# One-loop Effective Action in Chiral Einstein-Cartan gravity

Pratik Chattopadhyay

Department of Physics, Namtech,
IIT Gandhinagar campus, Gandhinagar 382421, India

(16th October 2023)

###### Abstract

In the chiral Einstein-Cartan gravity, a new gauge fixing procedure is
implemented recently, leading to a very economical perturbation expansion of
the action. Using this formulation and the relevant gauge-fixing, we develop
the ghost Lagrangian on an arbitrary Einstein background using the BRST
formalism. The novelty is the appearance of a new term quadratic in the tetrad
field. We next compute the heat-kernel coefficients and understand the
divergences arising in the gravitational one-loop effective action. In our
computation the arising heat kernel coefficients depend only on the self-dual
part of the Weyl curvature. We make a comparison between our results and what
has been obtained for metric GR.

##  1 Introduction

One of the major motivations of alternative formulations of General Relativity
(GR) is to simplify computations at the perturbative level. In almost all such
formulations, one considers more than one independent fields in contrast to
just the metric field in usual Einstein gravity. The equivalence of such
formulations to metric GR is shown by integrating out the additional fields by
using their equations of motion and putting them back to the action. In the
tetradic Palatini action [1], the tetrad and spin connection become
independent variables and the metric is understood as a second-order
construct. This gives rise to a polynomial action in the absence of the
cosmological constant. If one includes the cosmological constant, the action,
however, does not remain polynomial. The Einstein-Cartan first-order
formulation is the one which keeps the action polynomial when the cosmological
constant is nonzero, see [5]. A Hamiltonian analysis of first order Einstein-
Cartan has been carried out in [6], [7]. The Hamiltonian construct is rather
complicated and lead to second class constraints. Further, there has been one-
loop computations in this theory in the presence of fermions and the
cosmological constant [8]. When fermions are present, it leads to torsion and
this enables to enrich the theory by adding a Holst term to the action.
Inspite of successfully extending usual gravity in the presence of torsion,
the first order Einstein-Cartan suffers from its complicated character of the
Hamiltonian $(3+1)$ split. Further, at the perturbative level, one has to deal
with three propagators which significantly complicate any such perturbative
computation. All these difficulties are resolved in the chiral first-order
formulation. The original idea of a chiral reformulation of gravity was noted
by Plebanski, see [9] and was rediscovered by Capovilla et al [10]. In the
chiral Einstein-Cartan as is formulated in [15], one considers self-dual
projection of the curvature of the spin-connection. One can then construct an
action with just the self-dual projection instead of the full curvature. The
difference between the two is a total derivative term (Holst term), which does
not change the dynamics of the theory. Thus this action is equivalent to the
first-order Einstein-Cartan action. Further, one can rewrite the self-dual
part of the curvature of the spin connection as the curvature of the self-dual
part of the spin connection. This then gives rise to a better perturbation
theory where one of the propagators, namely the propagator of the connection
with itself, vanishes. Thus the algebraic complexity gets reduced
significantly in quantum computations.

All known reformulations of GR share the feature of non-renormalizability.
This is encoded in the negative-dimensional coupling constant, which sits in
front of the action. However, as explained in [16], the quantum theory of any
such reformulation may not behave in the same way as in the metric GR. Indeed,
in the pure connection formulation, the quantum behavior is quite different,
with the feature that the sign in front of the $R_{\mu\nu\rho\sigma}^{2}$ term
is opposite to that in the metric GR.

The purpose of this paper is to study the quantum effects of the chiral
Einstein-Cartan action. As we have mentioned, there are different formulations
of GR, which are equivalent at the classical level. But if one tries to
quantize these theories, one often finds, as in the pure-connection
formulation, that the quantum theories are rather different. This makes us ask
the question whether all these classically equivalent formulations lead to
different quantum theories of gravity. This work is a modest step towards that
direction. In particular, we want to understand the one-loop divergences
arising in the chiral Einstein-Cartan formalism and compare it with metric GR.
To do so, one needs to use the background field method and the relevant heat-
kernel technology. One can choose to work with any background, but then one
ends up with a very complicated differential operator in the bosonic as well
as the ghost sector, which is not minimal (Laplace-type). In an Einstein
background, some of these difficulties vanish, and the computation of the heat
kernel is simpler for the appearance of just the self-dual part of the Weyl
curvature in the differential operator that arises in the bosonic sector. In
this calculation, we therefore expand our action around an arbitrary Einstein
background, setting the cosmological constant to zero.

When the action is expanded in this background, one gets a novel quadratic
term of the tetrad field, which is not present in the flat background case.
Another aspect of our calculation is to impose the BRST gauge fixing
procedure. Gauge fixing is done in an analogous way to the flat background,
and the only difference is that the partial derivatives of [15] are replaced
by background covariant derivatives. The one-loop computation can be explored
using the standard heat-kernel technology as is outlined in [21]. One first
linearizes the action on an Einstein background and then uses the heat kernel
to obtain the regularized determinant of the arising operator. As we shall
see, the arising operator in the gravitational sector of the action is not of
Laplace type. To get a Laplace type operator we square the Dirac operator and
then proceed to find the heat-kernel coefficients. In the ghost sector, a
different issue arises because in this case the diffeomorphism ghost and the
Lorentz ghost are coupled in the Lagrangian. We thus decouple the ghost terms
and subsequently get a Laplace-type operator in this sector. All in all, as we
shall see, the coefficient of the one-loop divergence arising in this
formalism is different than in metric GR and thus it motivates us to perform
more non-trivial calculations at more than one-loop and analyze it against
what we obtain from metric GR.

The paper is organized as follows. In Section 2, we give a brief review of the
chiral Einstein-Cartan theory and outline the symmetries in this formalism. In
Section 3, we discuss the BRST formalism and explain the gauge-fixing
procedure. In Section 4, we detail out the heat kernel computation for both
the gravitational and the ghost sectors. We conclude with a discussion of the
results obtained. In the appendix, we outline the BRST closure for all the
relevant fields.

##  2 Review: Chiral Einstein-Cartan theory

We start with the chiral Einstein Cartan action for gravity, with zero
cosmological constant. In units $32\pi G=1$ we have

$\displaystyle S[e,\omega]=2i\int\Sigma^{AB}\wedge F_{AB}$ (1)

where $A,B=1,2$ are unprimed 2-component spinor indices and the self-dual
2-forms are

$\displaystyle\Sigma^{AB}=\frac{1}{2}e^{A}_{C^{\prime}}\wedge
e^{BC^{\prime}},$ (2)

where $e^{AA^{\prime}}$ is the soldering form and the curvature 2-form
$F^{AB}$is given by

$\displaystyle F^{AB}=d\omega^{AB}+\omega^{AC}\wedge\omega^{~{}B}_{C}.$ (3)

The object $\omega^{AB}$ is the self-dual part of the spin connection.
Locally, it takes values in the spin bundle of symmetric second-rank unprimed
spinors. The action in (1) is obtained by applying the chiral self-dual
projection to the first order Einstein-Cartan action in terms of the tetrad
$e^{AA^{\prime}}$ and the full spin connection.

###  2.1 Symmetries

The action in (1) is invariant under two classes of transformations. One is
the diffeomorphisms, which is similar to usual Einstein gravity. Another is
the local SL(2,C) gauge transformations. When the background is not flat, both
these transformations act on the independent fields i.e, the tetrad and the
spin connection. As we will see, it is possible to correct the diffeomorphism
by a gauge transformation and this results in a simpler set of rules for the
transformation of the fields.

####  2.1.1 Diffeomorphisms

Under the infinitesimal coordinate transformation

$\displaystyle x^{\prime\mu}=x^{\mu}+\epsilon\xi^{\mu}$ (4)

the tetrad field transforms (at order $\epsilon$) as follows

$\displaystyle\begin{split}\delta_{\xi}e^{i}_{\nu}&=\xi^{\mu}\partial_{\mu}e^{i}_{\nu}+e^{i}_{\mu}\partial_{\nu}\xi^{\mu}\\\
&=\xi^{\mu}\partial_{\mu}e^{i}_{\nu}-\xi^{\mu}\Gamma^{\lambda}_{\mu\nu}e^{i}_{\lambda}+e^{i}_{\mu}\partial_{\nu}\xi^{\mu}+e^{i}_{\mu}\Gamma^{\mu}_{\nu\lambda}\xi^{\lambda}\\\
&=\xi^{\mu}\nabla_{\mu}e^{i}_{\nu}+e^{i}_{\mu}\nabla_{\nu}\xi^{\mu}.\end{split}$
(5)

where in the second line of the above equation, we added and subtracted a term
with the Levi-Civita connection, which allows us to write the variation of the
tetrad in terms of the spacetime covariant derivative. We can also add and
subtract a term with the spin connection. We then have

$\displaystyle\begin{split}\delta_{\xi}e^{i}_{\nu}&=\xi^{\mu}\nabla_{\mu}e^{i}_{\nu}+e^{i}_{\mu}\nabla_{\nu}\xi^{\mu}+\xi^{\mu}\omega^{~{}i}_{\mu~{}j}e^{j}_{~{}\nu}-\xi^{\mu}\omega^{~{}i}_{\mu~{}j}e^{j}_{~{}\nu}\\\
&=\xi^{\mu}\Big{(}\nabla_{\mu}e^{i}_{\nu}+\xi^{\mu}\omega^{~{}i}_{\mu~{}j}e^{j}_{~{}\nu}\Big{)}+e^{i}_{\mu}\nabla_{\nu}\xi^{\mu}-\xi^{\mu}\omega^{~{}i}_{\mu~{}j}e^{j}_{~{}\nu}\\\
&=e^{i}_{\mu}\nabla_{\nu}\xi^{\mu}-\xi^{\mu}\omega^{~{}i}_{\mu~{}j}e^{j}_{~{}\nu}\\\
&=e^{i}_{\mu}\nabla_{\nu}\xi^{\mu}-\tilde{\phi}^{i}_{~{}j}e^{j}_{~{}\nu}\end{split}$
(6)

where $\xi^{\mu}\omega^{~{}i}_{\mu~{}~{}j}=\tilde{\phi}^{i}_{~{}j}$. Thus, we
arrive at a simple transformation rule for the tetrad under diffeomorphisms.
The first term is the covariant derivative of the parameter $\xi^{\mu}$ and
this is corrected by a gauge transformation. As we will see, this lets us to
get simpler rules for the total transformation when we add the local SL(2,C)
transformation to it. Let us now write the transformation rule for the
connection under diffeomorphisms. The connection is a one-form like the tetrad
and thus it will have a similar set of transformations. In particular,

$\displaystyle\begin{split}\delta_{\xi}\omega^{~{}i}_{\mu~{}j}&=\xi^{\nu}\partial_{\nu}\omega^{~{}i}_{\mu~{}j}+\omega^{i}_{\nu~{}j}\partial_{\mu}\xi^{\nu}.\\\
\end{split}$ (7)

We can also write the diffeomorphism transform of the connection in such a way
so that it does not contain explicit derivatives of the vector field which
generates it. In an analogous way like the tetrad case, we add and subtract
pair of terms with a spin connection. We then arrange terms in a way that the
vector field is inserted into the curvature of the connection. The remaining
term is a total covariant derivative. However, the total covariant derivative
term can be matched with the corresponding local SL(2,C) transform of the
connection and this results again in a simpler transformation rule. The
diffeomorphism of the connection thus reads

$\displaystyle\begin{split}\delta_{\xi}\omega^{~{}i}_{\mu~{}j}&=\xi^{\nu}F^{i}_{~{}\nu\mu
j}+\nabla^{T}_{\mu}(\xi^{\nu}\omega^{i}_{\nu~{}j})\end{split}.$ (8)

where $\nabla^{T}_{\mu}$ is the total covariant derivative and the last term
in (8) is of the form of a gauge transformation, which we can add to the local
SL(2,C) part of the transformation of the connection. It is computationally
simple to use this version of the transformation. Next, we write the local
SL(2,C) transformations of both the tetrad and the connection fields. We
consider one chiral half of the Lorentz to act on the fields, the other half
being set to zero from the beginning.

####  2.1.2 SL(2,C) transformations

SL(2,C) transformations act on both the tetrad and the connections fields.
They are given by

$\displaystyle\begin{split}\delta_{\phi}e^{i}_{~{}\mu}&=\phi^{i}_{~{}j}e^{j}_{~{}\mu},\\\
\delta_{\phi}\omega^{i}_{\mu~{}j}&=\nabla^{T}_{\mu}\phi^{i}_{~{}j}.\end{split}$
(9)

Thus the action of the SL(2,C) transformation on the tetrad amounts to a
Lorentz rotation given by the intfinitesimal parameter $\phi^{i}_{~{}j}$. The
action on the connection is given by the covariant derivative on the parameter
$\phi^{i}_{~{}j}$.

####  2.1.3 Total transformation

Note that the diffeomorphism of the tetrad is corrected by a gauge
transformation, with some parameter $\tilde{\phi}^{ij}$. Then it is possible
to absorb this into the local SL(2,C) transformation parameter $\phi^{ij}$ and
just set the total gauge transformation parameter to $\phi^{ij}$. Then the
full transformation of the tetrad reads

$\displaystyle\delta
e^{i}_{~{}\mu}=e^{i}_{~{}\nu}\nabla^{T}_{\mu}\xi^{\nu}-\phi^{i}_{~{}j}e^{j}_{~{}\mu}.$
(10)

Let us also write the full transformation for the spin connection. We add the
diffeomorphism and local Lorentz to get

$\displaystyle\delta\omega^{i}_{\mu~{}j}=\nabla^{T}_{\mu}\phi^{i}_{~{}j}+\xi^{\nu}F^{i}_{\nu\mu
j}.$ (11)

####  2.1.4 Total transformation in spinor notations

Let us introduce the spinor notations to express the above transformations. In
these notations, the total covariant derivative $\nabla^{T}_{\mu}$ becomes an
object $\nabla^{T}_{MM^{\prime}}$ which acts on the spinor version of both
Lorentz and spacetime indices, $\omega^{~{}i}_{\mu~{}j}$ becomes an object
$\omega^{AB}_{~{}~{}MM^{\prime}}$ where $(AB)$ is a pair of symmetrised
Lorentz indices and $MM^{\prime}$ corresponds to the spacetime index $\mu$.
The total transformation reads

$\displaystyle\begin{split}\delta
e^{AA^{\prime}}_{~{}~{}MM^{\prime}}&=e^{AA^{\prime}}_{~{}~{}~{}NN^{\prime}}\nabla^{T}_{MM^{\prime}}\xi^{NN^{\prime}}-\phi^{A}_{~{}~{}B}e^{BA^{\prime}}_{~{}~{}MM^{\prime}},\\\
\delta\omega^{AB}_{~{}~{}~{}CC^{\prime}}&=\nabla^{T}_{CC^{\prime}}\phi^{AB}+\xi^{MM^{\prime}}F^{AB}_{0~{}~{}MM^{\prime}CC^{\prime}}.\end{split}$
(12)

##  3 BRST formalism

The next step is to quantize the Einstein-Cartan action on an instanton
background. The system has two types of gauge symmetries, diffeomorphism and
local SL(2,C). It is essential to fix the gauges in order to obtain the
physical states of the theory. We will follow the well-developed BRST
formalism to gauge fix the theory. Thus, we need to introduce two kinds of
ghost fields which enter the BRST transformations. As we will see, the
transformation for the local Lorentz (SL(2,C)) ghost will have a term which
contains the diffeomorphism ghosts and thus the two ghosts are coupled to
generate the BRST complex. This is a nontrivial feature of our formalism. We
will gauge fix our theory on a general Einstein background and this makes the
BRST closure property more non-trivial to verify. Once we have gauge fixed the
theory, we can get the one-loop effective action using the heat-kernel
methods. Let us then begin to describe all this.

