On the quantum field theory of the gravitational interactions
Damiano Anselmi 0 1 2
Open Access 0 1 2
c The Authors. 0 1 2
0 INFN , Sezione di Pisa
1 Largo B. Pontecorvo 3 , 56127 Pisa , Italy
2 Dipartimento di Fisica “Enrico Fermi”, Universita` di Pisa
We study the main options for a unitary and renormalizable, local quantum field theory of the gravitational interactions. The first model is a Lee-Wick superrenormalizable higher-derivative gravity, formulated as a nonanalytically Wick rotated Euclidean theory. We show that, under certain conditions, the S matrix is unitary when the cosmological constant vanishes. The model is the simplest of its class. However, infinitely many similar options are allowed, which raises the issue of uniqueness. To deal with this problem, we propose a new quantization prescription, by doubling the unphysical poles of the higher-derivative propagators and turning them into Lee-Wick poles. The Lagrangian of the simplest theory of quantum gravity based on this idea is the linear combination of R, Rμν Rμν , R2 and the cosmological term. Only the graviton propagates in the cutting equations and, when the cosmological constant vanishes, the S matrix is unitary. The theory satisfies the locality of counterterms and is renormalizable by power counting. It is unique in the sense that it is the only one with a dimensionless gauge coupling.
Models of Quantum Gravity; Beyond Standard Model; Renormalization Reg-
1 Introduction 2 3 4
Superrenormalizable quantum gravity
Coupling to matter
The problem of uniqueness
Fake degrees of freedom
Quantum gravity with a dimensionless gauge coupling
The problem of quantum gravity is the compatibility between renormalizability and
unitarity. It is well known that the Hilbert-Einstein action is not renormalizable by power
counting [1–4]. However, if we include the infinitely many counterterms it generates,
multiplied by independent couplings, it is perturbatively unitary . An option to improve
the ultraviolet behavior of the loop integrals is to add quadratic terms with higher
derivatives. It is then possible to build higher-derivative theories of quantum gravity that are
renormalizable with finitely many couplings [6–9]. However, such theories are not unitary,
at least if they are formulated in the usual ways.
Higher-derivative theories must be formulated properly, because they are less trivial
than one would naively expect. For example, if they are defined directly in Minkowski
spacetime, i.e. by integrating the loop energies along the real axis of the complex energy
plane, they generate nonlocal, non-Hermitian divergences when the free propagators have
complex poles , which makes them unacceptable from the mathematical point of view.
On the other hand, the Wick rotation from Euclidean space is obstructed when the free
propagators have poles in the first or third quadrants of the complex energy plane. The
obstruction can actually be overcome by a nonanalytic procedure, which leads to a new
formulation  of an interesting subclass of higher-derivative theories, the Lee-Wick (LW)
models [12, 13].
Viewed as nonanalytically Wick rotated Euclidean theories, such models are
perturbatively unitary . Moreover, the new formulation is intrinsically equipped with all that
is needed to define the physical amplitudes properly, with no need of ad hoc prescriptions.
The complex energy hyperplane is divided into disjoint regions Ai of analyticity, which can
be connected to one another by a well defined, but nonanalytic procedure. It is necessary
to work in suitable subsets Oi of the regions Ai, in a generic Lorentz frame, and
analytically continue the results from Oi to Ai at the end. Finally, the nonanalytic behaviors of
the physical amplitudes suggest ways that may facilitate the experimental measurements
of the key parameters of the models.
Old formulations of the Lee-Wick models were based on ad hoc prescriptions, the
best known one being the CLOP prescription of ref. .1
Often, such approaches are
unambiguous in some loop diagrams, but ambiguous in others, and do not admit a clear
formulation at the Lagrangian level. In ref.  it has been shown that they may give
ambiguous results already at one loop.
In this paper, we investigate the main options for quantum gravity that are offered
by the nonanalytic Wick rotation of Euclidean higher-derivative theories, combined with
extra tools that we introduce anew. We begin with the superrenormalizable Lee-Wick
models, which are unitary when the cosmological constant vanishes. We investigate the
simplest representative of this class of models in detail and show that in various cases a
vanishing cosmological constant is consistent with the renormalization group, before and
after the coupling to matter. However, the theories with similar properties are infinitely
many, which raises the issue of uniqueness. A principle of maximum simplicity could be
used to single out the model studied here, but the principle itself would have to be justified
in its turn. For this reason, it is worth to move further on, in the search for a unique theory
of quantum gravity. We identify a candidate in a model whose Lagrangian contains the
of an additional trick, which consists of doubling the ghost poles of the free propagators
and treating the doubled versions as Lee-Wick poles. The perturbative unitarity of the
model then follows from the one of its Lee-Wick parent theory. The uniqueness of this
options relies on the fact that it is the only one whose gauge coupling is dimensionless
(according to the power counting of the high-energy limit).
