# Every Poincaré gauge theory is conformal: a compelling case for dynamical
vector torsion

Will Barker<EMAIL_ADDRESS>Astrophysics Group, Cavendish Laboratory, JJ
Thomson Avenue, Cambridge CB3 0HE, UK  Kavli Institute for Cosmology,
Madingley Road, Cambridge CB3 0HA, UK  Michael Hobson<EMAIL_ADDRESS>Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3
0HE, UK  Anthony Lasenby<EMAIL_ADDRESS>Astrophysics Group,
Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK  Kavli
Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK  Yun-Cherng Lin
<EMAIL_ADDRESS>Astrophysics Group, Cavendish Laboratory, JJ Thomson
Avenue, Cambridge CB3 0HE, UK  Kavli Institute for Cosmology, Madingley Road,
Cambridge CB3 0HA, UK  Zhiyuan Wei<EMAIL_ADDRESS>Astrophysics Group,
Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK  Institute for
Complex Quantum Systems, Ulm University, Albert-Einstein-Allee, 89069 Ulm, DE

###### Abstract

The Poincaré gauge theory (PGT) of gravity provides a viable formulation of
general relativity (Einstein–Cartan theory), and a popular model-building
framework for modified gravity with torsion. Notoriously, however, the PGT
terms which propagate vector torsion lead to strongly-coupled ghosts: the
modern view is that only scalar torsion can propagate. To fix this, we revisit
the concept of embedding explicit mass scales in scale-invariant theories,
showing how the Klein–Gordon theory naturally leads to a slowly-rolling
inflaton. We then show that the unique scale-invariant embedding of PGT leads
to two new terms, one of which is the Maxwell term for vector torsion. We
provide the full spectrum of quantum particles in the resulting theory. Our
result means that every PGT is conformal and – after a two-decade hiatus –
vector torsion is back on the menu.

## Introduction to torsion

General relativity (GR) offers a remarkably successful description of gravity
as spacetime curvature [1]. An open problem is the degree – if any – to which
spacetime _torsion_ also participates in the gravitational interaction. There
are excellent theoretical reasons to consider torsion: it was shown by Utiyama
[2], Kibble [3] and Sciama [4] that when fermions are coupled to gravity, the
natural result is a local gauge theory not only of spacetime translations
(i.e. the diffeomorphisms of GR), but also of spatial rotations and Lorentz
boosts. In this _Poincaré gauge theory_ (PGT) of gravity [5], the fundamental
fields can be the metric tensor $\tensor{g}{{}_{\mu\nu}}$ and an a priori
independent affine connection $\tensor{\Gamma}{{}^{\alpha}_{\mu}{}_{\nu}}$
which gives rise (as we will show later) to torsion and curvature as the field
strength tensors of spacetime translations and rotations, respectively:

$\displaystyle\tensor{T}{{}^{\alpha}_{\mu}{}_{\nu}}$ $\displaystyle\equiv
2\tensor{\Gamma}{{}^{\alpha}_{[\mu}{}_{\nu]}},$ (1a)
$\displaystyle\tensor{R}{{}^{\rho}_{\sigma}{}_{\mu\nu}}$ $\displaystyle\equiv
2\left(\tensor{\partial}{{}_{[\mu|}}\tensor{\Gamma}{{}^{\rho}_{|\nu]}{}_{\sigma}}+\tensor{\Gamma}{{}^{\rho}_{[\mu|}{}_{\alpha}}\tensor{\Gamma}{{}^{\alpha}_{|\nu]}{}_{\sigma}}\right).$
(1b)

To connect this to textbook GR, if $\tensor{T}{{}^{\alpha}_{\mu}{}_{\nu}}\to
0$ in Eq. 1a then $\tensor{\Gamma}{{}^{\mu}_{\nu}{}_{\rho}}$ loses its
independence from the metric111We will assume the metricity condition
throughout. and is fixed by the Christoffel formula
$\tensor{\Gamma}{{}^{\mu}_{\nu}{}_{\rho}}\to\tensor{\mathring{\Gamma}}{{}^{\mu}_{\nu\rho}}\equiv\tensor{g}{{}^{\mu\lambda}}\big{(}\tensor{\partial}{{}_{(\nu}}\tensor{g}{{}_{\rho)\lambda}}-\frac{1}{2}\tensor{\partial}{{}_{\lambda}}\tensor{g}{{}_{\nu\rho}}\big{)}$.
In this torsion-free limit, Eq. 1b defines the _Riemannian_ curvature
$\tensor{R}{{}^{\rho}_{\sigma}{}_{\mu\nu}}\to\tensor{\mathring{R}}{{}^{\rho}_{\sigma}{}_{\mu\nu}}$.
Despite concrete theoretical provenance from PGT, there is uncertainty over
what a signal for torsion might look like in the phenomena. This is in
contrast to curvature, whose presence can be inferred e.g. through the
geodesic motion of test particles. To understand this uncertainty, notice that
we can always do the field reparameterisation222Note
$\tensor{\Gamma}{{}^{\mu}_{\nu}{}_{\rho}}-\tensor{\mathring{\Gamma}}{{}^{\mu}_{\nu}{}_{\rho}}$
is sometimes called _contortion_ , and is linear in torsion.
$\tensor{\Gamma}{}\mapsto\tensor{\mathring{\Gamma}}{}+\tensor{T}{}$ (leading
to
$\tensor{R}{}\mapsto\tensor{\mathring{R}}{}+\tensor{\partial}{}\tensor{T}{}+\tensor{T}{}^{2}$
with indices suppressed). This _post-Riemannian expansion_ from variables
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{\Gamma}{{}^{\alpha}_{\mu}{}_{\nu}}\big{\\}}$
to
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{T}{{}^{\lambda}_{\mu\nu}}\big{\\}}$
reveals all PGT models to be nothing more than metric-based gravity coupled –
minimally or otherwise – to the field
$\tensor{T}{{}^{\lambda}_{\mu\nu}}\equiv\tensor{T}{{}^{\lambda}_{[\mu\nu]}}$
as if it were an exotic kind of matter. Notice how there are infinitely many
such models: if there are models for all experimental outcomes, then the
theoretical framework ceases to be useful. The classic example of a useful PGT
which establishes a predictive baseline is the minimal Einstein–Cartan theory
(ECT) [6, 7, 8, 9, 10, 11, 12]. ECT shares the Einstein–Hilbert action of GR

$S_{\text{ECT}}\equiv\int\mathrm{d}^{4}x\sqrt{-g}\Big{[}-\frac{{m_{\mathrm{P}}}{}^{2}}{2}\tensor{R}{}-{m_{\mathrm{P}}}{}^{2}\Lambda+\text{matter}\Big{]},$
(2)

where
$\tensor{R}{}\equiv\tensor{R}{{}^{\mu}_{\mu}}\equiv\tensor{R}{{}^{\mu\nu}_{\mu\nu}}$.
With $\Lambda\equiv 0$ and in the absence of matter, Eq. 2 implies the vacuum
Einstein equations $\tensor{R}{{}_{\mu\nu}}=0$ and the auxiliary equation
$\tensor{T}{{}^{\lambda}_{\mu\nu}}=0$. Inclusion of matter leads to a contact
spin-torsion coupling; the torsion integrates out to leave effective four-
Fermi interactions [3, 13, 14, 15, 16, 17, 18]. Such interactions have many
pheonomenological applications [14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 18, 55, 56, 57, 58], such as to fermionic dark
matter production. Thus, non-propagating torsion might reasonably be
constrained (albeit weakly) by dark matter abundances. Propagating torsion, on
the other hand, can come from any number of next-to-minimal
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{T}{{}^{\lambda}_{\mu\nu}}\big{\\}}$
models: it could be sufficiently heavy to evade current bounds, or
conveniently account for any new scalar or vector bosons that might be
observed (note that $\tensor{T}{{}^{\lambda}_{\mu\nu}}$ contains various
bosons up to spin-two). In other words, the phenomological richness of the
full
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{T}{{}^{\lambda}_{\mu\nu}}\big{\\}}$
theory-space, which has infinitely many parameters, renders it non-predictive.

