Effective quantum gravity, cosmological constant
and the Standard Model of particle physics

Breno L. Giacchini(a), Tibério de Paula Netto(a), Ilya L. Shapiro(b) 111Also
at Tomsk State Pedagogical University.

(a) Department of Physics, Southern University of Science and Technology,

Shenzhen, 518055, China

(b) Departamento de Física, ICE, Universidade Federal de Juiz de Fora,

Juiz de Fora, 36036-900, MG, Brazil

> Abstract. The renormalization group in effective quantum gravity can be
> consistently formulated using the Vilkovisky and DeWitt version of effective
> action and assuming a non-zero cosmological constant. Taking into account
> that the vacuum counterpart of the cosmological constant is dramatically
> different from the observed energy density of vacuum, the running of the
> last quantity in the late cosmology indicates strong constraints on the
> physics beyond the minimal Standard Model of particle physics.

##  1 Introduction

One of the main open issues in nowadays fundamental physics is the origin of
the so-called dark energy. The most likely candidate is the cosmological
constant because such a constant is an essential element of the consistent
quantum field theory (QFT) in curved space (see e.g. [1] for a recent
discussion and further references). On the other hand, there are many
theoretically interesting alternative models, assuming that the vacuum energy
may vary with time (see, e.g, [2, 3] as starting points). The question is how
one can expect to falsify these models using the existing observational data.
Indeed, the possibilities to do so are restricted to the observation of the
time-variable vacuum energy, or the effective equation of state [4].

Does the time dependence of the vacuum energy mean that there is an essence
more sophisticated than the cosmological constant, governing the accelerated
expansion of the Universe? The answer is not obvious because the proper
cosmological constant may be changing for at least two reasons. The observable
cosmological constant is a sum of the two contributions, namely, the
fundamental constant in the action of gravity and the induced counterpart
owing to the phase transition and symmetry breaking in the vacuum state [5]
(see also [6] for the QFT aspects of the problem). The induced component could
have different magnitude because of the typical energy scale at which the
symmetry breaking occurs or, almost equivalently, to restoration of the
symmetry in the hot vacuum of the early Universe. Furthermore, the observable
cosmological constant may change in the late Universe because of the
renormalization group running. The running of the vacuum component of the
cosmological constant in flat spacetime is a basic example of such a running
(see e.g. [7]), that can be extended to semiclassical [8] and quantum [9]
gravity.

The running cosmology models are based on the universal form of the scale-
dependent density of the energy of vacuum, that can be established using
covariance arguments [10] or the assumption of a standard quadratic decoupling
of massive degrees of freedom in the IR, in the semiclassical approach [6,
11],

$\displaystyle\rho_{\Lambda}=\rho^{0}_{\Lambda}+\frac{3\nu}{8\pi
G}\,\big{(}H^{2}-H^{2}_{0}\big{)}.$ (1)

Here $\rho^{0}_{\Lambda}$ and $H_{0}$ are the vacuum energy density and the
Hubble parameter at the reference point, e.g., in the present moment of time.
$\nu$ is a phenomenological parameter which cannot be calculated with the
known theoretical methods [10, 12]. Since the particle creation from vacuum in
the IR is suppressed [13], the energy conservation leads to the “traditional”
logarithmic form of the running for the Newton constant [14],

$\displaystyle
G(\mu)=\frac{G_{0}}{1+\nu\,\log\left(H^{2}/H_{0}^{2}\right)}\,.$ (2)

There is a possibility of a systematic scale-setting procedure in the
cosmological context [15], providing the identification of the Hubble
parameter $H$ with the scaling parameter $\mu$.

Let us stress that the aforementioned running in (1) and (2) is owing to the
quantum effects of massive particles in the IR, according to the corresponding
decoupling theorems [16, 17]. Usually, massless degrees of freedom do not
produce running of the dimensional parameters, such as $\rho_{\Lambda}$ and
$G$. On the other hand, (2) looks like the one-loop running of the
dimensionless coupling in the Minimal Subtraction scheme of renormalization,
regardless it is derived in a very different framework.

The models of running vacuum based on (1) were extensively explored and became
an active field of research (see e.g. [10, 12, 14, 18, 19] and further
references in the last work). These models give a well-motivated alternative
to the theories of modified gravity of all kinds, as in both cases one can
describe a slowly varying vacuum energy density.

From the theoretical side, one of the open questions is whether the
aforementioned universality of the running of vacuum energy (1) can be
extended into the full quantum gravity (QG). The definitive answer to this
question is unknown, as there is no completely consistent theory of quantum
gravity to derive the running. On the other hand, any kind of a purely metric
quantum gravity should have massless and massive degrees of freedom. The
typical dimensional constant in quantum gravity is the Planck mass $M_{P}$,
therefore we can expect that all massive modes have the mass of this magnitude
[20]. For the cosmological applications, what we need is an effective
formulation of QG, when the massive modes are assumed to decouple (see, e.g.,
[21, 22] and references therein). In this regime, we expect to meet the
universal IR theory of QG based on the quantum general relativity (GR).
Therefore, we need to account only for the massless modes [23, 24] and it
seems this can be a solid basis for deriving the QG-based running of
$\rho_{\Lambda}$ and $G$ that can be applied in the late running-vacuum
cosmology.

Now we are coming close to the main subject of this contribution, i.e., the
role of the “unique” effective action of Vilkovisky and DeWitt for the IR
running in the effective QG. In the framework of usual perturbative field
theory, the running of $\rho_{\Lambda}$ and $G$ is possible only if the
initial gravity theory has a non-zero cosmological constant. However, even in
this case, the individual running of both quantities depends on the gauge
fixing and on the parametrization of the quantum metric. Only the running of
the dimensionless ratio of these parameters is universal [9], but this is
insufficient for the cosmological applications. The construction of
Vilkovisky–DeWitt effective action resolves this difficulty at the one-loop
[25] and higher-loop [26] levels. This useful feature of the “unique”
effective action is owing to its geometric, covariant formulation in the space
of the fields. The renormalization group running of $\rho_{\Lambda}$ and $G$,
based on Vilkovisky’s construction has been explored in [27]. In the recent
works [28, 29] we performed an explicit verification of the gauge- and
parametrization-invariance of these renormalization group equations and proved
that, in the effective QG framework, these equations are exact, i.e., not
restricted by the one-loop approximation, as explained also in Sec. 3 below.

