# Developing a Theoretical Model for the Resummation of Infrared Effects
in the Pre- and Post-Reconstruction Galaxy Bispectra

Naonori Sugiyama<EMAIL_ADDRESS>National Astronomical Observatory
of Japan, Mitaka, Tokyo 181-8588, Japan

###### Abstract

While substantial progress has been made in studying the pre-reconstruction
galaxy bispectrum, investigations of the post-reconstruction bispectrum are
still in nascent stages. In this paper, we present a bispectrum model that
incorporates one-loop corrections in the Standard Perturbation Theory (SPT),
while simultaneously addressing infrared (IR) effects in the post-
reconstruction density fluctuations in a non-perturbative approach. This model
accurately captures the nonlinear behavior of the Baryon Acoustic Oscillation
(BAO) signal within the post-reconstruction bispectrum framework. Furthermore,
the incorporation of the one-loop correction extends its applicability to
smaller scales. Given that the approach to addressing IR effects remains
invariant before and after the reconstruction process, the resulting IR-
resummed bispectrum model exhibits uniformity across both scenarios.
Throughout this analysis, we achieve a comprehensive and unified description
of the bispectrum that seamlessly spans the pre- and post-reconstruction
phases.

##  I Introduction

The three-point correlation function (3PCF) of galaxies, or its corresponding
bispectrum in Fourier space, provides additional cosmological information
compared to the case only considering the galaxy two-point correlation
function (2PCF) and power spectrum. In particular, the isotropic component of
the three-point statistics, specifically the monopole component of the galaxy
3PCF or bispectrum, has been the primary focus for constraining standard
cosmological parameters [1, 2, 3, 4, 5] and examining primordial non-
Gaussianity [6, 7, 8]. Recently, by using the anisotropic components of the
three-point statistics, namely the quadrupole and hexadecapole components,
more improved constraints on standard cosmological parameters have been
achieved [9, 10, 11]. Furthermore, using the quadrupole 3PCF, tests on
modified gravity theories and the consistency relation of the Large-Scale
Structures (LSS) have been conducted [12, 13]. These data analysis techniques
are expected to further evolve in upcoming spectroscopic galaxy surveys such
as the Dark Energy Spectroscopic Instrument [DESI; 14]111http://desi.lbl.gov/,
the Subaru Prime Focus Spectrograph [PFS;
15]222https://pfs.ipmu.jp/index.html, and, Euclid [16]333www.euclid-ec.org.

One of the key factors affecting the constraining power of the galaxy
bispectrum is the non-Gaussian errors embedded within its covariance matrix.
Sugiyama  _et al._ [17] pointed out that non-Gaussian errors dominate over
Gaussian errors in the galaxy bispectrum. This dominance of non-Gaussian
errors is a unique statistical characteristic of the bispectrum, significantly
different from the power spectrum case, where Gaussian errors are more
prevalent. Specifically, the signal-to-noise ratio (S/N) of the bispectrum
decreases by a factor of 3 to 4 when non-Gaussian errors are considered, as
opposed to considering only Gaussian errors. This finding implies that the
presence of non-Gaussian errors significantly limits the constraining power of
cosmological parameters derived from the galaxy bispectrum.

The reconstruction of galaxy distributions [18] was originally developed to
enhance signals from baryon acoustic oscillations [BAO; 19, 20]. Recently, it
has also been recognized for its ability to minimize non-Gaussian errors in
the covariance matrix. Power spectrum analysis has confirmed that this
reduction of non-Gaussian errors significantly increases the constraining
power for standard cosmological parameters [21, 22]. This method of error
reduction through reconstruction is particularly effective in bispectrum
analysis, where Shirasaki  _et al._ [23] has shown that post-reconstruction
bispectrum analysis can improve constraints on primordial non-Gaussianity by a
factor of $3$ to $4$ compared to pre-reconstruction analysis. Given these
advancements, a joint analysis of the post-reconstruction power spectrum and
bispectrum is anticipated to become the norm in future research.

In the field of cosmology, when analyzing spectroscopic galaxy datasets, such
as data from the Baryon Oscillation Spectroscopic Survey [BOSS; 24, 25, 26,
27], we measure both power spectra and bispectra. Subsequently, these
measurements are compared with their respective theoretical models. For these
analyses, improved perturbation theories, which build upon Standard
Perturbation Theory (SPT) [e.g., 28], are frequently employed. The following
briefly reviews these improved perturbation theories across four scenarios:
pre-reconstruction power spectra, post-reconstruction power spectra, pre-
reconstruction bispectra, and post-reconstruction bispectra.

  1. 1.

For pre-reconstruction power spectra, various advanced theoretical models have
been proposed and applied in data analysis. These include Convolution
Lagrangian Perturbation Theory (CLPT) [29, 30], Renormalized Perturbation
Theory (RPT) [31], the TNS model [32], and the Effective Field Theory of
Large-scale Structure (EFT of LSS) [33, 34].

  2. 2.

Although there are fewer theoretical studies on pre-reconstruction bispectra
compared to those on the power spectrum, models based on EFT [4, 5, 10, 11]
and the resummation of infrared (IR) effects [35, 36, 9, 12, 13] have been
applied to actual galaxy data.

  3. 3.

Regarding post-reconstruction power spectra, theoretical calculations using
SPT [37, 38, 39] and the Zel’dovich approximation [40, 41, 42, 43] have been
conducted. However, models employed in actual data analyses have predominantly
been empirical, characterized by the nonlinear damping of BAO with a single
Gaussian damping function. More recently, Sugiyama [44] and Chen  _et al._
[45] have demonstrated that, by appropriately resumming IR effects on post-
reconstruction density fluctuations, the nonlinearity of the BAO signal in the
post-reconstruction power spectrum can indeed be described with a single
Gaussian damping function. This finding provides theoretical support for the
validity of previous post-reconstruction BAO analyses.

  4. 4.

Theoretical studies on post-reconstruction bispectra remain inadequate. While
some studies are based on SPT [37, 23], no theoretical calculations employing
improved perturbation theories beyond SPT have been conducted to date.
Furthermore, cosmological analyses using post-reconstruction bispectra have
yet to be carried out.

The aim of this paper is to develop a theoretical model for post-
reconstruction galaxy bispectra, with the intention of applying it to future
cosmological analyses. To this end, we introduce a model that incorporates
nonlinear IR effects, marking an initial step towards the development of
theoretical models for post-reconstruction bispectra that extend beyond SPT.
The emphasis on IR effects is motivated by their unique characteristic of
being amenable to non-perturbative treatment. Moreover, it is well-established
that these non-perturbative IR effects can accurately describe the nonlinear
damping of the BAO signal [46, 47, 48, 49]. Therefore, it is essential to
properly consider IR effects in reconstruction processes that are sensitive to
the BAO signal.

Sugiyama [44] has shown that IR effects can be represented as a coordinate
transformation of density fluctuations via a displacement vector, both before
and after reconstruction process. This finding suggests that IR effects can be
uniformly addressed, regardless of whether reconstruction has occurred.
Consequently, it is feasible to apply the IR-resummed model of the pre-
reconstruction bispectrum at the leading order (tree level), initially
proposed by Sugiyama  _et al._ [9], to the post-reconstruction bispectrum as
well. However, this application necessitates the substitution of certain
components, such as nonlinear kernel functions, to suit the post-
reconstruction scenario. Moreover, this paper presents the calculation of the
IR-resummed model including next-leading order terms, i.e., 1-loop correction
terms, which are relevant for scales smaller than those addressed by the tree-
level solution.

The calculation of the bispectrum is inherently complex, involving a variety
of kernel functions. This complexity is particularly prominent in the
resummation of IR effects, as this process entails dealing with infinite mode-
coupling integrals. In this paper, we develop a method to simplify the
calculation of IR effects on the pre-reconstruction bispectrum as performed by
Sugiyama  _et al._ [9], and apply this method to compute the 1-loop correction
terms.

The outline of this paper is as follows. In Section II, we review the methods
for handling the IR effects before and after reconstruction; Section III
reviews the general properties of the bispectrum. Section IV presents the IR-
resummed model at the tree-level. Section V then derives the main result of
this paper, an IR-resummed post-reconstruction bispectrum model incorporating
1-loop corrections. Section VI serves as the conclusion of this paper. In
Appendix A, we present the kernel functions representing the nonlinear effects
for the density fluctuations up to the fourth order, both before and after
reconstruction. Appendix B provides an explanation on the general
decomposition methods for the bispectrum using the $\Gamma$-expansion [50]. In
Appendix C, we give the expression for the smoothing factors characterizing
the exponential damping function that appears when resumming the IR effects,
specifically for the post-reconstruction case. Appendix D gives the solution
of the IR (high-k) limit of the 1-loop bispectrum for dark matter in real
space.

##  II IR effects on pre- and post-reconstruction density fluctuations

In this section, we briefly review the behavior of density fluctuations in the
infrared (IR) limit, before and after reconstruction. Our treatment of the IR
limit is based on Sugiyama [44], as well as foundational works [51, 52, 9].
For alternative approaches to handling the IR limit, see also [53, 54, 55, 35,
56, 36, 57].

###  II.1 Pre-reconstruction

In the Lagrangian description, the displacement vector connecting Eulerian
coordinates $\bm{x}$ to Lagrangian coordinates $\bm{q}$ is given by

$\displaystyle\bm{x}=\bm{q}+\bm{\Psi}(\bm{q})\;.$ (1)

When considering the redshift-space distortion (RSD) effect [58], it can be
expressed as

$\displaystyle\bm{\Psi}_{\rm
red}(\bm{q})=\bm{\Psi}(\bm{q})+\frac{a\dot{\bm{\Psi}}(\bm{q})\cdot\hat{n}}{aH}\hat{n}\;,$
(2)

where $\dot{\bm{\Psi}}(\bm{q})$ is the time-derivative of $\bm{\Psi}(\bm{q})$,
$\hat{n}$ is the unit vector in the line-of-sight direction, and $a$ and $H$
are the scale factor and the Hubble parameter, respectively. The galaxy
density fluctuation is then represented as [59]

$\displaystyle 1+\delta_{\rm g}(\bm{x},\hat{n})=\int d^{3}q\left[1+\delta_{\rm
b}(\bm{q})\right]\delta_{\rm D}\left(\bm{x}-\bm{q}-\bm{\Psi}_{\rm
red}(\bm{q},\hat{n})\right)\;,$ (3)

where $\delta_{\rm b}$ represents the biased density fluctuation in the
Lagrangian description, and $\delta_{\rm D}$ denotes the three-dimensional
delta function.

When considering the IR effects, we decompose the displacement vector into its
value evaluated at the origin $\bm{q}=\bm{0}$ and the other values:

$\displaystyle\bm{\Psi}_{\rm red}(\bm{q},\hat{n})=\overline{\bm{\Psi}}_{\rm
red}(\hat{n})+\bm{\Psi}_{\rm(S)red}(\bm{q},\hat{n})\;,$ (4)

where $\overline{\bm{\Psi}}_{\rm red}(\hat{n})=\bm{\Psi}_{\rm
red}(\bm{q}=\bm{0},\hat{n})$. Then, $\delta_{\rm g}$ can be formally rewritten
as [52, 9]

$\displaystyle\delta_{\rm
g}(\bm{x},\hat{n})=\delta_{\rm(S)g}(\bm{x}-\overline{\bm{\Psi}}_{\rm
red}(\hat{n}),\hat{n})\;,$ (5)

where the short-wavelength density fluctuation $\delta_{\rm(S)g}$ is obtained
by replacing $\bm{\Psi}_{\rm red}$ in Eq. (3) with $\bm{\Psi}_{\rm(S)red}$. In
Fourier space, Eq. (5) becomes

$\displaystyle\widetilde{\delta}_{\rm
g}(\bm{k},\hat{n})=e^{-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
red}(\hat{n})}\widetilde{\delta}_{\rm(S)g}(\bm{k},\hat{n})\;.$ (6)

Throughout this paper, the tilde denotes a Fourier-transformed quantity 444Our
convention for the Fourier transform is
$\displaystyle\widetilde{f}(\bm{k})=\int
d^{3}xe^{-i\bm{k}\cdot\bm{x}}f(\bm{x})\;.$ .