## No-go theorem for vector torsion

Happily, the literature has a convincing way to deal with this problem. Just
as PGT motivates torsion, so the usually assumed structure of its action also
endows torsionful model-building with predictivity. The logic is that PGT
should be restricted (see e.g. [59, 60, 61, 62, 63]) to a _Yang–Mills-type_
action

$\displaystyle
S_{\text{PGT}}\equiv\int\mathrm{d}^{4}x\sqrt{-g}\Big{[}-\frac{{m_{\mathrm{P}}}^{2}}{2}\tensor{R}{}-{m_{\mathrm{P}}}{}^{2}\Lambda+\tensor{\alpha}{{}_{1}}\tensor{R}{}^{2}+\tensor{R}{{}_{\mu\nu}}\big{(}\tensor{\alpha}{{}_{2}}\tensor{R}{{}^{\mu\nu}}$
$\displaystyle+\tensor{\alpha}{{}_{3}}\tensor{R}{{}^{\nu\mu}}\big{)}+\tensor{R}{{}_{\mu\nu\sigma\lambda}}\big{(}\tensor{\alpha}{{}_{4}}\tensor{R}{{}^{\mu\nu\sigma\lambda}}+\tensor{\alpha}{{}_{5}}\tensor{R}{{}^{\mu\sigma\nu\lambda}}+\tensor{\alpha}{{}_{6}}\tensor{R}{{}^{\sigma\lambda\mu\nu}}\big{)}$
$\displaystyle+\tensor{T}{{}_{\mu\nu\sigma}}\big{(}\tensor{\beta}{{}_{1}}\tensor{T}{{}^{\mu\nu\sigma}}+\tensor{\beta}{{}_{2}}\tensor{T}{{}^{\nu\mu\sigma}}\big{)}+\tensor{\beta}{{}_{3}}\tensor{T}{{}_{\mu}}\tensor{T}{{}^{\mu}}+\text{matter}\Big{]},$
(3)

where $\tensor{T}{{}_{\mu}}\equiv\tensor{T}{{}^{\nu}_{\nu\mu}}$, and No-go
theorem for vector torsion extends Eq. 2 by only admitting quadratic field
strength invariants333In this letter we omit the parity-odd invariants only
out of simplicity; there are no convincing theoretical grounds for excluding
them [64, 65].. Evidently, No-go theorem for vector torsion occupies just a
small corner of the
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{T}{{}^{\lambda}_{\mu\nu}}\big{\\}}$
theory-space: the Yang–Mills structure is similar to the gauge boson sector of
the standard model (SM), and it imposes surprising limits on the
phenomenology. Firstly, the parameters in No-go theorem for vector torsion
must be carefully tuned to eliminate ghosts and tachyons from the linearised
spectrum [66, 67, 68, 69, 61, 62, 5, 63, 70, 71, 72, 73, 74, 75, 76, 77, 78,
79, 80, 64, 81, 82, 83, 84, 85, 65, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95,
96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106]. Even then, inconsistencies
usually reappear at non-linear order due to strongly coupled particles [107,
108, 59, 109, 110, 110, 111, 112, 113, 114, 115, 62, 116, 117, 118, 119, 120,
121, 121, 122, 123, 124, 125, 126, 127, 128, 129, 103, 128, 55] (for general
strong coupling see [130, 131, 132, 118, 119, 120, 133, 134, 124]) and the
breaking of so-called ‘accidental’ symmetries which only survive linearly
[135, 136, 137, 138, 139, 61, 62, 140, 141, 142, 143, 144, 129] (see also [60,
67, 68, 62, 5, 79, 145, 146, 65, 29, 147, 98]). Weakly coupled, consistent
PGTs are very rare: the current consensus is that only spin-zero scalar
torsion is allowed to propagate, and then only with very specific non-linear
interactions [138, 139, 61, 62] (see e.g. [70, 71, 74, 76, 77, 78, 82, 86, 89,
91, 97, 101] for applications, [63, 75, 80, 81, 95] for reviews, and [88, 148,
149, 96, 99, 150, 100, 151, 152, 153] for similar analyses of spacetime non-
metricity). Spin-one vector torsion, on the other hand, is ruled out [61, 62].
Purely vector torsion can only be linearly propagated by the
$\tensor{R}{{}_{[\mu\nu]}}\tensor{R}{{}^{[\mu\nu]}}$ term, reached by setting
$\tensor{\alpha}{{}_{2}}+\tensor{\alpha}{{}_{3}}=0$ in No-go theorem for
vector torsion. However, that term carries in strongly-coupled ghosts, so the
resulting theory is inconsistent [61, 62]. In a recent attempt to evade this
no-go theorem, we introduced multiplier fields which pacify this strong
coupling [55]. A fair criticism of our approach, however, was that it
effectively reduced $\tensor{R}{{}_{[\mu\nu]}}\tensor{R}{{}^{[\mu\nu]}}$ to
the post-Riemannian Maxwell term
$\tensor{\partial}{{}_{[\mu}}\tensor{T}{{}_{\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}$
which – though non-linearly consistent – is absent from No-go theorem for
vector torsion. This Maxwell term can still be added by hand, as was done for
example in [104]. The problem with adding terms manually is that, since almost
any desired dynamics can then be fabricated, the resulting models are less
predictive and compelling. To summarise: every PGT in No-go theorem for vector
torsion has a post-Riemannian expansion, but not every covariant model of
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{T}{{}^{\lambda}_{\mu\nu}}\big{\\}}$
is the post-Riemannian expansion of a PGT in No-go theorem for vector torsion.

## In this letter

We show that the
$\tensor{\partial}{{}_{[\mu}}\tensor{T}{{}_{\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}$
term is a Yang–Mills-type term in disguise, without which No-go theorem for
vector torsion is actually _incomplete_. Our reasoning is that PGT is is not
the most fundamental gauge theory of gravity. Recently we developed an
_extended_ 444Given the considerations to be presented in this letter, perhaps
it would be more appropriate to replace ‘extended’ with ‘economical’.
Nonetheless, we retain the original nomenclature for consistency with the
existing literature. version of _Weyl gauge theory_ (eWGT), which gauges the
full conformal group [154, 155, 156, 157, 158]. We will show that the
covariant derivative of eWGT reduces precisely to that of PGT when it is
expressed in terms of scale-invariant variables (equivalent to fixing the
scale gauge, and so breaking the conformal symmetry to Poincaré symmetry). The
Yang–Mills-type action of eWGT reduces to No-go theorem for vector torsion
plus new terms (to be shown in Equivalence of PGT and eWGT). This provides a
concrete motivation for vector torsion, which was hitherto lacking. Before
obtaining these results, we introduce scale-invariant embeddings with a simple
toy model.

Figure 1: Massive scalar perturbations in Eq. 4 have a unique, non-linear,
scale-invariant embedding given in Scale-invariant embedding. This embedding
can lead to only one qualitative alteration in Eqs. 8a and 8b: the quadratic
potential (dashed, with a unit-circle illustrating the mass scale $m$)
develops a non-linear plateau (cyan) whose depth and height depend on model
parameters $\nu$ and $\sigma$. The plateau is consistent with current CMB
constraints on a slowly-rolling inflaton (arrow).

## Scale-invariant embedding

For our toy model, we work just with the matter sector. Let us suppose we know
that some scalar perturbations $\delta\varphi$ have a Klein–Gordon mass $m$

$S=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}\tensor{\partial}{{}_{\mu}}\delta\varphi\
\tensor{\partial}{{}^{\mu}}\delta\varphi-\frac{m^{2}}{2}\delta\varphi^{2}+\text{gravity}\right],$
(4)

where we allow the theory to be minimally coupled to a metric-based gravity
theory (e.g. GR). This situation already arises in cosmology: we typically
infer that the canonical inflaton potential is approximated by a mass term
$U(\varphi)=m^{2}\varphi^{2}/2+\dots$ in order to facilitate reheating into
the SM plasma after the end of inflation555Strictly, this is only true for
‘textbook’ slow-roll inflation [159, 160, 161]. We particularly note that
Higgs inflation does not require a reheating mass [162]. [163]. This inference
does not, however, place constraints on $U(\varphi)$ beyond the
$\delta\varphi$ linear regime. For a host of reasons the most favoured non-
linear completion envisages $\varphi$ rolling off a plateau in $U(\varphi)$
sufficiently slowly to drive $N\sim 55$ e-folds of inflation [164, 165, 166].
Quite how this plateau should be motivated through fundamental theory is not
yet settled. What we want to show here is that, starting purely from the
linear model in Eq. 4, there is a way to build up the non-linear plateau by
asking that $m$ be a dynamically acquired mass scale in a locally scale-
invariant ‘embedding’ theory (see Fig. 1). There is precedent here, since Eq.
4 also describes pions [167, 168, 169, 170]. Pion masses, however, emerge long
after reheating, through chiral and electroweak symmetry breaking [171, 172].
Moreover, the absence of explicit mass scales only makes the SM _globally_
scale-invariant. Under _local_ rescalings the metric deforms as
$\tensor{g}{{}_{\mu\nu}}\mapsto e^{2\rho}\tensor{g}{{}_{\mu\nu}}$ and the
scalar as $\varphi\mapsto e^{-\rho}\varphi$ for local $\rho\equiv\rho(x)$.
Considering the first term in Eq. 4, we must therefore introduce a _Weyl_
vector
$\tensor{B}{{}_{\mu}}\mapsto\tensor{B}{{}_{\mu}}-\tensor{\partial}{{}_{\mu}}\rho$
to define a new covariant derivative on scalars
$\tensor{\mathscr{D}}{{}_{\mu}}\varphi\equiv\big{(}\tensor{\partial}{{}_{\mu}}-\tensor{B}{{}_{\mu}}\big{)}\varphi$.
The kinematic d.o.f are thus increased from
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\varphi\big{\\}}$ to
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\varphi,\tensor{B}{{}_{\mu}}\big{\\}}$.
Next, considering the second term in Eq. 4, we must introduce a _compensator_
scalar $\phi\mapsto e^{-\rho}\phi$ to make up the mass dimension of $m$, which
now becomes a dimensionless parameter. In the effective field theory approach,
the non-linear completion of Eq. 4 includes all scale-invariant operators of
mass-dimension four in the new variables
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\varphi,\tensor{B}{{}_{\mu}},\phi\big{\\}}$,
namely

