YITP-22-16

Covariant BRST Quantization of Unimodular Gravity II
— Formulation with a vector antighost —

Taichiro Kugo,1,***e-mail address<EMAIL_ADDRESS>Ryuichi
Nakayama,2,†††e-mail address<EMAIL_ADDRESS>and
Nobuyoshi Ohta 3,4,‡‡‡e-mail address<EMAIL_ADDRESS>
1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
Japan

2 Division of Physics, Graduate School of Science, Hokkaido University,
Sapporo 060-0810, Japan

3 Department of Physics, National Central University, Zhongli, Taoyuan 320317,
Taiwan

and

4 Research Institute for Science and Technology, Kindai University, Higashi-
Osaka, Osaka 577-8502, Japan

Abstract

In our previous paper, we have presented a covariant BRST quantization of
unimodular gravity which may account for the smallness of the cosmological
constant, and we have shown that the physical degrees of freedom in the theory
are the same as general relativity. The formulation has been given by using
rank-2 antisymmetric tensor fields for both ghosts and antighosts. Here we
give an alternative formulation using a vector field for the antighost but
keeping the same structure for the ghosts. This gives a significantly simpler
covariant quantization with less ghosts and no tripole modes in the ghost
sector. We show that this also gives only two physical transverse modes as in
general relativity.

##  1 Introduction

In our previous paper [1], hereafter referred to as I, we have presented a
covariant local BRST quantization of unimodular gravity (UG), and have shown
that the physical degrees of freedom (dofs) in the theory are two transverse
modes as in general relativity (GR). UG is an interesting theory that may
explain why the cosmological constant is extremely small [2, 3, 4, 5, 6, 7].

UG can be formulated as GR with the constraint that the determinant of the
metric should be a fixed volume form in the general relativity:

$\displaystyle S_{\rm UG}=Z_{N}\int
d^{4}x\left[\sqrt{-g}R+\lambda(\sqrt{-g}-\omega)\right],$ (1.1)

where $Z_{N}=1/(16\pi G_{N})$ with $G_{N}$ being the Newton constant, and
$\lambda$ is a Lagrange multiplier field to impose the constraint

$\displaystyle\sqrt{-g}=\omega,$ (1.2)

with $\omega$ being a fixed volume form. Because of the unimodular constraint
$\sqrt{-g}=\omega$, we can derive only the traceless part of the Einstein
equation even if there may be a “cosmological constant” in the action. The
real cosmological constant may be introduced as an integration constant, and
thus is determined by the boundary condition, not by a constant term in the
action even if we have such a term.

The question how many physical dofs there exist in UG in the covariant
quantization is a nontrivial problem and there has been a lot of debate [8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. The reason is the
following. In the covariant BRST quantization of GR, there exist full
diffeomorphism, and we have four sets of ghosts and antighosts. This leaves
$10-8=2$ dofs in GR. However in UG, we have only transverse (or volume-
preserving) diffeomorphism TDiff:

$\displaystyle\delta_{{\rm B}}g^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\nabla^{\mu}c^{\nu}_{{\rm T}}-\nabla^{\nu}c^{\mu}_{{\rm T}},$
(1.3) $\displaystyle\delta_{{\rm B}}\lambda$ $\displaystyle=$ $\displaystyle
0,$ (1.4)

expressed in terms of diffeomorphism Faddeev-Popov (FP) ghosts, $c^{\mu}_{{\rm
T}}$, which satisfies a transversality condition:

$\nabla_{\mu}c^{\mu}_{{\rm T}}=0.$ (1.5)

The condition (1.5) eliminates one dof from the FP ghosts. We would also have
the same number of antighosts. Consequently the BRST quantization of this
system introduces only three sets of ghosts and antighosts. We also have
unimodular constraint, but it does not appear to introduce additional set of
ghost and antighost. Thus it seems that we are left with $10-6-1=3$ dofs, one
more dof than GR.

In order to quantize UG in the covariant manner, we have to realize the
symmetry (1.3) off shell, i.e. without using field equations. This is a
nontrivial task, and it may appear to require nonlocal projection operator
[14]. However this is not the case, and it has long been known in
supergravity[25, 26, 27] that a vector subject to transverse constraint can be
expressed by an unconstrained antisymmetric tensor even in the curved
spacetime. In our previous paper I, based on this idea, we have expressed the
reparametrization ghosts as

$c^{\mu}_{{\rm T}}=\nabla_{\nu}c^{\nu\mu},$ (1.6)

which automatically satisfy the transverse condition with rank-2 antisymmetric
tensor ghost $c^{\mu\nu}$. It turns out that after the first gauge fixing, the
ghost system needs the ghosts for ghosts [28, 29, 30]. The reason is clear:
The unconstrained rank-2 tensor has 6 dofs which are more than required to
express the transverse vector ghost modes with 3 dofs. We have found that this
redundancy manifests itself in the form of the gauge invariance in the ghost
system, and this further requires the gauge fixing and the introduction of
ghosts, reducing the number of dofs. Thus the ghost system becomes
significantly more complicated than usual. Since we must have the same number
of the antighosts, it is natural to introduce similar rank-2 tensor
antighosts, and this further requires the ghosts for ghosts. We have to
continue the gauge fixing and the introduction of the ghosts until there
remains no more gauge invariance. The important discovery in I is that after
all this gauge fixing of TDiff, the multiplier field $\lambda$ is actually
identified with a BRST daughter. This means that there exists a set of ghost
and antighost corresponding to the unimodular constraint after gauge fixing
only TDiff, contrary to the above naive expectation. This is the key
observation to get the correct number of dofs.

This formulation is nice in the sense that it gives a formulation symmetric in
ghosts and antighosts, but use of the rank-2 antisymmetric tensors for both
ghosts and antighosts gives complicated structure because of the necessity of
the ghosts for ghosts in both sectors. Here we note that what is really
required for the off-shell gauge fixing of TDiff is to use the antisymmetric
tensor fields only for the ghosts (not antighosts) to express the transverse
transformation parameter as in (1.6). We further notice that, in our other
paper [24] for the quantization of GR in unimodular gauge, we have actually
presented a general way of imposing such a transverse-vector gauge condition
by using a ($d$-component) vector antighost. It is realized at the price of
adding an extra scalar field BRST doublet, a set of BRST parent and daughter.
The variation of the action by this BRST daughter field impose the transverse
condition on shell on the vector antighost, thus leaving the necessary number
of dofs for antighosts. Because the structure in the antighost sector in this
formulation does not need the ghosts for ghosts, we expect that this
formulation gives considerably simpler covariant quantization of UG with less
ghosts. This is what we aim in this paper, and indeed we show that this
formulation successfully gives an alternative and simpler covariant BRST
quantization with correct number of dofs. It turns out that this formulation
also has the advantage that in the ghost and antighost sectors, there do not
exist tripole modes, which existed in I due to the use of the ghosts and
antighosts with derivatives (1.6). The formulation, however, gives the
asymmetric one in the ghost and antighost sectors.

This paper is organized as follows. In sect. 2, we start with the off-shell
gauge fixing of TDiff using rank-2 antisymmetric tensor ghosts and vector
antighosts. The structure in the ghost sector is basically the same as in I,
and we have the ghosts for ghosts phenomena. We gauge fix TDiff until there
remains no more invariance. The structure for the antighost sector is similar
to that in [24]. In sect. 3, to study the spectrum in the theory, we
concentrate on the theory at the linearized level. First, to check that we
have fully gauge fixed the gauge invariance, in subsect. 3.1, we show that
there indeed exist the propagators for the flat background for simplicity.
Next, in subsect. 3.2, we derive the equations of motion (EOMs) at the
linearized order. We can see that there is simplification in the antighost
sector, and there is no tripole field in ghosts and antighosts in contrast to
our pervious formulation in I though there is tripole in the graviton
excitation. In sect. 4, we identify which fields represent independent modes.
We use this result in sect. 5 to examine how most of the fields fall into the
BRST quartets and show that there remain only 2 physical dofs in the theory.
In sect. 6, we summarize our results and conclude the paper with some
discussions.

##  2 BRST quantization of unimodular gravity

The action (1.1) is invariant under the BRST transformation (1.3) and (1.4)
expressed in terms of diffeomorphism FP ghosts, $c^{\mu}_{{\rm T}}$, which
satisfies a transversality condition (1.5).

In our previous paper I, we have expressed this field in terms of an
unconstrained antisymmetric rank-2 tensor ghost $c^{\mu\nu}$ as (1.6). It was
shown in detail that imposing the nilpotency of the BRST transformation
automatically clarifies the existence of additional gauge invariance and
ghosts for ghosts [28, 29, 30]. Here we just summarize the result, referring
to I for the details. The BRST transformation laws are

$\displaystyle\delta_{{\rm B}}c^{\nu\mu}$ $\displaystyle=c^{\nu}_{{\rm
T}}c^{\mu}_{{\rm T}}+i\nabla_{\rho}d^{\rho\nu\mu},$ $\displaystyle\delta_{{\rm
B}}d^{\rho\nu\mu}$ $\displaystyle=ic^{\rho}_{{\rm T}}c^{\nu}_{{\rm
T}}c^{\mu}_{{\rm T}}-\nabla_{\sigma}t^{\sigma\rho\nu\mu},$ (2.1)
$\displaystyle\delta_{{\rm B}}t^{\sigma\rho\nu\mu}$
$\displaystyle=ic^{\sigma}_{{\rm T}}c^{\rho}_{{\rm T}}c^{\nu}_{{\rm
T}}c^{\mu}_{{\rm T}}.$

The field $d^{\rho\nu\mu}$ is a hermitian boson carrying double ghost number
$N_{\rm FP}=+2$ and denotes the ghost for the ghost corresponding to the gauge
transformation of $c^{\mu\nu}$ under which the “field strength” $c^{\mu}_{{\rm
T}}=\nabla_{\nu}c^{\nu\mu}$ is invariant. Another field $t^{\sigma\rho\nu\mu}$
is similarly introduced.