In this paper, three assumptions are made when taking the IR limit [44]:

  1. 1.

The dominant term in the IR limit arises from the contribution of
$\overline{\bm{\Psi}}_{\rm red}$.

  2. 2.

$\delta_{\rm(S)g}$ and $\overline{\bm{\Psi}}_{\rm red}$ are uncorrelated.

  3. 3.

$\delta_{\rm(S)g}$ is truncated at a finite perturbative order, and higher-
order perturbative effects than $\delta_{\rm(S)g}$ arise from
$\overline{\bm{\Psi}}_{\rm red}$.

###  II.2 Post-reconstruction

Using the galaxy density fluctuation, $\delta_{\rm g}(\bm{x})$, we construct
the displacement vector for reconstruction as [18]

$\displaystyle\bm{s}(\bm{x},\hat{n})=\int\frac{d^{3}p}{(2\pi)^{3}}e^{i\bm{p}\cdot\bm{x}}\left(\frac{i\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{1,\rm fid}}\right)\widetilde{\delta}_{\rm
g}(\bm{p},\hat{n})\;,$ (7)

where $b_{1,\rm fid}$ is an input fiducial linear bias parameter for
reconstruction, $W_{\rm G}(pR_{\rm s})=\exp\left(-p^{2}R_{\rm s}^{2}/2\right)$
is a Gaussian filter function, and $R_{\rm s}$ is an input smoothing scale. In
the Eulerian description, the density fluctuation after reconstruction is then
given by [23, 9]

$\displaystyle\delta_{\rm rec}(\bm{x},\hat{n})=\int d^{3}x^{\prime}\delta_{\rm
g}(\bm{x}^{\prime},\hat{n})\delta_{\rm
D}(\bm{x}-\bm{x}^{\prime}-\bm{s}(\bm{x}^{\prime},\hat{n}))\;.$ (8)

When taking the IR limit in the post-reconstruction density fluctuation, it is
important to consider the IR effects in the pre-reconstruction density
fluctuation that constitute $\bm{s}(\bm{x})$. By substituting Eq. (6) into Eq.
(7), we derive

$\displaystyle\bm{s}(\bm{x},\hat{n})=\bm{s}_{\rm(S)}(\bm{x}-\overline{\bm{\Psi}}_{\rm
red}(\hat{n}),\hat{n})\;,$ (9)

where $\bm{s}_{\rm(S)}$ is obtained by replacing $\widetilde{\delta}_{\rm g}$
in Eq. (7) with $\widetilde{\delta}_{\rm(S)g}$. Furthermore, in accordance
with Eq. (4), we decompose $\bm{s}_{\rm(S)}$ into its value at the origin and
the other values:

$\displaystyle\bm{s}_{\rm(S)}(\bm{x},\hat{n})=\overline{\bm{s}}(\hat{n})+\bm{s}_{\rm(SS)}(\bm{x},\hat{n})\;,$
(10)

where $\overline{\bm{s}}(\hat{n})=\bm{s}_{\rm(S)}(\bm{x}=\bm{0},\hat{n})$.
Eventually, this leads to

$\displaystyle\delta_{\rm
rec}(\bm{x},\hat{n})=\delta_{\rm(S)rec}(\bm{x}-\overline{\bm{\Psi}}_{\rm
rec}(\hat{n}),\hat{n})\;,$ (11)

where

$\displaystyle\delta_{\rm(S)rec}(\bm{x},\hat{n})$ $\displaystyle=$
$\displaystyle\int d^{3}x^{\prime}\delta_{\rm(S)g}(\bm{x}^{\prime},\hat{n})$
(12) $\displaystyle\times$ $\displaystyle\delta_{\rm
D}(\bm{x}-\bm{x}^{\prime}-\bm{s}_{\rm(SS)}(\bm{x}^{\prime},\hat{n}))\;.$

and

$\displaystyle\overline{\bm{\Psi}}_{\rm
rec}(\hat{n})=\overline{\bm{\Psi}}_{\rm
red}(\hat{n})+\overline{\bm{s}}(\hat{n})\;.$ (13)

In Fourier space, Eq. (11) becomes

$\displaystyle\widetilde{\delta}_{\rm
rec}(\bm{k},\hat{n})=e^{-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
rec}(\hat{n})}\widetilde{\delta}_{\rm(S)rec}(\bm{k},\hat{n})\;.$ (14)

Based on Eqs. (11) and (14) above, the IR effect still manifests as a
coordinate transformation within the short-wavelength density fluctuation
$\delta_{\rm(S)rec}$, even after reconstruction, similar to its behavior
before reconstruction. Thus, the treatment of the IR effect remains unchanged
before and after reconstruction. Although this paper primarily addresses the
post-reconstruction IR effect, it is important to note that out analysis also
includes pre-reconstruction scenarios as $R_{\rm s}\to\infty$ approaches
infinity.

###  II.3 Perturbation expansion

In cosmological perturbation theory, linear dark matter density fluctuations,
denoted as $\delta_{\rm lin}$, serve as the foundation for defining
perturbation expansions. Within this framework, the $n$th-order term of a
quantity $X$ is expressed as $X^{[n]}$, where $X^{[n]}={\cal O}([\delta_{\rm
lin}]^{n})$. In this paper, we use the superscript $[n]$ to denote the $n$th-
order term in the perturbation expansion and any associated quantities.

The linear dark matter density fluctuations $\delta_{\rm lin}$ can be
decomposed into their time and space dependences and expressed as $\delta_{\rm
lin}=D\delta_{\rm lin,0}$. Here, $D$ represents the linear growth factor that
characterizes the time dependence, and $\delta_{\rm lin,0}$ is the linear dark
matter density fluctuation at $z=0$. The linear growth rate function that
characterizes the first-order time derivative of the linear dark matter
fluctuations is then defined as $f=d\ln D/d\ln a$.

The $n$th-order post-reconstruction density fluctuation is given by

$\displaystyle\widetilde{\delta}_{\rm rec}^{\,[n]}(\bm{k},\hat{n})$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\int\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}\left(\bm{k}-\bm{p}_{[1,n]}\right)$ (15) $\displaystyle\times$
$\displaystyle Z_{\rm
rec}^{[n]}(\bm{p}_{1},\cdots,\bm{p}_{n},\hat{n})\widetilde{\delta}_{\rm
lin}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm lin}(\bm{p}_{n})\;,$

where $\bm{p}_{[1,n]}=\bm{p}_{1}+\cdots+\bm{p}_{n}$, and $Z_{\rm rec}^{[n]}$
is the kernel function, including the galaxy bias, RSD, and reconstruction
effects. At the first order, the kernel function remains the same before and
after reconstruction and is given by [58]

$\displaystyle Z_{\rm
rec}^{[1]}(\bm{p}_{1},\hat{n})=b_{1}+f(\hat{p}_{1}\cdot\hat{n})^{2}\;,$ (16)

where $b_{1}$ is the linear bias parameter, and $\hat{p}=\bm{p}/|\bm{p}|$. The
specific forms of the kernel functions up to the forth order, $Z_{\rm
rec}^{[2]}$, $Z_{\rm rec}^{[3]}$, and $Z_{\rm rec}^{[4]}$, are summarized in
Appendix A.

The $n$th-order displacement vector is expressed as

$\displaystyle\widetilde{\bm{\Psi}}^{\,[n]}(\bm{k})$ $\displaystyle=$
$\displaystyle
i\,\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\int\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}\left(\bm{k}-\bm{p}_{[1,n]}\right)$ (17) $\displaystyle\times$
$\displaystyle\bm{L}^{[n]}(\bm{p}_{1},\cdots,\bm{p}_{n})\,\widetilde{\delta}_{\rm
lin}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm lin}(\bm{p}_{n})\;,$

where $\bm{L}^{[n]}$ is the $n$th-order kernel function. In redshift space,
$\widetilde{\bm{\Psi}}_{\rm red}$ is represented by replacing $\bm{L}^{[n]}$
with $\bm{L}_{\rm red}^{[n]}$ given by [49]

$\displaystyle\bm{L}^{[n]}_{\rm
red}(\bm{p}_{1},\cdots,\bm{p}_{n},\hat{n})=\bm{R}^{[n]}(\hat{n})\cdot\bm{L}^{[n]}(\bm{p}_{1},\cdots,\bm{p}_{n})\;.$
(18)

Here, the three-dimensional transformation matrix $\bm{R}^{[n]}$ is

$\displaystyle[\bm{R}^{[n]}(\hat{n})]_{ij}=\bm{I}_{ij}+n\,f\,\hat{n}_{i}\hat{n}_{j}\;,$
(19)

where $i,j=1,2,3$, and $\bm{I}$ is the identity matrix.

As detailed in the subsequent section, we conduct 1-loop bispectrum
calculations. The perturbative orders of density fluctuations required for
these calculations are up to the fourth order. Therefore, we truncate the
short-wavelength density fluctuation at the fourth order. Below, we summarize
the relation between $\widetilde{\delta}_{\rm rec}$ and
$\widetilde{\delta}_{\rm(S)rec}$ up to the fourth order:

$\displaystyle\widetilde{\delta}_{\rm rec}^{\,[1]}$ $\displaystyle=$
$\displaystyle\widetilde{\delta}_{\rm(S)rec}^{\,[1]}\;,$
$\displaystyle\widetilde{\delta}_{\rm rec}^{\,[2]}$ $\displaystyle=$
$\displaystyle\widetilde{\delta}_{\rm(S)rec}^{\,[2]}+(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec})\widetilde{\delta}^{\,[1]}_{\rm(S)rec}\;,$
$\displaystyle\widetilde{\delta}_{\rm rec}^{\,[3]}$ $\displaystyle=$
$\displaystyle\widetilde{\delta}_{\rm(S)rec}^{\,[3]}+(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[2]}_{\rm
rec})\widetilde{\delta}^{\,[1]}_{\rm(S)rec}$ $\displaystyle+$
$\displaystyle(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec})\widetilde{\delta}^{\,[2]}_{\rm(S)rec}+\frac{1}{2}(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec})^{2}\widetilde{\delta}^{\,[1]}_{\rm(S)rec}\;,$
$\displaystyle\widetilde{\delta}_{\rm rec}^{\,[4]}$ $\displaystyle=$
$\displaystyle\widetilde{\delta}_{\rm(S)rec}^{\,[4]}+(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[3]}_{\rm
rec})\widetilde{\delta}^{\,[1]}_{\rm(S)rec}$ (20) $\displaystyle+$
$\displaystyle(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[2]}_{\rm
rec})\widetilde{\delta}^{\,[2]}_{\rm(S)rec}+(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec})\widetilde{\delta}^{\,[3]}_{\rm(S)rec}$ $\displaystyle+$
$\displaystyle(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec})(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[2]}_{\rm
rec})\widetilde{\delta}^{\,[1]}_{\rm(S)rec}$ $\displaystyle+$
$\displaystyle\frac{1}{2}(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec})^{2}\widetilde{\delta}^{\,[2]}_{\rm(S)rec}$ $\displaystyle+$
$\displaystyle\frac{1}{3!}(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec})^{3}\widetilde{\delta}^{\,[1]}_{\rm(S)rec}\;.$

Furthermore, in the coordinate transformation that describes the IR effect, we
consider only the first-order $\overline{\bm{\Psi}}_{\rm rec}$. This approach
is justified since terms in $\delta_{\rm(S)rec}$ with
$\overline{\bm{\Psi}}_{\rm rec}^{[n\geq 2]}$ do not contribute to the 1-loop
bispectrum, as demonstrated in the following section. Therefore, the post-
reconstruction density fluctuation discussed in this paper can be approximated
as follows:

$\displaystyle\delta_{\rm rec}(\bm{k},\hat{n})\approx
e^{-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}_{\rm
rec}(\hat{n})}\sum_{n=1}^{4}\widetilde{\delta}^{[n]}_{\rm(S)rec}(\bm{k},\hat{n})\;.$
(21)

##  III General properties in the bispectrum

###  III.1 Notational conventions

To simplify notation, the subscript “rec” used to represent reconstructed
quantities for density fluctuations and displacement vectors will be omitted
throughout the remainder of this paper. Furthermore, we do not explicitly
write the dependency on the line-of-sight direction, $\hat{n}$, due to the RSD
effect. Thus, we denote $\delta_{\rm rec}(\bm{x},\hat{n})$ and $\bm{\Psi}_{\rm
rec}(\bm{q},\hat{n})$ as $\delta(\bm{x})$ and $\bm{\Psi}(\bm{q})$,
respectively.