$\displaystyle S=\int\mathrm{d}^{4}x\sqrt{-g}\Big{[}$
$\displaystyle\frac{1}{2}\tensor{\mathscr{D}}{{}_{\mu}}\varphi\tensor{\mathscr{D}}{{}^{\mu}}\varphi+\frac{\sigma}{2}\tensor{\mathscr{D}}{{}_{\mu}}\varphi\tensor{\mathscr{D}}{{}^{\mu}}\phi+\frac{\nu}{2}\tensor{\mathscr{D}}{{}_{\mu}}\phi\tensor{\mathscr{D}}{{}^{\mu}}\phi$
$\displaystyle-\frac{\mu^{2}}{2}\phi^{2}\varphi^{2}-\frac{\xi}{16}\tensor{H}{{}_{\mu\nu}}\tensor{H}{{}^{\mu\nu}}+\text{gravity}\Big{]},$
(5)

where $\mu$, $\sigma$, $\nu$ and $\xi$ are dimensionless model parameters, and
$\tensor{H}{{}_{\mu\nu}}\equiv
2\tensor{\partial}{{}_{[\mu}}\tensor{B}{{}_{\nu]}}$ is the natural Weyl field-
strength tensor, i.e.
$\tensor{\mathscr{D}}{{}_{[\mu}}\tensor{\mathscr{D}}{{}_{\nu]}}\varphi\equiv-\frac{1}{2}\tensor{H}{{}_{\mu\nu}}\varphi$.
For brevity we omit the $\varphi^{4}$ and $\phi^{4}$ self-interaction terms,
which would not qualitatively alter our conclusions666The $\varphi^{4}$
operator deforms the potential without compromising the desired slow-roll
plateau; the $\phi^{4}$ operator provides a cosmological constant..

## Scale-invariant variables

One can compare $\tensor{B}{{}_{\mu}}$ to the electromagnetic four-vector
potential. But unlike in scalar electrodynamics, where the matter fields carry
a non-physical phase as well as a physical magnitude, the fields $\varphi$ and
$\phi$ each have only one real d.o.f which merges entirely with the choice of
scale gauge. This means that the field equations of Scale-invariant embedding
cannot simultaneously propagate $\varphi$ and $\phi$ from the initial data: we
must first declare a global solution for one of these fields. Remembering that
we want to recover Eq. 4, we may take the opportunity to eliminate $\phi$ by
assuming constant $\phi=\phi_{0}$. This is the _Einstein_ choice of scale
gauge. Rather than imposing the Einstein gauge, it is instead possible to
completely eliminate $\phi$ from Scale-invariant embedding through the field
reparameterisations
$\tensor{g}{{}_{\mu\nu}}\mapsto(\phi_{0}/\phi)^{2}\tensor{\hat{g}}{{}_{\mu\nu}}$
and $\varphi\mapsto\phi\hat{\varphi}/\phi_{0}$ and
$\tensor{B}{{}_{\mu}}\mapsto\tensor{\hat{B}}{{}_{\mu}}+\tensor{\partial}{{}_{\mu}}\ln\phi$,
resulting in

$\displaystyle S=\int\mathrm{d}^{4}x\sqrt{-\hat{g}}\Big{[}$
$\displaystyle\frac{1}{2}\left(\tensor{\partial}{{}_{\mu}}\hat{\varphi}-\left(\hat{\varphi}-\sigma\phi_{0}\right)\tensor{\hat{B}}{{}_{\mu}}\right)\left(\tensor{\partial}{{}^{\mu}}\hat{\varphi}-\hat{\varphi}\tensor{\hat{B}}{{}^{\mu}}\right)$
$\displaystyle-\frac{\mu^{2}\phi_{0}^{2}}{2}\hat{\varphi}^{2}-\frac{\xi}{16}\tensor{\hat{H}}{{}_{\mu\nu}}\tensor{\hat{H}}{{}^{\mu\nu}}+\frac{\nu\phi_{0}^{2}}{2}\tensor{\hat{B}}{{}_{\mu}}\tensor{\hat{B}}{{}^{\mu}}+\text{gravity}\Big{]},$
(6)

where we note $\tensor{\hat{H}}{{}_{\mu\nu}}\equiv
2\tensor{\partial}{{}_{[\mu}}\tensor{\hat{B}}{{}_{\nu]}}\equiv
2\tensor{\partial}{{}_{[\mu}}\tensor{B}{{}_{\nu]}}\equiv\tensor{H}{{}_{\mu\nu}}$.
By construction, the new fields
$\big{\\{}\tensor{\hat{g}}{{}_{\mu\nu}},\hat{\varphi},\tensor{\hat{B}}{{}_{\mu}}\big{\\}}$
are _scale-invariant variables_. The gauge symmetry in Scale-invariant
variables seems to be destroyed, but we now understand that the rescaling
transformations are still happening internally within the new fields. Scale-
invariant variables and the Einstein gauge choice both result in Scale-
invariant variables, a key feature of which is that the dimensionless
couplings are now accompanied by powers of $\phi_{0}$. We will return to the
natural values of these couplings later.

## Bootstrapping inflation

Dropping the hats, and for perturbations $\delta\varphi$ and
$\delta\tensor{B}{{}_{\mu}}$ around the vacuum
$\varphi=\tensor{B}{{}_{\mu}}=0$, we find that Scale-invariant variables
reduces to

$\displaystyle S$
$\displaystyle=\int\mathrm{d}^{4}x\sqrt{-g}\Big{[}\frac{1}{2}\tensor{\partial}{{}_{\mu}}\delta\varphi\tensor{\partial}{{}^{\mu}}\delta\varphi-\frac{m^{2}}{2}\delta\varphi^{2}$
$\displaystyle-\frac{\xi}{4}\tensor{\partial}{{}_{[\mu}}\delta\tensor{B}{{}_{\nu]}}\tensor{\partial}{{}^{[\mu}}\delta\tensor{B}{{}^{\nu]}}+\frac{\xi
M^{2}}{2}\delta\tensor{B}{{}_{\mu}}\delta\tensor{B}{{}^{\mu}}+\text{gravity}\Big{]},$
(7)

where $m^{2}\equiv 4\nu\mu^{2}\phi_{0}^{2}/(4\nu-\sigma^{2})$ and
$M^{2}\equiv\nu\phi_{0}^{2}/\xi$. This result is very interesting. Not only do
we recover Eq. 4 in the first line of Bootstrapping inflation, but we see how
local scale invariance additionally points to the presence of a new Proca
field. To compare, the Higgs mechanism pointed to the presence of a new
massive scalar, which was later observed experimentally [173]. Arguably, there
is no need for a new vector in particle physics or cosmology. But whilst
$\nu\to 0$ predicts two radiative d.o.f which are not observed, taking instead
$\xi\to 0$ nonetheless gives a heavy-enough Proca to evade all observational
bounds (even better, it may be a dark matter candidate [154]). Beyond the
perturbative regime, the Proca and Klein–Gordon fields acquire non-linear
interactions. For $\xi\to 0$ the Proca can be eliminated as
$\tensor{B}{{}_{\mu}}=\frac{1}{2}\tensor{\partial}{{}_{\mu}}\ln\left(\varphi^{2}-\sigma\phi_{0}\varphi+\nu\phi_{0}^{2}\right)$
on-shell, and Scale-invariant variables becomes

$\displaystyle
S=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}\tensor{\partial}{{}_{\mu}}\varphi\tensor{\partial}{{}^{\mu}}\varphi-U\left(\varphi\right)+\text{gravity}\right],$
(8a) $\displaystyle
U\left(\varphi\right)\equiv\frac{\mu^{2}\phi_{0}^{4}}{2}\left[\frac{\sigma}{2}+\sqrt{\nu-\frac{\sigma^{2}}{4}}\text{sinh}\left(\frac{(\varphi-c)/\phi_{0}}{\sqrt{\nu-\frac{\sigma^{2}}{4}}}\right)\right]^{2},$
(8b)

where $c$ is an integration constant. The non-linear potential (see Fig. 1) is
identical to that found in [174, 48], though the motivations are unrelated.
The predictions from the inflationary plateau approach the universal values
for the spectral index $n_{s}\approx 1-2/N$ and also the tensor-to-scalar
ratio $r\approx 12/N^{2}$. This agrees with current cosmic microwave
background (CMB) measurements [166, 175], pending future experiments [176,
177, 178, 179]. We hope to develop this toy model in future work, but we first
need to make the ‘$+\ \text{gravity}$’ term more concrete.