We now consider the BRST quantization of this UG system. Here the crucial
difference from our previous paper I is that we do not use the antisymmetric
tensor field as the multiplier BRST doublet field for fixing the
$(d-1)({=}3)$-component TDiff gauge invariance. Instead we use the method
developed in [24] to use a $d\,({=}4)$-component (unconstrained) vector
multiplier doublet field $(\bar{c}_{\mu},\,b_{\mu})$:

$\delta_{\rm B}\bar{c}_{\mu}=ib_{\mu}\,,$ (2.2)

for fixing the $(d-1)$-component TDiff gauge at the price of adding an extra
scalar BRST doublet field $(S,\,C_{S})$, transforming as

$\displaystyle\delta_{B}S$ $\displaystyle=$ $\displaystyle C_{S}\,.$ (2.3)

The gauge-fixing and FP ghost (GF+FP) terms in the first step are given by
[31, 32]

$\displaystyle{\cal L}_{\rm GF+FP,1}$ $\displaystyle=-i\delta_{{\rm
B}}\bigl{[}\bar{c}_{\mu}(\partial_{\lambda}\tilde{g}^{\lambda\mu}-\tilde{g}^{\mu\nu}\partial_{\nu}S)\bigr{]}$
$\displaystyle=b_{\mu}(\partial_{\lambda}\tilde{g}^{\lambda\mu}-\tilde{g}^{\mu\nu}\partial_{\nu}S)+i\bar{c}_{\mu}\left[\partial_{\nu}\delta_{{\rm
B}}(\tilde{g}^{\mu\nu})-\delta_{{\rm
B}}(\tilde{g}^{\mu\nu})\partial_{\nu}S-{\sqrt{-g}\,}\partial^{\mu}C_{S}\right],$
(2.4)

where $\tilde{g}^{\mu\nu}\equiv{\sqrt{-g}\,}g^{\mu\nu}$, and

$\displaystyle\delta_{\rm B}\tilde{g}^{\mu\nu}$
$\displaystyle=-{\sqrt{-g}\,}(\nabla^{\mu}c_{\rm T}^{\nu}+\nabla^{\nu}c_{\rm
T}^{\mu})+\tilde{g}^{\mu\nu}\nabla_{\lambda}c^{\lambda}_{\rm T}$
$\displaystyle=-{\sqrt{-g}\,}(\nabla^{\mu}\nabla_{\rho}c^{\rho\nu}+\nabla^{\nu}\nabla_{\rho}c^{\rho\mu}).$
(2.5)

Here in Eq. (2.4), we can see the double roles of the extra scalar $S$. First,
the gauge-fixing condition resulting from the variation of the multiplier
field $b_{\mu}$ is

$\partial_{\lambda}\tilde{g}^{\lambda\mu}-\tilde{g}^{\mu\nu}\partial_{\nu}S=0,$
(2.6)

which demands that the 4-component de Donder gauge condition
$\partial_{\lambda}\tilde{g}^{\lambda\mu}=0$ be satisfied aside from the
“longitudinal” component
$\tilde{g}^{\mu\nu}\partial_{\nu}S\equiv{\sqrt{-g}\,}\partial^{\mu}S$ which
remains arbitrary since $S$ is nowhere else specified. So the presence of
$\partial^{\mu}S$ term reduces the actual number of gauge conditions on the
metric field from 4 to 3. Second, the variation of the BRST partner $C_{S}$ of
$S$ gives the transverse constraint on the partner multiplier
$\bar{c}^{\mu}:=g^{\mu\nu}\bar{c}_{\nu}$ of $b^{\mu}:=g^{\mu\nu}b_{\nu}$:

$\partial_{\mu}({\sqrt{-g}\,}g^{\mu\nu}\bar{c}_{\nu})={\sqrt{-g}\,}\nabla_{\mu}\bar{c}^{\mu}=0.$
(2.7)

The variation of $S$ field itself yields the BRST transform of this equation
as the EOM:

$\delta_{\rm
B}[\partial_{\mu}({\sqrt{-g}\,}g^{\mu\nu}\bar{c}_{\nu})]={\sqrt{-g}\,}\nabla_{\mu}\delta_{\rm
B}\bigl{(}g^{\mu\nu}\bar{c}_{\nu}\bigr{)}={\sqrt{-g}\,}\bigl{(}\nabla_{\mu}b^{\mu}+\nabla_{\mu}\delta_{\rm
B}(g^{\mu\nu})\bar{c}_{\nu}\bigr{)}=0.$ (2.8)

Although $\delta_{\rm B}\bar{c}_{\nu}=b_{\nu}$, the contravariant multiplier
$b^{\mu}=g^{\mu\nu}b_{\nu}$ is required to be transverse on shell up to FP
ghost quadratic term $\nabla_{\mu}[\delta_{\rm
B}(\tilde{g}^{\mu\nu})\bar{c}_{\nu}]$.

This ghost Lagrangian (2.4) depends on $c^{\nu\mu}$ only through $c^{\mu}_{\rm
T}=\nabla_{\nu}c^{\nu\mu}$ and has the gauge invariance under the
transformations with rank-3 totally antisymmetric parameters
$\varepsilon^{\rho\nu\mu}$:

$\displaystyle\delta c^{\nu\mu}$
$\displaystyle=\nabla_{\rho}\varepsilon^{\rho\nu\mu}.$ (2.9)

This is just the gauge invariance already lifted in our BRST transformation
(2.1) with the ghost for ghost field $d^{\rho\nu\mu}$. We take the the
following gauge-fixing condition and introduce a multiplier BRST doublet to
impose it:

$\displaystyle\begin{array}[]{ccc}\text{gauge fixing cond.}&:&\text{multiplier
BRST doublet}\\\
\nabla^{[\rho}c^{\nu\mu]}=0&:&(\bar{d}_{\rho\nu\mu},\bar{c}_{\rho\nu\mu}),\quad\delta_{{\rm
B}}\bar{d}^{\rho\nu\mu}=\bar{c}^{\rho\nu\mu}.\end{array}$ (2.12)

Here and in what follows, the bracket $[\ ]$ attached to the indices means the
weight 1 antisymmetrization; e.g.,
$A_{[\mu}B_{\nu]}=(1/2)(A_{\mu}B_{\nu}-A_{\nu}B_{\mu})$. Similarly we will
also use $(\ )$ for the symmetrization with weight 1.

The GF+ FP terms in the second step are

$\displaystyle{\cal L}_{\rm GF+FP,2}$ $\displaystyle=$
$\displaystyle\frac{i}{2}{\sqrt{-g}\,}\delta_{{\rm
B}}\big{(}\bar{d}_{\rho\nu\mu}\nabla^{\rho}c^{\nu\mu}\big{)}{}=\frac{i}{2}{\sqrt{-g}\,}\delta_{\rm
B}\big{(}\bar{d}^{\rho\nu\mu}\nabla_{\rho}c_{\nu\mu}\big{)}$ $\displaystyle=$
$\displaystyle\frac{i}{2}{\sqrt{-g}\,}\big{[}\bar{c}^{\rho\nu\mu}\,\nabla_{\rho}c_{\nu\mu}-\nabla_{\rho}\bar{d}^{\rho\nu\mu}\cdot\delta_{{\rm
B}}(g_{\nu\sigma}g_{\mu\kappa})c^{\sigma\kappa}-\nabla^{\rho}\bar{d}_{\rho\sigma\kappa}\cdot(c^{\sigma}_{{\rm
T}}c^{\kappa}_{{\rm T}}+i\nabla_{\mu}d^{\mu\sigma\kappa})\big{]},$

where partial integrations have been performed in the second and third terms,
and use has been made of the commutativity $\delta_{\rm
B}\bigl{(}\nabla_{[\mu_{1}}A_{\mu_{2}\cdots\mu_{n}]}\bigr{)}=\nabla_{[\mu_{1}}\delta_{\rm
B}(A_{\mu_{2}\cdots\mu_{n}]})$ following from the equality
$\nabla_{[\mu_{1}}A_{\mu_{2}\cdots\mu_{n}]}=\partial_{[\mu_{1}}A_{\mu_{2}\cdots\mu_{n}]}$
valid for any totally antisymmetric tensor $A_{\mu_{2}\cdots\mu_{n}}$.

This action (LABEL:gfaction2) still has the gauge invariance under the
transformations [30]

$\displaystyle\delta d^{\rho\nu\mu}$
$\displaystyle=\nabla_{\sigma}\varepsilon^{\sigma\rho\nu\mu},$ (2.14)
$\displaystyle\delta\bar{d}^{\rho\nu\mu}$
$\displaystyle=\nabla_{\sigma}\bar{\varepsilon}^{\sigma\rho\nu\mu},$ (2.15)
$\displaystyle\delta\bar{c}^{\rho\nu\mu}$
$\displaystyle=\nabla_{\sigma}\bar{\theta}^{\sigma\rho\nu\mu},$ (2.16)

since it depends on these tensor fields only through their covariant
divergences like $\nabla_{\rho}d^{\rho\nu\mu}$, if partial integration is
performed when necessary. Here again, the first gauge invariance (2.14) is the
one already lifted in our BRST transformation (2.1) with the ghost for ghost
field $-t^{\sigma\rho\nu\mu}$. The second gauge transformation (2.15) for the
BRST parent field $\bar{d}^{\rho\nu\mu}$ is contained as a part of the
multiplier BRST transformation in Eq. (2.12). We fix the former two gauge
invariances by the following gauge-fixing conditions and introduce the
corresponding multiplier BRST doublets to impose them:

$\displaystyle\begin{array}[]{ccc}\text{gauge fixing cond.}&:&\text{multiplier
BRS doublet}\\\
\nabla^{[\sigma}d^{\rho\nu\mu]}=0&:&(\bar{t}_{\sigma\rho\nu\mu},\bar{d}_{\sigma\rho\nu\mu}),\quad\delta_{{\rm
B}}\bar{t}^{\sigma\rho\nu\mu}=i\bar{d}^{\sigma\rho\nu\mu},\\\
\nabla^{[\sigma}\bar{d}^{\rho\nu\mu]}=0&:&(c^{\sigma\rho\nu\mu},d^{\sigma\rho\nu\mu}),\quad\delta_{{\rm
B}}c^{\sigma\rho\nu\mu}=id^{\sigma\rho\nu\mu}.\end{array}$ (2.20)

The third gauge-invariance is automatically fixed by the gauge-fixing on
$\bar{d}_{\rho\nu\mu}$ of the second gauge-invariance because of $\delta_{\rm
B}\bar{d}^{\rho\nu\mu}=\bar{c}^{\rho\nu\mu}$.

