# The Effective Theory of Gravity and Dynamical Vacuum Energy

Emil Mottola<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics
and Astronomy, University of New Mexico
Albuquerque NM 87131

###### Abstract

Gravity and general relativity are considered as an Effective Field Theory
(EFT) at low energies and macroscopic distances. The effective action of the
conformal anomaly of light or massless quantum fields has significant effects
on macroscopic scales, due to associated light cone singularities that are not
captured by an expansion in local curvature invariants. A compact local form
for the Wess-Zumino effective action of the conformal anomaly and stress
tensor is given, requiring the introduction of a new light scalar field, which
it is argued should be included in the low energy effective action for
gravity. This scalar _conformalon_ couples to the conformal part of the
spacetime metric and allows the effective value of the vacuum energy,
described as a condensate of an exact $4$-form abelian gauge field strength
$F=dA$, to change in space and time. This is achieved by the identification of
the torsion dependent part of the Chern-Simons $3$-form of the Euler class
with the gauge potential $A$, which enters the effective action of the
conformal anomaly as a $J\cdot A$ interaction analogous to electromagnetism.
The conserved $3$-current $J$ describes the worldtube of $2$-surfaces that
separate regions of differing vacuum energy. The resulting EFT thus replaces
the fixed constant $\Lambda$ of classical gravity, and its apparently
unnaturally large sensitivity to UV physics, with a dynamical condensate whose
ground state value in empty flat space is $\Lambda_{\rm eff}=0$ identically.
By allowing $\Lambda_{\rm eff}$ to vary rapidly near the $2$-surface of a
black hole horizon, the proposed EFT of dynamical vacuum energy provides an
effective Lagrangian framework for gravitational condensate stars, as the
final state of complete gravitational collapse consistent with quantum theory.
The possible consequences of dynamical vacuum dark energy for cosmology, the
cosmic coincidence problem, and the role of conformal invariance for other
fine tuning issues in the Standard Model are discussed.

###### Contents

  1. I The Quantum Vacuum and Vacuum Energy in Black Holes and Cosmology
  2. II Relevance of the Conformal Anomaly to Macroscopic Gravity
  3. III The Effective Action of the Conformal Anomaly and Conformalon Scalar
  4. IV The Cosmological Term as a $4$-Form Gauge Field
  5. V The Chern-Simons $3$-Form and Anomaly Current Source for $F$
  6. VI The Effective Theory of Gravity in the Absence of Torsion
  7. VII The Effective Theory of Gravity in the Presence of Torsion
  8. VIII Gravitational Vacuum Condensate Stars in the EFT of Gravity
  9. IX Discussion and Outlook: Vacuum Energy as a Dynamical Condensate
  10. A Metrics, Tetrads, Differential Forms, and Hodge Star Dual
  11. B Topological and Torsional Susceptibility of the Gravitational Vacuum

##  I The Quantum Vacuum and Vacuum Energy in Black Holes and Cosmology

Observations of type Ia supernovae (SN) at moderately large redshifts indicate
that the expansion of the universe is accelerating [1, 2]. This is possible in
classical general relativity (GR) only if the dominant energy of the universe
has an effective mean eq. of state satisfying $\,\rho+3p<0$, i.e. assuming
positive energy density $\rho>0$, it must have negative pressure. As the
cosmological term $\Lambda$ enters Einstein’s eqs. as a constant with
$p_{\Lambda}=-\rho_{\Lambda}=-\Lambda/8\pi G_{\scriptstyle N}$ pervading all
space, the SN observations taken at face value imply a $\Lambda$ value of [3]

$\Lambda_{\mathrm{SN}}=\Omega_{\Lambda}\times
3\,\Bigg{(}\frac{H_{0}}{c}\Bigg{)}^{2}\simeq\left(\frac{\Omega_{\Lambda}}{0.70}\right)\left(\frac{H_{0}}{70{\,\rm
km/sec/Mpc}}\right)^{2}\ \left(\frac{3.1\times
10^{-122}}{L_{\mathrm{Pl}}^{2}}\ \right)\vspace{-3mm}$ (1)

when expressed in terms of the present Hubble expansion rate of $H_{0}\simeq
70\,$ km/sec/Mpc, or the microscopic Planck length
$L_{\mathrm{Pl}}=\sqrt{\hbar G_{\scriptstyle N}/c^{3}}\simeq 1.616\times
10^{-33}$ cm. respectively. Thus some $\Omega_{\Lambda}\simeq 70\%$ of the
energy in the present universe is in the form of $\Lambda$ dark energy, and is
the principal component of the current $\Lambda$CDM model of cosmology.

The contrast in (1) between the dimensionless value of the cosmological term
$\Omega_{\Lambda}$, of order unity in cosmological Hubble units, but of order
$10^{-122}$ in microscopic Planck units, is striking. From the time of W.
Pauli it has been thought that $\Lambda$ is related to the zero-point energy
density of the vacuum in quantum field theory (QFT), in which it appears as an
ultraviolet (UV) divergent sum over all field modes [4, 5, 6, 7]. If $\Lambda$
is such a UV sensitive quantity, and the short distance cutoff is of order of
$L_{\mathrm{Pl}}$, then the value of $\Lambda_{\mathrm{SN}}$ in Planck units
represents the most severe scale hierarchy problem in all of physics, clashing
with expectations of ‘naturalness’ developed over several decades of
successful application of Effective Field Theory (EFT) methods [8, 9, 10].

On the other hand, if one adopts the EFT hypothesis that macroscopic gravity
and the value of $\Lambda$ at cosmological scales should be decoupled from and
not require detailed knowledge of extreme UV physics, then (1) suggests
instead that the EFT of gravity is incomplete at _low energies_ , and one or
more additional EFT degrees of freedom are needed to account for a vacuum
energy naturally of order of the Hubble scale.

If the EFT of gravity relates $\Lambda_{\rm eff}$ to the cosmological Hubble
scale $3H_{0}^{2}$ rather than the microscopic Planck scale
$L_{\mathrm{Pl}}^{-2}$, the further implication is that $\Lambda_{\rm eff}$
would have to become a _dynamical_ quantity, i.e. dependent upon the content
and evolution of the universe, as the Hubble ‘constant’ itself is [6]. Related
to both possibilities of additional low energy gravitational degrees of
freedom other than the metric of classical GR and of $\Lambda_{\rm eff}$
becoming dynamical as a result, it is worth noting that current cosmological
models already require at least one additional scalar (inflaton) field of
unknown origin, in order to generate the present small ($\sim 10^{-5}$) CMB
anisotropies during a very early epoch of cosmic inflation [11]. This epoch is
assumed to have been dominated by a much larger effective $\Lambda_{\rm eff}$
vacuum energy, that is supposed to have dynamically ‘relaxed’ to its present
much smaller value.

Indications that the EFT of gravity may require some additional degree(s) of
freedom relevant at macroscopic scales come also from the quite different
domain of black hole (BH) physics. EFT methods in gravity for BHs have been
put into question by both the extreme blueshifting of energy scales in the
presence of horizons, invalidating the EFT assumption of decoupling of short
distance from long distance physics [12], and by the BH ‘information paradox’
[13, 14, 15, 16, 17, 18, 19, 20], and apparent conflict with unitary evolution
it entails [21]. The various forms of this paradox arise from ascribing an
enormous entropy to a BH, equal to $1/4$ its horizon area $A_{H}$ in Planck
units,

$S_{\mathrm{BH}}=k_{B}\,\mbox{\small$\displaystyle\frac{A_{H}}{4\,L_{\mathrm{Pl}}^{2}}$}\simeq
1.1\times
10^{77}\,k_{B}\,\left(\mbox{\small$\displaystyle\frac{M}{M_{\odot}}$}\right)^{2}\vspace{-3mm}$
(2)

despite the assumed classical nature of the BH horizon as a mathematical
causal boundary only, with no independent degrees of freedom of its own.
Significant quantum effects on the macroscopic scale of the BH horizon are
also in apparent conflict with the usual EFT approach to gravity, which relies
on an expansion in local curvature invariants [22, 23, 24], since these can
yield only negligibly small corrections for large BHs with small local
curvatures at their horizons. In fact, the enormous entropy (2) and Hawking
effect upon which it is predicated rely crucially upon the specification of
the quantum vacuum state, which (as always in quantum theory), requires _non-
local_ boundary conditions, that are not determined solely by the local
curvature [25, 26, 27].

In previous work it has been noted that large quantum effects on the horizon
follow quite generally from the stress tensor of the conformal anomaly in both
BH and cosmological spacetimes [28, 29, 30]. The importance of the conformal
anomaly in the near horizon behavior of the stress tensor is a consequence of
the conformal scaling behavior of the metric near the horizon and the extreme
blueshifting of local frequencies, which renders all finite mass scales
irrelevant there [30]. It is just this extreme blueshifting of frequencies,
hence energies, that can lead to effects not accounted for in local EFT
expansions based on the assumption of decoupling and strict separation of
scales. State dependent quantum vacuum entanglement and polarization effects
are contained in the effective action of the conformal anomaly, which by its
nature spans multiple scales.

That the physics of BHs and vacuum energy are related is inherent also in the
proposed resolution of the BH information paradox by the formation of a
gravitational vacuum condensate star with interior $\Lambda_{\rm eff}$ eq. of
state $p=-\rho$ [31, 32]. This $\Lambda_{\rm eff}$ eq. of state in the
interior of a _gravastar_ , collapsed to its gravitational radius
$r_{\\!{}_{M}}=2G_{\scriptstyle N}M/c^{2}$ prevents further collapse to a BH
singularity for the same reason that it causes the Hubble expansion of the
universe to accelerate, namely by defocusing (rather than the usual focusing)
of worldline geodesics, avoiding the classical singularity theorems [33, 34,
35]. The localized formation of such a $p=-\rho$ gravitational vacuum
condensate within an ultra compact star can occur only if there is at least
one additional degree of freedom in the low energy EFT of gravity, whose
variation allows $\Lambda_{\rm eff}$ to change abruptly at or near
$r=r_{\\!{}_{M}}$. This turns the BH horizon from a mathematical surface to a
physical phase boundary layer with a positive surface tension [36, 37, 38].

The purpose of this paper is to propose and develop the EFT of low energy
gravity, deduced from general principles of QFT in curved space and the
conformal anomaly, in which finite dynamical vacuum energy is consistently
described as a scalar vacuum condensate. The description of vacuum energy by
the scalar dual to an exact $4$-form abelian field strength $F=dA$ requires
$\Lambda_{\rm eff}\geq 0$ with $\Lambda_{\rm eff}=0$ the unique value of
lowest energy in flat space, independently of UV physics. When the $3$-form
potential $A$ is identified with the Chern-Simons $3$-form of the Euler class,
a $J\cdot A$ interaction is induced by the conformal anomaly effective action,
in analogy with electromagnetism. The $3$-current $J$ source for $F$ describes
the worldtube of $2$-surfaces that separate regions of differing vacuum energy
$\Lambda_{\rm eff}$, which therefore becomes spacetime dependent. The
observational implications of this EFT extension of classical GR and
identification of the relevant low energy degrees of freedom describing
$\Lambda_{\rm eff}$ as a dynamical gravitational vacuum condensate for both BH
physics and cosmological vacuum dark energy can then be studied in detail.

The metric and curvature conventions of the paper are those of MTW [39], while
the definitions and conventions for tetrads and differential forms used are
reviewed in appendix A. A second appendix B is devoted to the topological
aspects of the Euler density, associated Chern-Simons $3$-form and physical
interpretation of the new constant $\varkappa$ introduced in the EFT, as a
torsional topological susceptibility of the gravitational vacuum.

##  II Relevance of the Conformal Anomaly to Macroscopic Gravity

If one takes as the basic building block of a gravitational theory the
spacetime metric $g_{\mu\nu}(x)$, with the requirements that the field eqs.
must transform as tensor eqs. under general coordinate transformations, and be
no higher than second order in derivatives of the metric, one arrives at the
classical theory of general relativity (GR). This is described by the
classical action

$S\\!_{cl}=S_{\rm EH}-\mbox{\small$\displaystyle\frac{\Lambda}{8\pi
G_{\scriptstyle
N}}$}\int\\!d^{4}x\sqrt{-g}=\mbox{\small$\displaystyle\frac{1}{16\pi
G_{\scriptstyle
N}}$}\int\\!d^{4}x\sqrt{-g}\,\big{(}R-2\Lambda\big{)}\vspace{-4mm}$ (3)

namely the Einstein-Hilbert (EH) action involving the Ricci curvature scalar
$R$, second order in derivatives of the metric, or first order in derivatives
of the symmetric Christoffel connection

$\Gamma^{\lambda}_{\ \,\mu\nu}=\Gamma^{\lambda}_{\
\,\nu\mu}=\mbox{\small$\displaystyle\frac{1}{2}$}g^{\lambda\rho}\,\Big{(}\\!-\partial_{\rho}g_{\mu\nu}+\partial_{\mu}g_{\nu\rho}+\partial_{\nu}g_{\mu\rho}\Big{)}\,,\vspace{-3mm}$
(4)

together with the cosmological constant term $\Lambda$, involving no metric
derivatives. In modern terms the requirement of an action $S\\!_{cl}$ composed
of sums of all integrals of scalars that are invariant under general
coordinate transformations (up to possible surface terms) and yield field eqs.
no higher than second derivatives of the metric is just what is meant by a low
energy EFT of gravity, since local invariant terms higher order in derivatives
of the metric are negligible at low energies or long wavelengths. In (3) the
constants $G_{\scriptstyle N}$ and $\Lambda$ can be determined only by
experiment or astronomical observations. At this purely classical level, if
$\Lambda$ is given by observations to be (1) there is no naturalness problem,
for there is no other scale in the classical theory to which it can be
compared.

As É. Cartan pointed out soon after the appearance of GR [40, 41], the most
general setting of an affine geometry and differential manifold allows also
for a non-zero torsion, which is described by an anti-symmetric part of the
connection $\Gamma^{\lambda}_{\ \,[\mu\nu]}\neq 0$ that Einstein had assumed
to be vanishing, as the simplest realization of the Equivalence Principle. An
anti-symmetric part of $\Gamma^{\lambda}_{\ \,\mu\nu}$ drops out of the
geodesic eq. for the worldlines of freely falling point particles in any case.
Einstein-Cartan theory allows the connection $\Gamma^{\lambda}_{\ \,\mu\nu}$
and functions of it to be treated as dynamical variables in their own right, a
priori independent of the spacetime metric [41], a property that will be
exploited in Sec. V.

Let us note that the problems of reconciling classical GR with QFT first
appear with the stress-energy tensor $T^{\mu\nu}$, which is treated as a
completely classical source in Einstein’s eqs., whereas $\hat{T}^{\mu\nu}$ is
a UV divergent operator in QFT. Since matter and radiation in the Standard
Model (SM) are certainly quantum in nature, replacing the quantum operator
$\hat{T}^{\mu\nu}$ by its renormalized expectation value
$\langle\hat{T}^{\mu\nu}\rangle$ in a semi-classical approximation amounts to
two different assumptions, which should be recognized and distinguished at the
outset.

The first assumption is that UV divergences of QFT are to be removed by
counterterms involving up to dimension-four curvature invariants, such as
$R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu},R_{\alpha\beta}R^{\alpha\beta},R^{2}$,
by adding them to the effective action with finite renormalized coefficients,
leaving the low energy EFT unchanged. These terms
$\sim(\partial^{2}g_{\mu\nu})^{2}$, involve up to four derivatives of the
metric, in contrast to the dimension-two EH action. Since quantum theory
introduces a new scale $L_{\mathrm{Pl}}$, this standard renormalization
procedure amounts to the assumption that such higher derivative terms may be
important only on the corresponding Planck energy scale of
$M_{\mathrm{Pl}}c^{2}=1.221\times 10^{19}$ GeV. Since this scale is so much
higher than those generally encountered either in terrestrial accelerators or
astrophysics, the reasonable assumption of an EFT approach is that Planck
scale physics decouples from the low energy EFT, so that knowledge of the UV
completion or full quantum theory is not needed to describe macroscopic
gravitation.