The 3PCF is defined as the ensemble average of the product of density
fluctuations evaluated at three distinct points, $\bm{x}_{1}$, $\bm{x}_{2}$,
and $\bm{x}_{3}$. It is characterized by two relative distances,
$\bm{r}_{1}=\bm{x}_{1}-\bm{x}_{3}$ and $\bm{r}_{2}=\bm{x}_{2}-\bm{x}_{3}$, due
to statistical translational symmetry:

$\displaystyle\zeta(\bm{r}_{1},\bm{r}_{2})$ $\displaystyle=$
$\displaystyle\langle\delta(\bm{x}_{1})\delta(\bm{x}_{2})\delta(\bm{x}_{3})\rangle$
(22) $\displaystyle=$
$\displaystyle\langle\delta(\bm{x}_{1}-\bm{x}_{3})\delta(\bm{x}_{2}-\bm{x}_{3})\delta(\bm{0})\rangle\;.$

In Fourier space, the bispectrum is expressed as

$\displaystyle\langle\widetilde{\delta}(\bm{k}_{1})\widetilde{\delta}(\bm{k}_{2})\widetilde{\delta}(\bm{k}_{3})\rangle=(2\pi)^{3}\delta_{\rm
D}(\bm{k}_{123})B(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3})\;,$ (23)

where $\bm{k}_{123}=\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3}$. The condition
$\bm{k}_{123}=\bm{0}$, through the delta function, reflects the statistical
translational symmetry.

In the analytical expressions derived from perturbation theory, when
primordial non-Gaussianity is absent, the bispectrum is decomposed into three
permutations:

$\displaystyle B(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3})$ (24) $\displaystyle=$
$\displaystyle
B_{12}(\bm{k}_{1},\bm{k}_{2})+B_{13}(\bm{k}_{1},\bm{k}_{3})+B_{23}(\bm{k}_{2},\bm{k}_{3})\;.$

It is important to note that the exclusion of primordial non-Gaussianity
constitutes a fundamental assumption in this study. Throughout this paper, our
analysis will be confined to $B_{12}$ exclusively. For simplicity, the
subscript “12” will be omitted hereafter. Consequently, any term denoted as
$B(\bm{k}_{1},\bm{k}_{2})$ henceforth should be interpreted as referring the
first term on the right-hand side in Eq. (24).

In the framework of perturbation theory, we denote the bispectrum as
$B^{[nml]}$, calculated as the product of three density fluctuations of order
$n$, $m$, and $l$. For example, a bispectrum composed of two linear
fluctuations and one second-order fluctuation is denoted as $B^{[112]}$.

###  III.2 $\Gamma$-expansion

Figure 1:  Schematic diagram representing the bispectrum components decomposed
by the $G$-expansion method as given in Eq. (25). Among the lines connecting
the vertices corresponding to $\bm{k}_{1}$, $\bm{k}_{2}$, and $\bm{k}_{3}$,
closed loops indicate the presence of mode coupling integrals. Note that
although only one loop is illustrated, it implicitly shows an infinite number
of mode coupling integrals. In the IR limit, each vertex and the lines
connecting these vertices are associated with the functions ${\cal D}$ and
${\cal E}$ as defined in Eq. (35). This association enables a straightforward
derivation of the non-perturbative solution of each term in the IR limit as
given by Eq. (39).

The $\Gamma$-expansion (e.g., [50]) is a decomposition method that relies on
mode-coupling integrals. Using the $\Gamma$-expansion allows for the
decomposition of the bispectrum into five components [9], expressed as
follows:

$\displaystyle B(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$ $\displaystyle
B_{\rm GG}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm lin}(k_{2})$ (25)
$\displaystyle+$ $\displaystyle B_{\rm GM}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{1})$ $\displaystyle+$ $\displaystyle B_{\rm
MG}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})$ $\displaystyle+$ $\displaystyle
B_{\rm MM}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle+$ $\displaystyle B_{\rm
MMM}(\bm{k}_{1},\bm{k}_{2})\;,$

where $P_{\rm lin}$ denotes the linear dark matter power spectrum, calculated
as

$\displaystyle\langle\widetilde{\delta}_{\rm
lin}(\bm{k})\widetilde{\delta}_{\rm
lin}(\bm{k}^{\prime})\rangle=(2\pi)^{3}\delta_{\rm
D}(\bm{k}+\bm{k}^{\prime})P_{\rm lin}(k)\;.$ (26)

In Eq. (25), $B_{\rm GG}$ does not incorporate mode coupling integrals,
whereas both $B_{\rm GM}$ and $B_{\rm MG}$ partially include mode coupling
integrals. Conversely, $B_{\rm MM}$ and $B_{\rm MMM}$ are composed only of
mode-coupling integrals. For detailed expressions of $B_{\rm GG}$, $B_{\rm
GM}$, $B_{\rm MG}$, $B_{\rm MM}$, and $B_{\rm MMM}$, see Appendix B.

Figure 1 shows a schematic diagram representing the bispectrum components
decomposed by the $\Gamma$-expansion method. The bispectrum is represented by
three vertices corresponding to $\bm{k}_{1}$, $\bm{k}_{2}$, and $\bm{k}_{3}$,
and it can be decomposed into components based on how these wave vectors are
connected. Regarding $B_{\rm GG}$, $\bm{k}_{1}$ and $\bm{k}_{3}$, as well as
$\bm{k}_{2}$ and $\bm{k}_{3}$, are connected without the mode coupling
integral. As shown in Section IV, the tree-level bispectrum is classified
under this term. The $B_{\rm GM}$, $B_{\rm MG}$, and $B_{\rm MM}$ terms
respectively involve $\bm{k}_{1}$ and $\bm{k}_{3}$, $\bm{k}_{2}$ and
$\bm{k}_{3}$, or both being connected through the mode coupling integral. The
mode coupling integral is here represented as a closed loop. For simplicity,
only one loop is illustrated in the figure, but it is implied that an infinite
number of mode coupling integrals are considered. The $B_{\rm MMM}$ term
connects $\bm{k}_{1}$ and $\bm{k}_{2}$ to form a closed triangle diagram. It
is noted that the lines represented as straight lines in the $B_{\rm MMM}$
diagram include an infinite number of mode coupling integrals implicitly. For
the non-perturbative behavior in the IR limit of each term, see Sections IV.2
and V.2.

###  III.3 IR cancellation

If $\overline{\bm{\Psi}}$ and $\delta_{\rm(S)}$ are completely uncorrelated,
all IR effects induced by $\overline{\bm{\Psi}}$ are canceled out due to
statistical translational symmetry in the 3PCF calculation, yielding
contributions only from short-wavelength density fluctuations:

$\displaystyle\langle\delta(\bm{x}_{1})\delta(\bm{x}_{2})\delta(\bm{x}_{3})\rangle$
(27) $\displaystyle=$
$\displaystyle\langle\delta_{\rm(S)}(\bm{x}_{1}-\overline{\bm{\Psi}})\delta_{\rm(S)}(\bm{x}_{2}-\overline{\bm{\Psi}})\delta_{\rm(S)}(\bm{x}_{3}-\overline{\bm{\Psi}})\rangle$
$\displaystyle=$
$\displaystyle\langle\delta_{\rm(S)}(\bm{x}_{1})\delta_{\rm(S)}(\bm{x}_{2})\delta_{\rm(S)}(\bm{x}_{3})\rangle\;.$

In Fourier space, this relation can be expressed as

$\displaystyle\big{\langle}\widetilde{\delta}(\bm{k}_{1})\widetilde{\delta}(\bm{k}_{2})\widetilde{\delta}(\bm{k}_{3})\big{\rangle}$
(28) $\displaystyle=$
$\displaystyle\big{\langle}e^{-i(\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3})\cdot\overline{\bm{\Psi}}}\big{\rangle}\big{\langle}\widetilde{\delta}_{\rm(S)}(\bm{k}_{1})\widetilde{\delta}_{\rm(S)}(\bm{k}_{2})\widetilde{\delta}_{\rm(S)}(\bm{k}_{3})\big{\rangle}$
$\displaystyle=$
$\displaystyle\big{\langle}\widetilde{\delta}_{\rm(S)}(\bm{k}_{1})\widetilde{\delta}_{\rm(S)}(\bm{k}_{2})\widetilde{\delta}_{\rm(S)}(\bm{k}_{3})\big{\rangle}\;.$

The bispectrum resulting from short-wavelength density fluctuations is
represented as

$\displaystyle\langle\widetilde{\delta}_{\rm(S)}(\bm{k}_{1})\widetilde{\delta}_{\rm(S)}(\bm{k}_{2})\widetilde{\delta}_{\rm(S)}(\bm{k}_{3})\rangle$
(29) $\displaystyle=$ $\displaystyle(2\pi)^{3}\delta_{\rm
D}(\bm{k}_{123})\,[B_{\rm(S)}(\bm{k}_{1},\bm{k}_{2})+\mbox{2 perms.}]\;,$

and is denoted using the subscript ${\rm(S)}$.

##  IV Tree level bispectra

###  IV.1 SPT

The leading-order term, known as the tree-level, is represented by

$\displaystyle B_{\rm tree}(\bm{k}_{1},\bm{k}_{2})=B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm lin}(k_{2})\;,$
(30)

where

$\displaystyle B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})=2\,Z_{\rm
rec}^{[2]}(\bm{k}_{1},\bm{k}_{2})Z_{\rm rec}^{[1]}(\bm{k}_{1})Z_{\rm
rec}^{[1]}(\bm{k}_{2})\;.$ (31)

Consequently, the bispectrum at the tree level consists only of the $B_{\rm
GG}$ term.

Assuming the lack of correlation between $\delta_{\rm(S)}$ and
$\overline{\bm{\Psi}}$ in Eq. (20), we can show that the tree-level bispectrum
is the same as the short-wavelength one:

$\displaystyle B_{\rm(S)GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})\;.$ (32)

This is an expected result from the general discussion in Eq. (28).