The GF + FP Lagrangian in the third step is

$\displaystyle{\cal L}_{\rm GF+FP,3}$ $\displaystyle=-i\delta_{{\rm
B}}\Bigl{[}{\sqrt{-g}\,}\big{[}-\frac{1}{6}\bar{t}_{\sigma\rho\nu\mu}\bigl{(}\nabla^{\sigma}d^{\rho\nu\mu}+\frac{\alpha}{4}d^{\sigma\rho\nu\mu}\bigr{)}+\frac{1}{6}c^{\sigma\rho\nu\mu}\nabla_{\sigma}\bar{d}_{\rho\nu\mu}\big{]}\Bigr{]}$
$\displaystyle=\frac{1}{6}{\sqrt{-g}\,}\Big{[}-\bar{d}^{\sigma\rho\nu\mu}\nabla_{\sigma}d_{\rho\nu\mu}-\frac{\alpha}{4}\bar{d}^{\sigma\rho\nu\mu}d_{\sigma\rho\nu\mu}+i\nabla^{\sigma}\bar{t}_{\sigma\rho\nu\mu}\cdot\bigl{(}-\nabla_{\lambda}t^{\lambda\rho\nu\mu}+ic^{\rho}_{{\rm
T}}c^{\nu}_{{\rm T}}c^{\mu}_{{\rm T}}\bigr{)}$ $\displaystyle\hskip
40.00006pt{}+d^{\sigma\rho\nu\mu}\nabla_{\sigma}\bar{d}_{\rho\nu\mu}+ic^{\sigma\rho\nu\mu}\nabla_{\sigma}\bar{c}_{\rho\nu\mu}+i\nabla_{\sigma}\bar{t}^{\sigma\rho\nu\mu}\cdot
d^{\kappa\tau\lambda}\cdot\delta_{{\rm
B}}(g_{\rho\kappa}g_{\nu\tau}g_{\mu\lambda})$ $\displaystyle\hskip
30.00005pt{}-i\nabla_{\sigma}c^{\sigma\rho\nu\mu}\cdot\bar{d}^{\kappa\tau\lambda}\delta_{{\rm
B}}(g_{\rho\kappa}g_{\nu\tau}g_{\mu\lambda})\Big{]}.$ (2.21)

We have introduced a gauge parameter $\alpha$ for later convenience. The gauge
condition for the third gauge symmetry (2.16) follows from the terms
containing $c^{\sigma\rho\nu\mu}$ in (2.21). We see that our antighost system
is drastically simplified compared with that in I.

Now there remains no further invariance and we expect that the system is now
fully gauge fixed. To avoid too many tensor suffices, however, we rewrite the
antisymmetric tensor fields by their (Hodge) dual fields. Our sequence of
ghost fields, $c^{\mu\nu},\ d^{\mu\nu\rho}$ and $t^{\mu\nu\rho\sigma}$ are
expressed by their dual fields $C_{\mu\nu},\ D_{\mu},\ T$ (generally denoted
by the corresponding uppercase letters) as

$\displaystyle{\sqrt{-g}\,}c^{\mu\nu}$
$\displaystyle=\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}C_{\rho\sigma},$
$\displaystyle{\sqrt{-g}\,}d^{\mu\nu\rho}$
$\displaystyle=\varepsilon^{\mu\nu\rho\sigma}D_{\sigma},$
$\displaystyle{\sqrt{-g}\,}t^{\mu\nu\rho\sigma}$
$\displaystyle=\varepsilon^{\mu\nu\rho\sigma}T.$ (2.22)

Our convention for the $\varepsilon$ is $\varepsilon^{0123}=+1$ and
$\varepsilon_{0123}=-1$.

The $3$ multiplier BRST doublets are expressed by their duals as

$\displaystyle{\sqrt{-g}\,}\begin{pmatrix}\bar{d}^{\mu\nu\rho}\\\
\bar{c}^{\mu\nu\rho}\end{pmatrix}=-\varepsilon^{\mu\nu\rho\sigma}\begin{pmatrix}\bar{D}_{\sigma}\\\
\bar{C}_{\sigma}\end{pmatrix},\quad{\sqrt{-g}\,}\begin{pmatrix}\bar{t}^{\mu\nu\rho\sigma}\\\
\bar{d}^{\mu\nu\rho\sigma}\end{pmatrix}=-\varepsilon^{\mu\nu\rho\sigma}\begin{pmatrix}\bar{T}\\\
\bar{D}\end{pmatrix},$
$\displaystyle{\sqrt{-g}\,}\begin{pmatrix}c^{\mu\nu\rho\sigma}\\\
d^{\mu\nu\rho\sigma}\end{pmatrix}=-\varepsilon^{\mu\nu\rho\sigma}\begin{pmatrix}C\\\
D\end{pmatrix},$ (2.23)

Furthermore $c^{\mu}_{{\rm T}}$ should be understood to represent

$\displaystyle c^{\mu}_{{\rm T}}$
$\displaystyle=\nabla_{\nu}c^{\nu\mu}=-\frac{1}{2{\sqrt{-g}\,}}\varepsilon^{\mu\nu\rho\sigma}\partial_{\nu}C_{\rho\sigma}.$
(2.24)

In terms of these dual fields, the BRST transformations for the ghost fields
are rewritten as follows:

$\displaystyle\delta_{{\rm B}}C_{\mu\nu}$
$\displaystyle=-\frac{1}{2}{\sqrt{-g}\,}\varepsilon_{\mu\nu\rho\sigma}c^{\rho}_{{\rm
T}}c^{\sigma}_{{\rm T}}+i(\partial_{\mu}D_{\nu}-\partial_{\nu}D_{\mu}),$
$\displaystyle\delta_{{\rm B}}D_{\mu}$
$\displaystyle=\frac{i}{3!}{\sqrt{-g}\,}\varepsilon_{\mu\nu\rho\sigma}c^{\nu}_{{\rm
T}}c^{\rho}_{{\rm T}}c^{\sigma}_{{\rm T}}+\partial_{\mu}T,$
$\displaystyle\delta_{{\rm B}}T$
$\displaystyle=-\frac{i}{4!}{\sqrt{-g}\,}\varepsilon_{\mu\nu\rho\sigma}c^{\mu}_{{\rm
T}}c^{\nu}_{{\rm T}}c^{\rho}_{{\rm T}}c^{\sigma}_{{\rm T}}.$ (2.25)

The BRST transformations of multiplier BRST doublets are trivial for covariant
vectors and scalars:

$\displaystyle\delta_{{\rm B}}\bar{D}_{\mu}=\bar{C}_{\mu},\qquad\delta_{{\rm
B}}\bar{T}=i\bar{D},\qquad\delta_{{\rm B}}C=iD.$ (2.26)

By using these dual fields, the GF+FP ghost Lagrangians (2.4),
(LABEL:gfaction2) and (2.21) are rewritten as

$\displaystyle{\cal L}_{\rm GF+FP,1}$
$\displaystyle=b_{\mu}(\partial_{\lambda}\tilde{g}^{\lambda\mu}-\tilde{g}^{\mu\nu}\partial_{\nu}S)$
$\displaystyle\hskip
20.00003pt+i\bar{c}_{\mu}\Big{[}\partial_{\nu}\nabla^{(\mu}\big{(}\varepsilon^{\nu)\rho\sigma\lambda}\partial_{\rho}C_{\sigma\lambda}\big{)}-\delta_{{\rm
B}}(\tilde{g}^{\mu\nu})\partial_{\nu}S-\tilde{g}^{\mu\nu}\partial_{\nu}C_{S}\Big{]},$
(2.27) $\displaystyle{\cal L}_{\rm GF+FP,2}$ $\displaystyle=i\delta_{{\rm
B}}\Bigl{[}{\sqrt{-g}\,}\bar{D}_{\sigma}\nabla_{\rho}C^{\rho\sigma}\Bigr{]}$
$\displaystyle={\sqrt{-g}\,}\Big{[}i\bar{C}^{\sigma}\nabla^{\rho}C_{\rho\sigma}+\frac{3i}{4{\sqrt{-g}\,}}\varepsilon^{\kappa\mu\nu\lambda}\nabla^{\rho}\bar{D}^{\sigma}\cdot\nabla_{[\rho}C_{\kappa\sigma]}\cdot\nabla_{\mu}C_{\nu\lambda}$
$\displaystyle\hskip
10.00002pt{}+\big{(}\nabla^{\mu}\bar{D}^{\nu}-\nabla^{\nu}\bar{D}^{\mu}\big{)}\big{(}\nabla_{\mu}D_{\nu}-ig^{\rho\lambda}C_{\rho\nu}\delta_{{\rm
B}}g_{\mu\lambda}\big{)}\Big{]},$ (2.28) $\displaystyle{\cal L}_{\rm GF+FP,3}$
$\displaystyle=-i\delta_{{\rm
B}}\Bigl{[}{\sqrt{-g}\,}\bigl{(}\bar{T}\nabla_{\sigma}D^{\sigma}+\alpha\bar{T}D+C\nabla_{\sigma}\bar{D}^{\sigma}\bigr{)}\Bigr{]}$
$\displaystyle=\sqrt{-g}\Big{[}\bar{D}\nabla^{\mu}D_{\mu}+\alpha\bar{D}D+D\nabla_{\mu}\bar{D}^{\mu}+iC\nabla_{\mu}\bar{C}^{\mu}$
$\displaystyle\hskip
30.00005pt{}-\frac{1}{4}\nabla^{\sigma}\bar{T}\cdot\nabla_{[\sigma}C_{\nu\mu]}\big{(}\nabla^{\mu}C^{\lambda\rho}\cdot\nabla^{\nu}C_{\lambda\rho}-4\nabla_{\lambda}{C^{\mu}}_{\rho}\cdot\nabla^{\nu}C^{\lambda\rho}$
$\displaystyle\hskip
30.00005pt{}+2\nabla_{\lambda}{C^{\mu}}_{\rho}\cdot\nabla^{\lambda}C^{\nu\rho}-2\nabla_{\lambda}C^{\mu\rho}\cdot\nabla_{\rho}C^{\nu\lambda}\big{)}-i\nabla^{\mu}\bar{T}\cdot\nabla_{\mu}T$
$\displaystyle\hskip 30.00005pt{}+(i\nabla^{\mu}\bar{T}\cdot
D^{\nu}+i\nabla^{\mu}C\cdot\bar{D}^{\nu})\delta_{{\rm B}}g_{\mu\nu}\Big{]},$
(2.29)

where

$\displaystyle\delta_{{\rm B}}g_{\mu\nu}$
$\displaystyle=-({\sqrt{-g}\,})^{-1}g_{\lambda(\nu}\varepsilon^{\lambda\rho\sigma\tau}\nabla_{\mu)}\nabla_{\rho}C_{\sigma\tau}\
.$ (2.30)

is to be substituted in the above equations.

