\section{Introduction
\label{sec:Introduction}}

The large-scale structure of the Universe is an inexhaustible fountain
of the cosmological information and has been playing important roles
in cosmology. The various patterns in the spatial distribution of
galaxies have emerged from initial density fluctuations of the
Universe, which origin is believed to be generated by quantum
fluctuations in the inflationary phase of the very early Universe
\cite{Lyth:2009zz}. Considerable information on the evolution of the
Universe is also contained in the large-scale structure. For example,
the spatial clustering of galaxies is a promising probe to reveal the
nature of dark energy \cite{SDSS:2005xqv} and the initial condition in
the early Universe such as the primordial spectrum and non-Gaussianity
etc.~\cite{Dalal:2007cu}. While the large-scale structure is used as a
probe of the density field in the relatively late Universe, the
distribution of observable objects, such as galaxies, quasars, and
21cm lines, etc., are biased tracers of the underlying matter
distribution of the Universe. The difference between distributions of
galaxies and matter is referred to as ``galaxy bias''
\cite{Desjacques:2016bnm}. The phenomena of biasing are not restricted
to the galaxies, but are unavoidable when we use astronomical objects
as tracers of underlying matter distribution.

The number density of galaxies are primary target to probe the
properties of large-scale structure. Theoretical predictions for the
spatial distribution of galaxies are investigated by various methods,
including numerical simulations, analytic modeling, and the
perturbation theories. On one hand, the dynamical evolutions of the
mass density fields on relatively large scales are analytically
described by the perturbation theory
\cite{Hunter1964,Tomita1967,Peebles1980,Bernardeau:2001qr}, which is
valid as long as the density fluctuations are small and in the
quasi-linear regime. On the other hand, the dynamical structure
formation on relatively small scales, including the formation of
galaxies and other astronomical objects, is not analytically tractable
because of the full nonlinearity of the problem. One should resort to
numerical simulations and nonlinear modeling to describe and
understand the small-scale structure formation. For example, the halo
model \cite{Mo:1995cs,Mo:1996cn,Cooray:2002dia} statistically predicts
spatial distributions of halos depending on the initial conditions,
using simple assumptions based on the Press-Schechter theory
\cite{Press:1973iz} and its extensions \cite{Bond:1990iw,Sheth:1999mn}
of the nonlinear structure formation.

The biasing of tracers affects not only spatial distributions on small
scales, but also those on large scales. However, the biasing effects
are not so complicated on large scales as those on small scales. For
example, the density contrast $\delta_\mathrm{g}$ of biased tracers on
sufficiently large scales, where the dynamics are described by linear
theory, is proportional to the linear density contrast
$\delta_\mathrm{L}$, and we have a simple relation
$\delta_\mathrm{g} = b\,\delta_\mathrm{L}$, where a proportional
constant $b$ is called the linear bias parameter
\cite{Kaiser:1984sw,Bardeen:1985tr}. While nonlinear dynamics
complicate the situation, the nonlinear perturbation theory to
describe the spatial clustering of biased tracers has been also
developed \cite{Bernardeau:2001qr,Desjacques:2016bnm}. Since the
perturbation theory cannot describe the nonlinear structure formation
which takes place on small scales, it is inevitable to introduce some
unknown parameters or functions whose values strongly depend on the
small-scale nonlinear physics which cannot be predicted from the first
principle. These nuisance parameters are called bias parameters or
bias functions in the nonlinear perturbation theory. When a model of
bias is defined in some way, one can calculate the evolution of the
biased tracers by applying the nonlinear perturbation theory.

There are several different ways of characterizing the bias model in
the nonlinear perturbation theory \cite{Desjacques:2016bnm}. Among
others, the integrated perturbation theory (iPT)
\cite{Matsubara:2011ck,Matsubara:2012nc,Matsubara:2013ofa,Matsubara:2016wth},
which is based on the Lagrangian perturbation theory
\cite{Buchert:1989xx,Moutarde:1991evx} and orthogonal decompositions
of the bias \cite{Matsubara:1995wd}, provides a method to generically
include any model of bias formulated in Lagrangian space, irrespective
of whether the bias is local or nonlocal. In this formalism, the
concept of the renormalized bias functions \cite{Matsubara:2011ck} is
introduced. Dynamical evolutions on large scales which are described
by perturbation theory are separated from complicated nonlinear
processes of structure formation on small scales. The latter
complications are all encoded in the renormalized bias functions in
the formalism of iPT. The renormalized bias functions are given for
any models of bias, which can be nonlocal in general. For example, the
halo bias is one of the typical models of Lagrangian bias, and the
renormalized bias functions are uniquely determined and calculated
from the model \cite{Matsubara:2012nc}.

The number density of biased objects is not the only quantity we can
observe. For example, the position and shape of galaxies are
simultaneously observed in imaging surveys of galaxies, which are
essential in observations of weak lensing fields to probe the nature
of dark matter and dark energy \cite{Bartelmann:1999yn}. Before the
lensing effects, the galaxy shapes are more or less correlated to the
mass density field through, e.g., gravitational tidal fields and other
environmental structures in the underlying density field. The shapes
of galaxies, i.e., sizes and intrinsic alignments
\cite{Catelan:2000vm,Okumura:2008du,Joachimi:2015mma,Kogai:2018nse,Okumura:2019ned,Okumura:2019ozd,Okumura:2020hhr,Taruya:2020tdi,Okumura:2021xgc}
are expected to be promising probes of cosmological information
\cite{Chisari:2013dda} in near future when unprecedentedly large
imaging surveys are going to take place. For example, the shape
statistics of galaxies offer a new probe of particular features in
primordial non-Gaussianity generated during the inflation in the
presence of higher-spin fields \cite{Arkani-Hamed:2015bza}, which
features are dubbed ``cosmological collider physics.''

Motivated by recent progress in observational techniques of imaging
surveys, analytical modelings of galaxy shape statistics by the
nonlinear perturbation theory have recently been considered
\cite{Blazek:2015lfa,Blazek:2017wbz,Schmitz:2018rfw,Vlah:2019byq,Vlah:2020ovg,Taruya:2021jhg}.
Among others, Ref.~\cite{Vlah:2019byq} developed a formalism based on
spherical tensors to describe the galaxy shapes, with which rotation
and parity symmetries in the shape statistics are respected. They use
an approach of what is called Effective Field Theory of the
Large-Scale Structure (EFTofLSS) with the Eulerian perturbation
theory, and galaxy biasing are modeled by introducing a set of terms
that are generally allowed by conceivable symmetries of the problem
and the coefficients of these terms are free parameters in the theory.

The galaxy shape statistics are usually characterized by the
second-order moments of galaxy images in the two-dimensional sky,
which correspond to the projection of the three-dimensional
second-order moments of galaxy shapes to the two-dimensional sky. In
principle, the galaxy images contain information about other moments
higher than the second order. The higher-order moments in galaxy
images turn out to be possible probes for the higher-spin fields in
the context of cosmological collider physics through angle-dependent
primordial non-Gaussianity. Ref.~\cite{Kogai:2020vzz} showed that the
shape statistics of the order-$s$ moment turn out to be a probe for
spin-$s$ fields.

In this paper, inspired by the above developments, we consider a
generalization of the iPT formalism to predict correlations of tensor
fields of rank $l$ in general with the perturbation theory. In order
to have rotational symmetry apparent, we decompose the tensor fields
into irreducible tensors on the basis of spherical tensors, just
similarly in the pioneering work of
Refs.~\cite{Vlah:2019byq,Vlah:2020ovg}. However, unlike the previous
work, we do not fix the coordinates system to the directions of
wavevectors of Fourier modes, so as to make the theory fully covariant
with respect to the rotational symmetry in three-dimensional space. It
turns out that the generalizing iPT formalism on the basis of
irreducible tensors results in an elegant way of describing the
perturbation theory only with rotationally invariant quantities in the
presence of tensor-valued bias. Various methods developed for the
theory of angular momentum in quantum mechanics, such as the
$3nj$-symbols and their orthogonality relations, sum rules, the
Wigner-Eckart theorem, and so on, are effectively utilized in the
course of calculations for predicting statistical quantities of tensor
fields.

This paper defines a basic formalism of the tensor generalization of
the methods with iPT. Several applications of the fully general
formalism to relatively simple examples of calculating statistics are
also illustrated, i.e., the lowest-order predictions of the power
spectrum of tensor fields in real space and in redshift space,
lowest-order effects of primordial non-Gaussianity in the power
spectrum, and tree-level predictions of the bispectrum of tensor
fields in real space. The main purpose of this paper is to show the
fundamental formulation of tensor-valued iPT, and to give useful
methods and formulas for future applications to calculate statistics
of concrete tensor fields, such as galaxy spins, shapes, etc. Further
developments for calculating loop corrections in the
higher-order perturbation theory are described in a separate paper
in the series, Paper~II \cite{PaperII}, which follows this paper.

This paper is organized as follows. In Sec.~\ref{sec:SphericalBasis},
various notations regarding the spherical basis are introduced and
defined, and their fundamental properties are derived and summarized.
In Sec.~\ref{sec:iPTTensor}, the iPT formalism originally for
scalar-valued fields is generalized to those for tensor-valued fields,
and rotationally invariant components of renormalized bias functions
and higher-order propagators are identified. Additional symmetries in
them are also described. Explicit forms of several lower-order
propagators are calculated and presented. In
Sec.~\ref{sec:TensorSpectrum}, the calculation of the power spectrum
and bispectrum of tensor fields by our formalism is illustrated and
presented in several cases which are not relatively complicated. In
Sec.~\ref{sec:Semilocal}, we define a class of bias models, semi-local
models, in which the Lagrangian bias of the tensor field is determined
only by derivatives of the gravitational potential at the same
position in Lagrangian space. Conclusions are given in
Sec.~\ref{sec:Conclusions}. In Appendix~\ref{app:SphericalBasis},
explicit expressions of higher-order spherical tensor basis are
derived and given. Formulas regarding spherical harmonics which are
repeatedly used throughout this paper are summarized in
Appendix~\ref{app:SphericalHarmonics} for readers' convenience. In
Appendix~\ref{app:3njSymbols} is summarized some properties of
Wigner's $3j$-, $6j$- and $9j$-symbols which are useful in this
paper.

\section{Decomposition of tensors by spherical basis
  \label{sec:SphericalBasis}
}

For a given set of orthonormal basis vectors $\hat{\mathbf{e}}_i$
($i=1,2,3$) in a flat, three-dimensional space, the spherical basis
($\mathbf{e}_0$, $\mathbf{e}_\pm$) is defined by
\cite{Edmonds:1955fi,Khersonskii:1988krb}
\begin{equation}
  \mathbf{e}_0 = \hat{\mathbf{e}}_3, \quad
  \mathbf{e}_\pm = \mp
  \frac{\hat{\mathbf{e}}_1 \pm i\hat{\mathbf{e}}_2}{\sqrt{2}}.
  \label{eq:2-1}
\end{equation}
It is convenient to introduce the dual basis with an upper index by
taking complex conjugates of the spherical basis:
\begin{equation}
  \mathbf{e}^0 \equiv \mathbf{e}_0^* = \hat{\mathbf{e}}_3, \quad
  \mathbf{e}^\pm \equiv \mathbf{e}_\pm^*
  = \mp \frac{\hat{\mathbf{e}}_1 \mp i\hat{\mathbf{e}}_2}{\sqrt{2}}.
  \label{eq:2-2}
\end{equation}

The orthogonality relations among these bases are given by
\begin{equation}
  \mathbf{e}_m\cdot\mathbf{e}_{m'} = g^{(1)}_{mm'}, \quad
  \mathbf{e}_m\cdot\mathbf{e}^{m'} = \delta_m^{m'}, \quad
  \mathbf{e}^m\cdot\mathbf{e}^{m'} = g_{(1)}^{mm'},
  \label{eq:2-3}
\end{equation}
where $m,m'$ are azimuthal indices of the spherical basis which
represent one of $(-,0,+)$,
\begin{equation}
  g^{(1)}_{mm'} = g_{(1)}^{mm'} = (-1)^m\delta_{m,-m'},
  \label{eq:2-4}
\end{equation}
and $\delta_m^{m'}=\delta_{mm'}$ is the Kronecker's delta symbol which
is unity for $m=m'$ and zero for $m\ne m'$. The matrices
$g_{(1)}^{mm'}$ and $g^{(1)}_{mm'}$ are inverse matrices of each
other,
\begin{equation}
  g_{(1)}^{mm''} g^{(1)}_{m''m'} = \delta^m_{m'},
  \label{eq:2-5}
\end{equation}
where the repeated index $m''$ is assumed to be summed over, omitting
summation symbols as commonly adopted in the convention of tensor
analysis (Einstein summation convention). Throughout this paper, the
Einstein convention of summation is always assumed, unless otherwise
stated. They act as metric tensors of the spherical basis,
\begin{equation}
  \mathbf{e}^m = g_{(1)}^{mm'} \mathbf{e}_{m'}, \quad
  \mathbf{e}_m = g^{(1)}_{mm'} \mathbf{e}^{m'}.
  \label{eq:2-6}
\end{equation}
Any vector can be represented by
$\bm{V} \propto V^m\mathbf{e}_m = V_m\mathbf{e}^m$ up to normalization
constant, and thus $\mathbf{e}_m$ is considered as the basis of {\em
  contravariant} spherical vectors, and $\mathbf{e}^m$ is considered
as the basis of {\em covariant} spherical vectors.

One of the advantages of the spherical basis over the Cartesian basis
is that the Cartesian tensors are reduced into irreducible tensors on
the spherical basis under a rotation of coordinates system
\cite{Sakurai:2011zz}. We define the spherical basis of traceless,
symmetric tensors \cite{Desjacques:2010gz,Vlah:2019byq}
as\footnote{The normalization of the spherical tensors in this paper
  are different from those in
  Refs.~\cite{Desjacques:2010gz,Vlah:2019byq}. Our normalization is
  chosen so that Eq.~(\ref{eq:2-13}) below should hold with a usual
  normalization of the spherical harmonics. In addition, in the
  previous literature, the axis $\mathbf{e}^0$ of the coordinates
  system is specifically chosen to be parallel to the separation
  vector $\bm{r}$ of the correlation function
  \cite{Desjacques:2010gz}, or, to the wavevector $\bm{k}$ of Fourier
  modes \cite{Vlah:2019byq}, while our axis can be arbitrarily chosen,
  so that fully rotational symmetry is explicitly kept in our
  formalism, not choosing a special coordinates system.}
\begin{equation}
  \mathsf{Y}^{(0)} = \frac{1}{\sqrt{4\pi}}, \quad
  \mathsf{Y}^{(m)}_i = \sqrt{\frac{3}{4\pi}}\, {\mathrm{e}^m}_i
  \label{eq:2-8}
\end{equation}
for tensors of rank 0 and 1, and
\begin{align}
  &
    \mathsf{Y}^{(0)}_{ij} = \frac{1}{4} \sqrt{\frac{5}{\pi}}
    \left(
    3 {\mathrm{e}^0}_i\,{\mathrm{e}^0}_j - \delta_{ij}
    \right),
  \label{eq:2-9a}\\
  &
    \mathsf{Y}^{(\pm 1)}_{ij} = \frac{1}{2} \sqrt{\frac{15}{\pi}}
    {\mathrm{e}^0}_{(i}\,{\mathrm{e}^\pm}_{j)},
  \label{eq:2-9b}\\
  &
    \mathsf{Y}^{(\pm 2)}_{ij} = \frac{1}{2} \sqrt{\frac{15}{2\pi}}\,
    {\mathrm{e}^\pm}_i\,{\mathrm{e}^\pm}_j
  \label{eq:2-9c}
\end{align}
for tensors of rank 2, where ${\mathrm{e}^m}_i = [\mathbf{e}^m]_i$ is
the Cartesian components of the spherical basis, and round brackets in
the indices of the right-hand side (rhs) of
Eq.~(\ref{eq:2-9b}) indicate symmetrization with respect to the
indices inside the brackets, e.g.,
$x_{(i}y_{j)} = (x_iy_j + x_jy_i)/2$. These bases satisfy
orthogonality relations,
\begin{equation}
    \mathsf{Y}^{(0)}\mathsf{Y}^{(0)} = \frac{1}{4\pi}, \
    \mathsf{Y}^{(m)}_i \mathsf{Y}^{(m')}_i =
    \frac{3}{4\pi} g_{(1)}^{mm'},\
    \mathsf{Y}^{(m)}_{ij} \mathsf{Y}^{(m')}_{ij} =
    \frac{15}{8\pi} g_{(2)}^{mm'},
  \label{eq:2-10}
\end{equation}
where $g_{(2)}^{mm'} = (-1)^m \delta_{m,-m'}$ is the $5\times 5$
spherical metric, and the indices $m,m'$ run over the integers
$-2,-1,0,+1,+2$ in this case. The Einstein summation convention is also
applied for the Cartesian indices in the above, so that $i$ and/or $j$
are summed over on the left-hand side (lhs) of the above equations.
Taking complex conjugates of the spherical basis virtually lowers the
azimuthal indices,
\begin{equation}
  \mathsf{Y}^{(0)*} = \mathsf{Y}^{(0)},\quad
  \mathsf{Y}^{(m)*}_i = g^{(1)}_{mm'}\mathsf{Y}^{(m')}_i,\quad
  \mathsf{Y}^{(m)*}_{ij} = g^{(2)}_{mm'}\mathsf{Y}^{(m')}_{ij}.
  \label{eq:2-11}
\end{equation}
For a unit vector $\bm{n}$ whose direction is given by spherical
coordinates $(\theta,\phi)$, the spherical tensors are related to the
spherical harmonics $Y_{lm}(\theta,\phi)$ as
\begin{equation}
  Y_{00}(\theta,\phi) = \mathsf{Y}^{(0)*}, \
  Y_{1m}(\theta,\phi) = \mathsf{Y}^{(m)*}_i n_i, \
  Y_{2m}(\theta,\phi) = \mathsf{Y}^{(m)*}_{ij} n_i n_j.
  \label{eq:2-12}
\end{equation}
We have chosen the normalization of the spherical basis so that the
coefficients in the relations above are unity. In this paper, the
normalization convention of the spherical harmonics is given by a
standard one,
\begin{equation}
  Y_{lm}(\theta,\phi) =
  \sqrt{\frac{2l+1}{4\pi}}
  \sqrt{\frac{(l-m)!}{(l+m)!}}
  P_l^m(\cos\theta)\,e^{im\phi},
  \label{eq:2-12-1}
\end{equation}
where $P_l^m(x)$ is the associated Legendre polynomial.

Similarly, spherical bases for higher-rank symmetric tensors are
uniquely defined by imposing the similar properties of
Eq.~(\ref{eq:2-12}),
\begin{equation}
  Y_{lm}(\theta,\phi) =
  \mathsf{Y}^{(m)*}_{i_1i_2\cdots i_l} n_{i_1} n_{i_2}\cdots n_{i_l}.
  \label{eq:2-13}
\end{equation}
Explicit forms of the spherical bases of the rank 0 to 4 are derived
and given in Appendix~\ref{app:SphericalBasis}. The spherical bases
are irreducible representations of the rotation group SO(3) in the
same way as the spherical harmonics are. We consider a passive
rotation of the Cartesian basis,
\begin{equation}
  \hat{\mathbf{e}}_i \rightarrow \hat{\mathbf{e}}_i'
  = R_{ij}\hat{\mathbf{e}}_j,
  \label{eq:2-14}
\end{equation}
where $R_{ij} = \hat{\mathbf{e}}_i' \cdot \hat{\mathbf{e}}_j$ is
a real orthogonal matrix with $R^\mathrm{T}R = I$. Cartesian
components of a unit vector $\bm{n}$, where $|\bm{n}|=1$,
transform as
\begin{equation}
  n_i \rightarrow n_i' = (R^{-1})_{ij} n_j = n_jR_{ji},
  \label{eq:2-14-1}
\end{equation}
because we have $n_i \hat{\mathbf{e}}_i = n_i' \hat{\mathbf{e}}_i'$.
Denoting the spherical coordinates of $n_i$ and $n_i'$ as
$(\theta,\phi)$ and $(\theta',\phi')$, respectively, the spherical
harmonics transform as
\begin{equation}
  Y_{lm}(\theta',\phi') = Y_{lm'}(\theta,\phi)D_{(l)m}^{m'}(R),
  \label{eq:2-14-2}
\end{equation}
where $D_{(l)m}^{m'}(R)$ is the Wigner's rotation matrix of the
passive rotation $R$.
 Therefore, the construction of the tensor bases,
Eq.~(\ref{eq:2-13}), indicates that they transform as
\begin{align}
  \mathsf{Y}^{(m)}_{i_1i_2\cdots i_l} \rightarrow
  \mathsf{Y}^{(m)\prime}_{i_1i_2\cdots i_l}
  &=
    R_{i_1j_1} R_{i_2j_2}\cdots R_{i_lj_l}
    \mathsf{Y}^{(m)}_{j_1j_2\cdots j_l}
    \nonumber\\
  &=
    D^{m}_{(l)m'}(R^{-1}) \mathsf{Y}^{(m')}_{i_1i_2\cdots i_l},
  \label{eq:2-15}
\end{align}
where Wigner's matrix of the inverse rotation is given by
\begin{equation}
  D^{m}_{(l)m'}(R^{-1})  = D^{m'*}_{(l)m}(R).
  \label{eq:2-15-1}
\end{equation}

The orthogonality relations of Eq.~(\ref{eq:2-10}) are generalized for
higher-rank tensors, and are given by
\begin{equation}
  \mathsf{Y}^{(m)}_{i_1i_2\cdots i_l}
  \mathsf{Y}^{(m')}_{i_1i_2\cdots i_l} =
    \frac{(2l+1)!!}{4\pi l!} g_{(l)}^{mm'},
  \label{eq:2-16}
\end{equation}
where $(2l+1) \times (2l+1)$ matrix $g_{(l)}^{mm'}$ is defined just as
Eq.~(\ref{eq:2-4}) but the indices $m,m'$ run over integers from $-l$
to $+l$. The inverse matrix is similarly denoted by $g^{(l)}_{mm'}$,
which elements are the same as $g_{(l)}^{mm'}$. Thus, similar
relations as in Eqs.~(\ref{eq:2-4}) and (\ref{eq:2-5}) hold for
$g_{(l)}^{mm'}$ and $g^{(l)}_{mm'}$:
\begin{equation}
  g_{(l)}^{mm'} = g^{(l)}_{mm'} = (-1)^m \delta_{m,-m'}, \quad
  g_{(l)}^{mm''} g^{(l)}_{m''m'} = \delta^m_{m'}.
  \label{eq:2-16-1}
\end{equation}
The relations to the $1jm$-symbol introduced by Wigner \cite{Wigner93}
are given by
\begin{equation}
  (-1)^lg^{(l)}_{mm'} =
  \begin{pmatrix}
    l\\
    m\ \ m'
  \end{pmatrix}
  = (-1)^lg_{(l)}^{mm'}.
  \label{eq:2-16-2}
\end{equation}
Taking the complex conjugate of the basis virtually lowers the
azimuthal index as in Eq.~(\ref{eq:2-11}), i.e., we have
\begin{equation}
  \mathsf{Y}^{(m)*}_{i_1i_2\cdots i_l}
  = g^{(l)}_{mm'}\mathsf{Y}^{(m')}_{i_1i_2\cdots i_l}, \quad
  \mathsf{Y}^{(m)}_{i_1i_2\cdots i_l}
  = g_{(l)}^{mm'}\mathsf{Y}^{(m')*}_{i_1i_2\cdots i_l}.
  \label{eq:2-17}
\end{equation}
The same properties also apply to spherical harmonics:
\begin{equation}
  Y_{lm}^{*}(\theta,\phi)
  = g_{(l)}^{mm'} Y_{lm'}(\theta,\phi), \quad
  Y_{lm}(\theta,\phi)
  = g^{(l)}_{mm'} Y_{lm'}^{*}(\theta,\phi).
  \label{eq:2-17-1}
\end{equation}
Consequently, the orthogonality relations of Eq.~(\ref{eq:2-16}) is
also equivalent to
\begin{align}
  \mathsf{Y}^{(m)*}_{i_1i_2\cdots i_l}
  \mathsf{Y}^{(m')}_{i_1i_2\cdots i_l}
  &=
    \frac{(2l+1)!!}{4\pi l!} \delta_m^{m'},
  \label{eq:2-18a}\\
  \mathsf{Y}^{(m)*}_{i_1i_2\cdots i_l}
  \mathsf{Y}^{(m')*}_{i_1i_2\cdots i_l}
  &=
    \frac{(2l+1)!!}{4\pi l!} g^{(l)}_{mm'}.
  \label{eq:2-18b}
\end{align}

We decompose a scalar function $S$ and a vector function $V_i$ into
spherical components as
\begin{equation}
  S = 4\pi S_{00} \mathsf{Y}^{(0)}, \quad
  V_i = \frac{4\pi i}{3} V_{1m} \mathsf{Y}^{(m)}_i.
  \label{eq:2-19}
\end{equation}
The prefactors $4\pi$ and $4\pi i/3$ on the rhs of Eq.~(\ref{eq:2-19})
are just our normalization conventions that turn out to be convenient
for our purpose below. Due to the orthogonality relations of
Eq.~(\ref{eq:2-10}), the inverted relations of the above are given by
\begin{equation}
  S_{00} = S \mathsf{Y}^{(0)*}, \quad
  V_{1m} = -i V_i \mathsf{Y}^{(m)*}_i.
  \label{eq:2-20}
\end{equation}
Since the spherical basis is a traceless tensor, a symmetric rank-2
tensor function $T_{ij}$, which is not necessarily traceless, is
decomposed into spherical components of the trace part and traceless
part as
\begin{equation}
  T_{ij}
  = 4\pi T_{00} \frac{\delta_{ij}}{3} \mathsf{Y}^{(0)}
  - \frac{8\pi}{15} T_{2m} \mathsf{Y}^{(m)}_{ij},
  \label{eq:2-21}
\end{equation}
where
\begin{equation}
  T_{00} = T_{ii} \mathsf{Y}^{(0)*}, \quad
  T_{2m} = - T_{ij} \mathsf{Y}^{(m)*}_{ij}.
  \label{eq:2-22}
\end{equation}
The prefactor $-8\pi/15$ in front of the second term on the rhs of
Eq.~(\ref{eq:2-21}) is again our normalization convention.

Higher-rank tensors can be similarly decomposed into spherical
tensors. We first note that any symmetric tensor of rank $l$ can be
decomposed into \cite{Spencer1970}
\begin{equation}
  T_{i_1i_2\cdots i_l} =
  T^{(l)}_{i_1i_2\cdots i_l}
  + \frac{l(l-1)}{2(2l-1)}
  \delta_{(i_1i_2} T^{(l-2)}_{i_3\cdots i_l)}
  + \cdots,
  \label{eq:2-23}
\end{equation}
where $T^{(l)}_{i_1i_2\cdots i_l}$ is a symmetric traceless tensor of
rank $l$, $T^{(l-2)}_{i_3\cdots i_l}$ is traceless part of
$T_{jji_3i_4\cdots i_l}$, and so forth (one should note that the
Einstein summation convention is applied also to Cartesian indices,
and thus $j=1,2,3$ is summed over in the last tensor). The traceless
part of $T_{i_1\cdots i_l}$ is generally given by a formula
\cite{Spencer1970,Applequist1989}
\begin{multline}
  T^{(l)}_{i_1\cdots i_l} =
  \frac{l!}{(2l-1)!!}
  \sum_{k=0}^{[l/2]}
  \frac{(-1)^k(2l-2k-1)!!}{2^k k! (l-2k)!}
  \\ \times
  \delta_{(i_1i_2}\cdots\delta_{i_{2k-1}i_{2k}}
  T^{(l:k)}_{i_{2k+1}\cdots i_l)},
  \label{eq:2-23-1}
\end{multline}
where $[l/2]$ is the Gauss symbol, i.e., $[l/2] = l/2$ if $l$ is even
and $[l/2] = (l-1)/2$ if $l$ is odd, and
\begin{align}
  T^{(l:k)}_{i_{2k+1}\cdots i_l}
  &= \delta_{i_1i_2} \delta_{i_3i_4} \cdots \delta_{i_{2k-1}i_{2k}}
    T_{i_1\cdots i_{2k}i_{2k+1}\cdots i_l}
    \nonumber \\
  &=T_{j_1j_1j_2j_2\cdots j_kj_ki_{2k+1}\cdots i_l}
  \label{eq:2-23-2}
\end{align}
is the traces of the original tensor taken $k$ times. Tensors of rank
0 and 1 correspond to a scalar and a vector, respectively, and are
already traceless as they do not have any trace component from the
first place. The traceless part of the tensor components forms an
irreducible representation of the rotation group SO(3). Decomposition
of higher-rank tensors can be similarly obtained according to the
procedure described in Ref.~\cite{Spencer1970}. Recursively using
Eq.~(\ref{eq:2-23-1}), one obtains explicit expressions of
Eq.~(\ref{eq:2-23}). Up to the fourth rank, they are given by
\begin{align}
  T_{ij}
  &= T^{(2)}_{ij} + \frac{1}{3} \delta_{ij} T^{(0)},
  \label{eq:2-23-1a}\\
  T_{ijk}
  &= T^{(3)}_{ijk} + \frac{3}{5} \delta_{(ij}T^{(1)}_{k)},
  \label{eq:2-23-1b}\\
  T_{ijkl}
  &= T^{(4)}_{ijkl} + \frac{6}{7} \delta_{(ij} T^{(2)}_{kl)}
    + \frac{1}{5} \delta_{(ij}\delta_{kl)} T^{(0)}.
  \label{eq:2-23-1c}
\end{align}

The traceless part of a tensor is decomposed by spherical bases as
\begin{align}
  T^{(l)}_{i_1i_2\cdots i_l}
  &= i^l \alpha_l T_{lm} \mathsf{Y}^{(m)}_{i_1i_2\cdots i_l},
  \label{eq:2-24a}\\
  T_{lm}
  &= (-i)^l
  T^{(l)}_{i_1i_2\cdots i_l} \mathsf{Y}^{(m)*}_{i_1i_2\cdots i_l},
  \label{eq:2-24b}
\end{align}
where
\begin{equation}
  \alpha_l \equiv
  \frac{4\pi\,l!}{(2l+1)!!}.
  \label{eq:2-24-0}
\end{equation}
The prefactor $i^l\alpha_l$ on the rhs of Eq.~(\ref{eq:2-24a}) is our
normalization convention. The traceless part of a traced tensor is
similarly decomposed as
\begin{align}
  T^{(l-2k)}_{i_{2k+1}\cdots i_l}
  &= i^{l-2k} \alpha_{l-2k}T_{l-2k,m} \mathsf{Y}^{(m)}_{i_{2k+1}\cdots i_l},
  \label{eq:2-24-1a}\\
  T_{l-2k,m}
  &= (-i)^{l-2k}
  T^{(l)}_{i_{2k+1}\cdots i_l} \mathsf{Y}^{(m)*}_{i_{2k+1}\cdots i_l}.
  \label{eq:2-24-1b}
\end{align}

Under the passive rotation of the basis, Eq.~(\ref{eq:2-14}),
Cartesian tensors of rank $l$ transform as
\begin{equation}
  T^{(l)}_{i_1i_2\cdots i_l} \rightarrow
  {T'}^{(l)}_{i_1i_2\cdots i_l} =
  T^{(l)}_{j_1j_2\cdots j_l} R_{j_1i_1}\cdots R_{j_li_l},
  \label{eq:2-25}
\end{equation}
and the corresponding transformation of spherical tensors is given by
\begin{equation}
  T_{lm} \rightarrow
  T'_{lm} =
  T_{lm'} D^{m'}_{(l)m}(R),
  \label{eq:2-26}
\end{equation}
as derived from the transformation of the spherical base,
Eq.~(\ref{eq:2-15}). The contravariant components of the spherical
tensors are defined by
\begin{equation}
  T_{l}^{\,m} = g_{(l)}^{mm'} T_{lm'},
  \label{eq:2-27}
\end{equation}
and transform as
\begin{equation}
  T_{l}^{\,m} \rightarrow
  {T'}_{l}^{\,m} =
   D^{m}_{(l)m'}(R^{-1}) T_{l}^{\,m'}.
  \label{eq:2-28}
\end{equation}

\section{\label{sec:iPTTensor}
  The integrated perturbation theory of tensor fields
}

\subsection{\label{subsec:iPTTFormulation}
  Formulating the iPT of tensor fields
}

The integrated perturbation theory (iPT) is a systematic method of
perturbation theory to describe the quasi-nonlinear evolution of the
large-scale structure. This method is based on the Lagrangian scheme
of the biasing and nonlinear structure formation. The formalism of
this method is explicitly developed to describe the scalar fields,
such as the number density fields of galaxies or other biased objects
in the Universe \cite{Matsubara:2011ck,Matsubara:2013ofa}. In this
section, the formalism is generalized to be able to describe
quasi-nonlinear evolutions of objects which have generally tensor
values, such as the angular momentum of galaxies (vector),
the second moment of intrinsic alignment of galaxy shapes (rank-2
tensor), or higher moments of intrinsic alignment (rank-3 or higher
tensors), etc.

The fundamental formalism of iPT to calculate spatial correlations of
biased objects is described in Ref.~\cite{Matsubara:2011ck}. Basically
we can just follow the formalism with a generalization of assigning the
tensor values to the objects. We denote the observable objects $X$,
where $X$ is the class of objects selected in a given cosmological
surveys, such as a certain types of galaxies in galaxy surveys. We
consider each object of $a\in X$ has a tensor value
$F^a_{i_1i_2\cdots}$ in general, and define the tensor field by
\begin{equation}
  F_{X\,i_1i_2\cdots}(\bm{x}) =
  \frac{1}{\bar{n}_X}
  \sum_{a\in X} F^a_{i_1i_2\cdots}
  \delta_\mathrm{D}^3\left(\bm{x} - \bm{x}_a\right),
  \label{eq:3-1}
\end{equation}
where $\bm{x}_a$ is the location of a particular object $a$,
$\bar{n}_X$ is the mean number density of objects $X$, and
$\delta_\mathrm{D}^3$ is the 3-dimensional Dirac's delta function.
It is crucial to note that the above definition of the tensor field is
weighted by the number density of the objects $X$. In the above,
although the field really depends on the time
$F_{X\,i_1i_2\cdots}(\bm{x},t)$, the argument of time is omitted in
the notation for simplicity. Similarly, every field in this paper
depends on time, although they are not explicitly written in the
arguments.

