# Particle spectra of general Ricci-type Palatini or metric-affine theories

W. Barker<EMAIL_ADDRESS>Astrophysics Group, Cavendish Laboratory, JJ
Thomson Avenue, Cambridge CB3 0HE, UK  Kavli Institute for Cosmology,
Madingley Road, Cambridge CB3 0HA, UK  C. Marzo<EMAIL_ADDRESS>Laboratory for High Energy and Computational Physics, NICPB, Rävala 10,
Tallinn 10143, Estonia

###### Abstract

In the context of weak-field metric-affine (i.e. Palatini) gravity near
Minkowski spacetime, we compute the particle spectra in the simultaneous
presence of all independent contractions quadratic in Ricci-type tensors.
Apart from the full metric-affine geometry, we study kinematic limits with
vanishing torsion (i.e. a symmetric connection) and vanishing non-metricity
(i.e. a metric connection, which is physically indistinguishable from Poincaré
gauge theory at the level of the particle spectrum). We present a detailed
report on how spin-parity projection operators can be used to derive
systematically and unambiguously the character of the propagating states. The
unitarity constraints derived from the requirements of tachyon- and ghost-
freedom are obtained. We show that, even in the presence of all Ricci-type
operators, only a narrow selection of viable theories emerges by a tuning.

##  I Introduction

The success of the curvature-based geometrical formulation of gravity has
stimulated a search for the dynamical interpretation of geometrical properties
analogous to curvature. This program has been directed to solve, or mitigate,
some of the main shortcomings of the current status quo, mainly the lack of a
perturbatively renormalizable quantum theory of gravity, and the
phenomenological need for a dark sector. While hopes to address the
renormalization issue have, so far, had to rely on non-traditional routes [1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], the possibility of interesting
phenomenology from particles of geometrical origin is frequently used in
cosmology and dark-sector model-building [14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27]. Metric-affine gravity (MAG) [28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 31, 44, 45, 46] represents the principal
realization of this program. It broadens the matter content by considering the
affine connection as an independent three-index object, giving rise to torsion
and non-metricity (see Fig. 1) as dynamical fields. This results in a notable
growth in computational complexity, both in the number of allowed operators
and in the profiling of the multiple particle states carried in by the affine
connection. Such a broad parameter space is expected, as is often the case, to
narrow under the pressure of field-theoretical self-consistency constraints,
such as unitarity and elimination of tachyons [47, 48, 49, 33, 32, 28, 50, 45,
44, 51, 52, 53, 35, 54]. The imposition of these requirements is a highly non-
trivial task and has often necessitated severe simplifications to arrive at
positive scenarios. Also, the analysis often has to rely on a very indirect
route, without directly accessing the pole structure of the propagator.

Figure 1: The effects of metric-affine curvature
$\tensor{\mathcal{F}}{{}_{\mu\nu}^{\rho}{}_{\sigma}}$ in Eq. 2 – vector
rotation after parallel transport in a closed loop – torsion
$\tensor{\mathcal{T}}{{}_{\mu}^{\alpha}{}_{\nu}}$ in Eq. 1 – non-closure of
parallelograms formed from (infinitesimally) parallel-transported vectors –
and non-metricity $\tensor{\mathcal{Q}}{{}_{\lambda\mu\nu}}$ in Eq. 1 – change
in vector length under parallel transport.

In this paper, we use the arena of Ricci-type MAG to illustrate, in a detailed
step-by-step fashion, how the formalism of spin-parity projectors can
unambiguously and straightforwardly reveal the nature of the (tree-level)
particle spectrum. By _Ricci-type_ we refer to all the operators that may be
added to the Einstein–Hilbert Lagrangian which are quadratic in the rank-two
traces of metric-affine curvature (in MAG there are nine such operators,
whilst in standard GR there is only one). Building on early work (see e.g.
[55, 56, 57, 58, 59, 60]), this particular space of operators was first
properly charted by Annala and Räsänen [35]. The reparameterisation-based
methods used in that work are particularly innovative, but they are only able
to access a ‘punctuated’ bulk of the full parameter space due to certain
_degeneracy conditions_ which must be avoided. In the present work we build on
these foundations by applying the spin-parity projection formalism: our method
does not come with any restriction on the parameters of the model. Indeed, if
the set of projector operators is known, the approach adopted can be applied
to every tensor-valued Lagrangian admitting a Minkowskian background
expansion. We have authored a new Wolfram Language implementation of this
procedure for all such theories, including the MAG: _Particle Spectrum for any
Tensor Lagrangian_ (PSALTer). The PSALTer software will be properly presented
in a dedicated paper [61]111See [62] for a recent application of PSALTer to
theories proposed in [63, 64]., but in this paper we use it to confirm all our
results (see Appendices A, B and C). The body this paper is set out as
follows. In Section II we develop the underlying theory by setting out the MAG
conventions in Section II.1 and briefly presenting our spectral algorithm in
Section II.2 and the computer implementation in Section II.3. In Section III
we present all of the results, respectively for the vanishing non-metricity
kinematic limit in Section III.1 and the vanishing torsion kinematic limit in
Section III.2, and for the full MAG in Section III.3. Conclusions follow in
Section IV. We will use the ‘West Coast’ signature ${(+,-,-,-)}$, other
conventions will be introduced as needed.

##  II Theoretical development

The requirements of locality and Lorentz invariance select tensor fields as
the building-blocks for most of the theoretical speculations about high-energy
physics. The price to pay for using them is in the risk of uncontrolled
unitarity violations. It is a profound realization that the consistent
adjustments to recover unitarity restrict us to Maxwell’s theory, for rank-one
fields, as well as Einstein’s theory of gravity, when applied to models of
symmetric rank-two fields [65, 66, 67, 47, 68]. The continuation of this
program within the intricate scenarios of higher-rank fields, as well as
multi-field quadratic Lagrangians, is technically prohibitive. Many indirect
shortcuts have supported claims of healthy particle propagation, but the
proliferation of indices often prevents a direct approach to the pole
structure of the propagator. The spin-parity projection approach to spectral
analysis has been most thoroughly expostulated by Lin, Hobson and Lasenby in
[52], with follow-up papers in [69, 53]. Further information about the method
can also be found in (e.g.) [51, 28, 47, 48, 49, 70, 33] — but it is [52]
which provides the most concise introduction for our purposes.

###  II.1 Spin-parity kinematics of metric-affine gravity

To have full control over the particle spectrum we introduce operators which
project the Lorentz index structure onto the labels $J$ and $P$ of spin and
parity, enumerating the irreducible representation under the $\mathrm{SU}(2)$
little group. In general, further reduction to the $\mathrm{U}(1)$ little
group of massless particles and helicity states can be done. This results in
no practical improvement, making the decomposition into $\mathrm{SU}(2)$
representations generic enough. In the case of MAG, the needed set of
projection operators has been recently completed [28, 71], developing the past
studies of [72, 47, 32, 33, 48, 49, 51]. We present here a summary of the main
ideas regarding the use of projection operators for the computation of the
poles and residues of the propagator. We refer to [52, 48, 28, 73] for further
details.

The independent MAG connection $\tensor{A}{{}_{\mu}^{\nu}{}_{\rho}}$ carries
in 64 new degrees of freedom (d.o.f), which are not present in the derived
Levi–Civita connection
$\tensor{\Gamma}{{}_{\mu}^{\nu}{}_{\rho}}\equiv\tensor{\Gamma}{{}_{(\mu|}^{\nu}{}_{|\rho)}}\equiv\tensor{g}{{}^{\nu\lambda}}\big{(}\tensor{\partial}{{}_{(\mu}}\tensor{g}{{}_{\rho)\lambda}}-\frac{1}{2}\tensor{\partial}{{}_{\lambda}}\tensor{g}{{}_{\mu\rho}}\big{)}$.
These new d.o.f are often partitioned into the torsion and non-metricity
tensors

$\tensor{\mathcal{T}}{{}_{\mu}^{\alpha}{}_{\nu}}\equiv
2\tensor{A}{{}_{[\mu|}^{\alpha}{}_{|\nu]}},\quad\tensor{\mathcal{Q}}{{}_{\lambda\mu\nu}}\equiv-\tensor{\partial}{{}_{\lambda}}\tensor{g}{{}_{\mu\nu}}+2\tensor{A}{{}_{\lambda}^{\alpha}{}_{(\mu|}}\tensor{g}{{}_{\alpha|\nu)}}.$
(1)

The two tensors in Eq. 1 are geometric counterparts to the metric-affine (i.e.
non-Riemannian) curvature

$\tensor{\mathcal{F}}{{}_{\mu\nu}^{\rho}{}_{\sigma}}\equiv
2\left(\tensor{\partial}{{}_{[\mu}}\tensor{A}{{}_{\nu]}^{\rho}{}_{\sigma}}+\tensor{A}{{}_{[\mu|}^{\rho}{}_{\alpha}}\tensor{A}{{}_{|\nu]}^{\alpha}{}_{\sigma}}\right).$
(2)

The influences of these three geometric properties on vectors which are being
parallel-transported through the spacetime are shown in Fig. 1. We will be
particularly interested in the three _Ricci-type_ contractions of the metric-
affine curvature

$\displaystyle\tensor{\mathcal{F}}{{}_{\mu\nu}}$
$\displaystyle\equiv\tensor{\mathcal{F}}{{}_{\mu\nu\alpha}^{\alpha}},\quad\tensor{\mathcal{F}}{{}^{(14)}_{\mu\nu}}\equiv\tensor{\mathcal{F}}{{}_{\alpha\mu\nu}^{\alpha}},\quad\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}\equiv\tensor{\mathcal{F}}{{}_{\alpha\mu}^{\alpha}{}_{\nu}},$
(3)

by which we just mean the contractions with two free indices. The quantity
$\tensor{\mathcal{F}}{{}_{\mu\nu}}$ is the _homothetic_ curvature [74, 75],
while $\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}$ and
$\tensor{\mathcal{F}}{{}^{(14)}_{\mu\nu}}$ are variously _pseudo_ -Ricci
tensors [28], or $\tensor{\mathcal{F}}{{}^{(14)}_{\mu\nu}}$ is the _co_ -Ricci
[74]. The Riemannian curvature of course yields only one Ricci-type
contraction: the homothetic curvature vanishes identically in the absence of
non-metricity, and the (co-)Ricci tensors coincide in the absence of torsion.
We define
$\tensor{\mathcal{F}}{}\equiv\tensor{\mathcal{F}}{{}_{\mu\nu}^{\mu\nu}}$ as
the Ricci scalar: there is still only one such scalar in metric-affine
geometry.

