Renormalization of the nonprojectable Hořava theory

Jorge Bellorín1,a, Claudio Bórquez2,b and Byron Droguett1,c

1Department of Physics, Universidad de Antofagasta, 1240000 Antofagasta,
Chile.

2Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián,
Lago Panguipulli 1390, Puerto Montt, Chile.

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

> We present the proof of renormalization of the Hořava theory, in the
> nonprojectable version. We obtain a form of the quantum action that exhibits
> a manifest BRST-symmetry structure. Previous analysis have shown that the
> divergences produced by irregular loops cancel completely between them. The
> remaining divergences are local. The renormalization is achieved by using
> the approach developed by Barvinsky et al. with the background-field
> formalism.

##  1 Introduction

In this paper we present the proof of renormalization of the Hořava theory,
considering its nonprojectable version [1, 2]. This theory, whose gauge group
is given by the foliation-preserving diffeomorphisms (FDiff), is a proposal
for a quantum theory of gravitation. The theory is unitary [3], we are
presenting here its renormalization, and it might yield the classical dynamics
of general relativity at the large-distance limit.

We base the proof on three main aspects. First, the theory is quantized under
the Batalin-Fradkin-Vilkovisky (BFV) formalism, incorporating the second-class
constraints [4, 5, 6]. This formalism allows us to introduce a local gauge-
fixing condition which leads to regular propagators for most of the fields [7,
8], together with the measure of the second-class constraints [9, 10]. After
the integration on some ghost fields and the redefinition of the Becchi-Rouet-
Stora-Tyutin (BRST) symmetry transformations, we get a form of the quantum
action with manifest BRST-symmetry structure. Second, it is known from
previous studies [11, 12] that the only divergences produced by the
integration along the frequency (called irregular loops) cancel exactly
between them. In the integration on the spatial momentum, the behavior of the
irregular propagators is equivalent to the regular ones. Hence, all
divergences are local [13, 14]. The highest superficial degree of divergence
is equal to the order of the bare Lagrangian. Third, we use the approach
developed by Barvinsky et al. [15] to undertake the renormalization of gauge
theories, which is based on the background-field formalism [16, 17].

The proof of renormalization of the projectable case, which is a version of
the Hořava theory defined by the restriction that the lapse function depends
exclusively on time, is known [7, 15]. The need for an anisotropic gauge-
fixing condition that leads to regular propagators was identified in this
case. The resulting Lagrangian is local when it is expressed in terms of
canonical variables. However, a fundamental difference in the quantization of
the projectable and nonprojectable cases is the absence of second-class
constraints in the former. In the nonprojectable case, a similar quantization
can be carried on, with the analogous gauge-fixing condition. The second-class
constraints lead to a modification of the measure of the path integral. The
measure has the effect of yielding irregular propagators on some auxiliary
fields, despite the fact that the rest of quantum fields acquire regular
propagators. Since the regular structure is important for the control of
divergences [14], a careful study of the consequences of the irregular
propagators is required. For this reason it becomes essential the previously
mentioned analysis showing that, not only the divergences produced by the
irregular loops are canceled, but also the fact that these are the only
divergences in the direction of the frequency.

As most of the modern approaches of renormalization of gauge theories, the
proof relies on the BRST symmetry and the background-field formalism. The
Slavnov-Taylor and Ward identities are useful to determine the divergences of
the effective action. On the other hand, the BFV quantization is based on the
Hamiltonian formalism. Therefore, it is important to arrive at a form of the
quantum action being separated in the usual way: a sector invariant under the
FDiff gauge symmetry, and other sector fixing the gauge symmetry by means of
the BRST operator. We find such a BRST-symmetry structure. This allows us to
apply the background-field formalism of Ref. [15].

The analysis of renormalization requires to know the propagators. With the
vertices, the feature that we require is the highest order in spatial
derivatives, independently of their explicit form. For this reason, in this
study we need only to write explicitly the higher order terms in the
Lagrangian that contribute to the propagators.

