}\equiv
F^{a_{1}}_{I[a_{1}}F^{a_{2}}_{Ja_{2}}F^{a_{3}}_{Ka_{3}}\left(\alpha_{IJK}R_{a_{4}a_{5}]}{}^{a_{4}a_{5}}+\beta_{IJK}R^{a_{4}}{}_{a_{4}a_{5}]}{}^{a_{5}}\right)=0\,,$
(21)

where $I,J,K\in\\{1,2,3\\}$ (we do not include the index $0$ since it would
induce a delta function and therefore a trivial identity) and $4\leq I+J+K\leq
6$ (since the terms $\mathcal{R}_{IJ}$ are at most of weight $6$, i.e.
$I+J\leq 6$). Even if three $F_{I}$ appear in the above expressions, the
expansion of (21) and the contraction of all indices necessarily leads to
final terms with at most two $F_{I}$ contracted with the Riemann tensor, i.e.
exactly of the form (19).

Eventually we find only two non-trivial expressions from (21). The first,
corresponding to $(IJK)=(112)$, reads

$\displaystyle
F^{ab}_{2}F^{cd}_{2}R_{acbd}-2F^{ab}F^{cd}_{3}\left(R_{acbd}-R_{abcd}\right)+\frac{1}{2}f_{2}F^{ab}F^{cd}\left(R_{acbd}-R_{abcd}\right)+\frac{1}{4}g_{4}R=0$
(22)

and can be further simplified using (20) to yield

$F^{ab}_{2}F^{cd}_{2}R_{acbd}+2F^{ab}F^{cd}_{3}R_{acbd}-\frac{1}{2}f_{2}F^{ab}F^{cd}R_{acbd}+\frac{1}{4}g_{4}R=0\,.$
(23)

The second relation, corresponding to $(IJK)=(123)$, yields, after
simplification with (20)

$F^{ab}_{3}F^{cd}_{3}R_{acbd}+\frac{1}{2}f_{2}F^{ab}_{2}F^{cd}_{2}R_{acbd}+\frac{1}{4}g_{4}F^{ab}F^{cd}R_{acbd}+\frac{1}{2}g_{4}F^{ab}_{2}R_{ab}=0\,.$
(24)

The second relation tells us that the term $F^{ab}_{3}F^{cd}_{3}R_{acbd}$, of
weight 6, can always be replaced by terms of lower weight, up to some scalar
multiplicative coefficients777Of course, the total weight, when the weight of
the scalar coefficients is taken account, is unchanged., and therefore does
not have to be considered in the Lagrangian. Similarly, the relation (23)
tells us that one of the first two terms can be expressed in terms of the
other one and of lower weight terms.

In summary, all the possible Riemann-linear terms in the action can be written
as linear combinations of only four terms, with scalar coefficients that
depend on $f_{2}$ and $f_{4}$,

$\displaystyle\mathcal{R}_{1}\equiv R\;,\;\;\;\;\mathcal{R}_{2}\equiv
F^{ab}F^{cd}R_{abcd}\;,\;\;\;\;\mathcal{R}_{3}\equiv
F_{2}^{ab}R_{ab}\;,\;\;\;\;\mathcal{R}_{4}\equiv
F^{ab}F_{3}^{cd}R_{abcd}\;\;({\rm or}\;\;F_{2}^{ab}F_{2}^{cd}R_{acbd})\,.$
(25)

We have kept an alternative choice for the last term, but we will see later
that this term can be eliminated in favour of other terms in the Lagrangian,
thanks to boundary terms.

##  IV Quadratic terms in $\nabla F$

The goal of this section is to determine the minimal number of terms of the
form

$B^{abc,def}\nabla_{a}F_{bc}\nabla_{d}F_{ef}\,,$ (26)

taking into account the index symmetries, the Bianchi identities and the
dimensional identities.

For a picturial representation of these terms, one can imagine that each
$\nabla_{e}F_{ab}$ is a three-limb vertex, with a single leg corresponding to
the covariant derivative index $e$ and two arms associated with the indices of
the antisymmetric tensor $F_{ab}$. Each term in (26) thus corresponds to two
vertices whose limbs are all connected, and “dressed”, by “coloured” links
that consist of

$F_{I}^{ab}\quad\text{with}\qquad I\in\\{0,1,2,3\\}\,.$ (27)

In the following, we will call the indices $I$ “colour” indices to distinguish
them from the spacetime indices denotes by a lowercase latin letter.

###  IV.1 Two main families of terms

Among all possible scalar terms, it is useful to distinguish the family where
the two vertices share all three links, family which can be subdivided into
two subfamilies:

${\cal
A}_{IJK}=F_{I}^{ad}F_{J}^{bf}F_{K}^{ec}\nabla_{e}F_{ab}\nabla_{f}F_{cd}$ (28)

and

$\tilde{\cal
A}_{IJK}=F_{I}^{ad}F_{J}^{bc}F_{K}^{ef}\nabla_{e}F_{ab}\nabla_{f}F_{cd}\,.$
(29)

In the first subfamily, each leg is connected to one of the arms of the other
vertex, whereas in the second subfamily, the two vertex legs are
interconnected.

It is immediate to check that the above terms satisfy the following symmetries
under permutation of two colour indices:

${\cal A}_{IJK}=(-1)^{I+J+K}{\cal A}_{IKJ}\,,\qquad\tilde{\cal
A}_{IJK}=\tilde{\cal A}_{JIK}=(-1)^{I+J+K}\tilde{\cal A}_{JIK}\,,$ (30)

which implies that $I+J+K$ must be even to get a nonzero $\tilde{\cal A}$. As
a consequence of these symmetries, one finds 32 independent ${\cal A}$
terms888The nonzero independent terms are described by e.g. $J\leq K$ for $I$
even, which gives $10\times 2=20$ terms and $J<K$ for $I$ odd, which gives
$6\times 2=12$ terms. Hence a total of $32$., and 20 independent $\tilde{\cal
A}$ terms999The nonzero independent $\tilde{\cal A}$ terms are described by
$I\leq J$ with $I+J$ even for $K$ even, which gives $6\times 2=12$ terms, and
$I<J$ with $I+J$ odd for $K$ odd, which gives $4\times 2=8$ terms. Hence a
total of $20$..

For the other family of scalar terms, only one link is shared by the two
vertices, which leads to three subfamilies, depending whether this link
connects two arms, two legs or one leg with an arm, respectively. The
corresponding definitions are given by

${\cal
B}_{IJK}=F_{I}^{ad}F_{J}^{eb}F_{K}^{cf}\nabla_{e}F_{ab}\nabla_{f}F_{cd}$ (31)

and

$\tilde{\cal
B}_{IJK}=F_{I}^{ef}F_{J}^{ab}F_{K}^{cd}\nabla_{e}F_{ab}\nabla_{f}F_{cd}\,,\qquad\hat{\cal
B}_{IJK}=F_{I}^{bf}F_{J}^{ae}F_{K}^{cd}\nabla_{e}F_{ab}\nabla_{f}F_{cd}\,.$
(32)

These new terms satisfy the following symmetries under exchange of the colour
indices,

${\cal B}_{IJK}=(-1)^{I+J+K}{\cal B}_{IKJ},,\qquad\tilde{\cal
B}_{IJK}=(-1)^{I}\tilde{\cal B}_{IKJ}\,.$ (33)

Moreover, by exchanging the two arms of a vertex whenever they are
interconnected, one obtains the following identities,