In the _first-order_ or _Palatini_ parameterisation of MAG, we take the 10
d.o.f in $\tensor{g}{{}_{\mu\nu}}$ and the $64$ new d.o.f in
$\tensor{A}{{}_{\mu}^{\nu}{}_{\rho}}$ to be fundamental fields. An advantage
of the first-order parameterisation is that the MAG field strength tensor in
Eq. 2 is free from second derivatives. In the _second-order_ or _post-
Riemannian_ parameterisation, we keep $\tensor{g}{{}_{\mu\nu}}$ but treat the
tensor-valued difference
$\tensor{\Delta}{{}_{\mu}^{\nu}{}_{\rho}}\equiv\tensor{A}{{}_{\mu}^{\nu}{}_{\rho}}-\tensor{\Gamma}{{}_{\mu}^{\nu}{}_{\rho}}$
(sometimes termed the _distortion_) as fundamental, effectively partitioning
the 64 d.o.f among the $24$ d.o.f in
$\tensor{\mathcal{T}}{{}_{\mu}^{\alpha}{}_{\nu}}\equiv\tensor{\mathcal{T}}{{}_{[\mu|}^{\alpha}{}_{|\nu]}}$
plus the $40$ d.o.f in
$\tensor{\mathcal{Q}}{{}_{\lambda\mu\nu}}\equiv\tensor{\mathcal{Q}}{{}_{\lambda(\mu\nu)}}$
according to

$\tensor{\mathcal{T}}{{}_{\mu}^{\alpha}{}_{\nu}}\equiv
2\tensor{\Delta}{{}_{[\mu|}^{\alpha}{}_{|\nu]}},\quad\tensor{\mathcal{Q}}{{}_{\lambda\mu\nu}}\equiv
2\tensor{\Delta}{{}_{\lambda(\mu\nu)}}.$ (4)

In the second-order parameterisation Eq. 2 is expanded into the Riemannian
curvature (which naturally has second-derivatives in
$\tensor{g}{{}_{\mu\nu}}$) and many other terms which are (Levi–Civita)
covariant derivatives and second powers of
$\tensor{\mathcal{T}}{{}_{\mu}^{\alpha}{}_{\nu}}$ and
$\tensor{\mathcal{Q}}{{}_{\lambda\mu\nu}}$. The second-order parameterisation
has an advantage in revealing the true nature of all MAG-type theories: every
MAG theory is a (non-)minimal coupling of standard metric-based gravity to an
asymmetric rank-three matter field $\tensor{\Delta}{{}_{\mu}^{\rho}{}_{\nu}}$.
We are free to work with either set of variables due to re-parameterisation
invariance of the physics.

Working in the first-order formulation, the weak-field regime near to
Minkowski spacetime can be captured by an inherently perturbative
$\tensor{A}{{}_{\mu}^{\nu}{}_{\rho}}$ and a metric perturbation
$\tensor{g}{{}_{\mu\nu}}\equiv\tensor{\eta}{{}_{\mu\nu}}+\tensor{h}{{}_{\mu\nu}}$.
These perturbations carry multiple massive particle states

$\displaystyle A_{\mu\nu\rho}$
$\displaystyle\supset\Big{\\{}3_{1}^{-},2_{1}^{+},2_{2}^{+},2_{3}^{+},2_{1}^{-},2_{2}^{-},1^{+}_{1},1^{+}_{2},1^{+}_{3},1^{-}_{1},1^{-}_{2},$
$\displaystyle\hskip
25.0pt1^{-}_{3},1^{-}_{4},1^{-}_{5},1^{-}_{6},0^{+}_{1},0^{+}_{2},0^{+}_{3},0^{+}_{4},0^{-}_{1}\Big{\\}},$
(5a) $\displaystyle h_{\mu\nu}$
$\displaystyle\supset\Big{\\{}2_{4}^{+},1^{-}_{7},0^{+}_{5},0^{+}_{6}\Big{\\}},$
(5b)

where we used the compact notation $J^{P}_{j}$ in referring to the j-th
representation of spin $J$ and parity $P$. Enumeration convention is adopted
from [28]. From Eqs. 5a and 5b the 64 and 10 d.o.f can be recovered
respectively by summing the multiplicities $2J+1$ over all states. The PSALTer
notation for these states differs from the subscript notation in Eqs. 5a and
5b, and full definitions of these states are provided in Fig. C.1.

###  II.2 Saturated propagator and particle spectra

Following [73], we use a synthetic notation to describe the quadratic (i.e.
perturbative) action in momentum space

$\displaystyle
S[\Phi]=\frac{1}{2}\int\mathrm{d}^{4}k\,\bigg{(}\Phi(-k)\,\mathcal{\,}K(k)\,\Phi(k)+$
$\displaystyle\hskip
56.9055pt+\mathcal{J}(-k)\Phi(k)+\mathcal{J}(k)\Phi(-k)\bigg{)}\,,$ (6)

where $K(k)$ is the Fourier-transformed kinetic term (wave operator) and we
have introduced a linear coupling between the fields, collectively labelled as
$\Phi(k)$, and a source $\mathcal{J}(k)$. Connecting this to the specific
formulation in Section II.1, we identify $\Phi$ as the collection of
perturbative fields $\tensor{h}{{}_{\mu\nu}}$ and
$\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}$. Within the quadratic approximation, the
indices on these fields are raised and lowered using
$\tensor{\eta}{{}^{\mu\nu}}$ and $\tensor{\eta}{{}_{\mu\nu}}$, which are non-
dynamical, and the Greek indices refer to Cartesian coordinates on the
Minkowski background222Indeed, it should be explicitly stated that along with
these raising and lowering rules $\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}$ is a
dynamical _tensor_ field in the quadratic approximation. Geometrically,
$\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}$ is a connection, but the physics does
not know or care about the geometric foundations of the theory: only the
representations of the particle states are important. Of course,
$\tensor{\Delta}{{}_{\mu}^{\rho}{}_{\nu}}$ is already geometrically a tensor
in the second-order formulation of MAG. The re-parameterisation is shown
explicitly in Eq. 38, and the key point is that the linearised Levi–Civita
connection is also tensorial at lowest order in perturbations because the
partial derivative $\tensor{\partial}{{}_{\mu}}$ is covariant at that order..
Conjugate to $\Phi$, the fields $\mathcal{J}$ in MAG are the symmetric matter
stress-energy tensor $\tensor{T}{{}^{\mu\nu}}$ and the rank-three current
$\tensor{W}{{}^{\mu}_{\rho}{}^{\nu}}$. This latter current is sometimes called
the _hypermomentum_ [76, 77, 14, 27, 78, 79, 80]. In the second-order
formulation, a separate current must be defined as conjugate to
$\tensor{\Delta}{{}_{\mu}^{\rho}{}_{\nu}}$ — we do not bother to ascribe it
another symbol.