In general, any type of tensor can be decomposed into symmetric
tensors \cite{Spencer1970}, and therefore we assume that the tensor
field $F^a_{i_1i_2\cdots}$ is a symmetric tensor without loss of
generality. We decompose the symmetric tensor of each object
$F^a_{i_1i_2\cdots}$ into irreducible tensors $F^a_{lm}$ according to
the scheme described in the previous section. Accordingly,
corresponding components of Eq.~(\ref{eq:3-1}) are given by
\begin{equation}
  F_{Xlm}(\bm{x}) =
  \frac{1}{\bar{n}_X}
  \sum_{a\in X} F^a_{lm}
  \delta_\mathrm{D}^3\left(\bm{x} - \bm{x}_a\right),
  \label{eq:3-1-1}
\end{equation}
As this tensor field is a number-density weighted quantity by
construction, we also consider the unweighted field $G_{Xlm}(\bm{x})$
defined by
\begin{equation}
  F_{Xlm}(\bm{x}) = \left[1 + \delta_X(\bm{x})\right] G_{Xlm}(\bm{x}),
  \label{eq:3-2}
\end{equation}
where $\delta_X(\bm{x})$ is the number density contrast of the objects
$X$, defined by
\begin{equation}
  1 + \delta_X(\bm{x}) =
  \frac{1}{\bar{n}_X}
  \sum_{a\in X} \delta_\mathrm{D}^3\left(\bm{x} - \bm{x}_a\right).
  \label{eq:3-3}
\end{equation}
Substituting Eq.~(\ref{eq:3-3}) into Eq.~(\ref{eq:3-2}) and comparing
with Eq.~(\ref{eq:3-1-1}), the unweighted field satisfies
\begin{equation}
  G_{Xlm}(\bm{x}_a) = F^a_{Xlm}
  \label{eq:3-3-1}
\end{equation}
at each position of object $a$, as naturally expected. The function
$G_{Xlm}(\bm{x})$ can arbitrarily be interpolated where $\bm{x} \ne
\bm{x}_a$ for all $a \in X$, because the field $F_{Xlm}(\bm{x})$ is
not supported at those points.

The iPT utilizes the Lagrangian perturbation theory (LPT) for the
dynamical nonlinear evolutions (see Ref.~\cite{Matsubara:2015ipa} for
notations of LPT employed in this paper). The Eulerian coordinates
$\bm{x}$ and the Lagrangian coordinates $\bm{q}$ of a mass element are
related by
\begin{equation}
  \bm{x} = \bm{q} + \bm{\Psi}(\bm{q}),
  \label{eq:3-4}
\end{equation}
where $\bm{\Psi}(\bm{q})$ is the displacement field. The number
density of the objects in Lagrangian space $n^\mathrm{L}_X(\bm{q})$ is
given by the Eulerian counterpart by a continuity relation,
\begin{equation}
  n_X(\bm{x}) d^3\!x =  n^\mathrm{L}_X(\bm{q}) d^3\!q,
  \label{eq:3-5}
\end{equation}
i.e., the Lagrangian number density is defined so that the Eulerian
positions of objects are displaced back into the Lagrangian positions,
and the number density in Lagrangian space also depends on the time of
evaluating $n_X(\bm{x})$. The mean number density of objects
$\bar{n}_X$ is the same in Eulerian and Lagrangian spaces by
construction, thus we have
$1+\delta_X(\bm{x}) = [1+\delta^\mathrm{L}_X(\bm{q})]/J(\bm{q})$,
where $J(\bm{q})=|\partial(\bm{x})/\partial(\bm{q})|$ is the Jacobian
of the mapping from Lagrangian space to Eulerian space. The density
contrasts between these spaces are therefore related by
\begin{equation}
  1 + \delta_X(\bm{x}) =
  \int d^3\!q
  \left[1 + \delta^\mathrm{L}_X(\bm{q})\right]
  \delta_\mathrm{D}^3\left[\bm{x} - \bm{q} - \bm{\Psi}(\bm{q})\right].
  \label{eq:3-6}
\end{equation}

As in the Eulerian space, the tensor field in Lagrangian space is also
naturally defined with the weight of the number density of objects.
The unweighted tensor field $G^\mathrm{L}_{Xlm}(\bm{q})$ in Lagrangian
space is simply defined by taking the same value displaced back from
the Eulerian position of Eq.~(\ref{eq:3-2}),
\begin{equation}
  G^\mathrm{L}_{Xlm}(\bm{q}) \equiv G_{Xlm}(\bm{x}),
  \label{eq:3-7}
\end{equation}
where $\bm{x}$ on the right-hand side (rhs) is just given by
Eq.~(\ref{eq:3-4}) and the number density-weighted tensor field in
Lagrangian space, $F^\mathrm{L}_{lm}(\bm{q})$, is defined by
\begin{equation}
  F^\mathrm{L}_{Xlm}(\bm{q})
  = \left[1 + \delta^\mathrm{L}_X(\bm{q})\right]
  G^\mathrm{L}_{Xlm}(\bm{q}).
  \label{eq:3-8}
\end{equation}
As obviously derived from Eqs.~(\ref{eq:3-6}) and (\ref{eq:3-7}), the
number density-weighted tensor fields in Eulerian space and
Lagrangian space is related by
\begin{equation}
  F_{Xlm}(\bm{x}) =
  \int d^3\!q
  F^\mathrm{L}_{Xlm}(\bm{q})
  \delta_\mathrm{D}^3\left[\bm{x} - \bm{q} - \bm{\Psi}(\bm{q})\right].
  \label{eq:3-9}
\end{equation}

In cosmology, the properties of observable quantities are determined
by the initial condition of density fluctuations in the Universe, and
the growing mode of linear density contrast, $\delta_\mathrm{L}$, is a
representative of initial conditions. Therefore, the tensor field
$F_{Xlm}$ is considered as a functional of the linear density field
$\delta_\mathrm{L}$. The ensemble average of the $n$th functional
derivatives of the evolved field is called multi-point propagators
\cite{Matsubara:1995wd,Bernardeau:2008fa,Bernardeau:2010md}, which are
defined in equivalently various manners. Here we explain a
comprehensive definition according to Ref.~\cite{Matsubara:1995wd},
starting from a configuration space:
\begin{equation}
  \Gamma^{(n)}_{Xlm}\left(\bm{x}-\bm{x}_1,\ldots,\bm{x}-\bm{x}_n\right)
  \equiv
  \left\langle
    \frac{\delta^n {F}_{Xlm}(\bm{x})}{
      \delta\delta_\mathrm{L}(\bm{x}_1) \cdots \delta\delta_\mathrm{L}(\bm{x}_n)}
  \right\rangle.
  \label{eq:3-10}
\end{equation}
where $\delta_\mathrm{L}(\bm{x})$ is the linear density contrast,
$\delta/\delta\delta_\mathrm{L}(\bm{x})$ is the functional derivative
with respect to the linear density contrast, and
$\langle\cdots\rangle$ represents the ensemble average of the field.
Because of the translational invariance of the Universe in a
statistical sense, the rhs should be invariant under the homogeneous
translation, $\bm{x} \rightarrow \bm{x} + \bm{x}_0$ and
$\bm{x}_i \rightarrow \bm{x}_i + \bm{x}_0$ for $i=1,\ldots,n$ with any
fixed displacement $\bm{x}_0$, and thus the arguments on the lhs of
Eq.~(\ref{eq:3-10}) should be given as those in the expression.

We apply the Fourier transform of the linear density contrast and the
tensor field,
\begin{align}
  \tilde{\delta}_\mathrm{L}(\bm{k})
  &=
  \int d^3\!x\, e^{-i\bm{k}\cdot\bm{x}} \delta_\mathrm{L}(\bm{x}),
  \label{eq:3-11a}\\
  \tilde{F}_{Xlm}(\bm{k})
  &=
  \int d^3\!x\, e^{-i\bm{k}\cdot\bm{x}} F_{Xlm}(\bm{x}).
  \label{eq:3-11b}
\end{align}
and the functional derivative in Fourier space is given by
\begin{equation}
  \frac{\delta}{\delta\tilde{\delta}_\mathrm{L}(\bm{k})}
  = \frac{1}{(2\pi)^3} \int d^3\!x\,e^{i\bm{k}\cdot\bm{x}}
  \frac{\delta}{\delta\delta_\mathrm{L}(\bm{x})}.
  \label{eq:3-12}
\end{equation}

Using the above equations, the Fourier counterpart of the rhs of
Eq.~(\ref{eq:3-10}) is calculated, and we have
\begin{multline}
  \left\langle
    \frac{\delta^n \tilde{F}_{Xlm}(\bm{k})}{
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_1) \cdots
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_n)}
  \right\rangle
  \\
  = (2\pi)^{3-3n}
  \delta_\mathrm{D}^3(\bm{k}_1+\cdots +\bm{k}_n - \bm{k})
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n),
  \label{eq:3-13}
\end{multline}
where
\begin{multline}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n)
  = \int d^3\!x_1 \cdots  d^3\!x_n\,
  e^{-i(\bm{k}_1\cdot\bm{x}_1+\cdots +\bm{k}_n\cdot\bm{x}_n)}
  \\
  \times
  \Gamma^{(n)}_{Xlm}(\bm{x}_1,\ldots\bm{x}_n)
  \label{eq:3-14}
\end{multline}
is the Fourier transform of the propagator. The appearance of the
delta function on the rhs of Eq.~(\ref{eq:3-13}) is the consequence of
the statistical homogeneity of the Universe. Integrating
Eq.~(\ref{eq:3-13}), we have
\begin{equation}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n) =
  (2\pi)^{3n}
  \int \frac{d^3k}{(2\pi)^3}
  \left\langle
    \frac{\delta^n \tilde{F}_{Xlm}(\bm{k})}{
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_1) \cdots
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_n)}
  \right\rangle.
  \label{eq:3-15}
\end{equation}
The Fourier transform of Eq.~(\ref{eq:3-9}) is given by
\begin{equation}
  \tilde{F}_{Xlm}(\bm{k}) =
  \int d^3\!q\,e^{-i\bm{k}\cdot\bm{q} -i\bm{k}\cdot\bm{\Psi}(\bm{q})}
  F^\mathrm{L}_{Xlm}(\bm{q}).
  \label{eq:3-16}
\end{equation}

In the iPT, the displacement field $\bm{\Psi}(\bm{q})$ is expanded as
a perturbative series of Lagrangian perturbation theory (LPT),
\begin{multline}
  \bm{\Psi}(\bm{q}) =
  \sum_{n=1}^\infty \frac{i}{n!}
  \int \frac{d^3k_1}{(2\pi)^3} \cdots \frac{d^3k_n}{(2\pi)^3}
  e^{i(\bm{k}_1 + \cdots + \bm{k}_n)\cdot\bm{q}}
  \\ \times
  \bm{L}_n(\bm{k}_1,\ldots,\bm{k}_n)
  \tilde{\delta}_\mathrm{L}(\bm{k}_1) \cdots
  \tilde{\delta}_\mathrm{L}(\bm{k}_n).
  \label{eq:3-17}
\end{multline}
The perturbation kernels $\bm{L}_n$ are recursively obtained both for
the model of Einstein-de Sitter Universe \cite{Zheligovsky:2013eca}
and for general models \cite{Matsubara:2015ipa}. The propagators of
Eq.~(\ref{eq:3-15}) are systematically evaluated by applying
diagrammatic rules in the iPT. The detailed derivation of the
diagrammatic rules for the number density field $\delta_X$ is given in
Ref.~\cite{Matsubara:2011ck}. Exactly the same derivation applies to
the tensor field, simply replacing $1+\delta_X \rightarrow F_{Xlm}$
and $1+\delta^\mathrm{L}_X \rightarrow F^\mathrm{L}_{Xlm}$ in that
reference. In this method, the renormalized bias functions play
important roles. In the present case of the tensor field, the renormalized
bias functions $c^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n)$ in Fourier
space are defined by
\begin{multline}
  \left\langle
    \frac{\delta^n \tilde{F}^\mathrm{L}_{Xlm}(\bm{k})}{
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_1) \cdots
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_n)}
  \right\rangle
  \\
  = (2\pi)^{3-3n}
  \delta_\mathrm{D}^3(\bm{k}_1+\cdots +\bm{k}_n - \bm{k})
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n),
  \label{eq:3-18-0}
\end{multline}
or equivalently,
\begin{equation}
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n) =
  (2\pi)^{3n}
  \int \frac{d^3k}{(2\pi)^3}
  \left\langle
    \frac{\delta^n \tilde{F}^\mathrm{L}_{Xlm}(\bm{k})}{
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_1) \cdots
      \delta\tilde{\delta}_\mathrm{L}(\bm{k}_n)}
  \right\rangle.
  \label{eq:3-18}
\end{equation}
As can be seen by comparing the above definition with
Eqs.~(\ref{eq:3-13}) and (\ref{eq:3-15}), the renormalized bias
functions are the counterparts of the propagator for the biasing in
Lagrangian space.

The diagrammatic rules of iPT are summarized in the Appendix of
Ref.~\cite{Matsubara:2013ofa}. The same rules apply, simply replacing
$\Gamma_X^{(n)} \rightarrow \Gamma_{Xlm}^{(n)}$,
$c_X^{(n)} \rightarrow c_{Xlm}^{(n)}$, etc. For the terms without
$c^{(n)}_X$ in that reference, an additional factor $c^{(0)}_{Xlm}$
should be multiplied, which is unity in the original formulation for
scalar fields. The propagators are given in a form,
\begin{equation}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n) =
  \Pi(\bm{k})\,
  \hat{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n),
  \label{eq:3-19}
\end{equation}
where
\begin{equation}
  \Pi(\bm{k}) = \left\langle e^{-i\bm{k}\cdot\bm{\Psi}} \right\rangle
    =
    \exp\left[
      \sum_{n=2}^\infty \frac{(-i)^n}{n!}
      \left\langle (\bm{k}\cdot\bm{\Psi})^n \right\rangle_\mathrm{c}
    \right]
    \label{eq:3-19-1}
\end{equation}
is the vertex resummation factor in terms of the displacement field,
and $\langle\cdots\rangle_\mathrm{c}$ denotes the $n$-point connected
part of the random variables. The propagator
$\hat{\Gamma}^{(n)}_{Xlm}$ without the resummation factor
$\Pi(\bm{k})$ is called reduced propagators in the following.

Below, we give explicit forms of lower-order propagators, which can be
used to estimate power spectra, bispectra, etc. The vertex resummation
factor in the one-loop approximation of iPT is given by
\begin{equation}
  \Pi(\bm{k}) =
  \exp\left\{
    -\frac{1}{2} \int\frac{d^3p}{(2\pi)^3}
    \left[\bm{k}\cdot\bm{L}_1(\bm{p})\right]^2
    P_\mathrm{L}(p)
  \right\},
\label{eq:3-20}
\end{equation}
where $P_\mathrm{L}(k)$ is the linear power spectrum defined by
\begin{equation}
  \left\langle
    \tilde{\delta}_\mathrm{L}(\bm{k})
    \tilde{\delta}_\mathrm{L}(\bm{k})
  \right\rangle
  = (2\pi)^3 \delta_\mathrm{D}^3(\bm{k} + \bm{k}')
  P_\mathrm{L}(k),
  \label{eq:3-20-1}
\end{equation}
where the delta function on the rhs appears due to the statistical
homogeneity of the space. The reduced propagator of the first order,
up to the one-loop approximation, is given by
\begin{multline}
  \hat{\Gamma}_{Xlm}^{(1)}(\bm{k})
  = c_{Xlm}^{(1)}(\bm{k}) +
  \left[\bm{k}\cdot\bm{L}_1(\bm{k})\right] c_{Xlm}^{(0)}
\\
  + \int\frac{d^3p}{(2\pi)^3} P_\mathrm{L}(p)
  \biggl\{
    \left[\bm{k}\cdot\bm{L}_1(-\bm{p})\right]
    c_{Xlm}^{(2)}(\bm{k},\bm{p})
\\
      + \left[\bm{k}\cdot\bm{L}_1(-\bm{p})\right]
      \left[\bm{k}\cdot\bm{L}_1(\bm{k})\right]
      c_{Xlm}^{(1)}(\bm{p})
\\
      + \left[\bm{k}\cdot\bm{L}_2(\bm{k},-\bm{p})\right]
       c_{Xlm}^{(1)}(\bm{p})
\\
      + \frac{1}{2}
      \left[\bm{k}\cdot\bm{L}_3(\bm{k},\bm{p},-\bm{p})\right]
       c_{Xlm}^{(0)}
\\
      + \left[\bm{k}\cdot\bm{L}_1(\bm{p})\right]
      \left[\bm{k}\cdot\bm{L}_2(\bm{k},-\bm{p})\right]
       c_{Xlm}^{(0)}
  \biggr\}.
\label{eq:3-21}
\end{multline}
The first line of the above quation corresponds to the lowest, or
tree-level approximation. The reduced propagator of second order in
the tree-level approximation is given by
\begin{multline}
  \hat{\Gamma}^{(2)}_{Xlm}(\bm{k}_1,\bm{k}_2) =
  c^{(2)}_{Xlm}(\bm{k}_1,\bm{k}_2)
\\
  + \left[\bm{k}_{12}\cdot\bm{L}_1(\bm{k}_1)\right] c^{(1)}_{Xlm}(\bm{k}_2)
  + \left[\bm{k}_{12}\cdot\bm{L}_1(\bm{k}_2)\right] c^{(1)}_{Xlm}(\bm{k}_1)
\\
  +
  \left\{
    \left[\bm{k}_{12}\cdot\bm{L}_1(\bm{k}_1)\right]
    \left[\bm{k}_{12}\cdot\bm{L}_1(\bm{k}_2)\right]
    + \bm{k}_{12}\cdot\bm{L}_2(\bm{k}_1,\bm{k}_2)
  \right\} c^{(0)}_{Xlm},
\label{eq:3-22}
\end{multline}
where $\bm{k}_{12}=\bm{k}_1+\bm{k}_2$, and the reduced propagator of
the third order, again in the tree-level approximation, is given by
\begin{multline}
  \hat{\Gamma}^{(3)}_{Xlm}(\bm{k}_1,\bm{k}_2,\bm{k}_3) =
  c^{(3)}_{Xlm}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  \\
  + \left[\bm{k}_{123}\cdot\bm{L}_1(\bm{k}_1)\right]
  c^{(2)}_{Xlm}(\bm{k}_2,\bm{k}_3) +\,\mathrm{cyc.}
  \\
  +
  \Bigl\{
    \left[\bm{k}_{123}\cdot\bm{L}_1(\bm{k}_1)\right]
    \left[\bm{k}_{123}\cdot\bm{L}_1(\bm{k}_2)\right]
    +
    \left[\bm{k}_{123}\cdot\bm{L}_2(\bm{k}_1,\bm{k}_2)\right]
    \Bigr\}
    \\
    \hspace{10pc} \times
    c^{(1)}_{Xlm}(\bm{k}_3) +\,\mathrm{cyc.}
  \\
  +
  \Bigl\{
    \left[\bm{k}_{123}\cdot\bm{L}_1(\bm{k}_1)\right]
    \left[\bm{k}_{123}\cdot\bm{L}_1(\bm{k}_2)\right]
    \left[\bm{k}_{123}\cdot\bm{L}_1(\bm{k}_3)\right]
    \\
    +
    \left[\bm{k}_{123}\cdot\bm{L}_1(\bm{k}_1)\right]
    \left[\bm{k}_{123}\cdot\bm{L}_2(\bm{k}_2,\bm{k}_3)\right]
    +\,\mathrm{cyc.}
    \\
    + \bm{k}_{123}\cdot\bm{L}_3(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  \Bigr\} c^{(0)}_{Xlm},
\label{eq:3-23}
\end{multline}
where $\bm{k}_{123}=\bm{k}_1+\bm{k}_2+\bm{k}_3$, and $+\,\mathrm{cyc.}$
corresponds to the two terms which are added with cyclic permutations
of each previous term.

In real space, the kernels of LPT in the standard theory of gravity
(in the Newtonian limit) are given by
\cite{Catelan:1994ze,Catelan:1996hw}
\begin{align}
& \bm{L}_1(\bm{k}) = \frac{\bm{k}}{k^2},
\label{eq:3-24a}\\
& \bm{L}_2(\bm{k}_1,\bm{k}_2)
  =\frac37 \frac{\bm{k}_{12}}{{k_{12}}^2}
  \left[1 - \left(\frac{\bm{k}_1 \cdot \bm{k}_2}{k_1 k_2}\right)^2\right],
\label{eq:3-24b}\\
&  \bm{L}_3(\bm{k}_1,\bm{k}_2,\bm{k}_3) =
  \frac13
  \left[\tilde{\bm{L}}_3(\bm{k}_1,\bm{k}_2,\bm{k}_3) + \mathrm{cyc.}\right],
\label{eq:3-24c}\\
& \tilde{\bm{L}}_3(\bm{k}_1,\bm{k}_2,\bm{k}_3)
\nonumber\\
& \quad
  = \frac{\bm{k}_{123}}{{k_{123}}^2}
  \left\{
      \frac57
      \left[1 - \left(\frac{\bm{k}_1 \cdot \bm{k}_2}{k_1 k_2}\right)^2\right]
      \left[1 - \left(\frac{\bm{k}_{12} \cdot \bm{k}_3}
          {{k}_{12} k_3}\right)^2\right]
  \right.
\nonumber\\
& \qquad\quad
  \left.
  - \frac13
  \left[
      1 - 3\left(\frac{\bm{k}_1 \cdot \bm{k}_2}{k_1 k_2}\right)^2
      +\, 2 \frac{(\bm{k}_1 \cdot \bm{k}_2)(\bm{k}_2 \cdot \bm{k}_3)
        (\bm{k}_3 \cdot \bm{k}_1)}{{k_1}^2 {k_2}^2 {k_3}^2}
  \right]\right\}
\nonumber\\
  & \qquad

  + \frac{3}{7}\frac{\bm{k}_{123}}{{k_{123}}^2}
    \times
    \frac{(\bm{k}_1\times\bm{k}_{23})(\bm{k}_1\cdot\bm{k}_{23})}{{k_1}^2{k_{23}}^2}
    \left[1 - \left(\frac{\bm{k}_2 \cdot \bm{k}_3}{k_2 k_3}\right)^2\right].
\label{eq:3-24d}
\end{align}
In the above, weak dependencies on the time $t$ in the kernels are
neglected \cite{Bernardeau:2001qr,Matsubara:2015ipa}. In
Ref.~\cite{Matsubara:2015ipa}, complete expressions of the
displacement kernels of LPT up to the seventh order including the
transverse parts are explicitly given, together with a general way of
recursively deriving the kernels including weak dependencies on the
time in general cosmology and subleading growing modes. This method is
generalized to obtain kernels of LPT in modified theories of gravity
\cite{Aviles:2017aor}.

One of the benefits of the Lagrangian picture is that redshift-space
distortions are relatively easy to incorporate into the theory. The
displacement vector in redshift space $\bm{\Psi}^\mathrm{s}$ is
obtained from a linear mapping of that in real space by
$\bm{\Psi}^\mathrm{s} = \bm{\Psi} +
H^{-1}(\hat{\bm{z}}\cdot\dot{\bm{\Psi}})\hat{\bm{z}}$, where
$\dot{\bm{\Psi}} = \partial\bm{\Psi}/\partial t$ is the time
derivative of the displacement field, $\hat{\bm{z}}$ is a unit vector
directed to the line of sight, $H = \dot{a}/a$ is the time-dependent
Hubble parameter and $a(t)$ is the scale factor. A displacement kernel
in redshift space $\bm{L}^\mathrm{s}_n$ is simply related to the
kernel in real space in the same order by a linear mapping
\cite{Matsubara:2007wj}
\begin{equation}
  \bm{L}_n \rightarrow \bm{L}^\mathrm{s}_n = \bm{L}_n +
  nf\left(\hat{\bm{z}}\cdot\bm{L}_n\right)\hat{\bm{z}},
\label{eq:3-25}
\end{equation}
where $f=d\ln D/d\ln a = \dot{D}/HD$ is the linear growth rate, $D(t)$
is the linear growth factor. Throughout this paper, the
distant-observer approximation is assumed in redshift space and the
unit vector $\hat{\bm{z}}$ denotes the line-of-sight direction. The
expressions of Eqs.~(\ref{eq:3-20})--(\ref{eq:3-23}) apply as well in
redshift space when the displacement kernels $\bm{L}_n$ are replaced
by those in redshift space $\bm{L}^\mathrm{s}_n$.

\subsection{\label{subsec:PropSym}
  Symmetries
}

\subsubsection{\label{subsubsec:SymCC}
  Complex conjugate
}

We assume the Cartesian tensor field of Eq.~(\ref{eq:3-1}) is a real
tensor:
\begin{equation}
  F^*_{X\,i_1i_2\cdots i_l}(\bm{x}) = F_{X\,i_1i_2\cdots i_l}(\bm{x}),
  \label{eq:3-30}
\end{equation}
as it should be for physically observable quantities. For the spherical
decomposition of the traceless part, $F_{Xlm}(\bm{x})$,
Eqs.~(\ref{eq:2-17}), (\ref{eq:2-24b}), (\ref{eq:2-27}) and the above
equation indicates
\begin{equation}
  F^*_{Xlm}(\bm{x}) = (-1)^l g_{(l)}^{mm'} F_{Xlm'}(\bm{x})
   = (-1)^l F_{Xl}^{\phantom{X}m}(\bm{x}),
  \label{eq:3-31}
\end{equation}
and thus the Fourier transform of Eq.~(\ref{eq:3-11b}) satisfies
\begin{equation}
  \tilde{F}^*_{Xlm}(\bm{k})
  = (-1)^l g_{(l)}^{mm'}\tilde{F}_{Xlm'}(-\bm{k})
  = (-1)^l \tilde{F}_{Xl}^{\phantom{X}m}(-\bm{k}).
  \label{eq:3-32}
\end{equation}
That is, the complex conjugate of the irreducible tensor field in
Fourier space raises the azimuthal index, inverts the sign of
wavevector in the argument, and put a phase factor $(-1)^l$. The
necessity of the last phase factor comes from our convention of the
$i^l$ factor in Eq.~(\ref{eq:2-24a}). The linear density contrast
$\delta_\mathrm{L}(\bm{x})$ is also a real field, and its Fourier
transform satisfies
${\tilde{\delta}_\mathrm{L}}^{\,*}(\bm{k}) =
\tilde{\delta}_\mathrm{L}(-\bm{k})$. Therefore, the renormalized bias
functions satisfy
\begin{align}
  c^{(n)\,*}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n)
  &= (-1)^l
    g_{(l)}^{mm'}c^{(n)}_{Xlm'}(-\bm{k}_1,\ldots,-\bm{k}_n)
    \nonumber\\
  &= (-1)^l
    c^{(n)m}_{Xl}(-\bm{k}_1,\ldots,-\bm{k}_n).
  \label{eq:3-34}
\end{align}
Similarly, the tensor propagator of Eq.~(\ref{eq:3-15}) satisfies
\begin{align}
  \tilde{\Gamma}^{(n)\,*}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n)
  &= (-1)^l
    g_{(l)}^{mm'}\tilde{\Gamma}^{(n)}_{Xlm'}(-\bm{k}_1,\ldots,-\bm{k}_n)
    \nonumber\\
  &= (-1)^l
    \tilde{\Gamma}^{(n)m}_{Xl}(-\bm{k}_1,\ldots,-\bm{k}_n).
  \label{eq:3-33}
\end{align}
In redshift space, the propagators also depend on the direction of
lines of sight, $\hat{\bm{z}}$. The above property of the complex
conjugate equally applies to those in redshift space, and we have
\begin{align}
  \tilde{\Gamma}^{(n)*}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n;\hat{\bm{z}})
  &= (-1)^l
    g_{(l)}^{mm'}
    \tilde{\Gamma}^{(n)}_{Xlm'}(-\bm{k}_1,\ldots,-\bm{k}_n;\hat{\bm{z}})
    \nonumber\\
  &= (-1)^l
    \tilde{\Gamma}^{(n)m}_{Xl}(-\bm{k}_1,\ldots,-\bm{k}_n;\hat{\bm{z}}).
  \label{eq:3-33-1}
\end{align}

\subsubsection{\label{subsubsec:SymRotation}
  Rotation
}

Under the passive rotation of Eq.~(\ref{eq:2-14}), vector
components $V_i$ generally transform as
\begin{equation}
  V_i \rightarrow V_i' = (R^{-1})_{ij}V_j = V_jR_{ji}.
  \label{eq:3-35}
\end{equation}
We denote the rotation of the position and wavevector components as,
e.g., $\bm{x} \rightarrow \bm{x}' = R^{-1}\bm{x}$ and
$\bm{k} \rightarrow \bm{k}' = R^{-1}\bm{k}$, respectively. This
notation does not mean the physical rotations of vectors and does
mean the rotation of components,
$x_i \rightarrow x_i' = x_jR_{ji}$ and
$k_i \rightarrow k_i' = k_jR_{ji}$, respectively, as a result of the
passive rotation of Cartesian basis, Eq.~(\ref{eq:2-14}). The
Cartesian tensor field of Eq.~(\ref{eq:3-1}) transforms as
\begin{equation}
  F_{X\,i_1i_2\cdots}(\bm{x}) \rightarrow
  F_{X\,i_1i_2\cdots}'(\bm{x}') =
  F_{X\,j_1j_2\cdots}(\bm{x}) R_{j_1i_1} R_{j_2i_2}\cdots,
  \label{eq:3-36}
\end{equation}
and correspondingly the irreducible tensor of rank $l$ transforms as
\begin{equation}
  F_{Xlm}(\bm{x}) \rightarrow
  F_{Xlm}'(\bm{x}') = F_{Xlm'}(\bm{x}) D_{(l)m}^{m'}(R).
  \label{eq:3-37}
\end{equation}
The corresponding transformation in Fourier space,
Eq.~(\ref{eq:3-11b}), is given by
\begin{equation}
  \tilde{F}_{Xlm}(\bm{k}) \rightarrow
  \tilde{F}_{Xlm}'(\bm{k}') = \tilde{F}_{Xlm'}(\bm{k}) D_{(l)m}^{m'}(R),
  \label{eq:3-38}
\end{equation}
because the volume element $d^3\!x$ and the inner product
$\bm{k}\cdot\bm{x}$ are invariant under the rotation.

The same transformations apply to the tensor fields in
Lagrangian space, $F^\mathrm{L}_{Xlm}(\bm{q})$. Therefore, the
renormalized bias functions, Eq.~(\ref{eq:3-18}), transform as
\begin{multline}
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n) \rightarrow
  c^{(n)\prime}_{Xlm}(\bm{k}_1',\ldots\bm{k}_n')
  \\
  = c^{(n)}_{Xlm'}(\bm{k}_1,\ldots,\bm{k}_n) D_{(l)m}^{m'}(R).
  \label{eq:3-40}
\end{multline}
Similarly, the propagators of Eq.~(\ref{eq:3-15}) transform as
\begin{multline}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n) \rightarrow
  \tilde{\Gamma}^{(n)\prime}_{Xlm}(\bm{k}_1',\ldots\bm{k}_n')
  \\
  = \tilde{\Gamma}^{(n)}_{Xlm'}(\bm{k}_1,\ldots\bm{k}_n) D_{(l)m}^{m'}(R).
  \label{eq:3-39}
\end{multline}
In redshift space, the lines of sight also rotate,
$\hat{\bm{z}} \rightarrow \hat{\bm{z}}' = R^{-1}\hat{\bm{z}}$,
although simultaneous rotations of wavevectors and lines of sight
do not change the scalar products in the propagators, as in
Eqs.~(\ref{eq:3-20})--(\ref{eq:3-25}). Thus we have
\begin{multline}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n;\hat{\bm{z}})
  \rightarrow
  \tilde{\Gamma}^{(n)\prime}_{Xlm}(\bm{k}_1',\ldots\bm{k}_n';\hat{\bm{z}}')
    \\
    =
  \tilde{\Gamma}^{(n)}_{Xlm'}(\bm{k}_1,\ldots\bm{k}_n;\hat{\bm{z}})
  D_{(l)m}^{m'}(R).
  \label{eq:3-39-1}
\end{multline}

\subsubsection{\label{subsubsec:SymParity}
  Parity
}

Next, we consider the property of parity symmetry. Keeping the
left-handed coordinates system, we consider an active parity
transformation of the physical system, instead of flipping the axes of
coordinates system. With the active parity transformation, the field
values at a position $\bm{x}$ are mapped into those at $-\bm{x}$, and
the functional form of the tensor field is transformed as
\begin{equation}
  F_{X\,i_1\cdots i_l}(\bm{x}) \rightarrow
  F_{Xlm}'(\bm{x}) =
  (-1)^{s_X+l} F_{X\,i_1\cdots i_l}(-\bm{x}),
  \label{eq:3-41}
\end{equation}
where $s_X=0$ for ordinary tensors and $s_X=1$ for pseudotensors. The
angular momentum is a typical example of a pseudotensor of rank 1.
The corresponding transformation in Fourier space is given by
\begin{equation}
  \tilde{F}_{Xlm}(\bm{k}) \rightarrow
  \tilde{F}_{Xlm}'(\bm{k}) = (-1)^{s_X+l} \tilde{F}_{Xlm}(-\bm{k}),
  \label{eq:3-42}
\end{equation}
The same applies to tensor fields in Lagrangian space.
Accordingly, the renormalized bias functions transform as
\begin{multline}
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n) \rightarrow
  c^{(n)\prime}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n)
  \\ =
  (-1)^{s_X+l}\, c^{(n)}_{Xlm}(-\bm{k}_1,\ldots,-\bm{k}_n),
  \label{eq:3-43}
\end{multline}
for the linear density field transforms as
$\tilde{\delta}_\mathrm{L}(\bm{k}) \rightarrow
\tilde{\delta}_\mathrm{L}(-\bm{k})$.

Similarly, the propagators transform as
\begin{multline}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n) \rightarrow
  \tilde{\Gamma}^{(n)\prime}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n)
  \\
  = (-1)^{s_X+l}\,
  \tilde{\Gamma}^{(n)}_{Xlm}(-\bm{k}_1,\ldots,-\bm{k}_n).
  \label{eq:3-44}
\end{multline}
In redshift space, we have
\begin{multline}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n;\hat{\bm{z}})
  \rightarrow
  \tilde{\Gamma}^{(n)\prime}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n;\hat{\bm{z}})
  \\
  = (-1)^{s_X+l}\,
  \tilde{\Gamma}^{(n)}_{Xlm}(-\bm{k}_1,\ldots,-\bm{k}_n;-\hat{\bm{z}}).
  \label{eq:3-45}
\end{multline}

\subsubsection{\label{subsubsec:SymInterC}
  Interchange of arguments
}

An obvious symmetry of the renormalized bias functions and propagators
is that they are invariant under permutation of the wavevectors in the
argument. For the renormalized bias function, we have
\begin{equation}
  c^{(n)}_{Xlm}(\bm{k}_{\sigma(1)},\ldots,\bm{k}_{\sigma(n)}) =
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n),
  \label{eq:3-46}
\end{equation}
where $\sigma \in \mathcal{S}_n$ is a permutation of the symmetric
group $\mathcal{S}_n$ of order $n$. Any permutation can be realized by
a series of interchange operations of adjacent arguments:
\begin{equation}
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_i,\bm{k}_{i+1},\ldots\bm{k}_n)
  =
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_{i+1},\bm{k}_i,\ldots\bm{k}_n),
  \label{eq:3-47}
\end{equation}
with arbitrary $i=1,\ldots,n-1$. Similarly, we have
\begin{equation}
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_i,\bm{k}_{i+1},\ldots\bm{k}_n)
  =
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_{i+1},\bm{k}_i,\ldots\bm{k}_n),
  \label{eq:3-48}
\end{equation}
for the propagators in real space, and
\begin{multline}
  \tilde{\Gamma|}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_i,\bm{k}_{i+1},\ldots\bm{k}_n;\hat{\bm{z}})
  \\
  =
  \tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_{i+1},\bm{k}_i,\ldots\bm{k}_n;\hat{\bm{z}}),
  \label{eq:3-49}
\end{multline}
for the propagators in redshift space.