Less often noted is a second critical assumption in the replacement of the
stress tensor source in Einstein’s eqs. by a finite renormalized
$\langle\hat{T}^{\mu\nu}\rangle$, treated classically, namely that the
operator $\hat{T}^{\mu\nu}$ can be well approximated by its sharply peaked
mean value. It is easily verified in QFT that even after renormalization there
are quantum fluctuations from the mean, and e.g.
$\big{\langle}\hat{T}^{\alpha\beta}(x)\,\hat{T}^{\mu\nu}(y)\big{\rangle}-\big{\langle}\hat{T}^{\alpha\beta}(x)\big{\rangle}\,\big{\langle}\hat{T}^{\mu\nu}(y)\big{\rangle}\neq
0$ at one-loop order [42]. Connected higher point correlation functions of
this kind probe the polarization and entanglement properties of the quantum
vacuum even at macroscopic scales. These quantum correlators exhibit operator
product singularities as $x\to y$. All is well with these UV singularities
since dimensional analysis, as well as explicit calculations show that these
quantum correlations grow large as $L_{\mathrm{Pl}}^{2}/\ell^{2}$ relative to
$\langle\hat{T}^{\mu\nu}\rangle$ only for metric variations on the length
scale $\ell\lesssim L_{\mathrm{Pl}}$. Neglecting these short distance
correlations at scales $\ell\gg L_{\mathrm{Pl}}$ reduces then to the first
decoupling assumption of EFT.

There is however a second kinematic regime where the Lorentz invariant
distance $(x-y)^{2}\to 0$, even for $x\neq y$, and where two and higher point
quantum correlation functions of $\hat{T}^{\mu\nu}$ can become large, namely
on the light cone. Since light cones extend over arbitrarily large distances,
lightlike correlations are _not_ limited to the ultrashort $L_{\mathrm{Pl}}$,
but can lead to _macroscopic_ quantum effects, in particular on null horizons
[28], which can be relevant in both BH and cosmological spacetimes with
positive $\Lambda$, such as de Sitter space. The two very different sorts of
quantum effects, short distance UV vs. macroscopic lightlike correlations are
distinct and require two quite different EFT treatments.

The short distance UV quantum corrections to GR are taken into account by
adding to the action (3) of classical GR the expansion in ascending powers of
higher derivatives of local invariants, divided by appropriate powers of the
UV cutoff scale, expected to be the Planck scale $M_{\mathrm{Pl}}$ for
gravity. This is the most common EFT approach [22, 23, 24]. As already
mentioned, it amounts to the decoupling assumption common to all EFT
approaches, based on the decoupling theorem of massive states in the UV from
the low energy degrees of freedom [43].

On the other hand it has also been known for some time that QFT anomalies are
not captured by such an expansion in higher order local invariants, nor are
they suppressed by any UV scale. Anomalies are associated instead with the
fluctuations of massless fields which do not decouple, and which lead to
$1/k^{2}$ poles in momentum space correlation functions, that grow large on
the light cone $k^{2}\to 0$ rather than the extreme UV regime $k^{2}\sim
M_{\mathrm{Pl}}^{2}$. The prototype of this light cone pole is the Schwinger
model of $1+1$ dimensional massless electrodynamics and its chiral anomaly
[44], which extends to the two-dimensional conformal anomaly in curved space
[45, 46]. In $3+1$ dimensional flat space, light cone poles are found in
explicit calculations in the triangle anomaly diagrams of
$\langle\hat{J}_{5}^{\lambda}\hat{J}^{\alpha}\hat{J}^{\beta}\rangle,\langle\hat{T}^{\mu\nu}\hat{J}^{\alpha}\hat{J}^{\beta}\rangle$
in massless QED4 [29, 47], and in the stress tensor three-point correlator
$\langle\hat{T}^{\alpha\beta}\hat{T}^{\gamma\lambda}\hat{T}^{\mu\nu}\rangle$
of a general conformal field theory (CFT), by solution of the conformal Ward
Identities in momentum space [48, 49].

Quite contrary to the decoupling hypothesis, quantum anomalies lead instead to
the principle of _anomaly matching_ from UV to low energy EFT [50]. In the
strong interactions, the chiral anomaly of the UV theory, QCD, survives to low
energies, requiring a specific Wess-Zumino (WZ) addition to the low energy
meson EFT [51, 8], which is not suppressed by any high energy scale, and is
technically a marginally relevant operator in the infrared (IR). Indeed, as
befits being associated with light cone singularities, the chiral anomaly has
both UV and IR features. The successful prediction of the low energy
$\pi^{0}\to 2\gamma$ decay rate provides a window into the UV and evidence for
the $S\\!U(3)^{\rm color}$ group and fractional charge assignments of quarks
that helped establish QCD as the UV theory of the strong interactions [52, 53,
54].

For gravitational theory it is the conformal anomaly in the trace of the
stress-energy tensor $\hat{T}^{\mu\nu}$ that is associated with $1/k^{2}$
anomaly poles in higher point correlation functions, such as
$\langle\hat{T}^{\alpha\beta}\hat{T}^{\gamma\lambda}\hat{T}^{\mu\nu}\rangle$.
Such light cone singularities imply the existence of at least one additional
light (_a priori_ massless) degree of freedom in the low-energy EFT of
macroscopic gravity, that is not accounted for in the classical action of GR
(3), nor by an expansion in higher order local curvature invariants.

A representative of the light cone singularities and massless pole associated
with conformal anomalies is afforded by the effective action of two-
dimensional gravity coupled to conformal matter [45, 55]

$S_{\rm anom}^{\rm NL,\,2D}[g]=-\frac{c_{m}}{96\pi}\int d^{2}x\sqrt{-g(x)}\int
d^{2}y\sqrt{-g(y)}\ R(x)\,\big{(}\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule
height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,^{-1}\big{)}_{xy}\,R(y)\vspace{-1mm}$ (5)

where $c_{m}=N_{s}+N_{f}$ is the central charge, given by the sum of the
numbers of massless scalar and fermion fields, and $\big{(}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,^{-1}\big{)}_{xy}$ denotes the Green’s function inverse of the
scalar wave operator. This exhibits the light cone pole of the 2D conformal
anomaly, appearing already in the connected two-point function
$\langle\hat{T}^{\alpha\beta}(x)\hat{T}^{\mu\nu}(y)\rangle$ of the underlying
CFT. The massless scalar pole in (5) indicates that there is an additional
scalar degree of freedom in 2D gravity coupled to conformal matter.

The scalar degree of freedom can be made explicit by expressing the non-local
anomaly effective action (5) in the equivalent local form

$S_{{\cal A}}^{\rm 2D}[g;\varphi]=-\frac{c_{m}}{96\pi}\int
d^{2}x\sqrt{-g}\,\Big{\\{}g^{\mu\nu}\big{(}\nabla\\!_{\mu}\varphi\big{)}\big{(}\nabla\\!_{\nu}\varphi\big{)}-2R\varphi\Big{\\}}\vspace{-3mm}$
(6)

by the introduction of the scalar field $\varphi$ describing a collective
spin-0 degree of freedom, which is linearly coupled to $R$, and whose massless
propagator gives rise to the light cone singularities of the underlying
massless CFT [46]. Variation of (6) with respect to $\varphi$ gives its eq. of
motion $-\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern
7.22743pt\vrule} \hrule}$}\,\varphi=R$, which when solved for
$\varphi=-\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern
7.22743pt\vrule} \hrule}$}\,^{-1}R$ and substituted back into (6) returns the
non-local form of the effective action (5). At the same time variation of (6)
with respect to the metric $g_{\mu\nu}$ yields the stress tensor
$T^{\mu\nu}_{\cal A}[g;\varphi]$ whose trace is $-(c_{m}/24\pi)\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi=c_{m}R/24\pi$ which is the 2D conformal anomaly. Taking two
variations with respect to $g_{\alpha\beta}(x)$ and $g_{\mu\nu}(y)$ yields the
connected two-point CFT correlator
$\langle\hat{T}^{\alpha\beta}(x)\hat{T}^{\mu\nu}(y)\rangle$ which exhibits a
massless $1/k^{2}$ light cone pole in flat space [46]. Clearly such a massless
scalar degree of freedom affects the macroscopic behavior of 2D gravity [56],
but cannot be described by a local action in curvature invariants alone, as is
the very nature of an anomaly.

##  III The Effective Action of the Conformal Anomaly and Conformalon Scalar

The anomalous Ward identities for all higher point quantum correlation
functions of the stress tensor containing anomalous light cone singularities
can be derived by functional variation of the basic one-point expectation
value of $\langle\hat{T}^{\mu\nu}\rangle$ in a general curved space background
[49]. In $D=4$ this mean value, defined and renormalized by any method that
preserves its covariant conservation, results in it acquiring an anomalous
trace in background gravitational and gauge fields, the general form of which
is [57, 58, 59]

$\big{\langle}\hat{T}^{\mu}_{\ \ \mu}\big{\rangle}\equiv
g_{\mu\nu}\,\big{\langle}\hat{T}^{\mu\nu}\big{\rangle}=b\,C^{2}+b^{\prime}\,\left(E-\tfrac{2}{3}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)+\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}\ \beta_{i}\,{\cal L}_{i}\vspace{-2mm}$ (7)

even if the underlying QFT is conformally invariant at tree level, and one
might have expected this trace to vanish. In (7)

$E=R_{\alpha\beta\gamma\lambda}R^{\alpha\beta\gamma\lambda}-4R_{\alpha\beta}R^{\alpha\beta}+R^{2}\,,\qquad
C^{2}=R_{\alpha\beta\gamma\lambda}R^{\alpha\beta\gamma\lambda}-2R_{\alpha\beta}R^{\alpha\beta}+\mbox{\small$\displaystyle\frac{1}{3}$}R^{2}\vspace{-3mm}$
(8)

are the Euler-Gauss-Bonnet invariant and the square of the Weyl conformal
tensor respectively. The $b,b^{\prime},\beta_{i}$ coeffcients in (7) are
finite dimenionless coefficients (in units of $\hbar$) that depend only upon
the number and spin of the massless conformal fields contributing to the
anomaly, i.e.

$b=\mbox{\small$\displaystyle\frac{1}{(4\pi)^{2}}$}\,\mbox{\small$\displaystyle\frac{1}{120}$}\,\Big{(}N_{s}+6N_{f}+12N_{v}\Big{)}\,,\qquad
b^{\prime}=-\mbox{\small$\displaystyle\frac{1}{(4\pi)^{2}}$}\,\mbox{\small$\displaystyle\frac{1}{360}$}\,\Big{(}N_{s}+11N_{f}+62N_{v}\Big{)}\vspace{-2mm}$
(9)

where $(N_{s},N_{f},N_{v})$ are the number of free conformal scalar, Dirac
fermion, and gauge vector fields respectively. Interactions of the massless or
light QFT degrees of freedom are taken into account by the ${\cal L}_{i}$
terms in (7), which denote invariant Lagrangians to which these fields are
coupled, such as ${\cal L}_{F}=F_{\alpha\beta}F^{\alpha\beta}$ for light
charged particles coupled to electromagnetism, or ${\cal L}_{G}={\rm
tr}\,\big{\\{}G_{\alpha\beta}G^{\alpha\beta}\big{\\}}$ for light quarks
coupled to the $S\\!U(3)^{\rm color}$ gluonic gauge fields of QCD. The
$\beta_{i}$ are proportional to the $\beta$-functions of these couplings.

All terms in (7) are dependent only upon the low energy QFT particle content,
independently of a UV cutoff or the Planck scale. They are therefore
independent of UV physics and with fixed matter content cannot be removed in
any metric theory of gravity with a covariantly conserved stress tensor.
Successive variations of (7) with respect to the arbitrary metric background
yield the anomalous trace Ward identities that the stress tensor correlators
must satisfy, even in flat space. Note that these variations are independent
of the purely local counterterms needed to define a renormalized
$\big{\langle}\hat{T}^{\mu}_{\ \,\mu}\big{\rangle}$.

To construct the effective action corresponding to (7) one notes that upon
multiplying (7) by $\sqrt{-g}$, the various terms transform as

$\displaystyle\sqrt{-g}\,C^{2}$ $\displaystyle\to\sqrt{-g}\,C^{2}\,$ (10a)
$\displaystyle\sqrt{-g}\,{\cal L}_{i}$ $\displaystyle\to\sqrt{-g}\,{\cal
L}_{i}$ (10b) $\displaystyle\sqrt{-g}\,\left(E-\tfrac{2}{3}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)$ $\displaystyle\to\sqrt{-g}\,\left(E-\tfrac{2}{3}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)+4\,\sqrt{-g}\,\Delta_{4}\,\sigma\ $ (10c)

under the conformal variation of the metric $g_{\mu\nu}\to
e^{2\sigma}g_{\mu\nu}$. Here

$\Delta_{4}\equiv\nabla_{\mu}\left(\nabla^{\mu}\nabla^{\nu}+2R^{\mu\nu}-\tfrac{2}{3}Rg^{\mu\nu}\right)\nabla_{\nu}=\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,^{2}+2R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\tfrac{2}{3}R\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,+\tfrac{1}{3}(\nabla^{\mu}R)\nabla_{\nu}\vspace{-3mm}$ (11)

is the (unique) fourth order scalar differential operator that is conformally
covariant [60, 61]

$\sqrt{-g}\,\Delta_{4}\to\sqrt{-g}\,\Delta_{4}\vspace{-4.5mm}$ (12)

for arbitrary $\sigma(x)$. It is thus the four-dimensional analog of the
second order wave operator  which has the conformal property
$\sqrt{-g}\,\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule
height=7.22743pt\kern 7.22743pt\vrule} \hrule}$}\,\to\sqrt{-g}\,\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,$ in $D=2$.

Using these properties under conformal variations and the fact that the
anomaly density

${\cal A}\equiv\sqrt{-g}\,\big{\langle}\hat{T}^{\mu}_{\ \
\mu}\big{\rangle}=\frac{\delta S_{\rm
anom}}{\delta\sigma}=2\,g_{\mu\nu}\,\frac{\delta S_{\rm anom}}{\delta
g_{\mu\nu}}\vspace{-3mm}$ (13)

is the conformal variation of an effective action of the anomaly, it is
straightforward to find an action satisfying (13). The non-local form of this
anomaly effective action is [61, 62, 63, 64]

$\displaystyle S^{\mathrm{NL}}_{\rm anom}[g]$
$\displaystyle=\mbox{\small$\displaystyle\frac{1}{4}$}\int
d^{4}x\sqrt{-g_{x}}\,\left(E-\tfrac{2}{3}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)_{x}$ $\displaystyle\quad\times\int
d^{4}y\sqrt{-g_{y}}\,\big{(}\Delta_{4}^{-1}\big{)}_{xy}\,\bigg{\\{}\mbox{\small$\displaystyle\frac{b^{\prime}}{2}$}\left(E-\tfrac{2}{3}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)+b\,C^{2}+\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}\,\beta_{i}\,{\cal L}_{i}\bigg{\\}}_{y}$ (14)

where $\big{(}\Delta_{4}^{-1}\big{)}_{xy}$ denotes the Green’s function
inverse of the fourth order differential operator (11) between the spacetime
points $x$ and $y$, indicated by subscripts in (14), so that $\int
d^{4}x\,\big{(}\sqrt{-g}\Delta_{4}\big{)}_{x}\big{(}\Delta_{4}^{-1}\big{)}_{xy}=1$,
for all $y$. The non-local effective action (14) of the conformal anomaly in
$D=4$ is the analog of non-local effective action (5) in $D=2$ spacetime
dimensions. Note that due to the appearance of the fourth order curvature
invariants in (7) and (14) of $D=4$, a single light cone $1/k^{2}$ pole of the
anomaly first appears for _three_ variations of (14) with respect to the
metric, in the $3$-point correlation function
$\langle\hat{T}^{\alpha\beta}\hat{T}^{\gamma\lambda}\hat{T}^{\mu\nu}\rangle$
in flat space. This has been checked explicitly for a general CFT in [49].