##  3 Propagators and equations of motion at linear order

###  3.1 Propagators

Now the total Lagrangian of our UG system is given by

${\cal L}_{\rm UG}={\sqrt{-g}\,}R+\lambda({\sqrt{-g}\,}-\omega)+{\cal L}_{\rm
GF+FP,1}(\ref{gfaction1P})+{\cal L}_{\rm GF+FP,2}(\ref{gfaction2P})+{\cal
L}_{\rm GF+FP,3}(\ref{gfaction3P})\,.$ (3.1)

Let us check in detail if we get nonsingular fully gauge fixed action on the
flat background with $\omega=1$. We introduce a fluctuation $h^{\mu\nu}$
around the flat metric $\eta^{\mu\nu}$ defined by

$\tilde{g}^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu},$ (3.2)

and then to the linear order we have

$g_{\mu\nu}=\eta_{\mu\nu}-h_{\mu\nu}+\frac{1}{2}\eta_{\mu\nu}h+\cdots,\qquad\sqrt{-g}=1+\frac{1}{2}h+\cdots$
(3.3)

In what follows indices of the fields will be raised and lowered by using
$\eta^{\mu\nu}$ and $\eta_{\mu\nu}$, respectively. The quadratic terms in our
total action are given by

$\displaystyle{\cal L}_{\rm UG}\Bigr{|}_{\rm quadr}$ $\displaystyle=$
$\displaystyle{\cal L}_{N_{{\rm FP}}=0}+{\cal L}_{|N_{{\rm FP}}|=1}+{\cal
L}_{|N_{{\rm FP}}|=2}+{\cal L}_{|N_{{\rm FP}}|=3},$ $\displaystyle{\cal
L}_{N_{{\rm FP}}=0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}h_{\mu\nu}\square
h^{\mu\nu}+\frac{1}{2}(\partial_{\nu}h^{\mu\nu})^{2}-\frac{1}{8}h\square
h+\frac{1}{2}\lambda
h+b_{\mu}(\partial_{\lambda}h^{\mu\lambda}-\partial^{\mu}S),$
$\displaystyle{\cal L}_{|N_{{\rm FP}}|=1}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\varepsilon^{\mu\nu\rho\sigma}\bar{c}_{\mu}\square\partial_{\nu}C_{\rho\sigma}-i\bar{c}_{\mu}\partial^{\mu}C_{S}+iC\partial_{\mu}\bar{C}^{\mu}+i\bar{C}^{\mu}\partial^{\nu}C_{\nu\mu},$
$\displaystyle{\cal L}_{|N_{{\rm FP}}|=2}$ $\displaystyle=$
$\displaystyle-\bar{D}^{\mu}(\square
D_{\mu}-\partial^{\nu}\partial_{\mu}D_{\nu})+\bar{D}\,\partial^{\mu}D_{\mu}+\alpha\bar{D}D+\partial_{\mu}\bar{D}^{\mu}\cdot
D,$ $\displaystyle{\cal L}_{|N_{{\rm FP}}|=3}$ $\displaystyle=$ $\displaystyle
i\bar{T}\,\square T.$ (3.4)

We start with $N_{{\rm FP}}=0$ sector. The 2-point vertex
$\Gamma^{(2)}_{N_{{\rm FP}}=0}$ in momentum space is

$\displaystyle\Gamma^{(2)}_{N_{{\rm FP}}=0}=$
$\displaystyle\bordermatrix{&h_{\rho\sigma}&S&b_{\rho}&\lambda\cr
h_{\mu\nu}&\begin{matrix}-p^{2}\,\Bigl{[}\frac{1}{2}P^{(2)\mu\nu,\rho\sigma}-\frac{1}{12}d^{\mu\nu}d^{\rho\sigma}\\\
-\frac{1}{4}\left(d^{\mu\nu}e^{\rho\sigma}+e^{\mu\nu}d^{\rho\sigma}\right)-\frac{3}{4}e^{\mu\nu}e^{\rho\sigma}\Bigr{]}\end{matrix}&0&-ip^{(\mu}\eta^{\nu)\rho}&\frac{1}{2}\eta^{\mu\nu}\cr
S&0&0&ip^{\rho}&0\cr
b_{\mu}&ip^{(\rho}\eta^{\sigma)\mu}&-ip^{\mu}&0&0\cr\lambda&\frac{1}{2}\eta^{\rho\sigma}&0&0&0\cr},$
(3.5)

by using the projection operators

$\displaystyle d_{\mu\nu}=\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}},\qquad
e_{\mu\nu}=\frac{p_{\mu}p_{\nu}}{p^{2}},$ (3.6) $\displaystyle
P^{(2)}_{\mu\nu,\rho\sigma}=\frac{1}{2}\left(d_{\mu\rho}d_{\nu\sigma}+d_{\mu\sigma}d_{\nu\rho}-\frac{2}{3}d_{\mu\nu}d_{\rho\sigma}\right),$
(3.7)

which satisfy

$\displaystyle p^{\mu}d_{\mu\nu}=0,\quad d_{\mu\nu}\eta^{\mu\nu}=3,\quad
e_{\mu\nu}\eta^{\mu\nu}=1,$ (3.8) $\displaystyle
d_{\mu\alpha}d^{\alpha\nu}=d_{\mu}{}^{\nu},\quad
e_{\mu\alpha}e^{\alpha\nu}=e_{\mu}{}^{\nu},\quad
d_{\mu\alpha}e^{\alpha\nu}=0,$ (3.9) $\displaystyle
P^{(2)}_{\mu\nu,\alpha\beta}d^{\alpha\beta}=0,\quad
P^{(2)}_{\mu\nu,\alpha\beta}e^{\alpha\beta}=0,\quad
P^{(2)}_{\mu\nu,\alpha\beta}P^{(2)\alpha\beta,\rho\sigma}=P^{(2)}_{\mu\nu}{}^{\rho\sigma}.$
(3.10)

We can straightforwardly compute the inverse of the matrix,
${\Gamma^{(2)}}^{-1}_{N_{{\rm FP}}=0}$:

$\displaystyle{\Gamma^{(2)}}^{-1}_{N_{{\rm FP}}=0}=\frac{1}{-p^{2}}\times$
$\displaystyle\bordermatrix{&h_{\rho\sigma}&S&b_{\rho}&\lambda\cr
h_{\mu\nu}&\begin{matrix}\Bigl{[}2P^{(2)}_{\mu\nu,\rho\sigma}-\frac{1}{3}d_{\mu\nu}d_{\rho\sigma}\\\
+\left(d_{\mu\nu}e_{\rho\sigma}+e_{\mu\nu}d_{\rho\sigma}\right)-3e_{\mu\nu}e_{\rho\sigma}\Bigr{]}\end{matrix}&d_{\mu\nu}-3e_{\mu\nu}&2ip_{(\mu}d_{\nu)\rho}&-p^{2}(d_{\mu\nu}-e_{\mu\nu})\cr
S&d_{\rho\sigma}-3e_{\rho\sigma}&-3&-ip_{\rho}&p^{2}\cr
b_{\mu}&-2ip_{(\rho}d_{\sigma)\mu}&ip_{\mu}&0&0\cr\lambda&-p^{2}(d_{\rho\sigma}-e_{\rho\sigma})&p^{2}&0&0\cr}.~{}~{}$
(3.11)

The 2-point vertex $\Gamma^{(2)}_{|N_{{\rm FP}}|=1}$ in momentum space is

$\displaystyle\Gamma^{(2)}_{|N_{{\rm
FP}}|=1}=\bordermatrix{&C_{\rho\sigma}&C&C_{S}\cr\bar{c}_{\mu}&\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}p^{2}p_{\nu}&0&p^{\mu}\cr\bar{C}_{\mu}&-p^{[\rho}\eta^{\sigma]\mu}&-p^{\mu}&0\cr},$
(3.12)

the inverse of which is given by

${\Gamma^{(2)}}^{-1}_{|N_{{\rm
FP}}|=1}=\frac{1}{-p^{2}}\times\bordermatrix{&\bar{c}_{\rho}&\bar{C}_{\rho}\cr
C_{\mu\nu}&\varepsilon_{\mu\nu\rho\lambda}p^{\lambda}/p^{2}&2p_{[\mu}\eta_{\nu]\rho}\cr
C&0&p_{\rho}\cr C_{S}&-p_{\rho}&0\cr}.$ (3.13)

The 2-point vertex $\Gamma^{(2)}_{|N_{{\rm FP}}|=2}$ in momentum space is

$\displaystyle\Gamma^{(2)}_{|N_{{\rm
FP}}|=2}=\bordermatrix{&D_{\rho}&D\cr\bar{D}^{\mu}&p^{2}d_{\mu}{}^{\rho}&-ip_{\mu}\cr\bar{D}&ip^{\rho}&\alpha\cr},$
(3.14)

the inverse of which is given by

${\Gamma^{(2)}}^{-1}_{|N_{{\rm
FP}}|=2}=\frac{1}{-p^{2}}\times\bordermatrix{&\bar{D}^{\rho}&\bar{D}\cr
D_{\mu}&\left[-\delta_{\mu}{}^{\rho}+(\alpha+1)p_{\mu}p^{\rho}/p^{2}\right]&ip_{\mu}\cr
D&-ip^{\rho}&0\cr}.$ (3.15)

Finally the 2-point vertex $\Gamma^{(2)}_{|N_{{\rm FP}}|=3}$ for $T$ and
$\bar{T}$ in momentum space is given by

$\displaystyle\Gamma^{(2)}_{|N_{{\rm FP}}|=3}=-ip^{2},$ (3.16)

and the inverse of which is given by

${\Gamma^{(2)}}^{-1}_{|N_{{\rm FP}}|=3}=\frac{i}{p^{2}}.$ (3.17)

Thus we have confirmed that the propagator may be obtained and the system is
fully gauge fixed.