###  IV.2 IR cancellation

In Fourier space, the IR effect manifests as the exponential function in Eq.
(28). Assuming $\overline{\bm{\Psi}}\simeq\overline{\bm{\Psi}}^{[1]}$, it can
be represented as

$\displaystyle\big{\langle}e^{-i(\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3})\cdot\overline{\bm{\Psi}}^{[1]}}\big{\rangle}$
$\displaystyle=$ $\displaystyle{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}(\bm{k}_{3})$ (33) $\displaystyle\times$ $\displaystyle{\cal
E}(\bm{k}_{1},\bm{k}_{2}){\cal E}(\bm{k}_{2},\bm{k}_{3}){\cal
E}(\bm{k}_{1},\bm{k}_{3})\;.$

Here, ${\cal D}(\bm{k})$ and ${\cal E}(\bm{k},\bm{k}^{\prime})$ are defined as
follows 555The cumulants of a statistical quantity $X$, denoted $\langle
X^{n}\rangle_{\rm c}$, are obtained from a power series expansion of the
cumulant-generating function $\ln\langle e^{X}\rangle$:
$\displaystyle\ln\left(\langle
e^{X}\rangle\right)=\sum_{n=1}\frac{1}{n!}\langle X^{n}\rangle_{\rm c}\;.$
(34) :

$\displaystyle{\cal D}(\bm{k})$ $\displaystyle=$
$\displaystyle\exp\left(\frac{1}{2}\left\langle\left(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}\right)^{2}\right\rangle_{\rm
c}\right)\;,$ $\displaystyle{\cal E}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\exp\left(\left\langle\left(-i\bm{k}_{1}\cdot\overline{\bm{\Psi}}^{[1]}\right)\left(-i\bm{k}_{2}\cdot\overline{\bm{\Psi}}^{[1]}\right)\right\rangle_{\rm
c}\right)\;.$ (35)

The function ${\cal D}(\bm{k})$ is presented as a two-dimensional exponential
damping function, specifically calculated as

$\displaystyle{\cal
D}(\bm{k})=\exp\left(-\frac{k^{2}(1-\mu^{2})\sigma_{\perp}^{2}+k^{2}\mu^{2}\sigma^{2}_{\parallel}}{2}\right)\;,$
(36)

where $\mu=\hat{n}\cdot\hat{k}$. The radial and transverse components of the
smoothing factors, $\sigma_{\perp}^{2}$ and $\sigma_{\parallel}^{2}$, are
given by Eqs. (66) and (67). Furthermore, the function ${\cal
E}(\bm{k}_{1},\bm{k}_{2})$ satisfies the following relation:

$\displaystyle{\cal E}(\bm{k}_{1},\bm{k}_{2})={\cal D}^{-1}(\bm{k}_{1}){\cal
D}^{-1}(\bm{k}_{2}){\cal D}(\bm{k}_{3})$ (37)

under the condition $\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3}=\bm{0}$.

The function ${\cal E}(\bm{k},\bm{k}^{\prime})$ connects two different wave
vectors, $\bm{k}$ and $\bm{k}^{\prime}$. In other words, this operation
produces mode coupling integrals. On the other hand, the function ${\cal
D}(\bm{k})$ is only associated only with the corresponding wave vector
$\bm{k}$. Figure 1 shows how the functions ${\cal D}$ and ${\cal E}$ appear in
each term of the bispectrum decomposed by the $\Gamma$-expansion in the IR
limit. Taking this into account and substituting Eq. (32) and Eq. (33) into
Eq. (28), we derive

$\displaystyle B_{\rm GG,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}(\bm{k}_{3})B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})\;,$ $\displaystyle
B_{\rm GM,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$ $\displaystyle{\cal
D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\left({\cal
E}(\bm{k}_{2},\bm{k}_{3})-1\right)$ $\displaystyle\times$ $\displaystyle
B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})\;,$ $\displaystyle
B_{\rm MG,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$ $\displaystyle{\cal
D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\left({\cal
E}(\bm{k}_{1},\bm{k}_{3})-1\right)$ $\displaystyle\times$ $\displaystyle
B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})\;,$ $\displaystyle
B_{\rm MM,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$ $\displaystyle{\cal
D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})$ $\displaystyle\times$
$\displaystyle\left({\cal E}(\bm{k}_{2},\bm{k}_{3})-1\right)\left({\cal
E}(\bm{k}_{1},\bm{k}_{3})-1\right)$ $\displaystyle\times$ $\displaystyle
B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm
lin}(k_{2})\;,$ $\displaystyle B_{\rm MMM,\,IR}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}(\bm{k}_{3})\left({\cal E}(\bm{k}_{1},\bm{k}_{2})-1\right)$ (38)
$\displaystyle\times$ $\displaystyle{\cal E}(\bm{k}_{2},\bm{k}_{3}){\cal
E}(\bm{k}_{1},\bm{k}_{3})$ $\displaystyle\times$ $\displaystyle B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm lin}(k_{2})\;,$

where the subscript “IR” indicates a quantity in the IR limit. Furthermore,
using Eq. (37), we obtain

$\displaystyle B_{\rm GG,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}(\bm{k}_{3})B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})\;,$ $\displaystyle
B_{\rm GM,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\big{[}{\cal D}^{2}(\bm{k}_{1})-{\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\big{]}$ $\displaystyle\times$
$\displaystyle B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})\;,$
$\displaystyle B_{\rm MG,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\big{[}{\cal D}^{2}(\bm{k}_{2})-{\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\big{]}$ $\displaystyle\times$
$\displaystyle B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})\;,$
$\displaystyle B_{\rm MM,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\big{[}{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}^{-1}(\bm{k}_{3})-{\cal D}^{2}(\bm{k}_{1})$ $\displaystyle-{\cal
D}^{2}(\bm{k}_{2})+{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}(\bm{k}_{3})\big{]}$ $\displaystyle\times$ $\displaystyle B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm lin}(k_{2})\;,$
$\displaystyle B_{\rm MMM,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\big{[}1-{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}^{-1}(\bm{k}_{3})\big{]}$ (39) $\displaystyle\times$ $\displaystyle B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm lin}(k_{2})\;.$

Consequently, for all the terms in Eq. (39), the infinite perturbative orders
that emerge due to infrared (IR) effects, i.e., non-perturbative effects, can
be computed using the Gaussian damping function ${\cal D}$. Nonetheless, the
substitution of all these terms into Eq. (25) results in the cancellation of
all contributions from the IR effects, thereby isolating the standard tree-
level bispectrum.

$\displaystyle B_{\rm IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle B_{\rm GG,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm
lin}(k_{2})$ (40) $\displaystyle+$ $\displaystyle B_{\rm
GM,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})$ $\displaystyle+$
$\displaystyle B_{\rm MG,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})$
$\displaystyle+$ $\displaystyle B_{\rm MM,\,IR}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle+$ $\displaystyle B_{\rm MMM,\,IR}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle B_{\rm tree}(\bm{k}_{1},\bm{k}_{2})\;.$

This result serves as a specific example of tree-level perturbative
calculations for IR cancellation discussed in the general case in Section
III.3.

Eq. (39) exhibits the same form as that derived for the pre-reconstruction
bispectrum, as detailed in Eqs. (52)-(56) in Sugiyama  _et al._ [9]. This
similarity arises because the IR effect for the post-reconstruction case can
be addressed in a manner analogous to the pre-reconstruction case, as
discussed in Section II.2.

From Eq. (39), it becomes clear that the two terms consisting only of mode-
coupling integrals, $B_{\rm MM}$ and $B_{\rm MMM}$, tend to diverge in the
high-$k$ limit due to the presence of the ${\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}^{-1}(\bm{k}_{3})$ term. Nevertheless, by considering
the combination of $B_{\rm MM}$ and $B_{\rm MMM}$, i.e., $B_{\rm MM}+B_{\rm
MMM}$, the ${\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}^{-1}(\bm{k}_{3})$
term cancels out, resulting in a finite value at small scales, as demonstrated
in Figure $3$ in Sugiyama  _et al._ [9].

###  IV.3 IR-resummed model

It is noteworthy that among terms related to mode-coupling integrals, such as
$B_{\rm GM}$, $B_{\rm MG}$, $B_{\rm MM}$, and $B_{\rm MMM}$, in Eq. (39), the
linear power spectrum $P_{\rm lin}$ containing the BAO signal emerges.
However, the mode-coupling integral is expected to smooth out the BAO signal
in $P_{\rm lin}$. This expectation indicates that the assumption that
$\delta_{\rm(S)}$ and $\overline{\bm{\Psi}}$ being uncorrelated does not fully
hold, leading to incomplete IR cancellation due to the mode coupling effect.

To account for smoothing effect of the BAO signal, we decompose the linear
matter power spectrum into two components: the “wiggle” component with only
the BAO signal, and the “no-wiggle” component lacking the BAO signal. This
decomposition is expressed as $P_{\rm lin}=P_{\rm w}+P_{\rm nw}$ [60, 61, 62],
where the subscripts “w” and “nw” denote “wiggle” and “no-wiggle”,
respectively. Subsequently, we replace all $P_{\rm lin}$ in Eqs. (39) with
$P_{\rm nw}$.

Substituting Eq. (39) into Eq. (25) after the above replacement between
$P_{\rm lin}$ and $P_{\rm nw}$, we derive the post-reconstruction tree-level
bispectrum with IR resummation as

$\displaystyle B(\bm{k}_{1},\bm{k}_{2})$ (41) $\displaystyle=$ $\displaystyle
B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2}){\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})P_{\rm w}(k_{1})P_{\rm w}(k_{2})$
$\displaystyle+$ $\displaystyle B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2}){\cal
D}^{2}(\bm{k}_{1})P_{\rm w}(k_{1})P_{\rm nw}(k_{2})$ $\displaystyle+$
$\displaystyle B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2}){\cal
D}^{2}(\bm{k}_{2})P_{\rm nw}(k_{1})P_{\rm w}(k_{2})$ $\displaystyle+$
$\displaystyle B_{\rm tree,\,nw}(\bm{k}_{1},\bm{k}_{2})\;,$

where the no-wiggle version of the tree-level bispectrum is represented by

$\displaystyle B_{\rm tree,\,nw}(\bm{k}_{1},\bm{k}_{2})=B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm nw}(k_{1})P_{\rm nw}(k_{2})\;.$ (42)

This is the first main result of this paper. In our model, the nonlinearity of
the BAO signal can be divided into three distinct components: ${\cal
D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})P_{\rm w}(k_{1})P_{\rm
w}(k_{2})$, ${\cal D}^{2}(\bm{k}_{1})P_{\rm w}(k_{1})P_{\rm nw}(k_{2})$, and
${\cal D}^{2}(\bm{k}_{2})P_{\rm nw}(k_{1})P_{\rm w}(k_{2})$, which serve to
dampen the BAO signal. The shape of the bispectrum is represented through the
no-wiggle version of the tree-level solution in Eq. (42).

Eq. (41) is consistent with the pre-reconstruction bispectrum model with IR
resummation, as detailed in Eq. (1) of Sugiyama  _et al._ [9]. Note that the
post-reconstruction case requires the use of the post-reconstruction kernel
functions in Eq. (60) and the post-reconstruction Gaussian smoothing
parameters in Eq. (66).

It is noteworthy that the derivation of Eq. (41) benefits from the definition
of the function ${\cal E}$ in Eq. (35) during the process. This approach
allows us to avoid the complex calculations associated with kernel functions,
a methodology previously employed by Sugiyama  _et al._ [9]. Consequently,
this simplification significantly streamlines the computational process,
facilitating the straightforward inclusion of additional 1-loop correction
terms in the subsequent section.

