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

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 Poincar\'e 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 modelbuilding [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].Metric-affine gravity (MAG)  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 selfconsistency constraints, such as unitarity and elimination of tachyons [31,35,36,38,47,48,[53][54][55][56][57][58][59][60].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.
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 The effects of metric-affine curvature     in Eq. ( 2) -vector rotation after parallel transport in a closed loop -torsion     in Eq. ( 1) -non-closure of parallelograms formed from (infinitesimally) parallel-transported vectors -and non-metricity   in Eq. (1) -change in vector length under parallel transport.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.[61][62][63][64][65][66]), this particular space of operators was first properly charted by Annala and Räsänen [38].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 [67] 1 , but in this paper we use it to confirm all our results (see Appendices A to 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 A and briefly presenting our spectral algo-rithm in Section II B and the computer implementation in Section II C. In Section III we present all of the results, respectively for the vanishing non-metricity kinematic limit in Section III A and the vanishing torsion kinematic limit in Section III B, and for the full MAG in Section III C. 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 [53,[71][72][73][74].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 [58], with follow-up papers in [59,75].Further information about the method can also be found in (e.g.) [31, 36, 53-55, 57, 76] -but it is [58] which provides the most concise introduction for our purposes.

A. 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  and  of spin and parity, enumerating the irreducible representation under the SU(2) little group.In general, further reduction to the U(1) little group of massless particles and helicity states can be done.This results in no practical improvement, making the decomposition into SU(2) representations generic enough.In the case of MAG, the needed set of projection operators has been recently completed [31,77], developing the past studies of [35, 36, 53-55, 57, 78].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 [31,54,58,79] for further details.  .These new d.o.f are often partitioned into the torsion and non-metricity tensors The two tensors in Eq. ( 1) are geometric counterparts to the metric-affine (i.e.non-Riemannian) curvature 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 14)   ≡    ,  (13)   ≡     , (3) by which we just mean the contractions with two free indices.The quantity   is the homothetic curvature [80,81], while  (13)   and  (14)   are variously pseudo-Ricci tensors [31], or  (14)   is the co-Ricci [80].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  ≡    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   and the 64 new d.o.f in     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 In the second-order parameterisation Eq. ( 2) is expanded into the Riemannian curvature (which naturally has secondderivatives in   ) and many other terms which are (Levi-Civita) covariant derivatives and second powers of     and   .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 metricbased gravity to an asymmetric rank-three matter field Δ    .We are free to work with either set of variables due to reparameterisation 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      and a metric perturbation   ≡   + ℎ  .These perturbations carry multiple massive particle states where we used the compact notation    in referring to the j-th representation of spin  and parity  .Enumeration convention is adopted from [31].From Eqs. (5a) and (5b) the 64 and 10 d.o.f can be recovered respectively by summing the multiplicities 2 + 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

B. Saturated propagator and particle spectra
Following [79], we use a synthetic notation to describe the quadratic (i.e.perturbative) action in momentum space where () is the Fourier-transformed kinetic term (wave operator) and we have introduced a linear coupling between the fields, collectively labelled as Φ(), and a source  ().Connecting this to the specific formulation in Section II A, we identify Φ as the collection of perturbative fields ℎ  and     .Within the quadratic approximation, the indices on these fields are raised and lowered using   and   , which are nondynamical, and the Greek indices refer to Cartesian coordinates on the Minkowski background 2 .Conjugate to Φ, the fields  in MAG are the symmetric matter stress-energy tensor   and the rank-three current     .This latter current is sometimes called the hypermomentum [14,27,[82][83][84][85][86].In the second-order formulation, a separate current must be defined as conjugate to Δ    -we do not bother to ascribe it another symbol.
The propagating states will appear in the form of isolated poles for the propagator () defined through the equation () ⋅ () ≡ 1.The use of projection operators  , { , } drastically simplifies the solution of this inversion problem.By exploiting the defining properties 2 Indeed, it should be explicitly stated that along with these raising and lowering rules     is a dynamical tensor field in the quadratic approximation.Geometrically,     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, Δ    is already geometrically a tensor in the second-order formulation of MAG.The re-parameterisation is shown explicitly in Eq. (B1), and the key point is that the linearised Levi-Civita connection is also tensorial at lowest order in perturbations because the partial derivative   is covariant at that order.it is possible to decompose the kinetic term as where the tortuous fabric of the indices is reformulated in terms of the simpler spin-parity matrices  { , }

𝑖,𝑗
, obtained by tracing over the Lorentz indices The orthonormality of the projection operators in Eq. (7b) reduces the computation of the propagator () to an inversion problem for the matrices  , 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 J ()