###  3.2 Equations of motion at linear order

Let us denote the total action as $S$. The EOMs to linear order are given as
follows: for $\alpha=-1$,

$N_{\rm FP}=0$ sector $\displaystyle\frac{\delta S_{\rm
UG}}{\delta\lambda}:\quad$ $\displaystyle h(\equiv\eta_{\mu\nu}h^{\mu\nu})=0,$
(3.18) $\displaystyle\frac{\delta S_{\rm UG}}{\delta h^{\mu\nu}}:\quad$
$\displaystyle\frac{1}{2}\square
h_{\mu\nu}-\partial_{(\mu}(h_{\nu)}+b_{\nu)})+\frac{1}{2}\lambda\eta_{\mu\nu}=0,$
(3.19) $\displaystyle\frac{\delta S_{\rm UG}}{\delta b^{\mu}}:\quad$
$\displaystyle\partial^{\nu}h_{\mu\nu}-\partial_{\mu}S=0,$ (3.20)
$\displaystyle\frac{\delta S_{\rm UG}}{\delta S}:\quad$
$\displaystyle\partial^{\mu}b_{\mu}=0,$ (3.21) $N_{\rm FP}=\pm 1$ sector
$\displaystyle\frac{\delta S_{\rm UG}}{\delta\bar{c}^{\mu}}:\quad$
$\displaystyle\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\square\partial_{\nu}C_{\rho\sigma}-\partial^{\mu}C_{S}=0,$
(3.22) $\displaystyle\frac{\delta S_{\rm UG}}{\delta C_{\mu\nu}}:\quad$
$\displaystyle\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\square\partial_{\rho}\bar{c}_{\sigma}-\partial^{[\mu}\bar{C}^{\nu]}=0,$
(3.23) $\displaystyle\frac{\delta S_{\rm UG}}{\delta\bar{C}^{\mu}}:\quad$
$\displaystyle\partial^{\nu}C_{\nu\mu}+\partial_{\mu}C=0,$ (3.24)
$\displaystyle\frac{\delta S_{\rm UG}}{\delta C}:\quad$
$\displaystyle\partial^{\mu}\bar{C}_{\mu}=0,$ (3.25)
$\displaystyle\frac{\delta S_{\rm UG}}{\delta C_{S}}:\quad$
$\displaystyle\partial^{\mu}\bar{c}_{\mu}=0,$ (3.26) $N_{\rm FP}=\pm 2$ sector
$\displaystyle\frac{\delta S_{\rm UG}}{\delta\bar{D}},\ \frac{\delta S_{\rm
UG}}{\delta D}:\quad$
$\displaystyle\partial^{\mu}D_{\mu}=D,\quad\partial_{\mu}\bar{D}^{\mu}=\bar{D},\qquad\qquad$
(3.27) $\displaystyle\frac{\delta S_{\rm UG}}{\delta\bar{D}^{\mu}},\
\frac{\delta S_{\rm UG}}{\delta D_{\mu}}:\quad$ $\displaystyle\square
D_{\mu}=0,\quad\square\bar{D}^{\mu}=0,$ (3.28) $N_{\rm FP}=\pm 3$ sector
$\displaystyle\frac{\delta S_{\rm UG}}{\delta\bar{T}},\ \frac{\delta S_{\rm
UG}}{\delta T}:\quad$ $\displaystyle\square T=0,\quad\square\bar{T}=0.$ (3.29)

where $h_{\mu}\equiv\partial^{\nu}h_{\mu\nu}$. Note also that Eqs. (3.19),
(3.23) and (3.28) are already simplified by their preceding equations.

Taking the trace of (3.19) and using (3.18) and (3.21), we find

$\displaystyle 2\lambda=\partial^{\mu}\partial^{\nu}h_{\mu\nu}.$ (3.30)

The divergence of the gravity equation (3.19), combined with (3.30) and
(3.21), yields

$\displaystyle\square b_{\mu}=-\partial_{\mu}\lambda,$ (3.31)

which, together with (3.21), gives

$\displaystyle\square\lambda=0.$ (3.32)

Equation (3.20) gives

$\displaystyle-\square S+2\lambda=0\,.$ (3.33)

Equations (3.31) and (3.33)) imply that $b_{\mu}$ and $S$ fields satisfy
dipole equations $\square^{2}b_{\mu}=0$ and $\square^{2}S=0$, and their dipole
parts are supplied by the simple pole $\lambda$ field. Applying $\square$ to
the gravity field equation (3.19) and using Eqs. (3.31) and (3.33), we get

$\displaystyle\frac{1}{2}\square^{2}h_{\mu\nu}-\partial_{\mu}\partial_{\nu}\lambda=0.$
(3.34)

So, we see that $\tilde{h}_{\mu\nu}$ field is now a tripole field, and the
tripole part is supplied by the simple pole $\lambda$ field and the dipole
parts are supplied by simple pole parts of $S$ and $b_{\mu}$.

For the $N_{\rm FP}\neq 0$ ghost sector, we find from (3.22)

$\displaystyle\square C_{S}=0.$ (3.35)

The dual of (3.22) gives

$\displaystyle\square^{2}C_{\mu\nu}=0.$ (3.36)

Taking the $\partial_{\nu}$ divergence of (3.23) and using (3.25) gives

$\displaystyle\square\bar{C}^{\mu}=0.$ (3.37)

The dual of (3.23) yields

$\displaystyle\square^{2}\bar{c}_{\mu}=0.$ (3.38)

Equations (3.24), (3.28) and (3.29) show that all the other ghost fields are
of simple pole:

$\displaystyle\square C=\square D=\square\bar{D}=\square
D_{\mu}=\square\bar{D}^{\mu}=\square T=\square\bar{T}=0.$ (3.39)

There are no tripole ghost fields, in contrast to our paper I, though there
are in the graviton fluctuation. This is one of the simplifications that our
new formulation brings in.

##  4 Identifying the independent fields

We have not only the usual simple pole fields but also dipole and tripole
fields in this system. So the number of independent particle modes are
different from that of independent fields. To avoid the complication of
decomposing the multipole fields into simple pole modes, we also adopt here,
as in the previous paper I, the 4-dimensional Fourier expansion[33] of the
fields

$\phi(x)=\frac{1}{\sqrt{(2\pi)^{3}}}\int
d^{4}p\,\theta(p^{0})\left[\phi(p)e^{ipx}+\phi^{\dagger}(p)e^{-ipx}\right]$
(4.1)

to define annihilation and creation operators $\phi(p)$ and
$\phi^{\dagger}(p)$ for such general multipole fields. We analyze independent
4-dimensional Fourier modes for each ghost number $N_{\rm FP}$ sector
successively, in the Lorentz frame in which the 3-momentum ${\boldsymbol{p}}$
is along $x^{3}$ axis:

$p^{\mu}=\bigl{(}p^{0},0,0,p^{3}\bigr{)},\quad\text{i.e.,}\quad p^{i}=0\
(i=1,2),\ p^{3}=:|p|>0.$ (4.2)

Note that if the field $\phi$ is a massless simple pole field
$\phi(p)\propto\delta(p^{2})$, this reads

$p^{\mu}\,\phi(p^{2})=\bigl{(}|p|,0,0,|p|\bigr{)}\,\phi(p^{2}),\quad\text{i.e.,}\quad
p^{0}=p^{3}=|p|\ .$ (4.3)

We will show in the following analysis that the independent fields in each
ghost number sector are given by the list in Table 1.

Table 1: List of independent fields. $i$ denotes transverse directions 1 and 2. $N_{\rm FP}=0$ |  $h_{{\rm T}1},\ h_{{\rm T}2};\quad\chi^{0},\ \chi^{i}$ | ; |  $b_{0},\ b_{i}$
---|---|---|---
$|N_{\rm FP}|=1$ |  ${C}_{0i},\ {C}_{12},\ C$ | ; |  $\bar{c}_{0},\ \bar{c}_{i},\ \bar{C}^{0}$
$|N_{\rm FP}|=2$ |  $D_{0},\ D_{i},\ D$ | ; |  $\bar{D}^{0},\ \bar{D}^{i},\ \bar{D}$
$|N_{\rm FP}|=3$ | $T$ | ; |  $\bar{T}$

###  4.1 $N_{\rm FP}=0$ sector

We begin with the fields with ghost number $N_{\rm FP}=0$. We have 10
component gravity $h_{\mu\nu}$ field, 1 scalar field $S$, 1 unimodular
multiplier field $\lambda$, plus a 4 component vector NL field $b_{\mu}$;
thus, $10+1+1+4=16$ component fields in all. Let us count/identify the
independent fields among them, by using the EOMs (3.18) – (3.21).

These EOMs (3.18) – (3.21) in this $N_{\rm FP}=0$ sector take exactly the same
forms as those in GR theory in unimodular gauge, if we identify the unimodular
NL field $b$ there with the present unimodular multiplier $\lambda$. The same
counting there also applies here.