The LW models have been studied in QED , the standard model [17–20] and grand
unified theories [21, 22], besides quantum gravity [23–27]. Although the CLOP or other
ad hoc prescriptions have been advocated in such investigations, some conclusions may
survive once those prescriptions are removed in favor of the formulation of ref. .
We recall other options to make sense of quantum gravity that can be found in the
literature. A well known idea is asymptotic safety . If there exists an ultraviolet
interacting fixed point with a finite dimensional critical surface, then it is possible to
reduce the free parameters of quantum gravity to a finite number, by demanding that the
theory lie on the critical surface at high energies. The weakness of this approach is that it
is nonperturbative. Nevertheless, truncations and consistency checks can provide evidence
that ultraviolet fixed points may indeed exist and have good critical surfaces [29–32].
Nonlocal theories of quantum gravity have also been explored [33–39]. Some theories
of this class are claimed to have a simple, local renormalization [33–37]. This may be true
1See  for explicit calculations in this approach.
if they are defined in Euclidean space, but the results of  suggest that if they are defined
in Minkowski spacetime, they generate nonlocal divergences that cannot be removed by
any standard procedures. On the other hand, it is hard to Wick rotate such nonlocal
Euclidean theories, because their free propagators contain nonpolynomial functions that
have extremely involved behaviors at infinity in the first and third quadrants of the complex
energy plane. Finally, the usual proofs of perturbative unitarity [40–43] do not extend to
nonlocal theories straightforwardly .
Other possibilities to make sense of higher-derivative theories have been explored.
One is that the unphysical degrees of freedom, even if present, might be unobservable if
the renormalization group keeps their masses always above the running energy [44, 45].
Another possibility is that the unphysical degrees of freedom might be a blunder due to
the expansion around the wrong vacuum.
We also recall that it is possible to treat quantum gravity as a low energy effective
field theory with infinitely many couplings. In principle, this approach can even lead to
physical predictions beyond the low-energy regime, if we identify physical quantities that
just depend on a finite subset of parameters. For example, organizing the Lagrangian in
a convenient way , it can be proved that the Friedmann-Lemaˆıtre-Robertson-Walker
(FLRW) metrics are exact solutions of the complete field equations in arbitrary dimensions
with a homogeneous and isotropic matter distribution (after a perturbative field redefinition
of the metric tensor).
The paper is organized as follows. In section 2 we study the simplest
superrenormalizable model of quantum gravity and work out the conditions under which it is unitary.
In section 3 we extend the analysis to the coupling to matter. In section 4 we address the
uniqueness problem. In section 5 we introduce the concept of fake degree of freedom. By
turning ghosts into fakes, in section 6 we build the unique model of quantum gravity that
has a dimensionless gauge coupling and show that it is unitary up to “anomalous” effects
due to the cosmological constant. Section 7 contains our conclusions.
Superrenormalizable quantum gravity
The first option that we consider is a superrenormalizable higher-derivative gravity,
formulated by nonanalytically Wick rotating its Euclidean version. We focus on the simplest
representative of this class. Up to total derivatives, its most general Lagrangian LQG is
−2κ2µ ε √LQ−Gg = 2λC M 2 + ζR − M 2 RμνRμν +
of mass and M is the Lee-Wick mass scale. The last two lines contain a convenient basis
for the six independent scalars that can be built with three Riemann tensors.
the De Donder function
We complete the gauge-fixing following the steps of ref. , so as to obtain the gauge-fixed
Lgf = LQG +
under which it is perturbatively unitary.