The anomaly effective action (14) is one term in the full one-particle
irreducible (1PI) effective action obtained by integrating out all the
matter/radiation fields in a fixed but arbitrary metric and background gauge
fields. It is possible to classify all the terms in the full 1PI effective
action of QFT into three general classes, so that the full 1PI action may be
expressed as the sum [65]

$S\\!_{\rm 1PI}[g]=S\\!_{\rm local}[g]+S^{\\!\mathrm{NL}}_{\\!\rm
anom}[g]+S\\!_{\rm inv}[g]\vspace{-4mm}$ (15)

of (i) purely local, (ii) non-local anomalous, and (iii) non-local invariant
under the action of the local Weyl transformation $g_{\mu\nu}\to
e^{2\sigma}g_{\mu\nu}$. The classification of terms (15) for the possible
quantum corrections to the effective action for gravity and the conformal
anomaly effective action (14) applies in a general curved space background,
and hence is considerably more general than expansions around flat space [66,
67, 68]. The local terms $S\\!_{\rm local}$ are the ones usually considered in
EFT approaches including the clearly IR relevant cosmological constant and EH
term of (3) which scale as $e^{4\sigma_{0}}$ and $e^{2\sigma_{0}}$ under
global Weyl rescalings respectively, together with terms higher order in local
curvature invariants, divided by some high energy UV energy scale, which scale
as $e^{-2n\sigma_{0}}$ for $n\geq 0$. For $n>0$ these local terms are
irrelevant in the IR. The $n=0$ terms are marginal and require special care.

The local $R^{2}$ and $C^{2}$ actions are neutral under global Weyl
rescalings, while $S^{\\!\mathrm{NL}}_{\\!\rm anom}$ of (14) is unique (up to
the $b,b^{\prime},\beta_{i}$ coefficients and $S_{\rm inv}$) in scaling
_linearly_ with $\sigma_{0}$, i.e. logarithmically under the global rescalings
of the metric and distance scales. Hence the anomaly terms (14) can grow to
importance in the IR and are classified as marginally relevant. All other
$n=0$ terms satisfying $S\\!_{\rm inv}[e^{2\sigma}g]=S\\!_{\rm inv}[g]$ are
neutral under all Weyl rescalings, and hence do not grow in the IR. While
contributions to low energy gravity of the Weyl invariant terms $S\\!_{\rm
inv}[g]$ cannot be excluded, the minimal assumption is to take only the
relevant $n<0$ local terms of (3) and the logarithmic non-local (14) as the
basis for an EFT treatment of gravity. One could add to (7) a $\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R$ term with an arbitrary coefficient, but since this term is the
trace of the metric variation of a local $R^{2}$ action, it is classified with
the local terms $S\\!_{\rm local}$ and not as part of the true anomaly, which
is _not_ the trace of a variation of any local action [65].

Like the chiral anomaly in QCD, the conformal anomaly is intrinsically a non-
local quantum effect and like (5), $S^{\\!\mathrm{NL}}_{\\!\rm anom}[g]$ is
non-local in terms of the original metric and curvature variables. The non-
local 1PI effective action of the conformal anomaly (7) in 4D can nevertheless
be rendered in a compact local form [61, 69, 30, 64] by introducing an
additional local scalar field $\varphi$ and making the replacement

$\displaystyle S^{\mathrm{NL}}_{\rm anom}[g]\to S\\!_{\cal A}[g;\varphi]=$
$\displaystyle\mbox{\small$\displaystyle\frac{b^{\prime}}{2\,}$}\int
d^{4}x\sqrt{-g}\,\bigg{\\{}\\!-\left(\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi\right)^{2}+2\,\Big{(}R^{\mu\nu}-\tfrac{1}{3}Rg^{\mu\nu}\Big{)}\,(\nabla_{\mu}\varphi)\,(\nabla_{\nu}\varphi)\bigg{\\}}+\mbox{\small$\displaystyle\frac{1}{2}$}\int
d^{4}x\,{\cal A}\,\varphi\,.$ (16)

Since $S\\!_{\cal A}[g;\varphi]$ is quadratic in $\varphi$ with kinetic term
$-(b^{\prime}/2)\int d^{4}x\sqrt{-g}\,\varphi\Delta_{4}\varphi$, variation of
(16) with respect to $\varphi$ gives the linear eq.
$b^{\prime}\sqrt{-g}\,\Delta_{4}\varphi={\cal A}/2$, which when solved for
$\varphi$ and substituted back into (16) returns the non-local form of the
effective action (14), up to a surface term and a Weyl invariant term that can
be absorbed into $S\\!_{\rm inv}[g]$. The anomaly effective action (16) is the
4D analog of (6), and both are non-trivial solutions of the WZ consistency
condition [63, 65, 64]

$S\\!_{\cal A}[e^{-2\sigma}g;\varphi]=S\\!_{\cal
A}[g;\varphi+2\sigma]-S\\!_{\cal A}[g;2\sigma]\vspace{-4mm}$ (17)

under Weyl conformal transformations. The scalar degree of freedom $\varphi$
is therefore closely related (by the shift $\varphi\to\varphi+2\sigma$) to the
conformal factor of the spacetime metric to which it couples.

The general classification (15) does not preclude the possibility that $S_{\rm
inv}$ might contain additional light cone singular terms relevant in the IR,
implying additional low energy degrees of freedom besides $\varphi$. However
only a single scalar $\varphi$ is necessary to account for the known anomaly.
Although it is possible to introduce additional low energy degrees of freedom,
as was done in [69, 28], there is no general symmetry-based reason to do so,
and the simplest possibility is the minimal one of $S\\!_{\cal A}[g;\varphi]$,
with $\varphi$ the single scalar carrying all the conformal transformation
properties of the WZ effective action and identity (17). Thus (16) is the
minimal addition to the classical action (3) needed to take into account the
light cone singularities of correlation functions of the stress tensor of
quantum matter. Also although $\varphi$ resembles the dilaton of string theory
in some respects, it should be distinguished from it, because of the specific
WZ identity (17) required by the Weyl cohomology of the anomaly [65], which
reflects its quite different physical origin. The scalar $\varphi$ is a
_collective_ mode, composed of a quantum correlated pair of massless SM fields
contributing to $\langle\hat{T}^{\mu\nu}\rangle$ [29, 30], similar to the
Schwinger boson in QED2, or a $U(1)^{\rm ch}$ flavor singlet ($\eta$ or
$\eta^{\prime}$) meson in low energy QCD. For this reason the distinct term of
_conformalon_ is reserved for $\varphi$.

As an explicit solution of the WZ consistency condition (17), the effective
action (16) is a generating functional that reproduces all the anomalous
conformal Ward identities of a CFT. Since $\varphi$ has zero scaling
dimension, and its propagator
$\big{(}\Delta_{4}^{-1}\big{)}_{xy}\sim\log(x-y)^{2}$ in position space, (16)
scales logarithmically under global Weyl rescalings, and is a marginally IR
relevant operator under finite size renormalization group scaling [70]. In
$S\\!_{\cal A}[g;\varphi]$ the $b,b^{\prime}$ coefficients are to be fixed by
experiment, as they may receive contributions from the gravitational field(s)
themselves [63]. These coefficients may become dependent upon the energy scale
as well, since the number and spin of fields that can be considered light
enough to be effectively massless, and contributing to the conformal anomaly,
is scale dependent.

As in 2D the WZ effective action of the anomaly $S\\!_{\cal A}$ is not purely
a local functional of higher order curvature invariants, unlike the higher
dimensional quantum corrections to GR usually considered [71, 22, 24]. Being
derived from non-local quantum fluctuations of massless fields, the effects of
(16) need not be negligible on macroscopic scales far greater than
$L_{\mathrm{Pl}}$, and generally are relevant on null horizons, even when
local curvatures are small there [28, 30, 72]. The massless excitations
described by $\varphi$ do not generally decouple, so they can have physically
relevant effects at low energies, which must be studied on a case by case
basis, and particularly in cases where naive EFT decoupling arguments would
seem to fail. In fact the covariantly conserved stress tensor

$T\\!_{\cal A}^{\
\mu\nu}[g;\varphi]\equiv\mbox{\small$\displaystyle\frac{2\\!}{\\!\\!\sqrt{-g}}$}\,\mbox{\small$\displaystyle\frac{\delta}{\delta
g_{\mu\nu}}$}\ S\\!_{\cal A}[g;\varphi]\vspace{-4mm}$ (18)

derived from (16), whose trace is (7), generally grows without bound like
$(r-r_{\\!{}_{M}})^{-2}$, resp. $(r-r_{\\!{}_{H}})^{-2}$, as either the
Schwarzschild or de Sitter horizons at $r_{\\!{}_{M}}$ or $r_{\\!{}_{H}}$ are
approached [28], cf. section VIII.

The effective action $S\\!_{\cal A}$ thus amounts to a specific addition to
Einstein’s GR, consistent with, and in fact required by first principles of
QFT, general covariance, and the general form of the conformal anomaly (7). It
is a relevant addition in both the mathematical and physical sense [65],
capturing the macroscopic light cone singularities of anomalous correlation
functions. It is therefore a necessary part of the low energy EFT of gravity,
and should be added to (3) of classical GR, much as the WZ term must be added
to the low energy meson theory to account for the chiral anomaly of QCD.

##  IV The Cosmological Term as a $4$-Form Gauge Field

The action (16) of the conformal anomaly is the first essential element in the
EFT of gravity taking macroscopic quantum effects into account. Quite apart
from and independent of the anomaly, there is a second element that plays an
essential role in the characterization of vacuum energy and resolution of the
naturalness problem of $\Lambda$. This is the observation that the constant
$\Lambda$ term in (3) can be reformulated in terms of an abelian gauge theory
as follows [73, 74, 75, 76, 77]. Let

$F=\mbox{\small$\displaystyle\frac{1}{\,4!}$}\,F\\!_{\alpha\beta\gamma\lambda}\,dx^{\alpha}\wedge
dx^{\beta}\wedge dx^{\gamma}\wedge dx^{\lambda}\vspace{-3mm}$ (19)

be a $4$-form field strength which is the curl of a totally anti-symmetric
$3$-form gauge potential

$F\\!_{\alpha\beta\gamma\lambda}=4\,\partial_{[\alpha}A_{\beta\gamma\lambda]}=4\,\nabla_{[\alpha}A_{\beta\gamma\lambda]}=\nabla_{\alpha}A_{\beta\gamma\lambda}-\nabla_{\beta}A_{\alpha\gamma\lambda}+\nabla_{\gamma}A_{\alpha\beta\lambda}-\nabla_{\lambda}A_{\alpha\beta\gamma}\vspace{-2mm}$
(20)

so that

$F=dA\,,\qquad
A=\mbox{\small$\displaystyle\frac{1}{\,3!}$}\,A_{\alpha\beta\gamma}\
dx^{\alpha}\wedge dx^{\beta}\wedge dx^{\gamma}\vspace{-2mm}$ (21)

i.e. $F$ is an exact $4$-form. As a natural generalization of ordinary
electromagnetism where $F=dA$ is an exact $2$-form exterior derivative of the
$1$-form vector gauge potential $A=A_{\mu}\,dx^{\mu}$, let $F$ be provided
with the ‘Maxwell’ action

$S\\!_{F}=-\mbox{\small$\displaystyle\frac{1}{2\varkappa^{4}}$}\int
F\wedge{\star}F=-\mbox{\small$\displaystyle\frac{1}{\,48\,\varkappa^{4}}$}\int
d^{4}x\sqrt{-g}\
F\\!_{\alpha\beta\gamma\lambda}F^{\alpha\beta\gamma\lambda}=\mbox{\small$\displaystyle\frac{1}{2\varkappa^{4}}$}\int
d^{4}x\sqrt{-g}\,{\widetilde{F}}^{\,2}\vspace{-2mm}$ (22)

where

${\widetilde{F}}\equiv{\star}F=\mbox{\small$\displaystyle\frac{1}{\,4!}$}\,\varepsilon_{\alpha\beta\gamma\lambda}\,F^{\alpha\beta\gamma\lambda}\,,\qquad
F\\!_{\alpha\beta\gamma\lambda}=-\varepsilon_{\alpha\beta\gamma\lambda}\,{\widetilde{F}}\vspace{-2mm}$
(23)

is the scalar Hodge star ${\star}$ dual to $F$, cf. (78), and $\varkappa$ is a
free parameter whose significance as the topological susceptibility of the
gravitational vacuum is discussed in appendix B.

Now the point is that when the rank of the $D$-form $F$ is matched to the
number of $D=4$ spacetime dimensions, the free ‘Maxwell’ theory (22) has two
very special properties, namely:

  1. (i)

$F$ is constrained to be a constant, with no propagating degrees of freedom,
and

  2. (ii)

its stress tensor $T^{\mu\nu}_{F}$ is proportional to the metric $g^{\mu\nu}$,
hence equivalent to a cosmological term.

The simplest example of this is usual $2$-form electrodynamics in $D=2$
spacetime dimensions, where the classical Maxwell action is

$-\frac{1}{4e^{2}}\int
d^{2}x\sqrt{-g}\,F\\!_{\alpha\beta}F^{\alpha\beta}=\frac{1}{2e^{2}}\int
d^{2}x\sqrt{-g}\,{\widetilde{F}}^{\,2}\vspace{-1mm}$ (24)

with the dual
${\widetilde{F}}=\frac{1}{2}\varepsilon_{\alpha\beta}F^{\alpha\beta}=F^{01}$
the electric field in one spatial dimension. The stress tensor corresponding
to (24) is $-g^{\mu\nu}{\widetilde{F}}^{\,2}/2e^{2}$, provided
$F\\!_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$ is
taken to be independent of the metric, with spacetime indices raised by
$g^{\alpha\beta}$. Since the electric field ${\widetilde{F}}$ is constrained
by Maxwell’s eqs. $\partial_{\nu}F^{\mu\nu}=j^{\mu}$ in $D=2$, to be a
spacetime constant of integration, i.e. ${\widetilde{F}}=const$. in the
absence of sources $j^{\mu}=0$, the stress tensor of electric field energy,
proportional to $g^{\mu\nu}$ is equivalent to a cosmological vacuum energy.
Classical Maxwell theory contains no propagating degrees of freedom at all in
one space plus one time dimension, in the absence of sources, and the constant
${\widetilde{F}}$ simply parametrizes the energy of the vacuum. In $D=2$ the
electric charge $e$ has mass dimension one, while in $D=4$, the constant
$\varkappa$ also carries mass dimension one if the field strength tensor
$F\\!_{\alpha\beta\gamma\lambda}$ is mass dimension four.

The exactly analogous situation obtains in $D=4$ for (19)–(23), and in fact
may be generalized to any even $D$ spacetime dimension [76]. The equivalence
to $\Lambda$ in $D=4$ follows once again the fact that the energy-momentum-
stress tensor corresponding to (22)

$T^{\mu\nu}_{F}=\frac{2\\!}{\\!\\!\sqrt{-g}}\frac{\delta S\\!_{F}}{\delta
g_{\mu\nu}}=-\frac{1}{4!\,\varkappa^{4}}\left(\mbox{\small$\displaystyle\frac{1}{2}$}g^{\mu\nu}F^{\alpha\beta\gamma\lambda}F\\!_{\alpha\beta\gamma\lambda}-4F^{\mu\alpha\beta\gamma}F^{\nu}\\!_{\alpha\beta\gamma}\right)=-\frac{1}{2\varkappa^{4}}\,g^{\mu\nu}\,{\widetilde{F}}^{\,2}\vspace{-2mm}$
(25)

is proportional to the metric tensor, if the convention that
$F\\!_{\alpha\beta\gamma\lambda}$ with all lower indices is independent of the
metric is again adopted. Analogous to the $D=2$ case that ${\widetilde{F}}$ is
a constant follows from the sourcefree ‘Maxwell’ eq. obtained by variation of
(22) with respect to $A_{\alpha\beta\gamma}$, viz.