###  3.1 BRST complex

Consider two pairs of anti-commuting fields $c^{\mu},\bar{c}^{\mu}$ for
diffeomorphisms and $b^{ij},\bar{b}^{ij}$ for SL(2,C) transformations. The
total BRST transformation operator is

$\displaystyle s=s_{D}+s_{L},$ (13)

where $s_{D}$ is the operator corresponding to diffeomorphisms and $s_{L}$ is
for local SL(2,C) transformations. The transformation we use however is most
conveniently written in the condensed form using the total transformation
operator. Thus the splitting we described above is just to illustrate that it
is the total transformation parameter that is nilpotent, i.e $s^{2}=0$. The
individual transformation parameters are not nilpotent and this is the non
triviality of the complex in the present case. Let us now define the BRST
transformations

$\displaystyle\begin{split}se^{i}_{~{}\mu}&=e^{i}_{~{}\nu}\nabla^{T}_{\mu}c^{\nu^{\prime}}-b^{i}_{~{}j}e^{j}_{~{}\mu},\\\
sc^{\mu}&=c^{\nu}\nabla^{T}_{\nu}c^{\mu},\\\ s\bar{c}^{\mu}&=\lambda^{\mu},\\\
s\lambda^{\mu}&=0,\\\
s\omega^{ij}_{~{}~{}\mu}&=\nabla^{T}_{\mu}b^{ij}+c^{\nu}F^{ij}_{~{}~{}\nu\mu},\\\
sb^{ij}&=-\frac{1}{2}[b,b]^{ij}+\frac{1}{2}c^{\mu}c^{\nu}F^{ij}_{~{}~{}\mu\nu},\\\
s\bar{b}^{ij}&=\beta^{ij},\\\ s\beta^{ij}&=0.\end{split}$ (14)

As we can see, the transformation for the tetrad has two terms on it, one for
diffeomorphism ghost and another for the Lorentz ghost. The relative sign is
minus which is purely a convention. In a similar way, the spin connection
transformation admits two terms, one of which follows from its Lorentz
transformation and another gets added owing to diffeomorphisms. We now verify
that the BRST transformation $s$ is nilpotent, i.e $s^{2}=0$. For the third,
fourth, seventh, eighth transformations in (14), this condition trivially
follows because $s^{2}\lambda^{\mu}=s^{2}\beta^{ij}=0$ and
$s^{2}\bar{c}^{\mu}=s\lambda^{\mu}=0$, $s^{2}\bar{b}^{ij}=s\beta^{ij}=0$. For
the rest of the transformations, we detail out in Appendix.

###  3.2 Linearised action

The main objective of the development of BRST complex is to understand the
ghost contribution to the one-loop effective action in our chiral formalism.
The one loop computation is efficiently done using the heat kernel methods. To
employ it one needs to expand the action around an arbitrary background and
then compute the regularised determinant of the differential operator which
arises. However, in an arbitrary background, the metric need not satisfy the
Einstein condition and one cannot use the background field equation to
simplify the calculation of the heat kernel. The calculation thus becomes very
complicated at this level. Thus, in this paper, we expand the action in (1)
around an arbitrary Einstein background. This simplifies much of the
computational complexity which would otherwise had to be dealt with. We then
decompose the tetrad and the spin connection into a fixed background and
consider small fluctuations around this background.

$\displaystyle
e^{AA^{\prime}}_{~{}~{}~{}MM^{\prime}}=\tilde{e}^{AA^{\prime}}_{~{}~{}~{}MM^{\prime}}+h^{AA^{\prime}}_{~{}~{}~{}MM^{\prime}},$
(15)

where $\tilde{e}^{AA^{\prime}}_{~{}~{}~{}MM^{\prime}}$ is the background
tetrad corresponding to an Einstein background, so that the background metric
is

$\displaystyle\tilde{g}_{\mu\nu}=\tilde{e}^{AA^{\prime}}_{\mu}\tilde{e}_{\nu
AA^{\prime}}.$ (16)

The background metric satisfies Einstein equations. The fluctuations around
this metric are then considered and the full metric is decomposed as

$\displaystyle g_{\mu\nu}=\tilde{g}_{\mu\nu}+h_{\mu\nu}.$ (17)

Similarly, the connection can be split into

$\displaystyle\omega^{AB}_{T~{}~{}CC^{\prime}}=\omega^{AB}_{0~{}CC^{\prime}}+w^{AB}_{~{}~{}~{}CC^{\prime}}.$
(18)

where $\omega^{AB}_{0~{}CC^{\prime}}$ is the background connection. As a
result, the decomposition of the curvature is given by

$\displaystyle\begin{split}F_{T}&=F_{0}+\delta F,\\\
F^{AB}_{0}&=d\omega^{AB}_{0}+\omega_{0}^{AC}\wedge\omega_{0C}^{B},\\\ \delta
F&=D_{\omega_{0}}\delta\omega=D_{\omega_{0}}w,\end{split}$ (19)

where in the above, $F_{0}$ is the background curvature constructed from the
background connection, $\delta F$ is the linear order fluctuations around this
baackground. The fluctuation can be expressed as a covariant derivative of the
connection fluctuation with respect to the background connection. Let us now
write the self-dual two form in terms of the splitting form of the tetrad.
This will then be used to decompose the action into kinetic and interaction
terms in a convenient way. The self-dual two form can be written as

$\displaystyle\begin{split}\Sigma^{AB}&=\frac{1}{2}\Big{(}\tilde{e}^{A}_{~{}A^{\prime}}+h^{A}_{~{}A^{\prime}}\Big{)}\wedge\Big{(}\tilde{e}^{BA^{\prime}}+h^{BA^{\prime}}\Big{)}\\\
&=\frac{1}{2}\Big{(}\tilde{e}^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}+\tilde{e}^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}\Big{)}.\end{split}$ (20)

Linearisation of the action in (1) through the decomposition in (15) is given
by

$\displaystyle\begin{split}S&=i\int\Big{(}\tilde{e}^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}+\tilde{e}^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}\Big{)}\wedge\Big{(}d\omega_{0AB}+\omega_{0AC}\wedge\omega^{C}_{0~{}B}\Big{)}\\\
&+i\int\Big{(}\tilde{e}^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}+\tilde{e}^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}\Big{)}\wedge\Big{(}dw_{AB}+\omega_{0AC}\wedge w^{C}_{~{}B}\\\
&+w_{AC}\wedge w^{C}_{~{}B}\Big{)}.\end{split}$ (21)

where in the above steps we have decomposed the tetrad and the connection
fields and written the action in terms of this decomposition. We now expand
the wedge product between the tetrad field and the connection and collect all
individual terms which can arise. The terms which comprise of just the
background fields do not participate in the dynamical equations because they
are fixed. Thus we ignore such terms. The rest of the terms have both the
background and fluctuations in them. Let us first write the free part of the
action. It is given by three terms respectively.

$\displaystyle\begin{split}S_{free}&=i\int\tilde{e}^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}\wedge\Big{(}w_{AC}\wedge
w^{C}_{~{}B}\Big{)}+i\int h^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}\wedge\Big{(}d\omega_{0AB}+\omega_{0AC}\wedge\omega^{C}_{0~{}B}\Big{)}\\\
&+i\int\Big{(}\tilde{e}^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}\Big{)}\wedge
dw_{AB}.\end{split}$ (22)

The first term is quadratic in the connection fluctuations. The appearance of
this term makes the tetrad propagator non-zero. The second term is quadratic
in the tetrad fluctuations, giving a novel ’mass’ term for the gravitons while
the third term is a cross term where metric/tetrad fluctuations is wedged with
the connection fluctuation. Let us now write the interaction part of the
action.

$\displaystyle\begin{split}S_{interaction}&=2i\int\tilde{e}^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}\wedge w_{AC}\wedge w^{C}_{~{}B}\\\ &+i\int
h^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}\wedge\Big{(}d\omega_{AB}+\omega_{0AC}\wedge\omega^{C}_{~{}B}+\omega_{AC}\wedge\omega^{C}_{~{}B}\Big{)}.\end{split}$
(23)

We next expand the tetrad perturbation and the connection in terms of the
background tetrad. This is done to parameterise our tetrad and connection
fields in terms of the background and it appears convenient to write the
action in terms of the coefficients of the expansion. Since the background is
fixed once and for all, we can use it as a basis to expand the fluctuating
fields. This is quite similar in spirit to that of the background field method
for perturbation theory. The fluctuations of the tetrad and connection then
reads

$\displaystyle\begin{split}h^{AA^{\prime}}&=h^{AA^{\prime}}_{~{}~{}~{}MM^{\prime}}\tilde{e}^{MM^{\prime}},\\\
\omega^{AB}&=\omega^{AB}_{~{}~{}~{}MM^{\prime}}\tilde{e}^{MM^{\prime}},\\\
D\omega^{AB}&=\tilde{\nabla}_{MM^{\prime}}\omega^{AB}_{~{}~{}~{}NN^{\prime}}\tilde{e}^{MM^{\prime}}\wedge\tilde{e}^{NN^{\prime}}.\end{split}$
(24)

where $\tilde{\nabla}_{MM^{\prime}}$ is the background covariant derivative.
Using the above expansion, we write the kinetic part of the free action as
follows

$\displaystyle\begin{split}S_{kinetic}&=i\int\Big{(}\tilde{e}^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}+h^{A}_{~{}A^{\prime}}\wedge\tilde{e}^{BA^{\prime}}\Big{)}\wedge
d\omega_{AB}\\\
&=i\int(\tilde{e}^{A}_{A^{\prime}}\wedge\tilde{e}^{MM^{\prime}}\wedge\tilde{e}^{KK^{\prime}}\wedge\tilde{e}^{JJ^{\prime}})h^{BA^{\prime}}_{~{}~{}~{}MM^{\prime}}\tilde{\nabla}_{KK^{\prime}}\omega_{ABJJ^{\prime}}\\\
&+i\int(\tilde{e}^{MM^{\prime}}\wedge\tilde{e}^{BA^{\prime}}\wedge\tilde{e}^{KK^{\prime}}\wedge\tilde{e}^{JJ^{\prime}})h^{A}_{~{}MM^{\prime}A^{\prime}}\tilde{\nabla}_{KK^{\prime}}\omega_{ABJJ^{\prime}}.\end{split}$
(25)

In the above action, we have replaced the derivative on the spin connection by
the appropriate background covariant derivative, which acts on the coefficient
of the expansion of the connection on the basis of the background tetrads.
This then couples with the coefficient of the fluctuation of the tetrad field,
thus giving the resultant kinetic terms of the form $hd\omega$ with the
background covariant derivatives everywhere in place of the ordinary
derivatives. Next we identify the wedge product of four copies of tetrad as
the oriented volume form

$\displaystyle\tilde{e}^{AA^{\prime}}\wedge\tilde{e}^{BB^{\prime}}\wedge\tilde{e}^{CC^{\prime}}\wedge\tilde{e}^{DD^{\prime}}=\epsilon^{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}v.$
(26)

and use the spinor representation of the totally anti-symmetric tensor.

$\displaystyle\epsilon^{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}=i\Big{(}\epsilon^{AD}\epsilon^{BC}\epsilon^{A^{\prime}C^{\prime}}\epsilon^{B^{\prime}D^{\prime}}-\epsilon^{AC}\epsilon^{BD}\epsilon^{A^{\prime}D^{\prime}}\epsilon^{B^{\prime}C^{\prime}}\Big{)}.$
(27)

The validity of the spinor representation can be easily checked. For instance,
if we swap any pair of indices, say $AA^{\prime}\leftrightarrow BB^{\prime}$,
the right hand side picks up an overall minus sign, because the first term of
the right side goes to the second under the swap. In a similar way, for all
such pairs, an overall sign factor comes up. This shows the totally anti-
symmetric nature of the tensor
$\epsilon^{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}$. Let us then use it
and write the kinetic part in (25) as

$\displaystyle\begin{split}S_{kinetic}&=\int
d^{4}x\Big{[}h^{BJ^{\prime}JK^{\prime}}\tilde{\nabla}_{KK^{\prime}}\omega^{K}_{~{}BJJ^{\prime}}-h^{BK^{\prime}KJ^{\prime}}\tilde{\nabla}_{KK^{\prime}}\omega^{J}_{~{}BJJ^{\prime}}\Big{]}\\\
&+\int
d^{4}x\Big{[}h^{AKJ^{\prime}K^{\prime}}\tilde{\nabla}_{KK^{\prime}}\omega_{A~{}JJ^{\prime}}^{~{}J}-h^{AJK^{\prime}J^{\prime}}\tilde{\nabla}_{KK^{\prime}}\omega^{~{}K}_{~{}AJJ^{\prime}}\Big{]}\\\
&=-2\int
d^{4}x\Big{[}h^{BJ^{\prime}JK^{\prime}}\Big{(}\tilde{\nabla}_{JJ^{\prime}}\omega^{K}_{~{}BKK^{\prime}}-\tilde{\nabla}_{KK^{\prime}}\omega^{K}_{~{}BJJ^{\prime}}\Big{)}\Big{]}.\end{split}$
(28)

We now decompose the tetrad perturbation into its irreducible components. This
is reminiscent of the decomposition of a tensorial quantity, for instance the
Riemann tensor, in which we treat the tetrad field as an object with four
spinor indices. The decomposition reads

$\displaystyle
h^{AA^{\prime}MM^{\prime}}=h^{(AM)(A^{\prime}M^{\prime})}+h^{(AM)}\epsilon^{A^{\prime}M^{\prime}}+h^{(A^{\prime}M^{\prime})}\epsilon^{AM}+\epsilon^{AM}\epsilon^{A^{\prime}M^{\prime}}h.$
(29)

The first object on the right hand side above is a tetrad field with its
primed and unprimed indices symmetrised. The object with two spinor indices in
the second and third terms are the part of the tetrad perturbation is a
symmetric field which only propagate off-shell. It is then convenient to
combine the second and fourth terms and define a new field
$h^{AM}=h^{(AM)}+h\epsilon^{AM}$. This field is thus no more symmetric in its
pair of unprimed indices. This results in