\subsection{\label{subsec:RenBiasFn}
  Renormalized bias functions of tensor fields
}

In the following of this paper, we mostly work in Fourier space. For
notational simplicity, we omit the tildes above variables in Fourier
space, and denote $\delta_\mathrm{L}(\bm{k})$, $F_{Xlm}(\bm{k})$,
$\Gamma^{(n)}_{Xlm}(\bm{k}_1,\ldots)$ etc.~instead of
$\tilde{\delta}_\mathrm{L}(\bm{k})$, $\tilde{F}_{Xlm}(\bm{k})$,
$\tilde{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots)$, respectively, as long
as when they are not confusing.

One of the essential parts of iPT is the introduction of the
renormalized bias function defined by Eq.~(\ref{eq:3-18}). They
characterize complicated fully nonlinear processes in the formation of
astronomical objects, such as various types of galaxies and clusters,
etc. If these complicated processes are described by some kind of
models and the function $F^\mathrm{L}_{Xlm}$ in Lagrangian space is
analytically given in terms of the linear density field
$\delta_\mathrm{L}$, then we can analytically calculate the
renormalized bias function $c^{(n)}_{Xlm}$ according to the
model. In the case of the number density of halos, the Press-Schechter
model and its extensions are applied to the concrete calculations
along this line
\cite{Matsubara:2012nc,Matsubara:2013ofa,Matsubara:2016wth}. When this
kind of models are not known, the renormalized bias functions have an
infinite degrees of freedom which is difficult to calculate from the
fundamental level. However, the rotational symmetry considered in the
previous subsection places some constraints on the functional form of
the renormalized bias functions as we show below.

We consider the renormalized bias functions given in the form of
Eq.~(\ref{eq:3-18}). In the simplified notation without tildes,
\begin{equation}
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n) =
  (2\pi)^{3n}
  \int \frac{d^3k}{(2\pi)^3}
  \left\langle
    \frac{\delta^n F^\mathrm{L}_{Xlm}(\bm{k})}{
      \delta\delta_\mathrm{L}(\bm{k}_1) \cdots
      \delta\delta_\mathrm{L}(\bm{k}_n)}
  \right\rangle.
  \label{eq:3-50}
\end{equation}
The angular dependencies of the wavevectors in the arguments can be
decomposed in a series expansion with spherical harmonics as
\begin{multline}
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n) =
  \sum_{l_1,\ldots,l_n}
  Y_{l_1m_1}^*(\hat{\bm{k}}_1)\cdots Y_{l_nm_n}^*(\hat{\bm{k}}_n)
  \\ \times
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n),
  \label{eq:3-51}
\end{multline}
where $k_i = |\bm{k}_i|$ with $i=1,\ldots,n$ are absolute values of
the wavevectors, and $\hat{\bm{k}}_i = (\theta_i,\phi_i)$ represents
the spherical coordinates for the direction of wavevectors $\bm{k}_i$.
The repeated indices $m$, $m_1,\ldots,m_n$ in Eq.~(\ref{eq:3-51}) are
summed over according to the Einstein summation convention as in the
previous section. Due to the orthonormality relation for the spherical
harmonics, Eq.~(\ref{eq:a-5}), the coefficients are inversely given by
\begin{multline}
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n) =
  \int d^2\hat{k}_1 \cdots  d^2\hat{k}_n\,
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n)
  \\ \times
  Y_{l_1m_1}(\hat{\bm{k}}_1)\cdots Y_{l_nm_n}(\hat{\bm{k}}_n),
  \label{eq:3-52}
\end{multline}
where $d^2\hat{k}_i = \sin\theta_i\,d\theta_i\,d\phi_i$ represents the
angular integration of the wavevector $\bm{k}_i$. Using the property
of Eq.~(\ref{eq:3-34}), the complex conjugate of the above equation
satisfies
\begin{multline}
  c^{(n)\,l;l_1\cdots l_n\,*}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  =
  (-1)^{l+l_1+\cdots +l_n} g_{(l)}^{mm'}
  g_{(l_1)}^{m_1m_1'} \cdots g_{(l_2)}^{m_2m_2'}
  \\ \times
  c^{(n)\,l;l_1\cdots l_n}_{Xm';m_1'\cdots m_n'}(k_1,\ldots,k_n).
  \label{eq:3-55}
\end{multline}
Under the passive rotation of Eq.~(\ref{eq:2-14}), the function of
Eq.~(\ref{eq:3-52}) covariantly transforms as
\begin{multline}
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  \rightarrow
  c^{(n)\,l;l_1\cdots l_n}_{Xm';m_1'\cdots m_n'}(k_1,\ldots,k_n)
  \\ \times
  D_{(l)m}^{m'}(R) D_{(l_1)m_1}^{m_1'}(R) \cdots  D_{(l_n)m_n}^{m_n'}(R),
  \label{eq:3-56}
\end{multline}
as obviously indicated by the lower positions of the azimuthal indices
$m, m_1,\ldots m_n$. Under the active parity transformation of
Eq.~(\ref{eq:3-43}), the same function transforms as
\begin{multline}
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  \\
  \rightarrow
  (-1)^{s_X+l+l_1+\cdots +l_n}
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n),
  \label{eq:3-56-1}
\end{multline}
because of the parity property of the spherical harmonics,
Eq.~(\ref{eq:a-1}). The parity symmetry suggests that the renormalized
bias function is invariant under the above transformation, and thus we
have a condition to have a non-zero value of the function,
\begin{equation}
  s_X + l + l_1 + \cdots + l_n = \mathrm{even},
\label{eq:3-56-2}
\end{equation}
in the Universe with parity symmetry.

For the statistically isotropic Universe, the functions of
Eq.~(\ref{eq:3-52}) are rotationally invariant as these functions are
given by ensemble averages of the field variables as seen by
Eqs.~(\ref{eq:3-50}) and (\ref{eq:3-52}), and they should not depend
on the choice of coordinates system $\hat{\mathbf{e}}_i$ to describe
wavevectors $\bm{k}_1,\ldots,\bm{k}_n$ of the renormalized bias
functions $c^{(n)}_{Xlm}(\bm{k}_1,\ldots,\bm{k}_n)$. Therefore, they
are invariant under the transformation of Eq.~(\ref{eq:3-56}), and we
have
\begin{multline}
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  = c^{(n)\,l;l_1\cdots l_n}_{Xm';m_1'\cdots m_n'}(k_1,\ldots,k_n)
  \\ \times
  \frac{1}{8\pi^2} \int [dR]
  D_{(l)m}^{m'}(R) D_{(l_1)m_1}^{m_1'}(R) \cdots  D_{(l_n)m_n}^{m_n'}(R).
  \label{eq:3-57}
\end{multline}
where
\begin{equation}
  \int [dR] \cdots
  = \int_0^{2\pi}d\alpha \int_0^\pi d\beta\,\sin\beta
  \int_0^{2\pi} d\gamma \cdots
\label{eq:3-57-1}
\end{equation}
 represents the integrals over Euler angles
$(\alpha\beta\gamma)$ of the rotation $R$. A product of two Wigner's
rotation matrices of the same rotation reduces to a single matrix
through vector-coupling coefficients as \cite{Edmonds:1955fi}
\begin{multline}
  D_{(l_1)m_1}^{m_1'}(R) D_{(l_2)m_2}^{m_2'}(R)
  \\
  =
  \sum_{l,m,m'} (2l+1)
  \begin{pmatrix}
    l_1 & l_2 & l \\
    m_1' & m_2' & m'
  \end{pmatrix}
  \begin{pmatrix}
    l_1 & l_2 & l \\
    m_1 & m_2 & m
  \end{pmatrix}
  D_{(l)m}^{m'\,*}(R),
  \label{eq:3-58}
\end{multline}
using the Wigner's $3j$-symbol. The complex conjugate of the rotation
matrix is given by
$D_{(l)m_1}^{m_2*}(R) = g_{(l)}^{m_1m_1'}
g^{(l)}_{m_2m_2'}D_{(l)m_1'}^{m_2'}(R)$ using our metric tensor for
spherical basis.

In the following, we use a simplified notation for the $3j$-symbols,
\begin{equation}
  \left(l_1\,l_2\,l_3\right)_{m_1m_2m_3} \equiv
    \begin{pmatrix}
      l_1 & l_2 & l_3 \\
      m_1 & m_2 & m_3
    \end{pmatrix},
  \label{eq:3-59}
\end{equation}
which is non-zero only when $m_1+m_2+m_3=0$. We consider that the
azimuthal indices can be raised or lowered by spherical metric, e.g.,
\begin{align}
  \left(l_1\,l_2\,l_3\right)_{m_1}^{\phantom{m_1}m_2m_3}
  &= g_{(l_2)}^{m_2m_2'} g_{(l_3)}^{m_3m_3'}
    \left(l_1\,l_2\,l_3\right)_{m_1m_2'm_3'}
    \nonumber \\
  &= (-1)^{m_2+m_3}
    \begin{pmatrix}
      l_1 & l_2 & l_3 \\
      m_1 & -m_2 & -m_3
    \end{pmatrix},
  \label{eq:3-60}
\end{align}
and so forth. The properties and formulas for Wigner's $3j$-symbol
in our notation, which are repeatedly used in this paper below, are
summarized in Appendix~\ref{app:3njSymbols}.

With our notation of the $3j$-symbol, Eq.~(\ref{eq:3-58}) is
equivalent to an expression,
\begin{multline}
  D_{(l_1)m_1}^{m_1'}(R) D_{(l_2)m_2}^{m_2'}(R)
  = (-1)^{l_1+l_2}
  \sum_{l=0}^\infty (-1)^l(2l+1)
  \\ \times
  \left(l_1\,l_2\,l\right)_{m_1m_2}^{\phantom{m_1m_2}m}
  \left(l_1\,l_2\,l\right)^{m_1'm_2'}_{\phantom{m_1'm_2'}m'}
  D_{(l)m}^{m'}(R),
  \label{eq:3-64}
\end{multline}
in which the rotational covariance is explicit. We use a property of
Eq.~(\ref{eq:c-2}) for the $3j$-symbol to derive the above equation.

Consecutive applications of Eq.~(\ref{eq:3-64}) to
Eq.~(\ref{eq:3-57}), the dependence on the rotation $R$ in the
integrand on the rhs can be represented by a linear combination of a
product of less than or equal to three rotation matrices and the
coefficients of the expression are given by the products of
$3j$-symbols. The averages of products of rotation matrices of three
or fewer are given by
\begin{align}
  &
    \frac{1}{8\pi^2} \int [dR]
  D_{(l)m}^{m'}(R)
    = \delta_{l0} \delta_{m0} \delta_{m'0},
  \label{eq:3-65a}\\
  &
    \frac{1}{8\pi^2} \int [dR]
  D_{(l_1)m_1}^{m_1'}(R) D_{(l_2)m_2}^{m_2'}(R)
    = \frac{\delta_{l_1l_2}}{2l_1+1}
    g^{(l_1)}_{m_1m_2} g_{(l_2)}^{m_1'm_2'},
  \label{eq:3-65b}\\
  &
    \frac{1}{8\pi^2} \int [dR]
    D_{(l_1)m_1}^{m_1'}(R) D_{(l_2)m_2}^{m_2'}(R) D_{(l_3)m_3}^{m_3'}(R)
    \nonumber\\
  & \hspace{5pc}
    = (-1)^{l_1+l_2+l_3} \left(l_1\,l_2\,l_3\right)_{m_1m_2m_3}
     \left(l_1\,l_2\,l_3\right)^{m_1'm_2'm_3'}.
  \label{eq:3-65c}
\end{align}
Eqs.~(\ref{eq:3-65a}) and (\ref{eq:3-65b}) are also derived from
Eq.~(\ref{eq:3-65c}) noting $D_{(0)0}^0(R)=1$ and Eqs.~(\ref{eq:c-8}).
Also, Eqs.~(\ref{eq:3-65b}) and (\ref{eq:3-65c}) are derived from
further applications of Eq.~(\ref{eq:3-64}) to reduce the number of
products and finally using Eq.~(\ref{eq:3-65a}).

As a result of the above procedure, averaging over the rotation $R$,
the dependence of the indices of $m$'s in Eq.~(\ref{eq:3-57}) are all
represented by spherical metrics and $3j$-symbols. First, we
specifically derive the results for lower orders of $n$.
For $n=0$, using Eqs.~(\ref{eq:3-57}) and (\ref{eq:3-65a}), we derive
\begin{equation}
  c^{(0)}_{Xlm} =
  c^{(0)\,l}_{Xm} = c^{(0)\,0}_{X\,0}\delta_{l0} \delta_{m0}.
  \label{eq:3-66}
\end{equation}
This result is trivial because the corresponding function of
Eq.~(\ref{eq:3-50}) for $n=0$ does not depend on the direction of the
wavevector. For notational simplicity, we simply denote
$c^{(0)}_X \equiv c^{(0)\,0}_{X\,0}/\sqrt{4\pi}$, and we have
\begin{equation}
  c^{(0)}_{Xlm} =
  c^{(0)\,l}_{Xm} = \frac{\delta_{l0} \delta_{m0}}{\sqrt{4\pi}} c^{(0)}_{X}.
  \label{eq:3-67}
\end{equation}
For a scalar function, $F_X$, the coefficient $c^{(0)}_X$ just
corresponds to the mean value,
$c^{(0)}_X = \sqrt{4\pi} c^{(0)}_{X00} =\sqrt{4\pi} \langle F_{X00}
\rangle = 4\pi \langle F_{X00} \rangle \mathsf{Y}^{(0)}= \langle F_X
\rangle $, as seen from Eqs.~(\ref{eq:2-8}) and (\ref{eq:2-19}). The
parity symmetry of Eq.~(\ref{eq:3-56-2}) suggests that $s_X=0$, which
is trivial because the mean value of a scalar field
$\langle F_X \rangle$ vanish if the field is a pseudo-scalar in the
Universe with parity symmetry.

For $n=1$, using Eqs.~(\ref{eq:3-57}) and (\ref{eq:3-65b}), we derive
\begin{equation}
  c^{(1)\,l;l_1}_{Xm;m_1}(k_1)
  = \frac{\delta_{l\,l_1}}{2l+1}
  g^{(l)}_{mm_1} g_{(l)}^{m'm_1'} c^{(1)\,l;l}_{Xm';m_1'}(k_1),
  \label{eq:3-68}
\end{equation}
and therefore both indices $m,m_1$ appear only in the metric
$g^{(l)}_{mm_1}$. Defining a rotationally invariant function,
\begin{equation}
  c^{(1)}_{Xl}(k_1)
  \equiv \frac{g_{(l)}^{mm_1}}{2l+1}
   c^{(1)\,l;l}_{Xm;m_1}(k_1),
  \label{eq:3-69}
\end{equation}
Eq.~(\ref{eq:3-68}) reduces to a simple form,
\begin{equation}
  c^{(1)\,l;l_1}_{Xm;m_1}(k_1)
  = \delta_{ll_1} g^{(l)}_{mm_1} c^{(1)}_{Xl}(k_1).
  \label{eq:3-70}
\end{equation}
For the symmetry of complex conjugate, Eq.~(\ref{eq:3-55}), we have
\begin{equation}
  c^{(1)\,*}_{Xl}(k_1) = c^{(1)}_{Xl}(k_1),
  \label{eq:3-70-2}
\end{equation}
and thus the reduced function is a real function. The parity symmetry
of Eq.~(\ref{eq:3-56-2}) in this $n=1$ case suggests $s_X = 0$,
because of the constraint $l=l_1$. This means that pseudotensors do
not have the first-order renormalized bias function. This property is
partly because we consider the case that only scalar perturbations
$\delta_\mathrm{L}$ are responsible for the properties of tensor
fields. For example, the angular momenta are not generated by the
first-order effect in linear perturbation theory \cite{Peebles1969},
and so forth. Substituting the expression of Eq.~(\ref{eq:3-70}) into
Eq.~(\ref{eq:3-51}) with $n=1$, we derive
\begin{equation}
  c^{(1)}_{Xlm}(\bm{k}_1) = c^{(1)}_{Xl}(k_1) Y_{lm}(\hat{\bm{k}}_1).
  \label{eq:3-70-1}
\end{equation}
The invariant coefficient $c^{(1)}_{Xl}(k_1)$ can also be read off
from an expression of the renormalized bias function
$c^{(1)}_{Xlm}(\bm{k}_1)$ in which the angular dependence on
$\bm{k}_1$ is expanded by the spherical harmonics
$Y_{lm}(\hat{\bm{k}}_1)$. One can also explicitly invert the above
equation by using the orthonormality relation of spherical harmonics.

For $n=2$, using Eqs.~(\ref{eq:3-57}) and (\ref{eq:3-65c}), we derive
\begin{equation}
  c^{(2)\,l;l_1l_2}_{Xm;m_1m_2}(k_1,k_2)
  = (-1)^l \sqrt{2l+1}
  \left(l\,l_1\,l_2\right)_{mm_1m_2}
    c^{(2)\,l}_{Xl_1l_2}(k_1,k_2),
  \label{eq:3-71}
\end{equation}
where we define a rotationally invariant function,
\begin{equation}
  c^{(2)\,l}_{Xl_1l_2}(k_1,k_2)
  \equiv
  \frac{(-1)^{l_1+l_2}}{\sqrt{2l+1}}
    \left(l\,l_1\,l_2\right)^{mm_1m_2}
    c^{(2)\,l;l_1l_2}_{Xm;m_1m_2}(k_1,k_2).
  \label{eq:3-72}
\end{equation}
The above Eq.~(\ref{eq:3-71}) essentially corresponds to the
Wigner-Eckart theorem of representation theory and quantum mechanics
\cite{Sakurai:2011zz}. For the symmetry of complex conjugate,
Eq.~(\ref{eq:3-55}), we have
\begin{equation}
  c^{(2)\,l\,*}_{Xl_1l_2}(k_1,k_2)
  = c^{(2)\,l}_{Xl_1l_2}(k_1,k_2),
  \label{eq:3-72-1}
\end{equation}
and thus the reduced function is a real function. The parity symmetry
of Eq.~(\ref{eq:3-56-2}) in this case suggests
$s_X+l+l_1+l_2=\mathrm{even}$. Substituting Eq.~(\ref{eq:3-71}) into
Eq.~(\ref{eq:3-51}) with $n=2$, we derive
\begin{multline}
  c^{(2)}_{Xlm}(\bm{k}_1,\bm{k}_2)
  = (-1)^l \sqrt{2l+1}
  \sum_{l_1,l_2}
  c^{(2)\,l}_{Xl_1l_2}(k_1,k_2)
  \left(l\,l_1\,l_2\right)_{m}^{\phantom{m}m_1m_2}
  \\ \times
    Y_{l_1m_1}(\hat{\bm{k}}_1) Y_{l_2m_2}(\hat{\bm{k}}_2).
  \label{eq:3-73}
\end{multline}
The last expression contains the bipolar spherical harmonics
\cite{Khersonskii:1988krb}, defined by
\begin{multline}
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes Y_{l_2}(\hat{\bm{k}}_2)
  \right\}_{lm}
  \\
  \equiv (-1)^l \sqrt{2l+1}
  \left(l\,l_1\,l_2\right)_{m}^{\phantom{m}m_1m_2}
    Y_{l_1m_1}(\hat{\bm{k}}_1) Y_{l_2m_2}(\hat{\bm{k}}_2).
  \label{eq:3-74}
\end{multline}
Thus, Eq.~(\ref{eq:3-73}) is concisely given by
\begin{multline}
  c^{(2)}_{Xlm}(\bm{k}_1,\bm{k}_2)
  = \sum_{l_1,l_2}
    c^{(2)\,l}_{Xl_1l_2}(k_1,k_2)
    \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes Y_{l_2}(\hat{\bm{k}}_2)
    \right\}_{lm}.
  \label{eq:3-75}
\end{multline}
The interchange symmetry of Eq.~(\ref{eq:3-47}) in this case is given
by $c^{(2)}_{Xlm}(\bm{k}_1,\bm{k}_2) =
c^{(2)}_{Xlm}(\bm{k}_2,\bm{k}_1)$. Due to the symmetry of $3j$-symbols
of Eq.~(\ref{eq:c-4}), the bipolar spherical harmonics satisfy
\begin{equation}
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes Y_{l_2}(\hat{\bm{k}}_2)
  \right\}_{lm} = (-1)^{l+l_1+l_2}
  \left\{
    Y_{l_2}(\hat{\bm{k}}_2)\otimes Y_{l_1}(\hat{\bm{k}}_1)
  \right\}_{lm},
  \label{eq:3-75-1}
\end{equation}
and therefore the invariant function satisfies
\begin{equation}
  c^{(2)\,l}_{Xl_2l_1}(k_2,k_1)
  = (-1)^{l+l_1+l_2}
  c^{(2)\,l}_{Xl_1l_2}(k_1,k_2).
  \label{eq:3-75-2}
\end{equation}

For $n=3$, using Eqs.~(\ref{eq:3-57}), (\ref{eq:3-64}) and
(\ref{eq:3-65c}), we derive
\begin{multline}
  c^{(3)\,l;l_1l_2l_3}_{Xm;m_1m_2m_3}(k_1,k_2,k_3)
  = (-1)^l \sqrt{2l+1}
  \sum_L (-1)^L\sqrt{2L+1}
  \\ \times
    \left(l\,l_1\,L\right)_{mm_1M}
    \left(L\,l_2\,l_3\right)^{M}_{\phantom{M}m_2m_3}
    c^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3),
  \label{eq:3-76}
\end{multline}
where we define
\begin{multline}
  c^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3)
  \equiv
  \frac{(-1)^{l_1+l_2+l_3}}{\sqrt{2l+1}}
  (-1)^{L} \sqrt{2L+1}
    \left(l\,l_1\,L\right)^{mm_1}_{\phantom{mm_1}M}
    \\ \times
    \left(L\,l_2\,l_3\right)^{Mm_2m_3}
    c^{(3)\,l;l_1l_2l_3}_{Xm;m_1m_2m_3}(k_1,k_2,k_3).
  \label{eq:3-77}
\end{multline}
For the symmetry of complex conjugate, Eq.~(\ref{eq:3-55}), we have
\begin{equation}
  c^{(3)\,l;L\,*}_{Xl_1l_2l_3}(k_1,k_2,k_3)
  = c^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3),
  \label{eq:3-77-1}
\end{equation}
and thus the reduced function is a real function.
The parity symmetry of Eq.~(\ref{eq:3-56-2}) in this case suggests
$s_X+l+l_1+l_2+l_3=\mathrm{even}$.
Substituting Eq.~(\ref{eq:3-76}) into Eq.~(\ref{eq:3-51}) with $n=3$, we
derive
\begin{multline}
  c^{(3)}_{Xlm}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  = (-1)^l \sqrt{2l+1}
  \\ \times
  \sum_{l_1,l_2,l_3,L}
  (-1)^L\sqrt{2L+1}
  c^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3)
  \left(l\,l_1\,L\right)_{m}^{\phantom{m}m_1M}
  \\ \times
  \left(L\,l_2\,l_3\right)_{M}^{\phantom{M}m_2m_3}
  Y_{l_1m_1}(\hat{\bm{k}}_1) Y_{l_2m_2}(\hat{\bm{k}}_2)
  Y_{l_3m_3}(\hat{\bm{k}}_3).
  \label{eq:3-78}
\end{multline}
The last expression contains the tripolar spherical harmonics
\cite{Khersonskii:1988krb}, defined by
\begin{multline}
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{Y_{l_2}(\hat{\bm{k}}_2)
      \otimes Y_{l_3}(\hat{\bm{k}}_3)
    \right\}_L
  \right\}_{lm}
  \\
  = (-1)^{l+L} \sqrt{(2l+1)(2L+1)}
  \left(l\,l_1\,L\right)_{m}^{\phantom{m}m_1M}
  \\ \times
  \left(L\,l_2\,l_3\right)_{M}^{\phantom{M}m_2m_3}
  Y_{l_1m_1}(\hat{\bm{k}}_1) Y_{l_2m_2}(\hat{\bm{k}}_2)
  Y_{l_3m_3}(\hat{\bm{k}}_3).
  \label{eq:3-79}
\end{multline}
Thus, Eq.~(\ref{eq:3-78}) is concisely given by
\begin{multline}
  c^{(3)}_{Xlm}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  =
    \sum_{l_1,l_2,l_3,L}
    c^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3)
    \\ \times
    \left\{
      Y_{l_1}(\hat{\bm{k}}_1)\otimes
      \left\{
        Y_{l_2}(\hat{\bm{k}}_2) \otimes Y_{l_3}(\hat{\bm{k}}_3)
      \right\}_L
    \right\}_{lm}.
  \label{eq:3-80}
\end{multline}
The interchange symmetries of Eq.~(\ref{eq:3-47}) in terms of
invariant functions in this case are derived from properties of
interchanging arguments in tripolar spherical harmonics. For an
interchange of the last two arguments, $2 \leftrightarrow 3$, we have
\begin{multline}
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{Y_{l_2}(\hat{\bm{k}}_2)
      \otimes Y_{l_3}(\hat{\bm{k}}_3)
    \right\}_L
  \right\}_{lm}
  \\
  = (-1)^{l_2+l_3+L}
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{Y_{l_3}(\hat{\bm{k}}_3)
      \otimes Y_{l_2}(\hat{\bm{k}}_2)
    \right\}_L
  \right\}_{lm},
  \label{eq:3-80-1}
\end{multline}
just similarly in the case of Eq.~(\ref{eq:3-75-1}). For an
interchange of the first two arguments, $1 \leftrightarrow 2$,
however, a recoupling of the $3j$-symbols appears, and we have
\begin{multline}
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{Y_{l_2}(\hat{\bm{k}}_2)
      \otimes Y_{l_3}(\hat{\bm{k}}_3)
    \right\}_L
  \right\}_{lm}
  = (-1)^{l_1+l_2}
  \sum_{L'} (2L'+1)
  \\ \times
  \begin{Bmatrix}
    l_1 & l & L \\
    l_2 & l_3 & L'
  \end{Bmatrix}
  \left\{
    Y_{l_2}(\hat{\bm{k}}_2)\otimes
    \left\{Y_{l_1}(\hat{\bm{k}}_1)
      \otimes Y_{l_3}(\hat{\bm{k}}_3)
    \right\}_{L'}
  \right\}_{lm},
  \label{eq:3-80-2}
\end{multline}
where the factor in front of the tripolar spherical harmonics on the
rhs is a $6j$-symbol. This relation is derived by applying a sum rule
of the $3j$-symbols, Eq.~(\ref{eq:c-38}), to the definition of the
tripolar spherical harmonics, Eq.~(\ref{eq:3-79}). Correspondingly,
the interchange symmetries of invariant coefficients of
Eq.~(\ref{eq:3-80}) are given by
\begin{equation}
  c^{(3)l;L}_{Xl_ll_3l_2}(k_1,k_3,k_2)
  = (-1)^{l_2+l_3+L}
  c^{(3)l;L}_{Xl_ll_2l_3}(k_1,k_2,k_3),
  \label{eq:3-80-3}
\end{equation}
and
\begin{multline}
  c^{(3)l;L}_{Xl_2l_1l_3}(k_2,k_1,k_3)
  = (-1)^{l_1+l_2}
  \sum_{L'} (2L'+1)
  \begin{Bmatrix}
    l_1 & l & L \\
    l_2 & l_3 & L'
  \end{Bmatrix}
  \\ \times
  c^{(3)l;L'}_{Xl_ll_2l_3}(k_1,k_2,k_3).
  \label{eq:3-80-4}
\end{multline}
Combining Eqs.~(\ref{eq:3-80-3}) and (\ref{eq:3-80-4}), all the other
symmetries concerning the permutation of $(1,2,3)$ in the subscripts
of the arguments are straightforwardly obtained. A similar kind of
considerations of the interchange symmetry in a formulation of angular
trispectrum of the cosmic microwave background are found in
Ref.~\cite{Hu:2001fa} to enforce permutation symmetry in the angular
trispectrum.

In the same way, one derives the results for general orders $n > 3$.
Since the factors such as $2l+1$, $2L+1$ frequently appear in the
equations, we define a simplified notation,
\begin{equation}
  \{l\} \equiv 2l+1, \quad
  \{L\} \equiv 2L+1, \quad
  \mathrm{etc.}
  \label{eq:3-81-0}
\end{equation}
throughout the paper henceforth. The expansion coefficient of order
$n$ is given by
\begin{multline}
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  =(-1)^l \sqrt{\{l\}}
  \\ \times
  \sum_{L_2,\ldots,L_{n-1}}
  (-1)^{L_2+\cdots L_{n-1}}
  \sqrt{\{L_2\}\cdots \{L_{n-1}\}}
    \left(l\,l_1\,L_2\right)_{mm_1M_2}
    \\ \times
    \left(L_2\,l_2\,L_3\right)^{M_2}_{\phantom{M_2}m_2M_3}
    \cdots
    \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)^{M_{n-2}}_{\phantom{M_{n-2}}m_2M_{n-1}}
    \\ \times
    \left(L_{n-1}\,l_{n-1}\,l_n\right)^{M_{n-1}}_{\phantom{M_{n-1}}m_{n-1}m_n}
    c^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n),
  \label{eq:3-81}
\end{multline}
where we define
\begin{multline}
  c^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
  \equiv
  \frac{(-1)^{l_1+\cdots +l_n}}{\sqrt{\{l\}}}
  \\ \times
  (-1)^{L_2+\cdots +L_{n-1}}\sqrt{\{L_2\}\cdots\{L_{n-1}\}}
    \left(l\,l_1\,L_2\right)^{mm_1}_{\phantom{mm_1}M_2}
    \\ \times
    \left(L_2\,l_2\,L_3\right)^{M_2m_2}_{\phantom{M_2m_2}M_3}
    \cdots
    \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)^{M_{n-2}m_{n-2}}_{\phantom{M_{n-2}m_{n-2}}M_{n-1}}
    \\ \times
    \left(L_{n-1}\,l_{n-1}\,l_n\right)^{M_{n-1}m_{n-1}m_n}
    c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,,k_n).
  \label{eq:3-82}
\end{multline}
Thus the azimuthal indices of the decomposed renormalized bias
functions, Eq.~(\ref{eq:3-52}) are all represented by combinations of
$3j$-symbols, and the physical contents of the renormalized bias
functions are given by the invariant functions,
$c^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)$. The
fully rotational symmetry is the reason why the azimuthal indices of
decomposed renormalized bias function,
$c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(\ldots)$ are represented
by the combinations of $3j$-symbols. Due to the symmetry of
$3j$-symbols, we have $m+m_1+\cdots +m_n=0$, which is a manifestation
of the axial symmetry of the rotation around the third axis
$\hat{\mathbf{e}}_3$.
For the symmetry of complex conjugate, Eq.~(\ref{eq:3-55}), we have
\begin{equation}
  c^{(n)\,l;L_2\cdots L_{n-1}\,*}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
  = c^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n),
  \label{eq:3-82-1}
\end{equation}
and thus the reduced function is a real function.
The parity symmetry of Eq.~(\ref{eq:3-56-2}) in this case suggests
$s_X+l+l_1+\cdots +l_n=\mathrm{even}$.
The renormalized bias function of Eq.~(\ref{eq:3-51}) is given by
\begin{multline}
  c^{(n)}_{Xlm}(\bm{k}_1,\cdots,\bm{k}_n)
  = (-1)^l \sqrt{\{l\}}
  \\ \times
  \sum_{\substack{l_1,\ldots,l_n\\L_2,\ldots,L_{n-1}}}
  (-1)^{L_2 + \cdots + L_{n-1}}
  \sqrt{\{L_2\}\cdots\{L_{n-1}\}}
  \\ \times
  c^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
  \left(l\,l_1\,L_2\right)_{m}^{\phantom{m}m_1M_2}
  \\ \times
  \left(L_2\,l_2\,L_3\right)_{M_2}^{\phantom{M_2}m_2M_3} \cdots
  \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)_{M_{n-2}}^{\phantom{M_{n-2}}m_{n-2}M_{n-1}}
  \\ \times
  \left(L_{n-1}\,l_{n-1}\,l_n\right)_{M_{n-1}}^{\phantom{M_{n-1}}m_{n-1}m_n}
  Y_{l_1m_1}(\hat{\bm{k}}_1) \cdots Y_{l_nm_n}(\hat{\bm{k}}_n).
  \label{eq:3-83}
\end{multline}
We generally define polypolar spherical harmonics by
\begin{multline}
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{
      Y_{l_2}(\hat{\bm{k}}_2)\otimes
    \left\{ \cdots
      \otimes Y_{l_n}(\hat{\bm{k}}_n)
      \right\}_{L_{n-1}} \cdots \right\}_{L_2}
  \right\}_{lm}
  \\
  =
  (-1)^{l+L_2+\cdots +L_{n-1}}
  \sqrt{\{l\}\{L_2\}\cdots\{L_{n-1}\}}
  \left(l\,l_1\,L_2\right)_{m}^{\phantom{m}m_1M_2}
  \\ \times
  \left(L_2\,l_2\,L_3\right)_{M_2}^{\phantom{M_2}m_2M_3} \cdots
  \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)_{M_{n-2}}^{\phantom{M_{n-2}}m_{n-2}M_{n-1}}
  \\ \times
  \left(L_{n-1}\,l_{n-1}\,l_n\right)_{M_{n-1}}^{\phantom{M_{n-1}}m_{n-1}m_n}
  Y_{l_1m_1}(\hat{\bm{k}}_1) \cdots
  Y_{l_nm_n}(\hat{\bm{k}}_n),
  \label{eq:3-84}
\end{multline}
and Eq.~(\ref{eq:3-83}) is concisely represented by
\begin{multline}
  c^{(n)}_{Xlm}(\bm{k}_1,\cdots,\bm{k}_n)
  =
    \sum_{\substack{l_1,\ldots,l_n\\L_2,\ldots,L_{n-1}}}
    c^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\cdots,k_n)
    \\ \times
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{
      Y_{l_2}(\hat{\bm{k}}_2)\otimes
    \left\{ \cdots
      \otimes Y_{l_n}(\hat{\bm{k}}_n)
      \right\}_{L_{n-1}} \cdots \right\}_{L_2}
  \right\}_{lm}.
  \label{eq:3-85}
\end{multline}
The interchange symmetries of Eq.~(\ref{eq:3-47}) in terms of
invariant function in this case are also derived following a similar
way in the case of $n=3$. For the interchange of the last two
arguments, we have
\begin{multline}
  c^{(n)l;L_2\cdots L_{n-2}L_{n-1}}_{Xl_l\cdots l_{n-2}l_nl_{n-1}}
  (k_1,\cdots,k_{n-2},k_n,k_{n-1})
  \\
  = (-1)^{l_{n-1}+l_n+L_{n-1}}
  c^{(n)l;L_2\cdots L_{n-1}}_{Xl_l\cdots l_n}
  (k_1,\cdots,k_n),
  \label{eq:3-86}
\end{multline}
and for the interchange of the other adjacent arguments, $i
\leftrightarrow i+1$, we have
\begin{multline}
  c^{(n)\,l;L_2\cdots L_{i-1}L_{i+1}L_iL_{i+2}\cdots L_{n-1}}
  _{Xl_1\cdots l_{i-1}l_{i+1}l_il_{i+2}\cdots l_n}
  (k_1,\cdots,k_{i-1},k_{i+1},k_i,k_{i+2},\cdots,k_n)
  \\
  = (-1)^{l_i+l_{i+1}}
  \sum_{L'} (2L'+1)
  \begin{Bmatrix}
    l_i & L_i & L_{i+1} \\
    l_{i+1} & L_{i+2} & L'
  \end{Bmatrix}
  \\ \times
  c^{(n)\,l;L_2\cdots L_iL'L_{i+2}\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\cdots,k_n),
  \label{eq:3-87}
\end{multline}
where $i=1,\ldots,n-2$. Combining Eqs.~(\ref{eq:3-86}) and
(\ref{eq:3-87}), all the other symmetries concerning the permutation
of $(1,\ldots,n)$ in the subscripts of the arguments are
straightforwardly obtained.