The propagating states will appear in the form of isolated poles for the
propagator $\mathcal{D}(k)$ defined through the equation
$K(k)\cdot\mathcal{D}(k)\equiv\hat{1}$. The use of projection operators
$P^{i,i}_{\left\\{J,P\right\\}}$ drastically simplifies the solution of this
inversion problem. By exploiting the defining properties

$\displaystyle\sum_{J,P,i}P^{i,i}_{\left\\{J,P\right\\}}{}_{\mu_{1}\mu_{2}\cdots\mu_{n}}^{\quad\quad\nu_{1}\nu_{2}\cdots\nu_{n}}\equiv\displaystyle{\hat{1}}{}_{\mu_{1}\mu_{2}\cdots\mu_{n}}^{\quad\quad\nu_{1}\nu_{2}\cdots\nu_{n}},$
(7a) $\displaystyle
P^{i,k}_{\left\\{J,P\right\\}}{}_{\mu_{1}\mu_{2}\cdots\mu_{n}}^{\quad\quad\rho_{1}\rho_{2}\cdots\rho_{n}}\,\,P^{j,w}_{\left\\{J^{\prime},P^{\prime}\right\\}}{}_{\rho_{1}\rho_{2}\cdots\rho_{n}}^{\quad\quad\nu_{1}\nu_{2}\cdots\nu_{n}}$
$\displaystyle\hskip
50.0pt\equiv\delta_{k,j}\,\delta_{J,J^{\prime}}\,\delta_{P,P^{\prime}}\,P^{i,w}_{\left\\{J,P\right\\}}{}_{\mu_{1}\mu_{2}\cdots\mu_{n}}^{\quad\quad\nu_{1}\nu_{2}\cdots\nu_{n}},$
(7b) $\displaystyle
P^{i,j}_{\left\\{J,P\right\\}}{}_{\mu_{1}\mu_{2}\cdots\mu_{n}}^{\quad\quad\nu_{1}\nu_{2}\cdots\nu_{n}}\equiv\left(P^{j,i}_{\left\\{J,P\right\\}}{}^{\nu_{1}\nu_{2}\cdots\nu_{n}}_{\quad\quad\mu_{1}\mu_{2}\cdots\mu_{n}}\right)^{*},$
(7c)

it is possible to decompose the kinetic term as

$\displaystyle\int\mathrm{d}^{4}k\,\Phi(-k)\,K(k)\,\Phi(k)\,$
$\displaystyle=\int\mathrm{d}^{4}k\,\Phi(-k)\,\sum_{J,P,i,j}\bigg{(}a^{\left\\{J,P\right\\}}_{i,j}P^{i,j}_{\left\\{J,P\right\\}}\bigg{)}\,\Phi(k)\,\,,$
(8)

where the tortuous fabric of the indices is reformulated in terms of the
simpler spin-parity matrices $a^{\left\\{J,P\right\\}}_{i,j}$, obtained by
tracing over the Lorentz indices

$\displaystyle
a_{i,j}^{\left\\{J,P\right\\}}\equiv\frac{1}{2J-1}\,\,\mathrm{Tr}\,P^{i,j}_{\left\\{J,P\right\\}}\,\mathcal{K}(k)\,.$
(9)

The orthonormality of the projection operators in Eq. 7b reduces the
computation of the propagator $\mathcal{D}(k)$ to an inversion problem for the
matrices $a^{\left\\{J,P\right\\}}_{i,j}$

$\displaystyle\mathcal{D}(k)\equiv\sum_{J,P,i,j}b^{\left\\{J,P\right\\}}_{i,j}P^{i,j}_{\left\\{J,P\right\\}}\,,\quad
b^{\left\\{J,P\right\\}}_{i,j}\equiv\left(a^{\left\\{J,P\right\\}}_{i,j}\right)^{-1}\,.$
(10)

When the degeneracy of the spin-parity matrices hampers this inversion, the
model displays gauge symmetry. It is a bonus of this formalism that full
control over the gauge symmetries is provided by the inspection of such simple
objects as the null-vectors $X_{i}^{r=1,2,\cdots,n}$ of
$a^{\left\\{J,P\right\\}}_{i,j}$, so that manipulations of linear algebra
replace tensor operations. In turn, this promotes a code-friendly
implementation of the spectral problem. The matrix structure not only allows
the identification of symmetries which are already present in the model, but
it also allows parameter-tunings to be identified which lead to the emergence
of new symmetries. Finally, the gauge-invariant _saturated_ propagator is
obtained by restricting the inversion to the non-degenerate subspace in the
spin-parity matrices, and contracting the inverse matrix with constrained
sources $\tilde{J}(k)$

$\displaystyle\mathcal{D}_{S}(k)\equiv\tilde{J}^{*}\left(k\right)\bigg{(}\sum_{J,P,i,j}\tilde{b}^{\left\\{J,P\right\\}}_{i,j}P^{i,j}_{\left\\{J,P\right\\}}\bigg{)}\tilde{J}\left(k\right),$
(11a) $\displaystyle
X^{*}{}^{r}_{j}\,\,P^{i,j}_{\left\\{J,P\right\\}}\,\,\tilde{J}\left(k\right)\equiv
0\,,\quad\,\,r=1,2,\cdots,n\,\,.$ (11b)

###  II.3 Computer implementation

The saturated propagator in Eq. 11a represents the arrival point of our
computation. Previous efforts in MAG, even within the projection operators
approach, generally relied on an indirect determination for the signs of the
residues of the poles in $\mathcal{D}_{S}(k)$. Such methods avoid the
difficult computation of the source constraint equations Eq. 11b in the
massless limit. However, as detailed in [73, 61, 52], the constraint equations
can be fully resolved by choosing a suitable reference frame (with $k^{2}=0$
as a limit of $k^{2}>0$). This technique removes the main theoretical
challenges encountered in the computation of the spectrum leaving, as the only
limitation, the technical bounds in manipulating large expressions. The
simplifications afforded by the use of projection operators, as well as the
unambiguous computational procedure, have encouraged the development of
opportune tools to promptly and automatically tackle the particle spectrum.
Some of the most recent results on this subject [62, 54] and the core of this
work’s conclusions, are obtained with the use of PSALTer [61], a Wolfram
Language implementation of these ideas. The PSALTer software can automatically
return the spectrum of any tensor-valued field theory up to rank-three.

The PSALTer analyses of key theories to be considered in Section III are
presented in Figs. A.2, B.2, B.3, B.4, B.5, B.6, B.7, C.2 and C.3. These are
vector graphics which contain the following information;

  * •

The linearised (quadratic) action in Section II.2, in the position-space
representation. This expression is the only input to the PSALTer software, and
it is not necessary to perform any kind of decomposition of fields into
irreducible parts.

  * •

Automatically computed: the elements $a^{\left\\{J,P\right\\}}_{i,j}$ in Eq. 8
which encode the wave operator of the theory, provided as one matrix for each
spin sector. Because the theories considered here are not parity-violating, it
is reassuring to see that the matrices are always block-diagonal across
parity-even (red) and parity-odd (blue) sub-sectors: the mixed-parity (purple)
blocks are empty.

  * •

Automatically computed: the inverse $b^{\left\\{J,P\right\\}}_{i,j}$ matrices
in Eq. 10 which encode the saturated propagator of the theory. We believe our
implementation to be the first which uses Moore–Penrose inversion [81, 82]
(i.e. a uniquely defined gauge fixing) to obtain these coefficients.

  * •

Automatically computed: the source constraints $X_{i}^{r}$ in Eq. 11b, which
are guaranteed to encode all the gauge symmetries present in the theory.

  * •

Automatically computed: the spectrum of all massive and massless particles
present in the theory. This includes information about the particle spin $J$,
parity $P$, pole residue and mass. In the case of massless particles, there is
no physical notion of spin which survives, but the number of independent
polarisations is given.

  * •

Automatically computed: the overall unitarity conditions which must be imposed
on the Lagrangian coupling coefficients. These conditions are derived from the
above pole residues and masses, so as to support the no-ghost and no-tachyon
criteria. There is of course no guarantee that such conditions exist, so the
calculation is time-limited to ten seconds. In case of ‘timeout’, the masses
and residues provide all the relevant information for further tuning the
theory anyway.

Because these various outputs may be extremely cumbersome and have uncertain
dimensions after typesetting, PSALTer uses a rectangle-packing algorithm to
find the most economical layout for each theory: consequently some of the
formulae in Figs. A.2, B.2, B.3, B.4, B.5, B.6, B.7, C.2 and C.3 are rotated
on the page. The various $\mathrm{SO}(3)$ irrep definitions are provided
separately, in Figs. A.1, B.1 and C.1. Each of these figures defines a
‘kinematic module’ for the software: a declaration of the fundamental tensor
fields and their conjugate sources which are present in a class of theories.
Within each module, the spectral analysis of infinitely many distinct models
can be performed, depending on the admixture of operators in the quadratic
action.

Two steps of the analysis are computationally expensive: the Moore–Penrose
inversion and the evaluation of massless residues. When the theory contains
more than two or three independent Lagrangian couplings (parameters) and
tensor fields of rank three or more, these calculations start to pose a highly
non-trivial computer algebra problem. Consequently, many subroutines in
PSALTer are automatically parallelised, to take advantage of the available
infrastructure. For expedience, analysis of each theory in this work was
performed using a dedicated compute node consisting of 112 Intel® _Sapphire
Rapids_ CPUs, or 64 AMD® _Ryzen Threadripper_ CPUs, depending on availability.
The former setup is close to the current state of the art in high-performance
computing (HPC). The resulting throughput is very fast, and in fact each
theory would only have taken approximately 20 minutes to process on a modern
PC with four CPU cores.