We begin by studying the renormalization of the theory and then discuss the conditions
It is easy to see that the renormalization of a LW theory, formulated as the nonanalytic
Wick rotation of its Euclidean version, coincides with the renormalization of its Euclidean
version. Consider a Feynman diagram and integrate the loop energies by means of the
residue theorem, as usual. We recall [11, 14] that the nonanalytic behavior of the Wick
rotation is due to the pinching of LW poles and that the LW pinching conditions equate
denotes the loop space momenta. Now, the ultraviolet divergences are studied by keeping
the external momenta p fixed and letting k tend to infinity. In such a limit the LW
sum is fixed. For this reason, the LW pinching does not affect the renormalization of the
theory, which allows us to study the counterterms of LQG with the usual techniques.
By power counting, the counterterms have at most dimension four. Using the
dimensional regularization, we organize them as
2aC M 4 + aζM 2R − aγRμνRμν + (aγ − aη)R2 ,
counterterms proportional to the cosmological constant are present up to three loops, those
proportional to the Hilbert-Einstein term are present up to two loops and the counterterms
It is convenient to introduce the “fine structure constant of quantum gravity”
The structure of the coefficients that appear in formula (2.2) is
where the superscript “(1)” denotes the one-loop values, while
u = u1 +
v = v1 + v2
w = w1 +
z = z1 + z2
renormalization group relates the coefficients u2, v2, w2, w3 and z2 of the double and triple
The bare parameters read
From these expressions, we find the beta functions
βC = −2a(C1)α¯ − 2(2u1γ + 2v1η + 3w1α¯)α¯2,
of the divergences inside the beta functions gives u2, v2, w2, w3 and z2.
We have computed the one-loop counterterms in the most general case. However, due
to their involved expressions, we just report them in a simplified case that is enough for
+43200α2α3 + 77760α32 − 1920α4 + 9600α1α4 + 17920α2α4 + 34560α3α4
2The full beta functions can be downloaded in various formats from the website
http://renormalization.com at the link http://renormalization.com/Math/QG.
the parameters of LQG follow from unitarity, i.e. the very requirement that the theory is a
Lee-Wick model, which we then formulate by nonanalytically Wick rotating its Euclidean
First, the extra poles of the free propagators must not be located on the real axis, but
lie symmetrically with respect to it, as in
p2 − m2 + iǫ (p2 − µ 2)2 + M 4
Second, we need to have an identically vanishing cosmological constant. Indeed, when the
in the absence of matter. The proof of unitarity cannot be carried out to the very end
in that case, because it is not known how to build conventional asymptotic states and a
consistent scattering matrix S in nonflat spaces [such as (anti) de Sitter space], although
alternative approaches have been attempted [47, 48].
P (a, b, c, d) ≡ a(p2)3 + bM 2(p2)2 + cM 4p2 + dM 6.
hhμν (p)hρσ(−p)ifree = hhμν (p)hρσ(−p)ifηr=eeξ=0
as in (2.3). This happens (with m = 0), if
We must also ensure that the renormalization group is compatible with an identically
renormalization group invariance, by
Renormalitazion group invariance demands that the beta function of (2.9) be also zero,
In turn, the beta function of this relation must vanish, which implies
The beta function of this relation is identically zero, so the list of consistency conditions
that the right-hand side of (2.10) is zero. However, this condition requires knowledge about
the two-loop renormalization of the theory, because it involves u1 and z1. For this reason,
which means that the Lagrangian (2.1) must be completely finite.
Summarizing, to enforce finiteness, we must solve the system of equations
2 − 3
which is compatible with the unitarity bound (2.7). At this point, we must solve the system
quadratic equations in five unknowns. Acceptable solutions are easy to find algebraically,
(2.89114 . . . , −1.93684 . . .),
(0.800169 . . . , −0.368609 . . .),
(0.197062 . . . , −0.679314 . . .), (2.13)
In the end, a simple example of a consistent theory of pure quantum gravity is the one
with the Lagrangian
An example of acceptable solution is (rounding to four decimal places)
Once we have a model where the cosmological constant vanishes identically, the
asymptotic states and the S matrix can be defined in the usual way. Then, the proof of
perturbative unitarity can be worked out by combining the strategy of  (to show that
the Lee-Wick poles do not propagate through the cuts in the cutting equations), with the
strategy of ref.  (to show that the gauge degrees of freedom — i.e. those propagated by
through the cuts). In particular, we must work in a gauge that interpolates between the
Coulomb one and the one we used to derive the propagators (2.4) and (2.5). Artificial
masses mg for the graviton are introduced to have control on the infrared divergences.
arbitrarily without reaching it. After that, the limit mg → 0 can be safely taken in
suitable combinations of amplitudes where the infrared divergences mutually cancel out. For
details, see ref. .