$\nabla\\!_{\lambda}F_{\alpha\beta\gamma}\\!^{\,\lambda}=0\,,\qquad{\rm
for}\qquad J^{\alpha\beta\gamma}=0\vspace{-4mm}$ (26)

and $\partial_{\lambda}{\widetilde{F}}\\!=\\!0$, so that
${\widetilde{F}}\\!=\\!{\widetilde{F}}_{0}$ is a spacetime constant – in the
complete absence of any sources $J=0$.

Hence (19)–(23), and (25)–(26) are completely equivalent to a cosmological
term in Einstein’s eqs. in $D=4$ dimensions, with the identification

$\Lambda_{\rm eff}=\frac{4\pi G_{\scriptstyle
N}}{\varkappa^{4}}\,{\widetilde{F}}^{\,2}\geq 0\vspace{-3mm}$ (27)

the effective (necessarily non-negative) cosmological constant term for
$\varkappa$ and ${\widetilde{F}}={\widetilde{F}}_{0}$ real constants. In this
way one can freely trade a positive cosmological constant $\Lambda$ of
classical GR for a new fundamental constant $\varkappa$ of the low energy EFT,
together with an integration constant of the constraint
$\partial_{\lambda}{\widetilde{F}}=0$.

It may seem at first sight that little has been gained by this trade of an
equivalent reformulation of the $\Lambda$ term as a $4$-form gauge field at
the classical level. However the free integration constant
${\widetilde{F}}_{0}$ can then be fixed by a classical global boundary
condition in flat space, without any reference to quantum zero-point energy,
UV divergences, or cutoffs. A vanishing $\Lambda_{\rm eff}$ corresponds
instead to the vanishing of the sourcefree ‘electric’ field strength
${\widetilde{F}}=F^{0123}$ in infinite three-dimensional empty flat space,
analogous to the vanishing of ${\widetilde{F}}=F^{01}$ electric field of one
space dimension in the absence of sources. In either case this is simply the
classical state of lowest energy, as well as the unique state that is even
under the discrete symmetry of space parity inversion.

Moreover this setting of the value of the free constant
${\widetilde{F}}={\widetilde{F}}_{0}$, which is _a priori_ independent of
geometry, to zero in empty flat space is _required_ by the sourcefree
Einstein’s eqs.

$\left[R_{\mu\nu}-\mbox{\small$\displaystyle\frac{R}{2}$}g_{\mu\nu}\right]_{\rm
flat}=0=-\Lambda_{\rm eff}\Big{|}_{\rm flat}\,\eta_{\mu\nu}\vspace{-2mm}$ (28)

viewed as a low energy EFT. This shows already that flat space QFT estimates
of vacuum energy in any way dependent upon UV cutoffs or heavy mass scales are
_inconsistent_ with Einstein’s eqs. It is well-known that QFT in flat space is
sensitive only to _differences_ in energy. Hence the absolute value of quantum
zero point energy in flat space, and its dependence upon cutoffs or UV
regularization schemes is arbitrary and of no physical significance, a point
made before, e.g. [78], but well worth emphasizing. The value of $\Lambda_{\rm
eff}$ is significant only through its gravitational effects, and hence cannot
be evaluated in isolation, but only within the context of a gravitational EFT,
‘on shell’ as in (28), and only if each side of (28) can be evaluated
_independently_. This becomes possible only if $\Lambda_{\mathrm{eff}}$ is a
_free_ constant of integration, not a fixed parameter of the Lagrangian.

What has been gained then is that whereas in the usual treatment of $\Lambda$
as a fixed parameter of the classical theory (3), which receives quantum
corrections, dependent upon zero point energies and is apparently sensitive to
UV physics, treating $\Lambda_{\rm eff}$ instead as an integration constant of
the classical ‘Maxwell’ eq. $\partial_{\lambda}{\widetilde{F}}=0$ of a
$4$-form gauge field, the free constant ${\widetilde{F}}={\widetilde{F}}_{0}$
is independent of UV physics. Initially independent also of spacetime
curvature, its value is uniquely determined by evaluating both sides of (28)
in flat space. If $\Lambda_{\rm eff}$ is given by (27), ${\widetilde{F}}$ and
$\Lambda_{\rm eff}$ necessarily vanish in the flat space limit of GR, if
classical (or semi-classical) GR is to be a consistent approximation to the
low energy EFT of gravity.

Since the condition ${\widetilde{F}}_{0}=0$ in flat space holds for any value
of the EFT parameter $\varkappa$ in $D=4$, which remains arbitrary, the
condition on the value of the integration constant ${\widetilde{F}}_{0}$ at
the minimum of energy does not involve any fine tuning of fundamental
constants or naturalness problem in the EFT with the $\Lambda$ term replaced
by (22), any more than it does for setting the electric field strength
$F^{01}\\!=\\!0$ in $D=2$ classical Maxwell theory. Parameters of the EFT
Lagrangian, such as $\varkappa$ (or $\Lambda$), may receive UV divergent
contributions at higher loop order that require UV regularization and
renormalization, but the value of the classical ‘electric’ field and
integration constant $F^{0123}$ in flat space does not.

Thus the reformulation (27) of the cosmological constant $\Lambda_{\rm eff}$
in terms of ${\widetilde{F}}={\widetilde{F}}_{0}$ and $\varkappa$ shifts the
consideration of cosmological vacuum energy away from the UV divergences of
QFT to a macroscopic (IR) boundary condition solving the classical constraint
eq. (26) and minimization of energy in flat space. Although very simple
mathematically, and a completely equivalent parametrization of the
$\Lambda\to\Lambda_{\rm eff}$ term in the classical Einstein eqs. in the
absence of any sources for $F$, trading $\Lambda$ for ${\widetilde{F}}$ and a
boundary condition through (27) is a significant step conceptually. For in
addition to removing the fine tuning or naturalness problem of $\Lambda$,
introducing an independent $4$-form field $F$ in place of constant $\Lambda$
also allows for the introduction of sources in (26) that will enable $F$ (and
hence $\Lambda_{\rm eff}$) to change, departing from its zero value in
infinite sourcefree flat space in finite calculable ways, and eventually to
become a full-fledged dynamical variable of the low energy EFT in its own
right.

##  V The Chern-Simons $3$-Form and Anomaly Current Source for $F$

In section IV the $4$-form field strength $F=dA$ was postulated as an
independent degree of freedom, with the observation that it contributes to the
EFT of low energy gravity in the same way as an effective vacuum energy and
cosmological term according to (27). This has the advantage of reformulating
the naturalness problem of a vanishing cosmological term as simply the
solution of the sourcefree ‘Maxwell’ eq. (26) that minimizes the ‘electric’
energy in flat space, required by consistency of GR in its flat space limit
(28). In this section a fundamental geometric origin of $F$ is proposed by
identifying the abelian $3$-form potential $A$ with the possible torsion
dependent part of the Chern-Simons $3$-form defined by the topological Euler-
Gauss-Bonnet term in the the trace anomaly (7). This identification determines
the source current for the ‘Maxwell’ eq. (26) in terms of the conformalon
scalar $\varphi$, that allows $F$ and hence the vacuum energy $\Lambda_{\rm
eff}$ to change.

Of the several terms in the trace anomaly, the Euler-Gauss-Bonnet invariant
$E$ in (8) is distinguished by its topological character. Its integral is a
topological invariant insensitive to local variations, and therefore can be
related to global macroscopic effects, analogous to the index theorems
associated with the $\varepsilon_{\alpha\beta\mu\nu}F^{\alpha\beta}F^{\mu\nu}$
topological density of the axial anomaly [54]. Just as the axial anomaly
density can be expressed as the total divergence of a gauge dependent Chern-
Simons current, the topological character of $E$ implies that it is also is a
total divergence of a topological current, $E=\nabla\\!_{\mu}\Omega^{\mu}$,
with $\Omega^{\mu}$ dependent upon the choice of local Lorentz frame, through
the $S\\!O(3,1)$ gauge connection.

The explicit form of the topological $3$-form gauge field and $\Omega^{\mu}$
associated with $E$ follows from its relation to the $4$-form field strength

$\mathsf{F}\equiv\epsilon_{abcd}\,\mathsf{R}^{ab}\wedge\mathsf{R}^{cd}=\mbox{\small$\displaystyle\frac{1}{4}$}\epsilon_{abcd}\,R^{ab}_{\
\ \,\alpha\beta}\,R^{cd}_{\ \ \,\gamma\lambda}\,dx^{\alpha}\wedge
dx^{\beta}\wedge dx^{\gamma}\wedge dx^{\lambda}\vspace{-5mm}$ (29)

by the Hodge star ${\star}$ dual operation

$\displaystyle{\star}\mathsf{F}$
$\displaystyle=\varepsilon^{\alpha\beta\gamma\lambda}\left(\mbox{\small$\displaystyle\frac{1}{4}$}\epsilon_{abcd}\,R^{ab}_{\
\ \,\alpha\beta}\,R^{cd}_{\ \
\,\gamma\lambda}\right)=\mbox{\small$\displaystyle\frac{1}{4}$}\epsilon_{abcd}\,\epsilon^{mnrs}\,R^{ab}_{\
\ \ mn}R^{cd}_{\ \ \ rs}$
$\displaystyle=-\left(R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^{2}\right)=-E$ (30)

where the Latin $a,b,\dots$ are tangent space indices and the Greek
$\alpha,\beta,\dots$ are spacetime coordinate indices respectively. The
overall minus sign in (30) is the result of Lorentzian metric signature cf.
(83).

The curvature $2$-form $\mathsf{R}^{ab}$ in (29) is defined by the Cartan
structure eq.

$\mathsf{R}^{ab}=d\omega^{ab}+\omega^{ac}\wedge\omega^{\
\,b}_{c}\equiv\mbox{\small$\displaystyle\frac{1}{2}$}R^{ab}\\!_{\,\mu\nu}\,dx^{\mu}\wedge
dx^{\nu}\vspace{-3mm}$ (31)

that gives the Riemann curvature components $R^{ab}_{\ \ \,\mu\nu}$ in terms
of the affine connection $1$-form

$\omega^{ab}=-\omega^{ba}=\omega^{ab}\\!_{\,\mu}\,dx^{\mu}\vspace{-5mm}$ (32)

which specifies the law of parallel transport of orthonormal frames in tangent
space. Thus $\omega^{ab}$ may be regarded as a local gauge potential for the
Lie algebra of the ${\cal G}=S\\!O(3,1)$ Lorentz group, in close analogy to
Yang-Mills gauge potentials for any internal group ${\cal G}$, for which
$\mathsf{R}^{ab}$ would be the $2$-form field strength tensor.

The $4$-form $\mathsf{F}$ dual to $E$ by (30) is exact, i.e.
$\mathsf{F}=\mathsf{dA}$, where $\mathsf{A}$ is the $S\\!O(3,1)$ Lorentz frame
dependent Chern-Simons $3$-form [79]

$\mathsf{A}=\epsilon_{abcd}\left(\omega^{ab}\wedge
d\omega^{cd}+\mbox{\small$\displaystyle\frac{2}{3}$}\,\omega^{ab}\wedge\omega^{ce}\wedge\omega^{fd}\,\eta_{ef}\right)\vspace{-4mm}$
(33)

which has the spacetime coordinate components

$\mathsf{A}_{\alpha\beta\gamma}=3!\,\epsilon_{abcd}\left(\omega^{ab}\\!_{\,[\alpha}\,\partial\\!\\!_{\
\beta}\,\omega^{cd}\\!_{\,\gamma]}+\mbox{\small$\displaystyle\frac{2}{3}$}\,\omega^{ab}\\!_{\,[\alpha}\,\omega^{ce}\\!_{\,\beta}\,\omega^{fd}\\!_{\,\gamma]}\,\eta_{ef}\right)\vspace{-4mm}$
(34)

completely anti-symmetrized in its three indices $\alpha,\beta,\gamma$. These
relations imply, cf. (85)

$E=-\big{(}{\star}d\mathsf{A}\big{)}=\mbox{\small$\displaystyle\frac{1}{\,3!}$}\,\varepsilon^{\alpha\beta\gamma\mu}\partial_{\mu}\mathsf{A}_{\alpha\beta\gamma}=\mbox{\small$\displaystyle\frac{1}{\,3!}$}\,\,\mbox{\small$\displaystyle\frac{1}{\sqrt{-g}}$}\,\partial_{\mu}\left(\sqrt{-g}\,\varepsilon^{\alpha\beta\gamma\mu}\mathsf{A}_{\alpha\beta\gamma}\right)=\mbox{\small$\displaystyle\frac{1}{\,3!}$}\,\nabla\\!_{\mu}\left(\varepsilon^{\alpha\beta\gamma\mu}\mathsf{A}_{\alpha\beta\gamma}\right)\vspace{-2mm}$
(35)

demonstrating that the integrand $\sqrt{-g}\,E$ is in fact a total derivative
of a coordinate frame dependent abelian current dual to $\mathsf{A}$, with
$\Omega^{\mu}=\varepsilon^{\alpha\beta\gamma\mu}\mathsf{A}_{\alpha\beta\gamma}/3!$,
analogous to the topological density
$\varepsilon_{\alpha\beta\mu\nu}F^{\alpha\beta}F^{\mu\nu}$ of the axial
anomaly which is the total derivative of the gauge dependent Chern-Simons
current. Eq. (35) holds in the absence of torsion, since the it has been so
far implicitly assumed that the connection $\omega^{ab}$ is the usual
Riemannian or Levi-Civita connection of the metric $g_{\mu\nu}$.

To show that $\mathsf{A}$ is indeed an abelian gauge field, consider the
response of the gauge connection to an infinitesimal local $S\\!O(3,1)$
tangent frame rotation

$\delta_{\theta}\,\omega^{ab}=2\,\omega^{c[a}\,\theta^{\,b]}\\!_{\,c}+d\theta^{\,ab}\,,\qquad\delta_{\theta}\,\mathsf{R}^{ab}=2\,\mathsf{R}^{c[a}\,\theta^{\,b]}\\!_{\,c}\vspace{-4mm}$
(36)

where $\theta^{ab}(x)=-\theta^{ba}(x)$ are the $6$ functions of the local
Lorentz frame transformation. A short exercise then shows that the gauge
potential $3$-form $\mathsf{A}$ of (33) transforms under (36) as the exact
differential

$\delta_{\theta}\,\mathsf{A}=\epsilon_{abcd}\,d\theta^{\,ab}\wedge
d\omega^{cd}=d\left(\epsilon_{abcd}\,\theta^{\,ab}d\omega^{cd}\right)\vspace{-4mm}$
(37)

and hence $\delta_{\theta}\,\mathsf{F}=0$, so that the corresponding abelian
field strength tensor is invariant under the particular form of the local
$2$-form gauge parameter $\Theta=\epsilon_{abcd}\,\theta^{ab}\,d\omega^{cd}$.
This gauge transformation establishes the Chern-Simons $3$-form $\mathsf{A}$
as an abelian gauge potential. Since in order for the transformation (37) to
be non-null, $\Theta$ itself must be co-exact, i.e. not itself an exact
$2$-form, the gauge transformation (37) has just $3$ independent components.
This means that of the $4$ independent components of $\mathsf{A}$, three of
them are pure gauge, and only one is gauge invariant under (36). Hence the
single scalar ${\star}\mathsf{F}$ carries the full gauge invariant field
content of the $3$-form $\mathsf{A}$.