The status of the Vilkovisky–DeWitt effective action has been previously
discussed in the work [30] based on three interesting applications where the
gauge fixing and/or parametrization ambiguities in the usual effective action
do not enable one to achieve the desired qualitative output, while the
geometric methods provide the result. The general conclusion of this
consideration is that the Vilkovisky–DeWitt construction is not a panacea, nor
a placebo, but rather a useful tool for making calculations. In the present
contribution, we discuss the application of the effective QG-based running
[29] of $\rho_{\Lambda}$ and $G$ in cosmology and show that these effects may
provide dramatic consequences for particle physics. This can be viewed as one
more application where the Vilkovisky–DeWitt geometric approach yields a
nontrivial consequence. Since the technical details of the Vilkovisky
formalism are not the main topic of this paper, we postpone to the Appendix A
a brief review of it and a discussion about its possible limitations. However,
we would like to stress, from the very beginning, that this approach is an
additional independent input and that its consequences may be verified or
falsified only by means of experiments or observations.

The manuscript is organized as follows. In Sec. 2 we briefly review the
cosmological constant and the fine-tuning problem, from the standard
perspective of [5] and [6]. Sec. 3 discusses the exact effective QG running
[29] and explains why it might break down the fine-tuning for the cosmological
constant. In Sec. 4 we perform the numerical estimates of the mentioned
breaking for a few models of particle physics and arrive at severe constraints
on the physics beyond the standard model. Finally, in Sec. 5 we draw our
conclusions and add some extra discussion.

##  2 Cosmological constant problem and the running

As it was already mentioned above, the running of $\rho_{\Lambda}$ and $G$ in
GR-based QG is possible only because of the non-zero cosmological constant.
The higher loop contributions come with growing powers of the dimensionless
ratio $\rho_{\Lambda}G^{2}=\rho_{\Lambda}\,M_{P}^{-4}$. It is important to
stress that $\rho_{\Lambda}$ in this expression is not the observable vacuum
energy density $\rho^{obs}_{\Lambda}$. It is well-known [5] (see also [6])
that

$\displaystyle\rho^{obs}_{\Lambda}\,=\,\rho_{\Lambda}+\rho^{ind}_{\Lambda}\,,$
(3)

where $\rho_{\Lambda}$ and $\rho^{ind}_{\Lambda}$ are the vacuum and the
induced densities of cosmological constant, respectively. An independent
quantity $\rho_{\Lambda}$ is a necessary element of the renormalizable
semiclassical theory (see, e.g., [1] for an introduction). Loop corrections
without external lines of matter fields produce divergences, including those
without derivatives of the metric tensor. These divergences require
renormalization and, in particular, fixing the renormalization condition. As
usual with independent parameters, this procedure involves a measurement.
Thus, the value of $\rho_{\Lambda}$ can be defined only from the cosmological
observations of $\rho^{obs}_{\Lambda}$. After that, one has to subtract
$\rho^{ind}_{\Lambda}$ to arrive at the value of
$\rho_{\Lambda}=\rho^{vac}_{\Lambda}$.

From the theoretical side, the minimal magnitude of $\rho_{\Lambda}$ is
defined by its running in a semiclassical theory. For example, in the minimal
standard model (MSM) this indicates at a value of the order of the fourth
power of the Fermi mass, $M_{F}^{4}$. As $\rho_{\Lambda}$ is an independent
parameter, its value at the reference scale $\mu_{0}$ can be defined only from
the experimental or observational data. On the other hand,
$\rho^{ind}_{\Lambda}$ is, in principle, calculable from the underlying matter
fields model. If its origin is the spontaneous symmetry breaking (SSB) in the
MSM, we have the well-known relation with the VeV of the Higgs field,
$\rho^{ind}_{\Lambda}\sim\lambda v^{4}\approx 10^{8}\,\text{GeV}^{4}$. As the
value of $v$ is defined by the typical (Fermi) energy scale $M_{F}\approx
293\,\text{GeV}$, in what follows we shall associate $\rho^{ind}_{\Lambda}$
with $M_{F}^{4}$. In case there is another, similar, phase transition at
higher energy scale such as $M_{X}$, we have to replace $M_{F}$ by $M_{X}$ in
both $\rho_{\Lambda}$ and $\rho^{ind}_{\Lambda}$.

It is remarkable that the theoretical predictions for the two ingredients in
the r.h.s. of (3) give the same order of magnitude. At the same time, the
relation (3) is famous for the huge amount of fine-tuning required for the
cancelation in its r.h.s., providing a very small value for the observable
sum. In the particle physics units, the value of $\rho^{obs}_{\Lambda}$, it is
about $10^{-47}\,\text{GeV}^{4}$, such that even in the MSM we need about 56
orders of the fine-tuning in the choice of the renormalization condition
$\rho_{\Lambda}(\mu_{0})$ .

The cosmological constant problem is a real mystery, as the 56-order fine
tuning can be violated even by very small changes in the Yukawa couplings that
enter the game via the one-loop or higher-loop corrections (up to 21 loops).
It is worth noting that even a small mismatch in the choice of the condition
for $\rho_{\Lambda}(\mu_{0})$ may lead to either a negative, zero or too big
(more than 100 times greater) positive value of $\rho_{\Lambda}^{obs}$. All
three options contradict our own existence through the anthropic arguments
[31]. In the next sections we shall see that the well-defined running
$\rho_{\Lambda}(\mu)$ in effective QG imposes strong constraints on the
particle physics beyond MSM.