$\tilde{\cal B}_{IJK}=(-1)^{J+1}\tilde{\cal B}_{IJK}=(-1)^{K+1}\tilde{\cal
B}_{IJK}\,,\qquad\hat{\cal B}_{IJK}=(-1)^{K+1}\tilde{\cal B}_{IJK}\,,$ (34)

which simply state that the colour indices $J$ and $K$ must be odd in
$\tilde{\cal B}_{IJK}$, whereas $K$ must be odd in $\hat{\cal B}_{IJK}$. As a
consequence of these symmetries, we find 32 independent ${\cal B}$ terms
(since they have the same symmetries as the ${\cal A}$ terms), 8 independent
$\tilde{\cal B}$ terms101010For $I$ even, the independent $\tilde{\cal B}$
terms are described by $J\leq K$ , with $J$ and $K$ odd, which gives $2\times
3=6$ terms. For $I$ odd, the independent $\tilde{\cal B}$ terms are described
by $J<K$ , with $J$ and $K$ odd, which gives $2$ terms. Hence a total of $8$.
and 32 independent $\hat{\cal B}$ terms111111Since $K$ is odd, we have
$4\times 4\times 2=32$ terms.. It is easy to convince oneself that all
possible terms can be rewritten into one of the five forms above, up to some
trivial permutation of spacetime indices.

###  IV.2 Bianchi identities

Let us now exploit the Bianchi identities (16) satisfied by the Faraday
tensor, which can be written as

$\nabla_{e}F_{ab}+\nabla_{a}F_{be}+\nabla_{b}F_{ea}=0\,.$ (35)

Replacing $\nabla_{e}F_{ab}$ in the definition (28) of ${\cal A}$ by the above
relation, one obtains the sum of three terms, which are deduced from each
other by a “rotation”, i.e. a circular permutation, of the limbs of the first
vertex. After some straightforward reshuffling of the indices, this leads to
the relation

${\cal A}_{IJK}+{\cal A}_{KJI}-\tilde{\cal A}_{IKJ}=0\,.$ (36)

As a consequence, one finds that all the $\tilde{\cal A}$ terms can be
reexpressed in terms of the ${\cal A}$ terms via the relation

$\tilde{\cal A}_{IJK}={\cal A}_{IKJ}+{\cal A}_{JKI}\,.$ (37)

Moreover, for the indices $(IJK)$ such that $I+J+K$ is odd, we have seen that
$\tilde{\cal A}_{IJK}=0$. One thus obtains additional relations, which provide
10 independent constraints121212One can first consider the cases $I=J$ with
$K$ odd, for which (37) implies ${\cal A}_{IKI}=0$ and thus ${\cal
A}_{IIK}=0$. This yields the 4 constraints ${\cal A}_{001}={\cal
A}_{003}={\cal A}_{113}={\cal A}_{223}=0$. Let us now consider the cases
$I\neq J$. Since all the constraints are symmetric in the exchange of $I$ and
$J$, one can assume $I<J$. If $I+J$ is odd, and $K$ is thus even, one obtains
in principle 6 relations, but 2 are redundant with the previous ones, so only
4 remain: ${\cal A}_{120}={\cal A}_{012}$, ${\cal A}_{302}={\cal A}_{023}$,
${\cal A}_{201}=-{\cal A}_{102}$, ${\cal A}_{212}=0$. Finally, if $I+J$ is
even, and thus $K$ odd, we get 4 relations but 2 are redundant, so only 2
remain: ${\cal A}_{203}=-{\cal A}_{023}$ and ${\cal A}_{313}=0$. between the
${\cal A}$ terms themselves, leaving 22 independent ${\cal A}$ terms.

The Bianchi identities also provide some relations between the 1-limb sharing
terms. Indeed, starting from the ${\cal B}$ term and effecting a “rotation” of
the second vertex, one gets the relation

${\cal B}_{IJK}-(-1)^{K}{\cal B}_{IJK}+(-1)^{J}\hat{\cal B}_{IJK}=0\,.$ (38)

Since $K$ must be odd to have a nonzero $\hat{\cal B}_{IJK}$, this implies
that all the nonzero $\hat{\cal B}$ terms can be expressed as functions of the
${\cal B}$, according to the expression

$\hat{\cal B}_{IJK}=-2(-1)^{J}{\cal B}_{IJK}\quad\text{for}\;\;K\ {\rm
odd}\,.$ (39)

Similarly, a rotation of the first vertex in the expression of $\tilde{\cal
B}_{IJK}$ yields

$\tilde{\cal B}_{IJK}+(-1)^{J}\hat{\cal B}_{IJK}-\hat{\cal B}_{IJK}=0\,.$ (40)

For $J$ even, this relation does not bring anything new, since we already know
that $\tilde{\cal B}_{IJK}=0$ in this case. For $J$ odd, we get $\tilde{\cal
B}_{IJK}=2\hat{\cal B}_{IJK}$, which implies $\tilde{\cal B}_{IJK}=0$ for $K$
even, which we knew already. Finally, in the case where $J$ and $K$ are odd,
we obtain, after substitution of (39), the new relation

$\tilde{\cal B}_{IJK}=4{\cal B}_{IJK}\quad{\rm for}\ J\ {\rm and}\ K\ {\rm
odd}\,,$ (41)

which thus determines all the nonzero $\tilde{\cal B}$ in terms of the ${\cal
B}$.

In summary, taking into account the index symmetries and the Bianchi
identities, we have found that all possible terms of the form (26) can be
expressed as combinations of 22 ${\cal A}$ terms and 32 ${\cal B}$ terms.

###  IV.3 Dimensionally dependent identities

We now consider scalar DDIs of the form (18) that provide relations between
the ${\cal A}$ and ${\cal B}$ families of terms. To obtain such terms, the
tensor $\Lambda$ in (18) must contain two derivatives of $F$ and arbitrary
combinations of $F_{I}$ with possibly internal contractions such that the
whole tensor possesses 10 free indices. In other words, the $\Lambda$ tensor
can be seen as a (tensor) product of two vertices and of several $F_{I}$.

The systematic exploration of all possible $\Lambda$ tensors and corresponding
DDI relations is a bit involved and the details are explained in appendix
LABEL:app_DDI. Many of these identities, especially when the Bianchi
identities are taken into account, turn out to be redundant. We have thus
identified 32 independent DDIs, which can be obtained from just two specific
types of expressions for the $\Lambda$ tensor, which we give here explicitly.

In order to express the $\Lambda$ tensor with ten contravariant indices, we
introduce the partially antisymmetric tensor

$\Delta_{a_{1}a_{2}a_{3}a_{4}a_{5}b_{1}b_{2}b_{3}b_{4}b_{5}}=\delta_{a_{1}a_{2}a_{3}a_{4}a_{5}}^{c_{1}c_{2}c_{3}c_{4}c_{5}}\prod_{n=1}^{5}g_{c_{n}b_{n}},$
(42)

so that the scalar DDIs (18) are now written in the form

$\Lambda^{a_{1}a_{2}a_{3}a_{4}a_{5}b_{1}b_{2}b_{3}b_{4}b_{5}}\Delta_{a_{1}a_{2}a_{3}a_{4}a_{5}b_{1}b_{2}b_{3}b_{4}b_{5}}=0\,.$
(43)

Let us also introduce the “dressed” vertex tensors

$T^{abc}_{IJK}\equiv F_{I}^{ae}F_{J}^{bf}F_{K}^{cg}\nabla_{e}F_{fg}\,,$ (44)

which are useful for the systematic investigation of all possible DDIs, as
discussed in the appendix.