## Recap and extension

Collecting our results so far:

  1. 1.

Gravitational coupling to fermions naturally leads to PGT as a leading
formulation of gravity. PGT does not have local scale invariance.

  2. 2.

The
$\tensor{\partial}{{}_{[\mu}}\tensor{T}{{}_{\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}$
term for vector torsion $\tensor{T}{{}_{\mu}}$ is notoriously absent from the
PGT action in No-go theorem for vector torsion.

  3. 3.

Theories with mass scales have locally-scale-invariant embeddings which can
lead to excellent phenomenology (inflation) besides a new massive vector
$\tensor{B}{{}_{\mu}}$.

We will now connect these three observations by showing that eWGT is the
unique scale-invariant embedding of PGT. This will identify
$\tensor{T}{{}_{\mu}}/3$ with the vector $\tensor{B}{{}_{\mu}}$ when expressed
in scale-invariant variables, and thereby reveal
$\tensor{\partial}{{}_{[\mu}}\tensor{T}{{}_{\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}$
to be a Yang–Mills-type term.

## Poincaré gauge theory (PGT)

Gauge theories of gravity can be formulated on completely flat Minkowski
spacetime, just like the SM [154]. In this formulation, we use holonomic Greek
indices and Lorentz Roman indices. Translational gauge fields
$\tensor{b}{{}^{i}_{\mu}}$ and inverses $\tensor{h}{{}_{i}^{\mu}}$ obey
$\tensor{b}{{}^{i}_{\mu}}\tensor{h}{{}_{i}^{\nu}}\equiv\tensor{*}{\delta}{{}_{\mu}^{\nu}}$
and
$\tensor{b}{{}^{i}_{\mu}}\tensor{h}{{}_{j}^{\mu}}\equiv\tensor{*}{\delta}{{}_{j}^{i}}$,
and give rise to the curved-space metric $\tensor{g}{{}_{\mu\nu}}$ through
$\tensor{b}{{}^{i}_{\mu}}\tensor{b}{{}^{j}_{\nu}}\tensor{\eta}{{}_{ij}}\equiv\tensor{g}{{}_{\mu\nu}}$.
We assume contraction with these fields eliminates all coordinate indices of
general (i.e. not necessarily scalar) matter fields $\varphi$, which have only
suppressed Lorentz (or spinor) indices. A general $\varphi$ with Weyl weight
$w$ and belonging to the $\mathrm{SL}(2,\mathbb{C})$ representation with
Lorentz generators $\tensor{\Sigma}{{}_{ij}}\equiv\tensor{\Sigma}{{}_{[ij]}}$
transforms as $\varphi\mapsto
e^{w\rho+\tensor{\omega}{{}^{ij}}\tensor{\Sigma}{{}_{ij}}/2}\varphi$, where
$\tensor{\omega}{{}^{ij}}\equiv\tensor{\omega}{{}^{[ij]}}(x)$ are finite
angles. The rotational gauge field is
$\tensor{A}{{}^{ij}_{\mu}}\equiv\tensor{A}{{}^{[ij]}_{\mu}}$. The
transformations $\tensor{b}{{}^{i}_{\mu}}\mapsto
e^{\rho}\tensor{\Lambda}{{}^{i}_{j}}\tensor{b}{{}^{j}_{\mu}}$ and
$\tensor{A}{{}^{ij}_{\mu}}\mapsto\tensor{\Lambda}{{}^{i}_{k}}\tensor{\Lambda}{{}^{j}_{l}}\tensor{A}{{}^{kl}_{\mu}}-\tensor{\Lambda}{{}^{jk}}\tensor{\partial}{{}_{\mu}}\tensor{\Lambda}{{}^{i}_{k}}$
– where $\tensor{\Lambda}{}\equiv e^{\tensor{\omega}{}}$ is the Lorentz matrix
– ensure that the derivative
$\tensor{\mathscr{D}}{{}_{i}}\equiv\tensor{h}{{}_{i}^{\mu}}\tensor{\mathscr{D}}{{}_{\mu}}$
with

$\tensor{\mathscr{D}}{{}_{\mu}}\varphi\equiv\left(\tensor{\partial}{{}_{\mu}}+\frac{1}{2}\tensor{A}{{}^{ij}_{\mu}}\tensor{\Sigma}{{}_{ij}}\right)\varphi,$
(9)

is Lorentz-covariant. The field strengths from the commutator
$\tensor{b}{{}^{[i|}_{\mu}}\tensor{b}{{}^{|j]}_{\nu}}\tensor{\mathscr{D}}{{}_{[i}}\tensor{\mathscr{D}}{{}_{j]}}\varphi\equiv\frac{1}{4}\tensor{b}{{}^{i}_{\sigma}}\tensor{b}{{}^{j}_{\lambda}}\tensor{R}{{}^{\sigma\lambda}_{\mu\nu}}\tensor{\Sigma}{{}_{ij}}\varphi-\frac{1}{2}\tensor{T}{{}^{\lambda}_{\mu\nu}}\tensor{\mathscr{D}}{{}_{\lambda}}\varphi$
are

$\displaystyle\tensor{T}{{}^{\lambda}_{\mu\nu}}$ $\displaystyle\equiv
2\tensor{h}{{}_{i}^{\lambda}}\left(\tensor{\partial}{{}_{[\mu|}}\tensor{b}{{}^{i}_{|\nu]}}+\tensor{A}{{}^{i}_{j}{}_{[\mu|}}\tensor{b}{{}^{j}_{|\nu]}}\right),$
(10a) $\displaystyle\tensor{R}{{}^{\rho\sigma}_{\mu\nu}}$ $\displaystyle\equiv
2\tensor{h}{{}_{i}^{\rho}}\tensor{h}{{}_{j}^{\sigma}}\left(\tensor{\partial}{{}_{[\mu|}}\tensor{A}{{}^{ij}_{|\nu]}}+\tensor{A}{{}^{i}_{k}{}_{[\mu|}}\tensor{A}{{}^{kj}_{|\nu]}}\right),$
(10b)

where Eqs. 10a and 10b are equivalent to Eqs. 1a and 1b — the
reparameterisation from gauge-theoretic
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{A}{{}^{ij}_{\mu}}\big{\\}}$ to
geometric
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{\Gamma}{{}^{\alpha}_{\mu}{}_{\nu}}\big{\\}}$
is described in [5, 154].

## Weyl gauge theory (eWGT)

From our toy model, the Weyl-covariant extension of Eq. 9 is

$\tensor{\mathscr{D}}{{}_{\mu}}\varphi\equiv\left(\tensor{\partial}{{}_{\mu}}+\frac{1}{2}\tensor{A}{{}^{ij}_{\mu}}\tensor{\Sigma}{{}_{ij}}+w\tensor{B}{{}_{\mu}}\right)\varphi,$
(11)

and this construction defines the well-known _Weyl gauge theory_ (WGT) of
gravity [180, 181]. Whilst $\tensor{R}{{}^{\rho\sigma}_{\mu\nu}}$ in Eq. 10b
is already Weyl-covariant, we have under local dilations

$\tensor{T}{{}^{\lambda}_{\mu\nu}}\mapsto\tensor{T}{{}^{\lambda}_{\mu\nu}}-2\tensor{*}{\delta}{{}^{\lambda}_{[\mu}}\tensor{\partial}{{}_{\nu]}}\rho\implies\tensor{T}{{}_{\mu}}\mapsto\tensor{T}{{}_{\mu}}-3\tensor{\partial}{{}_{\mu}}\rho,$
(12)

so from Eq. 12 only the combination

$\tensor{\tilde{T}}{{}^{\lambda}_{\mu\nu}}\equiv\tensor{T}{{}^{\lambda}_{\mu\nu}}-2\tensor{*}{\delta}{{}^{\lambda}_{[\mu}}\tensor{B}{{}_{\nu]}},$
(13)

is covariant, indeed
$\tensor{b}{{}^{[i|}_{\mu}}\tensor{b}{{}^{|j]}_{\nu}}\tensor{\mathscr{D}}{{}_{[i}}\tensor{\mathscr{D}}{{}_{j]}}\varphi\equiv\frac{1}{4}\tensor{b}{{}^{i}_{\sigma}}\tensor{b}{{}^{j}_{\lambda}}\tensor{R}{{}^{\sigma\lambda}_{\mu\nu}}\tensor{\Sigma}{{}_{ij}}\varphi-\frac{1}{2}\tensor{\tilde{T}}{{}^{\lambda}_{\mu\nu}}\tensor{\mathscr{D}}{{}_{\lambda}}\varphi+\frac{1}{2}w\tensor{H}{{}_{\mu\nu}}\varphi$.
Following Scale-invariant embedding, the parity-even Yang–Mills-type embedding
of No-go theorem for vector torsion must then be strictly