##  V 1-loop level bispectra

###  V.1 SPT

The next-leading order term in SPT, the 1-loop correction, can be decomposed
into the following five components:

$\displaystyle B_{\rm 1\mathchar 45\relax loop}(\bm{k}_{1},\bm{k}_{2})$ (43)
$\displaystyle=$ $\displaystyle\left[B_{\rm
GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm lin}(k_{1})P_{\rm
lin}(k_{2})$ $\displaystyle+$ $\displaystyle B^{\rm[123-II]}_{\rm
GM}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})+B^{\rm[123-III]}_{\rm
MG}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})$ $\displaystyle+$ $\displaystyle
B_{\rm MMM}^{[222]}(\bm{k}_{1},\bm{k}_{2})\;,$

where

$\displaystyle B_{\rm GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle 12\,Z_{\rm rec}^{[1]}(\bm{k}_{1})Z_{\rm
rec}^{[1]}(\bm{k}_{2})\int\frac{d^{3}p}{(2\pi)^{3}}Z_{\rm
rec}^{[4]}(\bm{k}_{1},\bm{k}_{2},\bm{p},-\bm{p})P_{\rm lin}(p)\;,$
$\displaystyle B_{\rm GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle 6\,Z_{\rm
rec}^{[2]}(\bm{k}_{1},\bm{k}_{2})\int\frac{d^{3}p}{(2\pi)^{3}}\left[Z_{\rm
rec}^{[1]}(\bm{k}_{1})Z_{\rm rec}^{[3]}(\bm{k}_{2},\bm{p},-\bm{p})+Z_{\rm
rec}^{[1]}(\bm{k}_{2})Z_{\rm
rec}^{[3]}(\bm{k}_{1},\bm{p},-\bm{p})\right]P_{\rm lin}(p)\;,$ $\displaystyle
B_{\rm GM}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle 6\,Z_{\rm
rec}^{[1]}(\bm{k}_{1})\int\frac{d^{3}p}{(2\pi)^{3}}Z_{\rm
rec}^{[2]}(\bm{p},\bm{k}_{2}-\bm{p})Z_{\rm
rec}^{[3]}(\bm{k}_{1},\bm{p},\bm{k}_{2}-\bm{p})P_{\rm lin}(p)P_{\rm
lin}(|\bm{k}_{2}-\bm{p}|)\;,$ $\displaystyle B_{\rm
MG}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$ $\displaystyle
6\,Z_{\rm rec}^{[1]}(\bm{k}_{2})\int\frac{d^{3}p}{(2\pi)^{3}}Z_{\rm
rec}^{[2]}(\bm{p},\bm{k}_{1}-\bm{p})Z_{\rm
rec}^{[3]}(\bm{k}_{2},\bm{p},\bm{k}_{1}-\bm{p})P_{\rm lin}(p)P_{\rm
lin}(|\bm{k}_{1}-\bm{p}|)\;,$ $\displaystyle B_{\rm
MMM}^{[222]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\frac{8}{3}\,\int\frac{d^{3}p}{(2\pi)^{3}}Z_{\rm
rec}^{[2]}(\bm{p},\bm{k}_{1}-\bm{p})Z_{\rm
rec}^{[2]}(-\bm{p},\bm{k}_{2}+\bm{p})Z_{\rm
rec}^{[2]}(\bm{k}_{2}+\bm{p},\bm{k}_{1}-\bm{p})P_{\rm lin}(p)P_{\rm
lin}(|\bm{k}_{2}+\bm{p}|)P_{\rm lin}(|\bm{k}_{1}-\bm{p}|)\;.$

Here, it should be noted that the $B_{\rm MM}$ term arises from 2-loop and
higher-order corrections, and therefore, does not appear in the 1-loop
corrections.

The IR (high-$k$) limit solutions for the 1-loop bispectrum corrections are
obtained by substituting the post-reconstruction kernel functions in the IR
limit, as provided by Eq. (61), into Eq. (LABEL:Eq:B1loop). It is important to
note that the conditions for the IR limit in the bispectrum vary with each
term of the bispectrum: for $B_{\rm GG}^{[114]}$ and $B_{\rm
GG}^{\rm[123-I]}$, it is $p\to 0$; for $B_{\rm GM}^{\rm[123-II]}$, it is $p\to
0$ and $|\bm{k}_{2}-\bm{p}|\to 0$; $B_{\rm MG}^{\rm[123-III]}$, it is $p\to 0$
and $|\bm{k}_{1}-\bm{p}|\to 0$; for $B_{\rm MMM}^{[222]}$, it is $p\to 0$,
$|\bm{k}_{1}-\bm{p}|\to 0$, and $|\bm{k}_{2}+\bm{p}|\to 0$. Then, we obtain

$\displaystyle B_{\rm GG,\,IR}^{[114]}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle\left[\ln{\cal D}(\bm{k}_{3})\right]B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})\;,$ $\displaystyle B_{\rm
GG,\,IR}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\left[\ln\left({\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2})\right)\right]B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})\;,$
$\displaystyle B_{\rm GM,\,IR}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle\left[\ln\left({\cal D}(\bm{k}_{1}){\cal
D}^{-1}(\bm{k}_{2}){\cal D}^{-1}(\bm{k}_{3})\right)\right]$
$\displaystyle\times$ $\displaystyle B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})\;,$ $\displaystyle B_{\rm
MG,\,IR}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\left[\ln\left({\cal D}(\bm{k}_{2}){\cal
D}^{-1}(\bm{k}_{1}){\cal D}^{-1}(\bm{k}_{3})\right)\right]$
$\displaystyle\times$ $\displaystyle B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})\;,$ $\displaystyle B_{\rm
MMM,\,IR}^{[222]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\left[\ln\left({\cal D}(\bm{k}_{3}){\cal
D}^{-1}(\bm{k}_{1}){\cal D}^{-1}(\bm{k}_{2})\right)\right]$ (45)
$\displaystyle\times$ $\displaystyle B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm lin}(k_{2})\;.$

The expressions presented here are identical to those of Eq. (39) when
expanded to the 1-loop level. Moreover, the five components of the 1-loop
bispectrum cancel each other out in the IR limit:

$\displaystyle B_{\rm 1\mathchar 45\relax loop,\,IR}(\bm{k}_{1},\bm{k}_{2})$
(46) $\displaystyle=$ $\displaystyle\left[B_{\rm
GG,\,IR}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG,\,IR}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm lin}(k_{1})P_{\rm
lin}(k_{2})$ $\displaystyle+$ $\displaystyle B^{\rm[123-II]}_{\rm
GM,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})+B^{\rm[123-III]}_{\rm
MG,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})$ $\displaystyle+$
$\displaystyle B_{\rm MMM,\,IR}^{[222]}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle 0\;,$

This result indicates the IR cancellation of the bispectrum at the 1-loop
level.

Assuming the absence of correlation between $\delta_{\rm(S)}$ and
$\overline{\bm{\Psi}}$ in Eq. (20), each component of the short-wavelength
bispectrum at the 1-loop level is given by

$\displaystyle B_{\rm(S)GG}^{[114]}$ $\displaystyle=$ $\displaystyle B_{\rm
GG}^{[114]}-B_{\rm GG,\,IR}^{[114]}\;,$ $\displaystyle
B_{\rm(S)GG}^{\rm[123-I]}$ $\displaystyle=$ $\displaystyle B_{\rm
GG}^{\rm[123-I]}-B_{\rm GG,\,IR}^{\rm[123-I]}\;,$ $\displaystyle
B_{\rm(S)GM}^{\rm[123-II]}$ $\displaystyle=$ $\displaystyle B_{\rm
GM}^{\rm[123-II]}-B_{\rm GM,\,IR}^{\rm[123-II]}\;,$ $\displaystyle
B_{\rm(S)MG}^{\rm[123-III]}$ $\displaystyle=$ $\displaystyle B_{\rm
MG}^{\rm[123-III]}-B_{\rm MG,\,IR}^{\rm[123-III]}\;,$ $\displaystyle
B_{\rm(S)MMM}^{[222]}$ $\displaystyle=$ $\displaystyle B_{\rm
MMM}^{[222]}-B_{\rm MMM,\,IR}^{[222]}\;.$ (47)

In the 1-loop level, the components of the short-wavelength bispectrum are
defined as the normal components subtracted by the values at the IR limit.
However, it can be shown that the short-wavelength 1-loop bispectrum converges
to the normal bispectrum using Eq. (46):

$\displaystyle B_{\rm(S)1\mathchar 45\relax loop}(\bm{k}_{1},\bm{k}_{2})$ (48)
$\displaystyle=$
$\displaystyle\left[B_{\rm(S)GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{1})P_{\rm lin}(k_{2})$ $\displaystyle+$ $\displaystyle
B^{\rm[123-II]}_{\rm(S)GM}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{1})+B^{\rm[123-III]}_{\rm(S)MG}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{2})$ $\displaystyle+$ $\displaystyle
B_{\rm(S)MMM}^{[222]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$ $\displaystyle
B_{\rm 1\mathchar 45\relax loop}(\bm{k}_{1},\bm{k}_{2})\;.$

This result is what expected from Eq. (28), like the tree-level case in Eq.
(32).

Eq. (45) is a general expression that includes the effects of reconstruction,
RSD, and bias. However, it may be perceived as complex. To facilitate an
intuitive understanding for the reader, we summarize the solutions for the
high-$k$ limit of the 1-loop bispectrum in the simplest case, i.e., in the
context of dark matter in real space before reconstruction, in Appendix D.

###  V.2 IR cancellation

Consider that the bispectrum includes contributions up to the 1-loop
correction in SPT, and that terms higher than the 1-loop level arise from IR
effects. In this case, each component of the bispectrum, when decomposed by
the $\Gamma$-expansion method, can be calculated non-perturbatively by
applying Eqs. (33), (32), and (47) to Eq. (28):

$\displaystyle B_{\rm GG,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}(\bm{k}_{3})\left[B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]\;,$
$\displaystyle B_{\rm GM,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\big{[}{\cal D}^{2}(\bm{k}_{1})-{\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\big{]}\left[B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{2})$ $\displaystyle+$ $\displaystyle{\cal
D}^{2}(\bm{k}_{1})B_{\rm(S)GM}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})\;,$
$\displaystyle B_{\rm MG,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\big{[}{\cal D}^{2}(\bm{k}_{2})-{\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\big{]}\left[B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{1})$ $\displaystyle+$ $\displaystyle{\cal
D}^{2}(\bm{k}_{2})B_{\rm(S)MG}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})\;,$
$\displaystyle B_{\rm MM,\,IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\big{[}{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}^{-1}(\bm{k}_{3})-{\cal D}^{2}(\bm{k}_{1})-{\cal D}^{2}(\bm{k}_{2})+{\cal
D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\big{]}$
$\displaystyle\times$ $\displaystyle\left[B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{1})P_{\rm lin}(k_{2})$ $\displaystyle+$ $\displaystyle\big{[}{\cal
D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}^{-1}(\bm{k}_{3})-{\cal
D}^{2}(\bm{k}_{1})\big{]}B_{\rm(S)GM}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{1})$ $\displaystyle+$ $\displaystyle\big{[}{\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}^{-1}(\bm{k}_{3})-{\cal
D}^{2}(\bm{k}_{2})\big{]}B_{\rm(S)MG}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{2})\;,$ $\displaystyle B_{\rm MMM,\,IR}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle\big{[}1-{\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}^{-1}(\bm{k}_{3})\big{]}$ (49) $\displaystyle\times$
$\displaystyle\bigg{\\{}\left[B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm(S)GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{1})P_{\rm lin}(k_{2})$ $\displaystyle+$ $\displaystyle
B_{\rm(S)GM}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{1})+B_{\rm(S)MG}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{2})\bigg{\\}}+B_{\rm(S)MMM}^{[222]}(\bm{k}_{1},\bm{k}_{2})\;.$

Once again, substituting Eq. (49) into Eq. (25) leads to the 1-loop bispectrum
in SPT due to the IR cancellation:

$\displaystyle B_{\rm IR}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle B_{\rm GG,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})P_{\rm
lin}(k_{2})$ (50) $\displaystyle+$ $\displaystyle B_{\rm
GM,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{1})$ $\displaystyle+$
$\displaystyle B_{\rm MG,\,IR}(\bm{k}_{1},\bm{k}_{2})P_{\rm lin}(k_{2})$
$\displaystyle+$ $\displaystyle B_{\rm MM,\,IR}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle+$ $\displaystyle B_{\rm MMM,\,IR}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$ $\displaystyle B_{\rm tree}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
1\mathchar 45\relax loop}(\bm{k}_{1},\bm{k}_{2})\;.$

###  V.3 IR-resummed model

We now proceed to derive the IR-resummed bispectrum model at the 1-loop level.
To this end, analogous to the approach in Section IV.3, we replace the linear
matter power spectra, $P_{\rm lin}$, appearing in $B_{\rm GM}$, $B_{\rm MG}$,
$B_{\rm MM}$, and $B_{\rm MMM}$ in Eq. (49), with the no-wiggle power spectra,
$P_{\rm nw}$. Note that the components of the short-wavelength bispectrum at
the 1-loop level, i.e., $B_{\rm(S)GG}^{[114]}$, $B_{\rm(S)GG}^{\rm[123-I]}$,
$B_{\rm(S)GM}^{\rm[123-II]}$, $B_{\rm(S)MG}^{\rm[123-III]}$, and
$B_{\rm(S)MMM}^{[222]}$, are defined as the original components minus their
corresponding IR limit values. Consequently, $P_{\rm lin}$ appearing in
$B_{\rm GM,\,IR}^{\rm[123-II]}$, $B_{\rm MG,\,IR}^{\rm[123-III]}$, and $B_{\rm
MMM,\,IR}^{[222]}$ in Eq. (47) should also be replaced with $P_{\rm nw}$.