The two families of relevant DDI identities can then be obtained from

$\mathcal{D}_{IJKL}\equiv
T^{a_{1}a_{2}b_{1}}_{00I}T^{b_{2}b_{3}a_{3}}_{JK0}F_{L}^{b_{4}b_{5}}F^{a_{4}a_{5}}\Delta_{a_{1}a_{2}a_{3}a_{4}a_{5}b_{1}b_{2}b_{3}b_{4}b_{5}}$
(45)

where $L=1,3$, and

$\hat{\mathcal{D}}_{IJKL}\equiv
T_{0JI}^{a_{1}b_{1}e}T_{000}^{b_{2}a2f}g_{ef}F_{K}^{b_{3}b_{4}}F^{a_{3}a_{4}}F_{L}^{a_{5}b_{5}}\Delta_{a_{1}a_{2}a_{3}a_{4}a_{5}b_{1}b_{2}b_{3}b_{4}b_{5}}\,,$
(46)

where $L=1,3$, $M=1,2$.

As said above, only 32 DDIs of these classes can cover them all once the
Bianchi identity is taken into account. They can be chosen to be :

$\displaystyle\begin{split}\mathcal{D}_{0001}\,,\;\;\;\;\mathcal{D}_{0003}\,,\;\;\;\;\mathcal{D}_{0021}\,,\;\;\;\;\mathcal{D}_{0101}\,,\;\;\;\;\mathcal{D}_{0201}\,,\;\;\;\;\mathcal{D}_{1001}\,,\;\;\;\;\mathcal{D}_{1021}\,,\;\;\;\;\mathcal{D}_{1101}\,,\\\
\mathcal{D}_{2001}\,,\;\;\;\;\mathcal{D}_{2011}\,,\;\;\;\;\mathcal{D}_{2021}\,,\;\;\;\;\mathcal{D}_{2201}\,,\;\;\;\;\mathcal{D}_{3001}\,,\;\;\;\;\mathcal{D}_{3011}\,,\;\;\;\;\mathcal{D}_{3101}\,,\;\;\;\;\mathcal{D}_{3201}\,,\end{split}$
(47)

and

$\displaystyle\begin{split}\hat{\mathcal{D}}_{0012}\,,\;\;\;\;\hat{\mathcal{D}}_{0111}\,,\;\;\;\;\hat{\mathcal{D}}_{0112}\,,\;\;\;\;\hat{\mathcal{D}}_{0212}\,,\;\;\;\;\hat{\mathcal{D}}_{0311}\,,\;\;\;\;\hat{\mathcal{D}}_{1011}\,,\;\;\;\;\hat{\mathcal{D}}_{1112}\,,\;\;\;\;\hat{\mathcal{D}}_{1211}\,,\\\
\hat{\mathcal{D}}_{2012}\,,\;\;\;\;\hat{\mathcal{D}}_{2111}\,,\;\;\;\;\hat{\mathcal{D}}_{2112}\,,\;\;\;\;\hat{\mathcal{D}}_{2212}\,,\;\;\;\;\hat{\mathcal{D}}_{2311}\,,\;\;\;\;\hat{\mathcal{D}}_{3011}\,,\;\;\;\;\hat{\mathcal{D}}_{3112}\,,\;\;\;\;\hat{\mathcal{D}}_{3211}\,.\end{split}$
(48)

To give an explicit example, the first DDI of the first list above,
$\mathcal{D}_{0001}$, gives the relation

$\displaystyle f_{2}{\cal A}_{000}+4{\cal A}_{002}+2{\cal A}_{011}+4{\cal
A}_{101}+2{\cal A}_{200}-f_{2}\tilde{\cal A}_{000}-2\tilde{\cal
A}_{002}-4\tilde{\cal A}_{020}$ (49) $\displaystyle\qquad-2\tilde{\cal
A}_{110}+f_{2}{\cal B}_{000}+4{\cal B}_{002}+2{\cal B}_{011}-4{\cal
B}_{101}+2{\cal B}_{200}=0\,.$

After simplification using (37), this reduces to

$\displaystyle 2{\cal A}_{011}-4{\cal A}_{002}-2{\cal A}_{200}-f_{2}{\cal
A}_{000}+2{\cal B}_{011}$ (50) $\displaystyle\qquad+4{\cal B}_{002}-4{\cal
B}_{101}+2{\cal B}_{200}+f_{2}{\cal B}_{000}=0\,,$ (51)

which provides a new relation between the ${\cal A}$ and ${\cal B}$ of weight
2.

In summary, the 54 remaining Lagrangian terms quadratic in $\nabla F$, which
include 22 ${\cal A}$ terms and 32 ${\cal B}$ terms can be further reduced to
only 22 “seed” terms, since all the other terms can be recovered from these
ones. The choice of these 22 terms, together with the four Riemann terms
obtained in the previous section, constitutes a choice of basis. There is some
flexibility in the choice of the 22 representative terms, but one cannot
eliminate all the ${\cal A}$ or all the ${\cal B}$.

##  V Boundary terms

We now study whether it is possible to relate some of the 26 terms of the
algebraic basis obtained so far, via boundary terms, since the latter are
irrelevant in the equations of motion. Boundary terms are of the form
$\nabla_{a}B^{a}$ where $B$ is some vector. In our case, we are interested in
boundary terms $\nabla_{a}B^{a}$ that lead, when expanded, to terms quadratic
in $\nabla F$, discussed in the previous section, and therefore $B$ must
contain a single derivative of $F$.

###  V.1 Dressed vertex vectors

Vectors $B$ of this form are proportional to one of the following contracted
dressed vertices:

$V^{a}_{IJ}\equiv F_{I}^{ce}F_{J}^{ad}\nabla_{e}F_{cd}$ (52)

or

$\tilde{V}^{a}_{IJ}\equiv
F_{I}^{dc}F_{J}^{ae}\nabla_{e}F_{cd}=\frac{1}{I+1}F_{J}^{ae}\nabla_{e}(F^{d}_{I+1,d})$
(53)

where the last expression shows that $\tilde{V}_{IJ}$ vanishes if $I$ is even.

Replacing $\nabla_{e}F_{cd}$ in (53) by $\nabla_{[e}F_{cd]}=0$ yields the
relation

$V_{IJ}-(-1)^{I}V_{IJ}+\tilde{V}_{IJ}=0\,,$ (54)

which implies

$\tilde{V}_{IJ}=-2V_{IJ}\quad{\rm for}\ I\,{\rm
odd}\,,\qquad\tilde{V}_{IJ}=0\quad{\rm for}\ I\,{\rm even}\,,$ (55)

where we have omitted the spacetime index for all vector terms above to
simplify the notation. As a consequence, we need to consider only the vectors
$V_{IJ}$ since all the non-zero $\tilde{V}_{IJ}$ can be expressed in terms of
them.

###  V.2 Divergence of the vertex vectors

Let us now examine, for the above vectors, in which case their divergence
leads to only “permissible” terms. In other words, after the full expansion of
the vector $B$, one must find only terms of the form (26) and, for example, no
term of the form $\nabla_{a}\nabla_{b}F_{cd}$ which cannot be reduced to a
Riemann term.

For $I$ odd, the divergence of

$V^{a}_{IJ}=-\frac{1}{2}\tilde{V}^{a}_{IJ}=-\frac{1}{2(I+1)}F_{J}^{ae}\nabla_{e}(F^{d}_{I+1,d})$
(56)

leads to

$\nabla_{a}V^{a}_{IJ}=-\frac{1}{2(I+1)}\left[(\nabla_{a}F_{J}^{ae})\nabla_{e}(F^{d}_{I+1,d})+F_{J}^{ae}\nabla_{a}\nabla_{e}(F^{d}_{I+1,d})\right]\,$
(57)

where the last term on the right-hand side, which is not permissible, vanishes
if $I$ is odd. Therefore, each of the four terms $V_{11}$, $V_{13}$, $V_{31}$
and $V_{33}$ constitutes an appropriate boundary term.