$\displaystyle
S_{\text{WGT}}\equiv\int\mathrm{d}^{4}x\sqrt{-g}\Big{[}-\frac{\zeta\phi^{2}}{2}\tensor{R}{}-\lambda\phi^{4}+\tensor{\alpha}{{}_{1}}\tensor{R}{}^{2}+\tensor{R}{{}_{\mu\nu}}\big{(}\tensor{\alpha}{{}_{2}}\tensor{R}{{}^{\mu\nu}}$
$\displaystyle+\tensor{\alpha}{{}_{3}}\tensor{R}{{}^{\nu\mu}}\big{)}+\tensor{R}{{}_{\mu\nu\sigma\lambda}}\big{(}\tensor{\alpha}{{}_{4}}\tensor{R}{{}^{\mu\nu\sigma\lambda}}+\tensor{\alpha}{{}_{5}}\tensor{R}{{}^{\mu\sigma\nu\lambda}}+\tensor{\alpha}{{}_{6}}\tensor{R}{{}^{\sigma\lambda\mu\nu}}\big{)}$
$\displaystyle+\phi^{2}\tensor{\tilde{T}}{{}_{\mu\nu\sigma}}\big{(}\tensor{\gamma}{{}_{1}}\tensor{\tilde{T}}{{}^{\mu\nu\sigma}}+\tensor{\gamma}{{}_{2}}\tensor{\tilde{T}}{{}^{\nu\mu\sigma}}\big{)}+\phi^{2}\tensor{\gamma}{{}_{3}}\tensor{\tilde{T}}{{}_{\mu}}\tensor{\tilde{T}}{{}^{\mu}}+\frac{\nu}{2}\tensor{\mathscr{D}}{{}_{\mu}}\phi\tensor{\mathscr{D}}{{}^{\mu}}\phi$
$\displaystyle-\frac{\xi}{16}\tensor{H}{{}_{\mu\nu}}\tensor{H}{{}^{\mu\nu}}+\frac{\chi}{2}\tensor{R}{{}_{\mu\nu}}\tensor{H}{{}^{\mu\nu}}+\text{matter}\Big{]},$
(14)

with dimensionless $\zeta$, $\lambda$, $\tensor{\gamma}{{}_{1}}$,
$\tensor{\gamma}{{}_{2}}$, $\tensor{\gamma}{{}_{3}}$, $\nu$, $\xi$, $\chi$ and
$\alpha_{1}$ through $\alpha_{6}$. Crucially, everything is quadratic in field
strengths.

## Extended Weyl gauge theory (eWGT)

In moving from PGT to WGT, we have enlarged the kinematic space from
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{A}{{}^{ij}_{\mu}}\big{\\}}$ to
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{A}{{}^{ij}_{\mu}},\tensor{B}{{}_{\mu}},\phi\big{\\}}$,
just like in our toy model. Unlike for our toy model, however, the enlargement
proves unnecessary: it is more economical to recycle $\tensor{B}{{}_{\mu}}$
from the rotational gauge field. Let the traceless part of the latter field be
$\tensor{\tilde{A}}{{}^{ij}_{\mu}}\equiv\tensor{\tilde{A}}{{}^{[ij]}_{\mu}}$
with only 20 d.o.f, such that
$\tensor{h}{{}_{j}^{\nu}}\tensor{\tilde{A}}{{}^{ij}_{\nu}}\equiv 0$. Then,
using
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{\tilde{A}}{{}^{ij}_{\mu}},\tensor{B}{{}_{\mu}},\phi\big{\\}}$
there is one (and _only_ one) way to emulate the combined Poincaré and Weyl
covariance of Eq. 11, namely

$\displaystyle\tensor{\mathscr{D}}{{}_{\mu}}\varphi\equiv\Big{(}\tensor{\partial}{{}_{\mu}}+\frac{1}{2}\Big{[}$
$\displaystyle\tensor{\tilde{A}}{{}^{ij}_{\mu}}-\frac{2}{3}\tensor{b}{{}^{[i|}_{\mu}}\tensor{h}{{}^{|j]}{}^{\nu}}\big{(}3\tensor{B}{{}_{\nu}}$
$\displaystyle\ \ \
-2\tensor{h}{{}_{k}^{\lambda}}\tensor{\partial}{{}_{[\lambda|}}\tensor{b}{{}^{k}_{|\nu]}}\big{)}\Big{]}\Sigma{{}_{ij}}+w\tensor{B}{{}_{\mu}}\Big{)}\varphi.$
(15)

We will soon show that Eq. 15 is just one notational way to write the unique
eWGT covariant derivative from [154]. Because eWGT shares the symmetries of
WGT, it must also share the Yang–Mills-type action in Weyl gauge theory
(eWGT). The difference is that we must replace $\tensor{A}{{}^{ij}_{\mu}}$, as
when going from Eq. 11 to Eq. 15. After this change, we notice from Eq. 13
that $\tensor{\tilde{T}}{{}_{\mu}}\equiv 0$ identically. Thus,
$\tensor{\gamma}{{}_{3}}$ in Weyl gauge theory (eWGT) drops out of eWGT.

## Equivalence of PGT and eWGT

With the field reparameterisation
$\tensor{B}{{}_{\mu}}\mapsto\tensor{V}{{}_{\mu}}+\frac{2}{3}\tensor{h}{{}_{k}^{\lambda}}\tensor{\partial}{{}_{[\lambda|}}\tensor{b}{{}^{k}_{|\mu]}}$
we find Eq. 15 is

$\displaystyle\tensor{\mathscr{D}}{{}_{\mu}}\varphi\equiv\Big{(}\tensor{\partial}{{}_{\mu}}+\frac{1}{2}\Big{[}$
$\displaystyle\tensor{\tilde{A}}{{}^{ij}_{\mu}}-2\tensor{b}{{}^{[i|}_{\mu}}\tensor{h}{{}^{|j]}{}^{\nu}}\tensor{V}{{}_{\nu}}\Big{]}\Sigma{{}_{ij}}$
$\displaystyle\ \ \
+w\Big{[}\tensor{V}{{}_{\mu}}+\frac{2}{3}\tensor{h}{{}_{k}^{\lambda}}\tensor{\partial}{{}_{[\lambda|}}\tensor{b}{{}^{k}_{|\mu]}}\Big{]}\Big{)}\varphi.$
(16)

In this
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{\tilde{A}}{{}^{ij}_{\mu}},\tensor{V}{{}_{\mu}},\phi\big{\\}}$
formulation $\tensor{V}{{}_{\mu}}$ is a ‘Lorentz’ boson, not a ‘Weyl’ boson,
because
$\tensor{V}{{}_{\mu}}\mapsto\tensor{V}{{}_{\mu}}-\frac{1}{3}\tensor{h}{{}^{\lambda}_{l}}\tensor{b}{{}^{m}_{\mu}}\tensor{\Lambda}{{}_{k}^{l}}\tensor{\partial}{{}_{\lambda}}\tensor{\Lambda}{{}^{k}_{m}}$.
Relabelling
$\tensor{A}{{}^{ij}_{\mu}}\equiv\tensor{\tilde{A}}{{}^{ij}_{\mu}}-2\tensor{b}{{}^{[i|}_{\mu}}\tensor{h}{{}^{|j]}{}^{\nu}}\tensor{V}{{}_{\nu}}$,
where $\tensor{A}{{}^{ij}_{\mu}}\equiv\tensor{A}{{}^{[ij]}_{\mu}}$ carries 24
d.o.f, ensures the PGT transformation of $\tensor{A}{{}^{ij}_{\mu}}$. In this
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{A}{{}^{ij}_{\mu}},\phi\big{\\}}$
formulation Eq. 16 becomes

$\displaystyle\tensor{\mathscr{D}}{{}_{\mu}}\varphi$
$\displaystyle\equiv\Big{(}\tensor{\partial}{{}_{\mu}}+\frac{1}{2}\tensor{A}{{}^{ij}_{\mu}}\Sigma{{}_{ij}}+\frac{w}{3}\tensor{h}{{}_{k}^{\lambda}}\Big{[}2\tensor{\partial}{{}_{[\lambda|}}\tensor{b}{{}^{k}_{|\mu]}}+3\tensor{b}{{}_{l}{}_{\mu}}\tensor{A}{{}^{kl}_{\lambda}}\Big{]}\Big{)}\varphi$
$\displaystyle\equiv\Big{(}\tensor{\partial}{{}_{\mu}}+\frac{1}{2}\tensor{A}{{}^{ij}_{\mu}}\Sigma{{}_{ij}}+\frac{w}{3}\tensor{T}{{}_{\mu}}\Big{)}\varphi,$
(17)

where we used Eq. 10a in the last equality. New fields should not be
introduced where they are not needed: in hindsight it was obvious from Eq. 12
that PGT contained a vector $\tensor{T}{{}_{\mu}}/3$ which already performed
the function of $\tensor{B}{{}_{\mu}}$. This fact was observed by Obukhov in
[182], and the covariant derivative in Eq. 17 was first written down in [154]
(see also [183]). One could interpret Eq. 17 as a hint that PGT descends from
a scale-invariant theory. Explicit scales emerge in the
$\big{\\{}\tensor{\hat{b}}{{}^{i}_{\mu}},\tensor{\hat{A}}{{}^{ij}_{\mu}}\big{\\}}$
formulation, with scale-invariant variables
$\tensor{b}{{}^{i}_{\mu}}\mapsto\phi_{0}\tensor{\hat{b}}{{}^{i}_{\mu}}/\phi$
and $\tensor{A}{{}^{ij}_{\mu}}\mapsto\tensor{\hat{A}}{{}^{ij}_{\mu}}$.
Dropping the hats, and with the identifications