Now the essential point is that although $E$ is given in terms of the metric
and its derivatives in Riemannian spacetime by (8), the $3$-form potential
$\mathsf{A}$ given by (33) is defined in terms of the $S\\!O(3,1)$ spin
connection $\omega^{ab}$ in an orthonormal basis, and a priori independently
of the spacetime metric $g_{\mu\nu}$. This distinction becomes clear when one
considers Cartan’s second equation of structure

$\mathsf{T}^{a}=de^{a}+\omega^{a}\\!_{\,b}\wedge
e^{b}=\mbox{\small$\displaystyle\frac{1}{2}$}\,T^{a}\\!_{\,bc}\,e^{b}\wedge
e^{c}=\mbox{\small$\displaystyle\frac{1}{2}$}\,T^{a}\\!_{\,\mu\nu}\,dx^{\mu}\wedge
dx^{\nu}\ \vspace{-4mm}$ (38)

which defines the torsion $2$-form $\mathsf{T}^{a}$ [41, 80]. This definition
may be solved algebraically for the spin connection, viz. [41]

$\omega_{ab\,\mu}=-\omega_{ba\,\mu}=\upsilon^{\nu}\\!_{\,a}\,\eta_{bc}\,\big{(}\nabla_{\mu}e^{c}\\!_{\
\nu}\big{)}-K_{abc}\,e^{c}\\!_{\ \mu}\vspace{-2mm}$ (39)

whose first term is a purely Riemannian part in terms of the torsionless
covariant derivative

$\nabla\\!_{\mu}e^{c}_{\
\nu}=\partial_{\mu}e^{c}\\!_{\,\nu}-\Gamma^{\lambda}_{\ \,\mu\nu}e^{c}_{\
\lambda}\vspace{-5mm}$ (40)

with respect to the symmetric Levi-Civita connection $\Gamma^{\lambda}_{\
\,\mu\nu}$ of (4), which depends upon the metric and its derivatives. The
second part of (39) is dependent upon the torsion through the contorsion
tensor

$K_{abc}=\mbox{\small$\displaystyle\frac{1}{2}$}\,\big{(}T_{abc}+T_{bca}-T_{cab}\big{)}\vspace{-1mm}$
(41)

which is defined by (38) independently of the metric. The definitions and
properties of the vierbein field $e^{a}\\!_{\,\mu}$ and its inverse
$\upsilon^{\mu}\\!_{\,a}$ used here are given by eqs. (73)–(75) of appendix A.

It follows from (39) that if all components of torsion vanish,
$T^{a}\\!_{\,bc}=0$, then the affine connection $\omega_{ab\,\mu}$ reduces to
the Levi-Civita connection, which in holonomic coordinates is just the usual
symmetric Riemann-Christoffel symbol $\Gamma^{\lambda}\\!_{\ \mu\nu}$ of (4),
that is fully specified by the metric and its derivatives. In that case of
vanishing torsion the Chern-Simons $3$-form (33) is purely Riemannian
$\mathsf{A}=\mathsf{A_{R}}$ and metric dependent in the standard way.
Conversely, if torsion (38) is non-vanishing, both $\omega_{ab\,\mu}$ and
$\mathsf{A}$ will contain a torsion dependent part $\mathsf{A_{T}}$, which may
be treated as dynamical variables that can be varied independently of the
metric $g_{\mu\nu}$. In the general case (33) will contain both purely
Riemannian and torsion dependent terms, so that

$\mathsf{A}=\mathsf{A_{R}}+\mathsf{A_{T}}\,,\qquad
E=-({\star}d\mathsf{A_{R}})\vspace{-5mm}$ (42)

since the the spin connection (39) itself by which $\mathsf{A}$ is defined
contains purely Riemannian and torsional terms. In (42) $E$ is understood to
be the purely Riemannian, torsionless Euler-Gauss-Bonnet integrand of (8),
derived from $\mathsf{A_{R}}$ alone, so that $E=-({\star}d\mathsf{A_{R}})$
replaces (35) when torsion is present and $\mathsf{A_{T}}\neq 0$.

The independent variation of the affine connection is the basis of the first
order or Palatini formalism of GR [39, 81]. In the case of the EH action this
first order formalism leads to $T^{a}\\!_{\,bc}=0$ in the absence of spin
currents, and hence turns out to be equivalent to the more common approach to
GR where the connection is fixed to be the torsionless Christoffel connection
(4) from the start. For more general actions, including that of the conformal
anomaly, independent variation of the affine connection and the metric
generally leads to different Euler-Lagrange eqs., so that the resulting
Einstein-Cartan theory differs in general from the torsionless theory.

In the EFT based on the conformal anomaly it is only the particular dependence
on the spin connection through the $3$-form gauge field $\mathsf{A}$ of (33)
that enters, which has only $4$ (not the $24$ of $\omega_{ab\,\mu}$)
independent components. Hence rather than adopting the full first order
formalism, the proposal for the EFT is that the torsional parts of the Chern-
Simons $3$-form (33) and $4$-form (29) be identified with the corresponding
quantities introduced in Sec. IV, i.e.

$A=\mathsf{A_{T}}\qquad{\rm and}\qquad F=\mathsf{F_{T}}\vspace{-5mm}$ (43)

with $E$ replaced by $E-({\star}d\mathsf{A_{T}})=E-{\widetilde{F}}$ in the
anomaly effective action (16), and with $A=\mathsf{A_{T}}$ treated as an
independent variable, to be varied independently of the spacetime metric
$g_{\mu\nu}$. From (38)-(41) this independent variation is possible since (33)
holds also in a general Einstein-Cartan spacetime, if there is no a priori
condition on the torsion, which is defined independently of the metric.

With (35), the decomposition (42), and the identification (43), the term in
the anomaly effective action linear in $E$ and the conformalon $\varphi$ is
replaced by $(E-{\widetilde{F}})\varphi$, and the second term can be
integrated by parts, so that

$\mbox{\small$\displaystyle\frac{b^{\prime}}{2\,}$}\int
d^{4}x\sqrt{-g}\,\big{(}E-{\widetilde{F}}\big{)}\,\varphi=\mbox{\small$\displaystyle\frac{b^{\prime}}{2\,}$}\int
d^{4}x\sqrt{-g}\,E\,\varphi-\mbox{\small$\displaystyle\frac{b^{\prime}}{2}$}\mbox{\small$\displaystyle\frac{1}{\,3!}$}\,\int
d^{4}x\sqrt{-g}\,A_{\alpha\beta\gamma}\,\varepsilon^{\alpha\beta\gamma\mu}\,\partial_{\mu}\varphi\vspace{-3mm}$
(44)

up to a surface term which does not affect local variations and may be taken
to vanish for suitable boundary conditions. Then defining the $3$-current

$J^{\alpha\beta\gamma}\equiv-\mbox{\small$\displaystyle\frac{\,b^{\prime}}{2}$}\,\varepsilon^{\alpha\beta\gamma\mu}\,\partial_{\mu}\varphi\vspace{-5mm}$
(45)

the last term in (44) can be expressed in the form

$S_{\rm int}[\varphi,A]=\mbox{\small$\displaystyle\frac{1}{\,3!}$}\int
d^{4}x\sqrt{-g}\ J^{\alpha\beta\gamma}A_{\alpha\beta\gamma}\vspace{-4mm}$ (46)

analogous to a $J\cdot A$ interaction of ordinary electromagnetism.

The analogy with electromagnetism is apt because the current (45) is
covariantly conserved, i.e.

$\nabla\\!_{\gamma}J^{\alpha\beta\gamma}=-\mbox{\small$\displaystyle\frac{\,b^{\prime}}{2}$}\,\frac{1}{\sqrt{-g}}\
\partial_{\gamma}\left(\sqrt{-g}\,\varepsilon^{\alpha\beta\gamma\mu}\,\partial_{\mu}\varphi\right)=-\mbox{\small$\displaystyle\frac{\,b^{\prime}}{2}$}\,\varepsilon^{\alpha\beta\gamma\mu}\,\partial_{\gamma}\partial_{\mu}\varphi=0\vspace{-2mm}$
(47)

since $\sqrt{-g}\,\varepsilon^{\alpha\beta\gamma\mu}$ is independent of the
spacetime metric cf. (85). Thus the $J\cdot A$ interaction (46) is invariant
under the abelian gauge transformation

$\delta_{\Theta}A=d\Theta\,\qquad\delta_{\Theta}F=d^{2}\Theta=0\vspace{-4mm}$
(48)

where $\Theta$ is an arbitrary $2$-form, by another integration by parts. This
means that (45) is a candidate source term for the ‘Maxwell’ eq. (26),
consistent with abelian gauge invariance, in analogy with ordinary
electromagnetism.

With $A$ defined by the identification (43) in terms of the spin connection a
priori independent of the spacetime metric, it is a dynamical variable of the
EFT in its own right. Its variation independently of $g_{\mu\nu}$ is just what
is required to arrive at ‘Maxwell’ eqs. for $F$ with the conserved current
(45) as their source. Thus identifications of the $3$-form $A$ and $4$-form
$F=dA$ by (43), with the interaction (46) deduced from the anomaly effective
action have the consequence that the vacuum energy defined by $\Lambda_{\rm
eff}$ of (27) will change if and when $\varphi$ does and
$J^{\alpha\beta\gamma}\neq 0$, provided torsion (38) and $\mathsf{A_{T}}\neq
0$.

##  VI The Effective Theory of Gravity in the Absence of Torsion

Assembling the elements of the previous sections, the effective action for low
energy gravity in the absence of any torsional contribution to the Chern-
Simons $3$-form (33) is

$\displaystyle S^{\rm(I)}_{\rm
eff}[g;\varphi;A]=\mbox{\small$\displaystyle\frac{1}{16\pi G_{\scriptstyle
N}}$}\\!\int\\!d^{4}x\sqrt{-g}\,R+S\\!_{{\cal
A}}[g;\varphi]+S\\!_{F}[g;A]\hskip 28.45274pt\rm(I)$ (49)

where

$\displaystyle S\\!_{{\cal A}}[g;\varphi]$
$\displaystyle=\mbox{\small$\displaystyle\frac{b^{\prime}}{2\,}$}\int
d^{4}x\sqrt{-g}\,\bigg{\\{}\\!-\left(\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi\right)^{2}+2\,\Big{(}R^{\mu\nu}-\tfrac{1}{3}Rg^{\mu\nu}\Big{)}\,(\nabla_{\mu}\varphi)\,(\nabla_{\nu}\varphi)\bigg{\\}}$
$\displaystyle\quad+\mbox{\small$\displaystyle\frac{1}{2}$}\int
d^{4}x\sqrt{-g}\left\\{b\,C^{2}+b^{\prime}\left(E-\tfrac{2}{3}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)+\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}\,\beta_{i}\,{\cal L}_{i}\right\\}\varphi$ (50)

is the conformal anomaly effective action (16) and $S\\!_{F}$ is the ‘Maxwell’
action (22) of the $4$-form gauge field $F$, with (20). In this case (I) the
$3$-form gauge field $A$ and $4$-form field strength $F=dA$ are not coupled in
any way to the anomaly effective action $S\\!_{\cal A}[g;\varphi]$. Hence
${\widetilde{F}}={\widetilde{F}}_{0}$ is sourcefree and constant, as in (26),
and entirely equivalent to an effective cosmological term $\Lambda_{\rm eff}$
by (27).

With the condition that ${\widetilde{F}}_{0}=0$ in asymptotically flat space,
the lowest energy ground state, ${\widetilde{F}}$ and $S\\!_{F}[A;g]$ then
drop out entirely. In this case the EFT (I) is just classical GR with
$\Lambda_{\rm eff}\\!=\\!0$ and with the addition of the conformal anomaly
effective action $S\\!_{\cal A}[g;\varphi]$. Alternately, $S\\!_{F}[A;g]$ with
non-zero constant ${\widetilde{F}}$ may be retained to parametrize an
arbitrary positive constant vacuum energy $\Lambda_{\rm eff}$ in some
region(s) of space, its value to be determined by appropriate boundary
conditions, as in the application to gravitational condensate stars of section
VIII.

The classical Euler-Lagrange eqs. following from variation of (49) are

$\Delta_{4}\varphi\equiv\nabla_{\mu}\left(\nabla^{\mu}\nabla^{\nu}+2R^{\mu\nu}-\tfrac{2}{3}Rg^{\mu\nu}\right)\nabla_{\nu}\varphi=\mbox{\small$\displaystyle\frac{1}{2}$}\left(E-\mbox{\small$\displaystyle\frac{2}{3}$}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)+\mbox{\small$\displaystyle\frac{1}{2b^{\prime}}$}\,\left(bC^{2}+\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}\,\beta_{i}{\cal L}_{i}\right)\hskip 8.53581pt{\rm(I)}\vspace{-4mm}$ (51)

for the conformalon scalar $\varphi$, together with the semi-classical
Einstein eq.

$\hskip
28.45274ptR_{\mu\nu}-\mbox{\small$\displaystyle\frac{1}{2}$}Rg_{\mu\nu}=8\pi
G_{\scriptstyle N}\left(T\\!_{F\,\mu\nu}+T\\!_{{\cal
A}\,\mu\nu}[g;\varphi]+T^{cl}_{\mu\nu}\right)\vspace{-3mm}$ (52)

with $T^{\mu\nu}_{F}$ and $\Lambda_{\rm eff}$ given by (25) and (27), and with

$T_{{\cal
A}}^{\,\mu\nu}\,[g;\varphi]\equiv\mbox{\small$\displaystyle\frac{2}{\sqrt{-g}}$}\,\mbox{\small$\displaystyle\frac{\delta}{\delta
g_{\mu\nu}}$}\ S\\!_{\cal
A}[g;\varphi]=b^{\prime}E^{\mu\nu}+bC^{\mu\nu}+\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}\ \beta_{i}T^{(i)\,\mu\nu}\vspace{-3mm}$ (53)

the stress tensor resulting from the metric variation of $S\\!_{\cal
A}[g;\varphi]$. The first contribution here is

$\displaystyle E_{\mu\nu}$
$\displaystyle=-2\,(\nabla_{(\mu}\varphi)(\nabla_{\nu)}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi)+2\nabla^{\alpha}\big{[}(\nabla_{\alpha}\varphi)(\nabla_{\mu}\nabla_{\nu}\varphi)\big{]}-\tfrac{2}{3}\,\nabla_{\mu}\nabla_{\nu}\big{[}(\nabla_{\alpha}\varphi)(\nabla^{\alpha}\varphi)\big{]}$
$\displaystyle+\tfrac{2}{3}\,R_{\mu\nu}\,(\nabla_{\alpha}\varphi)(\nabla^{\alpha}\varphi)-4\,R^{\alpha}_{\
(\mu}\left[(\nabla_{\nu)}\varphi)(\nabla_{\alpha}\varphi)\right]+\tfrac{2}{3}\,R\,(\nabla_{(\mu}\varphi)(\nabla_{\nu)}\varphi)$
$\displaystyle+\tfrac{1}{6}\,g_{\mu\nu}\,\left\\{-3\,(\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi)^{2}+\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule
height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\big{[}(\nabla_{\alpha}\varphi)(\nabla^{\alpha}\varphi)\big{]}+2\,\big{(}3R^{\alpha\beta}-Rg^{\alpha\beta}\big{)}(\nabla_{\alpha}\varphi)(\nabla_{\beta}\varphi)\right\\}$
$\displaystyle-\tfrac{2}{3}\,\nabla_{\mu}\nabla_{\nu}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi-4\,C_{\mu\ \nu}^{\ \,\alpha\
\beta}\,\nabla_{\alpha}\nabla_{\beta}\varphi-4\,R^{\alpha}\\!_{(\mu}\nabla_{\nu)}\nabla_{\alpha}\varphi+\tfrac{8}{3}\,R_{\mu\nu}\,\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi+\tfrac{4}{3}\,R\,\nabla_{\mu}\nabla_{\nu}\varphi-\tfrac{2}{3}\left(\nabla\\!_{(\mu}R\right)\nabla_{\nu)}\varphi$
$\displaystyle+\tfrac{1}{3}\,g_{\mu\nu}\,\left[2\,\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,^{2}\varphi+6\,R^{\alpha\beta}\,\nabla_{\alpha}\nabla_{\beta}\varphi-4\,R\,\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\varphi+(\nabla^{\alpha}R)\nabla_{\alpha}\varphi\right]$ (54)

which is the metric variation of all the $b^{\prime}$ terms in (16), both
quadratic and linear in $\varphi$ [28, 30, 64], while

$\displaystyle C_{\mu\nu}\equiv-\frac{1}{\sqrt{-g}\ }\frac{\delta}{\delta
g^{\mu\nu}}\left\\{\int
d^{4}x\sqrt{-g}\,C^{2}\,\varphi\right\\}=-4\,\nabla_{\alpha}\nabla_{\beta}\,\Big{(}C_{(\mu\
\nu)}^{\ \ \alpha\ \ \beta}\,\varphi\Big{)}-2\,C_{\mu\ \,\nu}^{\ \,\alpha\
\,\beta}\,R_{\alpha\beta}\,\varphi$ (55a) $\displaystyle\hskip
108.12054ptT^{(i)}_{\mu\nu}\equiv-\frac{1}{\sqrt{-g}\ }\,\frac{\delta}{\delta
g^{\mu\nu}}\left\\{\int d^{4}x\sqrt{-g}\,{\cal L}_{i}\,\varphi\right\\}$ (55b)

are the metric variations of the last two $b$ and $\beta_{i}$ terms in (16),
both of which are only linear in $\varphi$.