For the other values of $I$ and $J$, it is instructive to expand the
divergence of $V_{IJ}$:

$\displaystyle\nabla_{a}V^{a}_{IJ}=F_{I}^{ce}F_{J}^{ad}\nabla_{a}\nabla_{e}F_{cd}+F_{I}^{ce}(\nabla_{a}F_{J}^{ad})\nabla_{e}F_{cd}+F_{J}^{ad}(\nabla_{a}F_{I}^{ce})\nabla_{e}F_{cd}$
(58)

The first term on the right hand side, which is not permissible, can however
be rewritten as

$F_{I}^{ce}F_{J}^{ad}\nabla_{a}\nabla_{e}F_{cd}=F_{I}^{ce}F_{J}^{ad}[\nabla_{a},\nabla_{e}]F_{cd}+F_{I}^{ce}F_{J}^{ad}\nabla_{e}\nabla_{a}F_{cd}\,.$
(59)

The last term on the right hand side can be slightly rewritten, after
exchanging the index labels $a$ and $e$ and the labels $c$ and $d$, so that
one gets

$F_{I}^{ce}F_{J}^{ad}\nabla_{a}\nabla_{e}F_{cd}=F_{I}^{ce}F_{J}^{ad}[\nabla_{a},\nabla_{e}]F_{cd}-(-1)^{I+J}F_{J}^{ce}F_{I}^{ad}\nabla_{a}\nabla_{e}F_{cd}$
(60)

Reexpressing the commutator of the covariant derivatives in terms of the
Riemann tensor, one finally obtains the relation

$F_{I}^{ce}F_{J}^{ad}\nabla_{a}\nabla_{e}F_{cd}+(-1)^{I+J}F_{J}^{ce}F_{I}^{ad}\nabla_{a}\nabla_{e}F_{cd}=F_{I}^{ce}F_{J}^{ad}(R_{eac}^{\
\ \ f}F_{df}+R_{ead}^{\ \ \ f}F_{fc})$ (61)

This shows that even if a vertex vector with $I$ even is not an acceptable
boundary term, the combination of two such vertices

$W_{IJ}\equiv\tilde{V}_{IJ}+(-1)^{I+J}\tilde{V}_{JI}$ (62)

provides an adequate boundary term. We thus find 7 additional boundary terms:
$W_{00}$, $W_{02}$, $W_{22}$, $W_{01}$, $W_{03}$, $W_{12}$ and $W_{23}$. In
total, we have thus identified 11 boundary terms. However, as we are going to
see below, some of these are redundant.

###  V.3 Supplementary identities

One can find relations between some of the 11 boundary terms obtained above,
as we now show. One such relation comes from the identity

$\nabla_{a}\nabla_{b}\left(\Phi(f_{2},f_{4})F^{ab}\right)=0$ (63)

which implies that the vector

$\nabla_{b}\left(\Phi(f_{2},f_{4})F^{ab}\right)=\Phi\nabla_{b}F^{ab}+2(\partial_{2}\Phi)F^{ab}F^{cd}\nabla_{b}F_{cd}+4(\partial_{4}\Phi)F^{ab}F_{3}^{cd}\nabla_{b}F_{cd}\,,$
(64)

is divergenceless. As a consequence, any boundary term constructed from
$V_{00}^{a}=-\nabla_{b}F^{ab}$ can be considered as redundant since it can be
replaced by other terms, according to the relation

$\nabla_{a}(\Phi
V_{00}^{a})=\nabla_{a}\left[4(\partial_{2}\Phi)V_{11}^{a}+8(\partial_{4}\Phi)V_{31}^{a}\right]\,,$
(65)

where we have used (55) to replace the $\tilde{V}$ terms on the right hand
side by $V$ terms.

Similarly, the identity

$\nabla_{a}\nabla_{b}\left(\Phi(f_{2},f_{4})F_{3}^{ab}\right)=0$ (66)

yields the relation

$\nabla_{a}\left(\Phi V_{02}^{a}+\Phi V_{11}^{a}+\Phi
V_{20}^{a}\right)=\nabla_{a}\left[4(\partial_{2}\Phi)V_{31}^{a}+8(\partial_{4}\Phi)V_{33}^{a}\right)]\,,$
(67)

which implies that one term among the three terms $V_{02}$, $V_{11}$ and
$V_{20}$ is redundant.

###  V.4 DDIs for the boundary terms

Finally, one can also find a few non trivial DDIs that provide additional
relations between the boundary terms. They are given by

$\Delta_{a_{1}a_{2}a_{3}a_{4}a_{5}b_{1}b_{2}b_{3}b_{4}b_{5}}\,\nabla^{b_{2}}\\!\left(\Phi
T^{a_{1}a_{2}b_{1}}_{00I}F^{b_{3}b_{4}}F^{a_{3}a_{4}}F^{b_{5}a_{5}}\right)=0\,.$
(68)

They provide four relations between the $V_{IJ}$ and the $\tilde{V}_{IJ}$.
After the replacement of the $\tilde{V}_{IJ}$ in terms of the $V_{IJ}$, one
gets the following relations

$\displaystyle W_{03}-W_{12}+\frac{1}{2}f_{2}W_{01}=0\,,$ (69) $\displaystyle
V_{13}+V_{31}-\frac{1}{2}W_{22}+\frac{1}{2}f_{2}V_{11}-\frac{1}{8}g_{4}W_{00}=0\,,$
(70) $\displaystyle W_{23}+\frac{1}{4}g_{4}W_{01}=0\,,$ (71) $\displaystyle
V_{33}+\frac{1}{4}f_{2}W_{22}+\frac{1}{4}g_{4}(V_{11}-W_{02})=0\,.$ (72)

In summary, the four above relations combined with the two relations discussed
in the previous subsection means that among the 11 initial boundary terms,
only 5 boundary terms are independent.

##  VI Basis of elementary Lagrangians

Thanks to the results of the previous sections, we are now able to rewrite any
action of the form (2) in terms of a linear combination of 21 terms (not
including $L_{0}$), with coefficients that are arbitrary functions of $f_{2}$
and $f_{4}$. One possible choice of these 21 terms is the following: the
Riemann-linear terms are

$\displaystyle\mathcal{R}_{1}\equiv R\;,\;\;\;\;\mathcal{R}_{2}\equiv
F^{ab}F^{cd}R_{abcd}\;,\;\;\;\;\mathcal{R}_{3}\equiv F_{2}^{ab}R_{ab}\,,$ (73)

while the 18 terms quadratic in $\nabla F$ chosen among the scalars ${\cal
A}_{IJK}$ and ${\cal B}_{IJK}$ and organised according to their weight (i.e.
the sum $I+J+K$), comprise one term of weight 0,