$\displaystyle
h^{AA^{\prime}MM^{\prime}}=h^{(AM)(A^{\prime}M^{\prime})}+h^{AM}\epsilon^{A^{\prime}M^{\prime}}+h^{(A^{\prime}M^{\prime})}\epsilon^{AM}.$
(30)

We get rid of the $h^{A^{\prime}M^{\prime}}$ part of the perturbation by
setting one chiral half of the Lorentz gauge to zero. This can be done because
this part of the perturbation does not appear in the free Lagrangian. Thus it
simplifies the computation to some extent. We put the decomposition

$\displaystyle
h^{AA^{\prime}MM^{\prime}}=h^{(AM)(A^{\prime}M^{\prime})}+h^{AM}\epsilon^{A^{\prime}M^{\prime}}$
(31)

in (28) to get for the kinetic part of the Lagrangian

$\displaystyle\begin{split}\mathcal{L}_{kinetic}&=2h^{JKJ^{\prime}K^{\prime}}\Big{(}\tilde{\nabla}_{KJ^{\prime}}\omega^{M}_{~{}JMK^{\prime}}-\tilde{\nabla}_{MK^{\prime}}\omega_{J~{}KJ^{\prime}}^{~{}M}\Big{)}\\\
&+2h^{JK}\Big{(}\tilde{\nabla}_{KJ^{\prime}}\omega^{M~{}~{}J^{\prime}}_{~{}JM}-\tilde{\nabla}_{MK^{\prime}}\omega_{J~{}K}^{~{}M~{}K^{\prime}}\Big{)}.\end{split}$
(32)

The kinetic term above can be written in a compact way. It is possible to
combine the parts of the tetrad perturbation into a single object and the
covariant derivative can then act on an appropriate combination of connection
perturbations. This is helpful because it gives rise to a familiar kinetic
term of the form $h\nabla\omega$ in a general background which is the one
suitable for a chiral first order formulation. Thus, re-written in this way,
we get

$\displaystyle\begin{split}\mathcal{L}_{kinetic}&=-2\Big{(}h^{JKJ^{\prime}K^{\prime}}-h^{(JK)}\epsilon^{J^{\prime}K^{\prime}}\Big{)}\tilde{\nabla}_{MJ^{\prime}}\Big{(}\omega_{J~{}KK^{\prime}}^{~{}M}+\epsilon^{M}_{~{}~{}K}\omega^{N}_{~{}JNK^{\prime}}\Big{)}\\\
&+4h^{JK}\tilde{\nabla}_{KK^{\prime}}\omega^{M~{}~{}K^{\prime}}_{~{}JM}.\end{split}$
(33)

We now redefine some fields as follows

$\displaystyle\begin{split}\omega^{JJ^{\prime}}&=\omega^{MJ~{}~{}J^{\prime}}_{~{}~{}~{}M},\\\
\Omega^{JKMJ^{\prime}}&=\omega^{JMKJ^{\prime}}+\epsilon^{M}_{~{}~{}K}\omega^{JJ^{\prime}}.\end{split}$
(34)

Thus the Lagrangian is written in terms of the new variables

$\displaystyle\begin{split}\mathcal{L}_{kinetic}&=-2\Big{[}h^{JKJ^{\prime}K^{\prime}}-h^{(JK)}\epsilon^{J^{\prime}K^{\prime}}\Big{]}\tilde{\nabla}_{MJ^{\prime}}\Omega^{~{}M}_{J~{}KK^{\prime}}+4h^{JK}\tilde{\nabla}_{KJ^{\prime}}\omega^{~{}J^{\prime}}_{J}.\end{split}$
(35)

As we see, there are two terms in the kinetic part above. However, the kinetic
term is degenerate. This is because the connection field
$\Omega^{JKMJ^{\prime}}$ has twelve independent components, while the other
field $\omega^{JJ^{\prime}}$ has four components. The tetrad perturbation
$h^{JKJ^{\prime}K^{\prime}}$ has nine components, which is conjugate to
$\Omega^{JKMJ^{\prime}}$ and therefore there is a mismatch in the number of
components, which makes it degenerate. To remove the degeneracy we need to
completely fix the gauge. We want to add a specific gauge fixing and ghost
term. This will be done by introducing a gauge fixing fermion in the BRST
formalism. Before doing that, let us write down the potential term for the
tetrad.

$\displaystyle\mathcal{L}_{tetrad-potential}=h^{A}_{~{}A^{\prime}}\wedge
h^{BA^{\prime}}\wedge F_{0AB}$ (36)

For brevity, let us also write the potential terms for the connection.
Following [15], we have

$\displaystyle\begin{split}\mathcal{L}_{connection-
potential}&=-\omega^{JKMJ^{\prime}}\omega_{JKMJ^{\prime}}+2\omega^{JK~{}J^{\prime}}_{~{}~{}K}\omega^{M}_{~{}JMJ^{\prime}}\\\
&=-\Omega^{JKMJ^{\prime}}\Omega_{JKMJ^{\prime}}+2\omega^{JJ^{\prime}}\omega_{JJ^{\prime}}\end{split}$

Let us now introduce the linearised BRST which we subsequently use to fix our
gauges.

###  3.3 Linearised BRST

The main purpose of this section is to disentangle the background from the
perturbations while defining the BRST transformations. This will allow us to
systematically use the transformations for the perturbations themselves. Let
us note how the perturbations of the metric and the connection transform under
diffeomorphisms and local SL(2,C) transformations. We begin with the
transformation for the general tetrad. It admits the following decomposing
into background and quantum fluctuations

$\displaystyle
e^{AA^{\prime}}_{~{}~{}~{}BB^{\prime}}=\tilde{e}^{AA^{\prime}}_{~{}~{}~{}BB^{\prime}}+h^{AA^{\prime}}_{~{}~{}~{}BB^{\prime}}.$
(37)

where the perturbation $h^{AA^{\prime}}_{~{}~{}~{}BB^{\prime}}$ is of order
$\epsilon$ and is very small compared to the background. Then, we get for the
total transformation of the tetrad perturbation

$\displaystyle\begin{split}\delta_{\xi}h^{AA^{\prime}}_{~{}~{}~{}BB^{\prime}}&=\tilde{e}^{AA^{\prime}}_{~{}~{}~{}NN^{\prime}}\tilde{\nabla}_{BB^{\prime}}\xi^{NN^{\prime}}-\phi^{A}_{~{}C}\tilde{e}^{CA^{\prime}}_{~{}~{}BB^{\prime}}.\end{split}$
(38)

where $\tilde{\nabla}_{BB^{\prime}}$ is the background total covariant
derivative. For the connection field, we similarly decompose it into the
background and perturbations

$\displaystyle\omega^{AB}_{T~{}~{}CC^{\prime}}=\omega^{AB}_{0~{}CC^{\prime}}+w^{AB}_{~{}~{}~{}CC^{\prime}}.$
(39)

where $\omega^{AB}_{0~{}CC^{\prime}}$ is the background and
$w^{AB}_{~{}~{}~{}CC^{\prime}}$ is the perturbation of order $\epsilon$. We
get for the transformation of the perturbation, under diffeomorphisms,

$\displaystyle\delta_{\xi}w^{AB}_{~{}~{}CC^{\prime}}$
$\displaystyle=\xi^{NN^{\prime}}F^{AB}_{0~{}~{}NN^{\prime}CC^{\prime}}+\tilde{\nabla}_{CC^{\prime}}\phi^{AB}.$
(40)

where $F^{AB}_{0~{}~{}NN^{\prime}CC^{\prime}}$ is the background curvature.
With these linearised versions of the diffeomorphisms and gauge
transformations, we are now ready to write the BRST transformations of these
fields. As usual, we have two pairs of ghost fields, each for diffeomorphisms
and local Lorentz transformations. Our linearised BRST is then given by

$\displaystyle\begin{split}sh^{AA^{\prime}}_{~{}~{}MM^{\prime}}&=\tilde{\nabla}_{MM^{\prime}}c^{AA^{\prime}}-b^{A}_{~{}~{}B}\tilde{e}^{BA^{\prime}}_{~{}~{}MM^{\prime}},\\\
sc^{MM^{\prime}}&=c^{LL^{\prime}}\nabla_{LL^{\prime}}c^{MM^{\prime}},\\\
s\bar{c}^{MM^{\prime}}&=\lambda^{MM^{\prime}},\\\
s\lambda^{LL^{\prime}}&=0,\\\
sw^{AB}_{~{}~{}~{}CC^{\prime}}&=c^{MM^{\prime}}F^{AB}_{0~{}~{}MM^{\prime}CC^{\prime}}+\tilde{\nabla}_{CC^{\prime}}b^{AB},\\\
sb^{AB}&=-\frac{1}{2}[b,b]^{AB}+\frac{1}{2}c^{MM^{\prime}}c^{NN^{\prime}}F^{AB}_{0~{}~{}MM^{\prime}NN^{\prime}},\\\
s\bar{b}^{AB}&=\beta^{AB},\\\ s\beta^{A^{\prime}B^{\prime}}&=0.\end{split}$
(41)

###  3.4 Gauge fixing

We follow the gauge fixing procedure which is already described in [15]. The
main difference is that we are dealing with a general Einstein background as
opposed to the flat background in [15]. Thus, instead of linear gauges, there
will be non-linearities present in our formalism. We also adopt the BRST gauge
fixing procedure, in which we write the gauge fixing fermion and take BRST
variations to produce the gauge fixing and ghost terms. This will help us to
generate the ghost Lagrangian in a general Einstein background. Let us then
begin to fix the gauges. One chiral half of the Lorentz was fixed by making
the $h^{A^{\prime}B^{\prime}}$ part of the perturbation vanish. The other
chiral half of the Lorentz can be fixed by imposing the non-linear version of
the Lorentz gauge fixing condition. Consider then the following gauge fixing
fermion

$\displaystyle\begin{split}\psi_{Lorentz}&=\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\Omega^{~{}~{}~{}MM^{\prime}}_{JK}.\end{split}$
(42)

Using (41), the BRST variation of the gauge fixing fermion gives

$\displaystyle\begin{split}s\psi_{Lorentz}&=s\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\Omega^{~{}~{}~{}MM^{\prime}}_{JK}-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}s\Omega^{~{}~{}~{}MM^{\prime}}_{JK}\\\
&=\beta^{JK}\tilde{\nabla}_{MM^{\prime}}\Omega^{~{}~{}~{}MM^{\prime}}_{JK}-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\tilde{\nabla}^{MM^{\prime}}b_{JK}\\\
&-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\Big{(}c^{CC^{\prime}}F^{MM^{\prime}}_{0~{}~{}JKCC^{\prime}}\Big{)}.\end{split}$
(43)

The variation of the Lorentz gauge fixing fermion results in a Lorentz gauge
fixing term of the form

$\displaystyle\mathcal{L}_{Lorentz~{}g.f}=-2\beta^{JK}\tilde{\nabla}_{MM^{\prime}}\Omega_{JK}^{~{}~{}~{}MM^{\prime}}.$
(44)

The above term can be written as
$-2\beta^{JK}\epsilon^{J^{\prime}K^{\prime}}\nabla_{MJ^{\prime}}\Omega_{J~{}KK^{\prime}}^{~{}M}$,
which when added to the Lagrangian in (35) along with the ghost terms results
in

$\displaystyle\begin{split}\mathcal{L}&=-2\Big{[}h^{JKJ^{\prime}K^{\prime}}+(\beta^{JK}-h^{(JK)})\epsilon^{J^{\prime}K^{\prime}}\Big{]}\tilde{\nabla}_{MJ^{\prime}}\Omega^{~{}M}_{J~{}KK^{\prime}}+4h^{JK}\tilde{\nabla}_{KJ^{\prime}}\omega^{~{}J^{\prime}}_{J}\\\
&-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\tilde{\nabla}^{MM^{\prime}}b_{JK}-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\Big{(}c^{CC^{\prime}}F^{MM^{\prime}}_{0~{}~{}JKCC^{\prime}}\Big{)}.\end{split}$
(45)

The remaining diffeomorphism gauge can be fixed by implementing a variant of
the de-Donder gauge fixing condition as follows. First we introduce a new name
for the combination of the terms in the bracketed expression of the first line
in (45)

$\displaystyle
H^{JKJ^{\prime}K^{\prime}}:=h^{JKJ^{\prime}K^{\prime}}+(\beta^{JK}-h^{(JK)})\epsilon^{J^{\prime}K^{\prime}}.$
(46)

Note that since the field $\beta^{JK}$ is independent, this results in the two
fields $H^{JKJ^{\prime}K^{\prime}}$ and $h^{JK}$ being independent of each
other. This then leads to decoupling of the kinetic part of the Lagrangian
into two sectors, $(H,\Omega)$ and $(h,\omega)$. We now construct the
diffeomorphism gauge-fixing fermion to be

$\displaystyle\psi_{diffeo}=2\bar{c}^{JJ^{\prime}}\Big{[}\tilde{\nabla}_{KK^{\prime}}\Big{(}h^{~{}K^{\prime}K}_{J~{}~{}~{}J^{\prime}}+h^{K~{}~{}~{}K^{\prime}}_{~{}J^{\prime}J}\Big{)}-2\tilde{\nabla}_{KJ^{\prime}}\beta^{~{}K}_{J}-2\beta_{JJ^{\prime}}\Big{]}.$
(47)

After using the redefinition in (46) we take the BRST variation of it. We have

$\displaystyle\begin{split}s\psi_{diffeo}&=4\beta^{JJ^{\prime}}\Big{[}\tilde{\nabla}^{KK^{\prime}}H_{JKJ^{\prime}K^{\prime}}-\tilde{\nabla}^{K}_{~{}J^{\prime}}H^{~{}~{}M^{\prime}}_{JK~{}M^{\prime}}+\tilde{\nabla}^{K}_{~{}J^{\prime}}h_{JK}-\beta_{JJ^{\prime}}\Big{]}\\\
&-4\bar{c}^{JJ^{\prime}}\Big{[}\tilde{\nabla}^{KK^{\prime}}\Big{(}\tilde{\nabla}_{KK^{\prime}}c_{JJ^{\prime}}-b_{JK}\epsilon_{J^{\prime}K^{\prime}}\Big{)}\Big{]}.\end{split}$
(48)

The variation produces a diffeomorphism gauge fixing term and a ghost term.
The gauge fixing term is an analogue of the variant of de-Donder gauge in
curved spacetime. The ghost term consists of a kinetic part for the
diffeomorphism ghost and a mixed part where the diffeomorphism and Lorentz
ghosts couple. Then, the total gauge fixing term becomes