Coupling to matter
Now we generalize the analysis of the previous section to the coupling to matter. In this
context, “matter” refers to every classical field but the graviton, including gauge vectors.
To begin with, we assume that the matter fields are massless and switch off all the matter
self interactions. In other words, we take free massless scalars, fermions and vectors, and
covariantize their actions to couple them to gravity. The matter Lagrangian Lm is given by
= − 4 Fμν F μν + iψ¯eaμγaDμψ + 2 (∂μϕ)gμν (∂ν ϕ) +
(1 + 2̟)Rϕ2,
and the counterterms are [49–51]
where c is known as “central charge” in conformal field theory, equal to
(ns + 6nf + 12nv).
Here ns denotes the number of real scalar fields, nf is the number of Dirac fermions plus
one half the number of Weyl fermions, while nv is the number of vector fields. Note that
reason being that the matter action is Weyl invariant in that case.
functions are obtained by making the replacements
In the end, we have
βC = − 2 (4ζ − 2γ2 + 2ηγ − 3η2)α¯ − 2(2u1γ + 2v1η + 3w1α¯)α¯2,
where the higher-loop contributions need not coincide with those of the pure theory.
We inquire when we can prove perturbative unitarity again. In the simple case
that make the solution renormalization group invariant can be obtained by making the
replacements (3.2) inside (2.12). The net result is again that the theory must be completely
finite. We must solve
c = −
At very large distances, the standard model loses the QCD sector, the massive vector
bosons as well as all the other massive particles. Only the free photon survives. If we couple
receives nontrivial corrections. In general, setting it to zero and imposing the consistency
conditions required by renormalization group invariance leads to a finite theory.
f = 0,
conditions (3.5) the renormalization group (RG) chain generated by f . In our cases, the
(assuming that such values do exist and are physically acceptable), which means that the
theory is finite. When the beta functions are particularly simple, there may be exceptions
where some parameters run after imposing the RG chain, as shown in the example (2.14).
In realistic models, the cosmological constant does not vanish identically, so we cannot
prove perturbative unitarity in a strict sense. It might be possible to prove (a generalized
notion of) perturbative unitarity in an unconventional approach, but we do not know this
for sure at present. The other option is that unitarity is anomalous in the universe and
the cosmological constant is the measure of such an anomaly.
It is worth noting that, under some assumptions, the proof of the cutting equations
the propagator acquires a sort of mass term. Although flat space is no longer a solution
of the classical field equations, we can still expand around it, since the physics does not
stays positive. In these cases, the derivation of the cutting equations formally extends to
the case of nonvanishing cosmological constant. Moreover, the examples of finite theories
contributions from the cosmological constant.
The problem of uniqueness
The theory with Lagrangian (2.1) is the simplest model belonging to the class of
superrenormalizable theories of quantum gravity. The other models can be obtained from (2.1)
by adding more and more higher derivatives and fulfilling the constraints due to unitarity
and renormalizability by power counting. The Lagrangians are
− 2M 2
RQn( c/M 2)R + V (R),
where c denotes the covariant D’Alembertian, Pn, Qn are real polynomials of degree n > 1
and V (R) is a linear combination of scalars that have dimensions ranging from 6 to 2n + 4
and are built with at least three Riemann tensors (or their covariant derivatives).
In the extended class, it is easier to set the cosmological constant to zero at all energies.
n−2R, plus a quadratic polynomial in the coefficients
n−1Rμν and R cn−1R. Moreover, ζ1, ζ2, γ and η do not run. Setting
in terms of the other three. The RG chain stops immediately. If the denominators of the
In the end, the superrenormalizable models of quantum gravity that are unitary are
infinitely many, which leads to a lack of uniqueness. The theory (2.1) is singled out among
the others if we accept a sort of “minimum principle”, stating that the right theory is just
the simplest one. However, it would be better to have a really unique answer.
A possibility would be a theory with a dimensionless gauge coupling, that is to say
a strictly renormalizable theory, which would make quantum gravity more similar to the
other gauge theories. A conventional strictly renormalizable Lee-Wick model of quantum
gravity in four dimensions does not exist, because the propagators would not have the
structure (2.3). For this reason, we need an improved approach to the problem.