C. 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   ().Such methods avoid the difficult computation of the source constraint equations Eq. (11b) in the massless limit.However, as detailed in [58,67,79], the constraint equations can be fully resolved by choosing a suitable reference frame (with  2 = 0 as a limit of  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 [60,68] and the core of this work's conclusions, are obtained with the use of PSALTer [67], a Wolfram Language implementation of these ideas.The PSALTer software can automatically return the spectrum of any tensorvalued 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 to B.7, C.2 and C.3.These are vector graphics which contain the following information; • The linearised (quadratic) action in Eq. ( 6), 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  { , } , 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  { , } , 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 [87,88] (i.e. a uniquely defined gauge fixing) to obtain these coefficients.
• Automatically computed: the source constraints    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  , parity  , 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 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 Riccitype MAG in the first-order formulation is where we borrow the numbering of Lagrangian couplings directly from [31] 3 .As emphasised in Section II A, re-parameterisation invariance leads to the equivalence [, ] ≅ [, Δ] with the second-order formalism.The computational algorithm sketched in Section II B can obviously handle both representations of the dynamical fields and our tests have adopted both approaches as a further selfconsistency 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).

A. 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 Δ  ≡ −Δ  nullifies   ≡ 0. This achieves a reduction of the spin-parity sectors of Eq. (5a) reflected by a further redundancy in the number of independent Ricci-type contractions The action Eq. ( 12) is therefore simplified into Note that we follow [31] in re-labelling the dimensionless coefficients from   to   when passing from Eq. ( 12) to Eq. ( 15) 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 B and III C) 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 D ().The shape of the residue and the position of the singularity will determine the nature of the propagating particles.Once the degeneracies of  {1,−} , and  {0,+} , are removed, we immediately find that no massive poles are present.We can therefore extract the known result [38]: 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  {1,−} , and  {0,+}

B. 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 spinparity 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 Δ  ≡ Δ  .
The available particle content in the symmetric distortion is then which causes   ≡ −2 (13)  [] , leaving seven independent combinations Note once again that we follow [31] in re-labelling the dimensionless coefficients from   to ℎ  when passing from Eq. ( 12) to Eq. ( 19) by kinematic restriction.The kinematic structures in Eqs.(5b) and ( 18 find that it is impossible to impose further gauge symmetries, unless  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 A, we can compute the positions of the zeroes for all the determinants.Already for the 2 + sector we find that a state of mass is allowed to propagate whenever ℎ 10 + ℎ 9 ≠ 0. The computation of the limit  2  →  2 2 + gives rise to the following pole residue, where we suppress the positive-definite quadratic form in the 2 + sources lim ] 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  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 , lim We can therefore explore the possibility of keeping such a state by adopting ℎ 10 = −ℎ 9 ≠ 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  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 The determinant in Eq. ( 23) can be read off the denominators of the  {1,−} , matrix elements 4 .Bona fide unitarity demands 4 Unfortunately, the  {1,−} , matrices are very large expressions, so PSALTer 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  4 in Eq. ( 23).The theories associated with this scan are presented in [89].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  2 in Eq. ( 23).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 [89].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 Eq. ( 23 Such a simple setup, which clearly presents a viable propagation, is only slightly modified when considering the branch ℎ 11 = ℎ 12 ≠ 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.
frequently suppresses them when attempting to typeset the results for publication.Although Eq. ( 23) 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 [89].

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 Eq. ( 12) 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  2 .Similarly, Eq. ( 23) is now of the form 5 and the dangerous dipole of Eq. ( 23) 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 [90] -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  = ( , 0, 0,  ) .We find, together with Eq. ( 16), the following reduction 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∕ 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 5 Again, the 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.

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  = ( , 0, 0, 0 ) and gives for diffeomorphism invariance, and for the extra Abelian symmetry.When testing the sign of each residue, as well as the masses, with the requirement  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  1 = 0 in Eq. (25a) and recomputing the residues for the remaining propagating states.Once more, no viable solutions are found (see [89]).Finally, we kill the massive 1 + propagation in Eq. (28b) by adding the further condition To coherently include both constraints we consider pairs of parameters which are solutions of the corresponding equation system.Twelve solutions are found (see [89]), 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  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 [91,92] -the extra massless vector in this case is identified with our 1 − state.

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  1 =  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  1 =  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 [89]), 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: producing the following propagator poles 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  2 1 − ∕ 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  = ( , 0, 0, 0 ) for some  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 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 lim   () 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  2  → 2 1 +   () > 0, which calls for  13 < 0.
Again, different choices in solving for  1 =  2 = 0 do affect the form of the residues, but no simultaneous solutions are found (see [89]).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 Lagrangian, which lends itself well to computer implementation.Once the needed operators are collected, the spectral problem is basically solved [58,59].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 [58,59,75], 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 [38] are able to extend their analyses to particle spec-tra 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 [93], but further applications to non-maximally-symmetric spacetimes are currently speculative [94].Secondly, the theories in Eqs. ( 12), ( 15) and ( 19) may propagate more 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 Eq. ( 19), for which the 1 + and 1 − torsion modes are strongly coupled near Minkowski spacetime [60].When this happens, it means that the model is inherently non-perturbative around Minkowski spacetime, so the quadratic approximation in Eq. ( 6) 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 [95].It is possible that the methods of [38] 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 [96], 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/PSALTer 6 .
we take the unusual approach of re-formulating the theory as a Poincaré gauge theory (PGT) [102] when presenting the PSALTer analysis.This allows us to recycle a pre-existing PGT kinematic module within PSALTer, and meanwhile the spectral analysis of Whilst our MAG conventions are designed to be identical to [31], our PGT conventions will be identical to those in [68,95,[103][104][105][106] ) , (A1a) ) . (A1b) The MAG torsion and curvature in Eqs. ( 1) and ( 2 (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-nonmetricity MAG and the PGT to be equivalent theories.