##  2 BFV Quantization of the Hořava theory

The initial assumption in the definition of he Hořava theory is the existence
of a foliation of spatial slices along a given direction of time with absolute
meaning. In the classical theory, the fields representing the gravitational
interaction are the Arnowitt-Deser-Misner (ADM) field variables
$N(t,\vec{x})$, $N^{i}(t,\vec{x})$ and $g_{ij}(t,\vec{x})$. We deal only with
the nonprojectable version, on which the lapse function $N$ can depend on time
and space. The corresponding gauge symmetry is given by the FDiff. In terms of
a given coordinate system $(t,\vec{x})$, the FDiff acts infinitesimally as
$\delta t=f(t)$ and $\delta x^{i}=-\zeta^{i}(t,\vec{x})$. In the quantum
theory we impose asymptotically-flat boundary conditions. Since $f(t)$ is
independent of the spatial point, the compatibility with the boundary
condition requires the restriction $f(t)=0$. After this restriction, the FDiff
transformations, which we denote by $\delta_{\zeta}$, are given by111The signs
of the FDiff transformations are the opposite of the standard diffeomorphisms.

$\displaystyle\delta_{\zeta}N=\zeta^{k}\partial_{k}N\,,$ (2.1)
$\displaystyle\delta_{\zeta}N^{i}=\zeta^{k}\partial_{k}N^{i}-N^{k}\partial_{k}\zeta^{i}+\dot{\zeta}^{i}\,,$
(2.2)
$\displaystyle\delta_{\zeta}g_{ij}=\zeta^{k}\partial_{k}g_{ij}+2g_{k(i}\partial_{j)}\zeta^{k}\,.$
(2.3)

It is important to compare with the spatial diffeomorphisms, since many
variables behave as spatial tensors that evolve in time, as the case of $N$,
$g_{ij}$, and the arbitrary parameter $\zeta^{i}$ itself. For the case of a
time-dependent spatial tensor $T^{ij\cdots}$, its FDiff transformation is
functionally identical to a spatial diffeomorphism:

$\delta_{\zeta}T^{ij\cdots}=\zeta^{k}\partial_{k}T^{ij\cdots}-T^{kj\cdots}\partial_{k}\zeta^{i}-T^{ik\cdots}\partial_{k}\zeta^{j}-\cdots\,,$
(2.4)

and the analogous standard formula for the case of a time-dependent spatial
tensor density. Throughout this study, the term FDiff gauge symmetry of the
Hořava theory refers to the transformations (2.1) – (2.3), and (2.4) for the
case of time-dependent spatial tensors (with the extension for densities).
Among the ADM variables, only the shift vector $N^{i}$ has a FDiff
transformation that is functionally different to a spatial diffeomorphism.

The primary classical Hamiltonian of the nonprojectable Hořava theory is given
by [18, 19, 20]

$H_{0}=\int
d^{3}x\sqrt{g}N\left(\frac{\pi^{ij}\pi_{ij}}{g}+\bar{\sigma}\frac{\pi^{2}}{g}+\mathcal{V}\right)\,.$
(2.5)

The classical canonical conjugate pairs are $(g_{ij},\pi^{ij})$ and $N$ with
its conjugate momentum. We denote the trace $\pi\equiv g^{ij}\pi_{ij}$. The
canonical momentum of $N$ is zero due to the constraints of the theory; we
discard it from the phase space. $\mathcal{V}=\mathcal{V}[g_{ij},a_{i}]$,
where $a_{i}=\partial_{i}N/N$, is called the potential. It contains all the
terms with spatial derivatives that are compatible with the FDiff gauge
symmetry, including the higher-order ones characteristic of the Hořava theory,
which in the $(3+1)$-dimensional theory are of sixth order. For the evaluation
of the propagators we require all the sixth-order terms that contribute to the
second order in perturbations, which are [21] 222The coupling constants of the
theory are $\lambda,\alpha_{3},\alpha_{4},\beta_{3},\beta_{4}$. We use the
shorthand $\bar{\sigma}=\lambda/(1-3\lambda)$.