Fake degrees of freedom
In this section we investigate the idea of doubling the ghost poles of the free propagators
and turn them into LW poles. An extra, fictitious LW scale E is introduced and removed
at the end. This leads to a new quantization prescription. In the next section we explore
the consequences of this idea in quantum gravity.
Start from the (massless) ϕ4 scalar field theory
in four dimensions and formulate it in Euclidean space. We write the Euclidean propagator
1/p2E as p2E/(p2E)2, where pE is the Euclidean momentum. Then, we deform the propagator
with the help of the fictitious LW scale E into
and consider the limit E → 0. If we first let E tend to zero and then Wick rotate, we obtain
the usual scalar field theory. On the other hand, if we first Wick rotate, then let the scale
E tend to zero, the S matrix is identically one. Indeed, at E > 0 we obtain a Lee-Wick
model. Formulated as a nonanalytically Wick rotated Euclidean theory, it is perturbatively
unitary and has no physical degree of freedom.
We point out that the prescription we have just defined does not give the principal
± iE2 above the integration path and the poles p0 = ∓pp2
For this reason, our construction defines a distribution of a new type. Flipping the
overall sign, after the Wick rotation we write it as
The subscript “LW” in the denominator is to remind us about the positions of the poles
with respect to the integration path (the right pair of poles being placed below and the
left pair being placed above).
A good check that (5.2) is well defined can be made by calculating the bubble diagram
explicitly with the technique explained in ref.  and then take the limit E → 0. The
calculation can be carried out to the very end and, after renormalizing the ultraviolet
divergence, gives for p real
where we have included the combinatorial factor 1/2. The complex energy plane is divided
p0 < −|p|. At p 6= 0 the half p0 plane with Re[p0] > 0 looks like
into three disjoint regions. The main region A0 is the one that contains the imaginary
axis, where the Wick rotation is analytic. The other two regions A1 and A′1 are symmetric
with respect to the imaginary axis and intersect the real axis in the half lines p0 > |p| and
The boundary separating the regions A0 and A1 can be deformed arbitrarily, as long as it
does not intersect the real axis anywhere but in the threshold p
0 = |p|. The real part of
the diagram vanishes for all real values of p, which confirms that the S matrix is identically
1. We may say that (5.2) turns the degree of freedom of the ϕ4 theory into a fake degree
Let us compare this result with the one given by the Feynman prescription, which is
The two expressions (5.3) and (5.5) coincide in the main region A0. However, the
Feynman prescription does not predict the region A1, but maximally extends A0, so the
With a similar procedure, we can turn the ghosts into fake degrees of freedom. Consider
the higher-derivative theory with Lagrangian
and its naive Wick rotation to Minkowski spacetime propagates a physical massless scalar
and a massive ghost. Let us deform (5.6) into
After the Wick rotation, we find (multiplying by −i)
ζ(p2 + iǫ) − ζ [(ζM 2 + γp2)2 + E4]LW
which just propagates a massless particle, since the poles at p2 = −(ζM 2
pensate each other in the cut propagators for every E > 0, then also for E → 0.
The procedure is very general and can be used to make sense of higher-derivative
quantum gravity, by turning its ghosts into fakes, as we explain in the next section.
with the renormalization of its Euclidean version. It is easy to show that this property
main region and the real axis is the interval −|p| < p0 < |p|.
survives the limit E → 0. We know that the complex energy plane is divided into disjoint
the Wick rotation is analytic. The other regions are related to the main one by means
of a nonanalytic procedure. Renormalizability holds because the divergent parts of the
amplitudes just concern the main region. In the limit E → 0 the intersection between the
For example, in the case of the bubble diagram, two LW pinchings have thresholds on
|p0| = pk2 + iE2 + p(k − p)2 − iE2,
where p is the external momentum and k is the loop momentum. This condition cannot be
solved for arbitrarily large loop space momentum k. Therefore, the ultraviolet divergences
are not affected by the LW pinching. The conclusion also holds at E → 0, where the
|p0| = |k| + |k − p|,
which is also bounded in |k| when the external momentum is fixed.
We can use the new distribution (5.2) to build theories with unforeseen properties. For
example, consider the theory with Lagrangian
L = 2 (∂μϕ)(∂μϕ) + 2 (∂μχ)(∂μχ) − 4!
ϕ − 4!
renormalizable, unitary, nonlocal theory of the self-interacting scalar field ϕ.