Conjugate to the tetrad perturbation 𝑓 𝜇
and the spin connection A   are the translational source (asymmetric stressenergy tensor)    and matter spin current    [17,21,109].The reduced-index 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 (  ) labels to identify them.Duplicate   states are distinguished by additional parallel (∥) and perpendicular (⟂) labels -but there is no significant meaning behind these auxiliary labels.

B. Zero torsion with PSALTer
Unlike in Appendix A, our PSALTer analysis corresponding to Section III B is fully grounded in the MAG formulation.The zero-torsion MAG kinematic module is displayed in To lowest order in perturbative fields, Eq. (B1) captures the transition from     to Δ    set out in Section II A. The notation is slightly abusive, because     on the RHS of Eq. (B1) is really Δ    .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 ℎ  and the affine connection     are the (symmetric) stress-energy tensor   and the current     which in MAG has become known as the hypermomentum.As with the PGT notation in Fig. A.2, the   states are labelled as such.To distinguish the duplicate   states, apart from the (∥) and (⟂) symbols, we use the letters (), () and () -once again there is no significant meaning behind these labels.Different labels (numerical subscripts) are used in Eq. (18).
We show the general analysis of the theory in Eq. (19) in Further theories considered in Section III B, whose matrices are too cumbersome for the appendices, are presented in [89]. maps to the antisymmetric distortion Δ  ≡ Δ  [𝜇𝜈] , and the asymmetric tetrad perturbation    contains at least the d.o.f in the symmetric metric perturbation ℎ  ≡ ℎ () .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 [99].

C. Generic case with PSALTer
As with Appendix B, our PSALTer analysis corresponding to Section III C is fully grounded in the MAG formulation.The generic MAG kinematic module is displayed in  18) with Eq. (5a) we see that the spin-zero and spinone matrices will have extra rows and columns.As with Ap-pendix B, the transition to the second-order formulation is made using Eq.(B1).The PSALTer labelling of duplicate   states is again different from that shown in Eqs.(5a) and (5b), and a new label () is introduced.
The full spectrum of the general theory in Eq. ( 12) using the first-order formulation is given in Further theories considered in Section III C, whose matrices are too cumbersome for the appendices, are presented in [89].

Wave operator and propagator
Massive and massless spectra Unitarity conditions α.  18), though the labelling of duplicate   states is different from that in [31].The key point is that the connection field carries an extra symmetry restriction     ≡   (| |) , 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. (B1), but from Eq. (4) we see that the effect will still be as desired if we keep using this kinematic module.As with Fig. C.1.
) 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 Eq. (19) displayed in Fig. B.2.After simple inspections of the determinants, we
Fig. B.1.The first-order analyses in Figs.B.2, B.4 and B.6 share all our notational conventions above: the fields ℎ  and     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 Figs. B.2 and B.3, respectively for the first-and secondorder formulations of the model.In Figs.B.4 to B.7 we consider tuned special-cases of the model in which the massive 1 − state is allowed to propagate.

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 A. 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 A   ≡ A[] 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 Eq. ( Fig. C.2.The equivalent result in the second-order formulation is given in Fig. C.3.

Figure A. 2 .Figure B. 1 .
Figure A.2.The full spectrum of the general theory in Eq. (15), but interpreted as a Poincaré gauge theory (PGT) in Eq. (A2).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.

Figure B. 3 .
Figure B.1.Kinematic structure of zero-torsion MAG, as studied throughout Section III B. The SO(3) irreps precisely correspond to those in Eqs.(5b) and (18), though the labelling of duplicate   states is different from that in[31].The key point is that the connection field carries an extra symmetry restriction     ≡   (| |) , 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. (B1), 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 to B.7.

Figure B. 4 .
Figure B.4.The spectrum of the theory in Eq. (19) with the additional constraints  9 =  10 =  11 =  = 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.

8 < . 7 Figure B. 5 .
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 C. 1 .C. 2 .Figure C. 3 .
Figure C.1.Kinematic structure of the unrestricted MAG, as studied throughout Section III C. The SO(3) irreps precisely correspond to those in Eqs.(5a) and (5b), though the labelling of duplicate   states is different from that in [31].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.