##  3 On the running in effective QG

The starting point in the discussion of the running in effective QG is the
gravitational action,

$\displaystyle S\,=\,-\,\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\,(R+2\Lambda),$
(4)

where $\rho_{\Lambda}=\frac{\Lambda}{8\pi G}$, together with the theory of
gauge-invariant renormalization (see e.g. [1] and references therein) and the
power counting formula for quantum GR,

$\displaystyle\omega(\mathcal{G})+d(\mathcal{G})=2+2L-2K_{\Lambda}.$ (5)

Here $L$ is the number of loops in the given diagram $\mathcal{G}$ and
$K_{\Lambda}$ is the number of vertices coming from the cosmological constant
term. For the logarithmic divergences $\omega(\mathcal{G})$ is zero and then
the last formula gives the number of derivatives $d(\mathcal{G})$ in the
corresponding counterterm.

It is easy to see that for $K_{\Lambda}=0$ we never get the renormalization of
the Einstein–Hilbert term, but only higher-derivative counterterms. However,
the situation is different in the case $K_{\Lambda}\neq 0$. One of the main
observations of [29] and of the present work is that $\Lambda$ in (4) does not
correspond to $\rho^{obs}_{\Lambda}$ but, instead, to the
$\rho_{\Lambda}=\frac{\Lambda}{8\pi G}$ in the vacuum part of (3). As we have
seen in the previous section, the two quantities $\rho^{obs}_{\Lambda}$ and
$\rho_{\Lambda}$ are dramatically different and this may change the game in
the effective QG-based running.

The important aspect in the effective QG is whether we are capable of
obtaining results which are free of ambiguities. For instance, the known
theorems about gauge-fixing and parametrization dependence [32, 33] tell us
that the one-loop divergences of the effective action,

$\displaystyle\Gamma^{(1)}_{div}=\frac{1}{\epsilon}\,\int
d^{4}x\sqrt{-g}\,\big{\\{}c_{1}\,R_{\mu\nu\alpha\beta}^{2}+c_{2}R_{\alpha\beta}^{2}+c_{3}R^{2}+c_{4}{\Box}R+c_{5}R+c_{6}\big{\\}},$
(6)

are universal only on the classical mass shell. Here
$\epsilon=(4\pi)^{2}(n-4)$ is the parameter of dimensional regularization.
Using the approach of [9, 34] we arrive at the relation

$\displaystyle\Gamma^{(1)}_{div}(\alpha_{i})\,-\,\Gamma^{(1)}_{div}(\alpha^{0}_{i})\,\,=\,\frac{1}{\epsilon}\,\int
d^{4}x\sqrt{-g}\,\Big{(}b_{1}R_{\mu\nu}+b_{2}Rg_{\mu\nu}\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\,\,b_{3}g_{\mu\nu}\Lambda+b_{4}g_{\mu\nu}\Box+b_{5}\nabla_{\mu}\nabla_{\nu}\Big{)}\,\varepsilon^{\mu\nu}\,,$

where $b_{k}=b_{k}(\alpha_{i})$ and $\alpha_{i}$ represent the full set of
parameters defining an arbitrary gauge-fixing and parametrization of the
metric, and $\varepsilon^{\mu\nu}=G^{\mu\nu}-\Lambda g^{\mu\nu}$ are the
classical equations of motion. The two invariant quantities are

$c_{1}\qquad\text{and}\qquad c_{\rm inv}\,=\,c_{6}-4\Lambda
c_{5}+4\Lambda^{2}c_{2}+16\Lambda^{2}c_{3},$ (7)

which means that the invariant running is possible only for the dimensionless
combination of $G$ and $\Lambda$ [9, 35].

The Vilkovisky [25] and DeWitt [26] definition of effective action is
different, as it is based on the covariant calculus in the space of physical
fields. A brief review of the formalism can be found in the Appendix A, where
we also discuss some subtleties involving its application to QG. More detailed
considerations can be found, e.g., in [36, 37, 27, 29, 28, 38, 39, 40, 30].

The renormalization group running of $G$ and $\Lambda$, in the effective QG
based on the Vilkovisky–DeWitt version of effective action, has the form [27,
29]

$\displaystyle
G(\mu)\,=\,G_{0}\,\Big{[}1+\frac{10}{(4\pi)^{2}}\,\gamma_{0}\,\mbox{ln}\,\frac{\mu}{\mu_{0}}\Big{]}^{-4/5}$
(8)

and

$\displaystyle\Lambda(\mu)\,=\,\Lambda_{0}\,\Big{[}1+\frac{10}{(4\pi)^{2}}\,\gamma_{0}\,\mbox{ln}\,\frac{\mu}{\mu_{0}}\Big{]}^{-1/5},$
(9)

where $\gamma=16\pi G\,\Lambda$ and the subscript label zero indicated the
value $\mu_{0}$ of the reference scale, e.g., the present epoch of the
Universe. Thus, $\gamma_{0}=16\pi
G_{0}\,\Lambda_{0}=128\pi^{2}G_{0}^{2}\rho_{\Lambda}^{0}=128\pi^{2}\rho^{0}_{\Lambda}M_{P}^{-4}$.

Let us stress that, differently from (1) and (2), Eqs. (8) and (9) come from a
real calculation, based solely on the assumption of applicability of the
Vilkovisky and DeWitt effective action in quantum field theory.

According to the power counting (5), the higher loop corrections to these
equations are proportional to higher powers of $\gamma$ and, for a
sufficiently small $\Lambda$, can be regarded as negligible. Starting from
this point and taking into account the arguments from Sec. 2, we can explore
the physical consequences of the running (8) and (9).