$\begin{gathered}{m_{\mathrm{P}}}{}^{2}\equiv\zeta\phi_{0}^{2},\quad\Lambda\equiv\lambda\phi_{0}^{2}/\zeta,\quad\tensor{\beta}{{}_{1}}\equiv\tensor{\gamma}{{}_{1}}\phi_{0}^{2},\\\
\tensor{\beta}{{}_{2}}\equiv\tensor{\gamma}{{}_{2}}\phi_{0}^{2},\quad\tensor{\beta}{{}_{3}}\equiv\big{(}\nu-6(2\tensor{\gamma}{{}_{1}}+\tensor{\gamma}{{}_{2}})\big{)}\phi_{0}^{2}/18,\end{gathered}$
(18)

we conclude from Weyl gauge theory (eWGT) that the only remaining difference
between PGT and eWGT is in the natural choice of action. Concretely, the
$\tensor{H}{{}_{\mu\nu}}$ terms in Weyl gauge theory (eWGT) yield a
correction:

$\displaystyle S_{\text{eWGT}}\equiv
S_{\text{PGT}}+\frac{1}{3}\int\mathrm{d}^{4}x\sqrt{-g}\Big{[}$
$\displaystyle\chi\tensor{R}{{}_{[\mu\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}$
$\displaystyle-\frac{\xi}{12}\tensor{\partial}{{}_{[\mu}}\tensor{T}{{}_{\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}\Big{]}.$
(19)

This is our central result: the relationship between Equivalence of PGT and
eWGT and No-go theorem for vector torsion is analogous to the relationship
between Eq. 8a and Eq. 4. Put another way, the traditional PGT action unfairly
omits two specific terms, which are naturally motivated by the unique, scale-
invariant embedding. This embedding allows us to claim that all PGT models are
ultimately conformal, irrespective of the explicit mass scales in their
spectra. We confirm in the supplemental material [184] that eWGT, as
formulated in Eqs. 13, Weyl gauge theory (eWGT) and 15, and the extension of
PGT in Equivalence of PGT and eWGT, No-go theorem for vector torsion, 10a and
10b, have completely equivalent spectra. In the expressions for masses and
pole residues, Eq. 18 maps the dimensionful couplings to their dimensionless
counterparts. As a final, parenthetic remark, we return to the natural values
of the dimensionless couplings. When we are concerned with physics at the
length-scale ${\ell_{\mathrm{Phy}}}{}$, it is helpful to work in units of
${\ell_{\mathrm{Phy}}}{}$. This can be done by taking
$\phi_{0}{\ell_{\mathrm{Phy}}}{}\equiv 1$ as a gauge choice, and in this
picture the dimensionless couplings ‘run’ with ${\ell_{\mathrm{Phy}}}{}$. If
we expect GR to be the low-energy limit of Equivalence of PGT and eWGT, then
the strength of gravity and the cosmological constant are the only known
scales:

$\zeta\equiv\left(\frac{{\ell_{\mathrm{Phy}}}{}}{{\ell_{\mathrm{P}}}{}}\right)^{2},\quad\frac{\lambda}{\zeta}\sim$1\text{\times}{10}^{-122}\text{\,}$\left(\frac{{\ell_{\mathrm{Phy}}}{}}{{\ell_{\mathrm{P}}}{}}\right)^{2}.$
(20)

A natural advantage of this convention, where
${\ell_{\mathrm{P}}}{}{m_{\mathrm{P}}}{}\equiv 1$, is that the cosmological
constant approaches $\lambda/\zeta\sim 1$ at the scale
${\ell_{\mathrm{Phy}}}{}\sim$10\text{\,}\mathrm{Gpc}$$, which is close to the
Hubbble horizon.

## Implications for the PGT spectrum

Since we have made a substantial correction to the PGT Lagrangian, it is
important to update the well-known PGT particle spectrum in [60]. To do this,
we take $\lambda=\Lambda=0$ and focus on the linearisation near Minkowski
spacetime. As shown in Appendix A and Fig. 4, the propagator of
$S_{\text{eWGT}}$ in Equivalence of PGT and eWGT contains a _quartic_ pole in
the parity-odd spin-one sector

$\displaystyle
2\Big{[}\big{(}2\tensor{\alpha}{{}_{2}}+4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}\big{)}\xi-\chi^{2}\Big{]}k^{4}+3\Big{[}96\tensor{\alpha}{{}_{2}}\big{(}2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+\tensor{\beta}{{}_{3}}\big{)}$
$\displaystyle+8\big{[}24\tensor{\alpha}{{}_{4}}\big{(}2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+\tensor{\beta}{{}_{3}}\big{)}+6\tensor{\alpha}{{}_{5}}\big{(}2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+\tensor{\beta}{{}_{3}}\big{)}$
$\displaystyle+\chi\big{(}4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}\big{)}\big{]}+\big{(}4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}\big{)}\xi\Big{]}k^{2}$
$\displaystyle+72\big{(}4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}\big{)}\big{(}2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+3\tensor{\beta}{{}_{3}}+{m_{\mathrm{P}}}{}^{2}\big{)}.$
(21)

The pole takes the familiar $k^{2}-M^{2}$ form when the $k^{4}$ coefficient is
removed from Implications for the PGT spectrum. For example, the branch
$2\tensor{\alpha}{{}_{2}}+4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}=\chi=0$
contains the Einstein–Proca theory (EPT)
$S_{\text{EPT}}\equiv\int\mathrm{d}^{4}x\sqrt{-g}\big{[}-\frac{1}{2}{m_{\mathrm{P}}}{}^{2}\tensor{\mathring{R}}{}-\frac{1}{4}\tensor{\partial}{{}_{[\mu}}\tensor{T}{{}_{\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}+\frac{1}{2}M^{2}\tensor{T}{{}_{\mu}}\tensor{T}{{}^{\mu}}+\text{matter}\big{]}$
in
$\big{\\{}\tensor{g}{{}_{\mu\nu}},\tensor{T}{{}^{\alpha}_{\mu}{}_{\nu}}\big{\\}}$
variables. EPT is evidently not strongly coupled, so this is the much-sought-
after _vector torsion_. The more general branch
$\tensor{\alpha}{{}_{5}}=\chi^{2}/\xi-2\left(\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{4}}\right)$
leads to the new mass spectrum in Eqs. 25a to 25f, as shown in Fig. 5. It is
also interesting to instead remove the _constant_ term in Implications for the
PGT spectrum. This produces a $k^{2}\big{(}k^{2}-M^{2}\big{)}$ pole, which
affects the massless spectrum. As shown in Fig. 6, we find an example case
$\tensor{\alpha}{{}_{2}}=\tensor{\alpha}{{}_{3}}-4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{6}}=4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}=4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}=0$
which propagates (without ghosts or tachyons) the Einstein graviton, one heavy
scalar, one heavy pseudoscalar, and a _one extra massless scalar_. To
understand where this scalar comes from, we strip away all the uninvolved
operators to show in Fig. 7 that the model

$S=\int\mathrm{d}^{4}x\sqrt{-g}\big{[}\chi\tensor{R}{{}_{[\mu\nu]}}\tensor{\partial}{{}^{[\mu}}\tensor{T}{{}^{\nu]}}+\tensor{\beta}{{}_{3}}\tensor{T}{{}_{\mu}}\tensor{T}{{}^{\mu}}\big{]},$
(22)

propagates, for $\chi\neq 0$, one massless scalar with the no-ghost condition
$\tensor{\beta}{{}_{3}}>0$. For $\chi=0$, the spectrum is empty. In fact Eq.
22 can be reduced even further by removing $\tensor{b}{{}^{i}_{\mu}}$, and
keeping only the axial-free part of $\tensor{A}{{}^{ij}_{\mu}}$. This part has
the index symmetries of a Curtright field $\tensor{C}{{}_{\mu\nu\sigma}}$,
i.e. $\tensor{C}{{}_{\mu\nu\sigma}}\equiv\tensor{C}{{}_{[\mu\nu]\sigma}}$ with
$\tensor{C}{{}_{[\mu\nu\sigma]}}\equiv 0$ [185], and Eq. 22 reduces (linearly)
to

$S=\int\mathrm{d}^{4}x\Big{[}\tensor{\beta}{{}_{3}}\tensor{C}{{}^{\mu}}\tensor{C}{{}_{\mu}}+\frac{\chi}{12}\tensor{C}{{}_{\mu\nu}}\Big{(}\tensor{C}{{}^{\mu\nu}}-\frac{4}{3}\tensor{\partial}{{}_{\sigma}}\tensor{C}{{}^{\mu(\nu\sigma)}}\Big{)}\Big{]},$
(23)

with trace $\tensor{C}{{}_{\mu}}\equiv\tensor{C}{{}^{\nu}_{\mu\nu}}$ and field
strength $\tensor{C}{{}_{\mu\nu}}\equiv
2\tensor{\partial}{{}_{[\mu}}\tensor{C}{{}_{\nu]}}$. As shown in Fig. 8, Eqs.
23 and 22 have the same spectra. Finally, in the branch $\xi=\chi=0$ we
recover the known spectrum of $S_{\text{PGT}}$ in No-go theorem for vector
torsion, as first presented in [60].