In (52) a classical matter/radiation stress tensor $T^{cl}_{\mu\nu}$
independent of $\varphi$ has been allowed as well. If any $\beta_{i}{\cal
L}_{i}$ or $T^{cl}_{\mu\nu}$ terms are non-zero, the Euler-Lagrange eqs. of
the fields (or fluid constitutive relations) upon which these additional
degrees of freedom depend must be appended to (51) and (52), to close the
system (51)–(55).

Note that the stress tensor $T\\!_{\cal A}^{\ \mu\nu}[g;\varphi]$ derived from
$S\\!_{\cal A}[g;\varphi]$ with $\varphi$ treated as a classical field, is the
finite renormalized stress tensor of the underlying quantum conformal QFT,
after all short distance UV divergences have been removed, with
$\big{\langle}T^{\mu\nu}\big{\rangle}_{\mathrm{flat}}$ in infinite flat space
defined to be zero, by the consistency condition (28), where $\varphi$ may be
taken to vanish as well. If boundary conditions different from infinite empty
flat space are considered, $T^{\mu\nu}_{F}$ and $\Lambda_{\rm eff}$ of (25)
parametrize any finite vacuum energy in place of $\Lambda$ by (25), and
$\varphi$ may be different from zero. For example the stress tensor (54) with
$\varphi\propto z^{2}/a^{2}$, $\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule
height=7.22743pt\kern 7.22743pt\vrule} \hrule}$}\,^{2}\varphi=0$ and
$\Lambda_{\rm eff}=0$ can account for the Casimir energy and force between two
parallel plates a distance $a$ apart in the $z$ direction in flat space, when
boundary conditions appropriate to that situation are imposed on $\varphi$
[28].

The scalar-tensor theory (49), based on first principles of QFT and the
conformal anomaly SM fields, is quite distinct from other modifications of GR,
such as Brans-Dicke theory, Hordenski theory or massive gravity. As the
possible anomaly sources for $\varphi$ in (51) are negligibly small in our
local neighborhood, this effective theory easily passes the most constraining
solar system tests, as well as the laboratory bounds on _ad hoc_ modified
gravity or other scalar-tensor theories [82, 64]. Being generally covariant,
the theory described by (49)–(55) is consistent with the Weak Equivalence
Principle (WEP) and local Lorentz invariance. The solutions of (51) propagate
at $c$, the speed of both light and of gravitational waves, consistent with
present gravitational wave observations.

There are nevertheless two situations where the effects of (53) are
significant at macroscopic scales and lead to effects qualitatively different
from the purely classical theory:

  1. (i)

In the vicinity of horizons, where there are large local blueshifts and the
light cone singularities of anomalies come to the fore, relevant for BHs and
cosmology, cf. section VIII;

  2. (ii)

When the source for $\varphi$ in (51) is non-gravitational in origin and
sufficiently strong, such as from the QCD trace anomaly ${\cal L}_{G}$ in
dense nuclear matter, where scalar gravitational waves may be generated in
compact binary mergers and in the hot, dense early universe [64].

Since the eq. of motion (51) for $\varphi$ involves the fourth order conformal
Panietz-Riegert operator $\Delta_{4}$ of (11) [60, 61], and typically,
differential eqs. higher than second order possess negative energy and/or
unstable solutions growing in time, a few additional comments are in order
here. Note first that $S\\!_{\cal A}$ and $\Delta_{4}$ do not occur in
isolation, but as part of the EFT of gravity, subject to the first class
constraints of diffeomorphism invariance. These constraints restrict the class
of physically allowed solutions both classically and quantum mechanically.
Since $S\\!_{\cal A}$ is quadratic in $\varphi$, a simple case in which the
specific effects of the higher derivative terms in the anomaly effective
action can be studied is the exactly solvable limit of $G_{\scriptstyle
N}^{-1}\to 0$, where the EH term is neglected, and on the product space
${\mathds{R}}\times{\mathds{S}}^{3}$, where $E,C^{2}$ and $\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R$ all vanish. The result of this analysis is that only a small
subset of solutions survive the constraints of diffeomorphism invariance, and
correspondingly, only a small subspace of physical states with positive norm,
and no propagating modes whatsoever, survive quantization [83].

When $G_{\scriptstyle N}^{-1}\neq 0$ and the EH term of classical GR is added,
the resulting EFT (49) becomes non-linear, but then can be studied in
linearized perturbation theory around flat space. The mixing of the scalar
$\varphi$ with the conformal factor of the metric turns this constrained mode
of classical GR into a propagating one. The result is that (49)–(54) predicts
the existence of scalar gravitational waves of positive energy [64], i.e. a
‘breather’ mode polarization in addition to the two transverse, traceless
gravitational wave modes of classical GR. The active linearized solutions of
(51) are those with $\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule
height=7.22743pt\kern 7.22743pt\vrule} \hrule}$}\,^{2}\varphi\\!=\\!0$, but
$\Upsilon\\!=\\!\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule
height=7.22743pt\kern 7.22743pt\vrule} \hrule}$}\,\varphi\neq 0$. In other
words, the modes with $\,\raise 0.8pt\hbox{$\vbox{\hrule\hbox{\vrule
height=7.22743pt\kern 7.22743pt\vrule} \hrule}$}\,\varphi\\!=\\!0$ decouple
entirely, and the remaining solutions satisfy the second order eq. $\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,\Upsilon\\!=\\!0$, with positive energy. Thus in linearization
around flat space, half of the solutionsof the fourth order $\Delta_{4}$
operator are eliminated by the constraints, and do not appear in the physical
asymptotic states and $S$-matrix of the EFT. There is no instability in this
case either, consistent with general results on the stability of flat space to
quantum corrections [84, 42].

Since $S\\!_{\cal A}$ is derived from the conformal anomaly of well-behaved SM
fields, each with a unitary $S$-matrix in flat space, the EFT incorporating
the anomaly is not expected to lead to unphysical instabilities in weakly
curved or asymptotically flat space at low energies, within its range of
validity. While this interesting issue certainly deserves further
investigation, and should be revisited now with the introduction of the
$4$-form gauge field, the detailed studies of the EFT obtained from the
conformal anomaly to date are consistent with this physical expectation.

##  VII The Effective Theory of Gravity in the Presence of Torsion

In the presence of torsion the Chern-Simons $3$-form potential (33) acquires a
torsional dependent term $\mathsf{A_{T}}$, which may be identified with the
$3$-form potential of Sec. IV and makes the additional contribution (46) to
the effective action, which in this case becomes

$S^{\rm(II)}_{\rm eff}[g;\varphi;A]=\mbox{\small$\displaystyle\frac{1}{16\pi
G_{\scriptstyle N}}$}\\!\int\\!d^{4}x\sqrt{-g}\,R+S\\!_{\cal
A}[g;\varphi]+S\\!_{F}[g;A]+S\\!_{\rm int}[\varphi;A]\hskip
28.45274pt\rm(II)\vspace{-3mm}$ (56)

in which $S\\!_{\rm int}$ is given by (46). Since $A=\mathsf{A_{T}}$ is
independent of the metric, the independent variables of (56) are
$(g_{\mu\nu},\varphi,A_{\alpha\beta\gamma})$.

A possible additional torsion dependent term arising from the conformal
anomaly of $N_{f}$ massless fermions is [68, 85, 86]

$S\\!_{W}[g;\varphi;W]=-\mbox{\small$\displaystyle\frac{N_{f}}{48\pi^{2}}$}\int
d^{4}x\sqrt{-g}\,\big{(}\nabla_{\mu}W^{\perp}_{\nu}-\nabla_{\nu}W^{\perp}_{\mu}\big{)}^{\\!2}\varphi\vspace{-5mm}$
(57)

in terms of the transverse part of the axial vector field

$W\\!_{\mu}=\tfrac{1}{4}\,\epsilon_{abcd}\,K^{abc}\,e^{d}_{\
\mu}=\tfrac{1}{8}\,\epsilon_{abcd}\,T^{abc}\,e^{d}_{\ \mu}\vspace{-4mm}$ (58)

dependent upon torsion. This term has been omitted from (56), assuming that it
can be varied independently of $(g_{\mu\nu},\varphi,A_{\alpha\beta\gamma})$,
and the resulting Euler-Lagrange eq.

$\nabla\\!_{\nu}\,\big{(}\nabla^{\nu}W^{\perp}_{\mu}\varphi\big{)}=0\vspace{1mm}$
(59)

admits the solution $W^{\perp}_{\mu}=0$.

Since $\sqrt{-g}\,\varepsilon^{\alpha\beta\gamma\mu}$ appearing in (46) is in
fact independent of the metric, cf. (85), the additional term $S_{\rm int}$
makes no contribution to the stress tensor, and variation of (56) with respect
to the metric gives the Einstein eqs., identical in form to (52).

Variation of (56) with respect to $\varphi$ yields

$\Delta_{4}\varphi=-\mbox{\small$\displaystyle\frac{1}{2}$}\,{\widetilde{F}}+\mbox{\small$\displaystyle\frac{1}{2}$}\left(E-\mbox{\small$\displaystyle\frac{2}{3}$}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)+\mbox{\small$\displaystyle\frac{1}{2b^{\prime}}$}\,\left(bC^{2}+\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}\,\beta_{i}{\cal L}_{i}\right)\hskip 34.14322pt{\rm(II)}\vspace{-2mm}$
(60)

instead of (51). Since the ‘Maxwell’ action (22) and interaction term (46) may
be varied independently with respect to $A_{\alpha\beta\gamma}$, with
$g_{\mu\nu}$ and $\varphi$ held fixed, the novel feature of (56) is the
‘Maxwell’ eq.

$\nabla_{\\!\lambda}F^{\alpha\beta\gamma\lambda}=\varkappa^{4}J^{\alpha\beta\gamma}=-\mbox{\small$\displaystyle\frac{\varkappa^{4}b^{\prime}}{2}$}\,\varepsilon^{\alpha\beta\gamma\lambda}\,\partial_{\lambda}\varphi\vspace{-2mm}$
(61)

with the source current (45). Upon taking its dual, with (23), this becomes

$\partial_{\lambda}\left({\widetilde{F}}-\mbox{\small$\displaystyle\frac{\varkappa^{4}b^{\prime}}{2}$}\,\varphi\right)=0\vspace{-3mm}$
(62)

which is an eq. of constraint that is immediately solved by

${\widetilde{F}}=\mbox{\small$\displaystyle\frac{\varkappa^{4}b^{\prime}}{2}$}\,\varphi+{\widetilde{F}}_{0}\vspace{-3mm}$
(63)

in which ${\widetilde{F}}_{0}$ is a spacetime constant. Thus ${\widetilde{F}}$
can be eliminated in favor of $\varphi$ and (60) becomes

$\Delta_{4}\varphi+\mbox{\small$\displaystyle\frac{\varkappa^{4}b^{\prime}}{4}$}\,\varphi=-\mbox{\small$\displaystyle\frac{{\widetilde{F}}_{0}}{2}$}+\mbox{\small$\displaystyle\frac{1}{2}$}\left(E-\mbox{\small$\displaystyle\frac{2}{3}$}\,\raise
0.8pt\hbox{$\vbox{\hrule\hbox{\vrule height=7.22743pt\kern 7.22743pt\vrule}
\hrule}$}\,R\right)+\mbox{\small$\displaystyle\frac{1}{2b^{\prime}}$}\,\left(bC^{2}+\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}\,\beta_{i}{\cal L}_{i}\right)\,.\hskip 14.22636pt\rm(II)\vspace{-2mm}$
(64)

For asymptotically flat boundary conditions ${\widetilde{F}}_{0}$ may be set
to zero, but is retained here for more general cases.

For the second form of the effective action (56), the result is that
${\widetilde{F}}$ given by (63) is no longer a constant, and will change, as
will the effective cosmological term $\Lambda_{\rm eff}$, when $\varphi$
changes according to (64). Eq. (64) for $\varphi$ and the Einstein eq. (52)
together with (25) and (53)-(55) is the form of the proposed EFT of low energy
gravity and dynamical vacuum energy in the presence of torsion.

In deciding which which form of the EFT (49) or (56) applies, the critical
question is whether spacetime acquires a non-vanishing torsion. This question
remains open at present. Since the vanishing of torsion is equivalent to the
vanishing of the covariant derivative of thevierbein, according to the
definition (38), it has been suggested that a natural place for torsion to
appear is where the vierbein vanishes, and the locking together of the
$S\\!O(3,1)$ tangent space gauge group and $GL(4,\mathds{R})$ group of
coordinates transformations is broken [87]. This hypothesis will be adopted in
the application of the following section.

##  VIII Gravitational Vacuum Condensate Stars in the EFT of Gravity

In [31, 32, 36] it was proposed that the solution of the multiple BH paradoxes
is that the final state of complete gravitational collapse is a gravitational
condensate star rather than a BH. The proposed gravastar is a compact object
with a physical surface of positive surface tension replacing the BH horizon,
and a static region of de Sitter space with the eq. of state $p\\!=\\!-\rho$
replacing the singular interior of a BH. Because such an object is both
horizonless and non-singular, with low entropy, it suffers from no information
paradox, and is consistent with quantum unitary evolution. The EFT of Sec. VII
provides a fundamental first principles Lagrangian basis for this proposal.

The requirements for a gravitational condensate star to be realized in
gravitational collapse are first, that quantum vacuum polarization effects can
grow large in the vicinity of a BH horizon and second, they can induce a phase
transition to a non-vanishing interior gravitational Bose-Einstein condensate
(GBEC) with $p\\!=\\!-\rho$, equivalent to a non-zero $\Lambda_{\rm eff}$.