$\displaystyle\mathcal{F}_{1}\equiv{\cal
B}_{000}=\nabla^{b}F_{ab}\nabla^{c}F_{c}{}^{\,a}\,,$ (74)

seven terms of weight 2,

$\displaystyle\begin{array}[]{ll}\mathcal{F}_{2}\equiv{\cal
A}_{002}=F_{2}^{ec}\nabla_{e}F_{ab}\nabla^{b}F_{c}^{\
a}\,,&\quad\mathcal{F}_{3}\equiv{\cal
A}_{200}=F_{2}^{ad}\nabla^{c}F_{ab}\nabla^{b}F_{cd}\,,\\\
\mathcal{F}_{4}\equiv{\cal
B}_{020}=F_{2}^{eb}\nabla_{e}F_{ab}\nabla_{c}F^{ca}\,,&\quad\mathcal{F}_{5}\equiv{\cal
B}_{200}=F_{2}^{ad}\nabla^{b}F_{ab}\nabla^{c}F_{cd}\,,\\\
\mathcal{F}_{6}\equiv{\cal
A}_{101}=F^{ad}F^{ec}\nabla_{e}F_{ab}\nabla^{b}F_{cd}\,,&\quad\mathcal{F}_{7}\equiv{\cal
B}_{011}=F^{eb}F^{cf}\nabla_{e}F_{ab}\nabla_{f}F_{c}^{\ a}\,,\\\
\mathcal{F}_{8}\equiv{\cal
B}_{110}=F^{ad}F^{eb}\nabla_{e}F_{ab}\nabla^{c}F_{cd}\,,&\end{array}$ (79)

one term of weight 3,

$\displaystyle\mathcal{F}_{9}\equiv{\cal
B}_{210}=F_{2}^{ad}F^{eb}\nabla_{e}F_{ab}\nabla^{c}F_{cd}\,,$ (80)

six terms of weight 4,

$\displaystyle\begin{array}[]{ll}\mathcal{F}_{10}\equiv{\cal
A}_{301}=F_{3}^{ad}F^{ec}\nabla_{e}F_{ab}\nabla^{b}F_{cd}\,,&\quad\mathcal{F}_{11}\equiv{\cal
B}_{310}=F_{3}^{ad}F^{eb}\nabla_{e}F_{ab}\nabla^{c}F_{cd}\,,\\\
\mathcal{F}_{12}\equiv{\cal
B}_{022}=F_{2}^{eb}F_{2}^{cf}\nabla_{e}F_{ab}\nabla_{f}F_{c}^{\
a}\,,&\quad\mathcal{F}_{13}\equiv{\cal
B}_{220}=F_{2}^{ad}F_{2}^{eb}\nabla_{e}F_{ab}\nabla^{c}F_{cd}\,,\\\
\mathcal{F}_{14}\equiv{\cal
B}_{112}=F^{ad}F^{eb}F_{2}^{cf}\nabla_{e}F_{ab}\nabla_{f}F_{cd}\,,&\quad\mathcal{F}_{15}\equiv{\cal
B}_{211}=F_{2}^{ad}F^{eb}F^{cf}\nabla_{e}F_{ab}\nabla_{f}F_{cd}\,,\end{array}$
(84)

two terms of weight 5,

$\displaystyle\begin{split}\mathcal{F}_{16}\equiv{\cal
B}_{320}=F_{3}^{ad}F_{2}^{eb}\nabla_{e}F_{ab}\nabla^{c}F_{cd}\,,\quad\mathcal{F}_{17}\equiv{\cal
B}_{212}=F_{2}^{ad}F^{eb}F_{2}^{cf}\nabla_{e}F_{ab}\nabla_{f}F_{cd}\,,\end{split}$
(85)

and one last term of weight 6,

$\displaystyle\mathcal{F}_{18}\equiv{\cal
B}_{312}=F_{3}^{ad}F^{eb}F_{2}^{cf}\nabla_{e}F_{ab}\nabla_{f}F_{cd}\,.$ (86)

Note that the Riemann-linear term $\mathcal{R}_{4}$ does not appear in the
basis. Indeed, using the relations obtained in the previous section from the
boundary terms, this term can be rewritten as a linear combination of the
terms in the above basis. Explicitly, one finds

$\displaystyle\begin{split}\Phi\mathcal{R}_{4}&=\Phi\left(2\mathcal{F}_{6}-\frac{1}{2}g_{4}\mathcal{R}_{1}\right)+f_{2}\,g_{4}\,(\partial_{4}\Phi)\mathcal{R}_{2}+2\lambda_{1}\left(\frac{1}{2}{f_{2}}\,\mathcal{R}_{3}-\mathcal{F}_{2}-\mathcal{F}_{3}+\mathcal{F}_{4}+\mathcal{F}_{5}\right)\\\
&+8f_{2}\,\lambda_{2}\left(2\mathcal{F}_{11}-\mathcal{F}_{13}-\frac{1}{4}{f_{2}\,g_{4}}\mathcal{F}_{1}\right)+8\lambda_{3}\left(\mathcal{F}_{14}-\mathcal{F}_{15}\right)-2(\lambda_{4}-\lambda_{5})\mathcal{F}_{7}+2(\lambda_{4}-2\lambda_{5})\mathcal{F}_{8}\end{split}$
(87)

where

$\displaystyle\begin{split}\lambda_{1}&=\Phi+2g_{4}(\partial_{4}\Phi)\,,\;\;\;\;\lambda_{2}=3(\partial_{4}\Phi)+2g_{4}(\partial_{44}\Phi)\,,\;\;\;\;\lambda_{3}=\partial_{2}\Phi+4f_{2}(\partial_{4}\Phi)+2f_{2}\,g_{4}(\partial_{44}\Phi)\\\
\lambda_{4}&=-\Phi+2f_{2}\left[\partial_{2}\Phi+2g_{4}\partial_{24}\Phi\right]+2\left(g_{4}-2f_{2}^{2}\right)\partial_{4}\Phi\,,\;\;\;\;\lambda_{5}=-\Phi-2f_{2}^{2}\left[3\partial_{4}\Phi+2g_{4}\partial_{44}\Phi\right]\end{split}$
(88)

In flat spacetime the term $\mathcal{R}_{4}$ vanishes (as well as the other
$\mathcal{R}_{i}$) and the above relation implies that one of the
$\mathcal{F}_{i}$, for example $\mathcal{F}_{6}$, becomes redundant. Hence,
any action of the form (2) in Minkowski can be written as a linear combination
of 17 terms, all quadratic in derivatives of $F_{ab}$, with arbitrary
functions of $f_{2}$ and $f_{4}$ as coefficients.

Finally, let us stress that it is straightforward to adapt our analysis to
classify all the Lagrangians of the form (2) where $F_{ab}$ is now an
arbitrary 2-form, i.e. not associated with a $U(1)$ gauge field. The only
difference is that one cannot use anymore the relations obtained from Bianchi
identities (16). The Riemann-linear terms (19) are the same and can be reduced
to a combination of the four terms (25).

As for the terms of the form (26), we again start with 124 terms, as discussed
in section IV.1, and use 88 independent DDIs to reduce these terms to 36
elementary ones. We then find that among the 14 possible boundary terms, only
8 of them are independent when we consider their DDIs and the supplementary
identities. We thus end up with 28 independent elementary quadratic
Lagrangians of the form (26). In conclusion, adding the 4 terms linear in the
Riemann tensor, we get a basis of 32 independent Lagrangians for the most
general action (2) when $F_{ab}$ is an arbitrary 2-form.

##  VII Conclusions

In this article, we have classified (and reduced) higher-order Einstein-
Maxwell theories described by Lagrangians that are linear in the Riemann
tensor and quadratic in derivatives of the field strength tensor. As we have
seen, classifying such Lagrangians turns out to be much more complex than for
scalar-tensor theories. The main reason is that we need to take into account
an important number of dimensionally dependent identities, which renders the
analysis of the equivalence between the Lagrangians, when boundary terms are
also included, much more subtle but also very interesting.