$\mathcal{V}=-\alpha_{3}\nabla^{2}R\nabla_{i}a^{i}-\alpha_{4}\nabla^{2}a_{i}\nabla^{2}a^{i}-\beta_{3}\nabla_{i}R_{jk}\nabla^{i}R^{jk}-\beta_{4}\nabla_{i}R\nabla^{i}R\,.$
(2.6)

The BFV quantization requires to identify the constraints that are involutive
under Dirac brackets [4, 5, 6]. In the case of the Hořava theory this is the
momentum constraint $\mathcal{H}_{i}=-2g_{ij}\nabla_{k}\pi^{kj}=0$. The
second-class constraints are given by the vanishing of the momentum conjugate
to $N$, which we have already considered as solved, and the constraint

$\begin{split}\theta_{1}\equiv\frac{N}{\sqrt{g}}\left(\pi^{ij}\pi_{ij}+\bar{\sigma}\pi^{2}\right)+\sqrt{g}N\mathcal{V}-\alpha_{3}\sqrt{g}\nabla^{2}(N\nabla^{2}R)+2\alpha_{4}\sqrt{g}\nabla^{i}\nabla^{2}(N\nabla^{2}a_{i})=0\,.\end{split}$
(2.7)

The primary Hamiltonian (2.5) is equivalent to the integral of this second-
class constraint, $H_{0}=\int d^{3}x\,\theta_{1}$. The BFV quantization adds
the canonical pair $(N^{i},\pi_{i})$ and the BFV ghost pairs
$(C^{i},\bar{\mathcal{P}}_{i})$, $(\bar{C}_{i},\mathcal{P}^{i})$. The quantum
action of the BFV path integral takes the form

$S=\int
dtd^{3}x\left(\pi^{ij}\dot{g_{ij}}+\pi_{i}\dot{N}^{i}+\bar{\mathcal{P}}_{i}\dot{C}^{i}+\mathcal{P}^{i}\dot{\bar{C}}_{i}-\mathcal{H}_{\Psi}+\mathcal{A}\theta_{1}-\bar{\eta}\frac{\delta\theta_{1}}{\delta
N}\eta\right)\,.$ (2.8)

The integration includes the auxiliary fields $\mathcal{A},\eta,\bar{\eta}$,
where $\mathcal{A}$ is a bosonic scalar and $\eta,\bar{\eta}$ is a pair of
scalar ghosts. The last two terms of the action (2.8) containing these
auxiliary fields are the contribution of the measure of the second-class
constraints (details can be found in Ref. [3]). The gauge-fixed Hamiltonian
density is defined by
$\mathcal{H}_{\Psi}=\mathcal{H}_{0}+\\{\Psi,\Omega\\}_{\text{D}}$, where
$\Omega$ is the generator of the BRST symmetry, given by

$\Omega=\int
d^{3}x\left(\mathcal{H}_{k}C^{k}+\pi_{k}\mathcal{P}^{k}-C^{k}\partial_{k}C^{l}\bar{\mathcal{P}}_{l}\right)\,.$
(2.9)

$\Psi$ is a gauge fermion and $\\{\,,\\}_{\text{D}}$ indicates Dirac brackets.

For the gauge fixing we may use the original BFV structure of the gauge
fermion [8, 22]. The gauge-fixing condition has the general form
$\dot{N}^{i}-\chi^{i}=0$, and its associated gauge fermion is
$\Psi=\bar{\mathcal{P}}_{i}N^{i}+\bar{C}_{i}\chi^{i}$, where $\chi^{i}$ is a
factor that must be chosen. To write $\chi^{i}$ explicitly, we introduce
perturbative variables around a flat background. The perturbation of the
metric tensor is denoted by $g_{ij}=\delta_{ij}+h_{ij}$. We choose $\chi^{i}$
to be the local expression333The coefficients in (2.10) are chosen to simplify
the resulting propagators (see Refs. [7, 8]).

$\chi^{i}=\rho\mathfrak{D}^{ij}\pi_{j}-2\rho\Delta^{2}\partial_{j}h_{ij}+2\rho\lambda(1+\kappa)\Delta^{2}\partial_{i}h-2\kappa\rho\Delta\partial_{i}\partial_{j}\partial_{k}h_{jk}\,,$
(2.10)

where
$\mathfrak{D}^{ij}=\delta_{ij}\Delta^{2}+\kappa\Delta\partial_{i}\partial_{j}$,
$\Delta=\partial_{k}\partial_{k}$ and $\rho,\kappa$ are independent constants.