##  III Results

Using the building-blocks in Eq. 3 the most general Ricci-type MAG in the
first-order formulation is

$\displaystyle S[g,$ $\displaystyle
A]=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}\bigg{[}-a_{0}\tensor{\mathcal{F}}{}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{(13)}{}^{\mu\nu}}\Big{(}c_{7}\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}+c_{8}\tensor{\mathcal{F}}{{}^{(13)}_{\nu\mu}}\Big{)}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{(14)}{}^{\mu\nu}}\Big{(}c_{9}\tensor{\mathcal{F}}{{}^{(14)}_{\mu\nu}}+c_{10}\tensor{\mathcal{F}}{{}^{(14)}_{\nu\mu}}\Big{)}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{(14)}{}^{\mu\nu}}\Big{(}c_{11}\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}+c_{12}\tensor{\mathcal{F}}{{}^{(13)}_{\nu\mu}}\Big{)}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{\mu\nu}}\Big{(}c_{13}\tensor{\mathcal{F}}{{}_{\mu\nu}}+c_{14}\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}+c_{15}\tensor{\mathcal{F}}{{}^{(14)}_{\mu\nu}}\Big{)}\bigg{]},$
(12)

where we borrow the numbering of Lagrangian couplings directly from [28]333In
that work, a more general MAG action is considered, in which a total of 28
invariant operators are present in the Lagrangian density.. As emphasised in
Section II.1, re-parameterisation invariance leads to the equivalence
$S[g,A]\cong S[g,\Delta]$ with the second-order formalism. The computational
algorithm sketched in Section II.2 can obviously handle both representations
of the dynamical fields and our tests have adopted both approaches as a
further self-consistency check. While the final outcome does not change, the
particular form of the intermediate steps does. In this regard, we find the
second-order basis more convenient for enumerating the spin-parity states in
kinematically restricted version of the MAG, due to the index symmetries in
Eq. 4.

###  III.1 Zero non-metricity

The imposition of zero non-metricity is easily realized in a formalism with
explicit distortion. From Eq. 4 it is clear that a two-index-antisymmetric
rank-three field
$\tensor{\Delta}{{}_{\lambda\mu\nu}}\equiv-\tensor{\Delta}{{}_{\lambda\nu\mu}}$
nullifies $Q_{\lambda\mu\nu}\equiv 0$. This achieves a reduction of the spin-
parity sectors of Eq. 5a

$\displaystyle\tensor{\Delta}{{}_{\mu\nu\rho}}\supset\Big{\\{}2_{3}^{+},2_{2}^{-},1^{+}_{2},1^{+}_{3},1^{-}_{3},1^{-}_{6},0^{+}_{3},0^{-}_{1}\Big{\\}},$
(13)

reflected by a further redundancy in the number of independent Ricci-type
contractions

$\displaystyle\tensor{\mathcal{F}}{{}_{\mu\nu}}\equiv
0,\quad\tensor{\mathcal{F}}{{}^{(14)}_{\mu\nu}}\equiv-\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}\,.$
(14)

The action Section III is therefore simplified into

$\displaystyle S[g,$ $\displaystyle
A]=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}\bigg{[}-a_{0}\tensor{\mathcal{F}}{}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{(13)}{}^{\mu\nu}}\Big{(}g_{7}\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}+g_{8}\tensor{\mathcal{F}}{{}^{(13)}_{\nu\mu}}\Big{)}\bigg{]}.$
(15)

Note that we follow [28] in re-labelling the dimensionless coefficients from
$c_{i}$ to $g_{i}$ when passing from Section III to Section III.1 by kinematic
restriction. The spectrum of this simple three-parameter model can be
promptly, and unambiguously profiled within our formalism. Just within this
section (but not within Sections III.2 and III.3) the accompanying PSALTer
analysis in Appendix A will be made in the Poincaré gauge theory (PGT)
formulation of zero-non-metricity MAG. The kinematic structure of PGT is more
extensive than its MAG counterpart due to extra _antisymmetric_ parts of the
tetrad fields (which are nullified by an extra Lorentz gauge symmetry). This
kinematic structure is presented in Fig. A.1, but is otherwise analogous to
that in Eqs. 5b and 13. For now, we proceed with our discussion as if we were
working in the MAG formulation. To access the singular structure of the
propagator we first identify the spin-parity sectors of the kinetic term. The
PGT matrices are shown in Appendix A, in Fig. A.2. We can immediately
recognize in the absence of the $1^{-}_{7}$ and $0^{+}_{6}$ sectors the
hallmark of diffeomorphism invariance. All the information concerning the
quadratic terms is encoded in such matrices and, from the arguments of Section
II, a direct link exists between the zeroes of their determinants and the
singularities of $\tilde{D}_{S}(k)$. The shape of the residue and the position
of the singularity will determine the nature of the propagating particles.
Once the degeneracies of $a^{\left\\{1,-\right\\}}_{i,j}$ and
$a^{\left\\{0,+\right\\}}_{i,j}$ are removed, we immediately find that no
massive poles are present. We can therefore extract the known result [35]:

_No massive states propagate in linearised zero-non-metricity Ricci-type
theories_.

The massless poles are present in the $2^{+}$ and $0^{+}$ sectors. Again,
these are known traits of graviton propagation. To confirm that the graviton
is present we explore the form of these massless poles in the final, gauge
invariant, propagator. The constraints to be imposed are read off the null
vectors of $a^{\left\\{1,-\right\\}}_{i,j}$ and
$a^{\left\\{0,+\right\\}}_{i,j}$. By choosing the light-like frame
$k^{\mu}=\big{(}\mathcal{E},0,0,\mathcal{E}\big{)}$ the sources are
constrained as

$\displaystyle T^{00}=T^{03},\quad T^{13}=T^{01},\quad T^{23}=T^{02},\quad
T^{33}=T^{03}.$ (16)

By substituting them into the propagator, we find the following structure

$\displaystyle\lim_{k^{2}\mapsto
0}\mathcal{D}_{S}(k)=\frac{1}{k^{2}}\left(\frac{2}{-a_{0}}\right)\sum_{i=1,2}|S_{i}|^{2}\,,$
(17)

where the $S_{i}$ are linearly independent combinations of the sources. Thus,
Eq. 17 reveals the two (helicity) states with a residue $\propto-1/a_{0}$. We
conclude by refining the previous statement

_Linearised zero-non-metricity Ricci-type theories propagate only a healthy
massless graviton with $a_{0}<0$_.

###  III.2 Zero torsion

The zero-torsion case offers a stronger challenge to our methods, displaying a
larger parameter space and non-trivial interplay among different operators. We
will show that the spin-parity formalism, followed by direct access to the
propagator, gives full access to the tree-level spectrum without imposing
simplifying restrictions. First, in terms of the distortion tensor, the zero-
torsion condition in Eq. 4 is achieved by working with a two-index-symmetric
rank-three field
$\tensor{\Delta}{{}_{\lambda\mu\nu}}\equiv\tensor{\Delta}{{}_{\nu\mu\lambda}}$.
The available particle content in the symmetric distortion is then

$\displaystyle\tensor{\Delta}{{}_{\mu\nu\rho}}\supset\Big{\\{}3_{1}^{-},2_{1}^{+},2_{2}^{+},2_{1}^{-},1^{+}_{1},1^{-}_{1},1^{-}_{2},$
$\displaystyle\hskip
40.0pt1^{-}_{4},1^{-}_{5},0^{+}_{1},0^{+}_{2},0^{+}_{4}\Big{\\}},$ (18)

which causes
$\tensor{\mathcal{F}}{{}_{\mu\nu}}\equiv-2\tensor{\mathcal{F}}{{}^{(13)}_{[\mu\nu]}}$,
leaving seven independent combinations

$\displaystyle S[g,$ $\displaystyle
A]=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}\bigg{[}-a_{0}\tensor{\mathcal{F}}{}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{(13)}{}^{\mu\nu}}\Big{(}h_{7}\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}+h_{8}\tensor{\mathcal{F}}{{}^{(13)}_{\nu\mu}}\Big{)}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{(14)}{}^{\mu\nu}}\Big{(}h_{9}\tensor{\mathcal{F}}{{}^{(14)}_{\mu\nu}}+h_{10}\tensor{\mathcal{F}}{{}^{(14)}_{\nu\mu}}\Big{)}$
$\displaystyle+\tensor{\mathcal{F}}{{}^{(14)}{}^{\mu\nu}}\Big{(}h_{11}\tensor{\mathcal{F}}{{}^{(13)}_{\mu\nu}}+h_{12}\tensor{\mathcal{F}}{{}^{(13)}_{\nu\mu}}\Big{)}\bigg{]}.$
(19)

Note once again that we follow [28] in re-labelling the dimensionless
coefficients from $c_{i}$ to $h_{i}$ when passing from Section III to Section
III.2 by kinematic restriction. The kinematic structures in Eqs. 5b and III.2
are reflected in the PSALTer definitions of the states, which are fully
defined in Fig. B.1. The computation of the spin-parity matrices produces the
remaining output in Appendix B, with the general case Section III.2 displayed
in Fig. B.2. After simple inspections of the determinants, we find that it is
impossible to impose further gauge symmetries, unless $a_{0}$, the only
dimensionful parameter, is set to zero. This would nullify the graviton
propagation and, therefore, we discard this option. Taking into account the
constraints of diffeomorphism invariance as illustrated in Section III.1, we
can compute the positions of the zeroes for all the determinants. Already for
the $2^{+}$ sector we find that a state of mass

$m^{2}_{2^{+}}=-\frac{a_{0}}{h_{10}+h_{9}},$ (20)

is allowed to propagate whenever $h_{10}+h_{9}\neq 0$. The computation of the
limit $k^{2}\mapsto m^{2}_{2^{+}}$ gives rise to the following pole residue,
where we suppress the positive-definite quadratic form in the $2^{+}$ sources