The 10 component $h_{\mu\nu}$ is subject to the 1 traceless condition (3.18)
and the 4-vector de Donder gauge condition (3.20), so has $10-1-4=5$
independent fields, as which we can take the same fields as in GR case. First,
we have two BRST invariant simple-pole (hence physical) fields

$\displaystyle h_{{\rm
T}1}(p):=(\kern-0.3pt1\kern-0.8pt/\kern-0.2pt2)\bigl{(}h^{11}-h^{22}\bigr{)}(p),\qquad
h_{{\rm T}2}(p):=h^{12}(p).$ (4.4)

These transverse modes are BRST invariant since the BRST transformation of
$h^{\mu\nu}(p)$ at linearized level is given by

$\delta_{\rm B}h^{\mu\nu}(p)=-ip^{\mu}c^{\nu}_{\rm T}(p)-ip^{\nu}c^{\mu}_{\rm
T}(p),$ (4.5)

while the transverse momentum components $p^{i}\ (i=1,2)$ vanish by
definition. Simple-pole property $\square h_{{\rm T}j}(p)=0$ also follows from
the EOM (3.19) and $p^{i}=0$. In addition to these two, we have a transverse
vector field (hence possessing 3 independent components):

$\displaystyle\chi^{0}(p):=\frac{1}{2p^{0}}\Bigl{(}h^{00}-\frac{1}{2}\bigl{(}h^{11}+h^{22}\bigr{)}\Bigr{)}(p)=\frac{1}{2p^{0}}\frac{1}{2}\bigl{(}h^{00}+h^{33}\bigr{)}(p),$
$\displaystyle\chi^{i}(p):=\frac{1}{p^{0}}h^{0i}(p),\ \ (i=1,2),$
$\displaystyle\chi^{3}(p):=\frac{1}{2p^{3}}\Bigl{(}h^{33}-\frac{1}{2}\bigl{(}h^{11}+h^{22}\bigr{)}\Bigr{)}(p)=\frac{1}{2p^{3}}\frac{1}{2}\bigl{(}h^{00}+h^{33}\bigr{)}(p),$
(4.6)

satisfying transversality
$p_{\mu}\chi^{\mu}(p)=p_{0}\chi^{0}(p)+p_{3}\chi^{3}(p)=0$. So we can forget
the redundant component $\chi^{3}(p)$ henceforth. The second equality for the
expression $\chi^{0}(p)$ (or $\chi^{3}(p)$) follows from the tracelessness Eq.
(3.18), $h:=\eta_{\mu\nu}h^{\mu\nu}=0$,

$\left(h^{11}+h^{22}\right)(p)=\left(h^{00}-h^{33}\right)(p).$ (4.7)

This $\chi^{\mu}(p)$ field has a very simple BRST transformation property

$\delta_{\rm B}\chi^{\mu}(p)=-ic^{\mu}_{\rm T}(p).$ (4.8)

Next, the gauge fixing NL field $b_{\mu}$ is subject to the transversality
(3.21), $p^{0}b_{0}(p)+p^{3}b_{3}(p)=0$, so we can take 3 fields $b_{0}$ and
$b_{i}(p)$ $(i=1,2)$ as its independent components. These 5 components
$h_{{\rm T}i}$, $\chi^{0}$ and $\chi^{i}$ from $h_{\mu\nu}$ and 3 components
$b_{0}$ and $b_{i}$ give the all of the independent fields listed in the
$N_{\rm FP}=0$ sector in Table 1.

The other fields $\lambda$ and $S$ as well as the other 5 dependent components
in $h_{\mu\nu}$

$\bigl{(}h^{11}+h^{22}\bigr{)}(p),\quad\bigl{(}h^{00}-h^{33}\bigr{)}(p),\quad
h^{3i}(p),\quad h^{03}(p).$ (4.9)

can be shown to be explicitly expressed by the above $5+3$ independent fields
by using EOMs (3.18) – (3.21) as follows.

In Eq. (3.31) which we already derived from those EOMs, the unimodular
multiplier $\lambda$ was identified with the dipole part of the gauge-fixing
multiplier (NL) field $b_{0}$:

$\lambda(p)=-i\frac{1}{p^{0}}\square b_{0}(p)\,.$ (4.10)

This is an important relation showing that the unimodular multiplier field
$\lambda$ becomes the BRST daughter field, hence a member of a BRST quartet.
As for the $S$ field, we can rewrite Eq. (3.19) into the following form by
substituting $h_{\mu}=\partial_{\mu}S$ of Eq. (3.20):

$\partial^{\mu}\partial^{\nu}S=\frac{1}{2}\square
h^{\mu\nu}-\partial^{(\mu}b^{\nu)}+\frac{1}{2}\lambda\eta^{\mu\nu}.$ (4.11)

Adding two equations with indices $(\mu,\nu)=(0,0)$ and $(3,3)$ and dividing
by $-(p_{0}^{2}+p_{3}^{2})$, we obtain

$S(p)=-\frac{2p^{0}}{p_{0}^{2}+p_{3}^{2}}\bigl{(}\square\chi^{0}-ib^{0}\bigr{)}(p).$
(4.12)

Now the 5 dependent components in $h_{\mu\nu}$ in Eq. (4.9). The first
dependent field $(h^{11}+h^{22})(p)$ is already given by the second field
$(h^{00}-h^{33})(p)$ in Eq. (4.7). The latter one is shown in Eq. (4.43) in
the previous paper I to be

$\displaystyle\left(h^{00}-h^{33}\right)(p)=\frac{4ip^{0}}{p_{0}^{2}+p_{3}^{2}}\,b_{0}(p)\,,$
(4.13)

which is actually the same equation as Eq. (4.12) if we substitute
$-(p_{0}^{2}+p_{3}^{2})S(p)=i(p^{0}h^{0}+p^{3}h^{3})(p)=p_{0}^{2}h^{00}(p)-p_{3}^{2}h^{33}(p)$
following from
$\partial^{\nu}\partial^{\mu}S=\partial^{\nu}h^{\mu}=\partial^{\nu}\partial_{\lambda}h^{\mu\lambda}$.
The rest two components $h^{3i}$ and $h^{03}$ follow from the de Donder gauge
condition (3.20), $\partial_{\nu}h^{\nu\mu}=\partial^{\mu}S$; the $\mu=i$
($i=1,2$) component gives

$h^{3i}(p)=\frac{p_{0}^{\,2}}{p^{3}}\,\chi^{i}(p)\,,$ (4.14)

and the vanishing difference
$0=\partial^{[\nu}\partial^{\mu]}S=\partial^{[\nu}\partial_{\lambda}h^{\mu]\lambda}$
with $(\mu,\nu)=(0,3)$ gives Eq. (4.41) in I:

$h^{03}(p)=\frac{4p_{0}^{2}p^{3}}{p_{0}^{2}+p_{3}^{2}}\chi^{0}(p).$ (4.15)

###  4.2 $N_{\rm FP}\not=0$ sectors

In $N_{\rm FP}=+1$ ghost sector we have 6 component $C_{\mu\nu}$ and 2
scalars, $C_{S}$ and $C$, so 8 components in all. In $N_{\rm FP}=-1$ antighost
sector we have two 4-component vectors $\bar{c}_{\mu}$ and $\bar{C}^{\mu}$, so
also 8 components in all.

Begin with the $N_{\rm FP}=+1$ ghost sector. The constraint equation (3.24),
$\partial^{\nu}C_{\nu\mu}+\partial_{\mu}C=0$, takes the same form as that in
the previous paper I, which implied that only 3 components $C_{0i}$ and
$C_{12}$ are independent among 6 $C_{\mu\nu}$ if the scalar $C$ is chosen as
another independent field; the other 3 components are expressed by these and
$C$ as the Eq. (4.52) in I with $\bar{C}^{\mu\nu}$ and $\bar{C}$ there
replaced by $C_{\mu\nu}$ and $C$:

$C_{03}=-C(p),\qquad C_{3i}=-\frac{p^{0}}{p^{3}}C_{0i}(p),\quad(i=1,2).$
(4.16)

These $3+1=4$ fields $C_{0i}$, $C_{12}$ and $C$ are the all of the independent
fields listed in the $N_{\rm FP}=1$ sector in Table 1. The other remaining
field $C_{S}$ is indeed expressed by the dipole part of the independent field
$C_{12}$; the $\mu=0$ component of EOM (3.22) gives

$C_{S}(p)=\frac{1}{2p^{0}}\varepsilon^{03ij}\square p_{3}C_{ij}(p)=\square
C_{12}(p),$ (4.17)

where in the second equality, $p_{3}=p^{0}$ has been used since $\square
C_{12}(p)$ is a massless simple pole field.

Next consider the $N_{\rm FP}=-1$ antighost sector, consisting of two vectors
$\bar{c}_{\mu}$ and $\bar{C}^{\mu}$. Equations (3.26) and (3.25) show that
these vectors are both transversal:

$\displaystyle p^{0}\bar{c}_{0}+p^{3}\bar{c}_{3}=0\ $
$\displaystyle\rightarrow\ \bar{c}_{3}=-\frac{p^{0}}{p^{3}}\bar{c}_{0},$
(4.18) $\displaystyle p_{0}\bar{C}^{0}+p_{3}\bar{C}^{3}=0\ $
$\displaystyle\rightarrow\
\bar{C}^{3}=\frac{p^{0}}{p^{3}}\bar{C}^{0}=\bar{C}^{0}\,,$ (4.19)

where $p^{0}=p^{3}$ has been used in front of the simple pole field
$\bar{C}^{\mu}$. Moreover, Eq. (3.23) gives constraint relations between
$\bar{c}_{\mu}$ and $\bar{C}^{\nu}$: noting that $\square\bar{c}_{\mu}$ is
also a simple pole field,

($\mu=0$, $\nu=3$) $\displaystyle:~{}p^{0}\bar{C}^{3}-p^{3}\bar{C}^{0}=0\
\rightarrow\ \bar{C}^{0}=\bar{C}^{3},$ (4.20) ($\mu=0$ or 3, $\nu=i$)
$\displaystyle:~{}\varepsilon^{ij}\square
p_{3}\bar{c}_{j}(p)+p^{0}\bar{C}^{i}=0\ \rightarrow\
\bar{C}^{i}(p)=-\varepsilon^{ij}\square\bar{c}_{j}(p)$ $\displaystyle\hskip
130.0002pt(i,j=1,2,\ \varepsilon^{ij}=-\varepsilon^{ji}),$ (4.21) ($\mu=1$,
$\nu=2$) $\displaystyle:~{}\square p_{0}\bar{c}_{3}-\square
p_{3}\bar{c}_{0}=0\ \rightarrow\ \square\bar{c}_{3}=-\square\bar{c}_{0}\,.$
(4.22)

Equations (4.20) and (4.22) merely give identical relations with the
transversality of vectors $\bar{c}_{\mu}$ and $\bar{C}^{\nu}$. Only the Eq.
(4.21) implies new relations that $\bar{C}^{i}(p)$ are given by the dipole
part of $\varepsilon^{ij}\bar{c}_{j}(p)$. We thus have shown that $3+1=4$
fields $\bar{c}_{0}$, $\bar{c}_{i}$ and $\bar{C}^{0}$ are all of the
independent fields in $N_{\rm FP}=-1$ antighost sector as listed in Table 1.