\subsection{\label{subsec:Propagators}
  Propagators of tensor fields
}

The renormalized bias functions describe the bias mechanisms in the
physical space, and thus there is not any corresponding concept of
redshift space. However, the apparent clustering in redshift space
introduces anisotropy in the propagators in Eulerian space. In the
following, we separately consider the propagators in real space and
redshift space.

\subsubsection{\label{subsec:PropReal}
  Real space
}

In real space, where the fully rotational symmetry is satisfied even
in Eulerian space, the reduced propagators $\hat{\Gamma}^{(n)}_{Xlm}$
are similarly decomposed into rotationally invariant coefficients as
in the case of renormalized bias functions explained in the previous
subsection. Although the description below is pretty much redundant
with the formulas for renormalized bias functions, we dare to write
down the corresponding formulas for the propagator in real space, just
for convenience of later use.

The decomposition of the reduced propagator is given by
\begin{multline}
  \hat{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n)
  = \sum_{l_1,\ldots,l_n} \hat{\Gamma}^{(n)\,l;l_1\cdots
    l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n) \\ \times
  Y_{l_1m_1}^*(\hat{\bm{k}}_1)\cdots Y_{l_nm_n}^*(\hat{\bm{k}}_n),
  \label{eq:3-100}
\end{multline}
and the inverse relation of the decomposition is given by
\begin{multline}
  \hat{\Gamma}^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n) =
  \int d^2\hat{k}_1 \cdots  d^2\hat{k}_n\,
  \hat{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n)
  \\ \times
  Y_{l_1m_1}(\hat{\bm{k}}_1)\cdots Y_{l_nm_n}(\hat{\bm{k}}_n).
  \label{eq:3-101}
\end{multline}
The symmetry of complex conjugate,
Eq.~(\ref{eq:3-33}) indicates
\begin{multline}
  \hat{\Gamma}^{(n)\,l;l_1\cdots l_n\,*}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  =
  (-1)^{l+l_1+\cdots +l_n} g_{(l)}^{mm'}
  g_{(l_1)}^{m_1m_1'} \cdots g_{(l_n)}^{m_nm_n'}
  \\ \times
  \hat{\Gamma}^{(n)\,l;l_1\cdots l_n}_{Xm';m_1'\cdots m_n'}(k_1,\ldots,k_n).
  \label{eq:3-102}
\end{multline}
The parity transformation of Eq.~(\ref{eq:3-44}) suggests that
\begin{equation}
  s_X + l + l_1 + \cdots + l_n = \mathrm{even}
  \label{eq:3-102-1}
\end{equation}
in the Universe with parity symmetry.

The considerations of the symmetric properties of the renormalized
bias functions $c^{(n)}_{Xlm}$ in the previous subsection are all
applicable to the case of propagators $\hat{\Gamma}^{(n)}_{Xlm}$ in
real space. Therefore, all the equations in the previous subsection
holds with replacements of $c^{(n)}_{Xlm}$ with
$\hat{\Gamma}^{(n)}_{Xlm}$, etc. For $n=1$, we have
\begin{equation}
  \hat{\Gamma}^{(1)\,l;l_1}_{Xm;m_1}(k)
  = \delta_{ll_1} g^{(l)}_{mm_1} \hat{\Gamma}^{(1)}_{Xl}(k).
  \label{eq:3-103}
\end{equation}
with
\begin{equation}
  \hat{\Gamma}^{(1)}_{Xl}(k)
  \equiv \frac{g_{(l)}^{mm_1}}{\{l\}}
  \hat{\Gamma}^{(1)\,l;l}_{Xm;m_1}(k).
  \label{eq:3-104}
\end{equation}
The symmetry of complex conjugate is given by
\begin{equation}
  \hat{\Gamma}^{(1)\,*}_{Xl}(k) = \hat{\Gamma}^{(1)}_{Xl}(k),
  \label{eq:3-104-1}
\end{equation}
and thus the reduced function is a real function. In the Universe with
parity symmetry, Eq.~(\ref{eq:3-102-1}) suggests $s_X=0$ in order that
the first-order propagator $\hat{\Gamma}^{(1)}_{Xl}(k)$ does not
vanish. The propagator is given by
\begin{equation}
  \hat{\Gamma}^{(1)}_{Xlm}(\bm{k})
  = \hat{\Gamma}^{(1)}_{Xl}(k) Y_{lm}(\hat{\bm{k}}).
  \label{eq:3-105}
\end{equation}

For $n=2$, we have
\begin{equation}
  \hat{\Gamma}^{(2)\,l;l_1l_2}_{Xm;m_1m_2}(k_1,k_2)
  = (-1)^l \sqrt{\{l\}}
  \left(l\,l_1\,l_2\right)_{mm_1m_2}
    \hat{\Gamma}^{(2)\,l}_{Xl_1l_2}(k_1,k_2),
  \label{eq:3-106}
\end{equation}
with
\begin{equation}
  \hat{\Gamma}^{(2)\,l}_{Xl_1l_2}(k_1,k_2)
  \equiv
  \frac{(-1)^{l_1+l_2}}{\sqrt{\{l\}}}
    \left(l\,l_1\,l_2\right)^{mm_1m_2}
    \hat{\Gamma}^{(2)\,l;l_1l_2}_{Xm;m_1m_2}(k_1,k_2).
  \label{eq:3-107}
\end{equation}
The symmetry of complex conjugate is given by
\begin{equation}
  \hat{\Gamma}^{(2)\,l\,*}_{Xl_1l_2}(k_1,k_2)
  = \hat{\Gamma}^{(2)\,l}_{Xl_1l_2}(k_1,k_2),
  \label{eq:3-107-1}
\end{equation}
and thus the reduced function is a real function.
The parity symmetry of Eq.~(\ref{eq:3-102-1}) in this case suggests
$s_X+l+l_1+l_2=\mathrm{even}$. The propagator is given by
\begin{equation}
  \hat{\Gamma}^{(2)}_{Xlm}(\bm{k}_1,\bm{k}_2)
  =
  \sum_{l_1,l_2}
    \hat{\Gamma}^{(2)\,l}_{Xl_1l_2}(k_1,k_2)
    \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes Y_{l_2}(\hat{\bm{k}}_2)
    \right\}_{lm}.
  \label{eq:3-108}
\end{equation}
Corresponding to Eq.~(\ref{eq:3-75-2}) of the renormalized bias
function, the interchange symmetry for the second-order propagator is
given by
\begin{equation}
  \hat{\Gamma}^{(2)\,l}_{Xl_2l_1}(k_2,k_1)
  = (-1)^{l+l_1+l_2}
  \hat{\Gamma}^{(2)\,l}_{Xl_1l_2}(k_1,k_2).
  \label{eq:3-108-1}
\end{equation}

For $n=3$, we have
\begin{multline}
  \hat{\Gamma}^{(3)\,l;l_1l_2l_3}_{Xm;m_1m_2m_3}(k_1,k_2,k_3)
  = (-1)^l \sqrt{\{l\}}
  \sum_L (-1)^L\sqrt{\{L\}}
  \\ \times
    \left(l\,l_1\,L\right)_{mm_1M}
    \left(L\,l_2\,l_3\right)^{M}_{\phantom{M}m_2m_3}
    \hat{\Gamma}^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3),
  \label{eq:3-109}
\end{multline}
with
\begin{multline}
  \hat{\Gamma}^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3)
  \equiv
  \frac{(-1)^{l_1+l_2+l_3}}{\sqrt{\{l\}}}
  (-1)^L \sqrt{\{L\}}
    \left(l\,l_1\,L\right)^{mm_1}_{\phantom{mm_1}M}
    \\ \times
    \left(L\,l_2\,l_3\right)^{Mm_2m_3}
    \hat{\Gamma}^{(3)\,l;l_1l_2l_3}_{Xm;m_1m_2m_3}(k_1,k_2,k_3).
  \label{eq:3-110}
\end{multline}
The symmetry of complex conjugate is given by
\begin{equation}
  \hat{\Gamma}^{(3)\,l;L\,*}_{Xl_1l_2l_3}(k_1,k_2,k_3)
  = \hat{\Gamma}^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3),
  \label{eq:3-110-1}
\end{equation}
and thus the reduced function is a real function.
The parity symmetry of Eq.~(\ref{eq:3-102-1}) in this case suggests
$s_X+l+l_1+l_2+l_3=\mathrm{even}$. The propagator is given by
\begin{multline}
  \hat{\Gamma}^{(3)}_{Xlm}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  =
    \sum_{l_1,l_2,l_3,L}
    \hat{\Gamma}^{(3)\,l;L}_{Xl_1l_2l_3}(k_1,k_2,k_3)
    \\ \times
    \left\{
      Y_{l_1}(\hat{\bm{k}}_1)\otimes
      \left\{
        Y_{l_2}(\hat{\bm{k}}_2) \otimes Y_{l_3}(\hat{\bm{k}}_3)
      \right\}_L
    \right\}_{lm}.
  \label{eq:3-111}
\end{multline}
Corresponding to Eqs.~(\ref{eq:3-80-3}) and (\ref{eq:3-80-4}) of the
renormalized bias function, the interchange symmetries for the
third-order propagator are given by
\begin{equation}
  \hat{\Gamma}^{(3)l;L}_{Xl_ll_3l_2}(k_1,k_3,k_2)
  = (-1)^{l_2+l_3+L}
  \hat{\Gamma}^{(3)l;L}_{Xl_ll_2l_3}(k_1,k_2,k_3),
  \label{eq:3-111-1}
\end{equation}
and
\begin{multline}
  \hat{\Gamma}^{(3)l;L}_{Xl_2l_1l_3}(k_2,k_1,k_3)
  = (-1)^{l_1+l_2}
  \sum_{L'} (2L'+1)
  \begin{Bmatrix}
    l_1 & l & L \\
    l_2 & l_3 & L'
  \end{Bmatrix}
  \\ \times
  \hat{\Gamma}^{(3)l;L'}_{Xl_ll_2l_3}(k_1,k_2,k_3).
  \label{eq:3-111-2}
\end{multline}

For $n > 3$, we have
\begin{multline}
  \hat{\Gamma}^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  =(-1)^l \sqrt{\{l\}}
  \sum_{L_2,\ldots,L_{n-1}}
  \\ \times
  (-1)^{L_2+\cdots L_{n-1}}
  \sqrt{\{L_2\}\cdots\{L_{n-1}\}}
    \left(l\,l_1\,L_2\right)_{mm_1M_2}
    \\ \times
    \left(L_2\,l_2\,L_3\right)^{M_2}_{\phantom{M_2}m_2M_3}
    \cdots
    \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)^{M_{n-2}}_{\phantom{M_{n-2}}m_2M_{n-1}}
    \\ \times
    \left(L_{n-1}\,l_{n-1}\,l_n\right)^{M_{n-1}}_{\phantom{M_{n-1}}m_{n-1}m_n}
    \hat{\Gamma}^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n),
  \label{eq:3-112}
\end{multline}
where we define
\begin{multline}
  \hat{\Gamma}^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
  \equiv
  \frac{(-1)^{l_1+\cdots +l_n}} {\sqrt{\{l\}}}
  \\ \times
  (-1)^{L_2+\cdots +L_{n-1}} \sqrt{\{L_2\}\cdots\{L_{n-1}\}}
    \left(l\,l_1\,L_2\right)^{mm_1}_{\phantom{mm_1}M_2}
    \\ \times
    \left(L_2\,l_2\,L_3\right)^{M_2m_2}_{\phantom{M_2m_2}M_3}
    \cdots
    \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)^{M_{n-2}m_{n-2}}_{\phantom{M_{n-2}m_{n-2}}M_{n-1}}
    \\ \times
    \left(L_{n-1}\,l_{n-1}\,l_n\right)^{M_{n-1}m_{n-1}m_n}
    \hat{\Gamma}^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,,k_n).
  \label{eq:3-113}
\end{multline}
The symmetry of complex conjugate is given by
\begin{equation}
  \hat{\Gamma}^{(n)\,l;L_2\cdots L_{n-1}\,*}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
  = \hat{\Gamma}^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n),
  \label{eq:3-110-2}
\end{equation}
and thus the reduced function is a real function.
The parity transformation of Eq.~(\ref{eq:3-44}) suggests that
\begin{equation}
  s_X + l + l_1 + \cdots + l_n = \mathrm{even}
  \label{eq:3-110-3}
\end{equation}
in the Universe with the parity symmetry, which is the same condition
with Eq.~(\ref{eq:3-102-1}). The propagator is given by
\begin{multline}
  \hat{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\cdots,\bm{k}_n)
  =
    \sum_{\substack{l_1,\ldots,l_n\\L_2,\ldots,L_{n-1}}}
    \hat{\Gamma}^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\cdots,k_n)
    \\ \times
  \left\{
    Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{
      Y_{l_2}(\hat{\bm{k}}_2)\otimes
    \left\{ \cdots
      \otimes Y_{l_n}(\hat{\bm{k}}_n)
      \right\}_{L_{n-1}} \cdots \right\}_{L_2}
  \right\}_{lm}.
  \label{eq:3-114}
\end{multline}
Corresponding to Eqs.~(\ref{eq:3-86}) and (\ref{eq:3-87}) of the
renormalized bias function, the interchange symmetries for the
higher-order propagators are given by
\begin{multline}
  \hat{\Gamma}^{(n)l;L_2\cdots L_{n-2}L_{n-1}}_{Xl_l\cdots l_{n-2}l_nl_{n-1}}
  (k_1,\cdots,k_{n-2},k_n,k_{n-1})
  \\
  = (-1)^{l_{n-1}+l_n+L_{n-1}}
  \hat{\Gamma}^{(n)l;L_2\cdots L_{n-1}}_{Xl_l\cdots l_n}
  (k_1,\cdots,k_n),
  \label{eq:3-114-1}
\end{multline}
and
\begin{multline}
  \hat{\Gamma}^{(n)\,l;L_2\cdots L_{i-1}L_{i+1}L_iL_{i+2}\cdots L_{n-1}}
  _{Xl_1\cdots l_{i-1}l_{i+1}l_il_{i+2}\cdots l_n}
  (k_1,\cdots,k_{i-1},k_{i+1},k_i,k_{i+2},\cdots,k_n)
  \\
  = (-1)^{l_i+l_{i+1}}
  \sum_{L'} (2L'+1)
  \begin{Bmatrix}
    l_i & L_i & L_{i+1} \\
    l_{i+1} & L_{i+2} & L'
  \end{Bmatrix}
  \\ \times
  \hat{\Gamma}^{(n)\,l;L_2\cdots L_iL'L_{i+2}\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\cdots,k_n).
  \label{eq:3-114-2}
\end{multline}

\subsubsection{\label{subsec:PropRed}
  Redshift space
}

In redshift space, the propagators also depend on the direction of the
line of sight, $\hat{\bm{z}}$. In this paper, we approximately
consider the direction of the line of sight is fixed in the observed
space. This approximation is usually called the plane-parallel, or
distant-observer approximation, and is known as a good approximation
for most of the practical purposes. In order to keep the rotational
covariance in the theory, the line of sight, $\hat{\bm{z}}$, is
considered to be arbitrarily directed in the coordinates system, i.e.,
not fixed in the third axis, $\hat{\bm{z}} \ne \hat{\mathbf{e}}_3$, in
general.

In this case, the dependence of the normalized propagators on the line
of sight $\hat{\bm{z}}$ is implicit in the decomposed propagators of
Eqs.~(\ref{eq:3-100}) and (\ref{eq:3-101}). We further decompose this
implicit dependence on the direction of the line of sight with the
spherical harmonics, $Y_{l_zm_z}(\hat{\bm{z}})$:
\begin{multline}
  \hat{\Gamma}^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots
    m_n}(k_1,\ldots,k_n;\hat{\bm{z}})
  \\
  =
  \sum_{l_z} \sqrt{\frac{4\pi}{\{l_z\}}}\,
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xmm_z;m_1\cdots
    m_n}(k_1,\ldots,k_n)
  Y_{l_zm_z}^*(\hat{\bm{z}}).
  \label{eq:3-120}
\end{multline}
The prefactor $\sqrt{4\pi/\{l_z\}}$ is added in the definition in
order that the expansion reduces to the Legendre expansion when
$m_z=0$, since
$Y_{l0}(\theta,\phi) = \sqrt{\{l\}/(4\pi)}\,\mathit{P}_l(\cos\theta)$.
The function $\sqrt{4\pi/\{l\}}\,Y_{lm}(\bm{n})$ is also known as the
spherical harmonics with Racah's normalization. With the above
normalization of Eq.~(\ref{eq:3-120}), the coefficient has the same
normalization in real space and redshift space when $l_z=m_z=0$.
Unless we consider some strange situations, physical quantities are
usually invariant under the flip of the direction of the line of
sight, $\hat{\bm{z}} \rightarrow - \hat{\bm{z}}$. In such normal
cases, the non-negative integer $l_z$ should take even numbers, as
immediately seen from an identity of the spherical harmonics,
$Y_{l_zm_z}(-\hat{\bm{z}}) = (-1)^{l_z}Y_{l_zm_z}(\hat{\bm{z}})$.
Below we do not exclude the possibility that $l_z$ can take strangely
odd numbers, just for generality.

The above decomposition is equivalent to
\begin{multline}
  \hat{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n;\hat{\bm{z}})
  \\
  =
  \sum_{l_z,l_1,\ldots,l_n}
   \sqrt{\frac{4\pi}{\{l_z\}}}\,
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xmm_z;m_1\cdots
    m_n}(k_1,\ldots,k_n)
  \\ \times
  Y_{l_zm_z}^*(\hat{\bm{z}})
  Y_{l_1m_1}^*(\hat{\bm{k}}_1)\cdots Y_{l_nm_n}^*(\hat{\bm{k}}_n).
  \label{eq:3-121}
\end{multline}
The inverse relation is given by
\begin{multline}
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xmm_z;m_1\cdots
    m_n}(k_1,\ldots,k_n)
  \\
  =
 \sqrt{\frac{\{l_z\}}{4\pi}}
  \int d^2\hat{z}\,d^2\hat{k}_1 \cdots d^2\hat{k}_n\,
  \hat{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n;\hat{\bm{z}})
  \\ \times
  Y_{l_zm_z}(\hat{\bm{z}})
  Y_{l_1m_1}(\hat{\bm{k}}_1)\cdots Y_{l_nm_n}(\hat{\bm{k}}_n).
  \label{eq:3-122}
\end{multline}
The complex conjugate of the coefficient is given by
\begin{multline}
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n\,*}_{Xmm_z;m_1\cdots m_n}(k_1,\ldots,k_n)
  =
  (-1)^{l+l_1+\cdots +l_n} g_{(l)}^{mm'} g_{(l_z)}^{m_zm_z'}
  \\ \times
  g_{(l_1)}^{m_1m_1'} \cdots g_{(l_n)}^{m_nm_n'}
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xm'm_z';m_1'\cdots m_n'}
  (k_1,\ldots,k_n).
  \label{eq:3-123}
\end{multline}
The parity transformation of Eq.~(\ref{eq:3-45}) suggests that
\begin{equation}
  s_X + l + l_z + l_1 + \cdots + l_n = \mathrm{even}
  \label{eq:3-123-1}
\end{equation}
in the Universe with parity symmetry. In normal situations, $l_z$ is
an even number and the above condition is the same as that of
Eq.~(\ref{eq:3-110-3}) in real space.

The transformation under the
passive rotation of Eq.~(\ref{eq:2-14}) is given by
\begin{multline}
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xmm_z;m_1\cdots m_n}(k_1,\ldots,k_n)
  \rightarrow
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xm'm_z';m_1'\cdots m_n'}
  (k_1,\ldots,k_n)
  \\ \times
  D_{(l)m}^{m'}(R)
  D_{(l_z)m_z}^{m_z'}(R)
  D_{(l_1)m_1}^{m_1'}(R) \cdots
  D_{(l_n)m_n}^{m_n'}(R).
  \label{eq:3-124}
\end{multline}
When we rotate the direction of the line of sight together with
the coordinate system, the propagators should be invariant under the
rotation of the coordinate system in the Universe which satisfies the
cosmological principle. Thus we have
\begin{multline}
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xmm_z;m_1\cdots m_n}(k_1,\ldots,k_n)
  = \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xm'm_z';m_1'\cdots m_n'}(k_1,\ldots,k_n)
  \\ \times
  \frac{1}{8\pi^2} \int [dR]
  D_{(l)m}^{m'}(R)
  D_{(l_z)m_z}^{m_z'}(R)
  D_{(l_1)m_1}^{m_1'}(R) \cdots D_{(l_n)m_n}^{m_n'}(R).
  \label{eq:3-125}
\end{multline}
Following the same procedure below Eq.~(\ref{eq:3-57}) in the
case of renormalized bias functions, one can represent the dependence
of propagators in redshift space by spherical metric, $3j$-symbols, and
invariant variables. One of the main differences is that an angular
dependence on the line of sight additionally appears in the
expansion.

For $n=1$, we have
\begin{equation}
  \hat{\Gamma}^{(1)\,l\,l_z;l_1}_{Xmm_z;m_1}(k_1)
  = (-1)^l\sqrt{\{l\}}\, \left(l\,l_z\,l_1\right)_{mm_zm_1}
    \hat{\Gamma}^{(1)\,l\,l_z}_{Xl_1}(k_1),
  \label{eq:3-126}
\end{equation}
with
\begin{equation}
  \hat{\Gamma}^{(1)l\,l_z}_{Xl_1}(k_1)
  \equiv
  \frac{(-1)^{l_z+l_1}}{\sqrt{\{l\}}}
  \left(l\,l_z\,l_1\right)^{mm_zm_1}
  \hat{\Gamma}^{(1)\,l\,l_z;l_1}_{Xmm_z;m_1}(k_1),
  \label{eq:3-127}
\end{equation}
The symmetry of complex conjugate is given by
\begin{equation}
  \hat{\Gamma}^{(1)l\,l_z\,*}_{Xl_1}(k_1)
  = (-1)^{l_z} \hat{\Gamma}^{(1)l\,l_z}_{Xl_1}(k_1),
  \label{eq:3-127-1}
\end{equation}
and thus the reduced function is a real function for normal cases of
$l_z=\mathrm{even}$ and a pure imaginary function for strange cases of
$l_z=\mathrm{odd}$. The parity symmetry of Eq.~(\ref{eq:3-123-1}) in
this case suggests $s_X+l+l_z+l_1=\mathrm{even}$. The propagator is
given by
\begin{equation}
  \hat{\Gamma}^{(1)}_{Xlm}(\bm{k}_1;\hat{\bm{z}})
  = \sum_{l_z,l_1}
   \sqrt{\frac{4\pi}{\{l_z\}}}\,
    \hat{\Gamma}^{(1)\,l\,l_z}_{Xl_1}(k_1)
    \left\{
    Y_{l_z}(\hat{\bm{z}})\otimes Y_{l_1}(\hat{\bm{k}}_1)
    \right\}_{lm}.
  \label{eq:3-128}
\end{equation}

For $n=2$, we have
\begin{multline}
  \hat{\Gamma}^{(2)\,l\,l_z;l_1l_2}_{Xmm_z;m_1m_2}(k_1,k_2)
  = (-1)^l\sqrt{\{l\}}
  \sum_L (-1)^L \sqrt{\{L\}}
    \left(l\,l_z\,L\right)_{mm_zM}
    \\ \times
    \left(L\,l_1\,l_2\right)^M_{\phantom{M}m_1m_2}
    \hat{\Gamma}^{(2)\,l\,l_z;L}_{Xl_1l_2}(k_1,k_2),
  \label{eq:3-129}
\end{multline}
with
\begin{multline}
  \hat{\Gamma}^{(2)\,l\,l_z;L}_{Xl_1l_2}(k_1,k_2)
  \equiv
  \frac{(-1)^{l_z+l_1+l_2}}{\sqrt{\{l\}}}
  (-1)^L \sqrt{\{L\}}
    \left(l\,l_z\,L\right)^{mm_z}_{\phantom{mm_z}M}
    \\ \times
    \left(L\,l_1\,l_2\right)^{Mm_1m_2}
    \hat{\Gamma}^{(2)\,l\,l_z;l_1l_2}_{Xmm_z;m_1m_2}(k_1,k_2).
  \label{eq:3-130}
\end{multline}
The symmetry of complex conjugate is given by
\begin{equation}
  \hat{\Gamma}^{(2)\,l\,l_z;L\,*}_{Xl_1l_2}(k_1,k_2)
  = (-1)^{l_z} \hat{\Gamma}^{(2)\,l\,l_z;L}_{Xl_1l_2}(k_1,k_2),
  \label{eq:3-130-1}
\end{equation}
and thus the reduced function is a real function for normal cases of
$l_z=\mathrm{even}$ and a pure imaginary function for strange cases of
$l_z=\mathrm{odd}$. The parity symmetry of Eq.~(\ref{eq:3-102-1}) in
this case suggests $s_X+l+l_z+l_1+l_2=\mathrm{even}$. The propagator
is given by
\begin{multline}
  \hat{\Gamma}^{(2)}_{Xlm}(\bm{k}_1,\bm{k}_2;\hat{\bm{z}})
  =
    \sum_{l_z,l_1,l_2,L}
   \sqrt{\frac{4\pi}{\{l_z\}}}\,
    \hat{\Gamma}^{(2)\,l\,l_z;L}_{Xl_1l_2}(k_1,k_2)
    \\ \times
    \left\{
      Y_{l_z}(\hat{\bm{z}})\otimes
      \left\{
        Y_{l_1}(\hat{\bm{k}}_1) \otimes Y_{l_2}(\hat{\bm{k}}_2)
      \right\}_L
    \right\}_{lm}.
  \label{eq:3-131}
\end{multline}
Due to the symmetry of the $3j$-symbol,
$(L\,l_1\,l_2)_{Mm_1m_2} = (-1)^{L+l_1+l_2}(L\,l_2\,l_1)_{Mm_2m_1}$,
the invariant propagator of the second order in redshift space
satisfies an interchange symmetry,
\begin{equation}
  \hat{\Gamma}^{(2)\,l\,l_z;L}_{Xl_2l_1}(k_2,k_1)
  = (-1)^{l_1+l_2+L}
  \hat{\Gamma}^{(2)\,l\,l_z;L}_{Xl_1l_2}(k_1,k_2).
  \label{eq:3-131-1}
\end{equation}

For $n \geq 3$, we have
\begin{multline}
  \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xmm_z;m_1\cdots m_n}(k_1,\ldots,k_n)
  = (-1)^l\sqrt{\{l\}}
  \\ \times
  \sum_{L_1,\ldots,L_{n-1}}
  (-1)^{L_1+\cdots +L_{n-1}}
    \sqrt{\{L_1\}\cdots\{L_{n-1}\}}
    \left(l\,l_z\,L_1\right)_{mm_zM_1}
    \\ \times
    \left(L_1\,l_1\,L_2\right)^{M_1}_{\phantom{M_1}m_1M_2}
    \cdots
    \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)^{M_{n-2}}_{\phantom{M_{n-2}}m_{n-2}M_{n-1}}
    \\ \times
    \left(L_{n-1}\,l_{n-1}\,l_n\right)^{M_{n-1}}_{\phantom{M_{n-1}}m_{n-1}m_n}
    \hat{\Gamma}^{(n)\,l\,l_z;L_1\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n),
  \label{eq:3-132}
\end{multline}
with
\begin{multline}
  \hat{\Gamma}^{(n)\,l\,l_z;L_1\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
  \equiv
  \frac{(-1)^{l_z+l_1+\cdots +l_n}}{\sqrt{\{l\}}}
  \\ \times
  (-1)^{L_1+\cdots +L_{n-1}} \sqrt{\{L_1\}\cdots\{L_{n-1}\}}
    \left(l\,l_z\,L_1\right)^{mm_z}_{\phantom{mm_z}M_1}
    \\ \times
    \left(L_1\,l_1\,L_2\right)^{M_1m_1}_{\phantom{M_1m_1}M_2}
    \cdots
    \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)^{M_{n-2}m_{n-2}}_{\phantom{M_{n-2}m_{n-2}}M_{n-1}}
    \\ \times
    \left(L_{n-1}\,l_{n-1}\,l_n\right)^{M_{n-1}m_{n-1}m_n}
    \hat{\Gamma}^{(n)\,l\,l_z;l_1\cdots l_n}_{Xmm_z;m_1\cdots m_n}(k_1,\ldots,k_n),
  \label{eq:3-133}
\end{multline}
The symmetry of complex conjugate is given by
\begin{equation}
  \hat{\Gamma}^{(n)\,l\,l_z;L_1\cdots L_{n-1}\,*}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
  = (-1)^{l_z}
  \hat{\Gamma}^{(n)\,l\,l_z;L_1\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n),
  \label{eq:3-133-1}
\end{equation}
and thus the reduced function is a real function for normal cases of
$l_z=\mathrm{even}$ and a pure imaginary function for strange cases of
$l_z=\mathrm{odd}$. The parity symmetry of Eq.~(\ref{eq:3-123-1}) in
this case suggests $s_X+l+l_z+l_1+\cdots +l_n=\mathrm{even}$. The
propagator is given by
\begin{multline}
  \hat{\Gamma}^{(n)}_{Xlm}(\bm{k}_1,\cdots,\bm{k}_n;\hat{\bm{z}})
  \\
  =
    \sum_{\substack{l_z,l_1,\ldots,l_n\\L_1,\ldots,L_{n-1}}}
   \sqrt{\frac{4\pi}{\{l_z\}}}\,
    \hat{\Gamma}^{(n)\,l\,l_z;L_1\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\cdots,k_n)
    \\ \times
  \left\{
    Y_{l_z}(\hat{\bm{z}})\otimes
    \left\{
      Y_{l_1}(\hat{\bm{k}}_1)\otimes
    \left\{ \cdots
      \otimes Y_{l_n}(\hat{\bm{k}}_n)
      \right\}_{L_{n-1}} \cdots \right\}_{L_1}
  \right\}_{lm}.
  \label{eq:3-134}
\end{multline}
Corresponding to Eqs.~(\ref{eq:3-86}) and (\ref{eq:3-87}) of the
renormalized bias function, the interchange symmetries for the
higher-order propagators are given by
\begin{multline}
  \hat{\Gamma}^{(n)l\,l_z;L_1\cdots L_{n-2}L_{n-1}}_{Xl_l\cdots l_{n-2}l_nl_{n-1}}
  (k_1,\cdots,k_{n-2},k_n,k_{n-1})
  \\
  = (-1)^{l_{n-1}+l_n+L_{n-1}}
  \hat{\Gamma}^{(n)l\,l_z;L_1\cdots L_{n-1}}_{Xl_l\cdots l_n}
  (k_1,\cdots,k_n),
  \label{eq:3-134-1}
\end{multline}
and
\begin{multline}
  \hat{\Gamma}^{(n)\,l\,l_z;L_1\cdots L_{i-1}L_{i+1}L_iL_{i+2}\cdots L_{n-1}}
  _{Xl_1\cdots l_{i-1}l_{i+1}l_il_{i+2}\cdots l_n}
  (k_1,\cdots,k_{i-1},k_{i+1},k_i,k_{i+2},\cdots,k_n)
  \\
  = (-1)^{l_i+l_{i+1}}
  \sum_{L'} (2L'+1)
  \begin{Bmatrix}
    l_i & L_i & L_{i+1} \\
    l_{i+1} & L_{i+2} & L'
  \end{Bmatrix}
  \\ \times
  \hat{\Gamma}^{(n)\,l\,l_z;L_1\cdots L_iL'L_{i+2}\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\cdots,k_n).
  \label{eq:3-134-2}
\end{multline}

\subsection{Sample calculations of decomposed propagators
  \label{subsec:LowProp}
}

One can straightforwardly derive explicit forms of reduced propagators
with the formal decompositions as explained above. A simple way to
derive the invariant coefficients is to represent the angular
dependence of the propagators in terms of the spherical harmonics and
polypolar spherical harmonics, and read off the coefficients from the
resulting expressions.

The simplest example is the tree-level approximation of first-order
propagator of Eq.~(\ref{eq:3-21}):
\begin{equation}
  \hat{\Gamma}_{Xlm}^{(1)}(\bm{k})
  = c_{Xlm}^{(1)}(\bm{k}) +
  \left[\bm{k}\cdot\bm{L}_1(\bm{k})\right] c_{Xlm}^{(0)}.
  \label{eq:3-200}
\end{equation}
In real space, $\bm{L}_1(\bm{k})$ is given by Eq.~(\ref{eq:3-24a}),
and we have $\bm{k}\cdot\bm{L}_1(\bm{k}) = 1$. The relevant
renormalized bias functions are represented by invariant quantities
through Eqs.~(\ref{eq:3-67}) and (\ref{eq:3-70-1}). Therefore the
propagator of Eq.~(\ref{eq:3-200}) is represented in a form of
expansion with spherical harmonics as
\begin{equation}
  \hat{\Gamma}_{Xlm}^{(1)}(\bm{k}) =
  \left[
    c^{(1)}_{Xl}(k) + \delta_{l0} c^{(0)}_X
  \right]
  Y_{lm}(\hat{\bm{k}}).
  \label{eq:3-202}
\end{equation}
Comparing this expression with Eq.~(\ref{eq:3-105}), we have
\begin{equation}
  \hat{\Gamma}^{(1)}_{Xl}(k) =
  c^{(1)}_{Xl}(k) + \delta_{l0} c^{(0)}_X.
  \label{eq:3-202-1}
\end{equation}
The same result can also be derived directly from the definition of
the invariant coefficient, i.e., Eq.~(\ref{eq:3-104}) with
Eq.~(\ref{eq:3-101}) for $n=1$. In the Universe with parity symmetry,
the rhs of Eq.~(\ref{eq:3-202-1}) is non-zero only for normal tensors,
$s_X=0$, as discussed in Sec.~\ref{subsec:RenBiasFn} and also the lhs
consistently vanishes as discussed in Sec.~\ref{subsec:Propagators}.