$\displaystyle\lim_{k^{2}\mapsto m_{2^{+}}^{2}}\mathcal{D}_{S}(k)\propto
2\bigg{[}$ $\displaystyle
4h_{10}^{2}+h_{11}^{2}+h_{12}^{2}+2h_{11}\big{(}h_{12}-2h_{9}\big{)}$
$\displaystyle-4h_{12}h_{9}+4h_{9}^{2}-2h_{9}a_{0}$
$\displaystyle-2h_{10}\big{(}2h_{11}+2h_{12}-4h_{9}+a_{0}\big{)}\bigg{]}$
$\displaystyle\times\bigg{[}\big{(}h_{10}+h_{9}\big{)}^{2}a_{0}\bigg{]}^{-1}.$
(21)

The positivity of the spin-two mass in Eq. 20 and the residue in Eq. 21 select
possible real values of the parameters involved. Among these, the requirement
is seen for a positive $a_{0}$. This shows that a massive spin-two is
incompatible with the healthy propagation of the graviton: we must discard it.
This, as can be seen from the determinants, can be accomplished by demanding
$h_{10}=-h_{9}$ or the stronger $h_{10}=h_{9}=0$. The consequences of these
different choices can be appreciated by observing that the theory also
propagates a massive spin-one particle whose mass and residue is given by

$\displaystyle
m^{2}_{1^{+}}=\frac{a_{0}}{h_{9}-h_{10}}\,,\quad\lim_{k^{2}\mapsto
m_{1^{+}}^{2}}\mathcal{D}_{S}(k)\propto\frac{4}{h_{10}-h_{9}}\,.$ (22)

We can therefore explore the possibility of keeping such a state by adopting
$h_{10}=-h_{9}\neq 0$ and studying the consequences for the rest of the
spectrum. It is easy to show that this requires $h_{10}=-h_{9}>0$ and
$a_{0}<0$, so that such propagation can indeed be afforded without spoiling
the gravitational priorities of the model.

A more alarming scenario is presented by the degenerate spin $1^{-}$ sector,
where the restricted determinant shows a quartic equation in the momentum.
Imposing, for instance, $h_{10}=-h_{9}$ we would find

$\displaystyle|a_{i,j}^{\\{1,-\\}}|=\frac{a_{0}^{2}}{32}\Big{(}5\left((h_{12}-h_{11})^{2}+8h_{9}(h_{8}-h_{7})\right)k^{4}+$
$\displaystyle-4a_{0}\left(3h_{11}-3h_{12}-h_{7}+h_{8}+12h_{9}\right)k^{2}+12a_{0}^{2}\Big{)}\,.$
(23)

The determinant in Section III.2 can be read off the denominators of the
$b^{\left\\{1,-\right\\}}_{i,j}$ matrix elements444Unfortunately, the
$b^{\left\\{1,-\right\\}}_{i,j}$ matrices are very large expressions, so
PSALTer frequently suppresses them when attempting to typeset the results for
publication. Although Section III.2 cannot therefore be confirmed from
Appendix B, the full results are available in the Wolfram _Mathematica_
notebook file from which PSALTer is run: this document, along with the source
script, is made available in the supplement [83].. Bona fide unitarity demands
the removal of the quartic term. The absence of a ghostly massive spin-two
particle and the spin-one dipole motivates the defining constraint over the
parameter space. Many (apparently) different solutions can be found by asking
to solve such constraints in terms of different subsets of the couplings. It
is a great advantage that the algorithmic disposition of the spin-parity
approach allows a simple scan over the broad space of solutions. We proceed by
considering the two separate branches obtained from $h_{10}=-h_{9}>0$ and, for
each of the two possibilities, gather the different solutions yielded by
nullifying the quadratic coefficient of $k^{4}$ in Section III.2. The theories
associated with this scan are presented in [83]. Collecting all the masses and
residues for, besides the graviton, the two massive spin-one states of
opposite parity we find, in all cases that

_Linearised zero-torsion Ricci-type theories do not admit simultaneous
propagation of massive spin-one states of opposite parity_.

We can naturally continue by asking for the dismissal of the full $1^{-}$
propagation by setting to zero the coefficient of $k^{2}$ in Section III.2.
Again, we do this by gathering all the relevant equations and solving them for
all the possible subsets of the free parameters, and again we refer to [83].
We find that healthy solutions are available, although not for all the given
cases. In the healthy scenarios, we rediscover the mass/residue ratio of Eq.
22 rephrased in terms of the available couplings.

Finally, we can investigate the case $h_{9}=h_{10}=0$ which removes the
propagation of the massive spin-one state of positive parity Eq. 22 and,
simultaneously, of the massive spin-two. Under such circumstances, the
cancellation of the dipole propagation simplifies to demanding
$(h_{11}-h_{12})^{2}=0$ in Section III.2, namely $h_{11}=h_{12}\neq 0$ (see
Figs. B.6 and B.7) and $h_{11}=h_{12}=0$ (see Figs. B.4 and B.5). For the
second, simpler scenario, we find in the second-order case Fig. B.5

$\displaystyle
m^{2}_{1^{-}}=\frac{-3a_{0}}{h_{7}-h_{8}}\,,\quad\lim_{k^{2}\mapsto
m_{1^{-}}^{2}}\mathcal{D}_{S}(k)\propto\frac{34}{h_{7}-h_{8}}\,.$ (24)

Such a simple setup, which clearly presents a viable propagation, is only
slightly modified when considering the branch $h_{11}=h_{12}\neq 0$, with only
the residue’s form being affected. Again, ghost- and tachyon-freedom can be
accounted for. We conclude, accordingly, stating that, besides the graviton:

_Either a healthy massive vector of negative parity or positive parity
propagates in linearised zero-torsion Ricci-type theories_.

###  III.3 Generic case

####  III.3.1 General properties

The transition to the case of an unconstrained affine connection presents an
obvious growth in computational complexity induced by the multiple components
of Eqs. 5a and 5b and, consequently, by the independence of all three Ricci-
type tensors. The challenges of the associated spectral problem are quite
visible in the cumbersome spin-parity matrices of Appendix C. The general
spectrum associated with Section III is shown in Figs. C.2 and C.3,
respectively for the first- and second-order formulations of the theory. The
inclusion of all the components considerably changes the nature of the
unconstrained spectrum. First, we notice how the massive state of spin-two is
no longer present, the determinant having a simple proportionality to $k^{2}$.
Similarly, Section III.2 is now of the form555Again, the
$b^{\left\\{1,-\right\\}}_{i,j}$ matrices are suppressed in Appendix C, for
full results see the supplement [83].

$\displaystyle|a_{i,j}^{\\{1,-\\}}|$
$\displaystyle=\tfrac{1}{4}a_{0}^{4}k^{2}\Bigl{(}f_{1}a_{0}-5f_{2}k^{2}\Bigr{)},$
(25a) $\displaystyle f_{0}$ $\displaystyle\equiv
c_{10}-c_{11}+c_{12}-16c_{13}+4c_{14}+4c_{15}$ $\displaystyle\ \ \
-c_{7}+c_{8}-c_{9}),$ (25b) $\displaystyle f_{1}$
$\displaystyle\equiv(c_{14}+c_{15})^{2}+4c_{13}(c_{10}-c_{11}+c_{12}$
$\displaystyle\ \ \ -c_{7}+c_{8}-c_{9}),$ (25c)

and the dangerous dipole of Section III.2 leaves space for a massless vector.
That this is indeed the case, and that the pole is not a spurious feature of
the determinant, is demonstrated by the direct computation of the saturated
propagator in the massless limit. For this computation we have to account for
a further, associated peculiarity encountered in this scenario. The rank of
the spin $0^{+}$ sector is now reduced by two, signalling the emergence of an
Abelian symmetry. The presence of this symmetry was predicted by Iosifidis and
Koivisto [84] — it appears whenever squares of the full metric-affine
curvature are added to the Einstein–Hilbert term, and is a remnant of the full
projective symmetry of that term. It is instructive to explicitly show what
this entails in terms of source constraints in the light-like frame
$k=\big{(}\mathcal{E},0,0,\mathcal{E}\big{)}$. We find, together with Eq. 16,
the following reduction

$\displaystyle W^{000}-W^{011}-W^{022}-W^{033}$ $\displaystyle\hskip
50.0pt=W^{300}-W^{311}-W^{322}-W^{333}\,.$ (26)

Accounting for Eq. 26 we recognize four independent states in the massless
limit of the saturated propagator. Two of these are precisely the helicity
states of the graviton, proportional to $-1/a_{0}$ and recognizable in Eq. 17.
The residue of the other two states, while signalling unambiguously the
propagation connected to the massless spin-one state, has an extremely
convoluted form due to the concurrence of many different parameters in its
definition. Nevertheless, the requirement for its positivity has the
manageable structure

$\displaystyle 16c_{13}-4(c_{14}+c_{15})+c_{11}-c_{12}$ $\displaystyle\hskip
30.0pt+c_{7}-c_{8}+c_{9}-c_{10}>0.$ (27)

When searching for massive propagation, similarly to the torsionless case,
both spin-one sectors source one state each, with masses

$\displaystyle m^{2}_{1^{-}}$
$\displaystyle=\Big{[}{a_{0}}(c_{10}-c_{11}+c_{12}-16c_{13}+4c_{14}+4c_{15}$
$\displaystyle\ \ \
-c_{7}+c_{8}-c_{9})\Big{]}\times\Big{[}5\big{(}4c_{13}(c_{10}-c_{11}+c_{12}$
$\displaystyle\ \ \
-c_{7}+c_{8}-c_{9})+(c_{14}+c_{15})^{2}\big{)}\Big{]}^{-1},$ (28a)
$\displaystyle m^{2}_{1^{+}}$
$\displaystyle=-\frac{{a_{0}}}{c_{10}-c_{11}+c_{12}-c_{7}+c_{8}-c_{9}}\,.$
(28b)

The overall survey of the propagating states points, therefore, to three
additional particles populating the spectrum besides the graviton. To solve
the spectral problem we analyze the conditions for their simultaneous
propagation.