Our results show that the most general higher-order Einstein-Maxwell
Lagrangians considered here can be written as a linear combination of 21 terms
whose coefficients are arbitrary functions of the two scalar invariants
constructed from the field-strength tensor. We also provide an explicit basis.
These results could also be useful to simplify, or more simply to compare,
effective actions that contain terms of the form (2), such as those appearing
in the derivative expansion of the one-loop effective action of QED
Gusynin:1998bt ; Navarro-Salas:2020oew ; Karbstein:2021obd . Furthermore, our
approach to find dimensionally dependent identities could be used in other
contexts where effective actions are used (see e.g. Knorr:2021slg  where the
derivative expansion of gravity has been investigated in the context of
Asymptotic Safety). As mentioned in the previous section, our analysis is
readily extendible to the case where the Lagrangians are seen as functionals
of a 2-form $F_{ab}$, thus relaxing the $U(1)$ gauge symmetry.

At this stage, we have not investigated the physical properties of the
theories (2), in particular the nature and the number of their degrees of
freedom. As the Lagrangians involve second derivatives of the gauge field
$A^{a}$, which in general cannot be eliminated by integrations by part, they
could contain unwanted Ostrogradski ghosts. To avoid these, one must look for
a degenerate family of Lagrangians, satisfying degeneracy conditions analogous
to those that have been identified in DHOST theories, to ensure that the
theory does not propagate Ostrogradski ghosts even though the equations of
motion are higher order.

Indeed, a simple preliminary analysis shows that theories of the form (2)
could propagate up to three Ostrogradski ghosts, in addition to the usual two
degrees of freedom of the photon and the two degrees of freedom of the
graviton. The problem of finding necessary and sufficient degeneracy
conditions to get rid of these ghosts is much more involved than in higher-
derivative scalar-tensor theories for the reason that a gauge field $A^{a}$
has a richer structure (with its four components) than a scalar field. A first
exploration of these aspects will be considered for the flat case in a future
work.

###### Acknowledgements.

This work was supported by the French National Research Agency (ANR) via Grant
No. ANR-22-CE31-0015-01 associated with the project StronG. A.C. was supported
by the Research Authority of the Open University of Israel (grant number
512343).

##  Appendix A Dimensionally dependent identities

In this section we classify dimensionally dependent identities built from the
quadratic combination $\nabla F\nabla F$ and arbitrary products of field
strength in four dimensions, so that our purpose is to identify all the
possible 10-index tensors which can appear in (18). Given that there can be at
most nine field strength together with $\nabla F\nabla F$ in the corresponding
scalar invariants, a rough estimate to enumerate possible DDIs is to choose
ten among 24 indices, resulting in around two million possibilities. Thus, an
other strategy than the brute force calculation should be used in order to
obtain some control over such problems.

###  A.1 Diagrammatic representation

In order to do so, we use a graphical representation of the various tensorial
quantities involved in the scalar invariants of our classifications. The
coloured field strength are represented by

(89)

while for clarity, we recall the definition of the main coloured tensor, given
by

(90)

Although we could introduce an orientation on these graphs in order to encode
the positions of antisymmetrised indices with respect to that of the covariant
derivative, we will refrain to do so, our purpose here being a classification
rather than actual calculations of DDIs.

The previous tensorial quantities are sufficient to define all the DDIs,
however it is quite convenient to construct other coloured tensors from these
building blocks. In particular, we construct and represent the traces of $T$
by

(91) (92)

and we will represent by a spring either cases

(95)

Let us remark that one can easily reconstruct, from these definitions, the
elements of the two families introduced in the section IV.1 as follows,

$\displaystyle g_{ab}V^{a}_{IJ}V^{b}_{KL}=-(-1)^{I+J}{\cal B}_{J+L,IK}\,,$
(96) $\displaystyle
g_{ab}\tilde{V}^{a}_{IJ}\tilde{V}^{b}_{KL}=(-1)^{J}\tilde{\cal B}_{J+L,IK}\,,$
(97) $\displaystyle g_{ab}\tilde{V}^{a}_{IJ}V^{b}_{KL}=-(-1)^{L}\hat{\cal
B}_{J+L,KI}\,,$ (98)

and

$\displaystyle
g_{ae}\,g_{bf}\,g_{cd}T^{abc}_{IJK}T^{def}_{LMN}=(-1)^{I+J+K}{\cal
A}_{J+N,K+L,I+M}\,,$ (99) $\displaystyle
g_{ad}\,g_{bf}\,g_{ce}T^{abc}_{IJK}T^{def}_{LMN}=(-1)^{I+J+K}\tilde{\cal
A}_{J+N,K+L,I+M}\,.$ (100)

This enables to list all the disconnected diagrams representing coloured
$\nabla F\nabla F$, built from tensorial products of the previous quantities :

$\displaystyle\left\\{S^{2},\tilde{V}V,ST,T^{2}\right\\}$ (101)

Before listing connected ones, we will show that if rank two tensors, such as
$\tilde{V}V$ and $S^{2}$, are used to construct DDIs, the result will end up
being redundant compare to the Cayley-Hamilton (CH) identity. In order to do
so, notice that this identity can be obtained from both

$\displaystyle\mathscr{D}^{a}{}_{b}\equiv\delta^{a}_{[b}F^{e_{1}}{}_{e_{1}}F^{e_{2}}{}_{e_{2}}F^{e_{3}}{}_{e_{3}}F^{e_{4}}{}_{e_{4}]}=\delta^{a}_{[b}F_{a_{1}a_{2}}F_{a_{3}a_{4}]}F^{a_{1}a_{2}}F^{a_{3}a_{4}}=0\,.$
(102)

Noticing further that

$\displaystyle\delta^{a}_{[b}F_{a_{1}a_{2}}F_{a_{3}}{}^{a_{3}}F_{a_{4}]}{}^{a_{4}}F^{a_{1}a_{2}}=0$
(103)

in any dimensions, this shows that any rank two DDIs involving products of the
field strength is either trivial or redundant compared to CH, given that any
higher weight can be obtained from $F_{I}^{b}{}_{c}\mathscr{D}^{a}{}_{b}$ and
that we took into consideration the three different ways to antisymmetrise
five indices from products of $F$. It is also possible to check that colouring
each $F$ in the previous equations makes it again redundant compared to CH.
Denoting $H$ an arbitrary rank two tensor (which for our purpose should be
constructed from $\nabla F\nabla F$), the DDIs built from it are divided into
two families. In the first, $H$ is fully antisymmetrised,

$\displaystyle
H_{[a_{1}a_{2}}\left(F_{I}F_{J}F_{K}F_{L}\right)_{a_{3}a_{4}a_{5}]}^{a_{1}a_{2}a_{3}a_{4}a_{5}}=H_{a_{1}a_{2}}\left(F_{I}F_{J}F_{K}F_{L}\right)_{a_{3}a_{4}a_{5}}^{[a_{1}a_{2}a_{3}a_{4}a_{5}]}$
(104)

where the notation with parenthesis means that the indices can be distributed
arbitrarily on the tensor product. We see from the right-hand-side that this
(scalar) DDI reduces to a rank two tensor involving only products of the
coloured field strength, which is redundant with CH, as claimed above.
Similarly, the second DDI is given by

$\displaystyle
H_{[a_{1}}^{a_{1}}\left(F_{I}F_{J}F_{K}F_{L}\right)_{a_{2}a_{3}a_{4}a_{5}]}^{a_{2}a_{3}a_{4}a_{5}}=H_{b_{1}}^{a_{1}}\delta_{[a_{1}}^{b_{1}}\left(F_{I}F_{J}F_{K}F_{L}\right)_{a_{2}a_{3}a_{4}a_{5}]}^{a_{2}a_{3}a_{4}a_{5}},$
(105)

which is redundant as well.