##  3 The BRST-symmetry structure

In the BFV formalism, the BRST symmetry transformations on the canonical
fields are generated by $\Omega$, according to the rule
$\delta_{\Omega}\Phi=\\{\Phi\,,\Omega\\}_{\text{D}}\,\epsilon$, where
$\epsilon$ is the fermionic parameter of the transformation. The auxiliary
fields $\mathcal{A},\eta,\bar{\eta}$ are not canonical. We define their BRST
transformation in such a way that the measure is left invariant. The required
transformations are FDiff along $C^{i}\epsilon$:
$\delta_{\Omega}\mathcal{A}=\delta_{C\epsilon}\mathcal{A}$,
$\delta_{\Omega}\eta=\delta_{C\epsilon}\eta$, and
$\delta_{\Omega}\bar{\eta}=\delta_{C\epsilon}\bar{\eta}$.

On the action (2.8) we perform the integration on the ghost fields
$\mathcal{P}^{i},\bar{\mathcal{P}}_{i}$. The resulting action can be grouped
in two sectors, $S=S_{0}[\varphi^{a}]+S_{\Omega}$, where

$\displaystyle S_{0}[\varphi^{a}]=\int
dtd^{3}x\left(\pi^{ij}\dot{g_{ij}}-\mathcal{H}_{0}-N^{i}\mathcal{H}_{i}+\mathcal{A}\theta_{1}-\bar{\eta}\frac{\delta\theta_{1}}{\delta
N}\eta\right)\,,$ (3.1) $\displaystyle S_{\Omega}=\int
dtd^{3}x\left[\pi_{i}\left(\dot{N}^{i}-\chi^{i}\right)-\dot{\bar{C}}_{i}\left(\dot{C}^{i}+C^{j}\partial_{j}N^{i}-N^{j}\partial_{j}C^{i}\right)-\bar{C}_{i}\\{\chi^{i},\mathcal{H}_{j}\\}C^{j}\right]\,.$
(3.2)

$S_{0}[\varphi^{a}]$ depends exclusively on the set of fields
$\varphi^{a}=\\{g_{ij},\pi^{ij},N,N^{i},\mathcal{A},\eta,\bar{\eta}\\}$. At
this point it is useful to clarify that all the fields of the quantum theory
transform as time-dependent spatial tensors/densities under FDiff
transformations, except for $N^{i}$. Moreover, $S_{0}$ is invariant under
arbitrary FDiff gauge transformations. We remark that, for a FDiff
transformation with a time-dependent vector parameter $\zeta^{i}$, the first
and third terms of (3.1) combine themselves to cancel time derivatives of
$\zeta^{i}$,
$\delta_{\zeta}\int\left(\pi^{ij}\dot{g_{ij}}-N^{i}\mathcal{H}_{i}\right)=0$,
as it is well known from the ADM formulation of general relativity. The rest
of terms in (3.1) contain no time derivatives and are independent of $N^{i}$.
Their invariance under FDiff is automatic since they are written totally in
terms of spatial tensors/densities. In contrast, $S_{\Omega}$ is the gauge-
fixing sector of this symmetry.

As a consequence of the previous integration, the BRST symmetry must be
revised. Specifically, the transformation of $N^{i}$ is affected since the BFV
rule yields $\delta_{\Omega}N^{i}=\mathcal{P}^{i}\epsilon$. The second term in
(3.2) is key to unfold the new transformation, since it has the form of a
FDiff (2.2) along $C^{i}$. Therefore, we define the new BRST transformation of
$N^{i}$ to be the FDiff
$\delta_{\Omega}N^{i}=\delta_{C\epsilon}N^{i}=\left(C^{j}\partial_{j}N^{i}-N^{j}\partial_{j}C^{i}+\dot{C}^{i}\right)\epsilon$.
This transformation is nilpotent. After the (re)definitions we have done, it
turns out that the BRST transformation of all the $\varphi^{a}$ fields
corresponds to a FDiff along $C^{i}\epsilon$. Therefore, the BRST invariance
of $S_{0}[\varphi^{a}]$ is automatic.