That the anomaly effective action $S\\!_{\cal A}[g;\varphi]$ and stress tensor
(54)–(55) of section VI can have substantial effects in spacetimes with
horizons, satisfying this first part of the gravastar hypothesis may be seen
in the case of the exterior static Schwarzschild spacetime

$ds^{2}=-f(r)\,dt^{2}+\mbox{\small$\displaystyle\frac{dr^{2}}{h(r)}$}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\vspace{-3mm}$
(65)

with

$f(r)=h(r)=1-\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}}{r}$}=1-\mbox{\small$\displaystyle\frac{2G_{\scriptstyle
N}M}{c^{2}\,r}$}\,.\vspace{-1mm}$ (66)

The general solution to (51) for
$\varphi\\!=\\!\varphi(r),\,{\widetilde{F}}\\!=\\!{\widetilde{F}}_{0}\\!=\\!0$
and ${\cal L}_{i}\\!=\\!0$ that is finite as $r\\!\to\\!\infty$ for the
Schwarzschild metric (65)–(66) was found previously to be [28, 30]

$\displaystyle\frac{d\varphi_{{}_{S}}}{\\!dr}$
$\displaystyle=\mbox{\small$\displaystyle\frac{c_{{}_{S}}r_{\\!{}_{M}}}{r(r-r_{\\!{}_{M}})}$}-\mbox{\small$\displaystyle\frac{2}{3r_{\\!{}_{M}}\\!\\!}$}\left(\mbox{\small$\displaystyle\frac{r}{r_{\\!{}_{M}}\\!\\!}$}+1+\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}\\!}{r}$}\right)\ln\left(1-\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}\\!}{r}$}\right)-\mbox{\small$\displaystyle\frac{\\!2}{3r_{\\!{}_{M}}\\!\\!}$}-\mbox{\small$\displaystyle\frac{1}{r}$}$
(67a) $\displaystyle\varphi_{{}_{S}}(r)$
$\displaystyle=c_{{}_{S}}\ln\left(1-\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}\\!}{r}$}\right)+\int_{r/r_{\\!{}_{M}}}^{\infty}dx\,\Bigg{\\{}\mbox{\small$\displaystyle\frac{2}{3x}$}\left(x^{2}+x+1\right)\ln\left(1-\mbox{\small$\displaystyle\frac{1}{x}$}\right)+\mbox{\small$\displaystyle\frac{2}{3}$}+\mbox{\small$\displaystyle\frac{1}{x}$}\Bigg{\\}}$
(67b)

in terms of the dimensionless integration constant $c_{{}_{S}}$. 111An
additional constant of integration $c_{\infty}$ in $\varphi_{\\!S}$ has been
dropped here and in (68), since it does not contribute to the anomaly stress
tensor (53). This solution has the limits

$\varphi_{{}_{S}}(r)\to\left\\{\begin{array}[]{lr}c_{{}_{S}}\ln\left(1-\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}\\!}{r}$}\right)+c_{1}-2\left(1-\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}\\!}{r}$}\right)\left[\ln\left(1-\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}\\!}{r}$}\right)-\mbox{\small$\displaystyle\frac{1}{6}$}\right]+\dots\,,&r\to
r_{\\!{}_{M}}\\\\[8.61108pt]
-\left(c_{{}_{S}}+\mbox{\small$\displaystyle\frac{11}{9}$}\right)\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}\\!}{r}$}-\left(2c_{{}_{S}}+\mbox{\small$\displaystyle\frac{13}{9}$}\right)\mbox{\small$\displaystyle\frac{r_{\\!{}_{M}}^{2}}{4r^{2}}$}+\dots\,,&r\to\infty\end{array}\right.\vspace{-2mm}$
(68)

where the constant $c_{1}$ is the finite integral in (67b) evaluated at the
lower limit $r=r_{\\!{}_{M}},x=1$. Substituting the solution (67) into the
anomaly stress tensor (53)–(54), one finds

$\big{(}T^{\mu}_{\ \,\nu}\big{)}_{\\!{\cal
A}}\to\frac{c_{{}_{S}}^{2}}{6r_{\\!{}_{M}}^{2}\\!}\
\frac{b^{\prime}}{(r-r_{\\!{}_{M}})^{2}}\ {\rm
diag}\,(-3,1,1,1)\to\infty\qquad{\rm as}\qquad r\to
r_{\\!{}_{M}}\vspace{-2mm}$ (69)

diverging on the Schwarzshild BH horizon for any $c_{{}_{S}}\\!\neq\\!0$.

This leading order $(r-r_{\\!{}_{M}})^{-2}$ divergence of the stress tensor on
the horizon may be understood from the kinematic blueshifting of local
frequencies in (66) near the horizon according to

$\omega_{\rm
loc}(r)=\,\mbox{\small$\displaystyle\frac{\omega_{\infty}\\!}{\\!\\!\\!\sqrt{f(r)}}$}\vspace{-2mm}$
(70)

relative to that at $r\\!=\\!\infty$. The corresponding energy
$\hbar\omega_{\rm loc}(r)$ diverges as $r\to r_{\\!{}_{M}}$ and therefore
becomes much greater than any finite mass scale. This is reflected in the fact
that the wave eq. for a quantum field of arbitrary finite mass and spin
becomes indistinguishable from that of a massless conformal field in the
horizon limit $r\to r_{\\!{}_{M}}$ [30], and hence the conformal anomaly
effects come to the fore. Since the stress tensor is a dimension four,
conformal weight four operator, it behaves generically as the fourth power of
$\omega_{\rm loc}$ in (69), i.e. $\propto f^{-2}$. Noting that $\varphi$ is a
scalar, as is the norm of the static Killing field $K=\partial_{t}$ of (65),
which is $\sqrt{-K^{\mu}K_{\mu}}=\sqrt{-g_{tt}}=\sqrt{f(r)}$, the divergence
of (69) depending on the inverse fourth power of this norm is also a
coordinate invariant scalar, and observer independent.

The diverging behavior of the local stress tensor (69) shows that the anomaly
stress tensor can become important near the horizon of a BH and even dominate
the classical terms in the Einstein eq. (52), the smallness of the curvature
tensor there notwithstanding. Even with $c_{{}_{S}}\\!=\\!0$ in (67), which
can be arranged by specific choice of the state of the underlying QFT, to
remove the leading $f^{-2}$ divergence in (69), there remain subleading
divergences proportional to $f^{-1},(\ln f)^{2}$ and $\ln f$. In fact, there
is no solution of (51) in Schwarzschild spacetime with $\varphi=\varphi(r)$
only, corresponding to a fully Killing time $t$ invariant and spherically
symmetric quantum state, with a finite stress tensor at both regular singular
points $r=r_{\\!{}_{M}}$ and $r=\infty$ of the differential eq. (51). This
result, following simply and directly from the conformal anomaly effective
action, confirms results of previous studies of the stress tensor expectation
value in specific states in Schwarzschild spacetime [27].

The divergences on either the future or past BH horizon (but not both) can be
cancelled by allowing linear time dependent solutions of (51), which give rise
to a Hawking flux $\langle T^{t}_{\ \,r}\rangle$, such as in the Unruh states;
or by relaxing the regularity condition at infinity which gives rise to a non-
zero stress tensor there, as in the Hartle-Hawking state. This thermal state
is both incompatible with asymptotically flat boundary conditions and
unstable. Usually the _assumption_ of regularity of the semi-classical
$\langle T^{\mu}_{\ \,\nu}\rangle$ on the horizon is used to argue for the
necessity of Hawking radiation flux $\langle T^{t}_{\ \,r}\rangle>0$ [88].
However the converse is also true: if a _truly static_ and stable
asymptotically flat solution of the final state of gravitational collapse is
sought with $\langle T^{t}_{\ \,r}\rangle=0$, then quantum effects at the
horizon, specifically due to the conformal anomaly $T\\!_{\cal A}^{\ \mu\nu}$
cannot be neglected, and imply instead the breakdown of regularity there.

A similar behavior is observed in de Sitter spacetime, and indeed in any
spherically symmetric static spacetime (65) with a horizon at which
$-K^{\mu}K_{\mu}=f(r)\to 0$. With $\Lambda$ positive, the static patch of de
Sitter space is of this form with $f(r)\propto h(r)=1-H^{2}r^{2}=1-\Lambda
r^{2}/3$. The general spherically symmetric static solution of (51) for
$\varphi=\varphi(r)$ which is regular at the origin in this case is [28, 30]

$\displaystyle\varphi_{dS}(r)$
$\displaystyle=\ln\left(1-H^{2}r^{2}\right)+c_{0}+\frac{q}{2}\ln\left(\frac{1-Hr}{1+Hr}\right)+\frac{2c_{{}_{H}}-2-q}{2Hr}\ln\left(\frac{1-Hr}{1+Hr}\right)$
$\displaystyle\to\Big{[}c_{{}_{H}}+\Big{(}c_{{}_{H}}-1-\mbox{\small$\displaystyle\frac{q}{2}$}\Big{)}\big{(}1-Hr\big{)}+\dots\
\Big{]}\,\ln\big{(}1-Hr\big{)}+c^{\prime}+{\cal O}\,(1-Hr)$ (71)

where the constant $c^{\prime}=c_{0}+(2-c_{{}_{H}})\ln 2$. Substituting this
into the anomaly stress tensor (53) gives

$\big{(}T^{\mu}_{\ \,\nu}\big{)}\\!_{\cal
A}\to\mbox{\small$\displaystyle\frac{2}{3}$}\,c_{{}_{H}}^{2}\,H^{4}\frac{b^{\prime}}{(1-Hr)^{2}}\
{\rm diag}\,(-3,1,1,1)\to\infty\qquad{\rm as}\qquad r\to r_{\\!{}_{H}}\equiv
H^{-1}\vspace{-2mm}$ (72)

which also diverges as $f^{-2}$ for any $c_{{}_{H}}\neq 0$ as the de Sitter
static horizon is approached.

As in the Schwarzschild case this divergence can be removed if a thermal state
is considered, but then only if the temperature is precisely matched to the
Hawking temperature $T_{{}_{H}}\\!=\\!H/2\pi$ associated with the horizon,
which in the de Sitter case leads to the maximally $O(4,1)$ symmetric state
[89, 90]. However, this state is not a vacuum state of QFT and is unstable to
particle pair creation, much as a uniform, constant electric field is [91, 92,
93, 94, 95], and for essentially the same reason. Due to the non-existence of
a global static time by which positive and negative frequency (particle and
anti-particle) solutions can be invariantly distinguished, a time independent
Hamiltonian bounded from below and stable vacuum state cannot be defined. The
horizon where $f\\!=\\!0$ and the Killing vector of time translation
$\partial_{t}$ becomes null in either Schwarzschild or de Sitter space is the
sign of this, so that $\Lambda$ cannot be globally constant and positive
everywhere in space in QFT. The conformal anomaly shows this through its
sensitivity to lightlike correlations on the horizon and non-local boundary
conditions on the quantum state. Only flat space with $\Lambda_{\rm
eff}\\!=\\!0$ can be a candidate stable ground state of the semi-classical
EFT, as the state of lowest energy with vanishing ‘electric’ field
${\widetilde{F}}=0$, a global static Killing time and no horizon, as also
required by the low energy EFT consistency condition (28).

With $\Lambda_{\rm eff}$ replaced by a dynamical condensate according to (27),
vacuum energy can be non-zero only if localized in space, within the static
patch $r<r_{\\!{}_{H}}=H^{-1}$ of de Sitter space. If one seeks a stable
spherically symmetric static solution of the EFT, the light cone enhanced
effects of the anomaly stress tensor (18) at both the Schwarzschild and de
Sitter horizons, (69) and (72) respectively, should be taken into account, and
the assumption that either horizon is a mathematical boundary only should be
re-examined. In the gravastar proposal [31, 32], the BH horizon is the
location of a physical surface phase boundary layer between two different
phases characterized by different values of the vacuum energy $\Lambda_{\rm
eff}$, regarded as a gravitational condensate. On general thermodynamic
grounds the Gibbs relation $\rho+p\\!=\\!sT+\mu n\\!=\\!0$ implies that the
eq. of state of a zero temperature condensate with no conserved particle
number should be $p\\!=\\!-\rho$. An argument based on non-relativistic
condensed matter analogs given in [96] reached a similar conclusion.

In [36] the gravitational condensate star was shown to follow directly from
Schwarzschild’s constant density interior solution in the limit
$r\\!\to\\!r_{\\!{}_{M}}$, with $f(r)=h(r)/4$ leading to equal and opposite
surface gravities of the surface at $r_{\\!{}_{M}}\\!=\\!r_{\\!{}_{H}}$. The
surface tension of this physical boundary layer replacing the Schwarzschild
and de Sitter horizons is determined from the $\delta$-function discontinuity
in gradients of the surface gravities there, and the First Law becomes a
purely mechanical relation of a gravastar with this surface tension, and the
relativistic analog of the Rayleigh surface tension of a fluid droplet.

The singular behavior of both $\varphi$ and the anomaly stress tensor at the
Schwarzschild and de Sitter horizons (68), (69) and (71), (72) is clearly
associated with the vanishing of the norm of the static Killing vector field
$\partial_{t}$ in each case. The coordinate singularities at these static
horizons coincide with the vanishing or divergence of the vierbein $e^{0}_{\
t}$ or $e^{1}_{\ r}$ respectively. This is exactly the locus of a possible
breakdown of the locking together of the $S\\!O(3,1)$ tangent space gauge
group and $GL(4,\mathds{R})$ group of coordinates transformations, where
torsion may be expected to arise [87]. Thus it is natural to describe this
region where $f(r),h(r)\\!\to\\!0$ by the EFT (II) of section VII. This EFT
(II) then provides a mechanism and Lagrangian description for the second part
of the gravitational condensate star proposal of [31, 32, 36], by allowing the
value of $\Lambda_{\rm eff}$ to change from exterior to interior of the
gravastar, through (27), (63) and (64).

The scalar ${\widetilde{F}}$, dual to the $4$-form field strength of sections
IV–V, is a classical coherent field that provides an explicit realization of a
gravitational condensate $\Lambda_{\rm eff}$ interior. When coupled to the
conformalon scalar through the $3$-form abelian current
$J^{\alpha\beta\gamma}$, concentrated on a three-dimensional extended world
tube of topology ${\mathds{R}}\times{\mathds{S}^{2}}$ where
$\partial_{\mu}\varphi$ grows large, $\varphi,{\widetilde{F}}$ and the
condensate $\Lambda_{\rm eff}$ all change rapidly in the radial direction.
This is precisely the appropriate description of a thin shell phase boundary
layer of a gravastar with $\mathds{S}^{2}$ spatial topology sweeping out a
tube in spacetime.

One may now search for static, rotationally invariant solutions of the EFT
eqs. of Secs. VI-VII which are asymptotically Schwarzshild-like with
$\Lambda_{\rm eff}$ vanishing in the exterior region, changing rapidly but
continuously near the Schwarzschild $r_{\\!{}_{M}}=2G_{\scriptstyle N}M/c^{2}$
or de Sitter $r_{\\!{}_{H}}=H^{-1}$ classical horizons by (63) and (64) in the
phase boundary region, and then remaining nearly constant $3H^{2}$ in the
interior region. The blueshifting of local frequencies $\sim f^{-\frac{1}{2}}$
as in (70) leads to the $\varphi$ field having an increasingly large radial
derivative in the vicinity of $r_{\\!{}_{M}}\simeq r_{\\!{}_{H}}$, so that
(45) and (53) and the torsional effects become significant there. The physical
thickness of the phase boundary surface layer where the anomaly stress tensor
(69) or (72) becomes significant and large enough to compete with the
classical terms and where the EFT (II) of VII must be used is of the order of
$\sqrt{r_{\\!{}_{M}}L_{Pl}}$. The effects of this surface layer and regular de
Sitter interior on binary BH mergers, gravitational waves, ringdown and
‘echoes’ can then be investigated in the EFT.

##  IX Discussion and Outlook: Vacuum Energy as a Dynamical Condensate

In this paper an EFT of gravity has been proposed taking account of the most
significant macroscopic light cone effects of the conformal anomaly of
massless or light SM fields. In this EFT $\Lambda$ is no longer a fundamental
constant, whose value appears to be sensitive to ultra high energy physics,
but rather a dynamical condensate described by a classical $4$-form field
strength and $3$-form abelian gauge potential of sections IV–V.

The EFT based on the conformal anomaly (7) introduces two relevant scalar
degrees of freedom to low energy gravity beyond classical GR. This follows
from the local form of the effective action of the anomaly $S\\!_{\cal
A}[g;\varphi]$ of (16) in terms of the scalar $\varphi$ conformalon field,
which satisfies a fourth order eq. of motion (51)-(60). Before the addition of
$S\\!_{\cal A}[g;\varphi]$, the conformal factor of the metric in GR is
constrained to be non-propagating by the classical diffeomorphism constraints
of Einstein’s eqs. Once $S\\!_{\cal A}[g;\varphi]$ is added to the effective
action, the conformal part of the metric mixes with one of the two $\varphi$
modes, and gives rise to scalar gravitational waves [64].

Likewise before the identification (43) of the potential $A$ of the $4$-form
gauge field (19) with the torsional part of the Chern-Simon $3$-form, $F$ is
constrained to be a constant and simply equivalent to a cosmological constant
term in Einstein’s eqs. by (27). Once the interaction $S\\!_{\rm
int}[\varphi,A]$ is added to the effective action, the $3$-form current $J$ of
(45) provides a source for $F$ through (61), and $F$, hence $\Lambda_{\rm
eff}$ becomes dynamical through $\varphi$. It is this second scalar degree of
freedom of $\varphi$ in $S\\!_{\cal A}[g;\varphi]$ that allows (and requires)
the vacuum energy to become a dynamical variable through (27) and (63).