####  III.3.2 Allowing the massless vector

The presence of a spin-one massless state can be included in our analysis. The
related phenomenological concerns can then be seen as suggesting
incompleteness, rather than an inconsistency: a mechanism to provide a mass
gap is expected. Under such a hypothesis, we can investigate the coexistence
of such a state with the others. The computation of the residues of the
massive states is carried through the constrained propagator. The explicit
effect of the different gauge symmetries on the sources is extracted in the
rest-mass frame $k=\big{(}m,0,0,0\big{)}$ and gives

$\displaystyle T^{00}=T^{01}=T^{02}=T^{03}=0,$ (29)

for diffeomorphism invariance, and

$\displaystyle W^{000}-W^{011}-W^{022}-W^{033}=0\,,$ (30)

for the extra Abelian symmetry. When testing the sign of each residue, as well
as the masses, with the requirement $a_{0}<0$, we immediately find an
obstruction within the $1^{-}$ sector. Having committed to retaining the
massless propagation, we must simplify the model by removing its massive
counterpart. We proceed, therefore, by considering all the eleven solutions of
$f_{1}=0$ in Eq. 25a and recomputing the residues for the remaining
propagating states. Once more, no viable solutions are found (see [83]).
Finally, we kill the massive $1^{+}$ propagation in Eq. 28b by adding the
further condition

$\displaystyle c_{10}-c_{11}+c_{12}-c_{7}+c_{8}-c_{9}=0.$ (31)

To coherently include both constraints we consider pairs of parameters which
are solutions of the corresponding equation system. Twelve solutions are found
(see [83]), all of them with positive residues for the surviving massless
sector. We can therefore draw the following:

_A healthy massless vector of negative parity propagates in linearised generic
Ricci-type theories_.

Once again, we can make contact with the literature. We notice that Eq. 31
does not eliminate $c_{13}$, which controls the square of the homothetic
curvature. It is known that when this operator is added to the
Einstein–Hilbert term in full MAG geometry, the resulting theory cannot be
distinguished from the vacuum Einstein–Maxwell theory [85, 86] — the extra
massless vector in this case is identified with our $1^{-}$ state.

####  III.3.3 Removing the massless vector

The only way to dispose of the massless vector is to introduce a further
degeneracy in the $1^{-}$ sector. This can be enforced by solving for
$f_{1}=f_{2}=0$ in Eq. 25a. Once more, the spin-parity approach grants us the
possibility to explore the results in a systematic way. The analysis is made
more complicated by the peculiar challenges met in this scenario, where each
solution of the $f_{1}=f_{2}=0$ system affects the form of the gauge symmetry,
thus necessitating, each time, a recomputation of all the main features of the
theory. Despite the demanding computational task (see [83]), the outcome
turned out to be the same for all the (twenty) different solutions defined in
terms of pairs of independent parameters. We can, consequently, present the
results for this scenario by focusing on a particular solution:

$\displaystyle c_{9}$
$\displaystyle=c_{10}-c_{11}+c_{12}+16c_{13}-c_{7}+c_{8},$ (32a)
$\displaystyle c_{15}$ $\displaystyle=8c_{13}-c_{14}\,,$ (32b)

producing the following propagator poles

$\displaystyle m^{2}_{1^{-}}=\frac{{a_{0}}}{20c_{13}},\quad
m^{2}_{1^{+}}=\frac{{a_{0}}}{16c_{13}}\,.$ (33)

The correlation among the masses of the two spin-one sectors is not an
accident of the chosen solution but illustrates a common feature: the strict
proportionality $m^{2}_{1^{-}}/m^{2}_{1^{+}}=4/5$. Such correlation signals
the impossibility of removing the propagation of one state without interfering
with the other. To assess the nature of the states we need to compute the
saturated propagator in the presence of the augmented gauge constraints
defined, for instance, by Eqs. 32a and 32b. For the massive limit of
$k=\big{(}m,0,0,0\big{)}$ for some $m$ we must consider, on top of Eq. 29 and
Eq. 30 the extra degeneracy of the $1^{-}$ sector. Being generated by a three-
dimensional vector, these source constraints take the form

$\displaystyle
W^{00i}-W^{0i0}=\frac{a_{0}}{m^{2}}\frac{W^{i00}-W^{i11}-W^{i22}-W^{i33}}{c_{11}-c_{12}-4c_{14}+2c_{7}-2c_{8}}\,.$
(34)

The odd appearance of couplings within the definitions of the symmetry is a
curious feature of this analysis. This, however, does not present any
theoretical downsides if we consider that such parameters will be normalized
to pure numbers in the quadratic, final Lagrangian. The massive limit confirms
the presence of the propagation of three states as expected for massive spin-
one particles. For the two different sectors, and the representative selection
of dependent parameters shown in Eqs. 32a and 32b, we find the corresponding
residue

$\displaystyle\lim_{k^{2}\mapsto m_{1^{-}}^{2}}$
$\displaystyle\mathcal{D}_{S}(k)\propto\frac{1}{50c_{13}}\Big{[}11+\Big{(}50(6c_{13}-c_{14})(c_{12}-c_{11}$
$\displaystyle\ \ \
+2(6c_{13}+c_{14}-c_{7}+c_{8}))\Big{)}\times\Big{(}(c_{12}-c_{11}$
$\displaystyle\ \ \ +4c_{14}-2c_{7}+2c_{8})^{2}\Big{)}^{-1}\Big{]},$ (35a)
$\displaystyle\lim_{k^{2}\mapsto m_{1^{+}}^{2}}$
$\displaystyle\mathcal{D}_{S}(k)\propto-\frac{1}{4096\,c_{13}^{3}}\Big{[}c_{11}^{2}-2c_{11}(c_{12}+2(8c_{13}-c_{7}$
$\displaystyle\ \ \
+c_{8}))+c_{12}^{2}+4c_{12}(8c_{13}-c_{7}+c_{8})+768c_{13}^{2}$
$\displaystyle\ \ \ +64c_{13}(c_{8}-c_{7})+4(c_{7}-c_{8})^{2}\Big{]}\,.$ (35b)

While the requirement of positivity of $\lim_{k^{2}\mapsto
m_{1^{-}}^{2}}\mathcal{D}_{S}(k)$ does not challenge the unitarity of the
graviton sector, nor the tachyon-freedom conditions over the vector masses,
this is not the case for $\lim_{k^{2}\mapsto
m_{1^{+}}^{2}}\mathcal{D}_{S}(k)>0$, which calls for $c_{13}<0$. Again,
different choices in solving for $f_{1}=f_{2}=0$ do affect the form of the
residues, but no simultaneous solutions are found (see [83]). The correlation
existing for the propagation of both massive vectors in Eq. 33 prohibits,
therefore, both massive states from appearing in a healthy spectrum.

##  IV Conclusions

It is difficult to overestimate the importance of accommodating the absence of
ghosts and tachyons in quantum field theory. Control over unitarity is key to
understanding the shape of possible new theories and future extensions of the
current models. Spin-parity projectors provide a computational framework for
fully controlling the propagation of quadratic Lagrangia, which lends itself
well to computer implementation. Once the needed operators are collected, the
spectral problem is basically solved [52, 53]. The output is unambiguous,
given the direct access to the propagator, and does not rely upon intricate
field redefinitions or the introduction of spurious fields to achieve
reductions to known cases. In this work, we have adopted the spin-parity
formalism to illustrate its reach and the capacity to tackle a broader set of
operators than previously possible. We have made a thorough survey of the
Ricci-type MAG operator space, but our analysis is not intended to be
exhaustive. The point we are making is that if further special cases turn out
to be of interest in the future, then it will be economical to test them using
our approach. Recently, some spectral analyses of the PGT and Weyl gauge
theory (WGT) have been made [52, 69, 53], which really _are_ exhaustive. The
trick to making exhaustive surveys is to recursively search over the root
system of the wave operator determinant. This would make an appealing (and
apparently straightforward) extension to our current PSALTer program, but we
defer it to future work.

There are two key limitations to our approach. Firstly, the authors of [35]
are able to extend their analyses to particle spectra on Friedmann
backgrounds: we _cannot_ do this. There is some hope for the extension of the
spectral algorithm to de Sitter backgrounds in the near future [87], but
further applications to non-maximally-symmetric spacetimes are currently
speculative. Secondly, the theories in Sections III, III.1 and III.2 may
propagate more species species in their full nonlinear dynamics than are
revealed in the spectral analysis. This is already known to happen in the case
of the theory in Section III.2, for which the $1^{+}$ and $1^{-}$ torsion
modes are strongly coupled near Minkowski spacetime [54]. When this happens,
it means that the model is inherently non-perturbative around Minkowski
spacetime, so the quadratic approximation in Section II.2 is just a fictional
model which has nothing to do with the actual physics. It is hard to see how
this cannot be a pathology (with or without ghost-tachyon-freedom of the
strongly coupled modes), and the only sure way to diagnose it is via a
nonlinear Hamiltonian analysis [88]. It is possible that the methods of [35]
are also sensitive to strong coupling, if for example propagating d.o.f are
lost as the Friedmann background is deformed into the Minkowski background.
However, it is not clear that such an approach would always detect the problem
when it exists. Attempts at computer algebra Hamiltonian analysis were made in
[89], but the implementation was not theory-agnostic (restricted to PGT). The
PSALTer software showcased here is theory-agnostic by design. It can be
downloaded from github.com/wevbarker/PSALTer666In the longer term we hope that
PSALTer will become an official contribution to the _xAct_ project [90, 91,
92, 93, 94]..