The quantum action (3.1) – (3.2) can be written in the standard notation of
the BRST symmetry. We denote by $\boldsymbol{s}$ the BRST operator. The action
of $\boldsymbol{s}$ on the $\varphi^{a}$ fields is a FDiff transformation with
vector parameter equal to $C^{i}$. $\bar{C}_{i}$ and $\pi_{i}$ are the usual
auxiliary fields of the BRST symmetry. The action of the BRST operator is

$\boldsymbol{s}\varphi^{a}=\delta_{C}\varphi^{a}\,,\quad\boldsymbol{s}C^{i}=-C^{j}\partial_{j}C^{i}\,,\quad\boldsymbol{s}\bar{C}_{i}=\pi_{i}\,,\quad\boldsymbol{s}\pi_{i}=0\,.$
(3.3)

The sector $S_{\Omega}$ (3.2) is equal to the action of the BRST operator on a
gauge fermion, $S_{\Omega}=\int\boldsymbol{s}\tilde{\Psi}$, where

$\tilde{\Psi}=\bar{C}_{i}\left(\dot{N}^{i}-\chi^{i}\right)\,,$ (3.4)

and $\chi^{i}$ is given in (2.10). Therefore, the quantum action (3.1) – (3.2)
has the BRST-invariant form

$S=S_{0}[\varphi^{a}]+\int dtd^{3}x\boldsymbol{s}\tilde{\Psi}\,.$ (3.5)

We highlight that the whole Lagrangian is completely local.

##  4 Propagators and locality of divergences

The propagators can be calculated from the action (3.1) – (3.2), expanded at
second order in perturbations. We obtain two classes of propagators: the
regular and the irregular propagators. The regularity condition is appropriate
for the study of ultraviolet divergences in Lorentz-violating theories
[14].444Throughout this study we assume that infrared divergences have been
regularized. The gauge condition (2.10) is intended to get regular propagators
for the quantum fields [7, 8]. Nevertheless, the persistence of irregular
propagators in the nonprojectable theory demands a careful study of the
divergences [11, 12]. Specifically, the irregular propagators are associated
with the fields $\mathcal{A},\eta,\bar{\eta}$. We remark that this effect is a
consequence of the measure of the second-class constraints.

The most important feature of the regular propagators of this theory is that
they are given in terms of products of the four factors (in Fourier space
$(\omega,\vec{k})$, after a Wick rotation):

$\mathcal{T}_{A}=\frac{1}{\omega^{2}+\sigma_{A}k^{6}}\,,\quad A=1,2,3,4\,,$
(4.1)

where $\sigma_{A}$ are combinations of the coupling constants, that must
satisfy the condition $\sigma_{A}>0$ in order to maintain the definition of
regular propagator. There are other polynomials in the numerators of the
regular propagators; their growth at the ultraviolet is subordinated by the
factors $\mathcal{T}_{A}$. On the other hand, the three irregular propagators
are given by

$\langle\mathcal{A}\mathcal{A}\rangle=\langle\mathcal{A}n\rangle=\langle\bar{\eta}\eta\rangle=-\frac{1}{\alpha_{4}k^{6}}\,.$
(4.2)

These propagators are independent of $\omega$, violating the condition of
regularity in the time direction. But, as long as $\alpha_{4}\neq 0$, they
maintain a strict regular dependence on the spatial momentum $\vec{k}$.

In the action (3.1) – (3.2), time derivatives arise uniquely in terms that are
of second order in perturbations. As a consequence, vertices do not depend on
the frequency $\omega$. Hence, for the integration on $\omega$ we only need to
consider propagators. We call irregular loop to a loop formed completely with
the irregular propagators (4.2). Since these propagators do not depend on
$\omega$, an irregular loop produces a divergence of the kind $\sim\int
d\omega$. Such a divergence multiplies any diagram containing (at least) one
irregular loop. In previous analysis [11, 12], we have shown that all the
diagrams with irregular loops cancel completely between them. Moreover, this
is the only divergence produced by the integration on $\omega$, due to the
fact that the regular propagators automatically render the integration on
$\omega$ finite. Let us suppose first a loop composed completely of regular
propagators. The regular propagators with the lowest scaling in $\omega^{-1}$
are of order $\sim\omega^{-1}$. If the loop consists only of one propagator of
this kind, then the integral is zero since these propagators are odd in
$\omega$. The next order is a product of two propagators of this kind, or a
single propagator with scaling $\sim\omega^{-2}$. In both cases the integral
in $\omega$ is finite. By increasing the number of regular propagators, the
convergence in the integration on $\omega$ becomes faster. Now consider the
presence of one or more irregular propagators (4.2) in the loop, but not all
since we know that irregular loops cancel completely. Since the irregular
propagators are independent of $\omega$, the analysis of the integration on
$\omega$ is identical to the previous case of a loop made exclusively of
regular propagators.