Therefore, we define only the connected diagrams representing coloured $\nabla
F\nabla F$ with ranks higher than two. There are three families of these given
by,

(106) (107) (108)

In summary, only five families of tensors,
$\\{T^{2},ST,\tilde{X},\hat{X},X\\}$, can be used to construct all the
dimensionally dependent identities and we will commonly denote them as $H$, a
tensor built from coloured $\nabla F\nabla F$. In order to consider quantities
involving a pair of five antisymmetrised indices, as required to construct
scalar DDIs in four dimensions, they should be completed by products of the
coloured field strength, which we will call the completions of $H$.

In order to present an explicit classification of DDIs, we now wish to
illustrate how they can be represented using the previous diagrams. Given a
coloured six-indices tensor $H$, we can represent

(109)

It is clear that any completion involving two coloured field strength that is
symmetric exchanging $a_{i}\leftrightarrow b_{i}$ come in two configurations :
the one shown above $F_{A_{1}}^{a_{4}b_{4}}F_{A_{2}}^{a_{5}b_{5}}$ and
$F_{A_{1}}^{a_{4}a_{5}}F_{A_{2}}^{b_{4}b_{5}}$, so that we represent these two
states in a compact way as follows :

(110)

Combining all these diagrams together, the classification presented in Fig.1
can be described as follows. There are two main types of DDIs, depending on
whether the $H$ tensor has six (first three rows) or four indices (last three
rows). The left column indicates the general form of the DDIs, with their most
general completions by coloured field strength and irrespectively of which
specific tensors $H$ is made of. The composition of the $H$ tensor, together
with the specific positions of indices involved in the antisymmetrisations are
filling the rest of the table. As we said above, it is divided into five
families of coloured tensors quadratic in the covariant derivative of the
field strength.

Figure 1: Classification of scalar dimensionally dependent identities
quadratic in $\nabla F$. First column : six types of DDIs with products of
coloured field strength. First three rows : class 1 $\\{T^{2}\\}$. Second
three rows : classes $\\{2,3,4,5\\}$, $\\{ST,\tilde{X},\hat{X},X\\}$.

###  A.2 Useful simplifications

Although the previous DDIs are in principle straightforward to compute, there
is a huge number of possibilities associated with the color indices, which we
call the valency of the DDI. Our purpose now is to establish general results
to reduce these valencies. For instance, in the previous example (109), a
naive estimate would give $4^{8}$ possibilities. However, it is clear that if
$A_{i}=0$ ($i=1,2$), the overall DDI would contain a trivial factor $(d-4)$.
Thus, we have more generally that $A_{i}=1,2,3$ for coloured field strength
entering the DDIs as $F_{A_{i}}^{a_{i}b_{i}}$. Similarly, when fully
antisymmetrised, $F_{A_{i}}^{a_{i}a_{i+1}}$, it is clear that $A_{i}=1,3$.

Another important constraint regards the global sum of the colors for a given
DDI. Given that every (scalar) DDI can be expended in terms of the 124 scalar
invariants classified in (28) and (31), it is clear that the total weight must
satisfy

$\displaystyle 0\leq\sum_{i}I_{i}+\sum_{j}J_{j}+\sum_{k}A_{k}\leq 9$ (111)

where $I_{i}$ ($J_{j}$) are the colors associated with the indices $b_{i}$
($a_{i}$) involved in the tensor $H$ and $A_{k}$ are those involved in the
completion (see the example (109)). Fortunately, other simplifications can be
used.

####  A.2.1 Local color bounds

The first one is the local counterpart of the previous bounds on the colors.
Using a Laplace expansion of the generalized Kronecker delta combines the
colors together, making apparent what colors in the initial diagram are
redundant, given that the only requirement is that each combination of colors
$K$ should have the largest possible valency (the maximal one being $0\leq
K\leq 3$ by the CH identity). For instance, if all the resulting diagrams in
the expansion possess an index with colors $K=I+J$ with $0\leq I\leq 3$ and
$0\leq J\leq 3$, it is clear that one can set $J=0$.

Still using (109) as an example, we can schematically Laplace expand twice the
generalized Kronecker delta in the lower left indices to obtain

(112)

where $\sigma$ (resp. $\rho$) is a permutation of $\\{1,2,3\\}$ (resp.
$\\{1,2\\}$) and the symbol $\approx$ means that we do not consider traces
arising from the completion of $H$, as we will see in the next paragraph, and
that we do not concern ourselves with signs and numerical factors. This
expansion of a four-dimensional DDI into a sum of two-dimensional ones shows
that the colors $J_{i}$ are redundant and can be set to zero, while the colors
$A_{i}$ are also redundant as far as the color valency attached to each index
is concerned. However, as we saw before, $A_{i}\neq 0$ to ensure a non-trivial
result. Thus, the only effect of the values $A_{i}$ is on the symmetry of
$F_{A_{i}}$, but they play no role in these specific diagrams, so that we can
choose its minimal value $A_{i}=1$.

In summary, our example (109) contains at most $4^{3}$ inequivalent DDIs, or
at most $60$ if the global bound (111) is taken into account. The same can be
done for all the six types of $H$ tensors, considering in addition the
different behaviours of the two states (110) in the expansions of generalized
Kronecker deltas.

####  A.2.2 Connected scalar diagrams

Let’s now explain why it is possible to set the traces $f_{2}\approx
f_{4}\approx 0$ in the expansion of generalized Kronecker deltas and
ultimately to find a basis of dimensional identities. In terms of diagrams,
this means that among all the scalar invariants resulting from Laplace
expansion, only those represented by (scalar) connected diagrams can be kept.
More precisely, it is clear that any DDI can be expanded to yield

$\displaystyle\mathcal{S}_{o}=\sum_{2\leq 2(i+2j)\leq o\leq
9}f_{2}^{i}f_{4}^{j}\widehat{\mathcal{S}}_{o-2(i+2j)}^{(i,j)}$ (113)

where $\mathcal{S}_{o}$ is a linear combination with constant coefficients of
the scalar invariants $\\{{\cal A}_{IJK},\tilde{\cal A}_{IJK},{\cal
B}_{IJK},\tilde{\cal B}_{IJK},\hat{\cal B}_{IJK}\\}$, defined by (28) and
(31), while $o=I+J+K$. It is possible to obtain a basis from a set of such
DDIs while assuming $f_{2}\approx f_{4}\approx 0$, unless two inequivalent
DDIs yield twice (113) with two different $\widehat{\mathcal{S}}$. Indeed,
denoting $\widetilde{\mathcal{S}}$ this difference of invariants, we could
construct in that case a DDI given by

$\displaystyle\sum_{2\leq 2(i+2j)\leq o\leq
9}f_{2}^{i}f_{4}^{j}\widetilde{\mathcal{S}}_{o-2(i+2j)}^{(i,j)}=0,$ (114)

which would trivialize assuming $f_{2}\approx f_{4}\approx 0$. Thus such DDI
would be missed making this assumption. However, the unicity of the Laplace
expansions (113) prevents this from happening, making it safe to neglect
quantities involving the traces $f_{2}$ and $f_{4}$, as long as one is not
interested in a precise decomposition of some invariants using DDIs.