Finally, the EOMs (3.27) – (3.29) in the $|N_{\rm FP}|=2,3$ sectors are
trivial. So we can immediately see that the independent fields in these
sectors can be chosen as written in the list of Table 1.

##  5 Identifying BRST quartets

Now that all the independent fields are listed up in Table 1, we can show that
all of them other than the two physical transverse modes $h_{{\rm T}1}$ and
$h_{{\rm T}2}$, fall into BRST quartets, decoupling properly from the physical
sector. In order to do so, we recall the BRST transformation at linearized
level (which is also the BRST transformation of the asymptotic fields under
the perturbative assumption):

$\displaystyle\delta_{\rm B}\lambda$ $\displaystyle=0,$ (5.1)
$\displaystyle\delta_{\rm B}h^{\mu\nu}$
$\displaystyle=-2\partial_{\phantom{T}}^{(\mu}c_{{\rm
T}}^{\nu)}=\partial^{(\mu}\varepsilon^{\nu)\tau\rho\sigma}\partial_{\tau}C_{\rho\sigma},$
(5.2) $\displaystyle\delta_{\rm B}C_{\mu\nu}$
$\displaystyle=i(\partial_{\mu}D_{\nu}-\partial_{\nu}D_{\mu}),$ (5.3)
$\displaystyle\delta_{\rm B}S$ $\displaystyle=C_{S},\hskip
10.00002pt\qquad\delta_{\rm B}C_{S}=0,$ (5.4) $\displaystyle\delta_{\rm
B}\bar{c}_{\mu}$ $\displaystyle=ib_{\mu},\hskip 10.00002pt\qquad\delta_{\rm
B}b_{\mu}=0,$ (5.5) $\displaystyle\delta_{\rm B}\bar{D}^{\mu}$
$\displaystyle=\bar{C}^{\mu},\hskip 10.00002pt\qquad\delta_{\rm
B}\bar{C}^{\mu}=0,$ (5.6) $\displaystyle\delta_{\rm B}D_{\mu}$
$\displaystyle=\partial_{\mu}T,\hskip 6.99997pt\qquad\delta_{\rm B}T=0,$ (5.7)
$\displaystyle\delta_{\rm B}\bar{T}$ $\displaystyle=i\bar{D},\hskip
11.00008pt\qquad\delta_{\rm B}\bar{D}=0,$ (5.8) $\displaystyle\delta_{\rm B}C$
$\displaystyle=iD,\hskip 11.00008pt\qquad\delta_{\rm B}D=0.$ (5.9)

The BRST quartet is generally a pair of the BRST doublets which satisfy the
properties schematically drawn as [34]

$\textstyle{A(p)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\text{B}}}$$\textstyle{\
C(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inner-
product$\textstyle{\ \bar{C}(p)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\text{B}}}$$\textstyle{\
iB(p).}$ (5.10)

We denote this BRST quartet described by this scheme by using the same
notation as used in I simply as

$\bigl{(}A(p)\rightarrow C(p);\ \bar{C}(p)\rightarrow iB(p)\bigr{)},$ (5.11)

This scheme means that a pair of BRST doublets satisfying (assuming $A(p)$ a
boson),

$\displaystyle\delta_{{\rm B}}A(p)$ $\displaystyle=[iQ_{\rm B},\,A(p)]=C(p),$
$\displaystyle\delta_{{\rm B}}\bar{C}(p)$ $\displaystyle=\\{iQ_{\rm
B},\,\bar{C}(p)\\}=iB(p),$ (5.12)

have nonvanishing inner-product with each other:

$\langle 0|\bar{C}(p)C^{\dagger}(q)|0\rangle=\langle 0|\bar{C}(p)iQ_{\rm
B}A^{\dagger}(q)|0\rangle=i\langle
0|B(p)A^{\dagger}(q)|0\rangle\propto\delta^{4}(p-q)\not=0,$ (5.13)

or, equivalently, in terms of commutation relation,

$\displaystyle 0$ $\displaystyle=\bigl{\\{}iQ_{\rm
B},\,[\bar{C}(p),\,A^{\dagger}(q)]\bigr{\\}}=[iB(p),\,A^{\dagger}(q)]-\\{\bar{C}(p),\,C^{\dagger}(q)\\}$
$\displaystyle\rightarrow\
[iB(p),\,A^{\dagger}(q)]=\\{\bar{C}(p),\,C^{\dagger}(q)\\}\propto\delta^{4}(p-q)\not=0.$
(5.14)

We should note that the existence of nonvanishing inner-products/commutators
can also be judged from the nonvanishing propagators. As explained in I,
generally, the expressions of commutation relations (CR) and the propagators
for the free fields $\phi_{i}$ have the following exact correspondence:

$\begin{array}[]{ccc}\text{propagator}\
\langle\phi_{i}\,\phi_{j}\rangle&&\text{CR}\
[\phi_{i}(p),\,\phi^{\dagger}_{j}(q)]\\\\[0.43057pt]
\hline\cr\phantom{\bigg{|}}\displaystyle\frac{1}{i}\Bigl{[}\frac{1}{p^{2}},\
\frac{1}{p^{4}},\
\frac{1}{p^{6}}\Bigr{]}&\leftrightarrow&\Bigl{[}\delta(p^{2}),\
-\delta^{\prime}(p^{2}),\
\frac{1}{2}\delta^{\prime\prime}(p^{2})\Bigr{]}\theta(p^{0})\delta^{4}(p-q),\end{array},$
(5.15)

where three terms on both sides correspond to the terms of simple-pole, dipole
and tripole parts, respectively. This correspondence also holds for the
fermion fields if the commutators are understood to be anticommutators. We use
an abbreviated notation for the propagator in momentum space:

$\langle\,\phi_{i}\ \phi_{j}\,\rangle_{p}=\int d^{4}x\,e^{-ipx}\langle 0|{\rm
T}\phi_{i}(x)\,\phi_{j}(0)|0\rangle,$ (5.16)

which is identical with $i$ times the inverse of the 2-point vertex,
$i{\Gamma^{(2)}}^{-1}_{ij}$, so can be read directly from the results given in
sect. 3.

Now let us start the task to identify the BRST quartets. Among the fields in
Table 1, we can list all the boson fields which are not BRST invariant. Then
they must be BRST parents of certain BRST doublets, which can be read from the
BRST transformation Eqs. (5.1) – (5.9). Thus we find all the BRST doublets
possessing boson parent as

$\displaystyle\delta_{\rm B}\chi^{0}(p)$ $\displaystyle=-ic^{0}_{\rm
T}(p)=-p_{3}C_{12}(p),$ (5.17) $\displaystyle\delta_{\rm B}\chi^{i}(p)$
$\displaystyle=-ic^{i}_{\rm
T}(p)=\varepsilon^{ij}\bigl{(}p_{3}C_{0j}-p_{0}C_{3j}\bigr{)}(p)=-\varepsilon^{ij}\frac{1}{p^{3}}\square
C_{0j}(p),$ (5.18) $\displaystyle\delta_{\rm B}D^{0}(p)$
$\displaystyle=ip^{0}T(p),$ (5.19) $\displaystyle\delta_{\rm B}\bar{D}^{0}(p)$
$\displaystyle=\bar{C}^{0}(p),$ (5.20) $\displaystyle\delta_{\rm
B}\bar{D}^{i}(p)$
$\displaystyle=\bar{C}^{i}(p)=-\varepsilon^{ij}\square\bar{c}_{j}(p),$ (5.21)

where in Eq. (5.18), use has been made of Eq. (4.16) to rewrite $C_{3j}(p)$ as
$-(p^{0}/p^{3})C_{0j}(p)$ in the last equality. We have used Eq. (4.21) in the
last line to rewrite $\bar{C}^{i}$ which does not appear in Table 1 as
independent fields. In the same way, all the BRST doublets possessing fermion
parent can be found as

$\displaystyle\delta_{\rm B}C_{0i}(p)$ $\displaystyle=-p_{0}D_{i}(p),$ (5.22)
$\displaystyle\delta_{\rm B}C(p)$ $\displaystyle=iD(p),$ (5.23)
$\displaystyle\delta_{\rm B}\bar{c}_{0}(p)$ $\displaystyle=ib_{0}(p),$ (5.24)
$\displaystyle\delta_{\rm B}\bar{c}_{i}(p)$ $\displaystyle=ib_{i}(p),$ (5.25)
$\displaystyle\delta_{\rm B}\bar{T}^{i}(p)$ $\displaystyle=i\bar{D}(p).$
(5.26)

At this stage we notice that all the independent fields in Table 1 other than
the physical transverse graviton modes already appear in these BRST doublets,
as the parent or daughter fields. Since the BRST doublets are known to
necessarily form BRST quartets which essentially decouple from physical
subspace, one is tempted to conclude that the unitarity proof is completed. We
must, however, be very careful about the dipole and tripole fields appearing
in the BRST parent position. For instance, if $A$ is a dipole and $B$ is a
simple pole for the BRST doublet $\delta_{\rm B}A=iB$, then this implies
$\delta_{\rm B}\square A=0$. That is, the simple pole field $\square A$,
representing the dipole part mode in the $A$ field is BRST invariant. So
$\square A$ must appear somewhere at BRST daughter position in the full list
of the BRST doublets. Otherwise, the unphysical $\square A$ mode becomes a
BRST singlet appearing in the physical subspace and violates unitarity.