In the integration on spatial momentum $k^{i}$, all the propagators have a
regular structure on this variable, including the ones of the fields
$\mathcal{A},\eta,\bar{\eta}$. According to the analysis of Lorentz-violating
theories [13, 14], the locality of divergences produced by the integration on
$k^{i}$ is ensured.

On the basis of the scaling of propagators and the maximal number of spatial
derivatives in the vertices, we may compute the superficial degree of diverge
$D_{\text{div}}$. The diagrams with the highest divergence (those without
external legs for $N^{i}$ and $\pi^{ij}$, and no spatial derivatives on the
external legs) have $D_{\text{div}}=6$. This order is equal to the order of
the bare Lagrangian, in agreement with the power-counting criterium used in
the formulation of the classical theory.

##  5 The background fields and the renormalization

The aim of introducing background fields is to get a background-gauge symmetry
in the gauge-fixed quantum action. This symmetry transforms simultaneously the
quantum fields $\varphi^{a}$ and the background fields $\phi^{a}$ in the form
of the original FDiff gauge transformations (2.1) – (2.4), with the same
parameter for both classes of fields. Specifically, one requires to handle the
subset of fields $\varphi^{a}$ involved in the gauge-fixing condition, in
terms of the linear combination $\varphi^{a}-\phi^{a}$. In our case, we
require to introduce background fields only for $g_{ij}$ and $N^{i}$, which we
denote by $\bar{g}_{ij}$ and $\bar{N}^{i}$ respectively (hence
$\phi^{a}=\\{\bar{g}_{ij},\bar{N}^{i}\\}$). We use a notation for the
difference of fields: $h_{ij}=g_{ij}-\bar{g}_{ij}$ and
$n^{i}=N^{i}-\bar{N}^{i}$.555Not to confuse $h_{ij}$ with the variable of
section 2. Due to the linearity of the gauge transformations on the parameter
and the fields, $h_{ij}$ and $n^{i}$ transform exactly as time-dependent
spatial tensors under background-gauge transformations.

The gauge fermion (3.4) is replaced by a background-dependent one,

$\begin{split}\Psi_{\text{bg}}=\,&\bar{C}_{i}\left(D_{t}n^{i}-\rho\Theta^{ijk}h_{jk}-\rho\mathcal{D}^{ij}(\pi_{j}/\sqrt{\bar{g}})\right)-\mathbb{T}^{ij}h_{ij}-\mathbb{K}_{ij}\pi^{ij}-\mathbb{T}_{i}n^{i}\\\\[4.30554pt]
&-\mathbb{T}N-\mathbb{S}\mathcal{A}-\bar{\mathbb{N}}\eta-\bar{\eta}\mathbb{N}+\bar{J}_{i}C^{i}\,,\end{split}$
(5.1)

where666In the definition of the operator $\Theta^{ijk}$ we have chosen the
simplest combination of fifth-order covariant derivatives that reproduces the
flat case (2.10). Unitarity requires the operator $\mathcal{D}^{ij}$ be
invertible.

$\displaystyle
D_{t}n^{i}=\dot{n}^{i}-\bar{N}^{k}\bar{\nabla}_{k}n^{i}+n^{k}\bar{\nabla}_{k}\bar{N}^{i}\,,$
(5.2)
$\displaystyle\Theta^{ijk}=-2\bar{g}^{ij}\bar{\nabla}^{4}\bar{\nabla}^{k}+2\lambda(1+\kappa)\bar{g}^{jk}\bar{\nabla}^{4}\bar{\nabla}^{i}-2\kappa\bar{\nabla}^{2}\bar{\nabla}^{i}\bar{\nabla}^{j}\bar{\nabla}^{k}\,,$
(5.3)
$\displaystyle\mathcal{D}^{ij}=\bar{g}^{ij}\bar{\nabla}^{4}+\kappa\bar{\nabla}^{2}\bar{\nabla}^{i}\bar{\nabla}^{j}\,.$
(5.4)