$\displaystyle\begin{split}\mathcal{L}_{g.f}&=4\beta^{JJ^{\prime}}\Big{[}\tilde{\nabla}^{KK^{\prime}}H_{JKJ^{\prime}K^{\prime}}-\tilde{\nabla}^{K}_{~{}J^{\prime}}H^{~{}~{}M^{\prime}}_{JK~{}M^{\prime}}+\tilde{\nabla}^{K}_{~{}J^{\prime}}h_{JK}-\beta_{JJ^{\prime}}\Big{]}\\\
&-2\beta^{JK}\tilde{\nabla}_{MM^{\prime}}\Omega_{JK}^{~{}~{}~{}MM^{\prime}}.\end{split}$
(49)

and the ghost Lagrangian is given by

$\displaystyle\begin{split}\mathcal{L}_{ghost}&=-4\bar{c}^{JJ^{\prime}}\tilde{\nabla}^{KK^{\prime}}\tilde{\nabla}_{KK^{\prime}}c_{JJ^{\prime}}+4\bar{c}^{JK^{\prime}}\tilde{\nabla}_{KK^{\prime}}b^{K}_{~{}J}-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\tilde{\nabla}^{MM^{\prime}}b_{JK}\\\
&-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\Big{(}c^{CC^{\prime}}F^{MM^{\prime}}_{0~{}~{}JKCC^{\prime}}\Big{)}.\end{split}$
(50)

The Lagrangian above has terms with Lorentz and diffeomorphism ghosts mixed.
To compute the ghost contribution to the one-loop effective action, we need to
diagonalize and rid of the mixed terms. First, we re-scale the fields

$\displaystyle\begin{split}c^{AA^{\prime}}&\rightarrow
2c^{AA^{\prime}},~{}~{}\bar{c}^{AA^{\prime}}\rightarrow
2\bar{c}^{AA^{\prime}}.\end{split}$ (51)

and rewrite the Lagrangian

$\displaystyle\begin{split}\mathcal{L}_{ghost}&=-\bar{c}^{JJ^{\prime}}\tilde{\nabla}^{KK^{\prime}}\tilde{\nabla}_{KK^{\prime}}c_{JJ^{\prime}}+2\bar{c}^{JK^{\prime}}\tilde{\nabla}_{KK^{\prime}}b^{K}_{~{}J}-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\tilde{\nabla}^{MM^{\prime}}b_{JK}\\\
&-\frac{1}{2}\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\Big{(}c^{CC^{\prime}}F^{MM^{\prime}}_{0~{}~{}JKCC^{\prime}}\Big{)}.\end{split}$
(52)

We now use the background field equation and write

$\displaystyle\begin{split}F_{0~{}~{}MM^{\prime}NN^{\prime}}^{AB}&=\psi^{AB}_{~{}~{}CD}\Sigma^{CD}_{0~{}~{}MM^{\prime}NN^{\prime}}=\psi^{AB}_{~{}~{}MN}\epsilon_{M^{\prime}N^{\prime}}.\end{split}$
(53)

where $\psi^{ABCD}$ is the self-dual part of the Weyl tensor and the self-dual
two-form $\Sigma_{0}$ has the particular representation in terms of the
epsilons, given in the second line of the above equation. When we contract all
the indices of $F_{0}^{AB}$, we end up with contracting all the indices of
$\psi^{ABCD}$, which implies vanishing of the self-dual part of the Weyl
tensor.

$\displaystyle\begin{split}F_{0}&=F_{0~{}~{}MM^{\prime}NN^{\prime}}^{AB}\epsilon^{N}_{~{}A}\epsilon^{M}_{B}\epsilon^{M^{\prime}N^{\prime}}\\\
&=\psi^{AB}_{~{}~{}MN}\epsilon_{M^{\prime}N^{\prime}}\epsilon^{N}_{~{}A}\epsilon^{M}_{~{}B}\epsilon^{M^{\prime}N^{\prime}}=0.\end{split}$
(54)

where we used the fact that the self-dual part of the Weyl tensor vanishes
upon index contraction. The Lagrangian in (52) can now be written by replacing
$F_{0}$ with the self-dual part of the Weyl tensor. We have

$\displaystyle\begin{split}\mathcal{L}_{ghost}&=-\bar{c}^{JJ^{\prime}}\tilde{\nabla}^{KK^{\prime}}\tilde{\nabla}_{KK^{\prime}}c_{JJ^{\prime}}+2\bar{c}^{JK^{\prime}}\tilde{\nabla}_{KK^{\prime}}b^{K}_{~{}J}-\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}\tilde{\nabla}^{MM^{\prime}}b_{JK}\\\
&-\frac{1}{2}\psi^{M}_{~{}~{}CJK}\bar{b}^{JK}\tilde{\nabla}_{MM^{\prime}}c^{CM^{\prime}}.\end{split}$
(55)

##  4 Heat Kernel computation

Let us consider a Laplace-type operator of the form
$\Delta=\mathcal{D}^{\mu}\mathcal{D}_{\mu}+E$. The operator acts on some
vector bundle, $E$ is an endomorphism on the fibre and $\mathcal{D}_{\mu}$ is
an appropriate covariant derivative. The interesting object to study is the
determinant $\textrm{det}(\Delta)$. We use the identity log det($\Delta$)=Tr
log($\Delta$) and then rewrite the logarithm of the determinant in terms of an
integral

$\displaystyle\textrm{log
det}(\Delta)=-\int^{\infty}_{0}\frac{dt}{t}\textrm{Tr}(e^{-t\Delta}).$ (56)

Then we have the well-known expansion of the trace under the integral, in
powers of the auxiliary variable $t$

$\displaystyle\textrm{Tr}(e^{-t\Delta})=\int d^{4}x\sqrt{g}\frac{1}{(4\pi
t)^{2}}\sum_{n=0}^{\infty}t^{n}a^{\mathcal{R}}_{n}(E).$ (57)

Our task is to compute the heat kernel coefficients $a^{R}_{n}(E)$. In the
chiral Einstein-Cartan theory around some Einstein background, we consider a
manifold $M$ without a boundary, over which we have a vector bundle $V$. The
Laplace operator $\mathcal{D}_{\mu}\mathcal{D}^{\mu}$ acts on $V$. Then there
exists an expansion of the heat kernel coefficients such that all the odd
indexed coefficients vanish and the even indexed coefficients are given by
geometric invariants. In particular, the divergent UV behavior is controlled
by the coefficient $a^{R}_{2}(E)$, which is given by

$\displaystyle\begin{split}a^{\mathcal{R}}_{2}(E)&=Tr_{\mathcal{R}}\Big{[}\frac{1}{6}\mathcal{D}^{2}E+\frac{1}{2}E^{2}+\frac{1}{6}RE+\frac{1}{12}\Omega_{\mu\nu}\Omega^{\mu\nu}\\\
&+\frac{1}{30}\mathcal{D}^{2}R+\frac{1}{72}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{180}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\Big{]}.\end{split}$
(58)

where $\Omega_{\mu\nu}$ is the curvature of the appropriate covariant
derivative operator.

###  4.1 One-loop effective action

In the background field method, the effective action is defined by considering
the quantum fluctuations around a fixed background field. The generating
functional is given by

$\displaystyle e^{iW(\xi_{0},J)}=\int\mathcal{D}\xi
e^{i(S(\xi+\xi_{0})+J\xi)}.$ (59)

So, the background field effective action is

$\displaystyle\Gamma[\xi_{0},\xi]=W(\xi_{0},J)-J\tilde{\xi},$ (60)

where

$\displaystyle\tilde{\xi}=\frac{\delta W(\xi_{0},J)}{\delta J}.$ (61)

Under the Wick rotation, we go to the Euclidean signature and then define the
Euclidean path integral analogous to (59). In the absence of sources, we put
$J=0$ and obtain

$\displaystyle\begin{split}e^{-\Gamma(\xi_{0})}&=\int\mathcal{D}\xi
e^{-S(\xi+\xi_{0})}\\\ &=e^{-S(\xi_{0})}\int\mathcal{D}\xi
e^{-\int\xi\Delta\xi}.\end{split}$ (62)

Using the Gaussian integration formula

$\displaystyle\int\Pi_{j}dx_{j}e^{-\frac{1}{2}x_{a}M_{ab}x_{b}}=[\textrm{det}(M)]^{-1/2}$
(63)

we find

$\displaystyle\begin{split}e^{-\Gamma(\xi_{0})}&=e^{-S(\xi_{0})}[\textrm{det}(\Delta)]^{-1/2}.\end{split}$
(64)

where $\Delta$ is a generalized Laplace operator obtained after linearizing
the action around the background field. We then have the one-loop effective
action

$\displaystyle\Gamma_{1-loop}=\frac{1}{2}\textrm{log det}(\Delta).$ (65)

The relevant expression for the one-loop effective action is given using (56)
and (57)

$\displaystyle\Gamma_{1-loop}=-\frac{1}{2(4\pi)^{2}}\int_{0}^{\infty}dt~{}t^{n-3}\sum_{n=0}^{\infty}\int
d^{4}x\sqrt{g}~{}a^{\mathcal{R}}_{n}(E).$ (66)

In a renormalization scale, which is free of any mass scale, the running of
the coupling constants is obtained from the logarithmically divergent part.
This part is extracted from the third term in the above expansion,
particularly

$\displaystyle\gamma=\frac{1}{(4\pi)^{2}}\int d^{4}x\sqrt{g}~{}a_{2}.$ (67)

which allows to write

$\displaystyle\Gamma^{\textrm{log}}_{1-loop}=-\frac{1}{2}\gamma\int_{0}^{\infty}\frac{dt}{t}=\gamma~{}\textrm{log}\frac{\delta}{\delta_{0}}.$
(68)

where we have regularized the integral of $t$ by introducing some cut-offs
$t_{max}\approx 1/\delta^{2}$, $t_{min}\approx 1/\delta_{0}^{2}$. Therefore,
we can write

$\displaystyle\frac{\partial\Gamma_{1-loop}^{\textrm{log}}}{\partial
log\delta}=\gamma.$ (69)

###  4.2 Gravitational sector

The free part of the gauge fixed Lagrangian reads

$\displaystyle\begin{split}\mathcal{L}_{2}&=-2H^{JkJ^{\prime}K^{\prime}}\tilde{\nabla}_{MJ^{\prime}}\Omega^{~{}M}_{J~{}KK^{\prime}}+4h^{JK}\tilde{\nabla}_{KJ^{\prime}}\omega^{~{}J^{\prime}}_{J}-\Omega^{JKMJ^{\prime}}\Omega_{JKMJ^{\prime}}+2\omega^{JJ^{\prime}}\omega_{JJ^{\prime}}\end{split}$
(70)

As can be seen, it decouples into two sectors $(H,\Omega)$ and $(h,\omega)$.
We will now proceed with the heat kernel computation in each of these sectors.
To this end, we first rewrite the Lagrangian of the sector $(H,\Omega)$

####  4.2.1 $(H,~{}\Omega)$ sector

We write the Lagrangian in this sector as

$\displaystyle\mathcal{L}_{H\Omega}=\frac{1}{2}\begin{pmatrix}H&\Omega\end{pmatrix}\begin{pmatrix}0&-2\tilde{\nabla}\\\
-2\tilde{\nabla}^{*}&2B\end{pmatrix}\begin{pmatrix}H\\\ \Omega\end{pmatrix},$
(71)

where $B$ is some endomorphism and is given by
$B=-\epsilon^{P}_{~{}J}\epsilon^{Q}_{~{}K}\epsilon^{R}_{~{}M}\epsilon^{P^{\prime}}_{~{}M^{\prime}}$.

Here

$\displaystyle\tilde{\nabla}:S^{2}_{+}\times S_{+}\times S_{-}\rightarrow
S^{2}_{+}\times S_{-}\times
S_{-},~{}~{}~{}~{}(\tilde{\nabla}\Omega)_{JKJ^{\prime}K^{\prime}}:=\tilde{\nabla}_{MJ^{\prime}}\Omega^{~{}~{}M}_{JK~{}K^{\prime}},$
(72)

is the chiral part of a Dirac type operator and

$\displaystyle\tilde{\nabla}^{*}:S^{2}_{+}\times S_{-}\times S_{-}\rightarrow
S^{2}_{+}\times S_{+}\times
S_{-},~{}~{}~{}~{}(\tilde{\nabla}^{*}H)_{JKMK^{\prime}}:=\tilde{\nabla}^{J^{\prime}}_{M}H_{JKJ^{\prime}K^{\prime}}$
(73)

is the adjoint operator. It is clear that the arising operator $D$ in (71) is
of Dirac type and the standard method is to square this operator so that it
becomes an operator of Laplace type. One can then use the heat kernel
technology to compute the effective action. Squaring $D$ we have

$\displaystyle
D^{2}=\begin{pmatrix}\tilde{\nabla}\tilde{\nabla}^{*}&-\tilde{\nabla}B\\\
-B\tilde{\nabla}^{*}&\tilde{\nabla}^{*}\tilde{\nabla}+B^{2}\mathbb{1}\end{pmatrix}.$
(74)

Let us compute the entries of the above matrix. We have

$\displaystyle\begin{split}(\tilde{\nabla}\tilde{\nabla}^{*}H)_{JKJ^{\prime}M^{\prime}}&=\tilde{\nabla}_{MJ^{\prime}}\tilde{\nabla}^{MK^{\prime}}H_{JKK^{\prime}M^{\prime}}\\\
&=-\frac{1}{2}\tilde{\nabla}^{NN^{\prime}}\tilde{\nabla}_{NN^{\prime}}H_{JKJ^{\prime}M^{\prime}}+2\tilde{\nabla}_{P^{\prime}(J}\tilde{\nabla}_{P)}^{~{}P^{\prime}}H^{P}_{~{}KJ^{\prime}M^{\prime}}+2\tilde{\nabla}_{P(J^{\prime}}\tilde{\nabla}_{P^{\prime})}^{~{}P}H^{~{}~{}~{}P^{\prime}}_{JK~{}M^{\prime}}\\\
&=-\frac{1}{2}\tilde{\nabla}^{NN^{\prime}}\tilde{\nabla}_{NN^{\prime}}H_{JKM^{\prime}J^{\prime}}+2\square_{JP}H^{P}_{~{}KJ^{\prime}M^{\prime}}+2\square_{J^{\prime}P^{\prime}}H^{~{}~{}P^{\prime}}_{JK~{}M^{\prime}}\\\
&=-\frac{1}{2}\tilde{\nabla}^{NN^{\prime}}\tilde{\nabla}_{NN^{\prime}}H_{JKM^{\prime}J^{\prime}}-12\Lambda
H_{JKJ^{\prime}M^{\prime}}\\\
&=-\frac{1}{2}\tilde{\nabla}^{NN^{\prime}}\tilde{\nabla}_{NN^{\prime}}H_{JKM^{\prime}J^{\prime}}\end{split}$

where we have used the Lichnerowicz formula and used the fact that when one of
the indices of the Weyl spinor is contracted, this gives zero. Thus the only
non-zero contribution comes from the cosmological constant, but this vanishes
in the setting where the cosmological constant is zero. Next,