Finally, after replacing $P_{\rm lin}$ and $P_{\rm nw}$, substituting Eq. (49)
into Eq. (25) leads to the post-reconstruction 1-loop bispectrum model with IR
resummation:

$\displaystyle B(\bm{k}_{1},\bm{k}_{2})$ (51) $\displaystyle=$
$\displaystyle\left[\left[1-\ln\left({\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})\right)\right]B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]{\cal D}(\bm{k}_{1}){\cal
D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})P_{\rm w}(k_{1})P_{\rm w}(k_{2})$
$\displaystyle+$ $\displaystyle\left[\left[1-\ln{\cal
D}^{2}(\bm{k}_{1})\right]B_{\rm GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})+\left(B_{\rm
GM}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})/P_{\rm nw}(k_{2})\right)\right]{\cal
D}^{2}(\bm{k}_{1})P_{\rm w}(k_{1})P_{\rm nw}(k_{2})$ $\displaystyle+$
$\displaystyle\left[\left[1-\ln{\cal D}^{2}(\bm{k}_{2})\right]B_{\rm
GG}^{[112]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})+\left(B_{\rm
MG}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})/P_{\rm nw}(k_{1})\right)\right]{\cal
D}^{2}(\bm{k}_{2})P_{\rm nw}(k_{1})P_{\rm w}(k_{2})$ $\displaystyle+$
$\displaystyle B_{\rm tree,nw}(\bm{k}_{1},\bm{k}_{2})+B_{\rm 1\mathchar
45\relax loop,nw}(\bm{k}_{1},\bm{k}_{2})\;,$

where the no-wiggle versions of the 1-loop bispectrum correction is given by

$\displaystyle B_{\rm 1\mathchar 45\relax loop,nw}(\bm{k}_{1},\bm{k}_{2})$
(52) $\displaystyle=$ $\displaystyle\left[B_{\rm
GG}^{[114]}(\bm{k}_{1},\bm{k}_{2})+B_{\rm
GG}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm nw}(k_{1})P_{\rm
nw}(k_{2})$ $\displaystyle+$ $\displaystyle B_{\rm
GM}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})P_{\rm nw}(k_{1})+B_{\rm
MG}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})P_{\rm nw}(k_{2})$ $\displaystyle+$
$\displaystyle B_{\rm MMM}^{[222]}(\bm{k}_{1},\bm{k}_{2})\;.$

This is the second main result of this paper. Compared to the IR-resummed
model at the tree-level, as given in Eq. (41), we can see that the SPT 1-loop
correction terms appear in the three terms that include $P_{\rm w}$ and
describe the nonlinear damping of the BAO signal. In addition, the overall
shape is represented by the no-wiggle version of the SPT 1-loop bispectrum.
Consequently, our model describes the non-perturbative behavior of the BAO
exponential damping while incorporating 1-loop correction terms in SPT,
enabling theoretical predictions of the bispectrum up to small scales.

When perturbatively expanded, Eq. (51) can be schematically expressed as the
SPT 1-loop solutions with higher-order terms arising from the IR effects:

$\displaystyle B(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$ $\displaystyle
B_{\rm tree}(\bm{k}_{1},\bm{k}_{2})+B_{\rm 1\mathchar 45\relax
loop}(\bm{k}_{1},\bm{k}_{2})$ (53) $\displaystyle+$
$\displaystyle\mbox{[higher order corrections]}\;.$

This result illustrates the essence of the IR-resummed model, which accurately
incorporates the SPT solution at a finite order and focuses only on IR effects
for the higher-order correction terms. It is particularly noteworthy that the
result in Eq. (51), where the no-wiggle version of the 1-loop solution shapes
the overall form of the bispectrum, is achieved by summing IR effects to an
infinite order. This has a fundamentally different implication than merely
truncating the SPT solution at the 1-loop level, despite the apparent
similarity in form.

###  V.4 Comparison with previous works

Eq. (51) represents an improvement in two aspects compared to the IR-resummed
model for the pre-reconstruction bispectrum presented in Sugiyama  _et al._
[9]. First, it includes the 1-loop correction terms in SPT, which means that
it can make more accurate theoretical predictions for smaller scales. Second,
it uses the post-reconstruction nonlinear kernel functions (60) and the post-
reconstruction Gaussian smoothing factors (66), making it a model for the
post-reconstruction bispectrum with IR resummation. To reiterate, the
functional form of the IR-resummed model remains unchanged before and after
reconstruction. Therefore, if one wants to compute the pre-reconstruction
bispectrum, it is sufficient to simply replace the nonlinear kernel functions
and smoothing parameters with those from before reconstruction.

Finally, it is noteworthy to mention that Ivanov and Sibiryakov [36] has
already presented an IR-resummed model for the pre-reconstruction bispectrum,
including the SPT 1-loop correction terms, RSD effects, and bias effects,
through a different approach from ours, the time-sliced perturbation theory
[35]. The form of the bispectrum in the Ivanov and Sibiryakov [36]’s model is
schematically given as follows (for details, see Eq. (7.19) and Appendix D in
[36]):

$\displaystyle B$ $\displaystyle=$ $\displaystyle B_{\rm tree}\left[P_{\rm
nw}+\left(1-\ln{\cal D}^{2}\right){\cal D}^{2}P_{\rm w}\right]$ (54)
$\displaystyle+$ $\displaystyle B_{\rm 1\mathchar 45\relax loop}\left[P_{\rm
nw}+{\cal D}^{2}P_{\rm w}\right]\;.$

This expression indicates that the linear power spectrum appearing within the
bispectrum is replaced with $P_{\rm nw}+{\cal D}^{2}P_{\rm w}$. In the tree-
level bispectrum, a factor of $(1-\ln{\cal D}^{2})$ appears to avoid double
counting of nonlinear contributions. While our result in Eq. (51) align
closely with their model, a slight difference arises in the exponential
damping functions form at the order of $(P_{\rm w})^{2}$. Specifically, the
Ivanov and Sibiryakov [36]’s model includes ${\cal D}^{2}(\bm{k}_{1}){\cal
D}^{2}(\bm{k}_{2})P_{\rm w}(k_{1})P_{\rm w}(k_{2})$, whereas our result is
${\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal D}(\bm{k}_{3})P_{\rm
w}(k_{1})P_{\rm w}(k_{2})$. For a more intuitive comparison, when considered
in real space as in Appendix D, the Ivanov and Sibiryakov [36]’s model
includes

$\displaystyle{\cal D}^{2}(\bm{k}_{1}){\cal D}^{2}(\bm{k}_{2})P_{\rm
w}(k_{1})P_{\rm w}(k_{2})$ (55) $\displaystyle=$ $\displaystyle
e^{-k_{1}^{2}\sigma_{\perp}^{2}}e^{-k_{2}^{2}\sigma_{\perp}^{2}}P_{\rm
w}(k_{1})P_{\rm w}(k_{2}),$

while our model is

$\displaystyle{\cal D}(\bm{k}_{1}){\cal D}(\bm{k}_{2}){\cal
D}(\bm{k}_{3})P_{\rm w}(k_{1})P_{\rm w}(k_{2})$ (56) $\displaystyle=$
$\displaystyle
e^{-k_{1}^{2}\sigma_{\perp}^{2}}e^{-k_{2}^{2}\sigma_{\perp}^{2}}e^{-(\bm{k}_{1}\cdot\bm{k}_{2})\sigma_{\perp}^{2}}P_{\rm
w}(k_{1})P_{\rm w}(k_{2})\;,$

differing by a factor of $e^{-(\bm{k}_{1}\cdot\bm{k}_{2})\sigma_{\perp}^{2}}$.
We conjecture that this discrepancy arises due to the approximation employed
by Ivanov and Sibiryakov [36], who ignored terms of ${\cal O}(P_{\rm w}^{2})$
during their derivations. However, a detailed mathematical comparison or a
discussion on numerical impacts exceeds the scope of this paper and is left
for future work.

##  VI Conclusions

In this paper, we have developed a bispectrum model that incorporates infrared
(IR) effects into post-reconstruction density fluctuations, including 1-loop
corrections within SPT. The main results are presented in Eqs. (41) and (51).
This model effectively captures the nonlinear dynamics of the BAO signal
within the post-reconstruction bispectrum. Moreover, it facilitates analysis
on smaller scales via the no-wiggle version of the 1-loop solution. Note that
the no-wiggle version of the SPT solution, which shapes the overall bispectrum
form, can only be derived by properly summing up the IR effects to an infinite
order. This approach fundamentally differs from merely truncating the SPT
solution at the 1-loop level. By setting the smoothing parameter $R_{\rm s}$,
required for reconstruction in Eq. (7), to approach infinity $(R_{\rm
s})\to\infty$, our model encompasses the pre-reconstruction case.

Sugiyama [44] demonstrated that the IR effect can be treated consistently
before and after reconstruction. Specifically, the IR effect can be
characterized as a coordinate transformation through the displacement vector
in short-wavelength density fluctuations in both cases. This fact suggests
that the resummation method for addressing the IR effect in the pre-
reconstruction tree-level bispectrum, as developed by Sugiyama  _et al._ [9],
can be directly applied to the post-reconstruction bispectrum. In this paper,
we build upon the framework established by Sugiyama  _et al._ [9] by
incorporating the 1-loop correction terms. Consequently, our work extends
previous research in two aspects: firstly, by incorporating the SPT 1-loop
correction terms, and secondly, by considering the post-reconstruction
scenario.

The characteristics of the IR-resummed bispectrum model handle terms that
include infinite-order mode-coupling integrals. Through these mode-coupling
terms, the bispectrum is decomposed into five components, as demonstrated in
Eq. (25). We calculate the non-perturbative effects arising from the IR
effects for each component. To streamline the overall computation and minimize
complex calculations involving kernel functions, we introduce the functions
${\cal D}$ and ${\cal E}$ in Eq. (35). In the IR limit, statistical
translational symmetry requires the cancellation of all IR effects.
Nonetheless, by considering an incompleteness in the IR cancellation, namely
the smoothing effect of the BAO signal in the mode-coupling integrals, we are
enable to formulate a model that describes the nonlinear damping of the BAO
signal while incorporating the 1-loop bispectrum correction terms.

The bispectrum model proposed in this paper encompasses 1-loop correction
terms, bias effects, and RSD effects, making it a suitable candidate for
cosmological analysis of the post-reconstruction bispectrum.

###### Acknowledgements.

NS acknowledges financial support from JSPS KAKENHI Grant Number 19K14703. NS
thanks Shi-Fan Stephen Chen for useful comments and discussion.

##  Appendix A Post-reconstruction kernel functions

The $n$th-order pre-reconstruction galaxy density fluctuation in Fourier space
is expressed as

$\displaystyle\widetilde{\delta}_{\rm g}^{\,[n]}(\bm{k})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\int\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}\left(\bm{k}-\bm{p}_{[1,n]}\right)$ (57) $\displaystyle\times$
$\displaystyle Z^{[n]}(\bm{p}_{1},\cdots,\bm{p}_{n})\widetilde{\delta}_{\rm
lin}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm lin}(\bm{p}_{n})\;,$

where the functions $Z^{[n\geq 2]}$ characterize the nonlinear effects on the
galaxy density fluctuation, including the RSD and bias effects. As in the main
text, for simplicity of notation, the line-of-sight dependence $\hat{n}$ due
to the RSD effect is omitted. This appendix shows the relation between
$Z^{[n]}$ in Eq. (57) and $Z^{[n]}_{\rm rec}$ in Eq. (15) up to the forth
order.