Let’s see some examples for lower weights, which we recall, is ranging from 0
to 9. First, it is clear that at least 10 indices are needed to build a four-
dimensional scalar DDI, so $o\geq 2$ in (113). At weight 2, all DDIs have the
form

$\displaystyle\mathcal{S}_{2}=f_{2}\widehat{\mathcal{S}}_{0}^{(1,0)}$ (115)

Thus, (114) would imply the existence of a dimensional identity between weight
scalars, $\widetilde{\mathcal{S}}_{0}=0$, which is impossible as we just said.
The same would happen at weight 3, so we move to $o=4$. At this weight, DDIs
have the form

$\displaystyle\mathcal{S}_{4}=f_{2}\widehat{\mathcal{S}}_{2}+f_{2}^{2}\widehat{\mathcal{S}}_{0}^{(2,0)}+f_{4}\widehat{\mathcal{S}}_{0}^{(0,1)}$
(116)

Assuming that we have used all the lower weight DDIs, given by (115), so that
$\widetilde{\mathcal{S}}_{0}^{(0,1)}\neq 0$, (114) would yield
$f_{2}\widetilde{\mathcal{S}}_{2}+f_{2}^{2}\widetilde{\mathcal{S}}_{0}^{(2,0)}+f_{4}\widetilde{\mathcal{S}}_{0}^{(0,1)}=0$.
Let us now choose a configuration of the two-form field $F$ which fixes one
out of its six components, such that $f_{2}=0$ and $f_{4}\neq 0$. The previous
equation becomes a DDI among weight 0 scalar invariants, which is impossible,
as said above, unless in two dimensions, in which two-forms have only one
independent component. This is not the case here, so again relations such as
(114) are impossible up to weight 4. It is clear that the same would apply for
higher weight terms.

## References

  * (1) G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional space,” Int. J. Theor. Phys. 10 (1974) 363–384.
  * (2) G. W. Horndeski, “Conservation of Charge and the Einstein-Maxwell Field Equations,” J. Math. Phys. 17 (1976) 1980–1987.
  * (3) D. Langlois and K. Noui, “Degenerate higher derivative theories beyond Horndeski: Evading the Ostrogradski instability,” Journal of Cosmology and Astroparticle Physics 2016 (Feb., 2016) 034–034, 1510.06930.
  * (4) D. Langlois and K. Noui, “Hamiltonian analysis of higher derivative scalar-tensor theories,” JCAP 1607 (2016), no. 07 016, 1512.06820.
  * (5) J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Healthy theories beyond Horndeski,” Phys. Rev. Lett. 114 (2015), no. 21 211101, 1404.6495.
  * (6) J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Exploring gravitational theories beyond Horndeski,” JCAP 02 (2015) 018, 1408.1952.
  * (7) J. B. Achour, D. Langlois, and K. Noui, “Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations,” Physical Review D 93 (June, 2016) 124005, 1602.08398.
  * (8) J. B. Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and G. Tasinato, “Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order,” Journal of High Energy Physics 2016 (Dec., 2016) 100, 1608.08135.
  * (9) David Langlois, “Dark Energy and Modified Gravity in Degenerate Higher-Order Scalar-Tensor (DHOST) theories: A review,” International Journal of Modern Physics D 28 (Apr., 2019) 1942006, 1811.06271.
  * (10) T. Kobayashi, “Horndeski theory and beyond: A review,” Reports on Progress in Physics 82 (July, 2019) 086901.
  * (11) L. Heisenberg, “Generalization of the Proca Action,” JCAP 05 (2014) 015, 1402.7026.
  * (12) E. Allys, P. Peter, and Y. Rodriguez, “Generalized Proca action for an Abelian vector field,” JCAP 02 (2016) 004, 1511.03101.
  * (13) J. Beltran Jimenez and L. Heisenberg, “Derivative self-interactions for a massive vector field,” Phys. Lett. B 757 (2016) 405–411, 1602.03410.
  * (14) A. E. Gumrukcuoglu and R. Namba, “Role of matter in gravitation: going beyond the Einstein-Maxwell theory,” Phys. Rev. D 100 (2019) 124064, 1907.12292.
  * (15) A. De Felice and A. Naruko, “On metric transformations with a $U(1)$ gauge field,” Phys. Rev. D 101 (2020), no. 8 084044, 1911.10960.
  * (16) C. Deffayet, A. E. Gümrükçüoğlu, S. Mukohyama, and Y. Wang, “A no-go theorem for generalized vector Galileons on flat spacetime,” JHEP 04 (2014) 082, 1312.6690.
  * (17) F. Bopp, “Eine lineare theorie des elektrons,” Annalen der Physik 430 (1940), no. 5 345–384.
  * (18) B. Podolsky, “A Generalized Electrodynamics Part I-Non-Quantum,” Phys. Rev. 62 (1942) 68–71.
  * (19) F. Bopp, “Lineare theorie des elektrons. ii,” Annalen der Physik 434 (1943), no. 7-8 573–608.
  * (20) B. Podolsky and C. Kikuchi, “A Generalized Electrodynamics Part II-Quantum,” Phys. Rev. 65 (1944) 228–235.
  * (21) H. W. Lee, P. Y. Pac, and H. K. Shin, “Derivative expansions in quantum electrodynamics,” Phys. Rev. D 40 (1989) 4202–4205.
  * (22) D. Cangemi, E. D’Hoker, and G. V. Dunne, “Derivative expansion of the effective action and vacuum instability for QED in (2+1)-dimensions,” Phys. Rev. D 51 (1995) R2513–R2516, hep-th/9409113.
  * (23) V. P. Gusynin and I. A. Shovkovy, “Derivative expansion for the one loop effective Lagrangian in QED,” Can. J. Phys. 74 (1996) 282–289, hep-ph/9509383.
  * (24) V. P. Gusynin and I. A. Shovkovy, “Derivative expansion of the effective action for QED in (2+1)-dimensions and (3+1)-dimensions,” J. Math. Phys. 40 (1999) 5406–5439, hep-th/9804143.
  * (25) J. Navarro-Salas and S. Pla, “$(\mathcal{F},\mathcal{G})$-summed form of the QED effective action,” Phys. Rev. D 103 (2021), no. 8 L081702, 2011.09743.
  * (26) F. Karbstein, “Derivative corrections to the Heisenberg-Euler effective action,” JHEP 09 (2021) 070, 2108.02068.
  * (27) I. T. Drummond and S. J. Hathrell, “QED Vacuum Polarization in a Background Gravitational Field and Its Effect on the Velocity of Photons,” Phys. Rev. D 22 (1980) 343.
  * (28) F. Bastianelli, U. Nucamendi, C. Schubert, and V. M. Villanueva, “Photon-graviton mixing in an electromagnetic field,” J. Phys. A 41 (2008) 164048, 0711.0992.
  * (29) F. Bastianelli, J. M. Davila, and C. Schubert, “Gravitational corrections to the Euler-Heisenberg Lagrangian,” JHEP 03 (2009) 086, 0812.4849.
  * (30) C. de Rham and A. J. Tolley, “Causality in curved spacetimes: The speed of light and gravity,” Phys. Rev. D 102 (2020), no. 8 084048, 2007.01847.
  * (31) D. Ejlli and V. R. Thandlam, “Graviton-photon mixing,” Phys. Rev. D 99 (2019), no. 4 044022, 1807.00171.
  * (32) D. Ejlli, “Graviton-photon mixing. Exact solution in a constant magnetic field,” JHEP 06 (2020) 029, 2004.02714.
  * (33) S. B. Edgar and A. Hoglund, “Dimensionally dependent tensor identities by double antisymmetrization,” J. Math. Phys. 43 (2002) 659–677, gr-qc/0105066.
  * (34) B. Knorr, “The derivative expansion in asymptotically safe quantum gravity: general setup and quartic order,” SciPost Phys. Core 4 (2021) 020, 2104.11336.