In redshift space, the tree-level approximation of the first-order
propagator is given by
\begin{equation}
  \hat{\Gamma}_{Xlm}^{(1)}(\bm{k})
  = c_{Xlm}^{(1)}(\bm{k}) +
  \left[\bm{k}\cdot\bm{L}^\mathrm{s}_1(\bm{k})\right] c_{Xlm}^{(0)},
  \label{eq:3-203}
\end{equation}
where the displacement kernel is replaced by the one in redshift space
as in Eq.~(\ref{eq:3-25}), and the relevant factor in the above
equation is given by
\begin{equation}
  \bm{k}\cdot\bm{L}^\mathrm{s}_1(\bm{k})
  = 1 + f (\hat{\bm{z}}\cdot\hat{\bm{k}})^2
  = 1 + \frac{f}{3}
  + \frac{2f}{3} \mathit{P}_2(\hat{\bm{z}}\cdot\hat{\bm{k}}),
  \label{eq:3-204}
\end{equation}
where $\mathit{P}_2(x)$ is one of the Legendre polynomials
$\mathit{P}_l(x)$ with $l=2$. The Legendre polynomial can be
decomposed into a superposition of spherical harmonics,
\begin{equation}
  \mathit{P}_l(\hat{\bm{z}}\cdot\hat{\bm{k}})
    = \frac{4\pi}{\{l\}}
    g_{(l)}^{mm'} Y_{lm}(\hat{\bm{z}}) Y_{lm'}(\hat{\bm{k}}).
  \label{eq:3-205}
\end{equation}
The invariant combination of the rhs of the above equation is
given by a special case of bipolar spherical harmonics as
\begin{equation}
  \left\{Y_l(\hat{\bm{z}})\otimes Y_l(\hat{\bm{k}})\right\}_{00} =
  \frac{(-1)^l}{\sqrt{\{l\}}}
    g_{(l)}^{mm'} Y_{lm}(\hat{\bm{z}}) Y_{lm'}(\hat{\bm{k}}),
  \label{eq:3-206}
\end{equation}
and thus we have
\begin{equation}
  \mathit{P}_l(\hat{\bm{z}}\cdot\hat{\bm{k}})
    = \frac{4\pi (-1)^l}{\sqrt{\{l\}}}
  \left\{Y_l(\hat{\bm{z}})\otimes Y_l(\hat{\bm{k}})\right\}_{00}.
  \label{eq:3-207}
\end{equation}
Combining Eqs.~(\ref{eq:3-203})--(\ref{eq:3-207}) above, we derive
\begin{multline}
  \hat{\Gamma}^{(1)}_{Xlm}(\bm{k})
  = c^{(1)}_{Xl}(k) Y_{lm}(\hat{\bm{k}})
  \\
  +
  \frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}} c^{(0)}_X
  \left[
    1 + \frac{f}{3} + \frac{8\pi}{3\sqrt{5}} f
  \left\{Y_2(\hat{\bm{z}})\otimes Y_2(\hat{\bm{k}})\right\}_{00}
  \right].
  \label{eq:3-208}
\end{multline}
Each term of the above equation can be represented by special cases of
bipolar spherical harmonics, because we have
\begin{align}
  \delta_{l0} \delta_{m0}
  &= 4\pi\delta_{l0}
  \left\{Y_0(\hat{\bm{z}})\otimes Y_0(\hat{\bm{k}})\right\}_{lm},
    \label{eq:3-209a}\\
  Y_{lm}(\hat{\bm{k}})
  &= \sqrt{4\pi}
    \left\{Y_0(\hat{\bm{z}})\otimes Y_l(\hat{\bm{k}})\right\}_{lm},
    \label{eq:3-209b}
\end{align}
as straightforwardly shown from the definition. One can also rewrite
the bipolar spherical harmonics appeared in Eq.~(\ref{eq:3-208}) as
\begin{equation}
  \delta_{l0} \delta_{m0}
  \left\{Y_2(\hat{\bm{z}})\otimes Y_2(\hat{\bm{k}})\right\}_{00}
  =
  \delta_{l0}
  \left\{Y_2(\hat{\bm{z}})\otimes Y_2(\hat{\bm{k}})\right\}_{lm}.
  \label{eq:3-210}
\end{equation}
Substituting the above identities into Eq.~(\ref{eq:3-208}), one can
readily read off the coefficient of the bipolar spherical harmonics
$\{Y_2(\hat{\bm{z}})\otimes Y_2(\hat{\bm{k}})\}_{lm}$ of
Eq.~(\ref{eq:3-128}) as
\begin{multline}
  \hat{\Gamma}^{(1)l\,l_z}_{Xl_1}(k) =
  \delta_{l_z0}\delta_{l_1l} c^{(1)}_{Xl}(k)
  \\
  + \delta_{l0} c^{(0)}_X
  \left[
    \left(1 + \frac{f}{3}\right)
    \delta_{l_z0}\delta_{l_10} +
    \frac{2f}{3} \delta_{l_z2}\delta_{l_12}
  \right].
  \label{eq:3-211}
\end{multline}
Putting $f=0$, $l_z=0$ and $l_1=l$ in the above equation recovers the
result in real space of Eq.~(\ref{eq:3-202}), as it should. In the
Universe with parity symmetry, the rhs of Eq.~(\ref{eq:3-211}) only
nonzero when $s_X=0$ as discussed in Sec.~\ref{subsec:RenBiasFn}, and
apparently the rhs is non-zero only for $l+l_z+l_1=\mathrm{even}$.
Thus the lhs is non-zero only when $s_x+l+l_z+l_1=\mathrm{even}$,
which is consistent with the discussion in
Sec.~\ref{subsec:Propagators}.

The same result of Eq.~(\ref{eq:3-211}) can also be derived more
straightforwardly by using an orthonormal relation of the bipolar
spherical harmonics,
\begin{multline}
  \int d^2\hat{z}\,d^2\hat{k}
  \left\{Y_{l_z}(\hat{\bm{z}})\otimes Y_{l_1}(\hat{\bm{k}})\right\}_{lm}
  \left\{Y_{l_z'}(\hat{\bm{z}})\otimes
    Y_{l_1'}(\hat{\bm{k}})\right\}_{l'm'}
  \\
  = \delta^{\triangle}_{ll_zl_1}
  \delta_{l_zl_z'} \delta_{l_1l_1'} \delta_{ll'} g^{(l)}_{mm'},
  \label{eq:3-212}
\end{multline}
where $\delta^{\triangle}_{l_1l_2l_3}=1$ only when $(l_1,l_2,l_3)$
satisfies the trangle inequality, and zero otherwise. The
Eq.~(\ref{eq:3-212}) can be shown from an orthonormal relation of the
spherical harmonies and an orthonormal relation of the $3j$-symbols.
According to the orthonormal relation above, the Eq.~(\ref{eq:3-128})
is inverted as
\begin{multline}
  \hat{\Gamma}^{(1)l\,l_z}_{Xl_1}(k)
  = \sqrt{\frac{\{l_z\}}{4\pi}}\,
  \int d^2\hat{z}\,d^2\hat{k}\,
  \hat{\Gamma}^{(1)}_{Xl0}(\bm{k})
  \left\{Y_{l_z}(\hat{\bm{z}})\otimes Y_{l_1}(\hat{\bm{k}})\right\}_{l0}.
  \label{eq:3-213}
\end{multline}
Substituting Eq.~(\ref{eq:3-208}) into the above equation, using
Eqs.~(\ref{eq:3-209a})--(\ref{eq:3-210}) with the orthonormal relation
of Eq.~(\ref{eq:3-212}), we derive the same result of
Eq.~(\ref{eq:3-211}) as well.

For another example of calculating the invariant propagators, we
consider the second-order propagator $\hat{\Gamma}^{(2)}_{Xlm}$ in
real space, Eq.~(\ref{eq:3-107}). The calculation in this case is
quite similar to the derivation of Eq.~(\ref{eq:3-211}). With the
tree-level approximation, the second-order propagator is given by
Eq.~(\ref{eq:3-22}). Applying a similar procedure as in the above
derivation using Eqs.~(\ref{eq:3-206}), (\ref{eq:3-207}) and
(\ref{eq:3-209a})--(\ref{eq:3-210}), the rhs of Eq.~(\ref{eq:3-22}) is
represented in a form of the rhs of Eq.~(\ref{eq:3-108}), and we can
read off the invariant coefficient from the resulting expression.

In the course of calculation, one needs to represent a product of
spherical harmonics of the same argument through the formula
\cite{Edmonds:1955fi},
\begin{multline}
  Y_{l_1m_1}(\hat{\bm{k}}) Y_{l_2m_2}(\hat{\bm{k}})
  = \sum_{l,m} \sqrt{\frac{\{l_1\}\{l_2\}\{l\}}{4\pi}}
  \begin{pmatrix}
    l_1 & l_2 & l \\
    0 & 0 & 0
  \end{pmatrix}
  \\ \times
  \begin{pmatrix}
    l_1 & l_2 & l \\
    m_1 & m_2 & m
  \end{pmatrix}
  Y_{lm}^*(\hat{\bm{k}}).
  \label{eq:3-214}
\end{multline}
The coefficient of the spherical harmonics on the rhs is known as the
Gaunt integral. Similarly to the simplified notation for the
$3j$-symbols we introduced above, it is useful to define a simplified
symbol for the Gaunt integral,
\begin{multline}
  \left[l_1\,l_2\,l_3\right]_{m_1m_2m_3}
  \equiv
  \int d^2\hat{k}\,
  Y_{l_1m_1}(\hat{\bm{k}}) Y_{l_2m_2}(\hat{\bm{k}})
  Y_{l_3m_3}(\hat{\bm{k}})
  \\
  =
  \sqrt{\frac{\{l_1\}\{l_2\}\{l_3\}}{4\pi}}
  \begin{pmatrix}
    l_1 & l_2 & l_3 \\
    0 & 0 & 0
  \end{pmatrix}
    \left(l_1\,l_2\,l_3\right)_{m_1m_2m_3}.
  \label{eq:3-215}
\end{multline}
We consider the azimuthal indices in the Gaunt integral can be raised
and lowered by the spherical metric, such as
\begin{equation}
  \left[l_1\,l_2\,l_3\right]^{m_1m_2}_{\phantom{m_1m_2}m_3}
  = g_{(l_1)}^{m_1m_1'} g_{(l_2)}^{m_2m_2'}
  \left[l_1\,l_2\,l_3\right]_{m_1'm_2'm_3},
  \label{eq:3-215-1}
\end{equation}
and so forth. The Gaunt integral of Eq.~(\ref{eq:3-215}) is non-zero
only when $l_1+l_2+l_3$ is an even number.
With this notation, Eq.~(\ref{eq:3-214}) is simply represented by
\begin{equation}
  Y_{l_1m_1}(\hat{\bm{k}})Y_{l_2m_2}(\hat{\bm{k}})
  = \sum_{l} \left[l_1\,l_2\,l\right]_{m_1m_2}^{\phantom{m_1m_2}m}
  Y_{lm}(\hat{\bm{k}}).
  \label{eq:3-216}
\end{equation}

Following the procedures above to derive the invariant propagator of
first order, and additionally using the formula of
Eq.~(\ref{eq:3-216}), the invariant propagator of second order can be
straightforwardly derived. The result in real space is given by
\begin{multline}
  \hat{\Gamma}^{(2)l}_{Xl_1l_2}(k_1,k_2) =
  c^{(2)l}_{Xl_1l_2}(k_1,k_2)
  \\ +
  \sqrt{4\pi} c^{(1)}_{Xl}(k_1)
  \left[
    \delta_{l_1l}\delta_{l_20} + \frac{(-1)^l}{\sqrt{3}}\
    \frac{k_1}{k_2} \delta_{l_21} \sqrt{\{l_1\}}
    \begin{pmatrix}
      l & l_1 & 1 \\
      0 & 0 & 0
    \end{pmatrix}
  \right]
  \\ +
  \sqrt{4\pi} c^{(1)}_{Xl}(k_2)
  \left[
    \delta_{l_10}\delta_{l_2l} + \frac{(-1)^l}{\sqrt{3}}\
    \frac{k_2}{k_1} \delta_{l_11} \sqrt{\{l_2\}}
    \begin{pmatrix}
      l & l_2 & 1 \\
      0 & 0 & 0
    \end{pmatrix}
  \right]
  \\
  + \sqrt{4\pi} \delta_{l0} c^{(0)}_X
  \left[
    \frac{34}{21}\delta_{l_10}\delta_{l_20}
    - \frac{1}{\sqrt{3}}
    \left( \frac{k_1}{k_2} + \frac{k_2}{k_1} \right)
    \delta_{l_11} \delta_{l_21}
    \right.
      \\
    \left.
    + \frac{8}{21\sqrt{5}} \delta_{l_12} \delta_{l_22}
    \right].
  \label{eq:3-217}
\end{multline}
In the Universe with parity symmetry, the terms on the rhs of
Eq.~(\ref{eq:3-217}) except the first term is non-zero only for normal
tensors, $s_X=0$, as discussed in Sec.~\ref{subsec:RenBiasFn}, and
apparently non-zero only for $l+l_1+l_2=\mathrm{even}$. The first term
on the rhs is non-zero only for $s_X+l+l_1+l_2=\mathrm{even}$. The lhs
is non-zero only for $s_X+l+l_1+l_2=\mathrm{even}$, and therefore the
properties are consistent with each other. The corresponding result in
redshift space is explicitly given in Paper~II \cite{PaperII}.

Other propagators to arbitrary orders, both in real space and
redshift space can be calculated in a similar way as illustrated
above, although the calculations are more or less demanding in the
case of calculation including higher orders.

\section{The spectra of tensor fields in perturbation theory
  \label{sec:TensorSpectrum}
}

In this section, we consider how one can calculate the power spectrum
and higher-order spectra such as the bispectrum and so forth, for
tensor fields in general.

\subsection{The power spectrum
  \label{subsec:PowerSpec}
}

\subsubsection{\label{subsubsec:InvPS}
  The invariant power spectrum
}

The power spectrum $P^{(l_1l_2)}_{X_1X_2\,m_1m_2}(\bm{k})$ of
irreducible components of tensor field $F_{Xlm}(\bm{k})$ may be
defined by
\begin{multline}
  \left\langle
    F_{X_1l_1m_1}(\bm{k}_1) F_{X_2l_2m_2}(\bm{k}_2)
  \right\rangle_\mathrm{c}
  \\
  =
  (2\pi)^3\delta_\mathrm{D}^3(\bm{k}_1+\bm{k}_2)
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k}_1),
  \label{eq:4-0-1}
\end{multline}
where $\langle\cdots\rangle_\mathrm{c}$ indicates the two-point
cumulant, or the connected part, and the appearance of the delta
function is due to translational symmetry. We consider generally the
cross power spectra between two fields, $X_1$ and $X_2$ with
irreducible components $l_1$ and $l_2$, respectively. The auto power
spectrum is straightforwardly obtained by putting $X_1 = X_2 = X$.

The power spectrum $P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k})$ defined above
depend on the coordinates system, as they have azimuthal indices and
depend on the direction of wavevector $\bm{k}$ in the arguments. One
can also construct reduced power spectra which are invariant under the
rotation, in a similar manner to introducing the invariant
propagators in Sec.~\ref{subsec:Propagators}.

First, we consider the power spectrum in real space. The directional
dependence on the wavevector can be expanded by spherical harmonics
as
\begin{equation}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k}) =
  \sum_l P^{l_1l_2;l}_{X_1X_2m_1m_2;m}(k)\,Y^*_{lm}(\hat{\bm{k}}).
  \label{eq:4-11}
\end{equation}
Due to the rotational transformations of Eqs.~(\ref{eq:3-38}) and
(\ref{eq:2-14-2}), and rotational invariance of the delta function in
the definition of the power spectrum, Eq.~(\ref{eq:4-0-1}), the
rotational transformation of the expansion coefficient is given by
\begin{equation}
  P^{l_1l_2;l}_{X_1X_2m_1m_2;m}(k) \rightarrow
  P^{l_1l_2;l}_{X_1X_2m_1'm_2';m'}(k)
  D^{m_1'}_{(l_1)m_1}(R) D^{m_2'}_{(l_2)m_2}(R) D^{m'}_{(l)m}(R).
  \label{eq:4-0-2}
\end{equation}
For the statistically isotropic Universe, the functional form of the
power spectrum should not depend on the choice of coordinates system.
Following the same procedure as in
Sec.~\ref{subsec:RenBiasFn}, the power spectrum in the statistically
isotropic Universe is shown to have a form,
\begin{multline}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k}) =
  (-1)^{l_1+l_2} \sqrt{\{l_1\}\{l_2\}}
  \\ \times
  \sum_l (-1)^l \sqrt{\{l\}}
  \left(l_1\,l_2\,l\right)_{m_1m_2}^{\phantom{m_1m_2}m}
  Y_{lm}(\hat{\bm{k}}) P^{l_1l_2;l}_{X_1X_2}(k),
    \label{eq:4-0-3}
\end{multline}
where
\begin{equation}
  P^{l_1l_2;l}_{X_1X_2}(k) =
  \frac{1}{\sqrt{\{l_1\}\{l_2\}\{l\}}}
  \left(l_1\,l_2\,l\right)^{m_1m_2m}
  P^{l_1l_2;l}_{X_1X_2m_1m_2;m}(k)
  \label{eq:4-0-4}
\end{equation}
is the invariant power spectrum. The physical degrees of freedom of
the power spectrum are represented by the last invariant spectrum,
$P^{l_1l_2;l}_{X_1X_2}(k)$. There are a finite number of independent
functions of power spectra which depend only on the modulus of the
wavevector $k=|\bm{k}|$ of Fourier modes, due to the triangle
inequality of the $3j$-symbol, $|l_1-l_2| \leq l \leq l_1+l_2$ for a
fixed set of integers $(l_1,l_2)$. For example, when $l_1=l_2=2$, only
$l=0,1,\ldots, 4$ are allowed and there are five independent
components in the power spectrum\footnote{In Ref.~\cite{Vlah:2019byq},
  the power spectrum $P_{22}^{(q)}(k)$ of intrinsic alignment is
  defined in a coordinate-dependent way, where $q=0,\pm 1,\pm2$. The
  degree of freedom of their power spectrum is five and is consistent
  with ours.}.

According to Eqs.~(\ref{eq:3-32}), (\ref{eq:4-0-1}) and the symmetry
of the $3j$-symbol, Eq.~(\ref{eq:c-2}), the symmetry of complex
conjugate is shown to be given by
\begin{equation}
  P^{l_1l_2;l\,*}_{X_1X_2}(k) = P^{l_1l_2;l}_{X_1X_2}(k),
  \label{eq:4-0-4-1}
\end{equation}
and thus the invariant power spectrum is a real function.
Additionally, the parity transformation of the power spectrum is given
by
\begin{equation}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k}) \rightarrow
  (-1)^{s_{X_1}+s_{X_2}+l_1+l_2} P^{(l_1l_2)}_{X_1X_2m_1m_2}(-\bm{k}),
  \label{eq:4-0-5}
\end{equation}
because of Eqs.~(\ref{eq:3-42}) and (\ref{eq:4-0-1}). Accordingly, the
parity transformation of the invariant power spectrum is given by
\begin{equation}
  P^{l_1l_2;l}_{X_1X_2}(k) \rightarrow
  (-1)^{s_{X_1}+s_{X_2}+l_1+l_2+l} P^{l_1l_2;l}_{X_1X_2}(k).
  \label{eq:4-0-6}
\end{equation}
In the Universe with parity symmetry, the power spectrum is
invariant under the parity transformation, we thus have
\begin{equation}
  s_{X_1} + s_{X_2} + l_1 + l_2 + l = \mathrm{even}.
  \label{eq:4-0-7}
\end{equation}
Together with the triangle inequality,
$|l_1-l_2| \leq l \leq l_1+l_2$, mentioned above, the numbers of
independent components in the invariant power spectrum
$P^{l_1l_2;l}_{X_1X_2}(k)$ are further reduced in the presence of
parity symmetry. For example, when $l_1=l_2=2$ and
$s_{X_1}=s_{X_2}=0$, only $l=0,2,4$ are allowed and there are three
independent components in the power spectrum\footnote{In
  Ref.~\cite{Vlah:2019byq}, their power spectrum of intrinsic
  alignment satisfies $P_{22}^{(q)}(k)=P_{22}^{(-q)}(k)$ in the
  presence of parity symmetry, and thus only three components,
  $q=0,1,2$, are independent. Again, the degree of freedom of theirs
  is consistent with ours.}. The power spectrum in real space has an
interchange symmetry,
\begin{equation}
  P^{(l_2l_1)}_{X_2X_1m_2m_1}(\bm{k})
  = P^{(l_1l_2)}_{X_1X_2m_1m_2}(-\bm{k}),
  \label{eq:4-0-7-1}
\end{equation}
and the corresponding symmetry for the invariant spectrum is given by
\begin{equation}
  P^{l_2l_1;l}_{X_2X_1}(k) =
  (-1)^{l_1+l_2} P^{l_1l_2;l}_{X_1X_2}(k).
  \label{eq:4-0-7-2}
\end{equation}

Next, we consider the power spectrum in redshift space. The directional
dependence of the wavevector and the lines of sight can be expanded by
spherical harmonics as
\begin{equation}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k};\hat{\bm{z}}) =
  \sum_{l,l_z} \sqrt{\frac{4\pi}{\{l_z\}}}
    P^{l_1l_2;l\,l_z}_{X_1X_2m_1m_2;mm_z}(k)\,Y^*_{l_zm_z}(\hat{\bm{z}})
    Y^*_{lm}(\hat{\bm{k}}).
  \label{eq:4-0-8}
\end{equation}
Following similar considerations of rotational symmetry in the rest of
this paper, e.g., in deriving Eq.~(\ref{eq:3-128}), the power spectrum
in the statistically isotropic Universe is shown to have a form,
\begin{multline}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k};\hat{\bm{z}}) =
  (-1)^{l_1+l_2} \sqrt{\{l_1\}\{l_2\}}
  \sum_L
  \left(l_1\,l_2\,L\right)_{m_1m_2}^{\phantom{m_1m_2}M}
  \\ \times
  \sum_{l,l_z} (-1)^{l+l_z} \sqrt{4\pi\{l\}}
  \left\{Y_{l_z}(\hat{\bm{z}}) \otimes Y_l(\hat{\bm{k}}) \right\}_{LM}
  P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k),
  \label{eq:4-0-9}
\end{multline}
where
\begin{multline}
  P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k) =
  \frac{(-1)^L \sqrt{\{L\}}}{\sqrt{\{l_1\}\{l_2\}{\{l\}\{l_z\}}}}
  \left(l_1\,l_2\,L\right)^{m_1m_2}_{\phantom{m_1m_2}M}
  \left(L\,l\,l_z\right)^{Mmm_z}
  \\ \times
  P^{l_1l_2;l\,l_z}_{X_1X_2m_1m_2;mm_z}(k)
  \label{eq:4-0-10}
\end{multline}
is the invariant power spectrum.

According to Eqs.~(\ref{eq:3-32}), (\ref{eq:4-0-1}) and the symmetry
of the $3j$-symbol, Eq.~(\ref{eq:c-2}), the symmetry of complex
conjugate is shown to be given by
\begin{equation}
  P^{l_1l_2;l\,l_z;L\,*}_{X_1X_2}(k) = (-1)^{l_z}P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k),
  \label{eq:4-0-10-1}
\end{equation}
and thus the invariant power spectrum is a real function for normal
cases of $l_z=\mathrm{even}$ and a pure imaginary function for strange
cases of $l_z=\mathrm{odd}$. Additionally, the parity transformation
of the power spectrum is given by
\begin{equation}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k};\hat{\bm{z}}) \rightarrow
  (-1)^{s_{X_1}+s_{X_2}+l_1+l_2}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(-\bm{k};-\hat{\bm{z}}),
  \label{eq:4-0-11}
\end{equation}
because of Eqs.~(\ref{eq:3-42}) and (\ref{eq:4-0-1}). Accordingly, the
transformation of the invariant power spectrum is given by
\begin{equation}
  P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k) \rightarrow
  (-1)^{s_{X_1}+s_{X_2}+l_1+l_2+l+l_z} P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k).
  \label{eq:4-0-12}
\end{equation}
In the Universe with parity symmetry, the power spectrum in
redshift space is invariant under the parity transformation, we thus
have
\begin{equation}
  s_{X_1} + s_{X_2} + l_1 + l_2 + l + l_z = \mathrm{even}.
  \label{eq:4-0-13}
\end{equation}
The power spectrum in redshift space has an interchange symmetry,
\begin{equation}
  P^{(l_2l_1)}_{X_2X_1m_2m_1}(\bm{k};\hat{\bm{z}})
  = P^{(l_1l_2)}_{X_1X_2m_1m_2}(-\bm{k};\hat{\bm{z}}),
  \label{eq:4-0-14}
\end{equation}
and the corresponding symmetry for the invariant spectrum is given by
\begin{equation}
  P^{l_2l_1;l\,l_z;L}_{X_2X_1}(k) =
  (-1)^{l_1+l_2+l+L} P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k).
  \label{eq:4-0-15}
\end{equation}

\subsubsection{The linear power spectra in real space and redshift space
  \label{subsubsec:LinearPS}
}

We consider the power spectrum of the lowest-order approximation, or
the linear power spectrum, which is the simplest application of our
formalism. The formal expression for the power spectrum of scalar
fields in the propagator formalism with Gaussian initial conditions
\cite{Matsubara:1995wd,Bernardeau:2008fa,Matsubara:2011ck} is
straightforwardly generalized to the case of tensor field. In the
lowest order, we simply have
\begin{equation}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k})
  = \Pi^2(\bm{k})
  \hat{\Gamma}^{(1)}_{X_1l_1m_1}(\bm{k})
  \hat{\Gamma}^{(1)}_{X_2l_2m_2}(-\bm{k})
  P_\mathrm{L}(k),
  \label{eq:4-2}
\end{equation}
where the parity symmetry for the vertex resummation factor,
$\Pi(-\bm{k})=\Pi(\bm{k})$ is taken into account.

The vertex resummation factor $\Pi(\bm{k})$ is given by
Eq.~(\ref{eq:3-20}). In the lowest order of the perturbation theory,
this factor can be approximated by $\Pi(\bm{k})=1$. This factor
strongly damps the power spectrum on small scales, where the linear
theory does not apply. However, sometimes it is useful to keep this
exponent even in the linear theory, especially in redshift space where
the peculiar velocities damp the power on small scales. When the
resummation factor is retained, the linear approximation for this
factor is derived from Eqs.~(\ref{eq:3-20}), (\ref{eq:3-24a}),
(\ref{eq:3-25}), and we have
\cite{Matsubara:2007wj,Matsubara:2011ck,Matsubara:2013ofa}
\begin{equation}
  \Pi(k) =
  \exp\left[
    -\frac{k^2}{12\pi^2}
    \int dp P_\mathrm{L}(p)
  \right],
  \label{eq:4-3-0}
\end{equation}
in real space, and
\begin{equation}
  \Pi(k,\mu) =
  \exp\left\{
    -\frac{k^2}{12\pi^2}
    \left[ 1 + f(f+2)\mu^2 \right]
    \int dp P_\mathrm{L}(p)
  \right\},
  \label{eq:4-3}
\end{equation}
in redshift space, where
\begin{equation}
  \mu \equiv \hat{\bm{z}}\cdot\hat{\bm{k}}
  \label{eq:4-3-1}
\end{equation}
is a direction cosine between the wavevector and the line of sight.
Substituting $f=0$ in the expression in redshift space reduces to
the expression in real space as a matter of course.

In real space, the first-order propagator in terms of the invariant
propagator is given by Eq.~(\ref{eq:3-105}). Thus Eq.~(\ref{eq:4-2})
reduces to
\begin{multline}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k})
  = \Pi^2(k)\, (-1)^{l_2}\,
  \hat{\Gamma}^{(1)}_{X_1l_1}(k)
  \hat{\Gamma}^{(1)}_{X_2l_2}(k)
  P_\mathrm{L}(k)
  \\ \times
  \sum_l \left[l_1\,l_2\,l\right]_{m_1m_2}^{\phantom{m_1m_2}m}
  Y_{lm}(\hat{\bm{k}}).
  \label{eq:4-4}
\end{multline}
The invariant power spectrum is easily read off from the above
expression and Eq.~(\ref{eq:4-0-3}), and we have
\begin{equation}
  P^{l_1l_2;l}_{X_1X_2}(k)
  = \frac{(-1)^{l_2}}{\sqrt{4\pi}}
    \begin{pmatrix}
      l_1 & l_2 & l \\
      0 & 0 & 0
    \end{pmatrix}
    \Pi^2(k)
    \hat{\Gamma}^{(1)}_{X_1l_1}(k)
    \hat{\Gamma}^{(1)}_{X_2l_2}(k)
    P_\mathrm{L}(k).
  \label{eq:4-4-1}
\end{equation}
Further substituting the lowest-order approximation of the first-order
propagator, Eq.~(\ref{eq:3-202-1}), the above equation is explicitly
given by
\begin{multline}
    P^{l_1l_2;l}_{X_1X_2}(k) =
    \frac{\Pi^2(k)}{\sqrt{4\pi}} P_\mathrm{L}(k)
    \\ \times
    \Biggl\{
    (-1)^{l_2}
    \begin{pmatrix}
      l_1 & l_2 & l \\ 0 & 0 & 0
    \end{pmatrix}
    c^{(1)}_{X_1l_1}(k) c^{(1)}_{X_2l_2}(k)
    + \delta_{l0} \delta_{l_10} \delta_{l_20}
    c^{(0)}_{X_1} c^{(0)}_{X_2}
    \\
    + \frac{1}{\sqrt{\{l\}}}
    \Biggl[
    (-1)^l \delta_{l_1l} \delta_{l_20}
    c^{(1)}_{X_1l_1}(k) c^{(0)}_{X_2}
    + \delta_{l_10} \delta_{l_2l}
    c^{(0)}_{X_1} c^{(1)}_{X_2l_2}(k)
    \Biggr]
  \Biggr\}.
  \label{eq:4-4-2}
\end{multline}
This is one of the generic predictions of our theory.

As a consistency check, we consider the scalar case, $l_1=l_2=0$ of
auto power spectrum, $X_1=X_2=X$. In this case, Eq.~(\ref{eq:4-4-2})
reduces to
\begin{equation}
  P^{00;0}_{XX}(k) =
  \frac{\Pi^2(k)}{\sqrt{4\pi}} \left[b_X(k)\right]^2
  P_\mathrm{L}(k),
  \label{eq:4-5}
\end{equation}
where
\begin{equation}
  b_X(k) \equiv c^{(0)}_X + c^{(1)}_{X0}(k)
  \label{eq:4-5-0}
\end{equation}
corresponds to the linear bias factor. In the linear order with
$\Pi(k)=1$, the auto power spectrum of a scalarly biased field in the
well-known linear theory is given by
$P_X(k) = [b_X(k)]^2 P_\mathrm{L}(k)$. Due to the normalization of a
spherical basis, $\mathsf{Y}^{0)}=(4\pi)^{-1/2}$, the scalar power
spectrum in the spherical basis is given by
$P_{X}(k) = 4\pi P^{(00)}_{XX00}(\bm{k}) = \sqrt{4\pi}
P^{00;0}_{XX}(k)$, and Eq.~(\ref{eq:4-5}) is consistent with the
result of the usual linear power spectrum of a scalarly biased field.

The Kronecker's symbols in the Eq.~(\ref{eq:4-4-2}) indicate that the
resulting expressions are quite different between scalar fields
($l_1=0$ or $l_2=0$) and non-scalar fields ($l_1\ge 1$ or $l_2\ge 1$).
When the fields $X_1$ and $X_2$ are both non-scalar fields with
$l_1\ne 0$ and $l_2 \ne 0$, Eq.~(\ref{eq:4-4-2}) simply reduces to
\begin{multline}
    P^{l_1l_2;l}_{X_1X_2}(k) =
    \frac{\Pi^2(k)}{\sqrt{4\pi}} P_\mathrm{L}(k)
    (-1)^{l_2}
    \begin{pmatrix}
      l_1 & l_2 & l \\ 0 & 0 & 0
    \end{pmatrix}
    c^{(1)}_{X_1l_1}(k) c^{(1)}_{X_2l_2}(k),
    \\
    (l_1\ne 0 \mathrm{\ \ and\ \ } l_2 \ne 0).
  \label{eq:4-5-1}
\end{multline}
When $X_1$ is a non-scalar field and $X_2$ is a scalar field with
$l_1\ne 0$ and $l_2=0$, Eq.~(\ref{eq:4-4-2}) reduces to
\begin{equation}
  P^{l_10;l}_{X_1X_2}(k) =
  \delta_{l_1l}  \frac{\Pi^2(k)}{\sqrt{4\pi}} P_\mathrm{L}(k)
  \frac{(-1)^{l_1}}{\sqrt{\{l_1\}}}
  c^{(1)}_{X_1l_1}(k) b_{X_2}(k),
  \quad
  (l_1\ne 0),
  \label{eq:4-5-2}
\end{equation}
where $b_X(k)$ is given by Eq.~(\ref{eq:4-5-0}). The above equation
survives only when $l=l_1$. When $X_1$ and $X_2$ are both scalar fields
with $l_1=l_2=0$, Eq.~(\ref{eq:4-4-2}) reduces to
\begin{equation}
    P^{00;l}_{X_1X_2}(k) =
     \delta_{l0} \frac{\Pi^2(k)}{\sqrt{4\pi}}
    P_\mathrm{L}(k) b_{X_1}(k) b_{X_2}(k),
    \label{eq:4-5-3}
\end{equation}
which survives only when $l=0$. The last equation coincides with
Eq.~(\ref{eq:4-5}) when $X_1=X_2=X$ and $l=0$.