###### Acknowledgements.

We are particularly indebted to Jaakko Annala and Syksy Räsänen for multiple
in-person discussions, and to Damianos Iosifidis, Roberto Percacci and
Sebastian Zell for useful correspondence. W. B. is grateful for the kind
hospitality of Leiden University and the Lorentz Institute, and the support of
Girton College, Cambridge. The work of C. M. was supported by the Estonian
Research Council grants PRG1677, RVTT3, RVTT7, and the CoE program TK202 “
_Fundamental Universe_ ”. This work used the Newton server, access to which
was provisioned by Will Handley, and funded through an ERC grant. This work
used the DiRAC Data Intensive service (CSD3 www.csd3.cam.ac.uk) at the
University of Cambridge, managed by the University of Cambridge University
Information Services on behalf of the STFC DiRAC HPC Facility
(www.dirac.ac.uk). The DiRAC component of CSD3 at Cambridge was funded by
BEIS, UKRI and STFC capital funding and STFC operations grants. DiRAC is part
of the UKRI Digital Research Infrastructure.

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##  Appendix A Zero non-metricity with PSALTer

The full spectrum of the general theory in Section III.1 is given in Fig. A.2.
Whilst the formulation in Section III.1 is consistent with the MAG conventions
set out in Section II.1, we take the unusual approach of re-formulating the
theory as a Poincaré gauge theory (PGT) [95] when presenting the PSALTer
analysis. This allows us to recycle a pre-existing PGT kinematic module within
PSALTer, and meanwhile the spectral analysis of Fig. A.2 is already well-known
anyway [54]. The PGT kinematic module is displayed in Fig. A.1.

Whilst our MAG conventions are designed to be identical to [28], our PGT
conventions will be identical to those in [62, 62, 96, 97, 88, 98, 99]. To
define the PGT, we introduce $\tensor{e}{{}^{i}_{\mu}}$ and
$\tensor{e}{{}_{i}^{\mu}}$ as the co-tetrad and tetrad components, which are
associated with Roman Lorentz (i.e. anholonomic) indices, so that we can
compare with the MAG metric
$\tensor{e}{{}^{i}_{\mu}}\tensor{e}{{}^{j}_{\nu}}\tensor{\eta}{{}_{ij}}\cong\tensor{g}{{}_{\mu\nu}}$
and inverse
$\tensor{e}{{}_{i}^{\mu}}\tensor{e}{{}_{j}^{\nu}}\tensor{\eta}{{}^{ij}}\cong\tensor{g}{{}^{\mu\nu}}$
with identities
$\tensor{e}{{}^{i}_{\mu}}\tensor{e}{{}_{i}^{\nu}}\equiv\tensor{*}{\delta}{{}_{\mu}^{\nu}}$
and
$\tensor{e}{{}^{i}_{\mu}}\tensor{e}{{}_{j}^{\mu}}\equiv\tensor{*}{\delta}{{}_{j}^{i}}$
as kinematic restrictions. There is also a spin connection
$\tensor{\mathscr{A}}{{}^{ij}_{\mu}}\equiv\tensor{\mathscr{A}}{{}^{[ij]}_{\mu}}$,
so that the PGT torsion and PGT curvature are

$\displaystyle\tensor{\mathcal{T}}{{}^{k}_{ij}}$ $\displaystyle\equiv
2\tensor{e}{{}_{i}^{\mu}}\tensor{e}{{}_{j}^{\nu}}\big{(}\tensor{\partial}{{}_{[\mu|}}\tensor{e}{{}^{k}_{|\nu]}}+\tensor{\mathscr{A}}{{}^{k}_{m[\mu|}}\tensor{e}{{}^{m}_{|\nu]}}\big{)},$
(36a) $\displaystyle\tensor{\mathcal{R}}{{}^{kl}_{ij}}$ $\displaystyle\equiv
2\tensor{e}{{}_{i}^{\mu}}\tensor{e}{{}_{j}^{\nu}}\big{(}\tensor{\partial}{{}_{[\mu|}}\tensor{\mathscr{A}}{{}^{kl}_{|\nu]}}+\tensor{\mathscr{A}}{{}^{k}_{m[\mu|}}\tensor{\mathscr{A}}{{}^{ml}_{|\nu]}}\big{)}.$
(36b)

The MAG torsion and curvature in Eqs. 1 and 2 are precisely analogous to the
PGT counterparts in Eqs. 36a and 36b respectively, through the relations
$\tensor{\mathcal{T}}{{}_{\mu}^{\alpha}{}_{\nu}}\cong\tensor{e}{{}^{i}_{\mu}}\tensor{e}{{}_{k}^{\alpha}}\tensor{e}{{}^{j}_{\nu}}\tensor{\mathcal{T}}{{}^{k}_{ij}}$
and
$\tensor{\mathcal{F}}{{}_{\mu\nu}^{\rho}{}_{\sigma}}\cong\tensor{e}{{}^{i}_{\mu}}\tensor{e}{{}^{j}_{\nu}}\tensor{e}{{}_{k}^{\rho}}\tensor{e}{{}_{l}{}_{\sigma}}\tensor{\mathcal{R}}{{}^{k}_{lij}}$,
where we pay attention to the different ordering of the indices according to
the two conventions. In terms of the PGT field strength tensors, the action in
Section III.1 corresponds to

$S[e,\mathscr{A}]=\int\mathrm{d}^{4}xe\bigg{[}\alpha_{0}\mathcal{R}+\tensor{\mathcal{R}}{{}^{ij}}\big{(}\alpha_{1}\tensor{\mathcal{R}}{{}_{ij}}+\alpha_{2}\tensor{\mathcal{R}}{{}_{ji}}\big{)}\bigg{]},$
(37)

where $S[e,\mathscr{A}]\cong S[g,A]$ and we adopt the dimensionful coupling
$\alpha_{0}$ and dimensionless couplings $\alpha_{1}$ and $\alpha_{2}$ in
place of $a_{0}$, $g_{1}$ and $g_{2}$. The contractions are defined
$\tensor{\mathcal{R}}{{}_{ij}}\equiv\tensor{\mathcal{R}}{{}^{l}_{ilj}}$ and
$\mathcal{R}\equiv\tensor{\mathcal{R}}{{}^{i}_{i}}$ with the measure
$e\equiv\det\big{(}\tensor{e}{{}^{i}_{\mu}}\big{)}\cong\sqrt{-g}$. In the
weak-field regime, we take $\tensor{\mathscr{A}}{{}^{ij}_{\mu}}$ to be
inherently perturbative, and we define the exact tetrad perturbation
$\tensor{e}{{}_{i}^{\mu}}\equiv\tensor{*}{\delta}{{}_{i}^{\mu}}+\tensor{f}{{}_{i}^{\mu}}$,
i.e. the ‘Kronecker’ choice of Minkowski vacuum [62, 100, 101].

To lowest order in the quadratic action, the Greek and Roman indices are then
interchangeable — indeed PSALTer only knows about one set of Lorentz indices
on the Minkowski background, and these are strictly associated with Greek
indices which represent Cartesian coordinates. There are 16 d.o.f in
$\tensor{f}{{}_{i}^{\mu}}$ and 24 d.o.f in
$\tensor{\mathscr{A}}{{}^{ij}_{\mu}}$. The latter can clearly be accounted
for, in the second-order formulation, by the d.o.f in Eq. 13. Most of the
former are accounted for by the metric d.o.f in Eq. 5b, but there are six
further d.o.f in the antisymmetric part of the tetrad which do not appear in
MAG. This is not a problem, because, these six d.o.f are immediately
eliminated by the six gauge generators of the Lorentz symmetry, part of the
Poincaré symmetry, which also is not visible in the MAG. As a consequence, the
spin-one matrices in Fig. A.2 have two rows and two columns more than they
would do in the MAG formulation, but the dimension of their null space also
increases by two. Kinematic extensions of the theory which are cancelled by
symmetries in this way do not alter the physics, and in this sense we
understand the zero-non-metricity MAG and the PGT to be equivalent theories.

Conjugate to the tetrad perturbation $\tensor{f}{{}_{i}^{\mu}}$ and the spin
connection $\tensor{\mathscr{A}}{{}^{ij}_{\mu}}$ are the translational source
(asymmetric stress-energy tensor) $\tensor{\tau}{{}^{i}_{\mu}}$ and matter
spin current $\tensor{\sigma}{{}_{ij}^{\mu}}$ [17, 102, 21]. The reduced-index
$\mathrm{SO}(3)$ irreducible parts of these fields and sources label the rows
and columns of the matrices in Fig. A.2, and have spin-parity ($J^{P}$) labels
to identify them. Duplicate $J^{P}$ states are distinguished by additional
parallel ($\parallel$) and perpendicular ($\perp$) labels — but there is no
significant meaning behind these auxiliary labels.