$\displaystyle\begin{split}(\tilde{\nabla}^{*}\tilde{\nabla}\Omega)_{JKNJ^{\prime}}&=\tilde{\nabla}^{K^{\prime}}_{~{}N}\tilde{\nabla}^{M}_{~{}K^{\prime}}\Omega_{JKMJ^{\prime}}\\\
&=-\frac{1}{2}\tilde{\nabla}^{NN^{\prime}}\tilde{\nabla}_{NN^{\prime}}\Omega_{JKNJ^{\prime}}+2\tilde{\nabla}_{Q^{\prime}(J}\tilde{\nabla}_{Q)}^{~{}Q^{\prime}}\Omega^{Q}_{~{}KNJ^{\prime}}+2\tilde{\nabla}_{Q(J^{\prime}}\tilde{\nabla}_{Q^{\prime})}^{~{}Q}\Omega^{~{}~{}~{}P^{\prime}}_{JKN}\\\
&=-\frac{1}{2}\tilde{\nabla}^{NN^{\prime}}\tilde{\nabla}_{NN^{\prime}}\Omega_{JKNJ^{\prime}}\end{split}$

Thus the matrix we have is given by

$\displaystyle-2D^{2}=\begin{pmatrix}\tilde{\nabla}^{\mu}\tilde{\nabla}_{\mu}&2\tilde{\nabla}B\\\
2B\tilde{\nabla}^{*}&\tilde{\nabla}^{\mu}\tilde{\nabla}_{\mu}-2B^{2}\mathbb{1}\end{pmatrix}.$
(75)

We can rewrite $-2D^{2}$ as $D_{\mu}D^{\mu}-E$ where $D_{\mu}$ is a new matrix
valued covariant derivative operator and $E$ is some endomorphism. The
structure of $D_{\mu}$ that we recover is

$\displaystyle
D_{MM^{\prime}}=\begin{pmatrix}\tilde{\nabla}_{MM^{\prime}}&B_{MM^{\prime}}\\\
B_{MM^{\prime}}&\tilde{\nabla}_{MM^{\prime}}\end{pmatrix},$ (76)

where we have suppressed the SL(2,C) indices in the spinorial object $B$, and

$\displaystyle E=\begin{pmatrix}B^{2}\mathbb{1}&0\\\
0&3B^{2}\mathbb{1}\end{pmatrix}.$ (77)

Multiplying $E$ with the Ricci scalar $R$, and then taking the trace with
respect to the SL(2,C) indices, we get

$\displaystyle-\frac{1}{2.6}\textrm{tr}_{\mathcal{R}}(RE)=0.$ (78)

where we have used the fact that we are working for Ricci-flat background. The
curvature of the new connection $D_{MM^{\prime}}$ is given by

$\displaystyle\mathcal{F}_{MM^{\prime}NN^{\prime}}=\frac{1}{2}\begin{pmatrix}\tilde{\nabla}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}+B_{[MM^{\prime}}B_{NN^{\prime}]}&-\tilde{\nabla}_{[MM^{\prime}}B_{NN^{\prime}]}-B_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}\\\
-\tilde{\nabla}_{[MM^{\prime}}B_{NN^{\prime}]}-B_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}&\tilde{\nabla}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}+B_{[MM^{\prime}}B_{NN^{\prime}]}\end{pmatrix}.$
(79)

Let us compute the components of the above matrix. The diagonal parts
contribute

$\displaystyle\begin{split}\tilde{\nabla}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}+B_{[MM^{\prime}}B_{NN^{\prime}]}&=F^{AB}_{~{}~{}~{}MM^{\prime}NN^{\prime}}\Sigma_{AB}+B_{[MM^{\prime}}B_{NN^{\prime}]}\\\
&=\psi^{AB}_{~{}~{}MN}\epsilon_{M^{\prime}N^{\prime}}\Sigma_{AB}+\epsilon^{R}_{~{}M}\epsilon^{P^{\prime}}_{~{}M^{\prime}}\epsilon^{S}_{~{}N}\epsilon^{S^{\prime}}_{~{}N^{\prime}}-\epsilon^{R}_{~{}N}\epsilon^{P^{\prime}}_{~{}N^{\prime}}\epsilon^{S}_{~{}M}\epsilon^{S^{\prime}}_{~{}M^{\prime}}.\end{split}$
(80)

We do not need to explicitly compute the off-diagonal elements as these
vanish. This is because these elements are both symmetric and antisymmetric
with respect to $MM^{\prime}$ and $NN^{\prime}$ and this is why there are
vanishing. We need to take the trace of the square of the curvature because
this is what is there in the heat kernel coefficient. It is straightforward to
work this out. To compute explicitly, we have

$\displaystyle\begin{split}&2\textrm{tr}_{\mathcal{R}}\Big{[}\psi^{AB}_{~{}~{}MN}\epsilon_{M^{\prime}N^{\prime}}\Sigma_{AB}\psi^{CDMN}\epsilon^{M^{\prime}N^{\prime}}\Sigma_{CD}\\\
&+2\psi^{AB}_{~{}~{}MN}\epsilon_{M^{\prime}N^{\prime}}\Sigma_{AB}(\epsilon^{RM}\epsilon^{P^{\prime}M^{\prime}}\epsilon^{SN}\epsilon^{S^{\prime}N^{\prime}}-\epsilon^{RN}\epsilon^{P^{\prime}N^{\prime}}\epsilon^{SM}\epsilon^{S^{\prime}M^{\prime}})\Big{]},\end{split}$
(81)

where we have neglected the square terms coming from the epsilon metrics
because these vanish. The second term on the above line vanishes because of
the symmetry property of the Weyl spinor. Thus we are left with the first
term. Evaluating the trace, we get

$\displaystyle\begin{split}\textrm{tr}_{\mathcal{R}}\Big{[}\frac{1}{12}\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}\Big{]}&=\frac{1}{12.4}\textrm{tr}_{\mathcal{R}}[4\psi^{ABMN}\psi_{CDMN}]\\\
&=\frac{1}{12}\psi^{ABMN}\psi_{ABMN}.\end{split}$

The next object to compute is $E^{2}$. We have

$\displaystyle
E^{{}^{\prime}2}=\Big{(}-\frac{1}{2}E\Big{)}^{2}=\begin{pmatrix}(\frac{1}{2}B^{2})^{2}&0\\\
0&(\frac{3}{2}B^{2})^{2}\end{pmatrix}$ (82)

Evaluating the trace of this operator, we get

$\displaystyle\frac{1}{2}\textrm{tr}_{\mathcal{R}}(E^{2})=0$ (83)

####  4.2.2 $(h,\omega)$ sector

The kinetic part of the Lagrangian in this sector is analogous to what we
found in the $(H,\Omega)$ sector. The only difference that arises is because
of factors of 2 which we absorb in the definition of the fields $(h,\omega)$.
Let us rewrite the free part of the Lagrangian as follows

$\displaystyle\mathcal{L}_{h\omega}=\frac{1}{2}\begin{pmatrix}h&\omega\end{pmatrix}\begin{pmatrix}0&4\tilde{\nabla}^{*}\\\
4\tilde{\nabla}&4\tilde{B}\end{pmatrix}\begin{pmatrix}h\\\
\omega\end{pmatrix}.$ (84)

Here

$\displaystyle\tilde{\nabla}^{*}:S_{-}\rightarrow
S_{+},~{}~{}~{}~{}(\tilde{\nabla}\omega)_{MJ}:=\tilde{\nabla}_{MJ^{\prime}}\omega^{J^{\prime}}_{~{}J},$
(85)

is the chiral part of a Dirac type operator and

$\displaystyle\tilde{\nabla}:S_{+}\rightarrow
S_{-},~{}~{}~{}~{}(\tilde{\nabla}^{*}h)_{KJ^{\prime}}:=\tilde{\nabla}^{J}_{~{}J^{\prime}}h_{JK}$
(86)

is the adjoint operator. It is transparent that the arising operator
$\tilde{D}$ in (84) is of Dirac type. The computation in this sector exactly
mimicks the one in the previous. We do not repeat this computation here but
outline the main results of interest.

$\displaystyle\tilde{D}^{2}=\begin{pmatrix}-2\tilde{\nabla}^{\mu}\tilde{\nabla}_{\mu}&4\tilde{\nabla}^{*}\tilde{B}\\\
4\tilde{B}\tilde{\nabla}&-2\tilde{\nabla}^{\mu}\tilde{\nabla}_{\mu}+4\tilde{B}^{2}\mathbb{1}\end{pmatrix}.$
(87)

We can rewrite $-\frac{1}{2}\tilde{D}^{2}$ as
$\tilde{D}_{\mu}\tilde{D}^{\mu}-\tilde{E}$ where $\tilde{D}_{\mu}$ is a new
matrix valued covariant derivative operator and $\tilde{E}$ is some
endomorphism.

$\displaystyle\tilde{D}_{MM^{\prime}}=\begin{pmatrix}\tilde{\nabla}_{MM^{\prime}}&-\tilde{B}_{MM^{\prime}}\\\
-\tilde{B}_{MM^{\prime}}&\tilde{\nabla}_{MM^{\prime}}\end{pmatrix},$ (88)

and

$\displaystyle\tilde{E}=\begin{pmatrix}-B^{2}\mathbb{1}&0\\\
0&-3B^{2}\mathbb{1}\end{pmatrix}.$ (89)

We can directly write the curvature of the new connection which is given by

$\displaystyle\mathcal{\tilde{F}}_{MM^{\prime}NN^{\prime}}=-2\begin{pmatrix}\tilde{\nabla}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}+\tilde{B}_{[MM^{\prime}}\tilde{B}_{NN^{\prime}]}&-\tilde{\nabla}_{[MM^{\prime}}\tilde{B}_{NN^{\prime}]}-\tilde{B}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}\\\
-\tilde{\nabla}_{[MM^{\prime}}\tilde{B}_{NN^{\prime}]}-\tilde{B}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}&\tilde{\nabla}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}+\tilde{B}_{[MM^{\prime}}\tilde{B}_{NN^{\prime}]}\end{pmatrix}.$
(90)

The trace of the square of above object is already computed explicitly in the
previous sector, upto some numerical factors. Restoring the correct factors,
we quote the result here

$\displaystyle\begin{split}\textrm{tr}_{\mathcal{R}}\Big{[}\frac{1}{12}\mathcal{\tilde{F}}_{\mu\nu}\mathcal{\tilde{F}}^{\mu\nu}\Big{]}&=\frac{4}{12}\textrm{tr}_{R}[4\psi^{ABMN}\psi_{CDMN}]\\\
&=\frac{4}{3}\psi^{ABMN}\psi_{ABMN}.\end{split}$

The trace of the endomorphism $E$ is given by

$\displaystyle\frac{2}{6}\textrm{tr}_{\mathcal{R}}(RE)=0.$ (91)

The next object to compute is $\tilde{E}^{2}$. We have

$\displaystyle\tilde{E}^{2}=\begin{pmatrix}(\tilde{B}^{2})^{2}&0\\\
0&(3\tilde{B}^{2})^{2}\end{pmatrix}.$ (92)

However, the trace of $\tilde{E}^{2}$ is zero. We convert the result of the
square of Weyl spinors in terms of $R^{2}$, $R_{\mu\nu}^{2}$ and
$R_{\mu\nu\rho\sigma}^{2}$. However, both $R^{2}$ and $R_{\mu\nu}$ vanish and
the heat kernel coefficient in the gravitational sector is given by

$\displaystyle
a_{2bosonic}^{\mathcal{R}}(E)=\frac{64}{45}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$
(93)

###  4.3 Ghost sector

Let us now return to the ghost sector. The Lagrangian in (55) is not diagonal
in the fields. It has mixed terms which arise from the coupling of
diffeomorphism and Lorentz ghosts. To do a heat kernel computation, it is
necessary to obtain a diagonalised Lagrangian such that the differential
operator which arises is of Laplace type. Thus, we now proceed to diagonalize
it. Let us first give a outline of the diagonalization procedure and then we
will implement it in our problem. Consider a differential operator of the form

$\displaystyle\partial^{\mu}\partial_{\mu}\mathds{1}+\partial_{\mu}A^{\mu}+B,$
(94)

where $A^{\mu}$ and $B$ are matrices in some matrix space. This operator can
in turn be written as a matrix which then acts on the column vector comprising
of fields $b,c$. The conjugate fields $\bar{b},\bar{c}$ are contracted from
the left and this produces the Lagrangian. It is then possible to absorb the
linear parts of the differential operator by a suitable redefinition such that
it results in an operator of the form

$\displaystyle\mathcal{D}^{\mu}\mathcal{D}_{\mu}+E,$ (95)

where $E$ is some endomorphism on the field space and $\mathcal{D}_{\mu}$ is
an appropriate covariant derivative operator. It is customary to first write
the Lagrangian in (55) in a matrix form, so that one can diagonalize the
matrix by suitable field re-definitions. We have

$\displaystyle\mathcal{L}_{ghost}$
$\displaystyle=-\begin{pmatrix}\bar{b}^{JK}&\bar{c}^{BB^{\prime}}\end{pmatrix}\begin{pmatrix}X&Y\\\
Z&X\end{pmatrix}\begin{pmatrix}b_{JK}\\\ c_{BB^{\prime}}\\\ \end{pmatrix},$
(96)

where

$\displaystyle\begin{split}X&=\tilde{\nabla}_{MM^{\prime}}\tilde{\nabla}^{MM^{\prime}},\\\
Y&=\frac{1}{2}\psi^{MB}_{~{}~{}~{}JK}\tilde{\nabla}^{~{}B^{\prime}}_{M},\\\
Z&=-\tilde{\nabla}^{K}_{~{}B^{\prime}}\epsilon^{J}_{~{}B}.\\\ \end{split}$
(97)

where in writing $Y$, we used the Bianchi identity
$\nabla_{MB^{\prime}}\psi^{M}_{~{}~{}BJK}=0$ and got rid of the other term.
The matrix in (96) can be split in the following way

$\displaystyle\begin{split}\begin{pmatrix}X&Y\\\
Z&X\end{pmatrix}&=\tilde{\nabla}_{MM^{\prime}}\tilde{\nabla}^{MM^{\prime}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\begin{pmatrix}0&\frac{1}{2}\epsilon^{M^{\prime}B^{\prime}}\psi^{MB}_{~{}~{}~{}JK}\\\
-\epsilon^{J}_{~{}B}\epsilon^{KM}\epsilon^{M^{\prime}}_{~{}B^{\prime}}&0\end{pmatrix}\tilde{\nabla}_{MM^{\prime}}\\\
~{}\\\
&+\tilde{\nabla}^{MM^{\prime}}\begin{pmatrix}0&\frac{1}{2}\epsilon_{M^{\prime}B^{\prime}}\psi^{B}_{~{}~{}MJK}\\\
-\epsilon^{J}_{~{}B}\epsilon^{K}_{~{}M}\epsilon_{~{}M^{\prime}B^{\prime}}&0\end{pmatrix}.\end{split}$
(98)

This can now be re-written by absorbing the first order derivative parts and
expressing it in the form of a Laplace type operator

$\displaystyle\begin{pmatrix}X&Y\\\ Z&X\end{pmatrix}$
$\displaystyle=\mathcal{D}^{MM^{\prime}}\mathcal{D}_{MM^{\prime}}+E,$ (99)

where we define the new connection

$\displaystyle\mathcal{D}^{MM^{\prime}}=\begin{pmatrix}\tilde{\nabla}^{MM^{\prime}}&\frac{1}{2}\epsilon^{M^{\prime}B^{\prime}}\psi^{MB}_{~{}~{}~{}JK}\\\
~{}\\\
\epsilon^{J}_{~{}B}\epsilon^{KM}\epsilon^{M^{\prime}}_{~{}B^{\prime}}&\tilde{\nabla}^{MM^{\prime}}\end{pmatrix}$
(100)

and

$\displaystyle E=-\begin{pmatrix}\psi^{KJ}_{~{}~{}~{}JK}&0\\\ ~{}\\\
0&\psi^{KJ}_{~{}~{}~{}JK}\end{pmatrix}.$ (101)

Upon index contractions, the self dual part of the Weyl tensor vanishes and
therefore we can set $E=0$ henceforth. We thus have the relevant Laplace type
operator, which in this case with $E=0$ is

$\displaystyle\Delta:=\mathcal{D}_{MM^{\prime}}\mathcal{D}^{MM^{\prime}}.$
(102)

with the new connection defined in (100).