The $n$th-order displacement vector for reconstruction in Fourier space is
given by

$\displaystyle\widetilde{\bm{s}}^{\,[n]}(\bm{k})=\left(\frac{i\bm{k}}{k^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{1,\rm fid}}\right)\widetilde{\delta}_{\rm
g}^{\,[n]}(\bm{k})\;.$ (58)

Furthermore, the Fourier transform of the post-reconstruction density
fluctuation in Eq. (8) is given by

$\displaystyle\widetilde{\delta}_{\rm rec}(\bm{k},\hat{n})=\int
d^{3}x^{\prime}e^{-i\bm{k}\cdot\bm{x}^{\prime}}e^{-i\bm{k}\cdot\bm{s}(\bm{x}^{\prime},\hat{n})}\delta_{\rm
g}(\bm{x}^{\prime},\hat{n})\;.$ (59)

By substituting Eqs. (57) and (58) into Eq. (59), we obtain

$\displaystyle Z_{\rm rec}^{[1]}(\bm{p}_{1})$ $\displaystyle=$ $\displaystyle
Z^{[1]}(\bm{p}_{1})\;,$ $\displaystyle Z_{\rm
rec}^{[2]}(\bm{p}_{1},\bm{p}_{2})$ $\displaystyle=$ $\displaystyle
Z^{[2]}(\bm{p}_{1},\bm{p}_{2})+\frac{1}{2}\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{1,\rm
fid}}\right)+\left(\frac{\bm{k}\cdot\bm{p}_{2}}{p_{2}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{2}R_{\rm s})}{b_{1,\rm
fid}}\right)\Bigg{\\}}Z^{[1]}(\bm{p}_{1})Z^{[1]}(\bm{p}_{2})\;,$
$\displaystyle Z_{\rm rec}^{[3]}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})$
$\displaystyle=$ $\displaystyle Z^{[3]}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})$
$\displaystyle+$
$\displaystyle\frac{1}{3}\Bigg{\\{}\left(\frac{\bm{k}\cdot(\bm{p}_{1}+\bm{p}_{2})}{|\bm{p}_{1}+\bm{p}_{2}|^{2}}\right)\left(-\frac{W_{\rm
G}(|\bm{p}_{1}+\bm{p}_{2}|R_{\rm s})}{b_{1,\rm
fid}}\right)Z^{[2]}(\bm{p}_{1},\bm{p}_{2})Z^{[1]}(\bm{p}_{3})+\mbox{2
perms.}\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{3}\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{1,\rm
fid}}\right)Z^{[1]}(\bm{p}_{1})Z^{[2]}(\bm{p}_{2},\bm{p}_{3})+\mbox{2
perms.}\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{3!}\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{1,\rm
fid}}\right)\left(\frac{\bm{k}\cdot\bm{p}_{2}}{p_{2}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{2}R_{\rm s})}{b_{1,\rm fid}}\right)+\mbox{2
perms.}\Bigg{\\}}Z^{[1]}(\bm{p}_{1})Z^{[1]}(\bm{p}_{2})Z^{[1]}(\bm{p}_{3})\;,$
$\displaystyle Z_{\rm rec}^{[4]}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4})$
$\displaystyle=$ $\displaystyle
Z^{[4]}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4})$ (60) $\displaystyle+$
$\displaystyle\frac{1}{4}\Bigg{\\{}\left(\frac{\bm{k}\cdot(\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3})}{|\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3}|^{2}}\right)\left(-\frac{W_{\rm
G}(|\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3}|R_{\rm s})}{b_{1,\rm
fid}}\right)Z^{[3]}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})Z^{[1]}(\bm{p}_{4})+\mbox{4
perms.}\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{4}\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{1,\rm
fid}}\right)Z^{[1]}(\bm{p}_{1})Z^{[3]}(\bm{p}_{2},\bm{p}_{3},\bm{p}_{4})+\mbox{4
perms.}\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{6}\Bigg{\\{}\left(\frac{\bm{k}\cdot(\bm{p}_{1}+\bm{p}_{2})}{|\bm{p}_{1}+\bm{p}_{2}|^{2}}\right)\left(-\frac{W_{\rm
G}(|\bm{p}_{1}+\bm{p}_{2}|R_{\rm s})}{b_{1,\rm
fid}}\right)Z^{[2]}(\bm{p}_{1},\bm{p}_{2})Z^{[2]}(\bm{p}_{3},\bm{p}_{4})+\mbox{5
perms.}\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{12}\Bigg{\\{}\left(\frac{\bm{k}\cdot(\bm{p}_{1}+\bm{p}_{2})}{|\bm{p}_{1}+\bm{p}_{2}|^{2}}\right)\left(-\frac{W_{\rm
G}(|\bm{p}_{1}+\bm{p}_{2}|R_{\rm s})}{b_{1,\rm
fid}}\right)\left(\frac{\bm{k}\cdot\bm{p}_{3}}{p_{3}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{3}R_{\rm s})}{b_{1,\rm fid}}\right)$ $\displaystyle\hskip
28.45274pt\times
Z^{[2]}(\bm{p}_{1},\bm{p}_{2})Z^{[1]}(\bm{p}_{3})Z^{[1]}(\bm{p}_{4})+\mbox{11
perms.}\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{12}\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{1,\rm
fid}}\right)\left(\frac{\bm{k}\cdot\bm{p}_{2}}{p_{2}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{2}R_{\rm s})}{b_{1,\rm
fid}}\right)Z^{[1]}(\bm{p}_{1})Z^{[1]}(\bm{p}_{2})Z^{[2]}(\bm{p}_{3},\bm{p}_{4})+\mbox{5
perms.}\Bigg{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{4!}\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{1,\rm
fid}}\right)\left(\frac{\bm{k}\cdot\bm{p}_{2}}{p_{2}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{2}R_{\rm s})}{b_{1,\rm
fid}}\right)\left(\frac{\bm{k}\cdot\bm{p}_{3}}{p_{3}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{3}R_{\rm s})}{b_{1,\rm fid}}\right)+\mbox{3 perms.}\Bigg{\\}}$
$\displaystyle\hskip 28.45274pt\times
Z^{[1]}(\bm{p}_{1})Z^{[1]}(\bm{p}_{2})Z^{[1]}(\bm{p}_{3})Z^{[1]}(\bm{p}_{4})\;,$

where $\bm{k}=\bm{p}_{1}+\cdots+\bm{p}_{n}$ for $Z^{[n]}$. In the IR limit,
these kernel functions become

$\displaystyle Z^{[2]}_{\rm rec}(\bm{k},\bm{p})$
$\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle\frac{1}{2}\left\\{\left(\frac{\bm{k}\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)+\left(\frac{\bm{k}\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{\rm
1,fid}}\right)Z^{[1]}(\bm{p})\right\\}Z^{[1]}(\bm{k})\;,$ $\displaystyle
Z^{[3]}_{\rm rec}(\bm{k}_{1},\bm{p},-\bm{p})$ $\displaystyle\xrightarrow[p\to
0]{}$
$\displaystyle-\frac{1}{3!}\left\\{\left(\frac{\bm{k}\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)+\left(\frac{\bm{k}\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{\rm
1,fid}}\right)Z^{[1]}(\bm{p})\right\\}^{2}Z^{[1]}(\bm{k})\;,$ $\displaystyle
Z^{[3]}_{\rm rec}(\bm{k}_{1},\bm{k}_{2},\bm{p})$
$\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle\frac{1}{3}\left\\{\left(\frac{(\bm{k}_{1}+\bm{k}_{2})\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)+\left(\frac{(\bm{k}_{1}+\bm{k}_{2})\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{\rm 1,fid}}\right)Z^{[1]}(\bm{p})\right\\}Z_{\rm
rec}^{[2]}(\bm{k}_{1},\bm{k}_{2})\;,$ $\displaystyle Z^{[4]}_{\rm
rec}(\bm{k}_{1},\bm{k}_{2},\bm{p},-\bm{p})$ $\displaystyle\xrightarrow[p\to
0]{}$
$\displaystyle-\frac{1}{12}\left\\{\left(\frac{(\bm{k}_{1}+\bm{k}_{2})\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)+\left(\frac{(\bm{k}_{1}+\bm{k}_{2})\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{\rm 1,fid}}\right)Z^{[1]}(\bm{p})\right\\}^{2}Z_{\rm
rec}^{[2]}(\bm{k}_{1},\bm{k}_{2})\;,$ (61)

where $R^{[1]}_{ij}=\delta_{ij}+f\hat{n}_{i}\hat{n}_{j}$ given in Eq. (19). In
the derivation of the above relations, we used

$\displaystyle Z^{[2]}(\bm{k},\bm{p})$ $\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle\frac{1}{2}\left(\frac{\bm{k}\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)Z^{[1]}(\bm{k})\;,$
$\displaystyle Z^{[3]}(\bm{k}_{1},\bm{p},-\bm{p})$
$\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle-\frac{1}{3!}\left\\{\left(\frac{\bm{k}\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)\right\\}^{2}Z^{[1]}(\bm{k})\;,$
$\displaystyle Z^{[3]}(\bm{k}_{1},\bm{k}_{2},\bm{p})$
$\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle\frac{1}{3}\left\\{\left(\frac{(\bm{k}_{1}+\bm{k}_{2})\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)\right\\}Z^{[2]}(\bm{k}_{1},\bm{k}_{2})\;,$
$\displaystyle Z^{[4]}(\bm{k}_{1},\bm{k}_{2},\bm{p},-\bm{p})$
$\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle-\frac{1}{12}\left\\{\left(\frac{(\bm{k}_{1}+\bm{k}_{2})\cdot\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)\right\\}^{2}Z^{[2]}(\bm{k}_{1},\bm{k}_{2})\;.$
(62)

##  Appendix B $\Gamma$-expansion

Using the $\Gamma$-expansion scheme, we show specific forms of $B_{\rm GG}$,
$B_{\rm GM}$, $B_{\rm MG}$, $B_{\rm MM}$, and $B_{\rm MMM}$ in Eq. (25) using
the nonlinear kernel functions $Z^{[n]}$. The results obtained here are also
valid using the post-reconstruction nonlinear kernel functions $Z^{[n]}_{\rm
rec}$.