In redshift space, the lowest-order propagator is given by
Eq.~(\ref{eq:3-128}), which can be substituted into
Eq.~(\ref{eq:4-2}). As a result, a product of bipolar spherical
harmonics appears. A product of bipolar spherical harmonics reduces to
a superposition of a single bipolar spherical harmonics using a
$9j$-symbols \cite{Khersonskii:1988krb}:
\begin{multline}
  \left\{
    Y_{l_{z1}}(\hat{\bm{z}})\otimes Y_{l_1'}(\hat{\bm{k}})
  \right\}_{l_1m_1}
  \left\{
    Y_{l_{z2}}(\hat{\bm{z}})\otimes Y_{l_2'}(\hat{\bm{k}})
  \right\}_{l_2m_2}
  \\
  =(-1)^{l_1+l_2}
  \frac{\sqrt{\{l_{z1}\}\{l_{z2}\}\{l_1'\}\{l_2'\}\{l_1\}\{l_2\}}}{4\pi}
  \sum_L \sqrt{\{L\}}
  \left(L\,l_1\,l_2\right)^M_{\phantom{M}m_1m_2}
  \\ \times
  \sum_{L_z,L'} (-1)^{L_z+L'} \sqrt{\{L_z\}\{L'\}
  }
  \begin{pmatrix}
    l_{z1} & l_{z2} & L_z \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_1' & l_2' & L' \\
    0 & 0 & 0
  \end{pmatrix}
  \\ \times
  \begin{Bmatrix}
    l_{z1} & l_{z2} & L_z \\
    l_1' & l_2' & L' \\
    l_1 & l_2 & L
  \end{Bmatrix}
  \left\{
    Y_{L_z}(\hat{\bm{z}})\otimes Y_{L'}(\hat{\bm{k}})
  \right\}_{LM}.
  \label{eq:4-6}
\end{multline}
Thus we have
\begin{multline}
  P^{(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k};\hat{\bm{z}})
  = \Pi^2(\bm{k}) P_\mathrm{L}(k)
   (-1)^{l_1+l_2}\sqrt{\{l_1\}\{l_2\}}
  \\ \times
  \sum_{l_{z1},l_{z2},l_1',l_2'} (-1)^{l_2'}
  \sqrt{\{l_1'\}\{l_2'\}}
  \hat{\Gamma}^{(1)l_1l_{z1}}_{X_1l_1'}(k)
  \hat{\Gamma}^{(1)l_2l_{z2}}_{X_2l_2'}(k)
  \\ \times
  \sum_L \sqrt{\{L\}}
  \left(l_1\,l_2\,L\right)_{m_1m_2}^{\phantom{m_1m_2}M}
  \sum_{l_z,l} (-1)^{l_z+l} \sqrt{\{l_z\}\{l\}}
  \\ \times
  \begin{pmatrix}
    l_{z1} & l_{z2} & l_z \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_1' & l_2' & l \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{Bmatrix}
    l_{z1} & l_{z2} & l_z \\
    l_1' & l_2' & l\\
    l_1 & l_2 & L
  \end{Bmatrix}
  \\ \times
  \left\{
    Y_{l_z}(\hat{\bm{z}})\otimes Y_{l}(\hat{\bm{k}})
  \right\}_{LM}.
  \label{eq:4-7}
\end{multline}
Substitution of Eq.~(\ref{eq:3-211}) into the above gives an explicit
result. The invariant power spectrum is easily read off from the
expression of Eq.~(\ref{eq:4-0-9}), and we have
\begin{multline}
  P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k,\mu) =
  \frac{1}{\sqrt{4\pi}}
  \Pi^2(k,\mu) P_\mathrm{L}(k)
  \sqrt{\{l_z\}\{L\}}
  \\ \times
  \sum_{l_{z1},l_{z2},l_1',l_2'} (-1)^{l_2'}
  \sqrt{\{l_1'\}\{l_2'\}}
  \begin{pmatrix}
    l_{z1} & l_{z2} & l_z \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_1' & l_2' & l \\
    0 & 0 & 0
  \end{pmatrix}
  \\ \times
  \begin{Bmatrix}
    l_{z1} & l_{z2} & l_z \\
    l_1' & l_2' & l \\
    l_1 & l_2 & L
  \end{Bmatrix}
  \hat{\Gamma}^{(1)l_1l_{z1}}_{X_1l_1'}(k)
  \hat{\Gamma}^{(1)l_2l_{z2}}_{X_2l_2'}(k).
  \label{eq:4-8}
\end{multline}
This is one of the generic predictions of our theory.

As a consistency check, one can recover the result of the power
spectrum in real space, Eqs.~(\ref{eq:4-4}) and (\ref{eq:4-4-1}), by
substituting $l_{z1}=l_{z2}=0$ into Eqs.~(\ref{eq:4-7}) and
(\ref{eq:4-8}) and using identities
$\hat{\Gamma}^{(1)l0}_{Xl'}(k) =
\delta_{l'l}\hat{\Gamma}^{(1)}_{Xl}(k)$, $(00l)_{000}=\delta_{l0}$,
and a special case of $9j$-symbols \cite{Khersonskii:1988krb},
\begin{equation}
  \begin{Bmatrix}
    0 & 0 & 0 \\
    l_1' & l_2' & l_3' \\
    l_1 & l_2 & l_3
  \end{Bmatrix}
  = \frac{\delta_{l_1l_1'}\delta_{l_2l_2'}\delta_{l_3l_3'}}
  {\sqrt{\{l_1\}\{l_2\}\{l_3\}}}.
  \label{eq:4-9}
\end{equation}

As another consistency check, one can also see if Eq.~(\ref{eq:4-7})
reproduces the well-known result for the scalar case, i.e., the
Kaiser's formula \cite{Kaiser:1987qv}. In fact, substituting
Eq.~(\ref{eq:3-211}) into Eq.~(\ref{eq:4-7}) in the case of
$l_1=l_2=m_1=m_2=0$, $X_1=X_2=X$, and using Eqs.~(\ref{eq:c-8}) and
(\ref{eq:4-9}), we derive
\begin{multline}
  P^{(00)}_{XX00}(\bm{k};\hat{\bm{z}}) =
  \frac{\Pi^2(k,\mu)}{4\pi} \left[b_X(k)\right]^2P_\mathrm{L}(k)
  \\ \times
  \left[
    \left(1 + \frac{2}{3} \beta + \frac{1}{5}\beta^2\right)
    \mathit{P}_0(\mu)
    + \left(\frac{4}{3}\beta + \frac{4}{7}\beta^2\right)
    \mathit{P}_2(\mu)
    \right.
    \\
    \left.
    + \frac{8}{35}\beta^2 \mathit{P}_4(\mu)
  \right],
  \label{eq:4-10}
\end{multline}
where $\beta \equiv c^{(0)}_Xf/b_X(k)$ is the redshift-space
distortion parameter, and $\mu \equiv \hat{\bm{z}}\cdot\hat{\bm{k}}$
is the direction cosine of the wavevector with respect to the line of
sight. Due to the normalization of the spherical basis, we have
$P_X(\bm{k}) = (4\pi)^{-1} P^{(00)}_{XX00}(\bm{k})$, and we recover
the well-known result of the linear power spectrum of a scalarly
biased field in redshift space.

In the lowest-order approximation of the first-order propagator in
redshift space, Eq.~(\ref{eq:3-211}), either $l$ or $l_z$ is zero in
each term. Therefore, at least two of the indices $l_1$, $l_2$,
$l_{z1}$, $l_{z2}$ are zero in the $9j$-symbol of Eq.~(\ref{eq:4-8}).
When two of the indices in the $9j$-symbol are zero, there are formulas
\cite{Khersonskii:1988krb},
\begin{align}
  \begin{Bmatrix}
    l_1 & l_2 & l_3 \\
    l_4 & l_5 & l_6 \\
    l_7 & 0 & 0
  \end{Bmatrix}
  &= \delta_{l_70}
  \frac{\delta_{l_1l_4}\delta_{l_2l_5}\delta_{l_3l_6}}
  {\sqrt{\{l_1\}\{l_2\}\{l_3\}}},
  \label{eq:4-11a}\\
  \begin{Bmatrix}
    l_1 & l_2 & l_3 \\
    l_4 & 0 & l_6 \\
    l_7 & l_8 & 0
  \end{Bmatrix}
  &= (-1)^{l_1-l_2-l_3}
  \frac{\delta_{l_4l_6}\delta_{l_2l_8}\delta_{l_3l_6}\delta_{l_7l_8}}
  {\sqrt{\{l_2\}\{l_3\}}},
  \label{eq:4-11b}
\end{align}
A $9j$-symbol, in which two of the indices are zero, can always be
represented in a form of the lhs of the above equations by means of
the symmetric properties of the $9j$-symbol,
Eqs.~(\ref{eq:c-31})--(\ref{eq:c-33}). Thus, with the lowest-order
approximation of the first-order propagator given by
Eq.~(\ref{eq:3-211}), the summation of Eq.~(\ref{eq:4-8}) is
explicitly expanded to give
\begin{align}
  &
    P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k,\mu) =
    \frac{\Pi^2(k,\mu)}{\sqrt{4\pi}} P_\mathrm{L}(k)
    \nonumber\\
  &
    \quad \times
    \Biggl\{
    \delta_{l_z0} \delta_{lL} (-1)^{l_2}
    \begin{pmatrix}
      l_1 & l_2 & l \\ 0 & 0 & 0
    \end{pmatrix}
    c^{(1)}_{X_1l_1}(k) c^{(1)}_{X_2l_2}(k) \nonumber\\
  & \hspace{3pc}
    + \left(1 + \frac{f}{3}\right)
    \frac{\delta_{l_z0} \delta_{lL}}{\sqrt{\{l\}}}
    \Biggl[
    (-1)^l \delta_{l_1l} \delta_{l_20}
    c^{(1)}_{X_1l_1}(k) c^{(0)}_{X_2} \nonumber\\
  & \hspace{11pc}
    + \delta_{l_10} \delta_{l_2l}
    c^{(0)}_{X_1} c^{(1)}_{X_2l_2}(k)
    \Biggr] \nonumber\\
  & \hspace{3pc}
    + \frac{2f}{3}
     \frac{\delta_{l_z2}}{\sqrt{5}}
    \Biggl[
    \delta_{l_1L} \delta_{l_20}
    \begin{pmatrix}
      2 & l_1 & l \\ 0 & 0 & 0
    \end{pmatrix}
    c^{(1)}_{X_1l_1}(k) c^{(0)}_{X_2} \nonumber\\
  & \hspace{7.5pc}
    + (-1)^l\delta_{l_10} \delta_{l_2L}
    \begin{pmatrix}
      2 & l_2 & l \\ 0 & 0 & 0
    \end{pmatrix}
    c^{(0)}_{X_1} c^{(1)}_{X_2l_2}(k)
    \Biggl] \nonumber\\
  & \hspace{3pc}
    + \delta_{L0} \delta_{l_10} \delta_{l_20} \delta_{l\,l_z}
    \Biggl[
    \delta_{l_z0} \left( 1 + \frac{2f}{3} + \frac{f^2}{5} \right)
    \nonumber\\
  & \hspace{6pc}
    + \frac{\delta_{l_z2}}{5} \left(\frac{4f}{3} + \frac{4f^2}{7} \right)
    + \frac{\delta_{l_z4}}{9} \frac{8f^2}{35}
    \Biggr]
    c^{(0)}_{X_1} c^{(0)}_{X_2}
  \Biggr\},
  \label{eq:4-12}
\end{align}
where an identity
\begin{equation}
  \begin{pmatrix}
    2 & 2 & l \\ 0 & 0 & 0
  \end{pmatrix}^2
  = \frac{1}{5} \delta_{l0}
  + \frac{2}{35} \delta_{l2}
  + \frac{2}{35} \delta_{l4}
  \label{eq:4-13}
\end{equation}
is used in deriving the above result. The Eq.~(\ref{eq:4-12}) is one
of the generic predictions of our theory.

The above Eq.~(\ref{eq:4-12}) is
non-zero only when $l_z=0,2,4$, and the redshift-space distortions in
linear theory contains only monopole ($l_z=0$), quadrupole ($l_z=2$)
and hexadecapole ($l_z=4$) components, as is well known in the scalar
perturbation theory, Eq.~(\ref{eq:4-10}), while higher-order corrections
contain higher-order multipoles for the redshift-space distortions.

In linear theory besides the resummation factor $\Pi(k,\mu)$,
higher-rank tensors are not affected by the redshift-space distortions
as the first-order propagator of Eq.~(\ref{eq:3-211}) does not
neither. Thus, if neither $l_1$ nor $l_2$ is zero, Eq.~(\ref{eq:4-12})
is simply given by
\begin{multline}
    P^{l_1l_2;l\,l_z;L}_{X_1X_2}(k,\mu) =
    \frac{\delta_{l_z0}\delta_{lL}}{\sqrt{4\pi}}
    \Pi^2(k,\mu) P_\mathrm{L}(k)
    (-1)^{l_2}
    \begin{pmatrix}
      l_1 & l_2 & l \\ 0 & 0 & 0
    \end{pmatrix}
    \\ \times
    c^{(1)}_{X_1l_1}(k) c^{(1)}_{X_2l_2}(k),
    \qquad (l_1 \ne 0 \mathrm{\ \ and\ \ } l_2\ne 0),
  \label{eq:4-14}
\end{multline}
which survives only when $l_z=0$ and $l=L$. The above equation exactly
coincides with the result in real space, Eq.~(\ref{eq:4-5-1}). Namely,
redshift-space distortions on higher-rank tensors are nonlinear
effects. In fact, nonlinear loop corrections in the power spectrum do
introduce the redshift-space distortion effects, as explicitly shown
in Paper~II \cite{PaperII}.

As in the case of real space, Kronecker's symbols in the
Eq.~(\ref{eq:4-12}) indicate that the resulting expression is
different between scalar fields ($l_1=0$ or $l_2=0$) and non-scalar
fields ($l_1\ge 1$ or $l_2\ge 1$). When the fields $X_1$ and $X_2$ are
both non-scalar fields with $l_1\ne 0$ and $l_2 \ne 0$, the expression
reduces to Eq.~(\ref{eq:4-14}) above. When $X_1$ is a non-scalar field
and $X_2$ is a scalar field with $l_1\ne 0$ and $l_2=0$,
Eq.~(\ref{eq:4-12}) reduces to
\begin{multline}
    P^{l_10;l\,l_z;L}_{X_1X_2}(k,\mu) =
    \frac{\delta_{Ll_1}}{\sqrt{4\pi}}
    \Pi^2(k,\mu) P_\mathrm{L}(k)
    \\ \times
    \Biggl\{
    \delta_{l_z0} \delta_{ll_1}
    \frac{(-1)^{l_1}}{\sqrt{\{l_1\}}}
    c^{(1)}_{X_1l_1}(k)
    \left[
    c^{(1)}_{X_20}(k)
    + \left(1 + \frac{f}{3}\right) c^{(0)}_{X_2}
  \right]
  \\
  + \delta_{l_z2}
  \frac{2f}{3}
  \frac{1}{\sqrt{5}}
  \begin{pmatrix}
    2 & l_1 & l \\ 0 & 0 & 0
  \end{pmatrix}
  c^{(1)}_{X_1l_1}(k) c^{(0)}_{X_2}
  \Biggr\},
  \quad (l_1\ne 0),
  \label{eq:4-15}
\end{multline}
which survives only when $L=l_1$. Using the linear bias factor of
Eq.~(\ref{eq:4-5-0}), the above equation is alternatively represented
by
\begin{multline}
    P^{l_10;l\,l_z;L}_{X_1X_2}(k,\mu) =
    \delta_{Ll_1}
    \frac{\Pi^2(k,\mu)}{\sqrt{4\pi}} P_\mathrm{L}(k)
    \\ \times
    \left[
    \delta_{l_z0} \delta_{ll_1}
    \frac{(-1)^{l_1}}{\sqrt{\{l_1\}}}
    c^{(1)}_{X_1l_1}(k)
    b_{X_2}(k)
    \left(
      1 + \frac{1}{3}\beta_2
    \right)
    \right.
    \\
    \left.
  + \delta_{l_z2}
  \frac{2f}{3}
  \frac{1}{\sqrt{5}}
  \begin{pmatrix}
    2 & l_1 & l \\ 0 & 0 & 0
  \end{pmatrix}
  c^{(1)}_{X_1l_1}(k) c^{(0)}_{X_2}
  \right],
  \label{eq:4-16}
\end{multline}
where $\beta_2 \equiv c^{(0)}_{X_2}f/b_{X_2}(k)$.
When $X_1$ and $X_2$ are both scalar
fields with $l_1=l_2=0$, Eq.~(\ref{eq:4-12}) reduces to
\begin{align}
  &
    P^{00;l\,l_z;L}_{X_1X_2}(k,\mu) =
    \frac{\delta_{L0}\delta_{ll_z}}{\sqrt{4\pi}}
    \Pi^2(k,\mu) P_\mathrm{L}(k)
    \nonumber\\
  &
    \quad \times
    \Biggl\{
    \delta_{l_z0}
    c^{(1)}_{X_10}(k) c^{(1)}_{X_20}(k) \nonumber\\
  & \hspace{2pc}
    + \left(1 + \frac{f}{3}\right)
    \delta_{l_z0}
    \Biggl[
    c^{(1)}_{X_10}(k) c^{(0)}_{X_2}
    + c^{(0)}_{X_1} c^{(1)}_{X_20}(k)
    \Biggr] \nonumber\\
  & \hspace{2pc}
    + \frac{2f}{3}
     \frac{\delta_{l_z2}}{5}
    \Biggl[
    c^{(1)}_{X_1l_1}(k) c^{(0)}_{X_2}
    + c^{(0)}_{X_1} c^{(1)}_{X_2l_2}(k)
    \Biggl] \nonumber\\
  & \hspace{2pc}
    +
    \Biggl[
    \delta_{l_z0} \left( 1 + \frac{2f}{3} + \frac{f^2}{5} \right)
    \nonumber\\
  & \hspace{5pc}
    + \frac{\delta_{l_z2}}{5} \left(\frac{4f}{3} + \frac{4f^2}{7} \right)
    + \frac{\delta_{l_z4}}{9} \frac{8f^2}{35}
    \Biggr]
    c^{(0)}_{X_1} c^{(0)}_{X_2}
  \Biggr\},
  \label{eq:4-17}
\end{align}
which survives only when $L=0$ and $l=l_z$. Using the linear bias
factor of Eq.~(\ref{eq:4-5-0}), the above equation is alternatively
represented by
\begin{multline}
    P^{00;l\,l_z;L}_{X_1X_2}(k,\mu) =
    \delta_{L0}\delta_{ll_z}
    \frac{\Pi^2(k,\mu)}{\sqrt{4\pi}}
    P_\mathrm{L}(k)
    b_{X_1}(k) b_{X_2}(k)
    \\
    \times
    \Biggl\{
    \delta_{l_z0}
    \left[
      1 + \frac{1}{3}\left(\beta_1 + \beta_2\right)
      + \frac{1}{5}\beta_1\beta_2
    \right]
    \\
    + \frac{\delta_{l_z2}}{5}
    \left\{
      \frac{2}{3}\left(\beta_1 + \beta_2\right)
      + \frac{4}{7} \beta_1 \beta_2
    \right\}
    + \frac{\delta_{l_z4}}{9} \frac{8}{35} \beta_1 \beta_2
  \Biggr\},
  \label{eq:4-18}
\end{multline}
where $\beta_1 \equiv c^{(0)}_{X_1}f/b_{X_1}(k)$. The last form is
consistent with Eq.~(\ref{eq:4-10}).

\subsection{\label{subsec:PowerSpecNG} Scale-dependent bias in the
  power spectrum with non-Gaussian initial conditions }

In the presence of bias, the local-type non-Gaussianity is known to
produce the scale-dependent bias in the power spectrum on very large
scales \cite{Dalal:2007cu}. The same effect appears even in the shape
correlations \cite{Schmidt:2015xka,Akitsu:2020jvx}, which is
considered to be a second-rank ($l=2$) tensor field in our context.
Higher moments of the shape correlations are also investigated
\cite{Kogai:2020vzz}.

In our formalism, these previous findings are elegantly reproduced as
shown below. The derivation of scale-dependent bias from the
primordial non-Gaussianity in the iPT formalism of scalar fields is
already given in Ref.~\cite{Matsubara:2012nc}. We can simply
generalize the last method to the case of higher-rank tensor fields.
We consider the clustering in real space below, while the
generalization to that in redshift space is straightforward.

The primordial bispectrum $B_\mathrm{L}(\bm{k}_1,\bm{k}_2,\bm{k}_3)$
is defined by three-point correlations of linear density contrast in
Fourier space,
\begin{equation}
  \left\langle
    \delta_\mathrm{L}(\bm{k}_1)
    \delta_\mathrm{L}(\bm{k}_2)
    \delta_\mathrm{L}(\bm{k}_3)
  \right\rangle_\mathrm{c} =
  (2\pi)^3 \delta_\mathrm{D}^3(\bm{k}_1+\bm{k}_2+\bm{k}_3)
  B_\mathrm{L}(\bm{k}_1,\bm{k}_2,\bm{k}_3),
  \label{eq:4-20}
\end{equation}
where $\langle\cdots\rangle_\mathrm{c}$ denotes the three-point
cumulant, or the connected part. Applying a straightforward
generalization of the corresponding result of iPT
\cite{Matsubara:2012nc} to tensor fields, the lowest-order
contribution of the primordial bispectrum to the power spectrum in the
formalism of iPT is given by
\begin{multline}
  P^{\mathrm{NG}(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k})
  = \frac{1}{2} \Gamma^{(1)}_{X_1l_1m_1}(-\bm{k})
  \\ \times
  \int \frac{d^3p}{(2\pi)^3}
  \Gamma^{(2)}_{X_2l_2m_2}(\bm{p},\bm{k}-\bm{p})
  B_\mathrm{L}(\bm{k},-\bm{p},\bm{p}-\bm{k})
  \\ + [(X_1l_1m_1) \leftrightarrow (X_2l_2m_2)],
  \label{eq:4-21}
\end{multline}
where we implicitly assume the parity symmetry of the Universe. The
linear power spectrum with the Gaussian initial condition,
Eq.~(\ref{eq:4-2}), is added to the above in order to obtain the total
power spectrum up to the linear order of the initial power spectrum
and bispectrum. However, the Gaussian contribution does not generate
scale-dependent bias in the limit of large scales, $k\rightarrow 0$,
and thus the non-Gaussian contribution of Eq.~(\ref{eq:4-21})
dominates in that limit.

The prefactor of the integral in Eq.~(\ref{eq:4-21}) is given by
Eq.~(\ref{eq:3-202}):
\begin{equation}
  \Gamma^{(1)}_{X_1l_1m_1}(\bm{k})
  =
  \left[
    c^{(1)}_{X_1l_1}(k)
    + c^{(0)}_{X_1} \delta_{l_10}
  \right]
  Y_{l_1m_1}(\hat{\bm{k}}),
  \label{eq:4-22}
\end{equation}
where we consider the lowest-order approximation and the resummation
factor $\Pi(k)$ is just replaced by unity. In order to evaluate the
integral in Eq.~(\ref{eq:4-21}) with the spherical basis, one can
insert a unity
$\int d^3p' \delta_\mathrm{D}^3(\bm{p}+\bm{p}'-\bm{k}) = 1$ in the
integral, and re-express the delta function by a Fourier integral, and
primordial bispectrum can be also expanded by spherical harmonics. In
this way, the angular integrations are analytically performed, leaving
only the radial integrals over $p$ and $p'$.

We are interested in the scale-dependent bias from the primordial
non-Gaussianity, and instead of deriving a general expression
according to the above procedure, we directly consider a limiting case
of $k\rightarrow 0$ in Eq.~(\ref{eq:4-21}). In this case, the integral
is approximately given by
\begin{equation}
  \int \frac{d^3p}{(2\pi)^3}
  \Gamma^{(2)}_{X_2l_2m_2}(\bm{p},-\bm{p})
  B_\mathrm{L}(\bm{k},-\bm{p},\bm{p}).
  \label{eq:4-23}
\end{equation}
The second-order propagator in the integral is given by
Eq.~(\ref{eq:3-22}) with $\bm{k}_{12}=\bm{0}$, $\bm{k}_1=\bm{p}$ and
$\bm{k}_2=-\bm{p}$, and thus we have
\begin{multline}
  \Gamma^{(2)}_{X_2l_2m_2}(\bm{p},-\bm{p}) =
  c^{(2)}_{X_2l_2m_2}(\bm{p},-\bm{p})
  \\ =
  \sum_{l_2',l_2''} (-1)^{l_2''} c^{(2)l_2}_{X_2l_2'l_2''}(p,p)
  \left\{
    Y_{l_2'}(\hat{\bm{p}})\otimes Y_{l_2''}(\hat{\bm{p}})
  \right\}_{l_2m_2},
\label{eq:4-24}
\end{multline}
where Eq.~(\ref{eq:3-75}) is substituted. According to
Eqs.~(\ref{eq:3-74}), (\ref{eq:3-216}) and (\ref{eq:c-5}), we have
\begin{equation}
  \left\{
    Y_{l_2'}(\hat{\bm{p}})\otimes Y_{l_2''}(\hat{\bm{p}})
  \right\}_{l_2m_2} =
  (-1)^{l_2}\sqrt{\frac{\{l_2'\}\{l_2''\}}{4\pi}}
    \begin{pmatrix}
    l_2' & l_2'' & l_2 \\
    0 & 0 & 0
  \end{pmatrix}
  Y_{l_2m_2}(\hat{\bm{p}}).
  \label{eq:4-25}
\end{equation}

We decompose the primordial bispectrum with the spherical harmonics as
\begin{multline}
  B_\mathrm{L}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  =
  \sum_{l_1,l_2,l_3}
  Y_{l_1m_1}(\hat{\bm{k}}_1)
  Y_{l_2m_2}(\hat{\bm{k}}_2)
  Y_{l_3m_3}(\hat{\bm{k}}_3)
  \\ \times
  \left(l_1\,l_2\,l_3\right)^{m_1m_2m_3}
  B_{\mathrm{L}}^{l_1l_2l_3}(k_1,k_2,k_3).
  \label{eq:4-26}
\end{multline}
The appearance of the $3j$-symbols is due to the rotational symmetry.
The inverse relation of the above is given by
\begin{multline}
  B_{\mathrm{L}}^{l_1l_2l_3}(k_1,k_2,k_3)
  =
  \left(l_1\,l_2\,l_3\right)^{m_1m_2m_3}
  \\ \times
  \int d^2\hat{k}_1 d^2\hat{k}_2 d^2\hat{k}_3
  Y_{l_1m_1}(\hat{\bm{k}}_1)
  Y_{l_2m_2}(\hat{\bm{k}}_2)
  Y_{l_3m_3}(\hat{\bm{k}}_3)
  \\ \times
  B_\mathrm{L}(\bm{k}_1,\bm{k}_2,\bm{k}_3).
  \label{eq:4-27}
\end{multline}
For the particular bispectrum in the integrand of Eq.~(\ref{eq:4-23}),
using Eqs.~(\ref{eq:4-26}) and (\ref{eq:4-27}), we have
\begin{multline}
  B_\mathrm{L}(\bm{k},-\bm{p},\bm{p})
  =
  \frac{1}{\sqrt{4\pi}}
  \sum_{l}
  Y_{lm}(\hat{\bm{k}})
  Y_{lm}^*(\hat{\bm{p}})
  \sum_{l',l''}
  (-1)^{l'}
  \\ \times
  \sqrt{\frac{\{l'\}\{l''\}}{\{l\}}}
  \begin{pmatrix}
    l' & l'' & l \\
    0 & 0 & 0
  \end{pmatrix}
  B_{\mathrm{L}}^{l\,l'l''}(k,p,p).
  \label{eq:4-28}
\end{multline}

Putting the above equations together for the integral of
Eq.~(\ref{eq:4-23}), the angular integration over $\hat{\bm{p}}$ can
be analytically performed by using the orthonormality relation of the
spherical harmonics, Eq.~(\ref{eq:a-5}). As a result,
Eq.~(\ref{eq:4-21}) reduces to
\begin{multline}
  P^{\mathrm{NG}(l_1l_2)}_{X_1X_2m_1m_2}(\bm{k}) =
  \frac{1}{2\,(4\pi)^2}
  \frac{1}{\sqrt{\{l_2\}}}
  \left[c^{(1)}_{X_1l_1}(k) + \delta_{l0} c^{(0)}_{X_1}\right]
  \\ \times
  \sum_l \left[l_1\,l_2\,l\right]_{m_1m_2}^{\phantom{m_1m_2}m}
  Y_{lm}(\hat{\bm{k}})
  \sum_{l_1',l_1'',l_2',l_2''}
  (-1)^{l_1''+l_2''}
  \\ \times
  \sqrt{\{l_1'\}\{l_1''\}\{l_2'\}\{l_2''\}}
  \begin{pmatrix}
    l_1' & l_1'' & l_2 \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_2' & l_2'' & l_2 \\
    0 & 0 & 0
  \end{pmatrix}
  \\ \times
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)l_2}_{X_2l_2'l_2''}(p,p)
  B_\mathrm{L}^{l_2l_1'l_1''}(k,p,p)
  \\ + [(X_1l_1m_1) \leftrightarrow (X_2l_2m_2)].
  \label{eq:4-29}
\end{multline}
Comparing the above equation with Eq.~(\ref{eq:4-0-3}), the
corresponding invariant power spectrum is given by
\begin{multline}
  P^{\mathrm{NG}\,l_1l_2;l}_{X_1X_2}(k) =
  \frac{1}{2\,(4\pi)^{5/2}}
  \frac{1}{\sqrt{\{l_2\}}}
  \begin{pmatrix}
    l_1 & l_2 & l \\
    0 & 0 & 0
  \end{pmatrix}
  \\ \times
  \left[c^{(1)}_{X_1l_1}(k) + \delta_{l_10} c^{(0)}_{X_1}\right]
  \sum_{l_1',l_1'',l_2',l_2''}
  (-1)^{l_1''+l_2''}
  \sqrt{\{l_1'\}\{l_1''\}\{l_2'\}\{l_2''\}}
  \\ \times
  \begin{pmatrix}
    l_1' & l_1'' & l_2 \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_2' & l_2'' & l_2 \\
    0 & 0 & 0
  \end{pmatrix}
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)l_2}_{X_2l_2'l_2''}(p,p)
  B_\mathrm{L}^{l_2l_1'l_1''}(k,p,p)
  \\
  + [(X_1l_1) \leftrightarrow (X_2l_2)].
  \label{eq:4-29-1}
\end{multline}
This is one of the generic predictions of our theory.

The scale-dependent bias from the primordial non-Gaussianity is
conveniently derived from the cross power spectrum between the linear
density field and biased objects. In particular, we define the
lowest-order of scale-dependent bias factor by
\begin{equation}
  \Delta b_{Xlm}(\bm{k}) \equiv
  \sqrt{4\pi}\,
  \frac{P^{\mathrm{NG}(l0)}_{X\delta\,m0}(\bm{k})}{P_\mathrm{L}(k)},
  \label{eq:4-30}
\end{equation}
where $X_2=\delta$ indicates that the biased field
$F_{X_2l_2m_2}(\bm{k})$ is replaced by the mass density field
$\delta(\bm{k})$ with $l_2=m_2=0$, and thus $c^{(0)}_{X_2} = 1$ and
$c^{(n)}_{X_2l_2m_2} = 0$ for $n\geq 1$. In the simplest case of
linear theory and linear bias,
$F_{\delta\,00}(\bm{k}) = (4\pi)^{-1/2} \delta_\mathrm{L}(\bm{k})$ and
$F_{Xlm}(\bm{k}) = b_{Xlm}(\bm{k}) \delta_\mathrm{L}(\bm{k})$, we have
$P^{(l0)}_{X\delta\,m0}(\bm{k}) = (4\pi)^{-1/2} b_{Xlm}(\bm{k})
P_\mathrm{L}(k)$. This observation explains the choice of
normalization in Eq.~(\ref{eq:4-30}). Due to the rotational symmetry,
and as explicitly seen from Eq.~(\ref{eq:4-0-3}), the directional
dependence on the wavevector on the rhs of Eq.~(\ref{eq:4-30}) should
be proportional to $Y_{lm}(\hat{\bm{k}})$, thus we naturally define an
invariant bias factor $\Delta b_{Xl}(k)$ by
\begin{equation}
  \Delta b_{Xlm}(\bm{k}) =
  \Delta b_{Xl}(k)\, Y_{lm}(\hat{\bm{k}}).
  \label{eq:4-30-1}
\end{equation}
Substituting Eq.~(\ref{eq:4-0-3}) into Eq.~(\ref{eq:4-30}), the above
definition is equivalent to
\begin{equation}
  \Delta b_{Xl}(k) =
  \sqrt{4\pi} (-1)^l\sqrt{\{l\}}
  \frac{P^{\mathrm{NG}\,l0;l}_{X\delta}(k)}{P_\mathrm{L}(k)}.
  \label{eq:4-30-2}
\end{equation}
Substituting $l_1=l$, $l_2=0$, $c^{(0)}_{X_2}=1$, and
$c^{(1)}_{X_2l_2} = c^{(2)l_2}_{X_2l_2'l_2''} = 0$ into
Eq.~(\ref{eq:4-29-1}), the invariant scale dependent bias of
Eq.~(\ref{eq:4-30-2}) reduces to
\begin{multline}
  \Delta b_{Xl}(k) =
  \frac{1}{2\,(4\pi)^2P_\mathrm{L}(k)}
  \sum_{l_1',l_1'',l_2',l_2''}
  (-1)^{l_1''+l_2''}
  \\ \times
  \sqrt{\{l_1'\}\{l_2'\}\{l_1''\}\{l_2''\}}
  \begin{pmatrix}
    l_1' & l_1'' & l \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_2' & l_2'' & l \\
    0 & 0 & 0
  \end{pmatrix}
  \\ \times
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)l}_{Xl_2'l_2''}(p,p)
  B_\mathrm{L}^{l\,l_1'l_1''}(k,p,p).
  \label{eq:4-32}
\end{multline}
This is one of the generic predictions of our theory.

The above formula is applicable for arbitrary models of primordial
non-Gaussianity with a given bispectrum at the lowest order. We
consider a particular model of non-Gaussianity, which is a
generalization of the so-called local-type model
\cite{Shiraishi:2013vja,Schmidt:2015xka}:
\begin{multline}
  B_\mathrm{L}(\bm{k}_1,\bm{k}_2,\bm{k}_3) =
  \frac{2\mathcal{M}(k_3)}{\mathcal{M}(k_1)\mathcal{M}(k_2)}
  \\ \times
  \sum_l f_\mathrm{NL}^{(l)}
  \mathit{P}_l(\hat{\bm{k}}_1\cdot\hat{\bm{k}}_2)
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2) + \mathrm{cyc.}
  \label{eq:4-33}
\end{multline}
The function $\mathcal{M}(k)$ in the prefactor is given by
\begin{equation}
  \mathcal{M}(k) =
  \frac{2}{3}
  \frac{D(z)}{(1+z_*)D(z_*)}
  \frac{k^2 T(k)}{{H_0}^2 \Omega_\mathrm{m0}},
  \label{eq:4-34}
\end{equation}
where $D(z)$ is the linear growth factor with an arbitrary
normalization, $z_*$ is an arbitrary redshift at the matter-dominated
epoch, $T(k)$ is the transfer function, $H_0$ is the Hubble constant,
and $\Omega_\mathrm{m0}$ is the density parameter of total matter. The
factor $(1+z_*)D(z_*)$ does not depend on the choice of $z_*$ as long
as $z_*$ is deep in the matter-dominated epoch. Some authors choose a
normalization of the linear growth factor by $(1+z_*)D(z_*)=1$ in which
case the expression of Eq.~(\ref{eq:4-34}) has the simplest form.
Substituting Eq.~(\ref{eq:4-33}) into Eq.~(\ref{eq:4-27}), we have
\begin{multline}
  B_{\mathrm{L}}^{l_1l_2l_3}(k_1,k_2,k_3)
  =
  \frac{2\mathcal{M}(k_3)}{\mathcal{M}(k_1)\mathcal{M}(k_2)}
  P_\mathrm{L}(k_1)P_\mathrm{L}(k_2)
  \\ \times
  (4\pi)^{3/2}
  \frac{(-1)^{l_1}}{\sqrt{\{l_1\}}} f_\mathrm{NL}^{(l_1)}
  \delta_{l_1l_2} \delta_{l_30}
  + \mathrm{cyc.}
  \label{eq:4-35}
\end{multline}
We need a special form of the bispectrum in Eqs.~(\ref{eq:4-29-1}) and
(\ref{eq:4-32}), and in the above model, we have
\begin{multline}
  B_\mathrm{L}^{l_2l_1'l_1''}(k,p,p)
  \simeq
  \frac{2 P_\mathrm{L}(k)}{\mathcal{M}(k)}
  P_\mathrm{L}(p)
  (4\pi)^{3/2}
  \frac{(-1)^{l_2}}{\sqrt{\{l_2\}}} f_\mathrm{NL}^{(l_2)}
  \\ \times
  \left(
    \delta_{l_1'l_2} \delta_{l_1''0} +
    \delta_{l_1'0} \delta_{l_1''l_2}
  \right),
  \label{eq:4-36}
\end{multline}
where only two terms in which $\mathcal{M}(k)$ appeared in the
denominator are retained because the other term is negligible in the
limit of $k\rightarrow 0$.