All the multipole fields which appear in the parent position in these BRST
doublets (5.17) – (5.26) are: $\chi^{0},\ \chi^{i},\ C_{0i},\ \bar{c}_{0},\
\bar{c}_{i}$. Let us examine the BRST doublets possessing these parent fields
in turn.

First is the doublet (5.17), $\delta_{\rm B}\chi^{0}(p)=-p_{3}C_{12}(p)$. The
parent $\chi^{0}(p)$ is tripole while the daughter $C_{12}(p)$ is dipole,
meaning that the tripole mode part $\square^{2}\chi^{0}(p)$ could be a
dangerous BRST singlet. But Eq. (3.34),
$(1/2)\square^{2}h_{\mu\nu}=\partial_{\mu}\partial_{\nu}\lambda$, tells us
$\square^{2}\chi^{0}(p)\propto\lambda(p)$ and the unimodular multiplier field
$\lambda(p)$ was already noted in Eq. (4.10) to be a safe BRST daughter field
$\propto\square b_{0}(p)$ and it indeed appears in the daughter position of
the BRST doublet (5.24). Note that this BRST doublet is a safe doublet, in
which both the parent $\bar{c}_{0}(p)$ and the daughter $b_{0}(p)$ are dipole
fields, implying that it actually represents two simple-pole BRST doublets:

$\delta_{\rm
B}\hat{\bar{c}}_{0}(p)=i\hat{b}_{0}(p)\quad\text{and}\quad\delta_{\rm
B}\square\bar{c}_{0}(p)=i\square b_{0}(p),$ (5.27)

where the hat symbol $\hat{\phi}$ means the simple-pole part in the multipole
field. Here we emphasize that the second equation of (5.27) means that the
multiplier field $\lambda$ imposing the unimodular constraint becomes the BRST
daughter $\square b_{0}=ip^{0}\lambda$ and has associated ghost and antighost.
This is the key how we get the correct number of remaining physical dofs.

Second is the doublet (5.18), $\delta_{\rm
B}\chi^{i}(p)=-\varepsilon^{ij}\square C_{0j}(p)/p^{3}$. Since the parent
$\chi^{i}(p)$ is a dipole field while $\square C_{0j}(p)$ is a simple-pole
field, the dipole part $\square\chi^{i}(p)$ is a potentially dangerous BRST
singlet. But EOMs (3.19) and (3.20) lead to $\square\chi^{i}(p)=\square
h^{0i}(p)/p^{0}=ib^{i}(p)$, which safely appears in the daughter position of
the doublet (5.25).

Third is the doublet (5.22), in which the parent $C_{0i}(p)$ is dipole while
the daughter $D_{i}(p)$ is simple-pole. So $\square C_{0i}(p)$ is a
potentially dangerous BRST singlet but it actually appears in the daughter
position of the doublet (5.18).

Lastly are the doublets (5.24) and (5.25); the former doublet (5.24) was
already noted in advance to represent a safe and double BRST doublets in Eq.
(5.27). In the latter doublet (5.25), the parent $\bar{c}_{i}(p)$ is dipole
while $b_{i}(p)$ is simple-pole. But the potentially dangerous BRST singlet
$\square\bar{c}_{i}(p)$ just appears in the daughter position of the doublet
(5.21).

We thus have finished to show that the list of the BRST doublets in Eqs.
(5.17) – (5.26) is complete; that is, it contains all the independent modes in
our UG theory other than the physical transverse modes and they all fall into
BRST doublets decoupling from the physical sector.

For the unitarity proof, this is enough. For completeness, however, we
identify how those BRST doublets form BRST quartets. The task for doing so is
only to find for each BRST doublet with boson parent in (5.17) – (5.21) a
partner BRST doublet with fermion parent from (5.22) – (5.26) which has
nonvanishing inner-product with each other. As explained before, the existence
of nonvanishing inner-products can be easily found by examining the
nonvanishing propagators given in sect. 3. In this way, we can identify the
following BRST quartets:

$\displaystyle\bigl{(}\ \chi^{0}(p)\ \rightarrow\ -p^{3}C_{12}(p);\ \
\bar{c}_{0}(p)\ \rightarrow\ ib_{0}(p)\ \bigr{)},$ (5.28)
$\displaystyle\bigl{(}\ \chi^{i}(p)\ \rightarrow\
-(1/p^{3})\varepsilon^{ij}\square C_{0j}(p);\ \ \bar{c}_{i}(p)\ \rightarrow\
ib_{i}(p)\ \bigr{)},\quad(i=1,2),$ (5.29) $\displaystyle\bigl{(}\ D^{0}(p)\
\rightarrow\ ip^{0}T(p);\ \bar{T}(p)\ \rightarrow\ i\bar{D}(p)\bigr{)},$
(5.30) $\displaystyle\bigl{(}\ \bar{D}^{0}(p)\ \rightarrow\ \bar{C}^{0}(p);\
C(p)\ \rightarrow\ iD(p)\bigr{)},$ (5.31) $\displaystyle\bigl{(}\
\bar{D}^{i}(p)\ \rightarrow\ -\varepsilon^{ij}\square\bar{c}_{j}(p);\
C_{0i}(p)\ \rightarrow\ -p_{0}D_{i}(p)\bigr{)}.$ (5.32)

The relevant propagators showing the existence of nonvanishing inner-products
between the pair doublets in these BRST quartets are the following:

$\displaystyle\langle\,ib_{0}\
\chi^{0}\,\rangle_{p}=p^{3}\langle\,\bar{c}_{0}\
C_{12}\,\rangle_{p}=i\frac{p_{3}^{2}}{(p^{2})^{2}},$ (5.33)
$\displaystyle\langle\,ib_{i}\
\chi^{i}\,\rangle_{p}=\frac{1}{p^{3}}\langle\,\bar{c}_{i}\
\varepsilon^{ij}\square C_{0j}\,\rangle_{p}=i\frac{1}{p^{2}},$ (5.34)
$\displaystyle\langle\,i\bar{D}\ D^{0}\,\rangle_{p}=-p^{0}\langle\,\bar{T}\
iT\,\rangle_{p}=-i\frac{p^{0}}{p^{2}},$ (5.35) $\displaystyle\langle\,iD\
\bar{D}^{0}\,\rangle_{p}=\langle\,C\
\bar{C}^{0}\,\rangle_{p}=-i\frac{p^{0}}{p^{2}},$ (5.36) $\displaystyle
p^{0}\langle\,D_{i}\ \bar{D}^{i}\,\rangle_{p}=-\langle\,C_{0i}\
\varepsilon^{ij}\square\bar{c}_{j}\,\rangle_{p}=i\frac{p^{0}}{p^{2}}.$ (5.37)

Here we note that the propagators in the first line are of dipole. It implies
that the first BRST quartet (5.28) in fact represents two BRST quartets each
consisting of simple-pole fields:

$\displaystyle\bigl{(}\ \square\chi^{0}(p)\ \rightarrow\ -p^{3}\square
C_{12}(p);\ \ \hat{\bar{c}}_{0}(p)\ \rightarrow\ i\hat{b}_{0}(p)\ \bigr{)},$
(5.38) $\displaystyle\bigl{(}\ \hat{\chi}^{0}(p)\ \rightarrow\
-p^{3}\hat{C}_{12}(p);\ \ \square\bar{c}_{0}(p)\ \rightarrow\ i\square
b_{0}(p)\ \bigr{)},$ (5.39)

where hat means the simple-pole part. This structure can be understood if we
note the following relevant propagators of simple-pole:

$\displaystyle\langle\,ib_{0}\
\square\chi^{0}\,\rangle_{p}=p^{3}\langle\,\bar{c}_{0}\ \square
C_{12}\,\rangle_{p}=-i\frac{p_{3}^{2}}{p^{2}},$ (5.40)
$\displaystyle\langle\,i\square b_{0}\
\chi^{0}\,\rangle_{p}=p^{3}\langle\,\square\bar{c}_{0}\
C_{12}\,\rangle_{p}=-i\frac{p_{3}^{2}}{p^{2}}.$ (5.41)

The nonvanishing inner-products exist between the simple-pole parts and the
dipole parts.

Thus we have confirmed that all the fields including multipole fields except
for the transverse modes fall into the BRST quartets and decouple from the
physical sector, leaving the two transverse graviton modes as physical states.

##  6 Conclusions

In this paper, combining the new ideas in I and [24], we have given a new
covariant BRST quantization of UG. First, we have used rank-2 antisymmetric
tensor fields for the ghosts which satisfy the transverse condition. This
gives further gauge invariance in the ghost sector and we have successfully
gauge fixed TDiff. For the antighosts, we need fields with the same number of
dofs. In I, we have used the same rank-2 antisymmetric tensor fields, but this
leads to a formulation with lots of ghosts and tripole modes in the ghost
sector. Here instead we have used vector antighost which satisfy transverse
condition on shell due to the additional scalar field $S$. We have shown that
this leads to a fully gauge-fixed theory, and all the modes except for the 2
transverse graviton modes fall into the BRST quartets and completely decouple
from the physical subspace. In this way we have succeeded in giving a
covariant BRST quantization of UG with the correct number of dofs, the same as
GR.

This formulation has two advantages compared with our previous formulation in
I. One is that the formulation contains less number of ghosts without those
originating from antighosts. The second is that we do not have tripole modes
in the ghost sector. These lead to considerable simplification in the
resulting system. A difference is that this formulation is asymmetric in ghost
and antighost sectors, while that in I is symmetric. Both formulations give
the correct number of dofs. Which formulation is more useful for covariant
study of UG remains to be seen, but we hope that our formulations should be
useful for further study of UG.

## Acknowledgment

T.K. is supported in part by the JSPS KAKENHI Grant Number JP18K03659. N.O. is
supported in part by the Grant-in-Aid for Scientific Research Fund of the JSPS
(C) No. 16K05331, No. 20K03980, and Taiwan Grant No. MOST 110-2811-M-008-526.

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