We define $\Omega_{\mu\nu}$ as

$\displaystyle\begin{split}\Omega_{\mu\nu}&=[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]\\\
&=\begin{pmatrix}\tilde{\nabla}_{\mu}&Y_{1\mu}\\\
Y_{2\mu}&\tilde{\nabla}_{\mu}\end{pmatrix}\begin{pmatrix}\tilde{\nabla}_{\nu}&Y_{1\nu}\\\
Y_{2\nu}&\tilde{\nabla}_{\nu}\end{pmatrix}-\begin{pmatrix}\tilde{\nabla}_{\nu}&Y_{1\nu}\\\
Y_{2\nu}&\tilde{\nabla}_{\nu}\end{pmatrix}\begin{pmatrix}\tilde{\nabla}_{\mu}&Y_{1\mu}\\\
Y_{2\mu}&\tilde{\nabla}_{\mu}\end{pmatrix}\\\
&=\begin{pmatrix}\tilde{\nabla}_{[\mu}\tilde{\nabla}_{\nu]}+Y_{1[\mu}Y_{2\nu]}&\tilde{\nabla}_{[\mu}Y_{1\nu]}+Y_{1[\mu}\tilde{\nabla}_{\nu]}\\\
~{}\\\
Y_{2[\mu}\tilde{\nabla}_{\nu]}+\tilde{\nabla}_{[\mu}Y_{2\nu]}&Y_{2[\mu}Y_{1\nu]}+\tilde{\nabla}_{[\mu}\tilde{\nabla}_{\nu]}\end{pmatrix}.\end{split}$
(103)

where
$Y^{MM^{\prime}}_{1}=\frac{1}{2}\epsilon^{M^{\prime}B^{\prime}}\psi^{MB}_{~{}~{}~{}JK}$
and
$Y^{MM^{\prime}}_{2}=\epsilon^{J}_{~{}B}\epsilon^{KM}\epsilon^{M^{\prime}}_{~{}B^{\prime}}$.
Let us compute the matrix elements of $\Omega_{\mu\nu}$ using spinor
notations, which will yield simpler expressions. The diagonal elements are
equal to the commutator of the total covariant derivative. This commutator can
act on the Lorentz ghost which has just the internal indices or the
diffeomorphism ghost which has only the spacetime indices. Whatever the ghost
field, the action of the commutator can only be proportional to the curvature
of the spin connection. This is because the curvature of the spin connection
is related to the curvature of the Levi-civita connection via the tetrad one-
forms and thus they are equivalent descriptions of the same entity. Further,
the background field equation expresses the curvature in terms of the Weyl
spinor. We already arrived at a simple form of the curvature in spinor
notations when expressed in terms of the Weyl spinor. Thus, overall the
diagonal part of $\Omega_{\mu\nu}$ in spinor notations contributes

$\displaystyle\begin{split}\tilde{\nabla}_{[MM^{\prime}}\tilde{\nabla}_{NN^{\prime}]}+Y_{1[MM^{\prime}}Y_{2NN^{\prime}]}&=\epsilon_{M^{\prime}N^{\prime}}\psi^{AB}_{~{}~{}MN}\Sigma_{AB},\end{split}$
(104)

where we used the fact that when two of the indices of the Weyl spinor are
contracted, it vanishes, i.e $\psi^{J}_{~{}MJK}=0$. Thus the contribution from
the second term of the left hand side above is zero. Let us now compute the
off-diagonal parts. The lower off-diagonal element when evaluated in spinor
notations give

$\displaystyle\begin{split}\tilde{\nabla}_{[\mu}Y_{2\nu]}+Y_{2[\mu}\tilde{\nabla}_{\nu]}&=\tilde{\nabla}_{MM^{\prime}}\epsilon^{J}_{~{}B}\epsilon^{K}_{~{}N}\epsilon_{N^{\prime}B^{\prime}}-\tilde{\nabla}_{NN^{\prime}}\epsilon^{J}_{~{}B}\epsilon^{K}_{~{}M}\epsilon_{M^{\prime}B^{\prime}}\\\
&+\epsilon^{J}_{~{}B}\epsilon^{K}_{~{}M}\epsilon_{M^{\prime}B^{\prime}}\tilde{\nabla}_{NN^{\prime}}-\epsilon^{J}_{~{}B}\epsilon^{K}_{~{}N}\epsilon_{N^{\prime}B^{\prime}}\tilde{\nabla}_{MM^{\prime}}\\\
&=0.\end{split}$ (105)

Let us also compute the upper right off-diagonal part. This is given by

$\displaystyle\begin{split}\tilde{\nabla}_{[\mu}Y_{1\nu]}+Y_{1[\mu}\tilde{\nabla}_{\nu]}&=\epsilon_{N^{\prime}C^{\prime}}\nabla_{MM^{\prime}}\psi^{JK}_{~{}~{}NC}-\epsilon_{M^{\prime}C^{\prime}}\nabla_{NN^{\prime}}\psi^{JK}_{~{}~{}MC}\\\
&+\psi^{JK}_{~{}~{}MC}\epsilon_{M^{\prime}C^{\prime}}\tilde{\nabla}_{NN^{\prime}}-\psi^{JK}_{~{}~{}NC}\epsilon_{N^{\prime}C^{\prime}}\tilde{\nabla}_{MM^{\prime}}.\end{split}$
(106)

We now need to compute $\Omega^{2}$. Let us first see the structure of the
off-diagonal elements which arise in it. Indeed, it is easy to check that the
lower left off-diagonal element must vanish and the upper left will be given
by

$\displaystyle\begin{split}&\psi^{MN}_{~{}~{}~{}AB}\epsilon^{M^{\prime}N^{\prime}}\Big{(}\epsilon_{N^{\prime}C^{\prime}}\nabla_{MM^{\prime}}\psi^{JK}_{~{}~{}NC}-\epsilon_{M^{\prime}C^{\prime}}\nabla_{NN^{\prime}}\psi^{JK}_{~{}~{}MC}\\\
&+\psi^{JK}_{~{}~{}MC}\epsilon_{M^{\prime}C^{\prime}}\tilde{\nabla}_{NN^{\prime}}-\psi^{JK}_{~{}~{}NC}\epsilon_{N^{\prime}C^{\prime}}\tilde{\nabla}_{MM^{\prime}}\Big{)}.\end{split}$
(107)

Due to index contractions of the Weyl spinor, the above term vanishes. Then,
the square of the curvature is given entirely by diagonal terms, which are
however equal and is given by

$\displaystyle\begin{split}\Omega_{MM^{\prime}NN^{\prime}}\Omega^{MM^{\prime}NN^{\prime}}&=\begin{pmatrix}2\psi_{ABMN}\psi^{CDMN}&0\\\
~{}\\\ 0&2\psi_{ABMN}\psi^{CDMN}\end{pmatrix}.\end{split}$ (108)

Substituting this in the heat kernel formula (58), we have the relevant for
the heat kernel coefficient as

$\displaystyle\begin{split}\frac{1}{12}\textrm{tr}_{\mathcal{R}}[\Omega_{\mu\nu}\Omega^{\mu\nu}]&=\frac{1}{3}\psi_{MNPS}\psi^{MNPS}\end{split}$
(109)

Let us now rewrite the above as a linear combination of the curvature squares.
Doing so, we have for the heat kernel coefficient

$\displaystyle
a_{2ghost}^{\mathcal{R}}(E)=\frac{61}{180}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
(110)

###  4.4 Final result and comparison with metric gravity

The final result for the heat kernel coefficient is given by

$\displaystyle
a_{2bosonic}^{\mathcal{R}}-2a_{2ghost}^{\mathcal{R}}=\frac{67}{90}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
(111)

Thus,

$\displaystyle\gamma_{chiral-EC}=\frac{1}{(4\pi)^{2}}\int
d^{4}x\sqrt{g}\Bigg{(}\frac{67}{90}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\Bigg{)}$
(112)

The analogous result in metric gravity is

$\displaystyle\gamma_{metric}=\frac{1}{(4\pi)^{2}}\int
d^{4}x\sqrt{g}\Bigg{(}\frac{53}{45}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\Bigg{)}$
(113)

Although the above results are total derivatives, we notice that the
coefficient of $R_{\mu\nu\rho\sigma}^{2}$ divergences changes. This alreads
tells us that the quantum theory for the chiral Einstein-Cartan gravity is
different than metric GR. It would be interesting to see what happens when we
couple chiral Einstein-Cartan gravity to matter or deal with a non-zero
cosmological constant.

##  5 Discussion

In this paper, we studied the chiral Einstein-Cartan action on an arbitrary
Einstein background for zero cosmological constant. We developed the gauge-
fixing procedure using the BRST formalism. The action and the fields appearing
in the first order formalism are all expressed in spinor notations to
conveniently perform the gauge fixing. The diffeomorphism and Lorentz gauge
fixing, when done on an Einstein background, remarkably lead to decoupling of
the free part of the Lagrangian into two sectors, mimicking the result of its
flat version [15]. The gauge fixing fermion that we chose to work with is
similar to the one in its flat space version, except that now it is non-linear
in the fields. Nevertheless, this non-linearity does not cause us problems and
the bosonic part decouples nicely into two sectors, namely $(H,\Omega)$ and
$(h,\omega)$.
However, the novelty of this result is the appearance of a new quadratic term
for the tetrad/metric field. This new term does not affect the gauge fixing
procedure and does not complicate calculations. Let us now comment on the BRST
complex in this first-order formalism. The complex is so designed that the
diffeomorphism ghosts appear in the transformation of the Lorentz ghost. As a
result, it leads to the coupling of diffeomorphism and Lorentz ghosts in the
ghost Lagrangian. Also, we massaged the transformation for the tetrad and the
spin-connection in such a way that the diffeomorphism transform does not
contain the derivatives of the vector field which generates it. For the spin-
connection, the vector field is inserted into the curvature of the spin-
connection and this is corrected by a gauge transformation. Using the gauge
fixed Lagrangian, we computed the heat kernel coefficients appearing in the
one-loop effective action by computing the determinant of the operators
arising in the bosonic and the ghost sectors. In the bosonic sector, this
operator is a version of the Dirac operator, but with background covariant
derivatives everywhere. We squared it to get the relevant Laplace type
operator, from which we extracted the heat kernel coefficient following the
standard technology. In the ghost sector, the operator arising in the
Lagrangian is not of Laplace type. By suitable field redefinitions, we
converted it into a Laplace-type operator. We found that the endomorphism part
vanishes because what appears here is the Weyl spinor with its indices
contracted. We found that the arising one-loop divergences are proportional to
the $R_{\mu\nu\rho\lambda}^{2}$ term. However, the coefficient of this term is
different from the corresponding result in metric GR. This is not unexpected
because, although on-shell, the chiral formulation is equivalent to metric GR,
in an off-shell calculation there is no guarantee for such a similarity.
Indeed, this is what we found by doing a one-loop computation. This is to be
distinguished from the result of the pure connection formalism, achieved in
[16], where the sign in front of the $R_{\mu\nu\rho\lambda}^{2}$ term was
different and there was a close relation with the metric GR. Our result is
important in the context of quantization of different formulations of gravity
and in understanding whether they lead to different quantum theories.

It is possible to understand the renormalization flow of any such equivalent
formulation and search for a UV-fixed point in the ultraviolet. Indeed, this
has been done in the pure-connection formulation. In our present work, we do
not analyze the flow but instead only compute the one-loop effective action. A
more general study of RG flow can be done in a setting where the cosmological
constant is non-zero and the background is completely arbitrary. Also, a two-
loop computation is possible in such a context. We leave this for future
research.

There are other interesting calculations that can be done in the present
context. It is possible to couple fermions to the chiral Einstein-Cartan
action, which would lead to a nonzero torsion. Then one can study the one-loop
divergences that arise in such a setting. In this paper, we developed the
ghost Lagrangian for an arbitrary Einstein background. It is possible to take
its flat space limit by replacing covariant derivatives with ordinary partial
derivatives everywhere. This can be used to compute ghost diagrams for loop
level scattering amplitudes. We hope to return to such questions in a future
publication.

##  6 Acknowledgements

I am grateful to Kirill Krasnov for suggesting this interesting work and also
for various discussions. I am also grateful to K P Yogendran for carefully
reading the manuscript and for his insightful comments on the work.