## Closing remarks

Poincaré gauge theory (PGT) strongly motivates spacetime torsion [2, 3, 4],
but the consensus since the turn of the millennium has been that PGT prohibits
vector torsion from propagating [138, 139, 61, 62, 70, 71, 74, 76, 77, 78, 82,
86, 89, 91, 97, 101]. Separately, the full conformal group is a more appealing
gauge group than the Poincaré group. With this context, our letter achieved
three objectives:

  1. 1.

Starting just with the inflaton mass, scale-invariant embedding leads to the
slow-roll plateau in Eqs. 8b and 1. This is an intriguing, stand-alone result.

  2. 2.

We showed that PGT has a unique scale-invariant embedding as _extended_ Weyl
gauge theory (eWGT) [154, 155, 156, 157, 158]. In other words, PGT descends
from a conformal theory. This also means that the natural Yang–Mills-type PGT
action is missing the two terms in Equivalence of PGT and eWGT.

  3. 3.

The new terms provide the first compelling argument for propagating vector
torsion. The updated mass spectrum of PGT is shown in Eqs. 25a to 25f. The new
terms reveal that the Curtright field in Eq. 23 is embedded in the PGT
dynamics.

Further work is needed to develop our inflaton model in the full eWGT-PGT
framework. One key question is whether the model is susceptible to the Weyl
anomaly [186]. Also, the implications of the new PGT terms for strong coupling
should be investigated [128]. Finally, the Yang–Mills-type actions in No-go
theorem for vector torsion and Weyl gauge theory (eWGT) are restricted to
parity-even terms for simplicity: the parity-odd extensions should be
considered.

###### Acknowledgements.

We are grateful for useful discussions with Carlo Marzo and Sebastian Zell. WB
is grateful for the kind hospitality of Leiden University and the Lorentz
Institute, and the support of Girton College, Cambridge. Y-CL acknowledges
support from the Ministry of Education of Taiwan and the Cambridge
Commonwealth, European & International Trust via a Taiwan Cambridge
Scholarship and financial support from Corpus Christi College, Cambridge. 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. This work also used the Newton
server, access to which was provisioned by Will Handley, and funded through an
ERC grant.

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##  Appendix A Detailed tree-level spectra

Figure 2:  Kinematic structure of the tetrad perturbation
$\tensor{f}{{}_{\mu\nu}}$, as used in No-go theorem for vector torsion and
Equivalence of PGT and eWGT. These definitions are used in Figs. 4 to 11.
Figure 3:  Kinematic structure of the spin connection perturbation
$\tensor{A}{{}_{\mu\nu\sigma}}$, as used in No-go theorem for vector torsion
and Equivalence of PGT and eWGT. 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 [187]. These
definitions are used in Figs. 4 to 11.

The particle spectra are obtained using the _Particle Spectrum for Any Tensor
Lagrangian_ (PSALTer) software [188, 106, 174, 189]. Recall that the kinematic
variables of PGT were
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{A}{{}^{ij}_{\mu}}\big{\\}}$, and
that the inverse $\tensor{h}{{}_{i}^{\mu}}$ is determined by
$\tensor{b}{{}^{i}_{\mu}}$. This means that we are free to use
$\big{\\{}\tensor{h}{{}_{i}^{\mu}},\tensor{A}{{}^{ij}_{\mu}}\big{\\}}$
instead. Working in the weak-field regime, we take $\tensor{A}{{}^{ij}_{\mu}}$
to be inherently perturbative. We define the exact perturbation
$\tensor{h}{{}_{i}^{\mu}}\equiv\tensor{*}{\delta}{{}_{i}^{\mu}}+\tensor{f}{{}_{i}^{\mu}}$,
where we make the ‘Kronecker’ choice of Minkowski vacuum (see alternative
vacua [188, 190, 191]). There are a priori 16 d.o.f in
$\tensor{f}{{}_{i}^{\mu}}$, and 24 d.o.f in $\tensor{A}{{}^{ij}_{\mu}}$.
Conjugate to $\tensor{f}{{}_{i}^{\mu}}$ and $\tensor{A}{{}^{ij}_{\mu}}$ are
the translational source (i.e. the asymmetric stress-energy tensor)
$\tensor{\tau}{{}^{i}_{\mu}}$, and the matter spin current
$\tensor{\sigma}{{}_{ij}^{\mu}}$ [18, 46, 17]. To lowest order in the field
perturbations, the Greek and Roman indices are interchangeable. For this
reason the final set of fields used in the linearised analysis is
$\big{\\{}\tensor{f}{{}_{\mu\nu}},\tensor{A}{{}_{\mu\nu\sigma}}\big{\\}}$,
with conjugate sources
$\big{\\{}\tensor{\tau}{{}^{\mu\nu}},\tensor{\sigma}{{}^{\mu\nu\sigma}}\big{\\}}$.
The various $\mathrm{SO}(3)$ irreducible parts of these quantities are
presented in Figs. 2 and 3, and they have spin-parity ($J^{P}$) labels to
identify them. Duplicate $J^{P}$ states can arise, and these are distinguished
by extra labels $\\#1$, $\\#2$, etc.

The actual analysis of the theory in Equivalence of PGT and eWGT was performed
across 64 AMD® _Ryzen Threadripper_ CPUs. For the completely unrestricted case
in Equivalence of PGT and eWGT the mass expressions associated with the roots
of the quartic pole are expected to be very cumbersome, so we abort the
computation before they are computed: the structure of the wave operator and
saturated propagator, without the mass spectrum, is shown in Fig. 4. Before
the evaluation of the ‘general’ branch for which the $k^{4}$ coefficient in
Implications for the PGT spectrum is removed, the condition
$\tensor{\alpha}{{}_{5}}=\chi^{2}/\xi-2\left(\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{4}}\right)$
must be imposed on the linearised action. To avoid Lagrangian couplings
appearing on the denominator of the linearised action777As a mild technical
limitation, coupling coefficients in the denominator can cause PSALTer to
become slow. we impose the constraint by introducing a new coupling $\theta$
and setting in Equivalence of PGT and eWGT

$\tensor{\alpha}{{}_{5}}\mapsto\theta^{2}\xi-2\left(\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{4}}\right),\quad\chi\mapsto\theta\xi.$
(24)

The results are shown in Fig. 5, we see that apart from the two massless
polarisations of the Einstein graviton, up to six torsion particles can
propagate with the following square masses, where we eliminate $\theta$ in
terms of $\chi$ and $\xi$ using Eq. 24

$\displaystyle\left(\tensor{m}{{}_{0^{+}}}\right)^{2}$
$\displaystyle\equiv\frac{{m_{\mathrm{P}}}{}^{2}\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+3\tensor{\beta}{{}_{3}}+{m_{\mathrm{P}}}{}^{2}\right)}{2\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+3\tensor{\beta}{{}_{3}}\right)\left(6\tensor{\alpha}{{}_{1}}+2\tensor{\alpha}{{}_{3}}-2\tensor{\alpha}{{}_{4}}+2\tensor{\alpha}{{}_{6}}+\chi^{2}/\xi\right)},$
(25a) $\displaystyle\left(\tensor{m}{{}_{0^{-}}}\right)^{2}$
$\displaystyle\equiv-\frac{8\tensor{\beta}{{}_{1}}-8\tensor{\beta}{{}_{2}}+{m_{\mathrm{P}}}{}^{2}}{4\tensor{\alpha}{{}_{2}}+12\tensor{\alpha}{{}_{4}}-2\chi^{2}/\xi},$
(25b) $\displaystyle\left(\tensor{m}{{}_{1^{+}}}\right)^{2}$
$\displaystyle\equiv\frac{-32\tensor{\beta}{{}_{1}}^{2}+16\tensor{\beta}{{}_{2}}^{2}-10\tensor{\beta}{{}_{2}}{m_{\mathrm{P}}}{}^{2}+{m_{\mathrm{P}}}{}^{4}+4\tensor{\beta}{{}_{1}}\left(4\tensor{\beta}{{}_{2}}+{m_{\mathrm{P}}}{}^{2}\right)}{4\left(\tensor{\alpha}{{}_{2}}-\tensor{\alpha}{{}_{3}}+4\tensor{\alpha}{{}_{4}}-4\tensor{\alpha}{{}_{6}}\right)\left(2\tensor{\beta}{{}_{1}}-\tensor{\beta}{{}_{2}}\right)},$
(25c) $\displaystyle\left(\tensor{m}{{}_{1^{-}}}\right)^{2}$
$\displaystyle\equiv-\frac{24\left(4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}\right)\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+3\tensor{\beta}{{}_{3}}+{m_{\mathrm{P}}}{}^{2}\right)}{\xi\left(48\tensor{\beta}{{}_{3}}\chi^{3}/\xi^{3}+4\tensor{\beta}{{}_{1}}\left(1+8\chi/\xi+24\chi^{2}/\xi^{2}\right)+2\tensor{\beta}{{}_{2}}\left(1+8\chi/\xi+24\chi^{2}/\xi^{2}\right)-{m_{\mathrm{P}}}{}^{2}-8{m_{\mathrm{P}}}{}^{2}\chi/\xi\right)},$
(25d) $\displaystyle\left(\tensor{m}{{}_{2^{+}}}\right)^{2}$
$\displaystyle\equiv\frac{\left(4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}\right){m_{\mathrm{P}}}{}^{2}}{4\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}\right)\left(3\tensor{\alpha}{{}_{2}}-\tensor{\alpha}{{}_{3}}+4\tensor{\alpha}{{}_{4}}-4\tensor{\alpha}{{}_{6}}-2\chi^{2}/\xi\right)},$
(25e) $\displaystyle\left(\tensor{m}{{}_{2^{-}}}\right)^{2}$
$\displaystyle\equiv\frac{4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}}{4\tensor{\alpha}{{}_{2}}-2\chi^{2}/\xi}.$
(25f)