##  4 Cosmology with running vacuum in effective QG

To evaluate $\rho_{\Lambda}$ and $\gamma_{0}$ we have to remember that
$\rho_{\Lambda}$ has approximately—and with a great precision—the same
absolute value as $\rho^{ind}_{\Lambda}$. It is clear that
$\rho_{\Lambda}\gg\rho^{obs}_{\Lambda}$, but numerically $\gamma_{0}$ is still
a small quantity. For instance, in the theory when the MSM is valid until the
Planck energy scale, we have $\gamma_{0}\sim 10^{-65}$, while for the SUSY GUT
this coefficient may be $\gamma_{0}\sim 10^{-12}$–$10^{-8}$. Do these small
numbers guarantee that the variations caused by (8) and (9) are irrelevant?

To evaluate the consequences of the running, let us consider the strongest
option, that is, the SUSY GUT case. Then, the value of $\rho_{\Lambda}$ should
be of the order of $M_{X}^{4}$, assuming $M_{X}\sim 10^{16}$ GeV. Accordingly,
the parameter $\gamma$ is of the order of $(M_{X}/M_{P})^{4}$. In this case,
we find the strong inequality $\frac{10\gamma_{0}}{(4\pi)^{2}}\ll 1$. After a
small algebra, (8) and (9) boil down to

$\displaystyle
G(\mu)\,=\,G_{0}\Big{[}1\,-\,\frac{8}{(4\pi)^{2}}\,\gamma_{0}\,\mbox{ln}\,\frac{\mu}{\mu_{0}}\Big{]}$
(10)

and

$\displaystyle\rho_{\Lambda}(\mu)\,=\,\rho^{0}_{\Lambda}\,\Big{[}1+\frac{6}{(4\pi)^{2}}\,\gamma_{0}\,\mbox{ln}\,\frac{\mu}{\mu_{0}}\Big{]}.$
(11)

The derivation of the last equation requires expanding the right-hand side of
both (8) and (9) up to the first order in the small parameter $\gamma_{0}$ and
replacing the result into the formula $\rho_{\Lambda}=\frac{\Lambda}{8\pi G}$.

Remember that the standard identification of scale in cosmology is $\mu\propto
H$ [6, 10, 15]. Taking this identification, the effects of (10) and (11) are
dramatically different. Indeed, while for $G(\mu/\mu_{0})=G(H/H_{0})$ there is
a usual slow logarithmic running that is not too relevant in cosmology, the
effect of the cosmological constant running (11) may be strong.

In the SUSY GUT case, a simple calculation gives

$\displaystyle\frac{6}{(4\pi)^{2}}\,\gamma_{0}\,\sim\,48\Big{(}\frac{M_{X}}{M_{P}}\Big{)}^{4}\,\approx\,10^{-11}.$
(12)

This is a really huge number, because it has to be multiplied not only by the
logarithmic factor but, at the first place, by the $\rho^{0}_{\Lambda}$, i.e.,
by the vacuum energy density in the r.h.s. of the main relation (3). In SUSY
GUT the value of $\rho^{0}_{\Lambda}$ is about 111 orders of magnitude greater
than the observed value $\rho^{obs}_{\Lambda}$. Thus, the running (12)
produces a discrepancy with the cosmological observations proportional to
$10^{100}$ (googol) for a change of about one order of magnitude in the
parameter $H$. Needless to say that this result contradicts the anthropic
calculations [31]. Thus, we have to give up either on the effective QG based
on the Vilkovisky “unique” effective action, or on the SUSY GUT and the
corresponding generation of induced vacuum energy in the r.h.s. of (3). As the
present report is devoted to the effective QG-based running, we conclude that
the SUSY GUT hypothesis fails in this framework.

Let us consider another extreme of the energy scale and assume that the MSM is
valid up to the Planck scale. In this case, instead of (12) we meet

$\displaystyle\frac{6}{(4\pi)^{2}}\,\gamma_{0}\,\sim\,48\Big{(}\frac{M_{F}}{M_{P}}\Big{)}^{4}\,\approx\,10^{-65}.$
(13)

When multiplied by $\rho^{0}_{\Lambda}\sim M_{F}^{4}$, we find the variation
of the observed cosmological constant given by
$\delta\rho^{obs}_{\Lambda}\approx
10^{-55}\,\mbox{ln}\,(H/H_{0})\,\text{GeV}^{4}$.

Taking the range of change of $H$ between the inflationary epoch with
$H_{infl}\leqslant 10^{15}\,\text{GeV}$ and the present-day Universe with
$H_{0}\approx 10^{-42}\,\text{GeV}$, the logarithmic factor is
$\,\mbox{ln}\,(H/H_{0})\approx 131$. Then the numerical estimate based on (13)
gives $\delta\rho^{obs}_{\Lambda}\approx 10^{-53}\,\text{GeV}^{4}$, that is
just six orders of magnitude smaller than the observed value
$\rho^{obs}_{\Lambda}$. At this point we can make two observations:

1\. Our model of the effective QG running is lucky enough to pass the test
related to MSM. This means, e.g., that the experimentally confirmed model of
particle physics does not contradict the anthropic arguments. The opposite
output would mean the disproval of the Vilkovisky and DeWitt approach.

2\. Since the result is proportional to $M_{F}^{8}$, we can state that the
existence of new physics based on the symmetry breaking beyond the energy
scale of $10M_{F}$ contradicts the effective running of the cosmological
constant. It is remarkable that the effective QG provides such a relation
between the cosmological constant problem and the particle physics.

##  5 Conclusions and discussions

The running derived in the effective QG based on the Vilkovisky–DeWitt
effective action enables one to formulate the link between particle physics
and cosmology. In particular, we find that the MSM with the corresponding SSB
leads to the running of the vacuum cosmological constant that does not
contradict the cosmological observations and, in particular, the anthropic
restrictions derived by Weinberg [31].