The $r$th order coefficient of the $\Gamma$-expansion is expressed as [63, 51]

$\displaystyle\Gamma^{(r)}(\bm{p}_{1},\dots,\bm{p}_{r})$ (63) $\displaystyle=$
$\displaystyle\frac{1}{r!}\sum_{s=0}^{\infty}\frac{(r+2s)!}{2^{s}s!}\prod_{i=1}^{s}\int\frac{d^{3}q_{i}}{(2\pi)^{3}}\,P_{\rm
lin}(q_{i})$ $\displaystyle\times$ $\displaystyle
Z^{[r+2s]}(\bm{p}_{1},\dots,\bm{p}_{r},\bm{q}_{1},-\bm{q}_{1},\dots,\bm{q}_{s},-\bm{q}_{s})\;,$

where the superscript $(r)$ means the $r$th order of the $\Gamma$-expansion.
Then, we obtain [9]

$\displaystyle B_{\rm GG}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle
2\,\Gamma^{(2)}\left(\bm{k}_{1},\bm{k}_{2}\right)\Gamma^{(1)}\left(\bm{k}_{1}\right)\Gamma^{(1)}\left(\bm{k}_{2}\right)\;,$
$\displaystyle B_{\rm GM}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\Gamma^{(1)}(\bm{k}_{1})\sum_{r=2}^{\infty}(r+1)!\prod_{i=1}^{r}\int\frac{d^{3}p_{i}}{(2\pi)^{3}}P_{\rm
lin}(p_{i})(2\pi)^{3}\delta_{\rm
D}\big{(}\bm{k}_{2}-\bm{p}_{[1,r]}\big{)}\Gamma^{(r)}(\bm{p}_{1},\dots,\bm{p}_{r})\Gamma^{(r+1)}(\bm{p}_{1},\dots,\bm{p}_{r},\bm{k}_{1})\;,$
$\displaystyle B_{\rm MG}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\Gamma^{(1)}(\bm{k}_{2})\sum_{r=2}^{\infty}(r+1)!\prod_{i=1}^{r}\int\frac{d^{3}p_{i}}{(2\pi)^{3}}P_{\rm
lin}(p_{i})(2\pi)^{3}\delta_{\rm
D}\big{(}\bm{k}_{1}-\bm{p}_{[1,r]}\big{)}\Gamma^{(r)}(\bm{p}_{1},\dots,\bm{p}_{r})\Gamma^{(r+1)}(\bm{p}_{1},\dots,\bm{p}_{r},\bm{k}_{2})\;,$
$\displaystyle B_{\rm MM}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\sum_{r=2}^{\infty}\sum_{s=2}^{\infty}(r+s)!\prod_{i=1}^{r}\int\frac{d^{3}p_{i}}{(2\pi)^{3}}P_{\rm
lin}(p_{i})\prod_{j=1}^{s}\int\frac{d^{3}p^{\prime}_{j}}{(2\pi)^{3}}P_{\rm
lin}(p^{\prime}_{j})(2\pi)^{3}\delta_{\rm
D}\big{(}\bm{k}_{1}-\bm{p}_{[1,r]}\big{)}(2\pi)^{3}\delta_{\rm
D}\big{(}\bm{k}_{2}-\bm{p}^{\prime}_{[1,s]}\big{)}$ $\displaystyle\times$
$\displaystyle\Gamma^{(r)}(\bm{p}_{1},\dots,\bm{p}_{r})\Gamma^{(s)}(\bm{p}^{\prime}_{1},\dots,\bm{p}^{\prime}_{s})\Gamma^{(r+s)}(\bm{p}_{1},\dots,\bm{p}_{r},\bm{p}^{\prime}_{1},\dots,\bm{p}^{\prime}_{s})\;,$
$\displaystyle B_{\rm MMM}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{3}\sum_{r=1}^{\infty}\sum_{s=1}^{\infty}\sum_{t=1}^{\infty}\frac{(r+t)!(r+s)!(s+t)!}{r!s!t!}\prod_{i=1}^{r}\int\frac{d^{3}p_{i}}{(2\pi)^{3}}P_{\rm
lin}(p_{i})\prod_{j=1}^{s}\int\frac{d^{3}p^{\prime}_{j}}{(2\pi)^{3}}P_{\rm
lin}(p^{\prime}_{j})\prod_{k=1}^{t}\int\frac{d^{3}p^{\prime\prime}_{k}}{(2\pi)^{3}}P_{\rm
lin}(p^{\prime\prime}_{k})$ (64) $\displaystyle\times$
$\displaystyle(2\pi)^{3}\delta_{\rm
D}\big{(}\bm{k}_{1}-\bm{p}_{[1,r]}-\bm{p}^{\prime\prime}_{[1,t]}\big{)}(2\pi)^{3}\delta_{\rm
D}\big{(}\bm{k}_{2}+\bm{p}_{[1,r]}+\bm{p}^{\prime}_{[1,s]}\big{)}$
$\displaystyle\times$
$\displaystyle\Gamma^{(r+t)}(\bm{p}_{1},\dots,\bm{p}_{r},\bm{p}^{\prime\prime}_{1},\dots,\bm{p}^{\prime\prime}_{t})\Gamma^{(r+s)}(-\bm{p}_{1},\dots,-\bm{p}_{r},-\bm{p}^{\prime}_{1},\dots,-\bm{p}^{\prime}_{s})$
$\displaystyle\times$
$\displaystyle\Gamma^{(s+t)}(-\bm{p}^{\prime}_{1},\dots,-\bm{p}^{\prime}_{s},\bm{p}^{\prime\prime}_{1},\dots,\bm{p}^{\prime\prime}_{t})\;,$

where $\bm{p}_{[1,n]}=\bm{p}_{1}+\cdots+\bm{p}_{n}$ for any integer $n$ with
$n\geq 1$.

##  Appendix C Smoothing factors

This appendix gives the specific forms of the smoothing factors
$\sigma^{2}_{\perp}$ and $\sigma^{2}_{\parallel}$ that characterize the
Gaussian damping function ${\cal D}(\bm{k})$ defined in Eq. (36).

The first-order of the post-reconstruction displacement vector is given by

$\displaystyle\overline{\bm{\Psi}}_{\rm rec}^{[1]}$ $\displaystyle=$
$\displaystyle
i\,\int\frac{d^{3}p}{(2\pi)^{3}}\Bigg{\\{}\left(\frac{\bm{R}^{[1]}\cdot\bm{p}}{p^{2}}\right)$
(65) $\displaystyle+\left(\frac{\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{\rm 1,fid}}\right)Z^{[1]}(\bm{p})\Bigg{\\}}\delta_{\rm
lin}(\bm{p})\;.$

By substituting Eq. (65) into Eq. (35), the radial and transverse components
of the smoothing factors are calculated as [44]

$\displaystyle\sigma_{\perp}^{2}$ $\displaystyle=$ $\displaystyle\sigma_{\rm
pp\,,\perp}^{2}+\sigma_{\rm ps\,,\perp}^{2}+\sigma_{\rm ss\,,\perp}^{2}$
$\displaystyle\sigma_{\parallel}^{2}$ $\displaystyle=$
$\displaystyle\sigma_{\rm pp\,,\parallel}^{2}+\sigma_{\rm
ps\,,\parallel}^{2}+\sigma_{\rm ss\,,\parallel}^{2}\;,$ (66)

and

$\displaystyle\sigma^{2}_{\rm pp,\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int\frac{dp}{2\pi^{2}}P_{\rm lin}(p),$
$\displaystyle\sigma^{2}_{\rm pp,\parallel}$ $\displaystyle=$
$\displaystyle(1+f)^{2}\,\sigma^{2}_{\rm pp,\perp},$
$\displaystyle\sigma^{2}_{\rm ps,\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int_{k_{\rm min}}^{k_{\rm
max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm s})}{b_{1,\rm
fid}}\right)\left[2\left(b_{1}+\frac{f}{5}\right)\right]P_{\rm lin}(p)\;,$
$\displaystyle\sigma^{2}_{\rm ps,\parallel}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int_{k_{\rm min}}^{k_{\rm
max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm s})}{b_{1,\rm
fid}}\right)\Bigg{[}2\left(1+f\right)\left(b_{1}+\frac{3}{5}f\right)\Bigg{]}P_{\rm
lin}(p)\;,$ $\displaystyle\sigma^{2}_{\rm ss,\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int_{k_{\rm min}}^{k_{\rm
max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm s})}{b_{1,\rm
fid}}\right)^{2}\left[\left(b_{1}^{2}+\frac{2}{5}b_{1}f+\frac{3}{35}f^{2}\right)P_{\rm
lin}(p)+\frac{1}{\bar{n}}\right]\;,$ $\displaystyle\sigma^{2}_{\rm
ss,\parallel}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\int_{k_{\rm
min}}^{k_{\rm max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm
s})}{b_{1,\rm
fid}}\right)^{2}\Bigg{[}\left(b_{1}^{2}+\frac{42}{35}b_{1}f+\frac{3}{7}f^{2}\right)P_{\rm
lin}(p)+\frac{1}{\bar{n}}\Bigg{]}\;.$ (67)

Here, the shot-noise term $1/\bar{n}$ with $\bar{n}$ being the mean number
density appears in $\sigma_{\rm ss,\perp}^{2}$ and $\sigma_{\rm
ss,\parallel}^{2}$ due to the discreteness effect in the displacement vector
for reconstruction [64].

##  Appendix D The high-$k$ limit solutions of the matter bispectrum in real
space

The second-order density fluctuation for dark matter in real space is
expressed as

$\displaystyle\widetilde{\delta}^{\,[2]}(\bm{k})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}\left(\bm{k}-\bm{p}_{1}-\bm{p}_{2}\right)$ (68) $\displaystyle\times$
$\displaystyle F^{[2]}(\bm{p}_{1},\bm{p}_{2})\widetilde{\delta}_{\rm
lin}(\bm{p}_{1})\widetilde{\delta}_{\rm lin}(\bm{p}_{2})\;,$

where

$\displaystyle F^{[2]}(\bm{p}_{1},\bm{p}_{2})$ (69) $\displaystyle=$
$\displaystyle\frac{17}{21}+\frac{1}{2}\left(\hat{p}_{1}\cdot\hat{p}_{2}\right)\left(\frac{p_{1}}{p_{2}}+\frac{p_{2}}{p_{1}}\right)+\frac{2}{7}\left(\left(\hat{p}_{1}\cdot\hat{p}_{2}\right)^{2}-\frac{1}{3}\right)\;.$

The tree-level bispectrum is then given by

$\displaystyle B_{\rm
tree}(\bm{k}_{1},\bm{k}_{2})=2\,F^{[2]}(\bm{k}_{1},\bm{k}_{2})P_{\rm
lin}(k_{1})P_{\rm lin}(k_{2})\;.$ (70)

From Eq. (67), the smoothing factor is calculated as

$\displaystyle\sigma_{\perp}^{2}=\frac{1}{3}\int\frac{dp}{2\pi^{2}}P_{\rm
lin}(p)\;.$ (71)

Consequently, Eq. (45) can be rewritten as

$\displaystyle B_{\rm GG,\,IR}^{[114]}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$
$\displaystyle\left[-\frac{1}{2}k_{1}^{2}\sigma^{2}_{\perp}-\frac{1}{2}k_{1}^{2}\sigma^{2}_{\perp}-(\bm{k}_{1}\cdot\bm{k}_{2})\sigma^{2}_{\perp}\right]\left[2F_{2}(\bm{k}_{,}\bm{k}_{2})\right]\;,$
$\displaystyle B_{\rm GG,\,IR}^{\rm[123-I]}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$
$\displaystyle\left[-\frac{1}{2}k_{1}^{2}\sigma^{2}_{\perp}-\frac{1}{2}k_{1}^{2}\sigma^{2}_{\perp}\right]\left[2F_{2}(\bm{k}_{,}\bm{k}_{2})\right]\;,$
$\displaystyle B_{\rm GM,\,IR}^{\rm[123-II]}(\bm{k}_{1},\bm{k}_{2})$
$\displaystyle=$
$\displaystyle\left[k_{2}^{2}\sigma^{2}_{\perp}+(\bm{k}_{1}\cdot\bm{k}_{2})\sigma^{2}_{\perp}\right]\left[2F_{2}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{2})\;,$ $\displaystyle B_{\rm
MG,\,IR}^{\rm[123-III]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\left[k_{1}^{2}\sigma^{2}_{\perp}+(\bm{k}_{1}\cdot\bm{k}_{2})\sigma^{2}_{\perp}\right]\left[2F_{2}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{1})\;,$ $\displaystyle B_{\rm
MMM,\,IR}^{[222]}(\bm{k}_{1},\bm{k}_{2})$ $\displaystyle=$
$\displaystyle\left[-\left(\bm{k}_{1}\cdot\bm{k}_{2}\right)\sigma^{2}_{\perp}\right]\left[2F_{2}(\bm{k}_{1},\bm{k}_{2})\right]P_{\rm
lin}(k_{1})P_{\rm lin}(k_{2})\;.$ (72)

From these equations, we can see that the high-$k$ solution of the 1-loop
bispectrum includes not only the squares of $k_{1}$ and $k_{2}$ but also the
product of $\bm{k}_{1}$ and $\bm{k}_{2}$.

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