All indices are raised and lowered with the background metric $\bar{g}_{ij}$,
and $\bar{\nabla}$ is its covariant derivative. In Eq. (5.1) we have inserted
external sources for the BRST transformations of the $(\varphi^{a}-\phi^{a})$
fields. We denote these sources collectively by
$\gamma_{a}=\\{\mathbb{T}^{ij},\mathbb{K}_{ij},\mathbb{T}_{i},\mathbb{T},\mathbb{S},\bar{\mathbb{N}},\mathbb{N}\\}$,
whereas $\bar{J}_{i}$ is the source for $\boldsymbol{s}C^{i}$. All these
sources transform as time-dependent spatial tensors/densities under FDiff.
$D_{t}n^{i}$ transforms as a spatial vector under background-gauge
transformations. The operators $\Theta^{ijk}$ and $\mathcal{D}^{ij}$ are made
completely of spatial covariant derivatives; hence $\Psi_{\text{bg}}$ is
invariant under background-gauge transformations.

To write the action in the background-field approach, one introduces the
operator
$\boldsymbol{Q}=\boldsymbol{s}+\Omega^{a}\frac{\delta}{\delta\phi^{a}}$, where
$\Omega^{a}=\\{\Omega_{ij},\Omega^{i}\\}$ are external Grassmann fields.
$\boldsymbol{Q}$ is a nilpotent operator. The quantum gauge-fixed action in
the presence of background fields takes the form

$\Sigma_{0}=S_{0}[\varphi^{a}]+\int dtd^{3}x\boldsymbol{Q}\Psi_{\text{bg}}\,.$
(5.5)

By following standard procedures, we may compute the identities on the
effective action $\Gamma$ due to the underlying gauge symmetry. These are the
Slavnov-Taylor identity, the Ward identity for the background-gauge symmetry,
and the field equation for the ghost field $\bar{C}_{i}$. $\Gamma$ is defined
in the standard way by means of a Legendre transformation on the generating
functional of connected diagrams $W$.

We collect the several results we have found here and in previous analysis:
the BRST-invariant form (3.5) of the quantum action, together with its
background-field extension (5.5); the completely local form of the gauge-fixed
Lagrangian; the regularity of all the propagators that do not involve the
fields $\mathcal{A},\eta,\bar{\eta}$; the cancellation of the irregular loops
formed by the propagators of these fields; the absence of divergences along
the $\omega$ direction, regardless of the presence of irregular propagators in
diagrams; the regular structure of all the propagators with respect to the
dependence on $k^{i}$, leading to the locality of the divergences, and the
superficial degree of divergence of all diagrams that is not greater than the
order of the bare Lagrangian. On the basis of these results, the
renormalization of the theory is achieved by following the procedure developed
in Ref. [15].

Under an inductive approach in the order in loops, in Ref. [15] it is found a
field redefinition that brings the action at $L$-th order in loops to the
BRST-invariant form

$\Sigma_{L}=S_{L}[\varphi^{a}]+\int dtd^{3}x\,\boldsymbol{Q}\Psi_{L}\,,$ (5.6)

where $S_{L}[\varphi^{a}]$ is a FDiff gauge invariant local functional. The
gauge fermion is invariant under background-gauge transformations and has the
form

$\Psi_{L}=\bar{C}_{i}\left(D_{t}n^{i}-\rho\Theta^{ijk}h_{jk}-\mathcal{D}^{ij}\pi_{j}\right)-\gamma_{a}(\varphi^{a}-\phi^{a})+\bar{J}_{i}C^{i}+\mathcal{O}(\hbar^{L+1})\,.$
(5.7)

In the generating functional $W$, the fields that couple to the external
sources, denoted by $\tilde{\varphi}^{a}$ and $\tilde{C}^{i}$, are given by
the gauge fermion in the form
$\tilde{\varphi}^{a}_{L}=\phi^{a}-\frac{\delta\Psi_{L}}{\delta\gamma_{a}}$ and
$\tilde{C}^{i}_{L}=\frac{\delta\Psi_{L}}{\delta\bar{J}_{i}}$. This functional
relationship is preserved by the field redefinition at the $L$-th order.