On the other hand, the mentioned running of the cosmological constant imposes
severe restrictions on the SSB and the generation of induced cosmological
constant in the physics beyond the MSM. Even assuming the symmetry breaking at
the $10\,\text{TeV}$ scale means we may run out of the scope with the
cosmological constant violating the fine tuning in (3). The energy scale below
$10\,\text{TeV}$ is explored in LHC, but this does not mean that new physics
beyond MSM is “forbidden” by the effective QG and the corresponding running.
The obtained restrictions leave a lot of space for constructing particle
physics models beyond MSM. Thus, it would be interesting to explore this
possibility in more details. The corresponding analysis is beyond the scope of
the present work and will be presented as a separate publication. Let us just
note that the limitation concerns the value of the new Higgs-like vacuum
expectation value (VeV) and not the masses of the particles. Anyway, the
preliminary result is that many (albeit not all) GUT models and supersymmetric
extensions of the MSM may be ruled out by the new criterion based on quantum
gravity. On the other hand, there are theories, e.g., based on the technicolor
approach, which may have rather large masses of the particles beyond the
standard model and still escape the restrictions discussed in the present
work.

Finally, the well-defined gauge and parametrization independent running (1),
originally described by Taylor and Veneziano [27] and explored in the
effective framework in [29], provides interesting hints concerning the
connection between different branches of Physics and also opens new horizons
for further work.

##  Appendix A Brief review and discussion of the Vilkovisky effective action
for QG

In this appendix we make a short presentation of the Vilkovisky effective
action and comment on the possible limitations of the formalism, with a focus
on the choice of metric in the space of fields in QG. This discussion is done
in more detail than in our previous publications [28, 29].

The Lagrangian quantization of gauge theories, including QG, involves fixing a
gauge which breaks the classical action’s gauge invariance. In DeWitt’s
background field method (see [41] for review and references) the resultant
effective action is covariant with respect to gauge transformations of the
background field, but it still depends on the choice of gauge fixing for the
quantum field. The difference between these two situations is sometimes
referred to as “gauge invariance” and “gauge-fixing dependence” [30].

In quantum GR, the background field method allows the evaluation of the
divergent part of the effective action. Owing to locality, the divergences are
scalars constructed with the curvature tensors and their covariant derivatives
[see Eq. (6)]. As discussed in Sec. 3, the expression (6) is invariant under
spacetime diffeomorphisms, although some of the coefficients $c_{i}$ depend on
the gauge chosen for the quantized field, i.e., they are gauge-fixing
dependent.

The gauge-fixing dependence of the effective action can be regarded as part of
the more fundamental dependence on the field parametrization, which also
affects non-gauge theories [25, 33]. At one-loop level this can be understood
by recalling that the Hessian of the classical action,
$\frac{\delta^{2}S}{\delta\varphi^{i}\delta\varphi^{j}}$, does not behave as a
tensor under redefinitions of the field $\varphi^{i}$. In the same spirit as
in Riemannian geometry, Vilkovisky [25] introduced an affine structure on the
configuration space $\mathscr{M}_{\perp}$ of physical fields, with a metric
$\bar{G}_{ij}$ and a connection $\mathscr{T}^{k}_{ij}$, and modified the
definition of the effective action such that it transforms in a covariant
manner under diffeomorphism, namely,

$\displaystyle\exp
i\Gamma(\varphi)\,=\,\int\mathcal{D}\varphi^{\prime}\mu(\varphi^{\prime})\,\exp\left\\{i\left[S(\varphi^{\prime})+\sigma^{i}(\varphi,\varphi^{\prime})\Gamma_{,i}(\varphi)\right]\right\\},$
(14)

where $\mu(\varphi^{\prime})$ is an invariant functional measure and
$\sigma_{i}(\varphi,\varphi^{\prime})$ is the derivative with respect to
$\varphi^{i}$ of the world function $\sigma(\varphi,\varphi^{\prime})$ [42,
43]. Note that the affine structure is defined on the space of physical
fields, which means that if $\mathscr{G}$ is a gauge group acting on a space
$\mathscr{M}$ of fields $\varphi^{i}$, then
$\mathscr{M}_{\perp}=\mathscr{M}/\mathscr{G}$. Therefore, since
$\sigma^{i}(\varphi,\varphi^{\prime})$ behaves as a vector with respect to
$\varphi^{i}$ and as a scalar with regard to $\varphi^{\prime i}$, the
effective action $\Gamma(\varphi)$ defined by (14) is gauge invariant and it
is independent of the parametrization and gauge fixing choices. Because of
this, the covariant effective action $\Gamma(\varphi)$ is also known as
“unique effective action”.

The metric $G_{ij}$ in the full configuration space $\mathscr{M}$ of fields is
obtained through the following criteria: (i) It should be an ultra-local
quantity and do not contain derivatives of the fields in order to not violate
the $S$-matrix theory. (ii) For quadratic non-interacting field theories,
$G_{ij}$ should provide a flat field space. (iii) It must be uniquely
determined by the classical action $S(\varphi)$ of the theory, namely, it
should be chosen as the local metric contained in the highest-derivative term
of the classical action after projecting out the gauge-dependent degrees of
freedom. This is known as Vilkovisky’s prescription for the choice of metric
in the space of fields [25].