Substituting Eq.~(\ref{eq:4-36}) into Eq.~(\ref{eq:4-29-1}), and using
Eqs.~(\ref{eq:c-8}) and (\ref{eq:c-13}), we have
\begin{multline}
  P^{\mathrm{NG}\,l_1l_2;l}_{X_1X_2}(k) =
  \frac{(-1)^l\left[1 + (-1)^{l_2}\right]}{4\pi}
  f_\mathrm{NL}^{(l_2)}
  \frac{P_\mathrm{L}(k)}{\mathcal{M}(k)}
  \begin{pmatrix}
    l_1 & l_2 & l \\
    0 & 0 & 0
  \end{pmatrix}
  \\ \times
  \left[
    c^{(1)}_{X_1l_1}(k) + \delta_{l_10} c^{(0)}_{X_1}
  \right]
  \sum_{l',l''} (-1)^{l''}
  \sqrt{\{l'\}\{l''\}}
  \\ \times
  \begin{pmatrix}
    l' & l'' & l_2 \\
    0 & 0 & 0
  \end{pmatrix}
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)l_2}_{X_2l'l''}(p,p) P_\mathrm{L}(p)
  \\ + [(X_1l_1) \leftrightarrow (X_2l_2)].
  \label{eq:4-37}
\end{multline}
Substituting $l_1=l$, $l_2=0$, $c^{(0)}_{X_2}=1$, and
$c^{(1)}_{X_2l_2} = c^{(2)l_2}_{X_2l_2'l_2''} = 0$ into the above
equation, we have
\begin{multline}
  P^{\mathrm{NG}\,l0;l}_{X\delta}(k) =
  \frac{1 + (-1)^l}{4\pi \sqrt{\{l\}}}
  \frac{P_\mathrm{L}(k)}{\mathcal{M}(k)}
  f_\mathrm{NL}^{(l)}
  \sum_{l',l''} (-1)^{l''}
  \sqrt{\{l'\}\{l''\}}
  \\ \times
  \begin{pmatrix}
    l' & l'' & l \\
    0 & 0 & 0
  \end{pmatrix}
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)l}_{Xl'l''}(p,p) P_\mathrm{L}(p).
  \label{eq:4-38}
\end{multline}
Apparently, the rhs survives only in the case of $l=\mathrm{even}$.
Substituting the above equation into Eq.~(\ref{eq:4-30-2}) in this
case, we have
\begin{multline}
  \Delta b_{Xl}(k) =
 \frac{1 + (-1)^l}{\sqrt{4\pi}}
  \frac{f_\mathrm{NL}^{(l)}}{\mathcal{M}(k)}
  \sum_{l',l''} (-1)^{l''}
  \sqrt{\{l'\}\{l''\}}
  \\ \times
  \begin{pmatrix}
    l' & l'' & l \\
    0 & 0 & 0
  \end{pmatrix}
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)l}_{Xl'l''}(p,p) P_\mathrm{L}(p).
  \label{eq:4-39}
\end{multline}
This proves that the scale-dependent bias of the rank-$l$ tensor field
is sensitive to the multipole moment $l$ of the Legendre polynomials
$P_l(\hat{\bm{k}}_1\cdot\hat{\bm{k}}_2)$ in the model of
Eq.~(\ref{eq:4-33}) for the primordial non-Gaussianity, as shown in
Ref.~\cite{Kogai:2020vzz} by quite different considerations and a
method.

As a consistency check, we can see if the known results of scalar
perturbations are derived from the general result above. In the scalar
case of $l=0$, Eq.~(\ref{eq:4-39}) reduces to
\begin{equation}
  \Delta b_{X0}(k) =
  \frac{1}{\sqrt{4\pi}}\,
  \frac{2f_\mathrm{NL}}{\mathcal{M}(k)}
  \sum_{l} \sqrt{\{l\}}
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)0}_{Xll}(p,p) P_\mathrm{L}(p),
  \label{eq:4-40}
\end{equation}
where $f_\mathrm{NL} \equiv f_\mathrm{NL}^{(0)}$ is the original
parameter of local-type non-Gaussianity. One can confirm that the last
equation is consistent with the previously known results for the
scale-dependent bias in the halo models. In fact, the second-order
renormalized bias function of a simple halo model is given by
\cite{Matsubara:2012nc}
\begin{equation}
  c^{(2)}_\mathrm{h}(\bm{k}_1,\bm{k}_2) =
  \frac{\delta_\mathrm{c}b^\mathrm{L}_1}{\sigma^2}
  W(k_1R) W(k_2R),
  \label{eq:4-41}
\end{equation}
where
\begin{equation}
  \sigma^2 = \int \frac{p^2dp}{2\pi^2}
   W^2(pR) P_\mathrm{L}(p),
  \label{eq:4-42}
\end{equation}
and $b^\mathrm{L}_1$ is a Lagrangian bias parameter of linear order.
Eq.~(\ref{eq:4-41}) is a result for the simplest case with a
high-peaks (or high-mass) limit of halos in the Press-Schechter mass
function (see Ref.~\cite{Matsubara:2012nc} for results of
other extended halo models). The Lagrangian bias parameter $b^\mathrm{L}_1$ is
related to the Eulerian bias parameter $b_1$ by
$b^\mathrm{L}_1 = b_1 - 1$. Comparing the above function with
Eq.~(\ref{eq:3-73}) or Eq.~(\ref{eq:3-75}), and noting
$c^{(2)}_{\mathrm{h}00}(k_1,k_2) = (4\pi)^{-1/2}
c^{(2)}_{\mathrm{h}}(k_1,k_2)$, the invariant coefficient in this case
is given by
\begin{equation}
  c^{(2)0}_{\mathrm{h}\,l_1l_2}(k_1,k_2) =
  \sqrt{4\pi}\, \delta_{l_10}\delta_{l_20}
  \frac{\delta_\mathrm{c}b^\mathrm{L}_1}{\sigma^2}
  W(k_1R) W(k_2R).
  \label{eq:4-43}
\end{equation}
Substituting the above expression into Eq.~(\ref{eq:4-40}), we have
\begin{equation}
  \Delta b_{\mathrm{h}0}(k) =
  \frac{2f_\mathrm{NL}\delta_\mathrm{c} b^\mathrm{L}_1}{\mathcal{M}(k)}.
  \label{eq:4-44}
\end{equation}
Due to the normalization of the spherical basis for the scalar
component, Eqs.~(\ref{eq:2-8}) and (\ref{eq:2-19}), the
scale-dependent bias of the halo number density is given by
$\Delta b_\mathrm{h}(k) = 4\pi\, \Delta b_{\mathrm{h}00}(k)
\mathsf{Y}^{(0)} = \sqrt{4\pi}\, \Delta
b_{\mathrm{h}0}(k)\mathsf{Y}^{(0)} = \Delta b_{\mathrm{h}0}$, and thus
we have
\begin{equation}
  \Delta b_\mathrm{h}(k)
  = \frac{2(b_1-1)f_\mathrm{NL}\delta_\mathrm{c}}{\mathcal{M}(k)}.
  \label{eq:4-45}
\end{equation}
This exactly reproduces a well-known result of the scale-dependent
bias in the simplest halo model \cite{Dalal:2007cu}.

\subsection{\label{subsec:Bispectrum}
  The bispectrum
}

\subsubsection{\label{subsubsec:InvBispec}
  The invariant bispectrum
}

For yet another example of applications of our formalism, we consider
the bispectrum of the tensor field. For an illustrative purpose, we only
consider the bispectrum in real space below, while generalizing the
results to those in redshift space is straightforward with
similar methods employed for the power spectrum in
Sec.~\ref{subsec:PowerSpec}.

The bispectrum
$B^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2,\bm{k}_3)$
of irreducible components of tensor field $F_{Xlm}(\bm{k})$ may be
defined by
\begin{multline}
  \left\langle
    F_{X_1l_1m_1}(\bm{k}_1) F_{X_2l_2m_2}(\bm{k}_2)
    F_{X_3l_3m_3}(\bm{k}_3)
  \right\rangle_\mathrm{c}
  \\
  =
  (2\pi)^3\delta_\mathrm{D}^3(\bm{k}_1+\bm{k}_2+\bm{k}_3)
  B^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2,\bm{k}_3),
  \label{eq:4-60}
\end{multline}
as a simple generalization of the definition of the power spectrum of
Eq.~(\ref{eq:4-0-1}). We consider generally the cross bispectrum among
three fields, $X_1$, $X_2$ and $X_3$ with irreducible components
$l_1$, $l_2$ and $l_3$, respectively. The auto bispectrum is
straightforwardly obtained by putting $X_1=X_2=X_3$.

The bispectrum
$B^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2,\bm{k}_3)$
defined above depends on the coordinates system. Generalizing the
method of constructing the invariant power spectrum, one can also
construct the invariant bispectrum as explicitly shown below. Due to
the presence of the delta function on the lhs of Eq.~(\ref{eq:4-60}),
the third argument $\bm{k}_3$ in the bispectrum can be replaced by
$-\bm{k}_1-\bm{k}_2$, and regarded as a function of only $\bm{k}_1$
and $\bm{k}_2$. However, the resulting expression loses apparent
symmetry of permutating the subscripts $(1,2,3)$ of $\bm{k}_i$, $l_i$
and $m_i$. As is usually the case in the analytic predictions in
perturbation theory, we can decompose the symmetric bispectrum into a
sum of asymmetric components as
\begin{equation}
  B^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  =
    \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2)
    + \mathrm{cyc.}
  \label{eq:4-61}
\end{equation}
Although the decomposition into asymmetric components is not unique,
it is always possible to specify the originally symmetric bispectrum by
giving the asymmetric bispectra, and the predictions of perturbation
theory are usually given in the above form.

The directional dependence of the asymmetric bispectrum on the
wavevectors can be expanded by spherical harmonics as
\begin{multline}
  \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3m_1m_2m_3}(\bm{k}_1,\bm{k}_2)
  =
  \sum_{l_1',l_2'}
  \tilde{B}^{l_1l_2l_3;l_1'l_2'}_{X_1X_2X_3m_1m_2m_3;m_1'm_2'}(k_1,k_2)\,
  \\ \times
  Y^*_{l_1'm_1'}(\hat{\bm{k}}_1) Y^*_{l_2'm_2'}(\hat{\bm{k}}_2).
  \label{eq:4-63}
\end{multline}
Due to the rotational transformations of Eqs.~(\ref{eq:3-38}) and
(\ref{eq:2-14-2}), and rotational invariance of the delta function in
the definition of the bispectrum, Eq.~(\ref{eq:4-60}), the rotational
transformation of the expansion coefficients is given by
\begin{multline}
  \tilde{B}^{l_1l_2l_3;l_1'l_2'}_{X_1X_2X_3m_1m_2m_3;m_1'm_2'}(k_1,k_2)
  \\
  \rightarrow
  \tilde{B}^{l_1l_2l_3;l_1'l_2'}_{X_1X_2X_3m_1''m_2''m_3'';m_1'''m_2'''}(k_1,k_2)
  D^{m_1''}_{(l_1)m_1}(R)
  \\ \times
  D^{m_2''}_{(l_2)m_2}(R)
  D^{m_3''}_{(l_2)m_3}(R) D^{m_1'''}_{(l)m_1'}(R) D^{m_2'''}_{(l)m_2'}(R).
  \label{eq:4-64}
\end{multline}
For the statistically isotropic Universe, the functional form of the
bispectrum should not depend on the choice of coordinates system.
Following exactly the same procedure as in
Sec.~\ref{subsec:RenBiasFn}, the bispectrum in the statistically
isotropic Universe is shown to have a form,
\begin{multline}
  \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3m_1m_2m_3}(\bm{k}_1,\bm{k}_2) =
  (-1)^{l_1+l_2+l_3} \sqrt{\{l_1\}\{l_2\}\{l_3\}}
  \\ \times
  \sum_{\substack{L,L'\\l_1',l_2'}}
   (-i)^{l_1'+l_2'} (-1)^{L}
  \sqrt{\{L\}\{l_1'\}\{l_2'\}}
  \left(l_1\,l_2\,L\right)_{m_1m_2}^{\phantom{m_1m_2}M}
  \left(L\,l_3\,L'\right)_{Mm_3}^{\phantom{Mm_3}M'}
  \\ \times
  \left\{
    Y_{l_1'}(\hat{\bm{k}}_1) \otimes Y_{l_2'}(\hat{\bm{k}}_2)
  \right\}_{L'M'}
  \tilde{B}^{l_1l_2l_3;l_1'l_2';LL'}_{X_1X_2X_3}(k_1,k_2),
    \label{eq:4-65}
\end{multline}
where
\begin{multline}
  \tilde{B}^{l_1l_2l_3;l_1'l_2';LL'}_{X_1X_2X_3}(k_1,k_2) =
  \frac{(-i)^{l_1'+l_2'}(-1)^{L+L'} \sqrt{\{L\}\{L'\}}}
  {\sqrt{\{l_1\}\{l_2\}\{l_3\}\{l_1'\}\{l_2'\}}}
  \\ \times
    \left(l_1\,l_2\,L\right)^{m_1m_2}_{\phantom{m_1m_2}M}
  \left(L\,l_3\,L'\right)^{Mm_3}_{\phantom{Mm_3}M'}
  \left(L'\,l_1'\,l_2'\right)^{M'm_1'm_2'}
  \\ \times
  \tilde{B}^{l_1l_2l_3;l_1'l_2'}_{X_1X_2X_3m_1m_2m_3;m_1'm_2'}(k_1,k_2)
  \label{eq:4-66}
\end{multline}
is the invariant bispectrum.

According to Eqs.~(\ref{eq:3-32}), (\ref{eq:4-60}) and the symmetry
of the $3j$-symbol, Eq.~(\ref{eq:c-2}), the symmetry of complex
conjugate is shown to be given by
\begin{equation}
  B^{l_1l_2l_3;l_1'l_2';LL'\,*}_{X_1X_2X_3}(k) =
  B^{l_1l_2l_3;l_1'l_2';LL'}_{X_1X_2X_3}(k),
  \label{eq:4-66-1}
\end{equation}
and thus the invariant bispectrum is a real function. Additionally,
the parity transformation of the bispectrum is given by
\begin{multline}
  \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3m_1m_2m_3}(\bm{k}_1,\bm{k}_2) \rightarrow
  (-1)^{s_{X_1}+s_{X_2}+s_{X_3}+l_1+l_2+l_3}
  \\ \times
  \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3m_1m_2m_3}(-\bm{k}_1,-\bm{k}_2),
  \label{eq:4-67}
\end{multline}
because of Eqs.~(\ref{eq:3-42}) and (\ref{eq:4-60}). Accordingly, the
parity transformation of the invariant bispectrum is given by
\begin{multline}
  \tilde{B}^{l_1l_2l_3;l_1'l_2'}_{X_1X_2X_3}(k_1,k_2) \rightarrow
  (-1)^{s_{X_1}+s_{X_2}+s_{X_3}+l_1+l_2+l_3+l_1'+l_2'}
  \\ \times
  \tilde{B}^{l_1l_2l_3;l_1'l_2'}_{X_1X_2X_3}(k_1,k_2).
  \label{eq:4-68}
\end{multline}
In the Universe with the parity symmetry, the bispectrum is invariant
under the parity transformation, we thus have
\begin{equation}
  s_{X_1} + s_{X_2} + s_{X_3} + l_1 + l_2 + l_3 + l_1' + l_2'
  = \mathrm{even}.
  \label{eq:4-69}
\end{equation}

\subsubsection{\label{subsubsec:TreeBispec}
  The lowest-order bispectrum
}

We consider the bispectrum of the lowest-order approximation or the
tree-level bispectrum. The formal expression of the lowest-order
bispectrum in terms of propagators is given by
\begin{multline}
  \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2)
  = \Gamma^{(1)}_{X_1l_1m_1}(\bm{k}_1) \Gamma^{(1)}_{X_2l_2m_2}(\bm{k}_2)
  \\ \times
  \Gamma^{(2)}_{X_3l_3m_3}(-\bm{k}_1,-\bm{k}_2)
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2).
  \label{eq:4-71}
\end{multline}
We do not use the higher-order resummation factor and substitute
$\Pi(k) = 1$. Substituting Eqs.~(\ref{eq:3-105}) and (\ref{eq:3-108}),
the above equation is represented in terms of invariant functions as
\begin{multline}
  \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2)
  =
  \hat{\Gamma}^{(1)}_{X_1l_1}(k_1)
  \hat{\Gamma}^{(1)}_{X_2l_2}(k_2)
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2)
  \\ \times
  \sum_{l_1',l_2',l_1'',l_2''}
  (-1)^{l_1''+l_2''}
  \left[l_1\,l_1'\,l_1''\right]_{m_1}^{\phantom{m_1}m_1'm_1''}
  \left[l_2\,l_2'\,l_2''\right]_{m_2}^{\phantom{m_2}m_2'm_2''}
  \\ \times
  Y_{l_1'm_1'}(\hat{\bm{k}}_1) Y_{l_2'm_2'}(\hat{\bm{k}}_2)
  \left(l_1''\,l_2''\,l_3\right)_{m_1''m_2''m_3}
  \hat{\Gamma}^{(2)\,l_3}_{X_3l_1''l_2''}(k_1,k_2).
  \label{eq:4-72}
\end{multline}
Substituting Eqs.~(\ref{eq:3-202}) and (\ref{eq:3-217}), the above
expression is given by invariant forms of renormalized bias functions,
$c^{(0)}_X$, $c^{(1)}_{X\,l}(k)$ and $c^{(2)\,l}_{Xl_1l_2}(k_1,k_2)$.

The sum of the products of three $3j$-symbols appeared in the above
expression is represented by a $9j$-symbol due to a formula
of Eq.~(\ref{eq:c-41}) in Appendix~\ref{app:3njSymbols}, or equivalently,
\begin{multline}
  (-1)^{l_1''+l_2''+l_3}
  \left(l_1\,l_1'\,l_1''\right)_{m_1}^{\phantom{m_1}m_1'm_1''}
  \left(l_2\,l_2'\,l_2''\right)_{m_2}^{\phantom{m_2}m_2'm_2''}
  \left(l_1''\,l_2''\,l_3\right)_{m_1''m_2''m_3}
  \\
  = \sum_{L,L'} \{L\}\{L'\}
  \left(l_1\,l_2\,L\right)_{m_1m_2}^{\phantom{m_1m_2}M}
  \left(L\,l_3\,L'\right)_{Mm_3}^{\phantom{Mm_2}M'}
  \\ \times
  \left(L'\,l_1'\,l_2'\right)_{M'}^{\phantom{M'}m_1'm_2'}
  \begin{Bmatrix}
    l_1 & l_1' & l_1'' \\
    l_2 & l_2' & l_2'' \\
    L & L' & l_3
  \end{Bmatrix}.
  \label{eq:4-73}
\end{multline}
Consequently, Eq.~(\ref{eq:4-72}) reduces to an alternative
expression,
\begin{multline}
  \tilde{B}^{(l_1l_2l_3)}_{X_1X_2X_3\,m_1m_2m_3}(\bm{k}_1,\bm{k}_2)
  =
  \frac{(-1)^{l_3}}{4\pi}
  \sqrt{\{l_1\}\{l_2\}}
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2)
  \\ \times
  \hat{\Gamma}^{(1)}_{X_1l_1}(k_1)
  \hat{\Gamma}^{(1)}_{X_2l_2}(k_2)
  \sum_{\substack{l_1',l_2'\\l_1'',l_2''}}
  \sqrt{\{l_1'\}\{l_2'\}\{l_1''\}\{l_2''\}}
  \\ \times
  \begin{pmatrix}
    l_1 & l_1' & l_1'' \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_2 & l_2' & l_2'' \\
    0 & 0 & 0
  \end{pmatrix}
  \hat{\Gamma}^{(2)\,l_3}_{X_3l_1''l_2''}(k_1,k_2)
  \\ \times
  \sum_{L,L'} \{L\} (-1)^{L'} \sqrt{\{L'\}}
  \left(l_1\,l_2\,L\right)_{m_1m_2}^{\phantom{m_1m_2}M}
  \left(L\,l_3\,L'\right)_{Mm_3}^{\phantom{Mm_3}M'}
  \\ \times
  \begin{Bmatrix}
    l_1 & l_1' & l_1'' \\
    l_2 & l_2' & l_2'' \\
    L & L' & l_3
  \end{Bmatrix}
  \left\{
    Y_{l_1'}(\hat{\bm{k}}_1) \otimes Y_{l_2'}(\hat{\bm{k}}_2)
  \right\}_{L'M'}.
  \label{eq:4-74}
\end{multline}
Comparing the above equation with Eq.~(\ref{eq:4-65}), the invariant
bispectrum is given by
\begin{multline}
  \tilde{B}^{l_1l_2l_3;l_1'l_2';LL'}_{X_1X_2X_3}(k_1,k_2)
  =
  \frac{(-i)^{l_1'+l_2'}(-1)^{L+L'} \sqrt{\{L\}\{L'\}}}{4\pi\sqrt{\{l_3\}}}
  \\ \times
  \hat{\Gamma}^{(1)}_{X_1l_1}(k_1)
  \hat{\Gamma}^{(1)}_{X_2l_2}(k_2)
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2)
  \sum_{l_1'',l_2''} (-1)^{l_1''+l_2''}
  \sqrt{\{l_1''\}\{l_2''\}}
  \\ \times
  \begin{pmatrix}
    l_1 & l_1' & l_1'' \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{pmatrix}
    l_2 & l_2' & l_2'' \\
    0 & 0 & 0
  \end{pmatrix}
  \begin{Bmatrix}
    l_1 & l_1' & l_1'' \\
    l_2 & l_2' & l_2'' \\
    L & L' & l_3
  \end{Bmatrix}
  \hat{\Gamma}^{(2)\,l_3}_{X_3l_1''l_2''}(k_1,k_2).
  \label{eq:4-75}
\end{multline}
This is one of the generic predictions of our theory.

As a consistency check, let us see if the bispectrum of the scalarly
biased field in the lowest-order perturbation theory can be correctly
reproduced. Considering the case of $l_1=l_2=l_3=m_1=m_2=m_3=0$ and
$X_1=X_2=X_3\equiv X$ in Eq.~(\ref{eq:4-72}), and using
Eqs.~(\ref{eq:c-8}) and (\ref{eq:c-13}), we straightforwardly derive
\begin{multline}
  \tilde{B}^{(000)}_{XXX\,000}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  =
  \frac{1}{(4\pi)^2}
  \sum_l (-1)^l \sqrt{\{l\}}\,
  \mathit{P}_l(\hat{\bm{k}}_1\cdot\hat{\bm{k}}_2)
  \\ \times
  \hat{\Gamma}^{(1)}_{X0}(k_1) \hat{\Gamma}^{(1)}_{X0}(k_2)
  \hat{\Gamma}^{(2)\,0}_{Xll}(k_1,k_2)
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2).
  \label{eq:4-76}
\end{multline}
The same result is also derived from Eqs.~(\ref{eq:4-74}) and
(\ref{eq:4-75}) with Eq.~(\ref{eq:4-9}). We substitute
Eqs.~(\ref{eq:3-202}) and (\ref{eq:3-217}) in the above equation, and
straightforward calculations derive the scalar bispectrum as
\begin{multline}
  B_X(\bm{k}_1,\bm{k}_2,\bm{k}_3) = (4\pi)^{3/2}
  B^{(000)}_{XXX\,000}(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  \\
  =
    b_1(k_1) b_1(k_2)
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2)
  \Biggl\{
  b^\mathrm{L}_2(\bm{k}_1,\bm{k}_2)
  + b_1(k_1) +  b_1(k_2)
  \\
    - \frac{4}{7}
    + \left[
      \frac{k_2}{k_1} b_1(k_1) +
      \frac{k_1}{k_2} b_1(k_2)
    \right]
    \frac{\bm{k}_1\cdot\bm{k}_2}{k_1k_2}
    \\
    + \frac{4}{7}
    \left(\frac{\bm{k}_1\cdot\bm{k}_2}{k_1k_2}\right)^2
    \Biggr\}
    + \mathrm{cyc.},
  \label{eq:4-77}
\end{multline}
where $b_1(k) \equiv c^{(0)}_X + c^{(1)}_{X0}(k)$
corresponds to the Eulerian linear bias and  is
the same as $b_X(k)$ defined in Eq.~(\ref{eq:4-5-0}), and
\begin{multline}
  b^\mathrm{L}_2(\bm{k}_1,\bm{k}_2)
  \equiv
    c^{(2)}_X(\bm{k}_1,\bm{k}_2)
    = 4\pi\,c^{(2)}_{X00}(\bm{k}_1,\bm{k}_2) \mathsf{Y}^{(0)}
   \\
   =
    \frac{1}{\sqrt{4\pi}} \sum_l (-1)^l \sqrt{\{l\}}\,
    \mathit{P}_l(\hat{\bm{k}}_1\cdot\hat{\bm{k}}_2)
    c^{(2)\,0}_{Xll}(k_1,k_2)
  \label{eq:4-78}
\end{multline}
corresponds to the Lagrangian (nonlocal) bias function of second order
\cite{Matsubara:2011ck}. More commonly known form of the bispectrum in
the lowest-order perturbation theory is represented by the Eulerian
bias parameters $b_1$, $b_2$, $b_{s^2}$
\cite{Matarrese:1997sk,McDonald:2009dh,Sheth:2012fc,Baldauf:2012hs}.
It is natural to take a normalization
$\langle F^\mathrm{L}_X\rangle = 1$ [c.f., Eq.~(\ref{eq:3-1})] and we
have $c^{(0)}_X=1$ in this scalar case. On the one hand, in the
special case of local bias, in which $b_1$ and $b^\mathrm{L}_2$ are
constants and do not depend on wavevectors, Eq.~(\ref{eq:4-77})
reduces to
\begin{multline}
  B_X(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  =
  {b_1}^2
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2)
  \\ \times
  \left[
  b^\mathrm{L}_2 + 2b_1
    - \frac{4}{7}
    + b_1
    \left(
      \frac{k_2}{k_1} + \frac{k_1}{k_2}
      \right)
    \frac{\bm{k}_1\cdot\bm{k}_2}{k_1k_2}
    + \frac{4}{7}
    \left(\frac{\bm{k}_1\cdot\bm{k}_2}{k_1k_2}\right)^2
    \right]
    \\
    + \mathrm{cyc.}
  \label{eq:4-79}
\end{multline}
On the other hand, the prediction of scalar bispectrum in the standard
(Eulerian) perturbation theory is given by \cite{Baldauf:2012hs},
\begin{multline}
  B_X(\bm{k}_1,\bm{k}_2,\bm{k}_3)
  =
  {b_1}^2
  P_\mathrm{L}(k_1) P_\mathrm{L}(k_2)
  \\ \times
  \left\{
    b_2 + 2b_{s^2}
    \left[
      \left(\frac{\bm{k}_1\cdot\bm{k}_2}{k_1k_2}\right)^2
      - \frac{1}{3}
    \right]
    + b_1 F_2(\bm{k}_1,\bm{k}_2)
    \right\}
    \\
    + \mathrm{cyc.},
  \label{eq:4-80}
\end{multline}
where
\begin{equation}
  F_2(\bm{k}_1,\bm{k}_2) =
  \frac{10}{7} +
  \left(
    \frac{k_2}{k_1} + \frac{k_1}{k_2}
  \right)
  \frac{\bm{k}_1\cdot\bm{k}_2}{k_1k_2}
  + \frac{4}{7}
  \left(\frac{\bm{k}_1\cdot\bm{k}_2}{k_1k_2}\right)^2
  \label{eq:4-81}
\end{equation}
is the second-order kernel\footnote{Our choice of the normalizations
  for kernel functions $F_n$ is different from those in many previous
  references, where our $F_n$ corresponds to $n!F_n$ in the latter.
  We believe our choice simplifies the expressions of equations in
  general \cite{Matsubara:2011ck}.} of the Eulerian perturbation
theory \cite{Bernardeau:2001qr}. Comparing the coefficients of
Eqs.~(\ref{eq:4-79}) and (\ref{eq:4-80}), we find the relations
between Lagrangian local bias parameters and Eulerian local bias
parameters as
\begin{equation}
  b_1 =  1 + b^\mathrm{L}_1, \quad
  b_2 =  b^\mathrm{L}_2 - \frac{8}{21} b^\mathrm{L}_1, \quad
  b_{s^2} =  - \frac{2}{7} b^\mathrm{L}_1.
    \label{eq:4-82}
\end{equation}
The above relations exactly agree with those which are found in the
literature \cite{Sheth:2012fc,Baldauf:2012hs}. It is also possible to include the
nonlocal Lagrangian bias $b^\mathrm{L}_{s^2}$ in
$b^\mathrm{L}_2(\bm{k}_1,\bm{k}_2)$, and derive consistent results
with those in the literature \cite{Sheth:2012fc,Baldauf:2012hs}.

\section{\label{sec:Semilocal}
  Semi-local models of bias for tensor fields
}

So far the formulation is general enough, and we have not assumed any
model of bias for the tensor fields so far. The bias model is
naturally specified in Lagrangian space through the renormalized bias
functions $c^{(n)}_{Xlm}$ defined by Eq.~(\ref{eq:3-18}), which is
evaluated once a model of bias $F^\mathrm{L}_{Xlm}(\bm{q})$ is
analytically given as a functional of the linear density field
$\delta_\mathrm{L}(\bm{q})$. In this section, we consider a category
of bias models which are commonly adopted in models of cosmological
structure formation. We call this category ``the semi-local models''
of bias.

The concept of the semi-local models is somehow similar to the
EFTofLSS approach in perturbation theory of biased tracers
\cite{Mirbabayi:2014zca,Desjacques:2016bnm}, where the biased field is
given by a finite set of semi-local operators made of spatial and
temporal derivatives of the Newtonian potential in the perturbative
expansions order by order. However, while the EFTofLSS approach is
based on phenomenologically perturbative expansions of the biased
field, our iPT approach does not assume the underlying operators are
perturbative, because we essentially use orthogonal expansions instead
of Taylor expansions to include the fully nonlinear biasing into the
cosmological perturbation theory through the renormalized bias
functions.

This category of semi-local models of biasing is defined so that the
field $F^\mathrm{L}_{Xlm}(\bm{q})$ is given by functions (instead of
functionals) of the spatial derivatives of the gravitational potential at
the same position $\bm{q}$, smoothed with (generally, multiple numbers
of) window functions $W^{(a)}(k)$:
\begin{equation}
  \chi^{(a)}_{i_1i_2\cdots i_{L_a}}(\bm{q}) =
  \partial_{i_1}\partial_{i_2}\cdots\partial_{i_{L_a}}
  \psi^{(a)}(\bm{q}),
  \label{eq:5-1}
\end{equation}
where
\begin{equation}
  \psi^{(a)}(\bm{q}) =
  \int \frac{d^3k}{(2\pi)^3} e^{i\bm{k}\cdot\bm{q}}
  \delta_\mathrm{L}(\bm{k})\, \frac{W^{(a)}(k)}{(ik)^{L_a}}
  \label{eq:5-2}
\end{equation}
is the smoothed linear density field with an isotropic window function
$(ik)^{-L_a} W^{(a)}(k)$. The label ``$a$'' distinguishes different
kinds of the linear tensor field with a particular rank $L_a$, i.e., the
label $a$ uniquely specifies the rank $L_a$ and the form of the window
function $W^{(a)}(k)$. The window function usually contains a
parameter of smoothing radius $R_a$, which can take a different value
for each window function $W^{(a)}$.
Typical window functions include the Gaussian window,
$W^{(a)}(k) = \exp(-k^2{R_a}^2/2)$, and the top-hat window,
$W^{(a)}(k) = 3j_1(kR_a)/(kR_a)$. When only a single linear tensor
field of a fixed window function is considered, one can omit the label
$a$ in the above. We include the possibility of using multiple numbers
of fields and window functions in the general formulation below.

For a concrete example of biasing using multiple numbers of fields and
window functions, see Ref.~\cite{Matsubara:2016wth}. For example, the
second derivatives of the (normalized and smoothed) linear
gravitational potential $\varphi$ correspond to
$\varphi = \Laplace^{-1} \delta_R = \psi^{(a)}$ with a rank of
$L_a=2$, and
$\chi^{(a)}_{ij} = \partial_i\partial_j\Laplace^{-1}\delta_R$ where
$\delta_R$ is a smoothed linear density field with a smoothing window
function $W^{(a)}(k) = W(kR)$.

The linear density contrast $\delta_\mathrm{L}$ is a real function in
configuration space, and thus we have
$\delta_\mathrm{L}^{*}(\bm{k}) = \delta_\mathrm{L}(-\bm{k})$ as shown
from Eq.~(\ref{eq:3-11a}). Thus we see the function
$\psi^{(a)}(\bm{q})$ of Eq.~(\ref{eq:5-2}) is a real function when
$L_a=\mathrm{even}$, and a pure imaginary function when
$L_a=\mathrm{odd}$, and so as the function
$\chi^{(a)}_{i_1\cdots i_{L_a}}(\bm{q})$ of Eq.~(\ref{eq:5-1}).