##  7 Appendix: BRST closure

###  7.1 Diffeomorphism ghost

Let us start with the second transformation in (14), which is that of the
diffeomorphism ghost. We have

$\displaystyle\begin{split}s^{2}c_{\mu}&=sc^{\lambda}\nabla^{T}_{\lambda}c_{\mu}-c^{\lambda}\nabla^{T}_{\lambda}sc^{\mu}\\\
&=c^{\gamma}\nabla_{\gamma}c^{\lambda}\nabla_{\lambda}c^{\mu}-c^{\lambda}\nabla_{\lambda}c^{\nu}\nabla_{\nu}c^{\mu}-c^{\lambda}c^{\nu}\nabla_{\lambda}\nabla_{\nu}c^{\mu}\\\
&=-c^{\lambda}c^{\nu}\nabla_{\lambda}\nabla_{\nu}c^{\mu}\\\
&=-\frac{1}{2}(c^{\lambda}c^{\nu}-c^{\nu}c^{\lambda})\nabla_{\lambda}\nabla_{\nu}c^{\mu}\\\
&=-\frac{1}{2}c^{\lambda}c^{\nu}[\nabla_{\lambda},\nabla_{\nu}]c^{\mu}\\\
&=-\frac{1}{2}c^{\lambda}c^{\nu}R^{\mu}_{~{}\alpha\nu\lambda}c^{\alpha}\\\
&=-\frac{1}{2}c^{\alpha}c^{\lambda}c^{\nu}R^{\mu}_{~{}\alpha\nu\lambda}\\\
&=-\frac{1}{6}c^{\alpha}c^{\lambda}c^{\nu}(R^{\mu}_{~{}\alpha\nu\lambda}+R^{\mu}_{~{}\nu\lambda\alpha}+R^{\mu}_{~{}\lambda\alpha\nu})=0.\end{split}$
(114)

where we have used the sign carefully to take care of the anti-commutativity
of the ghosts. In the third line of the above, we used this property to write
the single term as a combination of two terms, which then becomes the
commutator of the covariant derivatives. Finally, we express this commutator
in terms of the Riemann tensor and used the symmetry properties of it to
deduce that the expression vanishes due to the first Bianchi identity.

###  7.2 Tetrad

For the transformation of the tetrad, we have

$\displaystyle\begin{split}s^{2}h^{i}_{\mu}&=sc^{\nu}\nabla_{\nu}h^{i}_{\mu}(x)-c^{\nu}\nabla_{\nu}sh^{i}_{\mu}(x)+sh^{i}_{\nu}(x)\nabla_{\mu}c^{\nu}+h^{i}_{\nu}(x)\nabla_{\mu}sc^{\nu}\\\
&=c_{\gamma}\nabla^{\gamma}c^{\delta}\nabla_{\delta}h^{i}_{\mu}-c^{\nu}\nabla_{\nu}c^{\delta}\nabla_{\delta}h^{i}_{\mu}-c^{\nu}c^{\delta}\nabla_{\delta}\nabla_{\nu}h^{i}_{\mu}\\\
&-c^{\gamma}\nabla_{\gamma}h^{i}_{\nu}\nabla_{\mu}c^{\nu}-c^{\gamma}h^{i}_{\nu}\nabla_{\gamma}\nabla_{\mu}c^{\nu}+c^{\gamma}\nabla_{\gamma}h^{i}_{\nu}\nabla_{\mu}c^{\nu}+h^{i}_{\gamma}\nabla_{\nu}c^{\gamma}\nabla_{\mu}c^{\nu}\\\
&+h^{i}_{\nu}\nabla_{\mu}c^{\gamma}\nabla_{\gamma}c^{\nu}+h^{i}_{\nu}c^{\gamma}\nabla_{\mu}\nabla_{\gamma}c^{\nu}\\\
&=h^{i}_{\nu}c^{\gamma}\nabla_{\mu}\nabla_{\gamma}c^{\nu}-c^{\gamma}h^{i}_{\nu}\nabla_{\gamma}\nabla_{\mu}c^{\nu}-c^{\gamma}c^{\delta}\nabla_{\delta}\nabla_{\gamma}h^{i}_{\mu}\\\
&=h^{i}_{\nu}c^{\gamma}[\nabla_{\mu},\nabla_{\gamma}]c^{\nu}+\frac{1}{2}c^{\delta}c^{\gamma}[\nabla_{\delta},\nabla_{\gamma}]h^{i}_{\mu}\\\
&=h^{i}_{\nu}c^{\gamma}R^{\nu}_{~{}\alpha\gamma\mu}c^{\alpha}+\frac{1}{2}c^{\delta}c^{\gamma}R^{\beta}_{~{}\mu\gamma\delta}h^{i}_{\beta}=h^{i}_{\nu}c^{\gamma}c^{\alpha}R^{\nu}_{~{}\alpha\gamma\mu}+\frac{1}{2}c^{\delta}c^{\gamma}R^{\beta}_{~{}\mu\gamma\delta}h^{i}_{\beta}\\\
&=\frac{1}{2}h^{i}_{\nu}(c^{\gamma}c^{\alpha}-c^{\alpha}c^{\gamma})R^{\nu}_{~{}\alpha\gamma\mu}+\frac{1}{2}c^{\delta}c^{\gamma}R^{\beta}_{~{}\mu\gamma\delta}h^{i}_{\beta}\\\
&=\frac{1}{2}h^{i}_{\nu}c^{\gamma}c^{\alpha}R^{\nu}_{~{}\alpha\gamma\mu}+\frac{1}{2}h^{i}_{\nu}c^{\gamma}c^{\alpha}R^{\nu}_{~{}\gamma\mu\alpha}+\frac{1}{2}c^{\delta}c^{\gamma}R^{\beta}_{~{}\mu\gamma\delta}h^{i}_{\beta}\\\
&=\frac{1}{2}h^{i}_{\nu}c^{\gamma}c^{\alpha}(R^{\nu}_{~{}\alpha\gamma\mu}+R^{\nu}_{~{}\gamma\mu\alpha}+R^{\nu}_{~{}\mu\alpha\gamma})=0.\end{split}$
(115)

In the second line of the above, many terms get cancelled and we are left with
only three terms. These terms are rearranged in the form of the commutator of
covariant derivatives which are further expressed again in terms of the
Riemann tensor.

###  7.3 Lorentz ghost

For the transformation of the Lorentz ghost field, we find

$\displaystyle\begin{split}s^{2}b^{ij}&=-\frac{1}{2}s[b,b]^{ij}+\frac{1}{2}s(c^{\mu}c^{\nu}F^{ij}_{~{}~{}\mu\nu})\\\
&=-sb^{i}_{~{}k}b^{kj}+b^{i}_{~{}k}sb^{kj}+\frac{1}{2}sc^{\mu}c^{\nu}F^{ij}_{~{}~{}\mu\nu}\\\
&-\frac{1}{2}c^{\mu}sc^{\nu}F^{ij}_{~{}~{}\mu\nu}+\frac{1}{2}c^{\mu}c^{\nu}sF^{ij}_{~{}~{}\mu\nu}\\\
&=b^{i}_{~{}m}b^{m}_{~{}k}b^{kj}-\frac{1}{2}c^{\mu}c^{\nu}F^{i}_{~{}k\mu\nu}b^{kj}-b^{i}_{~{}k}b^{k}_{~{}m}b^{mj}\\\
&+\frac{1}{2}b^{i}_{~{}k}c^{\mu}c^{\nu}F^{kj}_{~{}~{}\mu\nu}+\frac{1}{2}c^{\lambda}\nabla^{T}_{\lambda}c^{\mu}c^{\nu}F^{ij}_{~{}~{}\mu\nu}-\frac{1}{2}c^{\mu}c^{\lambda}\nabla^{T}_{\lambda}c^{\nu}F^{ij}_{~{}~{}\mu\nu}\\\
&+\frac{1}{2}c^{\mu}c^{\nu}c^{\lambda}\nabla^{T}_{\lambda}F^{ij}_{~{}~{}\mu\nu}+\frac{1}{2}c^{\mu}c^{\nu}F^{ij}_{~{}~{}\mu\lambda}\nabla^{T}_{\nu}c^{\lambda}+\frac{1}{2}c^{\mu}c^{\nu}F^{ij}_{~{}~{}\lambda\nu}\nabla^{T}_{\mu}c^{\lambda}+\frac{1}{2}c^{\mu}c^{\nu}[F_{\mu\nu},b]^{ij}\\\
&=\frac{1}{2}c^{\mu}c^{\nu}c^{\lambda}\nabla^{T}_{\lambda}F^{ij}_{~{}~{}\mu\nu}=\frac{1}{6}c^{\mu}c^{\nu}c^{\lambda}(\nabla^{T}_{\lambda}F^{ij}_{~{}~{}\mu\nu}+\nabla^{T}_{\mu}F^{ij}_{~{}~{}\nu\lambda}+\nabla^{T}_{\nu}F^{ij}_{~{}~{}\lambda\mu})\\\
&=0.\end{split}$ (116)

where in the last line, we used the Bianchi identity and the fact that the
Lorentz ghost fields anti commute and hence can be appropriately permuted to
give rise to three such terms.

###  7.4 Connection

Let us now check the closure for the connection field. We have the following.

$\displaystyle\begin{split}s^{2}\omega^{ij}_{~{}\mu}&=\nabla^{T}_{\mu}sb^{ij}+sc^{\nu}F^{ij}_{~{}~{}\nu\mu}-c^{\nu}sF^{ij}_{~{}~{}\nu\mu}\\\
&=-\nabla^{T}_{\mu}(b^{i}_{~{}k}b^{kj})+\frac{1}{2}\nabla^{T}_{\mu}(c^{\mu}c^{\nu}F^{ij}_{~{}~{}\mu\nu})+c^{k}\nabla^{T}_{k}c^{\nu}F^{ij}_{~{}~{}\nu\mu}-c^{\nu}[F_{\nu\mu},b]^{ij}\\\
&-c^{\nu}c^{\lambda}\nabla^{T}_{\lambda}F^{ij}_{~{}~{}\nu\mu}-c^{\nu}F^{ij}_{~{}~{}\nu\lambda}\nabla^{T}_{\mu}c^{\lambda}-c^{\nu}F^{ij}_{~{}~{}\lambda\mu}\nabla^{T}_{\nu}c^{\lambda}\\\
&=-\nabla^{T}_{\mu}(b^{i}_{~{}k}b^{kj})-\frac{1}{2}c^{\nu}\nabla^{T}_{\mu}c^{\mu}F^{ij}_{~{}~{}\mu\nu}+\frac{1}{2}c^{\mu}\nabla^{T}_{\mu}c^{\nu}F^{ij}_{~{}~{}\mu\nu}\\\
&+\frac{1}{2}c^{\mu}c^{\nu}\nabla^{T}_{\mu}F^{ij}_{~{}~{}\mu\nu}+c^{k}\nabla^{T}_{k}c^{\nu}F^{ij}_{~{}~{}\nu\mu}-c^{\nu}c^{\lambda}\nabla^{T}_{\lambda}F^{ij}_{~{}~{}\nu\mu}\\\
&-c^{\nu}F^{ij}_{~{}~{}\nu\lambda}\nabla^{T}_{\mu}c^{\lambda}-c^{\nu}F^{ij}_{~{}~{}\lambda\mu}\nabla^{T}_{\nu}c^{\lambda}-c^{\nu}[F_{\nu\mu},b]^{ij}\\\
&=-\nabla^{T}_{\mu}(b^{i}_{~{}k}b^{kj})-c^{\nu}[F_{\nu\mu},b]^{ij}\\\
&-\frac{1}{2}c^{\nu}c^{\lambda}(\nabla^{T}_{\lambda}F^{ij}_{~{}~{}\nu\mu}+\nabla^{T}_{\nu}F^{ij}_{~{}~{}\mu\lambda}+\nabla^{T}_{\mu}F^{ij}_{~{}~{}\lambda\nu})\\\
&=0.\end{split}$ (117)

where in the above, we used the Bianchi identity for the curvature,
interchanged dummy indices and used the anti-commutative nature of the ghost
fields. This completes the check of the nilpotency of the BRST operator.

## References

  * [1] Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton A. Palatini Rend.Circ.Mat.Palermo 43 (1919) 203-212
  * [2] G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in the theory of gravitation,” Ann. Inst. H. Poincare Phys. Theor. A 20 (1974), 69-94
  * [3] M. H. Goroff and A. Sagnotti, “QUANTUM GRAVITY AT TWO LOOPS,” Phys. Lett. B 160 (1985), 81-86 doi:10.1016/0370-2693(85)91470-4
  * [4] M. H. Goroff and A. Sagnotti, “The Ultraviolet Behavior of Einstein Gravity,” Nucl. Phys. B 266 (1986), 709-736 doi:10.1016/0550-3213(86)90193-8
  * [5] F. Gronwald and F. W. Hehl, “On the gauge aspects of gravity,” [arXiv:gr-qc/9602013 [gr-qc]].
  * [6] A. M. Frolov, N. Kiriushcheva and S. V. Kuzmin, “The Hamiltonian Formulation of Tetrad Gravity: Three-Dimensional Case,” Grav. Cosmol. 16 (2010), 181-194 doi:10.1134/S0202289310030011 [arXiv:0902.0856 [gr-qc]].
  * [7] N. Kiriushcheva and S. V. Kuzmin, “The Hamiltonian formulation of N-bein, Einstein-Cartan, gravity in any dimension: The Progress Report,” [arXiv:0907.1553 [gr-qc]].
  * [8] I. L. Shapiro and P. M. Teixeira, “Quantum Einstein-Cartan theory with the Holst term,” Class. Quant. Grav. 31 (2014), 185002 doi:10.1088/0264-9381/31/18/185002 [arXiv:1402.4854 [hep-th]].
  * [9] J. F. Plebanski, “On the separation of Einsteinian substructures,” J. Math. Phys. 18 (1977), 2511-2520 doi:10.1063/1.523215
  * [10] R. Capovilla, T. Jacobson, J. Dell and L. J. Mason, “Selfdual two forms and gravity,” Class. Quant. Grav. 8 (1991), 41-57 doi:10.1088/0264-9381/8/1/009
  * [11] K. Krasnov, “Self-Dual Gravity,” Class. Quant. Grav. 34 (2017) no.9, 095001 doi:10.1088/1361-6382/aa65e5 [arXiv:1610.01457 [hep-th]].
  * [12] Z. Bern, L. J. Dixon, M. Perelstein and J. S. Rozowsky, “One loop n point helicity amplitudes in (selfdual) gravity,” Phys. Lett. B 444 (1998), 273-283 doi:10.1016/S0370-2693(98)01397-5 [arXiv:hep-th/9809160 [hep-th]].
  * [13] K. Krasnov and E. Skvortsov, “Flat Self-dual Gravity,” doi:10.1007/JHEP08(2021)082 [arXiv:2106.01397 [hep-th]].
  * [14] M. Abou-Zeid and C. M. Hull, “A Chiral perturbation expansion for gravity,” JHEP 02 (2006), 057 doi:10.1088/1126-6708/2006/02/057 [arXiv:hep-th/0511189 [hep-th]].
  * [15] K. Krasnov and Y. Shtanov, “Chiral perturbation theory for GR,” JHEP 09 (2020), 017 doi:10.1007 [arXiv:2007.00995 [hep-th]].
  * [16] K. Groh, K. Krasnov and C. F. Steinwachs, “Pure connection gravity at one loop: Instanton background,” JHEP 07 (2013), 187 doi:10.1007 [arXiv:1304.6946 [hep-th]].
  * [17] K. Krasnov, “One-loop beta-function for an infinite-parameter family of gauge theories,” JHEP 03 (2015), 030 doi:10.1007 [arXiv:1501.00849 [hep-th]].
  * [18] S. M. Christensen and M. J. Duff, “Quantizing Gravity with a Cosmological Constant,” Nucl. Phys. B 170 (1980), 480-506 doi:10.1016/0550-3213(80)90423-X
  * [19] R. Camporesi and A. Higuchi, “On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces,” J. Geom. Phys. 20 (1996), 1-18 doi:10.1016/0393-0440(95)00042-9 [arXiv:gr-qc/9505009 [gr-qc]].
  * [20] C. Rigouzzo and S. Zell, “Coupling Metric-Affine Gravity to the Standard Model and Dark Matter Fermions,” [arXiv:2306.13134 [gr-qc]].
  * [21] D. V. Vassilevich, “Heat kernel expansion: User’s manual,” Phys. Rept. 388 (2003), 279-360 doi:10.1016/j.physrep.2003.09.002 [arXiv:hep-th/0306138 [hep-th]].
  * [22] A. L. Besse, “Einstein Manifolds,” Springer-Verlag, 1987, ISBN 978-0-387-15279-0
  * [23] K. Krasnov, “Formulations of General Relativity,” doi:10.1017/9781108674652
  * [24] Penrose R, Rindler W., ”1984 Spinors and spacetime vol 1, UK: Cambridge University Press.”