The projection onto the space $\mathscr{M}_{\perp}$ is performed by the
operator [25, 37, 38]

$\displaystyle\Pi^{i}{}_{j}=\delta^{i}{}_{j}-R^{i}_{\alpha}(N^{-1})^{\alpha\beta}R^{k}_{\beta}G_{kj},$
(15)

where $R^{i}_{\alpha}$ are the generators of gauge transformations and
$(N^{-1})^{\alpha\beta}$ is the inverse of the metric $N_{\alpha\beta}$ on the
gauge group,

$\displaystyle N_{\alpha\beta}=R^{i}_{\alpha}G_{ij}R^{j}_{\beta}.$ (16)

Therefore, the projection of the metric on $\mathscr{M}_{\perp}$ is

$\displaystyle\bar{G}_{ij}\,=\,\Pi^{k}{}_{i}G_{k\ell}\Pi^{\ell}{}_{j}\,=\,G_{ij}-G_{ik}R^{k}_{\alpha}(N^{-1})^{\alpha\beta}R^{\ell}_{\beta}G_{\ell
j}.$ (17)

The affine connection $\mathscr{T}^{k}_{ij}$ can then be obtained by requiring
its compatibility with the physical field-space metric,
$\bar{\nabla}_{k}\bar{G}_{ij}=0$ (see, e.g., [25, 44, 37] for further comments
and explicit formulas).

Notice, however, that by projecting out the gauge-dependent part of the field
we obtain

$\displaystyle\varphi^{i}_{\perp}\equiv\Pi^{i}{}_{j}\varphi^{j}=\varphi^{i}-R^{i}_{\alpha}(N^{-1})^{\alpha\beta}R^{k}_{\beta}G_{kj}\varphi^{j}.$
(18)

Thus, in the so-called Landau–DeWitt gauge, defined by

$\displaystyle R^{k}_{\beta}G_{kj}\varphi^{j}=0,$ (19)

we have

$\displaystyle\varphi^{i}_{\perp}=\varphi^{i}.$ (20)

This means that Vilkovisky’s prescription for $G_{ij}$ is equivalent to
getting the metric which follows from the theory’s action using the
Landau–DeWitt gauge [38]. We stress that this is only due to the fact that in
the gauge (19) the identity (20) holds, and that at this stage one is not
choosing a specific gauge which could, potentially, simplify calculations.
Thus, the aforementioned detail does not reduce the generality of the scheme.

Let us discuss in more detail the choice of the metric $G_{ij}$ in the case of
QG. In metric theories of gravity, $\varphi^{i}=g_{\mu\nu}$ and there is a
one-parameter family of metrics on $\mathscr{M}$ in accordance with (i)-(ii),

$\displaystyle
G_{ij}=G^{\mu\nu,\alpha\beta}=\tfrac{1}{2}(\delta^{\mu\nu,\alpha\beta}+ag^{\mu\nu}g^{\alpha\beta}),\qquad\delta^{\mu\nu,\alpha\beta}=\tfrac{1}{2}(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha}),$
(21)

where $a\neq-1/4$ is a constant. The value $a=-1/4$ is discarded, as otherwise
the metric (21) has no inverse. In order to fix the parameter $a$, we shall
follow Vilkovisky’s prescription.

Particularizing the discussion for quantum GR, the expansion of the
Einstein–Hilbert action (4) in the background field method splitting

$\displaystyle
g_{\mu\nu}\,\,\longrightarrow\,\,g^{\prime}_{\mu\nu}\,=\,g_{\mu\nu}+\kappa
h_{\mu\nu},\qquad\kappa^{2}=16\pi G,$ (22)

where $g_{\mu\nu}$ is the background field and $h_{\mu\nu}$ is the quantum
field, gives the quadratic form

$\begin{split}S^{(2)}=&-\frac{1}{2}\int
d^{4}x\sqrt{-g}\big{\\{}h_{\mu\nu}\left[\tfrac{1}{2}(\delta^{\mu\nu,\alpha\beta}-\tfrac{1}{2}g^{\mu\nu}g^{\alpha\beta})\Box\right]h_{\alpha\beta}\\\
&+(\nabla_{\lambda}h^{\mu\lambda}-\tfrac{1}{2}\nabla^{\mu}h)^{2}\big{\\}}+\cdots.\end{split}$
(23)

Here the ellipsis stands for terms without derivatives of the quantum field,
which are unimportant for the present discussion. Since the generators of the
spacetime diffeomorphisms are

$\displaystyle R^{i}_{\alpha}\equiv
R_{\mu\nu,\alpha}=-(g_{\mu\alpha}\nabla_{\nu}+g_{\nu\alpha}\nabla_{\mu}),$
(24)

a simple calculation shows that the Landau–DeWitt gauge condition (19) reads

$\displaystyle
R^{k}_{\alpha}G_{kj}\varphi^{j}=\nabla_{\mu}h^{\mu}_{\alpha}+a\nabla_{\alpha}h=0\quad\Longrightarrow\quad\nabla_{\mu}h^{\alpha\mu}=-a\nabla^{\alpha}h.$
(25)

Substituting this condition into (23) and recalling (20), we find the bilinear
projected action on $\mathscr{M}_{\perp}$,

$\displaystyle\quad S^{(2)}_{\perp}\,=\,-\frac{1}{2}\int
d^{4}x\sqrt{-g}\,h_{\mu\nu}^{\perp}\Big{\\{}\frac{1}{2}\big{[}\delta^{\mu\nu,\alpha\beta}-\big{(}2a^{2}+2a+1\big{)}g^{\mu\nu}g^{\alpha\beta}\big{]}\,\Box\Big{\\}}h_{\alpha\beta}^{\perp}\,+\,\cdots.$
(26)

Thus, the requirement that the term between curly brackets in the last
expression equals the metric (21) leads to the algebraic equation [25, 38]

$\displaystyle-(2a^{2}+2a+1)\,=\,a,$ (27)

which has the solutions

$\displaystyle a=-\frac{1}{2}\qquad\text{and}\qquad a=-1.$ (28)

The value $a=-1$ is discarded because the metric on the gauge group

$\displaystyle
N_{\alpha\beta}=R^{i}_{\alpha}G_{ij}R^{j}_{\beta}=-\big{[}g_{\alpha\beta}\Box+(1+2a)\nabla_{\alpha}\nabla_{\beta}+R_{\alpha\beta}\big{]}$
(29)

becomes degenerate. Thus, for the QG based on GR, in the simplest
parametrization (22), the Vilkovisky’s prescription for the choice of metric
gives $a=-1/2$ in an unambiguous way.

