−
d
2
− 6d + 12
2 (d
2
− 7d + 14) (d − 2)
3
c
14
+
(d − 4)
(d
2
− 7d + 14) (d − 2)
3
c
16
−
d
3
− 8d
2
+ 20d − 8
2 (d
2
− 7d + 14) (d − 2)
3
c
18
. (2.11)
Such theories yield a field equation of the form appearing in quartic Lovelock gravity or
the more general quartic quasi-topological gravity [26]: a total derivative of a polynomial
in f(r). However, although there are still 13 free parameters after the two additional con-
straints (2.10) are imposed, given the constraints in appendix A and eqs. (2.10) and (2.11),
we find that only seven of these terms make non-trivial contributions to the field equations;
these are characterized by the constants c
1
, c
2
, c
3
, c
4
, c
5
, c
6
and c
7
. Of these seven non-
trivial theories, we know that one must correspond to quartic Lovelock gravity; i.e. there
must be a choice of constants that produces the eight dimensional Euler density. We find
that this to be
X
8
: c
1
=96, c
2
=−48, c
3
=96, c
4
=−48, c
5
=−6, c
6
=48, c
7
= −3,
c
8
=−384, c
10
=−192, c
11
=32, c
13
=−192, c
14
=192, c
16
=−192. (2.12)
Another known term ensuring a non-trivial field equation is the selection
Z
(1)
d
: c
1
= 0, c
2
= 8(d−2)(860−2113d+1959d
2
−810d
3
+102d
4
+30d
5
−11d
6
+d
7
)
c
3
= 0, c
4
= 0, c
6
= 0,
c
5
= −(d − 2)(1108 − 2723d + 2639d
2
− 1224d
3
+ 235d
4
+ 10d
5
− 10d
6
+ d
7
),
c
7
= −1292 + 2929d − 2741d
2
+ 1527d
3
− 684d
4
+ 276d
5
− 82d
6
+ 14d
7
− d
8
,
c
8
= 0, c
10
= 0, c
11
= 0, c
13
= 0,
c
14
= 16(d − 2)
3
(274 − 389d + 183d
2
− 34d
3
+ 2d
4
), c
16
= 0 , (2.13)
corresponding to quartic quasi-topological gravity [26].
We thus have five new quartic quasi-topological theories, which to our knowledge
have not been discussed in the literature to date. We therefore choose a simple basis for
these terms:
Z
(2)
d
: c
1
= 1, other c
i
=0 except those constrained in appendix A and eqs. (2.10) and (2.11),
Z
(3)
d
: c
2
= 1, other c
i
=0 except those constrained in appendix A and eqs. (2.10) and (2.11),
Z
(4)
d
: c
3
= 1, other c
i
=0 except those constrained in appendix A and eqs. (2.10) and (2.11),
Z
(5)
d
: c
4
= 1, other c
i
=0 except those constrained in appendix A and eqs. (2.10) and (2.11),
Z
(6)
d
: c
5
= 1, other c
i
=0 except those constrained in appendix A and eqs. (2.10) and (2.11).
(2.14)
The resulting expressions for the Lagrangian densities of the quasi-topological theories
listed above exhibit complicated dependence on the spacetime dimension. We have included
explicit expressions for these, valid in any dimension d > 4, in appendix B. Each of the
quasi-topological theories contributes to the field equations in dimensions d ≥ 5, but are
– 9 –