In the semi-local models of bias, the field value
$F^\mathrm{L}_{Xlm}(\bm{q})$ at a particular position is given by a
function of the field values $\chi^{(a)}_{i_1\cdots i_l}(\bm{q})$ at
the same position in Lagrangian space. This function is common among
all the positions. Therefore, without loss of generality, one can
consider the particular position at the origin, $\bm{q}=\bm{0}$, to
describe the relation in the semi-local model of bias, and thus the
field $F^\mathrm{L}_{Xlm}$ is a function of
\begin{equation}
  \chi^{(a)}_{i_1\cdots i_{L_a}} =
  \int \frac{d^3k}{(2\pi)^3}\,
  \hat{k}_{i_1}\cdots\hat{k}_{i_{L_a}}\,
  \delta_\mathrm{L}(\bm{k}) W^{(a)}(k).
  \label{eq:5-3}
\end{equation}
The tensor field $F^\mathrm{L}_{Xlm}$ can depend on multiple of local
tensors of various ranks, $\chi^{(a_0)}$, $\chi^{(a_1)}_i$,
$\chi^{(a_2)}_{ij}$, $\chi^{(a_3)}_{ijk}$, etc. A local tensor is
decomposed by an irreducible spherical basis according to the
the procedure explained in Sec.~\ref{sec:SphericalBasis}. The tensor of
Eq.~(\ref{eq:5-3}) is decomposed into traceless tensor of rank
$L_a,L_a-2,\ldots$, and each component of traceless tensor is
decomposed by a spherical basis, as in the
Eqs.~(\ref{eq:2-23})--(\ref{eq:2-24b}). The decomposition is given by
\begin{equation}
  \chi^{(a)}_{i_1\cdots i_l}
  = \chi^{(a,l)}_{i_1i_2\cdots i_l} +
  \frac{l(l-1)}{2(2l-1)}
  \delta_{(i_1i_2} \chi^{(a,l-2)}_{i_3\cdots i_l)}
  + \cdots,
  \label{eq:5-4}
\end{equation}
where we simply denote $L_a=l$ in the above, and
$\chi^{(a,l-2)}_{i_3\cdots i_l}$ is the traceless part of the first
trace $\chi^{(a)}_{jji_3\cdots i_l}$ and so forth. The decomposed
traceless tensors are further decomposed by spherical basis as
\begin{equation}
  \chi^{(a,l)}_{i_1i_2\cdots i_l}
  = i^l \alpha_l\, \chi^{(a[l])}_{lm} \mathsf{Y}^{(m)}_{i_1i_2\cdots i_l}, \ \
  \chi^{(a,l-2)}_{i_3\cdots i_l}
  = i^{l-2} \alpha_{l-2}\, \chi^{(a[l])}_{l-2,m}
  \mathsf{Y}^{(m)}_{i_3\cdots i_l},
  \label{eq:5-4-1}
\end{equation}
and so forth.
In our simplified notation with $l=L_a$, the rank of the linear tensor
field $a$ is not obvious if we write, e.g., $\chi^{(a)}_{l-2,m}$ for
the first trace of the original tensor, and thus we instead employ a
notations $\chi^{(a[l])}_{lm}$, $\chi^{(a[l])}_{l-2,m}$, etc.~to
remind the rank of the original field $a$ is $l$ in the lower-rank
trace parts of the tensor. Namely,
\begin{equation}
  \chi^{(a[l])}_{lm}
  = \left.\chi^{(a)}_{L_am}\right|_{L_a=l}, \quad
  \chi^{(a[l])}_{l-2,m}
  = \left.\chi^{(a)}_{L_a-2,m}\right|_{L_a=l},
  \label{eq:5-4-2}
\end{equation}
and so forth.

The spherical tensors are given by
\begin{align}
  \chi^{(a[l])}_{lm}
  &= (-i)^l
  \chi^{(a)}_{i_1\cdots i_l}
  \mathsf{Y}^{(m)*}_{i_1\cdots i_l}
    \nonumber\\
    &=
    (-i)^l
    \int \frac{d^3k}{(2\pi)^3}
    \delta_\mathrm{L}(\bm{k})
    Y_{lm}(\hat{\bm{k}}) W^{(a)}(k),
    \label{eq:5-5a}\\
  \chi^{(a[l])}_{l-2,m}
  &= (-i)^{l-2}
  \chi^{(a)}_{jji_3\cdots i_l}
    \mathsf{Y}^{(m)*}_{i_3\cdots i_l}
    \nonumber\\
  &= -(-i)^l
    \int \frac{d^3k}{(2\pi)^3}
    \delta_\mathrm{L}(\bm{k})
    Y_{l-2,m}(\hat{\bm{k}})
    W^{(a)}(k),
    \label{eq:5-5b}
\end{align}
and, in general,
\begin{align}
  \chi^{(a[l])}_{l-2p,m}
  &= (-i)^{l-2p}
    \chi^{(a)}_{j_1j_1\cdots j_pj_pi_{2p+1}\cdots i_l}
    \mathsf{Y}^{(m)*}_{i_{2p+1}\cdots i_l}
    \nonumber\\
  &= (-1)^p (-i)^l
    \int \frac{d^3k}{(2\pi)^3}
    \delta_\mathrm{L}(\bm{k})
    Y_{l-2p,m}(\hat{\bm{k}})
    W^{(a)}(k),
    \label{eq:5-5-1}
\end{align}
where $p=0,1,\cdots,[l/2]$. Changing the labels of ranks by
$l \rightarrow L$ and $l-2p \rightarrow l$, Eq.~(\ref{eq:5-5-1}) is
equivalently represented by
\begin{equation}
  \chi^{(a[L])}_{lm}
  = (-i)^l
  \int \frac{d^3k}{(2\pi)^3}
  \delta_\mathrm{L}(\bm{k})
  Y_{lm}(\hat{\bm{k}})
  W^{(a)}(k),
  \label{eq:5-5-2}
\end{equation}
where $l=L,L-2$, \ldots, (0 or 1) for a linear tensor field $a$ of
rank $L$. The smallest value of $l$ in the last equation is 0 if $L$
is even, and 1 if $L$ is odd. When the rank $L_a$ of the original
tensor $\chi^{(a)}_{i_1\cdots i_{L_a}}$ is obvious, one can employ a
simplified notation $\chi^{(a)}_{lm}$ instead of
$\chi^{(a[L])}_{lm}$, where $l = L, L-2,\ldots$

For a simple example, we consider the second derivatives of the
normalized potential $\varphi$, smoothed with a window function
$W(kR)$. In this case, we have
\begin{equation}
  \chi^{(\varphi)}_{ij} = \partial_i\partial_j\varphi
  = \int \frac{d^3k}{(2\pi)^3}\, \hat{k}_i \hat{k}_j
  \delta_\mathrm{L}(\bm{k}) W(kR).
  \label{eq:5-8-0}
\end{equation}
The corresponding window function is $W^{(\varphi)}(k) = W(kR)$. The
second-rank tensor is decomposed into irreducible components as
\begin{align}
  \chi^{(\varphi)}_{ij}
  &=
  \left(
    \partial_i\partial_j - \frac{\delta_{ij}}{3} \Laplace
  \right)
  \varphi +
    \frac{\delta_{ij}}{3} \Laplace \varphi
    \nonumber \\
  &=
    -\frac{8\pi}{15} \chi^{(\varphi)}_{2m}\, \mathsf{Y}^{(m)}_{ij} +
    4\pi \frac{\delta_{ij}}{3} \chi^{(\varphi)}_{00}\, \mathsf{Y}^{(0)},
  \label{eq:5-8-1}
\end{align}
where, corresponding to Eqs.~(\ref{eq:5-5a}) and (\ref{eq:5-5b}), we
have
\begin{align}
  \chi^{(\varphi[2])}_{2m}
  &=
  - \int \frac{d^3k}{(2\pi)^3}
  \delta_\mathrm{L}(\bm{k})
    Y_{2m}(\hat{\bm{k}})
    W(kR),
    \label{eq:5-8-2a}\\
  \chi^{(\varphi[2])}_{00}
  &=
    \int \frac{d^3k}{(2\pi)^3}
  \delta_\mathrm{L}(\bm{k})
    Y_{00}(\hat{\bm{k}})
    W(kR).
  \label{eq:5-8-2b}
\end{align}
The last variable is apparently given by
\begin{equation}
  \chi^{(\varphi[2])}_{00} =
  \frac{1}{\sqrt{4\pi}}
  \int \frac{d^3k}{(2\pi)^3}
  \delta_\mathrm{L}(\bm{k}) W(kR)
  \equiv
  \frac{\delta_R}{\sqrt{4\pi}},
  \label{eq:5-8-3}
\end{equation}
where $\delta_R$ is the smoothed density field with a smoothing
function $W(kR)$.

Similarly, the second derivatives of the
smoothed density field is given by
\begin{equation}
  \chi^{(\delta)}_{ij} = \partial_i\partial_j\delta_R
  = - \int \frac{d^3k}{(2\pi)^3}\,k^2\, \hat{k}_i \hat{k}_j
  \delta_\mathrm{L}(\bm{k}) W(kR).
  \label{eq:5-8-3-1}
\end{equation}
and corresponding window function is $W^{(\delta)}(k) = k^2 W(kR)$.
The second-rank tensor is decomposed into irreducible components as
\begin{align}
  \chi^{(\delta)}_{ij} = \partial_i\partial_j \delta_R
  &=
  \left(
    \partial_i\partial_j - \frac{\delta_{ij}}{3} \Laplace
  \right) \delta_R +
    \frac{\delta_{ij}}{3} \Laplace \delta_R
    \nonumber \\
  &=
    -\frac{8\pi}{15} \chi^{(\delta[2])}_{2m} \mathsf{Y}^{(m)}_{ij} +
    4\pi \frac{\delta_{ij}}{3} \chi^{(\delta[2])}_{00} \mathsf{Y}^{(0)},
  \label{eq:5-8-4}
\end{align}
where
\begin{align}
  \chi^{(\delta[2])}_{2m}
  &=
  \int \frac{d^3k}{(2\pi)^3}
  \delta_\mathrm{L}(\bm{k})
    Y_{2m}(\hat{\bm{k}})\,
    k^2 W(kR),
    \label{eq:5-8-5a}\\
  \chi^{(\delta[2])}_{00}
  &=
  - \int \frac{d^3k}{(2\pi)^3}
  \delta_\mathrm{L}(\bm{k})
    Y_{00}(\hat{\bm{k}})\,
    k^2 W(kR),
  \label{eq:5-8-5b}
\end{align}
and the last variable is apparently given by
\begin{equation}
  \chi^{(\delta[2])}_{00} =
  - \frac{1}{\sqrt{4\pi}}
  \int \frac{d^3k}{(2\pi)^3}
  \delta_\mathrm{L}(\bm{k})\,k^2 W(kR)
  = \frac{\Laplace\delta_R}{\sqrt{4\pi}},
  \label{eq:5-8-6}
\end{equation}
where $\Laplace\delta_R$ is the Laplacian of the smoothed density
field.

In the semi-local models of bias, the tensor field
$F^\mathrm{L}_{Xlm}$ is generally considered as a function of various
irreducible tensors
\begin{equation}
  F^\mathrm{L}_{Xlm} =
  F^\mathrm{L}_{Xlm}\left(\left\{\chi^{(a)}_{l'm'}\right\}\right)
  \label{eq:5-9}
\end{equation}
at every position in (Lagrangian) configuration space. In this case,
the functional derivative in the definition of renormalized bias
functions, Eq.~(\ref{eq:3-18}), can be replaced by
\begin{equation}
  (2\pi)^3 \frac{\delta}{\delta\delta_\mathrm{L}(\bm{k})}
  \rightarrow
  \sum_{a} W^{(a)}(k)
  \sum_{l=L_a,L_a-2,\ldots}
  (-i)^l
  Y_{lm}(\hat{\bm{k}})
  \frac{\partial}{\partial \chi^{(a)}_{lm}},
  \label{eq:5-10}
\end{equation}
where the integer $L_a$ is the original rank of the linear tensor
$\chi^{(a)}_{i_1\cdots i_{L_a}}$. The summation over $m$ in
Eq.~(\ref{eq:5-10}) is implicitly assumed just as in the rest of this
paper. Therefore, the renormalized bias functions of
Eq.~(\ref{eq:3-18}) are given by
\begin{multline}
  c^{(n)}_{Xlm}(\bm{k}_1,\ldots\bm{k}_n)
  = \sum_{a_1,\ldots,a_n}
  W^{(a_1)}(k_1) \cdots W^{(a_n)}(k_n)
  \\ \times
  \sum_{l_1,\ldots,l_n}
  (-i)^{l_1+\cdots +l_n}
  Y_{l_1m_1}(\hat{\bm{k}}_1)
  \cdots Y_{l_n,m_n}(\hat{\bm{k}}_n)
  \\ \times
  \left\langle
  \frac{\partial^n F^\mathrm{L}_{Xlm}}
  {\partial \chi^{(a_1)}_{l_1m_1} \cdots \partial
    \chi^{(a_n)}_{l_nm_n}}
  \right\rangle,
  \label{eq:5-11}
\end{multline}
where integers $l_i$ ($i=1,\ldots,n$) run over
$l_i = L_{a_i}, L_{a_i}-2, \ldots$, (0 or 1). Comparing the above
equation with Eq.~(\ref{eq:3-51}), we have
\begin{multline}
  c^{(n)\,l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)
  = (-i)^{l_1+\cdots +l_n}
  \\ \times
  \sum_{a_1,\ldots,a_n}
  W^{(a_1)}(k_1) \cdots W^{(a_n)}(k_n)
  \\ \times
  g^{(l_1)}_{m_1m_1'} \cdots g^{(l_n)}_{m_nm_n'}
  \left\langle
  \frac{\partial^n F^\mathrm{L}_{Xlm}}
  {\partial \chi^{(a_1)}_{l_1m_1'} \cdots \partial
    \chi^{(a_n)}_{l_nm_n'}}
  \right\rangle.
  \label{eq:5-11-1}
\end{multline}

The renormalized bias functions in the form of Eq.~(\ref{eq:5-11}) is
evaluated for a given model of the biased tensor field, once the
underlying statistics of the fields $\chi^{(a)}_{lm}$ are specified.
These fields just linearly depend on the linear density contrast
$\delta_\mathrm{L}$ through Eqs.~(\ref{eq:5-5a})--(\ref{eq:5-5-2}) and
therefore the statistics are straightforwardly given by those of
linear density field.

Using Eqs.~(\ref{eq:3-20-1}) and (\ref{eq:5-5-2}), and the orthonormal
relation of the spherical harmonics, Eq.~(\ref{eq:a-5}), the
covariance of the fields (at the same position, the same applies
hereafter) can be straightforwardly calculated and is given by
\begin{equation}
  \left\langle
    \chi^{(a)}_{lm} \chi^{(b)}_{l'm'}
  \right\rangle
  = \delta_{ll'} \gamma^{ab}_{(l)} g^{(l)}_{mm'}.
  \label{eq:5-12}
\end{equation}
where
\begin{equation}
  \gamma^{ab}_{(l)} \equiv
  \frac{1}{4\pi}
  \int \frac{k^2dk}{2\pi^2}
  W^{(a)}(k) W^{(b)}(k)
  P_\mathrm{L}(k).
  \label{eq:5-13}
\end{equation}
One can regard the set of parameters $\gamma^{ab}_{(l)}$
as a matrix element of a matrix $\bm{\gamma}_{(l)}$, which components
are given by $[\bm{\gamma}_{(l)}]_{ab} = \gamma^{ab}_{(l)}$. We denote
the matrix elements of the inverse matrix as
\begin{equation}
  \gamma^{(l)}_{ab} \equiv
  \left[
    \bm{\gamma}_{(l)}^{-1}
  \right]_{ab}.
  \label{eq:5-16}
\end{equation}
When one considers a set of variables $\chi^{(a)}_{lm}$ as a vector
with the set of indices $(a,l,m)$, the inverse of the covariance
matrix of Eq.~(\ref{eq:5-12}) is given by
$\delta_{ll'} \gamma^{(l)}_{ab} g_{(l)}^{mm'}$ with our notation.

When the initial density field $\delta_\mathrm{L}$ is a Gaussian
random field, the two-point covariance of Eq.~(\ref{eq:5-12}) contains
all the information for the statistical distribution of the variables.
In this case, the distribution function is given by a multivariate
Gaussian distribution function,
\begin{multline}
  P_\mathrm{G}\left(\left\{\chi^{(a)}_{lm}\right\}\right)
  = \frac{1}{\sqrt{(2\pi)^N \det \bm{\gamma}_{(l)}}}
  \\ \times
  \exp\left[
    -\frac{1}{2} \sum_l \gamma^{(l)}_{ab}\, g_{(l)}^{mm'}
    \chi^{(a)}_{lm} \chi^{(b)}_{lm'}
  \right],
  \label{eq:5-17}
\end{multline}
where the Einstein summation convention is applied also to the indices
$a$, $b$ as well as azimuthal indices $m$, $m'$ and thus the summation
over these indices are implicitly assumed in the exponent. While the
variables $\chi^{(a)}_{lm}$ are complex numbers, the exponent of
Eq.~(\ref{eq:5-17}) is a real number. This can be readily shown by
noting a property
$\chi^{(a)\,*}_{lm} = g_{(l)}^{mm'} \chi^{(a)}_{lm'}$ as shown from
Eq.~(\ref{eq:5-5-2}), thus we have
\begin{equation}
  \gamma^{(l)}_{ab}\, g_{(l)}^{mm'} \chi^{(a)}_{lm} \chi^{(b)}_{lm'} =
  \gamma^{(l)}_{ab}\,\chi^{(a)}_{lm} \chi^{(b)\,*}_{lm},
  \label{eq:5-17-1}
\end{equation}
which is a real number because the coefficients $\gamma^{(l)}_{ab}$
are real numbers and symmetric with respect to the indices $a$ and
$b$. Moreover, the above factor is positive as long as the matrix
$\bm{\gamma}_{(l)}$ is positive definite.

The expectation value of Eq.~(\ref{eq:5-11}) is calculated by
multivariate Gaussian integrals with the distribution function of
Eq.~(\ref{eq:5-17}), which evaluation is analytically possible in many
cases when the semi-local bias function of Eq.~(\ref{eq:5-9}) is given
by an analytic function of a finite number of variables. When there is
a small non-Gaussianity in the initial condition, and the linear
density field is not exactly a random Gaussian field, one can evaluate
non-Gaussian corrections to the Gaussian distribution function of
Eq.~(\ref{eq:5-17}). The procedure of deriving the non-Gaussian
corrections are found in
Ref.~\cite{Matsubara:1995wd,Matsubara:2020lyv}, which is
straightforward to apply in this case. However, for the illustrative
examples below of this paper, we do not need to include the
non-Gaussian corrections in the evaluations of renormalized bias
functions.

The expectation value on the rhs of Eqs.~(\ref{eq:5-11}) and
(\ref{eq:5-11-1}) should be represented by rotationally invariant
variables, just in the case of the renormalized bias function
$c^{(n)l;l_1\cdots l_n}_{Xm;m_1\cdots m_n}(k_1,\ldots,k_n)$, while the
arguments of $k_i$'s are not present here. Similarly to
Eqs.~(\ref{eq:3-70}), (\ref{eq:3-71}), (\ref{eq:3-76}),
(\ref{eq:3-81}), we have
\begin{align}
  &
  \left\langle
  \frac{\partial F^\mathrm{L}_{Xlm}}{\partial\chi^{(a_1)}_{l_1m_1}}
  \right\rangle
  = \delta_{ll_1} \delta_m^{m_1} i^{l_1} b^{(1:a_1)}_{Xl},
    \label{eq:5-18a}\\
  &
  \left\langle
    \frac{\partial^2 F^\mathrm{L}_{Xlm}}
    {\partial\chi^{(a_1)}_{l_1m_1}\partial\chi^{(a_2)}_{l_2m_2}}
  \right\rangle
  = (-1)^l \sqrt{\{l\}} \left(l\,l_1\,l_2\right)_m^{\phantom{m}m_1m_2}
    i^{l_1+l_2} b^{(2:a_1a_2)}_{Xl;l_1l_2},
  \label{eq:5-18b}\\
  &
  \left\langle
  \frac{\partial^3 F^\mathrm{L}_{Xlm}}
  {\partial\chi^{(a_1)}_{l_1m_1}
  \partial\chi^{(a_2)}_{l_2m_2}
  \partial\chi^{(a_3)}_{l_3m_3}}
  \right\rangle
    = (-1)^l \sqrt{\{l\}} \sum_L (-1)^L\sqrt{\{L\}}
    \nonumber\\
  & \hspace{4pc} \times
    \left(l\,l_1\,L\right)_m^{\phantom{m}m_1M}
    \left(L\,l_2\,l_3\right)_M^{\phantom{M}m_2m_3}
     i^{l_1+l_2+l_3} b^{(3:a_1a_2a_3)L}_{Xl;l_1l_2l_3},
  \label{eq:5-18c}
\end{align}
and
\begin{multline}
  \left\langle
    \frac{\partial^3 F^\mathrm{L}_{Xlm}}
    {\partial\chi^{(a_1)}_{l_1m_1}\cdots
      \partial\chi^{(a_n)}_{l_nm_n}}
  \right\rangle
  = (-1)^l \sqrt{\{l\}}
  \sum_{L_2,\ldots,L_{n-1}}
  (-1)^{L_2+\cdots L_{n-1}}
  \\ \times
  \sqrt{\{L_2\}\cdots\{L_{n-1}\}}
    \left(l\,l_1\,L_2\right)_m^{\phantom{m}m_1M_2}
    \\ \times
    \left(L_2\,l_2\,L_3\right)_{M_2}^{\phantom{M_2}m_2M_3}
    \cdots
    \left(L_{n-2}\,l_{n-2}\,L_{n-1}\right)_{M_{n-2}}^{\phantom{M_{n-2}}m_2M_{n-1}}
    \\ \times
    \left(L_{n-1}\,l_{n-1}\,l_n\right)_{M_{n-1}}^{\phantom{M_{n-1}}m_{n-1}m_n}
     i^{l_1+\cdots +l_n}b^{(n:a_1\cdots a_n)L_2\cdots L_{n-1}}_{Xl;l_1\cdots l_n}.
  \label{eq:5-19}
\end{multline}

Therefore, comparing Eq.~(\ref{eq:5-11-1}) with Eqs.~(\ref{eq:3-70}),
(\ref{eq:3-71}), (\ref{eq:3-75}) and (\ref{eq:3-81}), invariant
functions of the renormalized bias functions are given by
\begin{align}
    c^{(1)}_{Xl}(k)
  &
    = \sum_a b^{(1:a)}_{Xl}W^{(a)}(k),
  \label{eq:5-20a}\\
    c^{(2)\,l}_{Xl_1l_2}(k_1,k_2)
  &
    = \sum_{a_1,a_2} b^{(2:a_1a_2)}_{Xl;l_1l_2}
    W^{(a_1)}(k_1) W^{(a_2)}(k_2),
  \label{eq:5-20b}\\
    c^{(3)\,l;L}_{Xl;l_1l_2l_3}(k_1,k_2,k_3)
  &
    =
    \sum_{a_1,a_2,a_3}
    b^{(3:a_1a_2a_3)L}_{Xl;l_1l_2l_3}
    \nonumber \\
  & \qquad \times
    W^{(a_1)}(k_1) W^{(a_2)}(k_2) W^{(a_3)}(k_3),
  \label{eq:5-20c}
\end{align}
and
\begin{multline}
    c^{(n)\,l;L_2\cdots L_{n-1}}_{Xl_1\cdots l_n}(k_1,\ldots,k_n)
    =
    \sum_{a_1,\ldots,a_n}
    b^{(n:a_1\cdots a_n)L_2\cdots L_{n-1}}_{Xl;l_1\cdots l_n}
  \\ \times
    W^{(a_1)}(k_1) \cdots W^{(a_n)}(k_n).
  \label{eq:5-21}
\end{multline}
Thus, the renormalized bias functions are given by superpositions
of products of window functions with constant coefficients which are
determined by a given semi-local model of bias. The scale dependencies
in the invariant renormalized functions are all contained in the
window functions which are fixed by a construction of the semi-local
model with a finite number of scale-independent parameters. These
properties simplify the calculations of loop corrections in our
applications.

Corresponding to the interchange symmetries of the renormalized bias
functions with orders greater than two, Eqs.~(\ref{eq:3-75-2}),
(\ref{eq:3-80-3}), (\ref{eq:3-80-4}), (\ref{eq:3-86}) and
(\ref{eq:3-87}), the same symmetries for the constant coefficients of
bias are given by
\begin{equation}
  b^{(2:a_2a_1)}_{Xl;l_2l_1}
  = (-1)^{l+l_1+l_2}
  b^{(2:a_1a_2)}_{Xl;l_1l_2}
  \label{eq:5-21-1}
\end{equation}
for the second-order bias,
\begin{align}
  b^{(3:a_1a_3a_2)L}_{Xl;l_ll_3l_2}
  &= (-1)^{l_2+l_3+L}
  b^{(3:a_1a_2a_3)L}_{Xl;l_ll_2l_3},
  \label{eq:5-21-2a}\\
  b^{(3:a_2a_1a_3)L}_{Xl;l_2l_1l_3}
  &= (-1)^{l_1+l_2}
  \sum_{L'} (2L'+1)
  \begin{Bmatrix}
    l_1 & l & L \\
    l_2 & l_3 & L'
  \end{Bmatrix}
  b^{(3:a_1a_2a_3)L'}_{Xl;l_ll_2l_3}
  \label{eq:5-21-2b}
\end{align}
for the third-order bias, and
\begin{align}
  &
    b^{(n:a_1\cdots a_{n-2}a_na_{n-1})L_2\cdots L_{n-2}L_{n-1}}_{Xl;l_l\cdots l_{n-2}l_nl_{n-1}}
  = (-1)^{l_{n-1}+l_n+L_{n-1}}
  b^{(n)L_2\cdots L_{n-1}}_{Xl;l_l\cdots l_n},
  \label{eq:5-21-3a}\\
  &
    b^{(n:a_1\cdots a_{i-1}a_{i+1}a_ia_{i+2}\cdots a_n);L_2\cdots L_{i-1}L_{i+1}L_iL_{i+2}\cdots L_{n-1}}
    _{Xl_1\cdots l_{i-1}l_{i+1}l_il_{i+2}\cdots l_n}
    \nonumber\\
  &\qquad
  = (-1)^{l_i+l_{i+1}}
  \sum_{L'} (2L'+1)
  \begin{Bmatrix}
    l_i & L_i & L_{i+1} \\
    l_{i+1} & L_{i+2} & L'
  \end{Bmatrix}
    \nonumber\\
  &\hspace{8pc} \times
  b^{(n:a_1\cdots a_n);L_2\cdots L_iL'L_{i+2}\cdots L_{n-1}}_{Xl;l_1\cdots l_n}
  \label{eq:5-21-3b}
\end{align}
for higher-order bias with $n>3$, where $i=1,\cdots,n-2$.

As a specific example, there is a loop integral in the scale-dependent
bias of Eq.~(\ref{eq:4-39}). Applying an expression
Eq.~(\ref{eq:5-20b}) of semi-local models, the integral is given by
\begin{equation}
  \int \frac{p^2dp}{2\pi^2}
  c^{(2)l}_{Xl_1l_2}(p,p) P_\mathrm{L}(p)
  =
  \sum_{a_1,a_2} b^{(2:a_1a_2)}_{Xl;l_1l_2} \sigma^{(a_1a_2)},
  \label{eq:5-22}
\end{equation}
where
\begin{equation}
  \sigma^{(a_1a_2)} \equiv
  \int \frac{k^2dk}{2\pi^2} P_\mathrm{L}(k)
  W^{(a_1)}(k) W^{(a_2)}(k).
  \label{eq:5-23}
\end{equation}
Thus Eq.~(\ref{eq:4-39}) reduces to
\begin{multline}
  \Delta b_{Xl}(k) =
  \frac{1}{\sqrt{4\pi}}\,
  \frac{2f_\mathrm{NL}^{(l)}}{\mathcal{M}(k)}
  \frac{1}{\{l\}}
  \sum_{l_1,l_2}
  (-1)^{l_2}
  \sqrt{\{l_1\}\{l_2\}}
  \\ \times
  \begin{pmatrix}
    l_1 & l_2 & l \\
    0 & 0 & 0
  \end{pmatrix}
  \sum_{a_1,a_2} b^{(2:a_1a_2)}_{Xl;l_1l_2} \sigma^{(a_1a_2)}.
  \label{eq:5-24}
\end{multline}

For a simple case of scalar bias with the high-mass limit of the halo
model, comparison of Eq.~(\ref{eq:4-43}) with Eq.~(\ref{eq:5-20b})
shows that only a term with
$b^{(2)\,0}_{\mathrm{h}\delta\delta;00} = (4\pi)^{1/2} \sigma^{-2}
\delta_\mathrm{c} b^\mathrm{L}_1$ survives, and one can readily see
the result of Eq.~(\ref{eq:4-44}) is reproduced as a consistency
check.

If we consider a simple model that the tensor bias is a local function
of only the second-order derivatives of the gravitational potential
$\partial_i\partial_j\varphi$, the window function is given by
$W^{(\varphi)}(k) = W(kR)$ as shown in Eq.~(\ref{eq:5-8-0}). We can
omit the label $(\varphi)$ in this case, because the linear tensor
field consists of only a single tensor.
The Eq.~(\ref{eq:5-23}) simply reduces to
\begin{equation}
  \sigma^2 = \int \frac{k^2dk}{2\pi^2}
  P_\mathrm{L}(k) W^2(kR),
  \label{eq:5-26}
\end{equation}
which is the variance of the smoothed linear density field. The ranks
$l_1$ and $l_2$ take only values of 0 and 2 in the summation of
Eq.~(\ref{eq:5-24}) in this case, and due to the $3j$-symbol
in Eq.~(\ref{eq:5-24}), only the cases of $l=0,2,4$ are non-zero.
Substituting concrete numbers of the $3j$-symbols, Eq.~(\ref{eq:5-24})
is explicitly expanded in a finite number of terms and the result is
given by
\begin{multline}
  \Delta b_{Xl}(k) =
  \frac{2\sigma^2}{\sqrt{4\pi}\,\mathcal{M}(k)}
  \Biggl[
    \delta_{l0}
    f^{(0)}_\mathrm{NL}
    \left(
      b^{(2)}_{0;00} + \sqrt{5} b^{(2)}_{0;22}
    \right)
  \\
    + \delta_{l2}
    f^{(2)}_\mathrm{NL}
    \left(
      \frac{2}{5} b^{(2)}_{2;02} - \sqrt{\frac{2}{35}} b^{(2)}_{2;22}
    \right)
    \\
    + \delta_{l4} f^{(4)}_\mathrm{NL}\,
    \frac{5}{9}\sqrt{\frac{2}{35}}\,
    b^{(2)}_{4;22}
  \Biggr],
  \label{eq:5-27}
\end{multline}
where we use an interchange symmetry $b^{(2)}_{2;20} = b^{(2)}_{2;02}$
derived from Eq.~(\ref{eq:5-21-1}). Therefore, if the tensor bias is
modeled as a local function of only a second-rank linear tensor field,
the scale-dependent bias of the tensor field is non-zero only when the
rank of the biased tensor is 0, 2, or 4. Due to the rotational
symmetry, only a small number of bias parameters, $b^{(2)}_{0;00}$,
$b^{(2)}_{0;22}$, $b^{(2)}_{2;02}$, $b^{(2)}_{2;22}$, and
$b^{(2)}_{4;22}$ appear in this particular semi-local model.

The above examples probably do not sufficiently exhibit the practical
merits of semi-local models of bias. Their advantages are more
apparently shown in the calculation of loop corrections in the
higher-order perturbation theory. The present formalism with the full
use of the spherical basis is quite compatible with the FFT-PT
framework \cite{Schmittfull:2016jsw,Schmittfull:2016yqx} or the
FAST-PT framework \cite{McEwen:2016fjn,Fang:2016wcf}, which
dramatically reduces the dimensionality of multidimensional integrals
in the calculation of loop corrections in the higher-order
perturbation theory. The loop corrections in the present formalism
will be considered in a separate paper, Paper~II \cite{PaperII}.

\section{\label{sec:Conclusions}
  Conclusions
}

In this paper, the formalism iPT is generalized to calculate
statistics of generally tensor-valued objects, using the nonlinear
perturbation theory. Higher-rank tensors are conveniently decomposed
into spherical tensors which are irreducible representations of the
three-dimensional rotation group SO(3), and mathematical techniques
developed in the theory of angular momentum in quantum mechanics are
effectively applied.

The fundamental formalism of iPT can be recycled without essential
modification, and the only difference is that the biased field is
colored by tensor values. Because the original formalism of iPT is
supposed to be applied to scalar fields, the rotational symmetry in
this original version of the theory is relatively trivial. When the
theory is generalized to include tensor-valued fields, the statistical
properties of biasing are largely constrained by symmetry. In
particular, the renormalized bias functions, which are one of the
important ingredients in the iPT, can be represented by a set of
invariant functions under rotations of the coordinates system.
Explicit constructions of these invariants are derived and presented
in this paper. These invariants can also be seen as coefficients of
angular expansions by polypolar spherical harmonics.

The iPT offers a systematic way of calculating the propagators of
nonlinear perturbation theory, both in real space and in redshift
space. The methodology of deriving propagators of tensor-valued fields is
generally explained in detail and a few examples of lower-order
propagators are explicitly derived in this paper. For an illustrative
purpose, these examples of propagators are applied to simple cases of
predicting the correlation statistics, including the linear power
spectra in real space and redshift space, the lowest-order power
spectrum with primordial non-Gaussianity, the bispectrum in the
tree-level approximation, for generally tensor fields. As a
consistency check in each example, we confirm that each derived
formula reproduces the known result for scalar fields, just
substituting $l=m=0$ in the general formulas.

In the last section, the concept of semi-local models of bias is
introduced. The formalism of iPT does not assume any form of bias
models, and the biasing can be given by any functional of the
underlying field. The semi-local models of bias are defined so that
the bias functional should be a function of a finite number of variables
derived from the linear density field in Lagrangian space. Therefore,
the biasing is modeled by a finite number of parameters in this
category of models. Almost all the existing models of bias introduced
in the literature fall into this category, including the halo model,
the peak theory of bias, excursion set peaks, and so forth. In this
paper, only the formal definition of the semi-local models of bias is
described, and rotationally invariant parameters in the models are
identified. Their applications to individual models of concrete
targets will be presented in future work.

In realistic observations of galaxy spins and intrinsic alignments and
so forth, we can only observe projected components of tensors onto the
two-dimensional sky. In this paper, while predictions for the spatial
correlations of tensors only in three-dimensional space are given, it
should be straightforward mathematics to transform our results of the
invariant spectra into correlation statistics of projected tensors.
Explicit expressions will also be presented in future work.

In this paper, the power spectrum and bispectrum are only evaluated in
a lowest-order, or tree-level approximation. Some useful techniques in
calculating higher-order, or loop corrections are described in a
subsequent Paper~II \cite{PaperII}. We hope that the present formalism
offers a way to carve out the future of applying the cosmological
perturbation theory to the analysis of observations in the era of
precision cosmology.

\begin{acknowledgments}
  I thank Y.~Urakawa, K.~Kogai, K.~Akitsu, and A.~Taruya for useful
  discussions. This work was supported by JSPS KAKENHI Grants
  No.~JP19K03835 and No.~21H03403.
\end{acknowledgments}