Furthermore, since the metric on $\mathscr{M}$ transforms as a tensor under
field reparametrizations, Eq. (27) holds even for quantum metric
parametrizations more general than (22). Thus, the ambiguity represented by
the coefficient $a$ is fixed in the same way, as explicitly shown in [28]. In
this paper (as well as in our work [29]) we assume this choice of the metric.

This elementary exposition of the Vilkovisky formalism enables us to discuss
some of the usual criticism about it. First of all, we notice that if the
conditions (i) and (ii) for the field-space metric are satisfied, the
Vilkovisky effective action produces the same result as the ordinary
definition of the effective action at on-shell level. In particular, both
formalisms generate the same elements for the $S$-matrix [39]. Therefore, one
might argue that if the $S$-matrix contains all the information about physical
observables, then the Vilkovisky effective action gives no new predictions. As
shown in [30], however, there are physical quantities (e.g., critical
temperatures) that in principle cannot be directly obtained from scattering
amplitudes and they depend on the choice of gauge and parametrization of the
quantum fields, if calculated using the standard effective action. In this
respect, the Vilkovisky effective action can give unambiguous results which
may be verified by experiments. The examples shown in the main part of this
work represent another application in which this formalism can be useful.

Secondly, there are other ways of constructing covariant effective actions
besides (14). In particular, a generalization of the unique effective action
was presented by DeWitt [26], which is usually called “Vilkovisky–DeWitt
effective action”. This more intricate definition coincides with (14) in the
one-loop approximation, and has the advantage of yielding a perturbative
expansion in terms of one-particle irreducible diagrams (which was a major
problem in Vilkovisky’s original proposal) [45, 39, 40]. Since the
considerations in the present work are based on the one-loop renormalization
group equations (and in Sec. 3 we argued why higher-loop effects are
suppressed [29]), the definition (14) is sufficient for our purposes, and our
results also hold within the more elaborate construction by DeWitt.

Last but not least, a common reservation about the use of the
Vilkovisky–DeWitt effective action when applied specifically to gravitational
theories is because the metric in the space of fields (21) is not uniquely
defined without the prescription (iii). Notice that (iii) is not needed for
the invariance of the Vilkovisky–DeWitt effective action. Indeed, by using
just (21) it is possible to construct a whole family of effective actions
parametrized by the arbitrary constant $a$, each of them being gauge and
parametrization invariant. Vilkovisky’s prescription fixes $a$ in an uniquely
way which only depends on the gravitational theory in question. Whether it is
the correct prescription is still an open question that might be decided only
at the experimental level (in this regard, see also [30, 46]).

## Acknowledgments

I.Sh. is grateful to A. Belyaev for useful discussions of the particle physics
aspects of the restrictions on the SSB. The work of I.Sh. is partially
supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico -
CNPq under the grant 303635/2018-5, by Fundação de Amparo à Pesquisa de Minas
Gerais - FAPEMIG under the grant PPM-00604-18, and by Ministry of Education of
Russian Federation under the project No. FEWF-2020-0003.