Quantum gravity with a dimensionless gauge coupling
In this section, we consider the theory described by the Lagrangian
− 2κ2 √LQ−Gg = 2ΛC + ζR − M 2 RμνRμν +
theory of refs. [6–9], we want to quantize it in a new way.
2p2(ζM 2 + γp2) (ημρηνσ + ημσηνρ − ημν ηρσ).
To have perturbative unitarity in this case, we proceed as explained in the previous section,
which means that we convert (6.2) into
p2 + iǫ − [(ζM 2 + γp2)2 + E 4]LW
where E is the fake LW scale, which must tend to zero pretty much like the width ǫ.
hhμν (p)hρσ(−p)ifηree = hhμν (p)hρσ(−p)ifηr=ee0 − 2(p2)2 (ζM 2 + γp2)[ζM 2 + (γ − 3η)p2]
[(ζM 2 + γp2)2 + E 4]LW[(ζM 2 + (γ − 3η)p2)2 + E 4]LW
A similar procedure must be applied to the propagators of the Faddeev-Popov ghosts.
The theory (6.1) is the unique renormalizable higher-derivative theory of quantum
dimensionless constants. With the prescriptions just given, the theory is also perturbatively
unitary up to corrections due to the cosmological constant.
We know that in general the cosmological constant is turned on by the radiative
corrections, which prevents us from proving perturbative unitarity in a strict sense. Modified
models with an identically vanishing cosmological constant might exist. For example, it is
likely possible to build supersymmetric extensions of the theory (6.1) that have one-loop
exact beta functions or are even finite, because similar constructions are familiar in
supersymmetric theories of fields of spins 0, 1/2 and 1 . If such models are finite, the
cosmological constant can be switched off at no cost. If they have one-loop exact beta
functions, extra conditions have to be imposed in order to fulfill the renormalization group
admit nontrivial solutions, as we found in sections 2 and 3 for the superrenormalizable
theory (2.1) and its coupling to matter. In all such cases, we may provide examples of
theories quantum gravity (coupled to matter) with a dimensionless gauge coupling and an
identically vanishing cosmological constant, where the proof of unitarity can be carried out
to the very end. However, it is unlikely that the ultimate theory of nature will have an
identically vanishing cosmological constant, so we must be prepared to accept that there
may be a small unitarity anomaly in the universe.
In this paper we have studied the main options for a consistent, local quantum field theory
of the gravitational interactions. Superrenormalizable higher-derivatives theories of gravity
can be built as Lee-Wick models and formulated as nonanalytically Wick rotated Euclidean
theories. They are perturbatively unitary when the cosmological constant vanishes. The
simplest example is encoded in formula (2.1), provided the parameters satisfy suitable
restrictions. The other models of this class can be build by adding more higher-derivatives
and fulfilling the Lee-Wick unitarity conditions. The possibilities are infinitely many, which
raises the question of uniqueness.
A better possibility is the theory (6.1), because it has a dimensionless gauge constant,
which makes it unique and more similar to the gauge theories that describe the other
interactions of nature. The Lagrangian (6.1) cannot be quantized in the conventional
LeeWick way. This lead us to introduce a new concept, the fake degrees of freedom, and a
prescription different from the usual one. Taking advantage of the Lee-Wick idea, a ghost
(or a normal degree of freedom) can be turned into a fake degree of freedom, which does
not contribute to the physical spectrum and does not propagate through the cuts of the
cutting equations. So doing, the ghosts of higher-derivative gravity can be eliminated.
The renormalization of the theory (6.1) is obviously richer than the one of a
superrenormalizable theory like (2.1), because nontrivial radiative corrections to the beta functions
and the anomalous dimensions are expected to all orders. Again, this is similar to what
we know from the other gauge theories that successfully describe nature.
If we accept that the gauge couplings are dimensionless, then there is only one theory
of the four interactions of nature, made of the gauge sector of the standard model coupled
to the quantum gravity theory (6.1). Constraining the matter sector is obviously harder.
In several models it is possible to turn the cosmological constant off to all energies,
consistently with the renormalization group. However, the more realistic models have a
nonvanishing cosmological constant, which might be the signal of a small unitarity anomaly.
It can be interesting to study the phenomenological implications of the theory (6.1)
coupled to the standard model. Some arguments existing in the literature (see for example
ref. ) might survive after switching to the correct formulation of the theory, others
might have to be reconsidered.
We are grateful to U. Aglietti and M. Piva for useful discussions.
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