Thus in the EFT each of the two scalar conformalon degrees of freedom present
in the anomaly action for $\varphi$ mix with, and source the previously
classically constrained conformal factor of $g_{\mu\nu}$ and $F$, turning each
into full-fledged dynamical degrees of freedom. Assuming the SM, the only new
free parameter of the EFT is the constant $\varkappa$, a kind of gravitational
vacuum susceptibility cf. appendix B, determining the coupling of the current
$J$ to the gauge field $A$, field strength $F$, and $\Lambda_{\rm eff}$ of
(27).

For the longstanding problem of the cosmological term when QFT is coupled to
gravity, in either form of the EFT the classical state of minimum energy is
that of vanishing $4$-form condensate:
${\widetilde{F}}\\!=\\!{\widetilde{F}}_{0}\\!=\\!0$. By the identification of
$\Lambda_{\rm eff}$ in (27), this condition automatically sets the value of
the cosmological term to zero in infinite flat Minkowski space. By simply
allowing a flat Minkowski solution, this removes one oft-stated obstacle and
‘no-go theorem’ to solution of the ‘cosmological constant problem’ [4, 97].
Moreover $\Lambda_{\rm eff}\\!=\\!0$ is required by consistency with the
classical or semi-classical Einstein eqs. (28) or (52) in infinite flat
Minkowski space, in the complete absence of matter and radiation. The
condition (28) is independent of UV physics or cutoffs, as simply the unique
classical state of minimum energy with zero curvature and zero torsion, and
the stable ground state of the EFT of low energy gravity.

Empty flat space with zero external fields is also the unique classical state
where the total anomaly ${\cal A}=0$, so that the WZ anomaly action (16)–(17)
has an additional global shift symmetry under $\varphi\to\varphi+\varphi_{0}$
in that state, and $F^{0123}=0$ is also required by spatial parity invariance
of the ground state. These enhanced global or discrete symmetries may be
regarded as replacing the ’t Hooft naturalness criterion [50] for
$\Lambda_{\rm eff}=0$.

The consistency condition of $\Lambda_{\rm eff}=0$ on the classical condensate
in the low energy EFT of gravity in flat space indicates that estimates of
vacuum energy as sensitive to UV physics are not applicable if GR, with or
without the conformal anomaly addition, is to be a well-behaved EFT at
macroscopic scales. Arguments or estimates of the cosmological term in flat
space as proportional to the fourth power of a UV cutoff (which break Lorentz
invariance) or the fourth power of all masses
$\raisebox{-6.45831pt}{$\stackrel{{\scriptstyle\textstyle\sum}}{{\scriptstyle
i}}$}m_{i}^{4}\ln m_{i}$ in QFT (as in dimensional regularization) have no
physical meaning in the absence of gravitation, as well as being in conflict
with observations even on non-cosmological, solar system scales [7].

Instead a way around the ‘naturalness’ problem of the $\Lambda$ term is to
extend the EFT of gravity beyond classical GR to contain the additional scalar
conformalon degree(s) of freedom inherent in the conformal anomaly effective
action (16), and to replace the fixed parameter $\Lambda$ of the classical
theory by $\Lambda_{\rm eff}$ of (27) in terms of a $4$-form gauge field $F$.
As a result of the ‘Maxwell’ eq. $F$ satisfies, this abelian gauge field
becomes a fully dynamical degree of freedom of low energy gravity by (61) and
(63). The setting of an integration constant to zero at the minimum of energy
in flat space is a solution of the naturalness problem of the cosmological
term that involves no fine tuning of any fixed parameters of the EFT
Lagrangian (49) of macroscopic gravity.

At lowest order all fields $(g_{\mu\nu},\varphi,A)$ in (49)-(56) are treated
as classical. Quantum loop corrections are then to be computed by the usual
EFT method of appending local terms as needed to absorb UV divergences in an
expansion in powers of $1/M_{pl}^{2}$, maintaining the physical meaning of the
constants of the lowest order EFT. Thus at one-loop order the energy of the
vacuum will continue to be defined to be identically zero in flat space by the
consistency condition (28) on the constant ${\widetilde{F}}_{0}=0$, hence
$\Lambda_{\rm eff}=0$ in the absence of any sources, and with $\varphi=0$. All
formal divergences in $\langle\hat{T}^{\mu\nu}\rangle_{\mathrm{flat}}$ in
purely flat space, quartic or otherwise, are treated as without physical
significance and removed by this consistency condition on
${\widetilde{F}}_{0}$. There is no sensitivity of this free integration
constant, or $\Lambda_{\rm eff}$ defined in terms of it by (27), on UV
divergences or UV mass scales.

Following the usual logic of EFT, divergences of quantum loops require the
introduction of additional local terms which are higher order in powers of the
Riemann curvature tensor and its derivatives, divided by higher powers of a UV
scale, presumably the Planck mass scale $M_{\mathrm{Pl}}$ [22, 23, 24], which
ultimately limits the range of applicability of the low energy EFT. These
higher order effects and their renormalization should be defined so as not to
disturb the meaning of the low energy parameters at lowest order, such as
$G_{\scriptstyle N}$, or ground state boundary condition on the condensate,
$\Lambda_{\rm eff}\\!=\\!0$ in flat space.

Logarithmic divergences in curved space (regulated by any covariant method)
require the introduction of counterterms proportional to the local $R^{2}$ and
$C^{2}$ curvature invariants, together with the finite logarithmic running of
their dimensionless couplings. These terms and local terms involving still
higher numbers of derivatives are not treated as fundamental, but rather as
suppressed at energy scales far below the Planck energy
$M_{\mathrm{Pl}}c^{2}$, remaining negligibly small at macroscopic distance
scales much greater than $L_{\mathrm{Pl}}$, as consistent with existing EFT
results [22, 23, 24].

One may also turn the EFT logic around, to conjecture that the important role
of the conformal anomaly and anomalies in general as windows into the UV, and
exceptions to the usual decoupling hypothesis of EFT, may indicate that in the
fundamental theory all masses vanish and conformal invariance is restored –
broken perhaps only spontaneously, at asymptotically high energies.
Speculations of this kind for resolution of the naturalness problem of the
cosmological term and possible relation with that of the Higgs mass hierarchy
have been advanced by a number of authors, e.g. [98, 99, 100, 101, 102].
Although no clearly successful complete theory has emerged from these
speculations, the idea of fundamental conformal invariance and relation
between these large hierarchy problems and the properties of the quantum
vacuum rather than UV physics remains intriguing.

That both $\Lambda$ and Higgs hierarchies may be resolvable only by the
consistent inclusion of gravity receives some support from the EFT approach to
vacuum energy and the cosmological ‘constant’ as a dynamical condensate
proposed in this paper. To the extent that
$\partial_{\lambda}{\widetilde{F}}\neq 0$, this dynamical condensate
necessarily requires non-vanishing spacetime torsion. The role of relaxing the
torsionless condition of classical GR in describing the condensate by (43) as
relevant to resolving BH singularities was anticipated in [87], where specific
models generating torsion were proposed. The possible extensions of the EFT
proposed in this paper to generate torsion dynamically and self-consistently
remain to be explored. The coupling of fermions to a condensate with torsion
through the spin connection and possible relation to neutrino mass generation
is another intriguing direction for future research. The microscopic
constituents of the gravitational Bose-Einstein condensate (GBEC) described by
${\widetilde{F}}$ from which its superfluid nature is emergent remain to be
elucidated [103, 104].

As a more immediate matter, the description of $\Lambda_{\rm eff}$ as a
dynamical condensate of a $4$-form gauge field in the EFT of sections VI-VII
makes possible calculations of vacuum energy in numerous applications, first
and foremost for a gravitational condensate star interior and surface. For
cosmology, the universe as the interior of a gravastar realizes the hypothesis
made in section I of automatically relating the effective value of the vacuum
energy $\Lambda_{\rm eff}$ to $3H^{2}$, and hence to the horizon Hubble scale
$H^{-1}$, with no fine tuning. More realistic cosmological models, with
$\Omega_{\Lambda}<1$, require dynamical EFT solutions including matter and
radiation, rather than a purely static de Sitter vacuum condensate.

The tying of the value of $\Lambda_{\rm eff}$ to the Hubble scale is clearly
relevant to the ‘cosmic coincidence problem’ of the $\Lambda$CDM model, and
immediately suggests a rather different set of possibilities for cosmological
models, in which spatial inhomogeneities and/or boundary conditions at the
Hubble scale $H^{-1}$ play an important role. The EFT of the conformal anomaly
coupled to the $3$-form potential and $4$-form abelian field strength term
presented in sections VI and VII thus provides a distinctly new framework for
dynamical dark energy in cosmology based on fundamental theory.

From the form of the anomaly ${\cal A}$ in (7), any deviation from exact
homegeneity and isotropy will lead in general to $F_{\mu\nu}F^{\mu\nu}\neq 0$
for the photon radiation field, and ${\rm tr}\,\\{G_{\mu\nu}G^{\mu\nu}\\}\neq
0$ for the electroweak and QCD color gauge fields in the unconfined phase of
the early universe. This will induce changes in the conformalon field
$\varphi$ through its eq. of motion (51), which will then cause the field
strength ${\widetilde{F}}$ and hence the vacuum energy $\Lambda_{\rm eff}$ to
change. After the transition to the confining phase of QCD, baryonic matter
will still contain non-vanishing gluonic condensates and thus still act as a
source for $\varphi$, thereby coupling non-relativistic baryonic matter to
dynamical vacuum energy as well.

Thus although $\varphi$ is _not_ an inflaton, it is a dynamical scalar that is
well-grounded in QFT of the SM and can produce backreaction effects on the
vacuum energy when fluctuations away from exact homogeneity and isotropy are
admitted. It permits interaction between both radiation and matter with
dynamical dark energy, in which adiabaticity of the matter and radiation
components will no longer be satisfied in general, in effect introducing a
bulk viscosity into the cosmological fluid. If
$\Lambda_{\mathrm{eff}}\propto{\widetilde{F}}^{2}$ does not remain constant in
the de Sitter phase, deviations from the $\Lambda$CDM cosmological model are
to be expected, and evolution away from a pure de Sitter phase due to
cosmological horizon modes [105] becomes calculable, and testable by the
cosmological data of large scale structure.

In addition to removing the singularity and paradoxes of BHs, developing
detailed predictions from the EFT proposed will allow study of gravastar
stability, normal modes of oscillation, surface modes and ‘echoes’ that can be
tested with gravitational wave and multi-messenger signals from binary merger
events, in the increasing data samples expected in the future. The prediction
of scalar gravitational waves can also be tested by the coming global array of
gravitational wave antennae [106].

The effects of rotational angular momentum have been neglected in the simplest
gravastar solution, although the first steps in including those effects have
been taken in [37, 38]. The EFT solutions of static or stationary gravastars
also leaves unexamined the process of their formation, and in particular the
behavior of the stress tensor near the would-be horizon of collapsing matter,
which would have to activate the dynamical condensate terms of (56)–(60).
These and many other interesting questions remain to be addressed in the
context of the EFT of dynamical vacuum energy proposed in this paper.

Acknowledgements

The author expresses his appreciation to his colleague Prof. Pawel O. Mazur
for a critical reading of this paper, and for bringing reference [87] to the
author’s attention, to Prof. Ilya L. Shapiro for useful discussions and
bringing references [85, 107, 108, 86] to his attention, and to Joan Solà
Peracaula for bringing reference [78] to his attention as well.

References

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##  Appendix A Metrics, Tetrads, Differential Forms, and Hodge Star Dual

In this first Appendix conventions and mathematical details used in the text
are collected and catalogued. The metric and curvature conventions used in
this paper are those of Misner, Thorne & Wheeler [39]. Greek indices are four-
dimensional coordinate (holonomic) indices, while Latin indices refer to local
othonormal tangent space.

In the tetrad or vierbein formalism the metric line element is written

$ds^{2}=g_{\mu\nu}(x)\,dx^{\mu}dx^{\nu}=e^{a}\eta_{ab}e^{b}\vspace{-4mm}$ (73)

with $\eta_{ab}=$ diag $(-1,1,1,1)$ the flat spacetime tangent space Minkowski
metric tensor, and $e^{a}(x)$ are the $1$-forms

$\displaystyle e^{a}=e^{a}_{\ \mu}\,dx^{\mu}\qquad{\rm satisfying}\qquad
e^{a}_{\ \mu}\,e^{b}_{\ \nu}\,\eta_{ab}=g_{\mu\nu}\,,\quad
g^{\mu\nu}\,e^{a}_{\ \mu}\,e^{b}_{\ \nu}=\eta^{ab}\,.$ (74)

The dual basis of vectors ${\bm{\upsilon}}_{a}$ satisfy

$\displaystyle e^{a}_{\ \mu}\upsilon^{\mu}_{\ b}$ $\displaystyle=\delta^{a}_{\
b}$ (75a) $\displaystyle e^{a}_{\ \nu}\upsilon^{\mu}_{\ a}$
$\displaystyle=\delta^{\mu}_{\ \nu}$ (75b) $\displaystyle\upsilon^{\mu}_{\
a}\upsilon^{\nu}_{\ b}\,g_{\mu\nu}$ $\displaystyle=\eta_{ab}$ (75c)
$\displaystyle\upsilon^{\mu}_{\ a}\upsilon^{\nu}_{\ b}\eta^{ab}$
$\displaystyle=g^{\mu\nu}$ (75d)

defining the orthonormal basis in tangent space. Coordinate indices are
lowered (resp. raised) by the metric tensor $g_{\mu\nu}$ (resp. its inverse
$g^{\mu\nu}$). Tangent space indices are lowered or raised by the flat
Minkowski tensor $\eta_{ab}$ or $\eta^{ab}$. The covariant derivative of the
vierbein field in (40) is defined with respect to the torsionless Levi-Civita
connection, that in holonomic coordinates is the familiar Christoffel symbol
(4), which is specified entirely by the metric tensor and its first
derivatives.

The exterior derivative operator $d$ maps the general $p$-form

$Q^{(p)}=\mbox{\small$\displaystyle\frac{1}{p!}$}\,Q^{(p)}\\!_{\mu_{1}\dots\mu_{p}}\,dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{p}}=Q^{(p)}\\!_{[\mu_{1}\dots\mu_{p}]}\,dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{p}}\vspace{-4mm}$ (76)

into the $p+1$-form

$dQ^{(p)}=\frac{1}{p!}\,\frac{\partial
Q^{(p)}\\!_{\mu_{1}\dots\mu_{p}}}{\partial x^{\lambda}}\,dx^{\lambda}\wedge
dx^{\mu_{1}}\wedge\dots\wedge dx^{\mu_{p}}=\partial\\!_{\
[\lambda}Q^{(p)}\\!_{\mu_{1}\dots\mu_{p}]}\ {\partial x^{\lambda}}\wedge
dx^{\mu_{1}}\wedge\dots\wedge dx^{\mu_{p}}\vspace{-3mm}$ (77)

where the square brackets denote anti-symmetrization with respect to the
enclosed indices, and both $p$ and $p+1$ must be $\leq D$ in $D$ dimensions,
by the anti-symmetry of the wedge product.

The Hodge ${\star}$ dual operator maps the $p$-form (76) into the $(4-p)$-form

${\star}Q^{(p)}=\mbox{\small$\displaystyle\frac{1}{(4-p)!}$}\,\mbox{\small$\displaystyle\frac{1}{p!}$}\,\varepsilon_{\mu_{1}\dots\mu_{4}}\,g^{\mu_{1}\nu_{1}}\dots
g^{\mu_{p}\nu_{p}}\,Q^{(p)}\\!_{\nu_{1}\dots\nu_{p}}\,dx^{\mu_{p+1}}\wedge\dots\wedge
dx^{\mu_{4}}\vspace{-3mm}$ (78)