## References

  * [1] P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 084008 (2009) [arXiv:0901.3775 [hep-th]].
  * [2] D. Blas, O. Pujolas and S. Sibiryakov, Consistent Extension of Hořava Gravity, Phys. Rev. Lett. 104 181302 (2010) [arXiv:0909.3525 [hep-th]].
  * [3] J. Bellorín, C. Bórquez and B. Droguett, BRST symmetry and unitarity of the Hořava theory, Phys. Rev. D 107 044059 (2023) [arXiv:2212.14079 [hep-th]].
  * [4] E. S. Fradkin and G. A. Vilkovisky, Quantization of relativistic systems with constraints, Phys. Lett. B 55 224 (1975).
  * [5] I. A. Batalin and G. A. Vilkovisky, Relativistic S Matrix of Dynamical Systems with Boson and Fermion Constraints, Phys. Lett. B 69 309 (1977).
  * [6] E. S. Fradkin and T. E. Fradkina, Quantization of Relativistic Systems with Boson and Fermion First and Second Class Constraints, Phys. Lett. B 72 343 (1978).
  * [7] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov and C. F. Steinwachs, Renormalization of Hořava gravity, Phys. Rev. D 93 064022 (2016) [arXiv:1512.02250 [hep-th]].
  * [8] J. Bellorín, C. Bórquez and B. Droguett, Quantum Lagrangian of the Hořava theory and its nonlocalities, Phys. Rev. D 105 024065 (2022) [arXiv:2112.10295 [hep-th]].
  * [9] P. Senjanovic, Path Integral Quantization of Field Theories with Second Class Constraints, Annals Phys. 100 227 (1976) [erratum: Annals Phys. 209 248 (1991)].
  * [10] E. S. Fradkin, Acta Universitatis Wratislaviensis No. 207, in Proceedings of X-th Winter School of Theoretical Physics in Karpacz (Wydawnictwo Uniwersytetu Wroclawskiego Sp., Poland, 1973).
  * [11] J. Bellorín, C. Bórquez and B. Droguett, Cancellation of divergences in the nonprojectable Hořava theory, Phys. Rev. D 106 044055 (2022) [arXiv:2207.08938 [hep-th]].
  * [12] J. Bellorín, C. Bórquez and B. Droguett, Effective action of the Hořava theory: Cancellation of divergences, Phys. Rev. D 109 084007 (2024) [arXiv:2312.16327 [hep-th]].
  * [13] D. Anselmi and M. Halat, Renormalization of Lorentz violating theories, Phys. Rev. D 76 125011 (2007) [arXiv:0707.2480 [hep-th]].
  * [14] D. Anselmi, Weighted power counting and Lorentz violating gauge theories. I. General properties, Annals Phys. 324 874 (2009) [arXiv:0808.3470 [hep-th]].
  * [15] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov and C. F. Steinwachs, Renormalization of gauge theories in the background-field approach, JHEP 07 035 (2018) [arXiv:1705.03480 [hep-th]].
  * [16] B. S. DeWitt, Quantum Theory of Gravity. 2. The Manifestly Covariant Theory, Phys. Rev. 162 1195 (1967); Quantum Theory of Gravity. 3. Applications of the Covariant Theory, Phys. Rev. 162 1239 (1967).
  * [17] L. F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 33 (1982).
  * [18] J. Kluson, Note About Hamiltonian Formalism of Healthy Extended Hořava-Lifshitz Gravity, JHEP 07 038 (2010) [arXiv:1004.3428 [hep-th]].
  * [19] W. Donnelly and T. Jacobson, Hamiltonian structure of Hořava gravity, Phys. Rev. D 84 104019 (2011) [arXiv:1106.2131 [hep-th]].
  * [20] J. Bellorín and A. Restuccia, Consistency of the Hamiltonian formulation of the lowest-order effective action of the complete Hořava theory, Phys. Rev. D 84 104037 (2011) [arXiv:1106.5766 [hep-th]].
  * [21] M. Colombo, A. E. Gumrukcuoglu and T. P. Sotiriou, Hořava gravity with mixed derivative terms, Phys. Rev. D 91 044021 (2015) [arXiv:1410.6360 [hep-th]].
  * [22] J. Bellorín and B. Droguett, BFV quantization of the nonprojectable (2+1)-dimensional Hořava theory, Phys. Rev. D 103 064039 (2021) [arXiv:2102.04595 [hep-th]].