## References

  * [1] I.L. Buchbinder and I.L. Shapiro, Introduction to Quantum Field Theory with Applications to Quantum Gravity, (Oxford University Press, 2021).
  * [2] P.J.E. Peebles and B. Ratra, The Cosmological Constant and Dark Energy, Rev. Mod. Phys. 75 (2003) 559, astro-ph/0207347.
  * [3] E.J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15 (2006) 1753, hep-th/0603057.
  * [4] V. Sahni, T.D. Saini, A.A. Starobinsky and U. Alam, Statefinder: A New geometrical diagnostic of dark energy, JETP Lett. 77 (2003) 201, astro-ph/0201498.
  * [5] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 (1989) 1.
  * [6] I.L. Shapiro, J. Solà, Scaling behavior of the cosmological constant: Interface between quantum field theory and cosmology, JHEP 02 (2002) 006, hep-th/0012227.
  * [7] L.S. Brown, Quantum Field Theory, (Cambridge University Press, Cambridge, 1994).
  * [8] B.L. Nelson and P. Panangaden, Scaling behavior of interacting quantum fields in curved space-time, Phys. Rev. D25 (1982) 1019.
  * [9] E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B201 (1982) 469.
  * [10] I.L. Shapiro, Effective action of vacuum: semiclassical approach, Class. Quant. Grav. 25 (2008) 103001, arXiv:0801.0216.
  * [11] A. Babic, B. Guberina, R. Horvat, and H. Stefancic, Renormalization group running of the cosmological constant and its implication for the Higgs boson mass in the standard model, Phys. Rev. D65 (2002) 085002, hep-ph/0111207.
  * [12] I.L. Shapiro, J. Solà, On the possible running of the cosmological ’constant’, Phys. Lett. B682 (2009) 105, hep-th/0910.4925.
  * [13] R. Opher and A. Pelinson, Studying the decay of the vacuum energy with the observed density fluctuation spectrum, Phys. Rev. D 70 (2004) 063529, astro-ph/0405430.
  * [14] I.L. Shapiro, J. Solà and H. Stefancic, Running G and Lambda at low energies from physics at M(X): Possible cosmological and astrophysical implications, JCAP 01 (2005) 012, hep-ph/0410095.
  * [15] A. Babic, B. Guberina, R. Horvat, and H. Stefancic, Renormalization-group running cosmologies. A Scale-setting procedure, Phys. Rev. D71 (2005) 124041, astro-ph/0407572.
  * [16] T. Appelquist and J. Carazzone, Infrared Singularities and Massive Fields, Phys. Rev. D11 (1975) 2856.
  * [17] E.V. Gorbar and I.L. Shapiro, Renormalization group and decoupling in curved space, JHEP 02 (2003) 021, hep-ph/0210388; Renormalization group and decoupling in curved space: II. The Standard Model and Beyond, JHEP 06 (2003) 004, hep-ph/0303124.
  * [18] J. Grande, J. Solà, J.C. Fabris and I.L. Shapiro, Cosmic perturbations with running G and Lambda, Class. Quant. Grav. 27 (2010) 105004, arXiv:1001.0259.
  * [19] J.A. Agudelo Ruiz, J.C. Fabris, A.M. Velasquez-Toribio and I.L. Shapiro, Constraints from observational data for a running cosmological constant and warm dark matter with curvature, Grav. Cosmol. 26 (2020) 316, arXiv:2007.12636.
  * [20] A. Accioly, B.L. Giacchini and I.L. Shapiro, On the gravitational seesaw in higher-derivative gravity, Eur. Phys. J. C77 (2017) 540, gr-qc/1604.07348.
  * [21] J.F. Donoghue, The effective field theory treatment of quantum gravity, AIP Conf. Proc. 1483 (2012) 73, arXiv:1209.3511.
  * [22] C.P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel. 7 (2004) 5, gr-qc/0311082.
  * [23] D.A.R. Dalvit and F.D. Mazzitelli, Geodesics, gravitons and the gauge fixing problem, Phys. Rev. D56 (1997) 7779, hep-th/9708102.
  * [24] T. de Paula Netto, I.L. Shapiro, and L. Modesto, Universal leading quantum correction to the Newton potential, Eur. Phys. J. C82 (2022) 160, arXiv:2110.14263.
  * [25] G.A. Vilkovisky, The unique effective action in quantum field theory, Nucl. Phys. B234 (1984) 125.
  * [26] B.S. DeWitt, The effective action, in Quantum Field Theory and Quantum Statistics, Essays in honor of the sixtieth birthday of E.S. Fradkin, edited by C.J. Isham, I.A. Batalin and G.A. Vilkovisky, (Hilger, Bristol, 1987).
  * [27] T. Taylor and G. Veneziano, Quantum gravity at large distances and the cosmological constant, Nucl. Phys. B345 (1990) 210.
  * [28] B.L. Giacchini, T. de Paula Netto and I. L. Shapiro, Vilkovisky unique effective action in quantum gravity, Phys. Rev. D102 (2020) 106006, arXiv:2006.04217.
  * [29] B.L. Giacchini, T. de Paula Netto and I.L. Shapiro, On the Vilkovisky-DeWitt approach and renormalization group in effective quantum gravity, JHEP 10 (2020) 011, arXiv:2009.04122.
  * [30] K. Kobes, G. Kunstatter and D.J. Toms, The Vilkovisky-DeWitt Effective Action: Panacea or Placebo?, in TeV Physics: 12th Johns Hopkins Workshop on Current Problems in Particle Theory, edited by G. Domokos and S. Kovesi-Domokos, (World Scientific, Singapore, 1988).
  * [31] S. Weinberg, Anthropic bound on the cosmological constant, Phys. Rev. Lett. 59 (1987) 2607.
  * [32] I.Ya. Arefeva, A.A. Slavnov and L.D. Faddeev, Generating functional for the $S$-matrix in gauge-invariant theories, Theor. Math. Phys. 21 (1975) 1165 [Version in Russian: Teor. Mat. Fiz. 21 (1974) 311].
  * [33] B.L. Voronov, P.M. Lavrov and I.V. Tyutin, Canonical transformations and the gauge dependence in general gauge theories, Sov. J. Nucl. Phys. 36 (1982) 498.
  * [34] I.L. Shapiro and A.G. Jacksenaev, Gauge dependence in higher derivative quantum gravity and the conformal anomaly problem, Phys. Lett. B324 (1994) 286.
  * [35] J.D. Gonçalves, T. de Paula Netto and I.L. Shapiro, On the gauge and parametrization ambiguity in quantum gravity, Phys. Rev. D97 (2018) 026015, arXiv:1712.03338.
  * [36] A.O. Barvinsky and G.A. Vilkovisky, The generalized Schwinger-DeWitt technique in gauge theories and quantum gravity, Phys. Repts. 119 (1985) 1.
  * [37] S.R. Huggins, G. Kunstatter, H.P. Leivo and D.J. Toms, The Vilkovisky-de Witt effective action for quantum gravity, Nucl. Phys. B301 (1988) 627.
  * [38] E.S. Fradkin and A.A. Tseytlin, On the new definition of off-shell Effective Action, Nucl. Phys. B234 (1984) 509.
  * [39] A. Rebhan, Feynman Rules and $S$-Matrix Equivalence of the Vilkovisky-deWitt Effective Action, Nucl. Phys. B298 (1988) 726.
  * [40] P. Ellicott and D. Toms, On the New Effective Action in Quantum Field Theory, Nucl. Phys. B312 (1989) 700.
  * [41] L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B13 (1982) 33.
  * [42] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965).
  * [43] J.L. Synge, Relativity: the general theory (North-Holland, Amsterdam, 1960).
  * [44] G. Kunstatter, Vilkovisky’s Unique Effective Action: An introduction and explicit calculation, in Super Field Theories, proceedings of NATO Advanced Research Workshop on Superfield Theories, edited by H.C. Lee, V. Elias, G. Kunstatter, R.B. Mann and K.S. Viswanathan, NATO ASI Series B, Vol. 160 (Plenum, New York, 1987).
  * [45] A. Rebhan, The Vilkovisky-DeWitt Effective Action and its application to Yang-Mills Theories, Nucl. Phys. B288 (1987) 832.
  * [46] G.A. Vilkovisky, Effective action in quantum gravity, Class. Quant. Grav. 9 (1992) 895.