Figure A.1:  Kinematic structure of Poincaré gauge theory (PGT), as used in
Fig. A.2. We re-purpose the PGT kinematic module in PSALTer for the study of
zero-non-metricity MAG in Section III.1. Because of kinematic differences
between the PGT and MAG (which are nullified by the extra Lorentz symmetry in
PGT), the irreps displayed here do not completely map to those in Eqs. 5b and
13. The key point is that the spin connection
$\tensor{\mathscr{A}}{{}^{ij}_{\mu}}\equiv\tensor{\mathscr{A}}{{}^{[ij]}_{\mu}}$
maps to the antisymmetric distortion
$\tensor{\Delta}{{}_{\lambda\mu\nu}}\equiv\tensor{\Delta}{{}_{\lambda[\mu\nu]}}$,
and the asymmetric tetrad perturbation $\tensor{f}{{}_{i}^{\mu}}$ contains at
least the d.o.f in the symmetric metric perturbation
$\tensor{h}{{}_{\mu\nu}}\equiv\tensor{h}{{}_{(\mu\nu)}}$. Note that the
$2^{-}$ state has a hidden multi-term cyclic symmetry on all its indices,
which is not accommodated by the C language implementation of the
Butler–Portugal algorithm [92]. Figure A.2:  The full spectrum of the general
theory in Section III.1, but interpreted as a Poincaré gauge theory (PGT) in
Eq. 37. Kinematically, the 10 d.o.f of the metric are replaced with the 16
d.o.f of the tetrad field. Consequently however, the additional gauging of the
Lorentz group resusts in six extra gauge generatrs on top of the
diffeomorphism (translation) generators, so the formulations are not
physically distinguishable. All the quantities in this output are defined in
Fig. A.1.

##  Appendix B Zero torsion with PSALTer

Unlike in Appendix A, our PSALTer analysis corresponding to Section III.2 is
fully grounded in the MAG formulation. The zero-torsion MAG kinematic module
is displayed in Fig. B.1. The first-order analyses in Figs. B.2, B.4 and B.6
share all our notational conventions above: the fields
$\tensor{h}{{}_{\mu\nu}}$ and $\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}$ are
perturbative. To reach the second-order formulation, we only have to edit the
quadratic action before substituting into the ParticleSpectrum function (which
is the main function provided by the PSALTer package). The reparameterisation
used to transform the quadratic action is

$\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}\mapsto\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}+\frac{1}{2}\big{(}2\tensor{\partial}{{}_{(\mu|}}\tensor{h}{{}_{\lambda|\nu)}}-\tensor{\partial}{{}_{\lambda}}\tensor{h}{{}_{\mu\nu}}\big{)}.$
(38)

To lowest order in perturbative fields, Eq. 38 captures the transition from
$\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}$ to
$\tensor{\Delta}{{}_{\mu}^{\rho}{}_{\nu}}$ set out in Section II.1. The
notation is slightly abusive, because $\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}$ on
the RHS of Eq. 38 is really $\tensor{\Delta}{{}_{\mu}^{\rho}{}_{\nu}}$. But
since there is no advantage in defining a new kinematic module for PSALTer
just to avoid the notational conflict, we therefore lazily recycle the first-
order zero-torsion MAG module for all our second-order calculations.

Conjugate to the metric perturbation $\tensor{h}{{}_{\mu\nu}}$ and the affine
connection $\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}$ are the (symmetric) stress-
energy tensor $\tensor{T}{{}^{\mu\nu}}$ and the current
$\tensor{W}{{}^{\mu}_{\rho}{}^{\nu}}$ which in MAG has become known as the
_hypermomentum_. As with the PGT notation in Fig. A.2, the $J^{P}$ states are
labelled as such. To distinguish the duplicate $J^{P}$ states, apart from the
($\parallel$) and ($\perp$) symbols, we use the letters ($\mathsf{s}$),
($\mathsf{h}$) and ($\mathsf{t}$) — once again there is no significant meaning
behind these labels. Different labels (numerical subscripts) are used in
Section III.2.

We show the general analysis of the theory in Section III.2 in Figs. B.2 and
B.3, respectively for the first- and second-order formulations of the model.
In Figs. B.4, B.5, B.6 and B.7 we consider tuned special-cases of the model in
which the massive $1^{-}$ state is allowed to propagate.

Further theories considered in Section III.2, whose matrices are too
cumbersome for the appendices, are presented in [83].

Figure B.1:  Kinematic structure of zero-torsion MAG, as studied throughout
Section III.2. The $\mathrm{SO}(3)$ irreps precisely correspond to those in
Sections III.2 and 5b, though the labelling of duplicate $J^{P}$ states is
different from that in [28]. The key point is that the connection field
carries an extra symmetry restriction
$\tensor{A}{{}_{\mu}^{\rho}{}_{\nu}}\equiv\tensor{A}{{}_{(\mu|}^{\rho}{}_{|\nu)}}$,
and by referring to Eq. 1 we see that this kills off the torsion in the first-
order formulation. In moving to the second-order formulation, we make a slight
notational abuse in Eq. 38, but from Eq. 4 we see that the effect will still
be as desired if we keep using this kinematic module. As with Fig. A.1, the
$2^{-}$ and $3^{-}$ states have extra cyclic symmetries which are hidden.
These definitions are used in Figs. B.2, B.3, B.4, B.5, B.6 and B.7. Figure
B.2:  The full spectrum of the general theory in Section III.2. No general
unitarity conditions are obtained, without further tuning. All the quantities
in this output are defined in Fig. B.1. Figure B.3:  The results in Fig. B.2
repeated in the second-order formulation. Note that the quadratic action in
this case contains very many more operators than does Fig. B.2. The matrix
elements and the forms of the pole residues are expected to change, but the
mass spectrum is the same. Once again, the unitarity conditions are not
obtained for the full theory: despite the apparent changes to the residues,
such conditions should be invariant under re-parameterisations. All the
quantities in this output are defined in Fig. B.1. Figure B.4:  The spectrum
of the theory in Section III.2 with the additional constraints
$c_{9}=c_{10}=c_{11}=c_{12}=0$. These are sufficient to eliminate the $2^{+}$
and $1^{+}$ massive states, leaving only the massive $1^{-}$ state in Eq. 24.
The overall theory is clearly unitary. All the quantities in this output are
defined in Fig. B.1. Figure B.5:  The results in Fig. B.4 repeated in the
second-order formulation. Note the much larger quadratic expansion, but the
consistent mass spectrum and unitarity conditions. The results are used in Eq.
24. All the quantities in this output are defined in Fig. B.1. Figure B.6:
The spectrum of the theory in Section III.2 with the additional constraints
$c_{9}=c_{10}=c_{11}-c_{12}=0$. As with Fig. B.4 these are sufficient to
eliminate the $2^{+}$ and $1^{+}$ massive states, leaving only the massive
$1^{-}$ state in Eq. 24. The overall theory is clearly unitary. All the
quantities in this output are defined in Fig. B.1. Figure B.7:  The results in
Fig. B.6 repeated in the second-order formulation. Note the much larger
quadratic expansion, but the consistent mass spectrum and unitarity
conditions. All the quantities in this output are defined in Fig. B.1.

##  Appendix C Generic case with PSALTer

As with Appendix B, our PSALTer analysis corresponding to Section III.3 is
fully grounded in the MAG formulation. The generic MAG kinematic module is
displayed in Fig. C.1. The generic first-order MAG kinematic module is of
course larger than the zero-torsion counterpart in Fig. B.1. By comparing
Section III.2 with Eq. 5a we see that the spin-zero and spin-one matrices will
have extra rows and columns. As with Appendix B, the transition to the second-
order formulation is made using Eq. 38. The PSALTer labelling of duplicate
$J^{P}$ states is again different from that shown in Eqs. 5a and 5b, and a new
label ($\mathsf{a}$) is introduced.

The full spectrum of the general theory in Section III using the first-order
formulation is given in Fig. C.2. The equivalent result in the second-order
formulation is given in Fig. C.3.

Further theories considered in Section III.3, whose matrices are too
cumbersome for the appendices, are presented in [83].

Figure C.1:  Kinematic structure of the unrestricted MAG, as studied
throughout Section III.3. The $\mathrm{SO}(3)$ irreps precisely correspond to
those in Eqs. 5a and 5b, though the labelling of duplicate $J^{P}$ states is
different from that in [28]. As with Figs. A.1 and B.1, the $2^{-}$ and
$3^{-}$ states have extra cyclic symmetries which are hidden. These
definitions are used in Figs. C.2 and C.3. Figure C.2:  The full spectrum of
the general theory in Section III using the first-order formulation. As with
Figs. B.2 and B.3 completely general unitarity conditions cannot be found
automatically within one minute, but we investigate the conditions in Section
III.3 based on the masses and residues in these results. All the quantities in
this output are defined in Fig. C.1. Figure C.3:  The full spectrum of the
general theory in Section III using the second-order formulation. Note that
the matrix elements differ from Fig. C.2, and the form of the gauge symmetries
and pole residues seem to differ, but the mass spectrum is the same. Moreover,
the number of gauge generators is the same, and the pole residues are expected
to have the same implications for the unitarity. Note also the much larger
quadratic expansion of the action. All the quantities in this output are
defined in Fig. C.1.