By adding up the $1+1+3+3+5+5$ spin multiplicities of Eqs. 25a to 25f we
recover the $16+24$ kinematic d.o.f in
$\big{\\{}\tensor{b}{{}^{i}_{\mu}},\tensor{A}{{}^{ij}_{\mu}}\big{\\}}$, less
the $2\times(4+6)$ gauge d.o.f associated with Poincaré gauge symmetry — the
only surviving symmetry of the embedded theory. The branch
$2\tensor{\alpha}{{}_{2}}+4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}=\chi=0$
is likewise found by setting
$\tensor{\alpha}{{}_{5}}\mapsto-2\left(\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{4}}\right)$
and $\chi\mapsto 0$ in Equivalence of PGT and eWGT. The analysis is presented
in Fig. 9, and shows that the mass spectrum is fully consistent with imposing
$\chi\mapsto 0$ in Eqs. 25a to 25f. Finally, the case $\xi=\chi=0$ in
Equivalence of PGT and eWGT without any further restrictions leads in Fig. 10
to the general case of PGT in No-go theorem for vector torsion, which was
found previously in [60], namely

$\displaystyle\left(\tensor{m}{{}_{0^{+}}}\right)^{2}$
$\displaystyle\equiv\frac{{m_{\mathrm{P}}}{}^{2}\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+3\tensor{\beta}{{}_{3}}+{m_{\mathrm{P}}}{}^{2}\right)}{2\left(6\tensor{\alpha}{{}_{1}}+2\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{3}}+2\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}+2\tensor{\alpha}{{}_{6}}\right)\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+3\tensor{\beta}{{}_{3}}\right)},$
(26a) $\displaystyle\left(\tensor{m}{{}_{0^{-}}}\right)^{2}$
$\displaystyle\equiv-\frac{8\tensor{\beta}{{}_{1}}-8\tensor{\beta}{{}_{2}}+{m_{\mathrm{P}}}{}^{2}}{4\tensor{\alpha}{{}_{4}}-2\tensor{\alpha}{{}_{5}}},$
(26b) $\displaystyle\left(\tensor{m}{{}_{1^{-}}}\right)^{2}$
$\displaystyle\equiv-\frac{\left(4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}\right)\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+3\tensor{\beta}{{}_{3}}+{m_{\mathrm{P}}}{}^{2}\right)}{2\left(2\tensor{\alpha}{{}_{2}}+4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}\right)\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}+\tensor{\beta}{{}_{3}}\right)},$
(26c) $\displaystyle\left(\tensor{m}{{}_{2^{+}}}\right)^{2}$
$\displaystyle\equiv\frac{{m_{\mathrm{P}}}{}^{2}\left(-4\tensor{\beta}{{}_{1}}-2\tensor{\beta}{{}_{2}}+{m_{\mathrm{P}}}{}^{2}\right)}{4\left(\tensor{\alpha}{{}_{2}}+\tensor{\alpha}{{}_{3}}+4\tensor{\alpha}{{}_{4}}+2\tensor{\alpha}{{}_{5}}+4\tensor{\alpha}{{}_{6}}\right)\left(2\tensor{\beta}{{}_{1}}+\tensor{\beta}{{}_{2}}\right)},$
(26d) $\displaystyle\left(\tensor{m}{{}_{2^{-}}}\right)^{2}$
$\displaystyle\equiv\frac{-4\tensor{\beta}{{}_{1}}-2\tensor{\beta}{{}_{2}}+{m_{\mathrm{P}}}{}^{2}}{2\left(4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}\right)}.$
(26e)

Whilst we confirmed already that the $\chi\to 0$ limit of Eqs. 25a to 25f was
continuous, it is evident from Eq. 25d that once $\chi\to 0$ has been taken
the limit $\xi\to 0$ eliminates the $1^{-}$ vector in which we have been so
interested. In fact, this is not surprising, because taking $\chi\to 0$ at
finite $\xi$ corresponds to setting
$2\tensor{\alpha}{{}_{2}}+4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}=0$.
From Eq. 26c we see how this latter condition would equivalently kill the
$1^{-}$ mode in No-go theorem for vector torsion (we confirm this in Fig. 11),
and the No-go theorem for vector torsion theory-space is reached by $\xi\to
0$.

Figure 4:  Partial particle spectrograph of the completely general theory in
Equivalence of PGT and eWGT. From the $1^{-}$ sector of the saturated
propagator we may read off the quartic pole in Implications for the PGT
spectrum. All quantities are defined in Figs. 2 and 3. Figure 5:  Particle
spectrograph of the branch
$\tensor{\alpha}{{}_{5}}=\chi^{2}/\xi-2\left(\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{4}}\right)$
of Equivalence of PGT and eWGT, which gives rise to the mass spectra in Eqs.
25a to 25f. All quantities are defined in Figs. 2, 3 and 24. Figure 6:
Particle spectrograph of the branch
$\tensor{\alpha}{{}_{2}}=\tensor{\alpha}{{}_{3}}-4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{6}}=4\tensor{\alpha}{{}_{4}}+\tensor{\alpha}{{}_{5}}=4\tensor{\beta}{{}_{1}}+2\tensor{\beta}{{}_{2}}-{m_{\mathrm{P}}}{}^{2}=0$
of Equivalence of PGT and eWGT. All quantities are defined in Figs. 2 and 3.
Figure 7:  Particle spectrograph of Eq. 22, i.e. the minimal PGT realisation
of the novel massless scalar particle which appears in Fig. 6. All quantities
are defined in Figs. 2 and 3. Figure 8:  Particle spectrograph of Eq. 23, i.e.
the minimal model of the novel scalar particle which appears in Figs. 6 and 7.
We use the rotational gauge field perturbation
$\tensor{A}{{}_{\mu\nu\sigma}}\equiv\tensor{A}{{}_{[\mu\nu]\sigma}}$ as a
proxy for the Curtright field
$\tensor{C}{{}_{\mu\nu\sigma}}\equiv\tensor{C}{{}_{[\mu\nu]\sigma}}$ with
$\tensor{C}{{}_{[\mu\nu\sigma]}}\equiv 0$, because the latter multi-term
symmetry is a difficult attribute to declare in the PSALTer framework. It can
be confirmed (by applying transformation rules to the Lagrangian density in
this figure, and repeating the analysis) that the axial part
$\tensor{A}{{}_{[\mu\nu\sigma]}}$ does _not_ contribute to the spectrum. All
quantities are defined in Figs. 2 and 3. Figure 9:  Particle spectrograph of
the branch
$\tensor{\alpha}{{}_{5}}=-2\left(\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{4}}\right)$
with $\chi=0$ of Equivalence of PGT and eWGT, which gives rise to the
$\chi\mapsto 0$ limit of the mass spectra in Eqs. 25a to 25f (see Fig. 5). All
quantities are defined in Figs. 2, 3 and 24. Figure 10:  Particle spectrograph
of the branch $\xi=\chi=0$ of Equivalence of PGT and eWGT, which gives rise to
the mass spectra in Eqs. 26a to 26e, as first presented in [60]. All
quantities are defined in Figs. 2 and 3. Figure 11:  Particle spectrograph of
the branch
$\tensor{\alpha}{{}_{5}}=-2\left(\tensor{\alpha}{{}_{2}}+2\tensor{\alpha}{{}_{4}}\right)$
of No-go theorem for vector torsion, confirming that the $1^{-}$ mode is
removed. All quantities are defined in Figs. 2 and 3.