# Searches for $pp\rightarrow H^{\pm}h\rightarrow W^{\pm}hh$ signal in the
MSSM model

Tetiana Obikhod and Ievgenii Petrenko

(Institute for Nuclear Research NAS of Ukraine, Kyiv 03028, Ukraine
)

##  1 Abstract

We considered the new signature of a charged Higgs boson, $pp\rightarrow
H^{\pm}h\rightarrow W^{\pm}hh$, which is almost background free as the
promising to study of supersymmetry at the LHC with both $\sqrt{s}=13$ TeV and
$\sqrt{s}=14$ TeV. In the framework of MSSM model we used the latest
theoretical parameter space, which is consistent with the experimental
constraints from Large Hadron Collider and direct or indirect searches. In the
surviving regions with the help of Pythia 8.3 and Feynhiggs programs it was
found that production cross sections, $\sigma(pp\rightarrow H^{\pm}h)$, and
Branching Ratios, $BR(H^{\pm}\rightarrow W^{\pm}h)$, are the largest at
Benchmark Points 4 and 14 correspondingly.

##  2 Introduction

Experimental searches for supersymmetry, associated with a number of key
unresolved problems in the Standard Model (SM), are the most privileged
channel for the search for new physics beyond the Standard Model (BSM).
Searches for the extended Higgs boson sector as the lightest superparticles
with masses up to 1 TeV are the most promising in this aspect. This is
facilitated by both experimental data on searches for superparticles with
masses of up to 100 GeV [1, 2] and theoretical constructions. Among
supersymmetric models, the most popular is 2-Higgs Doublet Model (2HDM) with
few Benchmark Points (BPs) which pass all present constraints, both
theoretical and experimental. There were performed Monte Carlo (MC) analysis
of different supersymmetry search channels, one of which, $W^{\pm}+4\gamma$,
is very promising [3], as it can be connected with significant excesses at the
LHC.

Since 2HDM is a tree-level Minimal supersymmetry standard model (MSSM) [4],
the purpose of the article is to carry out calculations for the production
cross sections $\sigma(pp\rightarrow H^{\pm}h)$, and Branching Ratios (BR)
$BR(H^{\pm}\rightarrow W^{\pm}h)$ using the selected BP scenarios within the
extended supersymmetry model, MSSM. It is important to select the most
significant events based on the calculation results and find the corresponding
masses and momentum distributions of the Higgs bosons. In addition, it is
necessary to compare the obtained data with similar calculations within the
framework of 2HDM in order to select the most suitable scenarios for the
search for supersymmetry.

##  3 Theory of 2HDM and MSSM model

Searches for BSM physics, especially supersymmetry, is a privileged area of
modern experimental high-energy physics. The most popular theory of
supersymmetry MSSM. At tree-level this model goes to another model, 2HDM as
its low-energy limit with neutral CP-even scalars (h and H), CP-odd
pseudoscalar (A) and two charged Higgs bosons (H±). To show this, it must be
noted that in the MSSM, the terms contributing to the scalar Higgs potential
VH come from three different sources:
i) the D terms

$V_{D}=\frac{g^{2}_{2}}{8}\Biggl{[}4|H^{+}_{1}\cdot
H_{2}|^{2}-2|H_{1}|^{2}|H_{2}|^{2}+(|H_{1}|^{2})^{2}+(|H_{2}|^{2})^{2}\Biggr{]}+\frac{g^{2}_{1}}{8}(|H_{2}|^{2}-|H_{1}|^{2})^{2}$

ii) the F terms

$V_{F}=\mu^{2}(|H_{1}|^{2}+|H_{2}|^{2})$

iii) a piece originating from the soft SUSY–breaking scalar Higgs mass terms
and the bilinear term

$V_{soft}=m^{2}_{H_{1}}H^{+}_{1}H_{1}+m^{2}_{H_{2}}H^{+}_{2}H_{2}+B\mu(H_{2}\cdot
H_{1}+h.c.)$

The full scalar potential involving the Higgs fields is the sum of the three
terms [5]

$V_{H}=(|\mu|^{2}+m^{2}_{H_{1}})|H_{1}|^{2}+(|\mu|^{2}+m^{2}_{H_{2}})|H_{2}|^{2}-\mu
Be_{ij}(H^{i}_{1}H^{j}_{2}+h.c.)+$
$\frac{g^{2}_{2}+g^{2}_{1}}{8}(|H_{1}|^{2}-|H_{2}|^{2})^{2}+\frac{1}{2}g^{2}_{2}|H^{+}_{1}H_{2}|^{2}$
(1)

The most general supersymmetric potential for two Higgs doublets (Hu and
$H_{d}$) at tree-level reads [6]

$V^{LO}_{MSSM}=(m^{2}_{H_{d}}+\mu^{2})|H_{d}|^{2}+(m^{2}_{H_{u}}+\mu^{2})|H_{u}|^{2}$
$-B\mu\varepsilon_{ij}(H^{i}_{d}H^{j}_{u}+h.c.)$
$+\frac{g^{2}+g^{{}^{\prime}2}}{8}(|H_{d}|^{2}-|H_{u}|^{2})^{2}+\frac{g^{2}}{2}|H^{*}_{d}H_{u}|^{2}$
(2)

where $m^{2}_{H_{d}},\ m^{2}_{H_{u}},\ B\mu$ denote the soft-SUSY breaking
mass parameters, $g$ and $g^{{}^{\prime}}$ \- the SU(2)L and U(1)Y gauge
couplings. From comparison of formula (1) and (2) we see matching expressions,
which confirms the coincidence of 2HDM with the MSSM model at the tree-level.
The advantage of 2HDM model is the possibility to avoid large effects of
Flavour Changing Neutral Currents.

##  4 Results of calculations

Using article data [7] we’ll focus on the decay modes of the charged Higgs
boson $H^{\pm}\rightarrow W^{\pm*}h$ in the 2HDM Type-I satisfying the
condition $M_{H^{\pm}}<M_{t}+M_{b}$. It is assumed that the properties of
heavy scalar H are consistent with the LHC measurements, and h is lighter than
125 GeV. In this paper was emphasized that the production and decay process
$pp\rightarrow H^{\pm}h\rightarrow W^{\pm*}hh\rightarrow l^{\pm}\nu+4\gamma$
is background free, so $W^{\pm}+4\gamma$ signal would yield significant
excesses at the LHC with an integrated luminosity of L = 300 fb-1 at both
$\sqrt{s}$ = 13 and 14 TeV.

So, this search channel is of particular interest for exploiting a MC analysis
at a detector level by including parton shower, hadronisation and heavy
flavour decays, [8], so that significances only depend upon the signal cross
sections and the integrated luminosity. Moreover, the estimate of the signal
significance over the Type-I parameter space, could be useful for LHC
experimental groups.

To estimate the production cross section of considered signal
$\sigma(pp\rightarrow H^{\pm}h)$ we choose the input parameters of more
general MSSM model in the form of BP of [7], presented in Table 1

Table 1. Input parameters of MSSM model
BP $M_{h}$ $M_{A}$ $M_{H^{\pm}}$ tan$\beta$ BP3 45.34 162.07 128.00 7.57 BP4
53.59 126.09 91.49 8.00 BP5 63.13 85.59 104.99 18.09 BP6 65.43 111.43 142.15
11.52 BP7 67.82 79.83 114.09 8.94 BP9 73.18 108.69 97.34 8.06 BP14 81.53
225.76 168.69 9.75

Then with the help of Pythia 3 [9] we calculated production cross sections at
13 and 14 TeV presented in Table 2 and Table 3

Table 2. Production cross sections, $\sigma(pp\rightarrow H^{\pm}h)$, at 13
TeV
for the corresponding BPs
BP $\sigma$ (in pb) statist. err. BP3 5.235e-01 8.977e-03 BP4 1.104e+00
1.877e-02 BP5 6.715e-01 1.103e-02 BP6 2.813e-01 4.740e-03 BP7 6.287e-01
1.044e-02 BP9 1.506e-01 2.536e-03 BP14 0.000e+00 0.000e+00

Table 3. Production cross sections, $\sigma(pp\rightarrow H^{\pm}h)$, at 14
TeV
for the corresponding BPs
BP $\sigma$ (in pb) statist. err. BP3 5.756e-01 9.561e-03 BP4 1.232e+00
2.076e-02 BP5 7.049e-01 1.161e-02 BP6 3.278e-01 5.297e-03 BP7 7.149e-01
1.190e-02 BP9 1.668e-01 2.783e-03 BP14 0.000e+00 0.000e+00

Using FeynHiggs program [10] we calculated BR$(H^{\pm}\rightarrow W^{\pm}h)$,
presented in Table 4.

Table 4. Branching Ratios, $BR(H^{\pm}\rightarrow W^{\pm}h)$, for the
corresponding BPs
BP BR BP3 0.002885294 BP4 0.001397527 BP5 0.0005224388 BP6 0.000561978 BP7
0.00259184 BP9 0.001528386 BP14 0.02296814

We received the mass and momentum distributions for the maximal value of
production cross section of BP4 for h boson, Fig. 1

_Fig.1_ _Mass (left) and momentum (right) distributions of h boson at BP4
scenario._

As for H+ boson we obtained the data in Fig. 2

_Fig.2_ _Mass (left) and momentum (right) distributions of H + boson at BP4
scenario. _

From the data presented in Fig. 1 and Fig. 2, we can conclude that the mass of
the lightest Higgs boson is 50 GeV at a maximum momentum of 50 GeV and the
mass of a charged H+ boson is 140 GeV at a maximum momentum of 100 GeV. As for
BR, it is not changed with energy and has maximal value for BP14 parameter
space without change of the Mh and M${}_{H^{\pm}}$ parameters and is equal to
0.02296814.

It should be emphasized that the supersymmetric structure of the theory has
imposed very strong constraints on the Higgs spectrum, so for the calculations
we used six parameters which describe the MSSM Higgs sector, Mh, MH, MA,
M${}_{H^{\pm}}$, $\beta$, $\alpha$. Out of the six parameters which describe
the MSSM Higgs sector, only two parameters, tan$\beta$ and MA, are free
parameters at the tree-level. In this case, calculations according to Pythia
for two parameters, tan$\beta$ and MA give almost identical results for any BP
choice, which is presented in Table 5.

Table 5. Production cross sections, $\sigma(pp\rightarrow H^{\pm}h)$, at 13
TeV
for the corresponding BPs at tree-level of MSSM model
BP $\sigma$ (in pb) statist. err. BP3 2.152e-02 3.688e-04 BP4 2.090e-02
3.577e-04 BP5 2.065e-02 3.420e-04 BP6 2.128e-02 3.594e-04 BP7 2.106e-02
3.608e-04 BP9 2.124e-02 3.619e-04 BP14 0.000e+00 0.000e+00

##  5 Conclusions

We compared MSSM and 2HDM models and demonstrated their coincidence at tree-
level. We considered the restricted parameter space of more general MSSM model
with six parameters, connected with the experimental constraints from Large
Hadron Collider in the form of Benchmark Points. In the surviving regions with
the help of Pythia 8.3 and Feynhiggs programs it was found that production
cross sections, $\sigma(pp\rightarrow H^{\pm}h)$, and $BR(H^{\pm}\rightarrow
W^{\pm}h)$, are the largest at BP4 and BP14 correspondingly. As for the
calculations within 2HDM model [7], the largest values of production cross
sections are observed for the BP8 scenario at both energies of 13 and 14 TeV.
The corresponding mass distributions for the lightest Higgs boson is 50 GeV at
a maximum momentum of 50 GeV and the mass of a charged H+ boson is 140 GeV at
a maximum momentum of 100 GeV. As for BR, it has maximal value for BP14
parameter space without change of the Mh and M${}_{H^{\pm}}$ parameters and is
equal to 0.02296814.

## References

  * [1] Heinemeyer, S. et al. A 96 GeV Higgs Boson in the 2HDM plus Singlet, arXiv:2112.11958 [hep-ph].
  * [2] CMS Collaboration. Search for a standard model-like Higgs boson in the mass range between 70 and 110 GeV in the diphoton final state in proton-proton collisions at $\sqrt{s}$= 8 and 13 TeV. Phys. Lett. B 793, pp. 320 – 347 (2019).
  * [3] Arhrib, A. et al. Identifying a light charged Higgs boson at the LHC Run II. Phys. Lett. B 774,591-598 (2017), arXiv:1706.01964 [hep-ph].
  * [4] Branco, G.C. et al. Theory and phenomenology of two-Higgs-doublet models. Phys. Rept. 516, 1–102 (2012), arXiv:hep-ph/1106.0034.
  * [5] Gunion, J.F. and Haber, H.E. Errata for Higgs Bosons in Supersymmetric Models: I, II and III. Nucl. Phys. B272 (1986) 1; (E), hep-ph/9301205.
  * [6] Liebler, S. et al. The hMSSM approach for Higgs self-couplings revisited. Eur. Phys. J. C 79:65 (2019).
  * [7] Wang, Y. et al. Analysis of W${}^{\pm}+4\gamma$ in the 2HDM Type-I at the LHC. JHEP volume 2021, Article number: 21 (2021), arXiv:2107.01451 [hep-ph].
  * [8] Wang, Y. et al. Searching for H${}^{\pm}\rightarrow W^{\pm}+4\gamma$ signals in the 2HDM Type-I at the LHC, arXiv:2111.12286 [hep-ph].
  * [9] Sj$\ddot{o}$strand, T. et al. An Introduction to PYTHIA 8.2. Comput. Phys.Commun. 191 (2015) 159, arXiv:1410.3012 [hep-ph].
  * [10] FeynHiggs programm, URL:https://wwwth.mpp.mpg.de/members/heinemey/feynhiggs/cite.html.